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 /!\ '''Edit conflict  your version:'''  There is another algorithm, that uses the matrix of $T_2$ (which I'm computing and caching above!), which can compute the charpolys of many other $T_p". It's described here: http://sage.math.washington.edu/home/wstein/days/17/highweight/, along with a magma implementation (need to port to Sage). Thus it might be nice to implement this and run it, and get say all $T_{p,k}$ for $p,k \leq 1000$. == Arithmetic data about every weight 2 newform on Gamma0(N) for all N<5135 (and many more up to 7248) == http://modular.fas.harvard.edu/Tables/arith_of_factors/ This table is challenging to replicate/extend, but easy to move over. It would likely be better to continue our aplist computations, etc. up to 10000 (they only went to 3200 so far), and also take our saved decompositions and compute other data. Anyway, this table is more of a challenge. == Systems of Hecke Eigenvalues: qexpansions of Newforms == http://modular.fas.harvard.edu/Tables/aplist/ This is the point of these tables from last summer, which are much more comprehensive. http://sage.math.washington.edu/home/wstein/db/modsym/ Also, Tom Boothby has made some tables of traces of these. == Eigenforms on the Supersingular Basis == http://modular.fas.harvard.edu/Tables/supersingular.html A close analogue of the above table could likely be easily extended/recomputed using the SupersingularModule code.  /!\ '''End of edit conflict'''  
Specific Tables/Projects
Contents

Specific Tables/Projects
 Component Groups of J0(N)(R) and J1(N)(R)
 Cuspidal Subgroup
 Discriminants of Hecke Algebra
 Compute a table of semisimplications of reducible representations of elliptic curves
 Dimensions of modular forms spaces
 Compute the exact torsion subgroup of J0(N) for as many N as possible
 Characteristic polynomial of T2 on level 1 modular forms
 Characteristic polys of many Tp on level 1
 Arithmetic data about every weight 2 newform on Gamma0(N) for all N<5135 (and many more up to 7248)
 Systems of Hecke Eigenvalues: qexpansions of Newforms
 Eigenforms on the Supersingular Basis
The misc tables are listed here:
http://modular.fas.harvard.edu/Tables/
The Harvard URL is the best, since the http://wstein.org/Tables has none of the cgibin script dynamic data.
Component Groups of J0(N)(R) and J1(N)(R)
URL: http://wstein.org/Tables/real_tamagawa/ and http://wstein.org/Tables/compgrp/. The second page has much more extensive data and a conjecture.
 New Code:
This function computes the J_0(N) real component groups.
def f(N): M = ModularSymbols(N).cuspidal_subspace() d = M.dimension()//2 S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list())  1 return 2^(S.nullity()d)
For J_1(N) it is:
def f(N): M = ModularSymbols(Gamma1(N)).cuspidal_subspace() d = M.dimension()//2 S = matrix(GF(2),2*d,2*d, M.star_involution().matrix().list())  1 return 2^(S.nullity()d)
Future extension: one could replace Gamma1(N) by GammaH(N,...). One could also do the new subspace.
And note Frank's conjecture:
Conjecture (Frank Calegari):
Let m = #odd prime factors of N + {1, if N = 0 mod 8 {0, otherwise. Then the component group is isomorphic to (Z/2Z)^f, where f = 2^m  1.
the above conjecture is wrong, but the following matches our data (up to level N<=2723):
Conjecture (BoothbyStein): Let m = #odd prime factors of N  {1, if N != 0 mod 8 {0, otherwise. Then the component group is isomorphic to (Z/2Z)^f, where f = 2^m  1, unless N=1,2,4, in which case the component is
Soroosh  the prime level case is known. See Calegari <insert link> which *just* cites Agashe and Merel (http://wstein.org/home/wstein/days/17/Merel_Laccouplement_de_Weil_entre_le_sousgroup.pdf  page 12).
Cuspidal Subgroup
Computing the structure of the cuspidal subgroup of J0(N) and J1(N) (say).
URL: http://wstein.org/Tables/cuspgroup/ (the displayed formula is backwards at the top)
 New Sage code:
def cuspidal_subgroup_J0(N): J = J0(N) I = J.cuspidal_subgroup().invariants() # maybe pickle J return I
def cuspidal_subgroup_J1(N): J = J1(N) I = J.cuspidal_subgroup().invariants() # maybe pickle J return I
BUT WAIT  isn't there an a priori formula for this structure/order? Yes  Ligozat, but not really  that gives only rational cuspidal subgroup, and might be just as hard.
Anyway, I'm computing a few of these here, as a test of the modular symbols code, etc., since this is easy:
http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J0N/
http://sage.math.washington.edu/home/was/db/days17/cuspidal_subgroup_J1N/
Discriminants of Hecke Algebra
Computation of discriminants of various Hecke algebras.
Amazingly, it seems that there is "discriminants of Hecke algebras" implementation in Sage! Here is a straightforward algorithm:
The input is the level N.
Chose a random vector v in the space M of cuspidal modular symbols of level N.
Compute the sturm bound B.
Compute the products T_1(v), ..., T_B(v), and find a basis b_i for the ZZmodule they span.
Find Hecke operators S_1, ..., S_n such that S_i(v) = b_i. (This is linear algebra  inverting a matrix and a matrix multiply.)
Compute the determinant det ( Trace(S_i * S_j) ). That is the discriminant. This also gives a basis for the Hecke algebra, which is very useful for lots of things.
Note: See http://trac.sagemath.org/sage_trac/ticket/6768 for very slow code for computing a basis for the Hecke algebra.
Here is a more complicated algorithm, but it might suck because of hidden denseness!
The input is the level N.
If N is divisible by a prime p^3 and X_0(N/p^3) has positive genus, then the discriminant is 0, as one can see by taking images of forms of level N/p^3.
I think the above is an if and only if condition for when the discriminant is 0. See I think ColemanVoloch.
 The actual algorithm now.
Find a random Hecke operator t such that the charpoly of t has nonzero discriminant.
Choose a random vector v in the space of cuspidal modular symbols.
Let B be the Sturm bound.
Compute the images T_n(v) for n up to the Sturm Bound.
Compute a table of semisimplications of reducible representations of elliptic curves
Ralph Greenberg asked for a specific example of an elliptic curve with certain representation, and Soroosh and William found it. In order to do this, we developed a (mostly) efficient algorithm for computing the two characters eps and psi that define the semisimplication of an elliptic curve's Galois representation. This project is to fully implement the algorithm, then run it on curves in the Cremona database and all primes for which the Galois representation is reducible. There is relevant code here: http://nt.sagenb.org/home/pub/19/ and http://nt.sagenb.org/home/pub/20/
In fact, one can use the algorithm mentioned above to compute the semisimplication for any modular abelian variety! It would be good to do this for say every J0 modabvar of level up to say 3200 (since we have an ap table up that far): http://sage.math.washington.edu/home/wstein/db/modsym/
Dimensions of modular forms spaces
Currently http://wstein.org/Tables/dimensions/ has a couple of table with a kludgy and completely broken. These tables are nicer: http://wstein.org/Tables/dimensionsall.html. I think a static table that can do Gamma0, Gamma1, and character for all levels up to 100000 and weight 2 would be good to have. But its value would only be in having it easily usable, since there is no value in asking for an individual space. Anyway, compute the data. It would in fact by a good idea. Also, for each character, we should compute the dimensions of the modular, eisenstein spaces and the new cuspidal, and pnew cuspidal subspaces for each p dividing the level. The following session illustrates that in fact that would be quite valuable to have precomputed in a table:
sage: G = DirichletGroup(21000) sage: time C = G.galois_orbits() Time: CPU 2.21 s, Wall: 2.52 s sage: time z = [(e[0], dimension_cusp_forms(e[0], 2)) for e in C] Time: CPU 8.86 s, Wall: 9.79 s
I (=William) started a small calculation going of dimensions of spaces with character for weights <= 16 and for each space computing the dimension of the cuspidal, eisenstein, new, and pnew for each p parts. It's here: http://sage.math.washington.edu/home/wstein/db/days17/dimension_character/
Compute the exact torsion subgroup of J0(N) for as many N as possible
See http://nt.sagenb.org/home/pub/21/ for some work in this direction by Stein and Yazdani.
Characteristic polynomial of T2 on level 1 modular forms
See this table: http://modular.fas.harvard.edu/Tables/charpoly_level1/t2/
I have a straightforward algorithm to recompute these, which I'm running here:
http://sage.math.washington.edu/home/wstein/db/days17/level1/
Characteristic polys of many Tp on level 1
Next, see this table, which gives "Characteristic polynomials of T2, T3, ..., T997 for k<=128"
http://modular.fas.harvard.edu/Tables/charpoly_level1/upto997/
That exact range can likely be easily done using modular symbols.
Edit conflict  other version:
There is another algorithm, that uses the matrix of T_2 (which I'm computing and caching above!), which can compute the charpolys of many other T_p. It's described here: http://sage.math.washington.edu/home/wstein/days/17/highweight/, along with a magma implementation (need to port to Sage). Thus it might be nice to implement this and run it, and get say all T_{p,k} for p,k \leq 1000.
Edit conflict  your version:
There is another algorithm, that uses the matrix of T_2 (which I'm computing and caching above!), which can compute the charpolys of many other T_p". It's described here: http://sage.math.washington.edu/home/wstein/days/17/highweight/, along with a magma implementation (need to port to Sage). Thus it might be nice to implement this and run it, and get say all T_{p,k} for p,k \leq 1000$.
Arithmetic data about every weight 2 newform on Gamma0(N) for all N<5135 (and many more up to 7248)
http://modular.fas.harvard.edu/Tables/arith_of_factors/
This table is challenging to replicate/extend, but easy to move over. It would likely be better to continue our aplist computations, etc. up to 10000 (they only went to 3200 so far), and also take our saved decompositions and compute other data. Anyway, this table is more of a challenge.
Systems of Hecke Eigenvalues: qexpansions of Newforms
http://modular.fas.harvard.edu/Tables/aplist/
This is the point of these tables from last summer, which are much more comprehensive.
Also, Tom Boothby has made some tables of traces of these.
Eigenforms on the Supersingular Basis
http://modular.fas.harvard.edu/Tables/supersingular.html
A close analogue of the above table could likely be easily extended/recomputed using the SupersingularModule code.
End of edit conflict