Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [504,2,Mod(85,504)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(504, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.85");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.cs (of order \(6\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(4.02446026187\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(36\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
85.1 | −1.41413 | + | 0.0151305i | −1.72706 | + | 0.131454i | 1.99954 | − | 0.0427932i | −1.26304 | − | 0.729219i | 2.44030 | − | 0.212025i | 0.500000 | + | 0.866025i | −2.82697 | + | 0.0907694i | 2.96544 | − | 0.454056i | 1.79715 | + | 1.01210i |
85.2 | −1.39778 | − | 0.214989i | 0.0497101 | + | 1.73134i | 1.90756 | + | 0.601014i | 0.261066 | + | 0.150726i | 0.302736 | − | 2.43071i | 0.500000 | + | 0.866025i | −2.53713 | − | 1.25019i | −2.99506 | + | 0.172130i | −0.332507 | − | 0.266808i |
85.3 | −1.38589 | − | 0.281624i | 1.31821 | − | 1.12353i | 1.84138 | + | 0.780600i | 2.74638 | + | 1.58562i | −2.14331 | + | 1.18585i | 0.500000 | + | 0.866025i | −2.33211 | − | 1.60040i | 0.475349 | − | 2.96210i | −3.35962 | − | 2.97094i |
85.4 | −1.31153 | + | 0.529035i | 0.349968 | − | 1.69633i | 1.44024 | − | 1.38770i | −2.84022 | − | 1.63980i | 0.438422 | + | 2.40993i | 0.500000 | + | 0.866025i | −1.15479 | + | 2.58195i | −2.75505 | − | 1.18732i | 4.59255 | + | 0.648078i |
85.5 | −1.27427 | − | 0.613390i | 1.34597 | + | 1.09012i | 1.24750 | + | 1.56324i | −2.33508 | − | 1.34816i | −1.04645 | − | 2.21471i | 0.500000 | + | 0.866025i | −0.630773 | − | 2.75720i | 0.623266 | + | 2.93454i | 2.14857 | + | 3.15023i |
85.6 | −1.25626 | + | 0.649466i | 1.65525 | − | 0.510053i | 1.15639 | − | 1.63180i | 1.14688 | + | 0.662153i | −1.74816 | + | 1.71579i | 0.500000 | + | 0.866025i | −0.392930 | + | 2.80100i | 2.47969 | − | 1.68853i | −1.87083 | − | 0.0869767i |
85.7 | −1.20390 | + | 0.742040i | −1.50659 | + | 0.854512i | 0.898754 | − | 1.78668i | 3.20115 | + | 1.84819i | 1.17970 | − | 2.14670i | 0.500000 | + | 0.866025i | 0.243781 | + | 2.81790i | 1.53962 | − | 2.57480i | −5.22530 | + | 0.150350i |
85.8 | −1.14128 | − | 0.835149i | 1.40449 | − | 1.01361i | 0.605053 | + | 1.90628i | −1.92732 | − | 1.11274i | −2.44944 | − | 0.0161356i | 0.500000 | + | 0.866025i | 0.901492 | − | 2.68092i | 0.945172 | − | 2.84722i | 1.27031 | + | 2.87955i |
85.9 | −0.972917 | − | 1.02637i | −0.120790 | − | 1.72783i | −0.106865 | + | 1.99714i | 2.23744 | + | 1.29179i | −1.65588 | + | 1.80501i | 0.500000 | + | 0.866025i | 2.15378 | − | 1.83337i | −2.97082 | + | 0.417410i | −0.850996 | − | 3.55324i |
85.10 | −0.930716 | + | 1.06479i | −1.06868 | − | 1.36306i | −0.267536 | − | 1.98203i | 0.220108 | + | 0.127079i | 2.44600 | + | 0.130709i | 0.500000 | + | 0.866025i | 2.35943 | + | 1.55983i | −0.715859 | + | 2.91334i | −0.340170 | + | 0.116093i |
85.11 | −0.870602 | + | 1.11447i | −0.119319 | + | 1.72794i | −0.484104 | − | 1.94053i | −0.525102 | − | 0.303168i | −1.82186 | − | 1.63732i | 0.500000 | + | 0.866025i | 2.58413 | + | 1.14991i | −2.97153 | − | 0.412353i | 0.795028 | − | 0.321274i |
85.12 | −0.828761 | − | 1.14593i | 1.26004 | + | 1.18840i | −0.626309 | + | 1.89940i | 3.33304 | + | 1.92433i | 0.317546 | − | 2.42882i | 0.500000 | + | 0.866025i | 2.69564 | − | 0.856447i | 0.175418 | + | 2.99487i | −0.557147 | − | 5.41424i |
85.13 | −0.589905 | + | 1.28531i | 1.55592 | + | 0.760985i | −1.30402 | − | 1.51642i | −3.48956 | − | 2.01470i | −1.89595 | + | 1.55093i | 0.500000 | + | 0.866025i | 2.71831 | − | 0.781526i | 1.84180 | + | 2.36807i | 4.64802 | − | 3.29667i |
85.14 | −0.578023 | − | 1.29069i | −1.26004 | − | 1.18840i | −1.33178 | + | 1.49210i | −3.33304 | − | 1.92433i | −0.805523 | + | 2.31325i | 0.500000 | + | 0.866025i | 2.69564 | + | 0.856447i | 0.175418 | + | 2.99487i | −0.557147 | + | 5.41424i |
85.15 | −0.402403 | − | 1.35576i | 0.120790 | + | 1.72783i | −1.67614 | + | 1.09112i | −2.23744 | − | 1.29179i | 2.29391 | − | 0.859046i | 0.500000 | + | 0.866025i | 2.15378 | + | 1.83337i | −2.97082 | + | 0.417410i | −0.850996 | + | 3.55324i |
85.16 | −0.305084 | + | 1.38091i | 1.64418 | + | 0.544667i | −1.81385 | − | 0.842588i | 2.23227 | + | 1.28880i | −1.25375 | + | 2.10431i | 0.500000 | + | 0.866025i | 1.71692 | − | 2.24771i | 2.40668 | + | 1.79106i | −2.46075 | + | 2.68938i |
85.17 | −0.248810 | + | 1.39215i | −1.73065 | − | 0.0695272i | −1.87619 | − | 0.692764i | 1.22717 | + | 0.708510i | 0.527397 | − | 2.39204i | 0.500000 | + | 0.866025i | 1.43125 | − | 2.43958i | 2.99033 | + | 0.240655i | −1.29169 | + | 1.53213i |
85.18 | −0.152619 | − | 1.40595i | −1.40449 | + | 1.01361i | −1.95342 | + | 0.429150i | 1.92732 | + | 1.11274i | 1.63945 | + | 1.81995i | 0.500000 | + | 0.866025i | 0.901492 | + | 2.68092i | 0.945172 | − | 2.84722i | 1.27031 | − | 2.87955i |
85.19 | 0.105921 | − | 1.41024i | −1.34597 | − | 1.09012i | −1.97756 | − | 0.298749i | 2.33508 | + | 1.34816i | −1.67990 | + | 1.78267i | 0.500000 | + | 0.866025i | −0.630773 | + | 2.75720i | 0.623266 | + | 2.93454i | 2.14857 | − | 3.15023i |
85.20 | 0.109593 | + | 1.40996i | 0.416131 | − | 1.68132i | −1.97598 | + | 0.309044i | −0.226907 | − | 0.131005i | 2.41620 | + | 0.402467i | 0.500000 | + | 0.866025i | −0.652293 | − | 2.75218i | −2.65367 | − | 1.39930i | 0.159844 | − | 0.334288i |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
72.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 504.2.cs.a | ✓ | 72 |
8.b | even | 2 | 1 | inner | 504.2.cs.a | ✓ | 72 |
9.c | even | 3 | 1 | inner | 504.2.cs.a | ✓ | 72 |
72.n | even | 6 | 1 | inner | 504.2.cs.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.cs.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
504.2.cs.a | ✓ | 72 | 8.b | even | 2 | 1 | inner |
504.2.cs.a | ✓ | 72 | 9.c | even | 3 | 1 | inner |
504.2.cs.a | ✓ | 72 | 72.n | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 108 T_{5}^{70} + 6402 T_{5}^{68} - 262128 T_{5}^{66} + 8189076 T_{5}^{64} + \cdots + 112392565559296 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).