Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [504,2,Mod(115,504)]
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([3, 3, 4, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.115");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.bf (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: | \(180\) |
Relative dimension: | \(90\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
115.1 | −1.41416 | − | 0.0128144i | 1.65702 | + | 0.504266i | 1.99967 | + | 0.0362430i | 2.12331 | − | 3.67768i | −2.33682 | − | 0.734345i | 2.60163 | + | 0.481180i | −2.82738 | − | 0.0768778i | 2.49143 | + | 1.67116i | −3.04982 | + | 5.17360i |
115.2 | −1.41416 | + | 0.0128144i | 1.65702 | + | 0.504266i | 1.99967 | − | 0.0362430i | −2.12331 | + | 3.67768i | −2.34975 | − | 0.691877i | −2.60163 | − | 0.481180i | −2.82738 | + | 0.0768778i | 2.49143 | + | 1.67116i | 2.95556 | − | 5.22802i |
115.3 | −1.41229 | − | 0.0736946i | −1.11864 | − | 1.32236i | 1.98914 | + | 0.208157i | 1.36131 | − | 2.35786i | 1.48240 | + | 1.95000i | −1.25248 | + | 2.33051i | −2.79390 | − | 0.440567i | −0.497287 | + | 2.95850i | −2.09633 | + | 3.22967i |
115.4 | −1.41229 | + | 0.0736946i | −1.11864 | − | 1.32236i | 1.98914 | − | 0.208157i | −1.36131 | + | 2.35786i | 1.67730 | + | 1.78512i | 1.25248 | − | 2.33051i | −2.79390 | + | 0.440567i | −0.497287 | + | 2.95850i | 1.74881 | − | 3.43031i |
115.5 | −1.40172 | − | 0.187532i | 0.894629 | − | 1.48312i | 1.92966 | + | 0.525737i | −0.356494 | + | 0.617465i | −1.53216 | + | 1.91115i | −0.577640 | + | 2.58192i | −2.60626 | − | 1.09881i | −1.39928 | − | 2.65368i | 0.615501 | − | 0.798662i |
115.6 | −1.40172 | + | 0.187532i | 0.894629 | − | 1.48312i | 1.92966 | − | 0.525737i | 0.356494 | − | 0.617465i | −0.975890 | + | 2.24669i | 0.577640 | − | 2.58192i | −2.60626 | + | 1.09881i | −1.39928 | − | 2.65368i | −0.383911 | + | 0.932370i |
115.7 | −1.37865 | − | 0.315175i | −0.265334 | + | 1.71161i | 1.80133 | + | 0.869029i | −1.58477 | + | 2.74491i | 0.905257 | − | 2.27607i | 2.63076 | + | 0.281244i | −2.20950 | − | 1.76582i | −2.85920 | − | 0.908295i | 3.04997 | − | 3.28478i |
115.8 | −1.37865 | + | 0.315175i | −0.265334 | + | 1.71161i | 1.80133 | − | 0.869029i | 1.58477 | − | 2.74491i | −0.173654 | − | 2.44333i | −2.63076 | − | 0.281244i | −2.20950 | + | 1.76582i | −2.85920 | − | 0.908295i | −1.31972 | + | 4.28374i |
115.9 | −1.37396 | − | 0.335011i | −1.69738 | + | 0.344799i | 1.77554 | + | 0.920584i | 0.237272 | − | 0.410968i | 2.44765 | + | 0.0949016i | −2.23531 | − | 1.41542i | −2.13111 | − | 1.85967i | 2.76223 | − | 1.17051i | −0.463682 | + | 0.485165i |
115.10 | −1.37396 | + | 0.335011i | −1.69738 | + | 0.344799i | 1.77554 | − | 0.920584i | −0.237272 | + | 0.410968i | 2.21663 | − | 1.04238i | 2.23531 | + | 1.41542i | −2.13111 | + | 1.85967i | 2.76223 | − | 1.17051i | 0.188324 | − | 0.644143i |
115.11 | −1.32350 | − | 0.498350i | 0.336489 | + | 1.69905i | 1.50330 | + | 1.31913i | 0.402355 | − | 0.696900i | 0.401379 | − | 2.41638i | 0.321531 | − | 2.62614i | −1.33222 | − | 2.49503i | −2.77355 | + | 1.14342i | −0.879816 | + | 0.721832i |
115.12 | −1.32350 | + | 0.498350i | 0.336489 | + | 1.69905i | 1.50330 | − | 1.31913i | −0.402355 | + | 0.696900i | −1.29206 | − | 2.08100i | −0.321531 | + | 2.62614i | −1.33222 | + | 2.49503i | −2.77355 | + | 1.14342i | 0.185217 | − | 1.12286i |
115.13 | −1.29841 | − | 0.560472i | 1.69247 | − | 0.368153i | 1.37174 | + | 1.45545i | −0.863157 | + | 1.49503i | −2.40386 | − | 0.470569i | 2.59622 | + | 0.509573i | −0.965350 | − | 2.65859i | 2.72893 | − | 1.24618i | 1.95866 | − | 1.45739i |
115.14 | −1.29841 | + | 0.560472i | 1.69247 | − | 0.368153i | 1.37174 | − | 1.45545i | 0.863157 | − | 1.49503i | −1.99119 | + | 1.42660i | −2.59622 | − | 0.509573i | −0.965350 | + | 2.65859i | 2.72893 | − | 1.24618i | −0.282810 | + | 2.42494i |
115.15 | −1.16893 | − | 0.795991i | −0.801348 | − | 1.53553i | 0.732796 | + | 1.86092i | −1.06898 | + | 1.85154i | −0.285545 | + | 2.43279i | −2.52157 | − | 0.801064i | 0.624685 | − | 2.75858i | −1.71568 | + | 2.46098i | 2.72337 | − | 1.31341i |
115.16 | −1.16893 | + | 0.795991i | −0.801348 | − | 1.53553i | 0.732796 | − | 1.86092i | 1.06898 | − | 1.85154i | 2.15899 | + | 1.15706i | 2.52157 | + | 0.801064i | 0.624685 | + | 2.75858i | −1.71568 | + | 2.46098i | 0.224237 | + | 3.01522i |
115.17 | −1.15708 | − | 0.813118i | 1.26585 | + | 1.18221i | 0.677678 | + | 1.88169i | 0.495626 | − | 0.858449i | −0.503410 | − | 2.39720i | −1.90467 | + | 1.83636i | 0.745907 | − | 2.72830i | 0.204739 | + | 2.99301i | −1.27150 | + | 0.590294i |
115.18 | −1.15708 | + | 0.813118i | 1.26585 | + | 1.18221i | 0.677678 | − | 1.88169i | −0.495626 | + | 0.858449i | −2.42597 | − | 0.338635i | 1.90467 | − | 1.83636i | 0.745907 | + | 2.72830i | 0.204739 | + | 2.99301i | −0.124541 | − | 1.39630i |
115.19 | −1.14308 | − | 0.832691i | −1.17641 | + | 1.27125i | 0.613252 | + | 1.90366i | 1.57644 | − | 2.73047i | 2.40328 | − | 0.473549i | 1.25267 | + | 2.33041i | 0.884165 | − | 2.68668i | −0.232133 | − | 2.99101i | −4.07562 | + | 1.80845i |
115.20 | −1.14308 | + | 0.832691i | −1.17641 | + | 1.27125i | 0.613252 | − | 1.90366i | −1.57644 | + | 2.73047i | 0.286170 | − | 2.43272i | −1.25267 | − | 2.33041i | 0.884165 | + | 2.68668i | −0.232133 | − | 2.99101i | −0.471646 | − | 4.43382i |
See next 80 embeddings (of 180 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
63.t | odd | 6 | 1 | inner |
504.bf | 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.bf.b | ✓ | 180 |
7.d | odd | 6 | 1 | 504.2.cz.b | yes | 180 | |
8.d | odd | 2 | 1 | inner | 504.2.bf.b | ✓ | 180 |
9.c | even | 3 | 1 | 504.2.cz.b | yes | 180 | |
56.m | even | 6 | 1 | 504.2.cz.b | yes | 180 | |
63.t | odd | 6 | 1 | inner | 504.2.bf.b | ✓ | 180 |
72.p | odd | 6 | 1 | 504.2.cz.b | yes | 180 | |
504.bf | even | 6 | 1 | inner | 504.2.bf.b | ✓ | 180 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.bf.b | ✓ | 180 | 1.a | even | 1 | 1 | trivial |
504.2.bf.b | ✓ | 180 | 8.d | odd | 2 | 1 | inner |
504.2.bf.b | ✓ | 180 | 63.t | odd | 6 | 1 | inner |
504.2.bf.b | ✓ | 180 | 504.bf | even | 6 | 1 | inner |
504.2.cz.b | yes | 180 | 7.d | odd | 6 | 1 | |
504.2.cz.b | yes | 180 | 9.c | even | 3 | 1 | |
504.2.cz.b | yes | 180 | 56.m | even | 6 | 1 | |
504.2.cz.b | yes | 180 | 72.p | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{180} + 264 T_{5}^{178} + 36267 T_{5}^{176} + 3416808 T_{5}^{174} + 246532878 T_{5}^{172} + \cdots + 86\!\cdots\!04 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).