Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [784,2,Mod(165,784)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("784.165");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 784.x (of order \(12\), degree \(4\), not 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: | \(6.26027151847\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
165.1 | −1.40346 | − | 0.174081i | −0.548135 | − | 2.04567i | 1.93939 | + | 0.488631i | 0.207992 | − | 0.776238i | 0.413172 | + | 2.96643i | 0 | −2.63679 | − | 1.02339i | −1.28622 | + | 0.742602i | −0.427037 | + | 1.05321i | ||
165.2 | −1.40346 | − | 0.174081i | 0.548135 | + | 2.04567i | 1.93939 | + | 0.488631i | −0.207992 | + | 0.776238i | −0.413172 | − | 2.96643i | 0 | −2.63679 | − | 1.02339i | −1.28622 | + | 0.742602i | 0.427037 | − | 1.05321i | ||
165.3 | −1.28409 | + | 0.592547i | −0.349212 | − | 1.30328i | 1.29778 | − | 1.52177i | −0.743228 | + | 2.77376i | 1.22067 | + | 1.46660i | 0 | −0.764745 | + | 2.72308i | 1.02150 | − | 0.589763i | −0.689213 | − | 4.00216i | ||
165.4 | −1.28409 | + | 0.592547i | 0.349212 | + | 1.30328i | 1.29778 | − | 1.52177i | 0.743228 | − | 2.77376i | −1.22067 | − | 1.46660i | 0 | −0.764745 | + | 2.72308i | 1.02150 | − | 0.589763i | 0.689213 | + | 4.00216i | ||
165.5 | −0.947436 | + | 1.04994i | −0.0278645 | − | 0.103992i | −0.204729 | − | 1.98949i | 1.01195 | − | 3.77665i | 0.135585 | + | 0.0692696i | 0 | 2.28281 | + | 1.66997i | 2.58804 | − | 1.49420i | 3.00648 | + | 4.64061i | ||
165.6 | −0.947436 | + | 1.04994i | 0.0278645 | + | 0.103992i | −0.204729 | − | 1.98949i | −1.01195 | + | 3.77665i | −0.135585 | − | 0.0692696i | 0 | 2.28281 | + | 1.66997i | 2.58804 | − | 1.49420i | −3.00648 | − | 4.64061i | ||
165.7 | −0.843680 | − | 1.13499i | −0.164681 | − | 0.614599i | −0.576410 | + | 1.91514i | −0.389432 | + | 1.45338i | −0.558626 | + | 0.705436i | 0 | 2.65997 | − | 0.961542i | 2.24746 | − | 1.29757i | 1.97813 | − | 0.784185i | ||
165.8 | −0.843680 | − | 1.13499i | 0.164681 | + | 0.614599i | −0.576410 | + | 1.91514i | 0.389432 | − | 1.45338i | 0.558626 | − | 0.705436i | 0 | 2.65997 | − | 0.961542i | 2.24746 | − | 1.29757i | −1.97813 | + | 0.784185i | ||
165.9 | −0.646444 | − | 1.25782i | −0.729365 | − | 2.72203i | −1.16422 | + | 1.62622i | −0.691444 | + | 2.58050i | −2.95233 | + | 2.67705i | 0 | 2.79809 | + | 0.413120i | −4.27938 | + | 2.47070i | 3.69279 | − | 0.798439i | ||
165.10 | −0.646444 | − | 1.25782i | 0.729365 | + | 2.72203i | −1.16422 | + | 1.62622i | 0.691444 | − | 2.58050i | 2.95233 | − | 2.67705i | 0 | 2.79809 | + | 0.413120i | −4.27938 | + | 2.47070i | −3.69279 | + | 0.798439i | ||
165.11 | 0.0679220 | + | 1.41258i | −0.856173 | − | 3.19528i | −1.99077 | + | 0.191891i | 0.238507 | − | 0.890122i | 4.45544 | − | 1.42644i | 0 | −0.406279 | − | 2.79910i | −6.87872 | + | 3.97143i | 1.27357 | + | 0.276452i | ||
165.12 | 0.0679220 | + | 1.41258i | 0.856173 | + | 3.19528i | −1.99077 | + | 0.191891i | −0.238507 | + | 0.890122i | −4.45544 | + | 1.42644i | 0 | −0.406279 | − | 2.79910i | −6.87872 | + | 3.97143i | −1.27357 | − | 0.276452i | ||
165.13 | 0.464209 | − | 1.33586i | −0.0888704 | − | 0.331669i | −1.56902 | − | 1.24023i | 0.613128 | − | 2.28822i | −0.484316 | − | 0.0352455i | 0 | −2.38512 | + | 1.52026i | 2.49597 | − | 1.44105i | −2.77212 | − | 1.88126i | ||
165.14 | 0.464209 | − | 1.33586i | 0.0888704 | + | 0.331669i | −1.56902 | − | 1.24023i | −0.613128 | + | 2.28822i | 0.484316 | + | 0.0352455i | 0 | −2.38512 | + | 1.52026i | 2.49597 | − | 1.44105i | 2.77212 | + | 1.88126i | ||
165.15 | 0.581133 | + | 1.28930i | −0.554865 | − | 2.07078i | −1.32457 | + | 1.49850i | −0.213735 | + | 0.797669i | 2.34740 | − | 1.91879i | 0 | −2.70177 | − | 0.836931i | −1.38220 | + | 0.798013i | −1.15264 | + | 0.187984i | ||
165.16 | 0.581133 | + | 1.28930i | 0.554865 | + | 2.07078i | −1.32457 | + | 1.49850i | 0.213735 | − | 0.797669i | −2.34740 | + | 1.91879i | 0 | −2.70177 | − | 0.836931i | −1.38220 | + | 0.798013i | 1.15264 | − | 0.187984i | ||
165.17 | 0.977831 | − | 1.02169i | −0.757585 | − | 2.82735i | −0.0876948 | − | 1.99808i | −0.967109 | + | 3.60930i | −3.62946 | − | 1.99065i | 0 | −2.12716 | − | 1.86418i | −4.82188 | + | 2.78391i | 2.74191 | + | 4.51737i | ||
165.18 | 0.977831 | − | 1.02169i | 0.757585 | + | 2.82735i | −0.0876948 | − | 1.99808i | 0.967109 | − | 3.60930i | 3.62946 | + | 1.99065i | 0 | −2.12716 | − | 1.86418i | −4.82188 | + | 2.78391i | −2.74191 | − | 4.51737i | ||
165.19 | 1.22699 | + | 0.703197i | −0.269295 | − | 1.00502i | 1.01103 | + | 1.72564i | 0.290260 | − | 1.08326i | 0.376306 | − | 1.42253i | 0 | 0.0270652 | + | 2.82830i | 1.66052 | − | 0.958704i | 1.11790 | − | 1.12505i | ||
165.20 | 1.22699 | + | 0.703197i | 0.269295 | + | 1.00502i | 1.01103 | + | 1.72564i | −0.290260 | + | 1.08326i | −0.376306 | + | 1.42253i | 0 | 0.0270652 | + | 2.82830i | 1.66052 | − | 0.958704i | −1.11790 | + | 1.12505i | ||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
16.e | even | 4 | 1 | inner |
112.l | odd | 4 | 1 | inner |
112.w | even | 12 | 1 | inner |
112.x | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 784.2.x.p | 96 | |
7.b | odd | 2 | 1 | inner | 784.2.x.p | 96 | |
7.c | even | 3 | 1 | 784.2.m.l | ✓ | 48 | |
7.c | even | 3 | 1 | inner | 784.2.x.p | 96 | |
7.d | odd | 6 | 1 | 784.2.m.l | ✓ | 48 | |
7.d | odd | 6 | 1 | inner | 784.2.x.p | 96 | |
16.e | even | 4 | 1 | inner | 784.2.x.p | 96 | |
112.l | odd | 4 | 1 | inner | 784.2.x.p | 96 | |
112.w | even | 12 | 1 | 784.2.m.l | ✓ | 48 | |
112.w | even | 12 | 1 | inner | 784.2.x.p | 96 | |
112.x | odd | 12 | 1 | 784.2.m.l | ✓ | 48 | |
112.x | odd | 12 | 1 | inner | 784.2.x.p | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
784.2.m.l | ✓ | 48 | 7.c | even | 3 | 1 | |
784.2.m.l | ✓ | 48 | 7.d | odd | 6 | 1 | |
784.2.m.l | ✓ | 48 | 112.w | even | 12 | 1 | |
784.2.m.l | ✓ | 48 | 112.x | odd | 12 | 1 | |
784.2.x.p | 96 | 1.a | even | 1 | 1 | trivial | |
784.2.x.p | 96 | 7.b | odd | 2 | 1 | inner | |
784.2.x.p | 96 | 7.c | even | 3 | 1 | inner | |
784.2.x.p | 96 | 7.d | odd | 6 | 1 | inner | |
784.2.x.p | 96 | 16.e | even | 4 | 1 | inner | |
784.2.x.p | 96 | 112.l | odd | 4 | 1 | inner | |
784.2.x.p | 96 | 112.w | even | 12 | 1 | inner | |
784.2.x.p | 96 | 112.x | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):
\( T_{3}^{96} - 336 T_{3}^{92} + 69072 T_{3}^{88} - 8966400 T_{3}^{84} + 849400992 T_{3}^{80} + \cdots + 4294967296 \) |
\( T_{5}^{96} - 832 T_{5}^{92} + 426416 T_{5}^{88} - 139298048 T_{5}^{84} + 33435938208 T_{5}^{80} + \cdots + 60\!\cdots\!36 \) |