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: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 112) |
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 | −1.40268 | − | 0.180222i | −0.152579 | − | 0.569433i | 1.93504 | + | 0.505589i | 0.414228 | − | 1.54592i | 0.111396 | + | 0.826233i | 0 | −2.62313 | − | 1.05792i | 2.29710 | − | 1.32623i | −0.859639 | + | 2.09378i | ||
165.2 | −1.29835 | + | 0.560616i | 0.752181 | + | 2.80718i | 1.37142 | − | 1.45575i | −0.998394 | + | 3.72606i | −2.55034 | − | 3.22301i | 0 | −0.964462 | + | 2.65891i | −4.71639 | + | 2.72301i | −0.792625 | − | 5.39744i | ||
165.3 | −0.329210 | − | 1.37536i | −0.231031 | − | 0.862219i | −1.78324 | + | 0.905566i | 0.857449 | − | 3.20004i | −1.10981 | + | 0.601602i | 0 | 1.83254 | + | 2.15448i | 1.90803 | − | 1.10160i | −4.68350 | − | 0.125816i | ||
165.4 | 0.114575 | + | 1.40956i | −0.509227 | − | 1.90046i | −1.97375 | + | 0.323001i | −0.792364 | + | 2.95714i | 2.62048 | − | 0.935533i | 0 | −0.681431 | − | 2.74511i | −0.754366 | + | 0.435533i | −4.25907 | − | 0.778075i | ||
165.5 | 1.03822 | − | 0.960264i | 0.219607 | + | 0.819585i | 0.155788 | − | 1.99392i | −0.356864 | + | 1.33183i | 1.01502 | + | 0.640026i | 0 | −1.75295 | − | 2.21972i | 1.97458 | − | 1.14003i | 0.908409 | + | 1.72541i | ||
165.6 | 1.37745 | − | 0.320359i | −0.811002 | − | 3.02670i | 1.79474 | − | 0.882557i | 0.143894 | − | 0.537019i | −2.08675 | − | 3.90932i | 0 | 2.18943 | − | 1.79064i | −5.90511 | + | 3.40932i | 0.0261678 | − | 0.785815i | ||
373.1 | −1.35072 | − | 0.418990i | −0.819585 | − | 0.219607i | 1.64889 | + | 1.13188i | 1.33183 | − | 0.356864i | 1.01502 | + | 0.640026i | 0 | −1.75295 | − | 2.21972i | −1.97458 | − | 1.14003i | −1.94846 | − | 0.0760019i | ||
373.2 | −1.02649 | + | 0.972785i | 0.862219 | + | 0.231031i | 0.107378 | − | 1.99712i | −3.20004 | + | 0.857449i | −1.10981 | + | 0.601602i | 0 | 1.83254 | + | 2.15448i | −1.90803 | − | 1.10160i | 2.45071 | − | 3.99312i | ||
373.3 | −0.966164 | − | 1.03273i | 3.02670 | + | 0.811002i | −0.133053 | + | 1.99557i | −0.537019 | + | 0.143894i | −2.08675 | − | 3.90932i | 0 | 2.18943 | − | 1.79064i | 5.90511 | + | 3.40932i | 0.667452 | + | 0.415570i | ||
373.4 | 0.545265 | + | 1.30487i | 0.569433 | + | 0.152579i | −1.40537 | + | 1.42300i | −1.54592 | + | 0.414228i | 0.111396 | + | 0.826233i | 0 | −2.62313 | − | 1.05792i | −2.29710 | − | 1.32623i | −1.38345 | − | 1.79136i | ||
373.5 | 1.13468 | + | 0.844095i | −2.80718 | − | 0.752181i | 0.575008 | + | 1.91556i | 3.72606 | − | 0.998394i | −2.55034 | − | 3.22301i | 0 | −0.964462 | + | 2.65891i | 4.71639 | + | 2.72301i | 5.07063 | + | 2.01229i | ||
373.6 | 1.16343 | − | 0.804007i | 1.90046 | + | 0.509227i | 0.707146 | − | 1.87081i | 2.95714 | − | 0.792364i | 2.62048 | − | 0.935533i | 0 | −0.681431 | − | 2.74511i | 0.754366 | + | 0.435533i | 2.80337 | − | 3.29942i | ||
557.1 | −1.35072 | + | 0.418990i | −0.819585 | + | 0.219607i | 1.64889 | − | 1.13188i | 1.33183 | + | 0.356864i | 1.01502 | − | 0.640026i | 0 | −1.75295 | + | 2.21972i | −1.97458 | + | 1.14003i | −1.94846 | + | 0.0760019i | ||
557.2 | −1.02649 | − | 0.972785i | 0.862219 | − | 0.231031i | 0.107378 | + | 1.99712i | −3.20004 | − | 0.857449i | −1.10981 | − | 0.601602i | 0 | 1.83254 | − | 2.15448i | −1.90803 | + | 1.10160i | 2.45071 | + | 3.99312i | ||
557.3 | −0.966164 | + | 1.03273i | 3.02670 | − | 0.811002i | −0.133053 | − | 1.99557i | −0.537019 | − | 0.143894i | −2.08675 | + | 3.90932i | 0 | 2.18943 | + | 1.79064i | 5.90511 | − | 3.40932i | 0.667452 | − | 0.415570i | ||
557.4 | 0.545265 | − | 1.30487i | 0.569433 | − | 0.152579i | −1.40537 | − | 1.42300i | −1.54592 | − | 0.414228i | 0.111396 | − | 0.826233i | 0 | −2.62313 | + | 1.05792i | −2.29710 | + | 1.32623i | −1.38345 | + | 1.79136i | ||
557.5 | 1.13468 | − | 0.844095i | −2.80718 | + | 0.752181i | 0.575008 | − | 1.91556i | 3.72606 | + | 0.998394i | −2.55034 | + | 3.22301i | 0 | −0.964462 | − | 2.65891i | 4.71639 | − | 2.72301i | 5.07063 | − | 2.01229i | ||
557.6 | 1.16343 | + | 0.804007i | 1.90046 | − | 0.509227i | 0.707146 | + | 1.87081i | 2.95714 | + | 0.792364i | 2.62048 | + | 0.935533i | 0 | −0.681431 | + | 2.74511i | 0.754366 | − | 0.435533i | 2.80337 | + | 3.29942i | ||
765.1 | −1.40268 | + | 0.180222i | −0.152579 | + | 0.569433i | 1.93504 | − | 0.505589i | 0.414228 | + | 1.54592i | 0.111396 | − | 0.826233i | 0 | −2.62313 | + | 1.05792i | 2.29710 | + | 1.32623i | −0.859639 | − | 2.09378i | ||
765.2 | −1.29835 | − | 0.560616i | 0.752181 | − | 2.80718i | 1.37142 | + | 1.45575i | −0.998394 | − | 3.72606i | −2.55034 | + | 3.22301i | 0 | −0.964462 | − | 2.65891i | −4.71639 | − | 2.72301i | −0.792625 | + | 5.39744i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
16.e | even | 4 | 1 | inner |
112.w | even | 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.m | 24 | |
7.b | odd | 2 | 1 | 784.2.x.l | 24 | ||
7.c | even | 3 | 1 | 784.2.m.h | 12 | ||
7.c | even | 3 | 1 | inner | 784.2.x.m | 24 | |
7.d | odd | 6 | 1 | 112.2.m.d | ✓ | 12 | |
7.d | odd | 6 | 1 | 784.2.x.l | 24 | ||
16.e | even | 4 | 1 | inner | 784.2.x.m | 24 | |
28.f | even | 6 | 1 | 448.2.m.d | 12 | ||
56.j | odd | 6 | 1 | 896.2.m.g | 12 | ||
56.m | even | 6 | 1 | 896.2.m.h | 12 | ||
112.l | odd | 4 | 1 | 784.2.x.l | 24 | ||
112.v | even | 12 | 1 | 448.2.m.d | 12 | ||
112.v | even | 12 | 1 | 896.2.m.h | 12 | ||
112.w | even | 12 | 1 | 784.2.m.h | 12 | ||
112.w | even | 12 | 1 | inner | 784.2.x.m | 24 | |
112.x | odd | 12 | 1 | 112.2.m.d | ✓ | 12 | |
112.x | odd | 12 | 1 | 784.2.x.l | 24 | ||
112.x | odd | 12 | 1 | 896.2.m.g | 12 | ||
224.bc | odd | 24 | 2 | 7168.2.a.bj | 12 | ||
224.be | even | 24 | 2 | 7168.2.a.bi | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
112.2.m.d | ✓ | 12 | 7.d | odd | 6 | 1 | |
112.2.m.d | ✓ | 12 | 112.x | odd | 12 | 1 | |
448.2.m.d | 12 | 28.f | even | 6 | 1 | ||
448.2.m.d | 12 | 112.v | even | 12 | 1 | ||
784.2.m.h | 12 | 7.c | even | 3 | 1 | ||
784.2.m.h | 12 | 112.w | even | 12 | 1 | ||
784.2.x.l | 24 | 7.b | odd | 2 | 1 | ||
784.2.x.l | 24 | 7.d | odd | 6 | 1 | ||
784.2.x.l | 24 | 112.l | odd | 4 | 1 | ||
784.2.x.l | 24 | 112.x | odd | 12 | 1 | ||
784.2.x.m | 24 | 1.a | even | 1 | 1 | trivial | |
784.2.x.m | 24 | 7.c | even | 3 | 1 | inner | |
784.2.x.m | 24 | 16.e | even | 4 | 1 | inner | |
784.2.x.m | 24 | 112.w | even | 12 | 1 | inner | |
896.2.m.g | 12 | 56.j | odd | 6 | 1 | ||
896.2.m.g | 12 | 112.x | odd | 12 | 1 | ||
896.2.m.h | 12 | 56.m | even | 6 | 1 | ||
896.2.m.h | 12 | 112.v | even | 12 | 1 | ||
7168.2.a.bi | 12 | 224.be | even | 24 | 2 | ||
7168.2.a.bj | 12 | 224.bc | odd | 24 | 2 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):
\( T_{3}^{24} - 4 T_{3}^{23} + 8 T_{3}^{22} - 24 T_{3}^{21} - 28 T_{3}^{20} + 288 T_{3}^{19} - 640 T_{3}^{18} + \cdots + 4096 \) |
\( T_{5}^{24} - 4 T_{5}^{23} + 8 T_{5}^{22} - 56 T_{5}^{21} - 28 T_{5}^{20} + 712 T_{5}^{19} + \cdots + 5308416 \) |