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Properties

Label 784.2.x.l
Level $784$
Weight $2$
Character orbit 784.x
Analytic conductor $6.260$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 4 q^{5} + 8 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 4 q^{5} + 8 q^{6} - 8 q^{8} + 4 q^{10} - 8 q^{12} - 48 q^{15} - 10 q^{16} + 8 q^{17} - 40 q^{20} + 28 q^{22} + 8 q^{24} + 20 q^{26} + 8 q^{27} - 8 q^{29} + 28 q^{30} + 8 q^{31} - 12 q^{32} + 16 q^{34} - 32 q^{36} + 20 q^{37} - 16 q^{38} + 8 q^{40} + 32 q^{43} - 14 q^{44} - 40 q^{45} + 28 q^{46} - 16 q^{47} + 32 q^{48} + 88 q^{50} + 16 q^{51} + 16 q^{52} - 4 q^{53} - 64 q^{54} - 14 q^{58} + 16 q^{59} - 60 q^{60} + 20 q^{61} + 16 q^{62} - 36 q^{64} - 32 q^{65} - 12 q^{66} - 24 q^{67} + 28 q^{68} - 8 q^{69} - 6 q^{72} + 38 q^{74} + 40 q^{75} + 96 q^{76} - 152 q^{78} - 24 q^{79} - 24 q^{80} + 44 q^{81} + 16 q^{82} - 40 q^{83} - 16 q^{85} - 38 q^{86} + 14 q^{88} - 80 q^{90} + 64 q^{92} + 48 q^{93} + 24 q^{94} + 16 q^{96} + 96 q^{97} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.6
Significant digits:
Format:

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.l 24
7.b odd 2 1 784.2.x.m 24
7.c even 3 1 112.2.m.d ✓ 12
7.c even 3 1 inner 784.2.x.l 24
7.d odd 6 1 784.2.m.h 12
7.d odd 6 1 784.2.x.m 24
16.e even 4 1 inner 784.2.x.l 24
28.g odd 6 1 448.2.m.d 12
56.k odd 6 1 896.2.m.h 12
56.p even 6 1 896.2.m.g 12
112.l odd 4 1 784.2.x.m 24
112.u odd 12 1 448.2.m.d 12
112.u odd 12 1 896.2.m.h 12
112.w even 12 1 112.2.m.d ✓ 12
112.w even 12 1 inner 784.2.x.l 24
112.w even 12 1 896.2.m.g 12
112.x odd 12 1 784.2.m.h 12
112.x odd 12 1 784.2.x.m 24
224.bd even 24 2 7168.2.a.bj 12
224.bf odd 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.c even 3 1
112.2.m.d ✓ 12 112.w even 12 1
448.2.m.d 12 28.g odd 6 1
448.2.m.d 12 112.u odd 12 1
784.2.m.h 12 7.d odd 6 1
784.2.m.h 12 112.x odd 12 1
784.2.x.l 24 1.a even 1 1 trivial
784.2.x.l 24 7.c even 3 1 inner
784.2.x.l 24 16.e even 4 1 inner
784.2.x.l 24 112.w even 12 1 inner
784.2.x.m 24 7.b odd 2 1
784.2.x.m 24 7.d odd 6 1
784.2.x.m 24 112.l odd 4 1
784.2.x.m 24 112.x odd 12 1
896.2.m.g 12 56.p even 6 1
896.2.m.g 12 112.w even 12 1
896.2.m.h 12 56.k odd 6 1
896.2.m.h 12 112.u odd 12 1
7168.2.a.bi 12 224.bf odd 24 2
7168.2.a.bj 12 224.bd even 24 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{24} + 4 T_{3}^{23} + 8 T_{3}^{22} + 24 T_{3}^{21} - 28 T_{3}^{20} - 288 T_{3}^{19} - 640 T_{3}^{18} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{5}^{24} + 4 T_{5}^{23} + 8 T_{5}^{22} + 56 T_{5}^{21} - 28 T_{5}^{20} - 712 T_{5}^{19} + \cdots + 5308416 \) Copy content Toggle raw display