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Properties

Label 784.2.x.p
Level $784$
Weight $2$
Character orbit 784.x
Analytic conductor $6.260$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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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: \(96\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 4 q^{4} + 8 q^{11} + 64 q^{15} - 36 q^{16} - 20 q^{18} - 56 q^{22} - 32 q^{29} - 96 q^{30} - 40 q^{32} + 80 q^{36} + 16 q^{37} + 16 q^{43} - 4 q^{44} - 64 q^{46} - 56 q^{50} - 16 q^{53} + 20 q^{58} - 8 q^{60} + 88 q^{64} - 40 q^{67} + 196 q^{72} + 28 q^{74} + 112 q^{78} - 80 q^{79} + 48 q^{81} + 108 q^{86} + 100 q^{88} + 128 q^{95} - 80 q^{99}+O(q^{100}) \) 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 −1.40346 − 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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.24
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{96} - 336 T_{3}^{92} + 69072 T_{3}^{88} - 8966400 T_{3}^{84} + 849400992 T_{3}^{80} + \cdots + 4294967296 \) Copy content Toggle raw display
\( T_{5}^{96} - 832 T_{5}^{92} + 426416 T_{5}^{88} - 139298048 T_{5}^{84} + 33435938208 T_{5}^{80} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display