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Properties

Label 495.2.bi.a
Level $495$
Weight $2$
Character orbit 495.bi
Analytic conductor $3.953$
Analytic rank $0$
Dimension $192$
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] = [495,2,Mod(53,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 15, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.bi (of order \(20\), degree \(8\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 16 q^{7} + 16 q^{10} - 16 q^{13} + 48 q^{16} - 8 q^{22} + 48 q^{25} - 72 q^{28} - 32 q^{31} - 32 q^{37} - 32 q^{40} - 16 q^{43} - 48 q^{46} - 144 q^{52} - 88 q^{55} - 72 q^{58} - 48 q^{61} - 224 q^{67} - 24 q^{70} + 88 q^{73} + 96 q^{76} - 80 q^{82} + 56 q^{88} - 64 q^{91} + 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −2.46120 + 1.25405i 0 3.30932 − 4.55488i 1.98178 + 1.03564i 0 −1.21066 + 0.191749i −1.56864 + 9.90399i 0 −6.17630 − 0.0636822i
53.2 −2.26761 + 1.15540i 0 2.63152 − 3.62198i −2.06632 + 0.854596i 0 0.180965 − 0.0286621i −0.986162 + 6.22638i 0 3.69820 − 4.32532i
53.3 −2.11421 + 1.07724i 0 2.13386 − 2.93701i 2.23504 + 0.0676631i 0 3.72010 − 0.589207i −0.605170 + 3.82089i 0 −4.79824 + 2.26463i
53.4 −1.77097 + 0.902357i 0 1.14653 − 1.57807i −1.42823 + 1.72051i 0 1.46633 − 0.232243i 0.0153607 − 0.0969834i 0 0.976851 − 4.33575i
53.5 −1.64645 + 0.838909i 0 0.831465 − 1.14441i −0.0153116 − 2.23602i 0 2.31969 − 0.367403i 0.169229 − 1.06847i 0 1.90102 + 3.66865i
53.6 −1.60569 + 0.818140i 0 0.733316 − 1.00932i −1.25625 − 1.84982i 0 −1.16207 + 0.184054i 0.212113 − 1.33923i 0 3.53056 + 1.94244i
53.7 −1.38862 + 0.707536i 0 0.252082 − 0.346961i 1.68655 − 1.46818i 0 −4.99764 + 0.791549i 0.383043 − 2.41844i 0 −1.30318 + 3.23204i
53.8 −1.19043 + 0.606553i 0 −0.126360 + 0.173920i 2.10094 + 0.765545i 0 −3.15982 + 0.500466i 0.462939 − 2.92288i 0 −2.96536 + 0.363005i
53.9 −0.922157 + 0.469862i 0 −0.545968 + 0.751461i −0.213292 + 2.22587i 0 3.35427 − 0.531264i 0.474192 − 2.99393i 0 −0.849165 − 2.15282i
53.10 −0.470672 + 0.239819i 0 −1.01155 + 1.39228i −2.23587 + 0.0295723i 0 −0.591659 + 0.0937095i 0.307485 − 1.94138i 0 1.04527 − 0.550124i
53.11 −0.215878 + 0.109995i 0 −1.14107 + 1.57054i 2.22948 + 0.171548i 0 3.63510 − 0.575743i 0.149382 − 0.943161i 0 −0.500164 + 0.208198i
53.12 −0.146298 + 0.0745425i 0 −1.15972 + 1.59622i 0.757983 + 2.10368i 0 −1.82807 + 0.289538i 0.102050 − 0.644318i 0 −0.267705 − 0.251262i
53.13 0.146298 − 0.0745425i 0 −1.15972 + 1.59622i −0.757983 − 2.10368i 0 −1.82807 + 0.289538i −0.102050 + 0.644318i 0 −0.267705 − 0.251262i
53.14 0.215878 − 0.109995i 0 −1.14107 + 1.57054i −2.22948 − 0.171548i 0 3.63510 − 0.575743i −0.149382 + 0.943161i 0 −0.500164 + 0.208198i
53.15 0.470672 − 0.239819i 0 −1.01155 + 1.39228i 2.23587 − 0.0295723i 0 −0.591659 + 0.0937095i −0.307485 + 1.94138i 0 1.04527 − 0.550124i
53.16 0.922157 − 0.469862i 0 −0.545968 + 0.751461i 0.213292 − 2.22587i 0 3.35427 − 0.531264i −0.474192 + 2.99393i 0 −0.849165 − 2.15282i
53.17 1.19043 − 0.606553i 0 −0.126360 + 0.173920i −2.10094 − 0.765545i 0 −3.15982 + 0.500466i −0.462939 + 2.92288i 0 −2.96536 + 0.363005i
53.18 1.38862 − 0.707536i 0 0.252082 − 0.346961i −1.68655 + 1.46818i 0 −4.99764 + 0.791549i −0.383043 + 2.41844i 0 −1.30318 + 3.23204i
53.19 1.60569 − 0.818140i 0 0.733316 − 1.00932i 1.25625 + 1.84982i 0 −1.16207 + 0.184054i −0.212113 + 1.33923i 0 3.53056 + 1.94244i
53.20 1.64645 − 0.838909i 0 0.831465 − 1.14441i 0.0153116 + 2.23602i 0 2.31969 − 0.367403i −0.169229 + 1.06847i 0 1.90102 + 3.66865i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
11.c even 5 1 inner
15.e even 4 1 inner
33.h odd 10 1 inner
55.k odd 20 1 inner
165.v even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.bi.a ✓ 192
3.b odd 2 1 inner 495.2.bi.a ✓ 192
5.c odd 4 1 inner 495.2.bi.a ✓ 192
11.c even 5 1 inner 495.2.bi.a ✓ 192
15.e even 4 1 inner 495.2.bi.a ✓ 192
33.h odd 10 1 inner 495.2.bi.a ✓ 192
55.k odd 20 1 inner 495.2.bi.a ✓ 192
165.v even 20 1 inner 495.2.bi.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.bi.a ✓ 192 1.a even 1 1 trivial
495.2.bi.a ✓ 192 3.b odd 2 1 inner
495.2.bi.a ✓ 192 5.c odd 4 1 inner
495.2.bi.a ✓ 192 11.c even 5 1 inner
495.2.bi.a ✓ 192 15.e even 4 1 inner
495.2.bi.a ✓ 192 33.h odd 10 1 inner
495.2.bi.a ✓ 192 55.k odd 20 1 inner
495.2.bi.a ✓ 192 165.v even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(495, [\chi])\).