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Properties

Label 656.2.o.a
Level $656$
Weight $2$
Character orbit 656.o
Analytic conductor $5.238$
Analytic rank $0$
Dimension $164$
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] = [656,2,Mod(245,656)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(656, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("656.245");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 656 = 2^{4} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 656.o (of order \(4\), degree \(2\), 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: \(5.23818637260\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164 q - 4 q^{2} - 4 q^{4} - 4 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 164 q - 4 q^{2} - 4 q^{4} - 4 q^{5} - 16 q^{8} - 16 q^{10} - 4 q^{16} + 4 q^{18} - 4 q^{20} - 16 q^{21} - 32 q^{31} + 16 q^{32} - 8 q^{33} + 40 q^{36} - 4 q^{37} - 44 q^{40} + 52 q^{42} - 36 q^{43} + 28 q^{45} - 16 q^{46} + 132 q^{49} + 16 q^{50} + 8 q^{51} + 20 q^{59} - 4 q^{61} - 48 q^{62} + 68 q^{64} + 76 q^{66} + 20 q^{72} - 56 q^{74} - 32 q^{77} - 8 q^{78} - 100 q^{80} - 140 q^{81} + 44 q^{82} - 4 q^{83} - 40 q^{84} - 12 q^{86} - 8 q^{90} - 80 q^{91} + 112 q^{92} - 136 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1 −1.41162 + 0.0856576i −2.26252 − 2.26252i 1.98533 − 0.241831i −1.97321 − 1.97321i 3.38762 + 3.00001i 2.05040 −2.78180 + 0.511432i 7.23800i 2.95444 + 2.61640i
245.2 −1.41162 + 0.0856576i 2.26252 + 2.26252i 1.98533 − 0.241831i −1.97321 − 1.97321i −3.38762 − 3.00001i −2.05040 −2.78180 + 0.511432i 7.23800i 2.95444 + 2.61640i
245.3 −1.41115 + 0.0929645i −0.636852 − 0.636852i 1.98272 − 0.262375i −0.612020 − 0.612020i 0.957902 + 0.839492i 0.878659 −2.77353 + 0.554573i − 2.18884i 0.920552 + 0.806759i
245.4 −1.41115 + 0.0929645i 0.636852 + 0.636852i 1.98272 − 0.262375i −0.612020 − 0.612020i −0.957902 − 0.839492i −0.878659 −2.77353 + 0.554573i − 2.18884i 0.920552 + 0.806759i
245.5 −1.40380 − 0.171266i −0.592959 − 0.592959i 1.94134 + 0.480849i 2.80269 + 2.80269i 0.730845 + 0.933953i 2.86296 −2.64290 − 1.00750i − 2.29680i −3.45442 − 4.41443i
245.6 −1.40380 − 0.171266i 0.592959 + 0.592959i 1.94134 + 0.480849i 2.80269 + 2.80269i −0.730845 − 0.933953i −2.86296 −2.64290 − 1.00750i − 2.29680i −3.45442 − 4.41443i
245.7 −1.37799 + 0.318038i −1.64581 − 1.64581i 1.79770 − 0.876504i 1.05875 + 1.05875i 2.79134 + 1.74448i 0.937357 −2.19845 + 1.77955i 2.41740i −1.79567 − 1.12222i
245.8 −1.37799 + 0.318038i 1.64581 + 1.64581i 1.79770 − 0.876504i 1.05875 + 1.05875i −2.79134 − 1.74448i −0.937357 −2.19845 + 1.77955i 2.41740i −1.79567 − 1.12222i
245.9 −1.37043 − 0.349154i −1.41376 − 1.41376i 1.75618 + 0.956986i −1.38016 − 1.38016i 1.44384 + 2.43108i −4.21319 −2.07260 − 1.92467i 0.997411i 1.40953 + 2.37331i
245.10 −1.37043 − 0.349154i 1.41376 + 1.41376i 1.75618 + 0.956986i −1.38016 − 1.38016i −1.44384 − 2.43108i 4.21319 −2.07260 − 1.92467i 0.997411i 1.40953 + 2.37331i
245.11 −1.24247 + 0.675473i −1.75443 − 1.75443i 1.08747 − 1.67851i 1.93318 + 1.93318i 3.36490 + 0.994760i −4.11187 −0.217364 + 2.82006i 3.15605i −3.70774 − 1.09611i
245.12 −1.24247 + 0.675473i 1.75443 + 1.75443i 1.08747 − 1.67851i 1.93318 + 1.93318i −3.36490 − 0.994760i 4.11187 −0.217364 + 2.82006i 3.15605i −3.70774 − 1.09611i
245.13 −1.23625 + 0.686792i −0.0704098 − 0.0704098i 1.05663 − 1.69809i −2.73754 − 2.73754i 0.135401 + 0.0386873i −4.71049 −0.140026 + 2.82496i − 2.99008i 5.26441 + 1.50417i
245.14 −1.23625 + 0.686792i 0.0704098 + 0.0704098i 1.05663 − 1.69809i −2.73754 − 2.73754i −0.135401 − 0.0386873i 4.71049 −0.140026 + 2.82496i − 2.99008i 5.26441 + 1.50417i
245.15 −1.19791 − 0.751672i −2.20062 − 2.20062i 0.869979 + 1.80087i 1.30890 + 1.30890i 0.982001 + 4.29028i 0.358626 0.311506 − 2.81122i 6.68543i −0.584080 − 2.55180i
245.16 −1.19791 − 0.751672i 2.20062 + 2.20062i 0.869979 + 1.80087i 1.30890 + 1.30890i −0.982001 − 4.29028i −0.358626 0.311506 − 2.81122i 6.68543i −0.584080 − 2.55180i
245.17 −1.17278 − 0.790310i −0.846502 − 0.846502i 0.750820 + 1.85372i 0.372889 + 0.372889i 0.323761 + 1.66176i 3.63981 0.584465 − 2.76738i − 1.56687i −0.142618 − 0.732014i
245.18 −1.17278 − 0.790310i 0.846502 + 0.846502i 0.750820 + 1.85372i 0.372889 + 0.372889i −0.323761 − 1.66176i −3.63981 0.584465 − 2.76738i − 1.56687i −0.142618 − 0.732014i
245.19 −1.07956 + 0.913543i −0.375813 − 0.375813i 0.330879 − 1.97244i 0.756192 + 0.756192i 0.749032 + 0.0623897i −0.930632 1.44471 + 2.43163i − 2.71753i −1.50716 − 0.125537i
245.20 −1.07956 + 0.913543i 0.375813 + 0.375813i 0.330879 − 1.97244i 0.756192 + 0.756192i −0.749032 − 0.0623897i 0.930632 1.44471 + 2.43163i − 2.71753i −1.50716 − 0.125537i
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
41.b even 2 1 inner
656.o even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 656.2.o.a ✓ 164
16.e even 4 1 inner 656.2.o.a ✓ 164
41.b even 2 1 inner 656.2.o.a ✓ 164
656.o even 4 1 inner 656.2.o.a ✓ 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
656.2.o.a ✓ 164 1.a even 1 1 trivial
656.2.o.a ✓ 164 16.e even 4 1 inner
656.2.o.a ✓ 164 41.b even 2 1 inner
656.2.o.a ✓ 164 656.o even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(656, [\chi])\).