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Properties

Label 495.2.k.c
Level $495$
Weight $2$
Character orbit 495.k
Analytic conductor $3.953$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,2,Mod(208,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.208"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{5} + 8 q^{11} - 8 q^{16} + 40 q^{20} + 32 q^{22} + 32 q^{23} - 8 q^{25} - 16 q^{31} - 8 q^{37} + 56 q^{38} - 24 q^{53} - 40 q^{55} - 160 q^{56} - 24 q^{58} - 32 q^{67} - 8 q^{70} + 32 q^{71}+ \cdots + 88 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 −1.91746 − 1.91746i 0 5.35328i −2.23563 + 0.0440169i 0 −2.40424 − 2.40424i 6.42978 − 6.42978i 0 4.37113 + 4.20233i
208.2 −1.45644 − 1.45644i 0 2.24244i 1.57013 − 1.59207i 0 −1.85474 − 1.85474i 0.353101 − 0.353101i 0 −4.60556 + 0.0319594i
208.3 −1.24819 − 1.24819i 0 1.11598i 0.445397 − 2.19126i 0 2.93706 + 2.93706i −1.10343 + 1.10343i 0 −3.29106 + 2.17918i
208.4 −0.857692 − 0.857692i 0 − 0.528729i −2.20289 − 0.383740i 0 −1.82630 − 1.82630i −2.16887 + 2.16887i 0 1.56027 + 2.21854i
208.5 −0.824369 − 0.824369i 0 − 0.640833i −0.623976 + 2.14724i 0 −0.423225 − 0.423225i −2.17702 + 2.17702i 0 2.28451 − 1.25573i
208.6 −0.478466 − 0.478466i 0 − 1.54214i 1.04698 + 1.97581i 0 3.26171 + 3.26171i −1.69480 + 1.69480i 0 0.444416 − 1.44630i
208.7 0.478466 + 0.478466i 0 − 1.54214i 1.04698 + 1.97581i 0 −3.26171 − 3.26171i 1.69480 − 1.69480i 0 −0.444416 + 1.44630i
208.8 0.824369 + 0.824369i 0 − 0.640833i −0.623976 + 2.14724i 0 0.423225 + 0.423225i 2.17702 − 2.17702i 0 −2.28451 + 1.25573i
208.9 0.857692 + 0.857692i 0 − 0.528729i −2.20289 − 0.383740i 0 1.82630 + 1.82630i 2.16887 − 2.16887i 0 −1.56027 − 2.21854i
208.10 1.24819 + 1.24819i 0 1.11598i 0.445397 − 2.19126i 0 −2.93706 − 2.93706i 1.10343 − 1.10343i 0 3.29106 − 2.17918i
208.11 1.45644 + 1.45644i 0 2.24244i 1.57013 − 1.59207i 0 1.85474 + 1.85474i −0.353101 + 0.353101i 0 4.60556 − 0.0319594i
208.12 1.91746 + 1.91746i 0 5.35328i −2.23563 + 0.0440169i 0 2.40424 + 2.40424i −6.42978 + 6.42978i 0 −4.37113 − 4.20233i
307.1 −1.91746 + 1.91746i 0 − 5.35328i −2.23563 − 0.0440169i 0 −2.40424 + 2.40424i 6.42978 + 6.42978i 0 4.37113 − 4.20233i
307.2 −1.45644 + 1.45644i 0 − 2.24244i 1.57013 + 1.59207i 0 −1.85474 + 1.85474i 0.353101 + 0.353101i 0 −4.60556 − 0.0319594i
307.3 −1.24819 + 1.24819i 0 − 1.11598i 0.445397 + 2.19126i 0 2.93706 − 2.93706i −1.10343 − 1.10343i 0 −3.29106 − 2.17918i
307.4 −0.857692 + 0.857692i 0 0.528729i −2.20289 + 0.383740i 0 −1.82630 + 1.82630i −2.16887 − 2.16887i 0 1.56027 − 2.21854i
307.5 −0.824369 + 0.824369i 0 0.640833i −0.623976 − 2.14724i 0 −0.423225 + 0.423225i −2.17702 − 2.17702i 0 2.28451 + 1.25573i
307.6 −0.478466 + 0.478466i 0 1.54214i 1.04698 − 1.97581i 0 3.26171 − 3.26171i −1.69480 − 1.69480i 0 0.444416 + 1.44630i
307.7 0.478466 − 0.478466i 0 1.54214i 1.04698 − 1.97581i 0 −3.26171 + 3.26171i 1.69480 + 1.69480i 0 −0.444416 − 1.44630i
307.8 0.824369 − 0.824369i 0 0.640833i −0.623976 − 2.14724i 0 0.423225 − 0.423225i 2.17702 + 2.17702i 0 −2.28451 − 1.25573i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.k.c 24
3.b odd 2 1 165.2.j.a ✓ 24
5.c odd 4 1 inner 495.2.k.c 24
11.b odd 2 1 inner 495.2.k.c 24
15.d odd 2 1 825.2.j.c 24
15.e even 4 1 165.2.j.a ✓ 24
15.e even 4 1 825.2.j.c 24
33.d even 2 1 165.2.j.a ✓ 24
55.e even 4 1 inner 495.2.k.c 24
165.d even 2 1 825.2.j.c 24
165.l odd 4 1 165.2.j.a ✓ 24
165.l odd 4 1 825.2.j.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.j.a ✓ 24 3.b odd 2 1
165.2.j.a ✓ 24 15.e even 4 1
165.2.j.a ✓ 24 33.d even 2 1
165.2.j.a ✓ 24 165.l odd 4 1
495.2.k.c 24 1.a even 1 1 trivial
495.2.k.c 24 5.c odd 4 1 inner
495.2.k.c 24 11.b odd 2 1 inner
495.2.k.c 24 55.e even 4 1 inner
825.2.j.c 24 15.d odd 2 1
825.2.j.c 24 15.e even 4 1
825.2.j.c 24 165.d even 2 1
825.2.j.c 24 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 86T_{2}^{20} + 2023T_{2}^{16} + 16908T_{2}^{12} + 48055T_{2}^{8} + 47134T_{2}^{4} + 7921 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display