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Properties

Label 158.2.e.a
Level $158$
Weight $2$
Character orbit 158.e
Analytic conductor $1.262$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [158,2,Mod(21,158)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(158, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([18])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("158.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 158 = 2 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 158.e (of order \(13\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.26163635194\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{2} - 2 q^{3} - 4 q^{4} + 11 q^{5} - 2 q^{6} - 8 q^{7} - 4 q^{8} - 18 q^{9} - 2 q^{10} - 8 q^{11} - 2 q^{12} + 16 q^{13} + 5 q^{14} - 2 q^{15} - 4 q^{16} + 6 q^{17} - 18 q^{18} + 6 q^{19} - 2 q^{20}+ \cdots - 58 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.748511 + 0.663123i −0.972085 + 1.40831i 0.120537 − 0.992709i 4.21009 − 1.03769i −0.206265 − 1.69874i −0.541791 + 0.784920i 0.568065 + 0.822984i 0.0254326 + 0.0670603i −2.46318 + 3.56853i
21.2 −0.748511 + 0.663123i 0.0800968 − 0.116040i 0.120537 − 0.992709i −2.53849 + 0.625681i 0.0169956 + 0.139971i −2.24316 + 3.24977i 0.568065 + 0.822984i 1.05676 + 2.78646i 1.48518 − 2.15166i
21.3 −0.748511 + 0.663123i 0.298584 − 0.432573i 0.120537 − 0.992709i −0.874632 + 0.215577i 0.0633559 + 0.521783i 2.93311 − 4.24935i 0.568065 + 0.822984i 0.965847 + 2.54673i 0.511717 − 0.741350i
21.4 −0.748511 + 0.663123i 1.72953 − 2.50566i 0.120537 − 0.992709i 1.08849 − 0.268288i 0.366987 + 3.02241i −1.08132 + 1.56657i 0.568065 + 0.822984i −2.22324 − 5.86221i −0.636836 + 0.922617i
65.1 −0.970942 + 0.239316i −2.36412 − 2.09443i 0.885456 − 0.464723i 0.339740 + 0.492198i 2.79665 + 1.46780i −2.09534 − 1.85631i −0.748511 + 0.663123i 0.840827 + 6.92483i −0.447659 − 0.396591i
65.2 −0.970942 + 0.239316i −0.718589 − 0.636615i 0.885456 − 0.464723i −0.0135650 − 0.0196524i 0.850060 + 0.446146i 1.84669 + 1.63602i −0.748511 + 0.663123i −0.250518 − 2.06320i 0.0178740 + 0.0158350i
65.3 −0.970942 + 0.239316i 0.371738 + 0.329331i 0.885456 − 0.464723i −2.16941 − 3.14293i −0.439750 − 0.230799i −1.77830 − 1.57544i −0.748511 + 0.663123i −0.331880 − 2.73328i 2.85852 + 2.53243i
65.4 −0.970942 + 0.239316i 1.21395 + 1.07547i 0.885456 − 0.464723i 2.48863 + 3.60540i −1.43605 − 0.753697i −2.49643 − 2.21164i −0.748511 + 0.663123i −0.0445623 − 0.367004i −3.27914 − 2.90507i
67.1 0.120537 + 0.992709i −1.14543 + 3.02024i −0.970942 + 0.239316i 1.75811 + 0.922729i −3.13628 − 0.773025i 0.00805121 − 0.0212293i −0.354605 − 0.935016i −5.56431 − 4.92955i −0.704084 + 1.85652i
67.2 0.120537 + 0.992709i −0.410768 + 1.08311i −0.970942 + 0.239316i −3.00343 − 1.57632i −1.12472 − 0.277219i −1.51386 + 3.99172i −0.354605 − 0.935016i 1.24114 + 1.09956i 1.20280 − 3.17153i
67.3 0.120537 + 0.992709i 0.254860 − 0.672010i −0.970942 + 0.239316i 1.37518 + 0.721749i 0.697830 + 0.172000i −0.934864 + 2.46503i −0.354605 − 0.935016i 1.85889 + 1.64683i −0.550727 + 1.45215i
67.4 0.120537 + 0.992709i 0.592124 − 1.56130i −0.970942 + 0.239316i 1.43820 + 0.754826i 1.62129 + 0.399613i 1.16412 − 3.06952i −0.354605 − 0.935016i 0.158475 + 0.140397i −0.575967 + 1.51870i
87.1 −0.354605 + 0.935016i −1.44243 − 0.757048i −0.748511 − 0.663123i 0.0397927 − 0.327722i 1.21935 − 1.08025i −2.46280 − 1.29258i 0.885456 − 0.464723i −0.196700 − 0.284970i 0.292315 + 0.153419i
87.2 −0.354605 + 0.935016i 0.210806 + 0.110640i −0.748511 − 0.663123i 0.420007 − 3.45907i −0.178203 + 0.157874i 4.05051 + 2.12587i 0.885456 − 0.464723i −1.67200 − 2.42230i 3.08535 + 1.61932i
87.3 −0.354605 + 0.935016i 0.295387 + 0.155031i −0.748511 − 0.663123i −0.452351 + 3.72544i −0.249702 + 0.221217i −0.130033 − 0.0682466i 0.885456 − 0.464723i −1.64098 − 2.37736i −3.32295 − 1.74402i
87.4 −0.354605 + 0.935016i 2.70715 + 1.42082i −0.748511 − 0.663123i 0.0216091 − 0.177967i −2.28846 + 2.02740i −0.585393 − 0.307238i 0.885456 − 0.464723i 3.60574 + 5.22382i 0.158739 + 0.0833128i
89.1 −0.354605 − 0.935016i −1.44243 + 0.757048i −0.748511 + 0.663123i 0.0397927 + 0.327722i 1.21935 + 1.08025i −2.46280 + 1.29258i 0.885456 + 0.464723i −0.196700 + 0.284970i 0.292315 − 0.153419i
89.2 −0.354605 − 0.935016i 0.210806 − 0.110640i −0.748511 + 0.663123i 0.420007 + 3.45907i −0.178203 − 0.157874i 4.05051 − 2.12587i 0.885456 + 0.464723i −1.67200 + 2.42230i 3.08535 − 1.61932i
89.3 −0.354605 − 0.935016i 0.295387 − 0.155031i −0.748511 + 0.663123i −0.452351 − 3.72544i −0.249702 − 0.221217i −0.130033 + 0.0682466i 0.885456 + 0.464723i −1.64098 + 2.37736i −3.32295 + 1.74402i
89.4 −0.354605 − 0.935016i 2.70715 − 1.42082i −0.748511 + 0.663123i 0.0216091 + 0.177967i −2.28846 − 2.02740i −0.585393 + 0.307238i 0.885456 + 0.464723i 3.60574 − 5.22382i 0.158739 − 0.0833128i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.4
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.e even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 158.2.e.a ✓ 48
79.e even 13 1 inner 158.2.e.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
158.2.e.a ✓ 48 1.a even 1 1 trivial
158.2.e.a ✓ 48 79.e even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 2 T_{3}^{47} + 17 T_{3}^{46} + 33 T_{3}^{45} + 79 T_{3}^{44} + 262 T_{3}^{43} + \cdots + 32761 \) acting on \(S_{2}^{\mathrm{new}}(158, [\chi])\). Copy content Toggle raw display