Properties

Label 137.2.g.a
Level $137$
Weight $2$
Character orbit 137.g
Analytic conductor $1.094$
Analytic rank $0$
Dimension $352$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,2,Mod(2,137)] 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("137.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(68)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 137.g (of order \(68\), degree \(32\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.09395050769\)
Analytic rank: \(0\)
Dimension: \(352\)
Relative dimension: \(11\) over \(\Q(\zeta_{68})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{68}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 34 q^{2} - 28 q^{3} - 4 q^{4} - 30 q^{5} - 38 q^{6} - 34 q^{7} - 34 q^{8} - 34 q^{9} - 44 q^{10} - 34 q^{11} + 50 q^{12} - 34 q^{13} - 30 q^{14} + 10 q^{15} + 12 q^{16} - 34 q^{17} - 32 q^{18} - 34 q^{19}+ \cdots - 126 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.76732 + 1.93866i 3.24188 + 0.762484i −0.450437 − 4.86099i −0.0606567 + 1.31198i −7.20764 + 4.93735i −1.63506 − 0.465213i 6.03294 + 4.55586i 7.24295 + 3.60656i −2.43629 − 2.43629i
2.2 −1.42090 + 1.55865i −0.223352 − 0.0525320i −0.225904 − 2.43789i 0.137077 − 2.96493i 0.399240 − 0.273486i −3.84904 − 1.09515i 0.754595 + 0.569844i −2.63836 − 1.31375i 4.42652 + 4.42652i
2.3 −1.33401 + 1.46334i −0.875109 − 0.205824i −0.177242 − 1.91275i −0.146903 + 3.17747i 1.46859 − 1.00601i 0.652694 + 0.185708i −0.124927 − 0.0943407i −1.96204 − 0.976978i −4.45375 − 4.45375i
2.4 −1.01087 + 1.10887i −3.23440 − 0.760724i −0.0232020 − 0.250389i 0.0810487 − 1.75306i 4.11311 − 2.81754i 2.24615 + 0.639085i −2.09373 − 1.58111i 7.19714 + 3.58375i 1.86199 + 1.86199i
2.5 −0.723076 + 0.793176i 2.14102 + 0.503564i 0.0782466 + 0.844415i 0.173450 − 3.75167i −1.94754 + 1.33409i 3.91072 + 1.11269i −2.43937 − 1.84213i 1.64491 + 0.819069i 2.85032 + 2.85032i
2.6 −0.486794 + 0.533988i 1.33773 + 0.314631i 0.136362 + 1.47158i −0.0834871 + 1.80580i −0.819207 + 0.561170i −1.22942 − 0.349799i −2.00544 − 1.51444i −0.994967 − 0.495435i −0.923634 − 0.923634i
2.7 0.332416 − 0.364643i −0.732893 − 0.172375i 0.162073 + 1.74904i −0.0307378 + 0.664849i −0.306481 + 0.209944i 4.10840 + 1.16894i 1.47917 + 1.11702i −2.17807 − 1.08455i 0.232215 + 0.232215i
2.8 0.517741 − 0.567935i 2.17561 + 0.511700i 0.130043 + 1.40338i 0.0361720 − 0.782388i 1.41702 − 0.970679i −3.59710 − 1.02346i 2.09092 + 1.57899i 1.78597 + 0.889305i −0.425618 − 0.425618i
2.9 0.575561 − 0.631360i −2.89411 − 0.680688i 0.117191 + 1.26470i −0.166670 + 3.60501i −2.09549 + 1.43545i −4.10588 − 1.16822i 2.22948 + 1.68362i 5.22703 + 2.60275i 2.18013 + 2.18013i
2.10 1.04865 − 1.15032i −1.70367 − 0.400701i −0.0390197 − 0.421089i 0.184506 − 3.99081i −2.24749 + 1.53957i −0.540617 − 0.153819i 1.95903 + 1.47939i 0.0564555 + 0.0281115i −4.39721 − 4.39721i
2.11 1.48104 − 1.62462i 0.638895 + 0.150267i −0.261383 − 2.82077i −0.0676592 + 1.46345i 1.19035 − 0.815411i −0.261150 − 0.0743036i −1.46111 − 1.10338i −2.29988 − 1.14521i 2.27734 + 2.27734i
7.1 −2.51460 − 0.715465i −1.51917 − 2.21771i 4.11088 + 2.54535i 2.73258 + 1.52204i 2.23340 + 6.66357i 3.30951 − 0.306672i −4.99347 − 5.47758i −1.52666 + 3.94075i −5.78238 − 5.78238i
7.2 −2.34468 − 0.667119i 1.56927 + 2.29085i 3.35204 + 2.07549i 2.68486 + 1.49545i −2.15116 − 6.41819i −4.56839 + 0.423323i −3.19026 − 3.49955i −1.70166 + 4.39248i −5.29748 − 5.29748i
7.3 −2.15113 − 0.612048i 0.469849 + 0.685894i 2.55231 + 1.58032i −2.39978 − 1.33667i −0.590903 − 1.76302i 2.18504 − 0.202473i −1.50966 − 1.65602i 0.834032 − 2.15288i 4.34413 + 4.34413i
7.4 −1.51373 − 0.430693i −0.805638 − 1.17609i 0.405448 + 0.251043i −0.271618 − 0.151290i 0.712986 + 2.12726i −3.35849 + 0.311210i 1.61492 + 1.77148i 0.349598 − 0.902417i 0.345997 + 0.345997i
7.5 −0.785951 − 0.223622i 0.0781368 + 0.114066i −1.13272 − 0.701352i 1.58644 + 0.883640i −0.0359041 − 0.107123i 1.90322 − 0.176359i 1.83444 + 2.01228i 1.07682 − 2.77959i −1.04926 − 1.04926i
7.6 0.246889 + 0.0702460i 1.10982 + 1.62014i −1.64441 − 1.01818i 2.55302 + 1.42202i 0.160194 + 0.477955i −0.457807 + 0.0424221i −0.680324 − 0.746280i −0.309423 + 0.798714i 0.530422 + 0.530422i
7.7 0.497766 + 0.141627i −0.656102 − 0.957792i −1.47272 − 0.911870i −2.65418 − 1.47837i −0.190937 − 0.569678i 1.79613 − 0.166436i −1.30123 − 1.42738i 0.596831 − 1.54060i −1.11178 − 1.11178i
7.8 0.565305 + 0.160843i −1.75988 − 2.56911i −1.40673 − 0.871013i 1.59703 + 0.889543i −0.581647 − 1.73540i −4.38761 + 0.406572i −1.44706 − 1.58735i −2.41943 + 6.24526i 0.759735 + 0.759735i
7.9 1.31512 + 0.374185i 1.44062 + 2.10304i −0.110897 − 0.0686647i −1.33370 − 0.742868i 1.10766 + 3.30482i 1.40896 − 0.130559i −1.96247 − 2.15272i −1.26369 + 3.26195i −1.47602 − 1.47602i
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.g even 68 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 137.2.g.a ✓ 352
137.g even 68 1 inner 137.2.g.a ✓ 352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
137.2.g.a ✓ 352 1.a even 1 1 trivial
137.2.g.a ✓ 352 137.g even 68 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(137, [\chi])\).