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Properties

Label 371.2.w.a
Level $371$
Weight $2$
Character orbit 371.w
Analytic conductor $2.962$
Analytic rank $0$
Dimension $1632$
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] = [371,2,Mod(3,371)] 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("371.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(156)) chi = DirichletCharacter(H, H._module([26, 51])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.w (of order \(156\), degree \(48\), 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: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(1632\)
Relative dimension: \(34\) over \(\Q(\zeta_{156})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{156}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1632 q - 26 q^{2} - 72 q^{3} - 26 q^{4} - 72 q^{5} - 52 q^{7} - 92 q^{8} - 26 q^{9} - 66 q^{10} - 26 q^{11} - 90 q^{12} - 62 q^{14} - 104 q^{15} - 98 q^{16} - 78 q^{17} - 22 q^{18} - 84 q^{19} - 56 q^{21}+ \cdots - 408 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.29075 + 2.34339i 1.85546 − 2.18135i −2.75652 − 4.35909i −0.866195 − 0.624013i 2.71682 + 7.16366i 0.547449 + 2.58849i 8.43211 − 0.510049i −0.834304 − 5.13368i 2.58035 − 1.22439i
3.2 −1.20264 + 2.18342i −0.703815 + 0.827430i −2.25206 − 3.56134i 1.54638 + 1.11402i −0.960194 − 2.53182i 2.36157 + 1.19288i 5.50797 − 0.333171i 0.291949 + 1.79643i −4.29210 + 2.03663i
3.3 −1.14844 + 2.08502i 1.13924 − 1.33933i −1.95946 − 3.09864i 3.47835 + 2.50583i 1.48418 + 3.91347i −1.11314 − 2.40019i 3.95895 − 0.239472i −0.0147057 − 0.0904876i −9.21936 + 4.37464i
3.4 −1.07963 + 1.96010i −1.84526 + 2.16935i −1.60747 − 2.54201i −2.81761 − 2.02982i −2.25995 − 5.95900i −1.51908 + 2.16619i 2.25070 − 0.136143i −0.819870 − 5.04486i 7.02065 − 3.33134i
3.5 −1.03943 + 1.88712i −1.66665 + 1.95937i −1.41187 − 2.23269i −0.452012 − 0.325632i −1.96520 − 5.18180i 1.35827 − 2.27048i 1.37987 − 0.0834666i −0.580183 − 3.57001i 1.08434 − 0.514528i
3.6 −1.03883 + 1.88602i 0.613883 − 0.721703i −1.40899 − 2.22814i −0.771282 − 0.555636i 0.723429 + 1.90753i −2.47803 + 0.927015i 1.36748 − 0.0827175i 0.337231 + 2.07507i 1.84917 − 0.877444i
3.7 −0.888354 + 1.61283i 0.662432 − 0.778779i −0.743121 − 1.17515i −2.54539 − 1.83371i 0.667565 + 1.76022i 2.37151 + 1.17300i −1.12041 + 0.0677724i 0.313554 + 1.92937i 5.21868 − 2.47629i
3.8 −0.881669 + 1.60069i −0.111072 + 0.130580i −0.715952 − 1.13219i −1.75950 − 1.26756i −0.111090 − 0.292921i 0.726100 − 2.54417i −1.20471 + 0.0728715i 0.476520 + 2.93214i 3.58027 − 1.69886i
3.9 −0.755672 + 1.37194i 2.14046 − 2.51640i −0.242256 − 0.383098i −1.03603 − 0.746362i 1.83488 + 4.83817i −0.896190 − 2.48935i −2.41821 + 0.146275i −1.26947 − 7.81139i 1.80686 − 0.857368i
3.10 −0.698403 + 1.26797i −1.38831 + 1.63214i −0.0510495 − 0.0807283i 2.14115 + 1.54250i −1.09991 − 2.90023i −2.03035 − 1.69638i −2.75188 + 0.166458i −0.255262 − 1.57069i −3.45122 + 1.63762i
3.11 −0.630623 + 1.14491i 0.355488 − 0.417925i 0.155790 + 0.246362i 2.32954 + 1.67822i 0.254309 + 0.670557i 0.0855439 + 2.64437i −2.98974 + 0.180846i 0.432945 + 2.66402i −3.39048 + 1.60880i
3.12 −0.546116 + 0.991488i 1.42821 − 1.67906i 0.384126 + 0.607446i 1.45666 + 1.04939i 0.884796 + 2.33302i 2.64553 + 0.0343149i −3.07180 + 0.185810i −0.298209 − 1.83495i −1.83596 + 0.871172i
3.13 −0.393243 + 0.713943i −1.43102 + 1.68235i 0.713857 + 1.12887i 0.871870 + 0.628101i −0.638368 − 1.68324i 0.120048 + 2.64303i −2.71386 + 0.164158i −0.301272 − 1.85380i −0.791284 + 0.375469i
3.14 −0.330807 + 0.600590i 0.525679 − 0.618007i 0.817657 + 1.29302i −0.733831 − 0.528657i 0.197270 + 0.520159i −2.59350 − 0.523212i −2.41590 + 0.146135i 0.375640 + 2.31140i 0.560263 − 0.265848i
3.15 −0.222318 + 0.403625i −1.11757 + 1.31386i 0.955444 + 1.51091i −2.39653 − 1.72648i −0.281849 − 0.743174i −2.64500 + 0.0630413i −1.74218 + 0.105382i 0.00398118 + 0.0244972i 1.22964 − 0.583473i
3.16 −0.158733 + 0.288185i −0.875620 + 1.02941i 1.01108 + 1.59889i −2.09897 − 1.51211i −0.157670 − 0.415742i 2.63599 − 0.227036i −1.27809 + 0.0773099i 0.188259 + 1.15841i 0.768946 − 0.364869i
3.17 −0.0641783 + 0.116517i 0.222471 − 0.261544i 1.05947 + 1.67542i 1.56329 + 1.12621i 0.0161967 + 0.0427072i 0.356048 − 2.62168i −0.528772 + 0.0319848i 0.462322 + 2.84478i −0.231552 + 0.109873i
3.18 0.0954586 − 0.173308i 1.89529 − 2.22817i 1.04801 + 1.65729i 1.42279 + 1.02499i −0.205237 − 0.541166i −2.04081 + 1.68378i 0.782257 − 0.0473179i −0.891381 − 5.48489i 0.313456 − 0.148737i
3.19 0.157270 − 0.285528i −1.43550 + 1.68762i 1.01214 + 1.60057i 3.05314 + 2.19950i 0.256103 + 0.675288i 2.50529 − 0.850604i 1.26695 − 0.0766364i −0.306180 − 1.88400i 1.10819 − 0.525841i
3.20 0.311596 − 0.565712i 0.0105463 − 0.0123986i 0.845994 + 1.33783i −0.555215 − 0.399981i −0.00372785 − 0.00982952i −0.360414 + 2.62109i 2.30978 − 0.139716i 0.481191 + 2.96089i −0.399277 + 0.189459i
See next 80 embeddings (of 1632 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 3.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
53.f odd 52 1 inner
371.w even 156 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 371.2.w.a ✓ 1632
7.d odd 6 1 inner 371.2.w.a ✓ 1632
53.f odd 52 1 inner 371.2.w.a ✓ 1632
371.w even 156 1 inner 371.2.w.a ✓ 1632
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
371.2.w.a ✓ 1632 1.a even 1 1 trivial
371.2.w.a ✓ 1632 7.d odd 6 1 inner
371.2.w.a ✓ 1632 53.f odd 52 1 inner
371.2.w.a ✓ 1632 371.w even 156 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(371, [\chi])\).