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Properties

Label 71.2.g.a
Level $71$
Weight $2$
Character orbit 71.g
Analytic conductor $0.567$
Analytic rank $0$
Dimension $120$
Inner twists $2$

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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] = [71,2,Mod(2,71)] 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("71.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(71, base_ring=CyclotomicField(70)) chi = DirichletCharacter(H, H._module([6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 71.g (of order \(35\), degree \(24\), 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: \(0.566937854351\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(5\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 22 q^{2} - 20 q^{3} - 18 q^{4} - 20 q^{5} - 20 q^{6} - 27 q^{7} - 27 q^{8} - 11 q^{9} - 8 q^{10} - 27 q^{11} + 3 q^{12} - 31 q^{13} + 2 q^{14} + 12 q^{15} + 30 q^{16} + 9 q^{17} + 27 q^{18} - 31 q^{19}+ \cdots - 2 q^{99}+O(q^{100}) \) 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 −0.119796 + 2.66747i 0.812884 + 0.710195i −5.10911 − 0.459828i −0.490011 − 1.50810i −1.99181 + 2.08327i 3.65834 + 1.00964i 1.12178 − 8.28134i −0.246297 − 1.81824i 4.08151 − 1.12643i
2.2 −0.0669265 + 1.49023i −0.529115 − 0.462274i −0.224368 − 0.0201935i 0.774775 + 2.38451i 0.724308 − 0.757567i −1.36859 − 0.377708i −0.355372 + 2.62346i −0.336434 − 2.48366i −3.60533 + 0.995008i
2.3 −0.0135643 + 0.302033i −2.13142 − 1.86217i 1.90091 + 0.171085i −1.01225 − 3.11539i 0.591347 − 0.618501i 2.21035 + 0.610018i −0.158625 + 1.17102i 0.672593 + 4.96528i 0.954681 − 0.263475i
2.4 0.0417847 − 0.930409i 0.0446684 + 0.0390256i 1.12803 + 0.101525i 0.322385 + 0.992198i 0.0381763 − 0.0399293i −2.48776 − 0.686577i 0.391630 − 2.89113i −0.402228 − 2.96936i 0.936621 − 0.258491i
2.5 0.119211 − 2.65443i 2.16883 + 1.89485i −5.03985 − 0.453595i −0.171217 − 0.526952i 5.28830 − 5.53113i −1.94286 − 0.536196i −1.09150 + 8.05775i 0.710671 + 5.24638i −1.41917 + 0.391666i
3.1 −1.62563 − 1.42027i 1.58857 − 2.95205i 0.357037 + 2.63575i 0.0897385 + 0.276187i −6.77512 + 2.54274i 1.48806 + 3.48148i 0.784662 − 1.18871i −4.53837 − 6.87534i 0.246378 − 0.576429i
3.2 −1.35901 − 1.18733i −0.353888 + 0.657633i 0.168685 + 1.24528i −1.18074 − 3.63395i 1.26176 − 0.473548i −0.979999 − 2.29282i −0.739005 + 1.11954i 1.34545 + 2.03826i −2.71006 + 6.34050i
3.3 −0.800540 − 0.699411i −1.38033 + 2.56508i −0.116777 − 0.862086i 0.792184 + 2.43809i 2.89906 − 1.08803i 1.31864 + 3.08512i −1.68071 + 2.54617i −3.02165 − 4.57761i 1.07105 − 2.50585i
3.4 −0.0906858 − 0.0792298i 0.481090 − 0.894015i −0.266520 − 1.96753i 0.692006 + 2.12978i −0.114461 + 0.0429578i −1.03931 − 2.43158i −0.264397 + 0.400544i 1.08488 + 1.64352i 0.105987 − 0.247968i
3.5 1.25969 + 1.10056i −0.911759 + 1.69433i 0.107124 + 0.790822i −0.446644 − 1.37463i −3.01324 + 1.13089i −0.310278 − 0.725932i 1.10761 − 1.67795i −0.386768 − 0.585928i 0.950226 − 2.22316i
4.1 −2.38426 − 0.214587i −0.366835 − 2.70808i 3.67077 + 0.666146i −2.45585 + 1.78428i 0.293509 + 6.53548i −2.65232 − 1.58469i −3.99386 − 1.10223i −4.30726 + 1.18873i 6.23825 − 3.72718i
4.2 −1.63009 − 0.146711i 0.0206726 + 0.152611i 0.667817 + 0.121191i 2.68191 − 1.94852i −0.0113085 − 0.251803i 0.0621796 + 0.0371506i 2.08458 + 0.575306i 2.86903 − 0.791801i −4.65763 + 2.78280i
4.3 −1.13429 − 0.102088i 0.322965 + 2.38423i −0.691667 − 0.125519i −2.14496 + 1.55840i −0.122936 − 2.73737i −0.760045 − 0.454106i 2.96740 + 0.818951i −2.68833 + 0.741933i 2.59210 − 1.54871i
4.4 0.262709 + 0.0236442i −0.333427 − 2.46146i −1.89940 − 0.344691i 0.786557 − 0.571467i −0.0293950 − 0.654532i 2.26116 + 1.35098i −0.999371 − 0.275809i −3.05572 + 0.843326i 0.220148 − 0.131532i
4.5 1.52544 + 0.137292i 0.0564352 + 0.416621i 0.340254 + 0.0617469i −0.536725 + 0.389953i 0.0288896 + 0.643278i −2.10174 − 1.25573i −2.44226 − 0.674021i 2.72150 − 0.751086i −0.872278 + 0.521162i
6.1 −1.86975 + 1.95560i −1.83587 + 0.689014i −0.238696 − 5.31497i −1.14237 + 0.829980i 2.08518 − 4.87851i −0.196262 − 1.44886i 6.76522 + 5.91060i 0.636468 − 0.556065i 0.512831 − 3.78587i
6.2 −1.18839 + 1.24296i 1.54029 − 0.578081i −0.0429451 − 0.956246i 1.33583 − 0.970538i −1.11193 + 2.60150i 0.313169 + 2.31191i −1.35044 − 1.17984i −0.220897 + 0.192992i −0.381149 + 2.81376i
6.3 −0.199826 + 0.209002i −2.27196 + 0.852682i 0.0859784 + 1.91446i −0.567040 + 0.411978i 0.275786 − 0.645233i 0.218271 + 1.61134i −0.852820 − 0.745087i 2.17553 − 1.90070i 0.0272052 − 0.200837i
6.4 0.0749867 − 0.0784299i 1.72079 − 0.645822i 0.0892014 + 1.98622i −1.81548 + 1.31902i 0.0783843 − 0.183389i −0.608682 − 4.49347i 0.325899 + 0.284729i 0.284802 − 0.248824i −0.0326858 + 0.241297i
6.5 1.46643 − 1.53377i −0.822306 + 0.308617i −0.112290 − 2.50034i −0.577761 + 0.419768i −0.732509 + 1.71379i −0.0309991 − 0.228844i −0.803573 − 0.702060i −1.67827 + 1.46626i −0.203420 + 1.50171i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.g even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.2.g.a ✓ 120
3.b odd 2 1 639.2.v.a 120
71.g even 35 1 inner 71.2.g.a ✓ 120
71.g even 35 1 5041.2.a.s 60
71.h odd 70 1 5041.2.a.t 60
213.o odd 70 1 639.2.v.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.2.g.a ✓ 120 1.a even 1 1 trivial
71.2.g.a ✓ 120 71.g even 35 1 inner
639.2.v.a 120 3.b odd 2 1
639.2.v.a 120 213.o odd 70 1
5041.2.a.s 60 71.g even 35 1
5041.2.a.t 60 71.h odd 70 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(71, [\chi])\).