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Properties

Label 31.5.h.a
Level $31$
Weight $5$
Character orbit 31.h
Analytic conductor $3.204$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [31,5,Mod(3,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.3"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 31.h (of order \(30\), degree \(8\), 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: \(3.20446885560\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 6 q^{2} - 16 q^{3} - 138 q^{4} - q^{5} - 9 q^{6} + 288 q^{7} - 221 q^{8} - 182 q^{9} + 102 q^{10} - 49 q^{11} - 159 q^{12} + 527 q^{13} - 740 q^{14} + 1060 q^{15} - 1970 q^{16} + 848 q^{17} + 3 q^{18}+ \cdots + 75228 q^{99}+O(q^{100}) \) 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 −6.23929 + 4.53311i −12.8658 − 1.35225i 13.4354 − 41.3498i −16.0694 + 27.8330i 86.4033 − 49.8850i −1.96773 + 0.418254i 65.4850 + 201.542i 84.4701 + 17.9547i −25.9084 − 246.502i
3.2 −5.87290 + 4.26691i 13.9923 + 1.47065i 11.3402 − 34.9014i 14.4955 − 25.1070i −88.4507 + 51.0671i 44.8663 − 9.53663i 46.4297 + 142.896i 114.393 + 24.3149i 21.9985 + 209.302i
3.3 −3.52193 + 2.55883i 2.65826 + 0.279395i 0.912109 − 2.80718i −6.17859 + 10.7016i −10.0771 + 5.81804i −46.4227 + 9.86745i −17.5534 − 54.0239i −72.2417 − 15.3554i −5.62312 − 53.5004i
3.4 −2.99168 + 2.17358i −8.61730 − 0.905715i −0.718579 + 2.21156i 15.9893 − 27.6944i 27.7489 − 16.0208i 48.5707 − 10.3240i −20.9408 − 64.4491i −5.79241 − 1.23122i 12.3610 + 117.607i
3.5 0.105341 − 0.0765347i 13.2039 + 1.38779i −4.93903 + 15.2008i 0.860842 − 1.49102i 1.49713 − 0.864368i 6.58473 − 1.39963i 1.28689 + 3.96065i 93.1878 + 19.8077i −0.0234330 − 0.222950i
3.6 0.957995 − 0.696024i −4.87678 − 0.512570i −4.51097 + 13.8833i −19.9028 + 34.4727i −5.02869 + 2.90332i 75.9020 − 16.1335i 11.1964 + 34.4590i −55.7097 − 11.8415i 4.92703 + 46.8775i
3.7 1.59715 − 1.16040i −12.1013 − 1.27190i −3.73990 + 11.5102i 1.48190 − 2.56672i −20.8036 + 12.0109i −87.7529 + 18.6524i 17.1442 + 52.7645i 65.5938 + 13.9424i −0.611606 − 5.81905i
3.8 3.51941 − 2.55700i 0.0842863 + 0.00885885i 0.903727 − 2.78139i 21.4739 − 37.1939i 0.319290 − 0.184342i 33.2459 − 7.06663i 17.5773 + 54.0974i −79.2229 − 16.8394i −19.5294 − 185.810i
3.9 4.95024 − 3.59656i 7.69999 + 0.809302i 6.62534 − 20.3907i −7.33404 + 12.7029i 41.0275 − 23.6872i −30.9988 + 6.58900i −10.2862 − 31.6576i −20.5951 − 4.37762i 9.38158 + 89.2598i
3.10 6.18665 − 4.49486i −15.7819 − 1.65875i 13.1266 − 40.3994i −2.57606 + 4.46187i −105.093 + 60.6755i 54.5218 − 11.5890i −62.5710 − 192.574i 167.088 + 35.5156i 4.11831 + 39.1831i
11.1 −2.06958 − 6.36951i 1.99917 − 1.80007i −23.3433 + 16.9599i −16.3958 − 28.3984i −15.6030 − 9.00839i 2.86136 + 27.2240i 69.6453 + 50.6002i −7.71034 + 73.3590i −146.952 + 163.206i
11.2 −1.81716 − 5.59264i −8.09014 + 7.28440i −15.0313 + 10.9209i 13.9166 + 24.1043i 55.4401 + 32.0083i 0.302758 + 2.88055i 12.2726 + 8.91656i 3.92116 − 37.3073i 109.518 − 121.632i
11.3 −1.20822 − 3.71853i 11.2749 − 10.1519i 0.576617 − 0.418937i 8.94674 + 15.4962i −51.3729 − 29.6601i 4.13118 + 39.3055i −52.8652 − 38.4088i 15.5941 − 148.368i 46.8134 − 51.9916i
11.4 −0.848371 − 2.61102i 1.78506 − 1.60727i 6.84659 − 4.97434i 0.623819 + 1.08049i −5.71100 − 3.29725i −7.64750 − 72.7611i −54.3336 − 39.4757i −7.86370 + 74.8181i 2.29194 − 2.54546i
11.5 −0.287645 − 0.885280i −10.3597 + 9.32794i 12.2433 − 8.89527i −21.4191 − 37.0990i 11.2378 + 6.48813i −3.76934 − 35.8629i −23.4456 − 17.0342i 11.8467 − 112.714i −26.6819 + 29.6333i
11.6 0.0742929 + 0.228650i −5.53397 + 4.98281i 12.8975 − 9.37059i 11.1159 + 19.2533i −1.55045 − 0.895155i 8.89648 + 84.6444i 6.21280 + 4.51386i −2.67036 + 25.4068i −3.57644 + 3.97204i
11.7 0.774721 + 2.38435i 7.54548 − 6.79398i 7.85936 − 5.71016i −19.1726 − 33.2079i 22.0448 + 12.7276i 3.49477 + 33.2505i 52.1557 + 37.8934i 2.30929 − 21.9714i 64.3257 − 71.4409i
11.8 0.970390 + 2.98655i 3.96725 − 3.57213i 4.96643 − 3.60832i 13.2322 + 22.9188i 14.5181 + 8.38205i −5.15690 − 49.0646i 56.2440 + 40.8637i −5.48783 + 52.2132i −55.6079 + 61.7588i
11.9 1.83141 + 5.63649i −6.35706 + 5.72392i −15.4717 + 11.2408i −1.99371 − 3.45321i −43.9052 − 25.3487i 0.0475085 + 0.452013i −14.9788 − 10.8827i −0.817885 + 7.78165i 15.8127 − 17.5618i
11.10 2.38919 + 7.35316i 10.8941 − 9.80911i −35.4165 + 25.7316i 7.71154 + 13.3568i 98.1560 + 56.6704i 2.44380 + 23.2512i −173.745 − 126.233i 13.9964 − 133.167i −79.7902 + 88.6160i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.h odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.5.h.a ✓ 80
31.h odd 30 1 inner 31.5.h.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.5.h.a ✓ 80 1.a even 1 1 trivial
31.5.h.a ✓ 80 31.h odd 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(31, [\chi])\).