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Properties

Label 31.7.e.a
Level $31$
Weight $7$
Character orbit 31.e
Analytic conductor $7.132$
Analytic rank $0$
Dimension $30$
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] = [31,7,Mod(6,31)] 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("31.6"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 31.e (of order \(6\), degree \(2\), 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: \(7.13167659222\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 4 q^{2} - 3 q^{3} + 1024 q^{4} - 73 q^{5} - 1446 q^{6} + 723 q^{7} - 1496 q^{8} + 3478 q^{9} - 848 q^{10} + 957 q^{11} - 2382 q^{12} + 8817 q^{13} - 8654 q^{14} + 25288 q^{16} - 3123 q^{17} + 6422 q^{18}+ \cdots - 3308442 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −14.9108 41.4660 − 23.9404i 158.333 45.3517 − 78.5515i −618.292 + 356.971i 261.731 + 453.331i −1406.59 781.784 − 1354.09i −676.232 + 1171.27i
6.2 −13.1923 −17.7721 + 10.2607i 110.037 −101.000 + 174.937i 234.455 − 135.363i 248.626 + 430.633i −607.338 −153.934 + 266.622i 1332.42 − 2307.82i
6.3 −13.1264 −23.3339 + 13.4718i 108.303 78.5134 − 135.989i 306.290 − 176.837i −97.4471 − 168.783i −581.535 −1.51931 + 2.63152i −1030.60 + 1785.05i
6.4 −10.5635 15.0440 − 8.68564i 47.5872 −48.4606 + 83.9363i −158.917 + 91.7507i −235.536 − 407.961i 173.376 −213.619 + 369.999i 511.913 − 886.660i
6.5 −7.52027 19.5539 − 11.2894i −7.44548 27.3733 − 47.4120i −147.050 + 84.8996i −20.9598 − 36.3035i 537.290 −109.597 + 189.828i −205.855 + 356.551i
6.6 −3.78169 −34.1361 + 19.7085i −49.6989 −34.2022 + 59.2400i 129.092 − 74.5312i −48.4748 − 83.9608i 429.973 412.347 − 714.206i 129.342 − 224.027i
6.7 −2.90357 −6.52683 + 3.76826i −55.5693 74.3557 − 128.788i 18.9511 − 10.9414i 196.242 + 339.902i 347.178 −336.100 + 582.143i −215.897 + 373.945i
6.8 −1.05479 31.1253 − 17.9702i −62.8874 −116.201 + 201.267i −32.8307 + 18.9548i 166.580 + 288.525i 133.839 281.358 − 487.326i 122.568 − 212.294i
6.9 3.42334 34.1533 − 19.7184i −52.2808 71.1104 − 123.167i 116.918 − 67.5028i −71.7584 − 124.289i −398.068 413.132 − 715.565i 243.435 − 421.642i
6.10 5.19428 −4.64512 + 2.68186i −37.0195 −31.7235 + 54.9467i −24.1281 + 13.9303i −229.047 − 396.721i −524.723 −350.115 + 606.417i −164.781 + 285.409i
6.11 7.74164 −12.6461 + 7.30123i −4.06698 −28.4834 + 49.3346i −97.9016 + 56.5235i 304.101 + 526.719i −526.950 −257.884 + 446.668i −220.508 + 381.931i
6.12 8.52647 −42.4212 + 24.4919i 8.70070 113.492 − 196.574i −361.703 + 208.829i −45.5422 − 78.8814i −471.508 835.205 − 1446.62i 967.689 − 1676.09i
6.13 12.2712 32.1769 − 18.5773i 86.5820 −37.1226 + 64.2983i 394.849 − 227.966i 51.1148 + 88.5334i 277.108 325.735 − 564.190i −455.539 + 789.017i
6.14 13.5489 1.93146 − 1.11513i 119.571 64.7312 − 112.118i 26.1691 − 15.1087i −37.6068 − 65.1369i 752.930 −362.013 + 627.025i 877.034 − 1519.07i
6.15 14.3476 −35.4695 + 20.4783i 141.854 −114.235 + 197.860i −508.902 + 293.815i −80.5231 − 139.470i 1117.01 474.222 − 821.377i −1638.99 + 2838.82i
26.1 −14.9108 41.4660 + 23.9404i 158.333 45.3517 + 78.5515i −618.292 − 356.971i 261.731 − 453.331i −1406.59 781.784 + 1354.09i −676.232 − 1171.27i
26.2 −13.1923 −17.7721 − 10.2607i 110.037 −101.000 − 174.937i 234.455 + 135.363i 248.626 − 430.633i −607.338 −153.934 − 266.622i 1332.42 + 2307.82i
26.3 −13.1264 −23.3339 − 13.4718i 108.303 78.5134 + 135.989i 306.290 + 176.837i −97.4471 + 168.783i −581.535 −1.51931 − 2.63152i −1030.60 − 1785.05i
26.4 −10.5635 15.0440 + 8.68564i 47.5872 −48.4606 − 83.9363i −158.917 − 91.7507i −235.536 + 407.961i 173.376 −213.619 − 369.999i 511.913 + 886.660i
26.5 −7.52027 19.5539 + 11.2894i −7.44548 27.3733 + 47.4120i −147.050 − 84.8996i −20.9598 + 36.3035i 537.290 −109.597 − 189.828i −205.855 − 356.551i
See all 30 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 6.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.7.e.a ✓ 30
31.e odd 6 1 inner 31.7.e.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.7.e.a ✓ 30 1.a even 1 1 trivial
31.7.e.a ✓ 30 31.e odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(31, [\chi])\).