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Properties

Label 31.8.c.a
Level $31$
Weight $8$
Character orbit 31.c
Analytic conductor $9.684$
Analytic rank $0$
Dimension $36$
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] = [31,8,Mod(5,31)] 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("31.5"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 31.c (of order \(3\), 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: \(9.68393579001\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{2} - 2 q^{3} + 2352 q^{4} + 70 q^{5} - 528 q^{6} - 2158 q^{7} + 3084 q^{8} - 14552 q^{9} + 5942 q^{10} - 4666 q^{11} - 11134 q^{12} - 10502 q^{13} - 19440 q^{14} + 99588 q^{15} + 172360 q^{16}+ \cdots + 23529296 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −22.3097 16.6965 − 28.9193i 369.723 −15.2153 − 26.3537i −372.495 + 645.180i −91.5723 + 158.608i −5392.77 535.951 + 928.294i 339.450 + 587.944i
5.2 −18.7585 −44.9325 + 77.8253i 223.881 −253.099 − 438.380i 842.866 − 1459.89i −285.995 + 495.358i −1798.59 −2944.35 − 5099.77i 4747.75 + 8223.35i
5.3 −16.1614 −18.8943 + 32.7258i 133.192 123.239 + 213.456i 305.359 − 528.897i 532.681 − 922.630i −83.9137 379.513 + 657.335i −1991.72 − 3449.75i
5.4 −14.6394 −0.580185 + 1.00491i 86.3124 4.49399 + 7.78382i 8.49356 − 14.7113i −534.076 + 925.047i 610.282 1092.83 + 1892.83i −65.7894 − 113.951i
5.5 −13.6347 31.0389 − 53.7609i 57.9055 −215.254 − 372.830i −423.206 + 733.015i 570.196 − 987.609i 955.719 −833.324 − 1443.36i 2934.92 + 5083.43i
5.6 −12.8319 41.2939 − 71.5232i 36.6568 222.593 + 385.543i −529.878 + 917.776i −321.915 + 557.573i 1172.10 −2316.88 − 4012.95i −2856.29 − 4947.24i
5.7 −4.35658 19.2211 − 33.2919i −109.020 −69.9001 − 121.071i −83.7382 + 145.039i −374.980 + 649.484i 1032.60 354.598 + 614.182i 304.525 + 527.453i
5.8 −4.13326 −38.1019 + 65.9945i −110.916 202.717 + 351.116i 157.485 − 272.773i −662.316 + 1147.16i 987.504 −1810.01 − 3135.04i −837.882 − 1451.25i
5.9 −3.61931 −18.6256 + 32.2605i −114.901 −136.026 − 235.605i 67.4119 − 116.761i 198.437 − 343.704i 879.133 399.674 + 692.255i 492.322 + 852.727i
5.10 −1.22204 17.1728 − 29.7442i −126.507 191.092 + 330.980i −20.9858 + 36.3485i 783.421 − 1356.93i 311.016 503.689 + 872.414i −233.521 − 404.470i
5.11 5.36855 8.51912 − 14.7556i −99.1786 111.547 + 193.205i 45.7354 − 79.2160i −251.708 + 435.971i −1219.62 948.349 + 1642.59i 598.847 + 1037.23i
5.12 8.45475 36.7048 − 63.5746i −56.5171 −87.9329 − 152.304i 310.330 − 537.507i 48.3070 − 83.6702i −1560.05 −1600.99 − 2772.99i −743.451 − 1287.70i
5.13 8.95515 −35.4232 + 61.3548i −47.8054 −17.1498 − 29.7044i −317.220 + 549.441i 486.436 − 842.533i −1574.36 −1416.11 − 2452.77i −153.579 − 266.007i
5.14 12.0194 −10.4182 + 18.0448i 16.4660 −228.315 − 395.453i −125.220 + 216.888i −746.171 + 1292.41i −1340.57 876.423 + 1518.01i −2744.21 − 4753.11i
5.15 14.6433 −11.3125 + 19.5939i 86.4251 170.496 + 295.308i −165.652 + 286.918i −28.8903 + 50.0395i −608.792 837.554 + 1450.69i 2496.62 + 4324.27i
5.16 18.6853 10.7383 − 18.5992i 221.139 −113.140 − 195.965i 200.647 − 347.531i 601.784 − 1042.32i 1740.32 862.879 + 1494.55i −2114.06 − 3661.65i
5.17 19.6890 34.4069 − 59.5945i 259.657 170.282 + 294.936i 677.438 − 1173.36i −579.733 + 1004.13i 2592.20 −1274.17 − 2206.93i 3352.67 + 5807.00i
5.18 21.8515 −38.5041 + 66.6910i 349.486 −25.4271 − 44.0410i −841.370 + 1457.30i −422.907 + 732.496i 4839.80 −1871.62 − 3241.75i −555.619 − 962.361i
25.1 −22.3097 16.6965 + 28.9193i 369.723 −15.2153 + 26.3537i −372.495 − 645.180i −91.5723 − 158.608i −5392.77 535.951 − 928.294i 339.450 − 587.944i
25.2 −18.7585 −44.9325 − 77.8253i 223.881 −253.099 + 438.380i 842.866 + 1459.89i −285.995 − 495.358i −1798.59 −2944.35 + 5099.77i 4747.75 − 8223.35i
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 5.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.8.c.a ✓ 36
31.c even 3 1 inner 31.8.c.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.8.c.a ✓ 36 1.a even 1 1 trivial
31.8.c.a ✓ 36 31.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(31, [\chi])\).