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Properties

Label 31.10.c.a
Level $31$
Weight $10$
Character orbit 31.c
Analytic conductor $15.966$
Analytic rank $0$
Dimension $46$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,10,Mod(5,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.5");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 31.c (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.9661109211\)
Analytic rank: \(0\)
Dimension: \(46\)
Relative dimension: \(23\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 4 q^{2} + 13 q^{3} + 10576 q^{4} - 797 q^{5} + 96 q^{6} + 6359 q^{7} + 7740 q^{8} - 157910 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 4 q^{2} + 13 q^{3} + 10576 q^{4} - 797 q^{5} + 96 q^{6} + 6359 q^{7} + 7740 q^{8} - 157910 q^{9} + 53094 q^{10} + 55493 q^{11} + 53834 q^{12} + 145047 q^{13} + 318552 q^{14} - 384066 q^{15} + 1197768 q^{16} + 179439 q^{17} + 488 q^{18} + 9617 q^{19} - 851828 q^{20} + 2848093 q^{21} + 2280832 q^{22} - 4505848 q^{23} - 962944 q^{24} - 9362932 q^{25} + 834990 q^{26} - 10077578 q^{27} + 1101446 q^{28} - 12115804 q^{29} + 31833000 q^{30} + 104358 q^{31} + 7891236 q^{32} + 6707970 q^{33} + 31130650 q^{34} + 28584082 q^{35} - 21596628 q^{36} + 21441487 q^{37} - 36647116 q^{38} + 19218642 q^{39} - 2396706 q^{40} - 15950261 q^{41} - 24469246 q^{42} - 30909159 q^{43} - 43236754 q^{44} - 45030474 q^{45} + 55317080 q^{46} - 205812848 q^{47} + 82153070 q^{48} - 41962654 q^{49} - 139505168 q^{50} + 78805089 q^{51} + 17936780 q^{52} + 36657039 q^{53} + 356952728 q^{54} + 81882133 q^{55} + 186013408 q^{56} + 219050933 q^{57} - 208914824 q^{58} - 51947085 q^{59} - 873539036 q^{60} + 325741700 q^{61} - 501415564 q^{62} - 16524896 q^{63} - 957872560 q^{64} + 110929109 q^{65} + 2292342908 q^{66} + 566643397 q^{67} + 558441884 q^{68} - 775695912 q^{69} - 802574200 q^{70} - 590330101 q^{71} - 69879376 q^{72} - 360127505 q^{73} + 98625866 q^{74} + 138587320 q^{75} + 640315046 q^{76} + 2382415466 q^{77} - 1397220776 q^{78} + 809243295 q^{79} - 602458368 q^{80} - 2002193499 q^{81} + 1460652790 q^{82} - 202545909 q^{83} + 2809175820 q^{84} - 1091854514 q^{85} - 1179503176 q^{86} + 1270760164 q^{87} + 557016880 q^{88} - 129110620 q^{89} + 441119184 q^{90} - 8386987946 q^{91} + 3290496872 q^{92} + 1074021285 q^{93} - 8999704920 q^{94} - 4201168290 q^{95} + 3912194164 q^{96} + 6820480268 q^{97} + 1548029280 q^{98} + 644680976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −41.1094 −17.3728 + 30.0906i 1177.98 −841.722 − 1457.91i 714.187 − 1237.01i 1875.00 − 3247.60i −27378.1 9237.87 + 16000.5i 34602.7 + 59933.6i
5.2 −38.9465 −92.8346 + 160.794i 1004.83 589.937 + 1021.80i 3615.58 − 6262.37i −208.947 + 361.908i −19194.0 −7395.01 − 12808.5i −22976.0 − 39795.5i
5.3 −37.7012 64.9472 − 112.492i 909.379 898.817 + 1556.80i −2448.59 + 4241.07i 17.4833 − 30.2820i −14981.7 1405.23 + 2433.92i −33886.5 − 58693.1i
5.4 −35.4938 128.462 − 222.502i 747.811 −977.060 − 1692.32i −4559.59 + 7897.45i −4885.66 + 8462.21i −8369.84 −23163.3 − 40120.0i 34679.6 + 60066.8i
5.5 −28.2719 −53.6589 + 92.9399i 287.302 −165.373 − 286.434i 1517.04 − 2627.59i −5806.49 + 10057.1i 6352.64 4082.96 + 7071.89i 4675.41 + 8098.05i
5.6 −26.9389 64.7453 − 112.142i 213.704 −93.5033 − 161.952i −1744.17 + 3020.98i 4435.64 − 7682.75i 8035.77 1457.59 + 2524.62i 2518.87 + 4362.82i
5.7 −20.5049 −131.009 + 226.914i −91.5480 72.4069 + 125.412i 2686.33 − 4652.86i 4009.63 − 6944.89i 12375.7 −24485.2 − 42409.6i −1484.70 − 2571.57i
5.8 −19.3900 −44.3222 + 76.7682i −136.029 −936.920 − 1622.79i 859.406 − 1488.53i 2156.00 − 3734.31i 12565.3 5912.59 + 10240.9i 18166.9 + 31465.9i
5.9 −13.4097 −37.4624 + 64.8868i −332.179 1283.89 + 2223.76i 502.360 − 870.114i 640.002 − 1108.52i 11320.2 7034.64 + 12184.3i −17216.6 − 29820.1i
5.10 −13.0143 57.2297 − 99.1247i −342.627 153.733 + 266.273i −744.806 + 1290.04i −2137.39 + 3702.06i 11122.4 3291.02 + 5700.22i −2000.73 − 3465.37i
5.11 −1.70424 138.183 − 239.340i −509.096 209.544 + 362.940i −235.497 + 407.893i 3407.98 − 5902.80i 1740.19 −28347.6 − 49099.5i −357.113 − 618.537i
5.12 2.34760 50.3593 − 87.2248i −506.489 −1307.88 − 2265.32i 118.223 − 204.769i −1619.56 + 2805.17i −2391.00 4769.39 + 8260.82i −3070.38 − 5318.05i
5.13 4.02855 −112.657 + 195.128i −495.771 −618.220 − 1070.79i −453.846 + 786.084i −3386.19 + 5865.05i −4059.85 −15541.9 − 26919.4i −2490.53 − 4313.72i
5.14 8.18617 −59.5227 + 103.096i −444.987 411.796 + 713.252i −487.262 + 843.963i −512.114 + 887.007i −7834.05 2755.60 + 4772.85i 3371.03 + 5838.80i
5.15 10.7415 −17.2845 + 29.9377i −396.621 41.3396 + 71.6022i −185.661 + 321.575i 5171.52 − 8957.33i −9759.93 9243.99 + 16011.1i 444.048 + 769.114i
5.16 14.8577 82.0513 − 142.117i −291.249 1004.03 + 1739.04i 1219.09 − 2111.53i −5211.62 + 9026.79i −11934.4 −3623.33 − 6275.79i 14917.6 + 25838.1i
5.17 23.5868 62.8926 − 108.933i 44.3392 −451.538 − 782.086i 1483.44 − 2569.39i 1235.75 − 2140.39i −11030.6 1930.54 + 3343.80i −10650.4 − 18447.0i
5.18 31.0622 −33.8700 + 58.6645i 452.861 −124.424 − 215.508i −1052.08 + 1822.25i −4003.98 + 6935.09i −1836.98 7547.15 + 13072.0i −3864.87 − 6694.16i
5.19 31.3190 −121.812 + 210.984i 468.879 1025.46 + 1776.16i −3815.02 + 6607.81i 855.444 − 1481.67i −1350.49 −19834.7 − 34354.8i 32116.5 + 55627.4i
5.20 33.0653 65.9179 − 114.173i 581.315 1199.07 + 2076.84i 2179.60 − 3775.17i 4234.05 − 7333.59i 2291.93 1151.15 + 1993.86i 39647.5 + 68671.5i
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.23
Significant digits:
Format:

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 31.10.c.a ✓ 46
31.c even 3 1 inner 31.10.c.a ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.10.c.a ✓ 46 1.a even 1 1 trivial
31.10.c.a ✓ 46 31.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(31, [\chi])\).