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Properties

Label 31.10.d.a
Level $31$
Weight $10$
Character orbit 31.d
Analytic conductor $15.966$
Analytic rank $0$
Dimension $92$
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(2,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.2");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 31.d (of order \(5\), degree \(4\), 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: \(92\)
Relative dimension: \(23\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 3 q^{2} - 151 q^{3} - 6319 q^{4} + 1584 q^{5} - 5386 q^{6} + 15771 q^{7} - 12953 q^{8} - 157912 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q - 3 q^{2} - 151 q^{3} - 6319 q^{4} + 1584 q^{5} - 5386 q^{6} + 15771 q^{7} - 12953 q^{8} - 157912 q^{9} + 18318 q^{10} + 49351 q^{11} - 306946 q^{12} + 117143 q^{13} + 441366 q^{14} + 449822 q^{15} - 1855799 q^{16} + 613995 q^{17} - 1381746 q^{18} + 289461 q^{19} + 281705 q^{20} - 1715851 q^{21} + 7251232 q^{22} - 208705 q^{23} - 6260441 q^{24} + 25610952 q^{25} - 2001216 q^{26} + 459161 q^{27} - 11304828 q^{28} + 1566863 q^{29} - 24725646 q^{30} - 5804508 q^{31} - 26968914 q^{32} + 33992053 q^{33} + 31693419 q^{34} + 6408048 q^{35} + 41944304 q^{36} - 67071820 q^{37} - 46931954 q^{38} + 15891463 q^{39} - 86537994 q^{40} - 9404335 q^{41} - 7378792 q^{42} - 36242639 q^{43} + 127495740 q^{44} + 91781098 q^{45} - 50236575 q^{46} + 93590983 q^{47} + 17999968 q^{48} - 171477320 q^{49} + 348293382 q^{50} - 460241827 q^{51} + 213126917 q^{52} - 182127291 q^{53} + 184687970 q^{54} + 181333082 q^{55} + 377755680 q^{56} - 4747334 q^{57} + 373882383 q^{58} - 254498857 q^{59} - 491449043 q^{60} - 886376336 q^{61} + 543847936 q^{62} - 1295550188 q^{63} + 99931235 q^{64} + 577850928 q^{65} + 1063614202 q^{66} + 257099728 q^{67} + 537721458 q^{68} + 573163321 q^{69} - 1049281267 q^{70} - 608870707 q^{71} - 1184294518 q^{72} - 542939885 q^{73} + 713414309 q^{74} - 1169688653 q^{75} + 766438841 q^{76} + 1398843651 q^{77} - 2437636805 q^{78} + 451474297 q^{79} - 1164338982 q^{80} + 429565424 q^{81} - 2147668162 q^{82} + 1991935861 q^{83} + 1791837753 q^{84} + 4248448078 q^{85} + 3993957107 q^{86} + 1084208242 q^{87} - 3863172636 q^{88} + 649141667 q^{89} + 871054434 q^{90} + 2980495001 q^{91} - 6085362530 q^{92} + 3068495993 q^{93} - 7397877454 q^{94} - 2290877894 q^{95} + 7959812326 q^{96} - 2207813463 q^{97} + 4266357892 q^{98} - 639007372 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −34.2039 − 24.8506i −42.9734 + 31.2220i 394.138 + 1213.03i 778.569 2245.74 2039.74 + 6277.67i 9974.37 − 30698.0i −5210.48 + 16036.2i −26630.1 − 19347.9i
2.2 −31.4635 − 22.8596i 211.187 − 153.436i 309.175 + 951.542i 679.781 −10152.2 854.444 + 2629.71i 5870.91 − 18068.8i 14974.9 − 46087.9i −21388.3 − 15539.5i
2.3 −29.8196 − 21.6652i 34.0538 − 24.7415i 261.610 + 805.152i −1883.76 −1551.50 −1361.77 − 4191.10i 3810.96 − 11728.9i −5534.86 + 17034.6i 56172.8 + 40811.9i
2.4 −28.1273 − 20.4357i −187.075 + 135.918i 215.311 + 662.659i 471.762 8039.50 −2813.32 − 8658.52i 1985.01 − 6109.24i 10441.0 − 32134.2i −13269.4 − 9640.77i
2.5 −23.8834 − 17.3523i 35.6732 − 25.9181i 111.098 + 341.923i 2554.06 −1301.74 −800.611 − 2464.03i −1391.03 + 4281.14i −5481.55 + 16870.5i −60999.7 − 44318.9i
2.6 −22.1126 − 16.0657i −158.931 + 115.470i 72.6413 + 223.567i −1269.45 5369.49 3390.13 + 10433.8i −2339.00 + 7198.70i 5843.39 − 17984.1i 28070.8 + 20394.6i
2.7 −16.2968 − 11.8403i 115.015 − 83.5633i −32.8248 − 101.024i 68.6750 −2863.79 −2227.04 − 6854.13i −3848.32 + 11843.9i 163.250 − 502.433i −1119.18 − 813.132i
2.8 −14.8277 − 10.7729i 131.699 − 95.6849i −54.4131 − 167.466i −1154.13 −2983.60 2599.13 + 7999.30i −3897.08 + 11994.0i 2106.64 − 6483.58i 17113.0 + 12433.3i
2.9 −12.5485 − 9.11702i −73.2220 + 53.1989i −83.8718 − 258.131i 758.790 1403.84 633.852 + 1950.80i −3754.99 + 11556.7i −3551.04 + 10929.0i −9521.68 − 6917.90i
2.10 −5.94595 − 4.31999i −128.478 + 93.3449i −141.525 − 435.568i −2314.93 1167.17 −2750.04 − 8463.75i −2202.98 + 6780.07i 1711.00 − 5265.92i 13764.5 + 10000.5i
2.11 0.585851 + 0.425645i 197.879 − 143.767i −158.055 − 486.442i 649.185 177.121 −2781.92 − 8561.88i 229.028 − 704.877i 12404.6 − 38177.3i 380.325 + 276.322i
2.12 3.74252 + 2.71910i −188.239 + 136.764i −151.604 − 466.588i 1778.75 −1076.36 27.4584 + 84.5082i 1433.23 − 4411.04i 10647.2 − 32768.8i 6657.00 + 4836.59i
2.13 3.79892 + 2.76008i 29.0658 − 21.1175i −151.403 − 465.970i −1644.92 168.704 224.598 + 691.241i 1453.89 − 4474.60i −5683.51 + 17492.1i −6248.91 − 4540.10i
2.14 4.94824 + 3.59511i 138.202 − 100.410i −146.656 − 451.362i 2448.48 1044.84 3793.47 + 11675.1i 1864.71 − 5739.00i 2935.30 − 9033.92i 12115.6 + 8802.53i
2.15 5.07595 + 3.68790i −13.1772 + 9.57377i −146.052 − 449.502i 417.592 −102.194 −511.211 − 1573.35i 1909.05 − 5875.45i −6000.40 + 18467.3i 2119.68 + 1540.04i
2.16 15.9164 + 11.5639i −150.460 + 109.315i −38.6102 − 118.830i −1497.34 −3658.89 3126.33 + 9621.85i 3872.31 − 11917.8i 4605.91 − 14175.5i −23832.2 − 17315.1i
2.17 16.5154 + 11.9992i 154.050 − 111.924i −29.4370 − 90.5978i −1892.14 3887.19 64.5011 + 198.514i 3830.80 − 11790.0i 5122.02 − 15764.0i −31249.5 − 22704.1i
2.18 19.6227 + 14.2567i 18.3856 − 13.3579i 23.5795 + 72.5704i 1651.42 551.216 −3038.37 − 9351.13i 3265.62 − 10050.6i −5922.79 + 18228.5i 32405.2 + 23543.8i
2.19 24.0916 + 17.5036i −12.6846 + 9.21588i 115.813 + 356.435i 283.207 −466.902 2149.18 + 6614.51i 1262.74 − 3886.31i −6006.42 + 18485.8i 6822.90 + 4957.12i
2.20 24.8040 + 18.0211i −164.557 + 119.558i 132.259 + 407.050i −624.568 −6236.24 −2239.70 − 6893.08i 795.862 − 2449.41i 6702.65 − 20628.6i −15491.8 − 11255.4i
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.10.d.a ✓ 92
31.d even 5 1 inner 31.10.d.a ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.10.d.a ✓ 92 1.a even 1 1 trivial
31.10.d.a ✓ 92 31.d even 5 1 inner

Hecke kernels

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