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Properties

Label 31.7.h.a.3.4
Level $31$
Weight $7$
Character 31.3
Analytic conductor $7.132$
Analytic rank $0$
Dimension $120$
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(3,31)] 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("31.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
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, 7, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(7.13167659222\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(15\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label 3.4
Character \(\chi\) \(=\) 31.3
Dual form 31.7.h.a.21.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-6.15108 + 4.46902i) q^{2} +(31.4465 + 3.30516i) q^{3} +(-1.91345 + 5.88901i) q^{4} +(118.800 - 205.767i) q^{5} +(-208.201 + 120.205i) q^{6} +(-18.5129 + 3.93503i) q^{7} +(-164.917 - 507.561i) q^{8} +(264.889 + 56.3040i) q^{9} +(188.831 + 1796.61i) q^{10} +(1660.95 - 1495.53i) q^{11} +(-79.6356 + 178.864i) q^{12} +(1162.25 + 2610.46i) q^{13} +(96.2884 - 106.939i) q^{14} +(4415.92 - 6078.00i) q^{15} +(2962.11 + 2152.10i) q^{16} +(1822.95 + 1641.39i) q^{17} +(-1880.98 + 837.466i) q^{18} +(1141.12 + 508.060i) q^{19} +(984.445 + 1093.34i) q^{20} +(-595.171 + 62.5550i) q^{21} +(-3533.11 + 16622.0i) q^{22} +(-10085.0 + 3276.80i) q^{23} +(-3508.48 - 16506.1i) q^{24} +(-20414.2 - 35358.4i) q^{25} +(-18815.3 - 10863.0i) q^{26} +(-13778.8 - 4477.02i) q^{27} +(12.2501 - 116.552i) q^{28} +(7434.77 + 10233.1i) q^{29} +57121.1i q^{30} +(4060.43 + 29513.0i) q^{31} +6317.66 q^{32} +(57174.2 - 41539.5i) q^{33} +(-18548.6 - 1949.53i) q^{34} +(-1389.62 + 4276.82i) q^{35} +(-838.428 + 1452.20i) q^{36} +(46915.8 - 27086.8i) q^{37} +(-9289.67 + 1974.58i) q^{38} +(27920.8 + 85931.3i) q^{39} +(-124031. - 26363.7i) q^{40} +(-5631.51 - 53580.3i) q^{41} +(3381.39 - 3044.61i) q^{42} +(-4017.42 + 9023.28i) q^{43} +(5629.02 + 12643.0i) q^{44} +(43054.2 - 47816.6i) q^{45} +(47389.3 - 65225.7i) q^{46} +(-64233.8 - 46668.6i) q^{47} +(86034.9 + 77466.2i) q^{48} +(-107150. + 47706.5i) q^{49} +(283587. + 126261. i) q^{50} +(51900.4 + 57641.3i) q^{51} +(-17596.9 + 1849.51i) q^{52} +(-17809.0 + 83784.9i) q^{53} +(104763. - 34039.4i) q^{54} +(-110410. - 519437. i) q^{55} +(5050.35 + 8747.46i) q^{56} +(34205.1 + 19748.3i) q^{57} +(-91463.8 - 29718.4i) q^{58} +(-32064.2 + 305070. i) q^{59} +(27343.7 + 37635.4i) q^{60} -182239. i q^{61} +(-156870. - 163391. i) q^{62} -5125.42 q^{63} +(-228435. + 165968. i) q^{64} +(675221. + 70968.6i) q^{65} +(-166042. + 511025. i) q^{66} +(-19781.4 + 34262.4i) q^{67} +(-13154.3 + 7594.65i) q^{68} +(-327967. + 69711.5i) q^{69} +(-10565.5 - 32517.3i) q^{70} +(-188145. - 39991.5i) q^{71} +(-15106.9 - 143733. i) q^{72} +(-192396. + 173234. i) q^{73} +(-167531. + 376281. i) q^{74} +(-525089. - 1.17937e6i) q^{75} +(-5175.46 + 5747.93i) q^{76} +(-24864.1 + 34222.5i) q^{77} +(-555771. - 403792. i) q^{78} +(595545. + 536231. i) q^{79} +(794727. - 353835. i) q^{80} +(-598850. - 266625. i) q^{81} +(274091. + 304409. i) q^{82} +(-97586.8 + 10256.8i) q^{83} +(770.446 - 3624.66i) q^{84} +(554311. - 180106. i) q^{85} +(-15613.7 - 73456.9i) q^{86} +(199976. + 346368. i) q^{87} +(-1.03299e6 - 596397. i) q^{88} +(946417. + 307510. i) q^{89} +(-51136.8 + 486534. i) q^{90} +(-31788.9 - 43753.6i) q^{91} -65660.4i q^{92} +(30141.1 + 941501. i) q^{93} +603670. q^{94} +(240107. - 174448. i) q^{95} +(198668. + 20880.9i) q^{96} +(-509527. + 1.56816e6i) q^{97} +(445890. - 772304. i) q^{98} +(524173. - 302632. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 6 q^{2} - 7 q^{3} - 1034 q^{4} + 68 q^{5} + 1431 q^{6} - 1753 q^{7} + 811 q^{8} - 3488 q^{9} + 1158 q^{10} + 633 q^{11} + 12777 q^{12} - 10627 q^{13} + 16244 q^{14} + 8395 q^{15} - 4178 q^{16} + 1713 q^{17}+ \cdots - 3201573 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{30}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.15108 + 4.46902i −0.768885 + 0.558628i −0.901623 − 0.432524i \(-0.857623\pi\)
0.132738 + 0.991151i \(0.457623\pi\)
\(3\) 31.4465 + 3.30516i 1.16469 + 0.122413i 0.667077 − 0.744988i \(-0.267545\pi\)
0.497608 + 0.867402i \(0.334212\pi\)
\(4\) −1.91345 + 5.88901i −0.0298977 + 0.0920157i
\(5\) 118.800 − 205.767i 0.950397 − 1.64614i 0.205829 − 0.978588i \(-0.434011\pi\)
0.744567 − 0.667547i \(-0.232656\pi\)
\(6\) −208.201 + 120.205i −0.963893 + 0.556504i
\(7\) −18.5129 + 3.93503i −0.0539734 + 0.0114724i −0.234819 − 0.972039i \(-0.575450\pi\)
0.180846 + 0.983511i \(0.442116\pi\)
\(8\) −164.917 − 507.561i −0.322103 − 0.991330i
\(9\) 264.889 + 56.3040i 0.363360 + 0.0772345i
\(10\) 188.831 + 1796.61i 0.188831 + 1.79661i
\(11\) 1660.95 − 1495.53i 1.24790 − 1.12361i 0.260481 − 0.965479i \(-0.416119\pi\)
0.987418 − 0.158134i \(-0.0505479\pi\)
\(12\) −79.6356 + 178.864i −0.0460854 + 0.103510i
\(13\) 1162.25 + 2610.46i 0.529017 + 1.18819i 0.958494 + 0.285114i \(0.0920316\pi\)
−0.429476 + 0.903078i \(0.641302\pi\)
\(14\) 96.2884 − 106.939i 0.0350905 − 0.0389720i
\(15\) 4415.92 − 6078.00i 1.30842 − 1.80089i
\(16\) 2962.11 + 2152.10i 0.723171 + 0.525414i
\(17\) 1822.95 + 1641.39i 0.371047 + 0.334092i 0.833467 − 0.552569i \(-0.186352\pi\)
−0.462421 + 0.886661i \(0.653019\pi\)
\(18\) −1880.98 + 837.466i −0.322527 + 0.143598i
\(19\) 1141.12 + 508.060i 0.166369 + 0.0740721i 0.488231 − 0.872714i \(-0.337642\pi\)
−0.321863 + 0.946786i \(0.604309\pi\)
\(20\) 984.445 + 1093.34i 0.123056 + 0.136667i
\(21\) −595.171 + 62.5550i −0.0642664 + 0.00675467i
\(22\) −3533.11 + 16622.0i −0.331809 + 1.56104i
\(23\) −10085.0 + 3276.80i −0.828878 + 0.269319i −0.692573 − 0.721348i \(-0.743523\pi\)
−0.136305 + 0.990667i \(0.543523\pi\)
\(24\) −3508.48 − 16506.1i −0.253796 − 1.19402i
\(25\) −20414.2 − 35358.4i −1.30651 − 2.26294i
\(26\) −18815.3 − 10863.0i −1.07051 − 0.618060i
\(27\) −13778.8 − 4477.02i −0.700038 − 0.227456i
\(28\) 12.2501 − 116.552i 0.000558040 − 0.00530940i
\(29\) 7434.77 + 10233.1i 0.304841 + 0.419578i 0.933764 − 0.357890i \(-0.116504\pi\)
−0.628922 + 0.777468i \(0.716504\pi\)
\(30\) 57121.1i 2.11560i
\(31\) 4060.43 + 29513.0i 0.136297 + 0.990668i
\(32\) 6317.66 0.192800
\(33\) 57174.2 − 41539.5i 1.59096 − 1.15590i
\(34\) −18548.6 − 1949.53i −0.471925 − 0.0496013i
\(35\) −1389.62 + 4276.82i −0.0324110 + 0.0997508i
\(36\) −838.428 + 1452.20i −0.0179704 + 0.0311257i
\(37\) 46915.8 − 27086.8i 0.926219 − 0.534753i 0.0406054 − 0.999175i \(-0.487071\pi\)
0.885614 + 0.464422i \(0.153738\pi\)
\(38\) −9289.67 + 1974.58i −0.169297 + 0.0359852i
\(39\) 27920.8 + 85931.3i 0.470688 + 1.44863i
\(40\) −124031. − 26363.7i −1.93799 − 0.411932i
\(41\) −5631.51 − 53580.3i −0.0817097 − 0.777416i −0.956267 − 0.292494i \(-0.905515\pi\)
0.874558 − 0.484922i \(-0.161152\pi\)
\(42\) 3381.39 − 3044.61i 0.0456401 − 0.0410946i
\(43\) −4017.42 + 9023.28i −0.0505292 + 0.113490i −0.937046 − 0.349206i \(-0.886451\pi\)
0.886517 + 0.462696i \(0.153118\pi\)
\(44\) 5629.02 + 12643.0i 0.0660808 + 0.148420i
\(45\) 43054.2 − 47816.6i 0.472474 − 0.524736i
\(46\) 47389.3 − 65225.7i 0.486863 − 0.670109i
\(47\) −64233.8 − 46668.6i −0.618685 − 0.449501i 0.233777 − 0.972290i \(-0.424892\pi\)
−0.852462 + 0.522789i \(0.824892\pi\)
\(48\) 86034.9 + 77466.2i 0.777949 + 0.700468i
\(49\) −107150. + 47706.5i −0.910764 + 0.405498i
\(50\) 283587. + 126261.i 2.26869 + 1.01009i
\(51\) 51900.4 + 57641.3i 0.391255 + 0.434533i
\(52\) −17596.9 + 1849.51i −0.125149 + 0.0131537i
\(53\) −17809.0 + 83784.9i −0.119622 + 0.562779i 0.876988 + 0.480512i \(0.159549\pi\)
−0.996610 + 0.0822667i \(0.973784\pi\)
\(54\) 104763. − 34039.4i 0.665312 − 0.216173i
\(55\) −110410. − 519437.i −0.663620 − 3.12209i
\(56\) 5050.35 + 8747.46i 0.0287579 + 0.0498101i
\(57\) 34205.1 + 19748.3i 0.184700 + 0.106636i
\(58\) −91463.8 − 29718.4i −0.468776 − 0.152314i
\(59\) −32064.2 + 305070.i −0.156122 + 1.48540i 0.583354 + 0.812218i \(0.301740\pi\)
−0.739476 + 0.673183i \(0.764926\pi\)
\(60\) 27343.7 + 37635.4i 0.126591 + 0.174238i
\(61\) − 182239.i − 0.802883i −0.915885 − 0.401442i \(-0.868509\pi\)
0.915885 − 0.401442i \(-0.131491\pi\)
\(62\) −156870. − 163391.i −0.658211 − 0.685570i
\(63\) −5125.42 −0.0204978
\(64\) −228435. + 165968.i −0.871412 + 0.633118i
\(65\) 675221. + 70968.6i 2.45870 + 0.258420i
\(66\) −166042. + 511025.i −0.577546 + 1.77750i
\(67\) −19781.4 + 34262.4i −0.0657708 + 0.113918i −0.897036 − 0.441958i \(-0.854284\pi\)
0.831265 + 0.555877i \(0.187617\pi\)
\(68\) −13154.3 + 7594.65i −0.0418352 + 0.0241535i
\(69\) −327967. + 69711.5i −0.998350 + 0.212206i
\(70\) −10565.5 − 32517.3i −0.0308032 − 0.0948026i
\(71\) −188145. − 39991.5i −0.525676 − 0.111736i −0.0625731 − 0.998040i \(-0.519931\pi\)
−0.463103 + 0.886304i \(0.653264\pi\)
\(72\) −15106.9 − 143733.i −0.0404743 − 0.385087i
\(73\) −192396. + 173234.i −0.494569 + 0.445312i −0.878284 − 0.478140i \(-0.841311\pi\)
0.383715 + 0.923452i \(0.374645\pi\)
\(74\) −167531. + 376281.i −0.413428 + 0.928575i
\(75\) −525089. − 1.17937e6i −1.24466 − 2.79554i
\(76\) −5175.46 + 5747.93i −0.0117898 + 0.0130939i
\(77\) −24864.1 + 34222.5i −0.0544628 + 0.0749616i
\(78\) −555771. − 403792.i −1.17115 − 0.850890i
\(79\) 595545. + 536231.i 1.20791 + 1.08760i 0.993844 + 0.110785i \(0.0353364\pi\)
0.214063 + 0.976820i \(0.431330\pi\)
\(80\) 794727. − 353835.i 1.55220 − 0.691085i
\(81\) −598850. − 266625.i −1.12684 − 0.501702i
\(82\) 274091. + 304409.i 0.497111 + 0.552098i
\(83\) −97586.8 + 10256.8i −0.170670 + 0.0179381i −0.189478 − 0.981885i \(-0.560680\pi\)
0.0188080 + 0.999823i \(0.494013\pi\)
\(84\) 770.446 − 3624.66i 0.00129988 − 0.00611547i
\(85\) 554311. − 180106.i 0.902602 − 0.293273i
\(86\) −15613.7 − 73456.9i −0.0245478 − 0.115488i
\(87\) 199976. + 346368.i 0.303682 + 0.525993i
\(88\) −1.03299e6 − 596397.i −1.51582 − 0.875161i
\(89\) 946417. + 307510.i 1.34249 + 0.436203i 0.890161 − 0.455646i \(-0.150592\pi\)
0.452333 + 0.891849i \(0.350592\pi\)
\(90\) −51136.8 + 486534.i −0.0701465 + 0.667399i
\(91\) −31788.9 − 43753.6i −0.0421843 − 0.0580617i
\(92\) − 65660.4i − 0.0843218i
\(93\) 30141.1 + 941501.i 0.0374723 + 1.17050i
\(94\) 603670. 0.726802
\(95\) 240107. − 174448.i 0.280049 − 0.203467i
\(96\) 198668. + 20880.9i 0.224551 + 0.0236013i
\(97\) −509527. + 1.56816e6i −0.558280 + 1.71821i 0.128840 + 0.991665i \(0.458875\pi\)
−0.687120 + 0.726544i \(0.741125\pi\)
\(98\) 445890. − 772304.i 0.473750 − 0.820559i
\(99\) 524173. − 302632.i 0.540218 − 0.311895i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 31.7.h.a.3.4 ✓ 120
31.21 odd 30 inner 31.7.h.a.21.4 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.7.h.a.3.4 ✓ 120 1.1 even 1 trivial
31.7.h.a.21.4 yes 120 31.21 odd 30 inner