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Properties

Label 35.5.l.a.32.8
Level $35$
Weight $5$
Character 35.32
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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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] = [35,5,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), 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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 32.8
Character \(\chi\) \(=\) 35.32
Dual form 35.5.l.a.23.8

$q$-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+(0.132660 - 0.495094i) q^{2} +(-4.13434 - 15.4296i) q^{3} +(13.6289 + 7.86864i) q^{4} +(-24.9995 + 0.157376i) q^{5} -8.18755 q^{6} +(-32.6831 - 36.5078i) q^{7} +(11.5027 - 11.5027i) q^{8} +(-150.831 + 87.0822i) q^{9} +(-3.23852 + 12.3980i) q^{10} +(42.7420 - 74.0314i) q^{11} +(65.0633 - 242.819i) q^{12} +(131.873 - 131.873i) q^{13} +(-22.4105 + 11.3381i) q^{14} +(105.785 + 385.081i) q^{15} +(121.729 + 210.841i) q^{16} +(-186.561 + 49.9889i) q^{17} +(23.1047 + 86.2277i) q^{18} +(435.787 - 251.602i) q^{19} +(-341.954 - 194.567i) q^{20} +(-428.176 + 655.221i) q^{21} +(-30.9823 - 30.9823i) q^{22} +(220.079 + 58.9699i) q^{23} +(-225.037 - 129.925i) q^{24} +(624.950 - 7.86864i) q^{25} +(-47.7951 - 82.7835i) q^{26} +(1052.31 + 1052.31i) q^{27} +(-158.167 - 754.732i) q^{28} -592.118i q^{29} +(204.685 - 1.28852i) q^{30} +(244.720 - 423.868i) q^{31} +(371.942 - 99.6616i) q^{32} +(-1318.98 - 353.420i) q^{33} +98.9968i q^{34} +(822.806 + 907.533i) q^{35} -2740.87 q^{36} +(-20.3539 + 75.9617i) q^{37} +(-66.7551 - 249.133i) q^{38} +(-2579.94 - 1489.53i) q^{39} +(-285.751 + 289.371i) q^{40} -1391.63 q^{41} +(267.594 + 298.909i) q^{42} +(-838.944 + 838.944i) q^{43} +(1165.05 - 672.643i) q^{44} +(3756.99 - 2200.75i) q^{45} +(58.3913 - 101.137i) q^{46} +(-483.986 + 1806.26i) q^{47} +(2749.92 - 2749.92i) q^{48} +(-264.635 + 2386.37i) q^{49} +(79.0103 - 310.453i) q^{50} +(1542.61 + 2671.89i) q^{51} +(2834.93 - 759.618i) q^{52} +(-127.620 - 476.285i) q^{53} +(660.593 - 381.394i) q^{54} +(-1056.88 + 1857.47i) q^{55} +(-795.880 - 43.9944i) q^{56} +(-5683.80 - 5683.80i) q^{57} +(-293.154 - 78.5505i) q^{58} +(-704.535 - 406.763i) q^{59} +(-1588.34 + 6080.61i) q^{60} +(1169.51 + 2025.66i) q^{61} +(-177.390 - 177.390i) q^{62} +(8108.79 + 2660.38i) q^{63} +3697.97i q^{64} +(-3275.99 + 3317.50i) q^{65} +(-349.953 + 606.136i) q^{66} +(6748.40 - 1808.23i) q^{67} +(-2935.96 - 786.689i) q^{68} -3639.52i q^{69} +(558.468 - 286.973i) q^{70} -767.932 q^{71} +(-733.279 + 2736.63i) q^{72} +(-702.461 - 2621.62i) q^{73} +(34.9081 + 20.1542i) q^{74} +(-2705.17 - 9610.18i) q^{75} +7919.06 q^{76} +(-4099.66 + 859.156i) q^{77} +(-1079.71 + 1079.71i) q^{78} +(4386.38 - 2532.48i) q^{79} +(-3076.35 - 5251.77i) q^{80} +(4832.45 - 8370.05i) q^{81} +(-184.614 + 688.987i) q^{82} +(-7579.82 + 7579.82i) q^{83} +(-10991.3 + 5560.77i) q^{84} +(4656.07 - 1279.06i) q^{85} +(304.062 + 526.651i) q^{86} +(-9136.13 + 2448.02i) q^{87} +(-359.911 - 1343.21i) q^{88} +(12187.4 - 7036.42i) q^{89} +(-591.175 - 2152.01i) q^{90} +(-9124.37 - 504.374i) q^{91} +(2535.42 + 2535.42i) q^{92} +(-7551.85 - 2023.51i) q^{93} +(830.064 + 479.238i) q^{94} +(-10854.9 + 6358.51i) q^{95} +(-3075.47 - 5326.87i) q^{96} +(6312.04 + 6312.04i) q^{97} +(1146.37 + 447.595i) q^{98} +14888.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.132660 − 0.495094i 0.0331650 − 0.123774i −0.947359 − 0.320174i \(-0.896259\pi\)
0.980524 + 0.196401i \(0.0629253\pi\)
\(3\) −4.13434 − 15.4296i −0.459371 − 1.71440i −0.674909 − 0.737901i \(-0.735817\pi\)
0.215538 − 0.976495i \(-0.430849\pi\)
\(4\) 13.6289 + 7.86864i 0.851805 + 0.491790i
\(5\) −24.9995 + 0.157376i −0.999980 + 0.00629503i
\(6\) −8.18755 −0.227432
\(7\) −32.6831 − 36.5078i −0.667001 − 0.745057i
\(8\) 11.5027 − 11.5027i 0.179729 − 0.179729i
\(9\) −150.831 + 87.0822i −1.86211 + 1.07509i
\(10\) −3.23852 + 12.3980i −0.0323852 + 0.123980i
\(11\) 42.7420 − 74.0314i 0.353240 − 0.611830i −0.633575 − 0.773681i \(-0.718413\pi\)
0.986815 + 0.161852i \(0.0517466\pi\)
\(12\) 65.0633 − 242.819i 0.451828 − 1.68625i
\(13\) 131.873 − 131.873i 0.780311 − 0.780311i −0.199572 − 0.979883i \(-0.563955\pi\)
0.979883 + 0.199572i \(0.0639553\pi\)
\(14\) −22.4105 + 11.3381i −0.114339 + 0.0578473i
\(15\) 105.785 + 385.081i 0.470154 + 1.71147i
\(16\) 121.729 + 210.841i 0.475505 + 0.823599i
\(17\) −186.561 + 49.9889i −0.645540 + 0.172972i −0.566712 − 0.823916i \(-0.691785\pi\)
−0.0788284 + 0.996888i \(0.525118\pi\)
\(18\) 23.1047 + 86.2277i 0.0713107 + 0.266135i
\(19\) 435.787 − 251.602i 1.20717 − 0.696958i 0.245028 − 0.969516i \(-0.421203\pi\)
0.962139 + 0.272558i \(0.0878696\pi\)
\(20\) −341.954 − 194.567i −0.854884 − 0.486418i
\(21\) −428.176 + 655.221i −0.970921 + 1.48576i
\(22\) −30.9823 − 30.9823i −0.0640131 − 0.0640131i
\(23\) 220.079 + 58.9699i 0.416028 + 0.111474i 0.460760 − 0.887525i \(-0.347577\pi\)
−0.0447323 + 0.998999i \(0.514243\pi\)
\(24\) −225.037 − 129.925i −0.390690 − 0.225565i
\(25\) 624.950 − 7.86864i 0.999921 − 0.0125898i
\(26\) −47.7951 − 82.7835i −0.0707028 − 0.122461i
\(27\) 1052.31 + 1052.31i 1.44350 + 1.44350i
\(28\) −158.167 − 754.732i −0.201744 − 0.962668i
\(29\) − 592.118i − 0.704065i −0.935988 − 0.352032i \(-0.885491\pi\)
0.935988 − 0.352032i \(-0.114509\pi\)
\(30\) 204.685 − 1.28852i 0.227427 − 0.00143169i
\(31\) 244.720 − 423.868i 0.254652 − 0.441069i −0.710149 − 0.704051i \(-0.751373\pi\)
0.964801 + 0.262982i \(0.0847059\pi\)
\(32\) 371.942 − 99.6616i 0.363225 − 0.0973258i
\(33\) −1318.98 − 353.420i −1.21119 − 0.324536i
\(34\) 98.9968i 0.0856374i
\(35\) 822.806 + 907.533i 0.671678 + 0.740843i
\(36\) −2740.87 −2.11487
\(37\) −20.3539 + 75.9617i −0.0148677 + 0.0554870i −0.972961 − 0.230969i \(-0.925810\pi\)
0.958093 + 0.286456i \(0.0924771\pi\)
\(38\) −66.7551 − 249.133i −0.0462293 − 0.172530i
\(39\) −2579.94 − 1489.53i −1.69621 − 0.979310i
\(40\) −285.751 + 289.371i −0.178594 + 0.180857i
\(41\) −1391.63 −0.827858 −0.413929 − 0.910309i \(-0.635844\pi\)
−0.413929 + 0.910309i \(0.635844\pi\)
\(42\) 267.594 + 298.909i 0.151697 + 0.169450i
\(43\) −838.944 + 838.944i −0.453728 + 0.453728i −0.896590 − 0.442862i \(-0.853963\pi\)
0.442862 + 0.896590i \(0.353963\pi\)
\(44\) 1165.05 − 672.643i 0.601783 − 0.347440i
\(45\) 3756.99 − 2200.75i 1.85530 − 1.08679i
\(46\) 58.3913 − 101.137i 0.0275951 − 0.0477962i
\(47\) −483.986 + 1806.26i −0.219097 + 0.817683i 0.765586 + 0.643333i \(0.222449\pi\)
−0.984684 + 0.174350i \(0.944218\pi\)
\(48\) 2749.92 − 2749.92i 1.19354 − 1.19354i
\(49\) −264.635 + 2386.37i −0.110219 + 0.993907i
\(50\) 79.0103 − 310.453i 0.0316041 − 0.124181i
\(51\) 1542.61 + 2671.89i 0.593085 + 1.02725i
\(52\) 2834.93 − 759.618i 1.04842 − 0.280924i
\(53\) −127.620 − 476.285i −0.0454326 − 0.169557i 0.939482 − 0.342599i \(-0.111307\pi\)
−0.984914 + 0.173042i \(0.944640\pi\)
\(54\) 660.593 − 381.394i 0.226541 − 0.130793i
\(55\) −1056.88 + 1857.47i −0.349381 + 0.614041i
\(56\) −795.880 − 43.9944i −0.253788 − 0.0140288i
\(57\) −5683.80 − 5683.80i −1.74940 − 1.74940i
\(58\) −293.154 − 78.5505i −0.0871446 − 0.0233503i
\(59\) −704.535 − 406.763i −0.202394 − 0.116852i 0.395377 − 0.918519i \(-0.370614\pi\)
−0.597772 + 0.801666i \(0.703947\pi\)
\(60\) −1588.34 + 6080.61i −0.441204 + 1.68906i
\(61\) 1169.51 + 2025.66i 0.314301 + 0.544386i 0.979289 − 0.202469i \(-0.0648965\pi\)
−0.664987 + 0.746855i \(0.731563\pi\)
\(62\) −177.390 − 177.390i −0.0461472 − 0.0461472i
\(63\) 8108.79 + 2660.38i 2.04303 + 0.670290i
\(64\) 3697.97i 0.902825i
\(65\) −3275.99 + 3317.50i −0.775383 + 0.785207i
\(66\) −349.953 + 606.136i −0.0803381 + 0.139150i
\(67\) 6748.40 − 1808.23i 1.50332 − 0.402813i 0.589109 − 0.808053i \(-0.299479\pi\)
0.914210 + 0.405240i \(0.132812\pi\)
\(68\) −2935.96 − 786.689i −0.634940 − 0.170132i
\(69\) − 3639.52i − 0.764445i
\(70\) 558.468 − 286.973i 0.113973 − 0.0585659i
\(71\) −767.932 −0.152337 −0.0761686 − 0.997095i \(-0.524269\pi\)
−0.0761686 + 0.997095i \(0.524269\pi\)
\(72\) −733.279 + 2736.63i −0.141450 + 0.527900i
\(73\) −702.461 − 2621.62i −0.131818 − 0.491953i 0.868172 − 0.496263i \(-0.165295\pi\)
−0.999991 + 0.00430977i \(0.998628\pi\)
\(74\) 34.9081 + 20.1542i 0.00637474 + 0.00368046i
\(75\) −2705.17 − 9610.18i −0.480919 − 1.70848i
\(76\) 7919.06 1.37103
\(77\) −4099.66 + 859.156i −0.691459 + 0.144907i
\(78\) −1079.71 + 1079.71i −0.177468 + 0.177468i
\(79\) 4386.38 − 2532.48i 0.702833 − 0.405781i −0.105569 − 0.994412i \(-0.533666\pi\)
0.808402 + 0.588631i \(0.200333\pi\)
\(80\) −3076.35 − 5251.77i −0.480680 − 0.820589i
\(81\) 4832.45 − 8370.05i 0.736541 − 1.27573i
\(82\) −184.614 + 688.987i −0.0274559 + 0.102467i
\(83\) −7579.82 + 7579.82i −1.10028 + 1.10028i −0.105903 + 0.994376i \(0.533773\pi\)
−0.994376 + 0.105903i \(0.966227\pi\)
\(84\) −10991.3 + 5560.77i −1.55772 + 0.788091i
\(85\) 4656.07 − 1279.06i 0.644438 − 0.177032i
\(86\) 304.062 + 526.651i 0.0411117 + 0.0712075i
\(87\) −9136.13 + 2448.02i −1.20705 + 0.323427i
\(88\) −359.911 − 1343.21i −0.0464761 − 0.173451i
\(89\) 12187.4 − 7036.42i 1.53862 − 0.888325i 0.539704 − 0.841855i \(-0.318536\pi\)
0.998920 − 0.0464696i \(-0.0147971\pi\)
\(90\) −591.175 − 2152.01i −0.0729846 − 0.265681i
\(91\) −9124.37 − 504.374i −1.10184 − 0.0609074i
\(92\) 2535.42 + 2535.42i 0.299553 + 0.299553i
\(93\) −7551.85 − 2023.51i −0.873147 − 0.233959i
\(94\) 830.064 + 479.238i 0.0939411 + 0.0542369i
\(95\) −10854.9 + 6358.51i −1.20276 + 0.704544i
\(96\) −3075.47 − 5326.87i −0.333710 − 0.578002i
\(97\) 6312.04 + 6312.04i 0.670852 + 0.670852i 0.957912 − 0.287061i \(-0.0926781\pi\)
−0.287061 + 0.957912i \(0.592678\pi\)
\(98\) 1146.37 + 447.595i 0.119364 + 0.0466051i
\(99\) 14888.3i 1.51906i
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 35.5.l.a.32.8 yes 56
5.3 odd 4 inner 35.5.l.a.18.7 yes 56
7.2 even 3 inner 35.5.l.a.2.7 ✓ 56
35.23 odd 12 inner 35.5.l.a.23.8 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.7 ✓ 56 7.2 even 3 inner
35.5.l.a.18.7 yes 56 5.3 odd 4 inner
35.5.l.a.23.8 yes 56 35.23 odd 12 inner
35.5.l.a.32.8 yes 56 1.1 even 1 trivial