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Properties

Label 35.5.l.a.18.7
Level $35$
Weight $5$
Character 35.18
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 18.7
Character \(\chi\) \(=\) 35.18
Dual form 35.5.l.a.2.7

$q$-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+(-0.495094 - 0.132660i) q^{2} +(15.4296 - 4.13434i) q^{3} +(-13.6289 - 7.86864i) q^{4} +(12.6360 - 21.5715i) q^{5} -8.18755 q^{6} +(-36.5078 + 32.6831i) q^{7} +(11.5027 + 11.5027i) q^{8} +(150.831 - 87.0822i) q^{9} +(-9.11771 + 9.00363i) q^{10} +(42.7420 - 74.0314i) q^{11} +(-242.819 - 65.0633i) 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} +(49.9889 + 186.561i) q^{17} +(-86.2277 + 23.1047i) 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} +(-58.9699 + 220.079i) q^{23} +(225.037 + 129.925i) q^{24} +(-305.661 - 545.157i) q^{25} +(-47.7951 - 82.7835i) q^{26} +(1052.31 - 1052.31i) q^{27} +(754.732 - 158.167i) q^{28} +592.118i q^{29} +(-103.458 + 176.618i) q^{30} +(244.720 - 423.868i) q^{31} +(-99.6616 - 371.942i) q^{32} +(353.420 - 1318.98i) q^{33} -98.9968i q^{34} +(243.709 + 1200.51i) q^{35} -2740.87 q^{36} +(75.9617 + 20.3539i) q^{37} +(249.133 - 66.7551i) q^{38} +(2579.94 + 1489.53i) q^{39} +(393.478 - 102.782i) q^{40} -1391.63 q^{41} +(298.909 - 267.594i) q^{42} +(-838.944 - 838.944i) q^{43} +(-1165.05 + 672.643i) q^{44} +(27.4093 - 4354.02i) q^{45} +(58.3913 - 101.137i) q^{46} +(1806.26 + 483.986i) 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} +(-759.618 - 2834.93i) q^{52} +(476.285 - 127.620i) 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} +(78.5505 - 293.154i) q^{58} +(704.535 + 406.763i) q^{59} +(-4471.79 + 4415.84i) q^{60} +(1169.51 + 2025.66i) q^{61} +(-177.390 + 177.390i) q^{62} +(-2660.38 + 8108.79i) q^{63} -3697.97i q^{64} +(4511.04 - 1178.34i) q^{65} +(-349.953 + 606.136i) q^{66} +(-1808.23 - 6748.40i) q^{67} +(786.689 - 2935.96i) q^{68} +3639.52i q^{69} +(38.6010 - 626.697i) q^{70} -767.932 q^{71} +(2736.63 + 733.279i) q^{72} +(2621.62 - 702.461i) q^{73} +(-34.9081 - 20.1542i) q^{74} +(-6970.08 - 7147.84i) q^{75} +7919.06 q^{76} +(859.156 + 4099.66i) q^{77} +(-1079.71 - 1079.71i) q^{78} +(-4386.38 + 2532.48i) q^{79} +(6086.34 + 38.3145i) q^{80} +(4832.45 - 8370.05i) q^{81} +(688.987 + 184.614i) 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} +(2448.02 + 9136.13i) q^{87} +(1343.21 - 359.911i) 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} +(2023.51 - 7551.85i) q^{93} +(-830.064 - 479.238i) q^{94} +(-79.1921 + 12579.8i) q^{95} +(-3075.47 - 5326.87i) q^{96} +(6312.04 - 6312.04i) q^{97} +(-447.595 + 1146.37i) 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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.495094 − 0.132660i −0.123774 − 0.0331650i 0.196401 − 0.980524i \(-0.437075\pi\)
−0.320174 + 0.947359i \(0.603741\pi\)
\(3\) 15.4296 − 4.13434i 1.71440 − 0.459371i 0.737901 − 0.674909i \(-0.235817\pi\)
0.976495 + 0.215538i \(0.0691505\pi\)
\(4\) −13.6289 − 7.86864i −0.851805 − 0.491790i
\(5\) 12.6360 − 21.5715i 0.505442 − 0.862861i
\(6\) −8.18755 −0.227432
\(7\) −36.5078 + 32.6831i −0.745057 + 0.667001i
\(8\) 11.5027 + 11.5027i 0.179729 + 0.179729i
\(9\) 150.831 − 87.0822i 1.86211 − 1.07509i
\(10\) −9.11771 + 9.00363i −0.0911771 + 0.0900363i
\(11\) 42.7420 − 74.0314i 0.353240 − 0.611830i −0.633575 − 0.773681i \(-0.718413\pi\)
0.986815 + 0.161852i \(0.0517466\pi\)
\(12\) −242.819 − 65.0633i −1.68625 − 0.451828i
\(13\) 131.873 + 131.873i 0.780311 + 0.780311i 0.979883 − 0.199572i \(-0.0639553\pi\)
−0.199572 + 0.979883i \(0.563955\pi\)
\(14\) 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
\(17\) 49.9889 + 186.561i 0.172972 + 0.645540i 0.996888 + 0.0788284i \(0.0251179\pi\)
−0.823916 + 0.566712i \(0.808215\pi\)
\(18\) −86.2277 + 23.1047i −0.266135 + 0.0713107i
\(19\) −435.787 + 251.602i −1.20717 + 0.696958i −0.962139 − 0.272558i \(-0.912130\pi\)
−0.245028 + 0.969516i \(0.578797\pi\)
\(20\) −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
\(23\) −58.9699 + 220.079i −0.111474 + 0.416028i −0.998999 − 0.0447323i \(-0.985757\pi\)
0.887525 + 0.460760i \(0.152423\pi\)
\(24\) 225.037 + 129.925i 0.390690 + 0.225565i
\(25\) −305.661 − 545.157i −0.489057 − 0.872252i
\(26\) −47.7951 − 82.7835i −0.0707028 − 0.122461i
\(27\) 1052.31 − 1052.31i 1.44350 − 1.44350i
\(28\) 754.732 − 158.167i 0.962668 − 0.201744i
\(29\) 592.118i 0.704065i 0.935988 + 0.352032i \(0.114509\pi\)
−0.935988 + 0.352032i \(0.885491\pi\)
\(30\) −103.458 + 176.618i −0.114954 + 0.196242i
\(31\) 244.720 − 423.868i 0.254652 − 0.441069i −0.710149 − 0.704051i \(-0.751373\pi\)
0.964801 + 0.262982i \(0.0847059\pi\)
\(32\) −99.6616 − 371.942i −0.0973258 − 0.363225i
\(33\) 353.420 − 1318.98i 0.324536 − 1.21119i
\(34\) − 98.9968i − 0.0856374i
\(35\) 243.709 + 1200.51i 0.198946 + 0.980010i
\(36\) −2740.87 −2.11487
\(37\) 75.9617 + 20.3539i 0.0554870 + 0.0148677i 0.286456 − 0.958093i \(-0.407523\pi\)
−0.230969 + 0.972961i \(0.574190\pi\)
\(38\) 249.133 − 66.7551i 0.172530 − 0.0462293i
\(39\) 2579.94 + 1489.53i 1.69621 + 0.979310i
\(40\) 393.478 − 102.782i 0.245924 − 0.0642387i
\(41\) −1391.63 −0.827858 −0.413929 − 0.910309i \(-0.635844\pi\)
−0.413929 + 0.910309i \(0.635844\pi\)
\(42\) 298.909 − 267.594i 0.169450 − 0.151697i
\(43\) −838.944 − 838.944i −0.453728 − 0.453728i 0.442862 − 0.896590i \(-0.353963\pi\)
−0.896590 + 0.442862i \(0.853963\pi\)
\(44\) −1165.05 + 672.643i −0.601783 + 0.347440i
\(45\) 27.4093 − 4354.02i 0.0135354 − 2.15013i
\(46\) 58.3913 − 101.137i 0.0275951 − 0.0477962i
\(47\) 1806.26 + 483.986i 0.817683 + 0.219097i 0.643333 − 0.765586i \(-0.277551\pi\)
0.174350 + 0.984684i \(0.444218\pi\)
\(48\) 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
\(52\) −759.618 − 2834.93i −0.280924 − 1.04842i
\(53\) 476.285 − 127.620i 0.169557 − 0.0454326i −0.173042 − 0.984914i \(-0.555360\pi\)
0.342599 + 0.939482i \(0.388693\pi\)
\(54\) −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
\(58\) 78.5505 − 293.154i 0.0233503 − 0.0871446i
\(59\) 704.535 + 406.763i 0.202394 + 0.116852i 0.597772 − 0.801666i \(-0.296053\pi\)
−0.395377 + 0.918519i \(0.629386\pi\)
\(60\) −4471.79 + 4415.84i −1.24216 + 1.22662i
\(61\) 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
\(63\) −2660.38 + 8108.79i −0.670290 + 2.04303i
\(64\) − 3697.97i − 0.902825i
\(65\) 4511.04 − 1178.34i 1.06770 − 0.278898i
\(66\) −349.953 + 606.136i −0.0803381 + 0.139150i
\(67\) −1808.23 − 6748.40i −0.402813 − 1.50332i −0.808053 − 0.589109i \(-0.799479\pi\)
0.405240 − 0.914210i \(-0.367188\pi\)
\(68\) 786.689 − 2935.96i 0.170132 − 0.634940i
\(69\) 3639.52i 0.764445i
\(70\) 38.6010 − 626.697i 0.00787775 − 0.127897i
\(71\) −767.932 −0.152337 −0.0761686 − 0.997095i \(-0.524269\pi\)
−0.0761686 + 0.997095i \(0.524269\pi\)
\(72\) 2736.63 + 733.279i 0.527900 + 0.141450i
\(73\) 2621.62 − 702.461i 0.491953 − 0.131818i −0.00430977 − 0.999991i \(-0.501372\pi\)
0.496263 + 0.868172i \(0.334705\pi\)
\(74\) −34.9081 − 20.1542i −0.00637474 − 0.00368046i
\(75\) −6970.08 − 7147.84i −1.23913 − 1.27073i
\(76\) 7919.06 1.37103
\(77\) 859.156 + 4099.66i 0.144907 + 0.691459i
\(78\) −1079.71 − 1079.71i −0.177468 − 0.177468i
\(79\) −4386.38 + 2532.48i −0.702833 + 0.405781i −0.808402 − 0.588631i \(-0.799667\pi\)
0.105569 + 0.994412i \(0.466334\pi\)
\(80\) 6086.34 + 38.3145i 0.950991 + 0.00598664i
\(81\) 4832.45 − 8370.05i 0.736541 − 1.27573i
\(82\) 688.987 + 184.614i 0.102467 + 0.0274559i
\(83\) −7579.82 − 7579.82i −1.10028 − 1.10028i −0.994376 − 0.105903i \(-0.966227\pi\)
−0.105903 − 0.994376i \(-0.533773\pi\)
\(84\) 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
\(87\) 2448.02 + 9136.13i 0.323427 + 1.20705i
\(88\) 1343.21 − 359.911i 0.173451 − 0.0464761i
\(89\) −12187.4 + 7036.42i −1.53862 + 0.888325i −0.539704 + 0.841855i \(0.681464\pi\)
−0.998920 + 0.0464696i \(0.985203\pi\)
\(90\) −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
\(93\) 2023.51 − 7551.85i 0.233959 − 0.873147i
\(94\) −830.064 − 479.238i −0.0939411 − 0.0542369i
\(95\) −79.1921 + 12579.8i −0.00877475 + 1.39389i
\(96\) −3075.47 − 5326.87i −0.333710 − 0.578002i
\(97\) 6312.04 − 6312.04i 0.670852 − 0.670852i −0.287061 − 0.957912i \(-0.592678\pi\)
0.957912 + 0.287061i \(0.0926781\pi\)
\(98\) −447.595 + 1146.37i −0.0466051 + 0.119364i
\(99\) − 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.18.7 yes 56
5.2 odd 4 inner 35.5.l.a.32.8 yes 56
7.2 even 3 inner 35.5.l.a.23.8 yes 56
35.2 odd 12 inner 35.5.l.a.2.7 ✓ 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.7 ✓ 56 35.2 odd 12 inner
35.5.l.a.18.7 yes 56 1.1 even 1 trivial
35.5.l.a.23.8 yes 56 7.2 even 3 inner
35.5.l.a.32.8 yes 56 5.2 odd 4 inner