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Properties

Label 35.5.l.a.2.7
Level $35$
Weight $5$
Character 35.2
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 2.7
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.7

$q$-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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

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

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 −0.320174 − 0.947359i \(-0.603741\pi\)
0.196401 + 0.980524i \(0.437075\pi\)
\(3\) 15.4296 + 4.13434i 1.71440 + 0.459371i 0.976495 − 0.215538i \(-0.0691505\pi\)
0.737901 + 0.674909i \(0.235817\pi\)
\(4\) −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 0.986815 − 0.161852i \(-0.0517466\pi\)
−0.633575 + 0.773681i \(0.718413\pi\)
\(12\) −242.819 + 65.0633i −1.68625 + 0.451828i
\(13\) 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
\(17\) 49.9889 − 186.561i 0.172972 − 0.645540i −0.823916 − 0.566712i \(-0.808215\pi\)
0.996888 − 0.0788284i \(-0.0251179\pi\)
\(18\) −86.2277 − 23.1047i −0.266135 − 0.0713107i
\(19\) −435.787 − 251.602i −1.20717 − 0.696958i −0.245028 − 0.969516i \(-0.578797\pi\)
−0.962139 + 0.272558i \(0.912130\pi\)
\(20\) −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 0.887525 − 0.460760i \(-0.152423\pi\)
−0.998999 + 0.0447323i \(0.985757\pi\)
\(24\) 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 \(-0.885491\pi\)
0.935988 − 0.352032i \(-0.114509\pi\)
\(30\) −103.458 − 176.618i −0.114954 − 0.196242i
\(31\) 244.720 + 423.868i 0.254652 + 0.441069i 0.964801 − 0.262982i \(-0.0847059\pi\)
−0.710149 + 0.704051i \(0.751373\pi\)
\(32\) −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 −0.230969 − 0.972961i \(-0.574190\pi\)
0.286456 + 0.958093i \(0.407523\pi\)
\(38\) 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 −0.896590 − 0.442862i \(-0.853963\pi\)
0.442862 + 0.896590i \(0.353963\pi\)
\(44\) −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 0.174350 − 0.984684i \(-0.444218\pi\)
0.643333 + 0.765586i \(0.277551\pi\)
\(48\) 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 0.342599 − 0.939482i \(-0.388693\pi\)
−0.173042 + 0.984914i \(0.555360\pi\)
\(54\) −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 −0.395377 − 0.918519i \(-0.629386\pi\)
0.597772 + 0.801666i \(0.296053\pi\)
\(60\) −4471.79 − 4415.84i −1.24216 − 1.22662i
\(61\) 1169.51 − 2025.66i 0.314301 − 0.544386i −0.664987 − 0.746855i \(-0.731563\pi\)
0.979289 + 0.202469i \(0.0648965\pi\)
\(62\) −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 0.405240 + 0.914210i \(0.367188\pi\)
−0.808053 + 0.589109i \(0.799479\pi\)
\(68\) 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 0.496263 − 0.868172i \(-0.334705\pi\)
−0.00430977 + 0.999991i \(0.501372\pi\)
\(74\) −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 0.105569 − 0.994412i \(-0.466334\pi\)
−0.808402 + 0.588631i \(0.799667\pi\)
\(80\) 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 −0.105903 + 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
\(87\) 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 −0.998920 − 0.0464696i \(-0.985203\pi\)
−0.539704 − 0.841855i \(-0.681464\pi\)
\(90\) −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 0.957912 − 0.287061i \(-0.0926781\pi\)
−0.287061 + 0.957912i \(0.592678\pi\)
\(98\) −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.2.7 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.8 yes 56
7.4 even 3 inner 35.5.l.a.32.8 yes 56
35.18 odd 12 inner 35.5.l.a.18.7 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.7 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.7 yes 56 35.18 odd 12 inner
35.5.l.a.23.8 yes 56 5.3 odd 4 inner
35.5.l.a.32.8 yes 56 7.4 even 3 inner