Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(89,522)] 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("522.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label 251.1
Character \(\chi\) \(=\) 522.251
Dual form 522.2.r.b.287.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.846724 + 0.532032i) q^{2} +(0.433884 - 0.900969i) q^{4} +(-0.501312 + 2.19639i) q^{5} +(2.25090 - 1.08398i) q^{7} +(0.111964 + 0.993712i) q^{8} +(-0.744077 - 2.12645i) q^{10} +(0.538175 - 4.77643i) q^{11} +(2.83201 - 2.25845i) q^{13} +(-1.32918 + 2.11538i) q^{14} +(-0.623490 - 0.781831i) q^{16} +(4.36923 + 4.36923i) q^{17} +(-3.24187 + 1.13438i) q^{19} +(1.76137 + 1.40464i) q^{20} +(2.08553 + 4.33065i) q^{22} +(-3.41668 + 0.779834i) q^{23} +(-0.0679740 - 0.0327346i) q^{25} +(-1.19636 + 3.41900i) q^{26} -2.49831i q^{28} +(4.35841 + 3.16296i) q^{29} +(-0.420983 - 0.669991i) q^{31} +(0.943883 + 0.330279i) q^{32} +(-6.02410 - 1.37496i) q^{34} +(1.25243 + 5.48727i) q^{35} +(9.23475 - 1.04051i) q^{37} +(2.14144 - 2.68529i) q^{38} +(-2.23871 - 0.252242i) q^{40} +(6.04963 - 6.04963i) q^{41} +(9.20266 + 5.78242i) q^{43} +(-4.06991 - 2.55730i) q^{44} +(2.47809 - 2.47809i) q^{46} +(0.638377 + 0.0719278i) q^{47} +(-0.472875 + 0.592966i) q^{49} +(0.0749711 - 0.00844721i) q^{50} +(-0.806032 - 3.53146i) q^{52} +(-7.96912 - 1.81890i) q^{53} +(10.2211 + 3.57653i) q^{55} +(1.32918 + 2.11538i) q^{56} +(-5.37316 - 0.359342i) q^{58} +2.58241i q^{59} +(-2.66567 + 7.61804i) q^{61} +(0.712913 + 0.343321i) q^{62} +(-0.974928 + 0.222521i) q^{64} +(3.54072 + 7.35238i) q^{65} +(-8.89589 - 7.09424i) q^{67} +(5.83228 - 2.04080i) q^{68} +(-3.97987 - 3.97987i) q^{70} +(-2.78116 - 3.48746i) q^{71} +(-5.34252 + 8.50257i) q^{73} +(-7.26570 + 5.79420i) q^{74} +(-0.384554 + 3.41301i) q^{76} +(-3.96617 - 11.3347i) q^{77} +(0.530681 + 4.70992i) q^{79} +(2.02977 - 0.977486i) q^{80} +(-1.90377 + 8.34096i) q^{82} +(5.55526 - 11.5356i) q^{83} +(-11.7869 + 7.40619i) q^{85} -10.8685 q^{86} +4.80666 q^{88} +(10.3176 - 6.48297i) q^{89} +(3.92646 - 8.15338i) q^{91} +(-0.779834 + 3.41668i) q^{92} +(-0.578797 + 0.278734i) q^{94} +(-0.866353 - 7.68909i) q^{95} +(0.656654 + 1.87661i) q^{97} +(0.0849175 - 0.753664i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{9}{28}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.846724 + 0.532032i −0.598724 + 0.376203i
\(3\) 0 0
\(4\) 0.433884 − 0.900969i 0.216942 − 0.450484i
\(5\) −0.501312 + 2.19639i −0.224193 + 0.982256i 0.730090 + 0.683351i \(0.239478\pi\)
−0.954283 + 0.298904i \(0.903379\pi\)
\(6\) 0 0
\(7\) 2.25090 − 1.08398i 0.850761 − 0.409705i 0.0429010 − 0.999079i \(-0.486340\pi\)
0.807860 + 0.589374i \(0.200626\pi\)
\(8\) 0.111964 + 0.993712i 0.0395854 + 0.351330i
\(9\) 0 0
\(10\) −0.744077 − 2.12645i −0.235298 − 0.672443i
\(11\) 0.538175 − 4.77643i 0.162266 − 1.44015i −0.603671 − 0.797234i \(-0.706296\pi\)
0.765937 − 0.642916i \(-0.222275\pi\)
\(12\) 0 0
\(13\) 2.83201 − 2.25845i 0.785458 − 0.626381i −0.146392 − 0.989227i \(-0.546766\pi\)
0.931849 + 0.362845i \(0.118195\pi\)
\(14\) −1.32918 + 2.11538i −0.355239 + 0.565360i
\(15\) 0 0
\(16\) −0.623490 − 0.781831i −0.155872 − 0.195458i
\(17\) 4.36923 + 4.36923i 1.05969 + 1.05969i 0.998101 + 0.0615921i \(0.0196178\pi\)
0.0615921 + 0.998101i \(0.480382\pi\)
\(18\) 0 0
\(19\) −3.24187 + 1.13438i −0.743736 + 0.260244i −0.675434 − 0.737420i \(-0.736044\pi\)
−0.0683018 + 0.997665i \(0.521758\pi\)
\(20\) 1.76137 + 1.40464i 0.393854 + 0.314088i
\(21\) 0 0
\(22\) 2.08553 + 4.33065i 0.444637 + 0.923298i
\(23\) −3.41668 + 0.779834i −0.712426 + 0.162607i −0.563349 − 0.826219i \(-0.690487\pi\)
−0.149077 + 0.988826i \(0.547630\pi\)
\(24\) 0 0
\(25\) −0.0679740 − 0.0327346i −0.0135948 − 0.00654691i
\(26\) −1.19636 + 3.41900i −0.234626 + 0.670522i
\(27\) 0 0
\(28\) − 2.49831i − 0.472137i
\(29\) 4.35841 + 3.16296i 0.809336 + 0.587347i
\(30\) 0 0
\(31\) −0.420983 − 0.669991i −0.0756108 − 0.120334i 0.806778 − 0.590854i \(-0.201209\pi\)
−0.882389 + 0.470520i \(0.844066\pi\)
\(32\) 0.943883 + 0.330279i 0.166857 + 0.0583856i
\(33\) 0 0
\(34\) −6.02410 − 1.37496i −1.03312 − 0.235804i
\(35\) 1.25243 + 5.48727i 0.211700 + 0.927518i
\(36\) 0 0
\(37\) 9.23475 − 1.04051i 1.51818 − 0.171058i 0.686836 − 0.726812i \(-0.258999\pi\)
0.831347 + 0.555754i \(0.187570\pi\)
\(38\) 2.14144 − 2.68529i 0.347388 − 0.435611i
\(39\) 0 0
\(40\) −2.23871 − 0.252242i −0.353971 − 0.0398830i
\(41\) 6.04963 − 6.04963i 0.944793 − 0.944793i −0.0537604 − 0.998554i \(-0.517121\pi\)
0.998554 + 0.0537604i \(0.0171207\pi\)
\(42\) 0 0
\(43\) 9.20266 + 5.78242i 1.40339 + 0.881810i 0.999456 − 0.0329769i \(-0.0104988\pi\)
0.403937 + 0.914787i \(0.367642\pi\)
\(44\) −4.06991 − 2.55730i −0.613563 − 0.385527i
\(45\) 0 0
\(46\) 2.47809 − 2.47809i 0.365374 − 0.365374i
\(47\) 0.638377 + 0.0719278i 0.0931168 + 0.0104917i 0.158400 − 0.987375i \(-0.449366\pi\)
−0.0652831 + 0.997867i \(0.520795\pi\)
\(48\) 0 0
\(49\) −0.472875 + 0.592966i −0.0675536 + 0.0847095i
\(50\) 0.0749711 − 0.00844721i 0.0106025 − 0.00119462i
\(51\) 0 0
\(52\) −0.806032 − 3.53146i −0.111776 − 0.489725i
\(53\) −7.96912 − 1.81890i −1.09464 − 0.249845i −0.363175 − 0.931721i \(-0.618307\pi\)
−0.731468 + 0.681876i \(0.761164\pi\)
\(54\) 0 0
\(55\) 10.2211 + 3.57653i 1.37822 + 0.482259i
\(56\) 1.32918 + 2.11538i 0.177619 + 0.282680i
\(57\) 0 0
\(58\) −5.37316 − 0.359342i −0.705531 − 0.0471839i
\(59\) 2.58241i 0.336201i 0.985770 + 0.168101i \(0.0537633\pi\)
−0.985770 + 0.168101i \(0.946237\pi\)
\(60\) 0 0
\(61\) −2.66567 + 7.61804i −0.341304 + 0.975390i 0.637475 + 0.770471i \(0.279979\pi\)
−0.978779 + 0.204919i \(0.934307\pi\)
\(62\) 0.712913 + 0.343321i 0.0905401 + 0.0436018i
\(63\) 0 0
\(64\) −0.974928 + 0.222521i −0.121866 + 0.0278151i
\(65\) 3.54072 + 7.35238i 0.439172 + 0.911951i
\(66\) 0 0
\(67\) −8.89589 − 7.09424i −1.08681 − 0.866699i −0.0951312 − 0.995465i \(-0.530327\pi\)
−0.991675 + 0.128766i \(0.958899\pi\)
\(68\) 5.83228 − 2.04080i 0.707267 − 0.247484i
\(69\) 0 0
\(70\) −3.97987 − 3.97987i −0.475685 − 0.475685i
\(71\) −2.78116 − 3.48746i −0.330062 − 0.413885i 0.588915 − 0.808195i \(-0.299555\pi\)
−0.918978 + 0.394310i \(0.870984\pi\)
\(72\) 0 0
\(73\) −5.34252 + 8.50257i −0.625295 + 0.995151i 0.372452 + 0.928052i \(0.378517\pi\)
−0.997746 + 0.0670991i \(0.978626\pi\)
\(74\) −7.26570 + 5.79420i −0.844620 + 0.673562i
\(75\) 0 0
\(76\) −0.384554 + 3.41301i −0.0441114 + 0.391499i
\(77\) −3.96617 − 11.3347i −0.451987 − 1.29170i
\(78\) 0 0
\(79\) 0.530681 + 4.70992i 0.0597063 + 0.529908i 0.987592 + 0.157043i \(0.0501960\pi\)
−0.927886 + 0.372865i \(0.878375\pi\)
\(80\) 2.02977 − 0.977486i 0.226935 − 0.109286i
\(81\) 0 0
\(82\) −1.90377 + 8.34096i −0.210236 + 0.921105i
\(83\) 5.55526 − 11.5356i 0.609769 − 1.26620i −0.336154 − 0.941807i \(-0.609126\pi\)
0.945923 − 0.324391i \(-0.105159\pi\)
\(84\) 0 0
\(85\) −11.7869 + 7.40619i −1.27847 + 0.803314i
\(86\) −10.8685 −1.17199
\(87\) 0 0
\(88\) 4.80666 0.512391
\(89\) 10.3176 − 6.48297i 1.09366 − 0.687193i 0.140629 − 0.990062i \(-0.455087\pi\)
0.953032 + 0.302869i \(0.0979445\pi\)
\(90\) 0 0
\(91\) 3.92646 − 8.15338i 0.411605 − 0.854707i
\(92\) −0.779834 + 3.41668i −0.0813033 + 0.356213i
\(93\) 0 0
\(94\) −0.578797 + 0.278734i −0.0596983 + 0.0287492i
\(95\) −0.866353 − 7.68909i −0.0888859 − 0.788884i
\(96\) 0 0
\(97\) 0.656654 + 1.87661i 0.0666731 + 0.190541i 0.972400 − 0.233322i \(-0.0749596\pi\)
−0.905726 + 0.423863i \(0.860674\pi\)
\(98\) 0.0849175 − 0.753664i 0.00857796 − 0.0761315i
\(99\) 0 0
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 522.2.r.b.251.1 ✓ 72
3.2 odd 2 inner 522.2.r.b.251.6 yes 72
29.26 odd 28 inner 522.2.r.b.287.6 yes 72
87.26 even 28 inner 522.2.r.b.287.1 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.251.1 ✓ 72 1.1 even 1 trivial
522.2.r.b.251.6 yes 72 3.2 odd 2 inner
522.2.r.b.287.1 yes 72 87.26 even 28 inner
522.2.r.b.287.6 yes 72 29.26 odd 28 inner