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 251.6
Character \(\chi\) \(=\) 522.251
Dual form 522.2.r.b.287.6

$q$-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 \(-0.760522\pi\)
0.954283 − 0.298904i \(-0.0966212\pi\)
\(6\) 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 \(0.293704\pi\)
−0.765937 + 0.642916i \(0.777725\pi\)
\(12\) 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 \(-0.980382\pi\)
−0.0615921 − 0.998101i \(-0.519618\pi\)
\(18\) 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 0.149077 − 0.988826i \(-0.452370\pi\)
0.563349 + 0.826219i \(0.309513\pi\)
\(24\) 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 −0.998554 − 0.0537604i \(-0.982879\pi\)
0.0537604 + 0.998554i \(0.482879\pi\)
\(42\) 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 0.0652831 − 0.997867i \(-0.479205\pi\)
−0.158400 + 0.987375i \(0.550634\pi\)
\(48\) 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 0.731468 − 0.681876i \(-0.238836\pi\)
0.363175 + 0.931721i \(0.381693\pi\)
\(54\) 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 \(-0.946237\pi\)
0.985770 − 0.168101i \(-0.0537633\pi\)
\(60\) 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 0.918978 − 0.394310i \(-0.129016\pi\)
−0.588915 + 0.808195i \(0.700445\pi\)
\(72\) 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 \(0.390874\pi\)
−0.945923 + 0.324391i \(0.894841\pi\)
\(84\) 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 −0.953032 − 0.302869i \(-0.902055\pi\)
−0.140629 + 0.990062i \(0.544913\pi\)
\(90\) 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.251.6 yes 72
3.2 odd 2 inner 522.2.r.b.251.1 ✓ 72
29.26 odd 28 inner 522.2.r.b.287.1 yes 72
87.26 even 28 inner 522.2.r.b.287.6 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.251.1 ✓ 72 3.2 odd 2 inner
522.2.r.b.251.6 yes 72 1.1 even 1 trivial
522.2.r.b.287.1 yes 72 29.26 odd 28 inner
522.2.r.b.287.6 yes 72 87.26 even 28 inner