Properties

Label 1920.2.bl.a.289.18
Level $1920$
Weight $2$
Character 1920.289
Analytic conductor $15.331$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,2,Mod(289,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1920.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1920.bl (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.3312771881\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 289.18
Character \(\chi\) \(=\) 1920.289
Dual form 1920.2.bl.a.1249.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{3} +(2.06370 + 0.860885i) q^{5} +0.707398 q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(0.707107 - 0.707107i) q^{3} +(2.06370 + 0.860885i) q^{5} +0.707398 q^{7} -1.00000i q^{9} +(-1.79993 + 1.79993i) q^{11} +(-3.86348 + 3.86348i) q^{13} +(2.06800 - 0.850522i) q^{15} -0.244884i q^{17} +(-1.53863 - 1.53863i) q^{19} +(0.500206 - 0.500206i) q^{21} +6.92280 q^{23} +(3.51775 + 3.55323i) q^{25} +(-0.707107 - 0.707107i) q^{27} +(4.89882 + 4.89882i) q^{29} +7.60734 q^{31} +2.54548i q^{33} +(1.45986 + 0.608988i) q^{35} +(8.47863 + 8.47863i) q^{37} +5.46378i q^{39} +2.12118i q^{41} +(0.684507 + 0.684507i) q^{43} +(0.860885 - 2.06370i) q^{45} -4.47342i q^{47} -6.49959 q^{49} +(-0.173159 - 0.173159i) q^{51} +(-1.47026 - 1.47026i) q^{53} +(-5.26405 + 2.16499i) q^{55} -2.17595 q^{57} +(-5.86121 + 5.86121i) q^{59} +(-0.0537432 - 0.0537432i) q^{61} -0.707398i q^{63} +(-11.2991 + 4.64706i) q^{65} +(7.85550 - 7.85550i) q^{67} +(4.89516 - 4.89516i) q^{69} -2.08595i q^{71} +9.69951 q^{73} +(4.99994 + 0.0250827i) q^{75} +(-1.27326 + 1.27326i) q^{77} -7.34690 q^{79} -1.00000 q^{81} +(6.80291 - 6.80291i) q^{83} +(0.210817 - 0.505369i) q^{85} +6.92797 q^{87} -3.07483i q^{89} +(-2.73301 + 2.73301i) q^{91} +(5.37920 - 5.37920i) q^{93} +(-1.85069 - 4.49986i) q^{95} -1.39922i q^{97} +(1.79993 + 1.79993i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 8 q^{19} - 48 q^{31} - 24 q^{35} + 48 q^{49} - 8 q^{51} + 32 q^{59} - 16 q^{61} + 16 q^{65} + 16 q^{69} + 16 q^{75} - 96 q^{79} - 48 q^{81} + 32 q^{91} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See 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 0
\(3\) 0.707107 0.707107i 0.408248 0.408248i
\(4\) 0 0
\(5\) 2.06370 + 0.860885i 0.922917 + 0.385000i
\(6\) 0 0
\(7\) 0.707398 0.267371 0.133686 0.991024i \(-0.457319\pi\)
0.133686 + 0.991024i \(0.457319\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) −1.79993 + 1.79993i −0.542699 + 0.542699i −0.924319 0.381621i \(-0.875366\pi\)
0.381621 + 0.924319i \(0.375366\pi\)
\(12\) 0 0
\(13\) −3.86348 + 3.86348i −1.07154 + 1.07154i −0.0742996 + 0.997236i \(0.523672\pi\)
−0.997236 + 0.0742996i \(0.976328\pi\)
\(14\) 0 0
\(15\) 2.06800 0.850522i 0.533955 0.219604i
\(16\) 0 0
\(17\) 0.244884i 0.0593932i −0.999559 0.0296966i \(-0.990546\pi\)
0.999559 0.0296966i \(-0.00945410\pi\)
\(18\) 0 0
\(19\) −1.53863 1.53863i −0.352986 0.352986i 0.508233 0.861219i \(-0.330299\pi\)
−0.861219 + 0.508233i \(0.830299\pi\)
\(20\) 0 0
\(21\) 0.500206 0.500206i 0.109154 0.109154i
\(22\) 0 0
\(23\) 6.92280 1.44350 0.721752 0.692152i \(-0.243337\pi\)
0.721752 + 0.692152i \(0.243337\pi\)
\(24\) 0 0
\(25\) 3.51775 + 3.55323i 0.703551 + 0.710645i
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 4.89882 + 4.89882i 0.909688 + 0.909688i 0.996247 0.0865591i \(-0.0275871\pi\)
−0.0865591 + 0.996247i \(0.527587\pi\)
\(30\) 0 0
\(31\) 7.60734 1.36632 0.683159 0.730270i \(-0.260606\pi\)
0.683159 + 0.730270i \(0.260606\pi\)
\(32\) 0 0
\(33\) 2.54548i 0.443112i
\(34\) 0 0
\(35\) 1.45986 + 0.608988i 0.246761 + 0.102938i
\(36\) 0 0
\(37\) 8.47863 + 8.47863i 1.39388 + 1.39388i 0.816420 + 0.577459i \(0.195956\pi\)
0.577459 + 0.816420i \(0.304044\pi\)
\(38\) 0 0
\(39\) 5.46378i 0.874905i
\(40\) 0 0
\(41\) 2.12118i 0.331272i 0.986187 + 0.165636i \(0.0529677\pi\)
−0.986187 + 0.165636i \(0.947032\pi\)
\(42\) 0 0
\(43\) 0.684507 + 0.684507i 0.104386 + 0.104386i 0.757371 0.652985i \(-0.226483\pi\)
−0.652985 + 0.757371i \(0.726483\pi\)
\(44\) 0 0
\(45\) 0.860885 2.06370i 0.128333 0.307639i
\(46\) 0 0
\(47\) 4.47342i 0.652515i −0.945281 0.326258i \(-0.894212\pi\)
0.945281 0.326258i \(-0.105788\pi\)
\(48\) 0 0
\(49\) −6.49959 −0.928513
\(50\) 0 0
\(51\) −0.173159 0.173159i −0.0242472 0.0242472i
\(52\) 0 0
\(53\) −1.47026 1.47026i −0.201956 0.201956i 0.598881 0.800838i \(-0.295612\pi\)
−0.800838 + 0.598881i \(0.795612\pi\)
\(54\) 0 0
\(55\) −5.26405 + 2.16499i −0.709804 + 0.291927i
\(56\) 0 0
\(57\) −2.17595 −0.288212
\(58\) 0 0
\(59\) −5.86121 + 5.86121i −0.763065 + 0.763065i −0.976875 0.213810i \(-0.931413\pi\)
0.213810 + 0.976875i \(0.431413\pi\)
\(60\) 0 0
\(61\) −0.0537432 0.0537432i −0.00688112 0.00688112i 0.703658 0.710539i \(-0.251549\pi\)
−0.710539 + 0.703658i \(0.751549\pi\)
\(62\) 0 0
\(63\) 0.707398i 0.0891237i
\(64\) 0 0
\(65\) −11.2991 + 4.64706i −1.40148 + 0.576397i
\(66\) 0 0
\(67\) 7.85550 7.85550i 0.959702 0.959702i −0.0395170 0.999219i \(-0.512582\pi\)
0.999219 + 0.0395170i \(0.0125819\pi\)
\(68\) 0 0
\(69\) 4.89516 4.89516i 0.589308 0.589308i
\(70\) 0 0
\(71\) 2.08595i 0.247557i −0.992310 0.123778i \(-0.960499\pi\)
0.992310 0.123778i \(-0.0395012\pi\)
\(72\) 0 0
\(73\) 9.69951 1.13524 0.567621 0.823290i \(-0.307864\pi\)
0.567621 + 0.823290i \(0.307864\pi\)
\(74\) 0 0
\(75\) 4.99994 + 0.0250827i 0.577343 + 0.00289630i
\(76\) 0 0
\(77\) −1.27326 + 1.27326i −0.145102 + 0.145102i
\(78\) 0 0
\(79\) −7.34690 −0.826591 −0.413295 0.910597i \(-0.635622\pi\)
−0.413295 + 0.910597i \(0.635622\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 6.80291 6.80291i 0.746716 0.746716i −0.227145 0.973861i \(-0.572939\pi\)
0.973861 + 0.227145i \(0.0729391\pi\)
\(84\) 0 0
\(85\) 0.210817 0.505369i 0.0228663 0.0548149i
\(86\) 0 0
\(87\) 6.92797 0.742757
\(88\) 0 0
\(89\) 3.07483i 0.325931i −0.986632 0.162966i \(-0.947894\pi\)
0.986632 0.162966i \(-0.0521060\pi\)
\(90\) 0 0
\(91\) −2.73301 + 2.73301i −0.286498 + 0.286498i
\(92\) 0 0
\(93\) 5.37920 5.37920i 0.557797 0.557797i
\(94\) 0 0
\(95\) −1.85069 4.49986i −0.189877 0.461676i
\(96\) 0 0
\(97\) 1.39922i 0.142070i −0.997474 0.0710348i \(-0.977370\pi\)
0.997474 0.0710348i \(-0.0226301\pi\)
\(98\) 0 0
\(99\) 1.79993 + 1.79993i 0.180900 + 0.180900i
\(100\) 0 0
\(101\) −11.7916 + 11.7916i −1.17331 + 1.17331i −0.191895 + 0.981415i \(0.561463\pi\)
−0.981415 + 0.191895i \(0.938537\pi\)
\(102\) 0 0
\(103\) 2.29240 0.225877 0.112938 0.993602i \(-0.463974\pi\)
0.112938 + 0.993602i \(0.463974\pi\)
\(104\) 0 0
\(105\) 1.46290 0.601657i 0.142764 0.0587157i
\(106\) 0 0
\(107\) −9.49009 9.49009i −0.917442 0.917442i 0.0794010 0.996843i \(-0.474699\pi\)
−0.996843 + 0.0794010i \(0.974699\pi\)
\(108\) 0 0
\(109\) −11.9058 11.9058i −1.14037 1.14037i −0.988383 0.151984i \(-0.951434\pi\)
−0.151984 0.988383i \(-0.548566\pi\)
\(110\) 0 0
\(111\) 11.9906 1.13810
\(112\) 0 0
\(113\) 7.92806i 0.745809i 0.927870 + 0.372904i \(0.121638\pi\)
−0.927870 + 0.372904i \(0.878362\pi\)
\(114\) 0 0
\(115\) 14.2866 + 5.95974i 1.33223 + 0.555748i
\(116\) 0 0
\(117\) 3.86348 + 3.86348i 0.357179 + 0.357179i
\(118\) 0 0
\(119\) 0.173231i 0.0158800i
\(120\) 0 0
\(121\) 4.52052i 0.410956i
\(122\) 0 0
\(123\) 1.49990 + 1.49990i 0.135241 + 0.135241i
\(124\) 0 0
\(125\) 4.20068 + 10.3612i 0.375721 + 0.926733i
\(126\) 0 0
\(127\) 19.4466i 1.72561i 0.505536 + 0.862806i \(0.331295\pi\)
−0.505536 + 0.862806i \(0.668705\pi\)
\(128\) 0 0
\(129\) 0.968040 0.0852311
\(130\) 0 0
\(131\) 0.354049 + 0.354049i 0.0309334 + 0.0309334i 0.722404 0.691471i \(-0.243037\pi\)
−0.691471 + 0.722404i \(0.743037\pi\)
\(132\) 0 0
\(133\) −1.08842 1.08842i −0.0943783 0.0943783i
\(134\) 0 0
\(135\) −0.850522 2.06800i −0.0732013 0.177985i
\(136\) 0 0
\(137\) −7.30033 −0.623709 −0.311854 0.950130i \(-0.600950\pi\)
−0.311854 + 0.950130i \(0.600950\pi\)
\(138\) 0 0
\(139\) 8.85519 8.85519i 0.751087 0.751087i −0.223595 0.974682i \(-0.571779\pi\)
0.974682 + 0.223595i \(0.0717793\pi\)
\(140\) 0 0
\(141\) −3.16319 3.16319i −0.266388 0.266388i
\(142\) 0 0
\(143\) 13.9080i 1.16304i
\(144\) 0 0
\(145\) 5.89239 + 14.3270i 0.489337 + 1.18980i
\(146\) 0 0
\(147\) −4.59590 + 4.59590i −0.379064 + 0.379064i
\(148\) 0 0
\(149\) −5.41385 + 5.41385i −0.443520 + 0.443520i −0.893193 0.449673i \(-0.851540\pi\)
0.449673 + 0.893193i \(0.351540\pi\)
\(150\) 0 0
\(151\) 0.341548i 0.0277948i −0.999903 0.0138974i \(-0.995576\pi\)
0.999903 0.0138974i \(-0.00442382\pi\)
\(152\) 0 0
\(153\) −0.244884 −0.0197977
\(154\) 0 0
\(155\) 15.6993 + 6.54904i 1.26100 + 0.526032i
\(156\) 0 0
\(157\) 2.86465 2.86465i 0.228624 0.228624i −0.583494 0.812118i \(-0.698315\pi\)
0.812118 + 0.583494i \(0.198315\pi\)
\(158\) 0 0
\(159\) −2.07927 −0.164897
\(160\) 0 0
\(161\) 4.89717 0.385951
\(162\) 0 0
\(163\) 5.94216 5.94216i 0.465426 0.465426i −0.435003 0.900429i \(-0.643253\pi\)
0.900429 + 0.435003i \(0.143253\pi\)
\(164\) 0 0
\(165\) −2.19137 + 5.25312i −0.170598 + 0.408955i
\(166\) 0 0
\(167\) 14.4595 1.11891 0.559454 0.828861i \(-0.311011\pi\)
0.559454 + 0.828861i \(0.311011\pi\)
\(168\) 0 0
\(169\) 16.8529i 1.29638i
\(170\) 0 0
\(171\) −1.53863 + 1.53863i −0.117662 + 0.117662i
\(172\) 0 0
\(173\) 6.65096 6.65096i 0.505663 0.505663i −0.407529 0.913192i \(-0.633610\pi\)
0.913192 + 0.407529i \(0.133610\pi\)
\(174\) 0 0
\(175\) 2.48845 + 2.51354i 0.188109 + 0.190006i
\(176\) 0 0
\(177\) 8.28901i 0.623040i
\(178\) 0 0
\(179\) −17.2660 17.2660i −1.29052 1.29052i −0.934466 0.356052i \(-0.884123\pi\)
−0.356052 0.934466i \(-0.615877\pi\)
\(180\) 0 0
\(181\) −5.12242 + 5.12242i −0.380746 + 0.380746i −0.871371 0.490625i \(-0.836769\pi\)
0.490625 + 0.871371i \(0.336769\pi\)
\(182\) 0 0
\(183\) −0.0760044 −0.00561841
\(184\) 0 0
\(185\) 10.1983 + 24.7965i 0.749791 + 1.82308i
\(186\) 0 0
\(187\) 0.440774 + 0.440774i 0.0322326 + 0.0322326i
\(188\) 0 0
\(189\) −0.500206 0.500206i −0.0363846 0.0363846i
\(190\) 0 0
\(191\) −17.8040 −1.28825 −0.644124 0.764921i \(-0.722778\pi\)
−0.644124 + 0.764921i \(0.722778\pi\)
\(192\) 0 0
\(193\) 17.2222i 1.23968i −0.784727 0.619842i \(-0.787197\pi\)
0.784727 0.619842i \(-0.212803\pi\)
\(194\) 0 0
\(195\) −4.70369 + 11.2756i −0.336838 + 0.807465i
\(196\) 0 0
\(197\) 10.0764 + 10.0764i 0.717915 + 0.717915i 0.968178 0.250263i \(-0.0805171\pi\)
−0.250263 + 0.968178i \(0.580517\pi\)
\(198\) 0 0
\(199\) 5.74179i 0.407025i −0.979072 0.203513i \(-0.934764\pi\)
0.979072 0.203513i \(-0.0652358\pi\)
\(200\) 0 0
\(201\) 11.1094i 0.783593i
\(202\) 0 0
\(203\) 3.46541 + 3.46541i 0.243224 + 0.243224i
\(204\) 0 0
\(205\) −1.82609 + 4.37748i −0.127540 + 0.305737i
\(206\) 0 0
\(207\) 6.92280i 0.481168i
\(208\) 0 0
\(209\) 5.53885 0.383130
\(210\) 0 0
\(211\) 9.73318 + 9.73318i 0.670060 + 0.670060i 0.957730 0.287670i \(-0.0928806\pi\)
−0.287670 + 0.957730i \(0.592881\pi\)
\(212\) 0 0
\(213\) −1.47499 1.47499i −0.101065 0.101065i
\(214\) 0 0
\(215\) 0.823339 + 2.00190i 0.0561512 + 0.136529i
\(216\) 0 0
\(217\) 5.38141 0.365314
\(218\) 0 0
\(219\) 6.85859 6.85859i 0.463460 0.463460i
\(220\) 0 0
\(221\) 0.946104 + 0.946104i 0.0636419 + 0.0636419i
\(222\) 0 0
\(223\) 22.1037i 1.48017i −0.672511 0.740087i \(-0.734784\pi\)
0.672511 0.740087i \(-0.265216\pi\)
\(224\) 0 0
\(225\) 3.55323 3.51775i 0.236882 0.234517i
\(226\) 0 0
\(227\) −5.35387 + 5.35387i −0.355349 + 0.355349i −0.862095 0.506746i \(-0.830848\pi\)
0.506746 + 0.862095i \(0.330848\pi\)
\(228\) 0 0
\(229\) 18.3018 18.3018i 1.20942 1.20942i 0.238204 0.971215i \(-0.423441\pi\)
0.971215 0.238204i \(-0.0765587\pi\)
\(230\) 0 0
\(231\) 1.80067i 0.118475i
\(232\) 0 0
\(233\) −12.0546 −0.789721 −0.394861 0.918741i \(-0.629207\pi\)
−0.394861 + 0.918741i \(0.629207\pi\)
\(234\) 0 0
\(235\) 3.85110 9.23182i 0.251218 0.602217i
\(236\) 0 0
\(237\) −5.19504 + 5.19504i −0.337454 + 0.337454i
\(238\) 0 0
\(239\) −27.9193 −1.80595 −0.902974 0.429695i \(-0.858621\pi\)
−0.902974 + 0.429695i \(0.858621\pi\)
\(240\) 0 0
\(241\) 13.7118 0.883253 0.441626 0.897199i \(-0.354402\pi\)
0.441626 + 0.897199i \(0.354402\pi\)
\(242\) 0 0
\(243\) −0.707107 + 0.707107i −0.0453609 + 0.0453609i
\(244\) 0 0
\(245\) −13.4132 5.59540i −0.856940 0.357477i
\(246\) 0 0
\(247\) 11.8889 0.756474
\(248\) 0 0
\(249\) 9.62077i 0.609691i
\(250\) 0 0
\(251\) 9.07173 9.07173i 0.572602 0.572602i −0.360252 0.932855i \(-0.617309\pi\)
0.932855 + 0.360252i \(0.117309\pi\)
\(252\) 0 0
\(253\) −12.4605 + 12.4605i −0.783388 + 0.783388i
\(254\) 0 0
\(255\) −0.208279 0.506420i −0.0130430 0.0317132i
\(256\) 0 0
\(257\) 9.23416i 0.576011i −0.957629 0.288006i \(-0.907008\pi\)
0.957629 0.288006i \(-0.0929922\pi\)
\(258\) 0 0
\(259\) 5.99776 + 5.99776i 0.372683 + 0.372683i
\(260\) 0 0
\(261\) 4.89882 4.89882i 0.303229 0.303229i
\(262\) 0 0
\(263\) −18.5516 −1.14394 −0.571971 0.820274i \(-0.693821\pi\)
−0.571971 + 0.820274i \(0.693821\pi\)
\(264\) 0 0
\(265\) −1.76846 4.29992i −0.108636 0.264142i
\(266\) 0 0
\(267\) −2.17423 2.17423i −0.133061 0.133061i
\(268\) 0 0
\(269\) −14.3977 14.3977i −0.877844 0.877844i 0.115467 0.993311i \(-0.463163\pi\)
−0.993311 + 0.115467i \(0.963163\pi\)
\(270\) 0 0
\(271\) 19.2974 1.17224 0.586118 0.810226i \(-0.300656\pi\)
0.586118 + 0.810226i \(0.300656\pi\)
\(272\) 0 0
\(273\) 3.86507i 0.233924i
\(274\) 0 0
\(275\) −12.7272 0.0638476i −0.767482 0.00385016i
\(276\) 0 0
\(277\) −3.52269 3.52269i −0.211658 0.211658i 0.593314 0.804971i \(-0.297819\pi\)
−0.804971 + 0.593314i \(0.797819\pi\)
\(278\) 0 0
\(279\) 7.60734i 0.455439i
\(280\) 0 0
\(281\) 29.2076i 1.74238i 0.490945 + 0.871191i \(0.336652\pi\)
−0.490945 + 0.871191i \(0.663348\pi\)
\(282\) 0 0
\(283\) 9.84279 + 9.84279i 0.585093 + 0.585093i 0.936298 0.351205i \(-0.114228\pi\)
−0.351205 + 0.936298i \(0.614228\pi\)
\(284\) 0 0
\(285\) −4.49052 1.87325i −0.265996 0.110961i
\(286\) 0 0
\(287\) 1.50052i 0.0885726i
\(288\) 0 0
\(289\) 16.9400 0.996472
\(290\) 0 0
\(291\) −0.989400 0.989400i −0.0579997 0.0579997i
\(292\) 0 0
\(293\) −11.8238 11.8238i −0.690755 0.690755i 0.271643 0.962398i \(-0.412433\pi\)
−0.962398 + 0.271643i \(0.912433\pi\)
\(294\) 0 0
\(295\) −17.1416 + 7.04998i −0.998025 + 0.410466i
\(296\) 0 0
\(297\) 2.54548 0.147704
\(298\) 0 0
\(299\) −26.7461 + 26.7461i −1.54677 + 1.54677i
\(300\) 0 0
\(301\) 0.484219 + 0.484219i 0.0279099 + 0.0279099i
\(302\) 0 0
\(303\) 16.6759i 0.958004i
\(304\) 0 0
\(305\) −0.0646434 0.157177i −0.00370147 0.00899992i
\(306\) 0 0
\(307\) −16.6232 + 16.6232i −0.948733 + 0.948733i −0.998748 0.0500151i \(-0.984073\pi\)
0.0500151 + 0.998748i \(0.484073\pi\)
\(308\) 0 0
\(309\) 1.62097 1.62097i 0.0922139 0.0922139i
\(310\) 0 0
\(311\) 13.8069i 0.782920i 0.920195 + 0.391460i \(0.128030\pi\)
−0.920195 + 0.391460i \(0.871970\pi\)
\(312\) 0 0
\(313\) −7.40380 −0.418488 −0.209244 0.977864i \(-0.567100\pi\)
−0.209244 + 0.977864i \(0.567100\pi\)
\(314\) 0 0
\(315\) 0.608988 1.45986i 0.0343126 0.0822538i
\(316\) 0 0
\(317\) −1.80712 + 1.80712i −0.101498 + 0.101498i −0.756032 0.654534i \(-0.772865\pi\)
0.654534 + 0.756032i \(0.272865\pi\)
\(318\) 0 0
\(319\) −17.6350 −0.987372
\(320\) 0 0
\(321\) −13.4210 −0.749088
\(322\) 0 0
\(323\) −0.376786 + 0.376786i −0.0209650 + 0.0209650i
\(324\) 0 0
\(325\) −27.3186 0.137046i −1.51536 0.00760197i
\(326\) 0 0
\(327\) −16.8373 −0.931106
\(328\) 0 0
\(329\) 3.16449i 0.174464i
\(330\) 0 0
\(331\) 5.82711 5.82711i 0.320287 0.320287i −0.528590 0.848877i \(-0.677279\pi\)
0.848877 + 0.528590i \(0.177279\pi\)
\(332\) 0 0
\(333\) 8.47863 8.47863i 0.464626 0.464626i
\(334\) 0 0
\(335\) 22.9741 9.44875i 1.25521 0.516240i
\(336\) 0 0
\(337\) 25.5357i 1.39102i −0.718516 0.695510i \(-0.755178\pi\)
0.718516 0.695510i \(-0.244822\pi\)
\(338\) 0 0
\(339\) 5.60598 + 5.60598i 0.304475 + 0.304475i
\(340\) 0 0
\(341\) −13.6927 + 13.6927i −0.741499 + 0.741499i
\(342\) 0 0
\(343\) −9.54958 −0.515629
\(344\) 0 0
\(345\) 14.3163 5.88799i 0.770766 0.316999i
\(346\) 0 0
\(347\) −17.6028 17.6028i −0.944970 0.944970i 0.0535933 0.998563i \(-0.482933\pi\)
−0.998563 + 0.0535933i \(0.982933\pi\)
\(348\) 0 0
\(349\) 5.07562 + 5.07562i 0.271692 + 0.271692i 0.829781 0.558089i \(-0.188465\pi\)
−0.558089 + 0.829781i \(0.688465\pi\)
\(350\) 0 0
\(351\) 5.46378 0.291635
\(352\) 0 0
\(353\) 0.171535i 0.00912990i −0.999990 0.00456495i \(-0.998547\pi\)
0.999990 0.00456495i \(-0.00145307\pi\)
\(354\) 0 0
\(355\) 1.79576 4.30479i 0.0953093 0.228474i
\(356\) 0 0
\(357\) −0.122492 0.122492i −0.00648299 0.00648299i
\(358\) 0 0
\(359\) 17.1694i 0.906169i −0.891468 0.453084i \(-0.850324\pi\)
0.891468 0.453084i \(-0.149676\pi\)
\(360\) 0 0
\(361\) 14.2652i 0.750802i
\(362\) 0 0
\(363\) 3.19649 + 3.19649i 0.167772 + 0.167772i
\(364\) 0 0
\(365\) 20.0169 + 8.35016i 1.04773 + 0.437068i
\(366\) 0 0
\(367\) 23.3124i 1.21690i 0.793593 + 0.608450i \(0.208208\pi\)
−0.793593 + 0.608450i \(0.791792\pi\)
\(368\) 0 0
\(369\) 2.12118 0.110424
\(370\) 0 0
\(371\) −1.04006 1.04006i −0.0539973 0.0539973i
\(372\) 0 0
\(373\) −1.11817 1.11817i −0.0578964 0.0578964i 0.677566 0.735462i \(-0.263035\pi\)
−0.735462 + 0.677566i \(0.763035\pi\)
\(374\) 0 0
\(375\) 10.2968 + 4.35614i 0.531724 + 0.224950i
\(376\) 0 0
\(377\) −37.8529 −1.94953
\(378\) 0 0
\(379\) −22.3707 + 22.3707i −1.14910 + 1.14910i −0.162375 + 0.986729i \(0.551915\pi\)
−0.986729 + 0.162375i \(0.948085\pi\)
\(380\) 0 0
\(381\) 13.7509 + 13.7509i 0.704478 + 0.704478i
\(382\) 0 0
\(383\) 32.1168i 1.64109i 0.571579 + 0.820547i \(0.306331\pi\)
−0.571579 + 0.820547i \(0.693669\pi\)
\(384\) 0 0
\(385\) −3.72378 + 1.53151i −0.189781 + 0.0780528i
\(386\) 0 0
\(387\) 0.684507 0.684507i 0.0347955 0.0347955i
\(388\) 0 0
\(389\) 22.7409 22.7409i 1.15301 1.15301i 0.167062 0.985946i \(-0.446572\pi\)
0.985946 0.167062i \(-0.0534280\pi\)
\(390\) 0 0
\(391\) 1.69529i 0.0857342i
\(392\) 0 0
\(393\) 0.500701 0.0252570
\(394\) 0 0
\(395\) −15.1618 6.32484i −0.762874 0.318237i
\(396\) 0 0
\(397\) 21.6530 21.6530i 1.08673 1.08673i 0.0908707 0.995863i \(-0.471035\pi\)
0.995863 0.0908707i \(-0.0289650\pi\)
\(398\) 0 0
\(399\) −1.53926 −0.0770596
\(400\) 0 0
\(401\) 17.1600 0.856931 0.428466 0.903558i \(-0.359054\pi\)
0.428466 + 0.903558i \(0.359054\pi\)
\(402\) 0 0
\(403\) −29.3908 + 29.3908i −1.46406 + 1.46406i
\(404\) 0 0
\(405\) −2.06370 0.860885i −0.102546 0.0427777i
\(406\) 0 0
\(407\) −30.5218 −1.51291
\(408\) 0 0
\(409\) 17.9588i 0.888006i −0.896025 0.444003i \(-0.853558\pi\)
0.896025 0.444003i \(-0.146442\pi\)
\(410\) 0 0
\(411\) −5.16211 + 5.16211i −0.254628 + 0.254628i
\(412\) 0 0
\(413\) −4.14621 + 4.14621i −0.204022 + 0.204022i
\(414\) 0 0
\(415\) 19.8957 8.18267i 0.976642 0.401671i
\(416\) 0 0
\(417\) 12.5231i 0.613260i
\(418\) 0 0
\(419\) −23.4031 23.4031i −1.14332 1.14332i −0.987840 0.155476i \(-0.950309\pi\)
−0.155476 0.987840i \(-0.549691\pi\)
\(420\) 0 0
\(421\) 27.0016 27.0016i 1.31598 1.31598i 0.399043 0.916932i \(-0.369342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(422\) 0 0
\(423\) −4.47342 −0.217505
\(424\) 0 0
\(425\) 0.870129 0.861442i 0.0422075 0.0417861i
\(426\) 0 0
\(427\) −0.0380178 0.0380178i −0.00183981 0.00183981i
\(428\) 0 0
\(429\) −9.83441 9.83441i −0.474810 0.474810i
\(430\) 0 0
\(431\) −28.4043 −1.36819 −0.684093 0.729395i \(-0.739802\pi\)
−0.684093 + 0.729395i \(0.739802\pi\)
\(432\) 0 0
\(433\) 4.04564i 0.194421i 0.995264 + 0.0972106i \(0.0309920\pi\)
−0.995264 + 0.0972106i \(0.969008\pi\)
\(434\) 0 0
\(435\) 14.2973 + 5.96419i 0.685503 + 0.285961i
\(436\) 0 0
\(437\) −10.6516 10.6516i −0.509537 0.509537i
\(438\) 0 0
\(439\) 10.4690i 0.499660i 0.968290 + 0.249830i \(0.0803748\pi\)
−0.968290 + 0.249830i \(0.919625\pi\)
\(440\) 0 0
\(441\) 6.49959i 0.309504i
\(442\) 0 0
\(443\) 1.81582 + 1.81582i 0.0862722 + 0.0862722i 0.748926 0.662654i \(-0.230570\pi\)
−0.662654 + 0.748926i \(0.730570\pi\)
\(444\) 0 0
\(445\) 2.64707 6.34554i 0.125483 0.300807i
\(446\) 0 0
\(447\) 7.65634i 0.362132i
\(448\) 0 0
\(449\) −22.2064 −1.04799 −0.523993 0.851723i \(-0.675558\pi\)
−0.523993 + 0.851723i \(0.675558\pi\)
\(450\) 0 0
\(451\) −3.81797 3.81797i −0.179781 0.179781i
\(452\) 0 0
\(453\) −0.241511 0.241511i −0.0113472 0.0113472i
\(454\) 0 0
\(455\) −7.99294 + 3.28732i −0.374715 + 0.154112i
\(456\) 0 0
\(457\) 12.2280 0.572000 0.286000 0.958230i \(-0.407674\pi\)
0.286000 + 0.958230i \(0.407674\pi\)
\(458\) 0 0
\(459\) −0.173159 + 0.173159i −0.00808238 + 0.00808238i
\(460\) 0 0
\(461\) −0.723447 0.723447i −0.0336943 0.0336943i 0.690059 0.723753i \(-0.257585\pi\)
−0.723753 + 0.690059i \(0.757585\pi\)
\(462\) 0 0
\(463\) 5.42343i 0.252048i 0.992027 + 0.126024i \(0.0402217\pi\)
−0.992027 + 0.126024i \(0.959778\pi\)
\(464\) 0 0
\(465\) 15.7320 6.47021i 0.729552 0.300049i
\(466\) 0 0
\(467\) 13.3242 13.3242i 0.616571 0.616571i −0.328079 0.944650i \(-0.606401\pi\)
0.944650 + 0.328079i \(0.106401\pi\)
\(468\) 0 0
\(469\) 5.55696 5.55696i 0.256597 0.256597i
\(470\) 0 0
\(471\) 4.05122i 0.186671i
\(472\) 0 0
\(473\) −2.46413 −0.113301
\(474\) 0 0
\(475\) 0.0545788 10.8796i 0.00250425 0.499191i
\(476\) 0 0
\(477\) −1.47026 + 1.47026i −0.0673188 + 0.0673188i
\(478\) 0 0
\(479\) −1.25963 −0.0575541 −0.0287771 0.999586i \(-0.509161\pi\)
−0.0287771 + 0.999586i \(0.509161\pi\)
\(480\) 0 0
\(481\) −65.5140 −2.98718
\(482\) 0 0
\(483\) 3.46282 3.46282i 0.157564 0.157564i
\(484\) 0 0
\(485\) 1.20457 2.88758i 0.0546967 0.131118i
\(486\) 0 0
\(487\) 15.8116 0.716491 0.358246 0.933627i \(-0.383375\pi\)
0.358246 + 0.933627i \(0.383375\pi\)
\(488\) 0 0
\(489\) 8.40349i 0.380019i
\(490\) 0 0
\(491\) 15.6008 15.6008i 0.704054 0.704054i −0.261224 0.965278i \(-0.584126\pi\)
0.965278 + 0.261224i \(0.0841263\pi\)
\(492\) 0 0
\(493\) 1.19964 1.19964i 0.0540292 0.0540292i
\(494\) 0 0
\(495\) 2.16499 + 5.26405i 0.0973090 + 0.236601i
\(496\) 0 0
\(497\) 1.47560i 0.0661896i
\(498\) 0 0
\(499\) −25.6650 25.6650i −1.14892 1.14892i −0.986765 0.162159i \(-0.948154\pi\)
−0.162159 0.986765i \(-0.551846\pi\)
\(500\) 0 0
\(501\) 10.2244 10.2244i 0.456792 0.456792i
\(502\) 0 0
\(503\) 4.72004 0.210456 0.105228 0.994448i \(-0.466443\pi\)
0.105228 + 0.994448i \(0.466443\pi\)
\(504\) 0 0
\(505\) −34.4857 + 14.1832i −1.53459 + 0.631144i
\(506\) 0 0
\(507\) −11.9168 11.9168i −0.529244 0.529244i
\(508\) 0 0
\(509\) 11.8862 + 11.8862i 0.526848 + 0.526848i 0.919631 0.392783i \(-0.128488\pi\)
−0.392783 + 0.919631i \(0.628488\pi\)
\(510\) 0 0
\(511\) 6.86141 0.303531
\(512\) 0 0
\(513\) 2.17595i 0.0960706i
\(514\) 0 0
\(515\) 4.73084 + 1.97349i 0.208466 + 0.0869625i
\(516\) 0 0
\(517\) 8.05183 + 8.05183i 0.354119 + 0.354119i
\(518\) 0 0
\(519\) 9.40587i 0.412872i
\(520\) 0 0
\(521\) 5.01467i 0.219696i −0.993948 0.109848i \(-0.964964\pi\)
0.993948 0.109848i \(-0.0350365\pi\)
\(522\) 0 0
\(523\) −18.9502 18.9502i −0.828635 0.828635i 0.158693 0.987328i \(-0.449272\pi\)
−0.987328 + 0.158693i \(0.949272\pi\)
\(524\) 0 0
\(525\) 3.53694 + 0.0177435i 0.154365 + 0.000774388i
\(526\) 0 0
\(527\) 1.86292i 0.0811499i
\(528\) 0 0
\(529\) 24.9252 1.08370
\(530\) 0 0
\(531\) 5.86121 + 5.86121i 0.254355 + 0.254355i
\(532\) 0 0
\(533\) −8.19512 8.19512i −0.354970 0.354970i
\(534\) 0 0
\(535\) −11.4149 27.7546i −0.493508 1.19994i
\(536\) 0 0
\(537\) −24.4178 −1.05370
\(538\) 0 0
\(539\) 11.6988 11.6988i 0.503903 0.503903i
\(540\) 0 0
\(541\) 22.8366 + 22.8366i 0.981822 + 0.981822i 0.999838 0.0180153i \(-0.00573475\pi\)
−0.0180153 + 0.999838i \(0.505735\pi\)
\(542\) 0 0
\(543\) 7.24419i 0.310878i
\(544\) 0 0
\(545\) −14.3205 34.8195i −0.613423 1.49150i
\(546\) 0 0
\(547\) 0.00963023 0.00963023i 0.000411759 0.000411759i −0.706901 0.707313i \(-0.749907\pi\)
0.707313 + 0.706901i \(0.249907\pi\)
\(548\) 0 0
\(549\) −0.0537432 + 0.0537432i −0.00229371 + 0.00229371i
\(550\) 0 0
\(551\) 15.0749i 0.642214i
\(552\) 0 0
\(553\) −5.19718 −0.221006
\(554\) 0 0
\(555\) 24.7450 + 10.3225i 1.05037 + 0.438167i
\(556\) 0 0
\(557\) −25.6660 + 25.6660i −1.08750 + 1.08750i −0.0917192 + 0.995785i \(0.529236\pi\)
−0.995785 + 0.0917192i \(0.970764\pi\)
\(558\) 0 0
\(559\) −5.28916 −0.223707
\(560\) 0 0
\(561\) 0.623348 0.0263178
\(562\) 0 0
\(563\) 6.16719 6.16719i 0.259916 0.259916i −0.565104 0.825020i \(-0.691164\pi\)
0.825020 + 0.565104i \(0.191164\pi\)
\(564\) 0 0
\(565\) −6.82515 + 16.3612i −0.287136 + 0.688319i
\(566\) 0 0
\(567\) −0.707398 −0.0297079
\(568\) 0 0
\(569\) 4.05892i 0.170159i −0.996374 0.0850794i \(-0.972886\pi\)
0.996374 0.0850794i \(-0.0271144\pi\)
\(570\) 0 0
\(571\) 12.9113 12.9113i 0.540323 0.540323i −0.383301 0.923624i \(-0.625213\pi\)
0.923624 + 0.383301i \(0.125213\pi\)
\(572\) 0 0
\(573\) −12.5893 + 12.5893i −0.525925 + 0.525925i
\(574\) 0 0
\(575\) 24.3527 + 24.5983i 1.01558 + 1.02582i
\(576\) 0 0
\(577\) 10.4193i 0.433761i 0.976198 + 0.216880i \(0.0695882\pi\)
−0.976198 + 0.216880i \(0.930412\pi\)
\(578\) 0 0
\(579\) −12.1780 12.1780i −0.506099 0.506099i
\(580\) 0 0
\(581\) 4.81236 4.81236i 0.199650 0.199650i
\(582\) 0 0
\(583\) 5.29274 0.219203
\(584\) 0 0
\(585\) 4.64706 + 11.2991i 0.192132 + 0.467160i
\(586\) 0 0
\(587\) 27.7728 + 27.7728i 1.14631 + 1.14631i 0.987273 + 0.159033i \(0.0508376\pi\)
0.159033 + 0.987273i \(0.449162\pi\)
\(588\) 0 0
\(589\) −11.7049 11.7049i −0.482291 0.482291i
\(590\) 0 0
\(591\) 14.2502 0.586175
\(592\) 0 0
\(593\) 25.4466i 1.04496i −0.852650 0.522482i \(-0.825006\pi\)
0.852650 0.522482i \(-0.174994\pi\)
\(594\) 0 0
\(595\) 0.149132 0.357497i 0.00611380 0.0146559i
\(596\) 0 0
\(597\) −4.06006 4.06006i −0.166167 0.166167i
\(598\) 0 0
\(599\) 1.32051i 0.0539544i 0.999636 + 0.0269772i \(0.00858816\pi\)
−0.999636 + 0.0269772i \(0.991412\pi\)
\(600\) 0 0
\(601\) 23.8236i 0.971787i 0.874018 + 0.485893i \(0.161506\pi\)
−0.874018 + 0.485893i \(0.838494\pi\)
\(602\) 0 0
\(603\) −7.85550 7.85550i −0.319901 0.319901i
\(604\) 0 0
\(605\) −3.89165 + 9.32902i −0.158218 + 0.379279i
\(606\) 0 0
\(607\) 1.20300i 0.0488283i 0.999702 + 0.0244141i \(0.00777203\pi\)
−0.999702 + 0.0244141i \(0.992228\pi\)
\(608\) 0 0
\(609\) 4.90083 0.198592
\(610\) 0 0
\(611\) 17.2830 + 17.2830i 0.699193 + 0.699193i
\(612\) 0 0
\(613\) 4.84340 + 4.84340i 0.195623 + 0.195623i 0.798121 0.602498i \(-0.205828\pi\)
−0.602498 + 0.798121i \(0.705828\pi\)
\(614\) 0 0
\(615\) 1.80411 + 4.38659i 0.0727486 + 0.176884i
\(616\) 0 0
\(617\) 26.9719 1.08585 0.542924 0.839782i \(-0.317317\pi\)
0.542924 + 0.839782i \(0.317317\pi\)
\(618\) 0 0
\(619\) −19.9202 + 19.9202i −0.800660 + 0.800660i −0.983199 0.182539i \(-0.941568\pi\)
0.182539 + 0.983199i \(0.441568\pi\)
\(620\) 0 0
\(621\) −4.89516 4.89516i −0.196436 0.196436i
\(622\) 0 0
\(623\) 2.17513i 0.0871446i
\(624\) 0 0
\(625\) −0.250824 + 24.9987i −0.0100330 + 0.999950i
\(626\) 0 0
\(627\) 3.91656 3.91656i 0.156412 0.156412i
\(628\) 0 0
\(629\) 2.07628 2.07628i 0.0827868 0.0827868i
\(630\) 0 0
\(631\) 12.7975i 0.509462i −0.967012 0.254731i \(-0.918013\pi\)
0.967012 0.254731i \(-0.0819870\pi\)
\(632\) 0 0
\(633\) 13.7648 0.547101
\(634\) 0 0
\(635\) −16.7413 + 40.1321i −0.664360 + 1.59260i
\(636\) 0 0
\(637\) 25.1110 25.1110i 0.994934 0.994934i
\(638\) 0 0
\(639\) −2.08595 −0.0825190
\(640\) 0 0
\(641\) 10.9406 0.432129 0.216064 0.976379i \(-0.430678\pi\)
0.216064 + 0.976379i \(0.430678\pi\)
\(642\) 0 0
\(643\) −18.2613 + 18.2613i −0.720157 + 0.720157i −0.968637 0.248480i \(-0.920069\pi\)
0.248480 + 0.968637i \(0.420069\pi\)
\(644\) 0 0
\(645\) 1.99775 + 0.833371i 0.0786612 + 0.0328140i
\(646\) 0 0
\(647\) 10.5055 0.413014 0.206507 0.978445i \(-0.433790\pi\)
0.206507 + 0.978445i \(0.433790\pi\)
\(648\) 0 0
\(649\) 21.0995i 0.828228i
\(650\) 0 0
\(651\) 3.80523 3.80523i 0.149139 0.149139i
\(652\) 0 0
\(653\) 11.1836 11.1836i 0.437648 0.437648i −0.453572 0.891220i \(-0.649850\pi\)
0.891220 + 0.453572i \(0.149850\pi\)
\(654\) 0 0
\(655\) 0.425857 + 1.03545i 0.0166396 + 0.0404583i
\(656\) 0 0
\(657\) 9.69951i 0.378414i
\(658\) 0 0
\(659\) −2.60687 2.60687i −0.101549 0.101549i 0.654507 0.756056i \(-0.272876\pi\)
−0.756056 + 0.654507i \(0.772876\pi\)
\(660\) 0 0
\(661\) −15.9084 + 15.9084i −0.618766 + 0.618766i −0.945215 0.326449i \(-0.894148\pi\)
0.326449 + 0.945215i \(0.394148\pi\)
\(662\) 0 0
\(663\) 1.33799 0.0519634
\(664\) 0 0
\(665\) −1.30918 3.18319i −0.0507677 0.123439i
\(666\) 0 0
\(667\) 33.9135 + 33.9135i 1.31314 + 1.31314i
\(668\) 0 0
\(669\) −15.6297 15.6297i −0.604278 0.604278i
\(670\) 0 0
\(671\) 0.193468 0.00746874
\(672\) 0 0
\(673\) 27.1091i 1.04498i 0.852645 + 0.522490i \(0.174997\pi\)
−0.852645 + 0.522490i \(0.825003\pi\)
\(674\) 0 0
\(675\) 0.0250827 4.99994i 0.000965434 0.192448i
\(676\) 0 0
\(677\) −11.0637 11.0637i −0.425211 0.425211i 0.461782 0.886993i \(-0.347210\pi\)
−0.886993 + 0.461782i \(0.847210\pi\)
\(678\) 0 0
\(679\) 0.989807i 0.0379853i
\(680\) 0 0
\(681\) 7.57152i 0.290141i
\(682\) 0 0
\(683\) −17.4289 17.4289i −0.666898 0.666898i 0.290098 0.956997i \(-0.406312\pi\)
−0.956997 + 0.290098i \(0.906312\pi\)
\(684\) 0 0
\(685\) −15.0657 6.28474i −0.575631 0.240128i
\(686\) 0 0
\(687\) 25.8827i 0.987487i
\(688\) 0 0
\(689\) 11.3607 0.432807
\(690\) 0 0
\(691\) −26.0352 26.0352i −0.990426 0.990426i 0.00952887 0.999955i \(-0.496967\pi\)
−0.999955 + 0.00952887i \(0.996967\pi\)
\(692\) 0 0
\(693\) 1.27326 + 1.27326i 0.0483673 + 0.0483673i
\(694\) 0 0
\(695\) 25.8978 10.6512i 0.982359 0.404023i
\(696\) 0 0
\(697\) 0.519443 0.0196753
\(698\) 0 0
\(699\) −8.52387 + 8.52387i −0.322402 + 0.322402i
\(700\) 0 0
\(701\) −2.08057 2.08057i −0.0785822 0.0785822i 0.666723 0.745305i \(-0.267696\pi\)
−0.745305 + 0.666723i \(0.767696\pi\)
\(702\) 0 0
\(703\) 26.0910i 0.984039i
\(704\) 0 0
\(705\) −3.80474 9.25102i −0.143295 0.348414i
\(706\) 0 0
\(707\) −8.34137 + 8.34137i −0.313709 + 0.313709i
\(708\) 0 0
\(709\) 3.27484 3.27484i 0.122989 0.122989i −0.642933 0.765922i \(-0.722283\pi\)
0.765922 + 0.642933i \(0.222283\pi\)
\(710\) 0 0
\(711\) 7.34690i 0.275530i
\(712\) 0 0
\(713\) 52.6641 1.97229
\(714\) 0 0
\(715\) 11.9732 28.7019i 0.447771 1.07339i
\(716\) 0 0
\(717\) −19.7419 + 19.7419i −0.737275 + 0.737275i
\(718\) 0 0
\(719\) −2.03220 −0.0757882 −0.0378941 0.999282i \(-0.512065\pi\)
−0.0378941 + 0.999282i \(0.512065\pi\)
\(720\) 0 0
\(721\) 1.62164 0.0603930
\(722\) 0 0
\(723\) 9.69568 9.69568i 0.360586 0.360586i
\(724\) 0 0
\(725\) −0.173772 + 34.6394i −0.00645375 + 1.28648i
\(726\) 0 0
\(727\) −6.37077 −0.236279 −0.118139 0.992997i \(-0.537693\pi\)
−0.118139 + 0.992997i \(0.537693\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 0.167625 0.167625i 0.00619984 0.00619984i
\(732\) 0 0
\(733\) −6.95026 + 6.95026i −0.256714 + 0.256714i −0.823716 0.567002i \(-0.808103\pi\)
0.567002 + 0.823716i \(0.308103\pi\)
\(734\) 0 0
\(735\) −13.4411 + 5.52804i −0.495784 + 0.203905i
\(736\) 0 0
\(737\) 28.2787i 1.04166i
\(738\) 0 0
\(739\) −0.0483355 0.0483355i −0.00177805 0.00177805i 0.706217 0.707995i \(-0.250400\pi\)
−0.707995 + 0.706217i \(0.750400\pi\)
\(740\) 0 0
\(741\) 8.40674 8.40674i 0.308829 0.308829i
\(742\) 0 0
\(743\) 0.148848 0.00546069 0.00273035 0.999996i \(-0.499131\pi\)
0.00273035 + 0.999996i \(0.499131\pi\)
\(744\) 0 0
\(745\) −15.8333 + 6.51188i −0.580087 + 0.238577i
\(746\) 0 0
\(747\) −6.80291 6.80291i −0.248905 0.248905i
\(748\) 0 0
\(749\) −6.71327 6.71327i −0.245297 0.245297i
\(750\) 0 0
\(751\) 1.69304 0.0617798 0.0308899 0.999523i \(-0.490166\pi\)
0.0308899 + 0.999523i \(0.490166\pi\)
\(752\) 0 0
\(753\) 12.8294i 0.467528i
\(754\) 0 0
\(755\) 0.294034 0.704855i 0.0107010 0.0256523i
\(756\) 0 0
\(757\) −26.6876 26.6876i −0.969977 0.969977i 0.0295851 0.999562i \(-0.490581\pi\)
−0.999562 + 0.0295851i \(0.990581\pi\)
\(758\) 0 0
\(759\) 17.6219i 0.639633i
\(760\) 0 0
\(761\) 17.1363i 0.621189i −0.950542 0.310595i \(-0.899472\pi\)
0.950542 0.310595i \(-0.100528\pi\)
\(762\) 0 0
\(763\) −8.42212 8.42212i −0.304901 0.304901i
\(764\) 0 0
\(765\) −0.505369 0.210817i −0.0182716 0.00762211i
\(766\) 0 0
\(767\) 45.2893i 1.63530i
\(768\) 0 0
\(769\) −25.6406 −0.924622 −0.462311 0.886718i \(-0.652980\pi\)
−0.462311 + 0.886718i \(0.652980\pi\)
\(770\) 0 0
\(771\) −6.52954 6.52954i −0.235156 0.235156i
\(772\) 0 0
\(773\) 27.4452 + 27.4452i 0.987137 + 0.987137i 0.999918 0.0127818i \(-0.00406868\pi\)
−0.0127818 + 0.999918i \(0.504069\pi\)
\(774\) 0 0
\(775\) 26.7607 + 27.0306i 0.961274 + 0.970967i
\(776\) 0 0
\(777\) 8.48212 0.304294
\(778\) 0 0
\(779\) 3.26371 3.26371i 0.116934 0.116934i
\(780\) 0 0
\(781\) 3.75456 + 3.75456i 0.134349 + 0.134349i
\(782\) 0 0
\(783\) 6.92797i 0.247586i
\(784\) 0 0
\(785\) 8.37792 3.44565i 0.299021 0.122981i
\(786\) 0 0
\(787\) −9.65912 + 9.65912i −0.344310 + 0.344310i −0.857985 0.513675i \(-0.828284\pi\)
0.513675 + 0.857985i \(0.328284\pi\)
\(788\) 0 0
\(789\) −13.1180 + 13.1180i −0.467012 + 0.467012i
\(790\) 0 0
\(791\) 5.60829i 0.199408i
\(792\) 0 0
\(793\) 0.415271 0.0147467
\(794\) 0 0
\(795\) −4.29099 1.79001i −0.152186 0.0634851i
\(796\) 0 0
\(797\) −11.9978 + 11.9978i −0.424984 + 0.424984i −0.886916 0.461931i \(-0.847157\pi\)
0.461931 + 0.886916i \(0.347157\pi\)
\(798\) 0 0
\(799\) −1.09547 −0.0387549
\(800\) 0 0
\(801\) −3.07483 −0.108644
\(802\) 0 0
\(803\) −17.4584 + 17.4584i −0.616094 + 0.616094i
\(804\) 0 0
\(805\) 10.1063 + 4.21590i 0.356201 + 0.148591i
\(806\) 0 0
\(807\) −20.3614 −0.716756
\(808\) 0 0
\(809\) 36.2916i 1.27594i −0.770060 0.637972i \(-0.779774\pi\)
0.770060 0.637972i \(-0.220226\pi\)
\(810\) 0 0
\(811\) −17.0781 + 17.0781i −0.599693 + 0.599693i −0.940231 0.340538i \(-0.889391\pi\)
0.340538 + 0.940231i \(0.389391\pi\)
\(812\) 0 0
\(813\) 13.6453 13.6453i 0.478563 0.478563i
\(814\) 0 0
\(815\) 17.3784 7.14735i 0.608738 0.250361i
\(816\) 0 0
\(817\) 2.10641i 0.0736939i
\(818\) 0 0
\(819\) 2.73301 + 2.73301i 0.0954992 + 0.0954992i
\(820\) 0 0
\(821\) −21.9249 + 21.9249i −0.765183 + 0.765183i −0.977254 0.212071i \(-0.931979\pi\)
0.212071 + 0.977254i \(0.431979\pi\)
\(822\) 0 0
\(823\) −4.38382 −0.152810 −0.0764052 0.997077i \(-0.524344\pi\)
−0.0764052 + 0.997077i \(0.524344\pi\)
\(824\) 0 0
\(825\) −9.04467 + 8.95438i −0.314895 + 0.311751i
\(826\) 0 0
\(827\) −21.5889 21.5889i −0.750718 0.750718i 0.223895 0.974613i \(-0.428123\pi\)
−0.974613 + 0.223895i \(0.928123\pi\)
\(828\) 0 0
\(829\) −7.88066 7.88066i −0.273707 0.273707i 0.556884 0.830590i \(-0.311997\pi\)
−0.830590 + 0.556884i \(0.811997\pi\)
\(830\) 0 0
\(831\) −4.98183 −0.172818
\(832\) 0 0
\(833\) 1.59165i 0.0551473i
\(834\) 0 0
\(835\) 29.8401 + 12.4480i 1.03266 + 0.430779i
\(836\) 0 0
\(837\) −5.37920 5.37920i −0.185932 0.185932i
\(838\) 0 0
\(839\) 6.03706i 0.208423i −0.994555 0.104211i \(-0.966768\pi\)
0.994555 0.104211i \(-0.0332318\pi\)
\(840\) 0 0
\(841\) 18.9968i 0.655063i
\(842\) 0 0
\(843\) 20.6529 + 20.6529i 0.711324 + 0.711324i
\(844\) 0 0
\(845\) 14.5084 34.7794i 0.499105 1.19645i
\(846\) 0 0
\(847\) 3.19781i 0.109878i
\(848\) 0 0
\(849\) 13.9198 0.477727
\(850\) 0 0
\(851\) 58.6959 + 58.6959i 2.01207 + 2.01207i
\(852\) 0 0
\(853\) 12.6685 + 12.6685i 0.433763 + 0.433763i 0.889906 0.456144i \(-0.150770\pi\)
−0.456144 + 0.889906i \(0.650770\pi\)
\(854\) 0 0
\(855\) −4.49986 + 1.85069i −0.153892 + 0.0632924i
\(856\) 0 0
\(857\) 4.79580 0.163821 0.0819106 0.996640i \(-0.473898\pi\)
0.0819106 + 0.996640i \(0.473898\pi\)
\(858\) 0 0
\(859\) 34.0244 34.0244i 1.16090 1.16090i 0.176616 0.984280i \(-0.443485\pi\)
0.984280 0.176616i \(-0.0565152\pi\)
\(860\) 0 0
\(861\) 1.06102 + 1.06102i 0.0361596 + 0.0361596i
\(862\) 0 0
\(863\) 8.20297i 0.279232i 0.990206 + 0.139616i \(0.0445869\pi\)
−0.990206 + 0.139616i \(0.955413\pi\)
\(864\) 0 0
\(865\) 19.4513 7.99990i 0.661365 0.272005i
\(866\) 0 0
\(867\) 11.9784 11.9784i 0.406808 0.406808i
\(868\) 0 0
\(869\) 13.2239 13.2239i 0.448590 0.448590i
\(870\) 0 0
\(871\) 60.6991i 2.05671i
\(872\) 0 0
\(873\) −1.39922 −0.0473565
\(874\) 0 0
\(875\) 2.97155 + 7.32948i 0.100457 + 0.247782i
\(876\) 0 0
\(877\) 23.9525 23.9525i 0.808817 0.808817i −0.175638 0.984455i \(-0.556199\pi\)
0.984455 + 0.175638i \(0.0561987\pi\)
\(878\) 0 0
\(879\) −16.7214 −0.563999
\(880\) 0 0
\(881\) 27.4850 0.925993 0.462997 0.886360i \(-0.346774\pi\)
0.462997 + 0.886360i \(0.346774\pi\)
\(882\) 0 0
\(883\) 22.9566 22.9566i 0.772550 0.772550i −0.206001 0.978552i \(-0.566045\pi\)
0.978552 + 0.206001i \(0.0660452\pi\)
\(884\) 0 0
\(885\) −7.13588 + 17.1061i −0.239870 + 0.575014i
\(886\) 0 0
\(887\) 8.52038 0.286086 0.143043 0.989716i \(-0.454311\pi\)
0.143043 + 0.989716i \(0.454311\pi\)
\(888\) 0 0
\(889\) 13.7565i 0.461379i
\(890\) 0 0
\(891\) 1.79993 1.79993i 0.0602998 0.0602998i
\(892\) 0 0
\(893\) −6.88294 + 6.88294i −0.230329 + 0.230329i
\(894\) 0 0
\(895\) −20.7678 50.4958i −0.694192 1.68789i
\(896\) 0 0
\(897\) 37.8247i 1.26293i
\(898\) 0 0
\(899\) 37.2670 + 37.2670i 1.24292 + 1.24292i
\(900\) 0 0
\(901\) −0.360044 + 0.360044i −0.0119948 + 0.0119948i
\(902\) 0 0
\(903\) 0.684789 0.0227883
\(904\) 0 0
\(905\) −14.9810 + 6.16134i −0.497984 + 0.204810i
\(906\) 0 0
\(907\) 21.1812 + 21.1812i 0.703312 + 0.703312i 0.965120 0.261808i \(-0.0843188\pi\)
−0.261808 + 0.965120i \(0.584319\pi\)
\(908\) 0 0
\(909\) 11.7916 + 11.7916i 0.391104 + 0.391104i
\(910\) 0 0
\(911\) −12.3002 −0.407525 −0.203763 0.979020i \(-0.565317\pi\)
−0.203763 + 0.979020i \(0.565317\pi\)
\(912\) 0 0
\(913\) 24.4895i 0.810484i
\(914\) 0 0
\(915\) −0.156851 0.0654311i −0.00518532 0.00216308i
\(916\) 0 0
\(917\) 0.250454 + 0.250454i 0.00827070 + 0.00827070i
\(918\) 0 0
\(919\) 3.59430i 0.118565i −0.998241 0.0592826i \(-0.981119\pi\)
0.998241 0.0592826i \(-0.0188813\pi\)
\(920\) 0 0
\(921\) 23.5087i 0.774638i
\(922\) 0 0
\(923\) 8.05902 + 8.05902i 0.265266 + 0.265266i
\(924\) 0 0
\(925\) −0.300757 + 59.9522i −0.00988882 + 1.97122i
\(926\) 0 0
\(927\) 2.29240i 0.0752923i
\(928\) 0 0
\(929\) −35.8527 −1.17629 −0.588145 0.808756i \(-0.700142\pi\)
−0.588145 + 0.808756i \(0.700142\pi\)
\(930\) 0 0
\(931\) 10.0005 + 10.0005i 0.327752 + 0.327752i
\(932\) 0 0
\(933\) 9.76298 + 9.76298i 0.319626 + 0.319626i
\(934\) 0 0
\(935\) 0.530171 + 1.28908i 0.0173385 + 0.0421575i
\(936\) 0 0
\(937\) 40.7563 1.33145 0.665725 0.746198i \(-0.268123\pi\)
0.665725 + 0.746198i \(0.268123\pi\)
\(938\) 0 0
\(939\) −5.23528 + 5.23528i −0.170847 + 0.170847i
\(940\) 0 0
\(941\) −36.2143 36.2143i −1.18055 1.18055i −0.979602 0.200950i \(-0.935597\pi\)
−0.200950 0.979602i \(-0.564403\pi\)
\(942\) 0 0
\(943\) 14.6845i 0.478193i
\(944\) 0 0
\(945\) −0.601657 1.46290i −0.0195719 0.0475880i
\(946\) 0 0
\(947\) 36.5242 36.5242i 1.18688 1.18688i 0.208951 0.977926i \(-0.432995\pi\)
0.977926 0.208951i \(-0.0670050\pi\)
\(948\) 0 0
\(949\) −37.4738 + 37.4738i −1.21645 + 1.21645i
\(950\) 0 0
\(951\) 2.55566i 0.0828729i
\(952\) 0 0
\(953\) −16.0927 −0.521294 −0.260647 0.965434i \(-0.583936\pi\)
−0.260647 + 0.965434i \(0.583936\pi\)
\(954\) 0 0
\(955\) −36.7421 15.3272i −1.18895 0.495975i
\(956\) 0 0
\(957\) −12.4699 + 12.4699i −0.403093 + 0.403093i
\(958\) 0 0
\(959\) −5.16423 −0.166762
\(960\) 0 0
\(961\) 26.8716 0.866826
\(962\) 0 0
\(963\) −9.49009 + 9.49009i −0.305814 + 0.305814i
\(964\) 0 0
\(965\) 14.8264 35.5416i 0.477278 1.14413i
\(966\) 0 0
\(967\) 21.3561 0.686767 0.343384 0.939195i \(-0.388427\pi\)
0.343384 + 0.939195i \(0.388427\pi\)
\(968\) 0 0
\(969\) 0.532856i 0.0171178i
\(970\) 0 0
\(971\) 20.0626 20.0626i 0.643839 0.643839i −0.307658 0.951497i \(-0.599545\pi\)
0.951497 + 0.307658i \(0.0995454\pi\)
\(972\) 0 0
\(973\) 6.26414 6.26414i 0.200819 0.200819i
\(974\) 0 0
\(975\) −19.4140 + 19.2202i −0.621747 + 0.615540i
\(976\) 0 0
\(977\) 27.7769i 0.888661i 0.895863 + 0.444331i \(0.146558\pi\)
−0.895863 + 0.444331i \(0.853442\pi\)
\(978\) 0 0
\(979\) 5.53447 + 5.53447i 0.176882 + 0.176882i
\(980\) 0 0
\(981\) −11.9058 + 11.9058i −0.380122 + 0.380122i
\(982\) 0 0
\(983\) −0.428090 −0.0136540 −0.00682698 0.999977i \(-0.502173\pi\)
−0.00682698 + 0.999977i \(0.502173\pi\)
\(984\) 0 0
\(985\) 12.1201 + 29.4694i 0.386179 + 0.938973i
\(986\) 0 0
\(987\) −2.23763 2.23763i −0.0712245 0.0712245i
\(988\) 0 0
\(989\) 4.73871 + 4.73871i 0.150682 + 0.150682i
\(990\) 0 0
\(991\) −5.82267 −0.184963 −0.0924816 0.995714i \(-0.529480\pi\)
−0.0924816 + 0.995714i \(0.529480\pi\)
\(992\) 0 0
\(993\) 8.24078i 0.261513i
\(994\) 0 0
\(995\) 4.94303 11.8494i 0.156704 0.375650i
\(996\) 0 0
\(997\) 20.1044 + 20.1044i 0.636713 + 0.636713i 0.949743 0.313030i \(-0.101344\pi\)
−0.313030 + 0.949743i \(0.601344\pi\)
\(998\) 0 0
\(999\) 11.9906i 0.379366i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1920.2.bl.a.289.18 48
4.3 odd 2 1920.2.bl.b.289.7 48
5.4 even 2 inner 1920.2.bl.a.289.7 48
8.3 odd 2 960.2.bl.a.529.15 48
8.5 even 2 240.2.bl.a.109.5 48
16.3 odd 4 960.2.bl.a.49.9 48
16.5 even 4 inner 1920.2.bl.a.1249.7 48
16.11 odd 4 1920.2.bl.b.1249.18 48
16.13 even 4 240.2.bl.a.229.20 yes 48
20.19 odd 2 1920.2.bl.b.289.18 48
24.5 odd 2 720.2.bm.h.109.20 48
40.19 odd 2 960.2.bl.a.529.9 48
40.29 even 2 240.2.bl.a.109.20 yes 48
48.29 odd 4 720.2.bm.h.469.5 48
80.19 odd 4 960.2.bl.a.49.15 48
80.29 even 4 240.2.bl.a.229.5 yes 48
80.59 odd 4 1920.2.bl.b.1249.7 48
80.69 even 4 inner 1920.2.bl.a.1249.18 48
120.29 odd 2 720.2.bm.h.109.5 48
240.29 odd 4 720.2.bm.h.469.20 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.bl.a.109.5 48 8.5 even 2
240.2.bl.a.109.20 yes 48 40.29 even 2
240.2.bl.a.229.5 yes 48 80.29 even 4
240.2.bl.a.229.20 yes 48 16.13 even 4
720.2.bm.h.109.5 48 120.29 odd 2
720.2.bm.h.109.20 48 24.5 odd 2
720.2.bm.h.469.5 48 48.29 odd 4
720.2.bm.h.469.20 48 240.29 odd 4
960.2.bl.a.49.9 48 16.3 odd 4
960.2.bl.a.49.15 48 80.19 odd 4
960.2.bl.a.529.9 48 40.19 odd 2
960.2.bl.a.529.15 48 8.3 odd 2
1920.2.bl.a.289.7 48 5.4 even 2 inner
1920.2.bl.a.289.18 48 1.1 even 1 trivial
1920.2.bl.a.1249.7 48 16.5 even 4 inner
1920.2.bl.a.1249.18 48 80.69 even 4 inner
1920.2.bl.b.289.7 48 4.3 odd 2
1920.2.bl.b.289.18 48 20.19 odd 2
1920.2.bl.b.1249.7 48 80.59 odd 4
1920.2.bl.b.1249.18 48 16.11 odd 4