Properties

Label 1840.2.m.g.1839.10
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1839,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1839");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), 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: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1839.10
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.g.1839.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.20329 q^{3} +(-1.71768 + 1.43163i) q^{5} -0.642617i q^{7} +1.85447 q^{9} +O(q^{10})\) \(q-2.20329 q^{3} +(-1.71768 + 1.43163i) q^{5} -0.642617i q^{7} +1.85447 q^{9} +2.87003 q^{11} +0.405025i q^{13} +(3.78454 - 3.15430i) q^{15} -4.93021 q^{17} +2.74284 q^{19} +1.41587i q^{21} +(4.71132 - 0.896383i) q^{23} +(0.900844 - 4.91818i) q^{25} +2.52393 q^{27} -1.62882 q^{29} -4.55939i q^{31} -6.32349 q^{33} +(0.919993 + 1.10381i) q^{35} +2.00091 q^{37} -0.892386i q^{39} -4.79334 q^{41} +3.29086i q^{43} +(-3.18539 + 2.65493i) q^{45} -3.58875 q^{47} +6.58704 q^{49} +10.8627 q^{51} -0.263678 q^{53} +(-4.92979 + 4.10883i) q^{55} -6.04326 q^{57} +10.3753i q^{59} +9.19418i q^{61} -1.19172i q^{63} +(-0.579847 - 0.695702i) q^{65} -2.18044i q^{67} +(-10.3804 + 1.97499i) q^{69} -10.3200i q^{71} +4.43051i q^{73} +(-1.98482 + 10.8362i) q^{75} -1.84433i q^{77} -3.16051 q^{79} -11.1243 q^{81} +9.14874i q^{83} +(8.46852 - 7.05826i) q^{85} +3.58875 q^{87} +14.0324i q^{89} +0.260276 q^{91} +10.0456i q^{93} +(-4.71132 + 3.92674i) q^{95} -10.3665 q^{97} +5.32239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −2.20329 −1.27207 −0.636034 0.771661i \(-0.719426\pi\)
−0.636034 + 0.771661i \(0.719426\pi\)
\(4\) 0 0
\(5\) −1.71768 + 1.43163i −0.768169 + 0.640247i
\(6\) 0 0
\(7\) 0.642617i 0.242886i −0.992598 0.121443i \(-0.961248\pi\)
0.992598 0.121443i \(-0.0387522\pi\)
\(8\) 0 0
\(9\) 1.85447 0.618157
\(10\) 0 0
\(11\) 2.87003 0.865346 0.432673 0.901551i \(-0.357571\pi\)
0.432673 + 0.901551i \(0.357571\pi\)
\(12\) 0 0
\(13\) 0.405025i 0.112334i 0.998421 + 0.0561668i \(0.0178879\pi\)
−0.998421 + 0.0561668i \(0.982112\pi\)
\(14\) 0 0
\(15\) 3.78454 3.15430i 0.977164 0.814437i
\(16\) 0 0
\(17\) −4.93021 −1.19575 −0.597876 0.801589i \(-0.703989\pi\)
−0.597876 + 0.801589i \(0.703989\pi\)
\(18\) 0 0
\(19\) 2.74284 0.629250 0.314625 0.949216i \(-0.398121\pi\)
0.314625 + 0.949216i \(0.398121\pi\)
\(20\) 0 0
\(21\) 1.41587i 0.308968i
\(22\) 0 0
\(23\) 4.71132 0.896383i 0.982377 0.186909i
\(24\) 0 0
\(25\) 0.900844 4.91818i 0.180169 0.983636i
\(26\) 0 0
\(27\) 2.52393 0.485730
\(28\) 0 0
\(29\) −1.62882 −0.302464 −0.151232 0.988498i \(-0.548324\pi\)
−0.151232 + 0.988498i \(0.548324\pi\)
\(30\) 0 0
\(31\) 4.55939i 0.818890i −0.912335 0.409445i \(-0.865722\pi\)
0.912335 0.409445i \(-0.134278\pi\)
\(32\) 0 0
\(33\) −6.32349 −1.10078
\(34\) 0 0
\(35\) 0.919993 + 1.10381i 0.155507 + 0.186578i
\(36\) 0 0
\(37\) 2.00091 0.328947 0.164474 0.986381i \(-0.447407\pi\)
0.164474 + 0.986381i \(0.447407\pi\)
\(38\) 0 0
\(39\) 0.892386i 0.142896i
\(40\) 0 0
\(41\) −4.79334 −0.748593 −0.374297 0.927309i \(-0.622116\pi\)
−0.374297 + 0.927309i \(0.622116\pi\)
\(42\) 0 0
\(43\) 3.29086i 0.501852i 0.968006 + 0.250926i \(0.0807351\pi\)
−0.968006 + 0.250926i \(0.919265\pi\)
\(44\) 0 0
\(45\) −3.18539 + 2.65493i −0.474850 + 0.395773i
\(46\) 0 0
\(47\) −3.58875 −0.523473 −0.261736 0.965139i \(-0.584295\pi\)
−0.261736 + 0.965139i \(0.584295\pi\)
\(48\) 0 0
\(49\) 6.58704 0.941006
\(50\) 0 0
\(51\) 10.8627 1.52108
\(52\) 0 0
\(53\) −0.263678 −0.0362190 −0.0181095 0.999836i \(-0.505765\pi\)
−0.0181095 + 0.999836i \(0.505765\pi\)
\(54\) 0 0
\(55\) −4.92979 + 4.10883i −0.664732 + 0.554035i
\(56\) 0 0
\(57\) −6.04326 −0.800449
\(58\) 0 0
\(59\) 10.3753i 1.35074i 0.737478 + 0.675372i \(0.236017\pi\)
−0.737478 + 0.675372i \(0.763983\pi\)
\(60\) 0 0
\(61\) 9.19418i 1.17719i 0.808427 + 0.588597i \(0.200319\pi\)
−0.808427 + 0.588597i \(0.799681\pi\)
\(62\) 0 0
\(63\) 1.19172i 0.150142i
\(64\) 0 0
\(65\) −0.579847 0.695702i −0.0719212 0.0862913i
\(66\) 0 0
\(67\) 2.18044i 0.266383i −0.991090 0.133192i \(-0.957477\pi\)
0.991090 0.133192i \(-0.0425225\pi\)
\(68\) 0 0
\(69\) −10.3804 + 1.97499i −1.24965 + 0.237761i
\(70\) 0 0
\(71\) 10.3200i 1.22475i −0.790566 0.612376i \(-0.790214\pi\)
0.790566 0.612376i \(-0.209786\pi\)
\(72\) 0 0
\(73\) 4.43051i 0.518552i 0.965803 + 0.259276i \(0.0834839\pi\)
−0.965803 + 0.259276i \(0.916516\pi\)
\(74\) 0 0
\(75\) −1.98482 + 10.8362i −0.229187 + 1.25125i
\(76\) 0 0
\(77\) 1.84433i 0.210181i
\(78\) 0 0
\(79\) −3.16051 −0.355585 −0.177792 0.984068i \(-0.556896\pi\)
−0.177792 + 0.984068i \(0.556896\pi\)
\(80\) 0 0
\(81\) −11.1243 −1.23604
\(82\) 0 0
\(83\) 9.14874i 1.00420i 0.864808 + 0.502102i \(0.167440\pi\)
−0.864808 + 0.502102i \(0.832560\pi\)
\(84\) 0 0
\(85\) 8.46852 7.05826i 0.918540 0.765576i
\(86\) 0 0
\(87\) 3.58875 0.384754
\(88\) 0 0
\(89\) 14.0324i 1.48744i 0.668494 + 0.743718i \(0.266939\pi\)
−0.668494 + 0.743718i \(0.733061\pi\)
\(90\) 0 0
\(91\) 0.260276 0.0272843
\(92\) 0 0
\(93\) 10.0456i 1.04168i
\(94\) 0 0
\(95\) −4.71132 + 3.92674i −0.483371 + 0.402875i
\(96\) 0 0
\(97\) −10.3665 −1.05256 −0.526278 0.850312i \(-0.676413\pi\)
−0.526278 + 0.850312i \(0.676413\pi\)
\(98\) 0 0
\(99\) 5.32239 0.534920
\(100\) 0 0
\(101\) −12.2075 −1.21469 −0.607346 0.794437i \(-0.707766\pi\)
−0.607346 + 0.794437i \(0.707766\pi\)
\(102\) 0 0
\(103\) 11.3098i 1.11439i 0.830382 + 0.557195i \(0.188122\pi\)
−0.830382 + 0.557195i \(0.811878\pi\)
\(104\) 0 0
\(105\) −2.02701 2.43201i −0.197816 0.237340i
\(106\) 0 0
\(107\) 15.5211i 1.50048i 0.661167 + 0.750239i \(0.270062\pi\)
−0.661167 + 0.750239i \(0.729938\pi\)
\(108\) 0 0
\(109\) 13.6376i 1.30625i 0.757251 + 0.653124i \(0.226542\pi\)
−0.757251 + 0.653124i \(0.773458\pi\)
\(110\) 0 0
\(111\) −4.40857 −0.418443
\(112\) 0 0
\(113\) −7.47503 −0.703192 −0.351596 0.936152i \(-0.614361\pi\)
−0.351596 + 0.936152i \(0.614361\pi\)
\(114\) 0 0
\(115\) −6.80924 + 8.28458i −0.634965 + 0.772541i
\(116\) 0 0
\(117\) 0.751107i 0.0694399i
\(118\) 0 0
\(119\) 3.16824i 0.290432i
\(120\) 0 0
\(121\) −2.76295 −0.251177
\(122\) 0 0
\(123\) 10.5611 0.952262
\(124\) 0 0
\(125\) 5.49368 + 9.73753i 0.491369 + 0.870951i
\(126\) 0 0
\(127\) 0.380884 0.0337980 0.0168990 0.999857i \(-0.494621\pi\)
0.0168990 + 0.999857i \(0.494621\pi\)
\(128\) 0 0
\(129\) 7.25072i 0.638390i
\(130\) 0 0
\(131\) 14.8999i 1.30181i −0.759157 0.650907i \(-0.774389\pi\)
0.759157 0.650907i \(-0.225611\pi\)
\(132\) 0 0
\(133\) 1.76259i 0.152836i
\(134\) 0 0
\(135\) −4.33529 + 3.61334i −0.373123 + 0.310987i
\(136\) 0 0
\(137\) 8.37933 0.715894 0.357947 0.933742i \(-0.383477\pi\)
0.357947 + 0.933742i \(0.383477\pi\)
\(138\) 0 0
\(139\) 3.37882i 0.286588i 0.989680 + 0.143294i \(0.0457694\pi\)
−0.989680 + 0.143294i \(0.954231\pi\)
\(140\) 0 0
\(141\) 7.90704 0.665893
\(142\) 0 0
\(143\) 1.16243i 0.0972074i
\(144\) 0 0
\(145\) 2.79778 2.33187i 0.232343 0.193651i
\(146\) 0 0
\(147\) −14.5131 −1.19702
\(148\) 0 0
\(149\) 17.7280i 1.45233i 0.687519 + 0.726167i \(0.258700\pi\)
−0.687519 + 0.726167i \(0.741300\pi\)
\(150\) 0 0
\(151\) 16.9047i 1.37568i −0.725861 0.687841i \(-0.758559\pi\)
0.725861 0.687841i \(-0.241441\pi\)
\(152\) 0 0
\(153\) −9.14294 −0.739163
\(154\) 0 0
\(155\) 6.52738 + 7.83157i 0.524292 + 0.629046i
\(156\) 0 0
\(157\) 11.1520 0.890027 0.445013 0.895524i \(-0.353199\pi\)
0.445013 + 0.895524i \(0.353199\pi\)
\(158\) 0 0
\(159\) 0.580959 0.0460730
\(160\) 0 0
\(161\) −0.576031 3.02757i −0.0453976 0.238606i
\(162\) 0 0
\(163\) −2.50634 −0.196312 −0.0981559 0.995171i \(-0.531294\pi\)
−0.0981559 + 0.995171i \(0.531294\pi\)
\(164\) 0 0
\(165\) 10.8617 9.05293i 0.845585 0.704770i
\(166\) 0 0
\(167\) −1.83999 −0.142382 −0.0711912 0.997463i \(-0.522680\pi\)
−0.0711912 + 0.997463i \(0.522680\pi\)
\(168\) 0 0
\(169\) 12.8360 0.987381
\(170\) 0 0
\(171\) 5.08652 0.388976
\(172\) 0 0
\(173\) 11.2607i 0.856136i 0.903747 + 0.428068i \(0.140806\pi\)
−0.903747 + 0.428068i \(0.859194\pi\)
\(174\) 0 0
\(175\) −3.16051 0.578898i −0.238912 0.0437605i
\(176\) 0 0
\(177\) 22.8597i 1.71824i
\(178\) 0 0
\(179\) 0.613572i 0.0458605i −0.999737 0.0229303i \(-0.992700\pi\)
0.999737 0.0229303i \(-0.00729957\pi\)
\(180\) 0 0
\(181\) 0.527590i 0.0392154i −0.999808 0.0196077i \(-0.993758\pi\)
0.999808 0.0196077i \(-0.00624173\pi\)
\(182\) 0 0
\(183\) 20.2574i 1.49747i
\(184\) 0 0
\(185\) −3.43692 + 2.86457i −0.252687 + 0.210607i
\(186\) 0 0
\(187\) −14.1498 −1.03474
\(188\) 0 0
\(189\) 1.62192i 0.117977i
\(190\) 0 0
\(191\) −13.0548 −0.944609 −0.472304 0.881435i \(-0.656578\pi\)
−0.472304 + 0.881435i \(0.656578\pi\)
\(192\) 0 0
\(193\) 20.3324i 1.46356i 0.681541 + 0.731780i \(0.261310\pi\)
−0.681541 + 0.731780i \(0.738690\pi\)
\(194\) 0 0
\(195\) 1.27757 + 1.53283i 0.0914887 + 0.109768i
\(196\) 0 0
\(197\) 15.0465i 1.07202i −0.844213 0.536008i \(-0.819932\pi\)
0.844213 0.536008i \(-0.180068\pi\)
\(198\) 0 0
\(199\) −6.81716 −0.483255 −0.241628 0.970369i \(-0.577681\pi\)
−0.241628 + 0.970369i \(0.577681\pi\)
\(200\) 0 0
\(201\) 4.80413i 0.338857i
\(202\) 0 0
\(203\) 1.04671i 0.0734643i
\(204\) 0 0
\(205\) 8.23341 6.86231i 0.575047 0.479284i
\(206\) 0 0
\(207\) 8.73701 1.66232i 0.607264 0.115539i
\(208\) 0 0
\(209\) 7.87202 0.544519
\(210\) 0 0
\(211\) 2.73591i 0.188348i −0.995556 0.0941739i \(-0.969979\pi\)
0.995556 0.0941739i \(-0.0300210\pi\)
\(212\) 0 0
\(213\) 22.7378i 1.55797i
\(214\) 0 0
\(215\) −4.71132 5.65265i −0.321309 0.385507i
\(216\) 0 0
\(217\) −2.92994 −0.198897
\(218\) 0 0
\(219\) 9.76168i 0.659633i
\(220\) 0 0
\(221\) 1.99686i 0.134323i
\(222\) 0 0
\(223\) 19.6037 1.31276 0.656379 0.754431i \(-0.272087\pi\)
0.656379 + 0.754431i \(0.272087\pi\)
\(224\) 0 0
\(225\) 1.67059 9.12063i 0.111373 0.608042i
\(226\) 0 0
\(227\) 18.0945i 1.20097i 0.799635 + 0.600487i \(0.205026\pi\)
−0.799635 + 0.600487i \(0.794974\pi\)
\(228\) 0 0
\(229\) 27.1740i 1.79571i −0.440293 0.897854i \(-0.645125\pi\)
0.440293 0.897854i \(-0.354875\pi\)
\(230\) 0 0
\(231\) 4.06359i 0.267364i
\(232\) 0 0
\(233\) 4.25756i 0.278922i 0.990228 + 0.139461i \(0.0445370\pi\)
−0.990228 + 0.139461i \(0.955463\pi\)
\(234\) 0 0
\(235\) 6.16432 5.13778i 0.402116 0.335152i
\(236\) 0 0
\(237\) 6.96350 0.452328
\(238\) 0 0
\(239\) 18.5247i 1.19826i 0.800651 + 0.599131i \(0.204487\pi\)
−0.800651 + 0.599131i \(0.795513\pi\)
\(240\) 0 0
\(241\) 3.60856i 0.232448i 0.993223 + 0.116224i \(0.0370790\pi\)
−0.993223 + 0.116224i \(0.962921\pi\)
\(242\) 0 0
\(243\) 16.9384 1.08660
\(244\) 0 0
\(245\) −11.3144 + 9.43024i −0.722852 + 0.602476i
\(246\) 0 0
\(247\) 1.11092i 0.0706860i
\(248\) 0 0
\(249\) 20.1573i 1.27742i
\(250\) 0 0
\(251\) 3.81994 0.241113 0.120556 0.992706i \(-0.461532\pi\)
0.120556 + 0.992706i \(0.461532\pi\)
\(252\) 0 0
\(253\) 13.5216 2.57264i 0.850096 0.161741i
\(254\) 0 0
\(255\) −18.6586 + 15.5514i −1.16845 + 0.973865i
\(256\) 0 0
\(257\) 1.24356i 0.0775712i −0.999248 0.0387856i \(-0.987651\pi\)
0.999248 0.0387856i \(-0.0123489\pi\)
\(258\) 0 0
\(259\) 1.28582i 0.0798968i
\(260\) 0 0
\(261\) −3.02059 −0.186970
\(262\) 0 0
\(263\) 10.0562i 0.620094i −0.950721 0.310047i \(-0.899655\pi\)
0.950721 0.310047i \(-0.100345\pi\)
\(264\) 0 0
\(265\) 0.452914 0.377491i 0.0278223 0.0231891i
\(266\) 0 0
\(267\) 30.9175i 1.89212i
\(268\) 0 0
\(269\) −2.19489 −0.133825 −0.0669125 0.997759i \(-0.521315\pi\)
−0.0669125 + 0.997759i \(0.521315\pi\)
\(270\) 0 0
\(271\) 4.25997i 0.258775i 0.991594 + 0.129387i \(0.0413010\pi\)
−0.991594 + 0.129387i \(0.958699\pi\)
\(272\) 0 0
\(273\) −0.573462 −0.0347075
\(274\) 0 0
\(275\) 2.58545 14.1153i 0.155908 0.851185i
\(276\) 0 0
\(277\) 8.87597i 0.533305i 0.963793 + 0.266653i \(0.0859176\pi\)
−0.963793 + 0.266653i \(0.914082\pi\)
\(278\) 0 0
\(279\) 8.45526i 0.506203i
\(280\) 0 0
\(281\) 8.57906i 0.511784i 0.966705 + 0.255892i \(0.0823692\pi\)
−0.966705 + 0.255892i \(0.917631\pi\)
\(282\) 0 0
\(283\) 3.40195i 0.202225i 0.994875 + 0.101113i \(0.0322402\pi\)
−0.994875 + 0.101113i \(0.967760\pi\)
\(284\) 0 0
\(285\) 10.3804 8.65174i 0.614881 0.512485i
\(286\) 0 0
\(287\) 3.08028i 0.181823i
\(288\) 0 0
\(289\) 7.30700 0.429824
\(290\) 0 0
\(291\) 22.8403 1.33892
\(292\) 0 0
\(293\) −28.9057 −1.68869 −0.844344 0.535801i \(-0.820010\pi\)
−0.844344 + 0.535801i \(0.820010\pi\)
\(294\) 0 0
\(295\) −14.8536 17.8214i −0.864809 1.03760i
\(296\) 0 0
\(297\) 7.24374 0.420324
\(298\) 0 0
\(299\) 0.363057 + 1.90820i 0.0209962 + 0.110354i
\(300\) 0 0
\(301\) 2.11477 0.121893
\(302\) 0 0
\(303\) 26.8966 1.54517
\(304\) 0 0
\(305\) −13.1627 15.7926i −0.753694 0.904284i
\(306\) 0 0
\(307\) 3.66239 0.209024 0.104512 0.994524i \(-0.466672\pi\)
0.104512 + 0.994524i \(0.466672\pi\)
\(308\) 0 0
\(309\) 24.9188i 1.41758i
\(310\) 0 0
\(311\) 8.46706i 0.480123i 0.970758 + 0.240061i \(0.0771676\pi\)
−0.970758 + 0.240061i \(0.922832\pi\)
\(312\) 0 0
\(313\) 18.5693 1.04960 0.524800 0.851226i \(-0.324140\pi\)
0.524800 + 0.851226i \(0.324140\pi\)
\(314\) 0 0
\(315\) 1.70610 + 2.04699i 0.0961280 + 0.115335i
\(316\) 0 0
\(317\) 0.673248i 0.0378134i −0.999821 0.0189067i \(-0.993981\pi\)
0.999821 0.0189067i \(-0.00601854\pi\)
\(318\) 0 0
\(319\) −4.67475 −0.261736
\(320\) 0 0
\(321\) 34.1974i 1.90871i
\(322\) 0 0
\(323\) −13.5228 −0.752427
\(324\) 0 0
\(325\) 1.99198 + 0.364864i 0.110495 + 0.0202390i
\(326\) 0 0
\(327\) 30.0476i 1.66164i
\(328\) 0 0
\(329\) 2.30619i 0.127144i
\(330\) 0 0
\(331\) 11.3823i 0.625630i −0.949814 0.312815i \(-0.898728\pi\)
0.949814 0.312815i \(-0.101272\pi\)
\(332\) 0 0
\(333\) 3.71063 0.203341
\(334\) 0 0
\(335\) 3.12159 + 3.74530i 0.170551 + 0.204627i
\(336\) 0 0
\(337\) −2.41570 −0.131592 −0.0657959 0.997833i \(-0.520959\pi\)
−0.0657959 + 0.997833i \(0.520959\pi\)
\(338\) 0 0
\(339\) 16.4696 0.894508
\(340\) 0 0
\(341\) 13.0856i 0.708623i
\(342\) 0 0
\(343\) 8.73127i 0.471444i
\(344\) 0 0
\(345\) 15.0027 18.2533i 0.807718 0.982725i
\(346\) 0 0
\(347\) −20.0264 −1.07507 −0.537536 0.843241i \(-0.680645\pi\)
−0.537536 + 0.843241i \(0.680645\pi\)
\(348\) 0 0
\(349\) −22.9116 −1.22643 −0.613215 0.789916i \(-0.710124\pi\)
−0.613215 + 0.789916i \(0.710124\pi\)
\(350\) 0 0
\(351\) 1.02225i 0.0545638i
\(352\) 0 0
\(353\) 28.3834i 1.51070i 0.655324 + 0.755348i \(0.272532\pi\)
−0.655324 + 0.755348i \(0.727468\pi\)
\(354\) 0 0
\(355\) 14.7744 + 17.7264i 0.784144 + 0.940818i
\(356\) 0 0
\(357\) 6.98054i 0.369449i
\(358\) 0 0
\(359\) −13.5509 −0.715189 −0.357595 0.933877i \(-0.616403\pi\)
−0.357595 + 0.933877i \(0.616403\pi\)
\(360\) 0 0
\(361\) −11.4768 −0.604044
\(362\) 0 0
\(363\) 6.08756 0.319514
\(364\) 0 0
\(365\) −6.34287 7.61019i −0.332001 0.398336i
\(366\) 0 0
\(367\) 4.18607i 0.218511i −0.994014 0.109256i \(-0.965153\pi\)
0.994014 0.109256i \(-0.0348467\pi\)
\(368\) 0 0
\(369\) −8.88911 −0.462749
\(370\) 0 0
\(371\) 0.169444i 0.00879710i
\(372\) 0 0
\(373\) −13.2025 −0.683599 −0.341799 0.939773i \(-0.611036\pi\)
−0.341799 + 0.939773i \(0.611036\pi\)
\(374\) 0 0
\(375\) −12.1041 21.4546i −0.625055 1.10791i
\(376\) 0 0
\(377\) 0.659711i 0.0339768i
\(378\) 0 0
\(379\) −7.79915 −0.400616 −0.200308 0.979733i \(-0.564194\pi\)
−0.200308 + 0.979733i \(0.564194\pi\)
\(380\) 0 0
\(381\) −0.839197 −0.0429934
\(382\) 0 0
\(383\) 0.691775i 0.0353480i −0.999844 0.0176740i \(-0.994374\pi\)
0.999844 0.0176740i \(-0.00562611\pi\)
\(384\) 0 0
\(385\) 2.64041 + 3.16797i 0.134568 + 0.161454i
\(386\) 0 0
\(387\) 6.10282i 0.310224i
\(388\) 0 0
\(389\) 8.94180i 0.453367i −0.973968 0.226684i \(-0.927212\pi\)
0.973968 0.226684i \(-0.0727883\pi\)
\(390\) 0 0
\(391\) −23.2278 + 4.41936i −1.17468 + 0.223497i
\(392\) 0 0
\(393\) 32.8289i 1.65600i
\(394\) 0 0
\(395\) 5.42874 4.52469i 0.273149 0.227662i
\(396\) 0 0
\(397\) 7.00315i 0.351478i −0.984437 0.175739i \(-0.943768\pi\)
0.984437 0.175739i \(-0.0562315\pi\)
\(398\) 0 0
\(399\) 3.88350i 0.194418i
\(400\) 0 0
\(401\) 1.06280i 0.0530736i −0.999648 0.0265368i \(-0.991552\pi\)
0.999648 0.0265368i \(-0.00844792\pi\)
\(402\) 0 0
\(403\) 1.84666 0.0919889
\(404\) 0 0
\(405\) 19.1081 15.9260i 0.949487 0.791370i
\(406\) 0 0
\(407\) 5.74266 0.284653
\(408\) 0 0
\(409\) 5.85837 0.289678 0.144839 0.989455i \(-0.453734\pi\)
0.144839 + 0.989455i \(0.453734\pi\)
\(410\) 0 0
\(411\) −18.4621 −0.910666
\(412\) 0 0
\(413\) 6.66732 0.328077
\(414\) 0 0
\(415\) −13.0977 15.7146i −0.642939 0.771399i
\(416\) 0 0
\(417\) 7.44451i 0.364559i
\(418\) 0 0
\(419\) 30.7667 1.50305 0.751525 0.659705i \(-0.229319\pi\)
0.751525 + 0.659705i \(0.229319\pi\)
\(420\) 0 0
\(421\) 37.3043i 1.81810i −0.416687 0.909050i \(-0.636809\pi\)
0.416687 0.909050i \(-0.363191\pi\)
\(422\) 0 0
\(423\) −6.65524 −0.323589
\(424\) 0 0
\(425\) −4.44135 + 24.2477i −0.215437 + 1.17618i
\(426\) 0 0
\(427\) 5.90834 0.285924
\(428\) 0 0
\(429\) 2.56117i 0.123654i
\(430\) 0 0
\(431\) 18.2987 0.881416 0.440708 0.897651i \(-0.354727\pi\)
0.440708 + 0.897651i \(0.354727\pi\)
\(432\) 0 0
\(433\) 31.1511 1.49702 0.748512 0.663121i \(-0.230769\pi\)
0.748512 + 0.663121i \(0.230769\pi\)
\(434\) 0 0
\(435\) −6.16432 + 5.13778i −0.295557 + 0.246338i
\(436\) 0 0
\(437\) 12.9224 2.45863i 0.618161 0.117612i
\(438\) 0 0
\(439\) 8.87110i 0.423394i −0.977335 0.211697i \(-0.932101\pi\)
0.977335 0.211697i \(-0.0678991\pi\)
\(440\) 0 0
\(441\) 12.2155 0.581690
\(442\) 0 0
\(443\) 34.1696 1.62344 0.811722 0.584044i \(-0.198530\pi\)
0.811722 + 0.584044i \(0.198530\pi\)
\(444\) 0 0
\(445\) −20.0893 24.1032i −0.952325 1.14260i
\(446\) 0 0
\(447\) 39.0598i 1.84747i
\(448\) 0 0
\(449\) 16.3702 0.772558 0.386279 0.922382i \(-0.373760\pi\)
0.386279 + 0.922382i \(0.373760\pi\)
\(450\) 0 0
\(451\) −13.7570 −0.647792
\(452\) 0 0
\(453\) 37.2458i 1.74996i
\(454\) 0 0
\(455\) −0.447070 + 0.372620i −0.0209590 + 0.0174687i
\(456\) 0 0
\(457\) 34.4242 1.61030 0.805148 0.593074i \(-0.202086\pi\)
0.805148 + 0.593074i \(0.202086\pi\)
\(458\) 0 0
\(459\) −12.4435 −0.580812
\(460\) 0 0
\(461\) 19.6000 0.912861 0.456430 0.889759i \(-0.349128\pi\)
0.456430 + 0.889759i \(0.349128\pi\)
\(462\) 0 0
\(463\) −12.4580 −0.578970 −0.289485 0.957183i \(-0.593484\pi\)
−0.289485 + 0.957183i \(0.593484\pi\)
\(464\) 0 0
\(465\) −14.3817 17.2552i −0.666935 0.800190i
\(466\) 0 0
\(467\) 9.93867i 0.459907i 0.973202 + 0.229953i \(0.0738574\pi\)
−0.973202 + 0.229953i \(0.926143\pi\)
\(468\) 0 0
\(469\) −1.40119 −0.0647008
\(470\) 0 0
\(471\) −24.5711 −1.13217
\(472\) 0 0
\(473\) 9.44487i 0.434276i
\(474\) 0 0
\(475\) 2.47087 13.4898i 0.113371 0.618953i
\(476\) 0 0
\(477\) −0.488984 −0.0223890
\(478\) 0 0
\(479\) −39.2013 −1.79115 −0.895576 0.444908i \(-0.853236\pi\)
−0.895576 + 0.444908i \(0.853236\pi\)
\(480\) 0 0
\(481\) 0.810417i 0.0369518i
\(482\) 0 0
\(483\) 1.26916 + 6.67061i 0.0577489 + 0.303523i
\(484\) 0 0
\(485\) 17.8063 14.8410i 0.808542 0.673896i
\(486\) 0 0
\(487\) 31.2472 1.41594 0.707972 0.706241i \(-0.249610\pi\)
0.707972 + 0.706241i \(0.249610\pi\)
\(488\) 0 0
\(489\) 5.52219 0.249722
\(490\) 0 0
\(491\) 28.2058i 1.27291i 0.771313 + 0.636456i \(0.219600\pi\)
−0.771313 + 0.636456i \(0.780400\pi\)
\(492\) 0 0
\(493\) 8.03041 0.361672
\(494\) 0 0
\(495\) −9.14215 + 7.61971i −0.410909 + 0.342481i
\(496\) 0 0
\(497\) −6.63178 −0.297476
\(498\) 0 0
\(499\) 15.7217i 0.703800i 0.936038 + 0.351900i \(0.114464\pi\)
−0.936038 + 0.351900i \(0.885536\pi\)
\(500\) 0 0
\(501\) 4.05402 0.181120
\(502\) 0 0
\(503\) 10.7408i 0.478911i 0.970907 + 0.239455i \(0.0769689\pi\)
−0.970907 + 0.239455i \(0.923031\pi\)
\(504\) 0 0
\(505\) 20.9686 17.4767i 0.933090 0.777703i
\(506\) 0 0
\(507\) −28.2813 −1.25602
\(508\) 0 0
\(509\) −17.3015 −0.766874 −0.383437 0.923567i \(-0.625260\pi\)
−0.383437 + 0.923567i \(0.625260\pi\)
\(510\) 0 0
\(511\) 2.84712 0.125949
\(512\) 0 0
\(513\) 6.92272 0.305646
\(514\) 0 0
\(515\) −16.1915 19.4266i −0.713484 0.856040i
\(516\) 0 0
\(517\) −10.2998 −0.452985
\(518\) 0 0
\(519\) 24.8106i 1.08906i
\(520\) 0 0
\(521\) 28.2184i 1.23627i 0.786071 + 0.618136i \(0.212112\pi\)
−0.786071 + 0.618136i \(0.787888\pi\)
\(522\) 0 0
\(523\) 22.2167i 0.971467i 0.874107 + 0.485734i \(0.161448\pi\)
−0.874107 + 0.485734i \(0.838552\pi\)
\(524\) 0 0
\(525\) 6.96350 + 1.27548i 0.303912 + 0.0556664i
\(526\) 0 0
\(527\) 22.4788i 0.979190i
\(528\) 0 0
\(529\) 21.3930 8.44629i 0.930130 0.367230i
\(530\) 0 0
\(531\) 19.2406i 0.834972i
\(532\) 0 0
\(533\) 1.94142i 0.0840922i
\(534\) 0 0
\(535\) −22.2205 26.6602i −0.960676 1.15262i
\(536\) 0 0
\(537\) 1.35188i 0.0583377i
\(538\) 0 0
\(539\) 18.9050 0.814296
\(540\) 0 0
\(541\) 35.1024 1.50917 0.754585 0.656202i \(-0.227838\pi\)
0.754585 + 0.656202i \(0.227838\pi\)
\(542\) 0 0
\(543\) 1.16243i 0.0498847i
\(544\) 0 0
\(545\) −19.5241 23.4251i −0.836320 1.00342i
\(546\) 0 0
\(547\) 13.1595 0.562659 0.281329 0.959611i \(-0.409225\pi\)
0.281329 + 0.959611i \(0.409225\pi\)
\(548\) 0 0
\(549\) 17.0503i 0.727691i
\(550\) 0 0
\(551\) −4.46758 −0.190325
\(552\) 0 0
\(553\) 2.03100i 0.0863667i
\(554\) 0 0
\(555\) 7.57251 6.31147i 0.321435 0.267907i
\(556\) 0 0
\(557\) −22.6673 −0.960446 −0.480223 0.877146i \(-0.659444\pi\)
−0.480223 + 0.877146i \(0.659444\pi\)
\(558\) 0 0
\(559\) −1.33288 −0.0563749
\(560\) 0 0
\(561\) 31.1762 1.31626
\(562\) 0 0
\(563\) 40.9106i 1.72418i 0.506759 + 0.862088i \(0.330844\pi\)
−0.506759 + 0.862088i \(0.669156\pi\)
\(564\) 0 0
\(565\) 12.8397 10.7015i 0.540171 0.450216i
\(566\) 0 0
\(567\) 7.14870i 0.300217i
\(568\) 0 0
\(569\) 37.6655i 1.57902i 0.613737 + 0.789510i \(0.289665\pi\)
−0.613737 + 0.789510i \(0.710335\pi\)
\(570\) 0 0
\(571\) −20.1278 −0.842322 −0.421161 0.906986i \(-0.638377\pi\)
−0.421161 + 0.906986i \(0.638377\pi\)
\(572\) 0 0
\(573\) 28.7634 1.20161
\(574\) 0 0
\(575\) −0.164414 23.9786i −0.00685655 0.999976i
\(576\) 0 0
\(577\) 18.0377i 0.750918i 0.926839 + 0.375459i \(0.122515\pi\)
−0.926839 + 0.375459i \(0.877485\pi\)
\(578\) 0 0
\(579\) 44.7982i 1.86175i
\(580\) 0 0
\(581\) 5.87914 0.243908
\(582\) 0 0
\(583\) −0.756763 −0.0313419
\(584\) 0 0
\(585\) −1.07531 1.29016i −0.0444586 0.0533416i
\(586\) 0 0
\(587\) −32.6326 −1.34689 −0.673446 0.739236i \(-0.735187\pi\)
−0.673446 + 0.739236i \(0.735187\pi\)
\(588\) 0 0
\(589\) 12.5057i 0.515287i
\(590\) 0 0
\(591\) 33.1517i 1.36368i
\(592\) 0 0
\(593\) 25.4394i 1.04467i −0.852740 0.522336i \(-0.825061\pi\)
0.852740 0.522336i \(-0.174939\pi\)
\(594\) 0 0
\(595\) −4.53576 5.44202i −0.185948 0.223101i
\(596\) 0 0
\(597\) 15.0202 0.614734
\(598\) 0 0
\(599\) 9.29234i 0.379675i 0.981816 + 0.189837i \(0.0607961\pi\)
−0.981816 + 0.189837i \(0.939204\pi\)
\(600\) 0 0
\(601\) −28.2140 −1.15087 −0.575437 0.817846i \(-0.695168\pi\)
−0.575437 + 0.817846i \(0.695168\pi\)
\(602\) 0 0
\(603\) 4.04356i 0.164667i
\(604\) 0 0
\(605\) 4.74585 3.95553i 0.192946 0.160815i
\(606\) 0 0
\(607\) 4.57912 0.185861 0.0929303 0.995673i \(-0.470377\pi\)
0.0929303 + 0.995673i \(0.470377\pi\)
\(608\) 0 0
\(609\) 2.30619i 0.0934516i
\(610\) 0 0
\(611\) 1.45353i 0.0588036i
\(612\) 0 0
\(613\) −43.8068 −1.76934 −0.884671 0.466217i \(-0.845617\pi\)
−0.884671 + 0.466217i \(0.845617\pi\)
\(614\) 0 0
\(615\) −18.1406 + 15.1196i −0.731498 + 0.609682i
\(616\) 0 0
\(617\) 39.1574 1.57642 0.788208 0.615409i \(-0.211009\pi\)
0.788208 + 0.615409i \(0.211009\pi\)
\(618\) 0 0
\(619\) −41.9315 −1.68537 −0.842685 0.538407i \(-0.819026\pi\)
−0.842685 + 0.538407i \(0.819026\pi\)
\(620\) 0 0
\(621\) 11.8910 2.26241i 0.477170 0.0907872i
\(622\) 0 0
\(623\) 9.01749 0.361278
\(624\) 0 0
\(625\) −23.3770 8.86102i −0.935078 0.354441i
\(626\) 0 0
\(627\) −17.3443 −0.692665
\(628\) 0 0
\(629\) −9.86490 −0.393339
\(630\) 0 0
\(631\) 1.85343 0.0737840 0.0368920 0.999319i \(-0.488254\pi\)
0.0368920 + 0.999319i \(0.488254\pi\)
\(632\) 0 0
\(633\) 6.02799i 0.239591i
\(634\) 0 0
\(635\) −0.654237 + 0.545287i −0.0259626 + 0.0216391i
\(636\) 0 0
\(637\) 2.66792i 0.105707i
\(638\) 0 0
\(639\) 19.1381i 0.757090i
\(640\) 0 0
\(641\) 31.2121i 1.23280i 0.787432 + 0.616401i \(0.211410\pi\)
−0.787432 + 0.616401i \(0.788590\pi\)
\(642\) 0 0
\(643\) 37.3161i 1.47160i −0.677198 0.735801i \(-0.736806\pi\)
0.677198 0.735801i \(-0.263194\pi\)
\(644\) 0 0
\(645\) 10.3804 + 12.4544i 0.408727 + 0.490392i
\(646\) 0 0
\(647\) 20.8936 0.821412 0.410706 0.911768i \(-0.365282\pi\)
0.410706 + 0.911768i \(0.365282\pi\)
\(648\) 0 0
\(649\) 29.7773i 1.16886i
\(650\) 0 0
\(651\) 6.45550 0.253011
\(652\) 0 0
\(653\) 44.2036i 1.72982i 0.501927 + 0.864910i \(0.332625\pi\)
−0.501927 + 0.864910i \(0.667375\pi\)
\(654\) 0 0
\(655\) 21.3313 + 25.5933i 0.833482 + 1.00001i
\(656\) 0 0
\(657\) 8.21626i 0.320547i
\(658\) 0 0
\(659\) 40.9875 1.59665 0.798323 0.602229i \(-0.205721\pi\)
0.798323 + 0.602229i \(0.205721\pi\)
\(660\) 0 0
\(661\) 22.5000i 0.875150i 0.899182 + 0.437575i \(0.144163\pi\)
−0.899182 + 0.437575i \(0.855837\pi\)
\(662\) 0 0
\(663\) 4.39965i 0.170868i
\(664\) 0 0
\(665\) 2.52339 + 3.02757i 0.0978530 + 0.117404i
\(666\) 0 0
\(667\) −7.67387 + 1.46004i −0.297133 + 0.0565331i
\(668\) 0 0
\(669\) −43.1925 −1.66992
\(670\) 0 0
\(671\) 26.3875i 1.01868i
\(672\) 0 0
\(673\) 17.6567i 0.680615i 0.940314 + 0.340307i \(0.110531\pi\)
−0.940314 + 0.340307i \(0.889469\pi\)
\(674\) 0 0
\(675\) 2.27366 12.4131i 0.0875133 0.477781i
\(676\) 0 0
\(677\) −41.8438 −1.60819 −0.804093 0.594503i \(-0.797349\pi\)
−0.804093 + 0.594503i \(0.797349\pi\)
\(678\) 0 0
\(679\) 6.66168i 0.255652i
\(680\) 0 0
\(681\) 39.8673i 1.52772i
\(682\) 0 0
\(683\) 35.4804 1.35762 0.678809 0.734315i \(-0.262496\pi\)
0.678809 + 0.734315i \(0.262496\pi\)
\(684\) 0 0
\(685\) −14.3930 + 11.9961i −0.549928 + 0.458349i
\(686\) 0 0
\(687\) 59.8721i 2.28426i
\(688\) 0 0
\(689\) 0.106796i 0.00406861i
\(690\) 0 0
\(691\) 42.0356i 1.59911i 0.600592 + 0.799556i \(0.294932\pi\)
−0.600592 + 0.799556i \(0.705068\pi\)
\(692\) 0 0
\(693\) 3.42026i 0.129925i
\(694\) 0 0
\(695\) −4.83724 5.80373i −0.183487 0.220148i
\(696\) 0 0
\(697\) 23.6322 0.895132
\(698\) 0 0
\(699\) 9.38062i 0.354808i
\(700\) 0 0
\(701\) 41.2075i 1.55639i 0.628025 + 0.778193i \(0.283864\pi\)
−0.628025 + 0.778193i \(0.716136\pi\)
\(702\) 0 0
\(703\) 5.48817 0.206990
\(704\) 0 0
\(705\) −13.5818 + 11.3200i −0.511519 + 0.426336i
\(706\) 0 0
\(707\) 7.84476i 0.295032i
\(708\) 0 0
\(709\) 42.1883i 1.58442i −0.610251 0.792208i \(-0.708931\pi\)
0.610251 0.792208i \(-0.291069\pi\)
\(710\) 0 0
\(711\) −5.86107 −0.219807
\(712\) 0 0
\(713\) −4.08696 21.4807i −0.153058 0.804459i
\(714\) 0 0
\(715\) −1.66418 1.99668i −0.0622367 0.0746718i
\(716\) 0 0
\(717\) 40.8152i 1.52427i
\(718\) 0 0
\(719\) 34.9606i 1.30381i −0.758301 0.651905i \(-0.773970\pi\)
0.758301 0.651905i \(-0.226030\pi\)
\(720\) 0 0
\(721\) 7.26789 0.270670
\(722\) 0 0
\(723\) 7.95069i 0.295689i
\(724\) 0 0
\(725\) −1.46731 + 8.01081i −0.0544945 + 0.297514i
\(726\) 0 0
\(727\) 25.1715i 0.933560i −0.884373 0.466780i \(-0.845414\pi\)
0.884373 0.466780i \(-0.154586\pi\)
\(728\) 0 0
\(729\) −3.94700 −0.146185
\(730\) 0 0
\(731\) 16.2247i 0.600091i
\(732\) 0 0
\(733\) −9.42700 −0.348194 −0.174097 0.984728i \(-0.555701\pi\)
−0.174097 + 0.984728i \(0.555701\pi\)
\(734\) 0 0
\(735\) 24.9289 20.7775i 0.919517 0.766390i
\(736\) 0 0
\(737\) 6.25792i 0.230513i
\(738\) 0 0
\(739\) 43.5175i 1.60082i 0.599455 + 0.800409i \(0.295384\pi\)
−0.599455 + 0.800409i \(0.704616\pi\)
\(740\) 0 0
\(741\) 2.44767i 0.0899174i
\(742\) 0 0
\(743\) 35.2929i 1.29477i −0.762162 0.647386i \(-0.775862\pi\)
0.762162 0.647386i \(-0.224138\pi\)
\(744\) 0 0
\(745\) −25.3800 30.4510i −0.929852 1.11564i
\(746\) 0 0
\(747\) 16.9661i 0.620757i
\(748\) 0 0
\(749\) 9.97410 0.364446
\(750\) 0 0
\(751\) 23.7239 0.865699 0.432849 0.901466i \(-0.357508\pi\)
0.432849 + 0.901466i \(0.357508\pi\)
\(752\) 0 0
\(753\) −8.41642 −0.306712
\(754\) 0 0
\(755\) 24.2013 + 29.0368i 0.880776 + 1.05676i
\(756\) 0 0
\(757\) −5.88600 −0.213930 −0.106965 0.994263i \(-0.534113\pi\)
−0.106965 + 0.994263i \(0.534113\pi\)
\(758\) 0 0
\(759\) −29.7920 + 5.66827i −1.08138 + 0.205745i
\(760\) 0 0
\(761\) −9.12950 −0.330944 −0.165472 0.986214i \(-0.552915\pi\)
−0.165472 + 0.986214i \(0.552915\pi\)
\(762\) 0 0
\(763\) 8.76377 0.317270
\(764\) 0 0
\(765\) 15.7046 13.0894i 0.567803 0.473247i
\(766\) 0 0
\(767\) −4.20223 −0.151734
\(768\) 0 0
\(769\) 5.71892i 0.206230i −0.994669 0.103115i \(-0.967119\pi\)
0.994669 0.103115i \(-0.0328809\pi\)
\(770\) 0 0
\(771\) 2.73992i 0.0986758i
\(772\) 0 0
\(773\) −8.00711 −0.287996 −0.143998 0.989578i \(-0.545996\pi\)
−0.143998 + 0.989578i \(0.545996\pi\)
\(774\) 0 0
\(775\) −22.4239 4.10730i −0.805490 0.147538i
\(776\) 0 0
\(777\) 2.83303i 0.101634i
\(778\) 0 0
\(779\) −13.1473 −0.471053
\(780\) 0 0
\(781\) 29.6185i 1.05983i
\(782\) 0 0
\(783\) −4.11101 −0.146916
\(784\) 0 0
\(785\) −19.1556 + 15.9656i −0.683692 + 0.569837i
\(786\) 0 0
\(787\) 10.9386i 0.389917i 0.980811 + 0.194959i \(0.0624573\pi\)
−0.980811 + 0.194959i \(0.937543\pi\)
\(788\) 0 0
\(789\) 22.1568i 0.788801i
\(790\) 0 0
\(791\) 4.80358i 0.170796i
\(792\) 0 0
\(793\) −3.72387 −0.132238
\(794\) 0 0
\(795\) −0.997900 + 0.831720i −0.0353919 + 0.0294981i
\(796\) 0 0
\(797\) −23.0968 −0.818132 −0.409066 0.912505i \(-0.634145\pi\)
−0.409066 + 0.912505i \(0.634145\pi\)
\(798\) 0 0
\(799\) 17.6933 0.625944
\(800\) 0 0
\(801\) 26.0228i 0.919469i
\(802\) 0 0
\(803\) 12.7157i 0.448727i
\(804\) 0 0
\(805\) 5.32382 + 4.37573i 0.187640 + 0.154224i
\(806\) 0 0
\(807\) 4.83598 0.170235
\(808\) 0 0
\(809\) 16.5248 0.580980 0.290490 0.956878i \(-0.406182\pi\)
0.290490 + 0.956878i \(0.406182\pi\)
\(810\) 0 0
\(811\) 51.9334i 1.82363i 0.410604 + 0.911814i \(0.365318\pi\)
−0.410604 + 0.911814i \(0.634682\pi\)
\(812\) 0 0
\(813\) 9.38593i 0.329179i
\(814\) 0 0
\(815\) 4.30509 3.58817i 0.150801 0.125688i
\(816\) 0 0
\(817\) 9.02631i 0.315791i
\(818\) 0 0
\(819\) 0.482674 0.0168660
\(820\) 0 0
\(821\) 10.2235 0.356802 0.178401 0.983958i \(-0.442908\pi\)
0.178401 + 0.983958i \(0.442908\pi\)
\(822\) 0 0
\(823\) −13.5688 −0.472979 −0.236490 0.971634i \(-0.575997\pi\)
−0.236490 + 0.971634i \(0.575997\pi\)
\(824\) 0 0
\(825\) −5.69648 + 31.1001i −0.198326 + 1.08277i
\(826\) 0 0
\(827\) 44.4177i 1.54455i −0.635286 0.772277i \(-0.719118\pi\)
0.635286 0.772277i \(-0.280882\pi\)
\(828\) 0 0
\(829\) −23.7469 −0.824765 −0.412382 0.911011i \(-0.635303\pi\)
−0.412382 + 0.911011i \(0.635303\pi\)
\(830\) 0 0
\(831\) 19.5563i 0.678401i
\(832\) 0 0
\(833\) −32.4755 −1.12521
\(834\) 0 0
\(835\) 3.16051 2.63419i 0.109374 0.0911599i
\(836\) 0 0
\(837\) 11.5076i 0.397759i
\(838\) 0 0
\(839\) 5.77605 0.199412 0.0997058 0.995017i \(-0.468210\pi\)
0.0997058 + 0.995017i \(0.468210\pi\)
\(840\) 0 0
\(841\) −26.3470 −0.908516
\(842\) 0 0
\(843\) 18.9021i 0.651024i
\(844\) 0 0
\(845\) −22.0481 + 18.3764i −0.758476 + 0.632167i
\(846\) 0 0
\(847\) 1.77552i 0.0610075i
\(848\) 0 0
\(849\) 7.49548i 0.257244i
\(850\) 0 0
\(851\) 9.42691 1.79358i 0.323150 0.0614831i
\(852\) 0 0
\(853\) 29.8835i 1.02319i 0.859227 + 0.511595i \(0.170945\pi\)
−0.859227 + 0.511595i \(0.829055\pi\)
\(854\) 0 0
\(855\) −8.73701 + 7.28204i −0.298799 + 0.249040i
\(856\) 0 0
\(857\) 43.5873i 1.48891i −0.667670 0.744457i \(-0.732708\pi\)
0.667670 0.744457i \(-0.267292\pi\)
\(858\) 0 0
\(859\) 1.90651i 0.0650492i −0.999471 0.0325246i \(-0.989645\pi\)
0.999471 0.0325246i \(-0.0103547\pi\)
\(860\) 0 0
\(861\) 6.78674i 0.231292i
\(862\) 0 0
\(863\) −28.0711 −0.955551 −0.477775 0.878482i \(-0.658557\pi\)
−0.477775 + 0.878482i \(0.658557\pi\)
\(864\) 0 0
\(865\) −16.1212 19.3423i −0.548138 0.657658i
\(866\) 0 0
\(867\) −16.0994 −0.546765
\(868\) 0 0
\(869\) −9.07074 −0.307704
\(870\) 0 0
\(871\) 0.883132 0.0299238
\(872\) 0 0
\(873\) −19.2243 −0.650646
\(874\) 0 0
\(875\) 6.25751 3.53033i 0.211542 0.119347i
\(876\) 0 0
\(877\) 47.8183i 1.61471i 0.590066 + 0.807355i \(0.299102\pi\)
−0.590066 + 0.807355i \(0.700898\pi\)
\(878\) 0 0
\(879\) 63.6875 2.14813
\(880\) 0 0
\(881\) 13.1731i 0.443813i −0.975068 0.221906i \(-0.928772\pi\)
0.975068 0.221906i \(-0.0712279\pi\)
\(882\) 0 0
\(883\) 26.6396 0.896494 0.448247 0.893910i \(-0.352049\pi\)
0.448247 + 0.893910i \(0.352049\pi\)
\(884\) 0 0
\(885\) 32.7267 + 39.2656i 1.10010 + 1.31990i
\(886\) 0 0
\(887\) 15.8626 0.532615 0.266307 0.963888i \(-0.414196\pi\)
0.266307 + 0.963888i \(0.414196\pi\)
\(888\) 0 0
\(889\) 0.244763i 0.00820908i
\(890\) 0 0
\(891\) −31.9272 −1.06960
\(892\) 0 0
\(893\) −9.84336 −0.329395
\(894\) 0 0
\(895\) 0.878411 + 1.05392i 0.0293621 + 0.0352287i
\(896\) 0 0
\(897\) −0.799919 4.20431i −0.0267085 0.140378i
\(898\) 0 0
\(899\) 7.42641i 0.247684i
\(900\) 0 0
\(901\) 1.29999 0.0433089
\(902\) 0 0
\(903\) −4.65944 −0.155056
\(904\) 0 0
\(905\) 0.755316 + 0.906230i 0.0251076 + 0.0301241i
\(906\) 0 0
\(907\) 33.0630i 1.09784i −0.835875 0.548920i \(-0.815039\pi\)
0.835875 0.548920i \(-0.184961\pi\)
\(908\) 0 0
\(909\) −22.6385 −0.750871
\(910\) 0 0
\(911\) 56.1545 1.86048 0.930240 0.366951i \(-0.119598\pi\)
0.930240 + 0.366951i \(0.119598\pi\)
\(912\) 0 0
\(913\) 26.2571i 0.868984i
\(914\) 0 0
\(915\) 29.0012 + 34.7957i 0.958750 + 1.15031i
\(916\) 0 0
\(917\) −9.57496 −0.316193
\(918\) 0 0
\(919\) 34.7455 1.14615 0.573074 0.819503i \(-0.305751\pi\)
0.573074 + 0.819503i \(0.305751\pi\)
\(920\) 0 0
\(921\) −8.06929 −0.265892
\(922\) 0 0
\(923\) 4.17984 0.137581
\(924\) 0 0
\(925\) 1.80250 9.84082i 0.0592660 0.323564i
\(926\) 0 0
\(927\) 20.9737i 0.688868i
\(928\) 0 0
\(929\) 5.36655 0.176071 0.0880354 0.996117i \(-0.471941\pi\)
0.0880354 + 0.996117i \(0.471941\pi\)
\(930\) 0 0
\(931\) 18.0672 0.592128
\(932\) 0 0
\(933\) 18.6554i 0.610749i
\(934\) 0 0
\(935\) 24.3049 20.2574i 0.794855 0.662488i
\(936\) 0 0
\(937\) −38.4772 −1.25700 −0.628498 0.777811i \(-0.716330\pi\)
−0.628498 + 0.777811i \(0.716330\pi\)
\(938\) 0 0
\(939\) −40.9135 −1.33516
\(940\) 0 0
\(941\) 46.7397i 1.52367i 0.647770 + 0.761836i \(0.275702\pi\)
−0.647770 + 0.761836i \(0.724298\pi\)
\(942\) 0 0
\(943\) −22.5829 + 4.29667i −0.735401 + 0.139919i
\(944\) 0 0
\(945\) 2.32199 + 2.78594i 0.0755345 + 0.0906265i
\(946\) 0 0
\(947\) −50.6457 −1.64576 −0.822881 0.568213i \(-0.807635\pi\)
−0.822881 + 0.568213i \(0.807635\pi\)
\(948\) 0 0
\(949\) −1.79447 −0.0582508
\(950\) 0 0
\(951\) 1.48336i 0.0481012i
\(952\) 0 0
\(953\) −47.8691 −1.55063 −0.775316 0.631574i \(-0.782409\pi\)
−0.775316 + 0.631574i \(0.782409\pi\)
\(954\) 0 0
\(955\) 22.4239 18.6896i 0.725620 0.604783i
\(956\) 0 0
\(957\) 10.2998 0.332946
\(958\) 0 0
\(959\) 5.38470i 0.173881i
\(960\) 0 0
\(961\) 10.2120 0.329419
\(962\) 0 0
\(963\) 28.7834i 0.927532i
\(964\) 0 0
\(965\) −29.1086 34.9246i −0.937039 1.12426i
\(966\) 0 0
\(967\) 20.7095 0.665973 0.332987 0.942932i \(-0.391944\pi\)
0.332987 + 0.942932i \(0.391944\pi\)
\(968\) 0 0
\(969\) 29.7946 0.957139
\(970\) 0 0
\(971\) 47.4132 1.52156 0.760780 0.649009i \(-0.224816\pi\)
0.760780 + 0.649009i \(0.224816\pi\)
\(972\) 0 0
\(973\) 2.17129 0.0696083
\(974\) 0 0
\(975\) −4.38891 0.803900i −0.140558 0.0257454i
\(976\) 0 0
\(977\) −45.9036 −1.46859 −0.734293 0.678833i \(-0.762486\pi\)
−0.734293 + 0.678833i \(0.762486\pi\)
\(978\) 0 0
\(979\) 40.2735i 1.28715i
\(980\) 0 0
\(981\) 25.2906i 0.807466i
\(982\) 0 0
\(983\) 25.1206i 0.801222i 0.916248 + 0.400611i \(0.131202\pi\)
−0.916248 + 0.400611i \(0.868798\pi\)
\(984\) 0 0
\(985\) 21.5410 + 25.8450i 0.686354 + 0.823490i
\(986\) 0 0
\(987\) 5.08120i 0.161736i
\(988\) 0 0
\(989\) 2.94988 + 15.5043i 0.0938006 + 0.493008i
\(990\) 0 0
\(991\) 1.60466i 0.0509737i 0.999675 + 0.0254868i \(0.00811359\pi\)
−0.999675 + 0.0254868i \(0.991886\pi\)
\(992\) 0 0
\(993\) 25.0786i 0.795844i
\(994\) 0 0
\(995\) 11.7097 9.75968i 0.371222 0.309403i
\(996\) 0 0
\(997\) 23.0326i 0.729450i −0.931115 0.364725i \(-0.881163\pi\)
0.931115 0.364725i \(-0.118837\pi\)
\(998\) 0 0
\(999\) 5.05014 0.159779
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 1840.2.m.g.1839.10 yes 40
4.3 odd 2 inner 1840.2.m.g.1839.31 yes 40
5.4 even 2 inner 1840.2.m.g.1839.29 yes 40
20.19 odd 2 inner 1840.2.m.g.1839.12 yes 40
23.22 odd 2 inner 1840.2.m.g.1839.9 40
92.91 even 2 inner 1840.2.m.g.1839.32 yes 40
115.114 odd 2 inner 1840.2.m.g.1839.30 yes 40
460.459 even 2 inner 1840.2.m.g.1839.11 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.m.g.1839.9 40 23.22 odd 2 inner
1840.2.m.g.1839.10 yes 40 1.1 even 1 trivial
1840.2.m.g.1839.11 yes 40 460.459 even 2 inner
1840.2.m.g.1839.12 yes 40 20.19 odd 2 inner
1840.2.m.g.1839.29 yes 40 5.4 even 2 inner
1840.2.m.g.1839.30 yes 40 115.114 odd 2 inner
1840.2.m.g.1839.31 yes 40 4.3 odd 2 inner
1840.2.m.g.1839.32 yes 40 92.91 even 2 inner