Properties

Label 475.2.a.c.1.1
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 475.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} -4.00000 q^{11} +2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} -3.00000 q^{18} +1.00000 q^{19} -4.00000 q^{22} +6.00000 q^{23} +2.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +5.00000 q^{32} -4.00000 q^{34} +3.00000 q^{36} +10.0000 q^{37} +1.00000 q^{38} -10.0000 q^{41} -2.00000 q^{43} +4.00000 q^{44} +6.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -2.00000 q^{52} -10.0000 q^{53} +6.00000 q^{56} -6.00000 q^{58} +2.00000 q^{61} -4.00000 q^{62} +6.00000 q^{63} +7.00000 q^{64} -8.00000 q^{67} +4.00000 q^{68} +4.00000 q^{71} +9.00000 q^{72} -4.00000 q^{73} +10.0000 q^{74} -1.00000 q^{76} +8.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} -10.0000 q^{82} +18.0000 q^{83} -2.00000 q^{86} +12.0000 q^{88} -2.00000 q^{89} -4.00000 q^{91} -6.00000 q^{92} +6.00000 q^{94} -6.00000 q^{97} -3.00000 q^{98} +12.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −3.00000 −1.06066
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −3.00000 −0.707107
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 6.00000 0.755929
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 9.00000 1.06066
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −10.0000 −1.10432
\(83\) 18.0000 1.97576 0.987878 0.155230i \(-0.0496119\pi\)
0.987878 + 0.155230i \(0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 12.0000 1.27920
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −3.00000 −0.303046
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 6.00000 0.534522
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 12.0000 1.02899
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.00000 0.335673
\(143\) −8.00000 −0.668994
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 24.0000 1.95309 0.976546 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) −3.00000 −0.243332
\(153\) 12.0000 0.970143
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 9.00000 0.707107
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 2.00000 0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) −18.0000 −1.32698
\(185\) 0 0
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 12.0000 0.852803
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) −18.0000 −1.25109
\(208\) −2.00000 −0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) −16.0000 −1.04819 −0.524097 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 12.0000 0.762001
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) −6.00000 −0.377964
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) −12.0000 −0.741362
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −4.00000 −0.239904
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 20.0000 1.18056
\(288\) −15.0000 −0.883883
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −30.0000 −1.74371
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 24.0000 1.38104
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 12.0000 0.685994
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) −4.00000 −0.222566
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 0 0
\(328\) 30.0000 1.65647
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −18.0000 −0.987878
\(333\) −30.0000 −1.64399
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) −3.00000 −0.162221
\(343\) 20.0000 1.07990
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −20.0000 −1.06600
\(353\) −4.00000 −0.212899 −0.106449 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −18.0000 −0.946059
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) −6.00000 −0.312772
\(369\) 30.0000 1.56174
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −18.0000 −0.928279
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) 6.00000 0.304997
\(388\) 6.00000 0.304604
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) −18.0000 −0.884652
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 20.0000 0.973585
\(423\) −18.0000 −0.875190
\(424\) 30.0000 1.45693
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −8.00000 −0.380521
\(443\) 2.00000 0.0950229 0.0475114 0.998871i \(-0.484871\pi\)
0.0475114 + 0.998871i \(0.484871\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) −14.0000 −0.661438
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −16.0000 −0.741186
\(467\) 38.0000 1.75843 0.879215 0.476425i \(-0.158068\pi\)
0.879215 + 0.476425i \(0.158068\pi\)
\(468\) 6.00000 0.277350
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 30.0000 1.37361
\(478\) −16.0000 −0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −10.0000 −0.445878 −0.222939 0.974832i \(-0.571565\pi\)
−0.222939 + 0.974832i \(0.571565\pi\)
\(504\) −18.0000 −0.801784
\(505\) 0 0
\(506\) −24.0000 −1.06693
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) −20.0000 −0.878750
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 18.0000 0.787839
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) 0 0
\(536\) 24.0000 1.03664
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) −20.0000 −0.857493
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 12.0000 0.512615
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 16.0000 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) 12.0000 0.508001
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −26.0000 −1.09674
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) −18.0000 −0.755929
\(568\) −12.0000 −0.503509
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 20.0000 0.834784
\(575\) 0 0
\(576\) −21.0000 −0.875000
\(577\) −24.0000 −0.999133 −0.499567 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 12.0000 0.490716
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 4.00000 0.163028
\(603\) 24.0000 0.977356
\(604\) −24.0000 −0.976546
\(605\) 0 0
\(606\) 0 0
\(607\) 44.0000 1.78590 0.892952 0.450151i \(-0.148630\pi\)
0.892952 + 0.450151i \(0.148630\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) −12.0000 −0.485071
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −24.0000 −0.966988
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) 0 0
\(626\) 24.0000 0.959233
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −12.0000 −0.477334
\(633\) 0 0
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 24.0000 0.950169
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) −38.0000 −1.49857 −0.749287 0.662246i \(-0.769604\pi\)
−0.749287 + 0.662246i \(0.769604\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −30.0000 −1.17942 −0.589711 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) −27.0000 −1.06066
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −16.0000 −0.626128 −0.313064 0.949732i \(-0.601356\pi\)
−0.313064 + 0.949732i \(0.601356\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 12.0000 0.468165
\(658\) −12.0000 −0.467809
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −54.0000 −2.09561
\(665\) 0 0
\(666\) −30.0000 −1.16248
\(667\) −36.0000 −1.39393
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 0 0
\(682\) 16.0000 0.612672
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 3.00000 0.114708
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −6.00000 −0.228086
\(693\) −24.0000 −0.911685
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) −4.00000 −0.150542
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 6.00000 0.224860
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) 0 0
\(727\) −2.00000 −0.0741759 −0.0370879 0.999312i \(-0.511808\pi\)
−0.0370879 + 0.999312i \(0.511808\pi\)
\(728\) 12.0000 0.444750
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) 30.0000 1.10581
\(737\) 32.0000 1.17874
\(738\) 30.0000 1.10432
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 20.0000 0.734223
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6.00000 0.219676
\(747\) −54.0000 −1.97576
\(748\) −16.0000 −0.585018
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 4.00000 0.145382 0.0726912 0.997354i \(-0.476841\pi\)
0.0726912 + 0.997354i \(0.476841\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −12.0000 −0.434429
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −26.0000 −0.935760
\(773\) −46.0000 −1.65451 −0.827253 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) −24.0000 −0.858238
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) −36.0000 −1.27920
\(793\) 4.00000 0.142044
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 18.0000 0.635602
\(803\) 16.0000 0.564628
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 18.0000 0.633238
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −24.0000 −0.842754 −0.421377 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) −14.0000 −0.489499
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 42.0000 1.46403 0.732014 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) 48.0000 1.67216
\(825\) 0 0
\(826\) 0 0
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 18.0000 0.625543
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 14.0000 0.485363
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −30.0000 −1.03387
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) −18.0000 −0.618853
\(847\) −10.0000 −0.343604
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) −18.0000 −0.609557
\(873\) 18.0000 0.609208
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 9.00000 0.303046
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 2.00000 0.0671913
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −36.0000 −1.20605
\(892\) 16.0000 0.535720
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 0 0
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 40.0000 1.33259
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 20.0000 0.663723
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −72.0000 −2.38285
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) −2.00000 −0.0657241
\(927\) 48.0000 1.57653
\(928\) −30.0000 −0.984798
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 16.0000 0.524097
\(933\) 0 0
\(934\) 38.0000 1.24340
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) 20.0000 0.653372 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) −60.0000 −1.95387
\(944\) 0 0
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) −24.0000 −0.777844
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 30.0000 0.971286
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 20.0000 0.644826
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) −15.0000 −0.482118
\(969\) 0 0
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 14.0000 0.447900 0.223950 0.974601i \(-0.428105\pi\)
0.223950 + 0.974601i \(0.428105\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) −20.0000 −0.638226
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −20.0000 −0.635001
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) −32.0000 −1.01345 −0.506725 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) 4.00000 0.126618
\(999\) 0 0
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 475.2.a.c.1.1 1
3.2 odd 2 4275.2.a.e.1.1 1
4.3 odd 2 7600.2.a.l.1.1 1
5.2 odd 4 95.2.b.a.39.2 yes 2
5.3 odd 4 95.2.b.a.39.1 2
5.4 even 2 475.2.a.a.1.1 1
15.2 even 4 855.2.c.b.514.1 2
15.8 even 4 855.2.c.b.514.2 2
15.14 odd 2 4275.2.a.p.1.1 1
19.18 odd 2 9025.2.a.c.1.1 1
20.3 even 4 1520.2.d.b.609.1 2
20.7 even 4 1520.2.d.b.609.2 2
20.19 odd 2 7600.2.a.i.1.1 1
95.18 even 4 1805.2.b.c.1084.2 2
95.37 even 4 1805.2.b.c.1084.1 2
95.94 odd 2 9025.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.a.39.1 2 5.3 odd 4
95.2.b.a.39.2 yes 2 5.2 odd 4
475.2.a.a.1.1 1 5.4 even 2
475.2.a.c.1.1 1 1.1 even 1 trivial
855.2.c.b.514.1 2 15.2 even 4
855.2.c.b.514.2 2 15.8 even 4
1520.2.d.b.609.1 2 20.3 even 4
1520.2.d.b.609.2 2 20.7 even 4
1805.2.b.c.1084.1 2 95.37 even 4
1805.2.b.c.1084.2 2 95.18 even 4
4275.2.a.e.1.1 1 3.2 odd 2
4275.2.a.p.1.1 1 15.14 odd 2
7600.2.a.i.1.1 1 20.19 odd 2
7600.2.a.l.1.1 1 4.3 odd 2
9025.2.a.c.1.1 1 19.18 odd 2
9025.2.a.h.1.1 1 95.94 odd 2