Properties

Label 1575.2.d.e.1324.4
Level $1575$
Weight $2$
Character 1575.1324
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1324.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1324
Dual form 1575.2.d.e.1324.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155i q^{2} -4.56155 q^{4} -1.00000i q^{7} -6.56155i q^{8} +O(q^{10})\) \(q+2.56155i q^{2} -4.56155 q^{4} -1.00000i q^{7} -6.56155i q^{8} +1.56155 q^{11} -0.438447i q^{13} +2.56155 q^{14} +7.68466 q^{16} +0.438447i q^{17} +7.12311 q^{19} +4.00000i q^{22} +3.12311i q^{23} +1.12311 q^{26} +4.56155i q^{28} +6.68466 q^{29} +6.56155i q^{32} -1.12311 q^{34} +6.00000i q^{37} +18.2462i q^{38} -5.12311 q^{41} -0.876894i q^{43} -7.12311 q^{44} -8.00000 q^{46} +8.68466i q^{47} -1.00000 q^{49} +2.00000i q^{52} -5.12311i q^{53} -6.56155 q^{56} +17.1231i q^{58} -4.00000 q^{59} +15.3693 q^{61} -1.43845 q^{64} +10.2462i q^{67} -2.00000i q^{68} -8.00000 q^{71} +12.2462i q^{73} -15.3693 q^{74} -32.4924 q^{76} -1.56155i q^{77} +2.43845 q^{79} -13.1231i q^{82} +4.00000i q^{83} +2.24621 q^{86} -10.2462i q^{88} -1.12311 q^{89} -0.438447 q^{91} -14.2462i q^{92} -22.2462 q^{94} +5.80776i q^{97} -2.56155i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{4} + O(q^{10}) \) \( 4q - 10q^{4} - 2q^{11} + 2q^{14} + 6q^{16} + 12q^{19} - 12q^{26} + 2q^{29} + 12q^{34} - 4q^{41} - 12q^{44} - 32q^{46} - 4q^{49} - 18q^{56} - 16q^{59} + 12q^{61} - 14q^{64} - 32q^{71} - 12q^{74} - 64q^{76} + 18q^{79} - 24q^{86} + 12q^{89} - 10q^{91} - 56q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155i 1.81129i 0.424035 + 0.905646i \(0.360613\pi\)
−0.424035 + 0.905646i \(0.639387\pi\)
\(3\) 0 0
\(4\) −4.56155 −2.28078
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) − 6.56155i − 2.31986i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) − 0.438447i − 0.121603i −0.998150 0.0608017i \(-0.980634\pi\)
0.998150 0.0608017i \(-0.0193657\pi\)
\(14\) 2.56155 0.684604
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 0.438447i 0.106339i 0.998586 + 0.0531695i \(0.0169324\pi\)
−0.998586 + 0.0531695i \(0.983068\pi\)
\(18\) 0 0
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 3.12311i 0.651213i 0.945505 + 0.325606i \(0.105568\pi\)
−0.945505 + 0.325606i \(0.894432\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.12311 0.220259
\(27\) 0 0
\(28\) 4.56155i 0.862052i
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 6.56155i 1.15993i
\(33\) 0 0
\(34\) −1.12311 −0.192611
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 18.2462i 2.95993i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.12311 −0.800095 −0.400047 0.916494i \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 0 0
\(43\) − 0.876894i − 0.133725i −0.997762 0.0668626i \(-0.978701\pi\)
0.997762 0.0668626i \(-0.0212989\pi\)
\(44\) −7.12311 −1.07385
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 8.68466i 1.26679i 0.773830 + 0.633394i \(0.218339\pi\)
−0.773830 + 0.633394i \(0.781661\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) − 5.12311i − 0.703713i −0.936054 0.351856i \(-0.885551\pi\)
0.936054 0.351856i \(-0.114449\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.56155 −0.876824
\(57\) 0 0
\(58\) 17.1231i 2.24837i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 15.3693 1.96784 0.983920 0.178611i \(-0.0571605\pi\)
0.983920 + 0.178611i \(0.0571605\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.43845 −0.179806
\(65\) 0 0
\(66\) 0 0
\(67\) 10.2462i 1.25177i 0.779914 + 0.625887i \(0.215263\pi\)
−0.779914 + 0.625887i \(0.784737\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 12.2462i 1.43331i 0.697428 + 0.716655i \(0.254328\pi\)
−0.697428 + 0.716655i \(0.745672\pi\)
\(74\) −15.3693 −1.78665
\(75\) 0 0
\(76\) −32.4924 −3.72714
\(77\) − 1.56155i − 0.177955i
\(78\) 0 0
\(79\) 2.43845 0.274347 0.137173 0.990547i \(-0.456198\pi\)
0.137173 + 0.990547i \(0.456198\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 13.1231i − 1.44920i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.24621 0.242215
\(87\) 0 0
\(88\) − 10.2462i − 1.09225i
\(89\) −1.12311 −0.119049 −0.0595245 0.998227i \(-0.518958\pi\)
−0.0595245 + 0.998227i \(0.518958\pi\)
\(90\) 0 0
\(91\) −0.438447 −0.0459618
\(92\) − 14.2462i − 1.48527i
\(93\) 0 0
\(94\) −22.2462 −2.29452
\(95\) 0 0
\(96\) 0 0
\(97\) 5.80776i 0.589689i 0.955545 + 0.294845i \(0.0952679\pi\)
−0.955545 + 0.294845i \(0.904732\pi\)
\(98\) − 2.56155i − 0.258756i
\(99\) 0 0
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) − 5.56155i − 0.547996i −0.961730 0.273998i \(-0.911654\pi\)
0.961730 0.273998i \(-0.0883462\pi\)
\(104\) −2.87689 −0.282103
\(105\) 0 0
\(106\) 13.1231 1.27463
\(107\) − 13.3693i − 1.29246i −0.763142 0.646230i \(-0.776345\pi\)
0.763142 0.646230i \(-0.223655\pi\)
\(108\) 0 0
\(109\) −5.31534 −0.509117 −0.254559 0.967057i \(-0.581930\pi\)
−0.254559 + 0.967057i \(0.581930\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 7.68466i − 0.726132i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −30.4924 −2.83115
\(117\) 0 0
\(118\) − 10.2462i − 0.943240i
\(119\) 0.438447 0.0401924
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 39.3693i 3.56433i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 6.24621i − 0.554262i −0.960832 0.277131i \(-0.910616\pi\)
0.960832 0.277131i \(-0.0893835\pi\)
\(128\) 9.43845i 0.834249i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.876894 0.0766146 0.0383073 0.999266i \(-0.487803\pi\)
0.0383073 + 0.999266i \(0.487803\pi\)
\(132\) 0 0
\(133\) − 7.12311i − 0.617652i
\(134\) −26.2462 −2.26733
\(135\) 0 0
\(136\) 2.87689 0.246692
\(137\) 17.1231i 1.46293i 0.681881 + 0.731463i \(0.261162\pi\)
−0.681881 + 0.731463i \(0.738838\pi\)
\(138\) 0 0
\(139\) 15.1231 1.28273 0.641363 0.767238i \(-0.278369\pi\)
0.641363 + 0.767238i \(0.278369\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 20.4924i − 1.71969i
\(143\) − 0.684658i − 0.0572540i
\(144\) 0 0
\(145\) 0 0
\(146\) −31.3693 −2.59614
\(147\) 0 0
\(148\) − 27.3693i − 2.24974i
\(149\) 12.2462 1.00325 0.501624 0.865086i \(-0.332736\pi\)
0.501624 + 0.865086i \(0.332736\pi\)
\(150\) 0 0
\(151\) −6.93087 −0.564026 −0.282013 0.959411i \(-0.591002\pi\)
−0.282013 + 0.959411i \(0.591002\pi\)
\(152\) − 46.7386i − 3.79100i
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 20.2462i 1.61582i 0.589303 + 0.807912i \(0.299402\pi\)
−0.589303 + 0.807912i \(0.700598\pi\)
\(158\) 6.24621i 0.496922i
\(159\) 0 0
\(160\) 0 0
\(161\) 3.12311 0.246135
\(162\) 0 0
\(163\) 7.12311i 0.557925i 0.960302 + 0.278962i \(0.0899905\pi\)
−0.960302 + 0.278962i \(0.910010\pi\)
\(164\) 23.3693 1.82484
\(165\) 0 0
\(166\) −10.2462 −0.795260
\(167\) 6.93087i 0.536327i 0.963373 + 0.268163i \(0.0864167\pi\)
−0.963373 + 0.268163i \(0.913583\pi\)
\(168\) 0 0
\(169\) 12.8078 0.985213
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000i 0.304997i
\(173\) − 4.43845i − 0.337449i −0.985663 0.168724i \(-0.946035\pi\)
0.985663 0.168724i \(-0.0539648\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) − 2.87689i − 0.215632i
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −17.6155 −1.30935 −0.654676 0.755910i \(-0.727195\pi\)
−0.654676 + 0.755910i \(0.727195\pi\)
\(182\) − 1.12311i − 0.0832501i
\(183\) 0 0
\(184\) 20.4924 1.51072
\(185\) 0 0
\(186\) 0 0
\(187\) 0.684658i 0.0500672i
\(188\) − 39.6155i − 2.88926i
\(189\) 0 0
\(190\) 0 0
\(191\) 13.5616 0.981280 0.490640 0.871363i \(-0.336763\pi\)
0.490640 + 0.871363i \(0.336763\pi\)
\(192\) 0 0
\(193\) − 19.3693i − 1.39423i −0.716957 0.697117i \(-0.754466\pi\)
0.716957 0.697117i \(-0.245534\pi\)
\(194\) −14.8769 −1.06810
\(195\) 0 0
\(196\) 4.56155 0.325825
\(197\) − 1.12311i − 0.0800180i −0.999199 0.0400090i \(-0.987261\pi\)
0.999199 0.0400090i \(-0.0127387\pi\)
\(198\) 0 0
\(199\) 1.75379 0.124323 0.0621614 0.998066i \(-0.480201\pi\)
0.0621614 + 0.998066i \(0.480201\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 41.6155i 2.92806i
\(203\) − 6.68466i − 0.469171i
\(204\) 0 0
\(205\) 0 0
\(206\) 14.2462 0.992581
\(207\) 0 0
\(208\) − 3.36932i − 0.233620i
\(209\) 11.1231 0.769401
\(210\) 0 0
\(211\) 14.0540 0.967516 0.483758 0.875202i \(-0.339272\pi\)
0.483758 + 0.875202i \(0.339272\pi\)
\(212\) 23.3693i 1.60501i
\(213\) 0 0
\(214\) 34.2462 2.34102
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) − 13.6155i − 0.922160i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.192236 0.0129312
\(222\) 0 0
\(223\) 2.43845i 0.163291i 0.996661 + 0.0816453i \(0.0260175\pi\)
−0.996661 + 0.0816453i \(0.973983\pi\)
\(224\) 6.56155 0.438412
\(225\) 0 0
\(226\) 35.8617 2.38549
\(227\) − 11.3153i − 0.751026i −0.926817 0.375513i \(-0.877467\pi\)
0.926817 0.375513i \(-0.122533\pi\)
\(228\) 0 0
\(229\) −10.8769 −0.718765 −0.359383 0.933190i \(-0.617013\pi\)
−0.359383 + 0.933190i \(0.617013\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 43.8617i − 2.87966i
\(233\) 5.12311i 0.335626i 0.985819 + 0.167813i \(0.0536704\pi\)
−0.985819 + 0.167813i \(0.946330\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 18.2462 1.18773
\(237\) 0 0
\(238\) 1.12311i 0.0728001i
\(239\) 19.8078 1.28126 0.640629 0.767851i \(-0.278674\pi\)
0.640629 + 0.767851i \(0.278674\pi\)
\(240\) 0 0
\(241\) −4.24621 −0.273523 −0.136761 0.990604i \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(242\) − 21.9309i − 1.40977i
\(243\) 0 0
\(244\) −70.1080 −4.48820
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.12311i − 0.198718i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.87689 0.560305 0.280152 0.959956i \(-0.409615\pi\)
0.280152 + 0.959956i \(0.409615\pi\)
\(252\) 0 0
\(253\) 4.87689i 0.306608i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) 10.4924i 0.654499i 0.944938 + 0.327250i \(0.106122\pi\)
−0.944938 + 0.327250i \(0.893878\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) 2.24621i 0.138771i
\(263\) − 12.8769i − 0.794023i −0.917814 0.397012i \(-0.870047\pi\)
0.917814 0.397012i \(-0.129953\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 18.2462 1.11875
\(267\) 0 0
\(268\) − 46.7386i − 2.85502i
\(269\) −20.7386 −1.26446 −0.632228 0.774782i \(-0.717860\pi\)
−0.632228 + 0.774782i \(0.717860\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 3.36932i 0.204295i
\(273\) 0 0
\(274\) −43.8617 −2.64978
\(275\) 0 0
\(276\) 0 0
\(277\) − 0.246211i − 0.0147934i −0.999973 0.00739670i \(-0.997646\pi\)
0.999973 0.00739670i \(-0.00235446\pi\)
\(278\) 38.7386i 2.32339i
\(279\) 0 0
\(280\) 0 0
\(281\) −12.4384 −0.742016 −0.371008 0.928630i \(-0.620988\pi\)
−0.371008 + 0.928630i \(0.620988\pi\)
\(282\) 0 0
\(283\) 11.3153i 0.672627i 0.941750 + 0.336314i \(0.109180\pi\)
−0.941750 + 0.336314i \(0.890820\pi\)
\(284\) 36.4924 2.16543
\(285\) 0 0
\(286\) 1.75379 0.103704
\(287\) 5.12311i 0.302407i
\(288\) 0 0
\(289\) 16.8078 0.988692
\(290\) 0 0
\(291\) 0 0
\(292\) − 55.8617i − 3.26906i
\(293\) − 2.68466i − 0.156839i −0.996920 0.0784197i \(-0.975013\pi\)
0.996920 0.0784197i \(-0.0249874\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 39.3693 2.28830
\(297\) 0 0
\(298\) 31.3693i 1.81718i
\(299\) 1.36932 0.0791896
\(300\) 0 0
\(301\) −0.876894 −0.0505434
\(302\) − 17.7538i − 1.02162i
\(303\) 0 0
\(304\) 54.7386 3.13948
\(305\) 0 0
\(306\) 0 0
\(307\) − 19.3153i − 1.10238i −0.834378 0.551192i \(-0.814173\pi\)
0.834378 0.551192i \(-0.185827\pi\)
\(308\) 7.12311i 0.405877i
\(309\) 0 0
\(310\) 0 0
\(311\) −31.6155 −1.79275 −0.896376 0.443294i \(-0.853810\pi\)
−0.896376 + 0.443294i \(0.853810\pi\)
\(312\) 0 0
\(313\) 22.3002i 1.26048i 0.776400 + 0.630241i \(0.217044\pi\)
−0.776400 + 0.630241i \(0.782956\pi\)
\(314\) −51.8617 −2.92673
\(315\) 0 0
\(316\) −11.1231 −0.625724
\(317\) − 10.4924i − 0.589313i −0.955603 0.294657i \(-0.904795\pi\)
0.955603 0.294657i \(-0.0952053\pi\)
\(318\) 0 0
\(319\) 10.4384 0.584441
\(320\) 0 0
\(321\) 0 0
\(322\) 8.00000i 0.445823i
\(323\) 3.12311i 0.173774i
\(324\) 0 0
\(325\) 0 0
\(326\) −18.2462 −1.01056
\(327\) 0 0
\(328\) 33.6155i 1.85611i
\(329\) 8.68466 0.478801
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) − 18.2462i − 1.00139i
\(333\) 0 0
\(334\) −17.7538 −0.971444
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.50758i − 0.0821230i −0.999157 0.0410615i \(-0.986926\pi\)
0.999157 0.0410615i \(-0.0130740\pi\)
\(338\) 32.8078i 1.78451i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −5.75379 −0.310223
\(345\) 0 0
\(346\) 11.3693 0.611218
\(347\) − 7.12311i − 0.382388i −0.981552 0.191194i \(-0.938764\pi\)
0.981552 0.191194i \(-0.0612360\pi\)
\(348\) 0 0
\(349\) −10.4924 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.2462i 0.546125i
\(353\) 5.80776i 0.309116i 0.987984 + 0.154558i \(0.0493954\pi\)
−0.987984 + 0.154558i \(0.950605\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.12311 0.271524
\(357\) 0 0
\(358\) 51.2311i 2.70765i
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) − 45.1231i − 2.37162i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.68466i − 0.453335i −0.973972 0.226668i \(-0.927217\pi\)
0.973972 0.226668i \(-0.0727831\pi\)
\(368\) 24.0000i 1.25109i
\(369\) 0 0
\(370\) 0 0
\(371\) −5.12311 −0.265978
\(372\) 0 0
\(373\) − 4.63068i − 0.239768i −0.992788 0.119884i \(-0.961748\pi\)
0.992788 0.119884i \(-0.0382522\pi\)
\(374\) −1.75379 −0.0906863
\(375\) 0 0
\(376\) 56.9848 2.93877
\(377\) − 2.93087i − 0.150947i
\(378\) 0 0
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 34.7386i 1.77738i
\(383\) 6.24621i 0.319166i 0.987184 + 0.159583i \(0.0510150\pi\)
−0.987184 + 0.159583i \(0.948985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 49.6155 2.52536
\(387\) 0 0
\(388\) − 26.4924i − 1.34495i
\(389\) −24.9309 −1.26405 −0.632023 0.774950i \(-0.717775\pi\)
−0.632023 + 0.774950i \(0.717775\pi\)
\(390\) 0 0
\(391\) −1.36932 −0.0692493
\(392\) 6.56155i 0.331408i
\(393\) 0 0
\(394\) 2.87689 0.144936
\(395\) 0 0
\(396\) 0 0
\(397\) 27.5616i 1.38327i 0.722245 + 0.691637i \(0.243110\pi\)
−0.722245 + 0.691637i \(0.756890\pi\)
\(398\) 4.49242i 0.225185i
\(399\) 0 0
\(400\) 0 0
\(401\) −31.5616 −1.57611 −0.788054 0.615606i \(-0.788911\pi\)
−0.788054 + 0.615606i \(0.788911\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −74.1080 −3.68701
\(405\) 0 0
\(406\) 17.1231 0.849805
\(407\) 9.36932i 0.464420i
\(408\) 0 0
\(409\) −6.49242 −0.321030 −0.160515 0.987033i \(-0.551315\pi\)
−0.160515 + 0.987033i \(0.551315\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 25.3693i 1.24986i
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.87689 0.141051
\(417\) 0 0
\(418\) 28.4924i 1.39361i
\(419\) 26.2462 1.28221 0.641106 0.767453i \(-0.278476\pi\)
0.641106 + 0.767453i \(0.278476\pi\)
\(420\) 0 0
\(421\) −2.68466 −0.130842 −0.0654211 0.997858i \(-0.520839\pi\)
−0.0654211 + 0.997858i \(0.520839\pi\)
\(422\) 36.0000i 1.75245i
\(423\) 0 0
\(424\) −33.6155 −1.63251
\(425\) 0 0
\(426\) 0 0
\(427\) − 15.3693i − 0.743773i
\(428\) 60.9848i 2.94781i
\(429\) 0 0
\(430\) 0 0
\(431\) 19.8078 0.954106 0.477053 0.878874i \(-0.341705\pi\)
0.477053 + 0.878874i \(0.341705\pi\)
\(432\) 0 0
\(433\) − 8.24621i − 0.396288i −0.980173 0.198144i \(-0.936509\pi\)
0.980173 0.198144i \(-0.0634913\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 24.2462 1.16118
\(437\) 22.2462i 1.06418i
\(438\) 0 0
\(439\) −9.36932 −0.447173 −0.223587 0.974684i \(-0.571777\pi\)
−0.223587 + 0.974684i \(0.571777\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.492423i 0.0234221i
\(443\) − 2.63068i − 0.124988i −0.998045 0.0624938i \(-0.980095\pi\)
0.998045 0.0624938i \(-0.0199054\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −6.24621 −0.295767
\(447\) 0 0
\(448\) 1.43845i 0.0679602i
\(449\) −1.80776 −0.0853137 −0.0426568 0.999090i \(-0.513582\pi\)
−0.0426568 + 0.999090i \(0.513582\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 63.8617i 3.00380i
\(453\) 0 0
\(454\) 28.9848 1.36033
\(455\) 0 0
\(456\) 0 0
\(457\) − 17.1231i − 0.800985i −0.916300 0.400493i \(-0.868839\pi\)
0.916300 0.400493i \(-0.131161\pi\)
\(458\) − 27.8617i − 1.30189i
\(459\) 0 0
\(460\) 0 0
\(461\) 13.1231 0.611204 0.305602 0.952159i \(-0.401142\pi\)
0.305602 + 0.952159i \(0.401142\pi\)
\(462\) 0 0
\(463\) − 12.4924i − 0.580572i −0.956940 0.290286i \(-0.906250\pi\)
0.956940 0.290286i \(-0.0937505\pi\)
\(464\) 51.3693 2.38476
\(465\) 0 0
\(466\) −13.1231 −0.607916
\(467\) − 22.4384i − 1.03833i −0.854675 0.519164i \(-0.826243\pi\)
0.854675 0.519164i \(-0.173757\pi\)
\(468\) 0 0
\(469\) 10.2462 0.473126
\(470\) 0 0
\(471\) 0 0
\(472\) 26.2462i 1.20808i
\(473\) − 1.36932i − 0.0629613i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) 50.7386i 2.32073i
\(479\) 4.87689 0.222831 0.111415 0.993774i \(-0.464462\pi\)
0.111415 + 0.993774i \(0.464462\pi\)
\(480\) 0 0
\(481\) 2.63068 0.119949
\(482\) − 10.8769i − 0.495429i
\(483\) 0 0
\(484\) 39.0540 1.77518
\(485\) 0 0
\(486\) 0 0
\(487\) − 3.12311i − 0.141521i −0.997493 0.0707607i \(-0.977457\pi\)
0.997493 0.0707607i \(-0.0225427\pi\)
\(488\) − 100.847i − 4.56511i
\(489\) 0 0
\(490\) 0 0
\(491\) 41.1771 1.85830 0.929148 0.369708i \(-0.120542\pi\)
0.929148 + 0.369708i \(0.120542\pi\)
\(492\) 0 0
\(493\) 2.93087i 0.132000i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) −41.1771 −1.84334 −0.921670 0.387976i \(-0.873174\pi\)
−0.921670 + 0.387976i \(0.873174\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 22.7386i 1.01487i
\(503\) 38.9309i 1.73584i 0.496703 + 0.867921i \(0.334544\pi\)
−0.496703 + 0.867921i \(0.665456\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.4924 −0.555356
\(507\) 0 0
\(508\) 28.4924i 1.26415i
\(509\) −11.7538 −0.520978 −0.260489 0.965477i \(-0.583884\pi\)
−0.260489 + 0.965477i \(0.583884\pi\)
\(510\) 0 0
\(511\) 12.2462 0.541740
\(512\) − 50.4233i − 2.22842i
\(513\) 0 0
\(514\) −26.8769 −1.18549
\(515\) 0 0
\(516\) 0 0
\(517\) 13.5616i 0.596436i
\(518\) 15.3693i 0.675289i
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) − 40.4924i − 1.77061i −0.465011 0.885305i \(-0.653950\pi\)
0.465011 0.885305i \(-0.346050\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 32.9848 1.43821
\(527\) 0 0
\(528\) 0 0
\(529\) 13.2462 0.575922
\(530\) 0 0
\(531\) 0 0
\(532\) 32.4924i 1.40873i
\(533\) 2.24621i 0.0972942i
\(534\) 0 0
\(535\) 0 0
\(536\) 67.2311 2.90394
\(537\) 0 0
\(538\) − 53.1231i − 2.29030i
\(539\) −1.56155 −0.0672608
\(540\) 0 0
\(541\) −37.8078 −1.62548 −0.812741 0.582625i \(-0.802026\pi\)
−0.812741 + 0.582625i \(0.802026\pi\)
\(542\) − 40.9848i − 1.76045i
\(543\) 0 0
\(544\) −2.87689 −0.123346
\(545\) 0 0
\(546\) 0 0
\(547\) − 2.24621i − 0.0960411i −0.998846 0.0480205i \(-0.984709\pi\)
0.998846 0.0480205i \(-0.0152913\pi\)
\(548\) − 78.1080i − 3.33661i
\(549\) 0 0
\(550\) 0 0
\(551\) 47.6155 2.02849
\(552\) 0 0
\(553\) − 2.43845i − 0.103693i
\(554\) 0.630683 0.0267952
\(555\) 0 0
\(556\) −68.9848 −2.92561
\(557\) 13.1231i 0.556044i 0.960575 + 0.278022i \(0.0896788\pi\)
−0.960575 + 0.278022i \(0.910321\pi\)
\(558\) 0 0
\(559\) −0.384472 −0.0162614
\(560\) 0 0
\(561\) 0 0
\(562\) − 31.8617i − 1.34401i
\(563\) − 28.0000i − 1.18006i −0.807382 0.590030i \(-0.799116\pi\)
0.807382 0.590030i \(-0.200884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −28.9848 −1.21832
\(567\) 0 0
\(568\) 52.4924i 2.20253i
\(569\) −30.9848 −1.29895 −0.649476 0.760382i \(-0.725012\pi\)
−0.649476 + 0.760382i \(0.725012\pi\)
\(570\) 0 0
\(571\) 40.4924 1.69456 0.847278 0.531150i \(-0.178240\pi\)
0.847278 + 0.531150i \(0.178240\pi\)
\(572\) 3.12311i 0.130584i
\(573\) 0 0
\(574\) −13.1231 −0.547748
\(575\) 0 0
\(576\) 0 0
\(577\) − 24.0540i − 1.00138i −0.865627 0.500690i \(-0.833080\pi\)
0.865627 0.500690i \(-0.166920\pi\)
\(578\) 43.0540i 1.79081i
\(579\) 0 0
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) − 8.00000i − 0.331326i
\(584\) 80.3542 3.32508
\(585\) 0 0
\(586\) 6.87689 0.284082
\(587\) 26.2462i 1.08330i 0.840605 + 0.541649i \(0.182200\pi\)
−0.840605 + 0.541649i \(0.817800\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 46.1080i 1.89503i
\(593\) − 27.5616i − 1.13182i −0.824468 0.565909i \(-0.808525\pi\)
0.824468 0.565909i \(-0.191475\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −55.8617 −2.28819
\(597\) 0 0
\(598\) 3.50758i 0.143436i
\(599\) −11.8078 −0.482452 −0.241226 0.970469i \(-0.577550\pi\)
−0.241226 + 0.970469i \(0.577550\pi\)
\(600\) 0 0
\(601\) 6.49242 0.264831 0.132416 0.991194i \(-0.457727\pi\)
0.132416 + 0.991194i \(0.457727\pi\)
\(602\) − 2.24621i − 0.0915487i
\(603\) 0 0
\(604\) 31.6155 1.28642
\(605\) 0 0
\(606\) 0 0
\(607\) − 42.0540i − 1.70692i −0.521160 0.853459i \(-0.674500\pi\)
0.521160 0.853459i \(-0.325500\pi\)
\(608\) 46.7386i 1.89550i
\(609\) 0 0
\(610\) 0 0
\(611\) 3.80776 0.154046
\(612\) 0 0
\(613\) − 40.7386i − 1.64542i −0.568463 0.822709i \(-0.692462\pi\)
0.568463 0.822709i \(-0.307538\pi\)
\(614\) 49.4773 1.99674
\(615\) 0 0
\(616\) −10.2462 −0.412832
\(617\) − 32.2462i − 1.29818i −0.760710 0.649092i \(-0.775149\pi\)
0.760710 0.649092i \(-0.224851\pi\)
\(618\) 0 0
\(619\) −32.1080 −1.29053 −0.645264 0.763960i \(-0.723253\pi\)
−0.645264 + 0.763960i \(0.723253\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 80.9848i − 3.24720i
\(623\) 1.12311i 0.0449963i
\(624\) 0 0
\(625\) 0 0
\(626\) −57.1231 −2.28310
\(627\) 0 0
\(628\) − 92.3542i − 3.68533i
\(629\) −2.63068 −0.104892
\(630\) 0 0
\(631\) −11.8078 −0.470060 −0.235030 0.971988i \(-0.575519\pi\)
−0.235030 + 0.971988i \(0.575519\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) 0 0
\(634\) 26.8769 1.06742
\(635\) 0 0
\(636\) 0 0
\(637\) 0.438447i 0.0173719i
\(638\) 26.7386i 1.05859i
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) − 1.56155i − 0.0615816i −0.999526 0.0307908i \(-0.990197\pi\)
0.999526 0.0307908i \(-0.00980257\pi\)
\(644\) −14.2462 −0.561379
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) − 36.4924i − 1.43467i −0.696731 0.717333i \(-0.745363\pi\)
0.696731 0.717333i \(-0.254637\pi\)
\(648\) 0 0
\(649\) −6.24621 −0.245185
\(650\) 0 0
\(651\) 0 0
\(652\) − 32.4924i − 1.27250i
\(653\) − 33.2311i − 1.30043i −0.759750 0.650216i \(-0.774678\pi\)
0.759750 0.650216i \(-0.225322\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −39.3693 −1.53711
\(657\) 0 0
\(658\) 22.2462i 0.867248i
\(659\) 9.17708 0.357488 0.178744 0.983896i \(-0.442797\pi\)
0.178744 + 0.983896i \(0.442797\pi\)
\(660\) 0 0
\(661\) −5.12311 −0.199266 −0.0996329 0.995024i \(-0.531767\pi\)
−0.0996329 + 0.995024i \(0.531767\pi\)
\(662\) 30.7386i 1.19469i
\(663\) 0 0
\(664\) 26.2462 1.01855
\(665\) 0 0
\(666\) 0 0
\(667\) 20.8769i 0.808357i
\(668\) − 31.6155i − 1.22324i
\(669\) 0 0
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) − 31.8617i − 1.22818i −0.789236 0.614090i \(-0.789523\pi\)
0.789236 0.614090i \(-0.210477\pi\)
\(674\) 3.86174 0.148749
\(675\) 0 0
\(676\) −58.4233 −2.24705
\(677\) − 4.93087i − 0.189509i −0.995501 0.0947544i \(-0.969793\pi\)
0.995501 0.0947544i \(-0.0302066\pi\)
\(678\) 0 0
\(679\) 5.80776 0.222882
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 6.73863i − 0.257847i −0.991655 0.128923i \(-0.958848\pi\)
0.991655 0.128923i \(-0.0411521\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.56155 −0.0978005
\(687\) 0 0
\(688\) − 6.73863i − 0.256908i
\(689\) −2.24621 −0.0855738
\(690\) 0 0
\(691\) −24.4924 −0.931736 −0.465868 0.884854i \(-0.654258\pi\)
−0.465868 + 0.884854i \(0.654258\pi\)
\(692\) 20.2462i 0.769645i
\(693\) 0 0
\(694\) 18.2462 0.692617
\(695\) 0 0
\(696\) 0 0
\(697\) − 2.24621i − 0.0850813i
\(698\) − 26.8769i − 1.01731i
\(699\) 0 0
\(700\) 0 0
\(701\) −28.9309 −1.09270 −0.546352 0.837556i \(-0.683984\pi\)
−0.546352 + 0.837556i \(0.683984\pi\)
\(702\) 0 0
\(703\) 42.7386i 1.61192i
\(704\) −2.24621 −0.0846573
\(705\) 0 0
\(706\) −14.8769 −0.559899
\(707\) − 16.2462i − 0.611002i
\(708\) 0 0
\(709\) −27.1771 −1.02066 −0.510328 0.859980i \(-0.670476\pi\)
−0.510328 + 0.859980i \(0.670476\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7.36932i 0.276177i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −91.2311 −3.40946
\(717\) 0 0
\(718\) 20.4924i 0.764770i
\(719\) 8.38447 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(720\) 0 0
\(721\) −5.56155 −0.207123
\(722\) 81.3002i 3.02568i
\(723\) 0 0
\(724\) 80.3542 2.98634
\(725\) 0 0
\(726\) 0 0
\(727\) 52.4924i 1.94684i 0.229035 + 0.973418i \(0.426443\pi\)
−0.229035 + 0.973418i \(0.573557\pi\)
\(728\) 2.87689i 0.106625i
\(729\) 0 0
\(730\) 0 0
\(731\) 0.384472 0.0142202
\(732\) 0 0
\(733\) − 6.68466i − 0.246903i −0.992351 0.123452i \(-0.960604\pi\)
0.992351 0.123452i \(-0.0393964\pi\)
\(734\) 22.2462 0.821123
\(735\) 0 0
\(736\) −20.4924 −0.755361
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −34.9309 −1.28495 −0.642476 0.766305i \(-0.722093\pi\)
−0.642476 + 0.766305i \(0.722093\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 13.1231i − 0.481764i
\(743\) − 32.9848i − 1.21010i −0.796189 0.605048i \(-0.793154\pi\)
0.796189 0.605048i \(-0.206846\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 11.8617 0.434289
\(747\) 0 0
\(748\) − 3.12311i − 0.114192i
\(749\) −13.3693 −0.488504
\(750\) 0 0
\(751\) 17.0691 0.622861 0.311431 0.950269i \(-0.399192\pi\)
0.311431 + 0.950269i \(0.399192\pi\)
\(752\) 66.7386i 2.43371i
\(753\) 0 0
\(754\) 7.50758 0.273410
\(755\) 0 0
\(756\) 0 0
\(757\) 39.3693i 1.43090i 0.698663 + 0.715451i \(0.253779\pi\)
−0.698663 + 0.715451i \(0.746221\pi\)
\(758\) 42.2462i 1.53445i
\(759\) 0 0
\(760\) 0 0
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) 0 0
\(763\) 5.31534i 0.192428i
\(764\) −61.8617 −2.23808
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 1.75379i 0.0633256i
\(768\) 0 0
\(769\) 42.4924 1.53232 0.766158 0.642652i \(-0.222166\pi\)
0.766158 + 0.642652i \(0.222166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 88.3542i 3.17994i
\(773\) 36.9309i 1.32831i 0.747594 + 0.664156i \(0.231209\pi\)
−0.747594 + 0.664156i \(0.768791\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 38.1080 1.36800
\(777\) 0 0
\(778\) − 63.8617i − 2.28955i
\(779\) −36.4924 −1.30748
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) − 3.50758i − 0.125431i
\(783\) 0 0
\(784\) −7.68466 −0.274452
\(785\) 0 0
\(786\) 0 0
\(787\) − 49.1771i − 1.75297i −0.481426 0.876487i \(-0.659881\pi\)
0.481426 0.876487i \(-0.340119\pi\)
\(788\) 5.12311i 0.182503i
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) − 6.73863i − 0.239296i
\(794\) −70.6004 −2.50551
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) − 24.0540i − 0.852036i −0.904715 0.426018i \(-0.859916\pi\)
0.904715 0.426018i \(-0.140084\pi\)
\(798\) 0 0
\(799\) −3.80776 −0.134709
\(800\) 0 0
\(801\) 0 0
\(802\) − 80.8466i − 2.85479i
\(803\) 19.1231i 0.674840i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) − 106.600i − 3.75019i
\(809\) 16.5464 0.581740 0.290870 0.956763i \(-0.406055\pi\)
0.290870 + 0.956763i \(0.406055\pi\)
\(810\) 0 0
\(811\) 19.6155 0.688794 0.344397 0.938824i \(-0.388083\pi\)
0.344397 + 0.938824i \(0.388083\pi\)
\(812\) 30.4924i 1.07007i
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) − 6.24621i − 0.218527i
\(818\) − 16.6307i − 0.581478i
\(819\) 0 0
\(820\) 0 0
\(821\) 21.4233 0.747678 0.373839 0.927494i \(-0.378041\pi\)
0.373839 + 0.927494i \(0.378041\pi\)
\(822\) 0 0
\(823\) 36.4924i 1.27205i 0.771670 + 0.636023i \(0.219422\pi\)
−0.771670 + 0.636023i \(0.780578\pi\)
\(824\) −36.4924 −1.27127
\(825\) 0 0
\(826\) −10.2462 −0.356511
\(827\) 5.36932i 0.186709i 0.995633 + 0.0933547i \(0.0297591\pi\)
−0.995633 + 0.0933547i \(0.970241\pi\)
\(828\) 0 0
\(829\) −34.8769 −1.21132 −0.605662 0.795722i \(-0.707092\pi\)
−0.605662 + 0.795722i \(0.707092\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.630683i 0.0218650i
\(833\) − 0.438447i − 0.0151913i
\(834\) 0 0
\(835\) 0 0
\(836\) −50.7386 −1.75483
\(837\) 0 0
\(838\) 67.2311i 2.32246i
\(839\) −28.8769 −0.996941 −0.498471 0.866907i \(-0.666105\pi\)
−0.498471 + 0.866907i \(0.666105\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) − 6.87689i − 0.236993i
\(843\) 0 0
\(844\) −64.1080 −2.20669
\(845\) 0 0
\(846\) 0 0
\(847\) 8.56155i 0.294178i
\(848\) − 39.3693i − 1.35195i
\(849\) 0 0
\(850\) 0 0
\(851\) −18.7386 −0.642352
\(852\) 0 0
\(853\) 7.26137i 0.248624i 0.992243 + 0.124312i \(0.0396724\pi\)
−0.992243 + 0.124312i \(0.960328\pi\)
\(854\) 39.3693 1.34719
\(855\) 0 0
\(856\) −87.7235 −2.99833
\(857\) 15.7538i 0.538139i 0.963121 + 0.269070i \(0.0867162\pi\)
−0.963121 + 0.269070i \(0.913284\pi\)
\(858\) 0 0
\(859\) 16.4924 0.562714 0.281357 0.959603i \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 50.7386i 1.72816i
\(863\) − 25.7538i − 0.876669i −0.898812 0.438335i \(-0.855569\pi\)
0.898812 0.438335i \(-0.144431\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 21.1231 0.717792
\(867\) 0 0
\(868\) 0 0
\(869\) 3.80776 0.129170
\(870\) 0 0
\(871\) 4.49242 0.152220
\(872\) 34.8769i 1.18108i
\(873\) 0 0
\(874\) −56.9848 −1.92754
\(875\) 0 0
\(876\) 0 0
\(877\) − 40.2462i − 1.35902i −0.733667 0.679509i \(-0.762193\pi\)
0.733667 0.679509i \(-0.237807\pi\)
\(878\) − 24.0000i − 0.809961i
\(879\) 0 0
\(880\) 0 0
\(881\) 11.8617 0.399632 0.199816 0.979833i \(-0.435966\pi\)
0.199816 + 0.979833i \(0.435966\pi\)
\(882\) 0 0
\(883\) 8.49242i 0.285793i 0.989738 + 0.142896i \(0.0456416\pi\)
−0.989738 + 0.142896i \(0.954358\pi\)
\(884\) −0.876894 −0.0294931
\(885\) 0 0
\(886\) 6.73863 0.226389
\(887\) − 20.4924i − 0.688068i −0.938957 0.344034i \(-0.888206\pi\)
0.938957 0.344034i \(-0.111794\pi\)
\(888\) 0 0
\(889\) −6.24621 −0.209491
\(890\) 0 0
\(891\) 0 0
\(892\) − 11.1231i − 0.372429i
\(893\) 61.8617i 2.07012i
\(894\) 0 0
\(895\) 0 0
\(896\) 9.43845 0.315316
\(897\) 0 0
\(898\) − 4.63068i − 0.154528i
\(899\) 0 0
\(900\) 0 0
\(901\) 2.24621 0.0748321
\(902\) − 20.4924i − 0.682323i
\(903\) 0 0
\(904\) −91.8617 −3.05528
\(905\) 0 0
\(906\) 0 0
\(907\) − 24.1080i − 0.800491i −0.916408 0.400246i \(-0.868925\pi\)
0.916408 0.400246i \(-0.131075\pi\)
\(908\) 51.6155i 1.71292i
\(909\) 0 0
\(910\) 0 0
\(911\) 28.4924 0.943996 0.471998 0.881600i \(-0.343533\pi\)
0.471998 + 0.881600i \(0.343533\pi\)
\(912\) 0 0
\(913\) 6.24621i 0.206719i
\(914\) 43.8617 1.45082
\(915\) 0 0
\(916\) 49.6155 1.63934
\(917\) − 0.876894i − 0.0289576i
\(918\) 0 0
\(919\) −40.3002 −1.32938 −0.664690 0.747119i \(-0.731436\pi\)
−0.664690 + 0.747119i \(0.731436\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 33.6155i 1.10707i
\(923\) 3.50758i 0.115453i
\(924\) 0 0
\(925\) 0 0
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 43.8617i 1.43983i
\(929\) 22.1080 0.725338 0.362669 0.931918i \(-0.381866\pi\)
0.362669 + 0.931918i \(0.381866\pi\)
\(930\) 0 0
\(931\) −7.12311 −0.233450
\(932\) − 23.3693i − 0.765487i
\(933\) 0 0
\(934\) 57.4773 1.88071
\(935\) 0 0
\(936\) 0 0
\(937\) − 55.6695i − 1.81864i −0.416094 0.909322i \(-0.636601\pi\)
0.416094 0.909322i \(-0.363399\pi\)
\(938\) 26.2462i 0.856969i
\(939\) 0 0
\(940\) 0 0
\(941\) −43.8617 −1.42985 −0.714926 0.699200i \(-0.753540\pi\)
−0.714926 + 0.699200i \(0.753540\pi\)
\(942\) 0 0
\(943\) − 16.0000i − 0.521032i
\(944\) −30.7386 −1.00046
\(945\) 0 0
\(946\) 3.50758 0.114041
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) 5.36932 0.174295
\(950\) 0 0
\(951\) 0 0
\(952\) − 2.87689i − 0.0932407i
\(953\) − 33.1231i − 1.07296i −0.843912 0.536481i \(-0.819753\pi\)
0.843912 0.536481i \(-0.180247\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −90.3542 −2.92226
\(957\) 0 0
\(958\) 12.4924i 0.403612i
\(959\) 17.1231 0.552934
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 6.73863i 0.217262i
\(963\) 0 0
\(964\) 19.3693 0.623844
\(965\) 0 0
\(966\) 0 0
\(967\) − 35.1231i − 1.12948i −0.825268 0.564741i \(-0.808976\pi\)
0.825268 0.564741i \(-0.191024\pi\)
\(968\) 56.1771i 1.80560i
\(969\) 0 0
\(970\) 0 0
\(971\) −49.4773 −1.58780 −0.793901 0.608048i \(-0.791953\pi\)
−0.793901 + 0.608048i \(0.791953\pi\)
\(972\) 0 0
\(973\) − 15.1231i − 0.484825i
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 118.108 3.78054
\(977\) − 33.2311i − 1.06316i −0.847009 0.531578i \(-0.821599\pi\)
0.847009 0.531578i \(-0.178401\pi\)
\(978\) 0 0
\(979\) −1.75379 −0.0560513
\(980\) 0 0
\(981\) 0 0
\(982\) 105.477i 3.36591i
\(983\) 51.4233i 1.64015i 0.572257 + 0.820074i \(0.306068\pi\)
−0.572257 + 0.820074i \(0.693932\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −7.50758 −0.239090
\(987\) 0 0
\(988\) 14.2462i 0.453232i
\(989\) 2.73863 0.0870835
\(990\) 0 0
\(991\) 12.4924 0.396835 0.198417 0.980118i \(-0.436420\pi\)
0.198417 + 0.980118i \(0.436420\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −20.4924 −0.649980
\(995\) 0 0
\(996\) 0 0
\(997\) − 2.68466i − 0.0850240i −0.999096 0.0425120i \(-0.986464\pi\)
0.999096 0.0425120i \(-0.0135361\pi\)
\(998\) − 105.477i − 3.33882i
\(999\) 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 1575.2.d.e.1324.4 4
3.2 odd 2 175.2.b.b.99.1 4
5.2 odd 4 1575.2.a.p.1.1 2
5.3 odd 4 315.2.a.e.1.2 2
5.4 even 2 inner 1575.2.d.e.1324.1 4
12.11 even 2 2800.2.g.t.449.3 4
15.2 even 4 175.2.a.f.1.2 2
15.8 even 4 35.2.a.b.1.1 2
15.14 odd 2 175.2.b.b.99.4 4
20.3 even 4 5040.2.a.bt.1.1 2
21.20 even 2 1225.2.b.f.99.1 4
35.13 even 4 2205.2.a.x.1.2 2
60.23 odd 4 560.2.a.i.1.1 2
60.47 odd 4 2800.2.a.bi.1.2 2
60.59 even 2 2800.2.g.t.449.2 4
105.23 even 12 245.2.e.i.116.2 4
105.38 odd 12 245.2.e.h.226.2 4
105.53 even 12 245.2.e.i.226.2 4
105.62 odd 4 1225.2.a.s.1.2 2
105.68 odd 12 245.2.e.h.116.2 4
105.83 odd 4 245.2.a.d.1.1 2
105.104 even 2 1225.2.b.f.99.4 4
120.53 even 4 2240.2.a.bh.1.1 2
120.83 odd 4 2240.2.a.bd.1.2 2
165.98 odd 4 4235.2.a.m.1.2 2
195.38 even 4 5915.2.a.l.1.2 2
420.83 even 4 3920.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 15.8 even 4
175.2.a.f.1.2 2 15.2 even 4
175.2.b.b.99.1 4 3.2 odd 2
175.2.b.b.99.4 4 15.14 odd 2
245.2.a.d.1.1 2 105.83 odd 4
245.2.e.h.116.2 4 105.68 odd 12
245.2.e.h.226.2 4 105.38 odd 12
245.2.e.i.116.2 4 105.23 even 12
245.2.e.i.226.2 4 105.53 even 12
315.2.a.e.1.2 2 5.3 odd 4
560.2.a.i.1.1 2 60.23 odd 4
1225.2.a.s.1.2 2 105.62 odd 4
1225.2.b.f.99.1 4 21.20 even 2
1225.2.b.f.99.4 4 105.104 even 2
1575.2.a.p.1.1 2 5.2 odd 4
1575.2.d.e.1324.1 4 5.4 even 2 inner
1575.2.d.e.1324.4 4 1.1 even 1 trivial
2205.2.a.x.1.2 2 35.13 even 4
2240.2.a.bd.1.2 2 120.83 odd 4
2240.2.a.bh.1.1 2 120.53 even 4
2800.2.a.bi.1.2 2 60.47 odd 4
2800.2.g.t.449.2 4 60.59 even 2
2800.2.g.t.449.3 4 12.11 even 2
3920.2.a.bs.1.2 2 420.83 even 4
4235.2.a.m.1.2 2 165.98 odd 4
5040.2.a.bt.1.1 2 20.3 even 4
5915.2.a.l.1.2 2 195.38 even 4