Properties

Label 1521.2.b.k.1351.6
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.6
Root \(0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.k.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.69202i q^{2} -5.24698 q^{4} -1.04892i q^{5} +0.554958i q^{7} -8.74094i q^{8} +O(q^{10})\) \(q+2.69202i q^{2} -5.24698 q^{4} -1.04892i q^{5} +0.554958i q^{7} -8.74094i q^{8} +2.82371 q^{10} -2.91185i q^{11} -1.49396 q^{14} +13.0368 q^{16} -4.85086 q^{17} +0.753020i q^{19} +5.50365i q^{20} +7.83877 q^{22} +5.76271 q^{23} +3.89977 q^{25} -2.91185i q^{28} +1.91185 q^{29} +9.51573i q^{31} +17.6136i q^{32} -13.0586i q^{34} +0.582105 q^{35} +5.75302i q^{37} -2.02715 q^{38} -9.16852 q^{40} +4.91185i q^{41} +11.0978 q^{43} +15.2784i q^{44} +15.5133i q^{46} -0.753020i q^{47} +6.69202 q^{49} +10.4983i q^{50} +7.58211 q^{53} -3.05429 q^{55} +4.85086 q^{56} +5.14675i q^{58} -4.09783i q^{59} -3.42327 q^{61} -25.6165 q^{62} -21.3424 q^{64} +1.87263i q^{67} +25.4523 q^{68} +1.56704i q^{70} -10.5036i q^{71} +10.4765i q^{73} -15.4873 q^{74} -3.95108i q^{76} +1.61596 q^{77} +1.33513 q^{79} -13.6746i q^{80} -13.2228 q^{82} +2.64310i q^{83} +5.08815i q^{85} +29.8756i q^{86} -25.4523 q^{88} -9.92692i q^{89} -30.2368 q^{92} +2.02715 q^{94} +0.789856 q^{95} -17.0737i q^{97} +18.0151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 22q^{4} + O(q^{10}) \) \( 6q - 22q^{4} + 2q^{10} + 10q^{14} + 22q^{16} - 2q^{17} - 18q^{22} - 22q^{25} + 4q^{29} - 8q^{35} + 6q^{40} + 30q^{43} + 30q^{49} + 34q^{53} + 6q^{55} + 2q^{56} - 26q^{61} - 4q^{62} + 26q^{68} - 30q^{74} + 30q^{77} + 6q^{79} + 6q^{82} - 26q^{88} - 14q^{92} - 42q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69202i 1.90355i 0.306802 + 0.951773i \(0.400741\pi\)
−0.306802 + 0.951773i \(0.599259\pi\)
\(3\) 0 0
\(4\) −5.24698 −2.62349
\(5\) − 1.04892i − 0.469090i −0.972105 0.234545i \(-0.924640\pi\)
0.972105 0.234545i \(-0.0753600\pi\)
\(6\) 0 0
\(7\) 0.554958i 0.209754i 0.994485 + 0.104877i \(0.0334450\pi\)
−0.994485 + 0.104877i \(0.966555\pi\)
\(8\) − 8.74094i − 3.09039i
\(9\) 0 0
\(10\) 2.82371 0.892935
\(11\) − 2.91185i − 0.877957i −0.898498 0.438979i \(-0.855340\pi\)
0.898498 0.438979i \(-0.144660\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −1.49396 −0.399277
\(15\) 0 0
\(16\) 13.0368 3.25921
\(17\) −4.85086 −1.17651 −0.588253 0.808677i \(-0.700184\pi\)
−0.588253 + 0.808677i \(0.700184\pi\)
\(18\) 0 0
\(19\) 0.753020i 0.172755i 0.996262 + 0.0863774i \(0.0275291\pi\)
−0.996262 + 0.0863774i \(0.972471\pi\)
\(20\) 5.50365i 1.23065i
\(21\) 0 0
\(22\) 7.83877 1.67123
\(23\) 5.76271 1.20161 0.600804 0.799396i \(-0.294847\pi\)
0.600804 + 0.799396i \(0.294847\pi\)
\(24\) 0 0
\(25\) 3.89977 0.779954
\(26\) 0 0
\(27\) 0 0
\(28\) − 2.91185i − 0.550289i
\(29\) 1.91185 0.355022 0.177511 0.984119i \(-0.443195\pi\)
0.177511 + 0.984119i \(0.443195\pi\)
\(30\) 0 0
\(31\) 9.51573i 1.70908i 0.519389 + 0.854538i \(0.326159\pi\)
−0.519389 + 0.854538i \(0.673841\pi\)
\(32\) 17.6136i 3.11367i
\(33\) 0 0
\(34\) − 13.0586i − 2.23953i
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) 5.75302i 0.945791i 0.881119 + 0.472895i \(0.156791\pi\)
−0.881119 + 0.472895i \(0.843209\pi\)
\(38\) −2.02715 −0.328847
\(39\) 0 0
\(40\) −9.16852 −1.44967
\(41\) 4.91185i 0.767103i 0.923520 + 0.383551i \(0.125299\pi\)
−0.923520 + 0.383551i \(0.874701\pi\)
\(42\) 0 0
\(43\) 11.0978 1.69240 0.846202 0.532862i \(-0.178884\pi\)
0.846202 + 0.532862i \(0.178884\pi\)
\(44\) 15.2784i 2.30331i
\(45\) 0 0
\(46\) 15.5133i 2.28732i
\(47\) − 0.753020i − 0.109839i −0.998491 0.0549197i \(-0.982510\pi\)
0.998491 0.0549197i \(-0.0174903\pi\)
\(48\) 0 0
\(49\) 6.69202 0.956003
\(50\) 10.4983i 1.48468i
\(51\) 0 0
\(52\) 0 0
\(53\) 7.58211 1.04148 0.520741 0.853715i \(-0.325656\pi\)
0.520741 + 0.853715i \(0.325656\pi\)
\(54\) 0 0
\(55\) −3.05429 −0.411841
\(56\) 4.85086 0.648223
\(57\) 0 0
\(58\) 5.14675i 0.675802i
\(59\) − 4.09783i − 0.533493i −0.963767 0.266746i \(-0.914051\pi\)
0.963767 0.266746i \(-0.0859486\pi\)
\(60\) 0 0
\(61\) −3.42327 −0.438305 −0.219153 0.975691i \(-0.570329\pi\)
−0.219153 + 0.975691i \(0.570329\pi\)
\(62\) −25.6165 −3.25330
\(63\) 0 0
\(64\) −21.3424 −2.66780
\(65\) 0 0
\(66\) 0 0
\(67\) 1.87263i 0.228778i 0.993436 + 0.114389i \(0.0364910\pi\)
−0.993436 + 0.114389i \(0.963509\pi\)
\(68\) 25.4523 3.08655
\(69\) 0 0
\(70\) 1.56704i 0.187297i
\(71\) − 10.5036i − 1.24655i −0.782001 0.623277i \(-0.785801\pi\)
0.782001 0.623277i \(-0.214199\pi\)
\(72\) 0 0
\(73\) 10.4765i 1.22618i 0.790012 + 0.613091i \(0.210074\pi\)
−0.790012 + 0.613091i \(0.789926\pi\)
\(74\) −15.4873 −1.80036
\(75\) 0 0
\(76\) − 3.95108i − 0.453220i
\(77\) 1.61596 0.184155
\(78\) 0 0
\(79\) 1.33513 0.150213 0.0751067 0.997176i \(-0.476070\pi\)
0.0751067 + 0.997176i \(0.476070\pi\)
\(80\) − 13.6746i − 1.52886i
\(81\) 0 0
\(82\) −13.2228 −1.46022
\(83\) 2.64310i 0.290118i 0.989423 + 0.145059i \(0.0463373\pi\)
−0.989423 + 0.145059i \(0.953663\pi\)
\(84\) 0 0
\(85\) 5.08815i 0.551887i
\(86\) 29.8756i 3.22157i
\(87\) 0 0
\(88\) −25.4523 −2.71323
\(89\) − 9.92692i − 1.05225i −0.850407 0.526126i \(-0.823644\pi\)
0.850407 0.526126i \(-0.176356\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −30.2368 −3.15241
\(93\) 0 0
\(94\) 2.02715 0.209084
\(95\) 0.789856 0.0810375
\(96\) 0 0
\(97\) − 17.0737i − 1.73357i −0.498683 0.866784i \(-0.666183\pi\)
0.498683 0.866784i \(-0.333817\pi\)
\(98\) 18.0151i 1.81980i
\(99\) 0 0
\(100\) −20.4620 −2.04620
\(101\) 7.32304 0.728670 0.364335 0.931268i \(-0.381296\pi\)
0.364335 + 0.931268i \(0.381296\pi\)
\(102\) 0 0
\(103\) 4.21983 0.415792 0.207896 0.978151i \(-0.433338\pi\)
0.207896 + 0.978151i \(0.433338\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 20.4112i 1.98251i
\(107\) −6.39373 −0.618105 −0.309053 0.951045i \(-0.600012\pi\)
−0.309053 + 0.951045i \(0.600012\pi\)
\(108\) 0 0
\(109\) 3.46011i 0.331418i 0.986175 + 0.165709i \(0.0529913\pi\)
−0.986175 + 0.165709i \(0.947009\pi\)
\(110\) − 8.22223i − 0.783958i
\(111\) 0 0
\(112\) 7.23490i 0.683634i
\(113\) −9.35690 −0.880223 −0.440111 0.897943i \(-0.645061\pi\)
−0.440111 + 0.897943i \(0.645061\pi\)
\(114\) 0 0
\(115\) − 6.04461i − 0.563662i
\(116\) −10.0315 −0.931398
\(117\) 0 0
\(118\) 11.0315 1.01553
\(119\) − 2.69202i − 0.246777i
\(120\) 0 0
\(121\) 2.52111 0.229191
\(122\) − 9.21552i − 0.834334i
\(123\) 0 0
\(124\) − 49.9288i − 4.48374i
\(125\) − 9.33513i − 0.834959i
\(126\) 0 0
\(127\) 4.48188 0.397702 0.198851 0.980030i \(-0.436279\pi\)
0.198851 + 0.980030i \(0.436279\pi\)
\(128\) − 22.2271i − 1.96462i
\(129\) 0 0
\(130\) 0 0
\(131\) 9.21744 0.805331 0.402666 0.915347i \(-0.368084\pi\)
0.402666 + 0.915347i \(0.368084\pi\)
\(132\) 0 0
\(133\) −0.417895 −0.0362361
\(134\) −5.04115 −0.435489
\(135\) 0 0
\(136\) 42.4010i 3.63586i
\(137\) 7.46980i 0.638188i 0.947723 + 0.319094i \(0.103379\pi\)
−0.947723 + 0.319094i \(0.896621\pi\)
\(138\) 0 0
\(139\) −17.9976 −1.52654 −0.763269 0.646081i \(-0.776407\pi\)
−0.763269 + 0.646081i \(0.776407\pi\)
\(140\) −3.05429 −0.258135
\(141\) 0 0
\(142\) 28.2760 2.37287
\(143\) 0 0
\(144\) 0 0
\(145\) − 2.00538i − 0.166537i
\(146\) −28.2030 −2.33409
\(147\) 0 0
\(148\) − 30.1860i − 2.48127i
\(149\) 15.3351i 1.25630i 0.778091 + 0.628151i \(0.216188\pi\)
−0.778091 + 0.628151i \(0.783812\pi\)
\(150\) 0 0
\(151\) 2.53079i 0.205953i 0.994684 + 0.102977i \(0.0328367\pi\)
−0.994684 + 0.102977i \(0.967163\pi\)
\(152\) 6.58211 0.533879
\(153\) 0 0
\(154\) 4.35019i 0.350548i
\(155\) 9.98121 0.801710
\(156\) 0 0
\(157\) 17.2392 1.37584 0.687919 0.725787i \(-0.258524\pi\)
0.687919 + 0.725787i \(0.258524\pi\)
\(158\) 3.59419i 0.285938i
\(159\) 0 0
\(160\) 18.4752 1.46059
\(161\) 3.19806i 0.252043i
\(162\) 0 0
\(163\) 15.7071i 1.23027i 0.788420 + 0.615137i \(0.210899\pi\)
−0.788420 + 0.615137i \(0.789101\pi\)
\(164\) − 25.7724i − 2.01249i
\(165\) 0 0
\(166\) −7.11529 −0.552254
\(167\) − 5.39612i − 0.417565i −0.977962 0.208782i \(-0.933050\pi\)
0.977962 0.208782i \(-0.0669500\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −13.6974 −1.05054
\(171\) 0 0
\(172\) −58.2301 −4.44000
\(173\) 23.9420 1.82028 0.910138 0.414306i \(-0.135976\pi\)
0.910138 + 0.414306i \(0.135976\pi\)
\(174\) 0 0
\(175\) 2.16421i 0.163599i
\(176\) − 37.9614i − 2.86145i
\(177\) 0 0
\(178\) 26.7235 2.00301
\(179\) 18.4088 1.37594 0.687969 0.725740i \(-0.258502\pi\)
0.687969 + 0.725740i \(0.258502\pi\)
\(180\) 0 0
\(181\) 3.63342 0.270070 0.135035 0.990841i \(-0.456885\pi\)
0.135035 + 0.990841i \(0.456885\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) − 50.3715i − 3.71344i
\(185\) 6.03444 0.443661
\(186\) 0 0
\(187\) 14.1250i 1.03292i
\(188\) 3.95108i 0.288162i
\(189\) 0 0
\(190\) 2.12631i 0.154259i
\(191\) 21.1782 1.53240 0.766201 0.642601i \(-0.222145\pi\)
0.766201 + 0.642601i \(0.222145\pi\)
\(192\) 0 0
\(193\) − 17.6112i − 1.26768i −0.773464 0.633840i \(-0.781478\pi\)
0.773464 0.633840i \(-0.218522\pi\)
\(194\) 45.9627 3.29993
\(195\) 0 0
\(196\) −35.1129 −2.50806
\(197\) − 4.66248i − 0.332188i −0.986110 0.166094i \(-0.946884\pi\)
0.986110 0.166094i \(-0.0531155\pi\)
\(198\) 0 0
\(199\) −15.0368 −1.06593 −0.532967 0.846136i \(-0.678923\pi\)
−0.532967 + 0.846136i \(0.678923\pi\)
\(200\) − 34.0877i − 2.41036i
\(201\) 0 0
\(202\) 19.7138i 1.38706i
\(203\) 1.06100i 0.0744675i
\(204\) 0 0
\(205\) 5.15213 0.359840
\(206\) 11.3599i 0.791480i
\(207\) 0 0
\(208\) 0 0
\(209\) 2.19269 0.151671
\(210\) 0 0
\(211\) −0.460107 −0.0316751 −0.0158375 0.999875i \(-0.505041\pi\)
−0.0158375 + 0.999875i \(0.505041\pi\)
\(212\) −39.7832 −2.73232
\(213\) 0 0
\(214\) − 17.2121i − 1.17659i
\(215\) − 11.6407i − 0.793890i
\(216\) 0 0
\(217\) −5.28083 −0.358486
\(218\) −9.31468 −0.630870
\(219\) 0 0
\(220\) 16.0258 1.08046
\(221\) 0 0
\(222\) 0 0
\(223\) 16.3502i 1.09489i 0.836842 + 0.547445i \(0.184399\pi\)
−0.836842 + 0.547445i \(0.815601\pi\)
\(224\) −9.77479 −0.653106
\(225\) 0 0
\(226\) − 25.1890i − 1.67555i
\(227\) 6.56033i 0.435425i 0.976013 + 0.217712i \(0.0698595\pi\)
−0.976013 + 0.217712i \(0.930141\pi\)
\(228\) 0 0
\(229\) − 3.95539i − 0.261380i −0.991423 0.130690i \(-0.958281\pi\)
0.991423 0.130690i \(-0.0417192\pi\)
\(230\) 16.2722 1.07296
\(231\) 0 0
\(232\) − 16.7114i − 1.09716i
\(233\) 8.35690 0.547478 0.273739 0.961804i \(-0.411739\pi\)
0.273739 + 0.961804i \(0.411739\pi\)
\(234\) 0 0
\(235\) −0.789856 −0.0515245
\(236\) 21.5013i 1.39961i
\(237\) 0 0
\(238\) 7.24698 0.469752
\(239\) 20.1008i 1.30021i 0.759843 + 0.650107i \(0.225276\pi\)
−0.759843 + 0.650107i \(0.774724\pi\)
\(240\) 0 0
\(241\) − 19.0127i − 1.22471i −0.790581 0.612357i \(-0.790222\pi\)
0.790581 0.612357i \(-0.209778\pi\)
\(242\) 6.78687i 0.436277i
\(243\) 0 0
\(244\) 17.9618 1.14989
\(245\) − 7.01938i − 0.448452i
\(246\) 0 0
\(247\) 0 0
\(248\) 83.1764 5.28171
\(249\) 0 0
\(250\) 25.1304 1.58938
\(251\) −0.763774 −0.0482090 −0.0241045 0.999709i \(-0.507673\pi\)
−0.0241045 + 0.999709i \(0.507673\pi\)
\(252\) 0 0
\(253\) − 16.7802i − 1.05496i
\(254\) 12.0653i 0.757045i
\(255\) 0 0
\(256\) 17.1511 1.07194
\(257\) −13.0911 −0.816602 −0.408301 0.912847i \(-0.633879\pi\)
−0.408301 + 0.912847i \(0.633879\pi\)
\(258\) 0 0
\(259\) −3.19269 −0.198384
\(260\) 0 0
\(261\) 0 0
\(262\) 24.8135i 1.53299i
\(263\) −18.3773 −1.13320 −0.566598 0.823995i \(-0.691741\pi\)
−0.566598 + 0.823995i \(0.691741\pi\)
\(264\) 0 0
\(265\) − 7.95300i − 0.488549i
\(266\) − 1.12498i − 0.0689771i
\(267\) 0 0
\(268\) − 9.82563i − 0.600196i
\(269\) −23.6625 −1.44273 −0.721363 0.692557i \(-0.756484\pi\)
−0.721363 + 0.692557i \(0.756484\pi\)
\(270\) 0 0
\(271\) − 19.7530i − 1.19991i −0.800034 0.599955i \(-0.795185\pi\)
0.800034 0.599955i \(-0.204815\pi\)
\(272\) −63.2398 −3.83448
\(273\) 0 0
\(274\) −20.1089 −1.21482
\(275\) − 11.3556i − 0.684767i
\(276\) 0 0
\(277\) −1.77777 −0.106816 −0.0534081 0.998573i \(-0.517008\pi\)
−0.0534081 + 0.998573i \(0.517008\pi\)
\(278\) − 48.4499i − 2.90583i
\(279\) 0 0
\(280\) − 5.08815i − 0.304075i
\(281\) − 1.62133i − 0.0967207i −0.998830 0.0483603i \(-0.984600\pi\)
0.998830 0.0483603i \(-0.0153996\pi\)
\(282\) 0 0
\(283\) −4.98361 −0.296245 −0.148122 0.988969i \(-0.547323\pi\)
−0.148122 + 0.988969i \(0.547323\pi\)
\(284\) 55.1124i 3.27032i
\(285\) 0 0
\(286\) 0 0
\(287\) −2.72587 −0.160903
\(288\) 0 0
\(289\) 6.53079 0.384164
\(290\) 5.39852 0.317012
\(291\) 0 0
\(292\) − 54.9700i − 3.21688i
\(293\) 0.0717525i 0.00419183i 0.999998 + 0.00209591i \(0.000667151\pi\)
−0.999998 + 0.00209591i \(0.999333\pi\)
\(294\) 0 0
\(295\) −4.29829 −0.250256
\(296\) 50.2868 2.92286
\(297\) 0 0
\(298\) −41.2825 −2.39143
\(299\) 0 0
\(300\) 0 0
\(301\) 6.15883i 0.354989i
\(302\) −6.81295 −0.392041
\(303\) 0 0
\(304\) 9.81700i 0.563044i
\(305\) 3.59073i 0.205605i
\(306\) 0 0
\(307\) − 5.19806i − 0.296669i −0.988937 0.148335i \(-0.952609\pi\)
0.988937 0.148335i \(-0.0473912\pi\)
\(308\) −8.47889 −0.483130
\(309\) 0 0
\(310\) 26.8696i 1.52609i
\(311\) −22.5429 −1.27829 −0.639145 0.769087i \(-0.720711\pi\)
−0.639145 + 0.769087i \(0.720711\pi\)
\(312\) 0 0
\(313\) 22.6612 1.28088 0.640442 0.768006i \(-0.278751\pi\)
0.640442 + 0.768006i \(0.278751\pi\)
\(314\) 46.4083i 2.61897i
\(315\) 0 0
\(316\) −7.00538 −0.394083
\(317\) 26.3424i 1.47954i 0.672861 + 0.739769i \(0.265065\pi\)
−0.672861 + 0.739769i \(0.734935\pi\)
\(318\) 0 0
\(319\) − 5.56704i − 0.311694i
\(320\) 22.3864i 1.25144i
\(321\) 0 0
\(322\) −8.60925 −0.479775
\(323\) − 3.65279i − 0.203247i
\(324\) 0 0
\(325\) 0 0
\(326\) −42.2838 −2.34188
\(327\) 0 0
\(328\) 42.9342 2.37065
\(329\) 0.417895 0.0230393
\(330\) 0 0
\(331\) 11.2295i 0.617230i 0.951187 + 0.308615i \(0.0998655\pi\)
−0.951187 + 0.308615i \(0.900134\pi\)
\(332\) − 13.8683i − 0.761123i
\(333\) 0 0
\(334\) 14.5265 0.794854
\(335\) 1.96423 0.107317
\(336\) 0 0
\(337\) −2.30798 −0.125724 −0.0628618 0.998022i \(-0.520023\pi\)
−0.0628618 + 0.998022i \(0.520023\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 26.6974i − 1.44787i
\(341\) 27.7084 1.50049
\(342\) 0 0
\(343\) 7.59850i 0.410280i
\(344\) − 97.0055i − 5.23019i
\(345\) 0 0
\(346\) 64.4523i 3.46498i
\(347\) 9.13706 0.490503 0.245252 0.969459i \(-0.421129\pi\)
0.245252 + 0.969459i \(0.421129\pi\)
\(348\) 0 0
\(349\) 23.9758i 1.28340i 0.766957 + 0.641699i \(0.221770\pi\)
−0.766957 + 0.641699i \(0.778230\pi\)
\(350\) −5.82610 −0.311418
\(351\) 0 0
\(352\) 51.2881 2.73367
\(353\) − 27.1239i − 1.44366i −0.692070 0.721830i \(-0.743301\pi\)
0.692070 0.721830i \(-0.256699\pi\)
\(354\) 0 0
\(355\) −11.0175 −0.584746
\(356\) 52.0863i 2.76057i
\(357\) 0 0
\(358\) 49.5569i 2.61916i
\(359\) 26.0790i 1.37640i 0.725521 + 0.688200i \(0.241599\pi\)
−0.725521 + 0.688200i \(0.758401\pi\)
\(360\) 0 0
\(361\) 18.4330 0.970156
\(362\) 9.78123i 0.514090i
\(363\) 0 0
\(364\) 0 0
\(365\) 10.9890 0.575190
\(366\) 0 0
\(367\) 9.57434 0.499776 0.249888 0.968275i \(-0.419606\pi\)
0.249888 + 0.968275i \(0.419606\pi\)
\(368\) 75.1275 3.91629
\(369\) 0 0
\(370\) 16.2448i 0.844530i
\(371\) 4.20775i 0.218456i
\(372\) 0 0
\(373\) 28.1497 1.45754 0.728769 0.684760i \(-0.240093\pi\)
0.728769 + 0.684760i \(0.240093\pi\)
\(374\) −38.0248 −1.96621
\(375\) 0 0
\(376\) −6.58211 −0.339446
\(377\) 0 0
\(378\) 0 0
\(379\) − 16.0465i − 0.824255i −0.911126 0.412127i \(-0.864786\pi\)
0.911126 0.412127i \(-0.135214\pi\)
\(380\) −4.14436 −0.212601
\(381\) 0 0
\(382\) 57.0122i 2.91700i
\(383\) − 24.6165i − 1.25785i −0.777467 0.628923i \(-0.783496\pi\)
0.777467 0.628923i \(-0.216504\pi\)
\(384\) 0 0
\(385\) − 1.69501i − 0.0863855i
\(386\) 47.4097 2.41309
\(387\) 0 0
\(388\) 89.5852i 4.54800i
\(389\) −17.2198 −0.873080 −0.436540 0.899685i \(-0.643796\pi\)
−0.436540 + 0.899685i \(0.643796\pi\)
\(390\) 0 0
\(391\) −27.9541 −1.41370
\(392\) − 58.4946i − 2.95442i
\(393\) 0 0
\(394\) 12.5515 0.632335
\(395\) − 1.40044i − 0.0704636i
\(396\) 0 0
\(397\) − 2.03923i − 0.102346i −0.998690 0.0511730i \(-0.983704\pi\)
0.998690 0.0511730i \(-0.0162960\pi\)
\(398\) − 40.4795i − 2.02905i
\(399\) 0 0
\(400\) 50.8407 2.54203
\(401\) 1.46144i 0.0729806i 0.999334 + 0.0364903i \(0.0116178\pi\)
−0.999334 + 0.0364903i \(0.988382\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −38.4239 −1.91166
\(405\) 0 0
\(406\) −2.85623 −0.141752
\(407\) 16.7520 0.830364
\(408\) 0 0
\(409\) 29.9390i 1.48039i 0.672393 + 0.740194i \(0.265266\pi\)
−0.672393 + 0.740194i \(0.734734\pi\)
\(410\) 13.8696i 0.684973i
\(411\) 0 0
\(412\) −22.1414 −1.09083
\(413\) 2.27413 0.111902
\(414\) 0 0
\(415\) 2.77240 0.136092
\(416\) 0 0
\(417\) 0 0
\(418\) 5.90276i 0.288713i
\(419\) −6.64742 −0.324748 −0.162374 0.986729i \(-0.551915\pi\)
−0.162374 + 0.986729i \(0.551915\pi\)
\(420\) 0 0
\(421\) − 13.5646i − 0.661100i −0.943788 0.330550i \(-0.892766\pi\)
0.943788 0.330550i \(-0.107234\pi\)
\(422\) − 1.23862i − 0.0602950i
\(423\) 0 0
\(424\) − 66.2747i − 3.21858i
\(425\) −18.9172 −0.917620
\(426\) 0 0
\(427\) − 1.89977i − 0.0919364i
\(428\) 33.5478 1.62159
\(429\) 0 0
\(430\) 31.3370 1.51121
\(431\) 35.9463i 1.73147i 0.500501 + 0.865736i \(0.333149\pi\)
−0.500501 + 0.865736i \(0.666851\pi\)
\(432\) 0 0
\(433\) 32.4741 1.56061 0.780303 0.625402i \(-0.215065\pi\)
0.780303 + 0.625402i \(0.215065\pi\)
\(434\) − 14.2161i − 0.682395i
\(435\) 0 0
\(436\) − 18.1551i − 0.869472i
\(437\) 4.33944i 0.207583i
\(438\) 0 0
\(439\) −12.8321 −0.612441 −0.306221 0.951961i \(-0.599065\pi\)
−0.306221 + 0.951961i \(0.599065\pi\)
\(440\) 26.6974i 1.27275i
\(441\) 0 0
\(442\) 0 0
\(443\) −11.9608 −0.568273 −0.284137 0.958784i \(-0.591707\pi\)
−0.284137 + 0.958784i \(0.591707\pi\)
\(444\) 0 0
\(445\) −10.4125 −0.493601
\(446\) −44.0151 −2.08417
\(447\) 0 0
\(448\) − 11.8442i − 0.559584i
\(449\) 12.4789i 0.588915i 0.955665 + 0.294458i \(0.0951390\pi\)
−0.955665 + 0.294458i \(0.904861\pi\)
\(450\) 0 0
\(451\) 14.3026 0.673483
\(452\) 49.0954 2.30926
\(453\) 0 0
\(454\) −17.6606 −0.828851
\(455\) 0 0
\(456\) 0 0
\(457\) − 32.4523i − 1.51806i −0.651058 0.759028i \(-0.725674\pi\)
0.651058 0.759028i \(-0.274326\pi\)
\(458\) 10.6480 0.497549
\(459\) 0 0
\(460\) 31.7159i 1.47876i
\(461\) 24.4034i 1.13658i 0.822828 + 0.568290i \(0.192395\pi\)
−0.822828 + 0.568290i \(0.807605\pi\)
\(462\) 0 0
\(463\) − 33.1836i − 1.54217i −0.636731 0.771086i \(-0.719714\pi\)
0.636731 0.771086i \(-0.280286\pi\)
\(464\) 24.9245 1.15709
\(465\) 0 0
\(466\) 22.4969i 1.04215i
\(467\) −38.5206 −1.78252 −0.891261 0.453490i \(-0.850179\pi\)
−0.891261 + 0.453490i \(0.850179\pi\)
\(468\) 0 0
\(469\) −1.03923 −0.0479871
\(470\) − 2.12631i − 0.0980794i
\(471\) 0 0
\(472\) −35.8189 −1.64870
\(473\) − 32.3153i − 1.48586i
\(474\) 0 0
\(475\) 2.93661i 0.134741i
\(476\) 14.1250i 0.647417i
\(477\) 0 0
\(478\) −54.1118 −2.47502
\(479\) 8.34481i 0.381284i 0.981660 + 0.190642i \(0.0610570\pi\)
−0.981660 + 0.190642i \(0.938943\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 51.1825 2.33130
\(483\) 0 0
\(484\) −13.2282 −0.601282
\(485\) −17.9089 −0.813200
\(486\) 0 0
\(487\) 14.8586i 0.673309i 0.941628 + 0.336654i \(0.109295\pi\)
−0.941628 + 0.336654i \(0.890705\pi\)
\(488\) 29.9226i 1.35453i
\(489\) 0 0
\(490\) 18.8963 0.853648
\(491\) 4.99894 0.225599 0.112799 0.993618i \(-0.464018\pi\)
0.112799 + 0.993618i \(0.464018\pi\)
\(492\) 0 0
\(493\) −9.27413 −0.417686
\(494\) 0 0
\(495\) 0 0
\(496\) 124.055i 5.57023i
\(497\) 5.82908 0.261470
\(498\) 0 0
\(499\) 0.385371i 0.0172516i 0.999963 + 0.00862579i \(0.00274571\pi\)
−0.999963 + 0.00862579i \(0.997254\pi\)
\(500\) 48.9812i 2.19051i
\(501\) 0 0
\(502\) − 2.05610i − 0.0917681i
\(503\) −22.6179 −1.00848 −0.504241 0.863563i \(-0.668228\pi\)
−0.504241 + 0.863563i \(0.668228\pi\)
\(504\) 0 0
\(505\) − 7.68127i − 0.341812i
\(506\) 45.1726 2.00817
\(507\) 0 0
\(508\) −23.5163 −1.04337
\(509\) − 11.6039i − 0.514333i −0.966367 0.257166i \(-0.917211\pi\)
0.966367 0.257166i \(-0.0827888\pi\)
\(510\) 0 0
\(511\) −5.81402 −0.257197
\(512\) 1.71678i 0.0758715i
\(513\) 0 0
\(514\) − 35.2416i − 1.55444i
\(515\) − 4.42626i − 0.195044i
\(516\) 0 0
\(517\) −2.19269 −0.0964342
\(518\) − 8.59478i − 0.377633i
\(519\) 0 0
\(520\) 0 0
\(521\) 1.62671 0.0712675 0.0356337 0.999365i \(-0.488655\pi\)
0.0356337 + 0.999365i \(0.488655\pi\)
\(522\) 0 0
\(523\) 10.0718 0.440407 0.220203 0.975454i \(-0.429328\pi\)
0.220203 + 0.975454i \(0.429328\pi\)
\(524\) −48.3637 −2.11278
\(525\) 0 0
\(526\) − 49.4722i − 2.15709i
\(527\) − 46.1594i − 2.01074i
\(528\) 0 0
\(529\) 10.2088 0.443862
\(530\) 21.4097 0.929976
\(531\) 0 0
\(532\) 2.19269 0.0950650
\(533\) 0 0
\(534\) 0 0
\(535\) 6.70650i 0.289947i
\(536\) 16.3685 0.707012
\(537\) 0 0
\(538\) − 63.6999i − 2.74630i
\(539\) − 19.4862i − 0.839330i
\(540\) 0 0
\(541\) − 20.4674i − 0.879962i −0.898007 0.439981i \(-0.854985\pi\)
0.898007 0.439981i \(-0.145015\pi\)
\(542\) 53.1756 2.28409
\(543\) 0 0
\(544\) − 85.4408i − 3.66325i
\(545\) 3.62937 0.155465
\(546\) 0 0
\(547\) −27.5478 −1.17786 −0.588929 0.808185i \(-0.700450\pi\)
−0.588929 + 0.808185i \(0.700450\pi\)
\(548\) − 39.1939i − 1.67428i
\(549\) 0 0
\(550\) 30.5694 1.30348
\(551\) 1.43967i 0.0613318i
\(552\) 0 0
\(553\) 0.740939i 0.0315079i
\(554\) − 4.78581i − 0.203329i
\(555\) 0 0
\(556\) 94.4331 4.00485
\(557\) − 37.9855i − 1.60950i −0.593615 0.804749i \(-0.702300\pi\)
0.593615 0.804749i \(-0.297700\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 7.58881 0.320686
\(561\) 0 0
\(562\) 4.36467 0.184112
\(563\) −29.7724 −1.25476 −0.627378 0.778714i \(-0.715872\pi\)
−0.627378 + 0.778714i \(0.715872\pi\)
\(564\) 0 0
\(565\) 9.81461i 0.412904i
\(566\) − 13.4160i − 0.563916i
\(567\) 0 0
\(568\) −91.8117 −3.85234
\(569\) −21.9541 −0.920362 −0.460181 0.887825i \(-0.652216\pi\)
−0.460181 + 0.887825i \(0.652216\pi\)
\(570\) 0 0
\(571\) −2.46575 −0.103188 −0.0515942 0.998668i \(-0.516430\pi\)
−0.0515942 + 0.998668i \(0.516430\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 7.33811i − 0.306287i
\(575\) 22.4733 0.937199
\(576\) 0 0
\(577\) − 17.4547i − 0.726650i −0.931662 0.363325i \(-0.881641\pi\)
0.931662 0.363325i \(-0.118359\pi\)
\(578\) 17.5810i 0.731275i
\(579\) 0 0
\(580\) 10.5222i 0.436909i
\(581\) −1.46681 −0.0608536
\(582\) 0 0
\(583\) − 22.0780i − 0.914377i
\(584\) 91.5745 3.78938
\(585\) 0 0
\(586\) −0.193159 −0.00797934
\(587\) 6.26337i 0.258517i 0.991611 + 0.129259i \(0.0412597\pi\)
−0.991611 + 0.129259i \(0.958740\pi\)
\(588\) 0 0
\(589\) −7.16554 −0.295251
\(590\) − 11.5711i − 0.476374i
\(591\) 0 0
\(592\) 75.0012i 3.08253i
\(593\) − 22.8745i − 0.939345i −0.882841 0.469672i \(-0.844372\pi\)
0.882841 0.469672i \(-0.155628\pi\)
\(594\) 0 0
\(595\) −2.82371 −0.115761
\(596\) − 80.4631i − 3.29590i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.05621 0.0431557 0.0215778 0.999767i \(-0.493131\pi\)
0.0215778 + 0.999767i \(0.493131\pi\)
\(600\) 0 0
\(601\) −33.3236 −1.35930 −0.679650 0.733537i \(-0.737868\pi\)
−0.679650 + 0.733537i \(0.737868\pi\)
\(602\) −16.5797 −0.675739
\(603\) 0 0
\(604\) − 13.2790i − 0.540316i
\(605\) − 2.64443i − 0.107511i
\(606\) 0 0
\(607\) 16.2403 0.659172 0.329586 0.944125i \(-0.393091\pi\)
0.329586 + 0.944125i \(0.393091\pi\)
\(608\) −13.2634 −0.537901
\(609\) 0 0
\(610\) −9.66632 −0.391378
\(611\) 0 0
\(612\) 0 0
\(613\) − 11.0479i − 0.446219i −0.974793 0.223109i \(-0.928379\pi\)
0.974793 0.223109i \(-0.0716207\pi\)
\(614\) 13.9933 0.564723
\(615\) 0 0
\(616\) − 14.1250i − 0.569112i
\(617\) − 4.65950i − 0.187584i −0.995592 0.0937922i \(-0.970101\pi\)
0.995592 0.0937922i \(-0.0298989\pi\)
\(618\) 0 0
\(619\) 31.9259i 1.28321i 0.767036 + 0.641604i \(0.221731\pi\)
−0.767036 + 0.641604i \(0.778269\pi\)
\(620\) −52.3712 −2.10328
\(621\) 0 0
\(622\) − 60.6859i − 2.43328i
\(623\) 5.50902 0.220714
\(624\) 0 0
\(625\) 9.70709 0.388283
\(626\) 61.0043i 2.43822i
\(627\) 0 0
\(628\) −90.4538 −3.60950
\(629\) − 27.9071i − 1.11273i
\(630\) 0 0
\(631\) − 39.4413i − 1.57013i −0.619411 0.785067i \(-0.712628\pi\)
0.619411 0.785067i \(-0.287372\pi\)
\(632\) − 11.6703i − 0.464218i
\(633\) 0 0
\(634\) −70.9144 −2.81637
\(635\) − 4.70112i − 0.186558i
\(636\) 0 0
\(637\) 0 0
\(638\) 14.9866 0.593325
\(639\) 0 0
\(640\) −23.3144 −0.921583
\(641\) 45.4510 1.79521 0.897603 0.440804i \(-0.145307\pi\)
0.897603 + 0.440804i \(0.145307\pi\)
\(642\) 0 0
\(643\) 29.7469i 1.17310i 0.809912 + 0.586552i \(0.199515\pi\)
−0.809912 + 0.586552i \(0.800485\pi\)
\(644\) − 16.7802i − 0.661231i
\(645\) 0 0
\(646\) 9.83340 0.386890
\(647\) −16.9312 −0.665635 −0.332818 0.942991i \(-0.607999\pi\)
−0.332818 + 0.942991i \(0.607999\pi\)
\(648\) 0 0
\(649\) −11.9323 −0.468384
\(650\) 0 0
\(651\) 0 0
\(652\) − 82.4148i − 3.22761i
\(653\) 33.1976 1.29912 0.649561 0.760309i \(-0.274953\pi\)
0.649561 + 0.760309i \(0.274953\pi\)
\(654\) 0 0
\(655\) − 9.66833i − 0.377773i
\(656\) 64.0350i 2.50015i
\(657\) 0 0
\(658\) 1.12498i 0.0438564i
\(659\) 42.6571 1.66168 0.830842 0.556508i \(-0.187859\pi\)
0.830842 + 0.556508i \(0.187859\pi\)
\(660\) 0 0
\(661\) 38.6902i 1.50488i 0.658664 + 0.752438i \(0.271122\pi\)
−0.658664 + 0.752438i \(0.728878\pi\)
\(662\) −30.2301 −1.17493
\(663\) 0 0
\(664\) 23.1032 0.896578
\(665\) 0.438337i 0.0169980i
\(666\) 0 0
\(667\) 11.0175 0.426598
\(668\) 28.3134i 1.09548i
\(669\) 0 0
\(670\) 5.28775i 0.204283i
\(671\) 9.96807i 0.384813i
\(672\) 0 0
\(673\) 2.59419 0.0999986 0.0499993 0.998749i \(-0.484078\pi\)
0.0499993 + 0.998749i \(0.484078\pi\)
\(674\) − 6.21313i − 0.239321i
\(675\) 0 0
\(676\) 0 0
\(677\) −1.75302 −0.0673740 −0.0336870 0.999432i \(-0.510725\pi\)
−0.0336870 + 0.999432i \(0.510725\pi\)
\(678\) 0 0
\(679\) 9.47517 0.363624
\(680\) 44.4752 1.70555
\(681\) 0 0
\(682\) 74.5916i 2.85626i
\(683\) − 16.3351i − 0.625046i −0.949910 0.312523i \(-0.898826\pi\)
0.949910 0.312523i \(-0.101174\pi\)
\(684\) 0 0
\(685\) 7.83520 0.299368
\(686\) −20.4553 −0.780988
\(687\) 0 0
\(688\) 144.681 5.51590
\(689\) 0 0
\(690\) 0 0
\(691\) − 15.9105i − 0.605265i −0.953107 0.302632i \(-0.902135\pi\)
0.953107 0.302632i \(-0.0978655\pi\)
\(692\) −125.623 −4.77547
\(693\) 0 0
\(694\) 24.5972i 0.933696i
\(695\) 18.8780i 0.716083i
\(696\) 0 0
\(697\) − 23.8267i − 0.902500i
\(698\) −64.5435 −2.44301
\(699\) 0 0
\(700\) − 11.3556i − 0.429200i
\(701\) −20.8635 −0.788005 −0.394002 0.919109i \(-0.628910\pi\)
−0.394002 + 0.919109i \(0.628910\pi\)
\(702\) 0 0
\(703\) −4.33214 −0.163390
\(704\) 62.1460i 2.34222i
\(705\) 0 0
\(706\) 73.0182 2.74807
\(707\) 4.06398i 0.152842i
\(708\) 0 0
\(709\) 15.6485i 0.587691i 0.955853 + 0.293846i \(0.0949351\pi\)
−0.955853 + 0.293846i \(0.905065\pi\)
\(710\) − 29.6592i − 1.11309i
\(711\) 0 0
\(712\) −86.7706 −3.25187
\(713\) 54.8364i 2.05364i
\(714\) 0 0
\(715\) 0 0
\(716\) −96.5906 −3.60976
\(717\) 0 0
\(718\) −70.2054 −2.62004
\(719\) −27.1594 −1.01288 −0.506438 0.862276i \(-0.669038\pi\)
−0.506438 + 0.862276i \(0.669038\pi\)
\(720\) 0 0
\(721\) 2.34183i 0.0872143i
\(722\) 49.6219i 1.84674i
\(723\) 0 0
\(724\) −19.0645 −0.708525
\(725\) 7.45580 0.276901
\(726\) 0 0
\(727\) −31.7784 −1.17859 −0.589297 0.807916i \(-0.700595\pi\)
−0.589297 + 0.807916i \(0.700595\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 29.5826i 1.09490i
\(731\) −53.8340 −1.99112
\(732\) 0 0
\(733\) − 46.8907i − 1.73195i −0.500090 0.865973i \(-0.666700\pi\)
0.500090 0.865973i \(-0.333300\pi\)
\(734\) 25.7743i 0.951347i
\(735\) 0 0
\(736\) 101.502i 3.74141i
\(737\) 5.45281 0.200857
\(738\) 0 0
\(739\) 17.0479i 0.627115i 0.949569 + 0.313558i \(0.101521\pi\)
−0.949569 + 0.313558i \(0.898479\pi\)
\(740\) −31.6626 −1.16394
\(741\) 0 0
\(742\) −11.3274 −0.415840
\(743\) − 11.6324i − 0.426750i −0.976970 0.213375i \(-0.931554\pi\)
0.976970 0.213375i \(-0.0684455\pi\)
\(744\) 0 0
\(745\) 16.0853 0.589319
\(746\) 75.7797i 2.77449i
\(747\) 0 0
\(748\) − 74.1135i − 2.70986i
\(749\) − 3.54825i − 0.129650i
\(750\) 0 0
\(751\) 13.0295 0.475455 0.237727 0.971332i \(-0.423598\pi\)
0.237727 + 0.971332i \(0.423598\pi\)
\(752\) − 9.81700i − 0.357989i
\(753\) 0 0
\(754\) 0 0
\(755\) 2.65459 0.0966106
\(756\) 0 0
\(757\) 22.7899 0.828311 0.414156 0.910206i \(-0.364077\pi\)
0.414156 + 0.910206i \(0.364077\pi\)
\(758\) 43.1976 1.56901
\(759\) 0 0
\(760\) − 6.90408i − 0.250437i
\(761\) 38.3424i 1.38991i 0.719052 + 0.694956i \(0.244576\pi\)
−0.719052 + 0.694956i \(0.755424\pi\)
\(762\) 0 0
\(763\) −1.92021 −0.0695164
\(764\) −111.122 −4.02024
\(765\) 0 0
\(766\) 66.2683 2.39437
\(767\) 0 0
\(768\) 0 0
\(769\) 3.63879i 0.131218i 0.997845 + 0.0656091i \(0.0208990\pi\)
−0.997845 + 0.0656091i \(0.979101\pi\)
\(770\) 4.56299 0.164439
\(771\) 0 0
\(772\) 92.4055i 3.32575i
\(773\) − 39.3424i − 1.41505i −0.706689 0.707524i \(-0.749812\pi\)
0.706689 0.707524i \(-0.250188\pi\)
\(774\) 0 0
\(775\) 37.1092i 1.33300i
\(776\) −149.240 −5.35740
\(777\) 0 0
\(778\) − 46.3562i − 1.66195i
\(779\) −3.69873 −0.132521
\(780\) 0 0
\(781\) −30.5851 −1.09442
\(782\) − 75.2529i − 2.69104i
\(783\) 0 0
\(784\) 87.2428 3.11581
\(785\) − 18.0825i − 0.645392i
\(786\) 0 0
\(787\) − 25.4252i − 0.906310i −0.891432 0.453155i \(-0.850298\pi\)
0.891432 0.453155i \(-0.149702\pi\)
\(788\) 24.4639i 0.871492i
\(789\) 0 0
\(790\) 3.77000 0.134131
\(791\) − 5.19269i − 0.184631i
\(792\) 0 0
\(793\) 0 0
\(794\) 5.48965 0.194820
\(795\) 0 0
\(796\) 78.8980 2.79646
\(797\) 20.7138 0.733720 0.366860 0.930276i \(-0.380433\pi\)
0.366860 + 0.930276i \(0.380433\pi\)
\(798\) 0 0
\(799\) 3.65279i 0.129227i
\(800\) 68.6889i 2.42852i
\(801\) 0 0
\(802\) −3.93422 −0.138922
\(803\) 30.5060 1.07653
\(804\) 0 0
\(805\) 3.35450 0.118231
\(806\) 0 0
\(807\) 0 0
\(808\) − 64.0103i − 2.25187i
\(809\) −11.0978 −0.390179 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(810\) 0 0
\(811\) 4.84223i 0.170034i 0.996380 + 0.0850169i \(0.0270944\pi\)
−0.996380 + 0.0850169i \(0.972906\pi\)
\(812\) − 5.56704i − 0.195365i
\(813\) 0 0
\(814\) 45.0966i 1.58064i
\(815\) 16.4754 0.577109
\(816\) 0 0
\(817\) 8.35690i 0.292371i
\(818\) −80.5964 −2.81799
\(819\) 0 0
\(820\) −27.0331 −0.944037
\(821\) − 43.7381i − 1.52647i −0.646121 0.763235i \(-0.723610\pi\)
0.646121 0.763235i \(-0.276390\pi\)
\(822\) 0 0
\(823\) −12.0998 −0.421771 −0.210885 0.977511i \(-0.567635\pi\)
−0.210885 + 0.977511i \(0.567635\pi\)
\(824\) − 36.8853i − 1.28496i
\(825\) 0 0
\(826\) 6.12200i 0.213012i
\(827\) 35.3212i 1.22824i 0.789213 + 0.614120i \(0.210489\pi\)
−0.789213 + 0.614120i \(0.789511\pi\)
\(828\) 0 0
\(829\) −53.4946 −1.85794 −0.928971 0.370152i \(-0.879306\pi\)
−0.928971 + 0.370152i \(0.879306\pi\)
\(830\) 7.46335i 0.259057i
\(831\) 0 0
\(832\) 0 0
\(833\) −32.4620 −1.12474
\(834\) 0 0
\(835\) −5.66009 −0.195875
\(836\) −11.5050 −0.397908
\(837\) 0 0
\(838\) − 17.8950i − 0.618172i
\(839\) − 23.1521i − 0.799300i −0.916668 0.399650i \(-0.869132\pi\)
0.916668 0.399650i \(-0.130868\pi\)
\(840\) 0 0
\(841\) −25.3448 −0.873959
\(842\) 36.5163 1.25844
\(843\) 0 0
\(844\) 2.41417 0.0830993
\(845\) 0 0
\(846\) 0 0
\(847\) 1.39911i 0.0480739i
\(848\) 98.8467 3.39441
\(849\) 0 0
\(850\) − 50.9256i − 1.74673i
\(851\) 33.1530i 1.13647i
\(852\) 0 0
\(853\) 26.7265i 0.915097i 0.889185 + 0.457548i \(0.151272\pi\)
−0.889185 + 0.457548i \(0.848728\pi\)
\(854\) 5.11423 0.175005
\(855\) 0 0
\(856\) 55.8872i 1.91019i
\(857\) 42.6064 1.45541 0.727703 0.685892i \(-0.240588\pi\)
0.727703 + 0.685892i \(0.240588\pi\)
\(858\) 0 0
\(859\) 33.6079 1.14669 0.573344 0.819315i \(-0.305646\pi\)
0.573344 + 0.819315i \(0.305646\pi\)
\(860\) 61.0786i 2.08276i
\(861\) 0 0
\(862\) −96.7682 −3.29594
\(863\) 18.7047i 0.636715i 0.947971 + 0.318358i \(0.103131\pi\)
−0.947971 + 0.318358i \(0.896869\pi\)
\(864\) 0 0
\(865\) − 25.1132i − 0.853873i
\(866\) 87.4210i 2.97069i
\(867\) 0 0
\(868\) 27.7084 0.940485
\(869\) − 3.88769i − 0.131881i
\(870\) 0 0
\(871\) 0 0
\(872\) 30.2446 1.02421
\(873\) 0 0
\(874\) −11.6819 −0.395145
\(875\) 5.18060 0.175136
\(876\) 0 0
\(877\) − 36.8237i − 1.24345i −0.783236 0.621724i \(-0.786433\pi\)
0.783236 0.621724i \(-0.213567\pi\)
\(878\) − 34.5442i − 1.16581i
\(879\) 0 0
\(880\) −39.8183 −1.34228
\(881\) −41.1250 −1.38554 −0.692768 0.721161i \(-0.743609\pi\)
−0.692768 + 0.721161i \(0.743609\pi\)
\(882\) 0 0
\(883\) −30.7482 −1.03476 −0.517380 0.855756i \(-0.673093\pi\)
−0.517380 + 0.855756i \(0.673093\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 32.1987i − 1.08173i
\(887\) −7.58940 −0.254827 −0.127414 0.991850i \(-0.540668\pi\)
−0.127414 + 0.991850i \(0.540668\pi\)
\(888\) 0 0
\(889\) 2.48725i 0.0834198i
\(890\) − 28.0307i − 0.939592i
\(891\) 0 0
\(892\) − 85.7891i − 2.87243i
\(893\) 0.567040 0.0189753
\(894\) 0 0
\(895\) − 19.3093i − 0.645439i
\(896\) 12.3351 0.412088
\(897\) 0 0
\(898\) −33.5934 −1.12103
\(899\) 18.1927i 0.606760i
\(900\) 0 0
\(901\) −36.7797 −1.22531
\(902\) 38.5029i 1.28201i
\(903\) 0 0
\(904\) 81.7881i 2.72023i
\(905\) − 3.81115i − 0.126687i
\(906\) 0 0
\(907\) −19.3333 −0.641952 −0.320976 0.947087i \(-0.604011\pi\)
−0.320976 + 0.947087i \(0.604011\pi\)
\(908\) − 34.4219i − 1.14233i
\(909\) 0 0
\(910\) 0 0
\(911\) −22.7149 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(912\) 0 0
\(913\) 7.69633 0.254711
\(914\) 87.3624 2.88969
\(915\) 0 0
\(916\) 20.7539i 0.685727i
\(917\) 5.11529i 0.168922i
\(918\) 0 0
\(919\) 14.2911 0.471420 0.235710 0.971823i \(-0.424258\pi\)
0.235710 + 0.971823i \(0.424258\pi\)
\(920\) −52.8355 −1.74194
\(921\) 0 0
\(922\) −65.6945 −2.16353
\(923\) 0 0
\(924\) 0 0
\(925\) 22.4355i 0.737674i
\(926\) 89.3309 2.93560
\(927\) 0 0
\(928\) 33.6746i 1.10542i
\(929\) 32.7211i 1.07354i 0.843727 + 0.536772i \(0.180356\pi\)
−0.843727 + 0.536772i \(0.819644\pi\)
\(930\) 0 0
\(931\) 5.03923i 0.165154i
\(932\) −43.8485 −1.43630
\(933\) 0 0
\(934\) − 103.698i − 3.39311i
\(935\) 14.8159 0.484533
\(936\) 0 0
\(937\) −4.01400 −0.131132 −0.0655658 0.997848i \(-0.520885\pi\)
−0.0655658 + 0.997848i \(0.520885\pi\)
\(938\) − 2.79763i − 0.0913457i
\(939\) 0 0
\(940\) 4.14436 0.135174
\(941\) − 28.2669i − 0.921476i −0.887536 0.460738i \(-0.847585\pi\)
0.887536 0.460738i \(-0.152415\pi\)
\(942\) 0 0
\(943\) 28.3056i 0.921757i
\(944\) − 53.4228i − 1.73876i
\(945\) 0 0
\(946\) 86.9934 2.82840
\(947\) 37.8455i 1.22981i 0.788600 + 0.614906i \(0.210806\pi\)
−0.788600 + 0.614906i \(0.789194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −7.90541 −0.256485
\(951\) 0 0
\(952\) −23.5308 −0.762637
\(953\) −40.8256 −1.32247 −0.661236 0.750178i \(-0.729968\pi\)
−0.661236 + 0.750178i \(0.729968\pi\)
\(954\) 0 0
\(955\) − 22.2142i − 0.718834i
\(956\) − 105.469i − 3.41110i
\(957\) 0 0
\(958\) −22.4644 −0.725792
\(959\) −4.14542 −0.133863
\(960\) 0 0
\(961\) −59.5491 −1.92094
\(962\) 0 0
\(963\) 0 0
\(964\) 99.7591i 3.21302i
\(965\) −18.4727 −0.594656
\(966\) 0 0
\(967\) − 10.2798i − 0.330575i −0.986245 0.165288i \(-0.947145\pi\)
0.986245 0.165288i \(-0.0528552\pi\)
\(968\) − 22.0368i − 0.708291i
\(969\) 0 0
\(970\) − 48.2111i − 1.54796i
\(971\) 19.4805 0.625161 0.312580 0.949891i \(-0.398807\pi\)
0.312580 + 0.949891i \(0.398807\pi\)
\(972\) 0 0
\(973\) − 9.98792i − 0.320198i
\(974\) −39.9997 −1.28167
\(975\) 0 0
\(976\) −44.6286 −1.42853
\(977\) 47.0538i 1.50539i 0.658372 + 0.752693i \(0.271245\pi\)
−0.658372 + 0.752693i \(0.728755\pi\)
\(978\) 0 0
\(979\) −28.9057 −0.923831
\(980\) 36.8305i 1.17651i
\(981\) 0 0
\(982\) 13.4572i 0.429438i
\(983\) − 34.4295i − 1.09813i −0.835779 0.549065i \(-0.814984\pi\)
0.835779 0.549065i \(-0.185016\pi\)
\(984\) 0 0
\(985\) −4.89056 −0.155826
\(986\) − 24.9661i − 0.795084i
\(987\) 0 0
\(988\) 0 0
\(989\) 63.9536 2.03361
\(990\) 0 0
\(991\) 7.30798 0.232146 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(992\) −167.606 −5.32149
\(993\) 0 0
\(994\) 15.6920i 0.497721i
\(995\) 15.7724i 0.500019i
\(996\) 0 0
\(997\) −35.6915 −1.13036 −0.565181 0.824967i \(-0.691194\pi\)
−0.565181 + 0.824967i \(0.691194\pi\)
\(998\) −1.03743 −0.0328392
\(999\) 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 1521.2.b.k.1351.6 6
3.2 odd 2 507.2.b.f.337.1 6
13.5 odd 4 1521.2.a.s.1.3 3
13.8 odd 4 1521.2.a.n.1.1 3
13.12 even 2 inner 1521.2.b.k.1351.1 6
39.2 even 12 507.2.e.l.22.3 6
39.5 even 4 507.2.a.i.1.1 3
39.8 even 4 507.2.a.l.1.3 yes 3
39.11 even 12 507.2.e.i.22.1 6
39.17 odd 6 507.2.j.i.361.6 12
39.20 even 12 507.2.e.i.484.1 6
39.23 odd 6 507.2.j.i.316.1 12
39.29 odd 6 507.2.j.i.316.6 12
39.32 even 12 507.2.e.l.484.3 6
39.35 odd 6 507.2.j.i.361.1 12
39.38 odd 2 507.2.b.f.337.6 6
156.47 odd 4 8112.2.a.cp.1.1 3
156.83 odd 4 8112.2.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.1 3 39.5 even 4
507.2.a.l.1.3 yes 3 39.8 even 4
507.2.b.f.337.1 6 3.2 odd 2
507.2.b.f.337.6 6 39.38 odd 2
507.2.e.i.22.1 6 39.11 even 12
507.2.e.i.484.1 6 39.20 even 12
507.2.e.l.22.3 6 39.2 even 12
507.2.e.l.484.3 6 39.32 even 12
507.2.j.i.316.1 12 39.23 odd 6
507.2.j.i.316.6 12 39.29 odd 6
507.2.j.i.361.1 12 39.35 odd 6
507.2.j.i.361.6 12 39.17 odd 6
1521.2.a.n.1.1 3 13.8 odd 4
1521.2.a.s.1.3 3 13.5 odd 4
1521.2.b.k.1351.1 6 13.12 even 2 inner
1521.2.b.k.1351.6 6 1.1 even 1 trivial
8112.2.a.cg.1.3 3 156.83 odd 4
8112.2.a.cp.1.1 3 156.47 odd 4