Properties

Label 3380.2.a.r.1.6
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.0545075\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.0545075 q^{3} -1.00000 q^{5} -3.71818 q^{7} -2.99703 q^{9} +O(q^{10})\) \(q-0.0545075 q^{3} -1.00000 q^{5} -3.71818 q^{7} -2.99703 q^{9} -5.22019 q^{11} +0.0545075 q^{15} -2.97721 q^{17} -6.13337 q^{19} +0.202668 q^{21} +3.62333 q^{23} +1.00000 q^{25} +0.326883 q^{27} -2.03371 q^{29} -0.100239 q^{31} +0.284540 q^{33} +3.71818 q^{35} -7.32662 q^{37} +12.1830 q^{41} +8.18427 q^{43} +2.99703 q^{45} +8.41641 q^{47} +6.82483 q^{49} +0.162280 q^{51} -6.11335 q^{53} +5.22019 q^{55} +0.334315 q^{57} -1.53604 q^{59} +7.09884 q^{61} +11.1435 q^{63} -14.6928 q^{67} -0.197499 q^{69} -13.3972 q^{71} +11.5465 q^{73} -0.0545075 q^{75} +19.4096 q^{77} -10.0334 q^{79} +8.97327 q^{81} -4.21504 q^{83} +2.97721 q^{85} +0.110853 q^{87} +7.86394 q^{89} +0.00546375 q^{93} +6.13337 q^{95} +12.6051 q^{97} +15.6451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + O(q^{10}) \) \( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.0545075 −0.0314699 −0.0157350 0.999876i \(-0.505009\pi\)
−0.0157350 + 0.999876i \(0.505009\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.71818 −1.40534 −0.702669 0.711517i \(-0.748009\pi\)
−0.702669 + 0.711517i \(0.748009\pi\)
\(8\) 0 0
\(9\) −2.99703 −0.999010
\(10\) 0 0
\(11\) −5.22019 −1.57395 −0.786974 0.616986i \(-0.788353\pi\)
−0.786974 + 0.616986i \(0.788353\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0.0545075 0.0140738
\(16\) 0 0
\(17\) −2.97721 −0.722079 −0.361039 0.932551i \(-0.617578\pi\)
−0.361039 + 0.932551i \(0.617578\pi\)
\(18\) 0 0
\(19\) −6.13337 −1.40709 −0.703546 0.710650i \(-0.748401\pi\)
−0.703546 + 0.710650i \(0.748401\pi\)
\(20\) 0 0
\(21\) 0.202668 0.0442259
\(22\) 0 0
\(23\) 3.62333 0.755516 0.377758 0.925904i \(-0.376695\pi\)
0.377758 + 0.925904i \(0.376695\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.326883 0.0629087
\(28\) 0 0
\(29\) −2.03371 −0.377651 −0.188825 0.982011i \(-0.560468\pi\)
−0.188825 + 0.982011i \(0.560468\pi\)
\(30\) 0 0
\(31\) −0.100239 −0.0180034 −0.00900169 0.999959i \(-0.502865\pi\)
−0.00900169 + 0.999959i \(0.502865\pi\)
\(32\) 0 0
\(33\) 0.284540 0.0495320
\(34\) 0 0
\(35\) 3.71818 0.628486
\(36\) 0 0
\(37\) −7.32662 −1.20449 −0.602244 0.798312i \(-0.705727\pi\)
−0.602244 + 0.798312i \(0.705727\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.1830 1.90266 0.951329 0.308176i \(-0.0997185\pi\)
0.951329 + 0.308176i \(0.0997185\pi\)
\(42\) 0 0
\(43\) 8.18427 1.24809 0.624044 0.781389i \(-0.285488\pi\)
0.624044 + 0.781389i \(0.285488\pi\)
\(44\) 0 0
\(45\) 2.99703 0.446771
\(46\) 0 0
\(47\) 8.41641 1.22766 0.613829 0.789439i \(-0.289628\pi\)
0.613829 + 0.789439i \(0.289628\pi\)
\(48\) 0 0
\(49\) 6.82483 0.974976
\(50\) 0 0
\(51\) 0.162280 0.0227238
\(52\) 0 0
\(53\) −6.11335 −0.839733 −0.419867 0.907586i \(-0.637923\pi\)
−0.419867 + 0.907586i \(0.637923\pi\)
\(54\) 0 0
\(55\) 5.22019 0.703891
\(56\) 0 0
\(57\) 0.334315 0.0442810
\(58\) 0 0
\(59\) −1.53604 −0.199975 −0.0999876 0.994989i \(-0.531880\pi\)
−0.0999876 + 0.994989i \(0.531880\pi\)
\(60\) 0 0
\(61\) 7.09884 0.908914 0.454457 0.890769i \(-0.349833\pi\)
0.454457 + 0.890769i \(0.349833\pi\)
\(62\) 0 0
\(63\) 11.1435 1.40395
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −14.6928 −1.79501 −0.897507 0.441000i \(-0.854624\pi\)
−0.897507 + 0.441000i \(0.854624\pi\)
\(68\) 0 0
\(69\) −0.197499 −0.0237760
\(70\) 0 0
\(71\) −13.3972 −1.58995 −0.794975 0.606642i \(-0.792516\pi\)
−0.794975 + 0.606642i \(0.792516\pi\)
\(72\) 0 0
\(73\) 11.5465 1.35142 0.675708 0.737169i \(-0.263838\pi\)
0.675708 + 0.737169i \(0.263838\pi\)
\(74\) 0 0
\(75\) −0.0545075 −0.00629398
\(76\) 0 0
\(77\) 19.4096 2.21193
\(78\) 0 0
\(79\) −10.0334 −1.12885 −0.564424 0.825485i \(-0.690902\pi\)
−0.564424 + 0.825485i \(0.690902\pi\)
\(80\) 0 0
\(81\) 8.97327 0.997030
\(82\) 0 0
\(83\) −4.21504 −0.462661 −0.231330 0.972875i \(-0.574308\pi\)
−0.231330 + 0.972875i \(0.574308\pi\)
\(84\) 0 0
\(85\) 2.97721 0.322924
\(86\) 0 0
\(87\) 0.110853 0.0118846
\(88\) 0 0
\(89\) 7.86394 0.833576 0.416788 0.909004i \(-0.363156\pi\)
0.416788 + 0.909004i \(0.363156\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.00546375 0.000566565 0
\(94\) 0 0
\(95\) 6.13337 0.629270
\(96\) 0 0
\(97\) 12.6051 1.27986 0.639929 0.768434i \(-0.278964\pi\)
0.639929 + 0.768434i \(0.278964\pi\)
\(98\) 0 0
\(99\) 15.6451 1.57239
\(100\) 0 0
\(101\) 3.85725 0.383810 0.191905 0.981413i \(-0.438533\pi\)
0.191905 + 0.981413i \(0.438533\pi\)
\(102\) 0 0
\(103\) −1.09068 −0.107467 −0.0537337 0.998555i \(-0.517112\pi\)
−0.0537337 + 0.998555i \(0.517112\pi\)
\(104\) 0 0
\(105\) −0.202668 −0.0197784
\(106\) 0 0
\(107\) 5.55026 0.536563 0.268282 0.963340i \(-0.413544\pi\)
0.268282 + 0.963340i \(0.413544\pi\)
\(108\) 0 0
\(109\) 0.0361815 0.00346556 0.00173278 0.999998i \(-0.499448\pi\)
0.00173278 + 0.999998i \(0.499448\pi\)
\(110\) 0 0
\(111\) 0.399355 0.0379051
\(112\) 0 0
\(113\) −13.3944 −1.26004 −0.630018 0.776581i \(-0.716952\pi\)
−0.630018 + 0.776581i \(0.716952\pi\)
\(114\) 0 0
\(115\) −3.62333 −0.337877
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.0698 1.01477
\(120\) 0 0
\(121\) 16.2504 1.47731
\(122\) 0 0
\(123\) −0.664063 −0.0598765
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 11.2976 1.00250 0.501248 0.865303i \(-0.332874\pi\)
0.501248 + 0.865303i \(0.332874\pi\)
\(128\) 0 0
\(129\) −0.446104 −0.0392772
\(130\) 0 0
\(131\) 1.04287 0.0911160 0.0455580 0.998962i \(-0.485493\pi\)
0.0455580 + 0.998962i \(0.485493\pi\)
\(132\) 0 0
\(133\) 22.8049 1.97744
\(134\) 0 0
\(135\) −0.326883 −0.0281336
\(136\) 0 0
\(137\) −17.8330 −1.52358 −0.761789 0.647825i \(-0.775679\pi\)
−0.761789 + 0.647825i \(0.775679\pi\)
\(138\) 0 0
\(139\) −17.5662 −1.48995 −0.744973 0.667095i \(-0.767538\pi\)
−0.744973 + 0.667095i \(0.767538\pi\)
\(140\) 0 0
\(141\) −0.458757 −0.0386343
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.03371 0.168891
\(146\) 0 0
\(147\) −0.372004 −0.0306824
\(148\) 0 0
\(149\) −2.54599 −0.208575 −0.104288 0.994547i \(-0.533256\pi\)
−0.104288 + 0.994547i \(0.533256\pi\)
\(150\) 0 0
\(151\) −0.213212 −0.0173510 −0.00867548 0.999962i \(-0.502762\pi\)
−0.00867548 + 0.999962i \(0.502762\pi\)
\(152\) 0 0
\(153\) 8.92278 0.721364
\(154\) 0 0
\(155\) 0.100239 0.00805136
\(156\) 0 0
\(157\) −23.4111 −1.86841 −0.934205 0.356736i \(-0.883890\pi\)
−0.934205 + 0.356736i \(0.883890\pi\)
\(158\) 0 0
\(159\) 0.333223 0.0264263
\(160\) 0 0
\(161\) −13.4722 −1.06176
\(162\) 0 0
\(163\) 7.31629 0.573056 0.286528 0.958072i \(-0.407499\pi\)
0.286528 + 0.958072i \(0.407499\pi\)
\(164\) 0 0
\(165\) −0.284540 −0.0221514
\(166\) 0 0
\(167\) −13.3675 −1.03441 −0.517203 0.855863i \(-0.673027\pi\)
−0.517203 + 0.855863i \(0.673027\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 18.3819 1.40570
\(172\) 0 0
\(173\) −15.0657 −1.14542 −0.572711 0.819758i \(-0.694108\pi\)
−0.572711 + 0.819758i \(0.694108\pi\)
\(174\) 0 0
\(175\) −3.71818 −0.281068
\(176\) 0 0
\(177\) 0.0837256 0.00629320
\(178\) 0 0
\(179\) 16.1015 1.20348 0.601741 0.798691i \(-0.294474\pi\)
0.601741 + 0.798691i \(0.294474\pi\)
\(180\) 0 0
\(181\) 19.0534 1.41623 0.708116 0.706096i \(-0.249546\pi\)
0.708116 + 0.706096i \(0.249546\pi\)
\(182\) 0 0
\(183\) −0.386940 −0.0286034
\(184\) 0 0
\(185\) 7.32662 0.538664
\(186\) 0 0
\(187\) 15.5416 1.13651
\(188\) 0 0
\(189\) −1.21541 −0.0884079
\(190\) 0 0
\(191\) 4.29896 0.311062 0.155531 0.987831i \(-0.450291\pi\)
0.155531 + 0.987831i \(0.450291\pi\)
\(192\) 0 0
\(193\) −23.4460 −1.68768 −0.843839 0.536596i \(-0.819710\pi\)
−0.843839 + 0.536596i \(0.819710\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.687466 −0.0489799 −0.0244899 0.999700i \(-0.507796\pi\)
−0.0244899 + 0.999700i \(0.507796\pi\)
\(198\) 0 0
\(199\) 1.75780 0.124607 0.0623036 0.998057i \(-0.480155\pi\)
0.0623036 + 0.998057i \(0.480155\pi\)
\(200\) 0 0
\(201\) 0.800869 0.0564889
\(202\) 0 0
\(203\) 7.56170 0.530727
\(204\) 0 0
\(205\) −12.1830 −0.850895
\(206\) 0 0
\(207\) −10.8592 −0.754768
\(208\) 0 0
\(209\) 32.0174 2.21469
\(210\) 0 0
\(211\) 4.24474 0.292220 0.146110 0.989268i \(-0.453325\pi\)
0.146110 + 0.989268i \(0.453325\pi\)
\(212\) 0 0
\(213\) 0.730245 0.0500356
\(214\) 0 0
\(215\) −8.18427 −0.558162
\(216\) 0 0
\(217\) 0.372705 0.0253008
\(218\) 0 0
\(219\) −0.629371 −0.0425290
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.8505 0.994462 0.497231 0.867618i \(-0.334350\pi\)
0.497231 + 0.867618i \(0.334350\pi\)
\(224\) 0 0
\(225\) −2.99703 −0.199802
\(226\) 0 0
\(227\) −21.4849 −1.42600 −0.713002 0.701162i \(-0.752665\pi\)
−0.713002 + 0.701162i \(0.752665\pi\)
\(228\) 0 0
\(229\) 18.1462 1.19913 0.599566 0.800325i \(-0.295340\pi\)
0.599566 + 0.800325i \(0.295340\pi\)
\(230\) 0 0
\(231\) −1.05797 −0.0696092
\(232\) 0 0
\(233\) 12.3377 0.808271 0.404136 0.914699i \(-0.367572\pi\)
0.404136 + 0.914699i \(0.367572\pi\)
\(234\) 0 0
\(235\) −8.41641 −0.549026
\(236\) 0 0
\(237\) 0.546896 0.0355247
\(238\) 0 0
\(239\) −5.05214 −0.326795 −0.163398 0.986560i \(-0.552245\pi\)
−0.163398 + 0.986560i \(0.552245\pi\)
\(240\) 0 0
\(241\) −14.4408 −0.930215 −0.465108 0.885254i \(-0.653984\pi\)
−0.465108 + 0.885254i \(0.653984\pi\)
\(242\) 0 0
\(243\) −1.46976 −0.0942851
\(244\) 0 0
\(245\) −6.82483 −0.436023
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.229751 0.0145599
\(250\) 0 0
\(251\) −4.44614 −0.280638 −0.140319 0.990106i \(-0.544813\pi\)
−0.140319 + 0.990106i \(0.544813\pi\)
\(252\) 0 0
\(253\) −18.9145 −1.18914
\(254\) 0 0
\(255\) −0.162280 −0.0101624
\(256\) 0 0
\(257\) 1.38523 0.0864083 0.0432042 0.999066i \(-0.486243\pi\)
0.0432042 + 0.999066i \(0.486243\pi\)
\(258\) 0 0
\(259\) 27.2417 1.69271
\(260\) 0 0
\(261\) 6.09509 0.377277
\(262\) 0 0
\(263\) −27.6676 −1.70606 −0.853028 0.521866i \(-0.825236\pi\)
−0.853028 + 0.521866i \(0.825236\pi\)
\(264\) 0 0
\(265\) 6.11335 0.375540
\(266\) 0 0
\(267\) −0.428644 −0.0262326
\(268\) 0 0
\(269\) 19.4679 1.18698 0.593491 0.804841i \(-0.297749\pi\)
0.593491 + 0.804841i \(0.297749\pi\)
\(270\) 0 0
\(271\) 5.45218 0.331196 0.165598 0.986193i \(-0.447045\pi\)
0.165598 + 0.986193i \(0.447045\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.22019 −0.314789
\(276\) 0 0
\(277\) 26.9111 1.61693 0.808465 0.588545i \(-0.200299\pi\)
0.808465 + 0.588545i \(0.200299\pi\)
\(278\) 0 0
\(279\) 0.300418 0.0179856
\(280\) 0 0
\(281\) 18.2499 1.08870 0.544348 0.838860i \(-0.316777\pi\)
0.544348 + 0.838860i \(0.316777\pi\)
\(282\) 0 0
\(283\) 0.916102 0.0544566 0.0272283 0.999629i \(-0.491332\pi\)
0.0272283 + 0.999629i \(0.491332\pi\)
\(284\) 0 0
\(285\) −0.334315 −0.0198031
\(286\) 0 0
\(287\) −45.2984 −2.67388
\(288\) 0 0
\(289\) −8.13623 −0.478602
\(290\) 0 0
\(291\) −0.687074 −0.0402770
\(292\) 0 0
\(293\) −8.96873 −0.523959 −0.261979 0.965073i \(-0.584375\pi\)
−0.261979 + 0.965073i \(0.584375\pi\)
\(294\) 0 0
\(295\) 1.53604 0.0894316
\(296\) 0 0
\(297\) −1.70639 −0.0990149
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −30.4305 −1.75399
\(302\) 0 0
\(303\) −0.210249 −0.0120785
\(304\) 0 0
\(305\) −7.09884 −0.406479
\(306\) 0 0
\(307\) −31.1840 −1.77976 −0.889882 0.456192i \(-0.849213\pi\)
−0.889882 + 0.456192i \(0.849213\pi\)
\(308\) 0 0
\(309\) 0.0594500 0.00338199
\(310\) 0 0
\(311\) −10.1816 −0.577343 −0.288672 0.957428i \(-0.593214\pi\)
−0.288672 + 0.957428i \(0.593214\pi\)
\(312\) 0 0
\(313\) 30.3714 1.71670 0.858348 0.513068i \(-0.171491\pi\)
0.858348 + 0.513068i \(0.171491\pi\)
\(314\) 0 0
\(315\) −11.1435 −0.627864
\(316\) 0 0
\(317\) 8.20709 0.460956 0.230478 0.973078i \(-0.425971\pi\)
0.230478 + 0.973078i \(0.425971\pi\)
\(318\) 0 0
\(319\) 10.6164 0.594403
\(320\) 0 0
\(321\) −0.302530 −0.0168856
\(322\) 0 0
\(323\) 18.2603 1.01603
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −0.00197216 −0.000109061 0
\(328\) 0 0
\(329\) −31.2937 −1.72528
\(330\) 0 0
\(331\) −0.911191 −0.0500836 −0.0250418 0.999686i \(-0.507972\pi\)
−0.0250418 + 0.999686i \(0.507972\pi\)
\(332\) 0 0
\(333\) 21.9581 1.20330
\(334\) 0 0
\(335\) 14.6928 0.802755
\(336\) 0 0
\(337\) 3.54525 0.193122 0.0965611 0.995327i \(-0.469216\pi\)
0.0965611 + 0.995327i \(0.469216\pi\)
\(338\) 0 0
\(339\) 0.730093 0.0396532
\(340\) 0 0
\(341\) 0.523265 0.0283364
\(342\) 0 0
\(343\) 0.651306 0.0351672
\(344\) 0 0
\(345\) 0.197499 0.0106330
\(346\) 0 0
\(347\) 1.58496 0.0850849 0.0425424 0.999095i \(-0.486454\pi\)
0.0425424 + 0.999095i \(0.486454\pi\)
\(348\) 0 0
\(349\) 5.47813 0.293238 0.146619 0.989193i \(-0.453161\pi\)
0.146619 + 0.989193i \(0.453161\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.38823 0.233562 0.116781 0.993158i \(-0.462742\pi\)
0.116781 + 0.993158i \(0.462742\pi\)
\(354\) 0 0
\(355\) 13.3972 0.711047
\(356\) 0 0
\(357\) −0.603386 −0.0319346
\(358\) 0 0
\(359\) 14.2203 0.750521 0.375261 0.926919i \(-0.377553\pi\)
0.375261 + 0.926919i \(0.377553\pi\)
\(360\) 0 0
\(361\) 18.6182 0.979906
\(362\) 0 0
\(363\) −0.885769 −0.0464908
\(364\) 0 0
\(365\) −11.5465 −0.604372
\(366\) 0 0
\(367\) 9.30803 0.485875 0.242938 0.970042i \(-0.421889\pi\)
0.242938 + 0.970042i \(0.421889\pi\)
\(368\) 0 0
\(369\) −36.5127 −1.90077
\(370\) 0 0
\(371\) 22.7305 1.18011
\(372\) 0 0
\(373\) 30.8716 1.59847 0.799236 0.601018i \(-0.205238\pi\)
0.799236 + 0.601018i \(0.205238\pi\)
\(374\) 0 0
\(375\) 0.0545075 0.00281475
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 28.0188 1.43923 0.719616 0.694373i \(-0.244318\pi\)
0.719616 + 0.694373i \(0.244318\pi\)
\(380\) 0 0
\(381\) −0.615802 −0.0315485
\(382\) 0 0
\(383\) 23.9075 1.22162 0.610809 0.791778i \(-0.290844\pi\)
0.610809 + 0.791778i \(0.290844\pi\)
\(384\) 0 0
\(385\) −19.4096 −0.989205
\(386\) 0 0
\(387\) −24.5285 −1.24685
\(388\) 0 0
\(389\) −34.2391 −1.73599 −0.867996 0.496571i \(-0.834592\pi\)
−0.867996 + 0.496571i \(0.834592\pi\)
\(390\) 0 0
\(391\) −10.7874 −0.545542
\(392\) 0 0
\(393\) −0.0568443 −0.00286741
\(394\) 0 0
\(395\) 10.0334 0.504836
\(396\) 0 0
\(397\) 11.8187 0.593163 0.296581 0.955008i \(-0.404153\pi\)
0.296581 + 0.955008i \(0.404153\pi\)
\(398\) 0 0
\(399\) −1.24304 −0.0622299
\(400\) 0 0
\(401\) 8.98456 0.448667 0.224334 0.974512i \(-0.427979\pi\)
0.224334 + 0.974512i \(0.427979\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −8.97327 −0.445885
\(406\) 0 0
\(407\) 38.2464 1.89580
\(408\) 0 0
\(409\) 27.9072 1.37992 0.689962 0.723846i \(-0.257627\pi\)
0.689962 + 0.723846i \(0.257627\pi\)
\(410\) 0 0
\(411\) 0.972033 0.0479469
\(412\) 0 0
\(413\) 5.71126 0.281033
\(414\) 0 0
\(415\) 4.21504 0.206908
\(416\) 0 0
\(417\) 0.957489 0.0468885
\(418\) 0 0
\(419\) 25.6462 1.25290 0.626449 0.779462i \(-0.284508\pi\)
0.626449 + 0.779462i \(0.284508\pi\)
\(420\) 0 0
\(421\) 17.4613 0.851010 0.425505 0.904956i \(-0.360096\pi\)
0.425505 + 0.904956i \(0.360096\pi\)
\(422\) 0 0
\(423\) −25.2242 −1.22644
\(424\) 0 0
\(425\) −2.97721 −0.144416
\(426\) 0 0
\(427\) −26.3947 −1.27733
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.5131 −0.602735 −0.301368 0.953508i \(-0.597443\pi\)
−0.301368 + 0.953508i \(0.597443\pi\)
\(432\) 0 0
\(433\) 5.43503 0.261191 0.130595 0.991436i \(-0.458311\pi\)
0.130595 + 0.991436i \(0.458311\pi\)
\(434\) 0 0
\(435\) −0.110853 −0.00531497
\(436\) 0 0
\(437\) −22.2232 −1.06308
\(438\) 0 0
\(439\) 6.52958 0.311640 0.155820 0.987785i \(-0.450198\pi\)
0.155820 + 0.987785i \(0.450198\pi\)
\(440\) 0 0
\(441\) −20.4542 −0.974010
\(442\) 0 0
\(443\) 11.9530 0.567905 0.283952 0.958838i \(-0.408354\pi\)
0.283952 + 0.958838i \(0.408354\pi\)
\(444\) 0 0
\(445\) −7.86394 −0.372787
\(446\) 0 0
\(447\) 0.138775 0.00656385
\(448\) 0 0
\(449\) 22.0981 1.04288 0.521438 0.853289i \(-0.325396\pi\)
0.521438 + 0.853289i \(0.325396\pi\)
\(450\) 0 0
\(451\) −63.5974 −2.99469
\(452\) 0 0
\(453\) 0.0116217 0.000546033 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.0669 −1.12580 −0.562902 0.826524i \(-0.690315\pi\)
−0.562902 + 0.826524i \(0.690315\pi\)
\(458\) 0 0
\(459\) −0.973199 −0.0454250
\(460\) 0 0
\(461\) 41.8691 1.95004 0.975020 0.222117i \(-0.0712969\pi\)
0.975020 + 0.222117i \(0.0712969\pi\)
\(462\) 0 0
\(463\) −36.9718 −1.71823 −0.859113 0.511786i \(-0.828984\pi\)
−0.859113 + 0.511786i \(0.828984\pi\)
\(464\) 0 0
\(465\) −0.00546375 −0.000253376 0
\(466\) 0 0
\(467\) −24.1048 −1.11544 −0.557718 0.830030i \(-0.688323\pi\)
−0.557718 + 0.830030i \(0.688323\pi\)
\(468\) 0 0
\(469\) 54.6305 2.52260
\(470\) 0 0
\(471\) 1.27608 0.0587987
\(472\) 0 0
\(473\) −42.7235 −1.96443
\(474\) 0 0
\(475\) −6.13337 −0.281418
\(476\) 0 0
\(477\) 18.3219 0.838902
\(478\) 0 0
\(479\) 43.0487 1.96694 0.983472 0.181062i \(-0.0579535\pi\)
0.983472 + 0.181062i \(0.0579535\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0.734334 0.0334134
\(484\) 0 0
\(485\) −12.6051 −0.572370
\(486\) 0 0
\(487\) −7.62896 −0.345701 −0.172851 0.984948i \(-0.555298\pi\)
−0.172851 + 0.984948i \(0.555298\pi\)
\(488\) 0 0
\(489\) −0.398793 −0.0180340
\(490\) 0 0
\(491\) −20.6025 −0.929780 −0.464890 0.885369i \(-0.653906\pi\)
−0.464890 + 0.885369i \(0.653906\pi\)
\(492\) 0 0
\(493\) 6.05478 0.272694
\(494\) 0 0
\(495\) −15.6451 −0.703194
\(496\) 0 0
\(497\) 49.8130 2.23442
\(498\) 0 0
\(499\) −8.06951 −0.361241 −0.180620 0.983553i \(-0.557811\pi\)
−0.180620 + 0.983553i \(0.557811\pi\)
\(500\) 0 0
\(501\) 0.728627 0.0325527
\(502\) 0 0
\(503\) −19.6068 −0.874225 −0.437112 0.899407i \(-0.643999\pi\)
−0.437112 + 0.899407i \(0.643999\pi\)
\(504\) 0 0
\(505\) −3.85725 −0.171645
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.0479 −0.534015 −0.267008 0.963694i \(-0.586035\pi\)
−0.267008 + 0.963694i \(0.586035\pi\)
\(510\) 0 0
\(511\) −42.9320 −1.89920
\(512\) 0 0
\(513\) −2.00489 −0.0885182
\(514\) 0 0
\(515\) 1.09068 0.0480609
\(516\) 0 0
\(517\) −43.9353 −1.93227
\(518\) 0 0
\(519\) 0.821191 0.0360463
\(520\) 0 0
\(521\) −13.8557 −0.607028 −0.303514 0.952827i \(-0.598160\pi\)
−0.303514 + 0.952827i \(0.598160\pi\)
\(522\) 0 0
\(523\) −26.7263 −1.16866 −0.584329 0.811517i \(-0.698642\pi\)
−0.584329 + 0.811517i \(0.698642\pi\)
\(524\) 0 0
\(525\) 0.202668 0.00884517
\(526\) 0 0
\(527\) 0.298431 0.0129999
\(528\) 0 0
\(529\) −9.87149 −0.429195
\(530\) 0 0
\(531\) 4.60355 0.199777
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −5.55026 −0.239958
\(536\) 0 0
\(537\) −0.877652 −0.0378735
\(538\) 0 0
\(539\) −35.6269 −1.53456
\(540\) 0 0
\(541\) 25.7586 1.10745 0.553725 0.832700i \(-0.313206\pi\)
0.553725 + 0.832700i \(0.313206\pi\)
\(542\) 0 0
\(543\) −1.03856 −0.0445687
\(544\) 0 0
\(545\) −0.0361815 −0.00154985
\(546\) 0 0
\(547\) −14.2898 −0.610987 −0.305493 0.952194i \(-0.598821\pi\)
−0.305493 + 0.952194i \(0.598821\pi\)
\(548\) 0 0
\(549\) −21.2754 −0.908014
\(550\) 0 0
\(551\) 12.4735 0.531389
\(552\) 0 0
\(553\) 37.3060 1.58641
\(554\) 0 0
\(555\) −0.399355 −0.0169517
\(556\) 0 0
\(557\) −18.9439 −0.802679 −0.401340 0.915929i \(-0.631455\pi\)
−0.401340 + 0.915929i \(0.631455\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.847134 −0.0357660
\(562\) 0 0
\(563\) −5.58658 −0.235446 −0.117723 0.993046i \(-0.537560\pi\)
−0.117723 + 0.993046i \(0.537560\pi\)
\(564\) 0 0
\(565\) 13.3944 0.563505
\(566\) 0 0
\(567\) −33.3642 −1.40116
\(568\) 0 0
\(569\) −27.2514 −1.14244 −0.571219 0.820798i \(-0.693529\pi\)
−0.571219 + 0.820798i \(0.693529\pi\)
\(570\) 0 0
\(571\) −16.8253 −0.704118 −0.352059 0.935978i \(-0.614518\pi\)
−0.352059 + 0.935978i \(0.614518\pi\)
\(572\) 0 0
\(573\) −0.234325 −0.00978909
\(574\) 0 0
\(575\) 3.62333 0.151103
\(576\) 0 0
\(577\) 30.8911 1.28601 0.643007 0.765861i \(-0.277687\pi\)
0.643007 + 0.765861i \(0.277687\pi\)
\(578\) 0 0
\(579\) 1.27798 0.0531111
\(580\) 0 0
\(581\) 15.6723 0.650195
\(582\) 0 0
\(583\) 31.9129 1.32170
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.67152 −0.234089 −0.117044 0.993127i \(-0.537342\pi\)
−0.117044 + 0.993127i \(0.537342\pi\)
\(588\) 0 0
\(589\) 0.614800 0.0253324
\(590\) 0 0
\(591\) 0.0374720 0.00154139
\(592\) 0 0
\(593\) −6.11872 −0.251266 −0.125633 0.992077i \(-0.540096\pi\)
−0.125633 + 0.992077i \(0.540096\pi\)
\(594\) 0 0
\(595\) −11.0698 −0.453817
\(596\) 0 0
\(597\) −0.0958133 −0.00392138
\(598\) 0 0
\(599\) 39.6757 1.62111 0.810553 0.585665i \(-0.199166\pi\)
0.810553 + 0.585665i \(0.199166\pi\)
\(600\) 0 0
\(601\) −44.2642 −1.80558 −0.902788 0.430086i \(-0.858483\pi\)
−0.902788 + 0.430086i \(0.858483\pi\)
\(602\) 0 0
\(603\) 44.0348 1.79324
\(604\) 0 0
\(605\) −16.2504 −0.660673
\(606\) 0 0
\(607\) 21.5056 0.872884 0.436442 0.899732i \(-0.356238\pi\)
0.436442 + 0.899732i \(0.356238\pi\)
\(608\) 0 0
\(609\) −0.412169 −0.0167019
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 14.3306 0.578809 0.289404 0.957207i \(-0.406543\pi\)
0.289404 + 0.957207i \(0.406543\pi\)
\(614\) 0 0
\(615\) 0.664063 0.0267776
\(616\) 0 0
\(617\) 31.0397 1.24961 0.624806 0.780780i \(-0.285178\pi\)
0.624806 + 0.780780i \(0.285178\pi\)
\(618\) 0 0
\(619\) −0.892227 −0.0358616 −0.0179308 0.999839i \(-0.505708\pi\)
−0.0179308 + 0.999839i \(0.505708\pi\)
\(620\) 0 0
\(621\) 1.18440 0.0475285
\(622\) 0 0
\(623\) −29.2395 −1.17146
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.74519 −0.0696960
\(628\) 0 0
\(629\) 21.8129 0.869736
\(630\) 0 0
\(631\) 17.5918 0.700318 0.350159 0.936690i \(-0.386128\pi\)
0.350159 + 0.936690i \(0.386128\pi\)
\(632\) 0 0
\(633\) −0.231370 −0.00919613
\(634\) 0 0
\(635\) −11.2976 −0.448330
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 40.1517 1.58838
\(640\) 0 0
\(641\) −18.3751 −0.725772 −0.362886 0.931834i \(-0.618209\pi\)
−0.362886 + 0.931834i \(0.618209\pi\)
\(642\) 0 0
\(643\) −27.4520 −1.08260 −0.541300 0.840830i \(-0.682068\pi\)
−0.541300 + 0.840830i \(0.682068\pi\)
\(644\) 0 0
\(645\) 0.446104 0.0175653
\(646\) 0 0
\(647\) −39.3570 −1.54728 −0.773642 0.633623i \(-0.781567\pi\)
−0.773642 + 0.633623i \(0.781567\pi\)
\(648\) 0 0
\(649\) 8.01842 0.314750
\(650\) 0 0
\(651\) −0.0203152 −0.000796215 0
\(652\) 0 0
\(653\) −11.8592 −0.464088 −0.232044 0.972705i \(-0.574541\pi\)
−0.232044 + 0.972705i \(0.574541\pi\)
\(654\) 0 0
\(655\) −1.04287 −0.0407483
\(656\) 0 0
\(657\) −34.6052 −1.35008
\(658\) 0 0
\(659\) −23.3963 −0.911392 −0.455696 0.890136i \(-0.650610\pi\)
−0.455696 + 0.890136i \(0.650610\pi\)
\(660\) 0 0
\(661\) −9.17424 −0.356837 −0.178418 0.983955i \(-0.557098\pi\)
−0.178418 + 0.983955i \(0.557098\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −22.8049 −0.884338
\(666\) 0 0
\(667\) −7.36881 −0.285321
\(668\) 0 0
\(669\) −0.809463 −0.0312956
\(670\) 0 0
\(671\) −37.0573 −1.43058
\(672\) 0 0
\(673\) 9.18712 0.354138 0.177069 0.984198i \(-0.443338\pi\)
0.177069 + 0.984198i \(0.443338\pi\)
\(674\) 0 0
\(675\) 0.326883 0.0125817
\(676\) 0 0
\(677\) −20.7895 −0.799007 −0.399503 0.916732i \(-0.630817\pi\)
−0.399503 + 0.916732i \(0.630817\pi\)
\(678\) 0 0
\(679\) −46.8681 −1.79863
\(680\) 0 0
\(681\) 1.17109 0.0448762
\(682\) 0 0
\(683\) 16.0263 0.613229 0.306614 0.951834i \(-0.400804\pi\)
0.306614 + 0.951834i \(0.400804\pi\)
\(684\) 0 0
\(685\) 17.8330 0.681365
\(686\) 0 0
\(687\) −0.989101 −0.0377366
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −44.9971 −1.71177 −0.855885 0.517167i \(-0.826987\pi\)
−0.855885 + 0.517167i \(0.826987\pi\)
\(692\) 0 0
\(693\) −58.1711 −2.20974
\(694\) 0 0
\(695\) 17.5662 0.666324
\(696\) 0 0
\(697\) −36.2712 −1.37387
\(698\) 0 0
\(699\) −0.672498 −0.0254362
\(700\) 0 0
\(701\) 32.9469 1.24439 0.622193 0.782864i \(-0.286242\pi\)
0.622193 + 0.782864i \(0.286242\pi\)
\(702\) 0 0
\(703\) 44.9369 1.69483
\(704\) 0 0
\(705\) 0.458757 0.0172778
\(706\) 0 0
\(707\) −14.3419 −0.539383
\(708\) 0 0
\(709\) 45.9596 1.72605 0.863025 0.505162i \(-0.168567\pi\)
0.863025 + 0.505162i \(0.168567\pi\)
\(710\) 0 0
\(711\) 30.0704 1.12773
\(712\) 0 0
\(713\) −0.363197 −0.0136018
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.275379 0.0102842
\(718\) 0 0
\(719\) −5.25214 −0.195872 −0.0979358 0.995193i \(-0.531224\pi\)
−0.0979358 + 0.995193i \(0.531224\pi\)
\(720\) 0 0
\(721\) 4.05532 0.151028
\(722\) 0 0
\(723\) 0.787133 0.0292738
\(724\) 0 0
\(725\) −2.03371 −0.0755302
\(726\) 0 0
\(727\) 28.5596 1.05922 0.529608 0.848243i \(-0.322339\pi\)
0.529608 + 0.848243i \(0.322339\pi\)
\(728\) 0 0
\(729\) −26.8397 −0.994063
\(730\) 0 0
\(731\) −24.3663 −0.901219
\(732\) 0 0
\(733\) −9.25426 −0.341814 −0.170907 0.985287i \(-0.554670\pi\)
−0.170907 + 0.985287i \(0.554670\pi\)
\(734\) 0 0
\(735\) 0.372004 0.0137216
\(736\) 0 0
\(737\) 76.6994 2.82526
\(738\) 0 0
\(739\) 32.7910 1.20623 0.603117 0.797652i \(-0.293925\pi\)
0.603117 + 0.797652i \(0.293925\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.7645 0.651716 0.325858 0.945419i \(-0.394347\pi\)
0.325858 + 0.945419i \(0.394347\pi\)
\(744\) 0 0
\(745\) 2.54599 0.0932777
\(746\) 0 0
\(747\) 12.6326 0.462202
\(748\) 0 0
\(749\) −20.6368 −0.754053
\(750\) 0 0
\(751\) 38.7659 1.41459 0.707293 0.706920i \(-0.249916\pi\)
0.707293 + 0.706920i \(0.249916\pi\)
\(752\) 0 0
\(753\) 0.242348 0.00883165
\(754\) 0 0
\(755\) 0.213212 0.00775959
\(756\) 0 0
\(757\) −18.0592 −0.656374 −0.328187 0.944613i \(-0.606438\pi\)
−0.328187 + 0.944613i \(0.606438\pi\)
\(758\) 0 0
\(759\) 1.03098 0.0374222
\(760\) 0 0
\(761\) −20.6089 −0.747071 −0.373535 0.927616i \(-0.621855\pi\)
−0.373535 + 0.927616i \(0.621855\pi\)
\(762\) 0 0
\(763\) −0.134529 −0.00487028
\(764\) 0 0
\(765\) −8.92278 −0.322604
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 34.1017 1.22974 0.614869 0.788629i \(-0.289209\pi\)
0.614869 + 0.788629i \(0.289209\pi\)
\(770\) 0 0
\(771\) −0.0755054 −0.00271926
\(772\) 0 0
\(773\) 16.6649 0.599396 0.299698 0.954034i \(-0.403114\pi\)
0.299698 + 0.954034i \(0.403114\pi\)
\(774\) 0 0
\(775\) −0.100239 −0.00360068
\(776\) 0 0
\(777\) −1.48487 −0.0532696
\(778\) 0 0
\(779\) −74.7226 −2.67722
\(780\) 0 0
\(781\) 69.9357 2.50250
\(782\) 0 0
\(783\) −0.664786 −0.0237575
\(784\) 0 0
\(785\) 23.4111 0.835579
\(786\) 0 0
\(787\) −1.29371 −0.0461157 −0.0230579 0.999734i \(-0.507340\pi\)
−0.0230579 + 0.999734i \(0.507340\pi\)
\(788\) 0 0
\(789\) 1.50809 0.0536894
\(790\) 0 0
\(791\) 49.8026 1.77078
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −0.333223 −0.0118182
\(796\) 0 0
\(797\) 7.56095 0.267823 0.133911 0.990993i \(-0.457246\pi\)
0.133911 + 0.990993i \(0.457246\pi\)
\(798\) 0 0
\(799\) −25.0574 −0.886467
\(800\) 0 0
\(801\) −23.5685 −0.832751
\(802\) 0 0
\(803\) −60.2750 −2.12706
\(804\) 0 0
\(805\) 13.4722 0.474832
\(806\) 0 0
\(807\) −1.06115 −0.0373542
\(808\) 0 0
\(809\) −34.4654 −1.21174 −0.605869 0.795564i \(-0.707174\pi\)
−0.605869 + 0.795564i \(0.707174\pi\)
\(810\) 0 0
\(811\) −7.78668 −0.273427 −0.136714 0.990611i \(-0.543654\pi\)
−0.136714 + 0.990611i \(0.543654\pi\)
\(812\) 0 0
\(813\) −0.297185 −0.0104227
\(814\) 0 0
\(815\) −7.31629 −0.256279
\(816\) 0 0
\(817\) −50.1971 −1.75618
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 36.5174 1.27447 0.637234 0.770671i \(-0.280079\pi\)
0.637234 + 0.770671i \(0.280079\pi\)
\(822\) 0 0
\(823\) 46.9851 1.63780 0.818898 0.573939i \(-0.194585\pi\)
0.818898 + 0.573939i \(0.194585\pi\)
\(824\) 0 0
\(825\) 0.284540 0.00990640
\(826\) 0 0
\(827\) −36.9816 −1.28598 −0.642989 0.765876i \(-0.722306\pi\)
−0.642989 + 0.765876i \(0.722306\pi\)
\(828\) 0 0
\(829\) 12.0453 0.418351 0.209175 0.977878i \(-0.432922\pi\)
0.209175 + 0.977878i \(0.432922\pi\)
\(830\) 0 0
\(831\) −1.46685 −0.0508846
\(832\) 0 0
\(833\) −20.3189 −0.704010
\(834\) 0 0
\(835\) 13.3675 0.462600
\(836\) 0 0
\(837\) −0.0327663 −0.00113257
\(838\) 0 0
\(839\) −27.6041 −0.953000 −0.476500 0.879175i \(-0.658095\pi\)
−0.476500 + 0.879175i \(0.658095\pi\)
\(840\) 0 0
\(841\) −24.8640 −0.857380
\(842\) 0 0
\(843\) −0.994754 −0.0342612
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −60.4219 −2.07612
\(848\) 0 0
\(849\) −0.0499344 −0.00171375
\(850\) 0 0
\(851\) −26.5467 −0.910011
\(852\) 0 0
\(853\) 3.78477 0.129588 0.0647940 0.997899i \(-0.479361\pi\)
0.0647940 + 0.997899i \(0.479361\pi\)
\(854\) 0 0
\(855\) −18.3819 −0.628647
\(856\) 0 0
\(857\) 20.4897 0.699915 0.349958 0.936765i \(-0.386196\pi\)
0.349958 + 0.936765i \(0.386196\pi\)
\(858\) 0 0
\(859\) 25.6031 0.873568 0.436784 0.899567i \(-0.356117\pi\)
0.436784 + 0.899567i \(0.356117\pi\)
\(860\) 0 0
\(861\) 2.46910 0.0841468
\(862\) 0 0
\(863\) −20.7175 −0.705232 −0.352616 0.935768i \(-0.614708\pi\)
−0.352616 + 0.935768i \(0.614708\pi\)
\(864\) 0 0
\(865\) 15.0657 0.512248
\(866\) 0 0
\(867\) 0.443486 0.0150616
\(868\) 0 0
\(869\) 52.3764 1.77675
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −37.7779 −1.27859
\(874\) 0 0
\(875\) 3.71818 0.125697
\(876\) 0 0
\(877\) 43.2388 1.46007 0.730036 0.683409i \(-0.239503\pi\)
0.730036 + 0.683409i \(0.239503\pi\)
\(878\) 0 0
\(879\) 0.488863 0.0164889
\(880\) 0 0
\(881\) −56.7984 −1.91359 −0.956793 0.290768i \(-0.906089\pi\)
−0.956793 + 0.290768i \(0.906089\pi\)
\(882\) 0 0
\(883\) −50.6651 −1.70502 −0.852508 0.522715i \(-0.824919\pi\)
−0.852508 + 0.522715i \(0.824919\pi\)
\(884\) 0 0
\(885\) −0.0837256 −0.00281440
\(886\) 0 0
\(887\) −19.8733 −0.667279 −0.333640 0.942701i \(-0.608277\pi\)
−0.333640 + 0.942701i \(0.608277\pi\)
\(888\) 0 0
\(889\) −42.0063 −1.40885
\(890\) 0 0
\(891\) −46.8422 −1.56927
\(892\) 0 0
\(893\) −51.6209 −1.72743
\(894\) 0 0
\(895\) −16.1015 −0.538214
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.203856 0.00679899
\(900\) 0 0
\(901\) 18.2007 0.606354
\(902\) 0 0
\(903\) 1.65869 0.0551978
\(904\) 0 0
\(905\) −19.0534 −0.633358
\(906\) 0 0
\(907\) 24.3321 0.807935 0.403967 0.914773i \(-0.367631\pi\)
0.403967 + 0.914773i \(0.367631\pi\)
\(908\) 0 0
\(909\) −11.5603 −0.383430
\(910\) 0 0
\(911\) −31.5701 −1.04596 −0.522981 0.852344i \(-0.675180\pi\)
−0.522981 + 0.852344i \(0.675180\pi\)
\(912\) 0 0
\(913\) 22.0033 0.728203
\(914\) 0 0
\(915\) 0.386940 0.0127918
\(916\) 0 0
\(917\) −3.87758 −0.128049
\(918\) 0 0
\(919\) −27.9720 −0.922711 −0.461356 0.887215i \(-0.652637\pi\)
−0.461356 + 0.887215i \(0.652637\pi\)
\(920\) 0 0
\(921\) 1.69976 0.0560090
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −7.32662 −0.240898
\(926\) 0 0
\(927\) 3.26879 0.107361
\(928\) 0 0
\(929\) 30.0456 0.985765 0.492882 0.870096i \(-0.335943\pi\)
0.492882 + 0.870096i \(0.335943\pi\)
\(930\) 0 0
\(931\) −41.8592 −1.37188
\(932\) 0 0
\(933\) 0.554971 0.0181689
\(934\) 0 0
\(935\) −15.5416 −0.508265
\(936\) 0 0
\(937\) 35.2378 1.15117 0.575584 0.817743i \(-0.304775\pi\)
0.575584 + 0.817743i \(0.304775\pi\)
\(938\) 0 0
\(939\) −1.65547 −0.0540243
\(940\) 0 0
\(941\) −19.2578 −0.627785 −0.313892 0.949459i \(-0.601633\pi\)
−0.313892 + 0.949459i \(0.601633\pi\)
\(942\) 0 0
\(943\) 44.1429 1.43749
\(944\) 0 0
\(945\) 1.21541 0.0395372
\(946\) 0 0
\(947\) −43.7557 −1.42187 −0.710935 0.703257i \(-0.751728\pi\)
−0.710935 + 0.703257i \(0.751728\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −0.447348 −0.0145063
\(952\) 0 0
\(953\) −31.5328 −1.02145 −0.510724 0.859745i \(-0.670623\pi\)
−0.510724 + 0.859745i \(0.670623\pi\)
\(954\) 0 0
\(955\) −4.29896 −0.139111
\(956\) 0 0
\(957\) −0.578672 −0.0187058
\(958\) 0 0
\(959\) 66.3063 2.14114
\(960\) 0 0
\(961\) −30.9900 −0.999676
\(962\) 0 0
\(963\) −16.6343 −0.536032
\(964\) 0 0
\(965\) 23.4460 0.754753
\(966\) 0 0
\(967\) 24.8301 0.798483 0.399242 0.916846i \(-0.369273\pi\)
0.399242 + 0.916846i \(0.369273\pi\)
\(968\) 0 0
\(969\) −0.995324 −0.0319744
\(970\) 0 0
\(971\) 31.9493 1.02530 0.512651 0.858597i \(-0.328664\pi\)
0.512651 + 0.858597i \(0.328664\pi\)
\(972\) 0 0
\(973\) 65.3142 2.09388
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −54.9763 −1.75885 −0.879424 0.476040i \(-0.842072\pi\)
−0.879424 + 0.476040i \(0.842072\pi\)
\(978\) 0 0
\(979\) −41.0513 −1.31201
\(980\) 0 0
\(981\) −0.108437 −0.00346213
\(982\) 0 0
\(983\) 20.4505 0.652270 0.326135 0.945323i \(-0.394254\pi\)
0.326135 + 0.945323i \(0.394254\pi\)
\(984\) 0 0
\(985\) 0.687466 0.0219045
\(986\) 0 0
\(987\) 1.70574 0.0542943
\(988\) 0 0
\(989\) 29.6543 0.942952
\(990\) 0 0
\(991\) −22.7815 −0.723680 −0.361840 0.932240i \(-0.617851\pi\)
−0.361840 + 0.932240i \(0.617851\pi\)
\(992\) 0 0
\(993\) 0.0496667 0.00157613
\(994\) 0 0
\(995\) −1.75780 −0.0557260
\(996\) 0 0
\(997\) −41.2518 −1.30646 −0.653229 0.757161i \(-0.726586\pi\)
−0.653229 + 0.757161i \(0.726586\pi\)
\(998\) 0 0
\(999\) −2.39495 −0.0757727
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 3380.2.a.r.1.6 9
13.5 odd 4 3380.2.f.j.3041.12 18
13.8 odd 4 3380.2.f.j.3041.11 18
13.12 even 2 3380.2.a.s.1.6 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.6 9 1.1 even 1 trivial
3380.2.a.s.1.6 yes 9 13.12 even 2
3380.2.f.j.3041.11 18 13.8 odd 4
3380.2.f.j.3041.12 18 13.5 odd 4