Properties

Label 320.3.b.c.191.3
Level $320$
Weight $3$
Character 320.191
Analytic conductor $8.719$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.3
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 320.191
Dual form 320.3.b.c.191.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.35114i q^{3} +2.23607 q^{5} -5.25731i q^{7} +3.47214 q^{9} +O(q^{10})\) \(q+2.35114i q^{3} +2.23607 q^{5} -5.25731i q^{7} +3.47214 q^{9} -19.9192i q^{11} +8.47214 q^{13} +5.25731i q^{15} +11.8885 q^{17} +15.2169i q^{19} +12.3607 q^{21} -0.555029i q^{23} +5.00000 q^{25} +29.3238i q^{27} +10.9443 q^{29} -8.29451i q^{31} +46.8328 q^{33} -11.7557i q^{35} +18.3607 q^{37} +19.9192i q^{39} -14.5836 q^{41} -22.2703i q^{43} +7.76393 q^{45} +53.3902i q^{47} +21.3607 q^{49} +27.9516i q^{51} +66.3607 q^{53} -44.5407i q^{55} -35.7771 q^{57} -17.4370i q^{59} -90.1378 q^{61} -18.2541i q^{63} +18.9443 q^{65} -50.2220i q^{67} +1.30495 q^{69} -80.7868i q^{71} -5.55418 q^{73} +11.7557i q^{75} -104.721 q^{77} -13.8448i q^{79} -37.6950 q^{81} +76.2155i q^{83} +26.5836 q^{85} +25.7315i q^{87} -111.443 q^{89} -44.5407i q^{91} +19.5016 q^{93} +34.0260i q^{95} -92.8328 q^{97} -69.1621i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 16q^{13} - 24q^{17} - 40q^{21} + 20q^{25} + 8q^{29} + 80q^{33} - 16q^{37} - 112q^{41} + 40q^{45} - 4q^{49} + 176q^{53} - 128q^{61} + 40q^{65} - 120q^{69} + 264q^{73} - 240q^{77} - 276q^{81} + 160q^{85} - 88q^{89} + 400q^{93} - 264q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.35114i 0.783714i 0.920026 + 0.391857i \(0.128167\pi\)
−0.920026 + 0.391857i \(0.871833\pi\)
\(4\) 0 0
\(5\) 2.23607 0.447214
\(6\) 0 0
\(7\) − 5.25731i − 0.751044i −0.926813 0.375522i \(-0.877463\pi\)
0.926813 0.375522i \(-0.122537\pi\)
\(8\) 0 0
\(9\) 3.47214 0.385793
\(10\) 0 0
\(11\) − 19.9192i − 1.81084i −0.424522 0.905418i \(-0.639558\pi\)
0.424522 0.905418i \(-0.360442\pi\)
\(12\) 0 0
\(13\) 8.47214 0.651703 0.325851 0.945421i \(-0.394349\pi\)
0.325851 + 0.945421i \(0.394349\pi\)
\(14\) 0 0
\(15\) 5.25731i 0.350487i
\(16\) 0 0
\(17\) 11.8885 0.699326 0.349663 0.936876i \(-0.386296\pi\)
0.349663 + 0.936876i \(0.386296\pi\)
\(18\) 0 0
\(19\) 15.2169i 0.800890i 0.916321 + 0.400445i \(0.131144\pi\)
−0.916321 + 0.400445i \(0.868856\pi\)
\(20\) 0 0
\(21\) 12.3607 0.588604
\(22\) 0 0
\(23\) − 0.555029i − 0.0241317i −0.999927 0.0120659i \(-0.996159\pi\)
0.999927 0.0120659i \(-0.00384077\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) 29.3238i 1.08606i
\(28\) 0 0
\(29\) 10.9443 0.377389 0.188694 0.982036i \(-0.439574\pi\)
0.188694 + 0.982036i \(0.439574\pi\)
\(30\) 0 0
\(31\) − 8.29451i − 0.267565i −0.991011 0.133782i \(-0.957288\pi\)
0.991011 0.133782i \(-0.0427123\pi\)
\(32\) 0 0
\(33\) 46.8328 1.41918
\(34\) 0 0
\(35\) − 11.7557i − 0.335877i
\(36\) 0 0
\(37\) 18.3607 0.496235 0.248117 0.968730i \(-0.420188\pi\)
0.248117 + 0.968730i \(0.420188\pi\)
\(38\) 0 0
\(39\) 19.9192i 0.510748i
\(40\) 0 0
\(41\) −14.5836 −0.355697 −0.177849 0.984058i \(-0.556914\pi\)
−0.177849 + 0.984058i \(0.556914\pi\)
\(42\) 0 0
\(43\) − 22.2703i − 0.517915i −0.965889 0.258957i \(-0.916621\pi\)
0.965889 0.258957i \(-0.0833789\pi\)
\(44\) 0 0
\(45\) 7.76393 0.172532
\(46\) 0 0
\(47\) 53.3902i 1.13596i 0.823042 + 0.567981i \(0.192275\pi\)
−0.823042 + 0.567981i \(0.807725\pi\)
\(48\) 0 0
\(49\) 21.3607 0.435932
\(50\) 0 0
\(51\) 27.9516i 0.548071i
\(52\) 0 0
\(53\) 66.3607 1.25209 0.626044 0.779788i \(-0.284673\pi\)
0.626044 + 0.779788i \(0.284673\pi\)
\(54\) 0 0
\(55\) − 44.5407i − 0.809830i
\(56\) 0 0
\(57\) −35.7771 −0.627668
\(58\) 0 0
\(59\) − 17.4370i − 0.295543i −0.989022 0.147771i \(-0.952790\pi\)
0.989022 0.147771i \(-0.0472100\pi\)
\(60\) 0 0
\(61\) −90.1378 −1.47767 −0.738834 0.673887i \(-0.764623\pi\)
−0.738834 + 0.673887i \(0.764623\pi\)
\(62\) 0 0
\(63\) − 18.2541i − 0.289748i
\(64\) 0 0
\(65\) 18.9443 0.291450
\(66\) 0 0
\(67\) − 50.2220i − 0.749582i −0.927109 0.374791i \(-0.877715\pi\)
0.927109 0.374791i \(-0.122285\pi\)
\(68\) 0 0
\(69\) 1.30495 0.0189123
\(70\) 0 0
\(71\) − 80.7868i − 1.13784i −0.822392 0.568921i \(-0.807361\pi\)
0.822392 0.568921i \(-0.192639\pi\)
\(72\) 0 0
\(73\) −5.55418 −0.0760846 −0.0380423 0.999276i \(-0.512112\pi\)
−0.0380423 + 0.999276i \(0.512112\pi\)
\(74\) 0 0
\(75\) 11.7557i 0.156743i
\(76\) 0 0
\(77\) −104.721 −1.36002
\(78\) 0 0
\(79\) − 13.8448i − 0.175251i −0.996154 0.0876253i \(-0.972072\pi\)
0.996154 0.0876253i \(-0.0279278\pi\)
\(80\) 0 0
\(81\) −37.6950 −0.465371
\(82\) 0 0
\(83\) 76.2155i 0.918260i 0.888369 + 0.459130i \(0.151839\pi\)
−0.888369 + 0.459130i \(0.848161\pi\)
\(84\) 0 0
\(85\) 26.5836 0.312748
\(86\) 0 0
\(87\) 25.7315i 0.295765i
\(88\) 0 0
\(89\) −111.443 −1.25217 −0.626083 0.779757i \(-0.715343\pi\)
−0.626083 + 0.779757i \(0.715343\pi\)
\(90\) 0 0
\(91\) − 44.5407i − 0.489458i
\(92\) 0 0
\(93\) 19.5016 0.209694
\(94\) 0 0
\(95\) 34.0260i 0.358169i
\(96\) 0 0
\(97\) −92.8328 −0.957039 −0.478520 0.878077i \(-0.658826\pi\)
−0.478520 + 0.878077i \(0.658826\pi\)
\(98\) 0 0
\(99\) − 69.1621i − 0.698607i
\(100\) 0 0
\(101\) −64.1115 −0.634767 −0.317383 0.948297i \(-0.602804\pi\)
−0.317383 + 0.948297i \(0.602804\pi\)
\(102\) 0 0
\(103\) 137.769i 1.33757i 0.743458 + 0.668783i \(0.233184\pi\)
−0.743458 + 0.668783i \(0.766816\pi\)
\(104\) 0 0
\(105\) 27.6393 0.263232
\(106\) 0 0
\(107\) 51.3320i 0.479739i 0.970805 + 0.239869i \(0.0771046\pi\)
−0.970805 + 0.239869i \(0.922895\pi\)
\(108\) 0 0
\(109\) −133.469 −1.22449 −0.612243 0.790669i \(-0.709733\pi\)
−0.612243 + 0.790669i \(0.709733\pi\)
\(110\) 0 0
\(111\) 43.1685i 0.388906i
\(112\) 0 0
\(113\) 170.721 1.51081 0.755404 0.655259i \(-0.227441\pi\)
0.755404 + 0.655259i \(0.227441\pi\)
\(114\) 0 0
\(115\) − 1.24108i − 0.0107920i
\(116\) 0 0
\(117\) 29.4164 0.251422
\(118\) 0 0
\(119\) − 62.5018i − 0.525225i
\(120\) 0 0
\(121\) −275.774 −2.27912
\(122\) 0 0
\(123\) − 34.2881i − 0.278765i
\(124\) 0 0
\(125\) 11.1803 0.0894427
\(126\) 0 0
\(127\) − 198.637i − 1.56407i −0.623235 0.782035i \(-0.714182\pi\)
0.623235 0.782035i \(-0.285818\pi\)
\(128\) 0 0
\(129\) 52.3607 0.405897
\(130\) 0 0
\(131\) − 7.77041i − 0.0593161i −0.999560 0.0296580i \(-0.990558\pi\)
0.999560 0.0296580i \(-0.00944183\pi\)
\(132\) 0 0
\(133\) 80.0000 0.601504
\(134\) 0 0
\(135\) 65.5699i 0.485703i
\(136\) 0 0
\(137\) 0.832816 0.00607895 0.00303947 0.999995i \(-0.499033\pi\)
0.00303947 + 0.999995i \(0.499033\pi\)
\(138\) 0 0
\(139\) 237.658i 1.70977i 0.518817 + 0.854885i \(0.326373\pi\)
−0.518817 + 0.854885i \(0.673627\pi\)
\(140\) 0 0
\(141\) −125.528 −0.890269
\(142\) 0 0
\(143\) − 168.758i − 1.18013i
\(144\) 0 0
\(145\) 24.4721 0.168773
\(146\) 0 0
\(147\) 50.2220i 0.341646i
\(148\) 0 0
\(149\) 36.9706 0.248125 0.124062 0.992274i \(-0.460408\pi\)
0.124062 + 0.992274i \(0.460408\pi\)
\(150\) 0 0
\(151\) 282.723i 1.87234i 0.351552 + 0.936168i \(0.385654\pi\)
−0.351552 + 0.936168i \(0.614346\pi\)
\(152\) 0 0
\(153\) 41.2786 0.269795
\(154\) 0 0
\(155\) − 18.5471i − 0.119659i
\(156\) 0 0
\(157\) −204.748 −1.30413 −0.652063 0.758165i \(-0.726096\pi\)
−0.652063 + 0.758165i \(0.726096\pi\)
\(158\) 0 0
\(159\) 156.023i 0.981279i
\(160\) 0 0
\(161\) −2.91796 −0.0181240
\(162\) 0 0
\(163\) − 107.235i − 0.657885i −0.944350 0.328943i \(-0.893308\pi\)
0.944350 0.328943i \(-0.106692\pi\)
\(164\) 0 0
\(165\) 104.721 0.634675
\(166\) 0 0
\(167\) 33.2090i 0.198856i 0.995045 + 0.0994280i \(0.0317013\pi\)
−0.995045 + 0.0994280i \(0.968299\pi\)
\(168\) 0 0
\(169\) −97.2229 −0.575284
\(170\) 0 0
\(171\) 52.8352i 0.308978i
\(172\) 0 0
\(173\) 226.361 1.30844 0.654222 0.756303i \(-0.272996\pi\)
0.654222 + 0.756303i \(0.272996\pi\)
\(174\) 0 0
\(175\) − 26.2866i − 0.150209i
\(176\) 0 0
\(177\) 40.9969 0.231621
\(178\) 0 0
\(179\) − 224.337i − 1.25328i −0.779308 0.626641i \(-0.784429\pi\)
0.779308 0.626641i \(-0.215571\pi\)
\(180\) 0 0
\(181\) −86.2229 −0.476370 −0.238185 0.971220i \(-0.576552\pi\)
−0.238185 + 0.971220i \(0.576552\pi\)
\(182\) 0 0
\(183\) − 211.927i − 1.15807i
\(184\) 0 0
\(185\) 41.0557 0.221923
\(186\) 0 0
\(187\) − 236.810i − 1.26636i
\(188\) 0 0
\(189\) 154.164 0.815683
\(190\) 0 0
\(191\) − 31.0198i − 0.162407i −0.996698 0.0812036i \(-0.974124\pi\)
0.996698 0.0812036i \(-0.0258764\pi\)
\(192\) 0 0
\(193\) 110.223 0.571103 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(194\) 0 0
\(195\) 44.5407i 0.228414i
\(196\) 0 0
\(197\) 172.525 0.875760 0.437880 0.899033i \(-0.355729\pi\)
0.437880 + 0.899033i \(0.355729\pi\)
\(198\) 0 0
\(199\) − 272.208i − 1.36788i −0.729538 0.683940i \(-0.760265\pi\)
0.729538 0.683940i \(-0.239735\pi\)
\(200\) 0 0
\(201\) 118.079 0.587457
\(202\) 0 0
\(203\) − 57.5374i − 0.283436i
\(204\) 0 0
\(205\) −32.6099 −0.159073
\(206\) 0 0
\(207\) − 1.92714i − 0.00930984i
\(208\) 0 0
\(209\) 303.108 1.45028
\(210\) 0 0
\(211\) 205.266i 0.972826i 0.873729 + 0.486413i \(0.161695\pi\)
−0.873729 + 0.486413i \(0.838305\pi\)
\(212\) 0 0
\(213\) 189.941 0.891743
\(214\) 0 0
\(215\) − 49.7980i − 0.231618i
\(216\) 0 0
\(217\) −43.6068 −0.200953
\(218\) 0 0
\(219\) − 13.0586i − 0.0596285i
\(220\) 0 0
\(221\) 100.721 0.455753
\(222\) 0 0
\(223\) 235.731i 1.05709i 0.848905 + 0.528545i \(0.177262\pi\)
−0.848905 + 0.528545i \(0.822738\pi\)
\(224\) 0 0
\(225\) 17.3607 0.0771586
\(226\) 0 0
\(227\) 58.5165i 0.257782i 0.991659 + 0.128891i \(0.0411417\pi\)
−0.991659 + 0.128891i \(0.958858\pi\)
\(228\) 0 0
\(229\) −162.721 −0.710574 −0.355287 0.934757i \(-0.615617\pi\)
−0.355287 + 0.934757i \(0.615617\pi\)
\(230\) 0 0
\(231\) − 246.215i − 1.06586i
\(232\) 0 0
\(233\) 319.050 1.36931 0.684656 0.728867i \(-0.259953\pi\)
0.684656 + 0.728867i \(0.259953\pi\)
\(234\) 0 0
\(235\) 119.384i 0.508017i
\(236\) 0 0
\(237\) 32.5511 0.137346
\(238\) 0 0
\(239\) 236.810i 0.990837i 0.868654 + 0.495419i \(0.164985\pi\)
−0.868654 + 0.495419i \(0.835015\pi\)
\(240\) 0 0
\(241\) −0.917961 −0.00380897 −0.00190448 0.999998i \(-0.500606\pi\)
−0.00190448 + 0.999998i \(0.500606\pi\)
\(242\) 0 0
\(243\) 175.287i 0.721347i
\(244\) 0 0
\(245\) 47.7639 0.194955
\(246\) 0 0
\(247\) 128.920i 0.521942i
\(248\) 0 0
\(249\) −179.193 −0.719653
\(250\) 0 0
\(251\) 136.690i 0.544582i 0.962215 + 0.272291i \(0.0877813\pi\)
−0.962215 + 0.272291i \(0.912219\pi\)
\(252\) 0 0
\(253\) −11.0557 −0.0436985
\(254\) 0 0
\(255\) 62.5018i 0.245105i
\(256\) 0 0
\(257\) −274.944 −1.06982 −0.534911 0.844908i \(-0.679655\pi\)
−0.534911 + 0.844908i \(0.679655\pi\)
\(258\) 0 0
\(259\) − 96.5278i − 0.372694i
\(260\) 0 0
\(261\) 38.0000 0.145594
\(262\) 0 0
\(263\) 406.385i 1.54519i 0.634899 + 0.772596i \(0.281042\pi\)
−0.634899 + 0.772596i \(0.718958\pi\)
\(264\) 0 0
\(265\) 148.387 0.559951
\(266\) 0 0
\(267\) − 262.018i − 0.981339i
\(268\) 0 0
\(269\) 348.525 1.29563 0.647816 0.761797i \(-0.275683\pi\)
0.647816 + 0.761797i \(0.275683\pi\)
\(270\) 0 0
\(271\) − 247.849i − 0.914571i −0.889320 0.457286i \(-0.848822\pi\)
0.889320 0.457286i \(-0.151178\pi\)
\(272\) 0 0
\(273\) 104.721 0.383595
\(274\) 0 0
\(275\) − 99.5959i − 0.362167i
\(276\) 0 0
\(277\) 54.7539 0.197667 0.0988337 0.995104i \(-0.468489\pi\)
0.0988337 + 0.995104i \(0.468489\pi\)
\(278\) 0 0
\(279\) − 28.7997i − 0.103225i
\(280\) 0 0
\(281\) −50.3607 −0.179220 −0.0896098 0.995977i \(-0.528562\pi\)
−0.0896098 + 0.995977i \(0.528562\pi\)
\(282\) 0 0
\(283\) 147.336i 0.520621i 0.965525 + 0.260310i \(0.0838249\pi\)
−0.965525 + 0.260310i \(0.916175\pi\)
\(284\) 0 0
\(285\) −80.0000 −0.280702
\(286\) 0 0
\(287\) 76.6705i 0.267145i
\(288\) 0 0
\(289\) −147.663 −0.510943
\(290\) 0 0
\(291\) − 218.263i − 0.750045i
\(292\) 0 0
\(293\) −178.859 −0.610441 −0.305220 0.952282i \(-0.598730\pi\)
−0.305220 + 0.952282i \(0.598730\pi\)
\(294\) 0 0
\(295\) − 38.9904i − 0.132171i
\(296\) 0 0
\(297\) 584.105 1.96668
\(298\) 0 0
\(299\) − 4.70228i − 0.0157267i
\(300\) 0 0
\(301\) −117.082 −0.388977
\(302\) 0 0
\(303\) − 150.735i − 0.497475i
\(304\) 0 0
\(305\) −201.554 −0.660833
\(306\) 0 0
\(307\) 284.550i 0.926873i 0.886130 + 0.463436i \(0.153384\pi\)
−0.886130 + 0.463436i \(0.846616\pi\)
\(308\) 0 0
\(309\) −323.915 −1.04827
\(310\) 0 0
\(311\) 282.199i 0.907392i 0.891157 + 0.453696i \(0.149895\pi\)
−0.891157 + 0.453696i \(0.850105\pi\)
\(312\) 0 0
\(313\) −567.548 −1.81325 −0.906626 0.421935i \(-0.861351\pi\)
−0.906626 + 0.421935i \(0.861351\pi\)
\(314\) 0 0
\(315\) − 40.8174i − 0.129579i
\(316\) 0 0
\(317\) −161.141 −0.508331 −0.254165 0.967161i \(-0.581801\pi\)
−0.254165 + 0.967161i \(0.581801\pi\)
\(318\) 0 0
\(319\) − 218.001i − 0.683389i
\(320\) 0 0
\(321\) −120.689 −0.375978
\(322\) 0 0
\(323\) 180.907i 0.560083i
\(324\) 0 0
\(325\) 42.3607 0.130341
\(326\) 0 0
\(327\) − 313.805i − 0.959647i
\(328\) 0 0
\(329\) 280.689 0.853158
\(330\) 0 0
\(331\) − 331.966i − 1.00292i −0.865181 0.501459i \(-0.832797\pi\)
0.865181 0.501459i \(-0.167203\pi\)
\(332\) 0 0
\(333\) 63.7508 0.191444
\(334\) 0 0
\(335\) − 112.300i − 0.335223i
\(336\) 0 0
\(337\) −269.108 −0.798541 −0.399271 0.916833i \(-0.630737\pi\)
−0.399271 + 0.916833i \(0.630737\pi\)
\(338\) 0 0
\(339\) 401.390i 1.18404i
\(340\) 0 0
\(341\) −165.220 −0.484516
\(342\) 0 0
\(343\) − 369.908i − 1.07845i
\(344\) 0 0
\(345\) 2.91796 0.00845786
\(346\) 0 0
\(347\) 503.075i 1.44978i 0.688863 + 0.724892i \(0.258110\pi\)
−0.688863 + 0.724892i \(0.741890\pi\)
\(348\) 0 0
\(349\) 0.504658 0.00144601 0.000723006 1.00000i \(-0.499770\pi\)
0.000723006 1.00000i \(0.499770\pi\)
\(350\) 0 0
\(351\) 248.435i 0.707791i
\(352\) 0 0
\(353\) −335.994 −0.951824 −0.475912 0.879493i \(-0.657882\pi\)
−0.475912 + 0.879493i \(0.657882\pi\)
\(354\) 0 0
\(355\) − 180.645i − 0.508859i
\(356\) 0 0
\(357\) 146.950 0.411626
\(358\) 0 0
\(359\) 98.4859i 0.274334i 0.990548 + 0.137167i \(0.0437997\pi\)
−0.990548 + 0.137167i \(0.956200\pi\)
\(360\) 0 0
\(361\) 129.446 0.358576
\(362\) 0 0
\(363\) − 648.384i − 1.78618i
\(364\) 0 0
\(365\) −12.4195 −0.0340261
\(366\) 0 0
\(367\) 498.473i 1.35824i 0.734029 + 0.679118i \(0.237638\pi\)
−0.734029 + 0.679118i \(0.762362\pi\)
\(368\) 0 0
\(369\) −50.6362 −0.137226
\(370\) 0 0
\(371\) − 348.879i − 0.940374i
\(372\) 0 0
\(373\) −600.354 −1.60953 −0.804765 0.593594i \(-0.797709\pi\)
−0.804765 + 0.593594i \(0.797709\pi\)
\(374\) 0 0
\(375\) 26.2866i 0.0700975i
\(376\) 0 0
\(377\) 92.7214 0.245945
\(378\) 0 0
\(379\) − 303.490i − 0.800765i −0.916348 0.400383i \(-0.868877\pi\)
0.916348 0.400383i \(-0.131123\pi\)
\(380\) 0 0
\(381\) 467.023 1.22578
\(382\) 0 0
\(383\) − 332.583i − 0.868362i −0.900826 0.434181i \(-0.857038\pi\)
0.900826 0.434181i \(-0.142962\pi\)
\(384\) 0 0
\(385\) −234.164 −0.608218
\(386\) 0 0
\(387\) − 77.3256i − 0.199808i
\(388\) 0 0
\(389\) −392.354 −1.00862 −0.504312 0.863522i \(-0.668254\pi\)
−0.504312 + 0.863522i \(0.668254\pi\)
\(390\) 0 0
\(391\) − 6.59849i − 0.0168759i
\(392\) 0 0
\(393\) 18.2693 0.0464868
\(394\) 0 0
\(395\) − 30.9579i − 0.0783744i
\(396\) 0 0
\(397\) −334.190 −0.841789 −0.420895 0.907110i \(-0.638284\pi\)
−0.420895 + 0.907110i \(0.638284\pi\)
\(398\) 0 0
\(399\) 188.091i 0.471407i
\(400\) 0 0
\(401\) 121.003 0.301753 0.150877 0.988553i \(-0.451790\pi\)
0.150877 + 0.988553i \(0.451790\pi\)
\(402\) 0 0
\(403\) − 70.2722i − 0.174373i
\(404\) 0 0
\(405\) −84.2887 −0.208120
\(406\) 0 0
\(407\) − 365.730i − 0.898599i
\(408\) 0 0
\(409\) −607.410 −1.48511 −0.742555 0.669785i \(-0.766386\pi\)
−0.742555 + 0.669785i \(0.766386\pi\)
\(410\) 0 0
\(411\) 1.95807i 0.00476415i
\(412\) 0 0
\(413\) −91.6718 −0.221966
\(414\) 0 0
\(415\) 170.423i 0.410658i
\(416\) 0 0
\(417\) −558.768 −1.33997
\(418\) 0 0
\(419\) 466.760i 1.11398i 0.830518 + 0.556992i \(0.188045\pi\)
−0.830518 + 0.556992i \(0.811955\pi\)
\(420\) 0 0
\(421\) 73.0883 0.173606 0.0868031 0.996225i \(-0.472335\pi\)
0.0868031 + 0.996225i \(0.472335\pi\)
\(422\) 0 0
\(423\) 185.378i 0.438246i
\(424\) 0 0
\(425\) 59.4427 0.139865
\(426\) 0 0
\(427\) 473.882i 1.10979i
\(428\) 0 0
\(429\) 396.774 0.924881
\(430\) 0 0
\(431\) − 463.630i − 1.07571i −0.843038 0.537853i \(-0.819235\pi\)
0.843038 0.537853i \(-0.180765\pi\)
\(432\) 0 0
\(433\) −99.8359 −0.230568 −0.115284 0.993333i \(-0.536778\pi\)
−0.115284 + 0.993333i \(0.536778\pi\)
\(434\) 0 0
\(435\) 57.5374i 0.132270i
\(436\) 0 0
\(437\) 8.44582 0.0193268
\(438\) 0 0
\(439\) 374.086i 0.852133i 0.904692 + 0.426066i \(0.140101\pi\)
−0.904692 + 0.426066i \(0.859899\pi\)
\(440\) 0 0
\(441\) 74.1672 0.168180
\(442\) 0 0
\(443\) 290.100i 0.654854i 0.944877 + 0.327427i \(0.106182\pi\)
−0.944877 + 0.327427i \(0.893818\pi\)
\(444\) 0 0
\(445\) −249.193 −0.559985
\(446\) 0 0
\(447\) 86.9231i 0.194459i
\(448\) 0 0
\(449\) 299.921 0.667976 0.333988 0.942577i \(-0.391606\pi\)
0.333988 + 0.942577i \(0.391606\pi\)
\(450\) 0 0
\(451\) 290.493i 0.644109i
\(452\) 0 0
\(453\) −664.721 −1.46738
\(454\) 0 0
\(455\) − 99.5959i − 0.218892i
\(456\) 0 0
\(457\) 822.328 1.79941 0.899703 0.436503i \(-0.143783\pi\)
0.899703 + 0.436503i \(0.143783\pi\)
\(458\) 0 0
\(459\) 348.617i 0.759513i
\(460\) 0 0
\(461\) 456.885 0.991075 0.495537 0.868587i \(-0.334971\pi\)
0.495537 + 0.868587i \(0.334971\pi\)
\(462\) 0 0
\(463\) 400.249i 0.864469i 0.901761 + 0.432234i \(0.142275\pi\)
−0.901761 + 0.432234i \(0.857725\pi\)
\(464\) 0 0
\(465\) 43.6068 0.0937781
\(466\) 0 0
\(467\) − 913.145i − 1.95534i −0.210139 0.977672i \(-0.567392\pi\)
0.210139 0.977672i \(-0.432608\pi\)
\(468\) 0 0
\(469\) −264.033 −0.562969
\(470\) 0 0
\(471\) − 481.391i − 1.02206i
\(472\) 0 0
\(473\) −443.607 −0.937858
\(474\) 0 0
\(475\) 76.0845i 0.160178i
\(476\) 0 0
\(477\) 230.413 0.483047
\(478\) 0 0
\(479\) − 526.131i − 1.09840i −0.835692 0.549198i \(-0.814933\pi\)
0.835692 0.549198i \(-0.185067\pi\)
\(480\) 0 0
\(481\) 155.554 0.323397
\(482\) 0 0
\(483\) − 6.86054i − 0.0142040i
\(484\) 0 0
\(485\) −207.580 −0.428001
\(486\) 0 0
\(487\) 443.541i 0.910762i 0.890297 + 0.455381i \(0.150497\pi\)
−0.890297 + 0.455381i \(0.849503\pi\)
\(488\) 0 0
\(489\) 252.125 0.515594
\(490\) 0 0
\(491\) 287.163i 0.584854i 0.956288 + 0.292427i \(0.0944628\pi\)
−0.956288 + 0.292427i \(0.905537\pi\)
\(492\) 0 0
\(493\) 130.111 0.263918
\(494\) 0 0
\(495\) − 154.651i − 0.312427i
\(496\) 0 0
\(497\) −424.721 −0.854570
\(498\) 0 0
\(499\) − 810.936i − 1.62512i −0.582876 0.812561i \(-0.698073\pi\)
0.582876 0.812561i \(-0.301927\pi\)
\(500\) 0 0
\(501\) −78.0789 −0.155846
\(502\) 0 0
\(503\) 642.471i 1.27728i 0.769506 + 0.638639i \(0.220502\pi\)
−0.769506 + 0.638639i \(0.779498\pi\)
\(504\) 0 0
\(505\) −143.358 −0.283876
\(506\) 0 0
\(507\) − 228.585i − 0.450858i
\(508\) 0 0
\(509\) 915.050 1.79774 0.898870 0.438216i \(-0.144389\pi\)
0.898870 + 0.438216i \(0.144389\pi\)
\(510\) 0 0
\(511\) 29.2000i 0.0571429i
\(512\) 0 0
\(513\) −446.217 −0.869818
\(514\) 0 0
\(515\) 308.061i 0.598177i
\(516\) 0 0
\(517\) 1063.49 2.05704
\(518\) 0 0
\(519\) 532.206i 1.02544i
\(520\) 0 0
\(521\) 1006.98 1.93279 0.966396 0.257058i \(-0.0827533\pi\)
0.966396 + 0.257058i \(0.0827533\pi\)
\(522\) 0 0
\(523\) 774.173i 1.48025i 0.672467 + 0.740127i \(0.265235\pi\)
−0.672467 + 0.740127i \(0.734765\pi\)
\(524\) 0 0
\(525\) 61.8034 0.117721
\(526\) 0 0
\(527\) − 98.6096i − 0.187115i
\(528\) 0 0
\(529\) 528.692 0.999418
\(530\) 0 0
\(531\) − 60.5437i − 0.114018i
\(532\) 0 0
\(533\) −123.554 −0.231809
\(534\) 0 0
\(535\) 114.782i 0.214546i
\(536\) 0 0
\(537\) 527.449 0.982214
\(538\) 0 0
\(539\) − 425.487i − 0.789401i
\(540\) 0 0
\(541\) 259.115 0.478955 0.239477 0.970902i \(-0.423024\pi\)
0.239477 + 0.970902i \(0.423024\pi\)
\(542\) 0 0
\(543\) − 202.722i − 0.373337i
\(544\) 0 0
\(545\) −298.446 −0.547607
\(546\) 0 0
\(547\) 149.818i 0.273890i 0.990579 + 0.136945i \(0.0437284\pi\)
−0.990579 + 0.136945i \(0.956272\pi\)
\(548\) 0 0
\(549\) −312.971 −0.570074
\(550\) 0 0
\(551\) 166.538i 0.302247i
\(552\) 0 0
\(553\) −72.7864 −0.131621
\(554\) 0 0
\(555\) 96.5278i 0.173924i
\(556\) 0 0
\(557\) −511.698 −0.918668 −0.459334 0.888264i \(-0.651912\pi\)
−0.459334 + 0.888264i \(0.651912\pi\)
\(558\) 0 0
\(559\) − 188.677i − 0.337526i
\(560\) 0 0
\(561\) 556.774 0.992467
\(562\) 0 0
\(563\) − 490.726i − 0.871627i −0.900037 0.435814i \(-0.856461\pi\)
0.900037 0.435814i \(-0.143539\pi\)
\(564\) 0 0
\(565\) 381.745 0.675654
\(566\) 0 0
\(567\) 198.175i 0.349514i
\(568\) 0 0
\(569\) −232.748 −0.409047 −0.204523 0.978862i \(-0.565564\pi\)
−0.204523 + 0.978862i \(0.565564\pi\)
\(570\) 0 0
\(571\) 210.755i 0.369098i 0.982823 + 0.184549i \(0.0590824\pi\)
−0.982823 + 0.184549i \(0.940918\pi\)
\(572\) 0 0
\(573\) 72.9318 0.127281
\(574\) 0 0
\(575\) − 2.77515i − 0.00482634i
\(576\) 0 0
\(577\) 341.712 0.592222 0.296111 0.955154i \(-0.404310\pi\)
0.296111 + 0.955154i \(0.404310\pi\)
\(578\) 0 0
\(579\) 259.150i 0.447581i
\(580\) 0 0
\(581\) 400.689 0.689654
\(582\) 0 0
\(583\) − 1321.85i − 2.26733i
\(584\) 0 0
\(585\) 65.7771 0.112439
\(586\) 0 0
\(587\) − 618.412i − 1.05351i −0.850016 0.526756i \(-0.823408\pi\)
0.850016 0.526756i \(-0.176592\pi\)
\(588\) 0 0
\(589\) 126.217 0.214290
\(590\) 0 0
\(591\) 405.630i 0.686345i
\(592\) 0 0
\(593\) 120.663 0.203478 0.101739 0.994811i \(-0.467559\pi\)
0.101739 + 0.994811i \(0.467559\pi\)
\(594\) 0 0
\(595\) − 139.758i − 0.234888i
\(596\) 0 0
\(597\) 640.000 1.07203
\(598\) 0 0
\(599\) − 849.927i − 1.41891i −0.704751 0.709455i \(-0.748941\pi\)
0.704751 0.709455i \(-0.251059\pi\)
\(600\) 0 0
\(601\) −11.3576 −0.0188978 −0.00944890 0.999955i \(-0.503008\pi\)
−0.00944890 + 0.999955i \(0.503008\pi\)
\(602\) 0 0
\(603\) − 174.378i − 0.289183i
\(604\) 0 0
\(605\) −616.649 −1.01926
\(606\) 0 0
\(607\) − 1115.12i − 1.83710i −0.395305 0.918550i \(-0.629361\pi\)
0.395305 0.918550i \(-0.370639\pi\)
\(608\) 0 0
\(609\) 135.279 0.222132
\(610\) 0 0
\(611\) 452.329i 0.740309i
\(612\) 0 0
\(613\) −499.475 −0.814805 −0.407402 0.913249i \(-0.633565\pi\)
−0.407402 + 0.913249i \(0.633565\pi\)
\(614\) 0 0
\(615\) − 76.6705i − 0.124667i
\(616\) 0 0
\(617\) 545.935 0.884822 0.442411 0.896813i \(-0.354123\pi\)
0.442411 + 0.896813i \(0.354123\pi\)
\(618\) 0 0
\(619\) − 455.011i − 0.735075i −0.930009 0.367537i \(-0.880201\pi\)
0.930009 0.367537i \(-0.119799\pi\)
\(620\) 0 0
\(621\) 16.2755 0.0262086
\(622\) 0 0
\(623\) 585.889i 0.940432i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 712.650i 1.13660i
\(628\) 0 0
\(629\) 218.282 0.347030
\(630\) 0 0
\(631\) 267.706i 0.424257i 0.977242 + 0.212128i \(0.0680395\pi\)
−0.977242 + 0.212128i \(0.931960\pi\)
\(632\) 0 0
\(633\) −482.610 −0.762417
\(634\) 0 0
\(635\) − 444.165i − 0.699473i
\(636\) 0 0
\(637\) 180.971 0.284098
\(638\) 0 0
\(639\) − 280.503i − 0.438971i
\(640\) 0 0
\(641\) −418.571 −0.652997 −0.326499 0.945198i \(-0.605869\pi\)
−0.326499 + 0.945198i \(0.605869\pi\)
\(642\) 0 0
\(643\) − 439.339i − 0.683265i −0.939834 0.341633i \(-0.889020\pi\)
0.939834 0.341633i \(-0.110980\pi\)
\(644\) 0 0
\(645\) 117.082 0.181523
\(646\) 0 0
\(647\) 419.644i 0.648600i 0.945954 + 0.324300i \(0.105129\pi\)
−0.945954 + 0.324300i \(0.894871\pi\)
\(648\) 0 0
\(649\) −347.331 −0.535179
\(650\) 0 0
\(651\) − 102.526i − 0.157490i
\(652\) 0 0
\(653\) 370.085 0.566746 0.283373 0.959010i \(-0.408547\pi\)
0.283373 + 0.959010i \(0.408547\pi\)
\(654\) 0 0
\(655\) − 17.3752i − 0.0265270i
\(656\) 0 0
\(657\) −19.2849 −0.0293529
\(658\) 0 0
\(659\) 322.823i 0.489868i 0.969540 + 0.244934i \(0.0787664\pi\)
−0.969540 + 0.244934i \(0.921234\pi\)
\(660\) 0 0
\(661\) 812.735 1.22955 0.614777 0.788701i \(-0.289246\pi\)
0.614777 + 0.788701i \(0.289246\pi\)
\(662\) 0 0
\(663\) 236.810i 0.357180i
\(664\) 0 0
\(665\) 178.885 0.269001
\(666\) 0 0
\(667\) − 6.07439i − 0.00910703i
\(668\) 0 0
\(669\) −554.237 −0.828456
\(670\) 0 0
\(671\) 1795.47i 2.67581i
\(672\) 0 0
\(673\) 467.378 0.694469 0.347235 0.937778i \(-0.387121\pi\)
0.347235 + 0.937778i \(0.387121\pi\)
\(674\) 0 0
\(675\) 146.619i 0.217213i
\(676\) 0 0
\(677\) −548.237 −0.809803 −0.404902 0.914360i \(-0.632694\pi\)
−0.404902 + 0.914360i \(0.632694\pi\)
\(678\) 0 0
\(679\) 488.051i 0.718779i
\(680\) 0 0
\(681\) −137.580 −0.202027
\(682\) 0 0
\(683\) − 23.9663i − 0.0350898i −0.999846 0.0175449i \(-0.994415\pi\)
0.999846 0.0175449i \(-0.00558500\pi\)
\(684\) 0 0
\(685\) 1.86223 0.00271859
\(686\) 0 0
\(687\) − 382.581i − 0.556886i
\(688\) 0 0
\(689\) 562.217 0.815989
\(690\) 0 0
\(691\) 186.981i 0.270595i 0.990805 + 0.135298i \(0.0431990\pi\)
−0.990805 + 0.135298i \(0.956801\pi\)
\(692\) 0 0
\(693\) −363.607 −0.524685
\(694\) 0 0
\(695\) 531.420i 0.764633i
\(696\) 0 0
\(697\) −173.378 −0.248748
\(698\) 0 0
\(699\) 750.130i 1.07315i
\(700\) 0 0
\(701\) 706.636 1.00804 0.504020 0.863692i \(-0.331854\pi\)
0.504020 + 0.863692i \(0.331854\pi\)
\(702\) 0 0
\(703\) 279.393i 0.397429i
\(704\) 0 0
\(705\) −280.689 −0.398140
\(706\) 0 0
\(707\) 337.054i 0.476738i
\(708\) 0 0
\(709\) 188.597 0.266005 0.133002 0.991116i \(-0.457538\pi\)
0.133002 + 0.991116i \(0.457538\pi\)
\(710\) 0 0
\(711\) − 48.0710i − 0.0676104i
\(712\) 0 0
\(713\) −4.60369 −0.00645679
\(714\) 0 0
\(715\) − 377.354i − 0.527769i
\(716\) 0 0
\(717\) −556.774 −0.776533
\(718\) 0 0
\(719\) 156.085i 0.217086i 0.994092 + 0.108543i \(0.0346186\pi\)
−0.994092 + 0.108543i \(0.965381\pi\)
\(720\) 0 0
\(721\) 724.296 1.00457
\(722\) 0 0
\(723\) − 2.15825i − 0.00298514i
\(724\) 0 0
\(725\) 54.7214 0.0754777
\(726\) 0 0
\(727\) − 715.164i − 0.983719i −0.870675 0.491859i \(-0.836317\pi\)
0.870675 0.491859i \(-0.163683\pi\)
\(728\) 0 0
\(729\) −751.381 −1.03070
\(730\) 0 0
\(731\) − 264.762i − 0.362191i
\(732\) 0 0
\(733\) −1233.29 −1.68252 −0.841259 0.540632i \(-0.818185\pi\)
−0.841259 + 0.540632i \(0.818185\pi\)
\(734\) 0 0
\(735\) 112.300i 0.152789i
\(736\) 0 0
\(737\) −1000.38 −1.35737
\(738\) 0 0
\(739\) − 8.55656i − 0.0115786i −0.999983 0.00578928i \(-0.998157\pi\)
0.999983 0.00578928i \(-0.00184280\pi\)
\(740\) 0 0
\(741\) −303.108 −0.409053
\(742\) 0 0
\(743\) − 1010.56i − 1.36011i −0.733163 0.680053i \(-0.761957\pi\)
0.733163 0.680053i \(-0.238043\pi\)
\(744\) 0 0
\(745\) 82.6687 0.110965
\(746\) 0 0
\(747\) 264.631i 0.354258i
\(748\) 0 0
\(749\) 269.868 0.360305
\(750\) 0 0
\(751\) − 1104.31i − 1.47046i −0.677820 0.735228i \(-0.737075\pi\)
0.677820 0.735228i \(-0.262925\pi\)
\(752\) 0 0
\(753\) −321.378 −0.426796
\(754\) 0 0
\(755\) 632.188i 0.837335i
\(756\) 0 0
\(757\) 875.633 1.15671 0.578357 0.815783i \(-0.303694\pi\)
0.578357 + 0.815783i \(0.303694\pi\)
\(758\) 0 0
\(759\) − 25.9936i − 0.0342471i
\(760\) 0 0
\(761\) −647.207 −0.850470 −0.425235 0.905083i \(-0.639808\pi\)
−0.425235 + 0.905083i \(0.639808\pi\)
\(762\) 0 0
\(763\) 701.688i 0.919644i
\(764\) 0 0
\(765\) 92.3018 0.120656
\(766\) 0 0
\(767\) − 147.729i − 0.192606i
\(768\) 0 0
\(769\) 631.430 0.821106 0.410553 0.911837i \(-0.365336\pi\)
0.410553 + 0.911837i \(0.365336\pi\)
\(770\) 0 0
\(771\) − 646.433i − 0.838434i
\(772\) 0 0
\(773\) 421.522 0.545306 0.272653 0.962112i \(-0.412099\pi\)
0.272653 + 0.962112i \(0.412099\pi\)
\(774\) 0 0
\(775\) − 41.4725i − 0.0535129i
\(776\) 0 0
\(777\) 226.950 0.292086
\(778\) 0 0
\(779\) − 221.917i − 0.284874i
\(780\) 0 0
\(781\) −1609.21 −2.06044
\(782\) 0 0
\(783\) 320.927i 0.409869i
\(784\) 0 0
\(785\) −457.830 −0.583223
\(786\) 0 0
\(787\) − 838.633i − 1.06561i −0.846239 0.532804i \(-0.821138\pi\)
0.846239 0.532804i \(-0.178862\pi\)
\(788\) 0 0
\(789\) −955.469 −1.21099
\(790\) 0 0
\(791\) − 897.535i − 1.13468i
\(792\) 0 0
\(793\) −763.659 −0.963001
\(794\) 0 0
\(795\) 348.879i 0.438841i
\(796\) 0 0
\(797\) −1213.57 −1.52268 −0.761339 0.648354i \(-0.775458\pi\)
−0.761339 + 0.648354i \(0.775458\pi\)
\(798\) 0 0
\(799\) 634.732i 0.794408i
\(800\) 0 0
\(801\) −386.944 −0.483076
\(802\) 0 0
\(803\) 110.635i 0.137777i
\(804\) 0 0
\(805\) −6.52476 −0.00810529
\(806\) 0 0
\(807\) 819.431i 1.01540i
\(808\) 0 0
\(809\) −229.214 −0.283330 −0.141665 0.989915i \(-0.545246\pi\)
−0.141665 + 0.989915i \(0.545246\pi\)
\(810\) 0 0
\(811\) − 454.225i − 0.560080i −0.959988 0.280040i \(-0.909652\pi\)
0.959988 0.280040i \(-0.0903478\pi\)
\(812\) 0 0
\(813\) 582.728 0.716762
\(814\) 0 0
\(815\) − 239.785i − 0.294215i
\(816\) 0 0
\(817\) 338.885 0.414792
\(818\) 0 0
\(819\) − 154.651i − 0.188829i
\(820\) 0 0
\(821\) −1130.90 −1.37747 −0.688733 0.725015i \(-0.741833\pi\)
−0.688733 + 0.725015i \(0.741833\pi\)
\(822\) 0 0
\(823\) 780.148i 0.947931i 0.880543 + 0.473966i \(0.157178\pi\)
−0.880543 + 0.473966i \(0.842822\pi\)
\(824\) 0 0
\(825\) 234.164 0.283835
\(826\) 0 0
\(827\) − 209.175i − 0.252932i −0.991971 0.126466i \(-0.959636\pi\)
0.991971 0.126466i \(-0.0403635\pi\)
\(828\) 0 0
\(829\) 508.525 0.613419 0.306710 0.951803i \(-0.400772\pi\)
0.306710 + 0.951803i \(0.400772\pi\)
\(830\) 0 0
\(831\) 128.734i 0.154915i
\(832\) 0 0
\(833\) 253.947 0.304859
\(834\) 0 0
\(835\) 74.2575i 0.0889311i
\(836\) 0 0
\(837\) 243.226 0.290593
\(838\) 0 0
\(839\) − 274.028i − 0.326613i −0.986575 0.163306i \(-0.947784\pi\)
0.986575 0.163306i \(-0.0522159\pi\)
\(840\) 0 0
\(841\) −721.223 −0.857578
\(842\) 0 0
\(843\) − 118.405i − 0.140457i
\(844\) 0 0
\(845\) −217.397 −0.257275
\(846\) 0 0
\(847\) 1449.83i 1.71172i
\(848\) 0 0
\(849\) −346.407 −0.408018
\(850\) 0 0
\(851\) − 10.1907i − 0.0119750i
\(852\) 0 0
\(853\) 1583.28 1.85613 0.928066 0.372416i \(-0.121471\pi\)
0.928066 + 0.372416i \(0.121471\pi\)
\(854\) 0 0
\(855\) 118.143i 0.138179i
\(856\) 0 0
\(857\) −1007.38 −1.17547 −0.587735 0.809054i \(-0.699980\pi\)
−0.587735 + 0.809054i \(0.699980\pi\)
\(858\) 0 0
\(859\) 76.6086i 0.0891835i 0.999005 + 0.0445917i \(0.0141987\pi\)
−0.999005 + 0.0445917i \(0.985801\pi\)
\(860\) 0 0
\(861\) −180.263 −0.209365
\(862\) 0 0
\(863\) − 255.450i − 0.296002i −0.988987 0.148001i \(-0.952716\pi\)
0.988987 0.148001i \(-0.0472839\pi\)
\(864\) 0 0
\(865\) 506.158 0.585154
\(866\) 0 0
\(867\) − 347.175i − 0.400433i
\(868\) 0 0
\(869\) −275.777 −0.317350
\(870\) 0 0
\(871\) − 425.487i − 0.488504i
\(872\) 0 0
\(873\) −322.328 −0.369219
\(874\) 0 0
\(875\) − 58.7785i − 0.0671755i
\(876\) 0 0
\(877\) −601.522 −0.685886 −0.342943 0.939356i \(-0.611424\pi\)
−0.342943 + 0.939356i \(0.611424\pi\)
\(878\) 0 0
\(879\) − 420.523i − 0.478411i
\(880\) 0 0
\(881\) 237.850 0.269977 0.134989 0.990847i \(-0.456900\pi\)
0.134989 + 0.990847i \(0.456900\pi\)
\(882\) 0 0
\(883\) 1.30294i 0.00147559i 1.00000 0.000737794i \(0.000234847\pi\)
−1.00000 0.000737794i \(0.999765\pi\)
\(884\) 0 0
\(885\) 91.6718 0.103584
\(886\) 0 0
\(887\) − 536.353i − 0.604682i −0.953200 0.302341i \(-0.902232\pi\)
0.953200 0.302341i \(-0.0977682\pi\)
\(888\) 0 0
\(889\) −1044.30 −1.17469
\(890\) 0 0
\(891\) 750.855i 0.842710i
\(892\) 0 0
\(893\) −812.433 −0.909780
\(894\) 0 0
\(895\) − 501.634i − 0.560485i
\(896\) 0 0
\(897\) 11.0557 0.0123252
\(898\) 0 0
\(899\) − 90.7773i − 0.100976i
\(900\) 0 0
\(901\) 788.932 0.875618
\(902\) 0 0
\(903\) − 275.276i − 0.304846i
\(904\) 0 0
\(905\) −192.800 −0.213039
\(906\) 0 0
\(907\) − 332.159i − 0.366217i −0.983093 0.183108i \(-0.941384\pi\)
0.983093 0.183108i \(-0.0586159\pi\)
\(908\) 0 0
\(909\) −222.604 −0.244889
\(910\) 0 0
\(911\) − 1450.06i − 1.59172i −0.605478 0.795862i \(-0.707018\pi\)
0.605478 0.795862i \(-0.292982\pi\)
\(912\) 0 0
\(913\) 1518.15 1.66282
\(914\) 0 0
\(915\) − 473.882i − 0.517904i
\(916\) 0 0
\(917\) −40.8514 −0.0445490
\(918\) 0 0
\(919\) 814.405i 0.886186i 0.896476 + 0.443093i \(0.146119\pi\)
−0.896476 + 0.443093i \(0.853881\pi\)
\(920\) 0 0
\(921\) −669.017 −0.726403
\(922\) 0 0
\(923\) − 684.437i − 0.741535i
\(924\) 0 0
\(925\) 91.8034 0.0992469
\(926\) 0 0
\(927\) 478.353i 0.516023i
\(928\) 0 0
\(929\) 400.039 0.430612 0.215306 0.976547i \(-0.430925\pi\)
0.215306 + 0.976547i \(0.430925\pi\)
\(930\) 0 0
\(931\) 325.043i 0.349134i
\(932\) 0 0
\(933\) −663.489 −0.711135
\(934\) 0 0
\(935\) − 529.524i − 0.566335i
\(936\) 0 0
\(937\) 249.279 0.266039 0.133020 0.991113i \(-0.457533\pi\)
0.133020 + 0.991113i \(0.457533\pi\)
\(938\) 0 0
\(939\) − 1334.39i − 1.42107i
\(940\) 0 0
\(941\) −724.229 −0.769638 −0.384819 0.922992i \(-0.625736\pi\)
−0.384819 + 0.922992i \(0.625736\pi\)
\(942\) 0 0
\(943\) 8.09432i 0.00858358i
\(944\) 0 0
\(945\) 344.721 0.364785
\(946\) 0 0
\(947\) 1141.54i 1.20542i 0.797959 + 0.602712i \(0.205913\pi\)
−0.797959 + 0.602712i \(0.794087\pi\)
\(948\) 0 0
\(949\) −47.0557 −0.0495845
\(950\) 0 0
\(951\) − 378.865i − 0.398386i
\(952\) 0 0
\(953\) 1295.33 1.35921 0.679604 0.733579i \(-0.262152\pi\)
0.679604 + 0.733579i \(0.262152\pi\)
\(954\) 0 0
\(955\) − 69.3623i − 0.0726307i
\(956\) 0 0
\(957\) 512.551 0.535581
\(958\) 0 0
\(959\) − 4.37837i − 0.00456556i
\(960\) 0 0
\(961\) 892.201 0.928409
\(962\) 0 0
\(963\) 178.232i 0.185080i
\(964\) 0 0
\(965\) 246.466 0.255405
\(966\) 0 0
\(967\) − 398.477i − 0.412075i −0.978544 0.206037i \(-0.933943\pi\)
0.978544 0.206037i \(-0.0660569\pi\)
\(968\) 0 0
\(969\) −425.337 −0.438945
\(970\) 0 0
\(971\) 928.093i 0.955811i 0.878411 + 0.477906i \(0.158604\pi\)
−0.878411 + 0.477906i \(0.841396\pi\)
\(972\) 0 0
\(973\) 1249.44 1.28411
\(974\) 0 0
\(975\) 99.5959i 0.102150i
\(976\) 0 0
\(977\) −1378.05 −1.41049 −0.705247 0.708962i \(-0.749164\pi\)
−0.705247 + 0.708962i \(0.749164\pi\)
\(978\) 0 0
\(979\) 2219.85i 2.26747i
\(980\) 0 0
\(981\) −463.423 −0.472398
\(982\) 0 0
\(983\) 311.291i 0.316675i 0.987385 + 0.158337i \(0.0506134\pi\)
−0.987385 + 0.158337i \(0.949387\pi\)
\(984\) 0 0
\(985\) 385.777 0.391652
\(986\) 0 0
\(987\) 659.939i 0.668631i
\(988\) 0 0
\(989\) −12.3607 −0.0124982
\(990\) 0 0
\(991\) 961.147i 0.969876i 0.874549 + 0.484938i \(0.161158\pi\)
−0.874549 + 0.484938i \(0.838842\pi\)
\(992\) 0 0
\(993\) 780.498 0.786000
\(994\) 0 0
\(995\) − 608.676i − 0.611735i
\(996\) 0 0
\(997\) 1089.68 1.09296 0.546479 0.837473i \(-0.315968\pi\)
0.546479 + 0.837473i \(0.315968\pi\)
\(998\) 0 0
\(999\) 538.404i 0.538943i
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 320.3.b.c.191.3 4
3.2 odd 2 2880.3.e.e.2431.1 4
4.3 odd 2 inner 320.3.b.c.191.2 4
5.2 odd 4 1600.3.h.n.1599.5 8
5.3 odd 4 1600.3.h.n.1599.3 8
5.4 even 2 1600.3.b.s.1151.2 4
8.3 odd 2 20.3.b.a.11.3 4
8.5 even 2 20.3.b.a.11.4 yes 4
12.11 even 2 2880.3.e.e.2431.2 4
16.3 odd 4 1280.3.g.e.1151.5 8
16.5 even 4 1280.3.g.e.1151.6 8
16.11 odd 4 1280.3.g.e.1151.4 8
16.13 even 4 1280.3.g.e.1151.3 8
20.3 even 4 1600.3.h.n.1599.6 8
20.7 even 4 1600.3.h.n.1599.4 8
20.19 odd 2 1600.3.b.s.1151.3 4
24.5 odd 2 180.3.c.a.91.1 4
24.11 even 2 180.3.c.a.91.2 4
40.3 even 4 100.3.d.b.99.1 8
40.13 odd 4 100.3.d.b.99.7 8
40.19 odd 2 100.3.b.f.51.2 4
40.27 even 4 100.3.d.b.99.8 8
40.29 even 2 100.3.b.f.51.1 4
40.37 odd 4 100.3.d.b.99.2 8
120.29 odd 2 900.3.c.k.451.4 4
120.53 even 4 900.3.f.e.199.2 8
120.59 even 2 900.3.c.k.451.3 4
120.77 even 4 900.3.f.e.199.7 8
120.83 odd 4 900.3.f.e.199.8 8
120.107 odd 4 900.3.f.e.199.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.b.a.11.3 4 8.3 odd 2
20.3.b.a.11.4 yes 4 8.5 even 2
100.3.b.f.51.1 4 40.29 even 2
100.3.b.f.51.2 4 40.19 odd 2
100.3.d.b.99.1 8 40.3 even 4
100.3.d.b.99.2 8 40.37 odd 4
100.3.d.b.99.7 8 40.13 odd 4
100.3.d.b.99.8 8 40.27 even 4
180.3.c.a.91.1 4 24.5 odd 2
180.3.c.a.91.2 4 24.11 even 2
320.3.b.c.191.2 4 4.3 odd 2 inner
320.3.b.c.191.3 4 1.1 even 1 trivial
900.3.c.k.451.3 4 120.59 even 2
900.3.c.k.451.4 4 120.29 odd 2
900.3.f.e.199.1 8 120.107 odd 4
900.3.f.e.199.2 8 120.53 even 4
900.3.f.e.199.7 8 120.77 even 4
900.3.f.e.199.8 8 120.83 odd 4
1280.3.g.e.1151.3 8 16.13 even 4
1280.3.g.e.1151.4 8 16.11 odd 4
1280.3.g.e.1151.5 8 16.3 odd 4
1280.3.g.e.1151.6 8 16.5 even 4
1600.3.b.s.1151.2 4 5.4 even 2
1600.3.b.s.1151.3 4 20.19 odd 2
1600.3.h.n.1599.3 8 5.3 odd 4
1600.3.h.n.1599.4 8 20.7 even 4
1600.3.h.n.1599.5 8 5.2 odd 4
1600.3.h.n.1599.6 8 20.3 even 4
2880.3.e.e.2431.1 4 3.2 odd 2
2880.3.e.e.2431.2 4 12.11 even 2