Properties

Label 7600.2.a.x.1.1
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{3} +2.41421 q^{7} +2.82843 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} +2.41421 q^{7} +2.82843 q^{9} -1.41421 q^{11} +1.82843 q^{13} +1.00000 q^{17} +1.00000 q^{19} -5.82843 q^{21} -5.24264 q^{23} +0.414214 q^{27} +3.82843 q^{29} -3.41421 q^{31} +3.41421 q^{33} +5.17157 q^{37} -4.41421 q^{39} -7.07107 q^{41} -2.24264 q^{43} -8.00000 q^{47} -1.17157 q^{49} -2.41421 q^{51} -1.82843 q^{53} -2.41421 q^{57} -14.4142 q^{59} +3.41421 q^{61} +6.82843 q^{63} +6.07107 q^{67} +12.6569 q^{69} -3.07107 q^{71} +13.8284 q^{73} -3.41421 q^{77} +0.828427 q^{79} -9.48528 q^{81} +2.48528 q^{83} -9.24264 q^{87} -3.75736 q^{89} +4.41421 q^{91} +8.24264 q^{93} +7.65685 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{7} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{7} - 2q^{13} + 2q^{17} + 2q^{19} - 6q^{21} - 2q^{23} - 2q^{27} + 2q^{29} - 4q^{31} + 4q^{33} + 16q^{37} - 6q^{39} + 4q^{43} - 16q^{47} - 8q^{49} - 2q^{51} + 2q^{53} - 2q^{57} - 26q^{59} + 4q^{61} + 8q^{63} - 2q^{67} + 14q^{69} + 8q^{71} + 22q^{73} - 4q^{77} - 4q^{79} - 2q^{81} - 12q^{83} - 10q^{87} - 16q^{89} + 6q^{91} + 8q^{93} + 4q^{97} - 8q^{99} + O(q^{100}) \)

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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.41421 0.912487 0.456243 0.889855i \(-0.349195\pi\)
0.456243 + 0.889855i \(0.349195\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −5.82843 −1.27187
\(22\) 0 0
\(23\) −5.24264 −1.09317 −0.546583 0.837405i \(-0.684072\pi\)
−0.546583 + 0.837405i \(0.684072\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) 3.82843 0.710921 0.355461 0.934691i \(-0.384324\pi\)
0.355461 + 0.934691i \(0.384324\pi\)
\(30\) 0 0
\(31\) −3.41421 −0.613211 −0.306605 0.951837i \(-0.599193\pi\)
−0.306605 + 0.951837i \(0.599193\pi\)
\(32\) 0 0
\(33\) 3.41421 0.594338
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.17157 0.850201 0.425101 0.905146i \(-0.360239\pi\)
0.425101 + 0.905146i \(0.360239\pi\)
\(38\) 0 0
\(39\) −4.41421 −0.706840
\(40\) 0 0
\(41\) −7.07107 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(42\) 0 0
\(43\) −2.24264 −0.341999 −0.171000 0.985271i \(-0.554700\pi\)
−0.171000 + 0.985271i \(0.554700\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) −2.41421 −0.338058
\(52\) 0 0
\(53\) −1.82843 −0.251154 −0.125577 0.992084i \(-0.540078\pi\)
−0.125577 + 0.992084i \(0.540078\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.41421 −0.319770
\(58\) 0 0
\(59\) −14.4142 −1.87657 −0.938285 0.345862i \(-0.887587\pi\)
−0.938285 + 0.345862i \(0.887587\pi\)
\(60\) 0 0
\(61\) 3.41421 0.437145 0.218573 0.975821i \(-0.429860\pi\)
0.218573 + 0.975821i \(0.429860\pi\)
\(62\) 0 0
\(63\) 6.82843 0.860301
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.07107 0.741699 0.370849 0.928693i \(-0.379067\pi\)
0.370849 + 0.928693i \(0.379067\pi\)
\(68\) 0 0
\(69\) 12.6569 1.52371
\(70\) 0 0
\(71\) −3.07107 −0.364469 −0.182234 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(72\) 0 0
\(73\) 13.8284 1.61849 0.809247 0.587468i \(-0.199875\pi\)
0.809247 + 0.587468i \(0.199875\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.41421 −0.389086
\(78\) 0 0
\(79\) 0.828427 0.0932053 0.0466027 0.998914i \(-0.485161\pi\)
0.0466027 + 0.998914i \(0.485161\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 2.48528 0.272795 0.136398 0.990654i \(-0.456448\pi\)
0.136398 + 0.990654i \(0.456448\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.24264 −0.990915
\(88\) 0 0
\(89\) −3.75736 −0.398279 −0.199140 0.979971i \(-0.563815\pi\)
−0.199140 + 0.979971i \(0.563815\pi\)
\(90\) 0 0
\(91\) 4.41421 0.462735
\(92\) 0 0
\(93\) 8.24264 0.854722
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.65685 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 15.8995 1.58206 0.791029 0.611778i \(-0.209545\pi\)
0.791029 + 0.611778i \(0.209545\pi\)
\(102\) 0 0
\(103\) 9.89949 0.975426 0.487713 0.873004i \(-0.337831\pi\)
0.487713 + 0.873004i \(0.337831\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.41421 0.233391 0.116695 0.993168i \(-0.462770\pi\)
0.116695 + 0.993168i \(0.462770\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −12.4853 −1.18505
\(112\) 0 0
\(113\) −13.0711 −1.22962 −0.614811 0.788674i \(-0.710768\pi\)
−0.614811 + 0.788674i \(0.710768\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.17157 0.478112
\(118\) 0 0
\(119\) 2.41421 0.221311
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 17.0711 1.53925
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.1716 −0.991317 −0.495658 0.868518i \(-0.665073\pi\)
−0.495658 + 0.868518i \(0.665073\pi\)
\(128\) 0 0
\(129\) 5.41421 0.476695
\(130\) 0 0
\(131\) −21.6569 −1.89217 −0.946084 0.323921i \(-0.894999\pi\)
−0.946084 + 0.323921i \(0.894999\pi\)
\(132\) 0 0
\(133\) 2.41421 0.209339
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.3137 1.39377 0.696887 0.717181i \(-0.254568\pi\)
0.696887 + 0.717181i \(0.254568\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 19.3137 1.62651
\(142\) 0 0
\(143\) −2.58579 −0.216234
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.82843 0.233285
\(148\) 0 0
\(149\) −17.6569 −1.44651 −0.723253 0.690583i \(-0.757354\pi\)
−0.723253 + 0.690583i \(0.757354\pi\)
\(150\) 0 0
\(151\) −11.1716 −0.909130 −0.454565 0.890714i \(-0.650205\pi\)
−0.454565 + 0.890714i \(0.650205\pi\)
\(152\) 0 0
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.343146 0.0273860 0.0136930 0.999906i \(-0.495641\pi\)
0.0136930 + 0.999906i \(0.495641\pi\)
\(158\) 0 0
\(159\) 4.41421 0.350070
\(160\) 0 0
\(161\) −12.6569 −0.997500
\(162\) 0 0
\(163\) −2.92893 −0.229412 −0.114706 0.993400i \(-0.536593\pi\)
−0.114706 + 0.993400i \(0.536593\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.2132 −1.79629 −0.898146 0.439698i \(-0.855086\pi\)
−0.898146 + 0.439698i \(0.855086\pi\)
\(168\) 0 0
\(169\) −9.65685 −0.742835
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) 18.8284 1.43150 0.715749 0.698357i \(-0.246085\pi\)
0.715749 + 0.698357i \(0.246085\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 34.7990 2.61565
\(178\) 0 0
\(179\) 22.9706 1.71690 0.858450 0.512897i \(-0.171428\pi\)
0.858450 + 0.512897i \(0.171428\pi\)
\(180\) 0 0
\(181\) 0.485281 0.0360707 0.0180353 0.999837i \(-0.494259\pi\)
0.0180353 + 0.999837i \(0.494259\pi\)
\(182\) 0 0
\(183\) −8.24264 −0.609314
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.41421 −0.103418
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 11.2426 0.813489 0.406744 0.913542i \(-0.366664\pi\)
0.406744 + 0.913542i \(0.366664\pi\)
\(192\) 0 0
\(193\) 7.65685 0.551152 0.275576 0.961279i \(-0.411131\pi\)
0.275576 + 0.961279i \(0.411131\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.7279 1.47680 0.738402 0.674361i \(-0.235581\pi\)
0.738402 + 0.674361i \(0.235581\pi\)
\(198\) 0 0
\(199\) −16.0711 −1.13925 −0.569624 0.821905i \(-0.692911\pi\)
−0.569624 + 0.821905i \(0.692911\pi\)
\(200\) 0 0
\(201\) −14.6569 −1.03381
\(202\) 0 0
\(203\) 9.24264 0.648706
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −14.8284 −1.03065
\(208\) 0 0
\(209\) −1.41421 −0.0978232
\(210\) 0 0
\(211\) 24.8995 1.71415 0.857076 0.515190i \(-0.172279\pi\)
0.857076 + 0.515190i \(0.172279\pi\)
\(212\) 0 0
\(213\) 7.41421 0.508014
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.24264 −0.559547
\(218\) 0 0
\(219\) −33.3848 −2.25593
\(220\) 0 0
\(221\) 1.82843 0.122993
\(222\) 0 0
\(223\) −3.17157 −0.212384 −0.106192 0.994346i \(-0.533866\pi\)
−0.106192 + 0.994346i \(0.533866\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.2426 1.14443 0.572217 0.820102i \(-0.306083\pi\)
0.572217 + 0.820102i \(0.306083\pi\)
\(228\) 0 0
\(229\) −26.9706 −1.78226 −0.891132 0.453743i \(-0.850088\pi\)
−0.891132 + 0.453743i \(0.850088\pi\)
\(230\) 0 0
\(231\) 8.24264 0.542326
\(232\) 0 0
\(233\) −2.34315 −0.153505 −0.0767523 0.997050i \(-0.524455\pi\)
−0.0767523 + 0.997050i \(0.524455\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −2.27208 −0.146969 −0.0734843 0.997296i \(-0.523412\pi\)
−0.0734843 + 0.997296i \(0.523412\pi\)
\(240\) 0 0
\(241\) 5.65685 0.364390 0.182195 0.983262i \(-0.441680\pi\)
0.182195 + 0.983262i \(0.441680\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.82843 0.116340
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 27.5563 1.73934 0.869671 0.493632i \(-0.164331\pi\)
0.869671 + 0.493632i \(0.164331\pi\)
\(252\) 0 0
\(253\) 7.41421 0.466128
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.27208 0.204107 0.102053 0.994779i \(-0.467459\pi\)
0.102053 + 0.994779i \(0.467459\pi\)
\(258\) 0 0
\(259\) 12.4853 0.775798
\(260\) 0 0
\(261\) 10.8284 0.670263
\(262\) 0 0
\(263\) 4.34315 0.267810 0.133905 0.990994i \(-0.457248\pi\)
0.133905 + 0.990994i \(0.457248\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.07107 0.555140
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 21.3848 1.29903 0.649516 0.760348i \(-0.274971\pi\)
0.649516 + 0.760348i \(0.274971\pi\)
\(272\) 0 0
\(273\) −10.6569 −0.644982
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.3137 1.16045 0.580224 0.814457i \(-0.302965\pi\)
0.580224 + 0.814457i \(0.302965\pi\)
\(278\) 0 0
\(279\) −9.65685 −0.578141
\(280\) 0 0
\(281\) −12.2426 −0.730335 −0.365167 0.930942i \(-0.618988\pi\)
−0.365167 + 0.930942i \(0.618988\pi\)
\(282\) 0 0
\(283\) −26.4853 −1.57439 −0.787193 0.616706i \(-0.788467\pi\)
−0.787193 + 0.616706i \(0.788467\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −17.0711 −1.00767
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −18.4853 −1.08363
\(292\) 0 0
\(293\) −28.7990 −1.68245 −0.841227 0.540681i \(-0.818166\pi\)
−0.841227 + 0.540681i \(0.818166\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.585786 −0.0339908
\(298\) 0 0
\(299\) −9.58579 −0.554360
\(300\) 0 0
\(301\) −5.41421 −0.312070
\(302\) 0 0
\(303\) −38.3848 −2.20515
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 24.2843 1.38598 0.692988 0.720949i \(-0.256294\pi\)
0.692988 + 0.720949i \(0.256294\pi\)
\(308\) 0 0
\(309\) −23.8995 −1.35959
\(310\) 0 0
\(311\) −14.7574 −0.836813 −0.418407 0.908260i \(-0.637411\pi\)
−0.418407 + 0.908260i \(0.637411\pi\)
\(312\) 0 0
\(313\) −4.65685 −0.263221 −0.131610 0.991302i \(-0.542015\pi\)
−0.131610 + 0.991302i \(0.542015\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −32.7990 −1.84217 −0.921087 0.389356i \(-0.872698\pi\)
−0.921087 + 0.389356i \(0.872698\pi\)
\(318\) 0 0
\(319\) −5.41421 −0.303138
\(320\) 0 0
\(321\) −5.82843 −0.325311
\(322\) 0 0
\(323\) 1.00000 0.0556415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.0711 0.667532
\(328\) 0 0
\(329\) −19.3137 −1.06480
\(330\) 0 0
\(331\) 17.3848 0.955554 0.477777 0.878481i \(-0.341443\pi\)
0.477777 + 0.878481i \(0.341443\pi\)
\(332\) 0 0
\(333\) 14.6274 0.801578
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −12.7279 −0.693334 −0.346667 0.937988i \(-0.612687\pi\)
−0.346667 + 0.937988i \(0.612687\pi\)
\(338\) 0 0
\(339\) 31.5563 1.71391
\(340\) 0 0
\(341\) 4.82843 0.261474
\(342\) 0 0
\(343\) −19.7279 −1.06521
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.8284 −1.11813 −0.559064 0.829124i \(-0.688840\pi\)
−0.559064 + 0.829124i \(0.688840\pi\)
\(348\) 0 0
\(349\) −16.6274 −0.890045 −0.445023 0.895519i \(-0.646804\pi\)
−0.445023 + 0.895519i \(0.646804\pi\)
\(350\) 0 0
\(351\) 0.757359 0.0404248
\(352\) 0 0
\(353\) −26.1127 −1.38984 −0.694919 0.719088i \(-0.744560\pi\)
−0.694919 + 0.719088i \(0.744560\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −5.82843 −0.308473
\(358\) 0 0
\(359\) −8.07107 −0.425975 −0.212987 0.977055i \(-0.568319\pi\)
−0.212987 + 0.977055i \(0.568319\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 21.7279 1.14042
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −17.4558 −0.911188 −0.455594 0.890188i \(-0.650573\pi\)
−0.455594 + 0.890188i \(0.650573\pi\)
\(368\) 0 0
\(369\) −20.0000 −1.04116
\(370\) 0 0
\(371\) −4.41421 −0.229175
\(372\) 0 0
\(373\) 5.97056 0.309144 0.154572 0.987982i \(-0.450600\pi\)
0.154572 + 0.987982i \(0.450600\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.00000 0.360518
\(378\) 0 0
\(379\) −33.3848 −1.71486 −0.857430 0.514600i \(-0.827940\pi\)
−0.857430 + 0.514600i \(0.827940\pi\)
\(380\) 0 0
\(381\) 26.9706 1.38174
\(382\) 0 0
\(383\) −4.72792 −0.241586 −0.120793 0.992678i \(-0.538544\pi\)
−0.120793 + 0.992678i \(0.538544\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.34315 −0.322440
\(388\) 0 0
\(389\) −11.8995 −0.603328 −0.301664 0.953414i \(-0.597542\pi\)
−0.301664 + 0.953414i \(0.597542\pi\)
\(390\) 0 0
\(391\) −5.24264 −0.265132
\(392\) 0 0
\(393\) 52.2843 2.63739
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.6274 −1.13564 −0.567819 0.823154i \(-0.692213\pi\)
−0.567819 + 0.823154i \(0.692213\pi\)
\(398\) 0 0
\(399\) −5.82843 −0.291786
\(400\) 0 0
\(401\) −25.8995 −1.29336 −0.646680 0.762762i \(-0.723843\pi\)
−0.646680 + 0.762762i \(0.723843\pi\)
\(402\) 0 0
\(403\) −6.24264 −0.310968
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.31371 −0.362527
\(408\) 0 0
\(409\) 5.41421 0.267716 0.133858 0.991001i \(-0.457263\pi\)
0.133858 + 0.991001i \(0.457263\pi\)
\(410\) 0 0
\(411\) −39.3848 −1.94271
\(412\) 0 0
\(413\) −34.7990 −1.71235
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −9.65685 −0.472898
\(418\) 0 0
\(419\) −15.2132 −0.743214 −0.371607 0.928390i \(-0.621193\pi\)
−0.371607 + 0.928390i \(0.621193\pi\)
\(420\) 0 0
\(421\) −15.1421 −0.737983 −0.368991 0.929433i \(-0.620297\pi\)
−0.368991 + 0.929433i \(0.620297\pi\)
\(422\) 0 0
\(423\) −22.6274 −1.10018
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.24264 0.398889
\(428\) 0 0
\(429\) 6.24264 0.301398
\(430\) 0 0
\(431\) 2.58579 0.124553 0.0622765 0.998059i \(-0.480164\pi\)
0.0622765 + 0.998059i \(0.480164\pi\)
\(432\) 0 0
\(433\) −9.07107 −0.435928 −0.217964 0.975957i \(-0.569941\pi\)
−0.217964 + 0.975957i \(0.569941\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.24264 −0.250790
\(438\) 0 0
\(439\) 5.27208 0.251623 0.125811 0.992054i \(-0.459847\pi\)
0.125811 + 0.992054i \(0.459847\pi\)
\(440\) 0 0
\(441\) −3.31371 −0.157796
\(442\) 0 0
\(443\) 15.5563 0.739104 0.369552 0.929210i \(-0.379511\pi\)
0.369552 + 0.929210i \(0.379511\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 42.6274 2.01621
\(448\) 0 0
\(449\) 31.9411 1.50739 0.753697 0.657222i \(-0.228268\pi\)
0.753697 + 0.657222i \(0.228268\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 26.9706 1.26719
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.31371 −0.295343 −0.147671 0.989036i \(-0.547178\pi\)
−0.147671 + 0.989036i \(0.547178\pi\)
\(458\) 0 0
\(459\) 0.414214 0.0193338
\(460\) 0 0
\(461\) −32.7279 −1.52429 −0.762146 0.647406i \(-0.775854\pi\)
−0.762146 + 0.647406i \(0.775854\pi\)
\(462\) 0 0
\(463\) 6.14214 0.285449 0.142725 0.989762i \(-0.454414\pi\)
0.142725 + 0.989762i \(0.454414\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.10051 0.0971998 0.0485999 0.998818i \(-0.484524\pi\)
0.0485999 + 0.998818i \(0.484524\pi\)
\(468\) 0 0
\(469\) 14.6569 0.676791
\(470\) 0 0
\(471\) −0.828427 −0.0381719
\(472\) 0 0
\(473\) 3.17157 0.145829
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.17157 −0.236790
\(478\) 0 0
\(479\) 17.4558 0.797578 0.398789 0.917043i \(-0.369431\pi\)
0.398789 + 0.917043i \(0.369431\pi\)
\(480\) 0 0
\(481\) 9.45584 0.431149
\(482\) 0 0
\(483\) 30.5563 1.39036
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −12.8284 −0.581312 −0.290656 0.956828i \(-0.593873\pi\)
−0.290656 + 0.956828i \(0.593873\pi\)
\(488\) 0 0
\(489\) 7.07107 0.319765
\(490\) 0 0
\(491\) −26.2426 −1.18431 −0.592157 0.805823i \(-0.701723\pi\)
−0.592157 + 0.805823i \(0.701723\pi\)
\(492\) 0 0
\(493\) 3.82843 0.172424
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.41421 −0.332573
\(498\) 0 0
\(499\) −33.0711 −1.48046 −0.740232 0.672351i \(-0.765284\pi\)
−0.740232 + 0.672351i \(0.765284\pi\)
\(500\) 0 0
\(501\) 56.0416 2.50376
\(502\) 0 0
\(503\) −13.1005 −0.584123 −0.292061 0.956400i \(-0.594341\pi\)
−0.292061 + 0.956400i \(0.594341\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.3137 1.03540
\(508\) 0 0
\(509\) −1.65685 −0.0734388 −0.0367194 0.999326i \(-0.511691\pi\)
−0.0367194 + 0.999326i \(0.511691\pi\)
\(510\) 0 0
\(511\) 33.3848 1.47686
\(512\) 0 0
\(513\) 0.414214 0.0182880
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 11.3137 0.497576
\(518\) 0 0
\(519\) −45.4558 −1.99529
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) −22.2132 −0.971316 −0.485658 0.874149i \(-0.661420\pi\)
−0.485658 + 0.874149i \(0.661420\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.41421 −0.148725
\(528\) 0 0
\(529\) 4.48528 0.195012
\(530\) 0 0
\(531\) −40.7696 −1.76925
\(532\) 0 0
\(533\) −12.9289 −0.560014
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −55.4558 −2.39310
\(538\) 0 0
\(539\) 1.65685 0.0713658
\(540\) 0 0
\(541\) −18.0416 −0.775670 −0.387835 0.921729i \(-0.626777\pi\)
−0.387835 + 0.921729i \(0.626777\pi\)
\(542\) 0 0
\(543\) −1.17157 −0.0502770
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −43.9411 −1.87879 −0.939393 0.342841i \(-0.888611\pi\)
−0.939393 + 0.342841i \(0.888611\pi\)
\(548\) 0 0
\(549\) 9.65685 0.412144
\(550\) 0 0
\(551\) 3.82843 0.163096
\(552\) 0 0
\(553\) 2.00000 0.0850487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.6863 −0.876506 −0.438253 0.898852i \(-0.644403\pi\)
−0.438253 + 0.898852i \(0.644403\pi\)
\(558\) 0 0
\(559\) −4.10051 −0.173433
\(560\) 0 0
\(561\) 3.41421 0.144148
\(562\) 0 0
\(563\) 11.0294 0.464835 0.232418 0.972616i \(-0.425336\pi\)
0.232418 + 0.972616i \(0.425336\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −22.8995 −0.961688
\(568\) 0 0
\(569\) 24.9706 1.04682 0.523410 0.852081i \(-0.324660\pi\)
0.523410 + 0.852081i \(0.324660\pi\)
\(570\) 0 0
\(571\) 11.6985 0.489566 0.244783 0.969578i \(-0.421283\pi\)
0.244783 + 0.969578i \(0.421283\pi\)
\(572\) 0 0
\(573\) −27.1421 −1.13388
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −25.0000 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) −18.4853 −0.768222
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 2.58579 0.107092
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.38478 −0.180979 −0.0904895 0.995897i \(-0.528843\pi\)
−0.0904895 + 0.995897i \(0.528843\pi\)
\(588\) 0 0
\(589\) −3.41421 −0.140680
\(590\) 0 0
\(591\) −50.0416 −2.05844
\(592\) 0 0
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 38.7990 1.58794
\(598\) 0 0
\(599\) 2.44365 0.0998449 0.0499224 0.998753i \(-0.484103\pi\)
0.0499224 + 0.998753i \(0.484103\pi\)
\(600\) 0 0
\(601\) 31.5563 1.28721 0.643605 0.765358i \(-0.277438\pi\)
0.643605 + 0.765358i \(0.277438\pi\)
\(602\) 0 0
\(603\) 17.1716 0.699281
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −13.5147 −0.548546 −0.274273 0.961652i \(-0.588437\pi\)
−0.274273 + 0.961652i \(0.588437\pi\)
\(608\) 0 0
\(609\) −22.3137 −0.904197
\(610\) 0 0
\(611\) −14.6274 −0.591762
\(612\) 0 0
\(613\) 22.7279 0.917972 0.458986 0.888443i \(-0.348213\pi\)
0.458986 + 0.888443i \(0.348213\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.79899 0.313976 0.156988 0.987601i \(-0.449822\pi\)
0.156988 + 0.987601i \(0.449822\pi\)
\(618\) 0 0
\(619\) 30.3848 1.22127 0.610634 0.791913i \(-0.290915\pi\)
0.610634 + 0.791913i \(0.290915\pi\)
\(620\) 0 0
\(621\) −2.17157 −0.0871422
\(622\) 0 0
\(623\) −9.07107 −0.363425
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.41421 0.136351
\(628\) 0 0
\(629\) 5.17157 0.206204
\(630\) 0 0
\(631\) 38.9706 1.55139 0.775697 0.631106i \(-0.217399\pi\)
0.775697 + 0.631106i \(0.217399\pi\)
\(632\) 0 0
\(633\) −60.1127 −2.38927
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.14214 −0.0848745
\(638\) 0 0
\(639\) −8.68629 −0.343624
\(640\) 0 0
\(641\) −39.2132 −1.54883 −0.774414 0.632679i \(-0.781955\pi\)
−0.774414 + 0.632679i \(0.781955\pi\)
\(642\) 0 0
\(643\) −31.1716 −1.22929 −0.614643 0.788805i \(-0.710700\pi\)
−0.614643 + 0.788805i \(0.710700\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0711 −0.474563 −0.237281 0.971441i \(-0.576256\pi\)
−0.237281 + 0.971441i \(0.576256\pi\)
\(648\) 0 0
\(649\) 20.3848 0.800172
\(650\) 0 0
\(651\) 19.8995 0.779923
\(652\) 0 0
\(653\) −32.2843 −1.26338 −0.631691 0.775221i \(-0.717639\pi\)
−0.631691 + 0.775221i \(0.717639\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 39.1127 1.52593
\(658\) 0 0
\(659\) 8.89949 0.346675 0.173338 0.984862i \(-0.444545\pi\)
0.173338 + 0.984862i \(0.444545\pi\)
\(660\) 0 0
\(661\) 43.4853 1.69138 0.845691 0.533673i \(-0.179189\pi\)
0.845691 + 0.533673i \(0.179189\pi\)
\(662\) 0 0
\(663\) −4.41421 −0.171434
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.0711 −0.777155
\(668\) 0 0
\(669\) 7.65685 0.296031
\(670\) 0 0
\(671\) −4.82843 −0.186399
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.3431 −0.820284 −0.410142 0.912022i \(-0.634521\pi\)
−0.410142 + 0.912022i \(0.634521\pi\)
\(678\) 0 0
\(679\) 18.4853 0.709400
\(680\) 0 0
\(681\) −41.6274 −1.59517
\(682\) 0 0
\(683\) −23.3137 −0.892074 −0.446037 0.895014i \(-0.647165\pi\)
−0.446037 + 0.895014i \(0.647165\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 65.1127 2.48420
\(688\) 0 0
\(689\) −3.34315 −0.127364
\(690\) 0 0
\(691\) −7.45584 −0.283634 −0.141817 0.989893i \(-0.545294\pi\)
−0.141817 + 0.989893i \(0.545294\pi\)
\(692\) 0 0
\(693\) −9.65685 −0.366834
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −7.07107 −0.267836
\(698\) 0 0
\(699\) 5.65685 0.213962
\(700\) 0 0
\(701\) 32.9706 1.24528 0.622640 0.782508i \(-0.286060\pi\)
0.622640 + 0.782508i \(0.286060\pi\)
\(702\) 0 0
\(703\) 5.17157 0.195050
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 38.3848 1.44361
\(708\) 0 0
\(709\) 10.6863 0.401332 0.200666 0.979660i \(-0.435689\pi\)
0.200666 + 0.979660i \(0.435689\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) 17.8995 0.670341
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.48528 0.204852
\(718\) 0 0
\(719\) −30.4142 −1.13426 −0.567129 0.823629i \(-0.691946\pi\)
−0.567129 + 0.823629i \(0.691946\pi\)
\(720\) 0 0
\(721\) 23.8995 0.890064
\(722\) 0 0
\(723\) −13.6569 −0.507904
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.78680 0.0662686 0.0331343 0.999451i \(-0.489451\pi\)
0.0331343 + 0.999451i \(0.489451\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) −2.24264 −0.0829471
\(732\) 0 0
\(733\) −10.6274 −0.392533 −0.196266 0.980551i \(-0.562882\pi\)
−0.196266 + 0.980551i \(0.562882\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.58579 −0.316262
\(738\) 0 0
\(739\) −7.41421 −0.272736 −0.136368 0.990658i \(-0.543543\pi\)
−0.136368 + 0.990658i \(0.543543\pi\)
\(740\) 0 0
\(741\) −4.41421 −0.162160
\(742\) 0 0
\(743\) −14.2426 −0.522512 −0.261256 0.965270i \(-0.584137\pi\)
−0.261256 + 0.965270i \(0.584137\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.02944 0.257194
\(748\) 0 0
\(749\) 5.82843 0.212966
\(750\) 0 0
\(751\) −29.7574 −1.08586 −0.542931 0.839777i \(-0.682685\pi\)
−0.542931 + 0.839777i \(0.682685\pi\)
\(752\) 0 0
\(753\) −66.5269 −2.42438
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16.3431 −0.594002 −0.297001 0.954877i \(-0.595986\pi\)
−0.297001 + 0.954877i \(0.595986\pi\)
\(758\) 0 0
\(759\) −17.8995 −0.649711
\(760\) 0 0
\(761\) −4.02944 −0.146067 −0.0730335 0.997329i \(-0.523268\pi\)
−0.0730335 + 0.997329i \(0.523268\pi\)
\(762\) 0 0
\(763\) −12.0711 −0.437002
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −26.3553 −0.951636
\(768\) 0 0
\(769\) −30.7990 −1.11064 −0.555320 0.831637i \(-0.687404\pi\)
−0.555320 + 0.831637i \(0.687404\pi\)
\(770\) 0 0
\(771\) −7.89949 −0.284493
\(772\) 0 0
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −30.1421 −1.08134
\(778\) 0 0
\(779\) −7.07107 −0.253347
\(780\) 0 0
\(781\) 4.34315 0.155410
\(782\) 0 0
\(783\) 1.58579 0.0566714
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 22.7574 0.811212 0.405606 0.914048i \(-0.367060\pi\)
0.405606 + 0.914048i \(0.367060\pi\)
\(788\) 0 0
\(789\) −10.4853 −0.373286
\(790\) 0 0
\(791\) −31.5563 −1.12201
\(792\) 0 0
\(793\) 6.24264 0.221683
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.1421 1.24480 0.622399 0.782700i \(-0.286158\pi\)
0.622399 + 0.782700i \(0.286158\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) −10.6274 −0.375501
\(802\) 0 0
\(803\) −19.5563 −0.690129
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.1421 0.849843
\(808\) 0 0
\(809\) −8.51472 −0.299362 −0.149681 0.988734i \(-0.547825\pi\)
−0.149681 + 0.988734i \(0.547825\pi\)
\(810\) 0 0
\(811\) 42.0122 1.47525 0.737624 0.675212i \(-0.235948\pi\)
0.737624 + 0.675212i \(0.235948\pi\)
\(812\) 0 0
\(813\) −51.6274 −1.81065
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.24264 −0.0784601
\(818\) 0 0
\(819\) 12.4853 0.436271
\(820\) 0 0
\(821\) −51.5980 −1.80078 −0.900391 0.435082i \(-0.856719\pi\)
−0.900391 + 0.435082i \(0.856719\pi\)
\(822\) 0 0
\(823\) 47.5269 1.65668 0.828342 0.560223i \(-0.189284\pi\)
0.828342 + 0.560223i \(0.189284\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.8701 1.17778 0.588889 0.808214i \(-0.299566\pi\)
0.588889 + 0.808214i \(0.299566\pi\)
\(828\) 0 0
\(829\) 25.7696 0.895014 0.447507 0.894281i \(-0.352312\pi\)
0.447507 + 0.894281i \(0.352312\pi\)
\(830\) 0 0
\(831\) −46.6274 −1.61749
\(832\) 0 0
\(833\) −1.17157 −0.0405926
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.41421 −0.0488824
\(838\) 0 0
\(839\) 49.9411 1.72416 0.862080 0.506773i \(-0.169162\pi\)
0.862080 + 0.506773i \(0.169162\pi\)
\(840\) 0 0
\(841\) −14.3431 −0.494591
\(842\) 0 0
\(843\) 29.5563 1.01797
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −21.7279 −0.746580
\(848\) 0 0
\(849\) 63.9411 2.19445
\(850\) 0 0
\(851\) −27.1127 −0.929411
\(852\) 0 0
\(853\) −19.9411 −0.682771 −0.341386 0.939923i \(-0.610896\pi\)
−0.341386 + 0.939923i \(0.610896\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.0000 0.546550 0.273275 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 0 0
\(859\) −5.21320 −0.177872 −0.0889361 0.996037i \(-0.528347\pi\)
−0.0889361 + 0.996037i \(0.528347\pi\)
\(860\) 0 0
\(861\) 41.2132 1.40454
\(862\) 0 0
\(863\) 35.0711 1.19383 0.596917 0.802303i \(-0.296392\pi\)
0.596917 + 0.802303i \(0.296392\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 38.6274 1.31186
\(868\) 0 0
\(869\) −1.17157 −0.0397429
\(870\) 0 0
\(871\) 11.1005 0.376126
\(872\) 0 0
\(873\) 21.6569 0.732973
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −29.4853 −0.995647 −0.497824 0.867278i \(-0.665867\pi\)
−0.497824 + 0.867278i \(0.665867\pi\)
\(878\) 0 0
\(879\) 69.5269 2.34508
\(880\) 0 0
\(881\) 12.2843 0.413868 0.206934 0.978355i \(-0.433652\pi\)
0.206934 + 0.978355i \(0.433652\pi\)
\(882\) 0 0
\(883\) −15.4558 −0.520131 −0.260065 0.965591i \(-0.583744\pi\)
−0.260065 + 0.965591i \(0.583744\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.928932 −0.0311905 −0.0155952 0.999878i \(-0.504964\pi\)
−0.0155952 + 0.999878i \(0.504964\pi\)
\(888\) 0 0
\(889\) −26.9706 −0.904564
\(890\) 0 0
\(891\) 13.4142 0.449393
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 23.1421 0.772693
\(898\) 0 0
\(899\) −13.0711 −0.435945
\(900\) 0 0
\(901\) −1.82843 −0.0609137
\(902\) 0 0
\(903\) 13.0711 0.434978
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8.55635 −0.284109 −0.142054 0.989859i \(-0.545371\pi\)
−0.142054 + 0.989859i \(0.545371\pi\)
\(908\) 0 0
\(909\) 44.9706 1.49158
\(910\) 0 0
\(911\) 9.65685 0.319946 0.159973 0.987121i \(-0.448859\pi\)
0.159973 + 0.987121i \(0.448859\pi\)
\(912\) 0 0
\(913\) −3.51472 −0.116320
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −52.2843 −1.72658
\(918\) 0 0
\(919\) −46.8406 −1.54513 −0.772565 0.634936i \(-0.781026\pi\)
−0.772565 + 0.634936i \(0.781026\pi\)
\(920\) 0 0
\(921\) −58.6274 −1.93184
\(922\) 0 0
\(923\) −5.61522 −0.184827
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) 25.4853 0.836145 0.418072 0.908414i \(-0.362706\pi\)
0.418072 + 0.908414i \(0.362706\pi\)
\(930\) 0 0
\(931\) −1.17157 −0.0383968
\(932\) 0 0
\(933\) 35.6274 1.16639
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.3137 0.728957 0.364479 0.931212i \(-0.381247\pi\)
0.364479 + 0.931212i \(0.381247\pi\)
\(938\) 0 0
\(939\) 11.2426 0.366890
\(940\) 0 0
\(941\) 57.1421 1.86278 0.931390 0.364022i \(-0.118597\pi\)
0.931390 + 0.364022i \(0.118597\pi\)
\(942\) 0 0
\(943\) 37.0711 1.20720
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41.2548 1.34060 0.670301 0.742089i \(-0.266165\pi\)
0.670301 + 0.742089i \(0.266165\pi\)
\(948\) 0 0
\(949\) 25.2843 0.820762
\(950\) 0 0
\(951\) 79.1838 2.56771
\(952\) 0 0
\(953\) 19.2132 0.622377 0.311188 0.950348i \(-0.399273\pi\)
0.311188 + 0.950348i \(0.399273\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 13.0711 0.422528
\(958\) 0 0
\(959\) 39.3848 1.27180
\(960\) 0 0
\(961\) −19.3431 −0.623972
\(962\) 0 0
\(963\) 6.82843 0.220043
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −50.1421 −1.61246 −0.806231 0.591601i \(-0.798496\pi\)
−0.806231 + 0.591601i \(0.798496\pi\)
\(968\) 0 0
\(969\) −2.41421 −0.0775557
\(970\) 0 0
\(971\) 35.5980 1.14239 0.571197 0.820813i \(-0.306479\pi\)
0.571197 + 0.820813i \(0.306479\pi\)
\(972\) 0 0
\(973\) 9.65685 0.309585
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.61522 −0.0516756 −0.0258378 0.999666i \(-0.508225\pi\)
−0.0258378 + 0.999666i \(0.508225\pi\)
\(978\) 0 0
\(979\) 5.31371 0.169827
\(980\) 0 0
\(981\) −14.1421 −0.451524
\(982\) 0 0
\(983\) 53.7990 1.71592 0.857961 0.513715i \(-0.171731\pi\)
0.857961 + 0.513715i \(0.171731\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 46.6274 1.48417
\(988\) 0 0
\(989\) 11.7574 0.373862
\(990\) 0 0
\(991\) −10.5269 −0.334398 −0.167199 0.985923i \(-0.553472\pi\)
−0.167199 + 0.985923i \(0.553472\pi\)
\(992\) 0 0
\(993\) −41.9706 −1.33190
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 27.5563 0.872718 0.436359 0.899773i \(-0.356268\pi\)
0.436359 + 0.899773i \(0.356268\pi\)
\(998\) 0 0
\(999\) 2.14214 0.0677742
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 7600.2.a.x.1.1 2
4.3 odd 2 3800.2.a.p.1.2 2
5.2 odd 4 1520.2.d.d.609.4 4
5.3 odd 4 1520.2.d.d.609.1 4
5.4 even 2 7600.2.a.bc.1.2 2
20.3 even 4 760.2.d.c.609.4 yes 4
20.7 even 4 760.2.d.c.609.1 4
20.19 odd 2 3800.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.c.609.1 4 20.7 even 4
760.2.d.c.609.4 yes 4 20.3 even 4
1520.2.d.d.609.1 4 5.3 odd 4
1520.2.d.d.609.4 4 5.2 odd 4
3800.2.a.l.1.1 2 20.19 odd 2
3800.2.a.p.1.2 2 4.3 odd 2
7600.2.a.x.1.1 2 1.1 even 1 trivial
7600.2.a.bc.1.2 2 5.4 even 2