Properties

Label 4600.2.e.o.4049.2
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.o.4049.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155i q^{3} +0.561553 q^{9} +O(q^{10})\) \(q-1.56155i q^{3} +0.561553 q^{9} -3.12311 q^{11} -0.438447i q^{13} +5.12311i q^{17} +3.12311 q^{19} +1.00000i q^{23} -5.56155i q^{27} -3.56155 q^{29} -2.43845 q^{31} +4.87689i q^{33} +8.24621i q^{37} -0.684658 q^{39} -9.80776 q^{41} +8.00000i q^{43} -0.684658i q^{47} +7.00000 q^{49} +8.00000 q^{51} -2.00000i q^{53} -4.87689i q^{57} -10.2462 q^{59} -4.24621 q^{61} +3.12311i q^{67} +1.56155 q^{69} +13.5616 q^{71} +14.6847i q^{73} -3.12311 q^{79} -7.00000 q^{81} -14.2462i q^{83} +5.56155i q^{87} -11.3693 q^{89} +3.80776i q^{93} +11.3693i q^{97} -1.75379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + O(q^{10}) \) \( 4 q - 6 q^{9} + 4 q^{11} - 4 q^{19} - 6 q^{29} - 18 q^{31} + 22 q^{39} + 2 q^{41} + 28 q^{49} + 32 q^{51} - 8 q^{59} + 16 q^{61} - 2 q^{69} + 46 q^{71} + 4 q^{79} - 28 q^{81} + 4 q^{89} - 40 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.56155i − 0.901563i −0.892634 0.450781i \(-0.851145\pi\)
0.892634 0.450781i \(-0.148855\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) −3.12311 −0.941652 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(12\) 0 0
\(13\) − 0.438447i − 0.121603i −0.998150 0.0608017i \(-0.980634\pi\)
0.998150 0.0608017i \(-0.0193657\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.12311i 1.24254i 0.783598 + 0.621268i \(0.213382\pi\)
−0.783598 + 0.621268i \(0.786618\pi\)
\(18\) 0 0
\(19\) 3.12311 0.716490 0.358245 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.56155i − 1.07032i
\(28\) 0 0
\(29\) −3.56155 −0.661364 −0.330682 0.943742i \(-0.607279\pi\)
−0.330682 + 0.943742i \(0.607279\pi\)
\(30\) 0 0
\(31\) −2.43845 −0.437958 −0.218979 0.975730i \(-0.570273\pi\)
−0.218979 + 0.975730i \(0.570273\pi\)
\(32\) 0 0
\(33\) 4.87689i 0.848958i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24621i 1.35567i 0.735215 + 0.677834i \(0.237081\pi\)
−0.735215 + 0.677834i \(0.762919\pi\)
\(38\) 0 0
\(39\) −0.684658 −0.109633
\(40\) 0 0
\(41\) −9.80776 −1.53172 −0.765858 0.643010i \(-0.777685\pi\)
−0.765858 + 0.643010i \(0.777685\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.684658i − 0.0998677i −0.998753 0.0499338i \(-0.984099\pi\)
0.998753 0.0499338i \(-0.0159011\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.87689i − 0.645960i
\(58\) 0 0
\(59\) −10.2462 −1.33394 −0.666972 0.745083i \(-0.732410\pi\)
−0.666972 + 0.745083i \(0.732410\pi\)
\(60\) 0 0
\(61\) −4.24621 −0.543672 −0.271836 0.962344i \(-0.587631\pi\)
−0.271836 + 0.962344i \(0.587631\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.12311i 0.381548i 0.981634 + 0.190774i \(0.0610998\pi\)
−0.981634 + 0.190774i \(0.938900\pi\)
\(68\) 0 0
\(69\) 1.56155 0.187989
\(70\) 0 0
\(71\) 13.5616 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(72\) 0 0
\(73\) 14.6847i 1.71871i 0.511380 + 0.859355i \(0.329134\pi\)
−0.511380 + 0.859355i \(0.670866\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.12311 −0.351377 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) − 14.2462i − 1.56372i −0.623451 0.781862i \(-0.714270\pi\)
0.623451 0.781862i \(-0.285730\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.56155i 0.596261i
\(88\) 0 0
\(89\) −11.3693 −1.20515 −0.602573 0.798064i \(-0.705858\pi\)
−0.602573 + 0.798064i \(0.705858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.80776i 0.394847i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.3693i 1.15438i 0.816610 + 0.577190i \(0.195851\pi\)
−0.816610 + 0.577190i \(0.804149\pi\)
\(98\) 0 0
\(99\) −1.75379 −0.176262
\(100\) 0 0
\(101\) 12.2462 1.21854 0.609272 0.792961i \(-0.291462\pi\)
0.609272 + 0.792961i \(0.291462\pi\)
\(102\) 0 0
\(103\) 14.2462i 1.40372i 0.712314 + 0.701860i \(0.247647\pi\)
−0.712314 + 0.701860i \(0.752353\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.1231i 1.07531i 0.843165 + 0.537656i \(0.180690\pi\)
−0.843165 + 0.537656i \(0.819310\pi\)
\(108\) 0 0
\(109\) −13.1231 −1.25697 −0.628483 0.777824i \(-0.716324\pi\)
−0.628483 + 0.777824i \(0.716324\pi\)
\(110\) 0 0
\(111\) 12.8769 1.22222
\(112\) 0 0
\(113\) 2.87689i 0.270635i 0.990802 + 0.135318i \(0.0432055\pi\)
−0.990802 + 0.135318i \(0.956794\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 0.246211i − 0.0227622i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 0 0
\(123\) 15.3153i 1.38094i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.684658i 0.0607536i 0.999539 + 0.0303768i \(0.00967073\pi\)
−0.999539 + 0.0303768i \(0.990329\pi\)
\(128\) 0 0
\(129\) 12.4924 1.09990
\(130\) 0 0
\(131\) 3.31534 0.289663 0.144831 0.989456i \(-0.453736\pi\)
0.144831 + 0.989456i \(0.453736\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.87689i 0.587533i 0.955877 + 0.293766i \(0.0949088\pi\)
−0.955877 + 0.293766i \(0.905091\pi\)
\(138\) 0 0
\(139\) 3.31534 0.281204 0.140602 0.990066i \(-0.455096\pi\)
0.140602 + 0.990066i \(0.455096\pi\)
\(140\) 0 0
\(141\) −1.06913 −0.0900370
\(142\) 0 0
\(143\) 1.36932i 0.114508i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 10.9309i − 0.901563i
\(148\) 0 0
\(149\) 20.2462 1.65863 0.829317 0.558778i \(-0.188730\pi\)
0.829317 + 0.558778i \(0.188730\pi\)
\(150\) 0 0
\(151\) −10.4384 −0.849469 −0.424734 0.905318i \(-0.639633\pi\)
−0.424734 + 0.905318i \(0.639633\pi\)
\(152\) 0 0
\(153\) 2.87689i 0.232583i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 21.1231i 1.68581i 0.538064 + 0.842904i \(0.319156\pi\)
−0.538064 + 0.842904i \(0.680844\pi\)
\(158\) 0 0
\(159\) −3.12311 −0.247678
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 18.9309i − 1.48278i −0.671074 0.741390i \(-0.734167\pi\)
0.671074 0.741390i \(-0.265833\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 0 0
\(169\) 12.8078 0.985213
\(170\) 0 0
\(171\) 1.75379 0.134116
\(172\) 0 0
\(173\) 10.0000i 0.760286i 0.924928 + 0.380143i \(0.124125\pi\)
−0.924928 + 0.380143i \(0.875875\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.0000i 1.20263i
\(178\) 0 0
\(179\) 3.31534 0.247800 0.123900 0.992295i \(-0.460460\pi\)
0.123900 + 0.992295i \(0.460460\pi\)
\(180\) 0 0
\(181\) 3.75379 0.279017 0.139508 0.990221i \(-0.455448\pi\)
0.139508 + 0.990221i \(0.455448\pi\)
\(182\) 0 0
\(183\) 6.63068i 0.490154i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 16.0000i − 1.17004i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 25.3693 1.83566 0.917830 0.396974i \(-0.129940\pi\)
0.917830 + 0.396974i \(0.129940\pi\)
\(192\) 0 0
\(193\) 0.438447i 0.0315601i 0.999875 + 0.0157801i \(0.00502316\pi\)
−0.999875 + 0.0157801i \(0.994977\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 16.9309i − 1.20627i −0.797637 0.603137i \(-0.793917\pi\)
0.797637 0.603137i \(-0.206083\pi\)
\(198\) 0 0
\(199\) −11.1231 −0.788496 −0.394248 0.919004i \(-0.628995\pi\)
−0.394248 + 0.919004i \(0.628995\pi\)
\(200\) 0 0
\(201\) 4.87689 0.343990
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.561553i 0.0390306i
\(208\) 0 0
\(209\) −9.75379 −0.674684
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) − 21.1771i − 1.45103i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 22.9309 1.54952
\(220\) 0 0
\(221\) 2.24621 0.151097
\(222\) 0 0
\(223\) 28.4924i 1.90799i 0.299817 + 0.953997i \(0.403075\pi\)
−0.299817 + 0.953997i \(0.596925\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 6.24621i − 0.414576i −0.978280 0.207288i \(-0.933536\pi\)
0.978280 0.207288i \(-0.0664637\pi\)
\(228\) 0 0
\(229\) −14.8769 −0.983093 −0.491546 0.870851i \(-0.663568\pi\)
−0.491546 + 0.870851i \(0.663568\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0540i 1.05173i 0.850568 + 0.525865i \(0.176258\pi\)
−0.850568 + 0.525865i \(0.823742\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.87689i 0.316788i
\(238\) 0 0
\(239\) −26.0540 −1.68529 −0.842646 0.538468i \(-0.819003\pi\)
−0.842646 + 0.538468i \(0.819003\pi\)
\(240\) 0 0
\(241\) 6.49242 0.418214 0.209107 0.977893i \(-0.432944\pi\)
0.209107 + 0.977893i \(0.432944\pi\)
\(242\) 0 0
\(243\) − 5.75379i − 0.369106i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.36932i − 0.0871275i
\(248\) 0 0
\(249\) −22.2462 −1.40980
\(250\) 0 0
\(251\) −6.24621 −0.394257 −0.197129 0.980378i \(-0.563162\pi\)
−0.197129 + 0.980378i \(0.563162\pi\)
\(252\) 0 0
\(253\) − 3.12311i − 0.196348i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 5.31534i − 0.331562i −0.986163 0.165781i \(-0.946986\pi\)
0.986163 0.165781i \(-0.0530145\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 19.1231i 1.17918i 0.807702 + 0.589591i \(0.200711\pi\)
−0.807702 + 0.589591i \(0.799289\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 17.7538i 1.08651i
\(268\) 0 0
\(269\) −13.3153 −0.811851 −0.405925 0.913906i \(-0.633051\pi\)
−0.405925 + 0.913906i \(0.633051\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.8078i 1.06996i 0.844863 + 0.534982i \(0.179682\pi\)
−0.844863 + 0.534982i \(0.820318\pi\)
\(278\) 0 0
\(279\) −1.36932 −0.0819789
\(280\) 0 0
\(281\) 5.12311 0.305619 0.152809 0.988256i \(-0.451168\pi\)
0.152809 + 0.988256i \(0.451168\pi\)
\(282\) 0 0
\(283\) − 3.12311i − 0.185649i −0.995682 0.0928247i \(-0.970410\pi\)
0.995682 0.0928247i \(-0.0295896\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.24621 −0.543895
\(290\) 0 0
\(291\) 17.7538 1.04075
\(292\) 0 0
\(293\) 15.3693i 0.897885i 0.893561 + 0.448943i \(0.148199\pi\)
−0.893561 + 0.448943i \(0.851801\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 17.3693i 1.00787i
\(298\) 0 0
\(299\) 0.438447 0.0253561
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 19.1231i − 1.09859i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 18.2462i − 1.04137i −0.853750 0.520683i \(-0.825677\pi\)
0.853750 0.520683i \(-0.174323\pi\)
\(308\) 0 0
\(309\) 22.2462 1.26554
\(310\) 0 0
\(311\) 5.56155 0.315367 0.157683 0.987490i \(-0.449597\pi\)
0.157683 + 0.987490i \(0.449597\pi\)
\(312\) 0 0
\(313\) 15.3693i 0.868725i 0.900738 + 0.434363i \(0.143026\pi\)
−0.900738 + 0.434363i \(0.856974\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3.75379i − 0.210834i −0.994428 0.105417i \(-0.966382\pi\)
0.994428 0.105417i \(-0.0336177\pi\)
\(318\) 0 0
\(319\) 11.1231 0.622774
\(320\) 0 0
\(321\) 17.3693 0.969461
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.4924i 1.13323i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 31.4233 1.72718 0.863590 0.504194i \(-0.168211\pi\)
0.863590 + 0.504194i \(0.168211\pi\)
\(332\) 0 0
\(333\) 4.63068i 0.253760i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 21.6155i − 1.17747i −0.808325 0.588736i \(-0.799626\pi\)
0.808325 0.588736i \(-0.200374\pi\)
\(338\) 0 0
\(339\) 4.49242 0.243995
\(340\) 0 0
\(341\) 7.61553 0.412404
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 16.4924i − 0.885360i −0.896680 0.442680i \(-0.854028\pi\)
0.896680 0.442680i \(-0.145972\pi\)
\(348\) 0 0
\(349\) 34.6847 1.85663 0.928314 0.371798i \(-0.121259\pi\)
0.928314 + 0.371798i \(0.121259\pi\)
\(350\) 0 0
\(351\) −2.43845 −0.130155
\(352\) 0 0
\(353\) − 15.5616i − 0.828258i −0.910218 0.414129i \(-0.864086\pi\)
0.910218 0.414129i \(-0.135914\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.6155 1.24638 0.623190 0.782071i \(-0.285836\pi\)
0.623190 + 0.782071i \(0.285836\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) 0 0
\(363\) 1.94602i 0.102140i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 14.2462i − 0.743646i −0.928304 0.371823i \(-0.878733\pi\)
0.928304 0.371823i \(-0.121267\pi\)
\(368\) 0 0
\(369\) −5.50758 −0.286713
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.4924i 0.543277i 0.962399 + 0.271639i \(0.0875655\pi\)
−0.962399 + 0.271639i \(0.912434\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.56155i 0.0804241i
\(378\) 0 0
\(379\) 12.4924 0.641693 0.320846 0.947131i \(-0.396033\pi\)
0.320846 + 0.947131i \(0.396033\pi\)
\(380\) 0 0
\(381\) 1.06913 0.0547732
\(382\) 0 0
\(383\) − 9.75379i − 0.498395i −0.968453 0.249198i \(-0.919833\pi\)
0.968453 0.249198i \(-0.0801669\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.49242i 0.228363i
\(388\) 0 0
\(389\) 23.3693 1.18487 0.592436 0.805618i \(-0.298166\pi\)
0.592436 + 0.805618i \(0.298166\pi\)
\(390\) 0 0
\(391\) −5.12311 −0.259087
\(392\) 0 0
\(393\) − 5.17708i − 0.261149i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.4384i 0.825022i 0.910953 + 0.412511i \(0.135348\pi\)
−0.910953 + 0.412511i \(0.864652\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −28.2462 −1.41055 −0.705274 0.708935i \(-0.749176\pi\)
−0.705274 + 0.708935i \(0.749176\pi\)
\(402\) 0 0
\(403\) 1.06913i 0.0532572i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 25.7538i − 1.27657i
\(408\) 0 0
\(409\) −26.3002 −1.30046 −0.650230 0.759737i \(-0.725328\pi\)
−0.650230 + 0.759737i \(0.725328\pi\)
\(410\) 0 0
\(411\) 10.7386 0.529698
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5.17708i − 0.253523i
\(418\) 0 0
\(419\) −20.4924 −1.00112 −0.500560 0.865702i \(-0.666873\pi\)
−0.500560 + 0.865702i \(0.666873\pi\)
\(420\) 0 0
\(421\) −18.8769 −0.920004 −0.460002 0.887918i \(-0.652151\pi\)
−0.460002 + 0.887918i \(0.652151\pi\)
\(422\) 0 0
\(423\) − 0.384472i − 0.0186937i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.13826 0.103236
\(430\) 0 0
\(431\) 4.49242 0.216392 0.108196 0.994130i \(-0.465493\pi\)
0.108196 + 0.994130i \(0.465493\pi\)
\(432\) 0 0
\(433\) 24.7386i 1.18886i 0.804146 + 0.594431i \(0.202623\pi\)
−0.804146 + 0.594431i \(0.797377\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.12311i 0.149398i
\(438\) 0 0
\(439\) 26.0540 1.24349 0.621744 0.783220i \(-0.286424\pi\)
0.621744 + 0.783220i \(0.286424\pi\)
\(440\) 0 0
\(441\) 3.93087 0.187184
\(442\) 0 0
\(443\) 34.9309i 1.65962i 0.558049 + 0.829808i \(0.311550\pi\)
−0.558049 + 0.829808i \(0.688450\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 31.6155i − 1.49536i
\(448\) 0 0
\(449\) −8.24621 −0.389163 −0.194581 0.980886i \(-0.562335\pi\)
−0.194581 + 0.980886i \(0.562335\pi\)
\(450\) 0 0
\(451\) 30.6307 1.44234
\(452\) 0 0
\(453\) 16.3002i 0.765850i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.3693i 0.906058i 0.891496 + 0.453029i \(0.149657\pi\)
−0.891496 + 0.453029i \(0.850343\pi\)
\(458\) 0 0
\(459\) 28.4924 1.32991
\(460\) 0 0
\(461\) 9.80776 0.456793 0.228397 0.973568i \(-0.426652\pi\)
0.228397 + 0.973568i \(0.426652\pi\)
\(462\) 0 0
\(463\) 20.4924i 0.952364i 0.879347 + 0.476182i \(0.157980\pi\)
−0.879347 + 0.476182i \(0.842020\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.36932i 0.433560i 0.976220 + 0.216780i \(0.0695555\pi\)
−0.976220 + 0.216780i \(0.930445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 32.9848 1.51986
\(472\) 0 0
\(473\) − 24.9848i − 1.14880i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1.12311i − 0.0514235i
\(478\) 0 0
\(479\) 6.24621 0.285397 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(480\) 0 0
\(481\) 3.61553 0.164854
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 24.6847i 1.11857i 0.828976 + 0.559284i \(0.188924\pi\)
−0.828976 + 0.559284i \(0.811076\pi\)
\(488\) 0 0
\(489\) −29.5616 −1.33682
\(490\) 0 0
\(491\) −7.80776 −0.352359 −0.176180 0.984358i \(-0.556374\pi\)
−0.176180 + 0.984358i \(0.556374\pi\)
\(492\) 0 0
\(493\) − 18.2462i − 0.821768i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0540 −0.629142 −0.314571 0.949234i \(-0.601861\pi\)
−0.314571 + 0.949234i \(0.601861\pi\)
\(500\) 0 0
\(501\) −24.9848 −1.11624
\(502\) 0 0
\(503\) 17.3693i 0.774460i 0.921983 + 0.387230i \(0.126568\pi\)
−0.921983 + 0.387230i \(0.873432\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 20.0000i − 0.888231i
\(508\) 0 0
\(509\) −38.3002 −1.69763 −0.848813 0.528693i \(-0.822682\pi\)
−0.848813 + 0.528693i \(0.822682\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) − 17.3693i − 0.766874i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.13826i 0.0940406i
\(518\) 0 0
\(519\) 15.6155 0.685446
\(520\) 0 0
\(521\) −21.6155 −0.946993 −0.473497 0.880796i \(-0.657008\pi\)
−0.473497 + 0.880796i \(0.657008\pi\)
\(522\) 0 0
\(523\) − 22.2462i − 0.972759i −0.873748 0.486379i \(-0.838317\pi\)
0.873748 0.486379i \(-0.161683\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 12.4924i − 0.544178i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −5.75379 −0.249693
\(532\) 0 0
\(533\) 4.30019i 0.186262i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 5.17708i − 0.223408i
\(538\) 0 0
\(539\) −21.8617 −0.941652
\(540\) 0 0
\(541\) 7.06913 0.303926 0.151963 0.988386i \(-0.451441\pi\)
0.151963 + 0.988386i \(0.451441\pi\)
\(542\) 0 0
\(543\) − 5.86174i − 0.251551i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 26.5464i − 1.13504i −0.823359 0.567521i \(-0.807903\pi\)
0.823359 0.567521i \(-0.192097\pi\)
\(548\) 0 0
\(549\) −2.38447 −0.101767
\(550\) 0 0
\(551\) −11.1231 −0.473860
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 9.12311i − 0.386558i −0.981144 0.193279i \(-0.938088\pi\)
0.981144 0.193279i \(-0.0619123\pi\)
\(558\) 0 0
\(559\) 3.50758 0.148355
\(560\) 0 0
\(561\) −24.9848 −1.05486
\(562\) 0 0
\(563\) − 22.2462i − 0.937566i −0.883313 0.468783i \(-0.844693\pi\)
0.883313 0.468783i \(-0.155307\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.24621 −0.345699 −0.172850 0.984948i \(-0.555297\pi\)
−0.172850 + 0.984948i \(0.555297\pi\)
\(570\) 0 0
\(571\) −44.4924 −1.86195 −0.930975 0.365083i \(-0.881041\pi\)
−0.930975 + 0.365083i \(0.881041\pi\)
\(572\) 0 0
\(573\) − 39.6155i − 1.65496i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 12.9309i − 0.538319i −0.963096 0.269160i \(-0.913254\pi\)
0.963096 0.269160i \(-0.0867459\pi\)
\(578\) 0 0
\(579\) 0.684658 0.0284534
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.24621i 0.258692i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.0540i 0.580070i 0.957016 + 0.290035i \(0.0936669\pi\)
−0.957016 + 0.290035i \(0.906333\pi\)
\(588\) 0 0
\(589\) −7.61553 −0.313792
\(590\) 0 0
\(591\) −26.4384 −1.08753
\(592\) 0 0
\(593\) − 4.73863i − 0.194592i −0.995255 0.0972962i \(-0.968981\pi\)
0.995255 0.0972962i \(-0.0310194\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.3693i 0.710879i
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −18.1922 −0.742077 −0.371038 0.928618i \(-0.620998\pi\)
−0.371038 + 0.928618i \(0.620998\pi\)
\(602\) 0 0
\(603\) 1.75379i 0.0714198i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 34.7386i 1.41000i 0.709208 + 0.704999i \(0.249052\pi\)
−0.709208 + 0.704999i \(0.750948\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.300187 −0.0121442
\(612\) 0 0
\(613\) − 46.1080i − 1.86228i −0.364659 0.931141i \(-0.618814\pi\)
0.364659 0.931141i \(-0.381186\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 30.9848i − 1.24740i −0.781663 0.623701i \(-0.785628\pi\)
0.781663 0.623701i \(-0.214372\pi\)
\(618\) 0 0
\(619\) 4.49242 0.180566 0.0902829 0.995916i \(-0.471223\pi\)
0.0902829 + 0.995916i \(0.471223\pi\)
\(620\) 0 0
\(621\) 5.56155 0.223177
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 15.2311i 0.608270i
\(628\) 0 0
\(629\) −42.2462 −1.68447
\(630\) 0 0
\(631\) 42.7386 1.70140 0.850699 0.525653i \(-0.176179\pi\)
0.850699 + 0.525653i \(0.176179\pi\)
\(632\) 0 0
\(633\) 6.24621i 0.248265i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.06913i − 0.121603i
\(638\) 0 0
\(639\) 7.61553 0.301266
\(640\) 0 0
\(641\) 11.3693 0.449061 0.224531 0.974467i \(-0.427915\pi\)
0.224531 + 0.974467i \(0.427915\pi\)
\(642\) 0 0
\(643\) − 1.36932i − 0.0540006i −0.999635 0.0270003i \(-0.991404\pi\)
0.999635 0.0270003i \(-0.00859550\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.3002i 0.955339i 0.878540 + 0.477669i \(0.158518\pi\)
−0.878540 + 0.477669i \(0.841482\pi\)
\(648\) 0 0
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.0540i 0.471709i 0.971788 + 0.235854i \(0.0757888\pi\)
−0.971788 + 0.235854i \(0.924211\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.24621i 0.321715i
\(658\) 0 0
\(659\) 14.2462 0.554954 0.277477 0.960732i \(-0.410502\pi\)
0.277477 + 0.960732i \(0.410502\pi\)
\(660\) 0 0
\(661\) 16.2462 0.631904 0.315952 0.948775i \(-0.397676\pi\)
0.315952 + 0.948775i \(0.397676\pi\)
\(662\) 0 0
\(663\) − 3.50758i − 0.136223i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.56155i − 0.137904i
\(668\) 0 0
\(669\) 44.4924 1.72018
\(670\) 0 0
\(671\) 13.2614 0.511949
\(672\) 0 0
\(673\) 40.0540i 1.54397i 0.635642 + 0.771984i \(0.280735\pi\)
−0.635642 + 0.771984i \(0.719265\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.9848i 1.03711i 0.855044 + 0.518556i \(0.173530\pi\)
−0.855044 + 0.518556i \(0.826470\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −9.75379 −0.373766
\(682\) 0 0
\(683\) − 28.6847i − 1.09759i −0.835958 0.548794i \(-0.815087\pi\)
0.835958 0.548794i \(-0.184913\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 23.2311i 0.886320i
\(688\) 0 0
\(689\) −0.876894 −0.0334070
\(690\) 0 0
\(691\) −44.9848 −1.71130 −0.855652 0.517551i \(-0.826844\pi\)
−0.855652 + 0.517551i \(0.826844\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 50.2462i − 1.90321i
\(698\) 0 0
\(699\) 25.0691 0.948202
\(700\) 0 0
\(701\) −27.8617 −1.05232 −0.526162 0.850385i \(-0.676369\pi\)
−0.526162 + 0.850385i \(0.676369\pi\)
\(702\) 0 0
\(703\) 25.7538i 0.971323i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12.6307 0.474355 0.237178 0.971466i \(-0.423778\pi\)
0.237178 + 0.971466i \(0.423778\pi\)
\(710\) 0 0
\(711\) −1.75379 −0.0657722
\(712\) 0 0
\(713\) − 2.43845i − 0.0913206i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 40.6847i 1.51940i
\(718\) 0 0
\(719\) 28.4924 1.06259 0.531294 0.847187i \(-0.321706\pi\)
0.531294 + 0.847187i \(0.321706\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 10.1383i − 0.377046i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 14.6307i − 0.542622i −0.962492 0.271311i \(-0.912543\pi\)
0.962492 0.271311i \(-0.0874572\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) −40.9848 −1.51588
\(732\) 0 0
\(733\) − 24.6307i − 0.909755i −0.890554 0.454878i \(-0.849683\pi\)
0.890554 0.454878i \(-0.150317\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 9.75379i − 0.359285i
\(738\) 0 0
\(739\) −12.6847 −0.466613 −0.233306 0.972403i \(-0.574954\pi\)
−0.233306 + 0.972403i \(0.574954\pi\)
\(740\) 0 0
\(741\) −2.13826 −0.0785510
\(742\) 0 0
\(743\) − 17.7538i − 0.651323i −0.945486 0.325662i \(-0.894413\pi\)
0.945486 0.325662i \(-0.105587\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 8.00000i − 0.292705i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −24.9848 −0.911710 −0.455855 0.890054i \(-0.650666\pi\)
−0.455855 + 0.890054i \(0.650666\pi\)
\(752\) 0 0
\(753\) 9.75379i 0.355448i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 34.4924i − 1.25365i −0.779161 0.626824i \(-0.784354\pi\)
0.779161 0.626824i \(-0.215646\pi\)
\(758\) 0 0
\(759\) −4.87689 −0.177020
\(760\) 0 0
\(761\) −30.6847 −1.11232 −0.556159 0.831076i \(-0.687725\pi\)
−0.556159 + 0.831076i \(0.687725\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.49242i 0.162212i
\(768\) 0 0
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 0 0
\(771\) −8.30019 −0.298924
\(772\) 0 0
\(773\) − 27.3693i − 0.984406i −0.870480 0.492203i \(-0.836192\pi\)
0.870480 0.492203i \(-0.163808\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −30.6307 −1.09746
\(780\) 0 0
\(781\) −42.3542 −1.51555
\(782\) 0 0
\(783\) 19.8078i 0.707872i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 9.75379i − 0.347685i −0.984773 0.173843i \(-0.944382\pi\)
0.984773 0.173843i \(-0.0556184\pi\)
\(788\) 0 0
\(789\) 29.8617 1.06311
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.86174i 0.0661123i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 4.24621i − 0.150409i −0.997168 0.0752043i \(-0.976039\pi\)
0.997168 0.0752043i \(-0.0239609\pi\)
\(798\) 0 0
\(799\) 3.50758 0.124089
\(800\) 0 0
\(801\) −6.38447 −0.225584
\(802\) 0 0
\(803\) − 45.8617i − 1.61843i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 20.7926i 0.731935i
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 5.06913 0.178001 0.0890006 0.996032i \(-0.471633\pi\)
0.0890006 + 0.996032i \(0.471633\pi\)
\(812\) 0 0
\(813\) − 37.4773i − 1.31439i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 24.9848i 0.874109i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24.7386 0.863384 0.431692 0.902021i \(-0.357917\pi\)
0.431692 + 0.902021i \(0.357917\pi\)
\(822\) 0 0
\(823\) − 53.5616i − 1.86704i −0.358527 0.933519i \(-0.616721\pi\)
0.358527 0.933519i \(-0.383279\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.4924i − 0.990779i −0.868671 0.495389i \(-0.835025\pi\)
0.868671 0.495389i \(-0.164975\pi\)
\(828\) 0 0
\(829\) 3.75379 0.130374 0.0651872 0.997873i \(-0.479236\pi\)
0.0651872 + 0.997873i \(0.479236\pi\)
\(830\) 0 0
\(831\) 27.8078 0.964641
\(832\) 0 0
\(833\) 35.8617i 1.24254i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 13.5616i 0.468756i
\(838\) 0 0
\(839\) −33.7538 −1.16531 −0.582655 0.812720i \(-0.697986\pi\)
−0.582655 + 0.812720i \(0.697986\pi\)
\(840\) 0 0
\(841\) −16.3153 −0.562598
\(842\) 0 0
\(843\) − 8.00000i − 0.275535i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −4.87689 −0.167375
\(850\) 0 0
\(851\) −8.24621 −0.282676
\(852\) 0 0
\(853\) − 30.9848i − 1.06090i −0.847716 0.530450i \(-0.822023\pi\)
0.847716 0.530450i \(-0.177977\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.4384i 0.424889i 0.977173 + 0.212445i \(0.0681425\pi\)
−0.977173 + 0.212445i \(0.931857\pi\)
\(858\) 0 0
\(859\) −54.0540 −1.84430 −0.922149 0.386835i \(-0.873568\pi\)
−0.922149 + 0.386835i \(0.873568\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 37.1771i − 1.26552i −0.774347 0.632761i \(-0.781921\pi\)
0.774347 0.632761i \(-0.218079\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.4384i 0.490355i
\(868\) 0 0
\(869\) 9.75379 0.330875
\(870\) 0 0
\(871\) 1.36932 0.0463975
\(872\) 0 0
\(873\) 6.38447i 0.216082i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 34.9848i − 1.18135i −0.806908 0.590677i \(-0.798861\pi\)
0.806908 0.590677i \(-0.201139\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 13.1231 0.442129 0.221064 0.975259i \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(882\) 0 0
\(883\) 28.9848i 0.975418i 0.873006 + 0.487709i \(0.162167\pi\)
−0.873006 + 0.487709i \(0.837833\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.31534i 0.245625i 0.992430 + 0.122813i \(0.0391914\pi\)
−0.992430 + 0.122813i \(0.960809\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 21.8617 0.732396
\(892\) 0 0
\(893\) − 2.13826i − 0.0715542i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 0.684658i − 0.0228601i
\(898\) 0 0
\(899\) 8.68466 0.289650
\(900\) 0 0
\(901\) 10.2462 0.341351
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 51.1231i 1.69751i 0.528782 + 0.848757i \(0.322649\pi\)
−0.528782 + 0.848757i \(0.677351\pi\)
\(908\) 0 0
\(909\) 6.87689 0.228092
\(910\) 0 0
\(911\) −28.8769 −0.956734 −0.478367 0.878160i \(-0.658771\pi\)
−0.478367 + 0.878160i \(0.658771\pi\)
\(912\) 0 0
\(913\) 44.4924i 1.47248i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −54.2462 −1.78942 −0.894709 0.446650i \(-0.852617\pi\)
−0.894709 + 0.446650i \(0.852617\pi\)
\(920\) 0 0
\(921\) −28.4924 −0.938857
\(922\) 0 0
\(923\) − 5.94602i − 0.195716i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) −14.1922 −0.465632 −0.232816 0.972521i \(-0.574794\pi\)
−0.232816 + 0.972521i \(0.574794\pi\)
\(930\) 0 0
\(931\) 21.8617 0.716490
\(932\) 0 0
\(933\) − 8.68466i − 0.284323i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 32.2462i 1.05344i 0.850040 + 0.526719i \(0.176578\pi\)
−0.850040 + 0.526719i \(0.823422\pi\)
\(938\) 0 0
\(939\) 24.0000 0.783210
\(940\) 0 0
\(941\) 30.8769 1.00656 0.503279 0.864124i \(-0.332127\pi\)
0.503279 + 0.864124i \(0.332127\pi\)
\(942\) 0 0
\(943\) − 9.80776i − 0.319385i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.56155i 0.0507436i 0.999678 + 0.0253718i \(0.00807697\pi\)
−0.999678 + 0.0253718i \(0.991923\pi\)
\(948\) 0 0
\(949\) 6.43845 0.209001
\(950\) 0 0
\(951\) −5.86174 −0.190080
\(952\) 0 0
\(953\) − 27.7538i − 0.899033i −0.893272 0.449517i \(-0.851596\pi\)
0.893272 0.449517i \(-0.148404\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 17.3693i − 0.561470i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −25.0540 −0.808193
\(962\) 0 0
\(963\) 6.24621i 0.201281i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 2.05398i − 0.0660514i −0.999455 0.0330257i \(-0.989486\pi\)
0.999455 0.0330257i \(-0.0105143\pi\)
\(968\) 0 0
\(969\) 24.9848 0.802629
\(970\) 0 0
\(971\) 6.63068 0.212789 0.106394 0.994324i \(-0.466069\pi\)
0.106394 + 0.994324i \(0.466069\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 25.2311i 0.807213i 0.914933 + 0.403607i \(0.132244\pi\)
−0.914933 + 0.403607i \(0.867756\pi\)
\(978\) 0 0
\(979\) 35.5076 1.13483
\(980\) 0 0
\(981\) −7.36932 −0.235284
\(982\) 0 0
\(983\) 52.1080i 1.66199i 0.556283 + 0.830993i \(0.312227\pi\)
−0.556283 + 0.830993i \(0.687773\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −56.9848 −1.81018 −0.905092 0.425217i \(-0.860198\pi\)
−0.905092 + 0.425217i \(0.860198\pi\)
\(992\) 0 0
\(993\) − 49.0691i − 1.55716i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) 0 0
\(999\) 45.8617 1.45100
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 4600.2.e.o.4049.2 4
5.2 odd 4 4600.2.a.s.1.1 2
5.3 odd 4 184.2.a.e.1.2 2
5.4 even 2 inner 4600.2.e.o.4049.3 4
15.8 even 4 1656.2.a.j.1.2 2
20.3 even 4 368.2.a.i.1.1 2
20.7 even 4 9200.2.a.br.1.2 2
35.13 even 4 9016.2.a.w.1.1 2
40.3 even 4 1472.2.a.p.1.2 2
40.13 odd 4 1472.2.a.u.1.1 2
60.23 odd 4 3312.2.a.t.1.1 2
115.68 even 4 4232.2.a.o.1.2 2
460.183 odd 4 8464.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.2 2 5.3 odd 4
368.2.a.i.1.1 2 20.3 even 4
1472.2.a.p.1.2 2 40.3 even 4
1472.2.a.u.1.1 2 40.13 odd 4
1656.2.a.j.1.2 2 15.8 even 4
3312.2.a.t.1.1 2 60.23 odd 4
4232.2.a.o.1.2 2 115.68 even 4
4600.2.a.s.1.1 2 5.2 odd 4
4600.2.e.o.4049.2 4 1.1 even 1 trivial
4600.2.e.o.4049.3 4 5.4 even 2 inner
8464.2.a.bd.1.1 2 460.183 odd 4
9016.2.a.w.1.1 2 35.13 even 4
9200.2.a.br.1.2 2 20.7 even 4