Properties

Label 108.5.c.c.53.1
Level 108
Weight 5
Character 108.53
Analytic conductor 11.164
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.1
Root \(-1.00000i\) of \(x^{2} + 1\)
Character \(\chi\) \(=\) 108.53
Dual form 108.5.c.c.53.2

$q$-expansion

\(f(q)\) \(=\) \(q-9.00000i q^{5} +5.00000 q^{7} +O(q^{10})\) \(q-9.00000i q^{5} +5.00000 q^{7} -117.000i q^{11} -34.0000 q^{13} -450.000i q^{17} -64.0000 q^{19} -612.000i q^{23} +544.000 q^{25} -1062.00i q^{29} -697.000 q^{31} -45.0000i q^{35} -748.000 q^{37} -684.000i q^{41} +2618.00 q^{43} +2646.00i q^{47} -2376.00 q^{49} +1071.00i q^{53} -1053.00 q^{55} +5814.00i q^{59} +6404.00 q^{61} +306.000i q^{65} -5218.00 q^{67} +6570.00i q^{71} -4519.00 q^{73} -585.000i q^{77} +7502.00 q^{79} +5481.00i q^{83} -4050.00 q^{85} -8874.00i q^{89} -170.000 q^{91} +576.000i q^{95} +10571.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{7} + O(q^{10}) \) \( 2q + 10q^{7} - 68q^{13} - 128q^{19} + 1088q^{25} - 1394q^{31} - 1496q^{37} + 5236q^{43} - 4752q^{49} - 2106q^{55} + 12808q^{61} - 10436q^{67} - 9038q^{73} + 15004q^{79} - 8100q^{85} - 340q^{91} + 21142q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 9.00000i − 0.360000i −0.983667 0.180000i \(-0.942390\pi\)
0.983667 0.180000i \(-0.0576098\pi\)
\(6\) 0 0
\(7\) 5.00000 0.102041 0.0510204 0.998698i \(-0.483753\pi\)
0.0510204 + 0.998698i \(0.483753\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 117.000i − 0.966942i −0.875360 0.483471i \(-0.839376\pi\)
0.875360 0.483471i \(-0.160624\pi\)
\(12\) 0 0
\(13\) −34.0000 −0.201183 −0.100592 0.994928i \(-0.532074\pi\)
−0.100592 + 0.994928i \(0.532074\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 450.000i − 1.55709i −0.627587 0.778547i \(-0.715957\pi\)
0.627587 0.778547i \(-0.284043\pi\)
\(18\) 0 0
\(19\) −64.0000 −0.177285 −0.0886427 0.996063i \(-0.528253\pi\)
−0.0886427 + 0.996063i \(0.528253\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 612.000i − 1.15690i −0.815718 0.578450i \(-0.803658\pi\)
0.815718 0.578450i \(-0.196342\pi\)
\(24\) 0 0
\(25\) 544.000 0.870400
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1062.00i − 1.26278i −0.775464 0.631391i \(-0.782484\pi\)
0.775464 0.631391i \(-0.217516\pi\)
\(30\) 0 0
\(31\) −697.000 −0.725286 −0.362643 0.931928i \(-0.618126\pi\)
−0.362643 + 0.931928i \(0.618126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 45.0000i − 0.0367347i
\(36\) 0 0
\(37\) −748.000 −0.546384 −0.273192 0.961959i \(-0.588079\pi\)
−0.273192 + 0.961959i \(0.588079\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 684.000i − 0.406901i −0.979085 0.203450i \(-0.934784\pi\)
0.979085 0.203450i \(-0.0652155\pi\)
\(42\) 0 0
\(43\) 2618.00 1.41590 0.707950 0.706262i \(-0.249620\pi\)
0.707950 + 0.706262i \(0.249620\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2646.00i 1.19783i 0.800814 + 0.598914i \(0.204401\pi\)
−0.800814 + 0.598914i \(0.795599\pi\)
\(48\) 0 0
\(49\) −2376.00 −0.989588
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1071.00i 0.381274i 0.981661 + 0.190637i \(0.0610554\pi\)
−0.981661 + 0.190637i \(0.938945\pi\)
\(54\) 0 0
\(55\) −1053.00 −0.348099
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5814.00i 1.67021i 0.550091 + 0.835105i \(0.314593\pi\)
−0.550091 + 0.835105i \(0.685407\pi\)
\(60\) 0 0
\(61\) 6404.00 1.72104 0.860521 0.509414i \(-0.170138\pi\)
0.860521 + 0.509414i \(0.170138\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 306.000i 0.0724260i
\(66\) 0 0
\(67\) −5218.00 −1.16240 −0.581198 0.813762i \(-0.697416\pi\)
−0.581198 + 0.813762i \(0.697416\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6570.00i 1.30331i 0.758514 + 0.651656i \(0.225926\pi\)
−0.758514 + 0.651656i \(0.774074\pi\)
\(72\) 0 0
\(73\) −4519.00 −0.848002 −0.424001 0.905662i \(-0.639375\pi\)
−0.424001 + 0.905662i \(0.639375\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 585.000i − 0.0986676i
\(78\) 0 0
\(79\) 7502.00 1.20205 0.601025 0.799230i \(-0.294759\pi\)
0.601025 + 0.799230i \(0.294759\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5481.00i 0.795616i 0.917469 + 0.397808i \(0.130229\pi\)
−0.917469 + 0.397808i \(0.869771\pi\)
\(84\) 0 0
\(85\) −4050.00 −0.560554
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 8874.00i − 1.12031i −0.828387 0.560157i \(-0.810741\pi\)
0.828387 0.560157i \(-0.189259\pi\)
\(90\) 0 0
\(91\) −170.000 −0.0205289
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 576.000i 0.0638227i
\(96\) 0 0
\(97\) 10571.0 1.12350 0.561749 0.827307i \(-0.310129\pi\)
0.561749 + 0.827307i \(0.310129\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 13113.0i − 1.28546i −0.766092 0.642731i \(-0.777801\pi\)
0.766092 0.642731i \(-0.222199\pi\)
\(102\) 0 0
\(103\) −5830.00 −0.549533 −0.274767 0.961511i \(-0.588601\pi\)
−0.274767 + 0.961511i \(0.588601\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1089.00i − 0.0951175i −0.998868 0.0475587i \(-0.984856\pi\)
0.998868 0.0475587i \(-0.0151441\pi\)
\(108\) 0 0
\(109\) −5020.00 −0.422523 −0.211262 0.977430i \(-0.567757\pi\)
−0.211262 + 0.977430i \(0.567757\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 21384.0i − 1.67468i −0.546682 0.837340i \(-0.684109\pi\)
0.546682 0.837340i \(-0.315891\pi\)
\(114\) 0 0
\(115\) −5508.00 −0.416484
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 2250.00i − 0.158887i
\(120\) 0 0
\(121\) 952.000 0.0650229
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 10521.0i − 0.673344i
\(126\) 0 0
\(127\) 9227.00 0.572075 0.286038 0.958218i \(-0.407662\pi\)
0.286038 + 0.958218i \(0.407662\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 4275.00i − 0.249111i −0.992213 0.124556i \(-0.960249\pi\)
0.992213 0.124556i \(-0.0397505\pi\)
\(132\) 0 0
\(133\) −320.000 −0.0180903
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11322.0i − 0.603229i −0.953430 0.301614i \(-0.902474\pi\)
0.953430 0.301614i \(-0.0975255\pi\)
\(138\) 0 0
\(139\) 9812.00 0.507841 0.253921 0.967225i \(-0.418280\pi\)
0.253921 + 0.967225i \(0.418280\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3978.00i 0.194533i
\(144\) 0 0
\(145\) −9558.00 −0.454602
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16479.0i 0.742264i 0.928580 + 0.371132i \(0.121030\pi\)
−0.928580 + 0.371132i \(0.878970\pi\)
\(150\) 0 0
\(151\) −25555.0 −1.12078 −0.560392 0.828227i \(-0.689350\pi\)
−0.560392 + 0.828227i \(0.689350\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6273.00i 0.261103i
\(156\) 0 0
\(157\) 21164.0 0.858615 0.429307 0.903158i \(-0.358758\pi\)
0.429307 + 0.903158i \(0.358758\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 3060.00i − 0.118051i
\(162\) 0 0
\(163\) 33830.0 1.27329 0.636644 0.771158i \(-0.280322\pi\)
0.636644 + 0.771158i \(0.280322\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 27378.0i 0.981677i 0.871250 + 0.490839i \(0.163310\pi\)
−0.871250 + 0.490839i \(0.836690\pi\)
\(168\) 0 0
\(169\) −27405.0 −0.959525
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 46197.0i 1.54355i 0.635894 + 0.771777i \(0.280632\pi\)
−0.635894 + 0.771777i \(0.719368\pi\)
\(174\) 0 0
\(175\) 2720.00 0.0888163
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20781.0i 0.648575i 0.945959 + 0.324288i \(0.105125\pi\)
−0.945959 + 0.324288i \(0.894875\pi\)
\(180\) 0 0
\(181\) −19504.0 −0.595342 −0.297671 0.954669i \(-0.596210\pi\)
−0.297671 + 0.954669i \(0.596210\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6732.00i 0.196698i
\(186\) 0 0
\(187\) −52650.0 −1.50562
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 7038.00i − 0.192922i −0.995337 0.0964612i \(-0.969248\pi\)
0.995337 0.0964612i \(-0.0307524\pi\)
\(192\) 0 0
\(193\) 51527.0 1.38331 0.691656 0.722227i \(-0.256881\pi\)
0.691656 + 0.722227i \(0.256881\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 60435.0i 1.55724i 0.627495 + 0.778621i \(0.284080\pi\)
−0.627495 + 0.778621i \(0.715920\pi\)
\(198\) 0 0
\(199\) 45665.0 1.15313 0.576564 0.817052i \(-0.304393\pi\)
0.576564 + 0.817052i \(0.304393\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 5310.00i − 0.128855i
\(204\) 0 0
\(205\) −6156.00 −0.146484
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7488.00i 0.171425i
\(210\) 0 0
\(211\) 13124.0 0.294782 0.147391 0.989078i \(-0.452912\pi\)
0.147391 + 0.989078i \(0.452912\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 23562.0i − 0.509724i
\(216\) 0 0
\(217\) −3485.00 −0.0740088
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15300.0i 0.313261i
\(222\) 0 0
\(223\) −78850.0 −1.58559 −0.792797 0.609486i \(-0.791376\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 22302.0i − 0.432805i −0.976304 0.216402i \(-0.930568\pi\)
0.976304 0.216402i \(-0.0694323\pi\)
\(228\) 0 0
\(229\) 8870.00 0.169142 0.0845712 0.996417i \(-0.473048\pi\)
0.0845712 + 0.996417i \(0.473048\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26010.0i − 0.479103i −0.970884 0.239551i \(-0.923000\pi\)
0.970884 0.239551i \(-0.0770003\pi\)
\(234\) 0 0
\(235\) 23814.0 0.431218
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 64152.0i − 1.12309i −0.827446 0.561545i \(-0.810207\pi\)
0.827446 0.561545i \(-0.189793\pi\)
\(240\) 0 0
\(241\) 33422.0 0.575438 0.287719 0.957715i \(-0.407103\pi\)
0.287719 + 0.957715i \(0.407103\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 21384.0i 0.356252i
\(246\) 0 0
\(247\) 2176.00 0.0356669
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 110466.i − 1.75340i −0.481037 0.876700i \(-0.659740\pi\)
0.481037 0.876700i \(-0.340260\pi\)
\(252\) 0 0
\(253\) −71604.0 −1.11866
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 15174.0i − 0.229739i −0.993381 0.114869i \(-0.963355\pi\)
0.993381 0.114869i \(-0.0366449\pi\)
\(258\) 0 0
\(259\) −3740.00 −0.0557535
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 69948.0i − 1.01126i −0.862750 0.505631i \(-0.831260\pi\)
0.862750 0.505631i \(-0.168740\pi\)
\(264\) 0 0
\(265\) 9639.00 0.137259
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 50346.0i 0.695762i 0.937539 + 0.347881i \(0.113099\pi\)
−0.937539 + 0.347881i \(0.886901\pi\)
\(270\) 0 0
\(271\) 108323. 1.47497 0.737483 0.675366i \(-0.236014\pi\)
0.737483 + 0.675366i \(0.236014\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 63648.0i − 0.841626i
\(276\) 0 0
\(277\) 30716.0 0.400318 0.200159 0.979763i \(-0.435854\pi\)
0.200159 + 0.979763i \(0.435854\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 129258.i − 1.63699i −0.574517 0.818493i \(-0.694810\pi\)
0.574517 0.818493i \(-0.305190\pi\)
\(282\) 0 0
\(283\) −63976.0 −0.798811 −0.399406 0.916774i \(-0.630783\pi\)
−0.399406 + 0.916774i \(0.630783\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 3420.00i − 0.0415205i
\(288\) 0 0
\(289\) −118979. −1.42454
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 108630.i 1.26536i 0.774413 + 0.632681i \(0.218045\pi\)
−0.774413 + 0.632681i \(0.781955\pi\)
\(294\) 0 0
\(295\) 52326.0 0.601275
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 20808.0i 0.232749i
\(300\) 0 0
\(301\) 13090.0 0.144480
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 57636.0i − 0.619575i
\(306\) 0 0
\(307\) 6410.00 0.0680113 0.0340057 0.999422i \(-0.489174\pi\)
0.0340057 + 0.999422i \(0.489174\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 70992.0i − 0.733987i −0.930223 0.366994i \(-0.880387\pi\)
0.930223 0.366994i \(-0.119613\pi\)
\(312\) 0 0
\(313\) 160961. 1.64298 0.821489 0.570224i \(-0.193143\pi\)
0.821489 + 0.570224i \(0.193143\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 194733.i 1.93785i 0.247347 + 0.968927i \(0.420441\pi\)
−0.247347 + 0.968927i \(0.579559\pi\)
\(318\) 0 0
\(319\) −124254. −1.22104
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 28800.0i 0.276050i
\(324\) 0 0
\(325\) −18496.0 −0.175110
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13230.0i 0.122227i
\(330\) 0 0
\(331\) −33286.0 −0.303812 −0.151906 0.988395i \(-0.548541\pi\)
−0.151906 + 0.988395i \(0.548541\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 46962.0i 0.418463i
\(336\) 0 0
\(337\) −127690. −1.12434 −0.562169 0.827022i \(-0.690033\pi\)
−0.562169 + 0.827022i \(0.690033\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 81549.0i 0.701310i
\(342\) 0 0
\(343\) −23885.0 −0.203019
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 138807.i 1.15280i 0.817169 + 0.576398i \(0.195542\pi\)
−0.817169 + 0.576398i \(0.804458\pi\)
\(348\) 0 0
\(349\) 203792. 1.67316 0.836578 0.547848i \(-0.184553\pi\)
0.836578 + 0.547848i \(0.184553\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 32328.0i − 0.259436i −0.991551 0.129718i \(-0.958593\pi\)
0.991551 0.129718i \(-0.0414071\pi\)
\(354\) 0 0
\(355\) 59130.0 0.469193
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 185922.i 1.44259i 0.692630 + 0.721293i \(0.256452\pi\)
−0.692630 + 0.721293i \(0.743548\pi\)
\(360\) 0 0
\(361\) −126225. −0.968570
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 40671.0i 0.305281i
\(366\) 0 0
\(367\) 151079. 1.12169 0.560844 0.827922i \(-0.310477\pi\)
0.560844 + 0.827922i \(0.310477\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5355.00i 0.0389056i
\(372\) 0 0
\(373\) −6478.00 −0.0465611 −0.0232806 0.999729i \(-0.507411\pi\)
−0.0232806 + 0.999729i \(0.507411\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36108.0i 0.254051i
\(378\) 0 0
\(379\) 99008.0 0.689274 0.344637 0.938736i \(-0.388002\pi\)
0.344637 + 0.938736i \(0.388002\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 91062.0i 0.620783i 0.950609 + 0.310391i \(0.100460\pi\)
−0.950609 + 0.310391i \(0.899540\pi\)
\(384\) 0 0
\(385\) −5265.00 −0.0355203
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 95319.0i 0.629913i 0.949106 + 0.314956i \(0.101990\pi\)
−0.949106 + 0.314956i \(0.898010\pi\)
\(390\) 0 0
\(391\) −275400. −1.80140
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 67518.0i − 0.432738i
\(396\) 0 0
\(397\) −163438. −1.03698 −0.518492 0.855083i \(-0.673506\pi\)
−0.518492 + 0.855083i \(0.673506\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 284616.i − 1.76999i −0.465601 0.884994i \(-0.654162\pi\)
0.465601 0.884994i \(-0.345838\pi\)
\(402\) 0 0
\(403\) 23698.0 0.145916
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 87516.0i 0.528322i
\(408\) 0 0
\(409\) 107525. 0.642781 0.321390 0.946947i \(-0.395850\pi\)
0.321390 + 0.946947i \(0.395850\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 29070.0i 0.170430i
\(414\) 0 0
\(415\) 49329.0 0.286422
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 150282.i 0.856010i 0.903776 + 0.428005i \(0.140783\pi\)
−0.903776 + 0.428005i \(0.859217\pi\)
\(420\) 0 0
\(421\) 134420. 0.758402 0.379201 0.925314i \(-0.376199\pi\)
0.379201 + 0.925314i \(0.376199\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 244800.i − 1.35529i
\(426\) 0 0
\(427\) 32020.0 0.175617
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 237726.i − 1.27974i −0.768483 0.639871i \(-0.778988\pi\)
0.768483 0.639871i \(-0.221012\pi\)
\(432\) 0 0
\(433\) 112187. 0.598366 0.299183 0.954196i \(-0.403286\pi\)
0.299183 + 0.954196i \(0.403286\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 39168.0i 0.205101i
\(438\) 0 0
\(439\) −204643. −1.06186 −0.530931 0.847415i \(-0.678158\pi\)
−0.530931 + 0.847415i \(0.678158\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 50490.0i − 0.257275i −0.991692 0.128638i \(-0.958940\pi\)
0.991692 0.128638i \(-0.0410604\pi\)
\(444\) 0 0
\(445\) −79866.0 −0.403313
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 363528.i − 1.80321i −0.432565 0.901603i \(-0.642391\pi\)
0.432565 0.901603i \(-0.357609\pi\)
\(450\) 0 0
\(451\) −80028.0 −0.393449
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1530.00i 0.00739041i
\(456\) 0 0
\(457\) 6677.00 0.0319705 0.0159852 0.999872i \(-0.494912\pi\)
0.0159852 + 0.999872i \(0.494912\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 229347.i − 1.07917i −0.841930 0.539587i \(-0.818581\pi\)
0.841930 0.539587i \(-0.181419\pi\)
\(462\) 0 0
\(463\) 238799. 1.11396 0.556981 0.830525i \(-0.311960\pi\)
0.556981 + 0.830525i \(0.311960\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 263133.i 1.20654i 0.797537 + 0.603270i \(0.206136\pi\)
−0.797537 + 0.603270i \(0.793864\pi\)
\(468\) 0 0
\(469\) −26090.0 −0.118612
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 306306.i − 1.36909i
\(474\) 0 0
\(475\) −34816.0 −0.154309
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 9342.00i − 0.0407163i −0.999793 0.0203582i \(-0.993519\pi\)
0.999793 0.0203582i \(-0.00648066\pi\)
\(480\) 0 0
\(481\) 25432.0 0.109923
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 95139.0i − 0.404460i
\(486\) 0 0
\(487\) 331262. 1.39673 0.698367 0.715740i \(-0.253910\pi\)
0.698367 + 0.715740i \(0.253910\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 396297.i − 1.64383i −0.569608 0.821917i \(-0.692905\pi\)
0.569608 0.821917i \(-0.307095\pi\)
\(492\) 0 0
\(493\) −477900. −1.96627
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32850.0i 0.132991i
\(498\) 0 0
\(499\) 45050.0 0.180923 0.0904615 0.995900i \(-0.471166\pi\)
0.0904615 + 0.995900i \(0.471166\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 233172.i 0.921596i 0.887505 + 0.460798i \(0.152437\pi\)
−0.887505 + 0.460798i \(0.847563\pi\)
\(504\) 0 0
\(505\) −118017. −0.462766
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 163449.i 0.630880i 0.948946 + 0.315440i \(0.102152\pi\)
−0.948946 + 0.315440i \(0.897848\pi\)
\(510\) 0 0
\(511\) −22595.0 −0.0865308
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 52470.0i 0.197832i
\(516\) 0 0
\(517\) 309582. 1.15823
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 384606.i − 1.41690i −0.705759 0.708452i \(-0.749394\pi\)
0.705759 0.708452i \(-0.250606\pi\)
\(522\) 0 0
\(523\) −214642. −0.784714 −0.392357 0.919813i \(-0.628340\pi\)
−0.392357 + 0.919813i \(0.628340\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 313650.i 1.12934i
\(528\) 0 0
\(529\) −94703.0 −0.338417
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 23256.0i 0.0818617i
\(534\) 0 0
\(535\) −9801.00 −0.0342423
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 277992.i 0.956874i
\(540\) 0 0
\(541\) 545156. 1.86263 0.931314 0.364217i \(-0.118663\pi\)
0.931314 + 0.364217i \(0.118663\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 45180.0i 0.152108i
\(546\) 0 0
\(547\) −491422. −1.64240 −0.821202 0.570638i \(-0.806696\pi\)
−0.821202 + 0.570638i \(0.806696\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 67968.0i 0.223873i
\(552\) 0 0
\(553\) 37510.0 0.122658
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 445077.i 1.43458i 0.696775 + 0.717290i \(0.254618\pi\)
−0.696775 + 0.717290i \(0.745382\pi\)
\(558\) 0 0
\(559\) −89012.0 −0.284856
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 153765.i 0.485111i 0.970138 + 0.242555i \(0.0779856\pi\)
−0.970138 + 0.242555i \(0.922014\pi\)
\(564\) 0 0
\(565\) −192456. −0.602885
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 269712.i 0.833059i 0.909122 + 0.416529i \(0.136754\pi\)
−0.909122 + 0.416529i \(0.863246\pi\)
\(570\) 0 0
\(571\) 589718. 1.80872 0.904362 0.426767i \(-0.140347\pi\)
0.904362 + 0.426767i \(0.140347\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 332928.i − 1.00697i
\(576\) 0 0
\(577\) 184094. 0.552953 0.276476 0.961021i \(-0.410833\pi\)
0.276476 + 0.961021i \(0.410833\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 27405.0i 0.0811853i
\(582\) 0 0
\(583\) 125307. 0.368670
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 411543.i 1.19437i 0.802103 + 0.597185i \(0.203714\pi\)
−0.802103 + 0.597185i \(0.796286\pi\)
\(588\) 0 0
\(589\) 44608.0 0.128583
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 339966.i 0.966777i 0.875406 + 0.483388i \(0.160594\pi\)
−0.875406 + 0.483388i \(0.839406\pi\)
\(594\) 0 0
\(595\) −20250.0 −0.0571994
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 299574.i 0.834931i 0.908693 + 0.417465i \(0.137082\pi\)
−0.908693 + 0.417465i \(0.862918\pi\)
\(600\) 0 0
\(601\) −516115. −1.42889 −0.714443 0.699694i \(-0.753320\pi\)
−0.714443 + 0.699694i \(0.753320\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 8568.00i − 0.0234082i
\(606\) 0 0
\(607\) −486574. −1.32060 −0.660300 0.751002i \(-0.729571\pi\)
−0.660300 + 0.751002i \(0.729571\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 89964.0i − 0.240983i
\(612\) 0 0
\(613\) −189550. −0.504432 −0.252216 0.967671i \(-0.581159\pi\)
−0.252216 + 0.967671i \(0.581159\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 148248.i 0.389420i 0.980861 + 0.194710i \(0.0623766\pi\)
−0.980861 + 0.194710i \(0.937623\pi\)
\(618\) 0 0
\(619\) 11390.0 0.0297264 0.0148632 0.999890i \(-0.495269\pi\)
0.0148632 + 0.999890i \(0.495269\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 44370.0i − 0.114318i
\(624\) 0 0
\(625\) 245311. 0.627996
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 336600.i 0.850771i
\(630\) 0 0
\(631\) −6715.00 −0.0168650 −0.00843252 0.999964i \(-0.502684\pi\)
−0.00843252 + 0.999964i \(0.502684\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 83043.0i − 0.205947i
\(636\) 0 0
\(637\) 80784.0 0.199089
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 441108.i − 1.07357i −0.843720 0.536783i \(-0.819639\pi\)
0.843720 0.536783i \(-0.180361\pi\)
\(642\) 0 0
\(643\) 547448. 1.32410 0.662050 0.749459i \(-0.269687\pi\)
0.662050 + 0.749459i \(0.269687\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 42228.0i − 0.100877i −0.998727 0.0504385i \(-0.983938\pi\)
0.998727 0.0504385i \(-0.0160619\pi\)
\(648\) 0 0
\(649\) 680238. 1.61500
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 258993.i 0.607382i 0.952771 + 0.303691i \(0.0982190\pi\)
−0.952771 + 0.303691i \(0.901781\pi\)
\(654\) 0 0
\(655\) −38475.0 −0.0896801
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 313083.i − 0.720923i −0.932774 0.360461i \(-0.882619\pi\)
0.932774 0.360461i \(-0.117381\pi\)
\(660\) 0 0
\(661\) −686320. −1.57081 −0.785405 0.618982i \(-0.787545\pi\)
−0.785405 + 0.618982i \(0.787545\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2880.00i 0.00651252i
\(666\) 0 0
\(667\) −649944. −1.46091
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 749268.i − 1.66415i
\(672\) 0 0
\(673\) 214727. 0.474085 0.237043 0.971499i \(-0.423822\pi\)
0.237043 + 0.971499i \(0.423822\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 343638.i 0.749763i 0.927073 + 0.374881i \(0.122317\pi\)
−0.927073 + 0.374881i \(0.877683\pi\)
\(678\) 0 0
\(679\) 52855.0 0.114643
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 626238.i − 1.34245i −0.741254 0.671225i \(-0.765769\pi\)
0.741254 0.671225i \(-0.234231\pi\)
\(684\) 0 0
\(685\) −101898. −0.217162
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 36414.0i − 0.0767061i
\(690\) 0 0
\(691\) 684476. 1.43351 0.716757 0.697323i \(-0.245626\pi\)
0.716757 + 0.697323i \(0.245626\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 88308.0i − 0.182823i
\(696\) 0 0
\(697\) −307800. −0.633582
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 809523.i − 1.64738i −0.567042 0.823689i \(-0.691912\pi\)
0.567042 0.823689i \(-0.308088\pi\)
\(702\) 0 0
\(703\) 47872.0 0.0968659
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 65565.0i − 0.131170i
\(708\) 0 0
\(709\) −319648. −0.635886 −0.317943 0.948110i \(-0.602992\pi\)
−0.317943 + 0.948110i \(0.602992\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 426564.i 0.839083i
\(714\) 0 0
\(715\) 35802.0 0.0700318
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 55836.0i − 0.108008i −0.998541 0.0540041i \(-0.982802\pi\)
0.998541 0.0540041i \(-0.0171984\pi\)
\(720\) 0 0
\(721\) −29150.0 −0.0560748
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 577728.i − 1.09913i
\(726\) 0 0
\(727\) 863873. 1.63449 0.817243 0.576294i \(-0.195502\pi\)
0.817243 + 0.576294i \(0.195502\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 1.17810e6i − 2.20469i
\(732\) 0 0
\(733\) 207608. 0.386399 0.193200 0.981159i \(-0.438114\pi\)
0.193200 + 0.981159i \(0.438114\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 610506.i 1.12397i
\(738\) 0 0
\(739\) −803590. −1.47145 −0.735725 0.677280i \(-0.763159\pi\)
−0.735725 + 0.677280i \(0.763159\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 619650.i − 1.12245i −0.827662 0.561227i \(-0.810329\pi\)
0.827662 0.561227i \(-0.189671\pi\)
\(744\) 0 0
\(745\) 148311. 0.267215
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 5445.00i − 0.00970587i
\(750\) 0 0
\(751\) −1.06598e6 −1.89004 −0.945020 0.327011i \(-0.893959\pi\)
−0.945020 + 0.327011i \(0.893959\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 229995.i 0.403482i
\(756\) 0 0
\(757\) 750410. 1.30950 0.654752 0.755844i \(-0.272773\pi\)
0.654752 + 0.755844i \(0.272773\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 413208.i 0.713509i 0.934198 + 0.356754i \(0.116117\pi\)
−0.934198 + 0.356754i \(0.883883\pi\)
\(762\) 0 0
\(763\) −25100.0 −0.0431146
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 197676.i − 0.336019i
\(768\) 0 0
\(769\) −733381. −1.24016 −0.620079 0.784539i \(-0.712899\pi\)
−0.620079 + 0.784539i \(0.712899\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 231822.i − 0.387968i −0.981005 0.193984i \(-0.937859\pi\)
0.981005 0.193984i \(-0.0621410\pi\)
\(774\) 0 0
\(775\) −379168. −0.631289
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 43776.0i 0.0721375i
\(780\) 0 0
\(781\) 768690. 1.26023
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 190476.i − 0.309101i
\(786\) 0 0
\(787\) 502724. 0.811671 0.405836 0.913946i \(-0.366981\pi\)
0.405836 + 0.913946i \(0.366981\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 106920.i − 0.170886i
\(792\) 0 0
\(793\) −217736. −0.346245
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 596241.i 0.938653i 0.883025 + 0.469327i \(0.155503\pi\)
−0.883025 + 0.469327i \(0.844497\pi\)
\(798\) 0 0
\(799\) 1.19070e6 1.86513
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 528723.i 0.819968i
\(804\) 0 0
\(805\) −27540.0 −0.0424984
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 376038.i 0.574559i 0.957847 + 0.287280i \(0.0927509\pi\)
−0.957847 + 0.287280i \(0.907249\pi\)
\(810\) 0 0
\(811\) −331072. −0.503362 −0.251681 0.967810i \(-0.580983\pi\)
−0.251681 + 0.967810i \(0.580983\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 304470.i − 0.458384i
\(816\) 0 0
\(817\) −167552. −0.251018
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 274626.i 0.407432i 0.979030 + 0.203716i \(0.0653019\pi\)
−0.979030 + 0.203716i \(0.934698\pi\)
\(822\) 0 0
\(823\) −541195. −0.799013 −0.399507 0.916730i \(-0.630819\pi\)
−0.399507 + 0.916730i \(0.630819\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 358362.i − 0.523975i −0.965071 0.261988i \(-0.915622\pi\)
0.965071 0.261988i \(-0.0843780\pi\)
\(828\) 0 0
\(829\) 39626.0 0.0576595 0.0288298 0.999584i \(-0.490822\pi\)
0.0288298 + 0.999584i \(0.490822\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.06920e6i 1.54088i
\(834\) 0 0
\(835\) 246402. 0.353404
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.05543e6i 1.49936i 0.661801 + 0.749679i \(0.269792\pi\)
−0.661801 + 0.749679i \(0.730208\pi\)
\(840\) 0 0
\(841\) −420563. −0.594619
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 246645.i 0.345429i
\(846\) 0 0
\(847\) 4760.00 0.00663499
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 457776.i 0.632112i
\(852\) 0 0
\(853\) −33034.0 −0.0454008 −0.0227004 0.999742i \(-0.507226\pi\)
−0.0227004 + 0.999742i \(0.507226\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 456246.i 0.621209i 0.950539 + 0.310604i \(0.100531\pi\)
−0.950539 + 0.310604i \(0.899469\pi\)
\(858\) 0 0
\(859\) 343604. 0.465663 0.232832 0.972517i \(-0.425201\pi\)
0.232832 + 0.972517i \(0.425201\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 74556.0i 0.100106i 0.998747 + 0.0500531i \(0.0159391\pi\)
−0.998747 + 0.0500531i \(0.984061\pi\)
\(864\) 0 0
\(865\) 415773. 0.555679
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 877734.i − 1.16231i
\(870\) 0 0
\(871\) 177412. 0.233855
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 52605.0i − 0.0687086i
\(876\) 0 0
\(877\) −696094. −0.905042 −0.452521 0.891754i \(-0.649475\pi\)
−0.452521 + 0.891754i \(0.649475\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 672426.i 0.866349i 0.901310 + 0.433174i \(0.142607\pi\)
−0.901310 + 0.433174i \(0.857393\pi\)
\(882\) 0 0
\(883\) −1.44813e6 −1.85731 −0.928657 0.370938i \(-0.879036\pi\)
−0.928657 + 0.370938i \(0.879036\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.50964e6i 1.91879i 0.282071 + 0.959393i \(0.408978\pi\)
−0.282071 + 0.959393i \(0.591022\pi\)
\(888\) 0 0
\(889\) 46135.0 0.0583750
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 169344.i − 0.212357i
\(894\) 0 0
\(895\) 187029. 0.233487
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 740214.i 0.915879i
\(900\) 0 0
\(901\) 481950. 0.593680
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 175536.i 0.214323i
\(906\) 0 0
\(907\) 374828. 0.455635 0.227818 0.973704i \(-0.426841\pi\)
0.227818 + 0.973704i \(0.426841\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1.21149e6i − 1.45977i −0.683572 0.729883i \(-0.739575\pi\)
0.683572 0.729883i \(-0.260425\pi\)
\(912\) 0 0
\(913\) 641277. 0.769315
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 21375.0i − 0.0254195i
\(918\) 0 0
\(919\) 656777. 0.777655 0.388827 0.921311i \(-0.372880\pi\)
0.388827 + 0.921311i \(0.372880\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 223380.i − 0.262205i
\(924\) 0 0
\(925\) −406912. −0.475573
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 578988.i 0.670870i 0.942063 + 0.335435i \(0.108883\pi\)
−0.942063 + 0.335435i \(0.891117\pi\)
\(930\) 0 0
\(931\) 152064. 0.175439
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 473850.i 0.542023i
\(936\) 0 0
\(937\) 195173. 0.222301 0.111150 0.993804i \(-0.464547\pi\)
0.111150 + 0.993804i \(0.464547\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1.31932e6i − 1.48995i −0.667095 0.744973i \(-0.732462\pi\)
0.667095 0.744973i \(-0.267538\pi\)
\(942\) 0 0
\(943\) −418608. −0.470743
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 972621.i 1.08454i 0.840206 + 0.542268i \(0.182434\pi\)
−0.840206 + 0.542268i \(0.817566\pi\)
\(948\) 0 0
\(949\) 153646. 0.170604
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 534384.i − 0.588393i −0.955745 0.294197i \(-0.904948\pi\)
0.955745 0.294197i \(-0.0950520\pi\)
\(954\) 0 0
\(955\) −63342.0 −0.0694520
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 56610.0i − 0.0615540i
\(960\) 0 0
\(961\) −437712. −0.473960
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 463743.i − 0.497992i
\(966\) 0 0
\(967\) −783619. −0.838015 −0.419008 0.907983i \(-0.637622\pi\)
−0.419008 + 0.907983i \(0.637622\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 276777.i − 0.293556i −0.989169 0.146778i \(-0.953110\pi\)
0.989169 0.146778i \(-0.0468904\pi\)
\(972\) 0 0
\(973\) 49060.0 0.0518205
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 648234.i − 0.679114i −0.940585 0.339557i \(-0.889723\pi\)
0.940585 0.339557i \(-0.110277\pi\)
\(978\) 0 0
\(979\) −1.03826e6 −1.08328
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 290862.i − 0.301009i −0.988609 0.150505i \(-0.951910\pi\)
0.988609 0.150505i \(-0.0480899\pi\)
\(984\) 0 0
\(985\) 543915. 0.560607
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1.60222e6i − 1.63806i
\(990\) 0 0
\(991\) 881243. 0.897322 0.448661 0.893702i \(-0.351901\pi\)
0.448661 + 0.893702i \(0.351901\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 410985.i − 0.415126i
\(996\) 0 0
\(997\) −690166. −0.694326 −0.347163 0.937805i \(-0.612855\pi\)
−0.347163 + 0.937805i \(0.612855\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 108.5.c.c.53.1 2
3.2 odd 2 inner 108.5.c.c.53.2 yes 2
4.3 odd 2 432.5.e.d.161.1 2
9.2 odd 6 324.5.g.d.53.1 4
9.4 even 3 324.5.g.d.269.1 4
9.5 odd 6 324.5.g.d.269.2 4
9.7 even 3 324.5.g.d.53.2 4
12.11 even 2 432.5.e.d.161.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.5.c.c.53.1 2 1.1 even 1 trivial
108.5.c.c.53.2 yes 2 3.2 odd 2 inner
324.5.g.d.53.1 4 9.2 odd 6
324.5.g.d.53.2 4 9.7 even 3
324.5.g.d.269.1 4 9.4 even 3
324.5.g.d.269.2 4 9.5 odd 6
432.5.e.d.161.1 2 4.3 odd 2
432.5.e.d.161.2 2 12.11 even 2