Properties

Label 2415.2.a.g.1.1
Level $2415$
Weight $2$
Character 2415.1
Self dual yes
Analytic conductor $19.284$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2838720881\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2415.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +3.00000 q^{33} -1.00000 q^{35} -2.00000 q^{36} -10.0000 q^{37} -4.00000 q^{39} -3.00000 q^{41} -10.0000 q^{43} -6.00000 q^{44} -1.00000 q^{45} -3.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} -6.00000 q^{51} +8.00000 q^{52} -9.00000 q^{53} -3.00000 q^{55} -1.00000 q^{57} -3.00000 q^{59} +2.00000 q^{60} +11.0000 q^{61} +1.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -10.0000 q^{67} +12.0000 q^{68} -1.00000 q^{69} +2.00000 q^{73} +1.00000 q^{75} +2.00000 q^{76} +3.00000 q^{77} -10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{84} +6.00000 q^{85} +6.00000 q^{87} -4.00000 q^{91} +2.00000 q^{92} +8.00000 q^{93} +1.00000 q^{95} +2.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 4.00000 0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 8.00000 1.10940
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 2.00000 0.258199
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 12.0000 1.45521
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −0.218218
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 2.00000 0.208514
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) −2.00000 −0.200000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −2.00000 −0.192450
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) −12.0000 −1.11417
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −3.00000 −0.270501
\(124\) −16.0000 −1.43684
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) −6.00000 −0.522233
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000 0.169031
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 4.00000 0.333333
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 20.0000 1.64399
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 8.00000 0.640513
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 6.00000 0.468521
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 20.0000 1.52499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 12.0000 0.904534
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 11.0000 0.813143
\(184\) 0 0
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 6.00000 0.437595
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) −8.00000 −0.577350
\(193\) −25.0000 −1.79954 −0.899770 0.436365i \(-0.856266\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) −2.00000 −0.142857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −13.0000 −0.921546 −0.460773 0.887518i \(-0.652428\pi\)
−0.460773 + 0.887518i \(0.652428\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 12.0000 0.840168
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) −16.0000 −1.10940
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 18.0000 1.23625
\(213\) 0 0
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 6.00000 0.404520
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 2.00000 0.132453
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 6.00000 0.390567
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) −4.00000 −0.258199
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −22.0000 −1.40841
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) −2.00000 −0.125988
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 16.0000 1.00000
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) −8.00000 −0.496139
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 20.0000 1.22169
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 26.0000 1.57939 0.789694 0.613501i \(-0.210239\pi\)
0.789694 + 0.613501i \(0.210239\pi\)
\(272\) −24.0000 −1.45521
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) 3.00000 0.180907
\(276\) 2.00000 0.120386
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −4.00000 −0.234082
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) −2.00000 −0.115470
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) −4.00000 −0.229416
\(305\) −11.0000 −0.629858
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) −6.00000 −0.341882
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) 33.0000 1.87126 0.935629 0.352985i \(-0.114833\pi\)
0.935629 + 0.352985i \(0.114833\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 20.0000 1.12509
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 8.00000 0.447214
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) −2.00000 −0.111111
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) 4.00000 0.218218
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) −12.0000 −0.650791
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) −12.0000 −0.643268
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 8.00000 0.419314
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −19.0000 −0.991792 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) −4.00000 −0.208514
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) −16.0000 −0.829561
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) −2.00000 −0.102598
\(381\) −13.0000 −0.666010
\(382\) 0 0
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) −4.00000 −0.203069
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) 15.0000 0.756650
\(394\) 0 0
\(395\) 10.0000 0.503155
\(396\) −6.00000 −0.301511
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 4.00000 0.200000
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 2.00000 0.0985329
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 2.00000 0.0975900
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 11.0000 0.532327
\(428\) 24.0000 1.16008
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 27.0000 1.30054 0.650272 0.759701i \(-0.274655\pi\)
0.650272 + 0.759701i \(0.274655\pi\)
\(432\) 4.00000 0.192450
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 20.0000 0.957826
\(437\) 1.00000 0.0478365
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 20.0000 0.949158
\(445\) 0 0
\(446\) 0 0
\(447\) 3.00000 0.141895
\(448\) −8.00000 −0.377964
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 12.0000 0.564433
\(453\) −19.0000 −0.892698
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) −2.00000 −0.0932505
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −25.0000 −1.16185 −0.580924 0.813958i \(-0.697309\pi\)
−0.580924 + 0.813958i \(0.697309\pi\)
\(464\) 24.0000 1.11417
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 8.00000 0.369800
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) 17.0000 0.783319
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 12.0000 0.550019
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 40.0000 1.82384
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 4.00000 0.181818
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 32.0000 1.43684
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 2.00000 0.0894427
\(501\) −15.0000 −0.670151
\(502\) 0 0
\(503\) 18.0000 0.802580 0.401290 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 26.0000 1.15356
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 1.00000 0.0440653
\(516\) 20.0000 0.880451
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 0 0
\(523\) −31.0000 −1.35554 −0.677768 0.735276i \(-0.737052\pi\)
−0.677768 + 0.735276i \(0.737052\pi\)
\(524\) −30.0000 −1.31056
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 12.0000 0.522233
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 2.00000 0.0867110
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 2.00000 0.0860663
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) 11.0000 0.469469
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) 10.0000 0.424476
\(556\) 32.0000 1.35710
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) −4.00000 −0.169031
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 6.00000 0.252646
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 24.0000 1.00349
\(573\) −15.0000 −0.626634
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) −8.00000 −0.333333
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) −25.0000 −1.03896
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) 0 0
\(583\) −27.0000 −1.11823
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −40.0000 −1.64399
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) −6.00000 −0.245770
\(597\) −13.0000 −0.532055
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) 38.0000 1.54620
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 12.0000 0.485071
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 0 0
\(615\) 3.00000 0.120972
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 16.0000 0.642575
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) −16.0000 −0.640513
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) −34.0000 −1.35675
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) 13.0000 0.515889
\(636\) 18.0000 0.713746
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 0 0
\(643\) −1.00000 −0.0394362 −0.0197181 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) 2.00000 0.0788110
\(645\) 10.0000 0.393750
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) −22.0000 −0.861586
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) −15.0000 −0.586098
\(656\) −12.0000 −0.468521
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 6.00000 0.233550
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) 30.0000 1.16073
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 33.0000 1.27395
\(672\) 0 0
\(673\) 23.0000 0.886585 0.443292 0.896377i \(-0.353810\pi\)
0.443292 + 0.896377i \(0.353810\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) −6.00000 −0.230769
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 2.00000 0.0764719
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) −13.0000 −0.495981
\(688\) −40.0000 −1.52499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −12.0000 −0.456172
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) −2.00000 −0.0755929
\(701\) −51.0000 −1.92624 −0.963122 0.269066i \(-0.913285\pi\)
−0.963122 + 0.269066i \(0.913285\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) −24.0000 −0.904534
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) −3.00000 −0.112827
\(708\) 6.00000 0.225494
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) −24.0000 −0.896922
\(717\) −30.0000 −1.12037
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −4.00000 −0.149071
\(721\) −1.00000 −0.0372419
\(722\) 0 0
\(723\) 17.0000 0.632237
\(724\) 20.0000 0.743294
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 47.0000 1.74313 0.871567 0.490277i \(-0.163104\pi\)
0.871567 + 0.490277i \(0.163104\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 60.0000 2.21918
\(732\) −22.0000 −0.813143
\(733\) −10.0000 −0.369358 −0.184679 0.982799i \(-0.559125\pi\)
−0.184679 + 0.982799i \(0.559125\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) −20.0000 −0.735215
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 33.0000 1.21065 0.605326 0.795977i \(-0.293043\pi\)
0.605326 + 0.795977i \(0.293043\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 0 0
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) −24.0000 −0.874609
\(754\) 0 0
\(755\) 19.0000 0.691481
\(756\) −2.00000 −0.0727393
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 0 0
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) −45.0000 −1.63125 −0.815624 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 30.0000 1.08536
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 16.0000 0.577350
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) 50.0000 1.79954
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) −10.0000 −0.358748
\(778\) 0 0
\(779\) 3.00000 0.107486
\(780\) −8.00000 −0.286446
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 4.00000 0.142857
\(785\) −17.0000 −0.606756
\(786\) 0 0
\(787\) −25.0000 −0.891154 −0.445577 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(788\) −24.0000 −0.854965
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −44.0000 −1.56249
\(794\) 0 0
\(795\) 9.00000 0.319197
\(796\) 26.0000 0.921546
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 20.0000 0.705346
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) −36.0000 −1.26569 −0.632846 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −12.0000 −0.421117
\(813\) 26.0000 0.911860
\(814\) 0 0
\(815\) −11.0000 −0.385313
\(816\) −24.0000 −0.840168
\(817\) 10.0000 0.349856
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) −6.00000 −0.209529
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −7.00000 −0.244005 −0.122002 0.992530i \(-0.538932\pi\)
−0.122002 + 0.992530i \(0.538932\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) −3.00000 −0.104320 −0.0521601 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 2.00000 0.0695048
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −19.0000 −0.659103
\(832\) 32.0000 1.10940
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) 6.00000 0.207514
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) −10.0000 −0.344214
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −36.0000 −1.23625
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) 0 0
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) −20.0000 −0.681994
\(861\) −3.00000 −0.102240
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) −16.0000 −0.543075
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −4.00000 −0.135147
\(877\) 5.00000 0.168838 0.0844190 0.996430i \(-0.473097\pi\)
0.0844190 + 0.996430i \(0.473097\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) −12.0000 −0.404520
\(881\) −48.0000 −1.61716 −0.808581 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −48.0000 −1.61441
\(885\) 3.00000 0.100844
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 32.0000 1.07144
\(893\) 3.00000 0.100391
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) 48.0000 1.60089
\(900\) −2.00000 −0.0666667
\(901\) 54.0000 1.79900
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −48.0000 −1.59294
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) 0 0
\(915\) −11.0000 −0.363649
\(916\) 26.0000 0.859064
\(917\) 15.0000 0.495344
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) −1.00000 −0.0328443
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −12.0000 −0.393073
\(933\) 33.0000 1.08037
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 23.0000 0.751377 0.375689 0.926746i \(-0.377406\pi\)
0.375689 + 0.926746i \(0.377406\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) −6.00000 −0.195698
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) 3.00000 0.0976934
\(944\) −12.0000 −0.390567
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −42.0000 −1.36482 −0.682408 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(948\) 20.0000 0.649570
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −21.0000 −0.680257 −0.340128 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) 0 0
\(955\) 15.0000 0.485389
\(956\) 60.0000 1.94054
\(957\) 18.0000 0.581857
\(958\) 0 0
\(959\) 3.00000 0.0968751
\(960\) 8.00000 0.258199
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −34.0000 −1.09507
\(965\) 25.0000 0.804778
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −16.0000 −0.512936
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 44.0000 1.40841
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) −3.00000 −0.0954911
\(988\) −8.00000 −0.254514
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) 41.0000 1.30241 0.651204 0.758903i \(-0.274264\pi\)
0.651204 + 0.758903i \(0.274264\pi\)
\(992\) 0 0
\(993\) −7.00000 −0.222138
\(994\) 0 0
\(995\) 13.0000 0.412128
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 2415.2.a.g.1.1 1
3.2 odd 2 7245.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.g.1.1 1 1.1 even 1 trivial
7245.2.a.j.1.1 1 3.2 odd 2