Properties

Label 2450.2.a.ba.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +3.00000 q^{11} -5.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} -3.00000 q^{18} -5.00000 q^{19} +3.00000 q^{22} -7.00000 q^{23} -5.00000 q^{26} -4.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -3.00000 q^{36} +1.00000 q^{37} -5.00000 q^{38} +3.00000 q^{41} +2.00000 q^{43} +3.00000 q^{44} -7.00000 q^{46} -7.00000 q^{47} -5.00000 q^{52} +9.00000 q^{53} -4.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{67} -2.00000 q^{68} -6.00000 q^{71} -3.00000 q^{72} -16.0000 q^{73} +1.00000 q^{74} -5.00000 q^{76} +14.0000 q^{79} +9.00000 q^{81} +3.00000 q^{82} -6.00000 q^{83} +2.00000 q^{86} +3.00000 q^{88} +2.00000 q^{89} -7.00000 q^{92} -7.00000 q^{94} -12.0000 q^{97} -9.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −3.00000 −0.707107
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) −5.00000 −0.811107
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −7.00000 −1.03209
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −3.00000 −0.353553
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 3.00000 0.331295
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.00000 −0.729800
\(93\) 0 0
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) −9.00000 −0.904534
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 15.0000 1.38675
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −15.0000 −1.25436
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) −5.00000 −0.405554
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.00000 −0.718278 −0.359139 0.933284i \(-0.616930\pi\)
−0.359139 + 0.933284i \(0.616930\pi\)
\(158\) 14.0000 1.11378
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 15.0000 1.16073 0.580367 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 15.0000 1.14708
\(172\) 2.00000 0.152499
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 13.0000 0.971666 0.485833 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 0 0
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.00000 −0.516047
\(185\) 0 0
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) −7.00000 −0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 0 0
\(197\) −5.00000 −0.356235 −0.178118 0.984009i \(-0.557001\pi\)
−0.178118 + 0.984009i \(0.557001\pi\)
\(198\) −9.00000 −0.639602
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 21.0000 1.45960
\(208\) −5.00000 −0.346688
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) −16.0000 −1.09374
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0000 0.672673
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 15.0000 0.980581
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −9.00000 −0.579741 −0.289870 0.957066i \(-0.593612\pi\)
−0.289870 + 0.957066i \(0.593612\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 0 0
\(247\) 25.0000 1.59071
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) −21.0000 −1.32026
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) −1.00000 −0.0617802
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 16.0000 0.959616
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 9.00000 0.536895 0.268447 0.963294i \(-0.413489\pi\)
0.268447 + 0.963294i \(0.413489\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −15.0000 −0.886969
\(287\) 0 0
\(288\) −3.00000 −0.176777
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −16.0000 −0.936329
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 35.0000 2.02410
\(300\) 0 0
\(301\) 0 0
\(302\) −6.00000 −0.345261
\(303\) 0 0
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −9.00000 −0.507899
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) −6.00000 −0.329293
\(333\) −3.00000 −0.164399
\(334\) 15.0000 0.820763
\(335\) 0 0
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 15.0000 0.811107
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 13.0000 0.687071
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 26.0000 1.36653
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) −7.00000 −0.364900
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) −7.00000 −0.360997
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −6.00000 −0.304997
\(388\) −12.0000 −0.609208
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 14.0000 0.708010
\(392\) 0 0
\(393\) 0 0
\(394\) −5.00000 −0.251896
\(395\) 0 0
\(396\) −9.00000 −0.452267
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 18.0000 0.902258
\(399\) 0 0
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 21.0000 1.03209
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) −15.0000 −0.733674
\(419\) 35.0000 1.70986 0.854931 0.518742i \(-0.173599\pi\)
0.854931 + 0.518742i \(0.173599\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) −9.00000 −0.438113
\(423\) 21.0000 1.02105
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 0 0
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 35.0000 1.67428
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.0000 0.475651
\(443\) 30.0000 1.42534 0.712672 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) 5.00000 0.235965 0.117982 0.993016i \(-0.462357\pi\)
0.117982 + 0.993016i \(0.462357\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 16.0000 0.747631
\(459\) 0 0
\(460\) 0 0
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) −17.0000 −0.790057 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −8.00000 −0.370593
\(467\) 34.0000 1.57333 0.786666 0.617379i \(-0.211805\pi\)
0.786666 + 0.617379i \(0.211805\pi\)
\(468\) 15.0000 0.693375
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −27.0000 −1.23625
\(478\) 20.0000 0.914779
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) −9.00000 −0.409939
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 25.0000 1.12480
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.00000 0.223161
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −21.0000 −0.933564
\(507\) 0 0
\(508\) 7.00000 0.310575
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) −21.0000 −0.923579
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −27.0000 −1.18289 −0.591446 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 12.0000 0.525226
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −1.00000 −0.0436852
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −15.0000 −0.649722
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) −10.0000 −0.431131
\(539\) 0 0
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) −8.00000 −0.341743
\(549\) −18.0000 −0.768221
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 23.0000 0.974541 0.487271 0.873251i \(-0.337993\pi\)
0.487271 + 0.873251i \(0.337993\pi\)
\(558\) 6.00000 0.254000
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) 9.00000 0.379642
\(563\) 2.00000 0.0842900 0.0421450 0.999112i \(-0.486581\pi\)
0.0421450 + 0.999112i \(0.486581\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −15.0000 −0.628833 −0.314416 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −15.0000 −0.627182
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) 34.0000 1.40333 0.701665 0.712507i \(-0.252440\pi\)
0.701665 + 0.712507i \(0.252440\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 35.0000 1.43126
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) −6.00000 −0.244137
\(605\) 0 0
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) −5.00000 −0.202777
\(609\) 0 0
\(610\) 0 0
\(611\) 35.0000 1.41595
\(612\) 6.00000 0.242536
\(613\) −15.0000 −0.605844 −0.302922 0.953015i \(-0.597962\pi\)
−0.302922 + 0.953015i \(0.597962\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) 19.0000 0.763674 0.381837 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −9.00000 −0.359139
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) 14.0000 0.556890
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −12.0000 −0.475085
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) −5.00000 −0.197488 −0.0987441 0.995113i \(-0.531483\pi\)
−0.0987441 + 0.995113i \(0.531483\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) −27.0000 −1.06148 −0.530740 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 9.00000 0.353553
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 48.0000 1.87266
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 5.00000 0.194331
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −3.00000 −0.116248
\(667\) 28.0000 1.08416
\(668\) 15.0000 0.580367
\(669\) 0 0
\(670\) 0 0
\(671\) 18.0000 0.694882
\(672\) 0 0
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −17.0000 −0.653363 −0.326682 0.945134i \(-0.605930\pi\)
−0.326682 + 0.945134i \(0.605930\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −6.00000 −0.229752
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 15.0000 0.573539
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) −45.0000 −1.71436
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) −12.0000 −0.454207
\(699\) 0 0
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) −5.00000 −0.188579
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0 0
\(708\) 0 0
\(709\) 12.0000 0.450669 0.225335 0.974281i \(-0.427652\pi\)
0.225335 + 0.974281i \(0.427652\pi\)
\(710\) 0 0
\(711\) −42.0000 −1.57512
\(712\) 2.00000 0.0749532
\(713\) 14.0000 0.524304
\(714\) 0 0
\(715\) 0 0
\(716\) 13.0000 0.485833
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6.00000 0.223297
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) 0 0
\(727\) −29.0000 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) 41.0000 1.51437 0.757185 0.653201i \(-0.226574\pi\)
0.757185 + 0.653201i \(0.226574\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) −7.00000 −0.258023
\(737\) 6.00000 0.221013
\(738\) −9.00000 −0.331295
\(739\) −29.0000 −1.06678 −0.533391 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −21.0000 −0.770415 −0.385208 0.922830i \(-0.625870\pi\)
−0.385208 + 0.922830i \(0.625870\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 18.0000 0.658586
\(748\) −6.00000 −0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) −7.00000 −0.255264
\(753\) 0 0
\(754\) 20.0000 0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) −29.0000 −1.05333
\(759\) 0 0
\(760\) 0 0
\(761\) 1.00000 0.0362500 0.0181250 0.999836i \(-0.494230\pi\)
0.0181250 + 0.999836i \(0.494230\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 21.0000 0.758761
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) −29.0000 −1.04577 −0.522883 0.852404i \(-0.675144\pi\)
−0.522883 + 0.852404i \(0.675144\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 45.0000 1.61854 0.809269 0.587439i \(-0.199864\pi\)
0.809269 + 0.587439i \(0.199864\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 14.0000 0.500639
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) −5.00000 −0.178118
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −9.00000 −0.319801
\(793\) −30.0000 −1.06533
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 14.0000 0.495284
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −15.0000 −0.529668
\(803\) −48.0000 −1.69388
\(804\) 0 0
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) 0 0
\(808\) 0 0
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) −33.0000 −1.15879 −0.579393 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) 0 0
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 21.0000 0.729800
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.00000 −0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −15.0000 −0.518786
\(837\) 0 0
\(838\) 35.0000 1.20905
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −20.0000 −0.689246
\(843\) 0 0
\(844\) −9.00000 −0.309793
\(845\) 0 0
\(846\) 21.0000 0.721995
\(847\) 0 0
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) 0 0
\(851\) −7.00000 −0.239957
\(852\) 0 0
\(853\) −43.0000 −1.47229 −0.736146 0.676823i \(-0.763356\pi\)
−0.736146 + 0.676823i \(0.763356\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16.0000 −0.546869
\(857\) 8.00000 0.273275 0.136637 0.990621i \(-0.456370\pi\)
0.136637 + 0.990621i \(0.456370\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.00000 0.0681203
\(863\) −11.0000 −0.374444 −0.187222 0.982318i \(-0.559948\pi\)
−0.187222 + 0.982318i \(0.559948\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 28.0000 0.951479
\(867\) 0 0
\(868\) 0 0
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 2.00000 0.0677285
\(873\) 36.0000 1.21842
\(874\) 35.0000 1.18389
\(875\) 0 0
\(876\) 0 0
\(877\) 31.0000 1.04680 0.523398 0.852088i \(-0.324664\pi\)
0.523398 + 0.852088i \(0.324664\pi\)
\(878\) −28.0000 −0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 10.0000 0.336336
\(885\) 0 0
\(886\) 30.0000 1.00787
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 27.0000 0.904534
\(892\) −8.00000 −0.267860
\(893\) 35.0000 1.17123
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 5.00000 0.166852
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 9.00000 0.299667
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) 2.00000 0.0662630 0.0331315 0.999451i \(-0.489452\pi\)
0.0331315 + 0.999451i \(0.489452\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −32.0000 −1.05386
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) 0 0
\(926\) −17.0000 −0.558655
\(927\) 24.0000 0.788263
\(928\) −4.00000 −0.131306
\(929\) 21.0000 0.688988 0.344494 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −8.00000 −0.262049
\(933\) 0 0
\(934\) 34.0000 1.11251
\(935\) 0 0
\(936\) 15.0000 0.490290
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −28.0000 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) 0 0
\(943\) −21.0000 −0.683854
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 80.0000 2.59691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −27.0000 −0.874157
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −5.00000 −0.161206
\(963\) 48.0000 1.54678
\(964\) −9.00000 −0.289870
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −33.0000 −1.05902 −0.529510 0.848304i \(-0.677624\pi\)
−0.529510 + 0.848304i \(0.677624\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 24.0000 0.765871
\(983\) −29.0000 −0.924956 −0.462478 0.886631i \(-0.653040\pi\)
−0.462478 + 0.886631i \(0.653040\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 25.0000 0.795356
\(989\) −14.0000 −0.445174
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 2450.2.a.ba.1.1 1
5.2 odd 4 490.2.c.a.99.2 2
5.3 odd 4 490.2.c.a.99.1 2
5.4 even 2 2450.2.a.k.1.1 1
7.2 even 3 350.2.e.c.151.1 2
7.4 even 3 350.2.e.c.51.1 2
7.6 odd 2 2450.2.a.bb.1.1 1
35.2 odd 12 70.2.i.b.39.2 yes 4
35.3 even 12 490.2.i.a.79.2 4
35.4 even 6 350.2.e.j.51.1 2
35.9 even 6 350.2.e.j.151.1 2
35.12 even 12 490.2.i.a.459.2 4
35.13 even 4 490.2.c.d.99.1 2
35.17 even 12 490.2.i.a.79.1 4
35.18 odd 12 70.2.i.b.9.2 yes 4
35.23 odd 12 70.2.i.b.39.1 yes 4
35.27 even 4 490.2.c.d.99.2 2
35.32 odd 12 70.2.i.b.9.1 4
35.33 even 12 490.2.i.a.459.1 4
35.34 odd 2 2450.2.a.j.1.1 1
105.2 even 12 630.2.u.a.109.1 4
105.23 even 12 630.2.u.a.109.2 4
105.32 even 12 630.2.u.a.289.2 4
105.53 even 12 630.2.u.a.289.1 4
140.23 even 12 560.2.bw.d.529.2 4
140.67 even 12 560.2.bw.d.289.2 4
140.107 even 12 560.2.bw.d.529.1 4
140.123 even 12 560.2.bw.d.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.i.b.9.1 4 35.32 odd 12
70.2.i.b.9.2 yes 4 35.18 odd 12
70.2.i.b.39.1 yes 4 35.23 odd 12
70.2.i.b.39.2 yes 4 35.2 odd 12
350.2.e.c.51.1 2 7.4 even 3
350.2.e.c.151.1 2 7.2 even 3
350.2.e.j.51.1 2 35.4 even 6
350.2.e.j.151.1 2 35.9 even 6
490.2.c.a.99.1 2 5.3 odd 4
490.2.c.a.99.2 2 5.2 odd 4
490.2.c.d.99.1 2 35.13 even 4
490.2.c.d.99.2 2 35.27 even 4
490.2.i.a.79.1 4 35.17 even 12
490.2.i.a.79.2 4 35.3 even 12
490.2.i.a.459.1 4 35.33 even 12
490.2.i.a.459.2 4 35.12 even 12
560.2.bw.d.289.1 4 140.123 even 12
560.2.bw.d.289.2 4 140.67 even 12
560.2.bw.d.529.1 4 140.107 even 12
560.2.bw.d.529.2 4 140.23 even 12
630.2.u.a.109.1 4 105.2 even 12
630.2.u.a.109.2 4 105.23 even 12
630.2.u.a.289.1 4 105.53 even 12
630.2.u.a.289.2 4 105.32 even 12
2450.2.a.j.1.1 1 35.34 odd 2
2450.2.a.k.1.1 1 5.4 even 2
2450.2.a.ba.1.1 1 1.1 even 1 trivial
2450.2.a.bb.1.1 1 7.6 odd 2