Properties

Label 3150.2.g.h.2899.2
Level 3150
Weight 2
Character 3150.2899
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1050)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(-1.00000i\)
Character \(\chi\) = 3150.2899
Dual form 3150.2.g.h.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{11} -1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{22} +1.00000i q^{23} +1.00000 q^{26} +1.00000i q^{28} -5.00000 q^{29} +7.00000 q^{31} +1.00000i q^{32} -3.00000 q^{34} +2.00000i q^{37} -7.00000 q^{41} -11.0000i q^{43} +2.00000 q^{44} -1.00000 q^{46} +8.00000i q^{47} -1.00000 q^{49} +1.00000i q^{52} +1.00000i q^{53} -1.00000 q^{56} -5.00000i q^{58} -5.00000 q^{59} -3.00000 q^{61} +7.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} -3.00000i q^{68} -12.0000 q^{71} -6.00000i q^{73} -2.00000 q^{74} +2.00000i q^{77} -10.0000 q^{79} -7.00000i q^{82} +11.0000i q^{83} +11.0000 q^{86} +2.00000i q^{88} -10.0000 q^{89} -1.00000 q^{91} -1.00000i q^{92} -8.00000 q^{94} +2.00000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{11} + 2q^{14} + 2q^{16} + 2q^{26} - 10q^{29} + 14q^{31} - 6q^{34} - 14q^{41} + 4q^{44} - 2q^{46} - 2q^{49} - 2q^{56} - 10q^{59} - 6q^{61} - 2q^{64} - 24q^{71} - 4q^{74} - 20q^{79} + 22q^{86} - 20q^{89} - 2q^{91} - 16q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.00000i − 0.426401i
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) − 11.0000i − 1.67748i −0.544529 0.838742i \(-0.683292\pi\)
0.544529 0.838742i \(-0.316708\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) 1.00000i 0.137361i 0.997639 + 0.0686803i \(0.0218788\pi\)
−0.997639 + 0.0686803i \(0.978121\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) − 5.00000i − 0.656532i
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 7.00000i 0.889001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 7.00000i − 0.773021i
\(83\) 11.0000i 1.20741i 0.797209 + 0.603703i \(0.206309\pi\)
−0.797209 + 0.603703i \(0.793691\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) 2.00000i 0.213201i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) − 1.00000i − 0.104257i
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) − 11.0000i − 1.08386i −0.840423 0.541931i \(-0.817693\pi\)
0.840423 0.541931i \(-0.182307\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) − 2.00000i − 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) − 5.00000i − 0.460287i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 3.00000i − 0.271607i
\(123\) 0 0
\(124\) −7.00000 −0.628619
\(125\) 0 0
\(126\) 0 0
\(127\) 22.0000i 1.95218i 0.217357 + 0.976092i \(0.430256\pi\)
−0.217357 + 0.976092i \(0.569744\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 12.0000i − 1.00702i
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) − 1.00000i − 0.0783260i −0.999233 0.0391630i \(-0.987531\pi\)
0.999233 0.0391630i \(-0.0124692\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) −11.0000 −0.853766
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 11.0000i 0.838742i
\(173\) − 24.0000i − 1.82469i −0.409426 0.912343i \(-0.634271\pi\)
0.409426 0.912343i \(-0.365729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) − 10.0000i − 0.749532i
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) − 1.00000i − 0.0741249i
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.00000i − 0.438763i
\(188\) − 8.00000i − 0.583460i
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 17.0000i − 1.21120i −0.795769 0.605600i \(-0.792933\pi\)
0.795769 0.605600i \(-0.207067\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 12.0000i − 0.844317i
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 11.0000 0.766406
\(207\) 0 0
\(208\) − 1.00000i − 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) − 1.00000i − 0.0686803i
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 0 0
\(217\) − 7.00000i − 0.475191i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 9.00000i 0.602685i 0.953516 + 0.301342i \(0.0974347\pi\)
−0.953516 + 0.301342i \(0.902565\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 17.0000i − 1.12833i −0.825662 0.564165i \(-0.809198\pi\)
0.825662 0.564165i \(-0.190802\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.00000i 0.328266i
\(233\) − 24.0000i − 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5.00000 0.325472
\(237\) 0 0
\(238\) 3.00000i 0.194461i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 0 0
\(244\) 3.00000 0.192055
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) − 7.00000i − 0.444500i
\(249\) 0 0
\(250\) 0 0
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 0 0
\(253\) − 2.00000i − 0.125739i
\(254\) −22.0000 −1.38040
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.0000i 1.43470i 0.696713 + 0.717350i \(0.254645\pi\)
−0.696713 + 0.717350i \(0.745355\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 8.00000i 0.494242i
\(263\) 21.0000i 1.29492i 0.762101 + 0.647458i \(0.224168\pi\)
−0.762101 + 0.647458i \(0.775832\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) − 12.0000i − 0.733017i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 20.0000i − 1.19952i
\(279\) 0 0
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 7.00000i 0.413197i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 6.00000i 0.351123i
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 15.0000i 0.868927i
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) − 18.0000i − 1.03578i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) − 2.00000i − 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 3.00000i 0.168497i 0.996445 + 0.0842484i \(0.0268489\pi\)
−0.996445 + 0.0842484i \(0.973151\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 0 0
\(322\) 1.00000i 0.0557278i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 1.00000 0.0553849
\(327\) 0 0
\(328\) 7.00000i 0.386510i
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) − 11.0000i − 0.603703i
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) 27.0000i 1.47078i 0.677642 + 0.735392i \(0.263002\pi\)
−0.677642 + 0.735392i \(0.736998\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) −14.0000 −0.758143
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 0 0
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 2.00000i − 0.106600i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) − 10.0000i − 0.528516i
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 18.0000i − 0.946059i
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) − 23.0000i − 1.20059i −0.799779 0.600295i \(-0.795050\pi\)
0.799779 0.600295i \(-0.204950\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.00000 0.0519174
\(372\) 0 0
\(373\) 24.0000i 1.24267i 0.783544 + 0.621336i \(0.213410\pi\)
−0.783544 + 0.621336i \(0.786590\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 5.00000i 0.257513i
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3.00000i 0.153493i
\(383\) − 4.00000i − 0.204390i −0.994764 0.102195i \(-0.967413\pi\)
0.994764 0.102195i \(-0.0325866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) − 2.00000i − 0.101535i
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) 17.0000 0.856448
\(395\) 0 0
\(396\) 0 0
\(397\) − 13.0000i − 0.652451i −0.945292 0.326226i \(-0.894223\pi\)
0.945292 0.326226i \(-0.105777\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) 0 0
\(403\) − 7.00000i − 0.348695i
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) −5.00000 −0.248146
\(407\) − 4.00000i − 0.198273i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 11.0000i 0.541931i
\(413\) 5.00000i 0.246034i
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) 25.0000 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 13.0000i − 0.632830i
\(423\) 0 0
\(424\) 1.00000 0.0485643
\(425\) 0 0
\(426\) 0 0
\(427\) 3.00000i 0.145180i
\(428\) 2.00000i 0.0966736i
\(429\) 0 0
\(430\) 0 0
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 0 0
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 7.00000 0.336011
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.00000i 0.142695i
\(443\) 16.0000i 0.760183i 0.924949 + 0.380091i \(0.124107\pi\)
−0.924949 + 0.380091i \(0.875893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −9.00000 −0.426162
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 14.0000 0.659234
\(452\) − 6.00000i − 0.282216i
\(453\) 0 0
\(454\) 17.0000 0.797850
\(455\) 0 0
\(456\) 0 0
\(457\) − 33.0000i − 1.54367i −0.635820 0.771837i \(-0.719338\pi\)
0.635820 0.771837i \(-0.280662\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 33.0000i 1.52706i 0.645774 + 0.763529i \(0.276535\pi\)
−0.645774 + 0.763529i \(0.723465\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 5.00000i 0.230144i
\(473\) 22.0000i 1.01156i
\(474\) 0 0
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 22.0000i 1.00207i
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 3.00000i 0.135804i
\(489\) 0 0
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) − 15.0000i − 0.675566i
\(494\) 0 0
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 7.00000i − 0.312425i
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) − 22.0000i − 0.976092i
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −23.0000 −1.01449
\(515\) 0 0
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 2.00000i 0.0878750i
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −21.0000 −0.915644
\(527\) 21.0000i 0.914774i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.00000i 0.303204i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) − 10.0000i − 0.431131i
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) − 28.0000i − 1.20270i
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) 37.0000i 1.58201i 0.611812 + 0.791003i \(0.290441\pi\)
−0.611812 + 0.791003i \(0.709559\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 10.0000i 0.425243i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) 0 0
\(562\) − 32.0000i − 1.34984i
\(563\) 31.0000i 1.30649i 0.757145 + 0.653247i \(0.226594\pi\)
−0.757145 + 0.653247i \(0.773406\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) 12.0000i 0.503509i
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) 0 0
\(574\) −7.00000 −0.292174
\(575\) 0 0
\(576\) 0 0
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) 8.00000i 0.332756i
\(579\) 0 0
\(580\) 0 0
\(581\) 11.0000 0.456357
\(582\) 0 0
\(583\) − 2.00000i − 0.0828315i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) − 27.0000i − 1.11441i −0.830375 0.557205i \(-0.811874\pi\)
0.830375 0.557205i \(-0.188126\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) − 14.0000i − 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −15.0000 −0.614424
\(597\) 0 0
\(598\) 1.00000i 0.0408930i
\(599\) 5.00000 0.204294 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) − 11.0000i − 0.448327i
\(603\) 0 0
\(604\) 18.0000 0.732410
\(605\) 0 0
\(606\) 0 0
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 14.0000i 0.565455i 0.959200 + 0.282727i \(0.0912392\pi\)
−0.959200 + 0.282727i \(0.908761\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 38.0000i 1.52982i 0.644136 + 0.764911i \(0.277217\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000i 0.721734i
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 10.0000i 0.397779i
\(633\) 0 0
\(634\) −3.00000 −0.119145
\(635\) 0 0
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 10.0000i 0.395904i
\(639\) 0 0
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 34.0000i 1.34083i 0.741987 + 0.670415i \(0.233884\pi\)
−0.741987 + 0.670415i \(0.766116\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) − 22.0000i − 0.864909i −0.901656 0.432455i \(-0.857648\pi\)
0.901656 0.432455i \(-0.142352\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 1.00000i 0.0391630i
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7.00000 −0.273304
\(657\) 0 0
\(658\) 8.00000i 0.311872i
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 17.0000i 0.660724i
\(663\) 0 0
\(664\) 11.0000 0.426883
\(665\) 0 0
\(666\) 0 0
\(667\) − 5.00000i − 0.193601i
\(668\) − 18.0000i − 0.696441i
\(669\) 0 0
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 29.0000i 1.11787i 0.829212 + 0.558934i \(0.188789\pi\)
−0.829212 + 0.558934i \(0.811211\pi\)
\(674\) −27.0000 −1.04000
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 32.0000i − 1.22986i −0.788582 0.614930i \(-0.789184\pi\)
0.788582 0.614930i \(-0.210816\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) − 14.0000i − 0.536088i
\(683\) 46.0000i 1.76014i 0.474843 + 0.880071i \(0.342505\pi\)
−0.474843 + 0.880071i \(0.657495\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) − 11.0000i − 0.419371i
\(689\) 1.00000 0.0380970
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 24.0000i 0.912343i
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 0 0
\(697\) − 21.0000i − 0.795432i
\(698\) − 15.0000i − 0.567758i
\(699\) 0 0
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000i 0.374766i
\(713\) 7.00000i 0.262152i
\(714\) 0 0
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) 25.0000i 0.932992i
\(719\) −10.0000 −0.372937 −0.186469 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) 0 0
\(721\) −11.0000 −0.409661
\(722\) − 19.0000i − 0.707107i
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) 0 0
\(727\) − 13.0000i − 0.482143i −0.970507 0.241072i \(-0.922501\pi\)
0.970507 0.241072i \(-0.0774989\pi\)
\(728\) 1.00000i 0.0370625i
\(729\) 0 0
\(730\) 0 0
\(731\) 33.0000 1.22055
\(732\) 0 0
\(733\) − 11.0000i − 0.406294i −0.979148 0.203147i \(-0.934883\pi\)
0.979148 0.203147i \(-0.0651170\pi\)
\(734\) 23.0000 0.848945
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) − 24.0000i − 0.884051i
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.00000i 0.0367112i
\(743\) − 29.0000i − 1.06391i −0.846774 0.531953i \(-0.821458\pi\)
0.846774 0.531953i \(-0.178542\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) 0 0
\(748\) 6.00000i 0.219382i
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 0 0
\(754\) −5.00000 −0.182089
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 25.0000i 0.908041i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 5.00000i 0.180540i
\(768\) 0 0
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 14.0000i − 0.503871i
\(773\) − 24.0000i − 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 10.0000i 0.358517i
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) − 3.00000i − 0.107280i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) 17.0000i 0.605600i
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 3.00000i 0.106533i
\(794\) 13.0000 0.461353
\(795\) 0 0
\(796\) 0 0
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 8.00000i 0.282490i
\(803\) 12.0000i 0.423471i
\(804\) 0 0
\(805\) 0 0
\(806\) 7.00000 0.246564
\(807\) 0 0
\(808\) 12.0000i 0.422159i
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) − 5.00000i − 0.175466i
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 10.0000i 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 54.0000i 1.88232i 0.337959 + 0.941161i \(0.390263\pi\)
−0.337959 + 0.941161i \(0.609737\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) −5.00000 −0.173972
\(827\) − 2.00000i − 0.0695468i −0.999395 0.0347734i \(-0.988929\pi\)
0.999395 0.0347734i \(-0.0110710\pi\)
\(828\) 0 0
\(829\) 5.00000 0.173657 0.0868286 0.996223i \(-0.472327\pi\)
0.0868286 + 0.996223i \(0.472327\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) − 3.00000i − 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 25.0000i 0.863611i
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 22.0000i 0.758170i
\(843\) 0 0
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 7.00000i 0.240523i
\(848\) 1.00000i 0.0343401i
\(849\) 0 0
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 29.0000i 0.992941i 0.868054 + 0.496471i \(0.165371\pi\)
−0.868054 + 0.496471i \(0.834629\pi\)
\(854\) −3.00000 −0.102658
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) − 22.0000i − 0.751506i −0.926720 0.375753i \(-0.877384\pi\)
0.926720 0.375753i \(-0.122616\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.00000i 0.102180i
\(863\) 16.0000i 0.544646i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) 7.00000i 0.237595i
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) − 35.0000i − 1.18119i
\(879\) 0 0
\(880\) 0 0
\(881\) −7.00000 −0.235836 −0.117918 0.993023i \(-0.537622\pi\)
−0.117918 + 0.993023i \(0.537622\pi\)
\(882\) 0 0
\(883\) 39.0000i 1.31245i 0.754563 + 0.656227i \(0.227849\pi\)
−0.754563 + 0.656227i \(0.772151\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −16.0000 −0.537531
\(887\) 58.0000i 1.94745i 0.227728 + 0.973725i \(0.426870\pi\)
−0.227728 + 0.973725i \(0.573130\pi\)
\(888\) 0 0
\(889\) 22.0000 0.737856
\(890\) 0 0
\(891\) 0 0
\(892\) − 9.00000i − 0.301342i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 20.0000i 0.667409i
\(899\) −35.0000 −1.16732
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 14.0000i 0.466149i
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 17.0000i 0.564476i 0.959344 + 0.282238i \(0.0910767\pi\)
−0.959344 + 0.282238i \(0.908923\pi\)
\(908\) 17.0000i 0.564165i
\(909\) 0 0
\(910\) 0 0
\(911\) 53.0000 1.75597 0.877984 0.478690i \(-0.158888\pi\)
0.877984 + 0.478690i \(0.158888\pi\)
\(912\) 0 0
\(913\) − 22.0000i − 0.728094i
\(914\) 33.0000 1.09154
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 8.00000i − 0.264183i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 12.0000i − 0.395199i
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) − 5.00000i − 0.164133i
\(929\) −45.0000 −1.47640 −0.738201 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 24.0000i 0.786146i
\(933\) 0 0
\(934\) −33.0000 −1.07979
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0000i 1.37208i 0.727564 + 0.686040i \(0.240653\pi\)
−0.727564 + 0.686040i \(0.759347\pi\)
\(938\) 12.0000i 0.391814i
\(939\) 0 0
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) − 7.00000i − 0.227951i
\(944\) −5.00000 −0.162736
\(945\) 0 0
\(946\) −22.0000 −0.715282
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) − 3.00000i − 0.0972306i
\(953\) 16.0000i 0.518291i 0.965838 + 0.259145i \(0.0834409\pi\)
−0.965838 + 0.259145i \(0.916559\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) − 20.0000i − 0.646171i
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 2.00000i 0.0644826i
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 20.0000i 0.641171i
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −3.00000 −0.0960277
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) − 2.00000i − 0.0638226i
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 15.0000 0.477697
\(987\) 0 0
\(988\) 0 0
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 7.00000i 0.222250i
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) − 58.0000i − 1.83688i −0.395562 0.918439i \(-0.629450\pi\)
0.395562 0.918439i \(-0.370550\pi\)
\(998\) − 25.0000i − 0.791361i
\(999\) 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 3150.2.g.h.2899.2 2
3.2 odd 2 1050.2.g.i.799.1 2
5.2 odd 4 3150.2.a.n.1.1 1
5.3 odd 4 3150.2.a.y.1.1 1
5.4 even 2 inner 3150.2.g.h.2899.1 2
15.2 even 4 1050.2.a.r.1.1 yes 1
15.8 even 4 1050.2.a.b.1.1 1
15.14 odd 2 1050.2.g.i.799.2 2
60.23 odd 4 8400.2.a.ck.1.1 1
60.47 odd 4 8400.2.a.d.1.1 1
105.62 odd 4 7350.2.a.cb.1.1 1
105.83 odd 4 7350.2.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.b.1.1 1 15.8 even 4
1050.2.a.r.1.1 yes 1 15.2 even 4
1050.2.g.i.799.1 2 3.2 odd 2
1050.2.g.i.799.2 2 15.14 odd 2
3150.2.a.n.1.1 1 5.2 odd 4
3150.2.a.y.1.1 1 5.3 odd 4
3150.2.g.h.2899.1 2 5.4 even 2 inner
3150.2.g.h.2899.2 2 1.1 even 1 trivial
7350.2.a.bi.1.1 1 105.83 odd 4
7350.2.a.cb.1.1 1 105.62 odd 4
8400.2.a.d.1.1 1 60.47 odd 4
8400.2.a.ck.1.1 1 60.23 odd 4