Properties

Label 2450.2.a.q.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} -2.00000 q^{11} +3.00000 q^{12} +1.00000 q^{16} -4.00000 q^{17} -6.00000 q^{18} +6.00000 q^{19} +2.00000 q^{22} -3.00000 q^{23} -3.00000 q^{24} +9.00000 q^{27} +9.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +4.00000 q^{34} +6.00000 q^{36} +4.00000 q^{37} -6.00000 q^{38} +7.00000 q^{41} +5.00000 q^{43} -2.00000 q^{44} +3.00000 q^{46} +8.00000 q^{47} +3.00000 q^{48} -12.0000 q^{51} +2.00000 q^{53} -9.00000 q^{54} +18.0000 q^{57} -9.00000 q^{58} -10.0000 q^{59} -1.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} +9.00000 q^{67} -4.00000 q^{68} -9.00000 q^{69} +2.00000 q^{71} -6.00000 q^{72} -4.00000 q^{73} -4.00000 q^{74} +6.00000 q^{76} +10.0000 q^{79} +9.00000 q^{81} -7.00000 q^{82} -7.00000 q^{83} -5.00000 q^{86} +27.0000 q^{87} +2.00000 q^{88} -1.00000 q^{89} -3.00000 q^{92} +12.0000 q^{93} -8.00000 q^{94} -3.00000 q^{96} +14.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 3.00000 0.866025
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −6.00000 −1.41421
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) −9.00000 −1.18176
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) −4.00000 −0.485071
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −6.00000 −0.707107
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −7.00000 −0.773021
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) 27.0000 2.89470
\(88\) 2.00000 0.213201
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 12.0000 1.24434
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 12.0000 1.18818
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 9.00000 0.866025
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −18.0000 −1.68585
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 1.00000 0.0905357
\(123\) 21.0000 1.89351
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 15.0000 1.32068
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) −9.00000 −0.777482
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 9.00000 0.766131
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −6.00000 −0.486664
\(153\) −24.0000 −1.94029
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −10.0000 −0.795557
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 36.0000 2.75299
\(172\) 5.00000 0.381246
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) −27.0000 −2.04686
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −30.0000 −2.25494
\(178\) 1.00000 0.0749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −3.00000 −0.221766
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) −12.0000 −0.879883
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 3.00000 0.216506
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 12.0000 0.852803
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 27.0000 1.90443
\(202\) 3.00000 0.211079
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) −1.00000 −0.0696733
\(207\) −18.0000 −1.25109
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 2.00000 0.137361
\(213\) 6.00000 0.411113
\(214\) 3.00000 0.205076
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 0 0
\(218\) 9.00000 0.609557
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) −12.0000 −0.805387
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 18.0000 1.19208
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 30.0000 1.94871
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) −21.0000 −1.33891
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) −21.0000 −1.33082
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) −15.0000 −0.933859
\(259\) 0 0
\(260\) 0 0
\(261\) 54.0000 3.34252
\(262\) 8.00000 0.494242
\(263\) −5.00000 −0.308313 −0.154157 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) 9.00000 0.549762
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −9.00000 −0.541736
\(277\) −12.0000 −0.721010 −0.360505 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) −14.0000 −0.839664
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −24.0000 −1.42918
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −6.00000 −0.353553
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 42.0000 2.46208
\(292\) −4.00000 −0.234082
\(293\) 28.0000 1.63578 0.817889 0.575376i \(-0.195144\pi\)
0.817889 + 0.575376i \(0.195144\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) −18.0000 −1.04447
\(298\) −3.00000 −0.173785
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) −9.00000 −0.517036
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) 24.0000 1.37199
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 3.00000 0.170664
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 32.0000 1.79730 0.898650 0.438667i \(-0.144549\pi\)
0.898650 + 0.438667i \(0.144549\pi\)
\(318\) −6.00000 −0.336463
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −27.0000 −1.49310
\(328\) −7.00000 −0.386510
\(329\) 0 0
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) −7.00000 −0.384175
\(333\) 24.0000 1.31519
\(334\) 21.0000 1.14907
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 13.0000 0.707107
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) −36.0000 −1.94666
\(343\) 0 0
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) −8.00000 −0.430083
\(347\) −19.0000 −1.01997 −0.509987 0.860182i \(-0.670350\pi\)
−0.509987 + 0.860182i \(0.670350\pi\)
\(348\) 27.0000 1.44735
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 30.0000 1.59448
\(355\) 0 0
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 7.00000 0.367912
\(363\) −21.0000 −1.10221
\(364\) 0 0
\(365\) 0 0
\(366\) 3.00000 0.156813
\(367\) −11.0000 −0.574195 −0.287098 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) −3.00000 −0.156386
\(369\) 42.0000 2.18643
\(370\) 0 0
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 0 0
\(381\) −48.0000 −2.45911
\(382\) 18.0000 0.920960
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) 30.0000 1.52499
\(388\) 14.0000 0.710742
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) −24.0000 −1.21064
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) 31.0000 1.54807 0.774033 0.633145i \(-0.218236\pi\)
0.774033 + 0.633145i \(0.218236\pi\)
\(402\) −27.0000 −1.34664
\(403\) 0 0
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 12.0000 0.594089
\(409\) −3.00000 −0.148340 −0.0741702 0.997246i \(-0.523631\pi\)
−0.0741702 + 0.997246i \(0.523631\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) 1.00000 0.0492665
\(413\) 0 0
\(414\) 18.0000 0.884652
\(415\) 0 0
\(416\) 0 0
\(417\) 42.0000 2.05675
\(418\) 12.0000 0.586939
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 26.0000 1.26566
\(423\) 48.0000 2.33384
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 9.00000 0.433013
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.00000 −0.431022
\(437\) −18.0000 −0.861057
\(438\) 12.0000 0.573382
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −31.0000 −1.47285 −0.736427 0.676517i \(-0.763489\pi\)
−0.736427 + 0.676517i \(0.763489\pi\)
\(444\) 12.0000 0.569495
\(445\) 0 0
\(446\) −28.0000 −1.32584
\(447\) 9.00000 0.425685
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) −14.0000 −0.659234
\(452\) −2.00000 −0.0940721
\(453\) −48.0000 −2.25524
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 22.0000 1.02799
\(459\) −36.0000 −1.68034
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 19.0000 0.883005 0.441502 0.897260i \(-0.354446\pi\)
0.441502 + 0.897260i \(0.354446\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 30.0000 1.38233
\(472\) 10.0000 0.460287
\(473\) −10.0000 −0.459800
\(474\) −30.0000 −1.37795
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) −16.0000 −0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 1.00000 0.0452679
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 21.0000 0.946753
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 21.0000 0.941033
\(499\) −18.0000 −0.805791 −0.402895 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 0 0
\(501\) −63.0000 −2.81463
\(502\) 0 0
\(503\) −21.0000 −0.936344 −0.468172 0.883637i \(-0.655087\pi\)
−0.468172 + 0.883637i \(0.655087\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −39.0000 −1.73205
\(508\) −16.0000 −0.709885
\(509\) −1.00000 −0.0443242 −0.0221621 0.999754i \(-0.507055\pi\)
−0.0221621 + 0.999754i \(0.507055\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 54.0000 2.38416
\(514\) −8.00000 −0.352865
\(515\) 0 0
\(516\) 15.0000 0.660338
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) −54.0000 −2.36352
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 5.00000 0.218010
\(527\) −16.0000 −0.696971
\(528\) −6.00000 −0.261116
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −60.0000 −2.60378
\(532\) 0 0
\(533\) 0 0
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) 36.0000 1.55351
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) −6.00000 −0.257722
\(543\) −21.0000 −0.901196
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 33.0000 1.41098 0.705489 0.708721i \(-0.250727\pi\)
0.705489 + 0.708721i \(0.250727\pi\)
\(548\) −12.0000 −0.512615
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 9.00000 0.383065
\(553\) 0 0
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −24.0000 −1.01600
\(559\) 0 0
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) −2.00000 −0.0843649
\(563\) 17.0000 0.716465 0.358232 0.933632i \(-0.383380\pi\)
0.358232 + 0.933632i \(0.383380\pi\)
\(564\) 24.0000 1.01058
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −30.0000 −1.25546 −0.627730 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) 0 0
\(573\) −54.0000 −2.25588
\(574\) 0 0
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) −78.0000 −3.24157
\(580\) 0 0
\(581\) 0 0
\(582\) −42.0000 −1.74096
\(583\) −4.00000 −0.165663
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −28.0000 −1.15667
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 4.00000 0.164399
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 18.0000 0.738549
\(595\) 0 0
\(596\) 3.00000 0.122885
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) 54.0000 2.19905
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 9.00000 0.365600
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −24.0000 −0.970143
\(613\) −12.0000 −0.484675 −0.242338 0.970192i \(-0.577914\pi\)
−0.242338 + 0.970192i \(0.577914\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 0 0
\(617\) −44.0000 −1.77137 −0.885687 0.464283i \(-0.846312\pi\)
−0.885687 + 0.464283i \(0.846312\pi\)
\(618\) −3.00000 −0.120678
\(619\) 46.0000 1.84890 0.924448 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) 0 0
\(621\) −27.0000 −1.08347
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) −36.0000 −1.43770
\(628\) 10.0000 0.399043
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) −10.0000 −0.397779
\(633\) −78.0000 −3.10022
\(634\) −32.0000 −1.27088
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 18.0000 0.712627
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 5.00000 0.197488 0.0987441 0.995113i \(-0.468517\pi\)
0.0987441 + 0.995113i \(0.468517\pi\)
\(642\) 9.00000 0.355202
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) −11.0000 −0.432455 −0.216227 0.976343i \(-0.569375\pi\)
−0.216227 + 0.976343i \(0.569375\pi\)
\(648\) −9.00000 −0.353553
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 27.0000 1.05578
\(655\) 0 0
\(656\) 7.00000 0.273304
\(657\) −24.0000 −0.936329
\(658\) 0 0
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 0 0
\(661\) 11.0000 0.427850 0.213925 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) 32.0000 1.24372
\(663\) 0 0
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) −24.0000 −0.929981
\(667\) −27.0000 −1.04544
\(668\) −21.0000 −0.812514
\(669\) 84.0000 3.24763
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 8.00000 0.306336
\(683\) 37.0000 1.41577 0.707883 0.706330i \(-0.249650\pi\)
0.707883 + 0.706330i \(0.249650\pi\)
\(684\) 36.0000 1.37649
\(685\) 0 0
\(686\) 0 0
\(687\) −66.0000 −2.51806
\(688\) 5.00000 0.190623
\(689\) 0 0
\(690\) 0 0
\(691\) −22.0000 −0.836919 −0.418460 0.908235i \(-0.637430\pi\)
−0.418460 + 0.908235i \(0.637430\pi\)
\(692\) 8.00000 0.304114
\(693\) 0 0
\(694\) 19.0000 0.721230
\(695\) 0 0
\(696\) −27.0000 −1.02343
\(697\) −28.0000 −1.06058
\(698\) −35.0000 −1.32477
\(699\) −72.0000 −2.72329
\(700\) 0 0
\(701\) −47.0000 −1.77517 −0.887583 0.460648i \(-0.847617\pi\)
−0.887583 + 0.460648i \(0.847617\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) −30.0000 −1.12747
\(709\) −11.0000 −0.413114 −0.206557 0.978435i \(-0.566226\pi\)
−0.206557 + 0.978435i \(0.566226\pi\)
\(710\) 0 0
\(711\) 60.0000 2.25018
\(712\) 1.00000 0.0374766
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 48.0000 1.79259
\(718\) 4.00000 0.149279
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) −30.0000 −1.11571
\(724\) −7.00000 −0.260153
\(725\) 0 0
\(726\) 21.0000 0.779383
\(727\) −21.0000 −0.778847 −0.389423 0.921059i \(-0.627326\pi\)
−0.389423 + 0.921059i \(0.627326\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) −3.00000 −0.110883
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 11.0000 0.406017
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) −18.0000 −0.663039
\(738\) −42.0000 −1.54604
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) −42.0000 −1.53670
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) −30.0000 −1.08965
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 48.0000 1.73886
\(763\) 0 0
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −15.0000 −0.541972
\(767\) 0 0
\(768\) 3.00000 0.108253
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) −26.0000 −0.935760
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) −30.0000 −1.07833
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) 42.0000 1.50481
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −12.0000 −0.429119
\(783\) 81.0000 2.89470
\(784\) 0 0
\(785\) 0 0
\(786\) 24.0000 0.856052
\(787\) 31.0000 1.10503 0.552515 0.833503i \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −15.0000 −0.534014
\(790\) 0 0
\(791\) 0 0
\(792\) 12.0000 0.426401
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −31.0000 −1.09465
\(803\) 8.00000 0.282314
\(804\) 27.0000 0.952217
\(805\) 0 0
\(806\) 0 0
\(807\) −9.00000 −0.316815
\(808\) 3.00000 0.105540
\(809\) −51.0000 −1.79306 −0.896532 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 18.0000 0.631288
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) 30.0000 1.04957
\(818\) 3.00000 0.104893
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 36.0000 1.25564
\(823\) −19.0000 −0.662298 −0.331149 0.943578i \(-0.607436\pi\)
−0.331149 + 0.943578i \(0.607436\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 0 0
\(826\) 0 0
\(827\) 19.0000 0.660695 0.330347 0.943859i \(-0.392834\pi\)
0.330347 + 0.943859i \(0.392834\pi\)
\(828\) −18.0000 −0.625543
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) −36.0000 −1.24883
\(832\) 0 0
\(833\) 0 0
\(834\) −42.0000 −1.45434
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 36.0000 1.24434
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 19.0000 0.654783
\(843\) 6.00000 0.206651
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) −48.0000 −1.65027
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 6.00000 0.205557
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30.0000 1.02180
\(863\) 11.0000 0.374444 0.187222 0.982318i \(-0.440052\pi\)
0.187222 + 0.982318i \(0.440052\pi\)
\(864\) −9.00000 −0.306186
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 0 0
\(872\) 9.00000 0.304778
\(873\) 84.0000 2.84297
\(874\) 18.0000 0.608859
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −20.0000 −0.674967
\(879\) 84.0000 2.83325
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 31.0000 1.04147
\(887\) 29.0000 0.973725 0.486862 0.873479i \(-0.338141\pi\)
0.486862 + 0.873479i \(0.338141\pi\)
\(888\) −12.0000 −0.402694
\(889\) 0 0
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) 28.0000 0.937509
\(893\) 48.0000 1.60626
\(894\) −9.00000 −0.301005
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 33.0000 1.10122
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 14.0000 0.466149
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) 48.0000 1.59469
\(907\) −5.00000 −0.166022 −0.0830111 0.996549i \(-0.526454\pi\)
−0.0830111 + 0.996549i \(0.526454\pi\)
\(908\) −4.00000 −0.132745
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 18.0000 0.596040
\(913\) 14.0000 0.463332
\(914\) −32.0000 −1.05847
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) 36.0000 1.18818
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 0 0
\(921\) 21.0000 0.691974
\(922\) −14.0000 −0.461065
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −19.0000 −0.624379
\(927\) 6.00000 0.197066
\(928\) −9.00000 −0.295439
\(929\) −43.0000 −1.41078 −0.705392 0.708817i \(-0.749229\pi\)
−0.705392 + 0.708817i \(0.749229\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −24.0000 −0.786146
\(933\) 54.0000 1.76788
\(934\) 13.0000 0.425373
\(935\) 0 0
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 24.0000 0.783210
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) −30.0000 −0.977453
\(943\) −21.0000 −0.683854
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 10.0000 0.325128
\(947\) 25.0000 0.812391 0.406195 0.913786i \(-0.366855\pi\)
0.406195 + 0.913786i \(0.366855\pi\)
\(948\) 30.0000 0.974355
\(949\) 0 0
\(950\) 0 0
\(951\) 96.0000 3.11301
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −54.0000 −1.74557
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −37.0000 −1.18984 −0.594920 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(968\) 7.00000 0.224989
\(969\) −72.0000 −2.31297
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −12.0000 −0.383718
\(979\) 2.00000 0.0639203
\(980\) 0 0
\(981\) −54.0000 −1.72409
\(982\) 12.0000 0.382935
\(983\) 17.0000 0.542216 0.271108 0.962549i \(-0.412610\pi\)
0.271108 + 0.962549i \(0.412610\pi\)
\(984\) −21.0000 −0.669456
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −4.00000 −0.127000
\(993\) −96.0000 −3.04647
\(994\) 0 0
\(995\) 0 0
\(996\) −21.0000 −0.665410
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 18.0000 0.569780
\(999\) 36.0000 1.13899
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.q.1.1 1
5.2 odd 4 2450.2.c.a.99.1 2
5.3 odd 4 2450.2.c.a.99.2 2
5.4 even 2 490.2.a.e.1.1 1
7.3 odd 6 350.2.e.l.51.1 2
7.5 odd 6 350.2.e.l.151.1 2
7.6 odd 2 2450.2.a.b.1.1 1
15.14 odd 2 4410.2.a.h.1.1 1
20.19 odd 2 3920.2.a.bk.1.1 1
35.3 even 12 350.2.j.f.149.1 4
35.4 even 6 490.2.e.f.471.1 2
35.9 even 6 490.2.e.f.361.1 2
35.12 even 12 350.2.j.f.249.1 4
35.13 even 4 2450.2.c.s.99.2 2
35.17 even 12 350.2.j.f.149.2 4
35.19 odd 6 70.2.e.a.11.1 2
35.24 odd 6 70.2.e.a.51.1 yes 2
35.27 even 4 2450.2.c.s.99.1 2
35.33 even 12 350.2.j.f.249.2 4
35.34 odd 2 490.2.a.k.1.1 1
105.59 even 6 630.2.k.f.541.1 2
105.89 even 6 630.2.k.f.361.1 2
105.104 even 2 4410.2.a.r.1.1 1
140.19 even 6 560.2.q.i.81.1 2
140.59 even 6 560.2.q.i.401.1 2
140.139 even 2 3920.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.a.11.1 2 35.19 odd 6
70.2.e.a.51.1 yes 2 35.24 odd 6
350.2.e.l.51.1 2 7.3 odd 6
350.2.e.l.151.1 2 7.5 odd 6
350.2.j.f.149.1 4 35.3 even 12
350.2.j.f.149.2 4 35.17 even 12
350.2.j.f.249.1 4 35.12 even 12
350.2.j.f.249.2 4 35.33 even 12
490.2.a.e.1.1 1 5.4 even 2
490.2.a.k.1.1 1 35.34 odd 2
490.2.e.f.361.1 2 35.9 even 6
490.2.e.f.471.1 2 35.4 even 6
560.2.q.i.81.1 2 140.19 even 6
560.2.q.i.401.1 2 140.59 even 6
630.2.k.f.361.1 2 105.89 even 6
630.2.k.f.541.1 2 105.59 even 6
2450.2.a.b.1.1 1 7.6 odd 2
2450.2.a.q.1.1 1 1.1 even 1 trivial
2450.2.c.a.99.1 2 5.2 odd 4
2450.2.c.a.99.2 2 5.3 odd 4
2450.2.c.s.99.1 2 35.27 even 4
2450.2.c.s.99.2 2 35.13 even 4
3920.2.a.b.1.1 1 140.139 even 2
3920.2.a.bk.1.1 1 20.19 odd 2
4410.2.a.h.1.1 1 15.14 odd 2
4410.2.a.r.1.1 1 105.104 even 2