Properties

Label 1950.2.a.u.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} +2.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +8.00000 q^{43} +2.00000 q^{44} -6.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{51} +1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -2.00000 q^{57} -2.00000 q^{58} -14.0000 q^{59} +10.0000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} +6.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} +2.00000 q^{76} +8.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +8.00000 q^{86} +2.00000 q^{87} +2.00000 q^{88} -18.0000 q^{89} +4.00000 q^{91} -6.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 2.00000 0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 2.00000 0.324443
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 1.00000 0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) −2.00000 −0.264906
\(58\) −2.00000 −0.262613
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 8.00000 0.911685
\(78\) −1.00000 −0.113228
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 2.00000 0.214423
\(88\) 2.00000 0.213201
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −6.00000 −0.625543
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 9.00000 0.909137
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −4.00000 −0.396059
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 4.00000 0.377964
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 1.00000 0.0924500
\(118\) −14.0000 −1.28880
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 10.0000 0.905357
\(123\) 6.00000 0.541002
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −2.00000 −0.174078
\(133\) 8.00000 0.693688
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 6.00000 0.510754
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 8.00000 0.671345
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) −9.00000 −0.742307
\(148\) −6.00000 −0.493197
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) 4.00000 0.323381
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −8.00000 −0.636446
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −4.00000 −0.308607
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 8.00000 0.609994
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 14.0000 1.05230
\(178\) −18.0000 −1.34916
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 4.00000 0.296500
\(183\) −10.0000 −0.739221
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 26.0000 1.85242 0.926212 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 2.00000 0.142134
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −14.0000 −0.985037
\(203\) −8.00000 −0.561490
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) −6.00000 −0.417029
\(208\) 1.00000 0.0693375
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 10.0000 0.686803
\(213\) −8.00000 −0.548151
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −16.0000 −1.08615
\(218\) 12.0000 0.812743
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 6.00000 0.402694
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −2.00000 −0.132453
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) −2.00000 −0.131306
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 8.00000 0.519656
\(238\) 16.0000 1.03713
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 2.00000 0.127257
\(248\) −4.00000 −0.254000
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 4.00000 0.251976
\(253\) −12.0000 −0.754434
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −8.00000 −0.498058
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −10.0000 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) 18.0000 1.10158
\(268\) 4.00000 0.244339
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 4.00000 0.242536
\(273\) −4.00000 −0.242091
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 16.0000 0.959616
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −8.00000 −0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 10.0000 0.585206
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 32.0000 1.84445
\(302\) 8.00000 0.460348
\(303\) 14.0000 0.804279
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 8.00000 0.455842
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 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
\(316\) −8.00000 −0.450035
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −10.0000 −0.560772
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) −24.0000 −1.33747
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −12.0000 −0.663602
\(328\) −6.00000 −0.331295
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 18.0000 0.989369 0.494685 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) 12.0000 0.658586
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 1.00000 0.0543928
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 2.00000 0.108148
\(343\) 8.00000 0.431959
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 2.00000 0.107211
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 2.00000 0.106600
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 14.0000 0.744092
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) −16.0000 −0.846810
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 6.00000 0.315353
\(363\) 7.00000 0.367405
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 40.0000 2.07670
\(372\) 4.00000 0.207390
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −2.00000 −0.103005
\(378\) −4.00000 −0.205738
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 8.00000 0.406663
\(388\) 6.00000 0.304604
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 26.0000 1.30986
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 24.0000 1.20301
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −4.00000 −0.199502
\(403\) −4.00000 −0.199254
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) −12.0000 −0.594818
\(408\) −4.00000 −0.198030
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 6.00000 0.295599
\(413\) −56.0000 −2.75558
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −16.0000 −0.783523
\(418\) 4.00000 0.195646
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) −28.0000 −1.36302
\(423\) 8.00000 0.388973
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 40.0000 1.93574
\(428\) −8.00000 −0.386695
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) −12.0000 −0.574038
\(438\) −10.0000 −0.477818
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) 8.00000 0.380091 0.190046 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 4.00000 0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) −12.0000 −0.564433
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −16.0000 −0.747631
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) −8.00000 −0.372194
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) −16.0000 −0.740392 −0.370196 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) 1.00000 0.0462250
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −14.0000 −0.644402
\(473\) 16.0000 0.735681
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 10.0000 0.457869
\(478\) −12.0000 −0.548867
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) −30.0000 −1.36646
\(483\) 24.0000 1.09204
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 10.0000 0.452679
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 6.00000 0.270501
\(493\) −8.00000 −0.360302
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 32.0000 1.43540
\(498\) −12.0000 −0.537733
\(499\) 30.0000 1.34298 0.671492 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 18.0000 0.802580 0.401290 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) −1.00000 −0.0444116
\(508\) −18.0000 −0.798621
\(509\) −16.0000 −0.709188 −0.354594 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) 40.0000 1.76950
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 16.0000 0.703679
\(518\) −24.0000 −1.05450
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) −16.0000 −0.696971
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −14.0000 −0.607548
\(532\) 8.00000 0.346844
\(533\) −6.00000 −0.259889
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 16.0000 0.687259
\(543\) −6.00000 −0.257485
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 2.00000 0.0854358
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 6.00000 0.255377
\(553\) −32.0000 −1.36078
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) −4.00000 −0.169334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) −18.0000 −0.759284
\(563\) −16.0000 −0.674320 −0.337160 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 4.00000 0.167984
\(568\) 8.00000 0.335673
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) −6.00000 −0.248708
\(583\) 20.0000 0.828315
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −9.00000 −0.371154
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −26.0000 −1.06950
\(592\) −6.00000 −0.246598
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 0 0
\(597\) −24.0000 −0.982255
\(598\) −6.00000 −0.245358
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 32.0000 1.30422
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 2.00000 0.0811107
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 4.00000 0.161690
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) −6.00000 −0.241355
\(619\) 42.0000 1.68812 0.844061 0.536247i \(-0.180158\pi\)
0.844061 + 0.536247i \(0.180158\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 8.00000 0.320771
\(623\) −72.0000 −2.88462
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) −4.00000 −0.159745
\(628\) −10.0000 −0.399043
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) −8.00000 −0.318223
\(633\) 28.0000 1.11290
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 9.00000 0.356593
\(638\) −4.00000 −0.158362
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 8.00000 0.315735
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 1.00000 0.0392837
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −4.00000 −0.156652
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) 32.0000 1.24749
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 24.0000 0.933492 0.466746 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) 18.0000 0.699590
\(663\) −4.00000 −0.155347
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) −4.00000 −0.154303
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 12.0000 0.460857
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) −8.00000 −0.306336
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 16.0000 0.610438
\(688\) 8.00000 0.304997
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 10.0000 0.380143
\(693\) 8.00000 0.303895
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) −24.0000 −0.909065
\(698\) −28.0000 −1.05982
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −12.0000 −0.452589
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) −56.0000 −2.10610
\(708\) 14.0000 0.526152
\(709\) −32.0000 −1.20179 −0.600893 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −18.0000 −0.674579
\(713\) 24.0000 0.898807
\(714\) −16.0000 −0.598785
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 20.0000 0.746393
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) −15.0000 −0.558242
\(723\) 30.0000 1.11571
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) −10.0000 −0.369611
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 8.00000 0.294684
\(738\) −6.00000 −0.220863
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 40.0000 1.46845
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) 12.0000 0.439057
\(748\) 8.00000 0.292509
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) −2.00000 −0.0728357
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 26.0000 0.944363
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 18.0000 0.652071
\(763\) 48.0000 1.73772
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −14.0000 −0.505511
\(768\) −1.00000 −0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 6.00000 0.215945
\(773\) 14.0000 0.503545 0.251773 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 24.0000 0.860995
\(778\) −26.0000 −0.932145
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) −24.0000 −0.858238
\(783\) 2.00000 0.0714742
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 26.0000 0.926212
\(789\) 10.0000 0.356009
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 2.00000 0.0710669
\(793\) 10.0000 0.355110
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) −8.00000 −0.283197
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) −18.0000 −0.635602
\(803\) 20.0000 0.705785
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 14.0000 0.492823
\(808\) −14.0000 −0.492518
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) −8.00000 −0.280745
\(813\) −16.0000 −0.561144
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 16.0000 0.559769
\(818\) −2.00000 −0.0699284
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −56.0000 −1.94849
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) 42.0000 1.45872 0.729360 0.684130i \(-0.239818\pi\)
0.729360 + 0.684130i \(0.239818\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 1.00000 0.0346688
\(833\) 36.0000 1.24733
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 4.00000 0.138260
\(838\) 16.0000 0.552711
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 28.0000 0.964944
\(843\) 18.0000 0.619953
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −28.0000 −0.962091
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) −8.00000 −0.274075
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −40.0000 −1.36637 −0.683187 0.730243i \(-0.739407\pi\)
−0.683187 + 0.730243i \(0.739407\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) −8.00000 −0.272481
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) −16.0000 −0.543075
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 12.0000 0.406371
\(873\) 6.00000 0.203069
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) −30.0000 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) 32.0000 1.07995
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 9.00000 0.303046
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 8.00000 0.268765
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 6.00000 0.201347
\(889\) −72.0000 −2.41480
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −16.0000 −0.535720
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 6.00000 0.200334
\(898\) 6.00000 0.200223
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 40.0000 1.33259
\(902\) −12.0000 −0.399556
\(903\) −32.0000 −1.06489
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 12.0000 0.398234
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 24.0000 0.794284
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) −4.00000 −0.131733
\(923\) 8.00000 0.263323
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) 6.00000 0.197066
\(928\) −2.00000 −0.0656532
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −4.00000 −0.131024
\(933\) −8.00000 −0.261908
\(934\) −16.0000 −0.523536
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 16.0000 0.522697 0.261349 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) 16.0000 0.522419
\(939\) −8.00000 −0.261070
\(940\) 0 0
\(941\) 40.0000 1.30396 0.651981 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 10.0000 0.325818
\(943\) 36.0000 1.17232
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 8.00000 0.259828
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 16.0000 0.518563
\(953\) −48.0000 −1.55487 −0.777436 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 4.00000 0.129302
\(958\) −12.0000 −0.387702
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −6.00000 −0.193448
\(963\) −8.00000 −0.257796
\(964\) −30.0000 −0.966235
\(965\) 0 0
\(966\) 24.0000 0.772187
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) −7.00000 −0.224989
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 64.0000 2.05175
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 4.00000 0.127906
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 8.00000 0.255290
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) −32.0000 −1.01857
\(988\) 2.00000 0.0636285
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −4.00000 −0.127000
\(993\) −18.0000 −0.571213
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 30.0000 0.949633
\(999\) 6.00000 0.189832
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 1950.2.a.u.1.1 1
3.2 odd 2 5850.2.a.x.1.1 1
5.2 odd 4 390.2.e.b.79.2 yes 2
5.3 odd 4 390.2.e.b.79.1 2
5.4 even 2 1950.2.a.g.1.1 1
15.2 even 4 1170.2.e.b.469.1 2
15.8 even 4 1170.2.e.b.469.2 2
15.14 odd 2 5850.2.a.bd.1.1 1
20.3 even 4 3120.2.l.h.1249.2 2
20.7 even 4 3120.2.l.h.1249.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.b.79.1 2 5.3 odd 4
390.2.e.b.79.2 yes 2 5.2 odd 4
1170.2.e.b.469.1 2 15.2 even 4
1170.2.e.b.469.2 2 15.8 even 4
1950.2.a.g.1.1 1 5.4 even 2
1950.2.a.u.1.1 1 1.1 even 1 trivial
3120.2.l.h.1249.1 2 20.7 even 4
3120.2.l.h.1249.2 2 20.3 even 4
5850.2.a.x.1.1 1 3.2 odd 2
5850.2.a.bd.1.1 1 15.14 odd 2