Properties

Label 5070.2.a.d.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
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 \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} +5.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +5.00000 q^{29} +1.00000 q^{30} -11.0000 q^{31} -1.00000 q^{32} +5.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -11.0000 q^{43} -5.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} -5.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} -5.00000 q^{58} -15.0000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +11.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +16.0000 q^{67} -2.00000 q^{68} +1.00000 q^{69} +2.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} -3.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} +10.0000 q^{77} -11.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -2.00000 q^{85} +11.0000 q^{86} -5.00000 q^{87} +5.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} -1.00000 q^{92} +11.0000 q^{93} -9.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) 5.00000 1.06600
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 1.00000 0.182574
\(31\) −11.0000 −1.97566 −0.987829 0.155543i \(-0.950287\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.00000 0.870388
\(34\) 2.00000 0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −2.00000 −0.308607
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) −5.00000 −0.753778
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) −5.00000 −0.674200
\(56\) 2.00000 0.267261
\(57\) 2.00000 0.264906
\(58\) −5.00000 −0.656532
\(59\) −15.0000 −1.95283 −0.976417 0.215894i \(-0.930733\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 11.0000 1.39700
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −2.00000 −0.242536
\(69\) 1.00000 0.120386
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −3.00000 −0.348743
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) 10.0000 1.13961
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 2.00000 0.218218
\(85\) −2.00000 −0.216930
\(86\) 11.0000 1.18616
\(87\) −5.00000 −0.536056
\(88\) 5.00000 0.533002
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 11.0000 1.14065
\(94\) −9.00000 −0.928279
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 0.303046
\(99\) −5.00000 −0.502519
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −2.00000 −0.198030
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) −6.00000 −0.582772
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 5.00000 0.476731
\(111\) −3.00000 −0.284747
\(112\) −2.00000 −0.188982
\(113\) 11.0000 1.03479 0.517396 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) −2.00000 −0.187317
\(115\) −1.00000 −0.0932505
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) 15.0000 1.38086
\(119\) 4.00000 0.366679
\(120\) 1.00000 0.0912871
\(121\) 14.0000 1.27273
\(122\) −10.0000 −0.905357
\(123\) 2.00000 0.180334
\(124\) −11.0000 −0.987829
\(125\) 1.00000 0.0894427
\(126\) 2.00000 0.178174
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 5.00000 0.435194
\(133\) 4.00000 0.346844
\(134\) −16.0000 −1.38219
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) −11.0000 −0.939793 −0.469897 0.882721i \(-0.655709\pi\)
−0.469897 + 0.882721i \(0.655709\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) −2.00000 −0.169031
\(141\) −9.00000 −0.757937
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 5.00000 0.415227
\(146\) 6.00000 0.496564
\(147\) 3.00000 0.247436
\(148\) 3.00000 0.246598
\(149\) 17.0000 1.39269 0.696347 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) −2.00000 −0.161690
\(154\) −10.0000 −0.805823
\(155\) −11.0000 −0.883541
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 11.0000 0.875113
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 2.00000 0.157622
\(162\) −1.00000 −0.0785674
\(163\) −15.0000 −1.17489 −0.587445 0.809264i \(-0.699866\pi\)
−0.587445 + 0.809264i \(0.699866\pi\)
\(164\) −2.00000 −0.156174
\(165\) 5.00000 0.389249
\(166\) −6.00000 −0.465690
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) 2.00000 0.153393
\(171\) −2.00000 −0.152944
\(172\) −11.0000 −0.838742
\(173\) 20.0000 1.52057 0.760286 0.649589i \(-0.225059\pi\)
0.760286 + 0.649589i \(0.225059\pi\)
\(174\) 5.00000 0.379049
\(175\) −2.00000 −0.151186
\(176\) −5.00000 −0.376889
\(177\) 15.0000 1.12747
\(178\) −2.00000 −0.149906
\(179\) −13.0000 −0.971666 −0.485833 0.874052i \(-0.661484\pi\)
−0.485833 + 0.874052i \(0.661484\pi\)
\(180\) 1.00000 0.0745356
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 1.00000 0.0737210
\(185\) 3.00000 0.220564
\(186\) −11.0000 −0.806559
\(187\) 10.0000 0.731272
\(188\) 9.00000 0.656392
\(189\) 2.00000 0.145479
\(190\) 2.00000 0.145095
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −24.0000 −1.72756 −0.863779 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 5.00000 0.355335
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −16.0000 −1.12855
\(202\) −2.00000 −0.140720
\(203\) −10.0000 −0.701862
\(204\) 2.00000 0.140028
\(205\) −2.00000 −0.139686
\(206\) −10.0000 −0.696733
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −10.0000 −0.683586
\(215\) −11.0000 −0.750194
\(216\) 1.00000 0.0680414
\(217\) 22.0000 1.49346
\(218\) 2.00000 0.135457
\(219\) 6.00000 0.405442
\(220\) −5.00000 −0.337100
\(221\) 0 0
\(222\) 3.00000 0.201347
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) −11.0000 −0.731709
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 2.00000 0.132453
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 1.00000 0.0659380
\(231\) −10.0000 −0.657952
\(232\) −5.00000 −0.328266
\(233\) 1.00000 0.0655122 0.0327561 0.999463i \(-0.489572\pi\)
0.0327561 + 0.999463i \(0.489572\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) −15.0000 −0.976417
\(237\) 11.0000 0.714527
\(238\) −4.00000 −0.259281
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) −14.0000 −0.899954
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −3.00000 −0.191663
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 11.0000 0.698501
\(249\) −6.00000 −0.380235
\(250\) −1.00000 −0.0632456
\(251\) 25.0000 1.57799 0.788993 0.614402i \(-0.210603\pi\)
0.788993 + 0.614402i \(0.210603\pi\)
\(252\) −2.00000 −0.125988
\(253\) 5.00000 0.314347
\(254\) −2.00000 −0.125491
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −17.0000 −1.06043 −0.530215 0.847863i \(-0.677889\pi\)
−0.530215 + 0.847863i \(0.677889\pi\)
\(258\) −11.0000 −0.684830
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 1.00000 0.0617802
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) −5.00000 −0.307729
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) −2.00000 −0.122398
\(268\) 16.0000 0.977356
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 1.00000 0.0608581
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 11.0000 0.664534
\(275\) −5.00000 −0.301511
\(276\) 1.00000 0.0601929
\(277\) −11.0000 −0.660926 −0.330463 0.943819i \(-0.607205\pi\)
−0.330463 + 0.943819i \(0.607205\pi\)
\(278\) −2.00000 −0.119952
\(279\) −11.0000 −0.658553
\(280\) 2.00000 0.119523
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 9.00000 0.535942
\(283\) 19.0000 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −5.00000 −0.293610
\(291\) 2.00000 0.117242
\(292\) −6.00000 −0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −3.00000 −0.174964
\(295\) −15.0000 −0.873334
\(296\) −3.00000 −0.174371
\(297\) 5.00000 0.290129
\(298\) −17.0000 −0.984784
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 22.0000 1.26806
\(302\) −8.00000 −0.460348
\(303\) −2.00000 −0.114897
\(304\) −2.00000 −0.114708
\(305\) 10.0000 0.572598
\(306\) 2.00000 0.114332
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 10.0000 0.569803
\(309\) −10.0000 −0.568880
\(310\) 11.0000 0.624758
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 7.00000 0.395033
\(315\) −2.00000 −0.112687
\(316\) −11.0000 −0.618798
\(317\) 16.0000 0.898650 0.449325 0.893368i \(-0.351665\pi\)
0.449325 + 0.893368i \(0.351665\pi\)
\(318\) 6.00000 0.336463
\(319\) −25.0000 −1.39973
\(320\) 1.00000 0.0559017
\(321\) −10.0000 −0.558146
\(322\) −2.00000 −0.111456
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 15.0000 0.830773
\(327\) 2.00000 0.110600
\(328\) 2.00000 0.110432
\(329\) −18.0000 −0.992372
\(330\) −5.00000 −0.275241
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 6.00000 0.329293
\(333\) 3.00000 0.164399
\(334\) −3.00000 −0.164153
\(335\) 16.0000 0.874173
\(336\) 2.00000 0.109109
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) −11.0000 −0.597438
\(340\) −2.00000 −0.108465
\(341\) 55.0000 2.97842
\(342\) 2.00000 0.108148
\(343\) 20.0000 1.07990
\(344\) 11.0000 0.593080
\(345\) 1.00000 0.0538382
\(346\) −20.0000 −1.07521
\(347\) 34.0000 1.82522 0.912608 0.408836i \(-0.134065\pi\)
0.912608 + 0.408836i \(0.134065\pi\)
\(348\) −5.00000 −0.268028
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −15.0000 −0.797241
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) −4.00000 −0.211702
\(358\) 13.0000 0.687071
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) 16.0000 0.840941
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 10.0000 0.522708
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) −3.00000 −0.155963
\(371\) −12.0000 −0.623009
\(372\) 11.0000 0.570323
\(373\) −19.0000 −0.983783 −0.491891 0.870657i \(-0.663694\pi\)
−0.491891 + 0.870657i \(0.663694\pi\)
\(374\) −10.0000 −0.517088
\(375\) −1.00000 −0.0516398
\(376\) −9.00000 −0.464140
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) −2.00000 −0.102598
\(381\) −2.00000 −0.102463
\(382\) −4.00000 −0.204658
\(383\) 31.0000 1.58403 0.792013 0.610504i \(-0.209033\pi\)
0.792013 + 0.610504i \(0.209033\pi\)
\(384\) 1.00000 0.0510310
\(385\) 10.0000 0.509647
\(386\) 24.0000 1.22157
\(387\) −11.0000 −0.559161
\(388\) −2.00000 −0.101535
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 3.00000 0.151523
\(393\) 1.00000 0.0504433
\(394\) 8.00000 0.403034
\(395\) −11.0000 −0.553470
\(396\) −5.00000 −0.251259
\(397\) 3.00000 0.150566 0.0752828 0.997162i \(-0.476014\pi\)
0.0752828 + 0.997162i \(0.476014\pi\)
\(398\) −24.0000 −1.20301
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 16.0000 0.798007
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 1.00000 0.0496904
\(406\) 10.0000 0.496292
\(407\) −15.0000 −0.743522
\(408\) −2.00000 −0.0990148
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 2.00000 0.0987730
\(411\) 11.0000 0.542590
\(412\) 10.0000 0.492665
\(413\) 30.0000 1.47620
\(414\) 1.00000 0.0491473
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) −2.00000 −0.0979404
\(418\) −10.0000 −0.489116
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 2.00000 0.0975900
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) 4.00000 0.194717
\(423\) 9.00000 0.437595
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 10.0000 0.483368
\(429\) 0 0
\(430\) 11.0000 0.530467
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) −22.0000 −1.05603
\(435\) −5.00000 −0.239732
\(436\) −2.00000 −0.0957826
\(437\) 2.00000 0.0956730
\(438\) −6.00000 −0.286691
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 5.00000 0.238366
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −3.00000 −0.142374
\(445\) 2.00000 0.0948091
\(446\) −26.0000 −1.23114
\(447\) −17.0000 −0.804072
\(448\) −2.00000 −0.0944911
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 10.0000 0.470882
\(452\) 11.0000 0.517396
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −10.0000 −0.467269
\(459\) 2.00000 0.0933520
\(460\) −1.00000 −0.0466252
\(461\) −39.0000 −1.81641 −0.908206 0.418524i \(-0.862547\pi\)
−0.908206 + 0.418524i \(0.862547\pi\)
\(462\) 10.0000 0.465242
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 5.00000 0.232119
\(465\) 11.0000 0.510113
\(466\) −1.00000 −0.0463241
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) −9.00000 −0.415139
\(471\) 7.00000 0.322543
\(472\) 15.0000 0.690431
\(473\) 55.0000 2.52890
\(474\) −11.0000 −0.505247
\(475\) −2.00000 −0.0917663
\(476\) 4.00000 0.183340
\(477\) 6.00000 0.274721
\(478\) 20.0000 0.914779
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 7.00000 0.318841
\(483\) −2.00000 −0.0910032
\(484\) 14.0000 0.636364
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −10.0000 −0.452679
\(489\) 15.0000 0.678323
\(490\) 3.00000 0.135526
\(491\) −16.0000 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 2.00000 0.0901670
\(493\) −10.0000 −0.450377
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) −11.0000 −0.493915
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 1.00000 0.0447214
\(501\) −3.00000 −0.134030
\(502\) −25.0000 −1.11580
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 2.00000 0.0890871
\(505\) 2.00000 0.0889988
\(506\) −5.00000 −0.222277
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 29.0000 1.28540 0.642701 0.766117i \(-0.277814\pi\)
0.642701 + 0.766117i \(0.277814\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 12.0000 0.530849
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 17.0000 0.749838
\(515\) 10.0000 0.440653
\(516\) 11.0000 0.484248
\(517\) −45.0000 −1.97910
\(518\) 6.00000 0.263625
\(519\) −20.0000 −0.877903
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −5.00000 −0.218844
\(523\) −31.0000 −1.35554 −0.677768 0.735276i \(-0.737052\pi\)
−0.677768 + 0.735276i \(0.737052\pi\)
\(524\) −1.00000 −0.0436852
\(525\) 2.00000 0.0872872
\(526\) −21.0000 −0.915644
\(527\) 22.0000 0.958335
\(528\) 5.00000 0.217597
\(529\) −22.0000 −0.956522
\(530\) −6.00000 −0.260623
\(531\) −15.0000 −0.650945
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) 2.00000 0.0865485
\(535\) 10.0000 0.432338
\(536\) −16.0000 −0.691095
\(537\) 13.0000 0.560991
\(538\) −14.0000 −0.603583
\(539\) 15.0000 0.646096
\(540\) −1.00000 −0.0430331
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 13.0000 0.558398
\(543\) 16.0000 0.686626
\(544\) 2.00000 0.0857493
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −11.0000 −0.469897
\(549\) 10.0000 0.426790
\(550\) 5.00000 0.213201
\(551\) −10.0000 −0.426014
\(552\) −1.00000 −0.0425628
\(553\) 22.0000 0.935535
\(554\) 11.0000 0.467345
\(555\) −3.00000 −0.127343
\(556\) 2.00000 0.0848189
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 11.0000 0.465667
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) −10.0000 −0.422200
\(562\) 10.0000 0.421825
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −9.00000 −0.378968
\(565\) 11.0000 0.462773
\(566\) −19.0000 −0.798630
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) −4.00000 −0.166957
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 13.0000 0.540729
\(579\) 24.0000 0.997406
\(580\) 5.00000 0.207614
\(581\) −12.0000 −0.497844
\(582\) −2.00000 −0.0829027
\(583\) −30.0000 −1.24247
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 3.00000 0.123718
\(589\) 22.0000 0.906494
\(590\) 15.0000 0.617540
\(591\) 8.00000 0.329076
\(592\) 3.00000 0.123299
\(593\) 31.0000 1.27302 0.636509 0.771270i \(-0.280378\pi\)
0.636509 + 0.771270i \(0.280378\pi\)
\(594\) −5.00000 −0.205152
\(595\) 4.00000 0.163984
\(596\) 17.0000 0.696347
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 1.00000 0.0408248
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) −22.0000 −0.896653
\(603\) 16.0000 0.651570
\(604\) 8.00000 0.325515
\(605\) 14.0000 0.569181
\(606\) 2.00000 0.0812444
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 2.00000 0.0811107
\(609\) 10.0000 0.405220
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −29.0000 −1.17130 −0.585649 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) −10.0000 −0.402911
\(617\) 41.0000 1.65060 0.825299 0.564696i \(-0.191007\pi\)
0.825299 + 0.564696i \(0.191007\pi\)
\(618\) 10.0000 0.402259
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −11.0000 −0.441771
\(621\) 1.00000 0.0401286
\(622\) −20.0000 −0.801927
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −20.0000 −0.799361
\(627\) −10.0000 −0.399362
\(628\) −7.00000 −0.279330
\(629\) −6.00000 −0.239236
\(630\) 2.00000 0.0796819
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 11.0000 0.437557
\(633\) 4.00000 0.158986
\(634\) −16.0000 −0.635441
\(635\) 2.00000 0.0793676
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 25.0000 0.989759
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 10.0000 0.394669
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 2.00000 0.0788110
\(645\) 11.0000 0.433125
\(646\) −4.00000 −0.157378
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 75.0000 2.94401
\(650\) 0 0
\(651\) −22.0000 −0.862248
\(652\) −15.0000 −0.587445
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −1.00000 −0.0390732
\(656\) −2.00000 −0.0780869
\(657\) −6.00000 −0.234082
\(658\) 18.0000 0.701713
\(659\) −39.0000 −1.51922 −0.759612 0.650376i \(-0.774611\pi\)
−0.759612 + 0.650376i \(0.774611\pi\)
\(660\) 5.00000 0.194625
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 4.00000 0.155113
\(666\) −3.00000 −0.116248
\(667\) −5.00000 −0.193601
\(668\) 3.00000 0.116073
\(669\) −26.0000 −1.00522
\(670\) −16.0000 −0.618134
\(671\) −50.0000 −1.93023
\(672\) −2.00000 −0.0771517
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 2.00000 0.0770371
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −4.00000 −0.153732 −0.0768662 0.997041i \(-0.524491\pi\)
−0.0768662 + 0.997041i \(0.524491\pi\)
\(678\) 11.0000 0.422452
\(679\) 4.00000 0.153506
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) −55.0000 −2.10606
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −11.0000 −0.420288
\(686\) −20.0000 −0.763604
\(687\) −10.0000 −0.381524
\(688\) −11.0000 −0.419371
\(689\) 0 0
\(690\) −1.00000 −0.0380693
\(691\) −30.0000 −1.14125 −0.570627 0.821209i \(-0.693300\pi\)
−0.570627 + 0.821209i \(0.693300\pi\)
\(692\) 20.0000 0.760286
\(693\) 10.0000 0.379869
\(694\) −34.0000 −1.29062
\(695\) 2.00000 0.0758643
\(696\) 5.00000 0.189525
\(697\) 4.00000 0.151511
\(698\) −20.0000 −0.757011
\(699\) −1.00000 −0.0378235
\(700\) −2.00000 −0.0755929
\(701\) 13.0000 0.491003 0.245502 0.969396i \(-0.421047\pi\)
0.245502 + 0.969396i \(0.421047\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) −5.00000 −0.188445
\(705\) −9.00000 −0.338960
\(706\) 18.0000 0.677439
\(707\) −4.00000 −0.150435
\(708\) 15.0000 0.563735
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) −2.00000 −0.0749532
\(713\) 11.0000 0.411953
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −13.0000 −0.485833
\(717\) 20.0000 0.746914
\(718\) −12.0000 −0.447836
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 1.00000 0.0372678
\(721\) −20.0000 −0.744839
\(722\) 15.0000 0.558242
\(723\) 7.00000 0.260333
\(724\) −16.0000 −0.594635
\(725\) 5.00000 0.185695
\(726\) 14.0000 0.519589
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) 22.0000 0.813699
\(732\) −10.0000 −0.369611
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 16.0000 0.590571
\(735\) 3.00000 0.110657
\(736\) 1.00000 0.0368605
\(737\) −80.0000 −2.94684
\(738\) 2.00000 0.0736210
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 3.00000 0.110282
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 51.0000 1.87101 0.935504 0.353315i \(-0.114946\pi\)
0.935504 + 0.353315i \(0.114946\pi\)
\(744\) −11.0000 −0.403280
\(745\) 17.0000 0.622832
\(746\) 19.0000 0.695639
\(747\) 6.00000 0.219529
\(748\) 10.0000 0.365636
\(749\) −20.0000 −0.730784
\(750\) 1.00000 0.0365148
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 9.00000 0.328196
\(753\) −25.0000 −0.911051
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 2.00000 0.0727393
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 2.00000 0.0726433
\(759\) −5.00000 −0.181489
\(760\) 2.00000 0.0725476
\(761\) 20.0000 0.724999 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 2.00000 0.0724524
\(763\) 4.00000 0.144810
\(764\) 4.00000 0.144715
\(765\) −2.00000 −0.0723102
\(766\) −31.0000 −1.12008
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 43.0000 1.55062 0.775310 0.631581i \(-0.217594\pi\)
0.775310 + 0.631581i \(0.217594\pi\)
\(770\) −10.0000 −0.360375
\(771\) 17.0000 0.612240
\(772\) −24.0000 −0.863779
\(773\) 32.0000 1.15096 0.575480 0.817816i \(-0.304815\pi\)
0.575480 + 0.817816i \(0.304815\pi\)
\(774\) 11.0000 0.395387
\(775\) −11.0000 −0.395132
\(776\) 2.00000 0.0717958
\(777\) 6.00000 0.215249
\(778\) −5.00000 −0.179259
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 0 0
\(782\) −2.00000 −0.0715199
\(783\) −5.00000 −0.178685
\(784\) −3.00000 −0.107143
\(785\) −7.00000 −0.249841
\(786\) −1.00000 −0.0356688
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) −8.00000 −0.284988
\(789\) −21.0000 −0.747620
\(790\) 11.0000 0.391362
\(791\) −22.0000 −0.782230
\(792\) 5.00000 0.177667
\(793\) 0 0
\(794\) −3.00000 −0.106466
\(795\) −6.00000 −0.212798
\(796\) 24.0000 0.850657
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 4.00000 0.141598
\(799\) −18.0000 −0.636794
\(800\) −1.00000 −0.0353553
\(801\) 2.00000 0.0706665
\(802\) −36.0000 −1.27120
\(803\) 30.0000 1.05868
\(804\) −16.0000 −0.564276
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) −2.00000 −0.0703598
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −10.0000 −0.350931
\(813\) 13.0000 0.455930
\(814\) 15.0000 0.525750
\(815\) −15.0000 −0.525427
\(816\) 2.00000 0.0700140
\(817\) 22.0000 0.769683
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −51.0000 −1.77991 −0.889956 0.456046i \(-0.849265\pi\)
−0.889956 + 0.456046i \(0.849265\pi\)
\(822\) −11.0000 −0.383669
\(823\) −2.00000 −0.0697156 −0.0348578 0.999392i \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) −10.0000 −0.348367
\(825\) 5.00000 0.174078
\(826\) −30.0000 −1.04383
\(827\) −50.0000 −1.73867 −0.869335 0.494223i \(-0.835453\pi\)
−0.869335 + 0.494223i \(0.835453\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −6.00000 −0.208263
\(831\) 11.0000 0.381586
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 2.00000 0.0692543
\(835\) 3.00000 0.103819
\(836\) 10.0000 0.345857
\(837\) 11.0000 0.380216
\(838\) 4.00000 0.138178
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −4.00000 −0.137931
\(842\) 16.0000 0.551396
\(843\) 10.0000 0.344418
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) −28.0000 −0.962091
\(848\) 6.00000 0.206041
\(849\) −19.0000 −0.652078
\(850\) 2.00000 0.0685994
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) −49.0000 −1.67773 −0.838864 0.544341i \(-0.816780\pi\)
−0.838864 + 0.544341i \(0.816780\pi\)
\(854\) 20.0000 0.684386
\(855\) −2.00000 −0.0683986
\(856\) −10.0000 −0.341793
\(857\) −25.0000 −0.853984 −0.426992 0.904255i \(-0.640427\pi\)
−0.426992 + 0.904255i \(0.640427\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) −11.0000 −0.375097
\(861\) −4.00000 −0.136320
\(862\) −40.0000 −1.36241
\(863\) 39.0000 1.32758 0.663788 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) 1.00000 0.0340207
\(865\) 20.0000 0.680020
\(866\) −28.0000 −0.951479
\(867\) 13.0000 0.441503
\(868\) 22.0000 0.746729
\(869\) 55.0000 1.86575
\(870\) 5.00000 0.169516
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) −2.00000 −0.0676897
\(874\) −2.00000 −0.0676510
\(875\) −2.00000 −0.0676123
\(876\) 6.00000 0.202721
\(877\) −19.0000 −0.641584 −0.320792 0.947150i \(-0.603949\pi\)
−0.320792 + 0.947150i \(0.603949\pi\)
\(878\) 4.00000 0.134993
\(879\) −6.00000 −0.202375
\(880\) −5.00000 −0.168550
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 3.00000 0.101015
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 0 0
\(885\) 15.0000 0.504219
\(886\) −20.0000 −0.671913
\(887\) 3.00000 0.100730 0.0503651 0.998731i \(-0.483962\pi\)
0.0503651 + 0.998731i \(0.483962\pi\)
\(888\) 3.00000 0.100673
\(889\) −4.00000 −0.134156
\(890\) −2.00000 −0.0670402
\(891\) −5.00000 −0.167506
\(892\) 26.0000 0.870544
\(893\) −18.0000 −0.602347
\(894\) 17.0000 0.568565
\(895\) −13.0000 −0.434542
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) −55.0000 −1.83435
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) −10.0000 −0.332964
\(903\) −22.0000 −0.732114
\(904\) −11.0000 −0.365855
\(905\) −16.0000 −0.531858
\(906\) 8.00000 0.265782
\(907\) 19.0000 0.630885 0.315442 0.948945i \(-0.397847\pi\)
0.315442 + 0.948945i \(0.397847\pi\)
\(908\) 0 0
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 2.00000 0.0662266
\(913\) −30.0000 −0.992855
\(914\) −38.0000 −1.25693
\(915\) −10.0000 −0.330590
\(916\) 10.0000 0.330409
\(917\) 2.00000 0.0660458
\(918\) −2.00000 −0.0660098
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 1.00000 0.0329690
\(921\) 0 0
\(922\) 39.0000 1.28440
\(923\) 0 0
\(924\) −10.0000 −0.328976
\(925\) 3.00000 0.0986394
\(926\) −14.0000 −0.460069
\(927\) 10.0000 0.328443
\(928\) −5.00000 −0.164133
\(929\) −16.0000 −0.524943 −0.262471 0.964940i \(-0.584538\pi\)
−0.262471 + 0.964940i \(0.584538\pi\)
\(930\) −11.0000 −0.360704
\(931\) 6.00000 0.196642
\(932\) 1.00000 0.0327561
\(933\) −20.0000 −0.654771
\(934\) −6.00000 −0.196326
\(935\) 10.0000 0.327035
\(936\) 0 0
\(937\) −50.0000 −1.63343 −0.816714 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) 32.0000 1.04484
\(939\) −20.0000 −0.652675
\(940\) 9.00000 0.293548
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) −7.00000 −0.228072
\(943\) 2.00000 0.0651290
\(944\) −15.0000 −0.488208
\(945\) 2.00000 0.0650600
\(946\) −55.0000 −1.78820
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 11.0000 0.357263
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) −16.0000 −0.518836
\(952\) −4.00000 −0.129641
\(953\) −51.0000 −1.65205 −0.826026 0.563632i \(-0.809404\pi\)
−0.826026 + 0.563632i \(0.809404\pi\)
\(954\) −6.00000 −0.194257
\(955\) 4.00000 0.129437
\(956\) −20.0000 −0.646846
\(957\) 25.0000 0.808135
\(958\) 24.0000 0.775405
\(959\) 22.0000 0.710417
\(960\) −1.00000 −0.0322749
\(961\) 90.0000 2.90323
\(962\) 0 0
\(963\) 10.0000 0.322245
\(964\) −7.00000 −0.225455
\(965\) −24.0000 −0.772587
\(966\) 2.00000 0.0643489
\(967\) 20.0000 0.643157 0.321578 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) −14.0000 −0.449977
\(969\) −4.00000 −0.128499
\(970\) 2.00000 0.0642161
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) −15.0000 −0.479647
\(979\) −10.0000 −0.319601
\(980\) −3.00000 −0.0958315
\(981\) −2.00000 −0.0638551
\(982\) 16.0000 0.510581
\(983\) −3.00000 −0.0956851 −0.0478426 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −8.00000 −0.254901
\(986\) 10.0000 0.318465
\(987\) 18.0000 0.572946
\(988\) 0 0
\(989\) 11.0000 0.349780
\(990\) 5.00000 0.158910
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 11.0000 0.349250
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) −6.00000 −0.190117
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −28.0000 −0.886325
\(999\) −3.00000 −0.0949158
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 5070.2.a.d.1.1 1
13.3 even 3 390.2.i.f.61.1 2
13.5 odd 4 5070.2.b.h.1351.2 2
13.8 odd 4 5070.2.b.h.1351.1 2
13.9 even 3 390.2.i.f.211.1 yes 2
13.12 even 2 5070.2.a.p.1.1 1
39.29 odd 6 1170.2.i.a.451.1 2
39.35 odd 6 1170.2.i.a.991.1 2
65.3 odd 12 1950.2.z.e.1699.1 4
65.9 even 6 1950.2.i.d.601.1 2
65.22 odd 12 1950.2.z.e.1849.1 4
65.29 even 6 1950.2.i.d.451.1 2
65.42 odd 12 1950.2.z.e.1699.2 4
65.48 odd 12 1950.2.z.e.1849.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.f.61.1 2 13.3 even 3
390.2.i.f.211.1 yes 2 13.9 even 3
1170.2.i.a.451.1 2 39.29 odd 6
1170.2.i.a.991.1 2 39.35 odd 6
1950.2.i.d.451.1 2 65.29 even 6
1950.2.i.d.601.1 2 65.9 even 6
1950.2.z.e.1699.1 4 65.3 odd 12
1950.2.z.e.1699.2 4 65.42 odd 12
1950.2.z.e.1849.1 4 65.22 odd 12
1950.2.z.e.1849.2 4 65.48 odd 12
5070.2.a.d.1.1 1 1.1 even 1 trivial
5070.2.a.p.1.1 1 13.12 even 2
5070.2.b.h.1351.1 2 13.8 odd 4
5070.2.b.h.1351.2 2 13.5 odd 4