Properties

Label 4730.2.a.a.1.1
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} +2.00000 q^{29} +2.00000 q^{30} -1.00000 q^{32} -2.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} +2.00000 q^{47} -2.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -12.0000 q^{51} +2.00000 q^{52} +12.0000 q^{53} -4.00000 q^{54} +1.00000 q^{55} -12.0000 q^{57} -2.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} +6.00000 q^{61} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -12.0000 q^{69} +16.0000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} -2.00000 q^{75} +6.00000 q^{76} +4.00000 q^{78} +6.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +6.00000 q^{85} -1.00000 q^{86} -4.00000 q^{87} -1.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} +6.00000 q^{92} -2.00000 q^{94} +6.00000 q^{95} +2.00000 q^{96} -2.00000 q^{97} +7.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −6.00000 −0.973329
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) −6.00000 −0.884652
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) −2.00000 −0.288675
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) −12.0000 −1.68034
\(52\) 2.00000 0.277350
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −4.00000 −0.544331
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −2.00000 −0.258199
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) −2.00000 −0.230940
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) −1.00000 −0.107833
\(87\) −4.00000 −0.428845
\(88\) −1.00000 −0.106600
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 6.00000 0.615587
\(96\) 2.00000 0.204124
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 7.00000 0.707107
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 12.0000 1.18818
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 4.00000 0.384900
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 12.0000 1.12390
\(115\) 6.00000 0.559503
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −6.00000 −0.543214
\(123\) 12.0000 1.08200
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) −2.00000 −0.175412
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 4.00000 0.344265
\(136\) −6.00000 −0.514496
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 12.0000 1.02151
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) −16.0000 −1.34269
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) −2.00000 −0.165521
\(147\) 14.0000 1.15470
\(148\) −2.00000 −0.164399
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 2.00000 0.163299
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −6.00000 −0.486664
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −6.00000 −0.477334
\(159\) −24.0000 −1.90332
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) −18.0000 −1.40987 −0.704934 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) −6.00000 −0.468521
\(165\) −2.00000 −0.155700
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 6.00000 0.458831
\(172\) 1.00000 0.0762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 24.0000 1.80395
\(178\) 2.00000 0.149906
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) −6.00000 −0.442326
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −2.00000 −0.144338
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) −7.00000 −0.500000
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) −6.00000 −0.419058
\(206\) −6.00000 −0.418040
\(207\) 6.00000 0.417029
\(208\) 2.00000 0.138675
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 12.0000 0.824163
\(213\) −32.0000 −2.19260
\(214\) −12.0000 −0.820303
\(215\) 1.00000 0.0681994
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 1.00000 0.0674200
\(221\) 12.0000 0.807207
\(222\) −4.00000 −0.268462
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −12.0000 −0.794719
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −2.00000 −0.130744
\(235\) 2.00000 0.130466
\(236\) −12.0000 −0.781133
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) −2.00000 −0.129099
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 10.0000 0.641500
\(244\) 6.00000 0.384111
\(245\) −7.00000 −0.447214
\(246\) −12.0000 −0.765092
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −12.0000 −0.752947
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 2.00000 0.123797
\(262\) 6.00000 0.370681
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 2.00000 0.123091
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) −4.00000 −0.244339
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −4.00000 −0.243432
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 1.00000 0.0603023
\(276\) −12.0000 −0.722315
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 4.00000 0.238197
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 16.0000 0.949425
\(285\) −12.0000 −0.710819
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 4.00000 0.234484
\(292\) 2.00000 0.117041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −14.0000 −0.816497
\(295\) −12.0000 −0.698667
\(296\) 2.00000 0.116248
\(297\) 4.00000 0.232104
\(298\) 14.0000 0.810998
\(299\) 12.0000 0.693978
\(300\) −2.00000 −0.115470
\(301\) 0 0
\(302\) 12.0000 0.690522
\(303\) 24.0000 1.37876
\(304\) 6.00000 0.344124
\(305\) 6.00000 0.343559
\(306\) −6.00000 −0.342997
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 4.00000 0.226455
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) 20.0000 1.12331 0.561656 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(318\) 24.0000 1.34585
\(319\) 2.00000 0.111979
\(320\) 1.00000 0.0559017
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) 36.0000 2.00309
\(324\) −11.0000 −0.611111
\(325\) 2.00000 0.110940
\(326\) 18.0000 0.996928
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 2.00000 0.110096
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) −12.0000 −0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 9.00000 0.489535
\(339\) 24.0000 1.30350
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) 0 0
\(344\) −1.00000 −0.0539164
\(345\) −12.0000 −0.646058
\(346\) −6.00000 −0.322562
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −4.00000 −0.214423
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) −1.00000 −0.0533002
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −24.0000 −1.27559
\(355\) 16.0000 0.849192
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −2.00000 −0.105556 −0.0527780 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) −6.00000 −0.315353
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 12.0000 0.627250
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 6.00000 0.312772
\(369\) −6.00000 −0.312348
\(370\) 2.00000 0.103975
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −6.00000 −0.310253
\(375\) −2.00000 −0.103280
\(376\) −2.00000 −0.103142
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 6.00000 0.307794
\(381\) −24.0000 −1.22956
\(382\) 0 0
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 1.00000 0.0508329
\(388\) −2.00000 −0.101535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 4.00000 0.202548
\(391\) 36.0000 1.82060
\(392\) 7.00000 0.353553
\(393\) 12.0000 0.605320
\(394\) −6.00000 −0.302276
\(395\) 6.00000 0.301893
\(396\) 1.00000 0.0502519
\(397\) 24.0000 1.20453 0.602263 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 12.0000 0.594089
\(409\) −24.0000 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 6.00000 0.296319
\(411\) 24.0000 1.18383
\(412\) 6.00000 0.295599
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −4.00000 −0.196352
\(416\) −2.00000 −0.0980581
\(417\) 16.0000 0.783523
\(418\) −6.00000 −0.293470
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −10.0000 −0.486792
\(423\) 2.00000 0.0972433
\(424\) −12.0000 −0.582772
\(425\) 6.00000 0.291043
\(426\) 32.0000 1.55041
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −4.00000 −0.193122
\(430\) −1.00000 −0.0482243
\(431\) −22.0000 −1.05970 −0.529851 0.848091i \(-0.677752\pi\)
−0.529851 + 0.848091i \(0.677752\pi\)
\(432\) 4.00000 0.192450
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 0 0
\(437\) 36.0000 1.72211
\(438\) 4.00000 0.191127
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −7.00000 −0.333333
\(442\) −12.0000 −0.570782
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 4.00000 0.189832
\(445\) −2.00000 −0.0948091
\(446\) 0 0
\(447\) 28.0000 1.32435
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −6.00000 −0.282529
\(452\) −12.0000 −0.564433
\(453\) 24.0000 1.12762
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −10.0000 −0.467269
\(459\) 24.0000 1.12022
\(460\) 6.00000 0.279751
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) −2.00000 −0.0922531
\(471\) −36.0000 −1.65879
\(472\) 12.0000 0.552345
\(473\) 1.00000 0.0459800
\(474\) 12.0000 0.551178
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) −18.0000 −0.823301
\(479\) −10.0000 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 2.00000 0.0912871
\(481\) −4.00000 −0.182384
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.00000 −0.0908153
\(486\) −10.0000 −0.453609
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) −6.00000 −0.271607
\(489\) 36.0000 1.62798
\(490\) 7.00000 0.316228
\(491\) 14.0000 0.631811 0.315906 0.948791i \(-0.397692\pi\)
0.315906 + 0.948791i \(0.397692\pi\)
\(492\) 12.0000 0.541002
\(493\) 12.0000 0.540453
\(494\) −12.0000 −0.539906
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) −24.0000 −1.07224
\(502\) 12.0000 0.535586
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) −6.00000 −0.266733
\(507\) 18.0000 0.799408
\(508\) 12.0000 0.532414
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 12.0000 0.531369
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 24.0000 1.05963
\(514\) 8.00000 0.352865
\(515\) 6.00000 0.264392
\(516\) −2.00000 −0.0880451
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) −2.00000 −0.0877058
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) −12.0000 −0.521247
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −4.00000 −0.173097
\(535\) 12.0000 0.518805
\(536\) 4.00000 0.172774
\(537\) 48.0000 2.07135
\(538\) 18.0000 0.776035
\(539\) −7.00000 −0.301511
\(540\) 4.00000 0.172133
\(541\) 4.00000 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) −22.0000 −0.944981
\(543\) −12.0000 −0.514969
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −12.0000 −0.512615
\(549\) 6.00000 0.256074
\(550\) −1.00000 −0.0426401
\(551\) 12.0000 0.511217
\(552\) 12.0000 0.510754
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 4.00000 0.169791
\(556\) −8.00000 −0.339276
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) −2.00000 −0.0843649
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −4.00000 −0.168430
\(565\) −12.0000 −0.504844
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) −16.0000 −0.671345
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 12.0000 0.502625
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) −19.0000 −0.790296
\(579\) −4.00000 −0.166234
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) −4.00000 −0.165805
\(583\) 12.0000 0.496989
\(584\) −2.00000 −0.0827606
\(585\) 2.00000 0.0826898
\(586\) 6.00000 0.247858
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 14.0000 0.577350
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) −12.0000 −0.493614
\(592\) −2.00000 −0.0821995
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) −32.0000 −1.30967
\(598\) −12.0000 −0.490716
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 2.00000 0.0816497
\(601\) 24.0000 0.978980 0.489490 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −12.0000 −0.488273
\(605\) 1.00000 0.0406558
\(606\) −24.0000 −0.974933
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 4.00000 0.161823
\(612\) 6.00000 0.242536
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −28.0000 −1.12999
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 12.0000 0.482711
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 24.0000 0.963087
\(622\) −16.0000 −0.641542
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 24.0000 0.959233
\(627\) −12.0000 −0.479234
\(628\) 18.0000 0.718278
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −6.00000 −0.238667
\(633\) −20.0000 −0.794929
\(634\) −20.0000 −0.794301
\(635\) 12.0000 0.476205
\(636\) −24.0000 −0.951662
\(637\) −14.0000 −0.554700
\(638\) −2.00000 −0.0791808
\(639\) 16.0000 0.632950
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 24.0000 0.947204
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) −36.0000 −1.41640
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 11.0000 0.432121
\(649\) −12.0000 −0.471041
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −18.0000 −0.704934
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) −12.0000 −0.466393
\(663\) −24.0000 −0.932083
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 12.0000 0.464642
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 30.0000 1.15556
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) −24.0000 −0.921714
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) 32.0000 1.22445 0.612223 0.790685i \(-0.290275\pi\)
0.612223 + 0.790685i \(0.290275\pi\)
\(684\) 6.00000 0.229416
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) −20.0000 −0.763048
\(688\) 1.00000 0.0381246
\(689\) 24.0000 0.914327
\(690\) 12.0000 0.456832
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) −8.00000 −0.303457
\(696\) 4.00000 0.151620
\(697\) −36.0000 −1.36360
\(698\) −10.0000 −0.378506
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) −8.00000 −0.301941
\(703\) −12.0000 −0.452589
\(704\) 1.00000 0.0376889
\(705\) −4.00000 −0.150649
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 24.0000 0.901975
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −16.0000 −0.600469
\(711\) 6.00000 0.225018
\(712\) 2.00000 0.0749532
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −24.0000 −0.896922
\(717\) −36.0000 −1.34444
\(718\) 2.00000 0.0746393
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 8.00000 0.297523
\(724\) 6.00000 0.222988
\(725\) 2.00000 0.0742781
\(726\) 2.00000 0.0742270
\(727\) 20.0000 0.741759 0.370879 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −2.00000 −0.0740233
\(731\) 6.00000 0.221918
\(732\) −12.0000 −0.443533
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 2.00000 0.0738213
\(735\) 14.0000 0.516398
\(736\) −6.00000 −0.221163
\(737\) −4.00000 −0.147342
\(738\) 6.00000 0.220863
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 14.0000 0.512576
\(747\) −4.00000 −0.146352
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 2.00000 0.0730297
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 2.00000 0.0729325
\(753\) 24.0000 0.874609
\(754\) −4.00000 −0.145671
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −28.0000 −1.01701
\(759\) −12.0000 −0.435572
\(760\) −6.00000 −0.217643
\(761\) −32.0000 −1.16000 −0.580000 0.814617i \(-0.696947\pi\)
−0.580000 + 0.814617i \(0.696947\pi\)
\(762\) 24.0000 0.869428
\(763\) 0 0
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 28.0000 1.01168
\(767\) −24.0000 −0.866590
\(768\) −2.00000 −0.0721688
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 16.0000 0.576226
\(772\) 2.00000 0.0719816
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −36.0000 −1.28983
\(780\) −4.00000 −0.143223
\(781\) 16.0000 0.572525
\(782\) −36.0000 −1.28736
\(783\) 8.00000 0.285897
\(784\) −7.00000 −0.250000
\(785\) 18.0000 0.642448
\(786\) −12.0000 −0.428026
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 6.00000 0.213741
\(789\) −32.0000 −1.13923
\(790\) −6.00000 −0.213470
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 12.0000 0.426132
\(794\) −24.0000 −0.851728
\(795\) −24.0000 −0.851192
\(796\) 16.0000 0.567105
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −18.0000 −0.635602
\(803\) 2.00000 0.0705785
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) 36.0000 1.26726
\(808\) 12.0000 0.422159
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 11.0000 0.386501
\(811\) −46.0000 −1.61528 −0.807639 0.589677i \(-0.799255\pi\)
−0.807639 + 0.589677i \(0.799255\pi\)
\(812\) 0 0
\(813\) −44.0000 −1.54315
\(814\) 2.00000 0.0701000
\(815\) −18.0000 −0.630512
\(816\) −12.0000 −0.420084
\(817\) 6.00000 0.209913
\(818\) 24.0000 0.839140
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) −24.0000 −0.837096
\(823\) −26.0000 −0.906303 −0.453152 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) −6.00000 −0.209020
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 6.00000 0.208514
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 4.00000 0.138842
\(831\) 4.00000 0.138758
\(832\) 2.00000 0.0693375
\(833\) −42.0000 −1.45521
\(834\) −16.0000 −0.554035
\(835\) 12.0000 0.415277
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 0 0
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −10.0000 −0.344623
\(843\) −4.00000 −0.137767
\(844\) 10.0000 0.344214
\(845\) −9.00000 −0.309609
\(846\) −2.00000 −0.0687614
\(847\) 0 0
\(848\) 12.0000 0.412082
\(849\) −40.0000 −1.37280
\(850\) −6.00000 −0.205798
\(851\) −12.0000 −0.411355
\(852\) −32.0000 −1.09630
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) −12.0000 −0.410152
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 4.00000 0.136558
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) 22.0000 0.749323
\(863\) −28.0000 −0.953131 −0.476566 0.879139i \(-0.658119\pi\)
−0.476566 + 0.879139i \(0.658119\pi\)
\(864\) −4.00000 −0.136083
\(865\) 6.00000 0.204006
\(866\) 8.00000 0.271851
\(867\) −38.0000 −1.29055
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 4.00000 0.135613
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) −36.0000 −1.21772
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 22.0000 0.742464
\(879\) 12.0000 0.404750
\(880\) 1.00000 0.0337100
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 7.00000 0.235702
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 12.0000 0.403604
\(885\) 24.0000 0.806751
\(886\) −16.0000 −0.537531
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −4.00000 −0.134231
\(889\) 0 0
\(890\) 2.00000 0.0670402
\(891\) −11.0000 −0.368514
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) −28.0000 −0.936460
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) −2.00000 −0.0667409
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 72.0000 2.39867
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) 6.00000 0.199447
\(906\) −24.0000 −0.797347
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 4.00000 0.132745
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) −12.0000 −0.397360
\(913\) −4.00000 −0.132381
\(914\) −6.00000 −0.198462
\(915\) −12.0000 −0.396708
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) −24.0000 −0.792118
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) −6.00000 −0.197814
\(921\) −56.0000 −1.84526
\(922\) 32.0000 1.05386
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −20.0000 −0.657241
\(927\) 6.00000 0.197066
\(928\) −2.00000 −0.0656532
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) −10.0000 −0.327561
\(933\) −32.0000 −1.04763
\(934\) 6.00000 0.196326
\(935\) 6.00000 0.196221
\(936\) −2.00000 −0.0653720
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 48.0000 1.56642
\(940\) 2.00000 0.0652328
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) 36.0000 1.17294
\(943\) −36.0000 −1.17232
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −1.00000 −0.0325128
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) −12.0000 −0.389742
\(949\) 4.00000 0.129845
\(950\) −6.00000 −0.194666
\(951\) −40.0000 −1.29709
\(952\) 0 0
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) 18.0000 0.582162
\(957\) −4.00000 −0.129302
\(958\) 10.0000 0.323085
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) 12.0000 0.386695
\(964\) −4.00000 −0.128831
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −72.0000 −2.31297
\(970\) 2.00000 0.0642161
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) −34.0000 −1.08943
\(975\) −4.00000 −0.128103
\(976\) 6.00000 0.192055
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −36.0000 −1.15115
\(979\) −2.00000 −0.0639203
\(980\) −7.00000 −0.223607
\(981\) 0 0
\(982\) −14.0000 −0.446758
\(983\) −8.00000 −0.255160 −0.127580 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) −12.0000 −0.382546
\(985\) 6.00000 0.191176
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 6.00000 0.190789
\(990\) −1.00000 −0.0317821
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 8.00000 0.253490
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −20.0000 −0.633089
\(999\) −8.00000 −0.253109
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 4730.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.a.1.1 1 1.1 even 1 trivial