Properties

Label 1850.2.a.h.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(1\)
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
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{11} -3.00000 q^{12} -2.00000 q^{13} +1.00000 q^{16} -7.00000 q^{17} +6.00000 q^{18} +5.00000 q^{19} -1.00000 q^{22} +6.00000 q^{23} -3.00000 q^{24} -2.00000 q^{26} -9.00000 q^{27} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -7.00000 q^{34} +6.00000 q^{36} +1.00000 q^{37} +5.00000 q^{38} +6.00000 q^{39} -3.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -4.00000 q^{47} -3.00000 q^{48} -7.00000 q^{49} +21.0000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -9.00000 q^{54} -15.0000 q^{57} +4.00000 q^{59} -8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{66} -13.0000 q^{67} -7.00000 q^{68} -18.0000 q^{69} -6.00000 q^{71} +6.00000 q^{72} -7.00000 q^{73} +1.00000 q^{74} +5.00000 q^{76} +6.00000 q^{78} +14.0000 q^{79} +9.00000 q^{81} -3.00000 q^{82} +3.00000 q^{83} -4.00000 q^{86} -1.00000 q^{88} -7.00000 q^{89} +6.00000 q^{92} +12.0000 q^{93} -4.00000 q^{94} -3.00000 q^{96} -18.0000 q^{97} -7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −3.00000 −0.866025
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 6.00000 1.41421
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 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
\(33\) 3.00000 0.522233
\(34\) −7.00000 −1.20049
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 1.00000 0.164399
\(38\) 5.00000 0.811107
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −3.00000 −0.433013
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 21.0000 2.94059
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) 0 0
\(57\) −15.0000 −1.98680
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) −7.00000 −0.848875
\(69\) −18.0000 −2.16695
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 6.00000 0.707107
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −3.00000 −0.331295
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −7.00000 −0.741999 −0.370999 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 12.0000 1.24434
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) −7.00000 −0.707107
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 21.0000 2.07931
\(103\) −18.0000 −1.77359 −0.886796 0.462160i \(-0.847074\pi\)
−0.886796 + 0.462160i \(0.847074\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 5.00000 0.483368 0.241684 0.970355i \(-0.422300\pi\)
0.241684 + 0.970355i \(0.422300\pi\)
\(108\) −9.00000 −0.866025
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) −17.0000 −1.59923 −0.799613 0.600516i \(-0.794962\pi\)
−0.799613 + 0.600516i \(0.794962\pi\)
\(114\) −15.0000 −1.40488
\(115\) 0 0
\(116\) 0 0
\(117\) −12.0000 −1.10940
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −8.00000 −0.724286
\(123\) 9.00000 0.811503
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) −13.0000 −1.12303
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) −18.0000 −1.53226
\(139\) −1.00000 −0.0848189 −0.0424094 0.999100i \(-0.513503\pi\)
−0.0424094 + 0.999100i \(0.513503\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) −6.00000 −0.503509
\(143\) 2.00000 0.167248
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) −7.00000 −0.579324
\(147\) 21.0000 1.73205
\(148\) 1.00000 0.0821995
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 5.00000 0.405554
\(153\) −42.0000 −3.39550
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 12.0000 0.957704 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 14.0000 1.11378
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 3.00000 0.232845
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 30.0000 2.29416
\(172\) −4.00000 −0.304997
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −12.0000 −0.901975
\(178\) −7.00000 −0.524672
\(179\) 13.0000 0.971666 0.485833 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 12.0000 0.879883
\(187\) 7.00000 0.511891
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) −3.00000 −0.216506
\(193\) 25.0000 1.79954 0.899770 0.436365i \(-0.143734\pi\)
0.899770 + 0.436365i \(0.143734\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −6.00000 −0.426401
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 39.0000 2.75085
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 21.0000 1.47029
\(205\) 0 0
\(206\) −18.0000 −1.25412
\(207\) 36.0000 2.50217
\(208\) −2.00000 −0.138675
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 2.00000 0.137361
\(213\) 18.0000 1.23334
\(214\) 5.00000 0.341793
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 21.0000 1.41905
\(220\) 0 0
\(221\) 14.0000 0.941742
\(222\) −3.00000 −0.201347
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −17.0000 −1.13082
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) −15.0000 −0.993399
\(229\) 24.0000 1.58596 0.792982 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −12.0000 −0.784465
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) −42.0000 −2.72819
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) −10.0000 −0.636285
\(248\) −4.00000 −0.254000
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −1.00000 −0.0631194 −0.0315597 0.999502i \(-0.510047\pi\)
−0.0315597 + 0.999502i \(0.510047\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) 14.0000 0.863277 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) 21.0000 1.28518
\(268\) −13.0000 −0.794101
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) 15.0000 0.906183
\(275\) 0 0
\(276\) −18.0000 −1.08347
\(277\) −20.0000 −1.20168 −0.600842 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) −1.00000 −0.0599760
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 12.0000 0.714590
\(283\) −7.00000 −0.416107 −0.208053 0.978117i \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 6.00000 0.353553
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 54.0000 3.16554
\(292\) −7.00000 −0.409644
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 21.0000 1.22474
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 9.00000 0.522233
\(298\) −22.0000 −1.27443
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) −18.0000 −1.03578
\(303\) −18.0000 −1.03407
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) −42.0000 −2.40098
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 54.0000 3.07195
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 6.00000 0.339683
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) −15.0000 −0.837218
\(322\) 0 0
\(323\) −35.0000 −1.94745
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 1.00000 0.0553849
\(327\) 18.0000 0.995402
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 3.00000 0.164646
\(333\) 6.00000 0.328798
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) −31.0000 −1.68868 −0.844339 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) −9.00000 −0.489535
\(339\) 51.0000 2.76994
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 30.0000 1.62221
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 8.00000 0.430083
\(347\) 23.0000 1.23470 0.617352 0.786687i \(-0.288205\pi\)
0.617352 + 0.786687i \(0.288205\pi\)
\(348\) 0 0
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 0 0
\(351\) 18.0000 0.960769
\(352\) −1.00000 −0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −7.00000 −0.370999
\(357\) 0 0
\(358\) 13.0000 0.687071
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −2.00000 −0.105118
\(363\) 30.0000 1.57459
\(364\) 0 0
\(365\) 0 0
\(366\) 24.0000 1.25450
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 6.00000 0.312772
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 7.00000 0.361961
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 6.00000 0.306987
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 25.0000 1.27247
\(387\) −24.0000 −1.21999
\(388\) −18.0000 −0.913812
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) −42.0000 −2.12403
\(392\) −7.00000 −0.353553
\(393\) 36.0000 1.81596
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 23.0000 1.14857 0.574283 0.818657i \(-0.305281\pi\)
0.574283 + 0.818657i \(0.305281\pi\)
\(402\) 39.0000 1.94514
\(403\) 8.00000 0.398508
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −1.00000 −0.0495682
\(408\) 21.0000 1.03965
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) −45.0000 −2.21969
\(412\) −18.0000 −0.886796
\(413\) 0 0
\(414\) 36.0000 1.76930
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 3.00000 0.146911
\(418\) −5.00000 −0.244558
\(419\) −17.0000 −0.830504 −0.415252 0.909706i \(-0.636307\pi\)
−0.415252 + 0.909706i \(0.636307\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) 13.0000 0.632830
\(423\) −24.0000 −1.16692
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 18.0000 0.872103
\(427\) 0 0
\(428\) 5.00000 0.241684
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −38.0000 −1.83040 −0.915198 0.403005i \(-0.867966\pi\)
−0.915198 + 0.403005i \(0.867966\pi\)
\(432\) −9.00000 −0.433013
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 30.0000 1.43509
\(438\) 21.0000 1.00342
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) −42.0000 −2.00000
\(442\) 14.0000 0.665912
\(443\) 15.0000 0.712672 0.356336 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) −3.00000 −0.142374
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 66.0000 3.12169
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) 3.00000 0.141264
\(452\) −17.0000 −0.799613
\(453\) 54.0000 2.53714
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −15.0000 −0.702439
\(457\) 21.0000 0.982339 0.491169 0.871064i \(-0.336570\pi\)
0.491169 + 0.871064i \(0.336570\pi\)
\(458\) 24.0000 1.12145
\(459\) 63.0000 2.94059
\(460\) 0 0
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) −12.0000 −0.554700
\(469\) 0 0
\(470\) 0 0
\(471\) −36.0000 −1.65879
\(472\) 4.00000 0.184115
\(473\) 4.00000 0.183920
\(474\) −42.0000 −1.92912
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 12.0000 0.548867
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −25.0000 −1.13872
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −8.00000 −0.362143
\(489\) −3.00000 −0.135665
\(490\) 0 0
\(491\) 44.0000 1.98569 0.992846 0.119401i \(-0.0380974\pi\)
0.992846 + 0.119401i \(0.0380974\pi\)
\(492\) 9.00000 0.405751
\(493\) 0 0
\(494\) −10.0000 −0.449921
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −9.00000 −0.403300
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) −1.00000 −0.0446322
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 27.0000 1.19911
\(508\) 4.00000 0.177471
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −45.0000 −1.98680
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −35.0000 −1.53338 −0.766689 0.642019i \(-0.778097\pi\)
−0.766689 + 0.642019i \(0.778097\pi\)
\(522\) 0 0
\(523\) −45.0000 −1.96771 −0.983856 0.178960i \(-0.942727\pi\)
−0.983856 + 0.178960i \(0.942727\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) 28.0000 1.21970
\(528\) 3.00000 0.130558
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 21.0000 0.908759
\(535\) 0 0
\(536\) −13.0000 −0.561514
\(537\) −39.0000 −1.68297
\(538\) −4.00000 −0.172452
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 16.0000 0.687259
\(543\) 6.00000 0.257485
\(544\) −7.00000 −0.300123
\(545\) 0 0
\(546\) 0 0
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) 15.0000 0.640768
\(549\) −48.0000 −2.04859
\(550\) 0 0
\(551\) 0 0
\(552\) −18.0000 −0.766131
\(553\) 0 0
\(554\) −20.0000 −0.849719
\(555\) 0 0
\(556\) −1.00000 −0.0424094
\(557\) 40.0000 1.69485 0.847427 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) −24.0000 −1.01600
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −21.0000 −0.886621
\(562\) −22.0000 −0.928014
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −7.00000 −0.294232
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 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
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 32.0000 1.33102
\(579\) −75.0000 −3.11689
\(580\) 0 0
\(581\) 0 0
\(582\) 54.0000 2.23837
\(583\) −2.00000 −0.0828315
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 21.0000 0.866025
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 1.00000 0.0410997
\(593\) 41.0000 1.68367 0.841834 0.539736i \(-0.181476\pi\)
0.841834 + 0.539736i \(0.181476\pi\)
\(594\) 9.00000 0.369274
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) −48.0000 −1.96451
\(598\) −12.0000 −0.490716
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) 0 0
\(603\) −78.0000 −3.17641
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) −42.0000 −1.69775
\(613\) −42.0000 −1.69636 −0.848182 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 54.0000 2.17220
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) −54.0000 −2.16695
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 15.0000 0.599042
\(628\) 12.0000 0.478852
\(629\) −7.00000 −0.279108
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) 14.0000 0.556890
\(633\) −39.0000 −1.55011
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 14.0000 0.554700
\(638\) 0 0
\(639\) −36.0000 −1.42414
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −15.0000 −0.592003
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −35.0000 −1.37706
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) 9.00000 0.353553
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 1.00000 0.0391630
\(653\) −40.0000 −1.56532 −0.782660 0.622449i \(-0.786138\pi\)
−0.782660 + 0.622449i \(0.786138\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) −42.0000 −1.63858
\(658\) 0 0
\(659\) 5.00000 0.194772 0.0973862 0.995247i \(-0.468952\pi\)
0.0973862 + 0.995247i \(0.468952\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) −19.0000 −0.738456
\(663\) −42.0000 −1.63114
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) −2.00000 −0.0773823
\(669\) −48.0000 −1.85579
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −31.0000 −1.19408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 51.0000 1.95864
\(679\) 0 0
\(680\) 0 0
\(681\) −84.0000 −3.21889
\(682\) 4.00000 0.153168
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 30.0000 1.14708
\(685\) 0 0
\(686\) 0 0
\(687\) −72.0000 −2.74697
\(688\) −4.00000 −0.152499
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −5.00000 −0.190209 −0.0951045 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) 8.00000 0.304114
\(693\) 0 0
\(694\) 23.0000 0.873068
\(695\) 0 0
\(696\) 0 0
\(697\) 21.0000 0.795432
\(698\) −12.0000 −0.454207
\(699\) 42.0000 1.58859
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 18.0000 0.679366
\(703\) 5.00000 0.188579
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 24.0000 0.901339 0.450669 0.892691i \(-0.351185\pi\)
0.450669 + 0.892691i \(0.351185\pi\)
\(710\) 0 0
\(711\) 84.0000 3.15025
\(712\) −7.00000 −0.262336
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 13.0000 0.485833
\(717\) −36.0000 −1.34444
\(718\) −6.00000 −0.223918
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6.00000 0.223297
\(723\) 75.0000 2.78928
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 30.0000 1.11340
\(727\) 20.0000 0.741759 0.370879 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 28.0000 1.03562
\(732\) 24.0000 0.887066
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −26.0000 −0.959678
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 13.0000 0.478861
\(738\) −18.0000 −0.662589
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 30.0000 1.10208
\(742\) 0 0
\(743\) −54.0000 −1.98107 −0.990534 0.137268i \(-0.956168\pi\)
−0.990534 + 0.137268i \(0.956168\pi\)
\(744\) 12.0000 0.439941
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 18.0000 0.658586
\(748\) 7.00000 0.255945
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −4.00000 −0.145865
\(753\) 3.00000 0.109326
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) −19.0000 −0.690111
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) −12.0000 −0.434714
\(763\) 0 0
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) −8.00000 −0.288863
\(768\) −3.00000 −0.108253
\(769\) −21.0000 −0.757279 −0.378640 0.925544i \(-0.623608\pi\)
−0.378640 + 0.925544i \(0.623608\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 25.0000 0.899770
\(773\) 22.0000 0.791285 0.395643 0.918405i \(-0.370522\pi\)
0.395643 + 0.918405i \(0.370522\pi\)
\(774\) −24.0000 −0.862662
\(775\) 0 0
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) −8.00000 −0.286814
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) −42.0000 −1.50192
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) −8.00000 −0.284988
\(789\) −42.0000 −1.49524
\(790\) 0 0
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) 16.0000 0.568177
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) 0 0
\(801\) −42.0000 −1.48400
\(802\) 23.0000 0.812158
\(803\) 7.00000 0.247025
\(804\) 39.0000 1.37542
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 12.0000 0.422420
\(808\) 6.00000 0.211079
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −48.0000 −1.68343
\(814\) −1.00000 −0.0350500
\(815\) 0 0
\(816\) 21.0000 0.735147
\(817\) −20.0000 −0.699711
\(818\) 11.0000 0.384606
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −45.0000 −1.56956
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) −18.0000 −0.627060
\(825\) 0 0
\(826\) 0 0
\(827\) 33.0000 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) 36.0000 1.25109
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 60.0000 2.08138
\(832\) −2.00000 −0.0693375
\(833\) 49.0000 1.69775
\(834\) 3.00000 0.103882
\(835\) 0 0
\(836\) −5.00000 −0.172929
\(837\) 36.0000 1.24434
\(838\) −17.0000 −0.587255
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 24.0000 0.827095
\(843\) 66.0000 2.27316
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) −24.0000 −0.825137
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 21.0000 0.720718
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 18.0000 0.616670
\(853\) 20.0000 0.684787 0.342393 0.939557i \(-0.388762\pi\)
0.342393 + 0.939557i \(0.388762\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 5.00000 0.170896
\(857\) 9.00000 0.307434 0.153717 0.988115i \(-0.450876\pi\)
0.153717 + 0.988115i \(0.450876\pi\)
\(858\) −6.00000 −0.204837
\(859\) 41.0000 1.39890 0.699451 0.714681i \(-0.253428\pi\)
0.699451 + 0.714681i \(0.253428\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −38.0000 −1.29429
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −9.00000 −0.306186
\(865\) 0 0
\(866\) 11.0000 0.373795
\(867\) −96.0000 −3.26033
\(868\) 0 0
\(869\) −14.0000 −0.474917
\(870\) 0 0
\(871\) 26.0000 0.880976
\(872\) −6.00000 −0.203186
\(873\) −108.000 −3.65525
\(874\) 30.0000 1.01477
\(875\) 0 0
\(876\) 21.0000 0.709524
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) −32.0000 −1.07995
\(879\) −72.0000 −2.42850
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −42.0000 −1.41421
\(883\) 7.00000 0.235569 0.117784 0.993039i \(-0.462421\pi\)
0.117784 + 0.993039i \(0.462421\pi\)
\(884\) 14.0000 0.470871
\(885\) 0 0
\(886\) 15.0000 0.503935
\(887\) 52.0000 1.74599 0.872995 0.487730i \(-0.162175\pi\)
0.872995 + 0.487730i \(0.162175\pi\)
\(888\) −3.00000 −0.100673
\(889\) 0 0
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 16.0000 0.535720
\(893\) −20.0000 −0.669274
\(894\) 66.0000 2.20737
\(895\) 0 0
\(896\) 0 0
\(897\) 36.0000 1.20201
\(898\) 15.0000 0.500556
\(899\) 0 0
\(900\) 0 0
\(901\) −14.0000 −0.466408
\(902\) 3.00000 0.0998891
\(903\) 0 0
\(904\) −17.0000 −0.565412
\(905\) 0 0
\(906\) 54.0000 1.79403
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 28.0000 0.929213
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) −15.0000 −0.496700
\(913\) −3.00000 −0.0992855
\(914\) 21.0000 0.694618
\(915\) 0 0
\(916\) 24.0000 0.792982
\(917\) 0 0
\(918\) 63.0000 2.07931
\(919\) 14.0000 0.461817 0.230909 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 0 0
\(921\) 21.0000 0.691974
\(922\) 4.00000 0.131733
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) −108.000 −3.54719
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −35.0000 −1.14708
\(932\) −14.0000 −0.458585
\(933\) 72.0000 2.35717
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) −12.0000 −0.392232
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) −36.0000 −1.17294
\(943\) −18.0000 −0.586161
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −56.0000 −1.81976 −0.909878 0.414876i \(-0.863825\pi\)
−0.909878 + 0.414876i \(0.863825\pi\)
\(948\) −42.0000 −1.36410
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −9.00000 −0.291539 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 4.00000 0.129234
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −2.00000 −0.0644826
\(963\) 30.0000 0.966736
\(964\) −25.0000 −0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −10.0000 −0.321412
\(969\) 105.000 3.37309
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −43.0000 −1.37569 −0.687846 0.725857i \(-0.741444\pi\)
−0.687846 + 0.725857i \(0.741444\pi\)
\(978\) −3.00000 −0.0959294
\(979\) 7.00000 0.223721
\(980\) 0 0
\(981\) −36.0000 −1.14939
\(982\) 44.0000 1.40410
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −10.0000 −0.318142
\(989\) −24.0000 −0.763156
\(990\) 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
\(993\) 57.0000 1.80884
\(994\) 0 0
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 16.0000 0.506471
\(999\) −9.00000 −0.284747
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 1850.2.a.h.1.1 yes 1
5.2 odd 4 1850.2.b.a.149.2 2
5.3 odd 4 1850.2.b.a.149.1 2
5.4 even 2 1850.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.g.1.1 1 5.4 even 2
1850.2.a.h.1.1 yes 1 1.1 even 1 trivial
1850.2.b.a.149.1 2 5.3 odd 4
1850.2.b.a.149.2 2 5.2 odd 4