Properties

Label 1850.2.a.d.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: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -3.00000 q^{11} +4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +3.00000 q^{18} +3.00000 q^{22} +8.00000 q^{23} -4.00000 q^{26} -1.00000 q^{28} -3.00000 q^{29} -7.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} -3.00000 q^{36} +1.00000 q^{37} +11.0000 q^{41} -11.0000 q^{43} -3.00000 q^{44} -8.00000 q^{46} -4.00000 q^{47} -6.00000 q^{49} +4.00000 q^{52} -11.0000 q^{53} +1.00000 q^{56} +3.00000 q^{58} -12.0000 q^{59} -15.0000 q^{61} +7.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +3.00000 q^{68} +6.00000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -1.00000 q^{74} +3.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} -11.0000 q^{82} -12.0000 q^{83} +11.0000 q^{86} +3.00000 q^{88} -4.00000 q^{91} +8.00000 q^{92} +4.00000 q^{94} +1.00000 q^{97} +6.00000 q^{98} +9.00000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 3.00000 0.707107
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) 7.00000 0.889001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −11.0000 −1.21475
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 6.00000 0.606092
\(99\) 9.00000 0.904534
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) −12.0000 −1.10940
\(118\) 12.0000 1.10469
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 15.0000 1.35804
\(123\) 0 0
\(124\) −7.00000 −0.628619
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 1.00000 0.0848189 0.0424094 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −12.0000 −1.00349
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) −9.00000 −0.727607
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) −9.00000 −0.707107
\(163\) −3.00000 −0.234978 −0.117489 0.993074i \(-0.537485\pi\)
−0.117489 + 0.993074i \(0.537485\pi\)
\(164\) 11.0000 0.858956
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) −11.0000 −0.838742
\(173\) 13.0000 0.988372 0.494186 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −9.00000 −0.639602
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −24.0000 −1.66812
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) −11.0000 −0.755483
\(213\) 0 0
\(214\) −10.0000 −0.683586
\(215\) 0 0
\(216\) 0 0
\(217\) 7.00000 0.475191
\(218\) 11.0000 0.745014
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −11.0000 −0.736614 −0.368307 0.929704i \(-0.620063\pi\)
−0.368307 + 0.929704i \(0.620063\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −1.00000 −0.0665190
\(227\) −17.0000 −1.12833 −0.564165 0.825662i \(-0.690802\pi\)
−0.564165 + 0.825662i \(0.690802\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 3.00000 0.194461
\(239\) −17.0000 −1.09964 −0.549819 0.835284i \(-0.685303\pi\)
−0.549819 + 0.835284i \(0.685303\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) −15.0000 −0.960277
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 7.00000 0.444500
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 3.00000 0.188982
\(253\) −24.0000 −1.50887
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 10.0000 0.617802
\(263\) −31.0000 −1.91154 −0.955771 0.294112i \(-0.904976\pi\)
−0.955771 + 0.294112i \(0.904976\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) −16.0000 −0.975537 −0.487769 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −1.00000 −0.0599760
\(279\) 21.0000 1.25724
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −11.0000 −0.649309
\(288\) 3.00000 0.176777
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 15.0000 0.876309 0.438155 0.898900i \(-0.355632\pi\)
0.438155 + 0.898900i \(0.355632\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) 11.0000 0.634029
\(302\) 14.0000 0.805609
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 9.00000 0.514496
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −5.00000 −0.282166
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 3.00000 0.166155
\(327\) 0 0
\(328\) −11.0000 −0.607373
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) −12.0000 −0.658586
\(333\) −3.00000 −0.164399
\(334\) −14.0000 −0.766046
\(335\) 0 0
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 0 0
\(341\) 21.0000 1.13721
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) −13.0000 −0.698884
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 16.0000 0.840941
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 8.00000 0.417029
\(369\) −33.0000 −1.71791
\(370\) 0 0
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 17.0000 0.869796
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 33.0000 1.67748
\(388\) 1.00000 0.0507673
\(389\) 23.0000 1.16615 0.583073 0.812420i \(-0.301850\pi\)
0.583073 + 0.812420i \(0.301850\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 9.00000 0.452267
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −28.0000 −1.39478
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 24.0000 1.17954
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 1.00000 0.0486792
\(423\) 12.0000 0.583460
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) 0 0
\(427\) 15.0000 0.725901
\(428\) 10.0000 0.483368
\(429\) 0 0
\(430\) 0 0
\(431\) 23.0000 1.10787 0.553936 0.832560i \(-0.313125\pi\)
0.553936 + 0.832560i \(0.313125\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) −7.00000 −0.336011
\(435\) 0 0
\(436\) −11.0000 −0.526804
\(437\) 0 0
\(438\) 0 0
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) −12.0000 −0.570782
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 11.0000 0.520865
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) −33.0000 −1.55391
\(452\) 1.00000 0.0470360
\(453\) 0 0
\(454\) 17.0000 0.797850
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0000 1.73079 0.865393 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) −20.0000 −0.934539
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) −12.0000 −0.554700
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 33.0000 1.51734
\(474\) 0 0
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 33.0000 1.51097
\(478\) 17.0000 0.777562
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) −36.0000 −1.63132 −0.815658 0.578535i \(-0.803625\pi\)
−0.815658 + 0.578535i \(0.803625\pi\)
\(488\) 15.0000 0.679018
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −18.0000 −0.805791 −0.402895 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) 26.0000 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 1.00000 0.0439375
\(519\) 0 0
\(520\) 0 0
\(521\) −7.00000 −0.306676 −0.153338 0.988174i \(-0.549002\pi\)
−0.153338 + 0.988174i \(0.549002\pi\)
\(522\) −9.00000 −0.393919
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 31.0000 1.35166
\(527\) −21.0000 −0.914774
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 36.0000 1.56227
\(532\) 0 0
\(533\) 44.0000 1.90585
\(534\) 0 0
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 16.0000 0.689809
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −22.0000 −0.944981
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) −35.0000 −1.49649 −0.748246 0.663421i \(-0.769104\pi\)
−0.748246 + 0.663421i \(0.769104\pi\)
\(548\) −6.00000 −0.256307
\(549\) 45.0000 1.92055
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 1.00000 0.0424094
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) −21.0000 −0.889001
\(559\) −44.0000 −1.86100
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 19.0000 0.800755 0.400377 0.916350i \(-0.368879\pi\)
0.400377 + 0.916350i \(0.368879\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) −9.00000 −0.377964
\(568\) −6.00000 −0.251754
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) 11.0000 0.459131
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 33.0000 1.36672
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −15.0000 −0.619644
\(587\) −15.0000 −0.619116 −0.309558 0.950881i \(-0.600181\pi\)
−0.309558 + 0.950881i \(0.600181\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) −32.0000 −1.30858
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) −47.0000 −1.91717 −0.958585 0.284807i \(-0.908071\pi\)
−0.958585 + 0.284807i \(0.908071\pi\)
\(602\) −11.0000 −0.448327
\(603\) −12.0000 −0.488678
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) 42.0000 1.70473 0.852364 0.522949i \(-0.175168\pi\)
0.852364 + 0.522949i \(0.175168\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) −9.00000 −0.363803
\(613\) 15.0000 0.605844 0.302922 0.953015i \(-0.402038\pi\)
0.302922 + 0.953015i \(0.402038\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3.00000 −0.120289
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 5.00000 0.199522
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 37.0000 1.47295 0.736473 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) −21.0000 −0.834017
\(635\) 0 0
\(636\) 0 0
\(637\) −24.0000 −0.950915
\(638\) −9.00000 −0.356313
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) −25.0000 −0.987441 −0.493720 0.869621i \(-0.664363\pi\)
−0.493720 + 0.869621i \(0.664363\pi\)
\(642\) 0 0
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) −2.00000 −0.0786281 −0.0393141 0.999227i \(-0.512517\pi\)
−0.0393141 + 0.999227i \(0.512517\pi\)
\(648\) −9.00000 −0.353553
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) −3.00000 −0.117489
\(653\) −38.0000 −1.48705 −0.743527 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 11.0000 0.429478
\(657\) 6.00000 0.234082
\(658\) −4.00000 −0.155936
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −25.0000 −0.972387 −0.486194 0.873851i \(-0.661615\pi\)
−0.486194 + 0.873851i \(0.661615\pi\)
\(662\) −2.00000 −0.0777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) −24.0000 −0.929284
\(668\) 14.0000 0.541676
\(669\) 0 0
\(670\) 0 0
\(671\) 45.0000 1.73721
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 0 0
\(682\) −21.0000 −0.804132
\(683\) −13.0000 −0.497431 −0.248716 0.968577i \(-0.580008\pi\)
−0.248716 + 0.968577i \(0.580008\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) −11.0000 −0.419371
\(689\) −44.0000 −1.67627
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 13.0000 0.494186
\(693\) −9.00000 −0.341882
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 33.0000 1.24996
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −3.00000 −0.112906
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) 29.0000 1.08912 0.544559 0.838723i \(-0.316697\pi\)
0.544559 + 0.838723i \(0.316697\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) −56.0000 −2.09722
\(714\) 0 0
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) 14.0000 0.522475
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 4.00000 0.148250
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −33.0000 −1.22055
\(732\) 0 0
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −12.0000 −0.442026
\(738\) 33.0000 1.21475
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11.0000 −0.403823
\(743\) 13.0000 0.476924 0.238462 0.971152i \(-0.423357\pi\)
0.238462 + 0.971152i \(0.423357\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) 36.0000 1.31717
\(748\) −9.00000 −0.329073
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) 7.00000 0.253750 0.126875 0.991919i \(-0.459505\pi\)
0.126875 + 0.991919i \(0.459505\pi\)
\(762\) 0 0
\(763\) 11.0000 0.398227
\(764\) −17.0000 −0.615038
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −48.0000 −1.73318
\(768\) 0 0
\(769\) −44.0000 −1.58668 −0.793340 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) −33.0000 −1.18616
\(775\) 0 0
\(776\) −1.00000 −0.0358979
\(777\) 0 0
\(778\) −23.0000 −0.824590
\(779\) 0 0
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) −24.0000 −0.858238
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) −1.00000 −0.0355559
\(792\) −9.00000 −0.319801
\(793\) −60.0000 −2.13066
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 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
\(800\) 0 0
\(801\) 0 0
\(802\) −10.0000 −0.353112
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 28.0000 0.986258
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 3.00000 0.105279
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 4.00000 0.139857
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −14.0000 −0.488603 −0.244302 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) −24.0000 −0.834058
\(829\) −23.0000 −0.798823 −0.399412 0.916772i \(-0.630786\pi\)
−0.399412 + 0.916772i \(0.630786\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 30.0000 1.03387
\(843\) 0 0
\(844\) −1.00000 −0.0344214
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 2.00000 0.0687208
\(848\) −11.0000 −0.377742
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) −15.0000 −0.513289
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) 17.0000 0.580709 0.290354 0.956919i \(-0.406227\pi\)
0.290354 + 0.956919i \(0.406227\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −23.0000 −0.783383
\(863\) 39.0000 1.32758 0.663788 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 0 0
\(868\) 7.00000 0.237595
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 11.0000 0.372507
\(873\) −3.00000 −0.101535
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.00000 0.0337676 0.0168838 0.999857i \(-0.494625\pi\)
0.0168838 + 0.999857i \(0.494625\pi\)
\(878\) −7.00000 −0.236239
\(879\) 0 0
\(880\) 0 0
\(881\) 39.0000 1.31394 0.656972 0.753915i \(-0.271837\pi\)
0.656972 + 0.753915i \(0.271837\pi\)
\(882\) −18.0000 −0.606092
\(883\) −29.0000 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) −7.00000 −0.235037 −0.117518 0.993071i \(-0.537494\pi\)
−0.117518 + 0.993071i \(0.537494\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −27.0000 −0.904534
\(892\) −11.0000 −0.368307
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) 21.0000 0.700389
\(900\) 0 0
\(901\) −33.0000 −1.09939
\(902\) 33.0000 1.09878
\(903\) 0 0
\(904\) −1.00000 −0.0332595
\(905\) 0 0
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −17.0000 −0.564165
\(909\) −42.0000 −1.39305
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) −37.0000 −1.22385
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) 10.0000 0.330229
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.00000 −0.0987997
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 3.00000 0.0981630
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) 0 0
\(943\) 88.0000 2.86567
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −33.0000 −1.07292
\(947\) −37.0000 −1.20234 −0.601169 0.799122i \(-0.705298\pi\)
−0.601169 + 0.799122i \(0.705298\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) 3.00000 0.0972306
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −33.0000 −1.06841
\(955\) 0 0
\(956\) −17.0000 −0.549819
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −4.00000 −0.128965
\(963\) −30.0000 −0.966736
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 34.0000 1.09337 0.546683 0.837340i \(-0.315890\pi\)
0.546683 + 0.837340i \(0.315890\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −1.00000 −0.0320915 −0.0160458 0.999871i \(-0.505108\pi\)
−0.0160458 + 0.999871i \(0.505108\pi\)
\(972\) 0 0
\(973\) −1.00000 −0.0320585
\(974\) 36.0000 1.15351
\(975\) 0 0
\(976\) −15.0000 −0.480138
\(977\) −33.0000 −1.05576 −0.527882 0.849318i \(-0.677014\pi\)
−0.527882 + 0.849318i \(0.677014\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 33.0000 1.05361
\(982\) 36.0000 1.14881
\(983\) 11.0000 0.350846 0.175423 0.984493i \(-0.443871\pi\)
0.175423 + 0.984493i \(0.443871\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 0 0
\(989\) −88.0000 −2.79824
\(990\) 0 0
\(991\) −45.0000 −1.42947 −0.714736 0.699394i \(-0.753453\pi\)
−0.714736 + 0.699394i \(0.753453\pi\)
\(992\) 7.00000 0.222250
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 0 0
\(997\) 44.0000 1.39349 0.696747 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(998\) 18.0000 0.569780
\(999\) 0 0
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.d.1.1 1
5.2 odd 4 370.2.b.b.149.1 2
5.3 odd 4 370.2.b.b.149.2 yes 2
5.4 even 2 1850.2.a.l.1.1 1
15.2 even 4 3330.2.d.c.1999.2 2
15.8 even 4 3330.2.d.c.1999.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.b.149.1 2 5.2 odd 4
370.2.b.b.149.2 yes 2 5.3 odd 4
1850.2.a.d.1.1 1 1.1 even 1 trivial
1850.2.a.l.1.1 1 5.4 even 2
3330.2.d.c.1999.1 2 15.8 even 4
3330.2.d.c.1999.2 2 15.2 even 4