Properties

Label 3330.2.a.v.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
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 \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{28} -3.00000 q^{29} +5.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} -3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -6.00000 q^{46} -12.0000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -3.00000 q^{53} -3.00000 q^{55} -1.00000 q^{56} -3.00000 q^{58} -1.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} -1.00000 q^{70} -6.00000 q^{71} -16.0000 q^{73} +1.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -3.00000 q^{82} +12.0000 q^{83} -3.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -12.0000 q^{94} +2.00000 q^{95} +17.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 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
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −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 \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −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
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 5.00000 0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −1.00000 −0.0905357
\(123\) 0 0
\(124\) 5.00000 0.449013
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −3.00000 −0.230089
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 17.0000 1.22053
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −3.00000 −0.206041
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 11.0000 0.745014
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 17.0000 1.13840 0.569202 0.822198i \(-0.307252\pi\)
0.569202 + 0.822198i \(0.307252\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 27.0000 1.79205 0.896026 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 3.00000 0.194461
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 5.00000 0.317500
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) −13.0000 −0.779688
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 3.00000 0.177084
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −3.00000 −0.176166
\(291\) 0 0
\(292\) −16.0000 −0.936329
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) −34.0000 −1.94048 −0.970241 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 5.00000 0.283981
\(311\) 9.00000 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −13.0000 −0.733632
\(315\) 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
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 6.00000 0.334367
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 11.0000 0.609234
\(327\) 0 0
\(328\) −3.00000 −0.165647
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) −3.00000 −0.162698
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −16.0000 −0.837478
\(366\) 0 0
\(367\) 35.0000 1.82699 0.913493 0.406855i \(-0.133375\pi\)
0.913493 + 0.406855i \(0.133375\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) 3.00000 0.155752
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −3.00000 −0.153493
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 17.0000 0.863044
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) −3.00000 −0.148159
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −1.00000 −0.0482243
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −5.00000 −0.240008
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 17.0000 0.804973
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) 27.0000 1.26717
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 0 0
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 3.00000 0.137505
\(477\) 0 0
\(478\) 9.00000 0.411650
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 17.0000 0.771930
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 18.0000 0.800198
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 36.0000 1.58328
\(518\) −1.00000 −0.0439375
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) −15.0000 −0.653410
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −3.00000 −0.130312
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 11.0000 0.471188
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −3.00000 −0.124568
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) −16.0000 −0.662085
\(585\) 0 0
\(586\) −21.0000 −0.867502
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) 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
\(595\) 3.00000 0.122988
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 1.00000 0.0407570
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −1.00000 −0.0404888
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) −34.0000 −1.37213
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 35.0000 1.40677 0.703384 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) 5.00000 0.200805
\(621\) 0 0
\(622\) 9.00000 0.360867
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 21.0000 0.834017
\(635\) 20.0000 0.793676
\(636\) 0 0
\(637\) 24.0000 0.950915
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −13.0000 −0.512670 −0.256335 0.966588i \(-0.582515\pi\)
−0.256335 + 0.966588i \(0.582515\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 26.0000 1.01052
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −4.00000 −0.154074
\(675\) 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
\(679\) −17.0000 −0.652400
\(680\) −3.00000 −0.115045
\(681\) 0 0
\(682\) −15.0000 −0.574380
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −19.0000 −0.722794 −0.361397 0.932412i \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(692\) −15.0000 −0.570214
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −13.0000 −0.493118
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 21.0000 0.790345
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 4.00000 0.148250
\(729\) 0 0
\(730\) −16.0000 −0.592187
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) −31.0000 −1.14501 −0.572506 0.819901i \(-0.694029\pi\)
−0.572506 + 0.819901i \(0.694029\pi\)
\(734\) 35.0000 1.29187
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 11.0000 0.404642 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) 51.0000 1.87101 0.935504 0.353315i \(-0.114946\pi\)
0.935504 + 0.353315i \(0.114946\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 9.00000 0.329073
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) −11.0000 −0.398227
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 3.00000 0.108112
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 17.0000 0.610264
\(777\) 0 0
\(778\) −9.00000 −0.322666
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 18.0000 0.643679
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) −13.0000 −0.463990
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −6.00000 −0.211867
\(803\) 48.0000 1.69388
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) −20.0000 −0.704470
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 3.00000 0.105279
\(813\) 0 0
\(814\) −3.00000 −0.105150
\(815\) 11.0000 0.385313
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) 32.0000 1.11885
\(819\) 0 0
\(820\) −3.00000 −0.104765
\(821\) −36.0000 −1.25641 −0.628204 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 0 0
\(829\) 47.0000 1.63238 0.816189 0.577785i \(-0.196083\pi\)
0.816189 + 0.577785i \(0.196083\pi\)
\(830\) 12.0000 0.416526
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 5.00000 0.172107
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −3.00000 −0.103020
\(849\) 0 0
\(850\) −3.00000 −0.102899
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) 15.0000 0.512390 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 0 0
\(862\) −15.0000 −0.510902
\(863\) −39.0000 −1.32758 −0.663788 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) 0 0
\(865\) −15.0000 −0.510015
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −5.00000 −0.169711
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 11.0000 0.372507
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 35.0000 1.18119
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −30.0000 −1.00787
\(887\) −33.0000 −1.10803 −0.554016 0.832506i \(-0.686905\pi\)
−0.554016 + 0.832506i \(0.686905\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 17.0000 0.569202
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 9.00000 0.299667
\(903\) 0 0
\(904\) −9.00000 −0.299336
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 27.0000 0.896026
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −6.00000 −0.197814
\(921\) 0 0
\(922\) −33.0000 −1.08680
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) −45.0000 −1.47640 −0.738201 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) 0 0
\(934\) −27.0000 −0.883467
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) −33.0000 −1.07236 −0.536178 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) 0 0
\(949\) 64.0000 2.07753
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) 3.00000 0.0972306
\(953\) 60.0000 1.94359 0.971795 0.235826i \(-0.0757795\pi\)
0.971795 + 0.235826i \(0.0757795\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) −28.0000 −0.901819
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 17.0000 0.545837
\(971\) −57.0000 −1.82922 −0.914609 0.404341i \(-0.867501\pi\)
−0.914609 + 0.404341i \(0.867501\pi\)
\(972\) 0 0
\(973\) 13.0000 0.416761
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 45.0000 1.43968 0.719839 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) −51.0000 −1.62665 −0.813324 0.581811i \(-0.802344\pi\)
−0.813324 + 0.581811i \(0.802344\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 5.00000 0.158750
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 14.0000 0.443162
\(999\) 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 3330.2.a.v.1.1 1
3.2 odd 2 370.2.a.a.1.1 1
12.11 even 2 2960.2.a.j.1.1 1
15.2 even 4 1850.2.b.g.149.1 2
15.8 even 4 1850.2.b.g.149.2 2
15.14 odd 2 1850.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.a.1.1 1 3.2 odd 2
1850.2.a.o.1.1 1 15.14 odd 2
1850.2.b.g.149.1 2 15.2 even 4
1850.2.b.g.149.2 2 15.8 even 4
2960.2.a.j.1.1 1 12.11 even 2
3330.2.a.v.1.1 1 1.1 even 1 trivial