Properties

Label 3330.2.a.n.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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +5.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -3.00000 q^{28} -4.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{37} +1.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -12.0000 q^{43} -1.00000 q^{44} +5.00000 q^{46} +2.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -9.00000 q^{53} +1.00000 q^{55} -3.00000 q^{56} -4.00000 q^{58} +14.0000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -4.00000 q^{67} +1.00000 q^{68} +3.00000 q^{70} -8.00000 q^{71} -15.0000 q^{73} +1.00000 q^{74} +1.00000 q^{76} +3.00000 q^{77} -14.0000 q^{79} -1.00000 q^{80} -10.0000 q^{82} -17.0000 q^{83} -1.00000 q^{85} -12.0000 q^{86} -1.00000 q^{88} -7.00000 q^{89} -3.00000 q^{91} +5.00000 q^{92} -1.00000 q^{95} -8.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −15.0000 −1.75562 −0.877809 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) −17.0000 −1.86599 −0.932996 0.359886i \(-0.882816\pi\)
−0.932996 + 0.359886i \(0.882816\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) −12.0000 −1.29399
\(87\) 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
\(91\) −3.00000 −0.314485
\(92\) 5.00000 0.521286
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 2.00000 0.202031
\(99\) 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
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 13.0000 1.25676 0.628379 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) 0 0
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) −3.00000 −0.283473
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −5.00000 −0.466252
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 14.0000 1.28880
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.00000 −0.0877058
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) −15.0000 −1.24141
\(147\) 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
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −15.0000 −1.18217
\(162\) 0 0
\(163\) 15.0000 1.17489 0.587445 0.809264i \(-0.300134\pi\)
0.587445 + 0.809264i \(0.300134\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −17.0000 −1.31946
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −1.00000 −0.0766965
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) 17.0000 1.29249 0.646243 0.763132i \(-0.276339\pi\)
0.646243 + 0.763132i \(0.276339\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −7.00000 −0.524672
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −3.00000 −0.222375
\(183\) 0 0
\(184\) 5.00000 0.368605
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −1.00000 −0.0731272
\(188\) 0 0
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −7.00000 −0.498729 −0.249365 0.968410i \(-0.580222\pi\)
−0.249365 + 0.968410i \(0.580222\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) 13.0000 0.888662
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) −13.0000 −0.880471
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 1.00000 0.0672673
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) −5.00000 −0.329690
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.0000 0.911322
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) −4.00000 −0.254000
\(249\) 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
\(253\) −5.00000 −0.314347
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.0000 −0.686161 −0.343081 0.939306i \(-0.611470\pi\)
−0.343081 + 0.939306i \(0.611470\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) −1.00000 −0.0620174
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) −3.00000 −0.183942
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 17.0000 1.03651 0.518254 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) −7.00000 −0.417585 −0.208792 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) −9.00000 −0.534994 −0.267497 0.963559i \(-0.586197\pi\)
−0.267497 + 0.963559i \(0.586197\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) 30.0000 1.77084
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) −15.0000 −0.877809
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) 36.0000 2.07501
\(302\) 19.0000 1.09333
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −6.00000 −0.342438 −0.171219 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −15.0000 −0.835917
\(323\) 1.00000 0.0556415
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 15.0000 0.830773
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −24.0000 −1.31916 −0.659580 0.751635i \(-0.729266\pi\)
−0.659580 + 0.751635i \(0.729266\pi\)
\(332\) −17.0000 −0.932996
\(333\) 0 0
\(334\) 19.0000 1.03963
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 29.0000 1.57973 0.789865 0.613280i \(-0.210150\pi\)
0.789865 + 0.613280i \(0.210150\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) −1.00000 −0.0542326
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 17.0000 0.913926
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −3.00000 −0.160357
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −7.00000 −0.370999
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) 15.0000 0.785136
\(366\) 0 0
\(367\) −19.0000 −0.991792 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 5.00000 0.260643
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) 27.0000 1.40177
\(372\) 0 0
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) 13.0000 0.665138
\(383\) 35.0000 1.78842 0.894208 0.447651i \(-0.147739\pi\)
0.894208 + 0.447651i \(0.147739\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 20.0000 1.01797
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 2.00000 0.101015
\(393\) 0 0
\(394\) −7.00000 −0.352655
\(395\) 14.0000 0.704416
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −19.0000 −0.948815 −0.474407 0.880305i \(-0.657338\pi\)
−0.474407 + 0.880305i \(0.657338\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) −1.00000 −0.0495682
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 2.00000 0.0985329
\(413\) −42.0000 −2.06668
\(414\) 0 0
\(415\) 17.0000 0.834497
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −1.00000 −0.0489116
\(419\) 1.00000 0.0488532 0.0244266 0.999702i \(-0.492224\pi\)
0.0244266 + 0.999702i \(0.492224\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 13.0000 0.628379
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) 0 0
\(433\) 23.0000 1.10531 0.552655 0.833410i \(-0.313615\pi\)
0.552655 + 0.833410i \(0.313615\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −13.0000 −0.622587
\(437\) 5.00000 0.239182
\(438\) 0 0
\(439\) 18.0000 0.859093 0.429547 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 1.00000 0.0475651
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 7.00000 0.331832
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) −3.00000 −0.141737
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 10.0000 0.469323
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) −5.00000 −0.233126
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 14.0000 0.644402
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) 1.00000 0.0455961
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 11.0000 0.496423 0.248212 0.968706i \(-0.420157\pi\)
0.248212 + 0.968706i \(0.420157\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 1.00000 0.0449921
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −29.0000 −1.29822 −0.649109 0.760695i \(-0.724858\pi\)
−0.649109 + 0.760695i \(0.724858\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) −5.00000 −0.222277
\(507\) 0 0
\(508\) 7.00000 0.310575
\(509\) −13.0000 −0.576215 −0.288107 0.957598i \(-0.593026\pi\)
−0.288107 + 0.957598i \(0.593026\pi\)
\(510\) 0 0
\(511\) 45.0000 1.99068
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −11.0000 −0.485189
\(515\) −2.00000 −0.0881305
\(516\) 0 0
\(517\) 0 0
\(518\) −3.00000 −0.131812
\(519\) 0 0
\(520\) −1.00000 −0.0438529
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 9.00000 0.390935
\(531\) 0 0
\(532\) −3.00000 −0.130066
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) −13.0000 −0.562039
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 17.0000 0.732922
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 33.0000 1.41878 0.709390 0.704816i \(-0.248970\pi\)
0.709390 + 0.704816i \(0.248970\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) 13.0000 0.556859
\(546\) 0 0
\(547\) 35.0000 1.49649 0.748246 0.663421i \(-0.230896\pi\)
0.748246 + 0.663421i \(0.230896\pi\)
\(548\) 4.00000 0.170872
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 42.0000 1.78602
\(554\) −17.0000 −0.722261
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) −7.00000 −0.295277
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −9.00000 −0.378298
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −41.0000 −1.71881 −0.859405 0.511296i \(-0.829166\pi\)
−0.859405 + 0.511296i \(0.829166\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 0 0
\(574\) 30.0000 1.25218
\(575\) 5.00000 0.208514
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 51.0000 2.11584
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) −15.0000 −0.620704
\(585\) 0 0
\(586\) 3.00000 0.123929
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −14.0000 −0.576371
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 5.00000 0.204465
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 36.0000 1.46725
\(603\) 0 0
\(604\) 19.0000 0.773099
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −6.00000 −0.242140
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 21.0000 0.841347
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 20.0000 0.799361
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 1.00000 0.0398726
\(630\) 0 0
\(631\) −10.0000 −0.398094 −0.199047 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) −14.0000 −0.556890
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 0 0
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) −15.0000 −0.591083
\(645\) 0 0
\(646\) 1.00000 0.0393445
\(647\) −7.00000 −0.275198 −0.137599 0.990488i \(-0.543939\pi\)
−0.137599 + 0.990488i \(0.543939\pi\)
\(648\) 0 0
\(649\) −14.0000 −0.549548
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 15.0000 0.587445
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 37.0000 1.43913 0.719567 0.694423i \(-0.244340\pi\)
0.719567 + 0.694423i \(0.244340\pi\)
\(662\) −24.0000 −0.932786
\(663\) 0 0
\(664\) −17.0000 −0.659728
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) −20.0000 −0.774403
\(668\) 19.0000 0.735132
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 49.0000 1.88881 0.944406 0.328783i \(-0.106638\pi\)
0.944406 + 0.328783i \(0.106638\pi\)
\(674\) 29.0000 1.11704
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) −1.00000 −0.0383482
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) −22.0000 −0.841807 −0.420903 0.907106i \(-0.638287\pi\)
−0.420903 + 0.907106i \(0.638287\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 15.0000 0.572703
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −26.0000 −0.989087 −0.494543 0.869153i \(-0.664665\pi\)
−0.494543 + 0.869153i \(0.664665\pi\)
\(692\) 17.0000 0.646243
\(693\) 0 0
\(694\) 30.0000 1.13878
\(695\) 0 0
\(696\) 0 0
\(697\) −10.0000 −0.378777
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) −3.00000 −0.113389
\(701\) 28.0000 1.05755 0.528773 0.848763i \(-0.322652\pi\)
0.528773 + 0.848763i \(0.322652\pi\)
\(702\) 0 0
\(703\) 1.00000 0.0377157
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) −18.0000 −0.676960
\(708\) 0 0
\(709\) 21.0000 0.788672 0.394336 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) −7.00000 −0.262336
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) −4.00000 −0.149279
\(719\) 14.0000 0.522112 0.261056 0.965324i \(-0.415929\pi\)
0.261056 + 0.965324i \(0.415929\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) −18.0000 −0.669891
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −3.00000 −0.111187
\(729\) 0 0
\(730\) 15.0000 0.555175
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −28.0000 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) −19.0000 −0.701303
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 27.0000 0.991201
\(743\) −20.0000 −0.733729 −0.366864 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) −8.00000 −0.292901
\(747\) 0 0
\(748\) −1.00000 −0.0365636
\(749\) −39.0000 −1.42503
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) −19.0000 −0.691481
\(756\) 0 0
\(757\) −5.00000 −0.181728 −0.0908640 0.995863i \(-0.528963\pi\)
−0.0908640 + 0.995863i \(0.528963\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) 39.0000 1.41189
\(764\) 13.0000 0.470323
\(765\) 0 0
\(766\) 35.0000 1.26460
\(767\) 14.0000 0.505511
\(768\) 0 0
\(769\) 44.0000 1.58668 0.793340 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) −3.00000 −0.108112
\(771\) 0 0
\(772\) 20.0000 0.719816
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 5.00000 0.178800
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −7.00000 −0.249365
\(789\) 0 0
\(790\) 14.0000 0.498098
\(791\) −42.0000 −1.49335
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −19.0000 −0.670913
\(803\) 15.0000 0.529339
\(804\) 0 0
\(805\) 15.0000 0.528681
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −11.0000 −0.386739 −0.193370 0.981126i \(-0.561942\pi\)
−0.193370 + 0.981126i \(0.561942\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) −1.00000 −0.0350500
\(815\) −15.0000 −0.525427
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 15.0000 0.523504 0.261752 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(822\) 0 0
\(823\) 19.0000 0.662298 0.331149 0.943578i \(-0.392564\pi\)
0.331149 + 0.943578i \(0.392564\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) −42.0000 −1.46137
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 17.0000 0.590079
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −19.0000 −0.657522
\(836\) −1.00000 −0.0345857
\(837\) 0 0
\(838\) 1.00000 0.0345444
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 30.0000 1.03081
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) 1.00000 0.0342997
\(851\) 5.00000 0.171398
\(852\) 0 0
\(853\) 33.0000 1.12990 0.564949 0.825126i \(-0.308896\pi\)
0.564949 + 0.825126i \(0.308896\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 13.0000 0.444331
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 15.0000 0.511793 0.255897 0.966704i \(-0.417629\pi\)
0.255897 + 0.966704i \(0.417629\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 15.0000 0.510902
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −17.0000 −0.578017
\(866\) 23.0000 0.781572
\(867\) 0 0
\(868\) 12.0000 0.407307
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) −13.0000 −0.440236
\(873\) 0 0
\(874\) 5.00000 0.169128
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 18.0000 0.607471
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0 0
\(883\) 11.0000 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) −21.0000 −0.704317
\(890\) 7.00000 0.234641
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 10.0000 0.332964
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 15.0000 0.498067 0.249033 0.968495i \(-0.419887\pi\)
0.249033 + 0.968495i \(0.419887\pi\)
\(908\) 10.0000 0.331862
\(909\) 0 0
\(910\) 3.00000 0.0994490
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 17.0000 0.562618
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) −42.0000 −1.38545 −0.692726 0.721201i \(-0.743591\pi\)
−0.692726 + 0.721201i \(0.743591\pi\)
\(920\) −5.00000 −0.164845
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) −16.0000 −0.524943 −0.262471 0.964940i \(-0.584538\pi\)
−0.262471 + 0.964940i \(0.584538\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −4.00000 −0.131024
\(933\) 0 0
\(934\) −18.0000 −0.588978
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) −50.0000 −1.62822
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 26.0000 0.844886 0.422443 0.906389i \(-0.361173\pi\)
0.422443 + 0.906389i \(0.361173\pi\)
\(948\) 0 0
\(949\) −15.0000 −0.486921
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 0 0
\(955\) −13.0000 −0.420670
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −21.0000 −0.678479
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 1.00000 0.0322413
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) −20.0000 −0.643823
\(966\) 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
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 1.00000 0.0319928 0.0159964 0.999872i \(-0.494908\pi\)
0.0159964 + 0.999872i \(0.494908\pi\)
\(978\) 0 0
\(979\) 7.00000 0.223721
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 11.0000 0.351024
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 0 0
\(985\) 7.00000 0.223039
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −21.0000 −0.665077 −0.332538 0.943090i \(-0.607905\pi\)
−0.332538 + 0.943090i \(0.607905\pi\)
\(998\) −29.0000 −0.917979
\(999\) 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.n.1.1 yes 1
3.2 odd 2 3330.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3330.2.a.g.1.1 1 3.2 odd 2
3330.2.a.n.1.1 yes 1 1.1 even 1 trivial