Properties

Label 3360.2.j.b.1009.1
Level $3360$
Weight $2$
Character 3360.1009
Analytic conductor $26.830$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3360.1009
Dual form 3360.2.j.b.1009.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} -2.00000 q^{13} +(-2.00000 + 1.00000i) q^{15} +2.00000i q^{17} -8.00000i q^{19} -1.00000i q^{21} -4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{27} -2.00000i q^{29} +4.00000 q^{31} -4.00000i q^{33} +(1.00000 + 2.00000i) q^{35} +6.00000 q^{37} +2.00000 q^{39} -10.0000 q^{43} +(2.00000 - 1.00000i) q^{45} +6.00000i q^{47} -1.00000 q^{49} -2.00000i q^{51} +14.0000 q^{53} +(4.00000 + 8.00000i) q^{55} +8.00000i q^{57} -6.00000i q^{59} -2.00000i q^{61} +1.00000i q^{63} +(-4.00000 + 2.00000i) q^{65} +10.0000 q^{67} +4.00000i q^{69} +12.0000 q^{71} +6.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -4.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +(2.00000 + 4.00000i) q^{85} +2.00000i q^{87} +8.00000 q^{89} -2.00000i q^{91} -4.00000 q^{93} +(-8.00000 - 16.0000i) q^{95} +6.00000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} - 4 q^{13} - 4 q^{15} + 6 q^{25} - 2 q^{27} + 8 q^{31} + 2 q^{35} + 12 q^{37} + 4 q^{39} - 20 q^{43} + 4 q^{45} - 2 q^{49} + 28 q^{53} + 8 q^{55} - 8 q^{65} + 20 q^{67} + 24 q^{71} - 6 q^{75} - 8 q^{77} + 28 q^{79} + 2 q^{81} - 24 q^{83} + 4 q^{85} + 16 q^{89} - 8 q^{93} - 16 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.00000 1.00000i 0.894427 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 + 1.00000i −0.516398 + 0.258199i
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i −0.397360 0.917663i \(-0.630073\pi\)
0.397360 0.917663i \(-0.369927\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000i 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 1.00000 + 2.00000i 0.169031 + 0.338062i
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 2.00000 1.00000i 0.298142 0.149071i
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 4.00000 + 8.00000i 0.539360 + 1.07872i
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 6.00000i 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) −4.00000 + 2.00000i −0.496139 + 0.248069i
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) −3.00000 + 4.00000i −0.346410 + 0.461880i
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 2.00000 + 4.00000i 0.216930 + 0.433861i
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 2.00000i 0.209657i
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −8.00000 16.0000i −0.820783 1.64157i
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) −1.00000 2.00000i −0.0975900 0.195180i
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 16.0000i 1.53252i −0.642529 0.766261i \(-0.722115\pi\)
0.642529 0.766261i \(-0.277885\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) −4.00000 8.00000i −0.373002 0.746004i
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 18.0000i 1.57267i −0.617802 0.786334i \(-0.711977\pi\)
0.617802 0.786334i \(-0.288023\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) −2.00000 + 1.00000i −0.172133 + 0.0860663i
\(136\) 0 0
\(137\) 6.00000i 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) 12.0000i 1.01783i −0.860818 0.508913i \(-0.830047\pi\)
0.860818 0.508913i \(-0.169953\pi\)
\(140\) 0 0
\(141\) 6.00000i 0.505291i
\(142\) 0 0
\(143\) 8.00000i 0.668994i
\(144\) 0 0
\(145\) −2.00000 4.00000i −0.166091 0.332182i
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 10.0000i 0.819232i 0.912258 + 0.409616i \(0.134337\pi\)
−0.912258 + 0.409616i \(0.865663\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 8.00000 4.00000i 0.642575 0.321288i
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) −14.0000 −1.11027
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 0 0
\(165\) −4.00000 8.00000i −0.311400 0.622799i
\(166\) 0 0
\(167\) 6.00000i 0.464294i −0.972681 0.232147i \(-0.925425\pi\)
0.972681 0.232147i \(-0.0745750\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000i 0.611775i
\(172\) 0 0
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 0 0
\(175\) 4.00000 + 3.00000i 0.302372 + 0.226779i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 20.0000i 1.49487i −0.664335 0.747435i \(-0.731285\pi\)
0.664335 0.747435i \(-0.268715\pi\)
\(180\) 0 0
\(181\) 2.00000i 0.148659i 0.997234 + 0.0743294i \(0.0236816\pi\)
−0.997234 + 0.0743294i \(0.976318\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) 12.0000 6.00000i 0.882258 0.441129i
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) 0 0
\(195\) 4.00000 2.00000i 0.286446 0.143223i
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 32.0000 2.21349
\(210\) 0 0
\(211\) 4.00000i 0.275371i 0.990476 + 0.137686i \(0.0439664\pi\)
−0.990476 + 0.137686i \(0.956034\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) −20.0000 + 10.0000i −1.36399 + 0.681994i
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 4.00000i 0.269069i
\(222\) 0 0
\(223\) 4.00000i 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 3.00000 4.00000i 0.200000 0.266667i
\(226\) 0 0
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i −0.886581 0.462573i \(-0.846926\pi\)
0.886581 0.462573i \(-0.153074\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 6.00000 + 12.0000i 0.391397 + 0.782794i
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.00000 + 1.00000i −0.127775 + 0.0638877i
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 26.0000i 1.64111i 0.571571 + 0.820553i \(0.306334\pi\)
−0.571571 + 0.820553i \(0.693666\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) −2.00000 4.00000i −0.125245 0.250490i
\(256\) 0 0
\(257\) 10.0000i 0.623783i −0.950118 0.311891i \(-0.899037\pi\)
0.950118 0.311891i \(-0.100963\pi\)
\(258\) 0 0
\(259\) 6.00000i 0.372822i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) 16.0000i 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 0 0
\(265\) 28.0000 14.0000i 1.72003 0.860013i
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 0 0
\(269\) 18.0000i 1.09748i 0.835993 + 0.548740i \(0.184892\pi\)
−0.835993 + 0.548740i \(0.815108\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 0 0
\(275\) 16.0000 + 12.0000i 0.964836 + 0.723627i
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 8.00000 + 16.0000i 0.473879 + 0.947758i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 6.00000i 0.351726i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −6.00000 12.0000i −0.349334 0.698667i
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 8.00000i 0.462652i
\(300\) 0 0
\(301\) 10.0000i 0.576390i
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) −2.00000 4.00000i −0.114520 0.229039i
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 8.00000i 0.455104i
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i −0.918396 0.395663i \(-0.870515\pi\)
0.918396 0.395663i \(-0.129485\pi\)
\(314\) 0 0
\(315\) 1.00000 + 2.00000i 0.0563436 + 0.112687i
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) −6.00000 + 8.00000i −0.332820 + 0.443760i
\(326\) 0 0
\(327\) 16.0000i 0.884802i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 4.00000i 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 20.0000 10.0000i 1.09272 0.546358i
\(336\) 0 0
\(337\) 8.00000i 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 4.00000 + 8.00000i 0.215353 + 0.430706i
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 10.0000i 0.535288i −0.963518 0.267644i \(-0.913755\pi\)
0.963518 0.267644i \(-0.0862451\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 2.00000i 0.106449i −0.998583 0.0532246i \(-0.983050\pi\)
0.998583 0.0532246i \(-0.0169499\pi\)
\(354\) 0 0
\(355\) 24.0000 12.0000i 1.27379 0.636894i
\(356\) 0 0
\(357\) 2.00000 0.105851
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 6.00000 + 12.0000i 0.314054 + 0.628109i
\(366\) 0 0
\(367\) 12.0000i 0.626395i −0.949688 0.313197i \(-0.898600\pi\)
0.949688 0.313197i \(-0.101400\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.0000i 0.726844i
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) −2.00000 + 11.0000i −0.103280 + 0.568038i
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 0 0
\(379\) 36.0000i 1.84920i 0.380945 + 0.924598i \(0.375599\pi\)
−0.380945 + 0.924598i \(0.624401\pi\)
\(380\) 0 0
\(381\) 16.0000i 0.819705i
\(382\) 0 0
\(383\) 22.0000i 1.12415i −0.827087 0.562074i \(-0.810004\pi\)
0.827087 0.562074i \(-0.189996\pi\)
\(384\) 0 0
\(385\) −8.00000 + 4.00000i −0.407718 + 0.203859i
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) 0 0
\(389\) 38.0000i 1.92668i −0.268290 0.963338i \(-0.586458\pi\)
0.268290 0.963338i \(-0.413542\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 28.0000 14.0000i 1.40883 0.704416i
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 0 0
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 2.00000 1.00000i 0.0993808 0.0496904i
\(406\) 0 0
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 6.00000i 0.295958i
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −24.0000 + 12.0000i −1.17811 + 0.589057i
\(416\) 0 0
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) 26.0000i 1.27018i 0.772437 + 0.635092i \(0.219038\pi\)
−0.772437 + 0.635092i \(0.780962\pi\)
\(420\) 0 0
\(421\) 4.00000i 0.194948i −0.995238 0.0974740i \(-0.968924\pi\)
0.995238 0.0974740i \(-0.0310763\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 8.00000 + 6.00000i 0.388057 + 0.291043i
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 8.00000i 0.386244i
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 0 0
\(435\) 2.00000 + 4.00000i 0.0958927 + 0.191785i
\(436\) 0 0
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) 16.0000 8.00000i 0.758473 0.379236i
\(446\) 0 0
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.00000 0.0939682
\(454\) 0 0
\(455\) −2.00000 4.00000i −0.0937614 0.187523i
\(456\) 0 0
\(457\) 20.0000i 0.935561i 0.883845 + 0.467780i \(0.154946\pi\)
−0.883845 + 0.467780i \(0.845054\pi\)
\(458\) 0 0
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) 18.0000i 0.838344i −0.907907 0.419172i \(-0.862320\pi\)
0.907907 0.419172i \(-0.137680\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) −8.00000 + 4.00000i −0.370991 + 0.185496i
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 10.0000i 0.461757i
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) 40.0000i 1.83920i
\(474\) 0 0
\(475\) −32.0000 24.0000i −1.46826 1.10120i
\(476\) 0 0
\(477\) 14.0000 0.641016
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) −4.00000 −0.182006
\(484\) 0 0
\(485\) 6.00000 + 12.0000i 0.272446 + 0.544892i
\(486\) 0 0
\(487\) 8.00000i 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 0 0
\(489\) −22.0000 −0.994874
\(490\) 0 0
\(491\) 8.00000i 0.361035i 0.983572 + 0.180517i \(0.0577772\pi\)
−0.983572 + 0.180517i \(0.942223\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) 4.00000 + 8.00000i 0.179787 + 0.359573i
\(496\) 0 0
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) 20.0000i 0.895323i 0.894203 + 0.447661i \(0.147743\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(500\) 0 0
\(501\) 6.00000i 0.268060i
\(502\) 0 0
\(503\) 10.0000i 0.445878i −0.974832 0.222939i \(-0.928435\pi\)
0.974832 0.222939i \(-0.0715651\pi\)
\(504\) 0 0
\(505\) 6.00000 + 12.0000i 0.266996 + 0.533993i
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) 2.00000i 0.0886484i −0.999017 0.0443242i \(-0.985887\pi\)
0.999017 0.0443242i \(-0.0141135\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 8.00000i 0.353209i
\(514\) 0 0
\(515\) 8.00000 + 16.0000i 0.352522 + 0.705044i
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) −4.00000 3.00000i −0.174574 0.130931i
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −8.00000 + 4.00000i −0.345870 + 0.172935i
\(536\) 0 0
\(537\) 20.0000i 0.863064i
\(538\) 0 0
\(539\) 4.00000i 0.172292i
\(540\) 0 0
\(541\) 28.0000i 1.20381i 0.798566 + 0.601907i \(0.205592\pi\)
−0.798566 + 0.601907i \(0.794408\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) −16.0000 32.0000i −0.685365 1.37073i
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) 0 0
\(549\) 2.00000i 0.0853579i
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) 0 0
\(553\) 14.0000i 0.595341i
\(554\) 0 0
\(555\) −12.0000 + 6.00000i −0.509372 + 0.254686i
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 6.00000 + 12.0000i 0.252422 + 0.504844i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) −16.0000 12.0000i −0.667246 0.500435i
\(576\) 0 0
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) 0 0
\(579\) 16.0000i 0.664937i
\(580\) 0 0
\(581\) 12.0000i 0.497844i
\(582\) 0 0
\(583\) 56.0000i 2.31928i
\(584\) 0 0
\(585\) −4.00000 + 2.00000i −0.165380 + 0.0826898i
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 18.0000i 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) −4.00000 + 2.00000i −0.163984 + 0.0819920i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) −10.0000 + 5.00000i −0.406558 + 0.203279i
\(606\) 0 0
\(607\) 16.0000i 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000i 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 0 0
\(619\) 40.0000i 1.60774i −0.594808 0.803868i \(-0.702772\pi\)
0.594808 0.803868i \(-0.297228\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) −32.0000 −1.27796
\(628\) 0 0
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 16.0000 + 32.0000i 0.634941 + 1.26988i
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) 20.0000 10.0000i 0.787499 0.393750i
\(646\) 0 0
\(647\) 38.0000i 1.49393i 0.664861 + 0.746967i \(0.268491\pi\)
−0.664861 + 0.746967i \(0.731509\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 4.00000i 0.156772i
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) −18.0000 36.0000i −0.703318 1.40664i
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 24.0000i 0.934907i 0.884018 + 0.467454i \(0.154829\pi\)
−0.884018 + 0.467454i \(0.845171\pi\)
\(660\) 0 0
\(661\) 42.0000i 1.63361i 0.576913 + 0.816805i \(0.304257\pi\)
−0.576913 + 0.816805i \(0.695743\pi\)
\(662\) 0 0
\(663\) 4.00000i 0.155347i
\(664\) 0 0
\(665\) 16.0000 8.00000i 0.620453 0.310227i
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) 4.00000i 0.154649i
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 4.00000i 0.154189i 0.997024 + 0.0770943i \(0.0245643\pi\)
−0.997024 + 0.0770943i \(0.975436\pi\)
\(674\) 0 0
\(675\) −3.00000 + 4.00000i −0.115470 + 0.153960i
\(676\) 0 0
\(677\) 32.0000 1.22986 0.614930 0.788582i \(-0.289184\pi\)
0.614930 + 0.788582i \(0.289184\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −6.00000 12.0000i −0.229248 0.458496i
\(686\) 0 0
\(687\) 14.0000i 0.534133i
\(688\) 0 0
\(689\) −28.0000 −1.06672
\(690\) 0 0
\(691\) 20.0000i 0.760836i −0.924815 0.380418i \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −12.0000 24.0000i −0.455186 0.910372i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 18.0000i 0.679851i 0.940452 + 0.339925i \(0.110402\pi\)
−0.940452 + 0.339925i \(0.889598\pi\)
\(702\) 0 0
\(703\) 48.0000i 1.81035i
\(704\) 0 0
\(705\) −6.00000 12.0000i −0.225973 0.451946i
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 48.0000i 1.80268i 0.433114 + 0.901339i \(0.357415\pi\)
−0.433114 + 0.901339i \(0.642585\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) −8.00000 16.0000i −0.299183 0.598366i
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 10.0000 0.371904
\(724\) 0 0
\(725\) −8.00000 6.00000i −0.297113 0.222834i
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 20.0000i 0.739727i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 2.00000 1.00000i 0.0737711 0.0368856i
\(736\) 0 0
\(737\) 40.0000i 1.47342i
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) 16.0000i 0.587775i
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 10.0000 + 20.0000i 0.366372 + 0.732743i
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 4.00000i 0.146157i
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 26.0000i 0.947493i
\(754\) 0 0
\(755\) −4.00000 + 2.00000i −0.145575 + 0.0727875i
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −32.0000 −1.16000 −0.580000 0.814617i \(-0.696947\pi\)
−0.580000 + 0.814617i \(0.696947\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 2.00000 + 4.00000i 0.0723102 + 0.144620i
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 10.0000i 0.360141i
\(772\) 0 0
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 12.0000 16.0000i 0.431053 0.574737i
\(776\) 0 0
\(777\) 6.00000i 0.215249i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 48.0000i 1.71758i
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) −20.0000 + 10.0000i −0.713831 + 0.356915i
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 0 0
\(789\) 16.0000i 0.569615i
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) −28.0000 + 14.0000i −0.993058 + 0.496529i
\(796\) 0 0
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 8.00000 0.282666