Properties

Label 5850.2.e.a.5149.1
Level $5850$
Weight $2$
Character 5850.5149
Analytic conductor $46.712$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5149.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5850.5149
Dual form 5850.2.e.a.5149.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{11} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000 q^{19} +6.00000i q^{22} +1.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +7.00000i q^{37} +2.00000i q^{38} -1.00000i q^{43} +6.00000 q^{44} +3.00000i q^{47} +6.00000 q^{49} -1.00000i q^{52} -1.00000 q^{56} -6.00000i q^{58} -6.00000 q^{59} +8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -14.0000i q^{67} +3.00000i q^{68} +3.00000 q^{71} +2.00000i q^{73} +7.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -8.00000 q^{79} -12.0000i q^{83} -1.00000 q^{86} -6.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} +3.00000 q^{94} +10.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 12 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 12 q^{29} - 8 q^{31} - 6 q^{34} + 12 q^{44} + 12 q^{49} - 2 q^{56} - 12 q^{59} + 16 q^{61} - 2 q^{64} + 6 q^{71} + 14 q^{74} + 4 q^{76} - 16 q^{79} - 2 q^{86} - 12 q^{89} - 2 q^{91} + 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times\).

\(n\) \(2251\) \(3251\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000i 1.27920i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) − 1.00000i − 0.188982i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 2.00000i 0.324443i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.00000i − 0.138675i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 14.0000i − 1.71037i −0.518321 0.855186i \(-0.673443\pi\)
0.518321 0.855186i \(-0.326557\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) − 6.00000i − 0.683763i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) − 6.00000i − 0.639602i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 6.00000i 0.552345i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) − 8.00000i − 0.724286i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 20.0000i − 1.77471i −0.461084 0.887357i \(-0.652539\pi\)
0.461084 0.887357i \(-0.347461\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 0 0
\(133\) − 2.00000i − 0.173422i
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 3.00000i − 0.251754i
\(143\) − 6.00000i − 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) − 7.00000i − 0.575396i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000i 0.0762493i
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 1.00000i 0.0741249i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 18.0000i 1.31629i
\(188\) − 3.00000i − 0.218797i
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 12.0000i − 0.844317i
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 1.00000i 0.0693375i
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) − 7.00000i − 0.474100i
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) − 19.0000i − 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 27.0000i 1.76883i 0.466702 + 0.884414i \(0.345442\pi\)
−0.466702 + 0.884414i \(0.654558\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) − 3.00000i − 0.194461i
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) − 25.0000i − 1.60706i
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.00000i − 0.127257i
\(248\) − 4.00000i − 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.00000i 0.561405i 0.959795 + 0.280702i \(0.0905674\pi\)
−0.959795 + 0.280702i \(0.909433\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 0 0
\(261\) 0 0
\(262\) − 21.0000i − 1.29738i
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 14.0000i 0.855186i
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) − 13.0000i − 0.779688i
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) −3.00000 −0.178017
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) − 2.00000i − 0.117041i
\(293\) − 21.0000i − 1.22683i −0.789760 0.613417i \(-0.789795\pi\)
0.789760 0.613417i \(-0.210205\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) 6.00000i 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) − 17.0000i − 0.978240i
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.00000i − 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) 6.00000i 0.341882i
\(309\) 0 0
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) − 1.00000i − 0.0565233i −0.999601 0.0282617i \(-0.991003\pi\)
0.999601 0.0282617i \(-0.00899717\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) −36.0000 −2.01561
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000i 0.333849i
\(324\) 0 0
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 23.0000i − 1.25289i −0.779466 0.626445i \(-0.784509\pi\)
0.779466 0.626445i \(-0.215491\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 0 0
\(347\) 3.00000i 0.161048i 0.996753 + 0.0805242i \(0.0256594\pi\)
−0.996753 + 0.0805242i \(0.974341\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.00000i 0.319801i
\(353\) − 24.0000i − 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) − 3.00000i − 0.158555i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 20.0000i − 1.05118i
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) − 26.0000i − 1.35719i −0.734513 0.678594i \(-0.762589\pi\)
0.734513 0.678594i \(-0.237411\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 18.0000i − 0.920960i
\(383\) − 21.0000i − 1.07305i −0.843884 0.536525i \(-0.819737\pi\)
0.843884 0.536525i \(-0.180263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 0 0
\(388\) − 10.0000i − 0.507673i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000i 0.303046i
\(393\) 0 0
\(394\) 3.00000 0.151138
\(395\) 0 0
\(396\) 0 0
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) 2.00000i 0.100251i
\(399\) 0 0
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) − 42.0000i − 2.08186i
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.00000i 0.197066i
\(413\) − 6.00000i − 0.295241i
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) − 12.0000i − 0.586939i
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 13.0000i 0.632830i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 0 0
\(433\) − 25.0000i − 1.20142i −0.799466 0.600712i \(-0.794884\pi\)
0.799466 0.600712i \(-0.205116\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) 0 0
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3.00000i − 0.142695i
\(443\) − 21.0000i − 0.997740i −0.866677 0.498870i \(-0.833748\pi\)
0.866677 0.498870i \(-0.166252\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −19.0000 −0.899676
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 6.00000i − 0.282216i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) − 13.0000i − 0.607450i
\(459\) 0 0
\(460\) 0 0
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) 0 0
\(463\) − 40.0000i − 1.85896i −0.368875 0.929479i \(-0.620257\pi\)
0.368875 0.929479i \(-0.379743\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 27.0000 1.25075
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) − 6.00000i − 0.276172i
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) − 15.0000i − 0.686084i
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 10.0000i 0.455488i
\(483\) 0 0
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) 8.00000i 0.362143i
\(489\) 0 0
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 3.00000i 0.134568i
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 24.0000i 1.07117i
\(503\) 30.0000i 1.33763i 0.743427 + 0.668817i \(0.233199\pi\)
−0.743427 + 0.668817i \(0.766801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 20.0000i 0.887357i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 9.00000 0.396973
\(515\) 0 0
\(516\) 0 0
\(517\) − 18.0000i − 0.791639i
\(518\) 7.00000i 0.307562i
\(519\) 0 0
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) −21.0000 −0.917389
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000i 0.0867110i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 14.0000 0.604708
\(537\) 0 0
\(538\) − 24.0000i − 1.03471i
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) − 11.0000i − 0.472490i
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) − 17.0000i − 0.726868i −0.931620 0.363434i \(-0.881604\pi\)
0.931620 0.363434i \(-0.118396\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) −13.0000 −0.551323
\(557\) 3.00000i 0.127114i 0.997978 + 0.0635570i \(0.0202445\pi\)
−0.997978 + 0.0635570i \(0.979756\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) − 6.00000i − 0.253095i
\(563\) − 39.0000i − 1.64365i −0.569737 0.821827i \(-0.692955\pi\)
0.569737 0.821827i \(-0.307045\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 3.00000i 0.125877i
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 6.00000i 0.250873i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −21.0000 −0.867502
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 7.00000i 0.287698i
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) − 1.00000i − 0.0407570i
\(603\) 0 0
\(604\) −17.0000 −0.691720
\(605\) 0 0
\(606\) 0 0
\(607\) − 14.0000i − 0.568242i −0.958788 0.284121i \(-0.908298\pi\)
0.958788 0.284121i \(-0.0917018\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) 38.0000i 1.53481i 0.641165 + 0.767403i \(0.278451\pi\)
−0.641165 + 0.767403i \(0.721549\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) − 24.0000i − 0.966204i −0.875564 0.483102i \(-0.839510\pi\)
0.875564 0.483102i \(-0.160490\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 30.0000i − 1.20289i
\(623\) − 6.00000i − 0.240385i
\(624\) 0 0
\(625\) 0 0
\(626\) −1.00000 −0.0399680
\(627\) 0 0
\(628\) 14.0000i 0.558661i
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 36.0000i 1.42525i
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) − 6.00000i − 0.235884i −0.993020 0.117942i \(-0.962370\pi\)
0.993020 0.117942i \(-0.0376297\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000i 0.626608i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 3.00000i 0.116952i
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) − 8.00000i − 0.310929i
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) − 19.0000i − 0.732396i −0.930537 0.366198i \(-0.880659\pi\)
0.930537 0.366198i \(-0.119341\pi\)
\(674\) −23.0000 −0.885927
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) − 24.0000i − 0.919007i
\(683\) − 24.0000i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) − 1.00000i − 0.0381246i
\(689\) 0 0
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) − 19.0000i − 0.719161i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) − 14.0000i − 0.528020i
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6.00000i − 0.224860i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) 0 0
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 15.0000i 0.558242i
\(723\) 0 0
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) 0 0
\(727\) 10.0000i 0.370879i 0.982656 + 0.185440i \(0.0593710\pi\)
−0.982656 + 0.185440i \(0.940629\pi\)
\(728\) − 1.00000i − 0.0370625i
\(729\) 0 0
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) 23.0000i 0.849524i 0.905305 + 0.424762i \(0.139642\pi\)
−0.905305 + 0.424762i \(0.860358\pi\)
\(734\) −26.0000 −0.959678
\(735\) 0 0
\(736\) 0 0
\(737\) 84.0000i 3.09418i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 0 0
\(748\) − 18.0000i − 0.658145i
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 3.00000i 0.109399i
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) 16.0000i 0.581530i 0.956795 + 0.290765i \(0.0939098\pi\)
−0.956795 + 0.290765i \(0.906090\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 7.00000i 0.253417i
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −21.0000 −0.758761
\(767\) − 6.00000i − 0.216647i
\(768\) 0 0
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) 39.0000i 1.40273i 0.712801 + 0.701366i \(0.247426\pi\)
−0.712801 + 0.701366i \(0.752574\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) 0 0
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) 40.0000i 1.42585i 0.701242 + 0.712923i \(0.252629\pi\)
−0.701242 + 0.712923i \(0.747371\pi\)
\(788\) − 3.00000i − 0.106871i
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 8.00000i 0.284088i
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 0 0
\(802\) 36.0000i 1.27120i
\(803\) − 12.0000i − 0.423471i
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 12.0000i 0.422159i
\(809\) −33.0000 −1.16022 −0.580109 0.814539i \(-0.696990\pi\)
−0.580109 + 0.814539i \(0.696990\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 0 0
\(814\) −42.0000 −1.47210
\(815\) 0 0
\(816\) 0 0
\(817\) 2.00000i 0.0699711i
\(818\) 32.0000i 1.11885i
\(819\) 0 0
\(820\) 0 0
\(821\) 3.00000 0.104701 0.0523504 0.998629i \(-0.483329\pi\)
0.0523504 + 0.998629i \(0.483329\pi\)
\(822\) 0 0
\(823\) 14.0000i 0.488009i 0.969774 + 0.244005i \(0.0784612\pi\)
−0.969774 + 0.244005i \(0.921539\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 18.0000i 0.625921i 0.949766 + 0.312961i \(0.101321\pi\)
−0.949766 + 0.312961i \(0.898679\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.00000i − 0.0346688i
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) − 9.00000i − 0.310900i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 17.0000i − 0.585859i
\(843\) 0 0
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 25.0000i 0.859010i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 37.0000i − 1.26686i −0.773802 0.633428i \(-0.781647\pi\)
0.773802 0.633428i \(-0.218353\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 33.0000i − 1.12398i
\(863\) 45.0000i 1.53182i 0.642949 + 0.765909i \(0.277711\pi\)
−0.642949 + 0.765909i \(0.722289\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −25.0000 −0.849535
\(867\) 0 0
\(868\) 4.00000i 0.135769i
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 7.00000i 0.237050i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0000i 0.438979i 0.975615 + 0.219489i \(0.0704391\pi\)
−0.975615 + 0.219489i \(0.929561\pi\)
\(878\) 26.0000i 0.877457i
\(879\) 0 0
\(880\) 0 0
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) 0 0
\(883\) 29.0000i 0.975928i 0.872864 + 0.487964i \(0.162260\pi\)
−0.872864 + 0.487964i \(0.837740\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −21.0000 −0.705509
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) 19.0000i 0.636167i
\(893\) − 6.00000i − 0.200782i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) − 6.00000i − 0.200223i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 37.0000i 1.22856i 0.789086 + 0.614282i \(0.210554\pi\)
−0.789086 + 0.614282i \(0.789446\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 72.0000i 2.38285i
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) 21.0000i 0.693481i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 9.00000i 0.296399i
\(923\) 3.00000i 0.0987462i
\(924\) 0 0
\(925\) 0 0
\(926\) −40.0000 −1.31448
\(927\) 0 0
\(928\) − 6.00000i − 0.196960i
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) − 27.0000i − 0.884414i
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) 34.0000i 1.11073i 0.831606 + 0.555366i \(0.187422\pi\)
−0.831606 + 0.555366i \(0.812578\pi\)
\(938\) − 14.0000i − 0.457116i
\(939\) 0 0
\(940\) 0 0
\(941\) 21.0000 0.684580 0.342290 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 6.00000i 0.194974i 0.995237 + 0.0974869i \(0.0310804\pi\)
−0.995237 + 0.0974869i \(0.968920\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 3.00000i 0.0972306i
\(953\) − 15.0000i − 0.485898i −0.970039 0.242949i \(-0.921885\pi\)
0.970039 0.242949i \(-0.0781147\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −15.0000 −0.485135
\(957\) 0 0
\(958\) 21.0000i 0.678479i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 7.00000i 0.225689i
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 31.0000i 0.996893i 0.866921 + 0.498446i \(0.166096\pi\)
−0.866921 + 0.498446i \(0.833904\pi\)
\(968\) 25.0000i 0.803530i
\(969\) 0 0
\(970\) 0 0
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 0 0
\(973\) 13.0000i 0.416761i
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) − 54.0000i − 1.72761i −0.503824 0.863807i \(-0.668074\pi\)
0.503824 0.863807i \(-0.331926\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 0 0
\(982\) − 9.00000i − 0.287202i
\(983\) − 39.0000i − 1.24391i −0.783054 0.621953i \(-0.786339\pi\)
0.783054 0.621953i \(-0.213661\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) 2.00000i 0.0636285i
\(989\) 0 0
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 4.00000i 0.127000i
\(993\) 0 0
\(994\) 3.00000 0.0951542
\(995\) 0 0
\(996\) 0 0
\(997\) 46.0000i 1.45683i 0.685134 + 0.728417i \(0.259744\pi\)
−0.685134 + 0.728417i \(0.740256\pi\)
\(998\) − 40.0000i − 1.26618i
\(999\) 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 5850.2.e.a.5149.1 2
3.2 odd 2 650.2.b.d.599.2 2
5.2 odd 4 234.2.a.e.1.1 1
5.3 odd 4 5850.2.a.p.1.1 1
5.4 even 2 inner 5850.2.e.a.5149.2 2
15.2 even 4 26.2.a.a.1.1 1
15.8 even 4 650.2.a.j.1.1 1
15.14 odd 2 650.2.b.d.599.1 2
20.7 even 4 1872.2.a.q.1.1 1
40.27 even 4 7488.2.a.h.1.1 1
40.37 odd 4 7488.2.a.g.1.1 1
45.2 even 12 2106.2.e.ba.1405.1 2
45.7 odd 12 2106.2.e.b.1405.1 2
45.22 odd 12 2106.2.e.b.703.1 2
45.32 even 12 2106.2.e.ba.703.1 2
60.23 odd 4 5200.2.a.x.1.1 1
60.47 odd 4 208.2.a.a.1.1 1
65.12 odd 4 3042.2.a.a.1.1 1
65.47 even 4 3042.2.b.a.1351.2 2
65.57 even 4 3042.2.b.a.1351.1 2
105.2 even 12 1274.2.f.p.1145.1 2
105.17 odd 12 1274.2.f.r.79.1 2
105.32 even 12 1274.2.f.p.79.1 2
105.47 odd 12 1274.2.f.r.1145.1 2
105.62 odd 4 1274.2.a.d.1.1 1
120.77 even 4 832.2.a.d.1.1 1
120.107 odd 4 832.2.a.i.1.1 1
165.32 odd 4 3146.2.a.n.1.1 1
195.2 odd 12 338.2.e.a.147.1 4
195.17 even 12 338.2.c.a.315.1 2
195.32 odd 12 338.2.e.a.23.2 4
195.38 even 4 8450.2.a.c.1.1 1
195.47 odd 4 338.2.b.c.337.1 2
195.62 even 12 338.2.c.a.191.1 2
195.77 even 4 338.2.a.f.1.1 1
195.107 even 12 338.2.c.d.191.1 2
195.122 odd 4 338.2.b.c.337.2 2
195.137 odd 12 338.2.e.a.23.1 4
195.152 even 12 338.2.c.d.315.1 2
195.167 odd 12 338.2.e.a.147.2 4
240.77 even 4 3328.2.b.m.1665.2 2
240.107 odd 4 3328.2.b.j.1665.2 2
240.197 even 4 3328.2.b.m.1665.1 2
240.227 odd 4 3328.2.b.j.1665.1 2
255.152 even 4 7514.2.a.c.1.1 1
285.227 odd 4 9386.2.a.j.1.1 1
780.47 even 4 2704.2.f.d.337.1 2
780.467 odd 4 2704.2.a.f.1.1 1
780.707 even 4 2704.2.f.d.337.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.a.1.1 1 15.2 even 4
208.2.a.a.1.1 1 60.47 odd 4
234.2.a.e.1.1 1 5.2 odd 4
338.2.a.f.1.1 1 195.77 even 4
338.2.b.c.337.1 2 195.47 odd 4
338.2.b.c.337.2 2 195.122 odd 4
338.2.c.a.191.1 2 195.62 even 12
338.2.c.a.315.1 2 195.17 even 12
338.2.c.d.191.1 2 195.107 even 12
338.2.c.d.315.1 2 195.152 even 12
338.2.e.a.23.1 4 195.137 odd 12
338.2.e.a.23.2 4 195.32 odd 12
338.2.e.a.147.1 4 195.2 odd 12
338.2.e.a.147.2 4 195.167 odd 12
650.2.a.j.1.1 1 15.8 even 4
650.2.b.d.599.1 2 15.14 odd 2
650.2.b.d.599.2 2 3.2 odd 2
832.2.a.d.1.1 1 120.77 even 4
832.2.a.i.1.1 1 120.107 odd 4
1274.2.a.d.1.1 1 105.62 odd 4
1274.2.f.p.79.1 2 105.32 even 12
1274.2.f.p.1145.1 2 105.2 even 12
1274.2.f.r.79.1 2 105.17 odd 12
1274.2.f.r.1145.1 2 105.47 odd 12
1872.2.a.q.1.1 1 20.7 even 4
2106.2.e.b.703.1 2 45.22 odd 12
2106.2.e.b.1405.1 2 45.7 odd 12
2106.2.e.ba.703.1 2 45.32 even 12
2106.2.e.ba.1405.1 2 45.2 even 12
2704.2.a.f.1.1 1 780.467 odd 4
2704.2.f.d.337.1 2 780.47 even 4
2704.2.f.d.337.2 2 780.707 even 4
3042.2.a.a.1.1 1 65.12 odd 4
3042.2.b.a.1351.1 2 65.57 even 4
3042.2.b.a.1351.2 2 65.47 even 4
3146.2.a.n.1.1 1 165.32 odd 4
3328.2.b.j.1665.1 2 240.227 odd 4
3328.2.b.j.1665.2 2 240.107 odd 4
3328.2.b.m.1665.1 2 240.197 even 4
3328.2.b.m.1665.2 2 240.77 even 4
5200.2.a.x.1.1 1 60.23 odd 4
5850.2.a.p.1.1 1 5.3 odd 4
5850.2.e.a.5149.1 2 1.1 even 1 trivial
5850.2.e.a.5149.2 2 5.4 even 2 inner
7488.2.a.g.1.1 1 40.37 odd 4
7488.2.a.h.1.1 1 40.27 even 4
7514.2.a.c.1.1 1 255.152 even 4
8450.2.a.c.1.1 1 195.38 even 4
9386.2.a.j.1.1 1 285.227 odd 4