Properties

Label 350.2.c.b.99.2
Level $350$
Weight $2$
Character 350.99
Analytic conductor $2.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.2.c.b.99.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000i q^{22} -6.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} -10.0000i q^{37} +2.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +8.00000i q^{47} -1.00000 q^{49} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +8.00000 q^{59} -14.0000 q^{61} +8.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} -16.0000 q^{71} -3.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} -4.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} +2.00000i q^{82} -8.00000i q^{83} +4.00000 q^{86} -4.00000i q^{88} -10.0000 q^{89} +6.00000 q^{91} -8.00000 q^{94} +2.00000i q^{97} -1.00000i q^{98} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 6q^{9} + 8q^{11} + 2q^{14} + 2q^{16} - 12q^{26} - 12q^{29} + 16q^{31} - 4q^{34} - 6q^{36} + 4q^{41} - 8q^{44} - 2q^{49} - 2q^{56} + 16q^{59} - 28q^{61} - 2q^{64} - 32q^{71} + 20q^{74} + 16q^{79} + 18q^{81} + 8q^{86} - 20q^{89} + 12q^{91} - 16q^{94} + 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 3.00000i 0.707107i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 6.00000i − 0.832050i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 8.00000i 1.01600i
\(63\) − 3.00000i − 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 0 0
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 2.00000i 0.220863i
\(83\) − 8.00000i − 0.878114i −0.898459 0.439057i \(-0.855313\pi\)
0.898459 0.439057i \(-0.144687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) − 4.00000i − 0.426401i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 18.0000i 1.66410i
\(118\) 8.00000i 0.736460i
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 14.0000i − 1.26750i
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 16.0000i − 1.34269i
\(143\) 24.0000i 2.00698i
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 10.0000i 0.821995i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000i 0.707107i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000i 0.304997i
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) − 10.0000i − 0.749532i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 6.00000i 0.444750i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) − 8.00000i − 0.583460i
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 12.0000i 0.852803i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 6.00000i − 0.422159i
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 6.00000i 0.416025i
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) − 8.00000i − 0.543075i
\(218\) − 6.00000i − 0.406371i
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −18.0000 −1.17670
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 2.00000i 0.129641i
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) − 8.00000i − 0.508001i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 22.0000i − 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) − 16.0000i − 0.988483i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000i 0.733017i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) − 16.0000i − 0.959616i
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) − 32.0000i − 1.90220i −0.308879 0.951101i \(-0.599954\pi\)
0.308879 0.951101i \(-0.400046\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) − 2.00000i − 0.118056i
\(288\) 3.00000i 0.176777i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000i 0.117041i
\(293\) − 10.0000i − 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) − 6.00000i − 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) 4.00000i 0.227921i
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 22.0000i 1.23564i 0.786318 + 0.617822i \(0.211985\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) − 2.00000i − 0.110432i
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 8.00000i 0.439057i
\(333\) − 30.0000i − 1.64399i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) 0 0
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000i 0.213201i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 14.0000i − 0.735824i
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) − 14.0000i − 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) − 36.0000i − 1.85409i
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 24.0000i 1.22795i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) − 12.0000i − 0.609994i
\(388\) − 2.00000i − 0.101535i
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) 10.0000i 0.501886i 0.968002 + 0.250943i \(0.0807406\pi\)
−0.968002 + 0.250943i \(0.919259\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 48.0000i 2.39105i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) − 40.0000i − 1.98273i
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 16.0000i 0.788263i
\(413\) − 8.00000i − 0.393654i
\(414\) 0 0
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 24.0000i 1.16692i
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) 14.0000i 0.677507i
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) − 12.0000i − 0.570782i
\(443\) 20.0000i 0.950229i 0.879924 + 0.475114i \(0.157593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 2.00000i 0.0940721i
\(453\) 0 0
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 0 0
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 40.0000i − 1.85098i −0.378773 0.925490i \(-0.623654\pi\)
0.378773 0.925490i \(-0.376346\pi\)
\(468\) − 18.0000i − 0.832050i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) − 8.00000i − 0.368230i
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 6.00000i 0.274721i
\(478\) − 16.0000i − 0.731823i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 60.0000 2.73576
\(482\) 10.0000i 0.455488i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) 14.0000i 0.633750i
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 16.0000i 0.717698i
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000i 0.354943i
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) 0 0
\(517\) 32.0000i 1.40736i
\(518\) − 10.0000i − 0.439375i
\(519\) 0 0
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) − 18.0000i − 0.787839i
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 6.00000i 0.258678i
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −42.0000 −1.79252
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 22.0000i 0.932170i 0.884740 + 0.466085i \(0.154336\pi\)
−0.884740 + 0.466085i \(0.845664\pi\)
\(558\) 24.0000i 1.01600i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 26.0000i 1.09674i
\(563\) 16.0000i 0.674320i 0.941447 + 0.337160i \(0.109466\pi\)
−0.941447 + 0.337160i \(0.890534\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 32.0000 1.34506
\(567\) − 9.00000i − 0.377964i
\(568\) 16.0000i 0.671345i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) − 24.0000i − 1.00349i
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 10.0000 0.413096
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) − 10.0000i − 0.410997i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) − 4.00000i − 0.163028i
\(603\) − 36.0000i − 1.46603i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) − 6.00000i − 0.242536i
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) − 22.0000i − 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.0000i − 0.962312i
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) − 10.0000i − 0.399043i
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 0 0
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) − 24.0000i − 0.950169i
\(639\) −48.0000 −1.89885
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 32.0000i 1.26196i 0.775800 + 0.630978i \(0.217346\pi\)
−0.775800 + 0.630978i \(0.782654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000i 0.629025i 0.949253 + 0.314512i \(0.101841\pi\)
−0.949253 + 0.314512i \(0.898159\pi\)
\(648\) − 9.00000i − 0.353553i
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) − 4.00000i − 0.156652i
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) − 6.00000i − 0.234082i
\(658\) 8.00000i 0.311872i
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 4.00000i 0.155464i
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 30.0000 1.16248
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 32.0000i 1.22534i
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) − 4.00000i − 0.152499i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) − 22.0000i − 0.836315i
\(693\) − 12.0000i − 0.455842i
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) − 10.0000i − 0.378506i
\(699\) 0 0
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 10.0000i 0.374766i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) − 8.00000i − 0.298557i
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) − 19.0000i − 0.707107i
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) − 6.00000i − 0.222375i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) − 48.0000i − 1.76810i
\(738\) 6.00000i 0.220863i
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.00000i 0.0734223i
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) − 24.0000i − 0.878114i
\(748\) − 8.00000i − 0.292509i
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000i 0.799604i 0.916602 + 0.399802i \(0.130921\pi\)
−0.916602 + 0.399802i \(0.869079\pi\)
\(758\) 12.0000i 0.435860i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 6.00000i 0.217215i
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 48.0000i 1.73318i
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000i 0.0719816i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) − 14.0000i − 0.501924i
\(779\) 0 0
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) 0 0
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 8.00000i − 0.285169i −0.989783 0.142585i \(-0.954459\pi\)
0.989783 0.142585i \(-0.0455413\pi\)
\(788\) − 14.0000i − 0.498729i
\(789\) 0 0
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) − 12.0000i − 0.426401i
\(793\) − 84.0000i − 2.98293i
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) − 14.0000i − 0.494357i
\(803\) − 8.00000i − 0.282314i
\(804\) 0 0
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 0 0
\(808\) 6.00000i 0.211079i
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 30.0000i 1.04893i
\(819\) 18.0000 0.628971
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 8.00000i 0.278862i 0.990232 + 0.139431i \(0.0445274\pi\)
−0.990232 + 0.139431i \(0.955473\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 44.0000i 1.53003i 0.644013 + 0.765015i \(0.277268\pi\)
−0.644013 + 0.765015i \(0.722732\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 6.00000i − 0.208013i
\(833\) − 2.00000i − 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 24.0000i − 0.829066i
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 10.0000i − 0.344623i
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −24.0000 −0.825137
\(847\) − 5.00000i − 0.171802i
\(848\) 2.00000i 0.0686803i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46.0000i 1.57501i 0.616308 + 0.787505i \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) −14.0000 −0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) − 54.0000i − 1.84460i −0.386469 0.922302i \(-0.626305\pi\)
0.386469 0.922302i \(-0.373695\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 8.00000i 0.271538i
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) 6.00000i 0.203186i
\(873\) 6.00000i 0.203069i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000i 0.472746i 0.971662 + 0.236373i \(0.0759588\pi\)
−0.971662 + 0.236373i \(0.924041\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 12.0000i 0.403832i 0.979403 + 0.201916i \(0.0647168\pi\)
−0.979403 + 0.201916i \(0.935283\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) − 16.0000i − 0.537227i −0.963248 0.268614i \(-0.913434\pi\)
0.963248 0.268614i \(-0.0865655\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 36.0000 1.20605
\(892\) 16.0000i 0.535720i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 30.0000i 1.00111i
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 8.00000i 0.266371i
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) 0 0
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) − 32.0000i − 1.05905i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 16.0000i 0.528367i
\(918\) 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
\(922\) 10.0000i 0.329332i
\(923\) − 96.0000i − 3.15988i
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) − 48.0000i − 1.57653i
\(928\) − 6.00000i − 0.196960i
\(929\) −58.0000 −1.90292 −0.951459 0.307775i \(-0.900416\pi\)
−0.951459 + 0.307775i \(0.900416\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 6.00000i − 0.196537i
\(933\) 0 0
\(934\) 40.0000 1.30884
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) 50.0000i 1.63343i 0.577042 + 0.816714i \(0.304207\pi\)
−0.577042 + 0.816714i \(0.695793\pi\)
\(938\) − 12.0000i − 0.391814i
\(939\) 0 0
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 44.0000i 1.42981i 0.699223 + 0.714904i \(0.253530\pi\)
−0.699223 + 0.714904i \(0.746470\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) − 2.00000i − 0.0648204i
\(953\) 54.0000i 1.74923i 0.484817 + 0.874616i \(0.338886\pi\)
−0.484817 + 0.874616i \(0.661114\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 60.0000i 1.93448i
\(963\) 36.0000i 1.16008i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) 16.0000i 0.512936i
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) − 30.0000i − 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) 0 0
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 12.0000i 0.382935i
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) 12.0000i 0.379853i
\(999\) 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 350.2.c.b.99.2 2
3.2 odd 2 3150.2.g.c.2899.1 2
4.3 odd 2 2800.2.g.n.449.2 2
5.2 odd 4 350.2.a.b.1.1 1
5.3 odd 4 70.2.a.a.1.1 1
5.4 even 2 inner 350.2.c.b.99.1 2
7.6 odd 2 2450.2.c.k.99.2 2
15.2 even 4 3150.2.a.bj.1.1 1
15.8 even 4 630.2.a.d.1.1 1
15.14 odd 2 3150.2.g.c.2899.2 2
20.3 even 4 560.2.a.d.1.1 1
20.7 even 4 2800.2.a.m.1.1 1
20.19 odd 2 2800.2.g.n.449.1 2
35.3 even 12 490.2.e.c.471.1 2
35.13 even 4 490.2.a.h.1.1 1
35.18 odd 12 490.2.e.d.471.1 2
35.23 odd 12 490.2.e.d.361.1 2
35.27 even 4 2450.2.a.l.1.1 1
35.33 even 12 490.2.e.c.361.1 2
35.34 odd 2 2450.2.c.k.99.1 2
40.3 even 4 2240.2.a.q.1.1 1
40.13 odd 4 2240.2.a.n.1.1 1
55.43 even 4 8470.2.a.j.1.1 1
60.23 odd 4 5040.2.a.bm.1.1 1
105.83 odd 4 4410.2.a.b.1.1 1
140.83 odd 4 3920.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.a.a.1.1 1 5.3 odd 4
350.2.a.b.1.1 1 5.2 odd 4
350.2.c.b.99.1 2 5.4 even 2 inner
350.2.c.b.99.2 2 1.1 even 1 trivial
490.2.a.h.1.1 1 35.13 even 4
490.2.e.c.361.1 2 35.33 even 12
490.2.e.c.471.1 2 35.3 even 12
490.2.e.d.361.1 2 35.23 odd 12
490.2.e.d.471.1 2 35.18 odd 12
560.2.a.d.1.1 1 20.3 even 4
630.2.a.d.1.1 1 15.8 even 4
2240.2.a.n.1.1 1 40.13 odd 4
2240.2.a.q.1.1 1 40.3 even 4
2450.2.a.l.1.1 1 35.27 even 4
2450.2.c.k.99.1 2 35.34 odd 2
2450.2.c.k.99.2 2 7.6 odd 2
2800.2.a.m.1.1 1 20.7 even 4
2800.2.g.n.449.1 2 20.19 odd 2
2800.2.g.n.449.2 2 4.3 odd 2
3150.2.a.bj.1.1 1 15.2 even 4
3150.2.g.c.2899.1 2 3.2 odd 2
3150.2.g.c.2899.2 2 15.14 odd 2
3920.2.a.t.1.1 1 140.83 odd 4
4410.2.a.b.1.1 1 105.83 odd 4
5040.2.a.bm.1.1 1 60.23 odd 4
8470.2.a.j.1.1 1 55.43 even 4