Properties

Label 2300.2.c.d.1749.1
Level $2300$
Weight $2$
Character 2300.1749
Analytic conductor $18.366$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 460)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2300.1749
Dual form 2300.2.c.d.1749.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -4.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -4.00000i q^{7} +2.00000 q^{9} -6.00000 q^{11} +1.00000i q^{13} -2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} -5.00000i q^{27} -9.00000 q^{29} +5.00000 q^{31} +6.00000i q^{33} +2.00000i q^{37} +1.00000 q^{39} -9.00000 q^{41} +4.00000i q^{43} -3.00000i q^{47} -9.00000 q^{49} +6.00000i q^{53} +2.00000i q^{57} +2.00000 q^{61} -8.00000i q^{63} -10.0000i q^{67} -1.00000 q^{69} -3.00000 q^{71} +7.00000i q^{73} +24.0000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{87} +4.00000 q^{91} -5.00000i q^{93} +8.00000i q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} - 12q^{11} - 4q^{19} - 8q^{21} - 18q^{29} + 10q^{31} + 2q^{39} - 18q^{41} - 18q^{49} + 4q^{61} - 2q^{69} - 6q^{71} + 20q^{79} + 2q^{81} + 8q^{91} - 24q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 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 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 6.00000i 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) − 8.00000i − 1.00791i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.0000i − 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 24.0000i 2.73505i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.00000i 0.964901i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) − 5.00000i − 0.518476i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) −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
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 0 0
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) − 12.0000i − 1.12887i −0.825479 0.564433i \(-0.809095\pi\)
0.825479 0.564433i \(-0.190905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 9.00000i 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000i 0.976092i 0.872818 + 0.488046i \(0.162290\pi\)
−0.872818 + 0.488046i \(0.837710\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −23.0000 −1.95083 −0.975417 0.220366i \(-0.929275\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) − 6.00000i − 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 1.00000i 0.0783260i 0.999233 + 0.0391630i \(0.0124692\pi\)
−0.999233 + 0.0391630i \(0.987531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 24.0000i − 1.85718i −0.371113 0.928588i \(-0.621024\pi\)
0.371113 0.928588i \(-0.378976\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) − 2.00000i − 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −20.0000 −1.45479
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) 19.0000i 1.36765i 0.729646 + 0.683825i \(0.239685\pi\)
−0.729646 + 0.683825i \(0.760315\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.0000i 1.49619i 0.663593 + 0.748094i \(0.269031\pi\)
−0.663593 + 0.748094i \(0.730969\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) 36.0000i 2.52670i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.00000i − 0.139010i
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 3.00000i 0.205557i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 20.0000i − 1.35769i
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 24.0000 1.57908
\(232\) 0 0
\(233\) 9.00000i 0.589610i 0.955557 + 0.294805i \(0.0952546\pi\)
−0.955557 + 0.294805i \(0.904745\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 10.0000i − 0.649570i
\(238\) 0 0
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.00000i − 0.127257i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.00000i 0.187135i 0.995613 + 0.0935674i \(0.0298271\pi\)
−0.995613 + 0.0935674i \(0.970173\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.0000 1.64622 0.823110 0.567883i \(-0.192237\pi\)
0.823110 + 0.567883i \(0.192237\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) − 4.00000i − 0.242091i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 31.0000i − 1.86261i −0.364241 0.931305i \(-0.618672\pi\)
0.364241 0.931305i \(-0.381328\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) − 26.0000i − 1.54554i −0.634686 0.772770i \(-0.718871\pi\)
0.634686 0.772770i \(-0.281129\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 36.0000i 2.12501i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 30.0000i 1.74078i
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 32.0000i 1.82634i 0.407583 + 0.913168i \(0.366372\pi\)
−0.407583 + 0.913168i \(0.633628\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) 0 0
\(313\) − 32.0000i − 1.80875i −0.426742 0.904373i \(-0.640339\pi\)
0.426742 0.904373i \(-0.359661\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 54.0000 3.02342
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.0000i 1.10600i
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 35.0000 1.92377 0.961887 0.273447i \(-0.0881639\pi\)
0.961887 + 0.273447i \(0.0881639\pi\)
\(332\) 0 0
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 36.0000i − 1.93258i −0.257454 0.966291i \(-0.582883\pi\)
0.257454 0.966291i \(-0.417117\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 3.00000i 0.159674i 0.996808 + 0.0798369i \(0.0254400\pi\)
−0.996808 + 0.0798369i \(0.974560\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) − 25.0000i − 1.31216i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 9.00000i − 0.463524i
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 11.0000 0.563547
\(382\) 0 0
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 15.0000i 0.756650i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.0000i 1.15434i 0.816625 + 0.577168i \(0.195842\pi\)
−0.816625 + 0.577168i \(0.804158\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 5.00000i 0.249068i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 12.0000i − 0.594818i
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 23.0000i 1.12631i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) − 14.0000i − 0.672797i −0.941720 0.336399i \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000i 0.0956730i
\(438\) 0 0
\(439\) −11.0000 −0.525001 −0.262501 0.964932i \(-0.584547\pi\)
−0.262501 + 0.964932i \(0.584547\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) − 9.00000i − 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 54.0000 2.54276
\(452\) 0 0
\(453\) − 5.00000i − 0.234920i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 28.0000i − 1.30978i −0.755722 0.654892i \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 18.0000i − 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) 0 0
\(469\) −40.0000 −1.84703
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) − 24.0000i − 1.10352i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 4.00000i 0.182006i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 13.0000i − 0.589086i −0.955638 0.294543i \(-0.904833\pi\)
0.955638 0.294543i \(-0.0951675\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) 19.0000 0.850557 0.425278 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 12.0000i − 0.532939i
\(508\) 0 0
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) 28.0000 1.23865
\(512\) 0 0
\(513\) 10.0000i 0.441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 18.0000i 0.791639i
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 9.00000i − 0.389833i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.0000i 0.647298i
\(538\) 0 0
\(539\) 54.0000 2.32594
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) − 2.00000i − 0.0858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 1.00000i − 0.0427569i −0.999771 0.0213785i \(-0.993195\pi\)
0.999771 0.0213785i \(-0.00680549\pi\)
\(548\) 0 0
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) − 40.0000i − 1.70097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 24.0000i − 1.01691i −0.861088 0.508456i \(-0.830216\pi\)
0.861088 0.508456i \(-0.169784\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 4.00000i − 0.167984i
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 43.0000i − 1.79011i −0.445952 0.895057i \(-0.647135\pi\)
0.445952 0.895057i \(-0.352865\pi\)
\(578\) 0 0
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) 0 0
\(583\) − 36.0000i − 1.49097i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0000i 1.36206i 0.732257 + 0.681028i \(0.238467\pi\)
−0.732257 + 0.681028i \(0.761533\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 21.0000 0.863825
\(592\) 0 0
\(593\) 42.0000i 1.72473i 0.506284 + 0.862367i \(0.331019\pi\)
−0.506284 + 0.862367i \(0.668981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14.0000i 0.572982i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) − 20.0000i − 0.814463i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 4.00000i − 0.162355i −0.996700 0.0811775i \(-0.974132\pi\)
0.996700 0.0811775i \(-0.0258681\pi\)
\(608\) 0 0
\(609\) 36.0000 1.45879
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 0 0
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 12.0000i − 0.479234i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) − 20.0000i − 0.794929i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 9.00000i − 0.356593i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.0000i 0.825595i 0.910823 + 0.412798i \(0.135448\pi\)
−0.910823 + 0.412798i \(0.864552\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) 9.00000i 0.352197i 0.984373 + 0.176099i \(0.0563478\pi\)
−0.984373 + 0.176099i \(0.943652\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 1.00000i 0.0385472i 0.999814 + 0.0192736i \(0.00613535\pi\)
−0.999814 + 0.0192736i \(0.993865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) − 3.00000i − 0.114792i −0.998351 0.0573959i \(-0.981720\pi\)
0.998351 0.0573959i \(-0.0182797\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.00000i 0.305219i
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 48.0000i 1.82337i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000i 0.902613i
\(708\) 0 0
\(709\) −32.0000 −1.20179 −0.600893 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) − 5.00000i − 0.187251i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.00000i − 0.112037i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 22.0000i 0.818189i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 10.0000i − 0.370879i −0.982656 0.185440i \(-0.940629\pi\)
0.982656 0.185440i \(-0.0593710\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 40.0000i 1.47743i 0.674016 + 0.738717i \(0.264568\pi\)
−0.674016 + 0.738717i \(0.735432\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 60.0000i 2.21013i
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) − 12.0000i − 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.0000i 0.878114i
\(748\) 0 0
\(749\) −72.0000 −2.63082
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) − 12.0000i − 0.437304i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 34.0000i − 1.23575i −0.786276 0.617876i \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) 80.0000i 2.89619i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) 0 0
\(773\) − 24.0000i − 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 8.00000i − 0.286998i
\(778\) 0 0
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 45.0000i 1.60817i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 42.0000i − 1.48215i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 27.0000i − 0.950445i
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −19.0000 −0.667180 −0.333590 0.942718i \(-0.608260\pi\)
−0.333590 + 0.942718i \(0.608260\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 8.00000i − 0.279885i
\(818\) 0 0
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 31.0000i 1.08059i 0.841475 + 0.540296i \(0.181688\pi\)
−0.841475 + 0.540296i \(0.818312\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) −31.0000 −1.07538
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 25.0000i − 0.864126i
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 12.0000i 0.413302i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 100.000i − 3.43604i
\(848\) 0 0
\(849\) −26.0000 −0.892318
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) − 38.0000i − 1.30110i −0.759465 0.650548i \(-0.774539\pi\)
0.759465 0.650548i \(-0.225461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.0000i 1.33221i 0.745856 + 0.666107i \(0.232041\pi\)
−0.745856 + 0.666107i \(0.767959\pi\)
\(858\) 0 0
\(859\) −5.00000 −0.170598 −0.0852989 0.996355i \(-0.527185\pi\)
−0.0852989 + 0.996355i \(0.527185\pi\)
\(860\) 0 0
\(861\) 36.0000 1.22688
\(862\) 0 0
\(863\) 45.0000i 1.53182i 0.642949 + 0.765909i \(0.277711\pi\)
−0.642949 + 0.765909i \(0.722289\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 17.0000i − 0.577350i
\(868\) 0 0
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) 0 0
\(873\) 16.0000i 0.541518i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000i 0.472746i 0.971662 + 0.236373i \(0.0759588\pi\)
−0.971662 + 0.236373i \(0.924041\pi\)
\(878\) 0 0
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 3.00000i − 0.100730i −0.998731 0.0503651i \(-0.983962\pi\)
0.998731 0.0503651i \(-0.0160385\pi\)
\(888\) 0 0
\(889\) 44.0000 1.47571
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 0 0
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.00000i − 0.0333890i
\(898\) 0 0
\(899\) −45.0000 −1.50083
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 10.0000i − 0.332045i −0.986122 0.166022i \(-0.946908\pi\)
0.986122 0.166022i \(-0.0530924\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) − 72.0000i − 2.38285i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 60.0000i 1.98137i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) − 3.00000i − 0.0987462i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) −39.0000 −1.27955 −0.639774 0.768563i \(-0.720972\pi\)
−0.639774 + 0.768563i \(0.720972\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 0 0
\(933\) − 21.0000i − 0.687509i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 0 0
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) 9.00000i 0.293080i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.0000i 1.07236i 0.844105 + 0.536178i \(0.180132\pi\)
−0.844105 + 0.536178i \(0.819868\pi\)
\(948\) 0 0
\(949\) −7.00000 −0.227230
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 54.0000i − 1.74557i
\(958\) 0 0
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) − 36.0000i − 1.16008i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 23.0000i 0.739630i 0.929105 + 0.369815i \(0.120579\pi\)
−0.929105 + 0.369815i \(0.879421\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 92.0000i 2.94938i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −40.0000 −1.27710
\(982\) 0 0
\(983\) − 54.0000i − 1.72233i −0.508323 0.861166i \(-0.669735\pi\)
0.508323 0.861166i \(-0.330265\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 12.0000i 0.381964i
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) − 35.0000i − 1.11069i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 2300.2.c.d.1749.1 2
5.2 odd 4 2300.2.a.d.1.1 1
5.3 odd 4 460.2.a.c.1.1 1
5.4 even 2 inner 2300.2.c.d.1749.2 2
15.8 even 4 4140.2.a.f.1.1 1
20.3 even 4 1840.2.a.c.1.1 1
20.7 even 4 9200.2.a.y.1.1 1
40.3 even 4 7360.2.a.v.1.1 1
40.13 odd 4 7360.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.c.1.1 1 5.3 odd 4
1840.2.a.c.1.1 1 20.3 even 4
2300.2.a.d.1.1 1 5.2 odd 4
2300.2.c.d.1749.1 2 1.1 even 1 trivial
2300.2.c.d.1749.2 2 5.4 even 2 inner
4140.2.a.f.1.1 1 15.8 even 4
7360.2.a.i.1.1 1 40.13 odd 4
7360.2.a.v.1.1 1 40.3 even 4
9200.2.a.y.1.1 1 20.7 even 4