Properties

Label 2450.2.a.bo.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} +4.24264 q^{17} -3.00000 q^{18} +5.65685 q^{19} -4.00000 q^{22} -4.24264 q^{26} -4.00000 q^{29} -5.65685 q^{31} +1.00000 q^{32} +4.24264 q^{34} -3.00000 q^{36} -6.00000 q^{37} +5.65685 q^{38} -1.41421 q^{41} -12.0000 q^{43} -4.00000 q^{44} -4.24264 q^{52} -12.0000 q^{53} -4.00000 q^{58} -11.3137 q^{59} +7.07107 q^{61} -5.65685 q^{62} +1.00000 q^{64} -12.0000 q^{67} +4.24264 q^{68} +8.00000 q^{71} -3.00000 q^{72} +4.24264 q^{73} -6.00000 q^{74} +5.65685 q^{76} +9.00000 q^{81} -1.41421 q^{82} -16.9706 q^{83} -12.0000 q^{86} -4.00000 q^{88} -4.24264 q^{89} -4.24264 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} - 6q^{9} - 8q^{11} + 2q^{16} - 6q^{18} - 8q^{22} - 8q^{29} + 2q^{32} - 6q^{36} - 12q^{37} - 24q^{43} - 8q^{44} - 24q^{53} - 8q^{58} + 2q^{64} - 24q^{67} + 16q^{71} - 6q^{72} - 12q^{74} + 18q^{81} - 24q^{86} - 8q^{88} + 24q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.24264 1.02899 0.514496 0.857493i \(-0.327979\pi\)
0.514496 + 0.857493i \(0.327979\pi\)
\(18\) −3.00000 −0.707107
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.24264 −0.832050
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.24264 0.727607
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 5.65685 0.917663
\(39\) 0 0
\(40\) 0 0
\(41\) −1.41421 −0.220863 −0.110432 0.993884i \(-0.535223\pi\)
−0.110432 + 0.993884i \(0.535223\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −4.24264 −0.588348
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 7.07107 0.905357 0.452679 0.891674i \(-0.350468\pi\)
0.452679 + 0.891674i \(0.350468\pi\)
\(62\) −5.65685 −0.718421
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 4.24264 0.514496
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −3.00000 −0.353553
\(73\) 4.24264 0.496564 0.248282 0.968688i \(-0.420134\pi\)
0.248282 + 0.968688i \(0.420134\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 5.65685 0.648886
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −1.41421 −0.156174
\(83\) −16.9706 −1.86276 −0.931381 0.364047i \(-0.881395\pi\)
−0.931381 + 0.364047i \(0.881395\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −4.24264 −0.449719 −0.224860 0.974391i \(-0.572192\pi\)
−0.224860 + 0.974391i \(0.572192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.24264 −0.430775 −0.215387 0.976529i \(-0.569101\pi\)
−0.215387 + 0.976529i \(0.569101\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) 9.89949 0.985037 0.492518 0.870302i \(-0.336076\pi\)
0.492518 + 0.870302i \(0.336076\pi\)
\(102\) 0 0
\(103\) 16.9706 1.67216 0.836080 0.548608i \(-0.184842\pi\)
0.836080 + 0.548608i \(0.184842\pi\)
\(104\) −4.24264 −0.416025
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 12.7279 1.17670
\(118\) −11.3137 −1.04151
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 7.07107 0.640184
\(123\) 0 0
\(124\) −5.65685 −0.508001
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 16.9706 1.48272 0.741362 0.671105i \(-0.234180\pi\)
0.741362 + 0.671105i \(0.234180\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 4.24264 0.363803
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 5.65685 0.479808 0.239904 0.970797i \(-0.422884\pi\)
0.239904 + 0.970797i \(0.422884\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 16.9706 1.41915
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 4.24264 0.351123
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 5.65685 0.458831
\(153\) −12.7279 −1.02899
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.24264 0.338600 0.169300 0.985565i \(-0.445849\pi\)
0.169300 + 0.985565i \(0.445849\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −1.41421 −0.110432
\(165\) 0 0
\(166\) −16.9706 −1.31717
\(167\) −16.9706 −1.31322 −0.656611 0.754230i \(-0.728011\pi\)
−0.656611 + 0.754230i \(0.728011\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −16.9706 −1.29777
\(172\) −12.0000 −0.914991
\(173\) −4.24264 −0.322562 −0.161281 0.986909i \(-0.551563\pi\)
−0.161281 + 0.986909i \(0.551563\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −4.24264 −0.317999
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −15.5563 −1.15629 −0.578147 0.815933i \(-0.696224\pi\)
−0.578147 + 0.815933i \(0.696224\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −16.9706 −1.24101
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −24.0000 −1.72756 −0.863779 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −4.24264 −0.304604
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 12.0000 0.852803
\(199\) 11.3137 0.802008 0.401004 0.916076i \(-0.368661\pi\)
0.401004 + 0.916076i \(0.368661\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 9.89949 0.696526
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 16.9706 1.18240
\(207\) 0 0
\(208\) −4.24264 −0.294174
\(209\) −22.6274 −1.56517
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −12.0000 −0.812743
\(219\) 0 0
\(220\) 0 0
\(221\) −18.0000 −1.21081
\(222\) 0 0
\(223\) 16.9706 1.13643 0.568216 0.822879i \(-0.307634\pi\)
0.568216 + 0.822879i \(0.307634\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.9706 −1.12638 −0.563188 0.826329i \(-0.690425\pi\)
−0.563188 + 0.826329i \(0.690425\pi\)
\(228\) 0 0
\(229\) 15.5563 1.02799 0.513996 0.857792i \(-0.328165\pi\)
0.513996 + 0.857792i \(0.328165\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 12.7279 0.832050
\(235\) 0 0
\(236\) −11.3137 −0.736460
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −15.5563 −1.00207 −0.501036 0.865426i \(-0.667048\pi\)
−0.501036 + 0.865426i \(0.667048\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 7.07107 0.452679
\(245\) 0 0
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) −5.65685 −0.359211
\(249\) 0 0
\(250\) 0 0
\(251\) −5.65685 −0.357057 −0.178529 0.983935i \(-0.557134\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.7279 0.793946 0.396973 0.917830i \(-0.370061\pi\)
0.396973 + 0.917830i \(0.370061\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 16.9706 1.04844
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 21.2132 1.29339 0.646696 0.762748i \(-0.276150\pi\)
0.646696 + 0.762748i \(0.276150\pi\)
\(270\) 0 0
\(271\) −11.3137 −0.687259 −0.343629 0.939105i \(-0.611656\pi\)
−0.343629 + 0.939105i \(0.611656\pi\)
\(272\) 4.24264 0.257248
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 5.65685 0.339276
\(279\) 16.9706 1.01600
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 16.9706 1.00349
\(287\) 0 0
\(288\) −3.00000 −0.176777
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 4.24264 0.248282
\(293\) 4.24264 0.247858 0.123929 0.992291i \(-0.460451\pi\)
0.123929 + 0.992291i \(0.460451\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 5.65685 0.324443
\(305\) 0 0
\(306\) −12.7279 −0.727607
\(307\) 16.9706 0.968561 0.484281 0.874913i \(-0.339081\pi\)
0.484281 + 0.874913i \(0.339081\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.9706 0.962312 0.481156 0.876635i \(-0.340217\pi\)
0.481156 + 0.876635i \(0.340217\pi\)
\(312\) 0 0
\(313\) −12.7279 −0.719425 −0.359712 0.933063i \(-0.617125\pi\)
−0.359712 + 0.933063i \(0.617125\pi\)
\(314\) 4.24264 0.239426
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −1.41421 −0.0780869
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −16.9706 −0.931381
\(333\) 18.0000 0.986394
\(334\) −16.9706 −0.928588
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 5.00000 0.271964
\(339\) 0 0
\(340\) 0 0
\(341\) 22.6274 1.22534
\(342\) −16.9706 −0.917663
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −4.24264 −0.228086
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −24.0416 −1.28692 −0.643459 0.765480i \(-0.722502\pi\)
−0.643459 + 0.765480i \(0.722502\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −21.2132 −1.12906 −0.564532 0.825411i \(-0.690943\pi\)
−0.564532 + 0.825411i \(0.690943\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.24264 −0.224860
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) −15.5563 −0.817624
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.9706 0.885856 0.442928 0.896557i \(-0.353940\pi\)
0.442928 + 0.896557i \(0.353940\pi\)
\(368\) 0 0
\(369\) 4.24264 0.220863
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) −16.9706 −0.877527
\(375\) 0 0
\(376\) 0 0
\(377\) 16.9706 0.874028
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −24.0000 −1.22157
\(387\) 36.0000 1.82998
\(388\) −4.24264 −0.215387
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 12.0000 0.603023
\(397\) −29.6985 −1.49052 −0.745262 0.666772i \(-0.767676\pi\)
−0.745262 + 0.666772i \(0.767676\pi\)
\(398\) 11.3137 0.567105
\(399\) 0 0
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) 9.89949 0.492518
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −9.89949 −0.489499 −0.244749 0.969586i \(-0.578706\pi\)
−0.244749 + 0.969586i \(0.578706\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 16.9706 0.836080
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −4.24264 −0.208013
\(417\) 0 0
\(418\) −22.6274 −1.10674
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 29.6985 1.42722 0.713609 0.700544i \(-0.247059\pi\)
0.713609 + 0.700544i \(0.247059\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) 0 0
\(438\) 0 0
\(439\) −39.5980 −1.88991 −0.944954 0.327203i \(-0.893894\pi\)
−0.944954 + 0.327203i \(0.893894\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −18.0000 −0.856173
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16.9706 0.803579
\(447\) 0 0
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 5.65685 0.266371
\(452\) 0 0
\(453\) 0 0
\(454\) −16.9706 −0.796468
\(455\) 0 0
\(456\) 0 0
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) 15.5563 0.726900
\(459\) 0 0
\(460\) 0 0
\(461\) −21.2132 −0.987997 −0.493999 0.869463i \(-0.664465\pi\)
−0.493999 + 0.869463i \(0.664465\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 16.9706 0.785304 0.392652 0.919687i \(-0.371558\pi\)
0.392652 + 0.919687i \(0.371558\pi\)
\(468\) 12.7279 0.588348
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −11.3137 −0.520756
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 36.0000 1.64833
\(478\) −8.00000 −0.365911
\(479\) −16.9706 −0.775405 −0.387702 0.921785i \(-0.626731\pi\)
−0.387702 + 0.921785i \(0.626731\pi\)
\(480\) 0 0
\(481\) 25.4558 1.16069
\(482\) −15.5563 −0.708572
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 7.07107 0.320092
\(489\) 0 0
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 0 0
\(493\) −16.9706 −0.764316
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −5.65685 −0.254000
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −5.65685 −0.252478
\(503\) 33.9411 1.51336 0.756680 0.653785i \(-0.226820\pi\)
0.756680 + 0.653785i \(0.226820\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.7279 0.564155 0.282078 0.959392i \(-0.408976\pi\)
0.282078 + 0.959392i \(0.408976\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 12.7279 0.561405
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.7279 0.557620 0.278810 0.960346i \(-0.410060\pi\)
0.278810 + 0.960346i \(0.410060\pi\)
\(522\) 12.0000 0.525226
\(523\) −33.9411 −1.48414 −0.742071 0.670321i \(-0.766156\pi\)
−0.742071 + 0.670321i \(0.766156\pi\)
\(524\) 16.9706 0.741362
\(525\) 0 0
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 33.9411 1.47292
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 21.2132 0.914566
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −11.3137 −0.485965
\(543\) 0 0
\(544\) 4.24264 0.181902
\(545\) 0 0
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 6.00000 0.256307
\(549\) −21.2132 −0.905357
\(550\) 0 0
\(551\) −22.6274 −0.963960
\(552\) 0 0
\(553\) 0 0
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) 5.65685 0.239904
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 16.9706 0.718421
\(559\) 50.9117 2.15333
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) −33.9411 −1.43045 −0.715224 0.698895i \(-0.753675\pi\)
−0.715224 + 0.698895i \(0.753675\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −8.00000 −0.335377 −0.167689 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 16.9706 0.709575
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) −21.2132 −0.883117 −0.441559 0.897232i \(-0.645574\pi\)
−0.441559 + 0.897232i \(0.645574\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 48.0000 1.98796
\(584\) 4.24264 0.175562
\(585\) 0 0
\(586\) 4.24264 0.175262
\(587\) 16.9706 0.700450 0.350225 0.936666i \(-0.386105\pi\)
0.350225 + 0.936666i \(0.386105\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 12.7279 0.522673 0.261337 0.965248i \(-0.415837\pi\)
0.261337 + 0.965248i \(0.415837\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 26.8701 1.09605 0.548026 0.836461i \(-0.315379\pi\)
0.548026 + 0.836461i \(0.315379\pi\)
\(602\) 0 0
\(603\) 36.0000 1.46603
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 5.65685 0.229416
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −12.7279 −0.514496
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 16.9706 0.684876
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −45.2548 −1.81895 −0.909473 0.415764i \(-0.863514\pi\)
−0.909473 + 0.415764i \(0.863514\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.9706 0.680458
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −12.7279 −0.508710
\(627\) 0 0
\(628\) 4.24264 0.169300
\(629\) −25.4558 −1.01499
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 16.0000 0.633446
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) −16.9706 −0.667182 −0.333591 0.942718i \(-0.608260\pi\)
−0.333591 + 0.942718i \(0.608260\pi\)
\(648\) 9.00000 0.353553
\(649\) 45.2548 1.77641
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.41421 −0.0552158
\(657\) −12.7279 −0.496564
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −24.0416 −0.935111 −0.467556 0.883964i \(-0.654865\pi\)
−0.467556 + 0.883964i \(0.654865\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) −16.9706 −0.658586
\(665\) 0 0
\(666\) 18.0000 0.697486
\(667\) 0 0
\(668\) −16.9706 −0.656611
\(669\) 0 0
\(670\) 0 0
\(671\) −28.2843 −1.09190
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 5.00000 0.192308
\(677\) 21.2132 0.815290 0.407645 0.913141i \(-0.366350\pi\)
0.407645 + 0.913141i \(0.366350\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 22.6274 0.866449
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −16.9706 −0.648886
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) 50.9117 1.93958
\(690\) 0 0
\(691\) −5.65685 −0.215197 −0.107598 0.994194i \(-0.534316\pi\)
−0.107598 + 0.994194i \(0.534316\pi\)
\(692\) −4.24264 −0.161281
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) −24.0416 −0.909989
\(699\) 0 0
\(700\) 0 0
\(701\) −44.0000 −1.66186 −0.830929 0.556379i \(-0.812190\pi\)
−0.830929 + 0.556379i \(0.812190\pi\)
\(702\) 0 0
\(703\) −33.9411 −1.28011
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −21.2132 −0.798369
\(707\) 0 0
\(708\) 0 0
\(709\) −44.0000 −1.65245 −0.826227 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.24264 −0.159000
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −5.65685 −0.210965 −0.105483 0.994421i \(-0.533639\pi\)
−0.105483 + 0.994421i \(0.533639\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 13.0000 0.483810
\(723\) 0 0
\(724\) −15.5563 −0.578147
\(725\) 0 0
\(726\) 0 0
\(727\) 16.9706 0.629403 0.314702 0.949191i \(-0.398096\pi\)
0.314702 + 0.949191i \(0.398096\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −50.9117 −1.88304
\(732\) 0 0
\(733\) −12.7279 −0.470117 −0.235058 0.971981i \(-0.575528\pi\)
−0.235058 + 0.971981i \(0.575528\pi\)
\(734\) 16.9706 0.626395
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 4.24264 0.156174
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) 50.9117 1.86276
\(748\) −16.9706 −0.620505
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 16.9706 0.618031
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −46.6690 −1.69175 −0.845876 0.533380i \(-0.820922\pi\)
−0.845876 + 0.533380i \(0.820922\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 0 0
\(767\) 48.0000 1.73318
\(768\) 0 0
\(769\) 7.07107 0.254989 0.127495 0.991839i \(-0.459306\pi\)
0.127495 + 0.991839i \(0.459306\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −24.0000 −0.863779
\(773\) 38.1838 1.37337 0.686687 0.726953i \(-0.259064\pi\)
0.686687 + 0.726953i \(0.259064\pi\)
\(774\) 36.0000 1.29399
\(775\) 0 0
\(776\) −4.24264 −0.152302
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 50.9117 1.81481 0.907403 0.420262i \(-0.138062\pi\)
0.907403 + 0.420262i \(0.138062\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 12.0000 0.426401
\(793\) −30.0000 −1.06533
\(794\) −29.6985 −1.05396
\(795\) 0 0
\(796\) 11.3137 0.401004
\(797\) 4.24264 0.150282 0.0751410 0.997173i \(-0.476059\pi\)
0.0751410 + 0.997173i \(0.476059\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 12.7279 0.449719
\(802\) −8.00000 −0.282490
\(803\) −16.9706 −0.598878
\(804\) 0 0
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) 0 0
\(808\) 9.89949 0.348263
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 45.2548 1.58911 0.794556 0.607191i \(-0.207704\pi\)
0.794556 + 0.607191i \(0.207704\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) −67.8823 −2.37490
\(818\) −9.89949 −0.346128
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 16.9706 0.591198
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) 15.5563 0.540294 0.270147 0.962819i \(-0.412928\pi\)
0.270147 + 0.962819i \(0.412928\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.24264 −0.147087
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −22.6274 −0.782586
\(837\) 0 0
\(838\) 0 0
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −6.00000 −0.206774
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46.6690 1.59792 0.798959 0.601386i \(-0.205384\pi\)
0.798959 + 0.601386i \(0.205384\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 12.7279 0.434778 0.217389 0.976085i \(-0.430246\pi\)
0.217389 + 0.976085i \(0.430246\pi\)
\(858\) 0 0
\(859\) 5.65685 0.193009 0.0965047 0.995333i \(-0.469234\pi\)
0.0965047 + 0.995333i \(0.469234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 29.6985 1.00920
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 50.9117 1.72508
\(872\) −12.0000 −0.406371
\(873\) 12.7279 0.430775
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −39.5980 −1.33637
\(879\) 0 0
\(880\) 0 0
\(881\) −32.5269 −1.09586 −0.547930 0.836524i \(-0.684584\pi\)
−0.547930 + 0.836524i \(0.684584\pi\)
\(882\) 0 0
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) −18.0000 −0.605406
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −16.9706 −0.569816 −0.284908 0.958555i \(-0.591963\pi\)
−0.284908 + 0.958555i \(0.591963\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −36.0000 −1.20605
\(892\) 16.9706 0.568216
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) 22.6274 0.754667
\(900\) 0 0
\(901\) −50.9117 −1.69611
\(902\) 5.65685 0.188353
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −16.9706 −0.563188
\(909\) −29.6985 −0.985037
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 67.8823 2.24657
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) 15.5563 0.513996
\(917\) 0 0
\(918\) 0 0
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −21.2132 −0.698620
\(923\) −33.9411 −1.11719
\(924\) 0 0
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) −50.9117 −1.67216
\(928\) −4.00000 −0.131306
\(929\) −29.6985 −0.974376 −0.487188 0.873297i \(-0.661977\pi\)
−0.487188 + 0.873297i \(0.661977\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 16.9706 0.555294
\(935\) 0 0
\(936\) 12.7279 0.416025
\(937\) −4.24264 −0.138601 −0.0693005 0.997596i \(-0.522077\pi\)
−0.0693005 + 0.997596i \(0.522077\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 49.4975 1.61357 0.806786 0.590844i \(-0.201205\pi\)
0.806786 + 0.590844i \(0.201205\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −11.3137 −0.368230
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −18.0000 −0.584305
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 36.0000 1.16554
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −16.9706 −0.548294
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 25.4558 0.820729
\(963\) −36.0000 −1.16008
\(964\) −15.5563 −0.501036
\(965\) 0 0
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) 45.2548 1.45230 0.726148 0.687538i \(-0.241309\pi\)
0.726148 + 0.687538i \(0.241309\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) 7.07107 0.226339
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) 16.9706 0.542382
\(980\) 0 0
\(981\) 36.0000 1.14939
\(982\) −4.00000 −0.127645
\(983\) 16.9706 0.541277 0.270638 0.962681i \(-0.412765\pi\)
0.270638 + 0.962681i \(0.412765\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −16.9706 −0.540453
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 0 0
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) −5.65685 −0.179605
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −38.1838 −1.20929 −0.604646 0.796494i \(-0.706685\pi\)
−0.604646 + 0.796494i \(0.706685\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 2450.2.a.bo.1.1 2
5.2 odd 4 490.2.c.g.99.3 yes 4
5.3 odd 4 490.2.c.g.99.1 4
5.4 even 2 2450.2.a.bi.1.2 2
7.6 odd 2 inner 2450.2.a.bo.1.2 2
35.2 odd 12 490.2.i.d.459.4 8
35.3 even 12 490.2.i.d.79.3 8
35.12 even 12 490.2.i.d.459.3 8
35.13 even 4 490.2.c.g.99.2 yes 4
35.17 even 12 490.2.i.d.79.2 8
35.18 odd 12 490.2.i.d.79.4 8
35.23 odd 12 490.2.i.d.459.1 8
35.27 even 4 490.2.c.g.99.4 yes 4
35.32 odd 12 490.2.i.d.79.1 8
35.33 even 12 490.2.i.d.459.2 8
35.34 odd 2 2450.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.c.g.99.1 4 5.3 odd 4
490.2.c.g.99.2 yes 4 35.13 even 4
490.2.c.g.99.3 yes 4 5.2 odd 4
490.2.c.g.99.4 yes 4 35.27 even 4
490.2.i.d.79.1 8 35.32 odd 12
490.2.i.d.79.2 8 35.17 even 12
490.2.i.d.79.3 8 35.3 even 12
490.2.i.d.79.4 8 35.18 odd 12
490.2.i.d.459.1 8 35.23 odd 12
490.2.i.d.459.2 8 35.33 even 12
490.2.i.d.459.3 8 35.12 even 12
490.2.i.d.459.4 8 35.2 odd 12
2450.2.a.bi.1.1 2 35.34 odd 2
2450.2.a.bi.1.2 2 5.4 even 2
2450.2.a.bo.1.1 2 1.1 even 1 trivial
2450.2.a.bo.1.2 2 7.6 odd 2 inner