Properties

Label 304.2.a.b.1.1
Level $304$
Weight $2$
Character 304.1
Self dual yes
Analytic conductor $2.427$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.42745222145\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 304.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} +1.00000 q^{13} -5.00000 q^{17} -1.00000 q^{19} +3.00000 q^{21} +1.00000 q^{23} -5.00000 q^{25} +5.00000 q^{27} -3.00000 q^{29} -4.00000 q^{31} +2.00000 q^{33} +2.00000 q^{37} -1.00000 q^{39} -8.00000 q^{41} +8.00000 q^{43} +8.00000 q^{47} +2.00000 q^{49} +5.00000 q^{51} +9.00000 q^{53} +1.00000 q^{57} -1.00000 q^{59} +14.0000 q^{61} +6.00000 q^{63} -13.0000 q^{67} -1.00000 q^{69} -10.0000 q^{71} +9.00000 q^{73} +5.00000 q^{75} +6.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -10.0000 q^{83} +3.00000 q^{87} -12.0000 q^{89} -3.00000 q^{91} +4.00000 q^{93} +14.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 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
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.0000 −1.45010 −0.725052 0.688694i \(-0.758184\pi\)
−0.725052 + 0.688694i \(0.758184\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.00000 0.598050 0.299025 0.954245i \(-0.403339\pi\)
0.299025 + 0.954245i \(0.403339\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −8.00000 −0.655386 −0.327693 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) 0 0
\(153\) 10.0000 0.808452
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.00000 −0.309529 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 15.0000 1.13389
\(176\) 0 0
\(177\) 1.00000 0.0751646
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 10.0000 0.731272
\(188\) 0 0
\(189\) −15.0000 −1.09109
\(190\) 0 0
\(191\) −23.0000 −1.66422 −0.832111 0.554609i \(-0.812868\pi\)
−0.832111 + 0.554609i \(0.812868\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) 9.00000 0.637993 0.318997 0.947756i \(-0.396654\pi\)
0.318997 + 0.947756i \(0.396654\pi\)
\(200\) 0 0
\(201\) 13.0000 0.916949
\(202\) 0 0
\(203\) 9.00000 0.631676
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −21.0000 −1.44570 −0.722850 0.691005i \(-0.757168\pi\)
−0.722850 + 0.691005i \(0.757168\pi\)
\(212\) 0 0
\(213\) 10.0000 0.685189
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 10.0000 0.666667
\(226\) 0 0
\(227\) −9.00000 −0.597351 −0.298675 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.00000 −0.249513 −0.124757 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −32.0000 −1.97320 −0.986602 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 25.0000 1.51864 0.759321 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) 10.0000 0.603023
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 7.00000 0.408944 0.204472 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −10.0000 −0.580259
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 9.00000 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) 0 0
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 0 0
\(325\) −5.00000 −0.277350
\(326\) 0 0
\(327\) −7.00000 −0.387101
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −7.00000 −0.372572 −0.186286 0.982496i \(-0.559645\pi\)
−0.186286 + 0.982496i \(0.559645\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −15.0000 −0.793884
\(358\) 0 0
\(359\) −17.0000 −0.897226 −0.448613 0.893726i \(-0.648082\pi\)
−0.448613 + 0.893726i \(0.648082\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 16.0000 0.832927
\(370\) 0 0
\(371\) −27.0000 −1.40177
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −9.00000 −0.462299 −0.231149 0.972918i \(-0.574249\pi\)
−0.231149 + 0.972918i \(0.574249\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −16.0000 −0.813326
\(388\) 0 0
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 0 0
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −8.00000 −0.395575 −0.197787 0.980245i \(-0.563376\pi\)
−0.197787 + 0.980245i \(0.563376\pi\)
\(410\) 0 0
\(411\) −7.00000 −0.345285
\(412\) 0 0
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) 0 0
\(423\) −16.0000 −0.777947
\(424\) 0 0
\(425\) 25.0000 1.21268
\(426\) 0 0
\(427\) −42.0000 −2.03252
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.00000 −0.0478365
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.00000 0.378387
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) 22.0000 1.03365
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 0 0
\(459\) −25.0000 −1.16690
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 39.0000 1.80085
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 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
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 30.0000 1.34568
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 4.00000 0.178707
\(502\) 0 0
\(503\) 33.0000 1.47140 0.735699 0.677309i \(-0.236854\pi\)
0.735699 + 0.677309i \(0.236854\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −27.0000 −1.19441
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 5.00000 0.218635 0.109317 0.994007i \(-0.465134\pi\)
0.109317 + 0.994007i \(0.465134\pi\)
\(524\) 0 0
\(525\) −15.0000 −0.654654
\(526\) 0 0
\(527\) 20.0000 0.871214
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −28.0000 −1.19501
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 0 0
\(553\) −30.0000 −1.27573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −10.0000 −0.422200
\(562\) 0 0
\(563\) 44.0000 1.85438 0.927189 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 23.0000 0.960839
\(574\) 0 0
\(575\) −5.00000 −0.208514
\(576\) 0 0
\(577\) −29.0000 −1.20729 −0.603643 0.797255i \(-0.706285\pi\)
−0.603643 + 0.797255i \(0.706285\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 30.0000 1.24461
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 8.00000 0.329076
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.00000 −0.368345
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 26.0000 1.05880
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −42.0000 −1.70473 −0.852364 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) −2.00000 −0.0798723
\(628\) 0 0
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 21.0000 0.834675
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 20.0000 0.791188
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 46.0000 1.81406 0.907031 0.421063i \(-0.138343\pi\)
0.907031 + 0.421063i \(0.138343\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.00000 0.353827 0.176913 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) 0 0
\(649\) 2.00000 0.0785069
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) −19.0000 −0.740135 −0.370067 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) 0 0
\(663\) 5.00000 0.194184
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) −28.0000 −1.08093
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) 0 0
\(675\) −25.0000 −0.962250
\(676\) 0 0
\(677\) 27.0000 1.03769 0.518847 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 0 0
\(679\) −42.0000 −1.61181
\(680\) 0 0
\(681\) 9.00000 0.344881
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 0 0
\(693\) −12.0000 −0.455842
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 0 0
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 44.0000 1.66186 0.830929 0.556379i \(-0.187810\pi\)
0.830929 + 0.556379i \(0.187810\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 42.0000 1.57957
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −9.00000 −0.336111
\(718\) 0 0
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.0000 0.557086
\(726\) 0 0
\(727\) −7.00000 −0.259616 −0.129808 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.0000 0.957722
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 20.0000 0.731762
\(748\) 0 0
\(749\) 45.0000 1.64426
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −37.0000 −1.34125 −0.670624 0.741797i \(-0.733974\pi\)
−0.670624 + 0.741797i \(0.733974\pi\)
\(762\) 0 0
\(763\) −21.0000 −0.760251
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.00000 −0.0361079
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) 0 0
\(773\) −49.0000 −1.76241 −0.881204 0.472737i \(-0.843266\pi\)
−0.881204 + 0.472737i \(0.843266\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 0 0
\(783\) −15.0000 −0.536056
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.00000 0.249523 0.124762 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) 0 0
\(789\) 32.0000 1.13923
\(790\) 0 0
\(791\) 54.0000 1.92002
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) −18.0000 −0.635206
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) −27.0000 −0.948098 −0.474049 0.880498i \(-0.657208\pi\)
−0.474049 + 0.880498i \(0.657208\pi\)
\(812\) 0 0
\(813\) −25.0000 −0.876788
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 0 0
\(823\) 11.0000 0.383436 0.191718 0.981450i \(-0.438594\pi\)
0.191718 + 0.981450i \(0.438594\pi\)
\(824\) 0 0
\(825\) −10.0000 −0.348155
\(826\) 0 0
\(827\) −33.0000 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) 0 0
\(829\) 39.0000 1.35453 0.677263 0.735741i \(-0.263166\pi\)
0.677263 + 0.735741i \(0.263166\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −10.0000 −0.346479
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −20.0000 −0.691301
\(838\) 0 0
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 21.0000 0.721569
\(848\) 0 0
\(849\) −6.00000 −0.205919
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −44.0000 −1.50301 −0.751506 0.659727i \(-0.770672\pi\)
−0.751506 + 0.659727i \(0.770672\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −54.0000 −1.83818 −0.919091 0.394046i \(-0.871075\pi\)
−0.919091 + 0.394046i \(0.871075\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) −13.0000 −0.440488
\(872\) 0 0
\(873\) −28.0000 −0.947656
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −37.0000 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(878\) 0 0
\(879\) −7.00000 −0.236104
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) −46.0000 −1.54802 −0.774012 0.633171i \(-0.781753\pi\)
−0.774012 + 0.633171i \(0.781753\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) −45.0000 −1.49917
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.00000 0.166022 0.0830111 0.996549i \(-0.473546\pi\)
0.0830111 + 0.996549i \(0.473546\pi\)
\(908\) 0 0
\(909\) 28.0000 0.928701
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 43.0000 1.41844 0.709220 0.704988i \(-0.249047\pi\)
0.709220 + 0.704988i \(0.249047\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) −10.0000 −0.329154
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 1.00000 0.0328089 0.0164045 0.999865i \(-0.494778\pi\)
0.0164045 + 0.999865i \(0.494778\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) −9.00000 −0.294647
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 49.0000 1.60076 0.800380 0.599493i \(-0.204631\pi\)
0.800380 + 0.599493i \(0.204631\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) −15.0000 −0.488986 −0.244493 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 0 0
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) −21.0000 −0.678125
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 30.0000 0.966736
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) −5.00000 −0.160623
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) −36.0000 −1.15411
\(974\) 0 0
\(975\) 5.00000 0.160128
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) −10.0000 −0.318950 −0.159475 0.987202i \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) 0 0
\(993\) −25.0000 −0.793351
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −16.0000 −0.506725 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(998\) 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 304.2.a.b.1.1 1
3.2 odd 2 2736.2.a.k.1.1 1
4.3 odd 2 152.2.a.b.1.1 1
5.4 even 2 7600.2.a.o.1.1 1
8.3 odd 2 1216.2.a.f.1.1 1
8.5 even 2 1216.2.a.l.1.1 1
12.11 even 2 1368.2.a.g.1.1 1
19.18 odd 2 5776.2.a.l.1.1 1
20.3 even 4 3800.2.d.f.3649.2 2
20.7 even 4 3800.2.d.f.3649.1 2
20.19 odd 2 3800.2.a.d.1.1 1
28.27 even 2 7448.2.a.g.1.1 1
76.75 even 2 2888.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.b.1.1 1 4.3 odd 2
304.2.a.b.1.1 1 1.1 even 1 trivial
1216.2.a.f.1.1 1 8.3 odd 2
1216.2.a.l.1.1 1 8.5 even 2
1368.2.a.g.1.1 1 12.11 even 2
2736.2.a.k.1.1 1 3.2 odd 2
2888.2.a.b.1.1 1 76.75 even 2
3800.2.a.d.1.1 1 20.19 odd 2
3800.2.d.f.3649.1 2 20.7 even 4
3800.2.d.f.3649.2 2 20.3 even 4
5776.2.a.l.1.1 1 19.18 odd 2
7448.2.a.g.1.1 1 28.27 even 2
7600.2.a.o.1.1 1 5.4 even 2