Properties

Label 304.2.a.d.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 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -4.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -4.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} -4.00000 q^{15} +3.00000 q^{17} +1.00000 q^{19} -3.00000 q^{21} +1.00000 q^{23} +11.0000 q^{25} -5.00000 q^{27} -5.00000 q^{29} +8.00000 q^{31} -2.00000 q^{33} +12.0000 q^{35} -2.00000 q^{37} -1.00000 q^{39} -8.00000 q^{41} -4.00000 q^{43} +8.00000 q^{45} -8.00000 q^{47} +2.00000 q^{49} +3.00000 q^{51} -1.00000 q^{53} +8.00000 q^{55} +1.00000 q^{57} -15.0000 q^{59} +2.00000 q^{61} +6.00000 q^{63} +4.00000 q^{65} -3.00000 q^{67} +1.00000 q^{69} -2.00000 q^{71} +9.00000 q^{73} +11.0000 q^{75} +6.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -12.0000 q^{85} -5.00000 q^{87} +3.00000 q^{91} +8.00000 q^{93} -4.00000 q^{95} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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 \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 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 \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 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
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 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
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 8.00000 1.19257
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −15.0000 −1.95283 −0.976417 0.215894i \(-0.930733\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 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
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) −5.00000 −0.536056
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 7.00000 0.676716 0.338358 0.941018i \(-0.390129\pi\)
0.338358 + 0.941018i \(0.390129\pi\)
\(108\) 0 0
\(109\) −15.0000 −1.43674 −0.718370 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) 20.0000 1.72133
\(136\) 0 0
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 20.0000 1.66091
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −32.0000 −2.57030
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −33.0000 −2.49457
\(176\) 0 0
\(177\) −15.0000 −1.12747
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) 25.0000 1.77220 0.886102 0.463491i \(-0.153403\pi\)
0.886102 + 0.463491i \(0.153403\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 0 0
\(203\) 15.0000 1.05279
\(204\) 0 0
\(205\) 32.0000 2.23498
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −27.0000 −1.85876 −0.929378 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) 0 0
\(213\) −2.00000 −0.137038
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −22.0000 −1.46667
\(226\) 0 0
\(227\) 17.0000 1.12833 0.564165 0.825662i \(-0.309198\pi\)
0.564165 + 0.825662i \(0.309198\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 32.0000 2.08745
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) −8.00000 −0.511101
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) 0 0
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) −22.0000 −1.32665
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 60.0000 3.49334
\(296\) 0 0
\(297\) 10.0000 0.580259
\(298\) 0 0
\(299\) −1.00000 −0.0578315
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 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
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 0 0
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 0 0
\(315\) −24.0000 −1.35225
\(316\) 0 0
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 7.00000 0.390702
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −11.0000 −0.610170
\(326\) 0 0
\(327\) −15.0000 −0.829502
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 9.00000 0.479022 0.239511 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) −9.00000 −0.476331
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −36.0000 −1.88433
\(366\) 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
\(371\) 3.00000 0.155752
\(372\) 0 0
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 26.0000 1.32854 0.664269 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) −40.0000 −2.01262
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) −4.00000 −0.198762
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −17.0000 −0.838548
\(412\) 0 0
\(413\) 45.0000 2.21431
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 0 0
\(423\) 16.0000 0.777947
\(424\) 0 0
\(425\) 33.0000 1.60074
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 20.0000 0.958927
\(436\) 0 0
\(437\) 1.00000 0.0478365
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 26.0000 1.23530 0.617649 0.786454i \(-0.288085\pi\)
0.617649 + 0.786454i \(0.288085\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) −2.00000 −0.0939682
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −7.00000 −0.327446 −0.163723 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) −32.0000 −1.48396
\(466\) 0 0
\(467\) 2.00000 0.0925490 0.0462745 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 11.0000 0.504715
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −15.0000 −0.675566
\(494\) 0 0
\(495\) −16.0000 −0.719147
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −39.0000 −1.73892 −0.869462 0.494000i \(-0.835534\pi\)
−0.869462 + 0.494000i \(0.835534\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −27.0000 −1.19441
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) −24.0000 −1.05757
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) 0 0
\(525\) −33.0000 −1.44024
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 30.0000 1.30189
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) 60.0000 2.57012
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) −30.0000 −1.27573
\(554\) 0 0
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) 28.0000 1.18640 0.593199 0.805056i \(-0.297865\pi\)
0.593199 + 0.805056i \(0.297865\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −56.0000 −2.35594
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) −7.00000 −0.292429
\(574\) 0 0
\(575\) 11.0000 0.458732
\(576\) 0 0
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 8.00000 0.329076
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 36.0000 1.47586
\(596\) 0 0
\(597\) 25.0000 1.02318
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) 0 0
\(605\) 28.0000 1.13836
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 32.0000 1.29036
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) −2.00000 −0.0798723
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) −27.0000 −1.07315
\(634\) 0 0
\(635\) 72.0000 2.85723
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) −23.0000 −0.904223 −0.452112 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 0 0
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) 48.0000 1.87552
\(656\) 0 0
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) −5.00000 −0.194772 −0.0973862 0.995247i \(-0.531048\pi\)
−0.0973862 + 0.995247i \(0.531048\pi\)
\(660\) 0 0
\(661\) −23.0000 −0.894596 −0.447298 0.894385i \(-0.647614\pi\)
−0.447298 + 0.894385i \(0.647614\pi\)
\(662\) 0 0
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −5.00000 −0.193601
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 0 0
\(675\) −55.0000 −2.11695
\(676\) 0 0
\(677\) 13.0000 0.499631 0.249815 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 17.0000 0.651441
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 68.0000 2.59815
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) 1.00000 0.0380970
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 0 0
\(693\) −12.0000 −0.455842
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 32.0000 1.20519
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) −15.0000 −0.560185
\(718\) 0 0
\(719\) 5.00000 0.186469 0.0932343 0.995644i \(-0.470279\pi\)
0.0932343 + 0.995644i \(0.470279\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) −8.00000 −0.297523
\(724\) 0 0
\(725\) −55.0000 −2.04265
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 0 0
\(735\) −8.00000 −0.295084
\(736\) 0 0
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) −1.00000 −0.0367359
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) −21.0000 −0.767323
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −2.00000 −0.0728841
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 0 0
\(763\) 45.0000 1.62911
\(764\) 0 0
\(765\) 24.0000 0.867722
\(766\) 0 0
\(767\) 15.0000 0.541619
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) 0 0
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) 88.0000 3.16105
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 25.0000 0.893427
\(784\) 0 0
\(785\) 8.00000 0.285532
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −42.0000 −1.49335
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 4.00000 0.141865
\(796\) 0 0
\(797\) 3.00000 0.106265 0.0531327 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −18.0000 −0.635206
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 3.00000 0.105344 0.0526721 0.998612i \(-0.483226\pi\)
0.0526721 + 0.998612i \(0.483226\pi\)
\(812\) 0 0
\(813\) −7.00000 −0.245501
\(814\) 0 0
\(815\) −64.0000 −2.24182
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) −29.0000 −1.01088 −0.505438 0.862863i \(-0.668669\pi\)
−0.505438 + 0.862863i \(0.668669\pi\)
\(824\) 0 0
\(825\) −22.0000 −0.765942
\(826\) 0 0
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 0 0
\(829\) −15.0000 −0.520972 −0.260486 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −8.00000 −0.275535
\(844\) 0 0
\(845\) 48.0000 1.65125
\(846\) 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
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 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
\(865\) 24.0000 0.816024
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 72.0000 2.43404
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 0 0
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 0 0
\(885\) 60.0000 2.01688
\(886\) 0 0
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 0 0
\(889\) 54.0000 1.81110
\(890\) 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
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 0 0
\(903\) 12.0000 0.399335
\(904\) 0 0
\(905\) −88.0000 −2.92522
\(906\) 0 0
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) 0 0
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) 0 0
\(917\) 36.0000 1.18882
\(918\) 0 0
\(919\) −5.00000 −0.164935 −0.0824674 0.996594i \(-0.526280\pi\)
−0.0824674 + 0.996594i \(0.526280\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) −55.0000 −1.80449 −0.902246 0.431222i \(-0.858082\pi\)
−0.902246 + 0.431222i \(0.858082\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) −7.00000 −0.229170
\(934\) 0 0
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) 29.0000 0.946379
\(940\) 0 0
\(941\) 7.00000 0.228193 0.114097 0.993470i \(-0.463603\pi\)
0.114097 + 0.993470i \(0.463603\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) −60.0000 −1.95180
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −9.00000 −0.292152
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 0 0
\(955\) 28.0000 0.906059
\(956\) 0 0
\(957\) 10.0000 0.323254
\(958\) 0 0
\(959\) 51.0000 1.64688
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −14.0000 −0.451144
\(964\) 0 0
\(965\) 24.0000 0.772587
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −11.0000 −0.352282
\(976\) 0 0
\(977\) 8.00000 0.255943 0.127971 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 0 0
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) −32.0000 −1.01960
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) −17.0000 −0.539479
\(994\) 0 0
\(995\) −100.000 −3.17021
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 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.d.1.1 1
3.2 odd 2 2736.2.a.w.1.1 1
4.3 odd 2 38.2.a.b.1.1 1
5.4 even 2 7600.2.a.h.1.1 1
8.3 odd 2 1216.2.a.n.1.1 1
8.5 even 2 1216.2.a.g.1.1 1
12.11 even 2 342.2.a.d.1.1 1
19.18 odd 2 5776.2.a.d.1.1 1
20.3 even 4 950.2.b.c.799.1 2
20.7 even 4 950.2.b.c.799.2 2
20.19 odd 2 950.2.a.b.1.1 1
28.27 even 2 1862.2.a.f.1.1 1
44.43 even 2 4598.2.a.a.1.1 1
52.51 odd 2 6422.2.a.b.1.1 1
60.59 even 2 8550.2.a.u.1.1 1
76.3 even 18 722.2.e.d.389.1 6
76.7 odd 6 722.2.c.d.429.1 2
76.11 odd 6 722.2.c.d.653.1 2
76.15 even 18 722.2.e.d.415.1 6
76.23 odd 18 722.2.e.c.415.1 6
76.27 even 6 722.2.c.f.653.1 2
76.31 even 6 722.2.c.f.429.1 2
76.35 odd 18 722.2.e.c.389.1 6
76.43 odd 18 722.2.e.c.595.1 6
76.47 odd 18 722.2.e.c.423.1 6
76.51 even 18 722.2.e.d.245.1 6
76.55 odd 18 722.2.e.c.99.1 6
76.59 even 18 722.2.e.d.99.1 6
76.63 odd 18 722.2.e.c.245.1 6
76.67 even 18 722.2.e.d.423.1 6
76.71 even 18 722.2.e.d.595.1 6
76.75 even 2 722.2.a.b.1.1 1
228.227 odd 2 6498.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 4.3 odd 2
304.2.a.d.1.1 1 1.1 even 1 trivial
342.2.a.d.1.1 1 12.11 even 2
722.2.a.b.1.1 1 76.75 even 2
722.2.c.d.429.1 2 76.7 odd 6
722.2.c.d.653.1 2 76.11 odd 6
722.2.c.f.429.1 2 76.31 even 6
722.2.c.f.653.1 2 76.27 even 6
722.2.e.c.99.1 6 76.55 odd 18
722.2.e.c.245.1 6 76.63 odd 18
722.2.e.c.389.1 6 76.35 odd 18
722.2.e.c.415.1 6 76.23 odd 18
722.2.e.c.423.1 6 76.47 odd 18
722.2.e.c.595.1 6 76.43 odd 18
722.2.e.d.99.1 6 76.59 even 18
722.2.e.d.245.1 6 76.51 even 18
722.2.e.d.389.1 6 76.3 even 18
722.2.e.d.415.1 6 76.15 even 18
722.2.e.d.423.1 6 76.67 even 18
722.2.e.d.595.1 6 76.71 even 18
950.2.a.b.1.1 1 20.19 odd 2
950.2.b.c.799.1 2 20.3 even 4
950.2.b.c.799.2 2 20.7 even 4
1216.2.a.g.1.1 1 8.5 even 2
1216.2.a.n.1.1 1 8.3 odd 2
1862.2.a.f.1.1 1 28.27 even 2
2736.2.a.w.1.1 1 3.2 odd 2
4598.2.a.a.1.1 1 44.43 even 2
5776.2.a.d.1.1 1 19.18 odd 2
6422.2.a.b.1.1 1 52.51 odd 2
6498.2.a.y.1.1 1 228.227 odd 2
7600.2.a.h.1.1 1 5.4 even 2
8550.2.a.u.1.1 1 60.59 even 2