Properties

Label 5850.2.a.k.1.1
Level $5850$
Weight $2$
Character 5850.1
Self dual yes
Analytic conductor $46.712$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.7124851824\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1950)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} -5.00000 q^{22} +1.00000 q^{26} -1.00000 q^{28} +7.00000 q^{29} -9.00000 q^{31} -1.00000 q^{32} +5.00000 q^{34} -8.00000 q^{37} +2.00000 q^{41} +8.00000 q^{43} +5.00000 q^{44} +9.00000 q^{47} -6.00000 q^{49} -1.00000 q^{52} -11.0000 q^{53} +1.00000 q^{56} -7.00000 q^{58} -1.00000 q^{59} -7.00000 q^{61} +9.00000 q^{62} +1.00000 q^{64} -15.0000 q^{67} -5.00000 q^{68} +8.00000 q^{71} +4.00000 q^{73} +8.00000 q^{74} -5.00000 q^{77} -4.00000 q^{79} -2.00000 q^{82} +9.00000 q^{83} -8.00000 q^{86} -5.00000 q^{88} -16.0000 q^{89} +1.00000 q^{91} -9.00000 q^{94} +2.00000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −7.00000 −0.919145
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 9.00000 1.14300
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −15.0000 −1.83254 −0.916271 0.400559i \(-0.868816\pi\)
−0.916271 + 0.400559i \(0.868816\pi\)
\(68\) −5.00000 −0.606339
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −5.00000 −0.533002
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 0 0
\(101\) 7.00000 0.696526 0.348263 0.937397i \(-0.386772\pi\)
0.348263 + 0.937397i \(0.386772\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.00000 0.649934
\(117\) 0 0
\(118\) 1.00000 0.0920575
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 7.00000 0.633750
\(123\) 0 0
\(124\) −9.00000 −0.808224
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 15.0000 1.29580
\(135\) 0 0
\(136\) 5.00000 0.428746
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) −21.0000 −1.70896 −0.854478 0.519488i \(-0.826123\pi\)
−0.854478 + 0.519488i \(0.826123\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 5.00000 0.402911
\(155\) 0 0
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 11.0000 0.836315 0.418157 0.908375i \(-0.362676\pi\)
0.418157 + 0.908375i \(0.362676\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) 16.0000 1.19925
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −11.0000 −0.817624 −0.408812 0.912619i \(-0.634057\pi\)
−0.408812 + 0.912619i \(0.634057\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −25.0000 −1.82818
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −7.00000 −0.492518
\(203\) −7.00000 −0.491304
\(204\) 0 0
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −11.0000 −0.755483
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) 5.00000 0.336336
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −29.0000 −1.92480 −0.962399 0.271640i \(-0.912434\pi\)
−0.962399 + 0.271640i \(0.912434\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.00000 −0.459573
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.00000 −0.0650945
\(237\) 0 0
\(238\) −5.00000 −0.324102
\(239\) −1.00000 −0.0646846 −0.0323423 0.999477i \(-0.510297\pi\)
−0.0323423 + 0.999477i \(0.510297\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 9.00000 0.571501
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) −2.00000 −0.123560
\(263\) −26.0000 −1.60323 −0.801614 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −15.0000 −0.916271
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 2.00000 0.119952
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −22.0000 −1.27443
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 21.0000 1.20841
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −5.00000 −0.284901
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 21.0000 1.18699 0.593495 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(314\) 5.00000 0.282166
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 35.0000 1.95962
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 9.00000 0.493939
\(333\) 0 0
\(334\) −4.00000 −0.218870
\(335\) 0 0
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 0 0
\(341\) −45.0000 −2.43689
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −11.0000 −0.591364
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 32.0000 1.71292 0.856460 0.516213i \(-0.172659\pi\)
0.856460 + 0.516213i \(0.172659\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) −27.0000 −1.42501 −0.712503 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 11.0000 0.578147
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) −17.0000 −0.880227 −0.440113 0.897942i \(-0.645062\pi\)
−0.440113 + 0.897942i \(0.645062\pi\)
\(374\) 25.0000 1.29272
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) −7.00000 −0.360518
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 18.0000 0.920960
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −18.0000 −0.902258
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 9.00000 0.448322
\(404\) 7.00000 0.348263
\(405\) 0 0
\(406\) 7.00000 0.347404
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.00000 0.295599
\(413\) 1.00000 0.0492068
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00000 0.0977064 0.0488532 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) −16.0000 −0.778868
\(423\) 0 0
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) 0 0
\(427\) 7.00000 0.338754
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −9.00000 −0.432014
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.00000 −0.237826
\(443\) 32.0000 1.52037 0.760183 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 29.0000 1.36104
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −11.0000 −0.511213 −0.255607 0.966781i \(-0.582275\pi\)
−0.255607 + 0.966781i \(0.582275\pi\)
\(464\) 7.00000 0.324967
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 32.0000 1.48078 0.740392 0.672176i \(-0.234640\pi\)
0.740392 + 0.672176i \(0.234640\pi\)
\(468\) 0 0
\(469\) 15.0000 0.692636
\(470\) 0 0
\(471\) 0 0
\(472\) 1.00000 0.0460287
\(473\) 40.0000 1.83920
\(474\) 0 0
\(475\) 0 0
\(476\) 5.00000 0.229175
\(477\) 0 0
\(478\) 1.00000 0.0457389
\(479\) −29.0000 −1.32504 −0.662522 0.749043i \(-0.730514\pi\)
−0.662522 + 0.749043i \(0.730514\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) −1.00000 −0.0453143 −0.0226572 0.999743i \(-0.507213\pi\)
−0.0226572 + 0.999743i \(0.507213\pi\)
\(488\) 7.00000 0.316875
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) −35.0000 −1.57632
\(494\) 0 0
\(495\) 0 0
\(496\) −9.00000 −0.404112
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 11.0000 0.492428 0.246214 0.969216i \(-0.420813\pi\)
0.246214 + 0.969216i \(0.420813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −6.00000 −0.267793
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 27.0000 1.19092
\(515\) 0 0
\(516\) 0 0
\(517\) 45.0000 1.97910
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) −18.0000 −0.787085 −0.393543 0.919306i \(-0.628751\pi\)
−0.393543 + 0.919306i \(0.628751\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) 26.0000 1.13365
\(527\) 45.0000 1.96023
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) 0 0
\(536\) 15.0000 0.647901
\(537\) 0 0
\(538\) 9.00000 0.388018
\(539\) −30.0000 −1.29219
\(540\) 0 0
\(541\) 28.0000 1.20381 0.601907 0.798566i \(-0.294408\pi\)
0.601907 + 0.798566i \(0.294408\pi\)
\(542\) −7.00000 −0.300676
\(543\) 0 0
\(544\) 5.00000 0.214373
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22.0000 0.924729
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 9.00000 0.377300 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −5.00000 −0.209061
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) 0 0
\(576\) 0 0
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 0 0
\(583\) −55.0000 −2.27787
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) −21.0000 −0.854478
\(605\) 0 0
\(606\) 0 0
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) 0 0
\(626\) −21.0000 −0.839329
\(627\) 0 0
\(628\) −5.00000 −0.199522
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) −35.0000 −1.38566
\(639\) 0 0
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 9.00000 0.350857
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 4.00000 0.154765
\(669\) 0 0
\(670\) 0 0
\(671\) −35.0000 −1.35116
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 45.0000 1.72314
\(683\) −19.0000 −0.727015 −0.363507 0.931591i \(-0.618421\pi\)
−0.363507 + 0.931591i \(0.618421\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 11.0000 0.419067
\(690\) 0 0
\(691\) 7.00000 0.266293 0.133146 0.991096i \(-0.457492\pi\)
0.133146 + 0.991096i \(0.457492\pi\)
\(692\) 11.0000 0.418157
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) −10.0000 −0.378777
\(698\) −32.0000 −1.21122
\(699\) 0 0
\(700\) 0 0
\(701\) 25.0000 0.944237 0.472118 0.881535i \(-0.343489\pi\)
0.472118 + 0.881535i \(0.343489\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) −7.00000 −0.263262
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 16.0000 0.599625
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 0 0
\(718\) 27.0000 1.00763
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −11.0000 −0.408812
\(725\) 0 0
\(726\) 0 0
\(727\) −38.0000 −1.40934 −0.704671 0.709534i \(-0.748905\pi\)
−0.704671 + 0.709534i \(0.748905\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 36.0000 1.32878
\(735\) 0 0
\(736\) 0 0
\(737\) −75.0000 −2.76266
\(738\) 0 0
\(739\) 3.00000 0.110357 0.0551784 0.998477i \(-0.482427\pi\)
0.0551784 + 0.998477i \(0.482427\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11.0000 −0.403823
\(743\) −39.0000 −1.43077 −0.715386 0.698730i \(-0.753749\pi\)
−0.715386 + 0.698730i \(0.753749\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 17.0000 0.622414
\(747\) 0 0
\(748\) −25.0000 −0.914091
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 9.00000 0.328196
\(753\) 0 0
\(754\) 7.00000 0.254925
\(755\) 0 0
\(756\) 0 0
\(757\) 33.0000 1.19941 0.599703 0.800223i \(-0.295286\pi\)
0.599703 + 0.800223i \(0.295286\pi\)
\(758\) 15.0000 0.544825
\(759\) 0 0
\(760\) 0 0
\(761\) 8.00000 0.290000 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) 0 0
\(763\) −6.00000 −0.217215
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 0 0
\(767\) 1.00000 0.0361079
\(768\) 0 0
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 2.00000 0.0717035
\(779\) 0 0
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) −5.00000 −0.178231 −0.0891154 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 7.00000 0.248577
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) 13.0000 0.460484 0.230242 0.973133i \(-0.426048\pi\)
0.230242 + 0.973133i \(0.426048\pi\)
\(798\) 0 0
\(799\) −45.0000 −1.59199
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) 0 0
\(806\) −9.00000 −0.317011
\(807\) 0 0
\(808\) −7.00000 −0.246259
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 0 0
\(811\) −29.0000 −1.01833 −0.509164 0.860670i \(-0.670045\pi\)
−0.509164 + 0.860670i \(0.670045\pi\)
\(812\) −7.00000 −0.245652
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) 16.0000 0.558404 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) 0 0
\(823\) −28.0000 −0.976019 −0.488009 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) −1.00000 −0.0347945
\(827\) 21.0000 0.730242 0.365121 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) 0 0
\(829\) −37.0000 −1.28506 −0.642532 0.766259i \(-0.722116\pi\)
−0.642532 + 0.766259i \(0.722116\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −2.00000 −0.0690889
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) −28.0000 −0.964944
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) −11.0000 −0.377742
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) −7.00000 −0.239535
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −27.0000 −0.919091 −0.459545 0.888154i \(-0.651988\pi\)
−0.459545 + 0.888154i \(0.651988\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 9.00000 0.305480
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 15.0000 0.508256
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 40.0000 1.35070 0.675352 0.737496i \(-0.263992\pi\)
0.675352 + 0.737496i \(0.263992\pi\)
\(878\) −4.00000 −0.134993
\(879\) 0 0
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 5.00000 0.168168
\(885\) 0 0
\(886\) −32.0000 −1.07506
\(887\) −4.00000 −0.134307 −0.0671534 0.997743i \(-0.521392\pi\)
−0.0671534 + 0.997743i \(0.521392\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 36.0000 1.20134
\(899\) −63.0000 −2.10117
\(900\) 0 0
\(901\) 55.0000 1.83232
\(902\) −10.0000 −0.332964
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) 0 0
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) −29.0000 −0.962399
\(909\) 0 0
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 45.0000 1.48928
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) −2.00000 −0.0660458
\(918\) 0 0
\(919\) −50.0000 −1.64935 −0.824674 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) 11.0000 0.361482
\(927\) 0 0
\(928\) −7.00000 −0.229786
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −32.0000 −1.04707
\(935\) 0 0
\(936\) 0 0
\(937\) 51.0000 1.66610 0.833049 0.553200i \(-0.186593\pi\)
0.833049 + 0.553200i \(0.186593\pi\)
\(938\) −15.0000 −0.489767
\(939\) 0 0
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.00000 −0.0325472
\(945\) 0 0
\(946\) −40.0000 −1.30051
\(947\) 41.0000 1.33232 0.666160 0.745808i \(-0.267937\pi\)
0.666160 + 0.745808i \(0.267937\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) −5.00000 −0.162051
\(953\) −51.0000 −1.65205 −0.826026 0.563632i \(-0.809404\pi\)
−0.826026 + 0.563632i \(0.809404\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.00000 −0.0323423
\(957\) 0 0
\(958\) 29.0000 0.936947
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) −8.00000 −0.257930
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) −17.0000 −0.546683 −0.273342 0.961917i \(-0.588129\pi\)
−0.273342 + 0.961917i \(0.588129\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) 0 0
\(971\) 34.0000 1.09111 0.545556 0.838074i \(-0.316319\pi\)
0.545556 + 0.838074i \(0.316319\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) 1.00000 0.0320421
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) −80.0000 −2.55681
\(980\) 0 0
\(981\) 0 0
\(982\) 30.0000 0.957338
\(983\) 31.0000 0.988746 0.494373 0.869250i \(-0.335398\pi\)
0.494373 + 0.869250i \(0.335398\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 35.0000 1.11463
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) 9.00000 0.285750
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) −11.0000 −0.348373 −0.174187 0.984713i \(-0.555730\pi\)
−0.174187 + 0.984713i \(0.555730\pi\)
\(998\) −11.0000 −0.348199
\(999\) 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 5850.2.a.k.1.1 1
3.2 odd 2 1950.2.a.p.1.1 yes 1
5.2 odd 4 5850.2.e.be.5149.1 2
5.3 odd 4 5850.2.e.be.5149.2 2
5.4 even 2 5850.2.a.bu.1.1 1
15.2 even 4 1950.2.e.a.1249.2 2
15.8 even 4 1950.2.e.a.1249.1 2
15.14 odd 2 1950.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.l.1.1 1 15.14 odd 2
1950.2.a.p.1.1 yes 1 3.2 odd 2
1950.2.e.a.1249.1 2 15.8 even 4
1950.2.e.a.1249.2 2 15.2 even 4
5850.2.a.k.1.1 1 1.1 even 1 trivial
5850.2.a.bu.1.1 1 5.4 even 2
5850.2.e.be.5149.1 2 5.2 odd 4
5850.2.e.be.5149.2 2 5.3 odd 4