Properties

Label 7623.2.a.cd.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} +1.00000 q^{7} +5.22982 q^{8} +O(q^{10})\) \(q+2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} +1.00000 q^{7} +5.22982 q^{8} +6.39905 q^{10} +5.87189 q^{13} +2.47283 q^{14} +4.70265 q^{16} +7.51396 q^{17} +2.35793 q^{19} +10.6483 q^{20} -6.94567 q^{23} +1.69641 q^{25} +14.5202 q^{26} +4.11491 q^{28} -5.87189 q^{29} -3.66152 q^{31} +1.16924 q^{32} +18.5808 q^{34} +2.58774 q^{35} +3.30359 q^{37} +5.83076 q^{38} +13.5334 q^{40} +5.28415 q^{41} -7.40530 q^{43} -17.1755 q^{46} -7.53341 q^{47} +1.00000 q^{49} +4.19493 q^{50} +24.1623 q^{52} +4.22982 q^{53} +5.22982 q^{56} -14.5202 q^{58} +0.926221 q^{59} +2.00000 q^{61} -9.05433 q^{62} -6.51396 q^{64} +15.1949 q^{65} -10.1017 q^{67} +30.9193 q^{68} +6.39905 q^{70} -4.45963 q^{71} +2.12811 q^{73} +8.16924 q^{74} +9.70265 q^{76} +4.45963 q^{79} +12.1692 q^{80} +13.0668 q^{82} -10.6894 q^{83} +19.4442 q^{85} -18.3121 q^{86} -15.8913 q^{89} +5.87189 q^{91} -28.5808 q^{92} -18.6289 q^{94} +6.10170 q^{95} +10.1212 q^{97} +2.47283 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{2} + 6q^{4} - 4q^{5} + 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 2q^{2} + 6q^{4} - 4q^{5} + 3q^{7} + 3q^{8} + 11q^{10} + 4q^{13} + 2q^{14} - 4q^{16} + 8q^{17} + 8q^{19} + 3q^{20} - 10q^{23} + 15q^{25} + q^{26} + 6q^{28} - 4q^{29} - 2q^{31} + 8q^{32} - 4q^{34} - 4q^{35} + 13q^{38} + 18q^{40} + 14q^{41} + 14q^{43} - 28q^{46} + 3q^{49} - 19q^{50} + 29q^{52} + 3q^{56} - q^{58} + 6q^{61} - 38q^{62} - 5q^{64} + 14q^{65} - 4q^{67} + 42q^{68} + 11q^{70} + 12q^{71} + 20q^{73} + 29q^{74} + 11q^{76} - 12q^{79} + 41q^{80} - 6q^{82} + 6q^{83} + 6q^{85} - 24q^{86} - 26q^{89} + 4q^{91} - 26q^{92} - 35q^{94} - 8q^{95} - 4q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.47283 1.74856 0.874279 0.485424i \(-0.161335\pi\)
0.874279 + 0.485424i \(0.161335\pi\)
\(3\) 0 0
\(4\) 4.11491 2.05745
\(5\) 2.58774 1.15727 0.578637 0.815586i \(-0.303585\pi\)
0.578637 + 0.815586i \(0.303585\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 5.22982 1.84902
\(9\) 0 0
\(10\) 6.39905 2.02356
\(11\) 0 0
\(12\) 0 0
\(13\) 5.87189 1.62857 0.814284 0.580466i \(-0.197130\pi\)
0.814284 + 0.580466i \(0.197130\pi\)
\(14\) 2.47283 0.660893
\(15\) 0 0
\(16\) 4.70265 1.17566
\(17\) 7.51396 1.82240 0.911202 0.411960i \(-0.135156\pi\)
0.911202 + 0.411960i \(0.135156\pi\)
\(18\) 0 0
\(19\) 2.35793 0.540945 0.270473 0.962728i \(-0.412820\pi\)
0.270473 + 0.962728i \(0.412820\pi\)
\(20\) 10.6483 2.38104
\(21\) 0 0
\(22\) 0 0
\(23\) −6.94567 −1.44827 −0.724136 0.689657i \(-0.757761\pi\)
−0.724136 + 0.689657i \(0.757761\pi\)
\(24\) 0 0
\(25\) 1.69641 0.339281
\(26\) 14.5202 2.84765
\(27\) 0 0
\(28\) 4.11491 0.777644
\(29\) −5.87189 −1.09038 −0.545191 0.838312i \(-0.683543\pi\)
−0.545191 + 0.838312i \(0.683543\pi\)
\(30\) 0 0
\(31\) −3.66152 −0.657629 −0.328814 0.944395i \(-0.606649\pi\)
−0.328814 + 0.944395i \(0.606649\pi\)
\(32\) 1.16924 0.206694
\(33\) 0 0
\(34\) 18.5808 3.18658
\(35\) 2.58774 0.437408
\(36\) 0 0
\(37\) 3.30359 0.543108 0.271554 0.962423i \(-0.412463\pi\)
0.271554 + 0.962423i \(0.412463\pi\)
\(38\) 5.83076 0.945874
\(39\) 0 0
\(40\) 13.5334 2.13982
\(41\) 5.28415 0.825245 0.412623 0.910902i \(-0.364613\pi\)
0.412623 + 0.910902i \(0.364613\pi\)
\(42\) 0 0
\(43\) −7.40530 −1.12930 −0.564649 0.825331i \(-0.690988\pi\)
−0.564649 + 0.825331i \(0.690988\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −17.1755 −2.53239
\(47\) −7.53341 −1.09886 −0.549430 0.835540i \(-0.685155\pi\)
−0.549430 + 0.835540i \(0.685155\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.19493 0.593253
\(51\) 0 0
\(52\) 24.1623 3.35071
\(53\) 4.22982 0.581010 0.290505 0.956874i \(-0.406177\pi\)
0.290505 + 0.956874i \(0.406177\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 5.22982 0.698863
\(57\) 0 0
\(58\) −14.5202 −1.90660
\(59\) 0.926221 0.120584 0.0602918 0.998181i \(-0.480797\pi\)
0.0602918 + 0.998181i \(0.480797\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −9.05433 −1.14990
\(63\) 0 0
\(64\) −6.51396 −0.814245
\(65\) 15.1949 1.88470
\(66\) 0 0
\(67\) −10.1017 −1.23412 −0.617060 0.786916i \(-0.711676\pi\)
−0.617060 + 0.786916i \(0.711676\pi\)
\(68\) 30.9193 3.74951
\(69\) 0 0
\(70\) 6.39905 0.764833
\(71\) −4.45963 −0.529261 −0.264630 0.964350i \(-0.585250\pi\)
−0.264630 + 0.964350i \(0.585250\pi\)
\(72\) 0 0
\(73\) 2.12811 0.249077 0.124538 0.992215i \(-0.460255\pi\)
0.124538 + 0.992215i \(0.460255\pi\)
\(74\) 8.16924 0.949655
\(75\) 0 0
\(76\) 9.70265 1.11297
\(77\) 0 0
\(78\) 0 0
\(79\) 4.45963 0.501748 0.250874 0.968020i \(-0.419282\pi\)
0.250874 + 0.968020i \(0.419282\pi\)
\(80\) 12.1692 1.36056
\(81\) 0 0
\(82\) 13.0668 1.44299
\(83\) −10.6894 −1.17332 −0.586660 0.809834i \(-0.699557\pi\)
−0.586660 + 0.809834i \(0.699557\pi\)
\(84\) 0 0
\(85\) 19.4442 2.10902
\(86\) −18.3121 −1.97464
\(87\) 0 0
\(88\) 0 0
\(89\) −15.8913 −1.68448 −0.842239 0.539104i \(-0.818763\pi\)
−0.842239 + 0.539104i \(0.818763\pi\)
\(90\) 0 0
\(91\) 5.87189 0.615541
\(92\) −28.5808 −2.97975
\(93\) 0 0
\(94\) −18.6289 −1.92142
\(95\) 6.10170 0.626022
\(96\) 0 0
\(97\) 10.1212 1.02765 0.513824 0.857896i \(-0.328229\pi\)
0.513824 + 0.857896i \(0.328229\pi\)
\(98\) 2.47283 0.249794
\(99\) 0 0
\(100\) 6.98055 0.698055
\(101\) −13.4053 −1.33388 −0.666939 0.745113i \(-0.732396\pi\)
−0.666939 + 0.745113i \(0.732396\pi\)
\(102\) 0 0
\(103\) −4.71585 −0.464667 −0.232333 0.972636i \(-0.574636\pi\)
−0.232333 + 0.972636i \(0.574636\pi\)
\(104\) 30.7089 3.01125
\(105\) 0 0
\(106\) 10.4596 1.01593
\(107\) 7.07378 0.683848 0.341924 0.939728i \(-0.388921\pi\)
0.341924 + 0.939728i \(0.388921\pi\)
\(108\) 0 0
\(109\) −18.8370 −1.80426 −0.902129 0.431467i \(-0.857996\pi\)
−0.902129 + 0.431467i \(0.857996\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.70265 0.444359
\(113\) 9.02792 0.849276 0.424638 0.905363i \(-0.360401\pi\)
0.424638 + 0.905363i \(0.360401\pi\)
\(114\) 0 0
\(115\) −17.9736 −1.67605
\(116\) −24.1623 −2.24341
\(117\) 0 0
\(118\) 2.29039 0.210848
\(119\) 7.51396 0.688804
\(120\) 0 0
\(121\) 0 0
\(122\) 4.94567 0.447760
\(123\) 0 0
\(124\) −15.0668 −1.35304
\(125\) −8.54885 −0.764632
\(126\) 0 0
\(127\) 10.2298 0.907749 0.453875 0.891066i \(-0.350041\pi\)
0.453875 + 0.891066i \(0.350041\pi\)
\(128\) −18.4464 −1.63045
\(129\) 0 0
\(130\) 37.5745 3.29550
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 2.35793 0.204458
\(134\) −24.9798 −2.15793
\(135\) 0 0
\(136\) 39.2966 3.36966
\(137\) 10.8370 0.925868 0.462934 0.886393i \(-0.346797\pi\)
0.462934 + 0.886393i \(0.346797\pi\)
\(138\) 0 0
\(139\) −0.459630 −0.0389853 −0.0194927 0.999810i \(-0.506205\pi\)
−0.0194927 + 0.999810i \(0.506205\pi\)
\(140\) 10.6483 0.899947
\(141\) 0 0
\(142\) −11.0279 −0.925443
\(143\) 0 0
\(144\) 0 0
\(145\) −15.1949 −1.26187
\(146\) 5.26247 0.435525
\(147\) 0 0
\(148\) 13.5940 1.11742
\(149\) 11.0474 0.905036 0.452518 0.891755i \(-0.350526\pi\)
0.452518 + 0.891755i \(0.350526\pi\)
\(150\) 0 0
\(151\) 16.1212 1.31192 0.655960 0.754795i \(-0.272264\pi\)
0.655960 + 0.754795i \(0.272264\pi\)
\(152\) 12.3315 1.00022
\(153\) 0 0
\(154\) 0 0
\(155\) −9.47507 −0.761056
\(156\) 0 0
\(157\) 13.0668 1.04285 0.521423 0.853298i \(-0.325401\pi\)
0.521423 + 0.853298i \(0.325401\pi\)
\(158\) 11.0279 0.877335
\(159\) 0 0
\(160\) 3.02569 0.239202
\(161\) −6.94567 −0.547395
\(162\) 0 0
\(163\) −10.1017 −0.791227 −0.395613 0.918417i \(-0.629468\pi\)
−0.395613 + 0.918417i \(0.629468\pi\)
\(164\) 21.7438 1.69790
\(165\) 0 0
\(166\) −26.4332 −2.05162
\(167\) −10.4860 −0.811434 −0.405717 0.913999i \(-0.632978\pi\)
−0.405717 + 0.913999i \(0.632978\pi\)
\(168\) 0 0
\(169\) 21.4791 1.65224
\(170\) 48.0823 3.68774
\(171\) 0 0
\(172\) −30.4721 −2.32348
\(173\) 14.4985 1.10230 0.551151 0.834405i \(-0.314189\pi\)
0.551151 + 0.834405i \(0.314189\pi\)
\(174\) 0 0
\(175\) 1.69641 0.128236
\(176\) 0 0
\(177\) 0 0
\(178\) −39.2966 −2.94541
\(179\) −14.3510 −1.07264 −0.536321 0.844014i \(-0.680186\pi\)
−0.536321 + 0.844014i \(0.680186\pi\)
\(180\) 0 0
\(181\) 9.66152 0.718135 0.359068 0.933312i \(-0.383095\pi\)
0.359068 + 0.933312i \(0.383095\pi\)
\(182\) 14.5202 1.07631
\(183\) 0 0
\(184\) −36.3246 −2.67788
\(185\) 8.54885 0.628524
\(186\) 0 0
\(187\) 0 0
\(188\) −30.9993 −2.26086
\(189\) 0 0
\(190\) 15.0885 1.09463
\(191\) 9.55286 0.691220 0.345610 0.938378i \(-0.387672\pi\)
0.345610 + 0.938378i \(0.387672\pi\)
\(192\) 0 0
\(193\) 22.1212 1.59232 0.796158 0.605089i \(-0.206863\pi\)
0.796158 + 0.605089i \(0.206863\pi\)
\(194\) 25.0279 1.79690
\(195\) 0 0
\(196\) 4.11491 0.293922
\(197\) −26.2423 −1.86969 −0.934843 0.355061i \(-0.884460\pi\)
−0.934843 + 0.355061i \(0.884460\pi\)
\(198\) 0 0
\(199\) 1.05433 0.0747396 0.0373698 0.999302i \(-0.488102\pi\)
0.0373698 + 0.999302i \(0.488102\pi\)
\(200\) 8.87189 0.627337
\(201\) 0 0
\(202\) −33.1491 −2.33236
\(203\) −5.87189 −0.412126
\(204\) 0 0
\(205\) 13.6740 0.955034
\(206\) −11.6615 −0.812497
\(207\) 0 0
\(208\) 27.6134 1.91465
\(209\) 0 0
\(210\) 0 0
\(211\) 8.71585 0.600024 0.300012 0.953935i \(-0.403009\pi\)
0.300012 + 0.953935i \(0.403009\pi\)
\(212\) 17.4053 1.19540
\(213\) 0 0
\(214\) 17.4923 1.19575
\(215\) −19.1630 −1.30691
\(216\) 0 0
\(217\) −3.66152 −0.248560
\(218\) −46.5808 −3.15485
\(219\) 0 0
\(220\) 0 0
\(221\) 44.1212 2.96791
\(222\) 0 0
\(223\) −23.7438 −1.59000 −0.795000 0.606609i \(-0.792529\pi\)
−0.795000 + 0.606609i \(0.792529\pi\)
\(224\) 1.16924 0.0781231
\(225\) 0 0
\(226\) 22.3246 1.48501
\(227\) −13.1755 −0.874488 −0.437244 0.899343i \(-0.644045\pi\)
−0.437244 + 0.899343i \(0.644045\pi\)
\(228\) 0 0
\(229\) 13.3664 0.883277 0.441638 0.897193i \(-0.354397\pi\)
0.441638 + 0.897193i \(0.354397\pi\)
\(230\) −44.4457 −2.93066
\(231\) 0 0
\(232\) −30.7089 −2.01614
\(233\) 7.13659 0.467533 0.233767 0.972293i \(-0.424895\pi\)
0.233767 + 0.972293i \(0.424895\pi\)
\(234\) 0 0
\(235\) −19.4945 −1.27168
\(236\) 3.81131 0.248095
\(237\) 0 0
\(238\) 18.5808 1.20441
\(239\) −19.9930 −1.29324 −0.646621 0.762811i \(-0.723818\pi\)
−0.646621 + 0.762811i \(0.723818\pi\)
\(240\) 0 0
\(241\) 9.19493 0.592297 0.296149 0.955142i \(-0.404298\pi\)
0.296149 + 0.955142i \(0.404298\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 8.22982 0.526860
\(245\) 2.58774 0.165325
\(246\) 0 0
\(247\) 13.8455 0.880967
\(248\) −19.1491 −1.21597
\(249\) 0 0
\(250\) −21.1399 −1.33700
\(251\) 9.42474 0.594885 0.297442 0.954740i \(-0.403866\pi\)
0.297442 + 0.954740i \(0.403866\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 25.2966 1.58725
\(255\) 0 0
\(256\) −32.5870 −2.03669
\(257\) 0.0194469 0.00121307 0.000606533 1.00000i \(-0.499807\pi\)
0.000606533 1.00000i \(0.499807\pi\)
\(258\) 0 0
\(259\) 3.30359 0.205275
\(260\) 62.5257 3.87768
\(261\) 0 0
\(262\) 9.89134 0.611089
\(263\) 27.9930 1.72612 0.863062 0.505097i \(-0.168543\pi\)
0.863062 + 0.505097i \(0.168543\pi\)
\(264\) 0 0
\(265\) 10.9457 0.672387
\(266\) 5.83076 0.357507
\(267\) 0 0
\(268\) −41.5676 −2.53914
\(269\) −30.9193 −1.88518 −0.942590 0.333951i \(-0.891618\pi\)
−0.942590 + 0.333951i \(0.891618\pi\)
\(270\) 0 0
\(271\) 22.3579 1.35815 0.679074 0.734070i \(-0.262382\pi\)
0.679074 + 0.734070i \(0.262382\pi\)
\(272\) 35.3355 2.14253
\(273\) 0 0
\(274\) 26.7981 1.61893
\(275\) 0 0
\(276\) 0 0
\(277\) −24.7717 −1.48839 −0.744194 0.667964i \(-0.767166\pi\)
−0.744194 + 0.667964i \(0.767166\pi\)
\(278\) −1.13659 −0.0681681
\(279\) 0 0
\(280\) 13.5334 0.808776
\(281\) −1.19493 −0.0712835 −0.0356418 0.999365i \(-0.511348\pi\)
−0.0356418 + 0.999365i \(0.511348\pi\)
\(282\) 0 0
\(283\) 15.5723 0.925677 0.462839 0.886443i \(-0.346831\pi\)
0.462839 + 0.886443i \(0.346831\pi\)
\(284\) −18.3510 −1.08893
\(285\) 0 0
\(286\) 0 0
\(287\) 5.28415 0.311913
\(288\) 0 0
\(289\) 39.4596 2.32115
\(290\) −37.5745 −2.20645
\(291\) 0 0
\(292\) 8.75698 0.512464
\(293\) −9.74378 −0.569238 −0.284619 0.958641i \(-0.591867\pi\)
−0.284619 + 0.958641i \(0.591867\pi\)
\(294\) 0 0
\(295\) 2.39682 0.139548
\(296\) 17.2772 1.00422
\(297\) 0 0
\(298\) 27.3183 1.58251
\(299\) −40.7842 −2.35861
\(300\) 0 0
\(301\) −7.40530 −0.426834
\(302\) 39.8649 2.29397
\(303\) 0 0
\(304\) 11.0885 0.635969
\(305\) 5.17548 0.296347
\(306\) 0 0
\(307\) −6.35097 −0.362469 −0.181234 0.983440i \(-0.558009\pi\)
−0.181234 + 0.983440i \(0.558009\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −23.4303 −1.33075
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 16.9457 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(314\) 32.3121 1.82348
\(315\) 0 0
\(316\) 18.3510 1.03232
\(317\) −20.0125 −1.12401 −0.562007 0.827133i \(-0.689970\pi\)
−0.562007 + 0.827133i \(0.689970\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −16.8565 −0.942304
\(321\) 0 0
\(322\) −17.1755 −0.957152
\(323\) 17.7174 0.985821
\(324\) 0 0
\(325\) 9.96111 0.552543
\(326\) −24.9798 −1.38351
\(327\) 0 0
\(328\) 27.6351 1.52589
\(329\) −7.53341 −0.415330
\(330\) 0 0
\(331\) 7.54037 0.414456 0.207228 0.978293i \(-0.433556\pi\)
0.207228 + 0.978293i \(0.433556\pi\)
\(332\) −43.9861 −2.41405
\(333\) 0 0
\(334\) −25.9302 −1.41884
\(335\) −26.1406 −1.42821
\(336\) 0 0
\(337\) 12.6894 0.691238 0.345619 0.938375i \(-0.387669\pi\)
0.345619 + 0.938375i \(0.387669\pi\)
\(338\) 53.1142 2.88903
\(339\) 0 0
\(340\) 80.0111 4.33921
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −38.7283 −2.08809
\(345\) 0 0
\(346\) 35.8524 1.92744
\(347\) 1.13659 0.0610153 0.0305076 0.999535i \(-0.490288\pi\)
0.0305076 + 0.999535i \(0.490288\pi\)
\(348\) 0 0
\(349\) 14.7911 0.791752 0.395876 0.918304i \(-0.370441\pi\)
0.395876 + 0.918304i \(0.370441\pi\)
\(350\) 4.19493 0.224228
\(351\) 0 0
\(352\) 0 0
\(353\) 7.52092 0.400298 0.200149 0.979765i \(-0.435857\pi\)
0.200149 + 0.979765i \(0.435857\pi\)
\(354\) 0 0
\(355\) −11.5404 −0.612499
\(356\) −65.3914 −3.46574
\(357\) 0 0
\(358\) −35.4876 −1.87558
\(359\) −17.5962 −0.928693 −0.464346 0.885654i \(-0.653711\pi\)
−0.464346 + 0.885654i \(0.653711\pi\)
\(360\) 0 0
\(361\) −13.4402 −0.707378
\(362\) 23.8913 1.25570
\(363\) 0 0
\(364\) 24.1623 1.26645
\(365\) 5.50700 0.288250
\(366\) 0 0
\(367\) 14.3121 0.747084 0.373542 0.927613i \(-0.378143\pi\)
0.373542 + 0.927613i \(0.378143\pi\)
\(368\) −32.6630 −1.70268
\(369\) 0 0
\(370\) 21.1399 1.09901
\(371\) 4.22982 0.219601
\(372\) 0 0
\(373\) 16.4332 0.850880 0.425440 0.904987i \(-0.360119\pi\)
0.425440 + 0.904987i \(0.360119\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −39.3983 −2.03181
\(377\) −34.4791 −1.77576
\(378\) 0 0
\(379\) 19.5334 1.00336 0.501682 0.865052i \(-0.332715\pi\)
0.501682 + 0.865052i \(0.332715\pi\)
\(380\) 25.1079 1.28801
\(381\) 0 0
\(382\) 23.6226 1.20864
\(383\) −2.35097 −0.120129 −0.0600644 0.998195i \(-0.519131\pi\)
−0.0600644 + 0.998195i \(0.519131\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 54.7019 2.78426
\(387\) 0 0
\(388\) 41.6476 2.11434
\(389\) 1.74378 0.0884130 0.0442065 0.999022i \(-0.485924\pi\)
0.0442065 + 0.999022i \(0.485924\pi\)
\(390\) 0 0
\(391\) −52.1895 −2.63934
\(392\) 5.22982 0.264146
\(393\) 0 0
\(394\) −64.8929 −3.26925
\(395\) 11.5404 0.580659
\(396\) 0 0
\(397\) 6.54189 0.328328 0.164164 0.986433i \(-0.447507\pi\)
0.164164 + 0.986433i \(0.447507\pi\)
\(398\) 2.60719 0.130687
\(399\) 0 0
\(400\) 7.97760 0.398880
\(401\) −5.66152 −0.282723 −0.141361 0.989958i \(-0.545148\pi\)
−0.141361 + 0.989958i \(0.545148\pi\)
\(402\) 0 0
\(403\) −21.5000 −1.07099
\(404\) −55.1616 −2.74439
\(405\) 0 0
\(406\) −14.5202 −0.720626
\(407\) 0 0
\(408\) 0 0
\(409\) 5.48755 0.271342 0.135671 0.990754i \(-0.456681\pi\)
0.135671 + 0.990754i \(0.456681\pi\)
\(410\) 33.8135 1.66993
\(411\) 0 0
\(412\) −19.4053 −0.956030
\(413\) 0.926221 0.0455764
\(414\) 0 0
\(415\) −27.6615 −1.35785
\(416\) 6.86565 0.336616
\(417\) 0 0
\(418\) 0 0
\(419\) −8.24926 −0.403003 −0.201501 0.979488i \(-0.564582\pi\)
−0.201501 + 0.979488i \(0.564582\pi\)
\(420\) 0 0
\(421\) 32.2617 1.57234 0.786171 0.618009i \(-0.212061\pi\)
0.786171 + 0.618009i \(0.212061\pi\)
\(422\) 21.5529 1.04918
\(423\) 0 0
\(424\) 22.1212 1.07430
\(425\) 12.7467 0.618307
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 29.1079 1.40699
\(429\) 0 0
\(430\) −47.3869 −2.28520
\(431\) −23.2772 −1.12122 −0.560611 0.828079i \(-0.689434\pi\)
−0.560611 + 0.828079i \(0.689434\pi\)
\(432\) 0 0
\(433\) −15.9302 −0.765558 −0.382779 0.923840i \(-0.625033\pi\)
−0.382779 + 0.923840i \(0.625033\pi\)
\(434\) −9.05433 −0.434622
\(435\) 0 0
\(436\) −77.5125 −3.71218
\(437\) −16.3774 −0.783436
\(438\) 0 0
\(439\) −22.8565 −1.09088 −0.545439 0.838150i \(-0.683637\pi\)
−0.545439 + 0.838150i \(0.683637\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 109.104 5.18956
\(443\) −21.5962 −1.02607 −0.513034 0.858368i \(-0.671478\pi\)
−0.513034 + 0.858368i \(0.671478\pi\)
\(444\) 0 0
\(445\) −41.1227 −1.94940
\(446\) −58.7144 −2.78021
\(447\) 0 0
\(448\) −6.51396 −0.307756
\(449\) −18.6630 −0.880763 −0.440382 0.897811i \(-0.645157\pi\)
−0.440382 + 0.897811i \(0.645157\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 37.1491 1.74735
\(453\) 0 0
\(454\) −32.5808 −1.52909
\(455\) 15.1949 0.712349
\(456\) 0 0
\(457\) −36.4721 −1.70609 −0.853047 0.521834i \(-0.825248\pi\)
−0.853047 + 0.521834i \(0.825248\pi\)
\(458\) 33.0529 1.54446
\(459\) 0 0
\(460\) −73.9597 −3.44839
\(461\) 7.17548 0.334196 0.167098 0.985940i \(-0.446560\pi\)
0.167098 + 0.985940i \(0.446560\pi\)
\(462\) 0 0
\(463\) −0.452670 −0.0210373 −0.0105187 0.999945i \(-0.503348\pi\)
−0.0105187 + 0.999945i \(0.503348\pi\)
\(464\) −27.6134 −1.28192
\(465\) 0 0
\(466\) 17.6476 0.817509
\(467\) 9.68097 0.447982 0.223991 0.974591i \(-0.428091\pi\)
0.223991 + 0.974591i \(0.428091\pi\)
\(468\) 0 0
\(469\) −10.1017 −0.466453
\(470\) −48.2067 −2.22361
\(471\) 0 0
\(472\) 4.84396 0.222962
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 30.9193 1.41718
\(477\) 0 0
\(478\) −49.4395 −2.26131
\(479\) 24.3635 1.11319 0.556597 0.830782i \(-0.312107\pi\)
0.556597 + 0.830782i \(0.312107\pi\)
\(480\) 0 0
\(481\) 19.3983 0.884488
\(482\) 22.7375 1.03567
\(483\) 0 0
\(484\) 0 0
\(485\) 26.1909 1.18927
\(486\) 0 0
\(487\) −19.2702 −0.873217 −0.436609 0.899652i \(-0.643821\pi\)
−0.436609 + 0.899652i \(0.643821\pi\)
\(488\) 10.4596 0.473485
\(489\) 0 0
\(490\) 6.39905 0.289080
\(491\) 29.6421 1.33773 0.668864 0.743385i \(-0.266781\pi\)
0.668864 + 0.743385i \(0.266781\pi\)
\(492\) 0 0
\(493\) −44.1212 −1.98712
\(494\) 34.2376 1.54042
\(495\) 0 0
\(496\) −17.2188 −0.773149
\(497\) −4.45963 −0.200042
\(498\) 0 0
\(499\) 29.3719 1.31487 0.657434 0.753512i \(-0.271642\pi\)
0.657434 + 0.753512i \(0.271642\pi\)
\(500\) −35.1777 −1.57320
\(501\) 0 0
\(502\) 23.3058 1.04019
\(503\) 12.5947 0.561570 0.280785 0.959771i \(-0.409405\pi\)
0.280785 + 0.959771i \(0.409405\pi\)
\(504\) 0 0
\(505\) −34.6894 −1.54366
\(506\) 0 0
\(507\) 0 0
\(508\) 42.0947 1.86765
\(509\) 15.2144 0.674365 0.337183 0.941439i \(-0.390526\pi\)
0.337183 + 0.941439i \(0.390526\pi\)
\(510\) 0 0
\(511\) 2.12811 0.0941421
\(512\) −43.6894 −1.93082
\(513\) 0 0
\(514\) 0.0480890 0.00212112
\(515\) −12.2034 −0.537746
\(516\) 0 0
\(517\) 0 0
\(518\) 8.16924 0.358936
\(519\) 0 0
\(520\) 79.4667 3.48484
\(521\) −20.5180 −0.898909 −0.449454 0.893303i \(-0.648382\pi\)
−0.449454 + 0.893303i \(0.648382\pi\)
\(522\) 0 0
\(523\) 29.8844 1.30675 0.653376 0.757033i \(-0.273352\pi\)
0.653376 + 0.757033i \(0.273352\pi\)
\(524\) 16.4596 0.719042
\(525\) 0 0
\(526\) 69.2221 3.01823
\(527\) −27.5125 −1.19846
\(528\) 0 0
\(529\) 25.2423 1.09749
\(530\) 27.0668 1.17571
\(531\) 0 0
\(532\) 9.70265 0.420663
\(533\) 31.0279 1.34397
\(534\) 0 0
\(535\) 18.3051 0.791399
\(536\) −52.8300 −2.28191
\(537\) 0 0
\(538\) −76.4582 −3.29635
\(539\) 0 0
\(540\) 0 0
\(541\) −26.9582 −1.15902 −0.579511 0.814965i \(-0.696756\pi\)
−0.579511 + 0.814965i \(0.696756\pi\)
\(542\) 55.2874 2.37480
\(543\) 0 0
\(544\) 8.78562 0.376680
\(545\) −48.7453 −2.08802
\(546\) 0 0
\(547\) −26.0558 −1.11407 −0.557034 0.830490i \(-0.688061\pi\)
−0.557034 + 0.830490i \(0.688061\pi\)
\(548\) 44.5933 1.90493
\(549\) 0 0
\(550\) 0 0
\(551\) −13.8455 −0.589837
\(552\) 0 0
\(553\) 4.45963 0.189643
\(554\) −61.2563 −2.60253
\(555\) 0 0
\(556\) −1.89134 −0.0802105
\(557\) 8.01945 0.339795 0.169897 0.985462i \(-0.445656\pi\)
0.169897 + 0.985462i \(0.445656\pi\)
\(558\) 0 0
\(559\) −43.4831 −1.83914
\(560\) 12.1692 0.514244
\(561\) 0 0
\(562\) −2.95486 −0.124643
\(563\) 21.9736 0.926077 0.463038 0.886338i \(-0.346759\pi\)
0.463038 + 0.886338i \(0.346759\pi\)
\(564\) 0 0
\(565\) 23.3619 0.982844
\(566\) 38.5077 1.61860
\(567\) 0 0
\(568\) −23.3230 −0.978613
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −37.6740 −1.57661 −0.788304 0.615286i \(-0.789041\pi\)
−0.788304 + 0.615286i \(0.789041\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 13.0668 0.545398
\(575\) −11.7827 −0.491371
\(576\) 0 0
\(577\) −7.80908 −0.325096 −0.162548 0.986701i \(-0.551971\pi\)
−0.162548 + 0.986701i \(0.551971\pi\)
\(578\) 97.5771 4.05867
\(579\) 0 0
\(580\) −62.5257 −2.59624
\(581\) −10.6894 −0.443473
\(582\) 0 0
\(583\) 0 0
\(584\) 11.1296 0.460547
\(585\) 0 0
\(586\) −24.0947 −0.995345
\(587\) 43.2633 1.78567 0.892833 0.450388i \(-0.148714\pi\)
0.892833 + 0.450388i \(0.148714\pi\)
\(588\) 0 0
\(589\) −8.63360 −0.355741
\(590\) 5.92694 0.244008
\(591\) 0 0
\(592\) 15.5356 0.638511
\(593\) −29.7438 −1.22143 −0.610715 0.791850i \(-0.709118\pi\)
−0.610715 + 0.791850i \(0.709118\pi\)
\(594\) 0 0
\(595\) 19.4442 0.797134
\(596\) 45.4589 1.86207
\(597\) 0 0
\(598\) −100.853 −4.12417
\(599\) −11.5404 −0.471527 −0.235763 0.971810i \(-0.575759\pi\)
−0.235763 + 0.971810i \(0.575759\pi\)
\(600\) 0 0
\(601\) −7.06129 −0.288036 −0.144018 0.989575i \(-0.546002\pi\)
−0.144018 + 0.989575i \(0.546002\pi\)
\(602\) −18.3121 −0.746344
\(603\) 0 0
\(604\) 66.3370 2.69922
\(605\) 0 0
\(606\) 0 0
\(607\) 17.3859 0.705670 0.352835 0.935686i \(-0.385218\pi\)
0.352835 + 0.935686i \(0.385218\pi\)
\(608\) 2.75698 0.111810
\(609\) 0 0
\(610\) 12.7981 0.518180
\(611\) −44.2353 −1.78957
\(612\) 0 0
\(613\) −39.4611 −1.59382 −0.796910 0.604098i \(-0.793534\pi\)
−0.796910 + 0.604098i \(0.793534\pi\)
\(614\) −15.7049 −0.633798
\(615\) 0 0
\(616\) 0 0
\(617\) 4.40378 0.177290 0.0886448 0.996063i \(-0.471746\pi\)
0.0886448 + 0.996063i \(0.471746\pi\)
\(618\) 0 0
\(619\) 16.4163 0.659825 0.329913 0.944011i \(-0.392981\pi\)
0.329913 + 0.944011i \(0.392981\pi\)
\(620\) −38.9890 −1.56584
\(621\) 0 0
\(622\) −19.7827 −0.793213
\(623\) −15.8913 −0.636673
\(624\) 0 0
\(625\) −30.6042 −1.22417
\(626\) 41.9038 1.67481
\(627\) 0 0
\(628\) 53.7688 2.14561
\(629\) 24.8231 0.989761
\(630\) 0 0
\(631\) 44.4596 1.76991 0.884955 0.465677i \(-0.154189\pi\)
0.884955 + 0.465677i \(0.154189\pi\)
\(632\) 23.3230 0.927741
\(633\) 0 0
\(634\) −49.4876 −1.96540
\(635\) 26.4721 1.05051
\(636\) 0 0
\(637\) 5.87189 0.232653
\(638\) 0 0
\(639\) 0 0
\(640\) −47.7346 −1.88688
\(641\) −10.5419 −0.416380 −0.208190 0.978088i \(-0.566757\pi\)
−0.208190 + 0.978088i \(0.566757\pi\)
\(642\) 0 0
\(643\) −23.4053 −0.923015 −0.461507 0.887136i \(-0.652691\pi\)
−0.461507 + 0.887136i \(0.652691\pi\)
\(644\) −28.5808 −1.12624
\(645\) 0 0
\(646\) 43.8121 1.72376
\(647\) −11.4945 −0.451896 −0.225948 0.974139i \(-0.572548\pi\)
−0.225948 + 0.974139i \(0.572548\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 24.6322 0.966153
\(651\) 0 0
\(652\) −41.5676 −1.62791
\(653\) −19.8774 −0.777863 −0.388932 0.921267i \(-0.627156\pi\)
−0.388932 + 0.921267i \(0.627156\pi\)
\(654\) 0 0
\(655\) 10.3510 0.404446
\(656\) 24.8495 0.970210
\(657\) 0 0
\(658\) −18.6289 −0.726229
\(659\) 15.0738 0.587191 0.293596 0.955930i \(-0.405148\pi\)
0.293596 + 0.955930i \(0.405148\pi\)
\(660\) 0 0
\(661\) −4.74226 −0.184453 −0.0922263 0.995738i \(-0.529398\pi\)
−0.0922263 + 0.995738i \(0.529398\pi\)
\(662\) 18.6461 0.724701
\(663\) 0 0
\(664\) −55.9038 −2.16949
\(665\) 6.10170 0.236614
\(666\) 0 0
\(667\) 40.7842 1.57917
\(668\) −43.1491 −1.66949
\(669\) 0 0
\(670\) −64.6414 −2.49731
\(671\) 0 0
\(672\) 0 0
\(673\) −9.87885 −0.380802 −0.190401 0.981706i \(-0.560979\pi\)
−0.190401 + 0.981706i \(0.560979\pi\)
\(674\) 31.3789 1.20867
\(675\) 0 0
\(676\) 88.3844 3.39940
\(677\) −29.0279 −1.11563 −0.557817 0.829964i \(-0.688361\pi\)
−0.557817 + 0.829964i \(0.688361\pi\)
\(678\) 0 0
\(679\) 10.1212 0.388414
\(680\) 101.690 3.89962
\(681\) 0 0
\(682\) 0 0
\(683\) 37.9597 1.45249 0.726243 0.687438i \(-0.241265\pi\)
0.726243 + 0.687438i \(0.241265\pi\)
\(684\) 0 0
\(685\) 28.0434 1.07148
\(686\) 2.47283 0.0944132
\(687\) 0 0
\(688\) −34.8245 −1.32767
\(689\) 24.8370 0.946214
\(690\) 0 0
\(691\) −15.5264 −0.590654 −0.295327 0.955396i \(-0.595429\pi\)
−0.295327 + 0.955396i \(0.595429\pi\)
\(692\) 59.6601 2.26794
\(693\) 0 0
\(694\) 2.81060 0.106689
\(695\) −1.18940 −0.0451167
\(696\) 0 0
\(697\) 39.7049 1.50393
\(698\) 36.5761 1.38442
\(699\) 0 0
\(700\) 6.98055 0.263840
\(701\) −27.4317 −1.03608 −0.518041 0.855356i \(-0.673338\pi\)
−0.518041 + 0.855356i \(0.673338\pi\)
\(702\) 0 0
\(703\) 7.78963 0.293792
\(704\) 0 0
\(705\) 0 0
\(706\) 18.5980 0.699945
\(707\) −13.4053 −0.504158
\(708\) 0 0
\(709\) −44.5180 −1.67191 −0.835954 0.548800i \(-0.815085\pi\)
−0.835954 + 0.548800i \(0.815085\pi\)
\(710\) −28.5374 −1.07099
\(711\) 0 0
\(712\) −83.1087 −3.11463
\(713\) 25.4317 0.952425
\(714\) 0 0
\(715\) 0 0
\(716\) −59.0529 −2.20691
\(717\) 0 0
\(718\) −43.5125 −1.62387
\(719\) −17.3859 −0.648383 −0.324191 0.945992i \(-0.605092\pi\)
−0.324191 + 0.945992i \(0.605092\pi\)
\(720\) 0 0
\(721\) −4.71585 −0.175628
\(722\) −33.2353 −1.23689
\(723\) 0 0
\(724\) 39.7563 1.47753
\(725\) −9.96111 −0.369946
\(726\) 0 0
\(727\) −18.6506 −0.691711 −0.345855 0.938288i \(-0.612411\pi\)
−0.345855 + 0.938288i \(0.612411\pi\)
\(728\) 30.7089 1.13815
\(729\) 0 0
\(730\) 13.6179 0.504021
\(731\) −55.6431 −2.05804
\(732\) 0 0
\(733\) −16.3510 −0.603937 −0.301968 0.953318i \(-0.597644\pi\)
−0.301968 + 0.953318i \(0.597644\pi\)
\(734\) 35.3914 1.30632
\(735\) 0 0
\(736\) −8.12115 −0.299350
\(737\) 0 0
\(738\) 0 0
\(739\) 32.8370 1.20793 0.603964 0.797011i \(-0.293587\pi\)
0.603964 + 0.797011i \(0.293587\pi\)
\(740\) 35.1777 1.29316
\(741\) 0 0
\(742\) 10.4596 0.383985
\(743\) 20.9123 0.767198 0.383599 0.923500i \(-0.374685\pi\)
0.383599 + 0.923500i \(0.374685\pi\)
\(744\) 0 0
\(745\) 28.5877 1.04737
\(746\) 40.6366 1.48781
\(747\) 0 0
\(748\) 0 0
\(749\) 7.07378 0.258470
\(750\) 0 0
\(751\) −39.2633 −1.43274 −0.716368 0.697722i \(-0.754197\pi\)
−0.716368 + 0.697722i \(0.754197\pi\)
\(752\) −35.4270 −1.29189
\(753\) 0 0
\(754\) −85.2610 −3.10502
\(755\) 41.7174 1.51825
\(756\) 0 0
\(757\) −10.3454 −0.376011 −0.188006 0.982168i \(-0.560202\pi\)
−0.188006 + 0.982168i \(0.560202\pi\)
\(758\) 48.3029 1.75444
\(759\) 0 0
\(760\) 31.9108 1.15753
\(761\) 30.9582 1.12223 0.561116 0.827737i \(-0.310372\pi\)
0.561116 + 0.827737i \(0.310372\pi\)
\(762\) 0 0
\(763\) −18.8370 −0.681945
\(764\) 39.3091 1.42215
\(765\) 0 0
\(766\) −5.81355 −0.210052
\(767\) 5.43867 0.196379
\(768\) 0 0
\(769\) 28.8998 1.04215 0.521077 0.853510i \(-0.325530\pi\)
0.521077 + 0.853510i \(0.325530\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 91.0265 3.27612
\(773\) 26.7523 0.962212 0.481106 0.876663i \(-0.340235\pi\)
0.481106 + 0.876663i \(0.340235\pi\)
\(774\) 0 0
\(775\) −6.21142 −0.223121
\(776\) 52.9317 1.90014
\(777\) 0 0
\(778\) 4.31207 0.154595
\(779\) 12.4596 0.446413
\(780\) 0 0
\(781\) 0 0
\(782\) −129.056 −4.61503
\(783\) 0 0
\(784\) 4.70265 0.167952
\(785\) 33.8135 1.20686
\(786\) 0 0
\(787\) −22.1406 −0.789227 −0.394614 0.918847i \(-0.629122\pi\)
−0.394614 + 0.918847i \(0.629122\pi\)
\(788\) −107.985 −3.84679
\(789\) 0 0
\(790\) 28.5374 1.01532
\(791\) 9.02792 0.320996
\(792\) 0 0
\(793\) 11.7438 0.417034
\(794\) 16.1770 0.574100
\(795\) 0 0
\(796\) 4.33848 0.153773
\(797\) −8.18396 −0.289891 −0.144945 0.989440i \(-0.546301\pi\)
−0.144945 + 0.989440i \(0.546301\pi\)
\(798\) 0 0
\(799\) −56.6058 −2.00257
\(800\) 1.98351 0.0701275
\(801\) 0 0
\(802\) −14.0000 −0.494357
\(803\) 0 0
\(804\) 0 0
\(805\) −17.9736 −0.633486
\(806\) −53.1660 −1.87269
\(807\) 0 0
\(808\) −70.1072 −2.46636
\(809\) 17.6685 0.621191 0.310595 0.950542i \(-0.399472\pi\)
0.310595 + 0.950542i \(0.399472\pi\)
\(810\) 0 0
\(811\) 43.0210 1.51067 0.755335 0.655339i \(-0.227474\pi\)
0.755335 + 0.655339i \(0.227474\pi\)
\(812\) −24.1623 −0.847930
\(813\) 0 0
\(814\) 0 0
\(815\) −26.1406 −0.915665
\(816\) 0 0
\(817\) −17.4611 −0.610888
\(818\) 13.5698 0.474457
\(819\) 0 0
\(820\) 56.2673 1.96494
\(821\) 18.0364 0.629475 0.314737 0.949179i \(-0.398084\pi\)
0.314737 + 0.949179i \(0.398084\pi\)
\(822\) 0 0
\(823\) −8.21037 −0.286195 −0.143098 0.989709i \(-0.545706\pi\)
−0.143098 + 0.989709i \(0.545706\pi\)
\(824\) −24.6630 −0.859178
\(825\) 0 0
\(826\) 2.29039 0.0796929
\(827\) 6.83148 0.237554 0.118777 0.992921i \(-0.462103\pi\)
0.118777 + 0.992921i \(0.462103\pi\)
\(828\) 0 0
\(829\) −25.7438 −0.894118 −0.447059 0.894504i \(-0.647529\pi\)
−0.447059 + 0.894504i \(0.647529\pi\)
\(830\) −68.4023 −2.37428
\(831\) 0 0
\(832\) −38.2493 −1.32605
\(833\) 7.51396 0.260343
\(834\) 0 0
\(835\) −27.1352 −0.939051
\(836\) 0 0
\(837\) 0 0
\(838\) −20.3991 −0.704674
\(839\) 30.6002 1.05644 0.528219 0.849108i \(-0.322860\pi\)
0.528219 + 0.849108i \(0.322860\pi\)
\(840\) 0 0
\(841\) 5.47908 0.188934
\(842\) 79.7779 2.74933
\(843\) 0 0
\(844\) 35.8649 1.23452
\(845\) 55.5823 1.91209
\(846\) 0 0
\(847\) 0 0
\(848\) 19.8913 0.683071
\(849\) 0 0
\(850\) 31.5205 1.08115
\(851\) −22.9457 −0.786567
\(852\) 0 0
\(853\) 30.4596 1.04292 0.521459 0.853276i \(-0.325388\pi\)
0.521459 + 0.853276i \(0.325388\pi\)
\(854\) 4.94567 0.169237
\(855\) 0 0
\(856\) 36.9946 1.26445
\(857\) −41.2702 −1.40976 −0.704882 0.709325i \(-0.749000\pi\)
−0.704882 + 0.709325i \(0.749000\pi\)
\(858\) 0 0
\(859\) −28.9193 −0.986712 −0.493356 0.869827i \(-0.664230\pi\)
−0.493356 + 0.869827i \(0.664230\pi\)
\(860\) −78.8540 −2.68890
\(861\) 0 0
\(862\) −57.5606 −1.96052
\(863\) −37.0015 −1.25955 −0.629773 0.776779i \(-0.716852\pi\)
−0.629773 + 0.776779i \(0.716852\pi\)
\(864\) 0 0
\(865\) 37.5184 1.27566
\(866\) −39.3928 −1.33862
\(867\) 0 0
\(868\) −15.0668 −0.511401
\(869\) 0 0
\(870\) 0 0
\(871\) −59.3161 −2.00985
\(872\) −98.5140 −3.33611
\(873\) 0 0
\(874\) −40.4985 −1.36988
\(875\) −8.54885 −0.289004
\(876\) 0 0
\(877\) 6.12562 0.206847 0.103424 0.994637i \(-0.467020\pi\)
0.103424 + 0.994637i \(0.467020\pi\)
\(878\) −56.5202 −1.90746
\(879\) 0 0
\(880\) 0 0
\(881\) −25.4123 −0.856161 −0.428080 0.903741i \(-0.640810\pi\)
−0.428080 + 0.903741i \(0.640810\pi\)
\(882\) 0 0
\(883\) −17.0040 −0.572230 −0.286115 0.958195i \(-0.592364\pi\)
−0.286115 + 0.958195i \(0.592364\pi\)
\(884\) 181.554 6.10634
\(885\) 0 0
\(886\) −53.4039 −1.79414
\(887\) 35.2269 1.18280 0.591401 0.806378i \(-0.298575\pi\)
0.591401 + 0.806378i \(0.298575\pi\)
\(888\) 0 0
\(889\) 10.2298 0.343097
\(890\) −101.690 −3.40864
\(891\) 0 0
\(892\) −97.7034 −3.27135
\(893\) −17.7632 −0.594424
\(894\) 0 0
\(895\) −37.1366 −1.24134
\(896\) −18.4464 −0.616252
\(897\) 0 0
\(898\) −46.1506 −1.54007
\(899\) 21.5000 0.717067
\(900\) 0 0
\(901\) 31.7827 1.05883
\(902\) 0 0
\(903\) 0 0
\(904\) 47.2144 1.57033
\(905\) 25.0015 0.831079
\(906\) 0 0
\(907\) −9.72682 −0.322974 −0.161487 0.986875i \(-0.551629\pi\)
−0.161487 + 0.986875i \(0.551629\pi\)
\(908\) −54.2159 −1.79922
\(909\) 0 0
\(910\) 37.5745 1.24558
\(911\) 42.7717 1.41709 0.708545 0.705666i \(-0.249352\pi\)
0.708545 + 0.705666i \(0.249352\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −90.1895 −2.98320
\(915\) 0 0
\(916\) 55.0015 1.81730
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −6.26871 −0.206786 −0.103393 0.994641i \(-0.532970\pi\)
−0.103393 + 0.994641i \(0.532970\pi\)
\(920\) −93.9986 −3.09904
\(921\) 0 0
\(922\) 17.7438 0.584360
\(923\) −26.1865 −0.861938
\(924\) 0 0
\(925\) 5.60424 0.184266
\(926\) −1.11938 −0.0367850
\(927\) 0 0
\(928\) −6.86565 −0.225376
\(929\) −20.0194 −0.656817 −0.328408 0.944536i \(-0.606512\pi\)
−0.328408 + 0.944536i \(0.606512\pi\)
\(930\) 0 0
\(931\) 2.35793 0.0772779
\(932\) 29.3664 0.961929
\(933\) 0 0
\(934\) 23.9394 0.783322
\(935\) 0 0
\(936\) 0 0
\(937\) 11.2144 0.366358 0.183179 0.983080i \(-0.441361\pi\)
0.183179 + 0.983080i \(0.441361\pi\)
\(938\) −24.9798 −0.815621
\(939\) 0 0
\(940\) −80.2181 −2.61643
\(941\) 31.3400 1.02165 0.510827 0.859683i \(-0.329339\pi\)
0.510827 + 0.859683i \(0.329339\pi\)
\(942\) 0 0
\(943\) −36.7019 −1.19518
\(944\) 4.35569 0.141766
\(945\) 0 0
\(946\) 0 0
\(947\) 23.2841 0.756633 0.378317 0.925676i \(-0.376503\pi\)
0.378317 + 0.925676i \(0.376503\pi\)
\(948\) 0 0
\(949\) 12.4960 0.405638
\(950\) 9.89134 0.320917
\(951\) 0 0
\(952\) 39.2966 1.27361
\(953\) 26.8300 0.869110 0.434555 0.900645i \(-0.356906\pi\)
0.434555 + 0.900645i \(0.356906\pi\)
\(954\) 0 0
\(955\) 24.7203 0.799931
\(956\) −82.2695 −2.66079
\(957\) 0 0
\(958\) 60.2468 1.94648
\(959\) 10.8370 0.349945
\(960\) 0 0
\(961\) −17.5933 −0.567525
\(962\) 47.9689 1.54658
\(963\) 0 0
\(964\) 37.8363 1.21862
\(965\) 57.2438 1.84274
\(966\) 0 0
\(967\) −36.2034 −1.16422 −0.582112 0.813109i \(-0.697773\pi\)
−0.582112 + 0.813109i \(0.697773\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 64.7658 2.07950
\(971\) −6.06281 −0.194565 −0.0972824 0.995257i \(-0.531015\pi\)
−0.0972824 + 0.995257i \(0.531015\pi\)
\(972\) 0 0
\(973\) −0.459630 −0.0147351
\(974\) −47.6521 −1.52687
\(975\) 0 0
\(976\) 9.40530 0.301056
\(977\) 4.28263 0.137013 0.0685067 0.997651i \(-0.478177\pi\)
0.0685067 + 0.997651i \(0.478177\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 10.6483 0.340148
\(981\) 0 0
\(982\) 73.2999 2.33909
\(983\) 17.1894 0.548257 0.274128 0.961693i \(-0.411611\pi\)
0.274128 + 0.961693i \(0.411611\pi\)
\(984\) 0 0
\(985\) −67.9083 −2.16374
\(986\) −109.104 −3.47459
\(987\) 0 0
\(988\) 56.9729 1.81255
\(989\) 51.4347 1.63553
\(990\) 0 0
\(991\) 35.4945 1.12752 0.563760 0.825939i \(-0.309354\pi\)
0.563760 + 0.825939i \(0.309354\pi\)
\(992\) −4.28120 −0.135928
\(993\) 0 0
\(994\) −11.0279 −0.349785
\(995\) 2.72834 0.0864942
\(996\) 0 0
\(997\) −30.4068 −0.962993 −0.481497 0.876448i \(-0.659907\pi\)
−0.481497 + 0.876448i \(0.659907\pi\)
\(998\) 72.6319 2.29912
\(999\) 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 7623.2.a.cd.1.3 3
3.2 odd 2 2541.2.a.bg.1.1 3
11.10 odd 2 693.2.a.l.1.1 3
33.32 even 2 231.2.a.e.1.3 3
77.76 even 2 4851.2.a.bi.1.1 3
132.131 odd 2 3696.2.a.bo.1.1 3
165.164 even 2 5775.2.a.bp.1.1 3
231.230 odd 2 1617.2.a.t.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.3 3 33.32 even 2
693.2.a.l.1.1 3 11.10 odd 2
1617.2.a.t.1.3 3 231.230 odd 2
2541.2.a.bg.1.1 3 3.2 odd 2
3696.2.a.bo.1.1 3 132.131 odd 2
4851.2.a.bi.1.1 3 77.76 even 2
5775.2.a.bp.1.1 3 165.164 even 2
7623.2.a.cd.1.3 3 1.1 even 1 trivial