Properties

Label 7623.2.a.bq.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 2
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +6.85410 q^{10} -6.23607 q^{13} +1.61803 q^{14} -4.85410 q^{16} -4.47214 q^{17} -3.00000 q^{19} +2.61803 q^{20} -8.47214 q^{23} +12.9443 q^{25} -10.0902 q^{26} +0.618034 q^{28} -3.00000 q^{29} -3.38197 q^{32} -7.23607 q^{34} +4.23607 q^{35} +3.47214 q^{37} -4.85410 q^{38} -9.47214 q^{40} +1.52786 q^{41} -10.9443 q^{43} -13.7082 q^{46} +3.00000 q^{47} +1.00000 q^{49} +20.9443 q^{50} -3.85410 q^{52} -8.94427 q^{53} -2.23607 q^{56} -4.85410 q^{58} +1.47214 q^{59} -3.52786 q^{61} +4.23607 q^{64} -26.4164 q^{65} -8.70820 q^{67} -2.76393 q^{68} +6.85410 q^{70} -1.52786 q^{71} -12.2361 q^{73} +5.61803 q^{74} -1.85410 q^{76} +13.4164 q^{79} -20.5623 q^{80} +2.47214 q^{82} +6.00000 q^{83} -18.9443 q^{85} -17.7082 q^{86} +4.47214 q^{89} -6.23607 q^{91} -5.23607 q^{92} +4.85410 q^{94} -12.7082 q^{95} +2.00000 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 4q^{5} + 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 4q^{5} + 2q^{7} + 7q^{10} - 8q^{13} + q^{14} - 3q^{16} - 6q^{19} + 3q^{20} - 8q^{23} + 8q^{25} - 9q^{26} - q^{28} - 6q^{29} - 9q^{32} - 10q^{34} + 4q^{35} - 2q^{37} - 3q^{38} - 10q^{40} + 12q^{41} - 4q^{43} - 14q^{46} + 6q^{47} + 2q^{49} + 24q^{50} - q^{52} - 3q^{58} - 6q^{59} - 16q^{61} + 4q^{64} - 26q^{65} - 4q^{67} - 10q^{68} + 7q^{70} - 12q^{71} - 20q^{73} + 9q^{74} + 3q^{76} - 21q^{80} - 4q^{82} + 12q^{83} - 20q^{85} - 22q^{86} - 8q^{91} - 6q^{92} + 3q^{94} - 12q^{95} + 4q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 4.23607 1.89443 0.947214 0.320603i \(-0.103886\pi\)
0.947214 + 0.320603i \(0.103886\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 6.85410 2.16746
\(11\) 0 0
\(12\) 0 0
\(13\) −6.23607 −1.72957 −0.864787 0.502139i \(-0.832547\pi\)
−0.864787 + 0.502139i \(0.832547\pi\)
\(14\) 1.61803 0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 2.61803 0.585410
\(21\) 0 0
\(22\) 0 0
\(23\) −8.47214 −1.76656 −0.883281 0.468844i \(-0.844671\pi\)
−0.883281 + 0.468844i \(0.844671\pi\)
\(24\) 0 0
\(25\) 12.9443 2.58885
\(26\) −10.0902 −1.97885
\(27\) 0 0
\(28\) 0.618034 0.116797
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) −7.23607 −1.24098
\(35\) 4.23607 0.716026
\(36\) 0 0
\(37\) 3.47214 0.570816 0.285408 0.958406i \(-0.407871\pi\)
0.285408 + 0.958406i \(0.407871\pi\)
\(38\) −4.85410 −0.787439
\(39\) 0 0
\(40\) −9.47214 −1.49768
\(41\) 1.52786 0.238612 0.119306 0.992858i \(-0.461933\pi\)
0.119306 + 0.992858i \(0.461933\pi\)
\(42\) 0 0
\(43\) −10.9443 −1.66899 −0.834493 0.551019i \(-0.814239\pi\)
−0.834493 + 0.551019i \(0.814239\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −13.7082 −2.02116
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 20.9443 2.96197
\(51\) 0 0
\(52\) −3.85410 −0.534468
\(53\) −8.94427 −1.22859 −0.614295 0.789076i \(-0.710560\pi\)
−0.614295 + 0.789076i \(0.710560\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) −4.85410 −0.637375
\(59\) 1.47214 0.191656 0.0958279 0.995398i \(-0.469450\pi\)
0.0958279 + 0.995398i \(0.469450\pi\)
\(60\) 0 0
\(61\) −3.52786 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) −26.4164 −3.27655
\(66\) 0 0
\(67\) −8.70820 −1.06388 −0.531938 0.846783i \(-0.678536\pi\)
−0.531938 + 0.846783i \(0.678536\pi\)
\(68\) −2.76393 −0.335176
\(69\) 0 0
\(70\) 6.85410 0.819222
\(71\) −1.52786 −0.181324 −0.0906621 0.995882i \(-0.528898\pi\)
−0.0906621 + 0.995882i \(0.528898\pi\)
\(72\) 0 0
\(73\) −12.2361 −1.43212 −0.716062 0.698037i \(-0.754057\pi\)
−0.716062 + 0.698037i \(0.754057\pi\)
\(74\) 5.61803 0.653083
\(75\) 0 0
\(76\) −1.85410 −0.212680
\(77\) 0 0
\(78\) 0 0
\(79\) 13.4164 1.50946 0.754732 0.656033i \(-0.227767\pi\)
0.754732 + 0.656033i \(0.227767\pi\)
\(80\) −20.5623 −2.29894
\(81\) 0 0
\(82\) 2.47214 0.273002
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −18.9443 −2.05479
\(86\) −17.7082 −1.90952
\(87\) 0 0
\(88\) 0 0
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 0 0
\(91\) −6.23607 −0.653718
\(92\) −5.23607 −0.545898
\(93\) 0 0
\(94\) 4.85410 0.500662
\(95\) −12.7082 −1.30383
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.61803 0.163446
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) −16.9443 −1.68602 −0.843009 0.537899i \(-0.819218\pi\)
−0.843009 + 0.537899i \(0.819218\pi\)
\(102\) 0 0
\(103\) 6.47214 0.637719 0.318859 0.947802i \(-0.396700\pi\)
0.318859 + 0.947802i \(0.396700\pi\)
\(104\) 13.9443 1.36735
\(105\) 0 0
\(106\) −14.4721 −1.40566
\(107\) 9.76393 0.943915 0.471957 0.881621i \(-0.343548\pi\)
0.471957 + 0.881621i \(0.343548\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.85410 −0.458670
\(113\) 6.47214 0.608847 0.304424 0.952537i \(-0.401536\pi\)
0.304424 + 0.952537i \(0.401536\pi\)
\(114\) 0 0
\(115\) −35.8885 −3.34662
\(116\) −1.85410 −0.172149
\(117\) 0 0
\(118\) 2.38197 0.219278
\(119\) −4.47214 −0.409960
\(120\) 0 0
\(121\) 0 0
\(122\) −5.70820 −0.516797
\(123\) 0 0
\(124\) 0 0
\(125\) 33.6525 3.00997
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) −42.7426 −3.74878
\(131\) 2.47214 0.215992 0.107996 0.994151i \(-0.465557\pi\)
0.107996 + 0.994151i \(0.465557\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) −14.0902 −1.21721
\(135\) 0 0
\(136\) 10.0000 0.857493
\(137\) −2.94427 −0.251546 −0.125773 0.992059i \(-0.540141\pi\)
−0.125773 + 0.992059i \(0.540141\pi\)
\(138\) 0 0
\(139\) 0.944272 0.0800921 0.0400460 0.999198i \(-0.487250\pi\)
0.0400460 + 0.999198i \(0.487250\pi\)
\(140\) 2.61803 0.221264
\(141\) 0 0
\(142\) −2.47214 −0.207457
\(143\) 0 0
\(144\) 0 0
\(145\) −12.7082 −1.05536
\(146\) −19.7984 −1.63853
\(147\) 0 0
\(148\) 2.14590 0.176392
\(149\) 17.9443 1.47005 0.735026 0.678039i \(-0.237170\pi\)
0.735026 + 0.678039i \(0.237170\pi\)
\(150\) 0 0
\(151\) −17.4164 −1.41733 −0.708664 0.705547i \(-0.750702\pi\)
−0.708664 + 0.705547i \(0.750702\pi\)
\(152\) 6.70820 0.544107
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −13.4164 −1.07075 −0.535373 0.844616i \(-0.679829\pi\)
−0.535373 + 0.844616i \(0.679829\pi\)
\(158\) 21.7082 1.72701
\(159\) 0 0
\(160\) −14.3262 −1.13259
\(161\) −8.47214 −0.667698
\(162\) 0 0
\(163\) 7.29180 0.571138 0.285569 0.958358i \(-0.407818\pi\)
0.285569 + 0.958358i \(0.407818\pi\)
\(164\) 0.944272 0.0737352
\(165\) 0 0
\(166\) 9.70820 0.753503
\(167\) 18.9443 1.46595 0.732976 0.680255i \(-0.238131\pi\)
0.732976 + 0.680255i \(0.238131\pi\)
\(168\) 0 0
\(169\) 25.8885 1.99143
\(170\) −30.6525 −2.35094
\(171\) 0 0
\(172\) −6.76393 −0.515745
\(173\) 7.52786 0.572333 0.286166 0.958180i \(-0.407619\pi\)
0.286166 + 0.958180i \(0.407619\pi\)
\(174\) 0 0
\(175\) 12.9443 0.978495
\(176\) 0 0
\(177\) 0 0
\(178\) 7.23607 0.542366
\(179\) 16.4721 1.23119 0.615593 0.788065i \(-0.288917\pi\)
0.615593 + 0.788065i \(0.288917\pi\)
\(180\) 0 0
\(181\) −3.41641 −0.253940 −0.126970 0.991907i \(-0.540525\pi\)
−0.126970 + 0.991907i \(0.540525\pi\)
\(182\) −10.0902 −0.747933
\(183\) 0 0
\(184\) 18.9443 1.39659
\(185\) 14.7082 1.08137
\(186\) 0 0
\(187\) 0 0
\(188\) 1.85410 0.135224
\(189\) 0 0
\(190\) −20.5623 −1.49175
\(191\) −2.94427 −0.213040 −0.106520 0.994311i \(-0.533971\pi\)
−0.106520 + 0.994311i \(0.533971\pi\)
\(192\) 0 0
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 3.23607 0.232336
\(195\) 0 0
\(196\) 0.618034 0.0441453
\(197\) −1.05573 −0.0752175 −0.0376088 0.999293i \(-0.511974\pi\)
−0.0376088 + 0.999293i \(0.511974\pi\)
\(198\) 0 0
\(199\) −12.4721 −0.884126 −0.442063 0.896984i \(-0.645753\pi\)
−0.442063 + 0.896984i \(0.645753\pi\)
\(200\) −28.9443 −2.04667
\(201\) 0 0
\(202\) −27.4164 −1.92901
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) 6.47214 0.452034
\(206\) 10.4721 0.729628
\(207\) 0 0
\(208\) 30.2705 2.09888
\(209\) 0 0
\(210\) 0 0
\(211\) 1.41641 0.0975095 0.0487548 0.998811i \(-0.484475\pi\)
0.0487548 + 0.998811i \(0.484475\pi\)
\(212\) −5.52786 −0.379655
\(213\) 0 0
\(214\) 15.7984 1.07995
\(215\) −46.3607 −3.16177
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 27.8885 1.87599
\(222\) 0 0
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −3.38197 −0.225967
\(225\) 0 0
\(226\) 10.4721 0.696596
\(227\) −15.5279 −1.03062 −0.515310 0.857004i \(-0.672323\pi\)
−0.515310 + 0.857004i \(0.672323\pi\)
\(228\) 0 0
\(229\) −24.4721 −1.61716 −0.808582 0.588383i \(-0.799765\pi\)
−0.808582 + 0.588383i \(0.799765\pi\)
\(230\) −58.0689 −3.82895
\(231\) 0 0
\(232\) 6.70820 0.440415
\(233\) 19.8885 1.30294 0.651471 0.758674i \(-0.274152\pi\)
0.651471 + 0.758674i \(0.274152\pi\)
\(234\) 0 0
\(235\) 12.7082 0.828992
\(236\) 0.909830 0.0592249
\(237\) 0 0
\(238\) −7.23607 −0.469045
\(239\) 12.7082 0.822025 0.411013 0.911630i \(-0.365175\pi\)
0.411013 + 0.911630i \(0.365175\pi\)
\(240\) 0 0
\(241\) −12.7082 −0.818607 −0.409304 0.912398i \(-0.634228\pi\)
−0.409304 + 0.912398i \(0.634228\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.18034 −0.139582
\(245\) 4.23607 0.270632
\(246\) 0 0
\(247\) 18.7082 1.19037
\(248\) 0 0
\(249\) 0 0
\(250\) 54.4508 3.44377
\(251\) 16.4164 1.03619 0.518097 0.855322i \(-0.326641\pi\)
0.518097 + 0.855322i \(0.326641\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −3.23607 −0.203049
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −0.347524 −0.0216780 −0.0108390 0.999941i \(-0.503450\pi\)
−0.0108390 + 0.999941i \(0.503450\pi\)
\(258\) 0 0
\(259\) 3.47214 0.215748
\(260\) −16.3262 −1.01251
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 0.236068 0.0145566 0.00727829 0.999974i \(-0.497683\pi\)
0.00727829 + 0.999974i \(0.497683\pi\)
\(264\) 0 0
\(265\) −37.8885 −2.32747
\(266\) −4.85410 −0.297624
\(267\) 0 0
\(268\) −5.38197 −0.328756
\(269\) 19.5279 1.19063 0.595317 0.803491i \(-0.297026\pi\)
0.595317 + 0.803491i \(0.297026\pi\)
\(270\) 0 0
\(271\) 3.47214 0.210917 0.105459 0.994424i \(-0.466369\pi\)
0.105459 + 0.994424i \(0.466369\pi\)
\(272\) 21.7082 1.31625
\(273\) 0 0
\(274\) −4.76393 −0.287800
\(275\) 0 0
\(276\) 0 0
\(277\) 12.4721 0.749378 0.374689 0.927151i \(-0.377749\pi\)
0.374689 + 0.927151i \(0.377749\pi\)
\(278\) 1.52786 0.0916352
\(279\) 0 0
\(280\) −9.47214 −0.566068
\(281\) 16.4164 0.979321 0.489660 0.871913i \(-0.337121\pi\)
0.489660 + 0.871913i \(0.337121\pi\)
\(282\) 0 0
\(283\) 18.8885 1.12281 0.561404 0.827542i \(-0.310262\pi\)
0.561404 + 0.827542i \(0.310262\pi\)
\(284\) −0.944272 −0.0560322
\(285\) 0 0
\(286\) 0 0
\(287\) 1.52786 0.0901870
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) −20.5623 −1.20746
\(291\) 0 0
\(292\) −7.56231 −0.442550
\(293\) 9.88854 0.577695 0.288847 0.957375i \(-0.406728\pi\)
0.288847 + 0.957375i \(0.406728\pi\)
\(294\) 0 0
\(295\) 6.23607 0.363078
\(296\) −7.76393 −0.451269
\(297\) 0 0
\(298\) 29.0344 1.68192
\(299\) 52.8328 3.05540
\(300\) 0 0
\(301\) −10.9443 −0.630817
\(302\) −28.1803 −1.62160
\(303\) 0 0
\(304\) 14.5623 0.835206
\(305\) −14.9443 −0.855707
\(306\) 0 0
\(307\) −16.9443 −0.967061 −0.483530 0.875328i \(-0.660646\pi\)
−0.483530 + 0.875328i \(0.660646\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.8885 1.01437 0.507183 0.861838i \(-0.330687\pi\)
0.507183 + 0.861838i \(0.330687\pi\)
\(312\) 0 0
\(313\) −20.4721 −1.15715 −0.578577 0.815628i \(-0.696392\pi\)
−0.578577 + 0.815628i \(0.696392\pi\)
\(314\) −21.7082 −1.22506
\(315\) 0 0
\(316\) 8.29180 0.466450
\(317\) −30.4721 −1.71149 −0.855743 0.517401i \(-0.826899\pi\)
−0.855743 + 0.517401i \(0.826899\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 17.9443 1.00312
\(321\) 0 0
\(322\) −13.7082 −0.763928
\(323\) 13.4164 0.746509
\(324\) 0 0
\(325\) −80.7214 −4.47762
\(326\) 11.7984 0.653451
\(327\) 0 0
\(328\) −3.41641 −0.188640
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 3.70820 0.203514
\(333\) 0 0
\(334\) 30.6525 1.67723
\(335\) −36.8885 −2.01544
\(336\) 0 0
\(337\) −24.8328 −1.35273 −0.676365 0.736567i \(-0.736446\pi\)
−0.676365 + 0.736567i \(0.736446\pi\)
\(338\) 41.8885 2.27844
\(339\) 0 0
\(340\) −11.7082 −0.634967
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 24.4721 1.31945
\(345\) 0 0
\(346\) 12.1803 0.654819
\(347\) −3.05573 −0.164040 −0.0820200 0.996631i \(-0.526137\pi\)
−0.0820200 + 0.996631i \(0.526137\pi\)
\(348\) 0 0
\(349\) 22.7082 1.21554 0.607771 0.794112i \(-0.292064\pi\)
0.607771 + 0.794112i \(0.292064\pi\)
\(350\) 20.9443 1.11952
\(351\) 0 0
\(352\) 0 0
\(353\) −32.5967 −1.73495 −0.867475 0.497481i \(-0.834258\pi\)
−0.867475 + 0.497481i \(0.834258\pi\)
\(354\) 0 0
\(355\) −6.47214 −0.343505
\(356\) 2.76393 0.146488
\(357\) 0 0
\(358\) 26.6525 1.40863
\(359\) −19.0557 −1.00572 −0.502861 0.864367i \(-0.667719\pi\)
−0.502861 + 0.864367i \(0.667719\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −5.52786 −0.290538
\(363\) 0 0
\(364\) −3.85410 −0.202010
\(365\) −51.8328 −2.71305
\(366\) 0 0
\(367\) −22.8328 −1.19186 −0.595932 0.803035i \(-0.703217\pi\)
−0.595932 + 0.803035i \(0.703217\pi\)
\(368\) 41.1246 2.14377
\(369\) 0 0
\(370\) 23.7984 1.23722
\(371\) −8.94427 −0.464363
\(372\) 0 0
\(373\) −2.58359 −0.133773 −0.0668867 0.997761i \(-0.521307\pi\)
−0.0668867 + 0.997761i \(0.521307\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.70820 −0.345949
\(377\) 18.7082 0.963522
\(378\) 0 0
\(379\) −4.23607 −0.217592 −0.108796 0.994064i \(-0.534700\pi\)
−0.108796 + 0.994064i \(0.534700\pi\)
\(380\) −7.85410 −0.402907
\(381\) 0 0
\(382\) −4.76393 −0.243744
\(383\) −13.8885 −0.709671 −0.354836 0.934929i \(-0.615463\pi\)
−0.354836 + 0.934929i \(0.615463\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −19.4164 −0.988269
\(387\) 0 0
\(388\) 1.23607 0.0627518
\(389\) −28.3607 −1.43794 −0.718972 0.695039i \(-0.755387\pi\)
−0.718972 + 0.695039i \(0.755387\pi\)
\(390\) 0 0
\(391\) 37.8885 1.91611
\(392\) −2.23607 −0.112938
\(393\) 0 0
\(394\) −1.70820 −0.0860581
\(395\) 56.8328 2.85957
\(396\) 0 0
\(397\) 12.4721 0.625959 0.312979 0.949760i \(-0.398673\pi\)
0.312979 + 0.949760i \(0.398673\pi\)
\(398\) −20.1803 −1.01155
\(399\) 0 0
\(400\) −62.8328 −3.14164
\(401\) 10.3607 0.517388 0.258694 0.965959i \(-0.416708\pi\)
0.258694 + 0.965959i \(0.416708\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −10.4721 −0.521008
\(405\) 0 0
\(406\) −4.85410 −0.240905
\(407\) 0 0
\(408\) 0 0
\(409\) −2.58359 −0.127750 −0.0638752 0.997958i \(-0.520346\pi\)
−0.0638752 + 0.997958i \(0.520346\pi\)
\(410\) 10.4721 0.517182
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 1.47214 0.0724391
\(414\) 0 0
\(415\) 25.4164 1.24764
\(416\) 21.0902 1.03403
\(417\) 0 0
\(418\) 0 0
\(419\) −24.5279 −1.19826 −0.599132 0.800650i \(-0.704488\pi\)
−0.599132 + 0.800650i \(0.704488\pi\)
\(420\) 0 0
\(421\) −16.4164 −0.800087 −0.400043 0.916496i \(-0.631005\pi\)
−0.400043 + 0.916496i \(0.631005\pi\)
\(422\) 2.29180 0.111563
\(423\) 0 0
\(424\) 20.0000 0.971286
\(425\) −57.8885 −2.80801
\(426\) 0 0
\(427\) −3.52786 −0.170725
\(428\) 6.03444 0.291686
\(429\) 0 0
\(430\) −75.0132 −3.61746
\(431\) −18.5967 −0.895774 −0.447887 0.894090i \(-0.647823\pi\)
−0.447887 + 0.894090i \(0.647823\pi\)
\(432\) 0 0
\(433\) 9.41641 0.452524 0.226262 0.974067i \(-0.427349\pi\)
0.226262 + 0.974067i \(0.427349\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.4164 1.21583
\(438\) 0 0
\(439\) 9.36068 0.446761 0.223380 0.974731i \(-0.428291\pi\)
0.223380 + 0.974731i \(0.428291\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 45.1246 2.14636
\(443\) 7.88854 0.374796 0.187398 0.982284i \(-0.439995\pi\)
0.187398 + 0.982284i \(0.439995\pi\)
\(444\) 0 0
\(445\) 18.9443 0.898045
\(446\) 9.70820 0.459697
\(447\) 0 0
\(448\) 4.23607 0.200135
\(449\) −28.4721 −1.34368 −0.671842 0.740695i \(-0.734496\pi\)
−0.671842 + 0.740695i \(0.734496\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.00000 0.188144
\(453\) 0 0
\(454\) −25.1246 −1.17916
\(455\) −26.4164 −1.23842
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −39.5967 −1.85023
\(459\) 0 0
\(460\) −22.1803 −1.03416
\(461\) 26.9443 1.25492 0.627460 0.778649i \(-0.284095\pi\)
0.627460 + 0.778649i \(0.284095\pi\)
\(462\) 0 0
\(463\) −32.1246 −1.49296 −0.746479 0.665409i \(-0.768257\pi\)
−0.746479 + 0.665409i \(0.768257\pi\)
\(464\) 14.5623 0.676038
\(465\) 0 0
\(466\) 32.1803 1.49073
\(467\) −15.4721 −0.715965 −0.357983 0.933728i \(-0.616535\pi\)
−0.357983 + 0.933728i \(0.616535\pi\)
\(468\) 0 0
\(469\) −8.70820 −0.402107
\(470\) 20.5623 0.948468
\(471\) 0 0
\(472\) −3.29180 −0.151517
\(473\) 0 0
\(474\) 0 0
\(475\) −38.8328 −1.78177
\(476\) −2.76393 −0.126685
\(477\) 0 0
\(478\) 20.5623 0.940498
\(479\) −28.4721 −1.30093 −0.650463 0.759538i \(-0.725425\pi\)
−0.650463 + 0.759538i \(0.725425\pi\)
\(480\) 0 0
\(481\) −21.6525 −0.987268
\(482\) −20.5623 −0.936587
\(483\) 0 0
\(484\) 0 0
\(485\) 8.47214 0.384700
\(486\) 0 0
\(487\) 13.8885 0.629350 0.314675 0.949199i \(-0.398104\pi\)
0.314675 + 0.949199i \(0.398104\pi\)
\(488\) 7.88854 0.357098
\(489\) 0 0
\(490\) 6.85410 0.309637
\(491\) 20.5967 0.929518 0.464759 0.885437i \(-0.346141\pi\)
0.464759 + 0.885437i \(0.346141\pi\)
\(492\) 0 0
\(493\) 13.4164 0.604245
\(494\) 30.2705 1.36193
\(495\) 0 0
\(496\) 0 0
\(497\) −1.52786 −0.0685341
\(498\) 0 0
\(499\) −23.2918 −1.04268 −0.521342 0.853348i \(-0.674568\pi\)
−0.521342 + 0.853348i \(0.674568\pi\)
\(500\) 20.7984 0.930132
\(501\) 0 0
\(502\) 26.5623 1.18553
\(503\) −19.4164 −0.865735 −0.432867 0.901458i \(-0.642498\pi\)
−0.432867 + 0.901458i \(0.642498\pi\)
\(504\) 0 0
\(505\) −71.7771 −3.19404
\(506\) 0 0
\(507\) 0 0
\(508\) −1.23607 −0.0548416
\(509\) −26.3607 −1.16842 −0.584208 0.811604i \(-0.698595\pi\)
−0.584208 + 0.811604i \(0.698595\pi\)
\(510\) 0 0
\(511\) −12.2361 −0.541292
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) −0.562306 −0.0248022
\(515\) 27.4164 1.20811
\(516\) 0 0
\(517\) 0 0
\(518\) 5.61803 0.246842
\(519\) 0 0
\(520\) 59.0689 2.59034
\(521\) −14.2361 −0.623693 −0.311847 0.950132i \(-0.600948\pi\)
−0.311847 + 0.950132i \(0.600948\pi\)
\(522\) 0 0
\(523\) 3.00000 0.131181 0.0655904 0.997847i \(-0.479107\pi\)
0.0655904 + 0.997847i \(0.479107\pi\)
\(524\) 1.52786 0.0667451
\(525\) 0 0
\(526\) 0.381966 0.0166545
\(527\) 0 0
\(528\) 0 0
\(529\) 48.7771 2.12074
\(530\) −61.3050 −2.66292
\(531\) 0 0
\(532\) −1.85410 −0.0803855
\(533\) −9.52786 −0.412698
\(534\) 0 0
\(535\) 41.3607 1.78818
\(536\) 19.4721 0.841068
\(537\) 0 0
\(538\) 31.5967 1.36223
\(539\) 0 0
\(540\) 0 0
\(541\) −15.4164 −0.662803 −0.331402 0.943490i \(-0.607521\pi\)
−0.331402 + 0.943490i \(0.607521\pi\)
\(542\) 5.61803 0.241315
\(543\) 0 0
\(544\) 15.1246 0.648462
\(545\) 0 0
\(546\) 0 0
\(547\) −36.8328 −1.57486 −0.787429 0.616406i \(-0.788588\pi\)
−0.787429 + 0.616406i \(0.788588\pi\)
\(548\) −1.81966 −0.0777320
\(549\) 0 0
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) 13.4164 0.570524
\(554\) 20.1803 0.857380
\(555\) 0 0
\(556\) 0.583592 0.0247498
\(557\) −39.0000 −1.65248 −0.826242 0.563316i \(-0.809525\pi\)
−0.826242 + 0.563316i \(0.809525\pi\)
\(558\) 0 0
\(559\) 68.2492 2.88663
\(560\) −20.5623 −0.868916
\(561\) 0 0
\(562\) 26.5623 1.12046
\(563\) −2.47214 −0.104188 −0.0520941 0.998642i \(-0.516590\pi\)
−0.0520941 + 0.998642i \(0.516590\pi\)
\(564\) 0 0
\(565\) 27.4164 1.15342
\(566\) 30.5623 1.28463
\(567\) 0 0
\(568\) 3.41641 0.143349
\(569\) −8.11146 −0.340050 −0.170025 0.985440i \(-0.554385\pi\)
−0.170025 + 0.985440i \(0.554385\pi\)
\(570\) 0 0
\(571\) −22.9443 −0.960188 −0.480094 0.877217i \(-0.659397\pi\)
−0.480094 + 0.877217i \(0.659397\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 2.47214 0.103185
\(575\) −109.666 −4.57337
\(576\) 0 0
\(577\) −24.0000 −0.999133 −0.499567 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) 4.85410 0.201904
\(579\) 0 0
\(580\) −7.85410 −0.326124
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) 27.3607 1.13219
\(585\) 0 0
\(586\) 16.0000 0.660954
\(587\) 0.639320 0.0263876 0.0131938 0.999913i \(-0.495800\pi\)
0.0131938 + 0.999913i \(0.495800\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 10.0902 0.415406
\(591\) 0 0
\(592\) −16.8541 −0.692699
\(593\) −29.7771 −1.22280 −0.611399 0.791322i \(-0.709393\pi\)
−0.611399 + 0.791322i \(0.709393\pi\)
\(594\) 0 0
\(595\) −18.9443 −0.776639
\(596\) 11.0902 0.454271
\(597\) 0 0
\(598\) 85.4853 3.49575
\(599\) −1.41641 −0.0578729 −0.0289364 0.999581i \(-0.509212\pi\)
−0.0289364 + 0.999581i \(0.509212\pi\)
\(600\) 0 0
\(601\) −9.18034 −0.374474 −0.187237 0.982315i \(-0.559953\pi\)
−0.187237 + 0.982315i \(0.559953\pi\)
\(602\) −17.7082 −0.721733
\(603\) 0 0
\(604\) −10.7639 −0.437978
\(605\) 0 0
\(606\) 0 0
\(607\) 43.2492 1.75543 0.877716 0.479181i \(-0.159066\pi\)
0.877716 + 0.479181i \(0.159066\pi\)
\(608\) 10.1459 0.411470
\(609\) 0 0
\(610\) −24.1803 −0.979033
\(611\) −18.7082 −0.756853
\(612\) 0 0
\(613\) 17.5279 0.707944 0.353972 0.935256i \(-0.384831\pi\)
0.353972 + 0.935256i \(0.384831\pi\)
\(614\) −27.4164 −1.10644
\(615\) 0 0
\(616\) 0 0
\(617\) 29.8885 1.20327 0.601634 0.798772i \(-0.294517\pi\)
0.601634 + 0.798772i \(0.294517\pi\)
\(618\) 0 0
\(619\) 46.2492 1.85891 0.929457 0.368931i \(-0.120276\pi\)
0.929457 + 0.368931i \(0.120276\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 28.9443 1.16056
\(623\) 4.47214 0.179172
\(624\) 0 0
\(625\) 77.8328 3.11331
\(626\) −33.1246 −1.32393
\(627\) 0 0
\(628\) −8.29180 −0.330879
\(629\) −15.5279 −0.619136
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) −30.0000 −1.19334
\(633\) 0 0
\(634\) −49.3050 −1.95815
\(635\) −8.47214 −0.336206
\(636\) 0 0
\(637\) −6.23607 −0.247082
\(638\) 0 0
\(639\) 0 0
\(640\) 57.6869 2.28028
\(641\) 38.8328 1.53380 0.766902 0.641764i \(-0.221797\pi\)
0.766902 + 0.641764i \(0.221797\pi\)
\(642\) 0 0
\(643\) −1.41641 −0.0558577 −0.0279288 0.999610i \(-0.508891\pi\)
−0.0279288 + 0.999610i \(0.508891\pi\)
\(644\) −5.23607 −0.206330
\(645\) 0 0
\(646\) 21.7082 0.854098
\(647\) −4.88854 −0.192188 −0.0960942 0.995372i \(-0.530635\pi\)
−0.0960942 + 0.995372i \(0.530635\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −130.610 −5.12294
\(651\) 0 0
\(652\) 4.50658 0.176491
\(653\) 44.9443 1.75881 0.879403 0.476079i \(-0.157942\pi\)
0.879403 + 0.476079i \(0.157942\pi\)
\(654\) 0 0
\(655\) 10.4721 0.409180
\(656\) −7.41641 −0.289562
\(657\) 0 0
\(658\) 4.85410 0.189233
\(659\) 25.7639 1.00362 0.501810 0.864978i \(-0.332668\pi\)
0.501810 + 0.864978i \(0.332668\pi\)
\(660\) 0 0
\(661\) 30.8328 1.19926 0.599629 0.800278i \(-0.295315\pi\)
0.599629 + 0.800278i \(0.295315\pi\)
\(662\) 38.8328 1.50928
\(663\) 0 0
\(664\) −13.4164 −0.520658
\(665\) −12.7082 −0.492803
\(666\) 0 0
\(667\) 25.4164 0.984127
\(668\) 11.7082 0.453004
\(669\) 0 0
\(670\) −59.6869 −2.30591
\(671\) 0 0
\(672\) 0 0
\(673\) −33.3050 −1.28381 −0.641906 0.766784i \(-0.721856\pi\)
−0.641906 + 0.766784i \(0.721856\pi\)
\(674\) −40.1803 −1.54769
\(675\) 0 0
\(676\) 16.0000 0.615385
\(677\) 28.3607 1.08999 0.544995 0.838439i \(-0.316532\pi\)
0.544995 + 0.838439i \(0.316532\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 42.3607 1.62446
\(681\) 0 0
\(682\) 0 0
\(683\) 9.52786 0.364574 0.182287 0.983245i \(-0.441650\pi\)
0.182287 + 0.983245i \(0.441650\pi\)
\(684\) 0 0
\(685\) −12.4721 −0.476536
\(686\) 1.61803 0.0617768
\(687\) 0 0
\(688\) 53.1246 2.02536
\(689\) 55.7771 2.12494
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 4.65248 0.176861
\(693\) 0 0
\(694\) −4.94427 −0.187682
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −6.83282 −0.258811
\(698\) 36.7426 1.39073
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) 6.94427 0.262282 0.131141 0.991364i \(-0.458136\pi\)
0.131141 + 0.991364i \(0.458136\pi\)
\(702\) 0 0
\(703\) −10.4164 −0.392862
\(704\) 0 0
\(705\) 0 0
\(706\) −52.7426 −1.98500
\(707\) −16.9443 −0.637255
\(708\) 0 0
\(709\) 11.5836 0.435031 0.217515 0.976057i \(-0.430205\pi\)
0.217515 + 0.976057i \(0.430205\pi\)
\(710\) −10.4721 −0.393012
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 10.1803 0.380457
\(717\) 0 0
\(718\) −30.8328 −1.15067
\(719\) 24.7771 0.924029 0.462015 0.886872i \(-0.347127\pi\)
0.462015 + 0.886872i \(0.347127\pi\)
\(720\) 0 0
\(721\) 6.47214 0.241035
\(722\) −16.1803 −0.602170
\(723\) 0 0
\(724\) −2.11146 −0.0784717
\(725\) −38.8328 −1.44221
\(726\) 0 0
\(727\) 44.8328 1.66276 0.831379 0.555706i \(-0.187552\pi\)
0.831379 + 0.555706i \(0.187552\pi\)
\(728\) 13.9443 0.516809
\(729\) 0 0
\(730\) −83.8673 −3.10407
\(731\) 48.9443 1.81027
\(732\) 0 0
\(733\) −24.4721 −0.903899 −0.451949 0.892044i \(-0.649271\pi\)
−0.451949 + 0.892044i \(0.649271\pi\)
\(734\) −36.9443 −1.36364
\(735\) 0 0
\(736\) 28.6525 1.05614
\(737\) 0 0
\(738\) 0 0
\(739\) −51.3050 −1.88728 −0.943642 0.330969i \(-0.892624\pi\)
−0.943642 + 0.330969i \(0.892624\pi\)
\(740\) 9.09017 0.334161
\(741\) 0 0
\(742\) −14.4721 −0.531289
\(743\) −15.7639 −0.578323 −0.289161 0.957280i \(-0.593376\pi\)
−0.289161 + 0.957280i \(0.593376\pi\)
\(744\) 0 0
\(745\) 76.0132 2.78491
\(746\) −4.18034 −0.153053
\(747\) 0 0
\(748\) 0 0
\(749\) 9.76393 0.356766
\(750\) 0 0
\(751\) −18.2361 −0.665444 −0.332722 0.943025i \(-0.607967\pi\)
−0.332722 + 0.943025i \(0.607967\pi\)
\(752\) −14.5623 −0.531033
\(753\) 0 0
\(754\) 30.2705 1.10239
\(755\) −73.7771 −2.68502
\(756\) 0 0
\(757\) 24.4164 0.887429 0.443715 0.896168i \(-0.353660\pi\)
0.443715 + 0.896168i \(0.353660\pi\)
\(758\) −6.85410 −0.248952
\(759\) 0 0
\(760\) 28.4164 1.03077
\(761\) 6.36068 0.230574 0.115287 0.993332i \(-0.463221\pi\)
0.115287 + 0.993332i \(0.463221\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.81966 −0.0658330
\(765\) 0 0
\(766\) −22.4721 −0.811951
\(767\) −9.18034 −0.331483
\(768\) 0 0
\(769\) 14.1246 0.509347 0.254673 0.967027i \(-0.418032\pi\)
0.254673 + 0.967027i \(0.418032\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.41641 −0.266922
\(773\) 11.2918 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.47214 −0.160540
\(777\) 0 0
\(778\) −45.8885 −1.64518
\(779\) −4.58359 −0.164224
\(780\) 0 0
\(781\) 0 0
\(782\) 61.3050 2.19226
\(783\) 0 0
\(784\) −4.85410 −0.173361
\(785\) −56.8328 −2.02845
\(786\) 0 0
\(787\) 27.0000 0.962446 0.481223 0.876598i \(-0.340193\pi\)
0.481223 + 0.876598i \(0.340193\pi\)
\(788\) −0.652476 −0.0232435
\(789\) 0 0
\(790\) 91.9574 3.27170
\(791\) 6.47214 0.230123
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) 20.1803 0.716173
\(795\) 0 0
\(796\) −7.70820 −0.273210
\(797\) 14.8197 0.524939 0.262470 0.964940i \(-0.415463\pi\)
0.262470 + 0.964940i \(0.415463\pi\)
\(798\) 0 0
\(799\) −13.4164 −0.474638
\(800\) −43.7771 −1.54775
\(801\) 0 0
\(802\) 16.7639 0.591955
\(803\) 0 0
\(804\) 0 0
\(805\) −35.8885 −1.26490
\(806\) 0 0
\(807\) 0 0
\(808\) 37.8885 1.33291
\(809\) −31.3607 −1.10258 −0.551291 0.834313i \(-0.685865\pi\)
−0.551291 + 0.834313i \(0.685865\pi\)
\(810\) 0 0
\(811\) −55.8328 −1.96056 −0.980278 0.197625i \(-0.936677\pi\)
−0.980278 + 0.197625i \(0.936677\pi\)
\(812\) −1.85410 −0.0650662
\(813\) 0 0
\(814\) 0 0
\(815\) 30.8885 1.08198
\(816\) 0 0
\(817\) 32.8328 1.14867
\(818\) −4.18034 −0.146162
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −17.9443 −0.626259 −0.313130 0.949710i \(-0.601377\pi\)
−0.313130 + 0.949710i \(0.601377\pi\)
\(822\) 0 0
\(823\) −15.1803 −0.529153 −0.264577 0.964365i \(-0.585232\pi\)
−0.264577 + 0.964365i \(0.585232\pi\)
\(824\) −14.4721 −0.504161
\(825\) 0 0
\(826\) 2.38197 0.0828792
\(827\) −20.1246 −0.699801 −0.349901 0.936787i \(-0.613785\pi\)
−0.349901 + 0.936787i \(0.613785\pi\)
\(828\) 0 0
\(829\) −38.8328 −1.34872 −0.674360 0.738403i \(-0.735580\pi\)
−0.674360 + 0.738403i \(0.735580\pi\)
\(830\) 41.1246 1.42746
\(831\) 0 0
\(832\) −26.4164 −0.915824
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) 80.2492 2.77714
\(836\) 0 0
\(837\) 0 0
\(838\) −39.6869 −1.37096
\(839\) 6.05573 0.209067 0.104533 0.994521i \(-0.466665\pi\)
0.104533 + 0.994521i \(0.466665\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −26.5623 −0.915398
\(843\) 0 0
\(844\) 0.875388 0.0301321
\(845\) 109.666 3.77261
\(846\) 0 0
\(847\) 0 0
\(848\) 43.4164 1.49093
\(849\) 0 0
\(850\) −93.6656 −3.21270
\(851\) −29.4164 −1.00838
\(852\) 0 0
\(853\) −22.5836 −0.773247 −0.386624 0.922238i \(-0.626359\pi\)
−0.386624 + 0.922238i \(0.626359\pi\)
\(854\) −5.70820 −0.195331
\(855\) 0 0
\(856\) −21.8328 −0.746230
\(857\) −28.4721 −0.972590 −0.486295 0.873795i \(-0.661652\pi\)
−0.486295 + 0.873795i \(0.661652\pi\)
\(858\) 0 0
\(859\) 37.7771 1.28894 0.644469 0.764631i \(-0.277079\pi\)
0.644469 + 0.764631i \(0.277079\pi\)
\(860\) −28.6525 −0.977041
\(861\) 0 0
\(862\) −30.0902 −1.02488
\(863\) −7.63932 −0.260045 −0.130023 0.991511i \(-0.541505\pi\)
−0.130023 + 0.991511i \(0.541505\pi\)
\(864\) 0 0
\(865\) 31.8885 1.08424
\(866\) 15.2361 0.517743
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 54.3050 1.84005
\(872\) 0 0
\(873\) 0 0
\(874\) 41.1246 1.39106
\(875\) 33.6525 1.13766
\(876\) 0 0
\(877\) 2.83282 0.0956574 0.0478287 0.998856i \(-0.484770\pi\)
0.0478287 + 0.998856i \(0.484770\pi\)
\(878\) 15.1459 0.511149
\(879\) 0 0
\(880\) 0 0
\(881\) 26.4853 0.892312 0.446156 0.894955i \(-0.352793\pi\)
0.446156 + 0.894955i \(0.352793\pi\)
\(882\) 0 0
\(883\) 40.7082 1.36994 0.684970 0.728571i \(-0.259815\pi\)
0.684970 + 0.728571i \(0.259815\pi\)
\(884\) 17.2361 0.579712
\(885\) 0 0
\(886\) 12.7639 0.428813
\(887\) 7.41641 0.249019 0.124509 0.992218i \(-0.460264\pi\)
0.124509 + 0.992218i \(0.460264\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 30.6525 1.02747
\(891\) 0 0
\(892\) 3.70820 0.124160
\(893\) −9.00000 −0.301174
\(894\) 0 0
\(895\) 69.7771 2.33239
\(896\) 13.6180 0.454947
\(897\) 0 0
\(898\) −46.0689 −1.53734
\(899\) 0 0
\(900\) 0 0
\(901\) 40.0000 1.33259
\(902\) 0 0
\(903\) 0 0
\(904\) −14.4721 −0.481336
\(905\) −14.4721 −0.481070
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −9.59675 −0.318479
\(909\) 0 0
\(910\) −42.7426 −1.41690
\(911\) 21.3050 0.705865 0.352932 0.935649i \(-0.385185\pi\)
0.352932 + 0.935649i \(0.385185\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −42.0689 −1.39151
\(915\) 0 0
\(916\) −15.1246 −0.499731
\(917\) 2.47214 0.0816371
\(918\) 0 0
\(919\) 23.0557 0.760538 0.380269 0.924876i \(-0.375831\pi\)
0.380269 + 0.924876i \(0.375831\pi\)
\(920\) 80.2492 2.64574
\(921\) 0 0
\(922\) 43.5967 1.43578
\(923\) 9.52786 0.313613
\(924\) 0 0
\(925\) 44.9443 1.47776
\(926\) −51.9787 −1.70813
\(927\) 0 0
\(928\) 10.1459 0.333055
\(929\) 29.0689 0.953719 0.476860 0.878979i \(-0.341775\pi\)
0.476860 + 0.878979i \(0.341775\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 12.2918 0.402631
\(933\) 0 0
\(934\) −25.0344 −0.819152
\(935\) 0 0
\(936\) 0 0
\(937\) 29.4164 0.960992 0.480496 0.876997i \(-0.340457\pi\)
0.480496 + 0.876997i \(0.340457\pi\)
\(938\) −14.0902 −0.460060
\(939\) 0 0
\(940\) 7.85410 0.256173
\(941\) −22.3607 −0.728937 −0.364469 0.931216i \(-0.618749\pi\)
−0.364469 + 0.931216i \(0.618749\pi\)
\(942\) 0 0
\(943\) −12.9443 −0.421523
\(944\) −7.14590 −0.232579
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 76.3050 2.47696
\(950\) −62.8328 −2.03857
\(951\) 0 0
\(952\) 10.0000 0.324102
\(953\) −11.3607 −0.368009 −0.184004 0.982925i \(-0.558906\pi\)
−0.184004 + 0.982925i \(0.558906\pi\)
\(954\) 0 0
\(955\) −12.4721 −0.403589
\(956\) 7.85410 0.254020
\(957\) 0 0
\(958\) −46.0689 −1.48842
\(959\) −2.94427 −0.0950755
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −35.0344 −1.12956
\(963\) 0 0
\(964\) −7.85410 −0.252964
\(965\) −50.8328 −1.63637
\(966\) 0 0
\(967\) 29.5279 0.949552 0.474776 0.880107i \(-0.342529\pi\)
0.474776 + 0.880107i \(0.342529\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 13.7082 0.440144
\(971\) 23.4721 0.753257 0.376628 0.926364i \(-0.377083\pi\)
0.376628 + 0.926364i \(0.377083\pi\)
\(972\) 0 0
\(973\) 0.944272 0.0302720
\(974\) 22.4721 0.720054
\(975\) 0 0
\(976\) 17.1246 0.548145
\(977\) −57.7771 −1.84845 −0.924226 0.381845i \(-0.875289\pi\)
−0.924226 + 0.381845i \(0.875289\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.61803 0.0836300
\(981\) 0 0
\(982\) 33.3262 1.06348
\(983\) −41.8885 −1.33604 −0.668019 0.744145i \(-0.732857\pi\)
−0.668019 + 0.744145i \(0.732857\pi\)
\(984\) 0 0
\(985\) −4.47214 −0.142494
\(986\) 21.7082 0.691330
\(987\) 0 0
\(988\) 11.5623 0.367846
\(989\) 92.7214 2.94837
\(990\) 0 0
\(991\) −54.2361 −1.72287 −0.861433 0.507872i \(-0.830432\pi\)
−0.861433 + 0.507872i \(0.830432\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −2.47214 −0.0784114
\(995\) −52.8328 −1.67491
\(996\) 0 0
\(997\) 10.5836 0.335186 0.167593 0.985856i \(-0.446401\pi\)
0.167593 + 0.985856i \(0.446401\pi\)
\(998\) −37.6869 −1.19296
\(999\) 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.bq.1.2 2
3.2 odd 2 2541.2.a.r.1.1 2
11.10 odd 2 7623.2.a.bb.1.1 2
33.32 even 2 2541.2.a.ba.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.r.1.1 2 3.2 odd 2
2541.2.a.ba.1.2 yes 2 33.32 even 2
7623.2.a.bb.1.1 2 11.10 odd 2
7623.2.a.bq.1.2 2 1.1 even 1 trivial