Properties

Label 7623.2.a.bb.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
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: \(0\)
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 \(-0.618034\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.61803 q^{4} -0.236068 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+0.618034 q^{2} -1.61803 q^{4} -0.236068 q^{5} -1.00000 q^{7} -2.23607 q^{8} -0.145898 q^{10} +1.76393 q^{13} -0.618034 q^{14} +1.85410 q^{16} -4.47214 q^{17} +3.00000 q^{19} +0.381966 q^{20} +0.472136 q^{23} -4.94427 q^{25} +1.09017 q^{26} +1.61803 q^{28} +3.00000 q^{29} +5.61803 q^{32} -2.76393 q^{34} +0.236068 q^{35} -5.47214 q^{37} +1.85410 q^{38} +0.527864 q^{40} -10.4721 q^{41} -6.94427 q^{43} +0.291796 q^{46} +3.00000 q^{47} +1.00000 q^{49} -3.05573 q^{50} -2.85410 q^{52} +8.94427 q^{53} +2.23607 q^{56} +1.85410 q^{58} -7.47214 q^{59} +12.4721 q^{61} -0.236068 q^{64} -0.416408 q^{65} +4.70820 q^{67} +7.23607 q^{68} +0.145898 q^{70} -10.4721 q^{71} +7.76393 q^{73} -3.38197 q^{74} -4.85410 q^{76} +13.4164 q^{79} -0.437694 q^{80} -6.47214 q^{82} -6.00000 q^{83} +1.05573 q^{85} -4.29180 q^{86} -4.47214 q^{89} -1.76393 q^{91} -0.763932 q^{92} +1.85410 q^{94} -0.708204 q^{95} +2.00000 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 4q^{5} - 2q^{7} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 4q^{5} - 2q^{7} - 7q^{10} + 8q^{13} + q^{14} - 3q^{16} + 6q^{19} + 3q^{20} - 8q^{23} + 8q^{25} - 9q^{26} + q^{28} + 6q^{29} + 9q^{32} - 10q^{34} - 4q^{35} - 2q^{37} - 3q^{38} + 10q^{40} - 12q^{41} + 4q^{43} + 14q^{46} + 6q^{47} + 2q^{49} - 24q^{50} + q^{52} - 3q^{58} - 6q^{59} + 16q^{61} + 4q^{64} + 26q^{65} - 4q^{67} + 10q^{68} + 7q^{70} - 12q^{71} + 20q^{73} - 9q^{74} - 3q^{76} - 21q^{80} - 4q^{82} - 12q^{83} + 20q^{85} - 22q^{86} - 8q^{91} - 6q^{92} - 3q^{94} + 12q^{95} + 4q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) −0.236068 −0.105573 −0.0527864 0.998606i \(-0.516810\pi\)
−0.0527864 + 0.998606i \(0.516810\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) −0.145898 −0.0461370
\(11\) 0 0
\(12\) 0 0
\(13\) 1.76393 0.489227 0.244613 0.969621i \(-0.421339\pi\)
0.244613 + 0.969621i \(0.421339\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) −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 \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0.381966 0.0854102
\(21\) 0 0
\(22\) 0 0
\(23\) 0.472136 0.0984472 0.0492236 0.998788i \(-0.484325\pi\)
0.0492236 + 0.998788i \(0.484325\pi\)
\(24\) 0 0
\(25\) −4.94427 −0.988854
\(26\) 1.09017 0.213800
\(27\) 0 0
\(28\) 1.61803 0.305780
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) −2.76393 −0.474010
\(35\) 0.236068 0.0399028
\(36\) 0 0
\(37\) −5.47214 −0.899614 −0.449807 0.893126i \(-0.648507\pi\)
−0.449807 + 0.893126i \(0.648507\pi\)
\(38\) 1.85410 0.300775
\(39\) 0 0
\(40\) 0.527864 0.0834626
\(41\) −10.4721 −1.63547 −0.817736 0.575593i \(-0.804771\pi\)
−0.817736 + 0.575593i \(0.804771\pi\)
\(42\) 0 0
\(43\) −6.94427 −1.05899 −0.529496 0.848313i \(-0.677619\pi\)
−0.529496 + 0.848313i \(0.677619\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.291796 0.0430230
\(47\) 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
\(50\) −3.05573 −0.432145
\(51\) 0 0
\(52\) −2.85410 −0.395793
\(53\) 8.94427 1.22859 0.614295 0.789076i \(-0.289440\pi\)
0.614295 + 0.789076i \(0.289440\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 0 0
\(58\) 1.85410 0.243456
\(59\) −7.47214 −0.972789 −0.486395 0.873739i \(-0.661688\pi\)
−0.486395 + 0.873739i \(0.661688\pi\)
\(60\) 0 0
\(61\) 12.4721 1.59689 0.798447 0.602066i \(-0.205655\pi\)
0.798447 + 0.602066i \(0.205655\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) −0.416408 −0.0516490
\(66\) 0 0
\(67\) 4.70820 0.575199 0.287599 0.957751i \(-0.407143\pi\)
0.287599 + 0.957751i \(0.407143\pi\)
\(68\) 7.23607 0.877502
\(69\) 0 0
\(70\) 0.145898 0.0174382
\(71\) −10.4721 −1.24281 −0.621407 0.783488i \(-0.713439\pi\)
−0.621407 + 0.783488i \(0.713439\pi\)
\(72\) 0 0
\(73\) 7.76393 0.908700 0.454350 0.890823i \(-0.349872\pi\)
0.454350 + 0.890823i \(0.349872\pi\)
\(74\) −3.38197 −0.393146
\(75\) 0 0
\(76\) −4.85410 −0.556804
\(77\) 0 0
\(78\) 0 0
\(79\) 13.4164 1.50946 0.754732 0.656033i \(-0.227767\pi\)
0.754732 + 0.656033i \(0.227767\pi\)
\(80\) −0.437694 −0.0489357
\(81\) 0 0
\(82\) −6.47214 −0.714728
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 1.05573 0.114510
\(86\) −4.29180 −0.462796
\(87\) 0 0
\(88\) 0 0
\(89\) −4.47214 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) 0 0
\(91\) −1.76393 −0.184910
\(92\) −0.763932 −0.0796454
\(93\) 0 0
\(94\) 1.85410 0.191236
\(95\) −0.708204 −0.0726602
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0.618034 0.0624309
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) −0.944272 −0.0939586 −0.0469793 0.998896i \(-0.514959\pi\)
−0.0469793 + 0.998896i \(0.514959\pi\)
\(102\) 0 0
\(103\) −2.47214 −0.243587 −0.121793 0.992555i \(-0.538865\pi\)
−0.121793 + 0.992555i \(0.538865\pi\)
\(104\) −3.94427 −0.386768
\(105\) 0 0
\(106\) 5.52786 0.536914
\(107\) −14.2361 −1.37625 −0.688126 0.725591i \(-0.741567\pi\)
−0.688126 + 0.725591i \(0.741567\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.85410 −0.175196
\(113\) −2.47214 −0.232559 −0.116279 0.993217i \(-0.537097\pi\)
−0.116279 + 0.993217i \(0.537097\pi\)
\(114\) 0 0
\(115\) −0.111456 −0.0103933
\(116\) −4.85410 −0.450692
\(117\) 0 0
\(118\) −4.61803 −0.425124
\(119\) 4.47214 0.409960
\(120\) 0 0
\(121\) 0 0
\(122\) 7.70820 0.697868
\(123\) 0 0
\(124\) 0 0
\(125\) 2.34752 0.209969
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −11.3820 −1.00603
\(129\) 0 0
\(130\) −0.257354 −0.0225715
\(131\) 6.47214 0.565473 0.282737 0.959198i \(-0.408758\pi\)
0.282737 + 0.959198i \(0.408758\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 2.90983 0.251371
\(135\) 0 0
\(136\) 10.0000 0.857493
\(137\) 14.9443 1.27678 0.638388 0.769715i \(-0.279602\pi\)
0.638388 + 0.769715i \(0.279602\pi\)
\(138\) 0 0
\(139\) 16.9443 1.43719 0.718597 0.695427i \(-0.244785\pi\)
0.718597 + 0.695427i \(0.244785\pi\)
\(140\) −0.381966 −0.0322820
\(141\) 0 0
\(142\) −6.47214 −0.543130
\(143\) 0 0
\(144\) 0 0
\(145\) −0.708204 −0.0588131
\(146\) 4.79837 0.397116
\(147\) 0 0
\(148\) 8.85410 0.727803
\(149\) −0.0557281 −0.00456542 −0.00228271 0.999997i \(-0.500727\pi\)
−0.00228271 + 0.999997i \(0.500727\pi\)
\(150\) 0 0
\(151\) −9.41641 −0.766296 −0.383148 0.923687i \(-0.625160\pi\)
−0.383148 + 0.923687i \(0.625160\pi\)
\(152\) −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 \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(158\) 8.29180 0.659660
\(159\) 0 0
\(160\) −1.32624 −0.104848
\(161\) −0.472136 −0.0372095
\(162\) 0 0
\(163\) 20.7082 1.62199 0.810996 0.585052i \(-0.198926\pi\)
0.810996 + 0.585052i \(0.198926\pi\)
\(164\) 16.9443 1.32313
\(165\) 0 0
\(166\) −3.70820 −0.287812
\(167\) −1.05573 −0.0816947 −0.0408473 0.999165i \(-0.513006\pi\)
−0.0408473 + 0.999165i \(0.513006\pi\)
\(168\) 0 0
\(169\) −9.88854 −0.760657
\(170\) 0.652476 0.0500426
\(171\) 0 0
\(172\) 11.2361 0.856742
\(173\) −16.4721 −1.25235 −0.626177 0.779681i \(-0.715381\pi\)
−0.626177 + 0.779681i \(0.715381\pi\)
\(174\) 0 0
\(175\) 4.94427 0.373752
\(176\) 0 0
\(177\) 0 0
\(178\) −2.76393 −0.207165
\(179\) 7.52786 0.562659 0.281329 0.959611i \(-0.409225\pi\)
0.281329 + 0.959611i \(0.409225\pi\)
\(180\) 0 0
\(181\) 23.4164 1.74053 0.870264 0.492586i \(-0.163948\pi\)
0.870264 + 0.492586i \(0.163948\pi\)
\(182\) −1.09017 −0.0808088
\(183\) 0 0
\(184\) −1.05573 −0.0778293
\(185\) 1.29180 0.0949747
\(186\) 0 0
\(187\) 0 0
\(188\) −4.85410 −0.354022
\(189\) 0 0
\(190\) −0.437694 −0.0317537
\(191\) 14.9443 1.08133 0.540665 0.841238i \(-0.318173\pi\)
0.540665 + 0.841238i \(0.318173\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 1.23607 0.0887445
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) 18.9443 1.34972 0.674862 0.737944i \(-0.264203\pi\)
0.674862 + 0.737944i \(0.264203\pi\)
\(198\) 0 0
\(199\) −3.52786 −0.250084 −0.125042 0.992151i \(-0.539907\pi\)
−0.125042 + 0.992151i \(0.539907\pi\)
\(200\) 11.0557 0.781758
\(201\) 0 0
\(202\) −0.583592 −0.0410614
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) 2.47214 0.172661
\(206\) −1.52786 −0.106451
\(207\) 0 0
\(208\) 3.27051 0.226769
\(209\) 0 0
\(210\) 0 0
\(211\) 25.4164 1.74974 0.874869 0.484360i \(-0.160947\pi\)
0.874869 + 0.484360i \(0.160947\pi\)
\(212\) −14.4721 −0.993950
\(213\) 0 0
\(214\) −8.79837 −0.601444
\(215\) 1.63932 0.111801
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.88854 −0.530641
\(222\) 0 0
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) −5.61803 −0.375371
\(225\) 0 0
\(226\) −1.52786 −0.101632
\(227\) 24.4721 1.62427 0.812136 0.583468i \(-0.198305\pi\)
0.812136 + 0.583468i \(0.198305\pi\)
\(228\) 0 0
\(229\) −15.5279 −1.02611 −0.513055 0.858356i \(-0.671486\pi\)
−0.513055 + 0.858356i \(0.671486\pi\)
\(230\) −0.0688837 −0.00454206
\(231\) 0 0
\(232\) −6.70820 −0.440415
\(233\) 15.8885 1.04089 0.520447 0.853894i \(-0.325765\pi\)
0.520447 + 0.853894i \(0.325765\pi\)
\(234\) 0 0
\(235\) −0.708204 −0.0461981
\(236\) 12.0902 0.787003
\(237\) 0 0
\(238\) 2.76393 0.179159
\(239\) 0.708204 0.0458099 0.0229050 0.999738i \(-0.492708\pi\)
0.0229050 + 0.999738i \(0.492708\pi\)
\(240\) 0 0
\(241\) −0.708204 −0.0456194 −0.0228097 0.999740i \(-0.507261\pi\)
−0.0228097 + 0.999740i \(0.507261\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −20.1803 −1.29191
\(245\) −0.236068 −0.0150818
\(246\) 0 0
\(247\) 5.29180 0.336709
\(248\) 0 0
\(249\) 0 0
\(250\) 1.45085 0.0917598
\(251\) −10.4164 −0.657478 −0.328739 0.944421i \(-0.606624\pi\)
−0.328739 + 0.944421i \(0.606624\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.23607 0.0775578
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −31.6525 −1.97443 −0.987214 0.159403i \(-0.949043\pi\)
−0.987214 + 0.159403i \(0.949043\pi\)
\(258\) 0 0
\(259\) 5.47214 0.340022
\(260\) 0.673762 0.0417850
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 4.23607 0.261207 0.130604 0.991435i \(-0.458308\pi\)
0.130604 + 0.991435i \(0.458308\pi\)
\(264\) 0 0
\(265\) −2.11146 −0.129706
\(266\) −1.85410 −0.113682
\(267\) 0 0
\(268\) −7.61803 −0.465345
\(269\) 28.4721 1.73598 0.867988 0.496584i \(-0.165413\pi\)
0.867988 + 0.496584i \(0.165413\pi\)
\(270\) 0 0
\(271\) 5.47214 0.332409 0.166204 0.986091i \(-0.446849\pi\)
0.166204 + 0.986091i \(0.446849\pi\)
\(272\) −8.29180 −0.502764
\(273\) 0 0
\(274\) 9.23607 0.557971
\(275\) 0 0
\(276\) 0 0
\(277\) −3.52786 −0.211969 −0.105984 0.994368i \(-0.533799\pi\)
−0.105984 + 0.994368i \(0.533799\pi\)
\(278\) 10.4721 0.628077
\(279\) 0 0
\(280\) −0.527864 −0.0315459
\(281\) 10.4164 0.621391 0.310695 0.950510i \(-0.399438\pi\)
0.310695 + 0.950510i \(0.399438\pi\)
\(282\) 0 0
\(283\) 16.8885 1.00392 0.501960 0.864891i \(-0.332613\pi\)
0.501960 + 0.864891i \(0.332613\pi\)
\(284\) 16.9443 1.00546
\(285\) 0 0
\(286\) 0 0
\(287\) 10.4721 0.618151
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) −0.437694 −0.0257023
\(291\) 0 0
\(292\) −12.5623 −0.735153
\(293\) 25.8885 1.51242 0.756212 0.654326i \(-0.227048\pi\)
0.756212 + 0.654326i \(0.227048\pi\)
\(294\) 0 0
\(295\) 1.76393 0.102700
\(296\) 12.2361 0.711207
\(297\) 0 0
\(298\) −0.0344419 −0.00199516
\(299\) 0.832816 0.0481630
\(300\) 0 0
\(301\) 6.94427 0.400261
\(302\) −5.81966 −0.334884
\(303\) 0 0
\(304\) 5.56231 0.319020
\(305\) −2.94427 −0.168589
\(306\) 0 0
\(307\) −0.944272 −0.0538924 −0.0269462 0.999637i \(-0.508578\pi\)
−0.0269462 + 0.999637i \(0.508578\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −17.8885 −1.01437 −0.507183 0.861838i \(-0.669313\pi\)
−0.507183 + 0.861838i \(0.669313\pi\)
\(312\) 0 0
\(313\) −11.5279 −0.651593 −0.325797 0.945440i \(-0.605632\pi\)
−0.325797 + 0.945440i \(0.605632\pi\)
\(314\) 8.29180 0.467933
\(315\) 0 0
\(316\) −21.7082 −1.22118
\(317\) −21.5279 −1.20913 −0.604563 0.796558i \(-0.706652\pi\)
−0.604563 + 0.796558i \(0.706652\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.0557281 0.00311529
\(321\) 0 0
\(322\) −0.291796 −0.0162612
\(323\) −13.4164 −0.746509
\(324\) 0 0
\(325\) −8.72136 −0.483774
\(326\) 12.7984 0.708836
\(327\) 0 0
\(328\) 23.4164 1.29295
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 9.70820 0.532807
\(333\) 0 0
\(334\) −0.652476 −0.0357019
\(335\) −1.11146 −0.0607253
\(336\) 0 0
\(337\) −28.8328 −1.57062 −0.785312 0.619100i \(-0.787497\pi\)
−0.785312 + 0.619100i \(0.787497\pi\)
\(338\) −6.11146 −0.332419
\(339\) 0 0
\(340\) −1.70820 −0.0926404
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 15.5279 0.837206
\(345\) 0 0
\(346\) −10.1803 −0.547298
\(347\) 20.9443 1.12435 0.562174 0.827019i \(-0.309965\pi\)
0.562174 + 0.827019i \(0.309965\pi\)
\(348\) 0 0
\(349\) −9.29180 −0.497378 −0.248689 0.968583i \(-0.580000\pi\)
−0.248689 + 0.968583i \(0.580000\pi\)
\(350\) 3.05573 0.163336
\(351\) 0 0
\(352\) 0 0
\(353\) 16.5967 0.883356 0.441678 0.897174i \(-0.354383\pi\)
0.441678 + 0.897174i \(0.354383\pi\)
\(354\) 0 0
\(355\) 2.47214 0.131207
\(356\) 7.23607 0.383511
\(357\) 0 0
\(358\) 4.65248 0.245891
\(359\) 36.9443 1.94984 0.974922 0.222547i \(-0.0714370\pi\)
0.974922 + 0.222547i \(0.0714370\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 14.4721 0.760639
\(363\) 0 0
\(364\) 2.85410 0.149596
\(365\) −1.83282 −0.0959340
\(366\) 0 0
\(367\) 30.8328 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(368\) 0.875388 0.0456328
\(369\) 0 0
\(370\) 0.798374 0.0415055
\(371\) −8.94427 −0.464363
\(372\) 0 0
\(373\) 29.4164 1.52312 0.761562 0.648092i \(-0.224433\pi\)
0.761562 + 0.648092i \(0.224433\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.70820 −0.345949
\(377\) 5.29180 0.272541
\(378\) 0 0
\(379\) 0.236068 0.0121260 0.00606300 0.999982i \(-0.498070\pi\)
0.00606300 + 0.999982i \(0.498070\pi\)
\(380\) 1.14590 0.0587833
\(381\) 0 0
\(382\) 9.23607 0.472558
\(383\) 21.8885 1.11845 0.559226 0.829015i \(-0.311098\pi\)
0.559226 + 0.829015i \(0.311098\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.41641 0.377485
\(387\) 0 0
\(388\) −3.23607 −0.164286
\(389\) 16.3607 0.829519 0.414760 0.909931i \(-0.363866\pi\)
0.414760 + 0.909931i \(0.363866\pi\)
\(390\) 0 0
\(391\) −2.11146 −0.106781
\(392\) −2.23607 −0.112938
\(393\) 0 0
\(394\) 11.7082 0.589851
\(395\) −3.16718 −0.159358
\(396\) 0 0
\(397\) 3.52786 0.177058 0.0885292 0.996074i \(-0.471783\pi\)
0.0885292 + 0.996074i \(0.471783\pi\)
\(398\) −2.18034 −0.109291
\(399\) 0 0
\(400\) −9.16718 −0.458359
\(401\) −34.3607 −1.71589 −0.857945 0.513741i \(-0.828259\pi\)
−0.857945 + 0.513741i \(0.828259\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.52786 0.0760141
\(405\) 0 0
\(406\) −1.85410 −0.0920175
\(407\) 0 0
\(408\) 0 0
\(409\) 29.4164 1.45455 0.727274 0.686347i \(-0.240787\pi\)
0.727274 + 0.686347i \(0.240787\pi\)
\(410\) 1.52786 0.0754558
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 7.47214 0.367680
\(414\) 0 0
\(415\) 1.41641 0.0695287
\(416\) 9.90983 0.485869
\(417\) 0 0
\(418\) 0 0
\(419\) −33.4721 −1.63522 −0.817610 0.575772i \(-0.804702\pi\)
−0.817610 + 0.575772i \(0.804702\pi\)
\(420\) 0 0
\(421\) 10.4164 0.507665 0.253832 0.967248i \(-0.418309\pi\)
0.253832 + 0.967248i \(0.418309\pi\)
\(422\) 15.7082 0.764663
\(423\) 0 0
\(424\) −20.0000 −0.971286
\(425\) 22.1115 1.07256
\(426\) 0 0
\(427\) −12.4721 −0.603569
\(428\) 23.0344 1.11341
\(429\) 0 0
\(430\) 1.01316 0.0488587
\(431\) −30.5967 −1.47379 −0.736897 0.676005i \(-0.763710\pi\)
−0.736897 + 0.676005i \(0.763710\pi\)
\(432\) 0 0
\(433\) −17.4164 −0.836979 −0.418490 0.908222i \(-0.637440\pi\)
−0.418490 + 0.908222i \(0.637440\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.41641 0.0677560
\(438\) 0 0
\(439\) 35.3607 1.68767 0.843837 0.536600i \(-0.180292\pi\)
0.843837 + 0.536600i \(0.180292\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.87539 −0.231899
\(443\) −27.8885 −1.32502 −0.662512 0.749051i \(-0.730510\pi\)
−0.662512 + 0.749051i \(0.730510\pi\)
\(444\) 0 0
\(445\) 1.05573 0.0500463
\(446\) 3.70820 0.175589
\(447\) 0 0
\(448\) 0.236068 0.0111532
\(449\) −19.5279 −0.921577 −0.460788 0.887510i \(-0.652433\pi\)
−0.460788 + 0.887510i \(0.652433\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.00000 0.188144
\(453\) 0 0
\(454\) 15.1246 0.709833
\(455\) 0.416408 0.0195215
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −9.59675 −0.448427
\(459\) 0 0
\(460\) 0.180340 0.00840839
\(461\) −9.05573 −0.421767 −0.210884 0.977511i \(-0.567634\pi\)
−0.210884 + 0.977511i \(0.567634\pi\)
\(462\) 0 0
\(463\) 8.12461 0.377583 0.188791 0.982017i \(-0.439543\pi\)
0.188791 + 0.982017i \(0.439543\pi\)
\(464\) 5.56231 0.258224
\(465\) 0 0
\(466\) 9.81966 0.454887
\(467\) −6.52786 −0.302074 −0.151037 0.988528i \(-0.548261\pi\)
−0.151037 + 0.988528i \(0.548261\pi\)
\(468\) 0 0
\(469\) −4.70820 −0.217405
\(470\) −0.437694 −0.0201893
\(471\) 0 0
\(472\) 16.7082 0.769057
\(473\) 0 0
\(474\) 0 0
\(475\) −14.8328 −0.680576
\(476\) −7.23607 −0.331665
\(477\) 0 0
\(478\) 0.437694 0.0200197
\(479\) 19.5279 0.892251 0.446125 0.894970i \(-0.352804\pi\)
0.446125 + 0.894970i \(0.352804\pi\)
\(480\) 0 0
\(481\) −9.65248 −0.440115
\(482\) −0.437694 −0.0199364
\(483\) 0 0
\(484\) 0 0
\(485\) −0.472136 −0.0214386
\(486\) 0 0
\(487\) −21.8885 −0.991865 −0.495932 0.868361i \(-0.665174\pi\)
−0.495932 + 0.868361i \(0.665174\pi\)
\(488\) −27.8885 −1.26246
\(489\) 0 0
\(490\) −0.145898 −0.00659100
\(491\) 28.5967 1.29055 0.645277 0.763949i \(-0.276742\pi\)
0.645277 + 0.763949i \(0.276742\pi\)
\(492\) 0 0
\(493\) −13.4164 −0.604245
\(494\) 3.27051 0.147147
\(495\) 0 0
\(496\) 0 0
\(497\) 10.4721 0.469739
\(498\) 0 0
\(499\) −36.7082 −1.64328 −0.821642 0.570003i \(-0.806942\pi\)
−0.821642 + 0.570003i \(0.806942\pi\)
\(500\) −3.79837 −0.169868
\(501\) 0 0
\(502\) −6.43769 −0.287328
\(503\) −7.41641 −0.330681 −0.165341 0.986237i \(-0.552872\pi\)
−0.165341 + 0.986237i \(0.552872\pi\)
\(504\) 0 0
\(505\) 0.222912 0.00991947
\(506\) 0 0
\(507\) 0 0
\(508\) −3.23607 −0.143577
\(509\) 18.3607 0.813823 0.406911 0.913468i \(-0.366606\pi\)
0.406911 + 0.913468i \(0.366606\pi\)
\(510\) 0 0
\(511\) −7.76393 −0.343456
\(512\) 18.7082 0.826794
\(513\) 0 0
\(514\) −19.5623 −0.862856
\(515\) 0.583592 0.0257161
\(516\) 0 0
\(517\) 0 0
\(518\) 3.38197 0.148595
\(519\) 0 0
\(520\) 0.931116 0.0408322
\(521\) −9.76393 −0.427766 −0.213883 0.976859i \(-0.568611\pi\)
−0.213883 + 0.976859i \(0.568611\pi\)
\(522\) 0 0
\(523\) −3.00000 −0.131181 −0.0655904 0.997847i \(-0.520893\pi\)
−0.0655904 + 0.997847i \(0.520893\pi\)
\(524\) −10.4721 −0.457477
\(525\) 0 0
\(526\) 2.61803 0.114152
\(527\) 0 0
\(528\) 0 0
\(529\) −22.7771 −0.990308
\(530\) −1.30495 −0.0566835
\(531\) 0 0
\(532\) 4.85410 0.210452
\(533\) −18.4721 −0.800117
\(534\) 0 0
\(535\) 3.36068 0.145295
\(536\) −10.5279 −0.454734
\(537\) 0 0
\(538\) 17.5967 0.758650
\(539\) 0 0
\(540\) 0 0
\(541\) −11.4164 −0.490830 −0.245415 0.969418i \(-0.578924\pi\)
−0.245415 + 0.969418i \(0.578924\pi\)
\(542\) 3.38197 0.145268
\(543\) 0 0
\(544\) −25.1246 −1.07721
\(545\) 0 0
\(546\) 0 0
\(547\) −16.8328 −0.719719 −0.359860 0.933006i \(-0.617175\pi\)
−0.359860 + 0.933006i \(0.617175\pi\)
\(548\) −24.1803 −1.03293
\(549\) 0 0
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) −13.4164 −0.570524
\(554\) −2.18034 −0.0926338
\(555\) 0 0
\(556\) −27.4164 −1.16271
\(557\) 39.0000 1.65248 0.826242 0.563316i \(-0.190475\pi\)
0.826242 + 0.563316i \(0.190475\pi\)
\(558\) 0 0
\(559\) −12.2492 −0.518087
\(560\) 0.437694 0.0184960
\(561\) 0 0
\(562\) 6.43769 0.271558
\(563\) −6.47214 −0.272768 −0.136384 0.990656i \(-0.543548\pi\)
−0.136384 + 0.990656i \(0.543548\pi\)
\(564\) 0 0
\(565\) 0.583592 0.0245519
\(566\) 10.4377 0.438729
\(567\) 0 0
\(568\) 23.4164 0.982531
\(569\) 43.8885 1.83990 0.919952 0.392032i \(-0.128228\pi\)
0.919952 + 0.392032i \(0.128228\pi\)
\(570\) 0 0
\(571\) 5.05573 0.211576 0.105788 0.994389i \(-0.466264\pi\)
0.105788 + 0.994389i \(0.466264\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 6.47214 0.270142
\(575\) −2.33437 −0.0973499
\(576\) 0 0
\(577\) −24.0000 −0.999133 −0.499567 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) 1.85410 0.0771205
\(579\) 0 0
\(580\) 1.14590 0.0475808
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) −17.3607 −0.718390
\(585\) 0 0
\(586\) 16.0000 0.660954
\(587\) 45.3607 1.87224 0.936118 0.351687i \(-0.114392\pi\)
0.936118 + 0.351687i \(0.114392\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 1.09017 0.0448816
\(591\) 0 0
\(592\) −10.1459 −0.416994
\(593\) −41.7771 −1.71558 −0.857790 0.514001i \(-0.828163\pi\)
−0.857790 + 0.514001i \(0.828163\pi\)
\(594\) 0 0
\(595\) −1.05573 −0.0432806
\(596\) 0.0901699 0.00369350
\(597\) 0 0
\(598\) 0.514708 0.0210480
\(599\) 25.4164 1.03849 0.519243 0.854627i \(-0.326214\pi\)
0.519243 + 0.854627i \(0.326214\pi\)
\(600\) 0 0
\(601\) −13.1803 −0.537637 −0.268819 0.963191i \(-0.586633\pi\)
−0.268819 + 0.963191i \(0.586633\pi\)
\(602\) 4.29180 0.174921
\(603\) 0 0
\(604\) 15.2361 0.619947
\(605\) 0 0
\(606\) 0 0
\(607\) 37.2492 1.51190 0.755950 0.654630i \(-0.227175\pi\)
0.755950 + 0.654630i \(0.227175\pi\)
\(608\) 16.8541 0.683524
\(609\) 0 0
\(610\) −1.81966 −0.0736759
\(611\) 5.29180 0.214083
\(612\) 0 0
\(613\) −26.4721 −1.06920 −0.534600 0.845105i \(-0.679538\pi\)
−0.534600 + 0.845105i \(0.679538\pi\)
\(614\) −0.583592 −0.0235519
\(615\) 0 0
\(616\) 0 0
\(617\) −5.88854 −0.237064 −0.118532 0.992950i \(-0.537819\pi\)
−0.118532 + 0.992950i \(0.537819\pi\)
\(618\) 0 0
\(619\) −34.2492 −1.37659 −0.688296 0.725430i \(-0.741641\pi\)
−0.688296 + 0.725430i \(0.741641\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −11.0557 −0.443294
\(623\) 4.47214 0.179172
\(624\) 0 0
\(625\) 24.1672 0.966687
\(626\) −7.12461 −0.284757
\(627\) 0 0
\(628\) −21.7082 −0.866252
\(629\) 24.4721 0.975768
\(630\) 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
\(634\) −13.3050 −0.528407
\(635\) −0.472136 −0.0187361
\(636\) 0 0
\(637\) 1.76393 0.0698895
\(638\) 0 0
\(639\) 0 0
\(640\) 2.68692 0.106210
\(641\) −14.8328 −0.585861 −0.292930 0.956134i \(-0.594630\pi\)
−0.292930 + 0.956134i \(0.594630\pi\)
\(642\) 0 0
\(643\) 25.4164 1.00233 0.501163 0.865353i \(-0.332906\pi\)
0.501163 + 0.865353i \(0.332906\pi\)
\(644\) 0.763932 0.0301031
\(645\) 0 0
\(646\) −8.29180 −0.326236
\(647\) 30.8885 1.21435 0.607177 0.794567i \(-0.292302\pi\)
0.607177 + 0.794567i \(0.292302\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −5.39010 −0.211417
\(651\) 0 0
\(652\) −33.5066 −1.31222
\(653\) 27.0557 1.05877 0.529386 0.848381i \(-0.322422\pi\)
0.529386 + 0.848381i \(0.322422\pi\)
\(654\) 0 0
\(655\) −1.52786 −0.0596986
\(656\) −19.4164 −0.758083
\(657\) 0 0
\(658\) −1.85410 −0.0722804
\(659\) −30.2361 −1.17783 −0.588915 0.808195i \(-0.700445\pi\)
−0.588915 + 0.808195i \(0.700445\pi\)
\(660\) 0 0
\(661\) −22.8328 −0.888094 −0.444047 0.896004i \(-0.646458\pi\)
−0.444047 + 0.896004i \(0.646458\pi\)
\(662\) 14.8328 0.576494
\(663\) 0 0
\(664\) 13.4164 0.520658
\(665\) 0.708204 0.0274630
\(666\) 0 0
\(667\) 1.41641 0.0548435
\(668\) 1.70820 0.0660924
\(669\) 0 0
\(670\) −0.686918 −0.0265379
\(671\) 0 0
\(672\) 0 0
\(673\) −29.3050 −1.12962 −0.564811 0.825220i \(-0.691051\pi\)
−0.564811 + 0.825220i \(0.691051\pi\)
\(674\) −17.8197 −0.686388
\(675\) 0 0
\(676\) 16.0000 0.615385
\(677\) 16.3607 0.628792 0.314396 0.949292i \(-0.398198\pi\)
0.314396 + 0.949292i \(0.398198\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) −2.36068 −0.0905279
\(681\) 0 0
\(682\) 0 0
\(683\) 18.4721 0.706817 0.353408 0.935469i \(-0.385023\pi\)
0.353408 + 0.935469i \(0.385023\pi\)
\(684\) 0 0
\(685\) −3.52786 −0.134793
\(686\) −0.618034 −0.0235966
\(687\) 0 0
\(688\) −12.8754 −0.490870
\(689\) 15.7771 0.601059
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 26.6525 1.01318
\(693\) 0 0
\(694\) 12.9443 0.491358
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 46.8328 1.77392
\(698\) −5.74265 −0.217362
\(699\) 0 0
\(700\) −8.00000 −0.302372
\(701\) 10.9443 0.413359 0.206680 0.978409i \(-0.433734\pi\)
0.206680 + 0.978409i \(0.433734\pi\)
\(702\) 0 0
\(703\) −16.4164 −0.619157
\(704\) 0 0
\(705\) 0 0
\(706\) 10.2574 0.386041
\(707\) 0.944272 0.0355130
\(708\) 0 0
\(709\) 38.4164 1.44276 0.721379 0.692540i \(-0.243508\pi\)
0.721379 + 0.692540i \(0.243508\pi\)
\(710\) 1.52786 0.0573397
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.1803 −0.455201
\(717\) 0 0
\(718\) 22.8328 0.852113
\(719\) −46.7771 −1.74449 −0.872246 0.489068i \(-0.837337\pi\)
−0.872246 + 0.489068i \(0.837337\pi\)
\(720\) 0 0
\(721\) 2.47214 0.0920672
\(722\) −6.18034 −0.230008
\(723\) 0 0
\(724\) −37.8885 −1.40812
\(725\) −14.8328 −0.550877
\(726\) 0 0
\(727\) −8.83282 −0.327591 −0.163796 0.986494i \(-0.552374\pi\)
−0.163796 + 0.986494i \(0.552374\pi\)
\(728\) 3.94427 0.146184
\(729\) 0 0
\(730\) −1.13274 −0.0419247
\(731\) 31.0557 1.14864
\(732\) 0 0
\(733\) 15.5279 0.573535 0.286767 0.958000i \(-0.407419\pi\)
0.286767 + 0.958000i \(0.407419\pi\)
\(734\) 19.0557 0.703360
\(735\) 0 0
\(736\) 2.65248 0.0977716
\(737\) 0 0
\(738\) 0 0
\(739\) −11.3050 −0.415859 −0.207930 0.978144i \(-0.566673\pi\)
−0.207930 + 0.978144i \(0.566673\pi\)
\(740\) −2.09017 −0.0768362
\(741\) 0 0
\(742\) −5.52786 −0.202934
\(743\) 20.2361 0.742389 0.371195 0.928555i \(-0.378948\pi\)
0.371195 + 0.928555i \(0.378948\pi\)
\(744\) 0 0
\(745\) 0.0131556 0.000481985 0
\(746\) 18.1803 0.665630
\(747\) 0 0
\(748\) 0 0
\(749\) 14.2361 0.520175
\(750\) 0 0
\(751\) −13.7639 −0.502253 −0.251127 0.967954i \(-0.580801\pi\)
−0.251127 + 0.967954i \(0.580801\pi\)
\(752\) 5.56231 0.202836
\(753\) 0 0
\(754\) 3.27051 0.119105
\(755\) 2.22291 0.0809001
\(756\) 0 0
\(757\) −2.41641 −0.0878258 −0.0439129 0.999035i \(-0.513982\pi\)
−0.0439129 + 0.999035i \(0.513982\pi\)
\(758\) 0.145898 0.00529926
\(759\) 0 0
\(760\) 1.58359 0.0574429
\(761\) 38.3607 1.39057 0.695287 0.718732i \(-0.255277\pi\)
0.695287 + 0.718732i \(0.255277\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −24.1803 −0.874814
\(765\) 0 0
\(766\) 13.5279 0.488782
\(767\) −13.1803 −0.475914
\(768\) 0 0
\(769\) 26.1246 0.942078 0.471039 0.882112i \(-0.343879\pi\)
0.471039 + 0.882112i \(0.343879\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −19.4164 −0.698812
\(773\) 24.7082 0.888692 0.444346 0.895855i \(-0.353436\pi\)
0.444346 + 0.895855i \(0.353436\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4.47214 −0.160540
\(777\) 0 0
\(778\) 10.1115 0.362513
\(779\) −31.4164 −1.12561
\(780\) 0 0
\(781\) 0 0
\(782\) −1.30495 −0.0466650
\(783\) 0 0
\(784\) 1.85410 0.0662179
\(785\) −3.16718 −0.113042
\(786\) 0 0
\(787\) −27.0000 −0.962446 −0.481223 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) −30.6525 −1.09195
\(789\) 0 0
\(790\) −1.95743 −0.0696421
\(791\) 2.47214 0.0878990
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) 2.18034 0.0773774
\(795\) 0 0
\(796\) 5.70820 0.202322
\(797\) 37.1803 1.31700 0.658498 0.752583i \(-0.271192\pi\)
0.658498 + 0.752583i \(0.271192\pi\)
\(798\) 0 0
\(799\) −13.4164 −0.474638
\(800\) −27.7771 −0.982068
\(801\) 0 0
\(802\) −21.2361 −0.749872
\(803\) 0 0
\(804\) 0 0
\(805\) 0.111456 0.00392831
\(806\) 0 0
\(807\) 0 0
\(808\) 2.11146 0.0742808
\(809\) −13.3607 −0.469736 −0.234868 0.972027i \(-0.575466\pi\)
−0.234868 + 0.972027i \(0.575466\pi\)
\(810\) 0 0
\(811\) 2.16718 0.0761001 0.0380501 0.999276i \(-0.487885\pi\)
0.0380501 + 0.999276i \(0.487885\pi\)
\(812\) 4.85410 0.170346
\(813\) 0 0
\(814\) 0 0
\(815\) −4.88854 −0.171238
\(816\) 0 0
\(817\) −20.8328 −0.728848
\(818\) 18.1803 0.635661
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 0.0557281 0.00194492 0.000972462 1.00000i \(-0.499690\pi\)
0.000972462 1.00000i \(0.499690\pi\)
\(822\) 0 0
\(823\) 7.18034 0.250291 0.125145 0.992138i \(-0.460060\pi\)
0.125145 + 0.992138i \(0.460060\pi\)
\(824\) 5.52786 0.192572
\(825\) 0 0
\(826\) 4.61803 0.160682
\(827\) −20.1246 −0.699801 −0.349901 0.936787i \(-0.613785\pi\)
−0.349901 + 0.936787i \(0.613785\pi\)
\(828\) 0 0
\(829\) 14.8328 0.515165 0.257582 0.966256i \(-0.417074\pi\)
0.257582 + 0.966256i \(0.417074\pi\)
\(830\) 0.875388 0.0303852
\(831\) 0 0
\(832\) −0.416408 −0.0144363
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) 0.249224 0.00862474
\(836\) 0 0
\(837\) 0 0
\(838\) −20.6869 −0.714618
\(839\) 23.9443 0.826648 0.413324 0.910584i \(-0.364368\pi\)
0.413324 + 0.910584i \(0.364368\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 6.43769 0.221858
\(843\) 0 0
\(844\) −41.1246 −1.41557
\(845\) 2.33437 0.0803047
\(846\) 0 0
\(847\) 0 0
\(848\) 16.5836 0.569483
\(849\) 0 0
\(850\) 13.6656 0.468727
\(851\) −2.58359 −0.0885644
\(852\) 0 0
\(853\) 49.4164 1.69199 0.845993 0.533194i \(-0.179009\pi\)
0.845993 + 0.533194i \(0.179009\pi\)
\(854\) −7.70820 −0.263769
\(855\) 0 0
\(856\) 31.8328 1.08802
\(857\) 19.5279 0.667059 0.333530 0.942740i \(-0.391760\pi\)
0.333530 + 0.942740i \(0.391760\pi\)
\(858\) 0 0
\(859\) −33.7771 −1.15246 −0.576230 0.817288i \(-0.695477\pi\)
−0.576230 + 0.817288i \(0.695477\pi\)
\(860\) −2.65248 −0.0904487
\(861\) 0 0
\(862\) −18.9098 −0.644071
\(863\) −52.3607 −1.78238 −0.891189 0.453632i \(-0.850128\pi\)
−0.891189 + 0.453632i \(0.850128\pi\)
\(864\) 0 0
\(865\) 3.88854 0.132214
\(866\) −10.7639 −0.365773
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.30495 0.281403
\(872\) 0 0
\(873\) 0 0
\(874\) 0.875388 0.0296104
\(875\) −2.34752 −0.0793608
\(876\) 0 0
\(877\) 50.8328 1.71650 0.858251 0.513230i \(-0.171551\pi\)
0.858251 + 0.513230i \(0.171551\pi\)
\(878\) 21.8541 0.737540
\(879\) 0 0
\(880\) 0 0
\(881\) −58.4853 −1.97042 −0.985210 0.171353i \(-0.945186\pi\)
−0.985210 + 0.171353i \(0.945186\pi\)
\(882\) 0 0
\(883\) 27.2918 0.918442 0.459221 0.888322i \(-0.348129\pi\)
0.459221 + 0.888322i \(0.348129\pi\)
\(884\) 12.7639 0.429297
\(885\) 0 0
\(886\) −17.2361 −0.579057
\(887\) 19.4164 0.651939 0.325970 0.945380i \(-0.394309\pi\)
0.325970 + 0.945380i \(0.394309\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0.652476 0.0218710
\(891\) 0 0
\(892\) −9.70820 −0.325055
\(893\) 9.00000 0.301174
\(894\) 0 0
\(895\) −1.77709 −0.0594015
\(896\) 11.3820 0.380245
\(897\) 0 0
\(898\) −12.0689 −0.402744
\(899\) 0 0
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) 0 0
\(904\) 5.52786 0.183854
\(905\) −5.52786 −0.183752
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −39.5967 −1.31406
\(909\) 0 0
\(910\) 0.257354 0.00853121
\(911\) −41.3050 −1.36849 −0.684247 0.729250i \(-0.739869\pi\)
−0.684247 + 0.729250i \(0.739869\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 16.0689 0.531511
\(915\) 0 0
\(916\) 25.1246 0.830141
\(917\) −6.47214 −0.213729
\(918\) 0 0
\(919\) −40.9443 −1.35063 −0.675313 0.737531i \(-0.735992\pi\)
−0.675313 + 0.737531i \(0.735992\pi\)
\(920\) 0.249224 0.00821666
\(921\) 0 0
\(922\) −5.59675 −0.184319
\(923\) −18.4721 −0.608018
\(924\) 0 0
\(925\) 27.0557 0.889587
\(926\) 5.02129 0.165010
\(927\) 0 0
\(928\) 16.8541 0.553263
\(929\) −29.0689 −0.953719 −0.476860 0.878979i \(-0.658225\pi\)
−0.476860 + 0.878979i \(0.658225\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) −25.7082 −0.842100
\(933\) 0 0
\(934\) −4.03444 −0.132011
\(935\) 0 0
\(936\) 0 0
\(937\) −2.58359 −0.0844023 −0.0422011 0.999109i \(-0.513437\pi\)
−0.0422011 + 0.999109i \(0.513437\pi\)
\(938\) −2.90983 −0.0950093
\(939\) 0 0
\(940\) 1.14590 0.0373751
\(941\) −22.3607 −0.728937 −0.364469 0.931216i \(-0.618749\pi\)
−0.364469 + 0.931216i \(0.618749\pi\)
\(942\) 0 0
\(943\) −4.94427 −0.161008
\(944\) −13.8541 −0.450913
\(945\) 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
\(949\) 13.6950 0.444560
\(950\) −9.16718 −0.297423
\(951\) 0 0
\(952\) −10.0000 −0.324102
\(953\) −33.3607 −1.08066 −0.540329 0.841454i \(-0.681700\pi\)
−0.540329 + 0.841454i \(0.681700\pi\)
\(954\) 0 0
\(955\) −3.52786 −0.114159
\(956\) −1.14590 −0.0370610
\(957\) 0 0
\(958\) 12.0689 0.389928
\(959\) −14.9443 −0.482576
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −5.96556 −0.192337
\(963\) 0 0
\(964\) 1.14590 0.0369069
\(965\) −2.83282 −0.0911916
\(966\) 0 0
\(967\) −38.4721 −1.23718 −0.618590 0.785714i \(-0.712296\pi\)
−0.618590 + 0.785714i \(0.712296\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −0.291796 −0.00936901
\(971\) 14.5279 0.466221 0.233111 0.972450i \(-0.425110\pi\)
0.233111 + 0.972450i \(0.425110\pi\)
\(972\) 0 0
\(973\) −16.9443 −0.543208
\(974\) −13.5279 −0.433461
\(975\) 0 0
\(976\) 23.1246 0.740201
\(977\) 13.7771 0.440768 0.220384 0.975413i \(-0.429269\pi\)
0.220384 + 0.975413i \(0.429269\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.381966 0.0122015
\(981\) 0 0
\(982\) 17.6738 0.563992
\(983\) −6.11146 −0.194925 −0.0974626 0.995239i \(-0.531073\pi\)
−0.0974626 + 0.995239i \(0.531073\pi\)
\(984\) 0 0
\(985\) −4.47214 −0.142494
\(986\) −8.29180 −0.264065
\(987\) 0 0
\(988\) −8.56231 −0.272403
\(989\) −3.27864 −0.104255
\(990\) 0 0
\(991\) −49.7639 −1.58080 −0.790402 0.612589i \(-0.790128\pi\)
−0.790402 + 0.612589i \(0.790128\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 6.47214 0.205284
\(995\) 0.832816 0.0264020
\(996\) 0 0
\(997\) −37.4164 −1.18499 −0.592495 0.805574i \(-0.701857\pi\)
−0.592495 + 0.805574i \(0.701857\pi\)
\(998\) −22.6869 −0.718142
\(999\) 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.bb.1.2 2
3.2 odd 2 2541.2.a.ba.1.1 yes 2
11.10 odd 2 7623.2.a.bq.1.1 2
33.32 even 2 2541.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.r.1.2 2 33.32 even 2
2541.2.a.ba.1.1 yes 2 3.2 odd 2
7623.2.a.bb.1.2 2 1.1 even 1 trivial
7623.2.a.bq.1.1 2 11.10 odd 2