Properties

Label 7623.2.a.t.1.1
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 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} -1.00000 q^{7} -7.47214 q^{8} +O(q^{10})\) \(q-2.61803 q^{2} +4.85410 q^{4} -1.00000 q^{5} -1.00000 q^{7} -7.47214 q^{8} +2.61803 q^{10} -3.23607 q^{13} +2.61803 q^{14} +9.85410 q^{16} -8.09017 q^{17} +6.23607 q^{19} -4.85410 q^{20} +6.09017 q^{23} -4.00000 q^{25} +8.47214 q^{26} -4.85410 q^{28} -2.38197 q^{29} +0.236068 q^{31} -10.8541 q^{32} +21.1803 q^{34} +1.00000 q^{35} -2.47214 q^{37} -16.3262 q^{38} +7.47214 q^{40} +11.1803 q^{41} -7.56231 q^{43} -15.9443 q^{46} +4.38197 q^{47} +1.00000 q^{49} +10.4721 q^{50} -15.7082 q^{52} +4.61803 q^{53} +7.47214 q^{56} +6.23607 q^{58} +0.0901699 q^{59} +5.38197 q^{61} -0.618034 q^{62} +8.70820 q^{64} +3.23607 q^{65} +7.32624 q^{67} -39.2705 q^{68} -2.61803 q^{70} +4.90983 q^{71} +9.76393 q^{73} +6.47214 q^{74} +30.2705 q^{76} -8.61803 q^{79} -9.85410 q^{80} -29.2705 q^{82} -10.7082 q^{83} +8.09017 q^{85} +19.7984 q^{86} -0.145898 q^{89} +3.23607 q^{91} +29.5623 q^{92} -11.4721 q^{94} -6.23607 q^{95} +7.00000 q^{97} -2.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} - 6q^{8} + 3q^{10} - 2q^{13} + 3q^{14} + 13q^{16} - 5q^{17} + 8q^{19} - 3q^{20} + q^{23} - 8q^{25} + 8q^{26} - 3q^{28} - 7q^{29} - 4q^{31} - 15q^{32} + 20q^{34} + 2q^{35} + 4q^{37} - 17q^{38} + 6q^{40} + 5q^{43} - 14q^{46} + 11q^{47} + 2q^{49} + 12q^{50} - 18q^{52} + 7q^{53} + 6q^{56} + 8q^{58} - 11q^{59} + 13q^{61} + q^{62} + 4q^{64} + 2q^{65} - q^{67} - 45q^{68} - 3q^{70} + 21q^{71} + 24q^{73} + 4q^{74} + 27q^{76} - 15q^{79} - 13q^{80} - 25q^{82} - 8q^{83} + 5q^{85} + 15q^{86} - 7q^{89} + 2q^{91} + 39q^{92} - 14q^{94} - 8q^{95} + 14q^{97} - 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61803 −1.85123 −0.925615 0.378467i \(-0.876451\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) 0 0
\(4\) 4.85410 2.42705
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −7.47214 −2.64180
\(9\) 0 0
\(10\) 2.61803 0.827895
\(11\) 0 0
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 2.61803 0.699699
\(15\) 0 0
\(16\) 9.85410 2.46353
\(17\) −8.09017 −1.96215 −0.981077 0.193617i \(-0.937978\pi\)
−0.981077 + 0.193617i \(0.937978\pi\)
\(18\) 0 0
\(19\) 6.23607 1.43065 0.715326 0.698791i \(-0.246278\pi\)
0.715326 + 0.698791i \(0.246278\pi\)
\(20\) −4.85410 −1.08541
\(21\) 0 0
\(22\) 0 0
\(23\) 6.09017 1.26989 0.634944 0.772558i \(-0.281023\pi\)
0.634944 + 0.772558i \(0.281023\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 8.47214 1.66152
\(27\) 0 0
\(28\) −4.85410 −0.917339
\(29\) −2.38197 −0.442320 −0.221160 0.975238i \(-0.570984\pi\)
−0.221160 + 0.975238i \(0.570984\pi\)
\(30\) 0 0
\(31\) 0.236068 0.0423991 0.0211995 0.999775i \(-0.493251\pi\)
0.0211995 + 0.999775i \(0.493251\pi\)
\(32\) −10.8541 −1.91875
\(33\) 0 0
\(34\) 21.1803 3.63240
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.47214 −0.406417 −0.203208 0.979136i \(-0.565137\pi\)
−0.203208 + 0.979136i \(0.565137\pi\)
\(38\) −16.3262 −2.64847
\(39\) 0 0
\(40\) 7.47214 1.18145
\(41\) 11.1803 1.74608 0.873038 0.487652i \(-0.162147\pi\)
0.873038 + 0.487652i \(0.162147\pi\)
\(42\) 0 0
\(43\) −7.56231 −1.15324 −0.576620 0.817012i \(-0.695629\pi\)
−0.576620 + 0.817012i \(0.695629\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −15.9443 −2.35085
\(47\) 4.38197 0.639175 0.319588 0.947557i \(-0.396456\pi\)
0.319588 + 0.947557i \(0.396456\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 10.4721 1.48098
\(51\) 0 0
\(52\) −15.7082 −2.17834
\(53\) 4.61803 0.634336 0.317168 0.948369i \(-0.397268\pi\)
0.317168 + 0.948369i \(0.397268\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.47214 0.998506
\(57\) 0 0
\(58\) 6.23607 0.818836
\(59\) 0.0901699 0.0117391 0.00586956 0.999983i \(-0.498132\pi\)
0.00586956 + 0.999983i \(0.498132\pi\)
\(60\) 0 0
\(61\) 5.38197 0.689090 0.344545 0.938770i \(-0.388033\pi\)
0.344545 + 0.938770i \(0.388033\pi\)
\(62\) −0.618034 −0.0784904
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 3.23607 0.401385
\(66\) 0 0
\(67\) 7.32624 0.895042 0.447521 0.894273i \(-0.352307\pi\)
0.447521 + 0.894273i \(0.352307\pi\)
\(68\) −39.2705 −4.76225
\(69\) 0 0
\(70\) −2.61803 −0.312915
\(71\) 4.90983 0.582690 0.291345 0.956618i \(-0.405897\pi\)
0.291345 + 0.956618i \(0.405897\pi\)
\(72\) 0 0
\(73\) 9.76393 1.14278 0.571391 0.820678i \(-0.306404\pi\)
0.571391 + 0.820678i \(0.306404\pi\)
\(74\) 6.47214 0.752371
\(75\) 0 0
\(76\) 30.2705 3.47227
\(77\) 0 0
\(78\) 0 0
\(79\) −8.61803 −0.969605 −0.484802 0.874624i \(-0.661108\pi\)
−0.484802 + 0.874624i \(0.661108\pi\)
\(80\) −9.85410 −1.10172
\(81\) 0 0
\(82\) −29.2705 −3.23239
\(83\) −10.7082 −1.17538 −0.587689 0.809087i \(-0.699962\pi\)
−0.587689 + 0.809087i \(0.699962\pi\)
\(84\) 0 0
\(85\) 8.09017 0.877502
\(86\) 19.7984 2.13491
\(87\) 0 0
\(88\) 0 0
\(89\) −0.145898 −0.0154652 −0.00773258 0.999970i \(-0.502461\pi\)
−0.00773258 + 0.999970i \(0.502461\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) 29.5623 3.08208
\(93\) 0 0
\(94\) −11.4721 −1.18326
\(95\) −6.23607 −0.639807
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) −2.61803 −0.264461
\(99\) 0 0
\(100\) −19.4164 −1.94164
\(101\) 9.79837 0.974975 0.487487 0.873130i \(-0.337914\pi\)
0.487487 + 0.873130i \(0.337914\pi\)
\(102\) 0 0
\(103\) 2.14590 0.211442 0.105721 0.994396i \(-0.466285\pi\)
0.105721 + 0.994396i \(0.466285\pi\)
\(104\) 24.1803 2.37108
\(105\) 0 0
\(106\) −12.0902 −1.17430
\(107\) −10.7082 −1.03520 −0.517601 0.855622i \(-0.673175\pi\)
−0.517601 + 0.855622i \(0.673175\pi\)
\(108\) 0 0
\(109\) 10.4721 1.00305 0.501524 0.865144i \(-0.332773\pi\)
0.501524 + 0.865144i \(0.332773\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −9.85410 −0.931125
\(113\) −6.85410 −0.644780 −0.322390 0.946607i \(-0.604486\pi\)
−0.322390 + 0.946607i \(0.604486\pi\)
\(114\) 0 0
\(115\) −6.09017 −0.567911
\(116\) −11.5623 −1.07353
\(117\) 0 0
\(118\) −0.236068 −0.0217318
\(119\) 8.09017 0.741625
\(120\) 0 0
\(121\) 0 0
\(122\) −14.0902 −1.27566
\(123\) 0 0
\(124\) 1.14590 0.102905
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 14.9443 1.32609 0.663045 0.748580i \(-0.269264\pi\)
0.663045 + 0.748580i \(0.269264\pi\)
\(128\) −1.09017 −0.0963583
\(129\) 0 0
\(130\) −8.47214 −0.743055
\(131\) 1.05573 0.0922394 0.0461197 0.998936i \(-0.485314\pi\)
0.0461197 + 0.998936i \(0.485314\pi\)
\(132\) 0 0
\(133\) −6.23607 −0.540736
\(134\) −19.1803 −1.65693
\(135\) 0 0
\(136\) 60.4508 5.18362
\(137\) −0.326238 −0.0278724 −0.0139362 0.999903i \(-0.504436\pi\)
−0.0139362 + 0.999903i \(0.504436\pi\)
\(138\) 0 0
\(139\) 5.94427 0.504187 0.252093 0.967703i \(-0.418881\pi\)
0.252093 + 0.967703i \(0.418881\pi\)
\(140\) 4.85410 0.410246
\(141\) 0 0
\(142\) −12.8541 −1.07869
\(143\) 0 0
\(144\) 0 0
\(145\) 2.38197 0.197812
\(146\) −25.5623 −2.11555
\(147\) 0 0
\(148\) −12.0000 −0.986394
\(149\) −2.14590 −0.175799 −0.0878994 0.996129i \(-0.528015\pi\)
−0.0878994 + 0.996129i \(0.528015\pi\)
\(150\) 0 0
\(151\) −17.9443 −1.46028 −0.730142 0.683295i \(-0.760546\pi\)
−0.730142 + 0.683295i \(0.760546\pi\)
\(152\) −46.5967 −3.77950
\(153\) 0 0
\(154\) 0 0
\(155\) −0.236068 −0.0189614
\(156\) 0 0
\(157\) −15.8885 −1.26804 −0.634022 0.773315i \(-0.718597\pi\)
−0.634022 + 0.773315i \(0.718597\pi\)
\(158\) 22.5623 1.79496
\(159\) 0 0
\(160\) 10.8541 0.858092
\(161\) −6.09017 −0.479973
\(162\) 0 0
\(163\) 4.70820 0.368775 0.184387 0.982854i \(-0.440970\pi\)
0.184387 + 0.982854i \(0.440970\pi\)
\(164\) 54.2705 4.23781
\(165\) 0 0
\(166\) 28.0344 2.17589
\(167\) −2.47214 −0.191300 −0.0956498 0.995415i \(-0.530493\pi\)
−0.0956498 + 0.995415i \(0.530493\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) −21.1803 −1.62446
\(171\) 0 0
\(172\) −36.7082 −2.79897
\(173\) −17.6180 −1.33947 −0.669737 0.742598i \(-0.733593\pi\)
−0.669737 + 0.742598i \(0.733593\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0.381966 0.0286296
\(179\) 8.23607 0.615593 0.307796 0.951452i \(-0.400408\pi\)
0.307796 + 0.951452i \(0.400408\pi\)
\(180\) 0 0
\(181\) 19.4164 1.44321 0.721605 0.692305i \(-0.243405\pi\)
0.721605 + 0.692305i \(0.243405\pi\)
\(182\) −8.47214 −0.627996
\(183\) 0 0
\(184\) −45.5066 −3.35479
\(185\) 2.47214 0.181755
\(186\) 0 0
\(187\) 0 0
\(188\) 21.2705 1.55131
\(189\) 0 0
\(190\) 16.3262 1.18443
\(191\) −20.2361 −1.46423 −0.732115 0.681181i \(-0.761467\pi\)
−0.732115 + 0.681181i \(0.761467\pi\)
\(192\) 0 0
\(193\) −0.326238 −0.0234831 −0.0117416 0.999931i \(-0.503738\pi\)
−0.0117416 + 0.999931i \(0.503738\pi\)
\(194\) −18.3262 −1.31575
\(195\) 0 0
\(196\) 4.85410 0.346722
\(197\) −17.7082 −1.26166 −0.630829 0.775922i \(-0.717285\pi\)
−0.630829 + 0.775922i \(0.717285\pi\)
\(198\) 0 0
\(199\) −3.76393 −0.266818 −0.133409 0.991061i \(-0.542592\pi\)
−0.133409 + 0.991061i \(0.542592\pi\)
\(200\) 29.8885 2.11344
\(201\) 0 0
\(202\) −25.6525 −1.80490
\(203\) 2.38197 0.167181
\(204\) 0 0
\(205\) −11.1803 −0.780869
\(206\) −5.61803 −0.391427
\(207\) 0 0
\(208\) −31.8885 −2.21107
\(209\) 0 0
\(210\) 0 0
\(211\) −13.3820 −0.921253 −0.460626 0.887594i \(-0.652375\pi\)
−0.460626 + 0.887594i \(0.652375\pi\)
\(212\) 22.4164 1.53957
\(213\) 0 0
\(214\) 28.0344 1.91639
\(215\) 7.56231 0.515745
\(216\) 0 0
\(217\) −0.236068 −0.0160253
\(218\) −27.4164 −1.85687
\(219\) 0 0
\(220\) 0 0
\(221\) 26.1803 1.76108
\(222\) 0 0
\(223\) −27.0344 −1.81036 −0.905180 0.425028i \(-0.860264\pi\)
−0.905180 + 0.425028i \(0.860264\pi\)
\(224\) 10.8541 0.725220
\(225\) 0 0
\(226\) 17.9443 1.19364
\(227\) 13.0344 0.865126 0.432563 0.901604i \(-0.357609\pi\)
0.432563 + 0.901604i \(0.357609\pi\)
\(228\) 0 0
\(229\) 11.2361 0.742500 0.371250 0.928533i \(-0.378929\pi\)
0.371250 + 0.928533i \(0.378929\pi\)
\(230\) 15.9443 1.05133
\(231\) 0 0
\(232\) 17.7984 1.16852
\(233\) −2.58359 −0.169257 −0.0846284 0.996413i \(-0.526970\pi\)
−0.0846284 + 0.996413i \(0.526970\pi\)
\(234\) 0 0
\(235\) −4.38197 −0.285848
\(236\) 0.437694 0.0284915
\(237\) 0 0
\(238\) −21.1803 −1.37292
\(239\) −5.90983 −0.382275 −0.191138 0.981563i \(-0.561218\pi\)
−0.191138 + 0.981563i \(0.561218\pi\)
\(240\) 0 0
\(241\) 17.2705 1.11249 0.556246 0.831018i \(-0.312241\pi\)
0.556246 + 0.831018i \(0.312241\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 26.1246 1.67246
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −20.1803 −1.28404
\(248\) −1.76393 −0.112010
\(249\) 0 0
\(250\) −23.5623 −1.49021
\(251\) −23.0000 −1.45175 −0.725874 0.687828i \(-0.758564\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −39.1246 −2.45490
\(255\) 0 0
\(256\) −14.5623 −0.910144
\(257\) −11.5623 −0.721237 −0.360618 0.932713i \(-0.617434\pi\)
−0.360618 + 0.932713i \(0.617434\pi\)
\(258\) 0 0
\(259\) 2.47214 0.153611
\(260\) 15.7082 0.974181
\(261\) 0 0
\(262\) −2.76393 −0.170756
\(263\) 23.1246 1.42592 0.712962 0.701202i \(-0.247353\pi\)
0.712962 + 0.701202i \(0.247353\pi\)
\(264\) 0 0
\(265\) −4.61803 −0.283684
\(266\) 16.3262 1.00103
\(267\) 0 0
\(268\) 35.5623 2.17231
\(269\) −10.1459 −0.618606 −0.309303 0.950963i \(-0.600096\pi\)
−0.309303 + 0.950963i \(0.600096\pi\)
\(270\) 0 0
\(271\) −19.7984 −1.20267 −0.601333 0.798999i \(-0.705363\pi\)
−0.601333 + 0.798999i \(0.705363\pi\)
\(272\) −79.7214 −4.83382
\(273\) 0 0
\(274\) 0.854102 0.0515982
\(275\) 0 0
\(276\) 0 0
\(277\) 6.03444 0.362574 0.181287 0.983430i \(-0.441974\pi\)
0.181287 + 0.983430i \(0.441974\pi\)
\(278\) −15.5623 −0.933365
\(279\) 0 0
\(280\) −7.47214 −0.446546
\(281\) −2.81966 −0.168207 −0.0841034 0.996457i \(-0.526803\pi\)
−0.0841034 + 0.996457i \(0.526803\pi\)
\(282\) 0 0
\(283\) 5.94427 0.353350 0.176675 0.984269i \(-0.443466\pi\)
0.176675 + 0.984269i \(0.443466\pi\)
\(284\) 23.8328 1.41422
\(285\) 0 0
\(286\) 0 0
\(287\) −11.1803 −0.659955
\(288\) 0 0
\(289\) 48.4508 2.85005
\(290\) −6.23607 −0.366195
\(291\) 0 0
\(292\) 47.3951 2.77359
\(293\) −11.0000 −0.642627 −0.321313 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) 0 0
\(295\) −0.0901699 −0.00524990
\(296\) 18.4721 1.07367
\(297\) 0 0
\(298\) 5.61803 0.325444
\(299\) −19.7082 −1.13975
\(300\) 0 0
\(301\) 7.56231 0.435884
\(302\) 46.9787 2.70332
\(303\) 0 0
\(304\) 61.4508 3.52445
\(305\) −5.38197 −0.308170
\(306\) 0 0
\(307\) 0.819660 0.0467805 0.0233902 0.999726i \(-0.492554\pi\)
0.0233902 + 0.999726i \(0.492554\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.618034 0.0351020
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) 2.67376 0.151130 0.0755650 0.997141i \(-0.475924\pi\)
0.0755650 + 0.997141i \(0.475924\pi\)
\(314\) 41.5967 2.34744
\(315\) 0 0
\(316\) −41.8328 −2.35328
\(317\) 24.5623 1.37956 0.689778 0.724021i \(-0.257708\pi\)
0.689778 + 0.724021i \(0.257708\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.70820 −0.486803
\(321\) 0 0
\(322\) 15.9443 0.888540
\(323\) −50.4508 −2.80716
\(324\) 0 0
\(325\) 12.9443 0.718019
\(326\) −12.3262 −0.682687
\(327\) 0 0
\(328\) −83.5410 −4.61278
\(329\) −4.38197 −0.241586
\(330\) 0 0
\(331\) −16.1803 −0.889352 −0.444676 0.895692i \(-0.646681\pi\)
−0.444676 + 0.895692i \(0.646681\pi\)
\(332\) −51.9787 −2.85270
\(333\) 0 0
\(334\) 6.47214 0.354140
\(335\) −7.32624 −0.400275
\(336\) 0 0
\(337\) −8.05573 −0.438823 −0.219412 0.975632i \(-0.570414\pi\)
−0.219412 + 0.975632i \(0.570414\pi\)
\(338\) 6.61803 0.359974
\(339\) 0 0
\(340\) 39.2705 2.12974
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 56.5066 3.04663
\(345\) 0 0
\(346\) 46.1246 2.47967
\(347\) 33.6525 1.80656 0.903280 0.429052i \(-0.141152\pi\)
0.903280 + 0.429052i \(0.141152\pi\)
\(348\) 0 0
\(349\) −8.20163 −0.439023 −0.219511 0.975610i \(-0.570446\pi\)
−0.219511 + 0.975610i \(0.570446\pi\)
\(350\) −10.4721 −0.559759
\(351\) 0 0
\(352\) 0 0
\(353\) −12.9098 −0.687121 −0.343560 0.939131i \(-0.611633\pi\)
−0.343560 + 0.939131i \(0.611633\pi\)
\(354\) 0 0
\(355\) −4.90983 −0.260587
\(356\) −0.708204 −0.0375347
\(357\) 0 0
\(358\) −21.5623 −1.13960
\(359\) −13.3820 −0.706273 −0.353137 0.935572i \(-0.614885\pi\)
−0.353137 + 0.935572i \(0.614885\pi\)
\(360\) 0 0
\(361\) 19.8885 1.04677
\(362\) −50.8328 −2.67171
\(363\) 0 0
\(364\) 15.7082 0.823334
\(365\) −9.76393 −0.511068
\(366\) 0 0
\(367\) −20.8328 −1.08746 −0.543732 0.839259i \(-0.682989\pi\)
−0.543732 + 0.839259i \(0.682989\pi\)
\(368\) 60.0132 3.12840
\(369\) 0 0
\(370\) −6.47214 −0.336470
\(371\) −4.61803 −0.239756
\(372\) 0 0
\(373\) 35.5623 1.84135 0.920673 0.390334i \(-0.127641\pi\)
0.920673 + 0.390334i \(0.127641\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −32.7426 −1.68857
\(377\) 7.70820 0.396993
\(378\) 0 0
\(379\) −26.0689 −1.33907 −0.669534 0.742781i \(-0.733506\pi\)
−0.669534 + 0.742781i \(0.733506\pi\)
\(380\) −30.2705 −1.55284
\(381\) 0 0
\(382\) 52.9787 2.71063
\(383\) −14.5967 −0.745859 −0.372929 0.927860i \(-0.621647\pi\)
−0.372929 + 0.927860i \(0.621647\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.854102 0.0434726
\(387\) 0 0
\(388\) 33.9787 1.72501
\(389\) 11.8197 0.599281 0.299640 0.954052i \(-0.403133\pi\)
0.299640 + 0.954052i \(0.403133\pi\)
\(390\) 0 0
\(391\) −49.2705 −2.49172
\(392\) −7.47214 −0.377400
\(393\) 0 0
\(394\) 46.3607 2.33562
\(395\) 8.61803 0.433620
\(396\) 0 0
\(397\) −23.1803 −1.16339 −0.581694 0.813408i \(-0.697610\pi\)
−0.581694 + 0.813408i \(0.697610\pi\)
\(398\) 9.85410 0.493941
\(399\) 0 0
\(400\) −39.4164 −1.97082
\(401\) 26.4721 1.32196 0.660978 0.750406i \(-0.270142\pi\)
0.660978 + 0.750406i \(0.270142\pi\)
\(402\) 0 0
\(403\) −0.763932 −0.0380542
\(404\) 47.5623 2.36631
\(405\) 0 0
\(406\) −6.23607 −0.309491
\(407\) 0 0
\(408\) 0 0
\(409\) 4.29180 0.212216 0.106108 0.994355i \(-0.466161\pi\)
0.106108 + 0.994355i \(0.466161\pi\)
\(410\) 29.2705 1.44557
\(411\) 0 0
\(412\) 10.4164 0.513180
\(413\) −0.0901699 −0.00443697
\(414\) 0 0
\(415\) 10.7082 0.525645
\(416\) 35.1246 1.72213
\(417\) 0 0
\(418\) 0 0
\(419\) 5.88854 0.287674 0.143837 0.989601i \(-0.454056\pi\)
0.143837 + 0.989601i \(0.454056\pi\)
\(420\) 0 0
\(421\) 5.23607 0.255190 0.127595 0.991826i \(-0.459274\pi\)
0.127595 + 0.991826i \(0.459274\pi\)
\(422\) 35.0344 1.70545
\(423\) 0 0
\(424\) −34.5066 −1.67579
\(425\) 32.3607 1.56972
\(426\) 0 0
\(427\) −5.38197 −0.260452
\(428\) −51.9787 −2.51249
\(429\) 0 0
\(430\) −19.7984 −0.954762
\(431\) −7.14590 −0.344206 −0.172103 0.985079i \(-0.555056\pi\)
−0.172103 + 0.985079i \(0.555056\pi\)
\(432\) 0 0
\(433\) −38.0344 −1.82782 −0.913909 0.405918i \(-0.866952\pi\)
−0.913909 + 0.405918i \(0.866952\pi\)
\(434\) 0.618034 0.0296666
\(435\) 0 0
\(436\) 50.8328 2.43445
\(437\) 37.9787 1.81677
\(438\) 0 0
\(439\) −5.61803 −0.268134 −0.134067 0.990972i \(-0.542804\pi\)
−0.134067 + 0.990972i \(0.542804\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −68.5410 −3.26016
\(443\) 8.09017 0.384376 0.192188 0.981358i \(-0.438442\pi\)
0.192188 + 0.981358i \(0.438442\pi\)
\(444\) 0 0
\(445\) 0.145898 0.00691623
\(446\) 70.7771 3.35139
\(447\) 0 0
\(448\) −8.70820 −0.411424
\(449\) 3.32624 0.156975 0.0784874 0.996915i \(-0.474991\pi\)
0.0784874 + 0.996915i \(0.474991\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −33.2705 −1.56491
\(453\) 0 0
\(454\) −34.1246 −1.60155
\(455\) −3.23607 −0.151709
\(456\) 0 0
\(457\) 6.85410 0.320621 0.160311 0.987067i \(-0.448750\pi\)
0.160311 + 0.987067i \(0.448750\pi\)
\(458\) −29.4164 −1.37454
\(459\) 0 0
\(460\) −29.5623 −1.37835
\(461\) 19.5066 0.908512 0.454256 0.890871i \(-0.349905\pi\)
0.454256 + 0.890871i \(0.349905\pi\)
\(462\) 0 0
\(463\) 17.7639 0.825560 0.412780 0.910831i \(-0.364558\pi\)
0.412780 + 0.910831i \(0.364558\pi\)
\(464\) −23.4721 −1.08967
\(465\) 0 0
\(466\) 6.76393 0.313333
\(467\) −33.5066 −1.55050 −0.775250 0.631655i \(-0.782376\pi\)
−0.775250 + 0.631655i \(0.782376\pi\)
\(468\) 0 0
\(469\) −7.32624 −0.338294
\(470\) 11.4721 0.529170
\(471\) 0 0
\(472\) −0.673762 −0.0310124
\(473\) 0 0
\(474\) 0 0
\(475\) −24.9443 −1.14452
\(476\) 39.2705 1.79996
\(477\) 0 0
\(478\) 15.4721 0.707679
\(479\) 26.2361 1.19876 0.599378 0.800466i \(-0.295415\pi\)
0.599378 + 0.800466i \(0.295415\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) −45.2148 −2.05948
\(483\) 0 0
\(484\) 0 0
\(485\) −7.00000 −0.317854
\(486\) 0 0
\(487\) 16.5967 0.752070 0.376035 0.926605i \(-0.377287\pi\)
0.376035 + 0.926605i \(0.377287\pi\)
\(488\) −40.2148 −1.82044
\(489\) 0 0
\(490\) 2.61803 0.118271
\(491\) 28.8541 1.30217 0.651084 0.759006i \(-0.274315\pi\)
0.651084 + 0.759006i \(0.274315\pi\)
\(492\) 0 0
\(493\) 19.2705 0.867900
\(494\) 52.8328 2.37706
\(495\) 0 0
\(496\) 2.32624 0.104451
\(497\) −4.90983 −0.220236
\(498\) 0 0
\(499\) 1.09017 0.0488027 0.0244014 0.999702i \(-0.492232\pi\)
0.0244014 + 0.999702i \(0.492232\pi\)
\(500\) 43.6869 1.95374
\(501\) 0 0
\(502\) 60.2148 2.68752
\(503\) 18.8328 0.839714 0.419857 0.907590i \(-0.362080\pi\)
0.419857 + 0.907590i \(0.362080\pi\)
\(504\) 0 0
\(505\) −9.79837 −0.436022
\(506\) 0 0
\(507\) 0 0
\(508\) 72.5410 3.21849
\(509\) −31.7426 −1.40697 −0.703484 0.710711i \(-0.748373\pi\)
−0.703484 + 0.710711i \(0.748373\pi\)
\(510\) 0 0
\(511\) −9.76393 −0.431931
\(512\) 40.3050 1.78124
\(513\) 0 0
\(514\) 30.2705 1.33517
\(515\) −2.14590 −0.0945596
\(516\) 0 0
\(517\) 0 0
\(518\) −6.47214 −0.284369
\(519\) 0 0
\(520\) −24.1803 −1.06038
\(521\) 7.76393 0.340144 0.170072 0.985432i \(-0.445600\pi\)
0.170072 + 0.985432i \(0.445600\pi\)
\(522\) 0 0
\(523\) 3.56231 0.155769 0.0778844 0.996962i \(-0.475183\pi\)
0.0778844 + 0.996962i \(0.475183\pi\)
\(524\) 5.12461 0.223870
\(525\) 0 0
\(526\) −60.5410 −2.63971
\(527\) −1.90983 −0.0831935
\(528\) 0 0
\(529\) 14.0902 0.612616
\(530\) 12.0902 0.525163
\(531\) 0 0
\(532\) −30.2705 −1.31239
\(533\) −36.1803 −1.56714
\(534\) 0 0
\(535\) 10.7082 0.462956
\(536\) −54.7426 −2.36452
\(537\) 0 0
\(538\) 26.5623 1.14518
\(539\) 0 0
\(540\) 0 0
\(541\) 12.7082 0.546368 0.273184 0.961962i \(-0.411923\pi\)
0.273184 + 0.961962i \(0.411923\pi\)
\(542\) 51.8328 2.22641
\(543\) 0 0
\(544\) 87.8115 3.76489
\(545\) −10.4721 −0.448577
\(546\) 0 0
\(547\) 35.2705 1.50806 0.754029 0.656841i \(-0.228108\pi\)
0.754029 + 0.656841i \(0.228108\pi\)
\(548\) −1.58359 −0.0676477
\(549\) 0 0
\(550\) 0 0
\(551\) −14.8541 −0.632806
\(552\) 0 0
\(553\) 8.61803 0.366476
\(554\) −15.7984 −0.671209
\(555\) 0 0
\(556\) 28.8541 1.22369
\(557\) −14.2361 −0.603202 −0.301601 0.953434i \(-0.597521\pi\)
−0.301601 + 0.953434i \(0.597521\pi\)
\(558\) 0 0
\(559\) 24.4721 1.03506
\(560\) 9.85410 0.416412
\(561\) 0 0
\(562\) 7.38197 0.311389
\(563\) 14.9787 0.631278 0.315639 0.948879i \(-0.397781\pi\)
0.315639 + 0.948879i \(0.397781\pi\)
\(564\) 0 0
\(565\) 6.85410 0.288354
\(566\) −15.5623 −0.654133
\(567\) 0 0
\(568\) −36.6869 −1.53935
\(569\) −25.3050 −1.06084 −0.530419 0.847735i \(-0.677966\pi\)
−0.530419 + 0.847735i \(0.677966\pi\)
\(570\) 0 0
\(571\) 28.3050 1.18453 0.592263 0.805745i \(-0.298235\pi\)
0.592263 + 0.805745i \(0.298235\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 29.2705 1.22173
\(575\) −24.3607 −1.01591
\(576\) 0 0
\(577\) −21.6525 −0.901404 −0.450702 0.892674i \(-0.648826\pi\)
−0.450702 + 0.892674i \(0.648826\pi\)
\(578\) −126.846 −5.27610
\(579\) 0 0
\(580\) 11.5623 0.480099
\(581\) 10.7082 0.444251
\(582\) 0 0
\(583\) 0 0
\(584\) −72.9574 −3.01900
\(585\) 0 0
\(586\) 28.7984 1.18965
\(587\) −1.00000 −0.0412744 −0.0206372 0.999787i \(-0.506569\pi\)
−0.0206372 + 0.999787i \(0.506569\pi\)
\(588\) 0 0
\(589\) 1.47214 0.0606583
\(590\) 0.236068 0.00971876
\(591\) 0 0
\(592\) −24.3607 −1.00122
\(593\) −29.1246 −1.19600 −0.598002 0.801494i \(-0.704039\pi\)
−0.598002 + 0.801494i \(0.704039\pi\)
\(594\) 0 0
\(595\) −8.09017 −0.331665
\(596\) −10.4164 −0.426673
\(597\) 0 0
\(598\) 51.5967 2.10995
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −0.291796 −0.0119026 −0.00595130 0.999982i \(-0.501894\pi\)
−0.00595130 + 0.999982i \(0.501894\pi\)
\(602\) −19.7984 −0.806921
\(603\) 0 0
\(604\) −87.1033 −3.54418
\(605\) 0 0
\(606\) 0 0
\(607\) −44.7984 −1.81831 −0.909155 0.416458i \(-0.863271\pi\)
−0.909155 + 0.416458i \(0.863271\pi\)
\(608\) −67.6869 −2.74507
\(609\) 0 0
\(610\) 14.0902 0.570494
\(611\) −14.1803 −0.573675
\(612\) 0 0
\(613\) −34.8885 −1.40914 −0.704568 0.709637i \(-0.748859\pi\)
−0.704568 + 0.709637i \(0.748859\pi\)
\(614\) −2.14590 −0.0866014
\(615\) 0 0
\(616\) 0 0
\(617\) 17.4164 0.701158 0.350579 0.936533i \(-0.385985\pi\)
0.350579 + 0.936533i \(0.385985\pi\)
\(618\) 0 0
\(619\) −26.2918 −1.05676 −0.528378 0.849009i \(-0.677200\pi\)
−0.528378 + 0.849009i \(0.677200\pi\)
\(620\) −1.14590 −0.0460204
\(621\) 0 0
\(622\) 23.5623 0.944762
\(623\) 0.145898 0.00584528
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −7.00000 −0.279776
\(627\) 0 0
\(628\) −77.1246 −3.07761
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −35.8328 −1.42648 −0.713241 0.700919i \(-0.752773\pi\)
−0.713241 + 0.700919i \(0.752773\pi\)
\(632\) 64.3951 2.56150
\(633\) 0 0
\(634\) −64.3050 −2.55388
\(635\) −14.9443 −0.593045
\(636\) 0 0
\(637\) −3.23607 −0.128218
\(638\) 0 0
\(639\) 0 0
\(640\) 1.09017 0.0430928
\(641\) −0.347524 −0.0137264 −0.00686319 0.999976i \(-0.502185\pi\)
−0.00686319 + 0.999976i \(0.502185\pi\)
\(642\) 0 0
\(643\) 28.4164 1.12063 0.560317 0.828278i \(-0.310679\pi\)
0.560317 + 0.828278i \(0.310679\pi\)
\(644\) −29.5623 −1.16492
\(645\) 0 0
\(646\) 132.082 5.19670
\(647\) 16.1803 0.636115 0.318057 0.948071i \(-0.396970\pi\)
0.318057 + 0.948071i \(0.396970\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −33.8885 −1.32922
\(651\) 0 0
\(652\) 22.8541 0.895036
\(653\) −13.7639 −0.538624 −0.269312 0.963053i \(-0.586796\pi\)
−0.269312 + 0.963053i \(0.586796\pi\)
\(654\) 0 0
\(655\) −1.05573 −0.0412507
\(656\) 110.172 4.30150
\(657\) 0 0
\(658\) 11.4721 0.447230
\(659\) 22.5279 0.877561 0.438780 0.898594i \(-0.355411\pi\)
0.438780 + 0.898594i \(0.355411\pi\)
\(660\) 0 0
\(661\) −14.4377 −0.561561 −0.280781 0.959772i \(-0.590593\pi\)
−0.280781 + 0.959772i \(0.590593\pi\)
\(662\) 42.3607 1.64639
\(663\) 0 0
\(664\) 80.0132 3.10511
\(665\) 6.23607 0.241824
\(666\) 0 0
\(667\) −14.5066 −0.561697
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 19.1803 0.741001
\(671\) 0 0
\(672\) 0 0
\(673\) 33.4164 1.28811 0.644054 0.764980i \(-0.277251\pi\)
0.644054 + 0.764980i \(0.277251\pi\)
\(674\) 21.0902 0.812363
\(675\) 0 0
\(676\) −12.2705 −0.471943
\(677\) −2.72949 −0.104903 −0.0524514 0.998623i \(-0.516703\pi\)
−0.0524514 + 0.998623i \(0.516703\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) −60.4508 −2.31818
\(681\) 0 0
\(682\) 0 0
\(683\) −38.5066 −1.47341 −0.736707 0.676213i \(-0.763620\pi\)
−0.736707 + 0.676213i \(0.763620\pi\)
\(684\) 0 0
\(685\) 0.326238 0.0124649
\(686\) 2.61803 0.0999570
\(687\) 0 0
\(688\) −74.5197 −2.84104
\(689\) −14.9443 −0.569331
\(690\) 0 0
\(691\) −48.5410 −1.84659 −0.923294 0.384095i \(-0.874514\pi\)
−0.923294 + 0.384095i \(0.874514\pi\)
\(692\) −85.5197 −3.25097
\(693\) 0 0
\(694\) −88.1033 −3.34436
\(695\) −5.94427 −0.225479
\(696\) 0 0
\(697\) −90.4508 −3.42607
\(698\) 21.4721 0.812732
\(699\) 0 0
\(700\) 19.4164 0.733871
\(701\) −19.5066 −0.736753 −0.368377 0.929677i \(-0.620086\pi\)
−0.368377 + 0.929677i \(0.620086\pi\)
\(702\) 0 0
\(703\) −15.4164 −0.581441
\(704\) 0 0
\(705\) 0 0
\(706\) 33.7984 1.27202
\(707\) −9.79837 −0.368506
\(708\) 0 0
\(709\) 22.0344 0.827521 0.413760 0.910386i \(-0.364215\pi\)
0.413760 + 0.910386i \(0.364215\pi\)
\(710\) 12.8541 0.482406
\(711\) 0 0
\(712\) 1.09017 0.0408558
\(713\) 1.43769 0.0538421
\(714\) 0 0
\(715\) 0 0
\(716\) 39.9787 1.49407
\(717\) 0 0
\(718\) 35.0344 1.30747
\(719\) −32.8885 −1.22654 −0.613268 0.789875i \(-0.710145\pi\)
−0.613268 + 0.789875i \(0.710145\pi\)
\(720\) 0 0
\(721\) −2.14590 −0.0799174
\(722\) −52.0689 −1.93780
\(723\) 0 0
\(724\) 94.2492 3.50274
\(725\) 9.52786 0.353856
\(726\) 0 0
\(727\) 43.4508 1.61150 0.805751 0.592254i \(-0.201762\pi\)
0.805751 + 0.592254i \(0.201762\pi\)
\(728\) −24.1803 −0.896183
\(729\) 0 0
\(730\) 25.5623 0.946103
\(731\) 61.1803 2.26284
\(732\) 0 0
\(733\) −35.0557 −1.29481 −0.647406 0.762145i \(-0.724146\pi\)
−0.647406 + 0.762145i \(0.724146\pi\)
\(734\) 54.5410 2.01315
\(735\) 0 0
\(736\) −66.1033 −2.43660
\(737\) 0 0
\(738\) 0 0
\(739\) 13.6180 0.500947 0.250474 0.968123i \(-0.419414\pi\)
0.250474 + 0.968123i \(0.419414\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 12.0902 0.443844
\(743\) −26.0902 −0.957156 −0.478578 0.878045i \(-0.658848\pi\)
−0.478578 + 0.878045i \(0.658848\pi\)
\(744\) 0 0
\(745\) 2.14590 0.0786196
\(746\) −93.1033 −3.40875
\(747\) 0 0
\(748\) 0 0
\(749\) 10.7082 0.391269
\(750\) 0 0
\(751\) −21.3050 −0.777429 −0.388714 0.921358i \(-0.627081\pi\)
−0.388714 + 0.921358i \(0.627081\pi\)
\(752\) 43.1803 1.57462
\(753\) 0 0
\(754\) −20.1803 −0.734925
\(755\) 17.9443 0.653059
\(756\) 0 0
\(757\) −19.2361 −0.699147 −0.349573 0.936909i \(-0.613673\pi\)
−0.349573 + 0.936909i \(0.613673\pi\)
\(758\) 68.2492 2.47892
\(759\) 0 0
\(760\) 46.5967 1.69024
\(761\) −47.3050 −1.71480 −0.857402 0.514648i \(-0.827923\pi\)
−0.857402 + 0.514648i \(0.827923\pi\)
\(762\) 0 0
\(763\) −10.4721 −0.379117
\(764\) −98.2279 −3.55376
\(765\) 0 0
\(766\) 38.2148 1.38076
\(767\) −0.291796 −0.0105361
\(768\) 0 0
\(769\) −28.4377 −1.02549 −0.512745 0.858541i \(-0.671371\pi\)
−0.512745 + 0.858541i \(0.671371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.58359 −0.0569947
\(773\) −35.7639 −1.28634 −0.643170 0.765724i \(-0.722381\pi\)
−0.643170 + 0.765724i \(0.722381\pi\)
\(774\) 0 0
\(775\) −0.944272 −0.0339192
\(776\) −52.3050 −1.87764
\(777\) 0 0
\(778\) −30.9443 −1.10941
\(779\) 69.7214 2.49803
\(780\) 0 0
\(781\) 0 0
\(782\) 128.992 4.61274
\(783\) 0 0
\(784\) 9.85410 0.351932
\(785\) 15.8885 0.567086
\(786\) 0 0
\(787\) −0.437694 −0.0156021 −0.00780105 0.999970i \(-0.502483\pi\)
−0.00780105 + 0.999970i \(0.502483\pi\)
\(788\) −85.9574 −3.06211
\(789\) 0 0
\(790\) −22.5623 −0.802731
\(791\) 6.85410 0.243704
\(792\) 0 0
\(793\) −17.4164 −0.618475
\(794\) 60.6869 2.15370
\(795\) 0 0
\(796\) −18.2705 −0.647581
\(797\) −27.7639 −0.983449 −0.491724 0.870751i \(-0.663633\pi\)
−0.491724 + 0.870751i \(0.663633\pi\)
\(798\) 0 0
\(799\) −35.4508 −1.25416
\(800\) 43.4164 1.53500
\(801\) 0 0
\(802\) −69.3050 −2.44724
\(803\) 0 0
\(804\) 0 0
\(805\) 6.09017 0.214650
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) −73.2148 −2.57569
\(809\) −5.50658 −0.193601 −0.0968005 0.995304i \(-0.530861\pi\)
−0.0968005 + 0.995304i \(0.530861\pi\)
\(810\) 0 0
\(811\) −54.8328 −1.92544 −0.962720 0.270499i \(-0.912811\pi\)
−0.962720 + 0.270499i \(0.912811\pi\)
\(812\) 11.5623 0.405757
\(813\) 0 0
\(814\) 0 0
\(815\) −4.70820 −0.164921
\(816\) 0 0
\(817\) −47.1591 −1.64989
\(818\) −11.2361 −0.392860
\(819\) 0 0
\(820\) −54.2705 −1.89521
\(821\) −50.6656 −1.76824 −0.884121 0.467257i \(-0.845242\pi\)
−0.884121 + 0.467257i \(0.845242\pi\)
\(822\) 0 0
\(823\) 3.88854 0.135546 0.0677731 0.997701i \(-0.478411\pi\)
0.0677731 + 0.997701i \(0.478411\pi\)
\(824\) −16.0344 −0.558586
\(825\) 0 0
\(826\) 0.236068 0.00821386
\(827\) 29.5967 1.02918 0.514590 0.857436i \(-0.327944\pi\)
0.514590 + 0.857436i \(0.327944\pi\)
\(828\) 0 0
\(829\) −1.02129 −0.0354707 −0.0177354 0.999843i \(-0.505646\pi\)
−0.0177354 + 0.999843i \(0.505646\pi\)
\(830\) −28.0344 −0.973090
\(831\) 0 0
\(832\) −28.1803 −0.976978
\(833\) −8.09017 −0.280308
\(834\) 0 0
\(835\) 2.47214 0.0855518
\(836\) 0 0
\(837\) 0 0
\(838\) −15.4164 −0.532551
\(839\) 18.5967 0.642031 0.321016 0.947074i \(-0.395976\pi\)
0.321016 + 0.947074i \(0.395976\pi\)
\(840\) 0 0
\(841\) −23.3262 −0.804353
\(842\) −13.7082 −0.472416
\(843\) 0 0
\(844\) −64.9574 −2.23593
\(845\) 2.52786 0.0869612
\(846\) 0 0
\(847\) 0 0
\(848\) 45.5066 1.56270
\(849\) 0 0
\(850\) −84.7214 −2.90592
\(851\) −15.0557 −0.516104
\(852\) 0 0
\(853\) 46.0902 1.57810 0.789049 0.614331i \(-0.210574\pi\)
0.789049 + 0.614331i \(0.210574\pi\)
\(854\) 14.0902 0.482156
\(855\) 0 0
\(856\) 80.0132 2.73479
\(857\) −3.90983 −0.133557 −0.0667786 0.997768i \(-0.521272\pi\)
−0.0667786 + 0.997768i \(0.521272\pi\)
\(858\) 0 0
\(859\) 1.43769 0.0490535 0.0245267 0.999699i \(-0.492192\pi\)
0.0245267 + 0.999699i \(0.492192\pi\)
\(860\) 36.7082 1.25174
\(861\) 0 0
\(862\) 18.7082 0.637204
\(863\) 30.0557 1.02311 0.511554 0.859251i \(-0.329070\pi\)
0.511554 + 0.859251i \(0.329070\pi\)
\(864\) 0 0
\(865\) 17.6180 0.599031
\(866\) 99.5755 3.38371
\(867\) 0 0
\(868\) −1.14590 −0.0388943
\(869\) 0 0
\(870\) 0 0
\(871\) −23.7082 −0.803322
\(872\) −78.2492 −2.64985
\(873\) 0 0
\(874\) −99.4296 −3.36326
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 14.7082 0.496378
\(879\) 0 0
\(880\) 0 0
\(881\) −20.8541 −0.702593 −0.351296 0.936264i \(-0.614259\pi\)
−0.351296 + 0.936264i \(0.614259\pi\)
\(882\) 0 0
\(883\) −46.6312 −1.56926 −0.784632 0.619961i \(-0.787148\pi\)
−0.784632 + 0.619961i \(0.787148\pi\)
\(884\) 127.082 4.27423
\(885\) 0 0
\(886\) −21.1803 −0.711567
\(887\) 5.78522 0.194249 0.0971243 0.995272i \(-0.469036\pi\)
0.0971243 + 0.995272i \(0.469036\pi\)
\(888\) 0 0
\(889\) −14.9443 −0.501215
\(890\) −0.381966 −0.0128035
\(891\) 0 0
\(892\) −131.228 −4.39384
\(893\) 27.3262 0.914438
\(894\) 0 0
\(895\) −8.23607 −0.275301
\(896\) 1.09017 0.0364200
\(897\) 0 0
\(898\) −8.70820 −0.290597
\(899\) −0.562306 −0.0187540
\(900\) 0 0
\(901\) −37.3607 −1.24466
\(902\) 0 0
\(903\) 0 0
\(904\) 51.2148 1.70338
\(905\) −19.4164 −0.645423
\(906\) 0 0
\(907\) −20.2361 −0.671928 −0.335964 0.941875i \(-0.609062\pi\)
−0.335964 + 0.941875i \(0.609062\pi\)
\(908\) 63.2705 2.09971
\(909\) 0 0
\(910\) 8.47214 0.280849
\(911\) 22.8197 0.756049 0.378025 0.925796i \(-0.376603\pi\)
0.378025 + 0.925796i \(0.376603\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −17.9443 −0.593544
\(915\) 0 0
\(916\) 54.5410 1.80209
\(917\) −1.05573 −0.0348632
\(918\) 0 0
\(919\) −20.8885 −0.689049 −0.344525 0.938777i \(-0.611960\pi\)
−0.344525 + 0.938777i \(0.611960\pi\)
\(920\) 45.5066 1.50031
\(921\) 0 0
\(922\) −51.0689 −1.68186
\(923\) −15.8885 −0.522978
\(924\) 0 0
\(925\) 9.88854 0.325133
\(926\) −46.5066 −1.52830
\(927\) 0 0
\(928\) 25.8541 0.848702
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) 6.23607 0.204379
\(932\) −12.5410 −0.410795
\(933\) 0 0
\(934\) 87.7214 2.87033
\(935\) 0 0
\(936\) 0 0
\(937\) −37.7214 −1.23230 −0.616152 0.787628i \(-0.711309\pi\)
−0.616152 + 0.787628i \(0.711309\pi\)
\(938\) 19.1803 0.626260
\(939\) 0 0
\(940\) −21.2705 −0.693768
\(941\) −31.0689 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(942\) 0 0
\(943\) 68.0902 2.21732
\(944\) 0.888544 0.0289196
\(945\) 0 0
\(946\) 0 0
\(947\) −19.7082 −0.640431 −0.320215 0.947345i \(-0.603755\pi\)
−0.320215 + 0.947345i \(0.603755\pi\)
\(948\) 0 0
\(949\) −31.5967 −1.02567
\(950\) 65.3050 2.11877
\(951\) 0 0
\(952\) −60.4508 −1.95922
\(953\) −1.76393 −0.0571394 −0.0285697 0.999592i \(-0.509095\pi\)
−0.0285697 + 0.999592i \(0.509095\pi\)
\(954\) 0 0
\(955\) 20.2361 0.654824
\(956\) −28.6869 −0.927801
\(957\) 0 0
\(958\) −68.6869 −2.21917
\(959\) 0.326238 0.0105348
\(960\) 0 0
\(961\) −30.9443 −0.998202
\(962\) −20.9443 −0.675270
\(963\) 0 0
\(964\) 83.8328 2.70007
\(965\) 0.326238 0.0105020
\(966\) 0 0
\(967\) 28.4508 0.914918 0.457459 0.889231i \(-0.348760\pi\)
0.457459 + 0.889231i \(0.348760\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 18.3262 0.588420
\(971\) −52.2492 −1.67676 −0.838379 0.545088i \(-0.816496\pi\)
−0.838379 + 0.545088i \(0.816496\pi\)
\(972\) 0 0
\(973\) −5.94427 −0.190565
\(974\) −43.4508 −1.39226
\(975\) 0 0
\(976\) 53.0344 1.69759
\(977\) 33.1803 1.06153 0.530767 0.847518i \(-0.321904\pi\)
0.530767 + 0.847518i \(0.321904\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −4.85410 −0.155059
\(981\) 0 0
\(982\) −75.5410 −2.41061
\(983\) 14.6180 0.466243 0.233121 0.972448i \(-0.425106\pi\)
0.233121 + 0.972448i \(0.425106\pi\)
\(984\) 0 0
\(985\) 17.7082 0.564230
\(986\) −50.4508 −1.60668
\(987\) 0 0
\(988\) −97.9574 −3.11644
\(989\) −46.0557 −1.46449
\(990\) 0 0
\(991\) −34.2705 −1.08864 −0.544319 0.838878i \(-0.683212\pi\)
−0.544319 + 0.838878i \(0.683212\pi\)
\(992\) −2.56231 −0.0813533
\(993\) 0 0
\(994\) 12.8541 0.407707
\(995\) 3.76393 0.119325
\(996\) 0 0
\(997\) −26.8673 −0.850895 −0.425447 0.904983i \(-0.639883\pi\)
−0.425447 + 0.904983i \(0.639883\pi\)
\(998\) −2.85410 −0.0903450
\(999\) 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.t.1.1 2
3.2 odd 2 847.2.a.h.1.2 yes 2
11.10 odd 2 7623.2.a.bx.1.2 2
21.20 even 2 5929.2.a.s.1.2 2
33.2 even 10 847.2.f.l.323.1 4
33.5 odd 10 847.2.f.c.729.1 4
33.8 even 10 847.2.f.d.372.1 4
33.14 odd 10 847.2.f.j.372.1 4
33.17 even 10 847.2.f.l.729.1 4
33.20 odd 10 847.2.f.c.323.1 4
33.26 odd 10 847.2.f.j.148.1 4
33.29 even 10 847.2.f.d.148.1 4
33.32 even 2 847.2.a.d.1.1 2
231.230 odd 2 5929.2.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.d.1.1 2 33.32 even 2
847.2.a.h.1.2 yes 2 3.2 odd 2
847.2.f.c.323.1 4 33.20 odd 10
847.2.f.c.729.1 4 33.5 odd 10
847.2.f.d.148.1 4 33.29 even 10
847.2.f.d.372.1 4 33.8 even 10
847.2.f.j.148.1 4 33.26 odd 10
847.2.f.j.372.1 4 33.14 odd 10
847.2.f.l.323.1 4 33.2 even 10
847.2.f.l.729.1 4 33.17 even 10
5929.2.a.i.1.1 2 231.230 odd 2
5929.2.a.s.1.2 2 21.20 even 2
7623.2.a.t.1.1 2 1.1 even 1 trivial
7623.2.a.bx.1.2 2 11.10 odd 2