Properties

Label 7623.2.a.bm.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 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} -1.61803 q^{10} +5.47214 q^{13} -1.61803 q^{14} -4.85410 q^{16} +0.763932 q^{17} -6.70820 q^{19} -0.618034 q^{20} +7.70820 q^{23} -4.00000 q^{25} +8.85410 q^{26} -0.618034 q^{28} +5.00000 q^{29} -0.763932 q^{31} -3.38197 q^{32} +1.23607 q^{34} +1.00000 q^{35} -7.00000 q^{37} -10.8541 q^{38} +2.23607 q^{40} +6.47214 q^{41} +7.70820 q^{43} +12.4721 q^{46} +4.23607 q^{47} +1.00000 q^{49} -6.47214 q^{50} +3.38197 q^{52} -10.1803 q^{53} +2.23607 q^{56} +8.09017 q^{58} -11.1803 q^{59} -2.00000 q^{61} -1.23607 q^{62} +4.23607 q^{64} -5.47214 q^{65} -14.2361 q^{67} +0.472136 q^{68} +1.61803 q^{70} -6.47214 q^{71} -13.4721 q^{73} -11.3262 q^{74} -4.14590 q^{76} +5.52786 q^{79} +4.85410 q^{80} +10.4721 q^{82} +11.2361 q^{83} -0.763932 q^{85} +12.4721 q^{86} -4.47214 q^{89} -5.47214 q^{91} +4.76393 q^{92} +6.85410 q^{94} +6.70820 q^{95} -3.70820 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 2q^{5} - 2q^{7} - q^{10} + 2q^{13} - q^{14} - 3q^{16} + 6q^{17} + q^{20} + 2q^{23} - 8q^{25} + 11q^{26} + q^{28} + 10q^{29} - 6q^{31} - 9q^{32} - 2q^{34} + 2q^{35} - 14q^{37} - 15q^{38} + 4q^{41} + 2q^{43} + 16q^{46} + 4q^{47} + 2q^{49} - 4q^{50} + 9q^{52} + 2q^{53} + 5q^{58} - 4q^{61} + 2q^{62} + 4q^{64} - 2q^{65} - 24q^{67} - 8q^{68} + q^{70} - 4q^{71} - 18q^{73} - 7q^{74} - 15q^{76} + 20q^{79} + 3q^{80} + 12q^{82} + 18q^{83} - 6q^{85} + 16q^{86} - 2q^{91} + 14q^{92} + 7q^{94} + 6q^{97} + q^{98} + O(q^{100}) \)

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
\(5\) −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
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) −1.61803 −0.511667
\(11\) 0 0
\(12\) 0 0
\(13\) 5.47214 1.51770 0.758849 0.651267i \(-0.225762\pi\)
0.758849 + 0.651267i \(0.225762\pi\)
\(14\) −1.61803 −0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) 0 0
\(19\) −6.70820 −1.53897 −0.769484 0.638666i \(-0.779486\pi\)
−0.769484 + 0.638666i \(0.779486\pi\)
\(20\) −0.618034 −0.138197
\(21\) 0 0
\(22\) 0 0
\(23\) 7.70820 1.60727 0.803636 0.595121i \(-0.202896\pi\)
0.803636 + 0.595121i \(0.202896\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 8.85410 1.73643
\(27\) 0 0
\(28\) −0.618034 −0.116797
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −0.763932 −0.137206 −0.0686031 0.997644i \(-0.521854\pi\)
−0.0686031 + 0.997644i \(0.521854\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) 1.23607 0.211984
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −10.8541 −1.76077
\(39\) 0 0
\(40\) 2.23607 0.353553
\(41\) 6.47214 1.01078 0.505389 0.862892i \(-0.331349\pi\)
0.505389 + 0.862892i \(0.331349\pi\)
\(42\) 0 0
\(43\) 7.70820 1.17549 0.587745 0.809046i \(-0.300016\pi\)
0.587745 + 0.809046i \(0.300016\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.4721 1.83892
\(47\) 4.23607 0.617894 0.308947 0.951079i \(-0.400023\pi\)
0.308947 + 0.951079i \(0.400023\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −6.47214 −0.915298
\(51\) 0 0
\(52\) 3.38197 0.468994
\(53\) −10.1803 −1.39838 −0.699189 0.714937i \(-0.746455\pi\)
−0.699189 + 0.714937i \(0.746455\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 0 0
\(58\) 8.09017 1.06229
\(59\) −11.1803 −1.45556 −0.727778 0.685813i \(-0.759447\pi\)
−0.727778 + 0.685813i \(0.759447\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −1.23607 −0.156981
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) −5.47214 −0.678735
\(66\) 0 0
\(67\) −14.2361 −1.73921 −0.869606 0.493746i \(-0.835627\pi\)
−0.869606 + 0.493746i \(0.835627\pi\)
\(68\) 0.472136 0.0572549
\(69\) 0 0
\(70\) 1.61803 0.193392
\(71\) −6.47214 −0.768101 −0.384051 0.923312i \(-0.625471\pi\)
−0.384051 + 0.923312i \(0.625471\pi\)
\(72\) 0 0
\(73\) −13.4721 −1.57679 −0.788397 0.615167i \(-0.789089\pi\)
−0.788397 + 0.615167i \(0.789089\pi\)
\(74\) −11.3262 −1.31665
\(75\) 0 0
\(76\) −4.14590 −0.475567
\(77\) 0 0
\(78\) 0 0
\(79\) 5.52786 0.621933 0.310967 0.950421i \(-0.399347\pi\)
0.310967 + 0.950421i \(0.399347\pi\)
\(80\) 4.85410 0.542705
\(81\) 0 0
\(82\) 10.4721 1.15645
\(83\) 11.2361 1.23332 0.616659 0.787230i \(-0.288486\pi\)
0.616659 + 0.787230i \(0.288486\pi\)
\(84\) 0 0
\(85\) −0.763932 −0.0828601
\(86\) 12.4721 1.34491
\(87\) 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
\(91\) −5.47214 −0.573636
\(92\) 4.76393 0.496674
\(93\) 0 0
\(94\) 6.85410 0.706947
\(95\) 6.70820 0.688247
\(96\) 0 0
\(97\) −3.70820 −0.376511 −0.188256 0.982120i \(-0.560283\pi\)
−0.188256 + 0.982120i \(0.560283\pi\)
\(98\) 1.61803 0.163446
\(99\) 0 0
\(100\) −2.47214 −0.247214
\(101\) −4.18034 −0.415959 −0.207980 0.978133i \(-0.566689\pi\)
−0.207980 + 0.978133i \(0.566689\pi\)
\(102\) 0 0
\(103\) −9.41641 −0.927826 −0.463913 0.885881i \(-0.653555\pi\)
−0.463913 + 0.885881i \(0.653555\pi\)
\(104\) −12.2361 −1.19985
\(105\) 0 0
\(106\) −16.4721 −1.59992
\(107\) 0.236068 0.0228216 0.0114108 0.999935i \(-0.496368\pi\)
0.0114108 + 0.999935i \(0.496368\pi\)
\(108\) 0 0
\(109\) −7.23607 −0.693090 −0.346545 0.938033i \(-0.612645\pi\)
−0.346545 + 0.938033i \(0.612645\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.85410 0.458670
\(113\) −8.47214 −0.796992 −0.398496 0.917170i \(-0.630468\pi\)
−0.398496 + 0.917170i \(0.630468\pi\)
\(114\) 0 0
\(115\) −7.70820 −0.718794
\(116\) 3.09017 0.286915
\(117\) 0 0
\(118\) −18.0902 −1.66534
\(119\) −0.763932 −0.0700295
\(120\) 0 0
\(121\) 0 0
\(122\) −3.23607 −0.292980
\(123\) 0 0
\(124\) −0.472136 −0.0423991
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −18.6525 −1.65514 −0.827570 0.561363i \(-0.810277\pi\)
−0.827570 + 0.561363i \(0.810277\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) −8.85410 −0.776556
\(131\) −16.9443 −1.48043 −0.740214 0.672371i \(-0.765276\pi\)
−0.740214 + 0.672371i \(0.765276\pi\)
\(132\) 0 0
\(133\) 6.70820 0.581675
\(134\) −23.0344 −1.98987
\(135\) 0 0
\(136\) −1.70820 −0.146477
\(137\) −6.29180 −0.537544 −0.268772 0.963204i \(-0.586618\pi\)
−0.268772 + 0.963204i \(0.586618\pi\)
\(138\) 0 0
\(139\) −5.52786 −0.468867 −0.234434 0.972132i \(-0.575324\pi\)
−0.234434 + 0.972132i \(0.575324\pi\)
\(140\) 0.618034 0.0522334
\(141\) 0 0
\(142\) −10.4721 −0.878802
\(143\) 0 0
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) −21.7984 −1.80405
\(147\) 0 0
\(148\) −4.32624 −0.355615
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 0 0
\(151\) −8.18034 −0.665707 −0.332853 0.942979i \(-0.608011\pi\)
−0.332853 + 0.942979i \(0.608011\pi\)
\(152\) 15.0000 1.21666
\(153\) 0 0
\(154\) 0 0
\(155\) 0.763932 0.0613605
\(156\) 0 0
\(157\) 11.4164 0.911129 0.455564 0.890203i \(-0.349438\pi\)
0.455564 + 0.890203i \(0.349438\pi\)
\(158\) 8.94427 0.711568
\(159\) 0 0
\(160\) 3.38197 0.267368
\(161\) −7.70820 −0.607492
\(162\) 0 0
\(163\) −9.29180 −0.727790 −0.363895 0.931440i \(-0.618553\pi\)
−0.363895 + 0.931440i \(0.618553\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) 18.1803 1.41107
\(167\) 8.65248 0.669549 0.334774 0.942298i \(-0.391340\pi\)
0.334774 + 0.942298i \(0.391340\pi\)
\(168\) 0 0
\(169\) 16.9443 1.30341
\(170\) −1.23607 −0.0948021
\(171\) 0 0
\(172\) 4.76393 0.363246
\(173\) −10.4721 −0.796182 −0.398091 0.917346i \(-0.630327\pi\)
−0.398091 + 0.917346i \(0.630327\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −7.23607 −0.542366
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) 0 0
\(181\) −5.23607 −0.389194 −0.194597 0.980883i \(-0.562340\pi\)
−0.194597 + 0.980883i \(0.562340\pi\)
\(182\) −8.85410 −0.656310
\(183\) 0 0
\(184\) −17.2361 −1.27066
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 0 0
\(188\) 2.61803 0.190940
\(189\) 0 0
\(190\) 10.8541 0.787439
\(191\) 15.2361 1.10244 0.551222 0.834359i \(-0.314162\pi\)
0.551222 + 0.834359i \(0.314162\pi\)
\(192\) 0 0
\(193\) 16.6525 1.19867 0.599336 0.800498i \(-0.295431\pi\)
0.599336 + 0.800498i \(0.295431\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 0.618034 0.0441453
\(197\) −7.52786 −0.536338 −0.268169 0.963372i \(-0.586419\pi\)
−0.268169 + 0.963372i \(0.586419\pi\)
\(198\) 0 0
\(199\) −26.1803 −1.85588 −0.927938 0.372736i \(-0.878420\pi\)
−0.927938 + 0.372736i \(0.878420\pi\)
\(200\) 8.94427 0.632456
\(201\) 0 0
\(202\) −6.76393 −0.475909
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) −6.47214 −0.452034
\(206\) −15.2361 −1.06155
\(207\) 0 0
\(208\) −26.5623 −1.84176
\(209\) 0 0
\(210\) 0 0
\(211\) 21.4164 1.47437 0.737183 0.675693i \(-0.236155\pi\)
0.737183 + 0.675693i \(0.236155\pi\)
\(212\) −6.29180 −0.432122
\(213\) 0 0
\(214\) 0.381966 0.0261107
\(215\) −7.70820 −0.525695
\(216\) 0 0
\(217\) 0.763932 0.0518591
\(218\) −11.7082 −0.792980
\(219\) 0 0
\(220\) 0 0
\(221\) 4.18034 0.281200
\(222\) 0 0
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) 3.38197 0.225967
\(225\) 0 0
\(226\) −13.7082 −0.911856
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) 2.76393 0.182646 0.0913229 0.995821i \(-0.470890\pi\)
0.0913229 + 0.995821i \(0.470890\pi\)
\(230\) −12.4721 −0.822388
\(231\) 0 0
\(232\) −11.1803 −0.734025
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) 0 0
\(235\) −4.23607 −0.276331
\(236\) −6.90983 −0.449792
\(237\) 0 0
\(238\) −1.23607 −0.0801224
\(239\) −30.1246 −1.94860 −0.974300 0.225256i \(-0.927678\pi\)
−0.974300 + 0.225256i \(0.927678\pi\)
\(240\) 0 0
\(241\) −8.05573 −0.518915 −0.259458 0.965755i \(-0.583544\pi\)
−0.259458 + 0.965755i \(0.583544\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.23607 −0.0791311
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −36.7082 −2.33569
\(248\) 1.70820 0.108471
\(249\) 0 0
\(250\) 14.5623 0.921001
\(251\) 28.1246 1.77521 0.887605 0.460606i \(-0.152368\pi\)
0.887605 + 0.460606i \(0.152368\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −30.1803 −1.89368
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) −3.38197 −0.209741
\(261\) 0 0
\(262\) −27.4164 −1.69379
\(263\) 14.1246 0.870961 0.435480 0.900198i \(-0.356579\pi\)
0.435480 + 0.900198i \(0.356579\pi\)
\(264\) 0 0
\(265\) 10.1803 0.625373
\(266\) 10.8541 0.665508
\(267\) 0 0
\(268\) −8.79837 −0.537446
\(269\) −18.9443 −1.15505 −0.577526 0.816372i \(-0.695982\pi\)
−0.577526 + 0.816372i \(0.695982\pi\)
\(270\) 0 0
\(271\) −18.7082 −1.13644 −0.568221 0.822876i \(-0.692368\pi\)
−0.568221 + 0.822876i \(0.692368\pi\)
\(272\) −3.70820 −0.224843
\(273\) 0 0
\(274\) −10.1803 −0.615017
\(275\) 0 0
\(276\) 0 0
\(277\) −2.47214 −0.148536 −0.0742681 0.997238i \(-0.523662\pi\)
−0.0742681 + 0.997238i \(0.523662\pi\)
\(278\) −8.94427 −0.536442
\(279\) 0 0
\(280\) −2.23607 −0.133631
\(281\) 2.52786 0.150800 0.0753999 0.997153i \(-0.475977\pi\)
0.0753999 + 0.997153i \(0.475977\pi\)
\(282\) 0 0
\(283\) 18.2361 1.08402 0.542011 0.840371i \(-0.317663\pi\)
0.542011 + 0.840371i \(0.317663\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) −6.47214 −0.382038
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) −8.09017 −0.475071
\(291\) 0 0
\(292\) −8.32624 −0.487256
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 0 0
\(295\) 11.1803 0.650945
\(296\) 15.6525 0.909782
\(297\) 0 0
\(298\) 8.09017 0.468651
\(299\) 42.1803 2.43935
\(300\) 0 0
\(301\) −7.70820 −0.444293
\(302\) −13.2361 −0.761650
\(303\) 0 0
\(304\) 32.5623 1.86758
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 3.05573 0.174400 0.0871998 0.996191i \(-0.472208\pi\)
0.0871998 + 0.996191i \(0.472208\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.23607 0.0702039
\(311\) 25.8885 1.46800 0.734002 0.679147i \(-0.237650\pi\)
0.734002 + 0.679147i \(0.237650\pi\)
\(312\) 0 0
\(313\) −6.65248 −0.376020 −0.188010 0.982167i \(-0.560204\pi\)
−0.188010 + 0.982167i \(0.560204\pi\)
\(314\) 18.4721 1.04244
\(315\) 0 0
\(316\) 3.41641 0.192188
\(317\) −1.81966 −0.102202 −0.0511011 0.998693i \(-0.516273\pi\)
−0.0511011 + 0.998693i \(0.516273\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −4.23607 −0.236803
\(321\) 0 0
\(322\) −12.4721 −0.695045
\(323\) −5.12461 −0.285141
\(324\) 0 0
\(325\) −21.8885 −1.21416
\(326\) −15.0344 −0.832681
\(327\) 0 0
\(328\) −14.4721 −0.799090
\(329\) −4.23607 −0.233542
\(330\) 0 0
\(331\) 15.4164 0.847362 0.423681 0.905811i \(-0.360738\pi\)
0.423681 + 0.905811i \(0.360738\pi\)
\(332\) 6.94427 0.381116
\(333\) 0 0
\(334\) 14.0000 0.766046
\(335\) 14.2361 0.777799
\(336\) 0 0
\(337\) −14.1803 −0.772452 −0.386226 0.922404i \(-0.626222\pi\)
−0.386226 + 0.922404i \(0.626222\pi\)
\(338\) 27.4164 1.49126
\(339\) 0 0
\(340\) −0.472136 −0.0256052
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −17.2361 −0.929307
\(345\) 0 0
\(346\) −16.9443 −0.910930
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −28.4164 −1.52110 −0.760548 0.649282i \(-0.775069\pi\)
−0.760548 + 0.649282i \(0.775069\pi\)
\(350\) 6.47214 0.345950
\(351\) 0 0
\(352\) 0 0
\(353\) −33.4721 −1.78154 −0.890771 0.454452i \(-0.849835\pi\)
−0.890771 + 0.454452i \(0.849835\pi\)
\(354\) 0 0
\(355\) 6.47214 0.343505
\(356\) −2.76393 −0.146488
\(357\) 0 0
\(358\) 14.4721 0.764876
\(359\) −3.41641 −0.180311 −0.0901556 0.995928i \(-0.528736\pi\)
−0.0901556 + 0.995928i \(0.528736\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) −8.47214 −0.445286
\(363\) 0 0
\(364\) −3.38197 −0.177263
\(365\) 13.4721 0.705164
\(366\) 0 0
\(367\) 15.8885 0.829375 0.414688 0.909964i \(-0.363891\pi\)
0.414688 + 0.909964i \(0.363891\pi\)
\(368\) −37.4164 −1.95047
\(369\) 0 0
\(370\) 11.3262 0.588823
\(371\) 10.1803 0.528537
\(372\) 0 0
\(373\) 26.6525 1.38001 0.690006 0.723803i \(-0.257608\pi\)
0.690006 + 0.723803i \(0.257608\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.47214 −0.488488
\(377\) 27.3607 1.40915
\(378\) 0 0
\(379\) −8.81966 −0.453036 −0.226518 0.974007i \(-0.572734\pi\)
−0.226518 + 0.974007i \(0.572734\pi\)
\(380\) 4.14590 0.212680
\(381\) 0 0
\(382\) 24.6525 1.26133
\(383\) −15.0557 −0.769312 −0.384656 0.923060i \(-0.625680\pi\)
−0.384656 + 0.923060i \(0.625680\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 26.9443 1.37143
\(387\) 0 0
\(388\) −2.29180 −0.116348
\(389\) −28.9443 −1.46753 −0.733766 0.679402i \(-0.762239\pi\)
−0.733766 + 0.679402i \(0.762239\pi\)
\(390\) 0 0
\(391\) 5.88854 0.297796
\(392\) −2.23607 −0.112938
\(393\) 0 0
\(394\) −12.1803 −0.613637
\(395\) −5.52786 −0.278137
\(396\) 0 0
\(397\) −17.1246 −0.859460 −0.429730 0.902958i \(-0.641391\pi\)
−0.429730 + 0.902958i \(0.641391\pi\)
\(398\) −42.3607 −2.12335
\(399\) 0 0
\(400\) 19.4164 0.970820
\(401\) 16.2918 0.813573 0.406787 0.913523i \(-0.366649\pi\)
0.406787 + 0.913523i \(0.366649\pi\)
\(402\) 0 0
\(403\) −4.18034 −0.208238
\(404\) −2.58359 −0.128539
\(405\) 0 0
\(406\) −8.09017 −0.401508
\(407\) 0 0
\(408\) 0 0
\(409\) −38.9443 −1.92567 −0.962835 0.270090i \(-0.912947\pi\)
−0.962835 + 0.270090i \(0.912947\pi\)
\(410\) −10.4721 −0.517182
\(411\) 0 0
\(412\) −5.81966 −0.286714
\(413\) 11.1803 0.550149
\(414\) 0 0
\(415\) −11.2361 −0.551557
\(416\) −18.5066 −0.907360
\(417\) 0 0
\(418\) 0 0
\(419\) −21.1803 −1.03473 −0.517364 0.855766i \(-0.673087\pi\)
−0.517364 + 0.855766i \(0.673087\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 34.6525 1.68686
\(423\) 0 0
\(424\) 22.7639 1.10551
\(425\) −3.05573 −0.148225
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0.145898 0.00705225
\(429\) 0 0
\(430\) −12.4721 −0.601460
\(431\) −4.70820 −0.226786 −0.113393 0.993550i \(-0.536172\pi\)
−0.113393 + 0.993550i \(0.536172\pi\)
\(432\) 0 0
\(433\) −1.52786 −0.0734245 −0.0367122 0.999326i \(-0.511688\pi\)
−0.0367122 + 0.999326i \(0.511688\pi\)
\(434\) 1.23607 0.0593332
\(435\) 0 0
\(436\) −4.47214 −0.214176
\(437\) −51.7082 −2.47354
\(438\) 0 0
\(439\) 11.1803 0.533609 0.266804 0.963751i \(-0.414032\pi\)
0.266804 + 0.963751i \(0.414032\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.76393 0.321727
\(443\) −9.52786 −0.452682 −0.226341 0.974048i \(-0.572676\pi\)
−0.226341 + 0.974048i \(0.572676\pi\)
\(444\) 0 0
\(445\) 4.47214 0.212000
\(446\) −9.70820 −0.459697
\(447\) 0 0
\(448\) −4.23607 −0.200135
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −5.23607 −0.246284
\(453\) 0 0
\(454\) −3.23607 −0.151876
\(455\) 5.47214 0.256538
\(456\) 0 0
\(457\) −10.7639 −0.503516 −0.251758 0.967790i \(-0.581009\pi\)
−0.251758 + 0.967790i \(0.581009\pi\)
\(458\) 4.47214 0.208969
\(459\) 0 0
\(460\) −4.76393 −0.222119
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) −27.1803 −1.26318 −0.631589 0.775304i \(-0.717597\pi\)
−0.631589 + 0.775304i \(0.717597\pi\)
\(464\) −24.2705 −1.12673
\(465\) 0 0
\(466\) 4.76393 0.220685
\(467\) −6.81966 −0.315576 −0.157788 0.987473i \(-0.550436\pi\)
−0.157788 + 0.987473i \(0.550436\pi\)
\(468\) 0 0
\(469\) 14.2361 0.657361
\(470\) −6.85410 −0.316156
\(471\) 0 0
\(472\) 25.0000 1.15072
\(473\) 0 0
\(474\) 0 0
\(475\) 26.8328 1.23117
\(476\) −0.472136 −0.0216403
\(477\) 0 0
\(478\) −48.7426 −2.22944
\(479\) −1.70820 −0.0780498 −0.0390249 0.999238i \(-0.512425\pi\)
−0.0390249 + 0.999238i \(0.512425\pi\)
\(480\) 0 0
\(481\) −38.3050 −1.74656
\(482\) −13.0344 −0.593703
\(483\) 0 0
\(484\) 0 0
\(485\) 3.70820 0.168381
\(486\) 0 0
\(487\) −0.944272 −0.0427890 −0.0213945 0.999771i \(-0.506811\pi\)
−0.0213945 + 0.999771i \(0.506811\pi\)
\(488\) 4.47214 0.202444
\(489\) 0 0
\(490\) −1.61803 −0.0730953
\(491\) 22.1246 0.998470 0.499235 0.866467i \(-0.333614\pi\)
0.499235 + 0.866467i \(0.333614\pi\)
\(492\) 0 0
\(493\) 3.81966 0.172029
\(494\) −59.3951 −2.67231
\(495\) 0 0
\(496\) 3.70820 0.166503
\(497\) 6.47214 0.290315
\(498\) 0 0
\(499\) 2.23607 0.100100 0.0500501 0.998747i \(-0.484062\pi\)
0.0500501 + 0.998747i \(0.484062\pi\)
\(500\) 5.56231 0.248754
\(501\) 0 0
\(502\) 45.5066 2.03106
\(503\) 15.7082 0.700394 0.350197 0.936676i \(-0.386115\pi\)
0.350197 + 0.936676i \(0.386115\pi\)
\(504\) 0 0
\(505\) 4.18034 0.186023
\(506\) 0 0
\(507\) 0 0
\(508\) −11.5279 −0.511466
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 13.4721 0.595972
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) 11.3262 0.499579
\(515\) 9.41641 0.414937
\(516\) 0 0
\(517\) 0 0
\(518\) 11.3262 0.497646
\(519\) 0 0
\(520\) 12.2361 0.536587
\(521\) 24.3050 1.06482 0.532410 0.846487i \(-0.321287\pi\)
0.532410 + 0.846487i \(0.321287\pi\)
\(522\) 0 0
\(523\) −9.65248 −0.422073 −0.211037 0.977478i \(-0.567684\pi\)
−0.211037 + 0.977478i \(0.567684\pi\)
\(524\) −10.4721 −0.457477
\(525\) 0 0
\(526\) 22.8541 0.996486
\(527\) −0.583592 −0.0254217
\(528\) 0 0
\(529\) 36.4164 1.58332
\(530\) 16.4721 0.715504
\(531\) 0 0
\(532\) 4.14590 0.179747
\(533\) 35.4164 1.53405
\(534\) 0 0
\(535\) −0.236068 −0.0102061
\(536\) 31.8328 1.37497
\(537\) 0 0
\(538\) −30.6525 −1.32152
\(539\) 0 0
\(540\) 0 0
\(541\) 19.0557 0.819270 0.409635 0.912250i \(-0.365656\pi\)
0.409635 + 0.912250i \(0.365656\pi\)
\(542\) −30.2705 −1.30023
\(543\) 0 0
\(544\) −2.58359 −0.110771
\(545\) 7.23607 0.309959
\(546\) 0 0
\(547\) 38.8328 1.66037 0.830186 0.557487i \(-0.188234\pi\)
0.830186 + 0.557487i \(0.188234\pi\)
\(548\) −3.88854 −0.166110
\(549\) 0 0
\(550\) 0 0
\(551\) −33.5410 −1.42890
\(552\) 0 0
\(553\) −5.52786 −0.235069
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −3.41641 −0.144888
\(557\) −21.4721 −0.909804 −0.454902 0.890542i \(-0.650326\pi\)
−0.454902 + 0.890542i \(0.650326\pi\)
\(558\) 0 0
\(559\) 42.1803 1.78404
\(560\) −4.85410 −0.205123
\(561\) 0 0
\(562\) 4.09017 0.172533
\(563\) 3.34752 0.141081 0.0705407 0.997509i \(-0.477528\pi\)
0.0705407 + 0.997509i \(0.477528\pi\)
\(564\) 0 0
\(565\) 8.47214 0.356425
\(566\) 29.5066 1.24025
\(567\) 0 0
\(568\) 14.4721 0.607237
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −26.4721 −1.10782 −0.553912 0.832575i \(-0.686866\pi\)
−0.553912 + 0.832575i \(0.686866\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −10.4721 −0.437099
\(575\) −30.8328 −1.28582
\(576\) 0 0
\(577\) 38.6525 1.60912 0.804562 0.593869i \(-0.202400\pi\)
0.804562 + 0.593869i \(0.202400\pi\)
\(578\) −26.5623 −1.10485
\(579\) 0 0
\(580\) −3.09017 −0.128312
\(581\) −11.2361 −0.466151
\(582\) 0 0
\(583\) 0 0
\(584\) 30.1246 1.24657
\(585\) 0 0
\(586\) −25.8885 −1.06945
\(587\) 30.0132 1.23878 0.619388 0.785085i \(-0.287381\pi\)
0.619388 + 0.785085i \(0.287381\pi\)
\(588\) 0 0
\(589\) 5.12461 0.211156
\(590\) 18.0902 0.744761
\(591\) 0 0
\(592\) 33.9787 1.39652
\(593\) 30.8328 1.26615 0.633076 0.774090i \(-0.281792\pi\)
0.633076 + 0.774090i \(0.281792\pi\)
\(594\) 0 0
\(595\) 0.763932 0.0313182
\(596\) 3.09017 0.126578
\(597\) 0 0
\(598\) 68.2492 2.79092
\(599\) 34.4721 1.40849 0.704247 0.709955i \(-0.251285\pi\)
0.704247 + 0.709955i \(0.251285\pi\)
\(600\) 0 0
\(601\) −27.0000 −1.10135 −0.550676 0.834719i \(-0.685630\pi\)
−0.550676 + 0.834719i \(0.685630\pi\)
\(602\) −12.4721 −0.508326
\(603\) 0 0
\(604\) −5.05573 −0.205715
\(605\) 0 0
\(606\) 0 0
\(607\) −29.1803 −1.18439 −0.592197 0.805793i \(-0.701739\pi\)
−0.592197 + 0.805793i \(0.701739\pi\)
\(608\) 22.6869 0.920076
\(609\) 0 0
\(610\) 3.23607 0.131025
\(611\) 23.1803 0.937776
\(612\) 0 0
\(613\) 35.5967 1.43774 0.718870 0.695145i \(-0.244660\pi\)
0.718870 + 0.695145i \(0.244660\pi\)
\(614\) 4.94427 0.199535
\(615\) 0 0
\(616\) 0 0
\(617\) −23.5279 −0.947196 −0.473598 0.880741i \(-0.657045\pi\)
−0.473598 + 0.880741i \(0.657045\pi\)
\(618\) 0 0
\(619\) 14.0689 0.565476 0.282738 0.959197i \(-0.408757\pi\)
0.282738 + 0.959197i \(0.408757\pi\)
\(620\) 0.472136 0.0189614
\(621\) 0 0
\(622\) 41.8885 1.67958
\(623\) 4.47214 0.179172
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −10.7639 −0.430213
\(627\) 0 0
\(628\) 7.05573 0.281554
\(629\) −5.34752 −0.213220
\(630\) 0 0
\(631\) −0.360680 −0.0143584 −0.00717922 0.999974i \(-0.502285\pi\)
−0.00717922 + 0.999974i \(0.502285\pi\)
\(632\) −12.3607 −0.491681
\(633\) 0 0
\(634\) −2.94427 −0.116932
\(635\) 18.6525 0.740201
\(636\) 0 0
\(637\) 5.47214 0.216814
\(638\) 0 0
\(639\) 0 0
\(640\) −13.6180 −0.538300
\(641\) −20.5410 −0.811321 −0.405661 0.914024i \(-0.632959\pi\)
−0.405661 + 0.914024i \(0.632959\pi\)
\(642\) 0 0
\(643\) 45.9574 1.81238 0.906192 0.422866i \(-0.138976\pi\)
0.906192 + 0.422866i \(0.138976\pi\)
\(644\) −4.76393 −0.187725
\(645\) 0 0
\(646\) −8.29180 −0.326236
\(647\) −43.6525 −1.71616 −0.858078 0.513519i \(-0.828341\pi\)
−0.858078 + 0.513519i \(0.828341\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −35.4164 −1.38915
\(651\) 0 0
\(652\) −5.74265 −0.224899
\(653\) 27.0557 1.05877 0.529386 0.848381i \(-0.322422\pi\)
0.529386 + 0.848381i \(0.322422\pi\)
\(654\) 0 0
\(655\) 16.9443 0.662067
\(656\) −31.4164 −1.22660
\(657\) 0 0
\(658\) −6.85410 −0.267201
\(659\) −43.5410 −1.69612 −0.848059 0.529902i \(-0.822229\pi\)
−0.848059 + 0.529902i \(0.822229\pi\)
\(660\) 0 0
\(661\) −26.5410 −1.03233 −0.516163 0.856490i \(-0.672640\pi\)
−0.516163 + 0.856490i \(0.672640\pi\)
\(662\) 24.9443 0.969487
\(663\) 0 0
\(664\) −25.1246 −0.975024
\(665\) −6.70820 −0.260133
\(666\) 0 0
\(667\) 38.5410 1.49231
\(668\) 5.34752 0.206902
\(669\) 0 0
\(670\) 23.0344 0.889898
\(671\) 0 0
\(672\) 0 0
\(673\) 45.5967 1.75763 0.878813 0.477167i \(-0.158336\pi\)
0.878813 + 0.477167i \(0.158336\pi\)
\(674\) −22.9443 −0.883780
\(675\) 0 0
\(676\) 10.4721 0.402774
\(677\) −43.3050 −1.66434 −0.832172 0.554517i \(-0.812903\pi\)
−0.832172 + 0.554517i \(0.812903\pi\)
\(678\) 0 0
\(679\) 3.70820 0.142308
\(680\) 1.70820 0.0655066
\(681\) 0 0
\(682\) 0 0
\(683\) 39.0132 1.49280 0.746398 0.665499i \(-0.231781\pi\)
0.746398 + 0.665499i \(0.231781\pi\)
\(684\) 0 0
\(685\) 6.29180 0.240397
\(686\) −1.61803 −0.0617768
\(687\) 0 0
\(688\) −37.4164 −1.42649
\(689\) −55.7082 −2.12231
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) −6.47214 −0.246034
\(693\) 0 0
\(694\) 45.3050 1.71975
\(695\) 5.52786 0.209684
\(696\) 0 0
\(697\) 4.94427 0.187278
\(698\) −45.9787 −1.74032
\(699\) 0 0
\(700\) 2.47214 0.0934380
\(701\) 39.8885 1.50657 0.753285 0.657695i \(-0.228468\pi\)
0.753285 + 0.657695i \(0.228468\pi\)
\(702\) 0 0
\(703\) 46.9574 1.77103
\(704\) 0 0
\(705\) 0 0
\(706\) −54.1591 −2.03830
\(707\) 4.18034 0.157218
\(708\) 0 0
\(709\) 39.7214 1.49177 0.745883 0.666076i \(-0.232028\pi\)
0.745883 + 0.666076i \(0.232028\pi\)
\(710\) 10.4721 0.393012
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −5.88854 −0.220528
\(714\) 0 0
\(715\) 0 0
\(716\) 5.52786 0.206586
\(717\) 0 0
\(718\) −5.52786 −0.206298
\(719\) −12.2361 −0.456328 −0.228164 0.973623i \(-0.573272\pi\)
−0.228164 + 0.973623i \(0.573272\pi\)
\(720\) 0 0
\(721\) 9.41641 0.350685
\(722\) 42.0689 1.56564
\(723\) 0 0
\(724\) −3.23607 −0.120268
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 1.81966 0.0674875 0.0337437 0.999431i \(-0.489257\pi\)
0.0337437 + 0.999431i \(0.489257\pi\)
\(728\) 12.2361 0.453499
\(729\) 0 0
\(730\) 21.7984 0.806794
\(731\) 5.88854 0.217796
\(732\) 0 0
\(733\) 43.8885 1.62106 0.810530 0.585697i \(-0.199179\pi\)
0.810530 + 0.585697i \(0.199179\pi\)
\(734\) 25.7082 0.948907
\(735\) 0 0
\(736\) −26.0689 −0.960912
\(737\) 0 0
\(738\) 0 0
\(739\) 24.0689 0.885388 0.442694 0.896673i \(-0.354023\pi\)
0.442694 + 0.896673i \(0.354023\pi\)
\(740\) 4.32624 0.159036
\(741\) 0 0
\(742\) 16.4721 0.604711
\(743\) 25.1803 0.923777 0.461889 0.886938i \(-0.347172\pi\)
0.461889 + 0.886938i \(0.347172\pi\)
\(744\) 0 0
\(745\) −5.00000 −0.183186
\(746\) 43.1246 1.57890
\(747\) 0 0
\(748\) 0 0
\(749\) −0.236068 −0.00862574
\(750\) 0 0
\(751\) 44.2361 1.61420 0.807099 0.590417i \(-0.201037\pi\)
0.807099 + 0.590417i \(0.201037\pi\)
\(752\) −20.5623 −0.749830
\(753\) 0 0
\(754\) 44.2705 1.61224
\(755\) 8.18034 0.297713
\(756\) 0 0
\(757\) 37.7214 1.37101 0.685503 0.728070i \(-0.259582\pi\)
0.685503 + 0.728070i \(0.259582\pi\)
\(758\) −14.2705 −0.518328
\(759\) 0 0
\(760\) −15.0000 −0.544107
\(761\) −43.7771 −1.58692 −0.793459 0.608624i \(-0.791722\pi\)
−0.793459 + 0.608624i \(0.791722\pi\)
\(762\) 0 0
\(763\) 7.23607 0.261963
\(764\) 9.41641 0.340674
\(765\) 0 0
\(766\) −24.3607 −0.880187
\(767\) −61.1803 −2.20909
\(768\) 0 0
\(769\) 3.94427 0.142234 0.0711170 0.997468i \(-0.477344\pi\)
0.0711170 + 0.997468i \(0.477344\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.2918 0.370410
\(773\) −3.47214 −0.124884 −0.0624420 0.998049i \(-0.519889\pi\)
−0.0624420 + 0.998049i \(0.519889\pi\)
\(774\) 0 0
\(775\) 3.05573 0.109765
\(776\) 8.29180 0.297658
\(777\) 0 0
\(778\) −46.8328 −1.67904
\(779\) −43.4164 −1.55555
\(780\) 0 0
\(781\) 0 0
\(782\) 9.52786 0.340716
\(783\) 0 0
\(784\) −4.85410 −0.173361
\(785\) −11.4164 −0.407469
\(786\) 0 0
\(787\) 27.6525 0.985704 0.492852 0.870113i \(-0.335954\pi\)
0.492852 + 0.870113i \(0.335954\pi\)
\(788\) −4.65248 −0.165738
\(789\) 0 0
\(790\) −8.94427 −0.318223
\(791\) 8.47214 0.301234
\(792\) 0 0
\(793\) −10.9443 −0.388642
\(794\) −27.7082 −0.983327
\(795\) 0 0
\(796\) −16.1803 −0.573497
\(797\) 11.4721 0.406364 0.203182 0.979141i \(-0.434872\pi\)
0.203182 + 0.979141i \(0.434872\pi\)
\(798\) 0 0
\(799\) 3.23607 0.114484
\(800\) 13.5279 0.478282
\(801\) 0 0
\(802\) 26.3607 0.930828
\(803\) 0 0
\(804\) 0 0
\(805\) 7.70820 0.271678
\(806\) −6.76393 −0.238249
\(807\) 0 0
\(808\) 9.34752 0.328845
\(809\) 6.30495 0.221670 0.110835 0.993839i \(-0.464647\pi\)
0.110835 + 0.993839i \(0.464647\pi\)
\(810\) 0 0
\(811\) −28.7082 −1.00808 −0.504041 0.863680i \(-0.668154\pi\)
−0.504041 + 0.863680i \(0.668154\pi\)
\(812\) −3.09017 −0.108444
\(813\) 0 0
\(814\) 0 0
\(815\) 9.29180 0.325477
\(816\) 0 0
\(817\) −51.7082 −1.80904
\(818\) −63.0132 −2.20320
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 1.47214 0.0513779 0.0256889 0.999670i \(-0.491822\pi\)
0.0256889 + 0.999670i \(0.491822\pi\)
\(822\) 0 0
\(823\) 5.18034 0.180575 0.0902876 0.995916i \(-0.471221\pi\)
0.0902876 + 0.995916i \(0.471221\pi\)
\(824\) 21.0557 0.733511
\(825\) 0 0
\(826\) 18.0902 0.629438
\(827\) 43.6525 1.51795 0.758973 0.651122i \(-0.225702\pi\)
0.758973 + 0.651122i \(0.225702\pi\)
\(828\) 0 0
\(829\) −35.7771 −1.24259 −0.621295 0.783577i \(-0.713393\pi\)
−0.621295 + 0.783577i \(0.713393\pi\)
\(830\) −18.1803 −0.631049
\(831\) 0 0
\(832\) 23.1803 0.803634
\(833\) 0.763932 0.0264687
\(834\) 0 0
\(835\) −8.65248 −0.299431
\(836\) 0 0
\(837\) 0 0
\(838\) −34.2705 −1.18386
\(839\) −43.5410 −1.50320 −0.751601 0.659617i \(-0.770718\pi\)
−0.751601 + 0.659617i \(0.770718\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −21.0344 −0.724895
\(843\) 0 0
\(844\) 13.2361 0.455604
\(845\) −16.9443 −0.582901
\(846\) 0 0
\(847\) 0 0
\(848\) 49.4164 1.69697
\(849\) 0 0
\(850\) −4.94427 −0.169587
\(851\) −53.9574 −1.84964
\(852\) 0 0
\(853\) 2.58359 0.0884605 0.0442303 0.999021i \(-0.485916\pi\)
0.0442303 + 0.999021i \(0.485916\pi\)
\(854\) 3.23607 0.110736
\(855\) 0 0
\(856\) −0.527864 −0.0180420
\(857\) 35.8885 1.22593 0.612965 0.790110i \(-0.289977\pi\)
0.612965 + 0.790110i \(0.289977\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −4.76393 −0.162449
\(861\) 0 0
\(862\) −7.61803 −0.259471
\(863\) 38.7639 1.31954 0.659770 0.751468i \(-0.270654\pi\)
0.659770 + 0.751468i \(0.270654\pi\)
\(864\) 0 0
\(865\) 10.4721 0.356063
\(866\) −2.47214 −0.0840066
\(867\) 0 0
\(868\) 0.472136 0.0160253
\(869\) 0 0
\(870\) 0 0
\(871\) −77.9017 −2.63960
\(872\) 16.1803 0.547935
\(873\) 0 0
\(874\) −83.6656 −2.83003
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −31.4164 −1.06086 −0.530428 0.847730i \(-0.677969\pi\)
−0.530428 + 0.847730i \(0.677969\pi\)
\(878\) 18.0902 0.610514
\(879\) 0 0
\(880\) 0 0
\(881\) −19.1115 −0.643881 −0.321941 0.946760i \(-0.604335\pi\)
−0.321941 + 0.946760i \(0.604335\pi\)
\(882\) 0 0
\(883\) −37.1803 −1.25122 −0.625609 0.780137i \(-0.715149\pi\)
−0.625609 + 0.780137i \(0.715149\pi\)
\(884\) 2.58359 0.0868956
\(885\) 0 0
\(886\) −15.4164 −0.517924
\(887\) 15.2361 0.511577 0.255789 0.966733i \(-0.417665\pi\)
0.255789 + 0.966733i \(0.417665\pi\)
\(888\) 0 0
\(889\) 18.6525 0.625584
\(890\) 7.23607 0.242554
\(891\) 0 0
\(892\) −3.70820 −0.124160
\(893\) −28.4164 −0.950919
\(894\) 0 0
\(895\) −8.94427 −0.298974
\(896\) −13.6180 −0.454947
\(897\) 0 0
\(898\) −32.3607 −1.07989
\(899\) −3.81966 −0.127393
\(900\) 0 0
\(901\) −7.77709 −0.259092
\(902\) 0 0
\(903\) 0 0
\(904\) 18.9443 0.630077
\(905\) 5.23607 0.174053
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −1.23607 −0.0410204
\(909\) 0 0
\(910\) 8.85410 0.293511
\(911\) 41.4164 1.37219 0.686093 0.727513i \(-0.259324\pi\)
0.686093 + 0.727513i \(0.259324\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −17.4164 −0.576084
\(915\) 0 0
\(916\) 1.70820 0.0564406
\(917\) 16.9443 0.559549
\(918\) 0 0
\(919\) 9.59675 0.316567 0.158284 0.987394i \(-0.449404\pi\)
0.158284 + 0.987394i \(0.449404\pi\)
\(920\) 17.2361 0.568256
\(921\) 0 0
\(922\) −45.3050 −1.49204
\(923\) −35.4164 −1.16575
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) −43.9787 −1.44523
\(927\) 0 0
\(928\) −16.9098 −0.555092
\(929\) 22.8885 0.750949 0.375474 0.926833i \(-0.377480\pi\)
0.375474 + 0.926833i \(0.377480\pi\)
\(930\) 0 0
\(931\) −6.70820 −0.219853
\(932\) 1.81966 0.0596049
\(933\) 0 0
\(934\) −11.0344 −0.361058
\(935\) 0 0
\(936\) 0 0
\(937\) 4.11146 0.134315 0.0671577 0.997742i \(-0.478607\pi\)
0.0671577 + 0.997742i \(0.478607\pi\)
\(938\) 23.0344 0.752101
\(939\) 0 0
\(940\) −2.61803 −0.0853909
\(941\) −15.6393 −0.509827 −0.254914 0.966964i \(-0.582047\pi\)
−0.254914 + 0.966964i \(0.582047\pi\)
\(942\) 0 0
\(943\) 49.8885 1.62459
\(944\) 54.2705 1.76635
\(945\) 0 0
\(946\) 0 0
\(947\) −21.4164 −0.695940 −0.347970 0.937506i \(-0.613129\pi\)
−0.347970 + 0.937506i \(0.613129\pi\)
\(948\) 0 0
\(949\) −73.7214 −2.39310
\(950\) 43.4164 1.40861
\(951\) 0 0
\(952\) 1.70820 0.0553632
\(953\) −26.7771 −0.867395 −0.433697 0.901059i \(-0.642791\pi\)
−0.433697 + 0.901059i \(0.642791\pi\)
\(954\) 0 0
\(955\) −15.2361 −0.493028
\(956\) −18.6180 −0.602150
\(957\) 0 0
\(958\) −2.76393 −0.0892986
\(959\) 6.29180 0.203173
\(960\) 0 0
\(961\) −30.4164 −0.981174
\(962\) −61.9787 −1.99827
\(963\) 0 0
\(964\) −4.97871 −0.160354
\(965\) −16.6525 −0.536062
\(966\) 0 0
\(967\) −37.1935 −1.19606 −0.598031 0.801473i \(-0.704050\pi\)
−0.598031 + 0.801473i \(0.704050\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 8.12461 0.260731 0.130366 0.991466i \(-0.458385\pi\)
0.130366 + 0.991466i \(0.458385\pi\)
\(972\) 0 0
\(973\) 5.52786 0.177215
\(974\) −1.52786 −0.0489559
\(975\) 0 0
\(976\) 9.70820 0.310752
\(977\) 18.1803 0.581641 0.290820 0.956778i \(-0.406072\pi\)
0.290820 + 0.956778i \(0.406072\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.618034 −0.0197424
\(981\) 0 0
\(982\) 35.7984 1.14237
\(983\) −0.583592 −0.0186137 −0.00930685 0.999957i \(-0.502963\pi\)
−0.00930685 + 0.999957i \(0.502963\pi\)
\(984\) 0 0
\(985\) 7.52786 0.239858
\(986\) 6.18034 0.196822
\(987\) 0 0
\(988\) −22.6869 −0.721767
\(989\) 59.4164 1.88933
\(990\) 0 0
\(991\) −6.81966 −0.216634 −0.108317 0.994116i \(-0.534546\pi\)
−0.108317 + 0.994116i \(0.534546\pi\)
\(992\) 2.58359 0.0820291
\(993\) 0 0
\(994\) 10.4721 0.332156
\(995\) 26.1803 0.829973
\(996\) 0 0
\(997\) −9.05573 −0.286798 −0.143399 0.989665i \(-0.545803\pi\)
−0.143399 + 0.989665i \(0.545803\pi\)
\(998\) 3.61803 0.114527
\(999\) 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.bm.1.2 2
3.2 odd 2 2541.2.a.t.1.1 2
11.10 odd 2 693.2.a.f.1.1 2
33.32 even 2 231.2.a.c.1.2 2
77.76 even 2 4851.2.a.w.1.1 2
132.131 odd 2 3696.2.a.be.1.1 2
165.164 even 2 5775.2.a.be.1.1 2
231.230 odd 2 1617.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.c.1.2 2 33.32 even 2
693.2.a.f.1.1 2 11.10 odd 2
1617.2.a.p.1.2 2 231.230 odd 2
2541.2.a.t.1.1 2 3.2 odd 2
3696.2.a.be.1.1 2 132.131 odd 2
4851.2.a.w.1.1 2 77.76 even 2
5775.2.a.be.1.1 2 165.164 even 2
7623.2.a.bm.1.2 2 1.1 even 1 trivial