Properties

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

Embedding invariants

Embedding label 1.1
Root \(-0.618034\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.47214 q^{8} +O(q^{10})\) \(q+0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.47214 q^{8} -0.381966 q^{10} -1.23607 q^{13} +0.381966 q^{14} +3.14590 q^{16} -3.09017 q^{17} -1.76393 q^{19} +1.85410 q^{20} -5.09017 q^{23} -4.00000 q^{25} -0.472136 q^{26} -1.85410 q^{28} +4.61803 q^{29} -4.23607 q^{31} +4.14590 q^{32} -1.18034 q^{34} -1.00000 q^{35} +6.47214 q^{37} -0.673762 q^{38} +1.47214 q^{40} +11.1803 q^{41} -12.5623 q^{43} -1.94427 q^{46} +6.61803 q^{47} +1.00000 q^{49} -1.52786 q^{50} +2.29180 q^{52} +2.38197 q^{53} -1.47214 q^{56} +1.76393 q^{58} -11.0902 q^{59} -7.61803 q^{61} -1.61803 q^{62} -4.70820 q^{64} +1.23607 q^{65} -8.32624 q^{67} +5.72949 q^{68} -0.381966 q^{70} +16.0902 q^{71} -14.2361 q^{73} +2.47214 q^{74} +3.27051 q^{76} +6.38197 q^{79} -3.14590 q^{80} +4.27051 q^{82} -2.70820 q^{83} +3.09017 q^{85} -4.79837 q^{86} -6.85410 q^{89} -1.23607 q^{91} +9.43769 q^{92} +2.52786 q^{94} +1.76393 q^{95} +7.00000 q^{97} +0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 3q^{4} - 2q^{5} + 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + 3q^{2} + 3q^{4} - 2q^{5} + 2q^{7} + 6q^{8} - 3q^{10} + 2q^{13} + 3q^{14} + 13q^{16} + 5q^{17} - 8q^{19} - 3q^{20} + q^{23} - 8q^{25} + 8q^{26} + 3q^{28} + 7q^{29} - 4q^{31} + 15q^{32} + 20q^{34} - 2q^{35} + 4q^{37} - 17q^{38} - 6q^{40} - 5q^{43} + 14q^{46} + 11q^{47} + 2q^{49} - 12q^{50} + 18q^{52} + 7q^{53} + 6q^{56} + 8q^{58} - 11q^{59} - 13q^{61} - q^{62} + 4q^{64} - 2q^{65} - q^{67} + 45q^{68} - 3q^{70} + 21q^{71} - 24q^{73} - 4q^{74} - 27q^{76} + 15q^{79} - 13q^{80} - 25q^{82} + 8q^{83} - 5q^{85} + 15q^{86} - 7q^{89} + 2q^{91} + 39q^{92} + 14q^{94} + 8q^{95} + 14q^{97} + 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.381966 0.270091 0.135045 0.990839i \(-0.456882\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) 0 0
\(4\) −1.85410 −0.927051
\(5\) −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
\(8\) −1.47214 −0.520479
\(9\) 0 0
\(10\) −0.381966 −0.120788
\(11\) 0 0
\(12\) 0 0
\(13\) −1.23607 −0.342824 −0.171412 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) 0.381966 0.102085
\(15\) 0 0
\(16\) 3.14590 0.786475
\(17\) −3.09017 −0.749476 −0.374738 0.927131i \(-0.622267\pi\)
−0.374738 + 0.927131i \(0.622267\pi\)
\(18\) 0 0
\(19\) −1.76393 −0.404674 −0.202337 0.979316i \(-0.564854\pi\)
−0.202337 + 0.979316i \(0.564854\pi\)
\(20\) 1.85410 0.414590
\(21\) 0 0
\(22\) 0 0
\(23\) −5.09017 −1.06137 −0.530687 0.847568i \(-0.678066\pi\)
−0.530687 + 0.847568i \(0.678066\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −0.472136 −0.0925935
\(27\) 0 0
\(28\) −1.85410 −0.350392
\(29\) 4.61803 0.857547 0.428774 0.903412i \(-0.358946\pi\)
0.428774 + 0.903412i \(0.358946\pi\)
\(30\) 0 0
\(31\) −4.23607 −0.760820 −0.380410 0.924818i \(-0.624217\pi\)
−0.380410 + 0.924818i \(0.624217\pi\)
\(32\) 4.14590 0.732898
\(33\) 0 0
\(34\) −1.18034 −0.202427
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 6.47214 1.06401 0.532006 0.846740i \(-0.321438\pi\)
0.532006 + 0.846740i \(0.321438\pi\)
\(38\) −0.673762 −0.109299
\(39\) 0 0
\(40\) 1.47214 0.232765
\(41\) 11.1803 1.74608 0.873038 0.487652i \(-0.162147\pi\)
0.873038 + 0.487652i \(0.162147\pi\)
\(42\) 0 0
\(43\) −12.5623 −1.91573 −0.957867 0.287213i \(-0.907271\pi\)
−0.957867 + 0.287213i \(0.907271\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.94427 −0.286667
\(47\) 6.61803 0.965339 0.482670 0.875802i \(-0.339667\pi\)
0.482670 + 0.875802i \(0.339667\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.52786 −0.216073
\(51\) 0 0
\(52\) 2.29180 0.317815
\(53\) 2.38197 0.327188 0.163594 0.986528i \(-0.447691\pi\)
0.163594 + 0.986528i \(0.447691\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.47214 −0.196722
\(57\) 0 0
\(58\) 1.76393 0.231616
\(59\) −11.0902 −1.44382 −0.721909 0.691988i \(-0.756735\pi\)
−0.721909 + 0.691988i \(0.756735\pi\)
\(60\) 0 0
\(61\) −7.61803 −0.975389 −0.487695 0.873014i \(-0.662162\pi\)
−0.487695 + 0.873014i \(0.662162\pi\)
\(62\) −1.61803 −0.205491
\(63\) 0 0
\(64\) −4.70820 −0.588525
\(65\) 1.23607 0.153315
\(66\) 0 0
\(67\) −8.32624 −1.01721 −0.508606 0.860999i \(-0.669839\pi\)
−0.508606 + 0.860999i \(0.669839\pi\)
\(68\) 5.72949 0.694803
\(69\) 0 0
\(70\) −0.381966 −0.0456537
\(71\) 16.0902 1.90955 0.954776 0.297326i \(-0.0960949\pi\)
0.954776 + 0.297326i \(0.0960949\pi\)
\(72\) 0 0
\(73\) −14.2361 −1.66621 −0.833103 0.553118i \(-0.813438\pi\)
−0.833103 + 0.553118i \(0.813438\pi\)
\(74\) 2.47214 0.287380
\(75\) 0 0
\(76\) 3.27051 0.375153
\(77\) 0 0
\(78\) 0 0
\(79\) 6.38197 0.718027 0.359014 0.933332i \(-0.383113\pi\)
0.359014 + 0.933332i \(0.383113\pi\)
\(80\) −3.14590 −0.351722
\(81\) 0 0
\(82\) 4.27051 0.471599
\(83\) −2.70820 −0.297264 −0.148632 0.988893i \(-0.547487\pi\)
−0.148632 + 0.988893i \(0.547487\pi\)
\(84\) 0 0
\(85\) 3.09017 0.335176
\(86\) −4.79837 −0.517422
\(87\) 0 0
\(88\) 0 0
\(89\) −6.85410 −0.726533 −0.363267 0.931685i \(-0.618339\pi\)
−0.363267 + 0.931685i \(0.618339\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 9.43769 0.983948
\(93\) 0 0
\(94\) 2.52786 0.260729
\(95\) 1.76393 0.180976
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0.381966 0.0385844
\(99\) 0 0
\(100\) 7.41641 0.741641
\(101\) 14.7984 1.47249 0.736247 0.676713i \(-0.236596\pi\)
0.736247 + 0.676713i \(0.236596\pi\)
\(102\) 0 0
\(103\) 8.85410 0.872421 0.436210 0.899845i \(-0.356320\pi\)
0.436210 + 0.899845i \(0.356320\pi\)
\(104\) 1.81966 0.178432
\(105\) 0 0
\(106\) 0.909830 0.0883705
\(107\) −2.70820 −0.261812 −0.130906 0.991395i \(-0.541789\pi\)
−0.130906 + 0.991395i \(0.541789\pi\)
\(108\) 0 0
\(109\) −1.52786 −0.146343 −0.0731714 0.997319i \(-0.523312\pi\)
−0.0731714 + 0.997319i \(0.523312\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.14590 0.297259
\(113\) −0.145898 −0.0137249 −0.00686247 0.999976i \(-0.502184\pi\)
−0.00686247 + 0.999976i \(0.502184\pi\)
\(114\) 0 0
\(115\) 5.09017 0.474661
\(116\) −8.56231 −0.794990
\(117\) 0 0
\(118\) −4.23607 −0.389962
\(119\) −3.09017 −0.283275
\(120\) 0 0
\(121\) 0 0
\(122\) −2.90983 −0.263444
\(123\) 0 0
\(124\) 7.85410 0.705319
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 2.94427 0.261262 0.130631 0.991431i \(-0.458300\pi\)
0.130631 + 0.991431i \(0.458300\pi\)
\(128\) −10.0902 −0.891853
\(129\) 0 0
\(130\) 0.472136 0.0414091
\(131\) −18.9443 −1.65517 −0.827584 0.561341i \(-0.810285\pi\)
−0.827584 + 0.561341i \(0.810285\pi\)
\(132\) 0 0
\(133\) −1.76393 −0.152952
\(134\) −3.18034 −0.274740
\(135\) 0 0
\(136\) 4.54915 0.390086
\(137\) 15.3262 1.30941 0.654704 0.755885i \(-0.272793\pi\)
0.654704 + 0.755885i \(0.272793\pi\)
\(138\) 0 0
\(139\) 11.9443 1.01310 0.506550 0.862211i \(-0.330921\pi\)
0.506550 + 0.862211i \(0.330921\pi\)
\(140\) 1.85410 0.156700
\(141\) 0 0
\(142\) 6.14590 0.515752
\(143\) 0 0
\(144\) 0 0
\(145\) −4.61803 −0.383507
\(146\) −5.43769 −0.450027
\(147\) 0 0
\(148\) −12.0000 −0.986394
\(149\) 8.85410 0.725356 0.362678 0.931914i \(-0.381862\pi\)
0.362678 + 0.931914i \(0.381862\pi\)
\(150\) 0 0
\(151\) 0.0557281 0.00453509 0.00226754 0.999997i \(-0.499278\pi\)
0.00226754 + 0.999997i \(0.499278\pi\)
\(152\) 2.59675 0.210624
\(153\) 0 0
\(154\) 0 0
\(155\) 4.23607 0.340249
\(156\) 0 0
\(157\) 19.8885 1.58728 0.793639 0.608389i \(-0.208184\pi\)
0.793639 + 0.608389i \(0.208184\pi\)
\(158\) 2.43769 0.193933
\(159\) 0 0
\(160\) −4.14590 −0.327762
\(161\) −5.09017 −0.401162
\(162\) 0 0
\(163\) −8.70820 −0.682079 −0.341040 0.940049i \(-0.610779\pi\)
−0.341040 + 0.940049i \(0.610779\pi\)
\(164\) −20.7295 −1.61870
\(165\) 0 0
\(166\) −1.03444 −0.0802883
\(167\) −6.47214 −0.500829 −0.250414 0.968139i \(-0.580567\pi\)
−0.250414 + 0.968139i \(0.580567\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 1.18034 0.0905279
\(171\) 0 0
\(172\) 23.2918 1.77598
\(173\) 15.3820 1.16947 0.584735 0.811225i \(-0.301199\pi\)
0.584735 + 0.811225i \(0.301199\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −2.61803 −0.196230
\(179\) 3.76393 0.281329 0.140665 0.990057i \(-0.455076\pi\)
0.140665 + 0.990057i \(0.455076\pi\)
\(180\) 0 0
\(181\) −7.41641 −0.551257 −0.275629 0.961264i \(-0.588886\pi\)
−0.275629 + 0.961264i \(0.588886\pi\)
\(182\) −0.472136 −0.0349970
\(183\) 0 0
\(184\) 7.49342 0.552422
\(185\) −6.47214 −0.475841
\(186\) 0 0
\(187\) 0 0
\(188\) −12.2705 −0.894919
\(189\) 0 0
\(190\) 0.673762 0.0488798
\(191\) −15.7639 −1.14064 −0.570319 0.821423i \(-0.693180\pi\)
−0.570319 + 0.821423i \(0.693180\pi\)
\(192\) 0 0
\(193\) −15.3262 −1.10321 −0.551603 0.834107i \(-0.685984\pi\)
−0.551603 + 0.834107i \(0.685984\pi\)
\(194\) 2.67376 0.191965
\(195\) 0 0
\(196\) −1.85410 −0.132436
\(197\) 4.29180 0.305778 0.152889 0.988243i \(-0.451142\pi\)
0.152889 + 0.988243i \(0.451142\pi\)
\(198\) 0 0
\(199\) −8.23607 −0.583839 −0.291920 0.956443i \(-0.594294\pi\)
−0.291920 + 0.956443i \(0.594294\pi\)
\(200\) 5.88854 0.416383
\(201\) 0 0
\(202\) 5.65248 0.397707
\(203\) 4.61803 0.324122
\(204\) 0 0
\(205\) −11.1803 −0.780869
\(206\) 3.38197 0.235633
\(207\) 0 0
\(208\) −3.88854 −0.269622
\(209\) 0 0
\(210\) 0 0
\(211\) 15.6180 1.07519 0.537595 0.843203i \(-0.319333\pi\)
0.537595 + 0.843203i \(0.319333\pi\)
\(212\) −4.41641 −0.303320
\(213\) 0 0
\(214\) −1.03444 −0.0707130
\(215\) 12.5623 0.856742
\(216\) 0 0
\(217\) −4.23607 −0.287563
\(218\) −0.583592 −0.0395258
\(219\) 0 0
\(220\) 0 0
\(221\) 3.81966 0.256938
\(222\) 0 0
\(223\) 2.03444 0.136236 0.0681182 0.997677i \(-0.478301\pi\)
0.0681182 + 0.997677i \(0.478301\pi\)
\(224\) 4.14590 0.277009
\(225\) 0 0
\(226\) −0.0557281 −0.00370698
\(227\) 16.0344 1.06424 0.532122 0.846668i \(-0.321395\pi\)
0.532122 + 0.846668i \(0.321395\pi\)
\(228\) 0 0
\(229\) 6.76393 0.446973 0.223487 0.974707i \(-0.428256\pi\)
0.223487 + 0.974707i \(0.428256\pi\)
\(230\) 1.94427 0.128201
\(231\) 0 0
\(232\) −6.79837 −0.446335
\(233\) 29.4164 1.92713 0.963566 0.267469i \(-0.0861873\pi\)
0.963566 + 0.267469i \(0.0861873\pi\)
\(234\) 0 0
\(235\) −6.61803 −0.431713
\(236\) 20.5623 1.33849
\(237\) 0 0
\(238\) −1.18034 −0.0765101
\(239\) 17.0902 1.10547 0.552736 0.833357i \(-0.313584\pi\)
0.552736 + 0.833357i \(0.313584\pi\)
\(240\) 0 0
\(241\) 16.2705 1.04808 0.524038 0.851695i \(-0.324425\pi\)
0.524038 + 0.851695i \(0.324425\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 14.1246 0.904236
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 2.18034 0.138732
\(248\) 6.23607 0.395991
\(249\) 0 0
\(250\) 3.43769 0.217419
\(251\) −23.0000 −1.45175 −0.725874 0.687828i \(-0.758564\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.12461 0.0705644
\(255\) 0 0
\(256\) 5.56231 0.347644
\(257\) 8.56231 0.534102 0.267051 0.963682i \(-0.413951\pi\)
0.267051 + 0.963682i \(0.413951\pi\)
\(258\) 0 0
\(259\) 6.47214 0.402159
\(260\) −2.29180 −0.142131
\(261\) 0 0
\(262\) −7.23607 −0.447046
\(263\) 17.1246 1.05595 0.527974 0.849260i \(-0.322952\pi\)
0.527974 + 0.849260i \(0.322952\pi\)
\(264\) 0 0
\(265\) −2.38197 −0.146323
\(266\) −0.673762 −0.0413110
\(267\) 0 0
\(268\) 15.4377 0.943007
\(269\) −16.8541 −1.02761 −0.513806 0.857906i \(-0.671765\pi\)
−0.513806 + 0.857906i \(0.671765\pi\)
\(270\) 0 0
\(271\) −4.79837 −0.291480 −0.145740 0.989323i \(-0.546556\pi\)
−0.145740 + 0.989323i \(0.546556\pi\)
\(272\) −9.72136 −0.589444
\(273\) 0 0
\(274\) 5.85410 0.353659
\(275\) 0 0
\(276\) 0 0
\(277\) 23.0344 1.38401 0.692003 0.721895i \(-0.256729\pi\)
0.692003 + 0.721895i \(0.256729\pi\)
\(278\) 4.56231 0.273629
\(279\) 0 0
\(280\) 1.47214 0.0879770
\(281\) 25.1803 1.50213 0.751067 0.660226i \(-0.229540\pi\)
0.751067 + 0.660226i \(0.229540\pi\)
\(282\) 0 0
\(283\) 11.9443 0.710013 0.355007 0.934864i \(-0.384479\pi\)
0.355007 + 0.934864i \(0.384479\pi\)
\(284\) −29.8328 −1.77025
\(285\) 0 0
\(286\) 0 0
\(287\) 11.1803 0.659955
\(288\) 0 0
\(289\) −7.45085 −0.438285
\(290\) −1.76393 −0.103582
\(291\) 0 0
\(292\) 26.3951 1.54466
\(293\) 11.0000 0.642627 0.321313 0.946973i \(-0.395876\pi\)
0.321313 + 0.946973i \(0.395876\pi\)
\(294\) 0 0
\(295\) 11.0902 0.645695
\(296\) −9.52786 −0.553796
\(297\) 0 0
\(298\) 3.38197 0.195912
\(299\) 6.29180 0.363864
\(300\) 0 0
\(301\) −12.5623 −0.724079
\(302\) 0.0212862 0.00122489
\(303\) 0 0
\(304\) −5.54915 −0.318266
\(305\) 7.61803 0.436207
\(306\) 0 0
\(307\) −23.1803 −1.32297 −0.661486 0.749958i \(-0.730074\pi\)
−0.661486 + 0.749958i \(0.730074\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.61803 0.0918982
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) 18.3262 1.03586 0.517930 0.855423i \(-0.326703\pi\)
0.517930 + 0.855423i \(0.326703\pi\)
\(314\) 7.59675 0.428709
\(315\) 0 0
\(316\) −11.8328 −0.665648
\(317\) 4.43769 0.249246 0.124623 0.992204i \(-0.460228\pi\)
0.124623 + 0.992204i \(0.460228\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 4.70820 0.263197
\(321\) 0 0
\(322\) −1.94427 −0.108350
\(323\) 5.45085 0.303293
\(324\) 0 0
\(325\) 4.94427 0.274259
\(326\) −3.32624 −0.184223
\(327\) 0 0
\(328\) −16.4590 −0.908795
\(329\) 6.61803 0.364864
\(330\) 0 0
\(331\) 6.18034 0.339702 0.169851 0.985470i \(-0.445671\pi\)
0.169851 + 0.985470i \(0.445671\pi\)
\(332\) 5.02129 0.275579
\(333\) 0 0
\(334\) −2.47214 −0.135269
\(335\) 8.32624 0.454911
\(336\) 0 0
\(337\) 25.9443 1.41327 0.706637 0.707576i \(-0.250211\pi\)
0.706637 + 0.707576i \(0.250211\pi\)
\(338\) −4.38197 −0.238348
\(339\) 0 0
\(340\) −5.72949 −0.310725
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 18.4934 0.997099
\(345\) 0 0
\(346\) 5.87539 0.315863
\(347\) −2.34752 −0.126022 −0.0630108 0.998013i \(-0.520070\pi\)
−0.0630108 + 0.998013i \(0.520070\pi\)
\(348\) 0 0
\(349\) 32.7984 1.75566 0.877828 0.478975i \(-0.158992\pi\)
0.877828 + 0.478975i \(0.158992\pi\)
\(350\) −1.52786 −0.0816678
\(351\) 0 0
\(352\) 0 0
\(353\) −24.0902 −1.28219 −0.641095 0.767461i \(-0.721520\pi\)
−0.641095 + 0.767461i \(0.721520\pi\)
\(354\) 0 0
\(355\) −16.0902 −0.853978
\(356\) 12.7082 0.673533
\(357\) 0 0
\(358\) 1.43769 0.0759845
\(359\) 15.6180 0.824288 0.412144 0.911119i \(-0.364780\pi\)
0.412144 + 0.911119i \(0.364780\pi\)
\(360\) 0 0
\(361\) −15.8885 −0.836239
\(362\) −2.83282 −0.148889
\(363\) 0 0
\(364\) 2.29180 0.120123
\(365\) 14.2361 0.745150
\(366\) 0 0
\(367\) 32.8328 1.71386 0.856930 0.515434i \(-0.172369\pi\)
0.856930 + 0.515434i \(0.172369\pi\)
\(368\) −16.0132 −0.834743
\(369\) 0 0
\(370\) −2.47214 −0.128520
\(371\) 2.38197 0.123666
\(372\) 0 0
\(373\) −15.4377 −0.799334 −0.399667 0.916661i \(-0.630874\pi\)
−0.399667 + 0.916661i \(0.630874\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.74265 −0.502439
\(377\) −5.70820 −0.293987
\(378\) 0 0
\(379\) 32.0689 1.64727 0.823634 0.567122i \(-0.191943\pi\)
0.823634 + 0.567122i \(0.191943\pi\)
\(380\) −3.27051 −0.167774
\(381\) 0 0
\(382\) −6.02129 −0.308076
\(383\) 34.5967 1.76781 0.883906 0.467665i \(-0.154905\pi\)
0.883906 + 0.467665i \(0.154905\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.85410 −0.297966
\(387\) 0 0
\(388\) −12.9787 −0.658894
\(389\) 34.1803 1.73301 0.866506 0.499167i \(-0.166360\pi\)
0.866506 + 0.499167i \(0.166360\pi\)
\(390\) 0 0
\(391\) 15.7295 0.795475
\(392\) −1.47214 −0.0743541
\(393\) 0 0
\(394\) 1.63932 0.0825878
\(395\) −6.38197 −0.321112
\(396\) 0 0
\(397\) −0.819660 −0.0411376 −0.0205688 0.999788i \(-0.506548\pi\)
−0.0205688 + 0.999788i \(0.506548\pi\)
\(398\) −3.14590 −0.157690
\(399\) 0 0
\(400\) −12.5836 −0.629180
\(401\) 17.5279 0.875300 0.437650 0.899145i \(-0.355811\pi\)
0.437650 + 0.899145i \(0.355811\pi\)
\(402\) 0 0
\(403\) 5.23607 0.260827
\(404\) −27.4377 −1.36508
\(405\) 0 0
\(406\) 1.76393 0.0875425
\(407\) 0 0
\(408\) 0 0
\(409\) −17.7082 −0.875614 −0.437807 0.899069i \(-0.644245\pi\)
−0.437807 + 0.899069i \(0.644245\pi\)
\(410\) −4.27051 −0.210905
\(411\) 0 0
\(412\) −16.4164 −0.808778
\(413\) −11.0902 −0.545712
\(414\) 0 0
\(415\) 2.70820 0.132941
\(416\) −5.12461 −0.251255
\(417\) 0 0
\(418\) 0 0
\(419\) −29.8885 −1.46015 −0.730075 0.683367i \(-0.760515\pi\)
−0.730075 + 0.683367i \(0.760515\pi\)
\(420\) 0 0
\(421\) 0.763932 0.0372318 0.0186159 0.999827i \(-0.494074\pi\)
0.0186159 + 0.999827i \(0.494074\pi\)
\(422\) 5.96556 0.290399
\(423\) 0 0
\(424\) −3.50658 −0.170294
\(425\) 12.3607 0.599581
\(426\) 0 0
\(427\) −7.61803 −0.368663
\(428\) 5.02129 0.242713
\(429\) 0 0
\(430\) 4.79837 0.231398
\(431\) 13.8541 0.667329 0.333664 0.942692i \(-0.391715\pi\)
0.333664 + 0.942692i \(0.391715\pi\)
\(432\) 0 0
\(433\) −8.96556 −0.430857 −0.215429 0.976520i \(-0.569115\pi\)
−0.215429 + 0.976520i \(0.569115\pi\)
\(434\) −1.61803 −0.0776681
\(435\) 0 0
\(436\) 2.83282 0.135667
\(437\) 8.97871 0.429510
\(438\) 0 0
\(439\) 3.38197 0.161412 0.0807062 0.996738i \(-0.474282\pi\)
0.0807062 + 0.996738i \(0.474282\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.45898 0.0693966
\(443\) −3.09017 −0.146818 −0.0734092 0.997302i \(-0.523388\pi\)
−0.0734092 + 0.997302i \(0.523388\pi\)
\(444\) 0 0
\(445\) 6.85410 0.324916
\(446\) 0.777088 0.0367962
\(447\) 0 0
\(448\) −4.70820 −0.222442
\(449\) −12.3262 −0.581711 −0.290856 0.956767i \(-0.593940\pi\)
−0.290856 + 0.956767i \(0.593940\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.270510 0.0127237
\(453\) 0 0
\(454\) 6.12461 0.287442
\(455\) 1.23607 0.0579478
\(456\) 0 0
\(457\) −0.145898 −0.00682482 −0.00341241 0.999994i \(-0.501086\pi\)
−0.00341241 + 0.999994i \(0.501086\pi\)
\(458\) 2.58359 0.120723
\(459\) 0 0
\(460\) −9.43769 −0.440035
\(461\) 18.5066 0.861937 0.430969 0.902367i \(-0.358172\pi\)
0.430969 + 0.902367i \(0.358172\pi\)
\(462\) 0 0
\(463\) 22.2361 1.03340 0.516699 0.856167i \(-0.327161\pi\)
0.516699 + 0.856167i \(0.327161\pi\)
\(464\) 14.5279 0.674439
\(465\) 0 0
\(466\) 11.2361 0.520501
\(467\) 4.50658 0.208540 0.104270 0.994549i \(-0.466749\pi\)
0.104270 + 0.994549i \(0.466749\pi\)
\(468\) 0 0
\(469\) −8.32624 −0.384470
\(470\) −2.52786 −0.116602
\(471\) 0 0
\(472\) 16.3262 0.751476
\(473\) 0 0
\(474\) 0 0
\(475\) 7.05573 0.323739
\(476\) 5.72949 0.262611
\(477\) 0 0
\(478\) 6.52786 0.298578
\(479\) −21.7639 −0.994419 −0.497210 0.867630i \(-0.665642\pi\)
−0.497210 + 0.867630i \(0.665642\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 6.21478 0.283076
\(483\) 0 0
\(484\) 0 0
\(485\) −7.00000 −0.317854
\(486\) 0 0
\(487\) −32.5967 −1.47710 −0.738550 0.674199i \(-0.764489\pi\)
−0.738550 + 0.674199i \(0.764489\pi\)
\(488\) 11.2148 0.507669
\(489\) 0 0
\(490\) −0.381966 −0.0172555
\(491\) −22.1459 −0.999430 −0.499715 0.866190i \(-0.666562\pi\)
−0.499715 + 0.866190i \(0.666562\pi\)
\(492\) 0 0
\(493\) −14.2705 −0.642711
\(494\) 0.832816 0.0374702
\(495\) 0 0
\(496\) −13.3262 −0.598366
\(497\) 16.0902 0.721743
\(498\) 0 0
\(499\) −10.0902 −0.451698 −0.225849 0.974162i \(-0.572516\pi\)
−0.225849 + 0.974162i \(0.572516\pi\)
\(500\) −16.6869 −0.746262
\(501\) 0 0
\(502\) −8.78522 −0.392103
\(503\) 34.8328 1.55312 0.776559 0.630044i \(-0.216963\pi\)
0.776559 + 0.630044i \(0.216963\pi\)
\(504\) 0 0
\(505\) −14.7984 −0.658519
\(506\) 0 0
\(507\) 0 0
\(508\) −5.45898 −0.242203
\(509\) 10.7426 0.476159 0.238080 0.971246i \(-0.423482\pi\)
0.238080 + 0.971246i \(0.423482\pi\)
\(510\) 0 0
\(511\) −14.2361 −0.629767
\(512\) 22.3050 0.985749
\(513\) 0 0
\(514\) 3.27051 0.144256
\(515\) −8.85410 −0.390158
\(516\) 0 0
\(517\) 0 0
\(518\) 2.47214 0.108619
\(519\) 0 0
\(520\) −1.81966 −0.0797974
\(521\) 12.2361 0.536072 0.268036 0.963409i \(-0.413625\pi\)
0.268036 + 0.963409i \(0.413625\pi\)
\(522\) 0 0
\(523\) 16.5623 0.724219 0.362110 0.932136i \(-0.382057\pi\)
0.362110 + 0.932136i \(0.382057\pi\)
\(524\) 35.1246 1.53443
\(525\) 0 0
\(526\) 6.54102 0.285202
\(527\) 13.0902 0.570217
\(528\) 0 0
\(529\) 2.90983 0.126514
\(530\) −0.909830 −0.0395205
\(531\) 0 0
\(532\) 3.27051 0.141795
\(533\) −13.8197 −0.598596
\(534\) 0 0
\(535\) 2.70820 0.117086
\(536\) 12.2574 0.529437
\(537\) 0 0
\(538\) −6.43769 −0.277549
\(539\) 0 0
\(540\) 0 0
\(541\) 0.708204 0.0304481 0.0152240 0.999884i \(-0.495154\pi\)
0.0152240 + 0.999884i \(0.495154\pi\)
\(542\) −1.83282 −0.0787262
\(543\) 0 0
\(544\) −12.8115 −0.549290
\(545\) 1.52786 0.0654465
\(546\) 0 0
\(547\) −1.72949 −0.0739477 −0.0369738 0.999316i \(-0.511772\pi\)
−0.0369738 + 0.999316i \(0.511772\pi\)
\(548\) −28.4164 −1.21389
\(549\) 0 0
\(550\) 0 0
\(551\) −8.14590 −0.347027
\(552\) 0 0
\(553\) 6.38197 0.271389
\(554\) 8.79837 0.373807
\(555\) 0 0
\(556\) −22.1459 −0.939195
\(557\) 9.76393 0.413711 0.206856 0.978371i \(-0.433677\pi\)
0.206856 + 0.978371i \(0.433677\pi\)
\(558\) 0 0
\(559\) 15.5279 0.656759
\(560\) −3.14590 −0.132938
\(561\) 0 0
\(562\) 9.61803 0.405712
\(563\) 31.9787 1.34774 0.673871 0.738849i \(-0.264630\pi\)
0.673871 + 0.738849i \(0.264630\pi\)
\(564\) 0 0
\(565\) 0.145898 0.00613798
\(566\) 4.56231 0.191768
\(567\) 0 0
\(568\) −23.6869 −0.993881
\(569\) −37.3050 −1.56390 −0.781952 0.623338i \(-0.785776\pi\)
−0.781952 + 0.623338i \(0.785776\pi\)
\(570\) 0 0
\(571\) 34.3050 1.43562 0.717809 0.696240i \(-0.245145\pi\)
0.717809 + 0.696240i \(0.245145\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4.27051 0.178248
\(575\) 20.3607 0.849099
\(576\) 0 0
\(577\) 9.65248 0.401838 0.200919 0.979608i \(-0.435607\pi\)
0.200919 + 0.979608i \(0.435607\pi\)
\(578\) −2.84597 −0.118377
\(579\) 0 0
\(580\) 8.56231 0.355530
\(581\) −2.70820 −0.112355
\(582\) 0 0
\(583\) 0 0
\(584\) 20.9574 0.867225
\(585\) 0 0
\(586\) 4.20163 0.173568
\(587\) −1.00000 −0.0412744 −0.0206372 0.999787i \(-0.506569\pi\)
−0.0206372 + 0.999787i \(0.506569\pi\)
\(588\) 0 0
\(589\) 7.47214 0.307884
\(590\) 4.23607 0.174396
\(591\) 0 0
\(592\) 20.3607 0.836819
\(593\) −11.1246 −0.456833 −0.228417 0.973564i \(-0.573355\pi\)
−0.228417 + 0.973564i \(0.573355\pi\)
\(594\) 0 0
\(595\) 3.09017 0.126685
\(596\) −16.4164 −0.672442
\(597\) 0 0
\(598\) 2.40325 0.0982763
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 13.7082 0.559169 0.279585 0.960121i \(-0.409803\pi\)
0.279585 + 0.960121i \(0.409803\pi\)
\(602\) −4.79837 −0.195567
\(603\) 0 0
\(604\) −0.103326 −0.00420426
\(605\) 0 0
\(606\) 0 0
\(607\) 20.2016 0.819959 0.409979 0.912095i \(-0.365536\pi\)
0.409979 + 0.912095i \(0.365536\pi\)
\(608\) −7.31308 −0.296585
\(609\) 0 0
\(610\) 2.90983 0.117816
\(611\) −8.18034 −0.330941
\(612\) 0 0
\(613\) −0.888544 −0.0358879 −0.0179440 0.999839i \(-0.505712\pi\)
−0.0179440 + 0.999839i \(0.505712\pi\)
\(614\) −8.85410 −0.357322
\(615\) 0 0
\(616\) 0 0
\(617\) −9.41641 −0.379090 −0.189545 0.981872i \(-0.560701\pi\)
−0.189545 + 0.981872i \(0.560701\pi\)
\(618\) 0 0
\(619\) −39.7082 −1.59601 −0.798004 0.602653i \(-0.794111\pi\)
−0.798004 + 0.602653i \(0.794111\pi\)
\(620\) −7.85410 −0.315428
\(621\) 0 0
\(622\) −3.43769 −0.137839
\(623\) −6.85410 −0.274604
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 7.00000 0.279776
\(627\) 0 0
\(628\) −36.8754 −1.47149
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 17.8328 0.709913 0.354957 0.934883i \(-0.384496\pi\)
0.354957 + 0.934883i \(0.384496\pi\)
\(632\) −9.39512 −0.373718
\(633\) 0 0
\(634\) 1.69505 0.0673190
\(635\) −2.94427 −0.116840
\(636\) 0 0
\(637\) −1.23607 −0.0489748
\(638\) 0 0
\(639\) 0 0
\(640\) 10.0902 0.398849
\(641\) −31.6525 −1.25020 −0.625099 0.780546i \(-0.714941\pi\)
−0.625099 + 0.780546i \(0.714941\pi\)
\(642\) 0 0
\(643\) 1.58359 0.0624508 0.0312254 0.999512i \(-0.490059\pi\)
0.0312254 + 0.999512i \(0.490059\pi\)
\(644\) 9.43769 0.371897
\(645\) 0 0
\(646\) 2.08204 0.0819167
\(647\) −6.18034 −0.242974 −0.121487 0.992593i \(-0.538766\pi\)
−0.121487 + 0.992593i \(0.538766\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1.88854 0.0740748
\(651\) 0 0
\(652\) 16.1459 0.632322
\(653\) −18.2361 −0.713632 −0.356816 0.934175i \(-0.616138\pi\)
−0.356816 + 0.934175i \(0.616138\pi\)
\(654\) 0 0
\(655\) 18.9443 0.740214
\(656\) 35.1722 1.37324
\(657\) 0 0
\(658\) 2.52786 0.0985464
\(659\) −31.4721 −1.22598 −0.612990 0.790091i \(-0.710033\pi\)
−0.612990 + 0.790091i \(0.710033\pi\)
\(660\) 0 0
\(661\) −34.5623 −1.34432 −0.672159 0.740407i \(-0.734633\pi\)
−0.672159 + 0.740407i \(0.734633\pi\)
\(662\) 2.36068 0.0917504
\(663\) 0 0
\(664\) 3.98684 0.154720
\(665\) 1.76393 0.0684023
\(666\) 0 0
\(667\) −23.5066 −0.910178
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 3.18034 0.122867
\(671\) 0 0
\(672\) 0 0
\(673\) −6.58359 −0.253779 −0.126889 0.991917i \(-0.540499\pi\)
−0.126889 + 0.991917i \(0.540499\pi\)
\(674\) 9.90983 0.381712
\(675\) 0 0
\(676\) 21.2705 0.818097
\(677\) 36.2705 1.39399 0.696994 0.717077i \(-0.254520\pi\)
0.696994 + 0.717077i \(0.254520\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) −4.54915 −0.174452
\(681\) 0 0
\(682\) 0 0
\(683\) −0.493422 −0.0188803 −0.00944014 0.999955i \(-0.503005\pi\)
−0.00944014 + 0.999955i \(0.503005\pi\)
\(684\) 0 0
\(685\) −15.3262 −0.585585
\(686\) 0.381966 0.0145835
\(687\) 0 0
\(688\) −39.5197 −1.50668
\(689\) −2.94427 −0.112168
\(690\) 0 0
\(691\) 18.5410 0.705334 0.352667 0.935749i \(-0.385275\pi\)
0.352667 + 0.935749i \(0.385275\pi\)
\(692\) −28.5197 −1.08416
\(693\) 0 0
\(694\) −0.896674 −0.0340373
\(695\) −11.9443 −0.453072
\(696\) 0 0
\(697\) −34.5492 −1.30864
\(698\) 12.5279 0.474187
\(699\) 0 0
\(700\) 7.41641 0.280314
\(701\) −18.5066 −0.698984 −0.349492 0.936939i \(-0.613646\pi\)
−0.349492 + 0.936939i \(0.613646\pi\)
\(702\) 0 0
\(703\) −11.4164 −0.430578
\(704\) 0 0
\(705\) 0 0
\(706\) −9.20163 −0.346308
\(707\) 14.7984 0.556550
\(708\) 0 0
\(709\) −7.03444 −0.264184 −0.132092 0.991237i \(-0.542169\pi\)
−0.132092 + 0.991237i \(0.542169\pi\)
\(710\) −6.14590 −0.230651
\(711\) 0 0
\(712\) 10.0902 0.378145
\(713\) 21.5623 0.807515
\(714\) 0 0
\(715\) 0 0
\(716\) −6.97871 −0.260807
\(717\) 0 0
\(718\) 5.96556 0.222633
\(719\) 2.88854 0.107725 0.0538623 0.998548i \(-0.482847\pi\)
0.0538623 + 0.998548i \(0.482847\pi\)
\(720\) 0 0
\(721\) 8.85410 0.329744
\(722\) −6.06888 −0.225860
\(723\) 0 0
\(724\) 13.7508 0.511044
\(725\) −18.4721 −0.686038
\(726\) 0 0
\(727\) −12.4508 −0.461776 −0.230888 0.972980i \(-0.574163\pi\)
−0.230888 + 0.972980i \(0.574163\pi\)
\(728\) 1.81966 0.0674411
\(729\) 0 0
\(730\) 5.43769 0.201258
\(731\) 38.8197 1.43580
\(732\) 0 0
\(733\) 52.9443 1.95554 0.977771 0.209677i \(-0.0672413\pi\)
0.977771 + 0.209677i \(0.0672413\pi\)
\(734\) 12.5410 0.462897
\(735\) 0 0
\(736\) −21.1033 −0.777879
\(737\) 0 0
\(738\) 0 0
\(739\) −11.3820 −0.418692 −0.209346 0.977842i \(-0.567134\pi\)
−0.209346 + 0.977842i \(0.567134\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 0.909830 0.0334009
\(743\) 14.9098 0.546989 0.273494 0.961874i \(-0.411821\pi\)
0.273494 + 0.961874i \(0.411821\pi\)
\(744\) 0 0
\(745\) −8.85410 −0.324389
\(746\) −5.89667 −0.215893
\(747\) 0 0
\(748\) 0 0
\(749\) −2.70820 −0.0989556
\(750\) 0 0
\(751\) 41.3050 1.50724 0.753620 0.657311i \(-0.228306\pi\)
0.753620 + 0.657311i \(0.228306\pi\)
\(752\) 20.8197 0.759215
\(753\) 0 0
\(754\) −2.18034 −0.0794033
\(755\) −0.0557281 −0.00202815
\(756\) 0 0
\(757\) −14.7639 −0.536604 −0.268302 0.963335i \(-0.586463\pi\)
−0.268302 + 0.963335i \(0.586463\pi\)
\(758\) 12.2492 0.444912
\(759\) 0 0
\(760\) −2.59675 −0.0941939
\(761\) −15.3050 −0.554804 −0.277402 0.960754i \(-0.589473\pi\)
−0.277402 + 0.960754i \(0.589473\pi\)
\(762\) 0 0
\(763\) −1.52786 −0.0553124
\(764\) 29.2279 1.05743
\(765\) 0 0
\(766\) 13.2148 0.477469
\(767\) 13.7082 0.494975
\(768\) 0 0
\(769\) 48.5623 1.75120 0.875601 0.483035i \(-0.160466\pi\)
0.875601 + 0.483035i \(0.160466\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 28.4164 1.02273
\(773\) −40.2361 −1.44719 −0.723595 0.690224i \(-0.757512\pi\)
−0.723595 + 0.690224i \(0.757512\pi\)
\(774\) 0 0
\(775\) 16.9443 0.608656
\(776\) −10.3050 −0.369926
\(777\) 0 0
\(778\) 13.0557 0.468071
\(779\) −19.7214 −0.706591
\(780\) 0 0
\(781\) 0 0
\(782\) 6.00813 0.214850
\(783\) 0 0
\(784\) 3.14590 0.112354
\(785\) −19.8885 −0.709853
\(786\) 0 0
\(787\) 20.5623 0.732967 0.366484 0.930425i \(-0.380562\pi\)
0.366484 + 0.930425i \(0.380562\pi\)
\(788\) −7.95743 −0.283472
\(789\) 0 0
\(790\) −2.43769 −0.0867293
\(791\) −0.145898 −0.00518754
\(792\) 0 0
\(793\) 9.41641 0.334386
\(794\) −0.313082 −0.0111109
\(795\) 0 0
\(796\) 15.2705 0.541249
\(797\) −32.2361 −1.14186 −0.570930 0.820999i \(-0.693417\pi\)
−0.570930 + 0.820999i \(0.693417\pi\)
\(798\) 0 0
\(799\) −20.4508 −0.723499
\(800\) −16.5836 −0.586319
\(801\) 0 0
\(802\) 6.69505 0.236410
\(803\) 0 0
\(804\) 0 0
\(805\) 5.09017 0.179405
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) −21.7852 −0.766401
\(809\) −32.5066 −1.14287 −0.571435 0.820647i \(-0.693613\pi\)
−0.571435 + 0.820647i \(0.693613\pi\)
\(810\) 0 0
\(811\) 1.16718 0.0409854 0.0204927 0.999790i \(-0.493477\pi\)
0.0204927 + 0.999790i \(0.493477\pi\)
\(812\) −8.56231 −0.300478
\(813\) 0 0
\(814\) 0 0
\(815\) 8.70820 0.305035
\(816\) 0 0
\(817\) 22.1591 0.775247
\(818\) −6.76393 −0.236495
\(819\) 0 0
\(820\) 20.7295 0.723905
\(821\) −56.6656 −1.97764 −0.988822 0.149100i \(-0.952362\pi\)
−0.988822 + 0.149100i \(0.952362\pi\)
\(822\) 0 0
\(823\) −31.8885 −1.11156 −0.555782 0.831328i \(-0.687581\pi\)
−0.555782 + 0.831328i \(0.687581\pi\)
\(824\) −13.0344 −0.454076
\(825\) 0 0
\(826\) −4.23607 −0.147392
\(827\) 19.5967 0.681446 0.340723 0.940164i \(-0.389328\pi\)
0.340723 + 0.940164i \(0.389328\pi\)
\(828\) 0 0
\(829\) −47.9787 −1.66637 −0.833185 0.552995i \(-0.813485\pi\)
−0.833185 + 0.552995i \(0.813485\pi\)
\(830\) 1.03444 0.0359060
\(831\) 0 0
\(832\) 5.81966 0.201760
\(833\) −3.09017 −0.107068
\(834\) 0 0
\(835\) 6.47214 0.223978
\(836\) 0 0
\(837\) 0 0
\(838\) −11.4164 −0.394373
\(839\) −30.5967 −1.05632 −0.528159 0.849146i \(-0.677117\pi\)
−0.528159 + 0.849146i \(0.677117\pi\)
\(840\) 0 0
\(841\) −7.67376 −0.264612
\(842\) 0.291796 0.0100560
\(843\) 0 0
\(844\) −28.9574 −0.996756
\(845\) 11.4721 0.394653
\(846\) 0 0
\(847\) 0 0
\(848\) 7.49342 0.257325
\(849\) 0 0
\(850\) 4.72136 0.161941
\(851\) −32.9443 −1.12932
\(852\) 0 0
\(853\) −34.9098 −1.19529 −0.597645 0.801761i \(-0.703897\pi\)
−0.597645 + 0.801761i \(0.703897\pi\)
\(854\) −2.90983 −0.0995723
\(855\) 0 0
\(856\) 3.98684 0.136268
\(857\) 15.0902 0.515470 0.257735 0.966216i \(-0.417024\pi\)
0.257735 + 0.966216i \(0.417024\pi\)
\(858\) 0 0
\(859\) 21.5623 0.735696 0.367848 0.929886i \(-0.380095\pi\)
0.367848 + 0.929886i \(0.380095\pi\)
\(860\) −23.2918 −0.794244
\(861\) 0 0
\(862\) 5.29180 0.180239
\(863\) 47.9443 1.63204 0.816021 0.578022i \(-0.196175\pi\)
0.816021 + 0.578022i \(0.196175\pi\)
\(864\) 0 0
\(865\) −15.3820 −0.523003
\(866\) −3.42454 −0.116371
\(867\) 0 0
\(868\) 7.85410 0.266586
\(869\) 0 0
\(870\) 0 0
\(871\) 10.2918 0.348724
\(872\) 2.24922 0.0761683
\(873\) 0 0
\(874\) 3.42956 0.116007
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 1.29180 0.0435960
\(879\) 0 0
\(880\) 0 0
\(881\) −14.1459 −0.476587 −0.238294 0.971193i \(-0.576588\pi\)
−0.238294 + 0.971193i \(0.576588\pi\)
\(882\) 0 0
\(883\) 31.6312 1.06447 0.532237 0.846595i \(-0.321351\pi\)
0.532237 + 0.846595i \(0.321351\pi\)
\(884\) −7.08204 −0.238195
\(885\) 0 0
\(886\) −1.18034 −0.0396543
\(887\) −57.2148 −1.92108 −0.960542 0.278134i \(-0.910284\pi\)
−0.960542 + 0.278134i \(0.910284\pi\)
\(888\) 0 0
\(889\) 2.94427 0.0987477
\(890\) 2.61803 0.0877567
\(891\) 0 0
\(892\) −3.77206 −0.126298
\(893\) −11.6738 −0.390648
\(894\) 0 0
\(895\) −3.76393 −0.125814
\(896\) −10.0902 −0.337089
\(897\) 0 0
\(898\) −4.70820 −0.157115
\(899\) −19.5623 −0.652439
\(900\) 0 0
\(901\) −7.36068 −0.245220
\(902\) 0 0
\(903\) 0 0
\(904\) 0.214782 0.00714353
\(905\) 7.41641 0.246530
\(906\) 0 0
\(907\) −15.7639 −0.523433 −0.261716 0.965145i \(-0.584289\pi\)
−0.261716 + 0.965145i \(0.584289\pi\)
\(908\) −29.7295 −0.986608
\(909\) 0 0
\(910\) 0.472136 0.0156512
\(911\) 45.1803 1.49689 0.748446 0.663196i \(-0.230800\pi\)
0.748446 + 0.663196i \(0.230800\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.0557281 −0.00184332
\(915\) 0 0
\(916\) −12.5410 −0.414367
\(917\) −18.9443 −0.625595
\(918\) 0 0
\(919\) −14.8885 −0.491128 −0.245564 0.969380i \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(920\) −7.49342 −0.247051
\(921\) 0 0
\(922\) 7.06888 0.232801
\(923\) −19.8885 −0.654639
\(924\) 0 0
\(925\) −25.8885 −0.851210
\(926\) 8.49342 0.279111
\(927\) 0 0
\(928\) 19.1459 0.628495
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) −1.76393 −0.0578105
\(932\) −54.5410 −1.78655
\(933\) 0 0
\(934\) 1.72136 0.0563246
\(935\) 0 0
\(936\) 0 0
\(937\) −51.7214 −1.68966 −0.844832 0.535032i \(-0.820299\pi\)
−0.844832 + 0.535032i \(0.820299\pi\)
\(938\) −3.18034 −0.103842
\(939\) 0 0
\(940\) 12.2705 0.400220
\(941\) −27.0689 −0.882420 −0.441210 0.897404i \(-0.645451\pi\)
−0.441210 + 0.897404i \(0.645451\pi\)
\(942\) 0 0
\(943\) −56.9098 −1.85324
\(944\) −34.8885 −1.13553
\(945\) 0 0
\(946\) 0 0
\(947\) −6.29180 −0.204456 −0.102228 0.994761i \(-0.532597\pi\)
−0.102228 + 0.994761i \(0.532597\pi\)
\(948\) 0 0
\(949\) 17.5967 0.571215
\(950\) 2.69505 0.0874389
\(951\) 0 0
\(952\) 4.54915 0.147439
\(953\) 6.23607 0.202006 0.101003 0.994886i \(-0.467795\pi\)
0.101003 + 0.994886i \(0.467795\pi\)
\(954\) 0 0
\(955\) 15.7639 0.510109
\(956\) −31.6869 −1.02483
\(957\) 0 0
\(958\) −8.31308 −0.268583
\(959\) 15.3262 0.494910
\(960\) 0 0
\(961\) −13.0557 −0.421153
\(962\) −3.05573 −0.0985206
\(963\) 0 0
\(964\) −30.1672 −0.971620
\(965\) 15.3262 0.493369
\(966\) 0 0
\(967\) 27.4508 0.882760 0.441380 0.897320i \(-0.354489\pi\)
0.441380 + 0.897320i \(0.354489\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −2.67376 −0.0858493
\(971\) 28.2492 0.906561 0.453280 0.891368i \(-0.350254\pi\)
0.453280 + 0.891368i \(0.350254\pi\)
\(972\) 0 0
\(973\) 11.9443 0.382916
\(974\) −12.4508 −0.398951
\(975\) 0 0
\(976\) −23.9656 −0.767119
\(977\) 10.8197 0.346152 0.173076 0.984909i \(-0.444629\pi\)
0.173076 + 0.984909i \(0.444629\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.85410 0.0592271
\(981\) 0 0
\(982\) −8.45898 −0.269937
\(983\) 12.3820 0.394923 0.197462 0.980311i \(-0.436730\pi\)
0.197462 + 0.980311i \(0.436730\pi\)
\(984\) 0 0
\(985\) −4.29180 −0.136748
\(986\) −5.45085 −0.173590
\(987\) 0 0
\(988\) −4.04257 −0.128611
\(989\) 63.9443 2.03331
\(990\) 0 0
\(991\) −0.729490 −0.0231730 −0.0115865 0.999933i \(-0.503688\pi\)
−0.0115865 + 0.999933i \(0.503688\pi\)
\(992\) −17.5623 −0.557604
\(993\) 0 0
\(994\) 6.14590 0.194936
\(995\) 8.23607 0.261101
\(996\) 0 0
\(997\) −55.8673 −1.76933 −0.884667 0.466224i \(-0.845614\pi\)
−0.884667 + 0.466224i \(0.845614\pi\)
\(998\) −3.85410 −0.121999
\(999\) 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.bx.1.1 2
3.2 odd 2 847.2.a.d.1.2 2
11.10 odd 2 7623.2.a.t.1.2 2
21.20 even 2 5929.2.a.i.1.2 2
33.2 even 10 847.2.f.j.323.1 4
33.5 odd 10 847.2.f.d.729.1 4
33.8 even 10 847.2.f.c.372.1 4
33.14 odd 10 847.2.f.l.372.1 4
33.17 even 10 847.2.f.j.729.1 4
33.20 odd 10 847.2.f.d.323.1 4
33.26 odd 10 847.2.f.l.148.1 4
33.29 even 10 847.2.f.c.148.1 4
33.32 even 2 847.2.a.h.1.1 yes 2
231.230 odd 2 5929.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.d.1.2 2 3.2 odd 2
847.2.a.h.1.1 yes 2 33.32 even 2
847.2.f.c.148.1 4 33.29 even 10
847.2.f.c.372.1 4 33.8 even 10
847.2.f.d.323.1 4 33.20 odd 10
847.2.f.d.729.1 4 33.5 odd 10
847.2.f.j.323.1 4 33.2 even 10
847.2.f.j.729.1 4 33.17 even 10
847.2.f.l.148.1 4 33.26 odd 10
847.2.f.l.372.1 4 33.14 odd 10
5929.2.a.i.1.2 2 21.20 even 2
5929.2.a.s.1.1 2 231.230 odd 2
7623.2.a.t.1.2 2 11.10 odd 2
7623.2.a.bx.1.1 2 1.1 even 1 trivial