Properties

Label 8004.2.a.i.1.12
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(2.30095\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.30095 q^{5} -4.14667 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.30095 q^{5} -4.14667 q^{7} +1.00000 q^{9} +1.07535 q^{11} +7.08498 q^{13} -2.30095 q^{15} +6.59628 q^{17} -6.26448 q^{19} +4.14667 q^{21} +1.00000 q^{23} +0.294386 q^{25} -1.00000 q^{27} +1.00000 q^{29} -5.73160 q^{31} -1.07535 q^{33} -9.54128 q^{35} +9.95867 q^{37} -7.08498 q^{39} -9.49984 q^{41} +10.2369 q^{43} +2.30095 q^{45} +6.32540 q^{47} +10.1948 q^{49} -6.59628 q^{51} -11.2635 q^{53} +2.47432 q^{55} +6.26448 q^{57} +12.4884 q^{59} +0.914556 q^{61} -4.14667 q^{63} +16.3022 q^{65} +0.117437 q^{67} -1.00000 q^{69} +5.86395 q^{71} +11.6237 q^{73} -0.294386 q^{75} -4.45910 q^{77} -9.13043 q^{79} +1.00000 q^{81} -2.02819 q^{83} +15.1777 q^{85} -1.00000 q^{87} -5.19189 q^{89} -29.3791 q^{91} +5.73160 q^{93} -14.4143 q^{95} +11.6087 q^{97} +1.07535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.30095 1.02902 0.514509 0.857485i \(-0.327974\pi\)
0.514509 + 0.857485i \(0.327974\pi\)
\(6\) 0 0
\(7\) −4.14667 −1.56729 −0.783646 0.621208i \(-0.786642\pi\)
−0.783646 + 0.621208i \(0.786642\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.07535 0.324229 0.162115 0.986772i \(-0.448169\pi\)
0.162115 + 0.986772i \(0.448169\pi\)
\(12\) 0 0
\(13\) 7.08498 1.96502 0.982510 0.186208i \(-0.0596200\pi\)
0.982510 + 0.186208i \(0.0596200\pi\)
\(14\) 0 0
\(15\) −2.30095 −0.594104
\(16\) 0 0
\(17\) 6.59628 1.59983 0.799916 0.600112i \(-0.204877\pi\)
0.799916 + 0.600112i \(0.204877\pi\)
\(18\) 0 0
\(19\) −6.26448 −1.43717 −0.718585 0.695439i \(-0.755210\pi\)
−0.718585 + 0.695439i \(0.755210\pi\)
\(20\) 0 0
\(21\) 4.14667 0.904877
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0.294386 0.0588772
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −5.73160 −1.02943 −0.514713 0.857363i \(-0.672102\pi\)
−0.514713 + 0.857363i \(0.672102\pi\)
\(32\) 0 0
\(33\) −1.07535 −0.187194
\(34\) 0 0
\(35\) −9.54128 −1.61277
\(36\) 0 0
\(37\) 9.95867 1.63719 0.818597 0.574368i \(-0.194752\pi\)
0.818597 + 0.574368i \(0.194752\pi\)
\(38\) 0 0
\(39\) −7.08498 −1.13451
\(40\) 0 0
\(41\) −9.49984 −1.48363 −0.741813 0.670607i \(-0.766034\pi\)
−0.741813 + 0.670607i \(0.766034\pi\)
\(42\) 0 0
\(43\) 10.2369 1.56111 0.780556 0.625086i \(-0.214936\pi\)
0.780556 + 0.625086i \(0.214936\pi\)
\(44\) 0 0
\(45\) 2.30095 0.343006
\(46\) 0 0
\(47\) 6.32540 0.922654 0.461327 0.887230i \(-0.347373\pi\)
0.461327 + 0.887230i \(0.347373\pi\)
\(48\) 0 0
\(49\) 10.1948 1.45640
\(50\) 0 0
\(51\) −6.59628 −0.923663
\(52\) 0 0
\(53\) −11.2635 −1.54716 −0.773582 0.633696i \(-0.781537\pi\)
−0.773582 + 0.633696i \(0.781537\pi\)
\(54\) 0 0
\(55\) 2.47432 0.333638
\(56\) 0 0
\(57\) 6.26448 0.829751
\(58\) 0 0
\(59\) 12.4884 1.62585 0.812924 0.582369i \(-0.197874\pi\)
0.812924 + 0.582369i \(0.197874\pi\)
\(60\) 0 0
\(61\) 0.914556 0.117097 0.0585484 0.998285i \(-0.481353\pi\)
0.0585484 + 0.998285i \(0.481353\pi\)
\(62\) 0 0
\(63\) −4.14667 −0.522431
\(64\) 0 0
\(65\) 16.3022 2.02204
\(66\) 0 0
\(67\) 0.117437 0.0143472 0.00717360 0.999974i \(-0.497717\pi\)
0.00717360 + 0.999974i \(0.497717\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 5.86395 0.695923 0.347961 0.937509i \(-0.386874\pi\)
0.347961 + 0.937509i \(0.386874\pi\)
\(72\) 0 0
\(73\) 11.6237 1.36045 0.680224 0.733004i \(-0.261883\pi\)
0.680224 + 0.733004i \(0.261883\pi\)
\(74\) 0 0
\(75\) −0.294386 −0.0339928
\(76\) 0 0
\(77\) −4.45910 −0.508162
\(78\) 0 0
\(79\) −9.13043 −1.02725 −0.513627 0.858014i \(-0.671698\pi\)
−0.513627 + 0.858014i \(0.671698\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.02819 −0.222622 −0.111311 0.993786i \(-0.535505\pi\)
−0.111311 + 0.993786i \(0.535505\pi\)
\(84\) 0 0
\(85\) 15.1777 1.64626
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −5.19189 −0.550340 −0.275170 0.961396i \(-0.588734\pi\)
−0.275170 + 0.961396i \(0.588734\pi\)
\(90\) 0 0
\(91\) −29.3791 −3.07976
\(92\) 0 0
\(93\) 5.73160 0.594339
\(94\) 0 0
\(95\) −14.4143 −1.47887
\(96\) 0 0
\(97\) 11.6087 1.17869 0.589344 0.807882i \(-0.299386\pi\)
0.589344 + 0.807882i \(0.299386\pi\)
\(98\) 0 0
\(99\) 1.07535 0.108076
\(100\) 0 0
\(101\) −10.5553 −1.05029 −0.525146 0.851012i \(-0.675989\pi\)
−0.525146 + 0.851012i \(0.675989\pi\)
\(102\) 0 0
\(103\) 0.940511 0.0926713 0.0463356 0.998926i \(-0.485246\pi\)
0.0463356 + 0.998926i \(0.485246\pi\)
\(104\) 0 0
\(105\) 9.54128 0.931134
\(106\) 0 0
\(107\) −2.02555 −0.195817 −0.0979087 0.995195i \(-0.531215\pi\)
−0.0979087 + 0.995195i \(0.531215\pi\)
\(108\) 0 0
\(109\) −10.5747 −1.01287 −0.506437 0.862277i \(-0.669038\pi\)
−0.506437 + 0.862277i \(0.669038\pi\)
\(110\) 0 0
\(111\) −9.95867 −0.945235
\(112\) 0 0
\(113\) 2.23699 0.210438 0.105219 0.994449i \(-0.466446\pi\)
0.105219 + 0.994449i \(0.466446\pi\)
\(114\) 0 0
\(115\) 2.30095 0.214565
\(116\) 0 0
\(117\) 7.08498 0.655007
\(118\) 0 0
\(119\) −27.3525 −2.50740
\(120\) 0 0
\(121\) −9.84363 −0.894875
\(122\) 0 0
\(123\) 9.49984 0.856572
\(124\) 0 0
\(125\) −10.8274 −0.968432
\(126\) 0 0
\(127\) −3.23810 −0.287334 −0.143667 0.989626i \(-0.545890\pi\)
−0.143667 + 0.989626i \(0.545890\pi\)
\(128\) 0 0
\(129\) −10.2369 −0.901308
\(130\) 0 0
\(131\) −4.86048 −0.424662 −0.212331 0.977198i \(-0.568106\pi\)
−0.212331 + 0.977198i \(0.568106\pi\)
\(132\) 0 0
\(133\) 25.9767 2.25247
\(134\) 0 0
\(135\) −2.30095 −0.198035
\(136\) 0 0
\(137\) −0.469986 −0.0401536 −0.0200768 0.999798i \(-0.506391\pi\)
−0.0200768 + 0.999798i \(0.506391\pi\)
\(138\) 0 0
\(139\) −4.81811 −0.408667 −0.204333 0.978901i \(-0.565503\pi\)
−0.204333 + 0.978901i \(0.565503\pi\)
\(140\) 0 0
\(141\) −6.32540 −0.532695
\(142\) 0 0
\(143\) 7.61882 0.637117
\(144\) 0 0
\(145\) 2.30095 0.191084
\(146\) 0 0
\(147\) −10.1948 −0.840856
\(148\) 0 0
\(149\) 10.6908 0.875825 0.437912 0.899018i \(-0.355718\pi\)
0.437912 + 0.899018i \(0.355718\pi\)
\(150\) 0 0
\(151\) 9.84163 0.800901 0.400450 0.916319i \(-0.368854\pi\)
0.400450 + 0.916319i \(0.368854\pi\)
\(152\) 0 0
\(153\) 6.59628 0.533277
\(154\) 0 0
\(155\) −13.1881 −1.05930
\(156\) 0 0
\(157\) 0.109545 0.00874268 0.00437134 0.999990i \(-0.498609\pi\)
0.00437134 + 0.999990i \(0.498609\pi\)
\(158\) 0 0
\(159\) 11.2635 0.893255
\(160\) 0 0
\(161\) −4.14667 −0.326803
\(162\) 0 0
\(163\) −2.53117 −0.198256 −0.0991282 0.995075i \(-0.531605\pi\)
−0.0991282 + 0.995075i \(0.531605\pi\)
\(164\) 0 0
\(165\) −2.47432 −0.192626
\(166\) 0 0
\(167\) −21.1472 −1.63642 −0.818209 0.574921i \(-0.805033\pi\)
−0.818209 + 0.574921i \(0.805033\pi\)
\(168\) 0 0
\(169\) 37.1970 2.86131
\(170\) 0 0
\(171\) −6.26448 −0.479057
\(172\) 0 0
\(173\) 3.94554 0.299974 0.149987 0.988688i \(-0.452077\pi\)
0.149987 + 0.988688i \(0.452077\pi\)
\(174\) 0 0
\(175\) −1.22072 −0.0922778
\(176\) 0 0
\(177\) −12.4884 −0.938684
\(178\) 0 0
\(179\) 10.7894 0.806440 0.403220 0.915103i \(-0.367891\pi\)
0.403220 + 0.915103i \(0.367891\pi\)
\(180\) 0 0
\(181\) 14.6702 1.09042 0.545212 0.838298i \(-0.316449\pi\)
0.545212 + 0.838298i \(0.316449\pi\)
\(182\) 0 0
\(183\) −0.914556 −0.0676059
\(184\) 0 0
\(185\) 22.9144 1.68470
\(186\) 0 0
\(187\) 7.09329 0.518712
\(188\) 0 0
\(189\) 4.14667 0.301626
\(190\) 0 0
\(191\) 11.8055 0.854217 0.427108 0.904200i \(-0.359532\pi\)
0.427108 + 0.904200i \(0.359532\pi\)
\(192\) 0 0
\(193\) 10.0127 0.720729 0.360365 0.932812i \(-0.382652\pi\)
0.360365 + 0.932812i \(0.382652\pi\)
\(194\) 0 0
\(195\) −16.3022 −1.16743
\(196\) 0 0
\(197\) 2.11966 0.151019 0.0755096 0.997145i \(-0.475942\pi\)
0.0755096 + 0.997145i \(0.475942\pi\)
\(198\) 0 0
\(199\) −2.88360 −0.204413 −0.102206 0.994763i \(-0.532590\pi\)
−0.102206 + 0.994763i \(0.532590\pi\)
\(200\) 0 0
\(201\) −0.117437 −0.00828335
\(202\) 0 0
\(203\) −4.14667 −0.291039
\(204\) 0 0
\(205\) −21.8587 −1.52668
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −6.73649 −0.465973
\(210\) 0 0
\(211\) 6.61196 0.455186 0.227593 0.973756i \(-0.426914\pi\)
0.227593 + 0.973756i \(0.426914\pi\)
\(212\) 0 0
\(213\) −5.86395 −0.401791
\(214\) 0 0
\(215\) 23.5546 1.60641
\(216\) 0 0
\(217\) 23.7670 1.61341
\(218\) 0 0
\(219\) −11.6237 −0.785455
\(220\) 0 0
\(221\) 46.7345 3.14370
\(222\) 0 0
\(223\) 22.5569 1.51052 0.755262 0.655423i \(-0.227509\pi\)
0.755262 + 0.655423i \(0.227509\pi\)
\(224\) 0 0
\(225\) 0.294386 0.0196257
\(226\) 0 0
\(227\) 13.7717 0.914062 0.457031 0.889451i \(-0.348913\pi\)
0.457031 + 0.889451i \(0.348913\pi\)
\(228\) 0 0
\(229\) −15.5754 −1.02925 −0.514624 0.857416i \(-0.672069\pi\)
−0.514624 + 0.857416i \(0.672069\pi\)
\(230\) 0 0
\(231\) 4.45910 0.293388
\(232\) 0 0
\(233\) −20.8619 −1.36671 −0.683356 0.730086i \(-0.739480\pi\)
−0.683356 + 0.730086i \(0.739480\pi\)
\(234\) 0 0
\(235\) 14.5544 0.949428
\(236\) 0 0
\(237\) 9.13043 0.593085
\(238\) 0 0
\(239\) 10.3213 0.667630 0.333815 0.942639i \(-0.391664\pi\)
0.333815 + 0.942639i \(0.391664\pi\)
\(240\) 0 0
\(241\) 5.34079 0.344031 0.172015 0.985094i \(-0.444972\pi\)
0.172015 + 0.985094i \(0.444972\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 23.4578 1.49867
\(246\) 0 0
\(247\) −44.3837 −2.82407
\(248\) 0 0
\(249\) 2.02819 0.128531
\(250\) 0 0
\(251\) −6.13104 −0.386988 −0.193494 0.981101i \(-0.561982\pi\)
−0.193494 + 0.981101i \(0.561982\pi\)
\(252\) 0 0
\(253\) 1.07535 0.0676065
\(254\) 0 0
\(255\) −15.1777 −0.950466
\(256\) 0 0
\(257\) −9.88476 −0.616594 −0.308297 0.951290i \(-0.599759\pi\)
−0.308297 + 0.951290i \(0.599759\pi\)
\(258\) 0 0
\(259\) −41.2953 −2.56596
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 20.5660 1.26815 0.634077 0.773270i \(-0.281380\pi\)
0.634077 + 0.773270i \(0.281380\pi\)
\(264\) 0 0
\(265\) −25.9168 −1.59206
\(266\) 0 0
\(267\) 5.19189 0.317739
\(268\) 0 0
\(269\) −7.11530 −0.433827 −0.216914 0.976191i \(-0.569599\pi\)
−0.216914 + 0.976191i \(0.569599\pi\)
\(270\) 0 0
\(271\) 28.8149 1.75038 0.875189 0.483781i \(-0.160737\pi\)
0.875189 + 0.483781i \(0.160737\pi\)
\(272\) 0 0
\(273\) 29.3791 1.77810
\(274\) 0 0
\(275\) 0.316567 0.0190897
\(276\) 0 0
\(277\) 7.38971 0.444005 0.222002 0.975046i \(-0.428741\pi\)
0.222002 + 0.975046i \(0.428741\pi\)
\(278\) 0 0
\(279\) −5.73160 −0.343142
\(280\) 0 0
\(281\) −0.755545 −0.0450720 −0.0225360 0.999746i \(-0.507174\pi\)
−0.0225360 + 0.999746i \(0.507174\pi\)
\(282\) 0 0
\(283\) 28.5928 1.69967 0.849833 0.527053i \(-0.176703\pi\)
0.849833 + 0.527053i \(0.176703\pi\)
\(284\) 0 0
\(285\) 14.4143 0.853828
\(286\) 0 0
\(287\) 39.3927 2.32528
\(288\) 0 0
\(289\) 26.5109 1.55946
\(290\) 0 0
\(291\) −11.6087 −0.680516
\(292\) 0 0
\(293\) 11.4820 0.670787 0.335393 0.942078i \(-0.391131\pi\)
0.335393 + 0.942078i \(0.391131\pi\)
\(294\) 0 0
\(295\) 28.7352 1.67303
\(296\) 0 0
\(297\) −1.07535 −0.0623980
\(298\) 0 0
\(299\) 7.08498 0.409735
\(300\) 0 0
\(301\) −42.4490 −2.44672
\(302\) 0 0
\(303\) 10.5553 0.606386
\(304\) 0 0
\(305\) 2.10435 0.120495
\(306\) 0 0
\(307\) −11.8578 −0.676762 −0.338381 0.941009i \(-0.609879\pi\)
−0.338381 + 0.941009i \(0.609879\pi\)
\(308\) 0 0
\(309\) −0.940511 −0.0535038
\(310\) 0 0
\(311\) 17.6246 0.999397 0.499699 0.866199i \(-0.333444\pi\)
0.499699 + 0.866199i \(0.333444\pi\)
\(312\) 0 0
\(313\) −18.7639 −1.06060 −0.530300 0.847810i \(-0.677921\pi\)
−0.530300 + 0.847810i \(0.677921\pi\)
\(314\) 0 0
\(315\) −9.54128 −0.537590
\(316\) 0 0
\(317\) 15.2215 0.854922 0.427461 0.904034i \(-0.359408\pi\)
0.427461 + 0.904034i \(0.359408\pi\)
\(318\) 0 0
\(319\) 1.07535 0.0602079
\(320\) 0 0
\(321\) 2.02555 0.113055
\(322\) 0 0
\(323\) −41.3223 −2.29923
\(324\) 0 0
\(325\) 2.08572 0.115695
\(326\) 0 0
\(327\) 10.5747 0.584783
\(328\) 0 0
\(329\) −26.2293 −1.44607
\(330\) 0 0
\(331\) −4.49941 −0.247310 −0.123655 0.992325i \(-0.539462\pi\)
−0.123655 + 0.992325i \(0.539462\pi\)
\(332\) 0 0
\(333\) 9.95867 0.545732
\(334\) 0 0
\(335\) 0.270217 0.0147635
\(336\) 0 0
\(337\) −23.2540 −1.26672 −0.633362 0.773855i \(-0.718326\pi\)
−0.633362 + 0.773855i \(0.718326\pi\)
\(338\) 0 0
\(339\) −2.23699 −0.121496
\(340\) 0 0
\(341\) −6.16346 −0.333770
\(342\) 0 0
\(343\) −13.2479 −0.715319
\(344\) 0 0
\(345\) −2.30095 −0.123879
\(346\) 0 0
\(347\) 27.7632 1.49040 0.745202 0.666839i \(-0.232353\pi\)
0.745202 + 0.666839i \(0.232353\pi\)
\(348\) 0 0
\(349\) −22.4824 −1.20346 −0.601728 0.798701i \(-0.705521\pi\)
−0.601728 + 0.798701i \(0.705521\pi\)
\(350\) 0 0
\(351\) −7.08498 −0.378168
\(352\) 0 0
\(353\) 6.35029 0.337992 0.168996 0.985617i \(-0.445948\pi\)
0.168996 + 0.985617i \(0.445948\pi\)
\(354\) 0 0
\(355\) 13.4927 0.716117
\(356\) 0 0
\(357\) 27.3525 1.44765
\(358\) 0 0
\(359\) −26.3409 −1.39022 −0.695110 0.718903i \(-0.744644\pi\)
−0.695110 + 0.718903i \(0.744644\pi\)
\(360\) 0 0
\(361\) 20.2437 1.06546
\(362\) 0 0
\(363\) 9.84363 0.516657
\(364\) 0 0
\(365\) 26.7455 1.39992
\(366\) 0 0
\(367\) 9.66726 0.504627 0.252314 0.967646i \(-0.418809\pi\)
0.252314 + 0.967646i \(0.418809\pi\)
\(368\) 0 0
\(369\) −9.49984 −0.494542
\(370\) 0 0
\(371\) 46.7061 2.42486
\(372\) 0 0
\(373\) −14.3987 −0.745535 −0.372768 0.927925i \(-0.621591\pi\)
−0.372768 + 0.927925i \(0.621591\pi\)
\(374\) 0 0
\(375\) 10.8274 0.559124
\(376\) 0 0
\(377\) 7.08498 0.364895
\(378\) 0 0
\(379\) 28.8818 1.48356 0.741780 0.670644i \(-0.233982\pi\)
0.741780 + 0.670644i \(0.233982\pi\)
\(380\) 0 0
\(381\) 3.23810 0.165893
\(382\) 0 0
\(383\) 12.2556 0.626230 0.313115 0.949715i \(-0.398627\pi\)
0.313115 + 0.949715i \(0.398627\pi\)
\(384\) 0 0
\(385\) −10.2602 −0.522908
\(386\) 0 0
\(387\) 10.2369 0.520371
\(388\) 0 0
\(389\) 30.5051 1.54667 0.773335 0.633997i \(-0.218587\pi\)
0.773335 + 0.633997i \(0.218587\pi\)
\(390\) 0 0
\(391\) 6.59628 0.333588
\(392\) 0 0
\(393\) 4.86048 0.245179
\(394\) 0 0
\(395\) −21.0087 −1.05706
\(396\) 0 0
\(397\) 19.6015 0.983770 0.491885 0.870660i \(-0.336308\pi\)
0.491885 + 0.870660i \(0.336308\pi\)
\(398\) 0 0
\(399\) −25.9767 −1.30046
\(400\) 0 0
\(401\) −11.8523 −0.591876 −0.295938 0.955207i \(-0.595632\pi\)
−0.295938 + 0.955207i \(0.595632\pi\)
\(402\) 0 0
\(403\) −40.6083 −2.02284
\(404\) 0 0
\(405\) 2.30095 0.114335
\(406\) 0 0
\(407\) 10.7090 0.530827
\(408\) 0 0
\(409\) 10.5742 0.522862 0.261431 0.965222i \(-0.415806\pi\)
0.261431 + 0.965222i \(0.415806\pi\)
\(410\) 0 0
\(411\) 0.469986 0.0231827
\(412\) 0 0
\(413\) −51.7851 −2.54818
\(414\) 0 0
\(415\) −4.66676 −0.229082
\(416\) 0 0
\(417\) 4.81811 0.235944
\(418\) 0 0
\(419\) 27.5181 1.34435 0.672174 0.740394i \(-0.265361\pi\)
0.672174 + 0.740394i \(0.265361\pi\)
\(420\) 0 0
\(421\) −29.8114 −1.45292 −0.726460 0.687208i \(-0.758836\pi\)
−0.726460 + 0.687208i \(0.758836\pi\)
\(422\) 0 0
\(423\) 6.32540 0.307551
\(424\) 0 0
\(425\) 1.94185 0.0941936
\(426\) 0 0
\(427\) −3.79236 −0.183525
\(428\) 0 0
\(429\) −7.61882 −0.367840
\(430\) 0 0
\(431\) 29.5427 1.42302 0.711511 0.702675i \(-0.248011\pi\)
0.711511 + 0.702675i \(0.248011\pi\)
\(432\) 0 0
\(433\) −16.5886 −0.797198 −0.398599 0.917125i \(-0.630504\pi\)
−0.398599 + 0.917125i \(0.630504\pi\)
\(434\) 0 0
\(435\) −2.30095 −0.110322
\(436\) 0 0
\(437\) −6.26448 −0.299671
\(438\) 0 0
\(439\) 0.326091 0.0155635 0.00778174 0.999970i \(-0.497523\pi\)
0.00778174 + 0.999970i \(0.497523\pi\)
\(440\) 0 0
\(441\) 10.1948 0.485468
\(442\) 0 0
\(443\) −32.6424 −1.55089 −0.775444 0.631417i \(-0.782474\pi\)
−0.775444 + 0.631417i \(0.782474\pi\)
\(444\) 0 0
\(445\) −11.9463 −0.566309
\(446\) 0 0
\(447\) −10.6908 −0.505658
\(448\) 0 0
\(449\) 0.219120 0.0103409 0.00517045 0.999987i \(-0.498354\pi\)
0.00517045 + 0.999987i \(0.498354\pi\)
\(450\) 0 0
\(451\) −10.2156 −0.481035
\(452\) 0 0
\(453\) −9.84163 −0.462400
\(454\) 0 0
\(455\) −67.5998 −3.16913
\(456\) 0 0
\(457\) 28.6459 1.34000 0.669999 0.742362i \(-0.266295\pi\)
0.669999 + 0.742362i \(0.266295\pi\)
\(458\) 0 0
\(459\) −6.59628 −0.307888
\(460\) 0 0
\(461\) −26.9950 −1.25728 −0.628641 0.777696i \(-0.716388\pi\)
−0.628641 + 0.777696i \(0.716388\pi\)
\(462\) 0 0
\(463\) 35.8655 1.66681 0.833405 0.552663i \(-0.186388\pi\)
0.833405 + 0.552663i \(0.186388\pi\)
\(464\) 0 0
\(465\) 13.1881 0.611586
\(466\) 0 0
\(467\) 11.9580 0.553350 0.276675 0.960964i \(-0.410767\pi\)
0.276675 + 0.960964i \(0.410767\pi\)
\(468\) 0 0
\(469\) −0.486971 −0.0224862
\(470\) 0 0
\(471\) −0.109545 −0.00504759
\(472\) 0 0
\(473\) 11.0082 0.506158
\(474\) 0 0
\(475\) −1.84418 −0.0846166
\(476\) 0 0
\(477\) −11.2635 −0.515721
\(478\) 0 0
\(479\) 32.6505 1.49184 0.745919 0.666037i \(-0.232011\pi\)
0.745919 + 0.666037i \(0.232011\pi\)
\(480\) 0 0
\(481\) 70.5570 3.21712
\(482\) 0 0
\(483\) 4.14667 0.188680
\(484\) 0 0
\(485\) 26.7111 1.21289
\(486\) 0 0
\(487\) 32.9040 1.49102 0.745510 0.666494i \(-0.232206\pi\)
0.745510 + 0.666494i \(0.232206\pi\)
\(488\) 0 0
\(489\) 2.53117 0.114463
\(490\) 0 0
\(491\) −3.13387 −0.141429 −0.0707147 0.997497i \(-0.522528\pi\)
−0.0707147 + 0.997497i \(0.522528\pi\)
\(492\) 0 0
\(493\) 6.59628 0.297081
\(494\) 0 0
\(495\) 2.47432 0.111213
\(496\) 0 0
\(497\) −24.3158 −1.09071
\(498\) 0 0
\(499\) −3.37944 −0.151285 −0.0756423 0.997135i \(-0.524101\pi\)
−0.0756423 + 0.997135i \(0.524101\pi\)
\(500\) 0 0
\(501\) 21.1472 0.944786
\(502\) 0 0
\(503\) 5.30322 0.236459 0.118229 0.992986i \(-0.462278\pi\)
0.118229 + 0.992986i \(0.462278\pi\)
\(504\) 0 0
\(505\) −24.2872 −1.08077
\(506\) 0 0
\(507\) −37.1970 −1.65198
\(508\) 0 0
\(509\) 34.1426 1.51334 0.756672 0.653794i \(-0.226824\pi\)
0.756672 + 0.653794i \(0.226824\pi\)
\(510\) 0 0
\(511\) −48.1995 −2.13222
\(512\) 0 0
\(513\) 6.26448 0.276584
\(514\) 0 0
\(515\) 2.16407 0.0953604
\(516\) 0 0
\(517\) 6.80200 0.299152
\(518\) 0 0
\(519\) −3.94554 −0.173190
\(520\) 0 0
\(521\) 33.1152 1.45080 0.725402 0.688325i \(-0.241654\pi\)
0.725402 + 0.688325i \(0.241654\pi\)
\(522\) 0 0
\(523\) 0.226623 0.00990954 0.00495477 0.999988i \(-0.498423\pi\)
0.00495477 + 0.999988i \(0.498423\pi\)
\(524\) 0 0
\(525\) 1.22072 0.0532766
\(526\) 0 0
\(527\) −37.8072 −1.64691
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.4884 0.541950
\(532\) 0 0
\(533\) −67.3062 −2.91536
\(534\) 0 0
\(535\) −4.66070 −0.201500
\(536\) 0 0
\(537\) −10.7894 −0.465598
\(538\) 0 0
\(539\) 10.9630 0.472209
\(540\) 0 0
\(541\) 8.87188 0.381432 0.190716 0.981645i \(-0.438919\pi\)
0.190716 + 0.981645i \(0.438919\pi\)
\(542\) 0 0
\(543\) −14.6702 −0.629557
\(544\) 0 0
\(545\) −24.3319 −1.04227
\(546\) 0 0
\(547\) 26.1436 1.11782 0.558909 0.829229i \(-0.311220\pi\)
0.558909 + 0.829229i \(0.311220\pi\)
\(548\) 0 0
\(549\) 0.914556 0.0390323
\(550\) 0 0
\(551\) −6.26448 −0.266876
\(552\) 0 0
\(553\) 37.8608 1.61001
\(554\) 0 0
\(555\) −22.9144 −0.972663
\(556\) 0 0
\(557\) −7.38317 −0.312835 −0.156418 0.987691i \(-0.549995\pi\)
−0.156418 + 0.987691i \(0.549995\pi\)
\(558\) 0 0
\(559\) 72.5282 3.06762
\(560\) 0 0
\(561\) −7.09329 −0.299479
\(562\) 0 0
\(563\) 2.98968 0.126000 0.0629999 0.998014i \(-0.479933\pi\)
0.0629999 + 0.998014i \(0.479933\pi\)
\(564\) 0 0
\(565\) 5.14720 0.216545
\(566\) 0 0
\(567\) −4.14667 −0.174144
\(568\) 0 0
\(569\) 27.9097 1.17004 0.585018 0.811021i \(-0.301088\pi\)
0.585018 + 0.811021i \(0.301088\pi\)
\(570\) 0 0
\(571\) −45.5825 −1.90757 −0.953783 0.300495i \(-0.902848\pi\)
−0.953783 + 0.300495i \(0.902848\pi\)
\(572\) 0 0
\(573\) −11.8055 −0.493182
\(574\) 0 0
\(575\) 0.294386 0.0122767
\(576\) 0 0
\(577\) 0.987488 0.0411097 0.0205548 0.999789i \(-0.493457\pi\)
0.0205548 + 0.999789i \(0.493457\pi\)
\(578\) 0 0
\(579\) −10.0127 −0.416113
\(580\) 0 0
\(581\) 8.41021 0.348914
\(582\) 0 0
\(583\) −12.1122 −0.501636
\(584\) 0 0
\(585\) 16.3022 0.674014
\(586\) 0 0
\(587\) −43.1055 −1.77915 −0.889577 0.456785i \(-0.849001\pi\)
−0.889577 + 0.456785i \(0.849001\pi\)
\(588\) 0 0
\(589\) 35.9055 1.47946
\(590\) 0 0
\(591\) −2.11966 −0.0871910
\(592\) 0 0
\(593\) 16.0276 0.658174 0.329087 0.944300i \(-0.393259\pi\)
0.329087 + 0.944300i \(0.393259\pi\)
\(594\) 0 0
\(595\) −62.9369 −2.58016
\(596\) 0 0
\(597\) 2.88360 0.118018
\(598\) 0 0
\(599\) 24.8181 1.01404 0.507021 0.861934i \(-0.330747\pi\)
0.507021 + 0.861934i \(0.330747\pi\)
\(600\) 0 0
\(601\) −13.1161 −0.535018 −0.267509 0.963555i \(-0.586200\pi\)
−0.267509 + 0.963555i \(0.586200\pi\)
\(602\) 0 0
\(603\) 0.117437 0.00478240
\(604\) 0 0
\(605\) −22.6497 −0.920842
\(606\) 0 0
\(607\) 13.5553 0.550194 0.275097 0.961416i \(-0.411290\pi\)
0.275097 + 0.961416i \(0.411290\pi\)
\(608\) 0 0
\(609\) 4.14667 0.168031
\(610\) 0 0
\(611\) 44.8153 1.81303
\(612\) 0 0
\(613\) 4.84990 0.195886 0.0979428 0.995192i \(-0.468774\pi\)
0.0979428 + 0.995192i \(0.468774\pi\)
\(614\) 0 0
\(615\) 21.8587 0.881428
\(616\) 0 0
\(617\) 20.4968 0.825170 0.412585 0.910919i \(-0.364626\pi\)
0.412585 + 0.910919i \(0.364626\pi\)
\(618\) 0 0
\(619\) 17.0019 0.683364 0.341682 0.939816i \(-0.389003\pi\)
0.341682 + 0.939816i \(0.389003\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 21.5290 0.862543
\(624\) 0 0
\(625\) −26.3853 −1.05541
\(626\) 0 0
\(627\) 6.73649 0.269030
\(628\) 0 0
\(629\) 65.6901 2.61924
\(630\) 0 0
\(631\) 33.4415 1.33129 0.665643 0.746270i \(-0.268157\pi\)
0.665643 + 0.746270i \(0.268157\pi\)
\(632\) 0 0
\(633\) −6.61196 −0.262802
\(634\) 0 0
\(635\) −7.45071 −0.295672
\(636\) 0 0
\(637\) 72.2302 2.86187
\(638\) 0 0
\(639\) 5.86395 0.231974
\(640\) 0 0
\(641\) −43.8861 −1.73339 −0.866697 0.498834i \(-0.833762\pi\)
−0.866697 + 0.498834i \(0.833762\pi\)
\(642\) 0 0
\(643\) −29.4699 −1.16218 −0.581090 0.813839i \(-0.697374\pi\)
−0.581090 + 0.813839i \(0.697374\pi\)
\(644\) 0 0
\(645\) −23.5546 −0.927462
\(646\) 0 0
\(647\) −16.5982 −0.652544 −0.326272 0.945276i \(-0.605793\pi\)
−0.326272 + 0.945276i \(0.605793\pi\)
\(648\) 0 0
\(649\) 13.4293 0.527148
\(650\) 0 0
\(651\) −23.7670 −0.931504
\(652\) 0 0
\(653\) −46.7391 −1.82904 −0.914520 0.404540i \(-0.867432\pi\)
−0.914520 + 0.404540i \(0.867432\pi\)
\(654\) 0 0
\(655\) −11.1837 −0.436985
\(656\) 0 0
\(657\) 11.6237 0.453483
\(658\) 0 0
\(659\) −25.2475 −0.983505 −0.491752 0.870735i \(-0.663643\pi\)
−0.491752 + 0.870735i \(0.663643\pi\)
\(660\) 0 0
\(661\) −42.9629 −1.67106 −0.835531 0.549444i \(-0.814840\pi\)
−0.835531 + 0.549444i \(0.814840\pi\)
\(662\) 0 0
\(663\) −46.7345 −1.81502
\(664\) 0 0
\(665\) 59.7712 2.31783
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −22.5569 −0.872102
\(670\) 0 0
\(671\) 0.983465 0.0379662
\(672\) 0 0
\(673\) 35.3303 1.36188 0.680942 0.732338i \(-0.261571\pi\)
0.680942 + 0.732338i \(0.261571\pi\)
\(674\) 0 0
\(675\) −0.294386 −0.0113309
\(676\) 0 0
\(677\) 33.8478 1.30088 0.650439 0.759559i \(-0.274585\pi\)
0.650439 + 0.759559i \(0.274585\pi\)
\(678\) 0 0
\(679\) −48.1375 −1.84735
\(680\) 0 0
\(681\) −13.7717 −0.527734
\(682\) 0 0
\(683\) 10.3826 0.397278 0.198639 0.980073i \(-0.436348\pi\)
0.198639 + 0.980073i \(0.436348\pi\)
\(684\) 0 0
\(685\) −1.08142 −0.0413188
\(686\) 0 0
\(687\) 15.5754 0.594237
\(688\) 0 0
\(689\) −79.8019 −3.04021
\(690\) 0 0
\(691\) −31.9223 −1.21438 −0.607190 0.794557i \(-0.707703\pi\)
−0.607190 + 0.794557i \(0.707703\pi\)
\(692\) 0 0
\(693\) −4.45910 −0.169387
\(694\) 0 0
\(695\) −11.0862 −0.420525
\(696\) 0 0
\(697\) −62.6636 −2.37355
\(698\) 0 0
\(699\) 20.8619 0.789071
\(700\) 0 0
\(701\) −20.8227 −0.786462 −0.393231 0.919440i \(-0.628643\pi\)
−0.393231 + 0.919440i \(0.628643\pi\)
\(702\) 0 0
\(703\) −62.3859 −2.35293
\(704\) 0 0
\(705\) −14.5544 −0.548152
\(706\) 0 0
\(707\) 43.7693 1.64611
\(708\) 0 0
\(709\) 34.8372 1.30834 0.654168 0.756349i \(-0.273019\pi\)
0.654168 + 0.756349i \(0.273019\pi\)
\(710\) 0 0
\(711\) −9.13043 −0.342418
\(712\) 0 0
\(713\) −5.73160 −0.214650
\(714\) 0 0
\(715\) 17.5305 0.655605
\(716\) 0 0
\(717\) −10.3213 −0.385456
\(718\) 0 0
\(719\) −5.81381 −0.216819 −0.108409 0.994106i \(-0.534576\pi\)
−0.108409 + 0.994106i \(0.534576\pi\)
\(720\) 0 0
\(721\) −3.89998 −0.145243
\(722\) 0 0
\(723\) −5.34079 −0.198626
\(724\) 0 0
\(725\) 0.294386 0.0109332
\(726\) 0 0
\(727\) −10.4648 −0.388117 −0.194059 0.980990i \(-0.562165\pi\)
−0.194059 + 0.980990i \(0.562165\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 67.5254 2.49752
\(732\) 0 0
\(733\) −50.7216 −1.87345 −0.936723 0.350071i \(-0.886157\pi\)
−0.936723 + 0.350071i \(0.886157\pi\)
\(734\) 0 0
\(735\) −23.4578 −0.865255
\(736\) 0 0
\(737\) 0.126285 0.00465178
\(738\) 0 0
\(739\) −34.0674 −1.25319 −0.626594 0.779346i \(-0.715552\pi\)
−0.626594 + 0.779346i \(0.715552\pi\)
\(740\) 0 0
\(741\) 44.3837 1.63048
\(742\) 0 0
\(743\) −12.4315 −0.456067 −0.228034 0.973653i \(-0.573230\pi\)
−0.228034 + 0.973653i \(0.573230\pi\)
\(744\) 0 0
\(745\) 24.5990 0.901239
\(746\) 0 0
\(747\) −2.02819 −0.0742075
\(748\) 0 0
\(749\) 8.39928 0.306903
\(750\) 0 0
\(751\) 14.2789 0.521045 0.260522 0.965468i \(-0.416105\pi\)
0.260522 + 0.965468i \(0.416105\pi\)
\(752\) 0 0
\(753\) 6.13104 0.223428
\(754\) 0 0
\(755\) 22.6451 0.824141
\(756\) 0 0
\(757\) −4.57983 −0.166457 −0.0832284 0.996531i \(-0.526523\pi\)
−0.0832284 + 0.996531i \(0.526523\pi\)
\(758\) 0 0
\(759\) −1.07535 −0.0390326
\(760\) 0 0
\(761\) −11.9792 −0.434246 −0.217123 0.976144i \(-0.569667\pi\)
−0.217123 + 0.976144i \(0.569667\pi\)
\(762\) 0 0
\(763\) 43.8498 1.58747
\(764\) 0 0
\(765\) 15.1777 0.548752
\(766\) 0 0
\(767\) 88.4800 3.19483
\(768\) 0 0
\(769\) −35.1733 −1.26838 −0.634191 0.773177i \(-0.718667\pi\)
−0.634191 + 0.773177i \(0.718667\pi\)
\(770\) 0 0
\(771\) 9.88476 0.355991
\(772\) 0 0
\(773\) 6.45121 0.232034 0.116017 0.993247i \(-0.462987\pi\)
0.116017 + 0.993247i \(0.462987\pi\)
\(774\) 0 0
\(775\) −1.68730 −0.0606097
\(776\) 0 0
\(777\) 41.2953 1.48146
\(778\) 0 0
\(779\) 59.5116 2.13222
\(780\) 0 0
\(781\) 6.30578 0.225639
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 0.252059 0.00899637
\(786\) 0 0
\(787\) −21.6976 −0.773436 −0.386718 0.922198i \(-0.626391\pi\)
−0.386718 + 0.922198i \(0.626391\pi\)
\(788\) 0 0
\(789\) −20.5660 −0.732169
\(790\) 0 0
\(791\) −9.27604 −0.329818
\(792\) 0 0
\(793\) 6.47961 0.230098
\(794\) 0 0
\(795\) 25.9168 0.919175
\(796\) 0 0
\(797\) −27.2348 −0.964708 −0.482354 0.875976i \(-0.660218\pi\)
−0.482354 + 0.875976i \(0.660218\pi\)
\(798\) 0 0
\(799\) 41.7241 1.47609
\(800\) 0 0
\(801\) −5.19189 −0.183447
\(802\) 0 0
\(803\) 12.4995 0.441097
\(804\) 0 0
\(805\) −9.54128 −0.336286
\(806\) 0 0
\(807\) 7.11530 0.250470
\(808\) 0 0
\(809\) 52.7793 1.85562 0.927811 0.373049i \(-0.121688\pi\)
0.927811 + 0.373049i \(0.121688\pi\)
\(810\) 0 0
\(811\) 29.3021 1.02893 0.514467 0.857510i \(-0.327990\pi\)
0.514467 + 0.857510i \(0.327990\pi\)
\(812\) 0 0
\(813\) −28.8149 −1.01058
\(814\) 0 0
\(815\) −5.82410 −0.204009
\(816\) 0 0
\(817\) −64.1288 −2.24358
\(818\) 0 0
\(819\) −29.3791 −1.02659
\(820\) 0 0
\(821\) 26.7232 0.932645 0.466323 0.884615i \(-0.345579\pi\)
0.466323 + 0.884615i \(0.345579\pi\)
\(822\) 0 0
\(823\) −12.5153 −0.436257 −0.218129 0.975920i \(-0.569995\pi\)
−0.218129 + 0.975920i \(0.569995\pi\)
\(824\) 0 0
\(825\) −0.316567 −0.0110215
\(826\) 0 0
\(827\) 13.8437 0.481394 0.240697 0.970600i \(-0.422624\pi\)
0.240697 + 0.970600i \(0.422624\pi\)
\(828\) 0 0
\(829\) −39.9325 −1.38691 −0.693456 0.720499i \(-0.743913\pi\)
−0.693456 + 0.720499i \(0.743913\pi\)
\(830\) 0 0
\(831\) −7.38971 −0.256346
\(832\) 0 0
\(833\) 67.2479 2.33000
\(834\) 0 0
\(835\) −48.6587 −1.68390
\(836\) 0 0
\(837\) 5.73160 0.198113
\(838\) 0 0
\(839\) −43.1482 −1.48964 −0.744821 0.667264i \(-0.767465\pi\)
−0.744821 + 0.667264i \(0.767465\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 0.755545 0.0260224
\(844\) 0 0
\(845\) 85.5885 2.94433
\(846\) 0 0
\(847\) 40.8182 1.40253
\(848\) 0 0
\(849\) −28.5928 −0.981302
\(850\) 0 0
\(851\) 9.95867 0.341379
\(852\) 0 0
\(853\) 40.5150 1.38721 0.693604 0.720357i \(-0.256022\pi\)
0.693604 + 0.720357i \(0.256022\pi\)
\(854\) 0 0
\(855\) −14.4143 −0.492958
\(856\) 0 0
\(857\) 53.5208 1.82823 0.914117 0.405450i \(-0.132885\pi\)
0.914117 + 0.405450i \(0.132885\pi\)
\(858\) 0 0
\(859\) 38.4144 1.31068 0.655341 0.755334i \(-0.272525\pi\)
0.655341 + 0.755334i \(0.272525\pi\)
\(860\) 0 0
\(861\) −39.3927 −1.34250
\(862\) 0 0
\(863\) −38.9820 −1.32696 −0.663481 0.748193i \(-0.730922\pi\)
−0.663481 + 0.748193i \(0.730922\pi\)
\(864\) 0 0
\(865\) 9.07850 0.308678
\(866\) 0 0
\(867\) −26.5109 −0.900356
\(868\) 0 0
\(869\) −9.81838 −0.333066
\(870\) 0 0
\(871\) 0.832038 0.0281925
\(872\) 0 0
\(873\) 11.6087 0.392896
\(874\) 0 0
\(875\) 44.8976 1.51782
\(876\) 0 0
\(877\) 30.2074 1.02003 0.510015 0.860166i \(-0.329640\pi\)
0.510015 + 0.860166i \(0.329640\pi\)
\(878\) 0 0
\(879\) −11.4820 −0.387279
\(880\) 0 0
\(881\) 15.5901 0.525245 0.262622 0.964899i \(-0.415413\pi\)
0.262622 + 0.964899i \(0.415413\pi\)
\(882\) 0 0
\(883\) −4.09079 −0.137666 −0.0688330 0.997628i \(-0.521928\pi\)
−0.0688330 + 0.997628i \(0.521928\pi\)
\(884\) 0 0
\(885\) −28.7352 −0.965923
\(886\) 0 0
\(887\) 19.8859 0.667702 0.333851 0.942626i \(-0.391652\pi\)
0.333851 + 0.942626i \(0.391652\pi\)
\(888\) 0 0
\(889\) 13.4273 0.450337
\(890\) 0 0
\(891\) 1.07535 0.0360255
\(892\) 0 0
\(893\) −39.6254 −1.32601
\(894\) 0 0
\(895\) 24.8260 0.829841
\(896\) 0 0
\(897\) −7.08498 −0.236561
\(898\) 0 0
\(899\) −5.73160 −0.191160
\(900\) 0 0
\(901\) −74.2973 −2.47520
\(902\) 0 0
\(903\) 42.4490 1.41261
\(904\) 0 0
\(905\) 33.7553 1.12207
\(906\) 0 0
\(907\) −34.6632 −1.15097 −0.575486 0.817811i \(-0.695187\pi\)
−0.575486 + 0.817811i \(0.695187\pi\)
\(908\) 0 0
\(909\) −10.5553 −0.350097
\(910\) 0 0
\(911\) −9.37318 −0.310547 −0.155274 0.987871i \(-0.549626\pi\)
−0.155274 + 0.987871i \(0.549626\pi\)
\(912\) 0 0
\(913\) −2.18100 −0.0721807
\(914\) 0 0
\(915\) −2.10435 −0.0695677
\(916\) 0 0
\(917\) 20.1548 0.665570
\(918\) 0 0
\(919\) −56.8751 −1.87614 −0.938068 0.346450i \(-0.887387\pi\)
−0.938068 + 0.346450i \(0.887387\pi\)
\(920\) 0 0
\(921\) 11.8578 0.390729
\(922\) 0 0
\(923\) 41.5460 1.36750
\(924\) 0 0
\(925\) 2.93169 0.0963934
\(926\) 0 0
\(927\) 0.940511 0.0308904
\(928\) 0 0
\(929\) −11.2034 −0.367571 −0.183785 0.982966i \(-0.558835\pi\)
−0.183785 + 0.982966i \(0.558835\pi\)
\(930\) 0 0
\(931\) −63.8654 −2.09310
\(932\) 0 0
\(933\) −17.6246 −0.577002
\(934\) 0 0
\(935\) 16.3213 0.533764
\(936\) 0 0
\(937\) 5.59354 0.182733 0.0913665 0.995817i \(-0.470877\pi\)
0.0913665 + 0.995817i \(0.470877\pi\)
\(938\) 0 0
\(939\) 18.7639 0.612337
\(940\) 0 0
\(941\) 53.5483 1.74562 0.872812 0.488056i \(-0.162294\pi\)
0.872812 + 0.488056i \(0.162294\pi\)
\(942\) 0 0
\(943\) −9.49984 −0.309357
\(944\) 0 0
\(945\) 9.54128 0.310378
\(946\) 0 0
\(947\) 44.5448 1.44751 0.723756 0.690056i \(-0.242414\pi\)
0.723756 + 0.690056i \(0.242414\pi\)
\(948\) 0 0
\(949\) 82.3535 2.67331
\(950\) 0 0
\(951\) −15.2215 −0.493589
\(952\) 0 0
\(953\) 23.7799 0.770307 0.385154 0.922852i \(-0.374148\pi\)
0.385154 + 0.922852i \(0.374148\pi\)
\(954\) 0 0
\(955\) 27.1639 0.879004
\(956\) 0 0
\(957\) −1.07535 −0.0347610
\(958\) 0 0
\(959\) 1.94888 0.0629325
\(960\) 0 0
\(961\) 1.85126 0.0597181
\(962\) 0 0
\(963\) −2.02555 −0.0652725
\(964\) 0 0
\(965\) 23.0387 0.741643
\(966\) 0 0
\(967\) −52.5573 −1.69013 −0.845064 0.534665i \(-0.820438\pi\)
−0.845064 + 0.534665i \(0.820438\pi\)
\(968\) 0 0
\(969\) 41.3223 1.32746
\(970\) 0 0
\(971\) 44.1202 1.41588 0.707942 0.706270i \(-0.249624\pi\)
0.707942 + 0.706270i \(0.249624\pi\)
\(972\) 0 0
\(973\) 19.9791 0.640500
\(974\) 0 0
\(975\) −2.08572 −0.0667965
\(976\) 0 0
\(977\) 58.0608 1.85753 0.928765 0.370670i \(-0.120872\pi\)
0.928765 + 0.370670i \(0.120872\pi\)
\(978\) 0 0
\(979\) −5.58309 −0.178436
\(980\) 0 0
\(981\) −10.5747 −0.337625
\(982\) 0 0
\(983\) −3.12163 −0.0995645 −0.0497823 0.998760i \(-0.515853\pi\)
−0.0497823 + 0.998760i \(0.515853\pi\)
\(984\) 0 0
\(985\) 4.87723 0.155401
\(986\) 0 0
\(987\) 26.2293 0.834888
\(988\) 0 0
\(989\) 10.2369 0.325514
\(990\) 0 0
\(991\) −19.6463 −0.624085 −0.312043 0.950068i \(-0.601013\pi\)
−0.312043 + 0.950068i \(0.601013\pi\)
\(992\) 0 0
\(993\) 4.49941 0.142785
\(994\) 0 0
\(995\) −6.63503 −0.210344
\(996\) 0 0
\(997\) 3.23323 0.102397 0.0511987 0.998688i \(-0.483696\pi\)
0.0511987 + 0.998688i \(0.483696\pi\)
\(998\) 0 0
\(999\) −9.95867 −0.315078
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 8004.2.a.i.1.12 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.12 16 1.1 even 1 trivial