Properties

Label 7920.2.a.bo.1.2
Level $7920$
Weight $2$
Character 7920.1
Self dual yes
Analytic conductor $63.242$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.2415184009\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 660)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 7920.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.60555 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +2.60555 q^{7} +1.00000 q^{11} +2.60555 q^{13} +0.605551 q^{17} -7.21110 q^{19} +1.00000 q^{25} -8.00000 q^{29} +5.21110 q^{31} -2.60555 q^{35} +11.2111 q^{37} -8.00000 q^{41} +10.6056 q^{43} +9.21110 q^{47} -0.211103 q^{49} -2.00000 q^{53} -1.00000 q^{55} +8.00000 q^{59} -7.21110 q^{61} -2.60555 q^{65} +4.00000 q^{67} +14.4222 q^{71} -6.60555 q^{73} +2.60555 q^{77} -3.21110 q^{79} -3.39445 q^{83} -0.605551 q^{85} -6.00000 q^{89} +6.78890 q^{91} +7.21110 q^{95} +16.4222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{25} - 16 q^{29} - 4 q^{31} + 2 q^{35} + 8 q^{37} - 16 q^{41} + 14 q^{43} + 4 q^{47} + 14 q^{49} - 4 q^{53} - 2 q^{55} + 16 q^{59} + 2 q^{65} + 8 q^{67} - 6 q^{73} - 2 q^{77} + 8 q^{79} - 14 q^{83} + 6 q^{85} - 12 q^{89} + 28 q^{91} + 4 q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.60555 0.984806 0.492403 0.870367i \(-0.336119\pi\)
0.492403 + 0.870367i \(0.336119\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.60555 0.722650 0.361325 0.932440i \(-0.382325\pi\)
0.361325 + 0.932440i \(0.382325\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.605551 0.146868 0.0734339 0.997300i \(-0.476604\pi\)
0.0734339 + 0.997300i \(0.476604\pi\)
\(18\) 0 0
\(19\) −7.21110 −1.65434 −0.827170 0.561951i \(-0.810051\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 5.21110 0.935942 0.467971 0.883744i \(-0.344985\pi\)
0.467971 + 0.883744i \(0.344985\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.60555 −0.440419
\(36\) 0 0
\(37\) 11.2111 1.84309 0.921547 0.388267i \(-0.126926\pi\)
0.921547 + 0.388267i \(0.126926\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 10.6056 1.61733 0.808666 0.588268i \(-0.200190\pi\)
0.808666 + 0.588268i \(0.200190\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.21110 1.34358 0.671789 0.740743i \(-0.265526\pi\)
0.671789 + 0.740743i \(0.265526\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −7.21110 −0.923287 −0.461644 0.887066i \(-0.652740\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.60555 −0.323179
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.4222 1.71160 0.855800 0.517306i \(-0.173065\pi\)
0.855800 + 0.517306i \(0.173065\pi\)
\(72\) 0 0
\(73\) −6.60555 −0.773121 −0.386561 0.922264i \(-0.626337\pi\)
−0.386561 + 0.922264i \(0.626337\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.60555 0.296930
\(78\) 0 0
\(79\) −3.21110 −0.361277 −0.180639 0.983550i \(-0.557816\pi\)
−0.180639 + 0.983550i \(0.557816\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.39445 −0.372589 −0.186295 0.982494i \(-0.559648\pi\)
−0.186295 + 0.982494i \(0.559648\pi\)
\(84\) 0 0
\(85\) −0.605551 −0.0656813
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 6.78890 0.711670
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.21110 0.739844
\(96\) 0 0
\(97\) 16.4222 1.66742 0.833711 0.552201i \(-0.186212\pi\)
0.833711 + 0.552201i \(0.186212\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.21110 0.916539 0.458269 0.888813i \(-0.348469\pi\)
0.458269 + 0.888813i \(0.348469\pi\)
\(102\) 0 0
\(103\) 13.2111 1.30173 0.650864 0.759194i \(-0.274407\pi\)
0.650864 + 0.759194i \(0.274407\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.81665 −0.562317 −0.281159 0.959661i \(-0.590719\pi\)
−0.281159 + 0.959661i \(0.590719\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.2111 1.43094 0.715470 0.698643i \(-0.246213\pi\)
0.715470 + 0.698643i \(0.246213\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.57779 0.144636
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.39445 0.123737 0.0618687 0.998084i \(-0.480294\pi\)
0.0618687 + 0.998084i \(0.480294\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −21.2111 −1.85322 −0.926611 0.376021i \(-0.877292\pi\)
−0.926611 + 0.376021i \(0.877292\pi\)
\(132\) 0 0
\(133\) −18.7889 −1.62920
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 12.4222 1.05364 0.526819 0.849978i \(-0.323385\pi\)
0.526819 + 0.849978i \(0.323385\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.60555 0.217887
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.21110 −0.426910 −0.213455 0.976953i \(-0.568472\pi\)
−0.213455 + 0.976953i \(0.568472\pi\)
\(150\) 0 0
\(151\) 4.78890 0.389715 0.194857 0.980832i \(-0.437576\pi\)
0.194857 + 0.980832i \(0.437576\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.21110 −0.418566
\(156\) 0 0
\(157\) −11.2111 −0.894743 −0.447372 0.894348i \(-0.647640\pi\)
−0.447372 + 0.894348i \(0.647640\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −9.21110 −0.721469 −0.360735 0.932669i \(-0.617474\pi\)
−0.360735 + 0.932669i \(0.617474\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.81665 0.140577 0.0702884 0.997527i \(-0.477608\pi\)
0.0702884 + 0.997527i \(0.477608\pi\)
\(168\) 0 0
\(169\) −6.21110 −0.477777
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −21.8167 −1.65869 −0.829345 0.558737i \(-0.811286\pi\)
−0.829345 + 0.558737i \(0.811286\pi\)
\(174\) 0 0
\(175\) 2.60555 0.196961
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 8.78890 0.653274 0.326637 0.945150i \(-0.394085\pi\)
0.326637 + 0.945150i \(0.394085\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −11.2111 −0.824257
\(186\) 0 0
\(187\) 0.605551 0.0442823
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.4222 −0.754124 −0.377062 0.926188i \(-0.623066\pi\)
−0.377062 + 0.926188i \(0.623066\pi\)
\(192\) 0 0
\(193\) −0.183346 −0.0131975 −0.00659877 0.999978i \(-0.502100\pi\)
−0.00659877 + 0.999978i \(0.502100\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.6056 1.46808 0.734042 0.679104i \(-0.237631\pi\)
0.734042 + 0.679104i \(0.237631\pi\)
\(198\) 0 0
\(199\) −18.4222 −1.30592 −0.652958 0.757394i \(-0.726472\pi\)
−0.652958 + 0.757394i \(0.726472\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −20.8444 −1.46299
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.21110 −0.498802
\(210\) 0 0
\(211\) 8.78890 0.605053 0.302526 0.953141i \(-0.402170\pi\)
0.302526 + 0.953141i \(0.402170\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.6056 −0.723293
\(216\) 0 0
\(217\) 13.5778 0.921721
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.57779 0.106134
\(222\) 0 0
\(223\) 5.21110 0.348961 0.174481 0.984661i \(-0.444175\pi\)
0.174481 + 0.984661i \(0.444175\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.81665 0.386065 0.193032 0.981192i \(-0.438168\pi\)
0.193032 + 0.981192i \(0.438168\pi\)
\(228\) 0 0
\(229\) 20.4222 1.34954 0.674769 0.738029i \(-0.264243\pi\)
0.674769 + 0.738029i \(0.264243\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.39445 0.222378 0.111189 0.993799i \(-0.464534\pi\)
0.111189 + 0.993799i \(0.464534\pi\)
\(234\) 0 0
\(235\) −9.21110 −0.600866
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.2111 1.37203 0.686016 0.727586i \(-0.259358\pi\)
0.686016 + 0.727586i \(0.259358\pi\)
\(240\) 0 0
\(241\) −0.788897 −0.0508174 −0.0254087 0.999677i \(-0.508089\pi\)
−0.0254087 + 0.999677i \(0.508089\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.211103 0.0134868
\(246\) 0 0
\(247\) −18.7889 −1.19551
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.4222 1.16280 0.581400 0.813618i \(-0.302505\pi\)
0.581400 + 0.813618i \(0.302505\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.78890 0.548236 0.274118 0.961696i \(-0.411614\pi\)
0.274118 + 0.961696i \(0.411614\pi\)
\(258\) 0 0
\(259\) 29.2111 1.81509
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.0278 1.17330 0.586651 0.809840i \(-0.300446\pi\)
0.586651 + 0.809840i \(0.300446\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −8.42221 −0.513511 −0.256755 0.966476i \(-0.582653\pi\)
−0.256755 + 0.966476i \(0.582653\pi\)
\(270\) 0 0
\(271\) 19.2111 1.16699 0.583496 0.812116i \(-0.301685\pi\)
0.583496 + 0.812116i \(0.301685\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −18.2389 −1.09587 −0.547933 0.836522i \(-0.684585\pi\)
−0.547933 + 0.836522i \(0.684585\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 2.60555 0.154884 0.0774420 0.996997i \(-0.475325\pi\)
0.0774420 + 0.996997i \(0.475325\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.8444 −1.23041
\(288\) 0 0
\(289\) −16.6333 −0.978430
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.81665 0.106130 0.0530650 0.998591i \(-0.483101\pi\)
0.0530650 + 0.998591i \(0.483101\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 27.6333 1.59276
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.21110 0.412907
\(306\) 0 0
\(307\) −1.39445 −0.0795854 −0.0397927 0.999208i \(-0.512670\pi\)
−0.0397927 + 0.999208i \(0.512670\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.57779 0.0894685 0.0447343 0.998999i \(-0.485756\pi\)
0.0447343 + 0.998999i \(0.485756\pi\)
\(312\) 0 0
\(313\) 23.2111 1.31197 0.655985 0.754774i \(-0.272254\pi\)
0.655985 + 0.754774i \(0.272254\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.21110 0.180353 0.0901767 0.995926i \(-0.471257\pi\)
0.0901767 + 0.995926i \(0.471257\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.36669 −0.242969
\(324\) 0 0
\(325\) 2.60555 0.144530
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 21.2111 1.16587 0.582934 0.812520i \(-0.301905\pi\)
0.582934 + 0.812520i \(0.301905\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 22.2389 1.21143 0.605714 0.795683i \(-0.292888\pi\)
0.605714 + 0.795683i \(0.292888\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.21110 0.282197
\(342\) 0 0
\(343\) −18.7889 −1.01451
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.02776 0.377270 0.188635 0.982047i \(-0.439594\pi\)
0.188635 + 0.982047i \(0.439594\pi\)
\(348\) 0 0
\(349\) −23.2111 −1.24246 −0.621231 0.783628i \(-0.713367\pi\)
−0.621231 + 0.783628i \(0.713367\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) −14.4222 −0.765451
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.2111 1.75281 0.876407 0.481570i \(-0.159933\pi\)
0.876407 + 0.481570i \(0.159933\pi\)
\(360\) 0 0
\(361\) 33.0000 1.73684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.60555 0.345750
\(366\) 0 0
\(367\) −21.2111 −1.10721 −0.553605 0.832779i \(-0.686748\pi\)
−0.553605 + 0.832779i \(0.686748\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.21110 −0.270547
\(372\) 0 0
\(373\) −2.60555 −0.134910 −0.0674552 0.997722i \(-0.521488\pi\)
−0.0674552 + 0.997722i \(0.521488\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.8444 −1.07354
\(378\) 0 0
\(379\) −10.4222 −0.535353 −0.267676 0.963509i \(-0.586256\pi\)
−0.267676 + 0.963509i \(0.586256\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 33.2111 1.69701 0.848504 0.529189i \(-0.177504\pi\)
0.848504 + 0.529189i \(0.177504\pi\)
\(384\) 0 0
\(385\) −2.60555 −0.132791
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 26.8444 1.36107 0.680533 0.732718i \(-0.261748\pi\)
0.680533 + 0.732718i \(0.261748\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.21110 0.161568
\(396\) 0 0
\(397\) −0.422205 −0.0211899 −0.0105949 0.999944i \(-0.503373\pi\)
−0.0105949 + 0.999944i \(0.503373\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.42221 0.420585 0.210292 0.977639i \(-0.432558\pi\)
0.210292 + 0.977639i \(0.432558\pi\)
\(402\) 0 0
\(403\) 13.5778 0.676358
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.2111 0.555714
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 20.8444 1.02569
\(414\) 0 0
\(415\) 3.39445 0.166627
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −4.78890 −0.233397 −0.116698 0.993167i \(-0.537231\pi\)
−0.116698 + 0.993167i \(0.537231\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.605551 0.0293736
\(426\) 0 0
\(427\) −18.7889 −0.909258
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.4222 −1.46539 −0.732693 0.680559i \(-0.761737\pi\)
−0.732693 + 0.680559i \(0.761737\pi\)
\(432\) 0 0
\(433\) −20.4222 −0.981429 −0.490714 0.871321i \(-0.663264\pi\)
−0.490714 + 0.871321i \(0.663264\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.7889 0.512596 0.256298 0.966598i \(-0.417497\pi\)
0.256298 + 0.966598i \(0.417497\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.8444 1.83318 0.916591 0.399827i \(-0.130930\pi\)
0.916591 + 0.399827i \(0.130930\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.78890 −0.318268
\(456\) 0 0
\(457\) 2.60555 0.121883 0.0609413 0.998141i \(-0.480590\pi\)
0.0609413 + 0.998141i \(0.480590\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5.21110 −0.242705 −0.121353 0.992609i \(-0.538723\pi\)
−0.121353 + 0.992609i \(0.538723\pi\)
\(462\) 0 0
\(463\) 13.2111 0.613972 0.306986 0.951714i \(-0.400680\pi\)
0.306986 + 0.951714i \(0.400680\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 10.4222 0.481253
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.6056 0.487644
\(474\) 0 0
\(475\) −7.21110 −0.330868
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −29.2111 −1.33469 −0.667345 0.744749i \(-0.732569\pi\)
−0.667345 + 0.744749i \(0.732569\pi\)
\(480\) 0 0
\(481\) 29.2111 1.33191
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.4222 −0.745694
\(486\) 0 0
\(487\) −34.4222 −1.55982 −0.779910 0.625892i \(-0.784735\pi\)
−0.779910 + 0.625892i \(0.784735\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −34.4222 −1.55345 −0.776726 0.629838i \(-0.783121\pi\)
−0.776726 + 0.629838i \(0.783121\pi\)
\(492\) 0 0
\(493\) −4.84441 −0.218181
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 37.5778 1.68559
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.8167 1.15111 0.575554 0.817764i \(-0.304787\pi\)
0.575554 + 0.817764i \(0.304787\pi\)
\(504\) 0 0
\(505\) −9.21110 −0.409889
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.422205 −0.0187139 −0.00935696 0.999956i \(-0.502978\pi\)
−0.00935696 + 0.999956i \(0.502978\pi\)
\(510\) 0 0
\(511\) −17.2111 −0.761374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −13.2111 −0.582151
\(516\) 0 0
\(517\) 9.21110 0.405104
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20.4222 −0.894713 −0.447357 0.894356i \(-0.647634\pi\)
−0.447357 + 0.894356i \(0.647634\pi\)
\(522\) 0 0
\(523\) 22.2389 0.972437 0.486219 0.873837i \(-0.338376\pi\)
0.486219 + 0.873837i \(0.338376\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.15559 0.137460
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −20.8444 −0.902872
\(534\) 0 0
\(535\) 5.81665 0.251476
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.211103 −0.00909283
\(540\) 0 0
\(541\) 32.4222 1.39394 0.696970 0.717101i \(-0.254531\pi\)
0.696970 + 0.717101i \(0.254531\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 35.8167 1.53141 0.765705 0.643192i \(-0.222390\pi\)
0.765705 + 0.643192i \(0.222390\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 57.6888 2.45763
\(552\) 0 0
\(553\) −8.36669 −0.355788
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 43.0278 1.82314 0.911572 0.411140i \(-0.134869\pi\)
0.911572 + 0.411140i \(0.134869\pi\)
\(558\) 0 0
\(559\) 27.6333 1.16876
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.81665 0.0765628 0.0382814 0.999267i \(-0.487812\pi\)
0.0382814 + 0.999267i \(0.487812\pi\)
\(564\) 0 0
\(565\) −15.2111 −0.639936
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.6333 −0.655382 −0.327691 0.944785i \(-0.606271\pi\)
−0.327691 + 0.944785i \(0.606271\pi\)
\(570\) 0 0
\(571\) 16.4222 0.687248 0.343624 0.939107i \(-0.388345\pi\)
0.343624 + 0.939107i \(0.388345\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 43.2111 1.79890 0.899451 0.437022i \(-0.143967\pi\)
0.899451 + 0.437022i \(0.143967\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −8.84441 −0.366928
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −22.4222 −0.925463 −0.462732 0.886498i \(-0.653131\pi\)
−0.462732 + 0.886498i \(0.653131\pi\)
\(588\) 0 0
\(589\) −37.5778 −1.54837
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.0278 0.945637 0.472818 0.881160i \(-0.343237\pi\)
0.472818 + 0.881160i \(0.343237\pi\)
\(594\) 0 0
\(595\) −1.57779 −0.0646833
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −23.2111 −0.946801 −0.473400 0.880847i \(-0.656974\pi\)
−0.473400 + 0.880847i \(0.656974\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −41.0278 −1.66527 −0.832633 0.553826i \(-0.813167\pi\)
−0.832633 + 0.553826i \(0.813167\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −0.183346 −0.00740528 −0.00370264 0.999993i \(-0.501179\pi\)
−0.00370264 + 0.999993i \(0.501179\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.2111 1.57858 0.789290 0.614021i \(-0.210449\pi\)
0.789290 + 0.614021i \(0.210449\pi\)
\(618\) 0 0
\(619\) −23.6333 −0.949903 −0.474951 0.880012i \(-0.657534\pi\)
−0.474951 + 0.880012i \(0.657534\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −15.6333 −0.626335
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.78890 0.270691
\(630\) 0 0
\(631\) 20.8444 0.829803 0.414901 0.909866i \(-0.363816\pi\)
0.414901 + 0.909866i \(0.363816\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.39445 −0.0553370
\(636\) 0 0
\(637\) −0.550039 −0.0217933
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.8444 1.53426 0.767131 0.641490i \(-0.221684\pi\)
0.767131 + 0.641490i \(0.221684\pi\)
\(642\) 0 0
\(643\) −43.6333 −1.72073 −0.860365 0.509679i \(-0.829764\pi\)
−0.860365 + 0.509679i \(0.829764\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.7889 1.21044 0.605218 0.796060i \(-0.293086\pi\)
0.605218 + 0.796060i \(0.293086\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.2111 −0.751789 −0.375894 0.926663i \(-0.622664\pi\)
−0.375894 + 0.926663i \(0.622664\pi\)
\(654\) 0 0
\(655\) 21.2111 0.828786
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −41.2666 −1.60509 −0.802543 0.596595i \(-0.796520\pi\)
−0.802543 + 0.596595i \(0.796520\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 18.7889 0.728602
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.21110 −0.278382
\(672\) 0 0
\(673\) −18.6056 −0.717191 −0.358596 0.933493i \(-0.616744\pi\)
−0.358596 + 0.933493i \(0.616744\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.6056 −0.484471 −0.242235 0.970218i \(-0.577881\pi\)
−0.242235 + 0.970218i \(0.577881\pi\)
\(678\) 0 0
\(679\) 42.7889 1.64209
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −45.2111 −1.72995 −0.864977 0.501811i \(-0.832667\pi\)
−0.864977 + 0.501811i \(0.832667\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.21110 −0.198527
\(690\) 0 0
\(691\) 42.4222 1.61382 0.806908 0.590677i \(-0.201139\pi\)
0.806908 + 0.590677i \(0.201139\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.4222 −0.471201
\(696\) 0 0
\(697\) −4.84441 −0.183495
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17.5778 −0.663904 −0.331952 0.943296i \(-0.607707\pi\)
−0.331952 + 0.943296i \(0.607707\pi\)
\(702\) 0 0
\(703\) −80.8444 −3.04910
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000 0.902613
\(708\) 0 0
\(709\) −37.6333 −1.41335 −0.706674 0.707539i \(-0.749805\pi\)
−0.706674 + 0.707539i \(0.749805\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −2.60555 −0.0974421
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.42221 0.0903330 0.0451665 0.998979i \(-0.485618\pi\)
0.0451665 + 0.998979i \(0.485618\pi\)
\(720\) 0 0
\(721\) 34.4222 1.28195
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) −28.8444 −1.06978 −0.534890 0.844922i \(-0.679647\pi\)
−0.534890 + 0.844922i \(0.679647\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.42221 0.237534
\(732\) 0 0
\(733\) −14.6056 −0.539468 −0.269734 0.962935i \(-0.586936\pi\)
−0.269734 + 0.962935i \(0.586936\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 45.2666 1.66516 0.832580 0.553905i \(-0.186863\pi\)
0.832580 + 0.553905i \(0.186863\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.6056 1.63642 0.818209 0.574920i \(-0.194967\pi\)
0.818209 + 0.574920i \(0.194967\pi\)
\(744\) 0 0
\(745\) 5.21110 0.190920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15.1556 −0.553773
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.78890 −0.174286
\(756\) 0 0
\(757\) −27.2111 −0.989004 −0.494502 0.869176i \(-0.664650\pi\)
−0.494502 + 0.869176i \(0.664650\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.6333 0.566707 0.283353 0.959016i \(-0.408553\pi\)
0.283353 + 0.959016i \(0.408553\pi\)
\(762\) 0 0
\(763\) −26.0555 −0.943273
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.8444 0.752648
\(768\) 0 0
\(769\) −28.7889 −1.03815 −0.519077 0.854727i \(-0.673724\pi\)
−0.519077 + 0.854727i \(0.673724\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −24.0555 −0.865217 −0.432608 0.901582i \(-0.642407\pi\)
−0.432608 + 0.901582i \(0.642407\pi\)
\(774\) 0 0
\(775\) 5.21110 0.187188
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 57.6888 2.06692
\(780\) 0 0
\(781\) 14.4222 0.516067
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.2111 0.400141
\(786\) 0 0
\(787\) −2.97224 −0.105949 −0.0529745 0.998596i \(-0.516870\pi\)
−0.0529745 + 0.998596i \(0.516870\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 39.6333 1.40920
\(792\) 0 0
\(793\) −18.7889 −0.667213
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.0555 1.13546 0.567732 0.823213i \(-0.307821\pi\)
0.567732 + 0.823213i \(0.307821\pi\)
\(798\) 0 0
\(799\) 5.57779 0.197328
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.60555 −0.233105
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 36.4222 1.27896 0.639478 0.768809i \(-0.279150\pi\)
0.639478 + 0.768809i \(0.279150\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.21110 0.322651
\(816\) 0 0
\(817\) −76.4777 −2.67562
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −44.8444 −1.56508 −0.782540 0.622600i \(-0.786076\pi\)
−0.782540 + 0.622600i \(0.786076\pi\)
\(822\) 0 0
\(823\) −26.7889 −0.933802 −0.466901 0.884310i \(-0.654630\pi\)
−0.466901 + 0.884310i \(0.654630\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.60555 0.160151 0.0800754 0.996789i \(-0.474484\pi\)
0.0800754 + 0.996789i \(0.474484\pi\)
\(828\) 0 0
\(829\) 49.6333 1.72384 0.861918 0.507048i \(-0.169263\pi\)
0.861918 + 0.507048i \(0.169263\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.127833 −0.00442917
\(834\) 0 0
\(835\) −1.81665 −0.0628679
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.57779 0.0544715 0.0272358 0.999629i \(-0.491330\pi\)
0.0272358 + 0.999629i \(0.491330\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.21110 0.213668
\(846\) 0 0
\(847\) 2.60555 0.0895278
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.2389 0.350572 0.175286 0.984518i \(-0.443915\pi\)
0.175286 + 0.984518i \(0.443915\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.0278 0.376701 0.188350 0.982102i \(-0.439686\pi\)
0.188350 + 0.982102i \(0.439686\pi\)
\(858\) 0 0
\(859\) −42.7889 −1.45994 −0.729969 0.683480i \(-0.760466\pi\)
−0.729969 + 0.683480i \(0.760466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.63331 −0.123679 −0.0618396 0.998086i \(-0.519697\pi\)
−0.0618396 + 0.998086i \(0.519697\pi\)
\(864\) 0 0
\(865\) 21.8167 0.741788
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −3.21110 −0.108929
\(870\) 0 0
\(871\) 10.4222 0.353143
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.60555 −0.0880837
\(876\) 0 0
\(877\) 26.2389 0.886023 0.443012 0.896516i \(-0.353910\pi\)
0.443012 + 0.896516i \(0.353910\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.8444 1.03917 0.519587 0.854417i \(-0.326086\pi\)
0.519587 + 0.854417i \(0.326086\pi\)
\(882\) 0 0
\(883\) 22.4222 0.754567 0.377284 0.926098i \(-0.376858\pi\)
0.377284 + 0.926098i \(0.376858\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.0278 −0.773196 −0.386598 0.922248i \(-0.626350\pi\)
−0.386598 + 0.922248i \(0.626350\pi\)
\(888\) 0 0
\(889\) 3.63331 0.121857
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −66.4222 −2.22273
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −41.6888 −1.39040
\(900\) 0 0
\(901\) −1.21110 −0.0403477
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.78890 −0.292153
\(906\) 0 0
\(907\) −38.0555 −1.26361 −0.631806 0.775126i \(-0.717686\pi\)
−0.631806 + 0.775126i \(0.717686\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10.4222 −0.345303 −0.172652 0.984983i \(-0.555233\pi\)
−0.172652 + 0.984983i \(0.555233\pi\)
\(912\) 0 0
\(913\) −3.39445 −0.112340
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −55.2666 −1.82506
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 37.5778 1.23689
\(924\) 0 0
\(925\) 11.2111 0.368619
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 30.8444 1.01197 0.505986 0.862542i \(-0.331129\pi\)
0.505986 + 0.862542i \(0.331129\pi\)
\(930\) 0 0
\(931\) 1.52228 0.0498908
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.605551 −0.0198036
\(936\) 0 0
\(937\) 57.0278 1.86302 0.931508 0.363721i \(-0.118494\pi\)
0.931508 + 0.363721i \(0.118494\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −38.7889 −1.26448 −0.632241 0.774772i \(-0.717865\pi\)
−0.632241 + 0.774772i \(0.717865\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.6333 1.28791 0.643955 0.765064i \(-0.277293\pi\)
0.643955 + 0.765064i \(0.277293\pi\)
\(948\) 0 0
\(949\) −17.2111 −0.558696
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −29.4500 −0.953978 −0.476989 0.878909i \(-0.658272\pi\)
−0.476989 + 0.878909i \(0.658272\pi\)
\(954\) 0 0
\(955\) 10.4222 0.337255
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 15.6333 0.504826
\(960\) 0 0
\(961\) −3.84441 −0.124013
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.183346 0.00590212
\(966\) 0 0
\(967\) −4.18335 −0.134527 −0.0672637 0.997735i \(-0.521427\pi\)
−0.0672637 + 0.997735i \(0.521427\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 32.3667 1.03763
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −48.4777 −1.54620 −0.773100 0.634285i \(-0.781295\pi\)
−0.773100 + 0.634285i \(0.781295\pi\)
\(984\) 0 0
\(985\) −20.6056 −0.656547
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −7.63331 −0.242480 −0.121240 0.992623i \(-0.538687\pi\)
−0.121240 + 0.992623i \(0.538687\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.4222 0.584023
\(996\) 0 0
\(997\) −21.0278 −0.665956 −0.332978 0.942935i \(-0.608053\pi\)
−0.332978 + 0.942935i \(0.608053\pi\)
\(998\) 0 0
\(999\) 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 7920.2.a.bo.1.2 2
3.2 odd 2 2640.2.a.bc.1.2 2
4.3 odd 2 1980.2.a.h.1.1 2
12.11 even 2 660.2.a.e.1.1 2
20.3 even 4 9900.2.c.q.5149.3 4
20.7 even 4 9900.2.c.q.5149.2 4
20.19 odd 2 9900.2.a.bl.1.2 2
60.23 odd 4 3300.2.c.l.1849.2 4
60.47 odd 4 3300.2.c.l.1849.3 4
60.59 even 2 3300.2.a.w.1.2 2
132.131 odd 2 7260.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
660.2.a.e.1.1 2 12.11 even 2
1980.2.a.h.1.1 2 4.3 odd 2
2640.2.a.bc.1.2 2 3.2 odd 2
3300.2.a.w.1.2 2 60.59 even 2
3300.2.c.l.1849.2 4 60.23 odd 4
3300.2.c.l.1849.3 4 60.47 odd 4
7260.2.a.w.1.2 2 132.131 odd 2
7920.2.a.bo.1.2 2 1.1 even 1 trivial
9900.2.a.bl.1.2 2 20.19 odd 2
9900.2.c.q.5149.2 4 20.7 even 4
9900.2.c.q.5149.3 4 20.3 even 4