Properties

Label 8280.2.a.bf.1.2
Level $8280$
Weight $2$
Character 8280.1
Self dual yes
Analytic conductor $66.116$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1161328736\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 8280.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.56155 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.56155 q^{7} +2.00000 q^{11} +0.561553 q^{13} -5.56155 q^{17} -2.00000 q^{19} +1.00000 q^{23} +1.00000 q^{25} -0.123106 q^{29} -8.12311 q^{31} +1.56155 q^{35} -3.56155 q^{37} +4.12311 q^{41} -10.2462 q^{43} -3.68466 q^{47} -4.56155 q^{49} -4.43845 q^{53} +2.00000 q^{55} +5.56155 q^{59} -9.12311 q^{61} +0.561553 q^{65} -11.5616 q^{67} +5.00000 q^{71} +3.43845 q^{73} +3.12311 q^{77} -9.12311 q^{79} +4.68466 q^{83} -5.56155 q^{85} -8.00000 q^{89} +0.876894 q^{91} -2.00000 q^{95} -3.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - q^{7} + 4 q^{11} - 3 q^{13} - 7 q^{17} - 4 q^{19} + 2 q^{23} + 2 q^{25} + 8 q^{29} - 8 q^{31} - q^{35} - 3 q^{37} - 4 q^{43} + 5 q^{47} - 5 q^{49} - 13 q^{53} + 4 q^{55} + 7 q^{59} - 10 q^{61} - 3 q^{65} - 19 q^{67} + 10 q^{71} + 11 q^{73} - 2 q^{77} - 10 q^{79} - 3 q^{83} - 7 q^{85} - 16 q^{89} + 10 q^{91} - 4 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 1.56155 0.590211 0.295106 0.955465i \(-0.404645\pi\)
0.295106 + 0.955465i \(0.404645\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.56155 −1.34887 −0.674437 0.738332i \(-0.735614\pi\)
−0.674437 + 0.738332i \(0.735614\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.123106 −0.0228601 −0.0114301 0.999935i \(-0.503638\pi\)
−0.0114301 + 0.999935i \(0.503638\pi\)
\(30\) 0 0
\(31\) −8.12311 −1.45895 −0.729476 0.684006i \(-0.760236\pi\)
−0.729476 + 0.684006i \(0.760236\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.56155 0.263951
\(36\) 0 0
\(37\) −3.56155 −0.585516 −0.292758 0.956187i \(-0.594573\pi\)
−0.292758 + 0.956187i \(0.594573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.12311 0.643921 0.321960 0.946753i \(-0.395658\pi\)
0.321960 + 0.946753i \(0.395658\pi\)
\(42\) 0 0
\(43\) −10.2462 −1.56253 −0.781266 0.624198i \(-0.785426\pi\)
−0.781266 + 0.624198i \(0.785426\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.68466 −0.537463 −0.268731 0.963215i \(-0.586604\pi\)
−0.268731 + 0.963215i \(0.586604\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.43845 −0.609668 −0.304834 0.952406i \(-0.598601\pi\)
−0.304834 + 0.952406i \(0.598601\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.56155 0.724053 0.362026 0.932168i \(-0.382085\pi\)
0.362026 + 0.932168i \(0.382085\pi\)
\(60\) 0 0
\(61\) −9.12311 −1.16809 −0.584047 0.811720i \(-0.698532\pi\)
−0.584047 + 0.811720i \(0.698532\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.561553 0.0696521
\(66\) 0 0
\(67\) −11.5616 −1.41247 −0.706234 0.707978i \(-0.749607\pi\)
−0.706234 + 0.707978i \(0.749607\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.00000 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) 3.43845 0.402440 0.201220 0.979546i \(-0.435509\pi\)
0.201220 + 0.979546i \(0.435509\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.12311 0.355911
\(78\) 0 0
\(79\) −9.12311 −1.02643 −0.513215 0.858260i \(-0.671546\pi\)
−0.513215 + 0.858260i \(0.671546\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.68466 0.514208 0.257104 0.966384i \(-0.417232\pi\)
0.257104 + 0.966384i \(0.417232\pi\)
\(84\) 0 0
\(85\) −5.56155 −0.603235
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0.876894 0.0919235
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −3.12311 −0.317103 −0.158552 0.987351i \(-0.550682\pi\)
−0.158552 + 0.987351i \(0.550682\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.80776 0.975909 0.487954 0.872869i \(-0.337743\pi\)
0.487954 + 0.872869i \(0.337743\pi\)
\(102\) 0 0
\(103\) 2.24621 0.221326 0.110663 0.993858i \(-0.464703\pi\)
0.110663 + 0.993858i \(0.464703\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.6847 −1.03292 −0.516462 0.856310i \(-0.672751\pi\)
−0.516462 + 0.856310i \(0.672751\pi\)
\(108\) 0 0
\(109\) 7.12311 0.682270 0.341135 0.940014i \(-0.389189\pi\)
0.341135 + 0.940014i \(0.389189\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.56155 −0.899475 −0.449738 0.893161i \(-0.648483\pi\)
−0.449738 + 0.893161i \(0.648483\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.68466 −0.796121
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0.807764 0.0716775 0.0358387 0.999358i \(-0.488590\pi\)
0.0358387 + 0.999358i \(0.488590\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.93087 −0.867664 −0.433832 0.900994i \(-0.642839\pi\)
−0.433832 + 0.900994i \(0.642839\pi\)
\(132\) 0 0
\(133\) −3.12311 −0.270808
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.6155 1.67587 0.837934 0.545772i \(-0.183763\pi\)
0.837934 + 0.545772i \(0.183763\pi\)
\(138\) 0 0
\(139\) 14.3693 1.21879 0.609395 0.792867i \(-0.291412\pi\)
0.609395 + 0.792867i \(0.291412\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.12311 0.0939188
\(144\) 0 0
\(145\) −0.123106 −0.0102234
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.12311 −0.747394 −0.373697 0.927551i \(-0.621910\pi\)
−0.373697 + 0.927551i \(0.621910\pi\)
\(150\) 0 0
\(151\) −2.56155 −0.208456 −0.104228 0.994553i \(-0.533237\pi\)
−0.104228 + 0.994553i \(0.533237\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.12311 −0.652464
\(156\) 0 0
\(157\) −3.31534 −0.264593 −0.132297 0.991210i \(-0.542235\pi\)
−0.132297 + 0.991210i \(0.542235\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.56155 0.123068
\(162\) 0 0
\(163\) 5.68466 0.445257 0.222628 0.974903i \(-0.428536\pi\)
0.222628 + 0.974903i \(0.428536\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −12.6847 −0.975743
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.12311 −0.0853881 −0.0426941 0.999088i \(-0.513594\pi\)
−0.0426941 + 0.999088i \(0.513594\pi\)
\(174\) 0 0
\(175\) 1.56155 0.118042
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 19.6847 1.47130 0.735650 0.677362i \(-0.236877\pi\)
0.735650 + 0.677362i \(0.236877\pi\)
\(180\) 0 0
\(181\) −17.1231 −1.27275 −0.636375 0.771380i \(-0.719567\pi\)
−0.636375 + 0.771380i \(0.719567\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.56155 −0.261851
\(186\) 0 0
\(187\) −11.1231 −0.813402
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.12311 −0.370695 −0.185347 0.982673i \(-0.559341\pi\)
−0.185347 + 0.982673i \(0.559341\pi\)
\(192\) 0 0
\(193\) −9.93087 −0.714840 −0.357420 0.933944i \(-0.616343\pi\)
−0.357420 + 0.933944i \(0.616343\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.80776 0.200045 0.100022 0.994985i \(-0.468109\pi\)
0.100022 + 0.994985i \(0.468109\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.192236 −0.0134923
\(204\) 0 0
\(205\) 4.12311 0.287970
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 18.9309 1.30325 0.651627 0.758539i \(-0.274087\pi\)
0.651627 + 0.758539i \(0.274087\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.2462 −0.698786
\(216\) 0 0
\(217\) −12.6847 −0.861091
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.12311 −0.210083
\(222\) 0 0
\(223\) 27.1231 1.81630 0.908149 0.418648i \(-0.137496\pi\)
0.908149 + 0.418648i \(0.137496\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.75379 −0.116403 −0.0582015 0.998305i \(-0.518537\pi\)
−0.0582015 + 0.998305i \(0.518537\pi\)
\(228\) 0 0
\(229\) −22.2462 −1.47007 −0.735036 0.678029i \(-0.762835\pi\)
−0.735036 + 0.678029i \(0.762835\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.3002 1.65747 0.828735 0.559641i \(-0.189061\pi\)
0.828735 + 0.559641i \(0.189061\pi\)
\(234\) 0 0
\(235\) −3.68466 −0.240361
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.1231 1.04292 0.521459 0.853277i \(-0.325388\pi\)
0.521459 + 0.853277i \(0.325388\pi\)
\(240\) 0 0
\(241\) 14.4924 0.933539 0.466769 0.884379i \(-0.345418\pi\)
0.466769 + 0.884379i \(0.345418\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.56155 −0.291427
\(246\) 0 0
\(247\) −1.12311 −0.0714615
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.876894 −0.0553491 −0.0276745 0.999617i \(-0.508810\pi\)
−0.0276745 + 0.999617i \(0.508810\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.6847 −0.728869 −0.364434 0.931229i \(-0.618738\pi\)
−0.364434 + 0.931229i \(0.618738\pi\)
\(258\) 0 0
\(259\) −5.56155 −0.345578
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.6847 1.64545 0.822723 0.568442i \(-0.192454\pi\)
0.822723 + 0.568442i \(0.192454\pi\)
\(264\) 0 0
\(265\) −4.43845 −0.272652
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.49242 −0.334879 −0.167439 0.985882i \(-0.553550\pi\)
−0.167439 + 0.985882i \(0.553550\pi\)
\(270\) 0 0
\(271\) −21.1771 −1.28642 −0.643208 0.765691i \(-0.722397\pi\)
−0.643208 + 0.765691i \(0.722397\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) −24.5616 −1.47576 −0.737880 0.674932i \(-0.764173\pi\)
−0.737880 + 0.674932i \(0.764173\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.876894 0.0523111 0.0261556 0.999658i \(-0.491673\pi\)
0.0261556 + 0.999658i \(0.491673\pi\)
\(282\) 0 0
\(283\) −24.9309 −1.48199 −0.740993 0.671513i \(-0.765645\pi\)
−0.740993 + 0.671513i \(0.765645\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.43845 0.380050
\(288\) 0 0
\(289\) 13.9309 0.819463
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.4384 −0.609821 −0.304910 0.952381i \(-0.598626\pi\)
−0.304910 + 0.952381i \(0.598626\pi\)
\(294\) 0 0
\(295\) 5.56155 0.323806
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.561553 0.0324754
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.12311 −0.522388
\(306\) 0 0
\(307\) 9.36932 0.534735 0.267368 0.963595i \(-0.413846\pi\)
0.267368 + 0.963595i \(0.413846\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.8078 1.40672 0.703360 0.710834i \(-0.251682\pi\)
0.703360 + 0.710834i \(0.251682\pi\)
\(312\) 0 0
\(313\) 14.9309 0.843943 0.421971 0.906609i \(-0.361338\pi\)
0.421971 + 0.906609i \(0.361338\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.630683 −0.0354227 −0.0177113 0.999843i \(-0.505638\pi\)
−0.0177113 + 0.999843i \(0.505638\pi\)
\(318\) 0 0
\(319\) −0.246211 −0.0137852
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11.1231 0.618906
\(324\) 0 0
\(325\) 0.561553 0.0311493
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.75379 −0.317217
\(330\) 0 0
\(331\) 29.4924 1.62105 0.810525 0.585704i \(-0.199182\pi\)
0.810525 + 0.585704i \(0.199182\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.5616 −0.631675
\(336\) 0 0
\(337\) 4.87689 0.265661 0.132831 0.991139i \(-0.457593\pi\)
0.132831 + 0.991139i \(0.457593\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.2462 −0.879782
\(342\) 0 0
\(343\) −18.0540 −0.974823
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.24621 −0.120583 −0.0602915 0.998181i \(-0.519203\pi\)
−0.0602915 + 0.998181i \(0.519203\pi\)
\(348\) 0 0
\(349\) 4.12311 0.220705 0.110352 0.993893i \(-0.464802\pi\)
0.110352 + 0.993893i \(0.464802\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −24.8078 −1.32038 −0.660192 0.751097i \(-0.729525\pi\)
−0.660192 + 0.751097i \(0.729525\pi\)
\(354\) 0 0
\(355\) 5.00000 0.265372
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.87689 −0.151837 −0.0759183 0.997114i \(-0.524189\pi\)
−0.0759183 + 0.997114i \(0.524189\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.43845 0.179977
\(366\) 0 0
\(367\) −31.8078 −1.66035 −0.830176 0.557502i \(-0.811760\pi\)
−0.830176 + 0.557502i \(0.811760\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.93087 −0.359833
\(372\) 0 0
\(373\) −24.7386 −1.28092 −0.640459 0.767992i \(-0.721256\pi\)
−0.640459 + 0.767992i \(0.721256\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.0691303 −0.00356039
\(378\) 0 0
\(379\) 32.9848 1.69432 0.847159 0.531340i \(-0.178311\pi\)
0.847159 + 0.531340i \(0.178311\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.9309 −0.762932 −0.381466 0.924383i \(-0.624581\pi\)
−0.381466 + 0.924383i \(0.624581\pi\)
\(384\) 0 0
\(385\) 3.12311 0.159168
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 27.1231 1.37520 0.687598 0.726092i \(-0.258665\pi\)
0.687598 + 0.726092i \(0.258665\pi\)
\(390\) 0 0
\(391\) −5.56155 −0.281260
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.12311 −0.459033
\(396\) 0 0
\(397\) 24.1771 1.21341 0.606706 0.794926i \(-0.292490\pi\)
0.606706 + 0.794926i \(0.292490\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −16.7386 −0.835887 −0.417944 0.908473i \(-0.637249\pi\)
−0.417944 + 0.908473i \(0.637249\pi\)
\(402\) 0 0
\(403\) −4.56155 −0.227227
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.12311 −0.353079
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.68466 0.427344
\(414\) 0 0
\(415\) 4.68466 0.229961
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −19.6155 −0.958281 −0.479141 0.877738i \(-0.659052\pi\)
−0.479141 + 0.877738i \(0.659052\pi\)
\(420\) 0 0
\(421\) −39.1231 −1.90674 −0.953372 0.301798i \(-0.902413\pi\)
−0.953372 + 0.301798i \(0.902413\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.56155 −0.269775
\(426\) 0 0
\(427\) −14.2462 −0.689422
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13.7538 −0.662497 −0.331248 0.943544i \(-0.607470\pi\)
−0.331248 + 0.943544i \(0.607470\pi\)
\(432\) 0 0
\(433\) −1.31534 −0.0632113 −0.0316056 0.999500i \(-0.510062\pi\)
−0.0316056 + 0.999500i \(0.510062\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) 16.8078 0.802191 0.401095 0.916036i \(-0.368630\pi\)
0.401095 + 0.916036i \(0.368630\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.0540 0.715236 0.357618 0.933868i \(-0.383589\pi\)
0.357618 + 0.933868i \(0.383589\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.6847 −1.44810 −0.724049 0.689748i \(-0.757721\pi\)
−0.724049 + 0.689748i \(0.757721\pi\)
\(450\) 0 0
\(451\) 8.24621 0.388299
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.876894 0.0411094
\(456\) 0 0
\(457\) 10.9309 0.511325 0.255662 0.966766i \(-0.417706\pi\)
0.255662 + 0.966766i \(0.417706\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.17708 −0.287695 −0.143848 0.989600i \(-0.545948\pi\)
−0.143848 + 0.989600i \(0.545948\pi\)
\(462\) 0 0
\(463\) −20.8769 −0.970232 −0.485116 0.874450i \(-0.661223\pi\)
−0.485116 + 0.874450i \(0.661223\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.68466 −0.124231 −0.0621156 0.998069i \(-0.519785\pi\)
−0.0621156 + 0.998069i \(0.519785\pi\)
\(468\) 0 0
\(469\) −18.0540 −0.833655
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −20.4924 −0.942243
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.12311 −0.141813
\(486\) 0 0
\(487\) −11.6847 −0.529482 −0.264741 0.964319i \(-0.585287\pi\)
−0.264741 + 0.964319i \(0.585287\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −28.6155 −1.29140 −0.645700 0.763591i \(-0.723434\pi\)
−0.645700 + 0.763591i \(0.723434\pi\)
\(492\) 0 0
\(493\) 0.684658 0.0308355
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.80776 0.350226
\(498\) 0 0
\(499\) 9.73863 0.435961 0.217981 0.975953i \(-0.430053\pi\)
0.217981 + 0.975953i \(0.430053\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19.5616 −0.872207 −0.436103 0.899897i \(-0.643642\pi\)
−0.436103 + 0.899897i \(0.643642\pi\)
\(504\) 0 0
\(505\) 9.80776 0.436440
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.5616 1.26597 0.632984 0.774165i \(-0.281830\pi\)
0.632984 + 0.774165i \(0.281830\pi\)
\(510\) 0 0
\(511\) 5.36932 0.237525
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.24621 0.0989799
\(516\) 0 0
\(517\) −7.36932 −0.324102
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 33.8617 1.48351 0.741755 0.670671i \(-0.233994\pi\)
0.741755 + 0.670671i \(0.233994\pi\)
\(522\) 0 0
\(523\) −6.24621 −0.273128 −0.136564 0.990631i \(-0.543606\pi\)
−0.136564 + 0.990631i \(0.543606\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 45.1771 1.96794
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.31534 0.100289
\(534\) 0 0
\(535\) −10.6847 −0.461938
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.12311 −0.392960
\(540\) 0 0
\(541\) −21.0540 −0.905181 −0.452591 0.891718i \(-0.649500\pi\)
−0.452591 + 0.891718i \(0.649500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.12311 0.305120
\(546\) 0 0
\(547\) −19.6847 −0.841655 −0.420828 0.907141i \(-0.638260\pi\)
−0.420828 + 0.907141i \(0.638260\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.246211 0.0104890
\(552\) 0 0
\(553\) −14.2462 −0.605811
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −32.3002 −1.36860 −0.684301 0.729199i \(-0.739893\pi\)
−0.684301 + 0.729199i \(0.739893\pi\)
\(558\) 0 0
\(559\) −5.75379 −0.243359
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.8078 1.34054 0.670269 0.742118i \(-0.266179\pi\)
0.670269 + 0.742118i \(0.266179\pi\)
\(564\) 0 0
\(565\) −9.56155 −0.402258
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.73863 0.198654 0.0993269 0.995055i \(-0.468331\pi\)
0.0993269 + 0.995055i \(0.468331\pi\)
\(570\) 0 0
\(571\) 18.7386 0.784187 0.392094 0.919925i \(-0.371751\pi\)
0.392094 + 0.919925i \(0.371751\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −19.6847 −0.819483 −0.409742 0.912202i \(-0.634381\pi\)
−0.409742 + 0.912202i \(0.634381\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.31534 0.303492
\(582\) 0 0
\(583\) −8.87689 −0.367643
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −43.3002 −1.78719 −0.893595 0.448874i \(-0.851825\pi\)
−0.893595 + 0.448874i \(0.851825\pi\)
\(588\) 0 0
\(589\) 16.2462 0.669413
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −48.3542 −1.98567 −0.992834 0.119504i \(-0.961870\pi\)
−0.992834 + 0.119504i \(0.961870\pi\)
\(594\) 0 0
\(595\) −8.68466 −0.356036
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −40.8617 −1.66679 −0.833393 0.552682i \(-0.813605\pi\)
−0.833393 + 0.552682i \(0.813605\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 36.0000 1.46119 0.730597 0.682808i \(-0.239242\pi\)
0.730597 + 0.682808i \(0.239242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.06913 −0.0837081
\(612\) 0 0
\(613\) 31.6155 1.27694 0.638470 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.3002 −1.86398 −0.931988 0.362490i \(-0.881927\pi\)
−0.931988 + 0.362490i \(0.881927\pi\)
\(618\) 0 0
\(619\) 11.5076 0.462529 0.231264 0.972891i \(-0.425714\pi\)
0.231264 + 0.972891i \(0.425714\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.4924 −0.500498
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 19.8078 0.789787
\(630\) 0 0
\(631\) −34.7386 −1.38292 −0.691462 0.722413i \(-0.743033\pi\)
−0.691462 + 0.722413i \(0.743033\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.807764 0.0320551
\(636\) 0 0
\(637\) −2.56155 −0.101492
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.1231 −0.439336 −0.219668 0.975575i \(-0.570497\pi\)
−0.219668 + 0.975575i \(0.570497\pi\)
\(642\) 0 0
\(643\) 23.1771 0.914015 0.457007 0.889463i \(-0.348921\pi\)
0.457007 + 0.889463i \(0.348921\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 34.1771 1.34364 0.671820 0.740715i \(-0.265513\pi\)
0.671820 + 0.740715i \(0.265513\pi\)
\(648\) 0 0
\(649\) 11.1231 0.436620
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.4233 1.26882 0.634411 0.772996i \(-0.281243\pi\)
0.634411 + 0.772996i \(0.281243\pi\)
\(654\) 0 0
\(655\) −9.93087 −0.388031
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11.6155 −0.452477 −0.226238 0.974072i \(-0.572643\pi\)
−0.226238 + 0.974072i \(0.572643\pi\)
\(660\) 0 0
\(661\) 44.2462 1.72098 0.860489 0.509469i \(-0.170158\pi\)
0.860489 + 0.509469i \(0.170158\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.12311 −0.121109
\(666\) 0 0
\(667\) −0.123106 −0.00476667
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −18.2462 −0.704387
\(672\) 0 0
\(673\) −36.8078 −1.41884 −0.709418 0.704788i \(-0.751042\pi\)
−0.709418 + 0.704788i \(0.751042\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.68466 0.256912 0.128456 0.991715i \(-0.458998\pi\)
0.128456 + 0.991715i \(0.458998\pi\)
\(678\) 0 0
\(679\) −4.87689 −0.187158
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21.9309 0.839161 0.419581 0.907718i \(-0.362177\pi\)
0.419581 + 0.907718i \(0.362177\pi\)
\(684\) 0 0
\(685\) 19.6155 0.749471
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.49242 −0.0949537
\(690\) 0 0
\(691\) 32.4924 1.23607 0.618035 0.786151i \(-0.287929\pi\)
0.618035 + 0.786151i \(0.287929\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.3693 0.545059
\(696\) 0 0
\(697\) −22.9309 −0.868569
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26.2462 −0.991306 −0.495653 0.868521i \(-0.665071\pi\)
−0.495653 + 0.868521i \(0.665071\pi\)
\(702\) 0 0
\(703\) 7.12311 0.268653
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.3153 0.575993
\(708\) 0 0
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.12311 −0.304213
\(714\) 0 0
\(715\) 1.12311 0.0420018
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.68466 −0.323883 −0.161942 0.986800i \(-0.551776\pi\)
−0.161942 + 0.986800i \(0.551776\pi\)
\(720\) 0 0
\(721\) 3.50758 0.130629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.123106 −0.00457203
\(726\) 0 0
\(727\) 15.5616 0.577146 0.288573 0.957458i \(-0.406819\pi\)
0.288573 + 0.957458i \(0.406819\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 56.9848 2.10766
\(732\) 0 0
\(733\) −5.17708 −0.191220 −0.0956099 0.995419i \(-0.530480\pi\)
−0.0956099 + 0.995419i \(0.530480\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23.1231 −0.851751
\(738\) 0 0
\(739\) −31.4924 −1.15847 −0.579234 0.815161i \(-0.696648\pi\)
−0.579234 + 0.815161i \(0.696648\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29.7538 1.09156 0.545780 0.837928i \(-0.316233\pi\)
0.545780 + 0.837928i \(0.316233\pi\)
\(744\) 0 0
\(745\) −9.12311 −0.334245
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.6847 −0.609644
\(750\) 0 0
\(751\) −46.3542 −1.69149 −0.845744 0.533589i \(-0.820843\pi\)
−0.845744 + 0.533589i \(0.820843\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.56155 −0.0932245
\(756\) 0 0
\(757\) 44.5464 1.61907 0.809533 0.587074i \(-0.199720\pi\)
0.809533 + 0.587074i \(0.199720\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 43.9848 1.59445 0.797225 0.603683i \(-0.206301\pi\)
0.797225 + 0.603683i \(0.206301\pi\)
\(762\) 0 0
\(763\) 11.1231 0.402683
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.12311 0.112769
\(768\) 0 0
\(769\) 33.6155 1.21221 0.606103 0.795386i \(-0.292732\pi\)
0.606103 + 0.795386i \(0.292732\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.8769 0.750890 0.375445 0.926845i \(-0.377490\pi\)
0.375445 + 0.926845i \(0.377490\pi\)
\(774\) 0 0
\(775\) −8.12311 −0.291791
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.24621 −0.295451
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.31534 −0.118330
\(786\) 0 0
\(787\) 20.3002 0.723624 0.361812 0.932251i \(-0.382158\pi\)
0.361812 + 0.932251i \(0.382158\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.9309 −0.530881
\(792\) 0 0
\(793\) −5.12311 −0.181927
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −45.8078 −1.62259 −0.811297 0.584634i \(-0.801238\pi\)
−0.811297 + 0.584634i \(0.801238\pi\)
\(798\) 0 0
\(799\) 20.4924 0.724970
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.87689 0.242680
\(804\) 0 0
\(805\) 1.56155 0.0550375
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.1922 −1.06150 −0.530751 0.847528i \(-0.678090\pi\)
−0.530751 + 0.847528i \(0.678090\pi\)
\(810\) 0 0
\(811\) −10.3693 −0.364116 −0.182058 0.983288i \(-0.558276\pi\)
−0.182058 + 0.983288i \(0.558276\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.68466 0.199125
\(816\) 0 0
\(817\) 20.4924 0.716939
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.50758 −0.192216 −0.0961079 0.995371i \(-0.530639\pi\)
−0.0961079 + 0.995371i \(0.530639\pi\)
\(822\) 0 0
\(823\) 12.1771 0.424466 0.212233 0.977219i \(-0.431926\pi\)
0.212233 + 0.977219i \(0.431926\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −40.6847 −1.41474 −0.707372 0.706841i \(-0.750119\pi\)
−0.707372 + 0.706841i \(0.750119\pi\)
\(828\) 0 0
\(829\) −16.4384 −0.570931 −0.285465 0.958389i \(-0.592148\pi\)
−0.285465 + 0.958389i \(0.592148\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 25.3693 0.878995
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.61553 −0.262917 −0.131459 0.991322i \(-0.541966\pi\)
−0.131459 + 0.991322i \(0.541966\pi\)
\(840\) 0 0
\(841\) −28.9848 −0.999477
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.6847 −0.436366
\(846\) 0 0
\(847\) −10.9309 −0.375589
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.56155 −0.122088
\(852\) 0 0
\(853\) 18.4924 0.633168 0.316584 0.948564i \(-0.397464\pi\)
0.316584 + 0.948564i \(0.397464\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29.5464 −1.00929 −0.504643 0.863328i \(-0.668376\pi\)
−0.504643 + 0.863328i \(0.668376\pi\)
\(858\) 0 0
\(859\) 24.3693 0.831470 0.415735 0.909486i \(-0.363524\pi\)
0.415735 + 0.909486i \(0.363524\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −21.6847 −0.738154 −0.369077 0.929399i \(-0.620326\pi\)
−0.369077 + 0.929399i \(0.620326\pi\)
\(864\) 0 0
\(865\) −1.12311 −0.0381867
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.2462 −0.618960
\(870\) 0 0
\(871\) −6.49242 −0.219987
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.56155 0.0527901
\(876\) 0 0
\(877\) 51.4773 1.73826 0.869132 0.494580i \(-0.164678\pi\)
0.869132 + 0.494580i \(0.164678\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.7386 0.900847 0.450424 0.892815i \(-0.351273\pi\)
0.450424 + 0.892815i \(0.351273\pi\)
\(882\) 0 0
\(883\) −20.9848 −0.706196 −0.353098 0.935586i \(-0.614872\pi\)
−0.353098 + 0.935586i \(0.614872\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.17708 0.0730992 0.0365496 0.999332i \(-0.488363\pi\)
0.0365496 + 0.999332i \(0.488363\pi\)
\(888\) 0 0
\(889\) 1.26137 0.0423049
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.36932 0.246605
\(894\) 0 0
\(895\) 19.6847 0.657986
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.00000 0.0333519
\(900\) 0 0
\(901\) 24.6847 0.822365
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −17.1231 −0.569191
\(906\) 0 0
\(907\) 37.4233 1.24262 0.621310 0.783565i \(-0.286601\pi\)
0.621310 + 0.783565i \(0.286601\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14.8769 −0.492894 −0.246447 0.969156i \(-0.579263\pi\)
−0.246447 + 0.969156i \(0.579263\pi\)
\(912\) 0 0
\(913\) 9.36932 0.310079
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.5076 −0.512105
\(918\) 0 0
\(919\) 12.4924 0.412087 0.206043 0.978543i \(-0.433941\pi\)
0.206043 + 0.978543i \(0.433941\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.80776 0.0924187
\(924\) 0 0
\(925\) −3.56155 −0.117103
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −49.7386 −1.63187 −0.815936 0.578142i \(-0.803778\pi\)
−0.815936 + 0.578142i \(0.803778\pi\)
\(930\) 0 0
\(931\) 9.12311 0.298998
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −11.1231 −0.363764
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.492423 0.0160525 0.00802626 0.999968i \(-0.497445\pi\)
0.00802626 + 0.999968i \(0.497445\pi\)
\(942\) 0 0
\(943\) 4.12311 0.134267
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.4384 −0.436691 −0.218345 0.975872i \(-0.570066\pi\)
−0.218345 + 0.975872i \(0.570066\pi\)
\(948\) 0 0
\(949\) 1.93087 0.0626787
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 27.7538 0.899033 0.449517 0.893272i \(-0.351596\pi\)
0.449517 + 0.893272i \(0.351596\pi\)
\(954\) 0 0
\(955\) −5.12311 −0.165780
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 30.6307 0.989116
\(960\) 0 0
\(961\) 34.9848 1.12854
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −9.93087 −0.319686
\(966\) 0 0
\(967\) −36.1771 −1.16338 −0.581688 0.813412i \(-0.697608\pi\)
−0.581688 + 0.813412i \(0.697608\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −34.1080 −1.09458 −0.547288 0.836944i \(-0.684340\pi\)
−0.547288 + 0.836944i \(0.684340\pi\)
\(972\) 0 0
\(973\) 22.4384 0.719344
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −41.3153 −1.32179 −0.660897 0.750476i \(-0.729824\pi\)
−0.660897 + 0.750476i \(0.729824\pi\)
\(978\) 0 0
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 41.0388 1.30894 0.654468 0.756090i \(-0.272893\pi\)
0.654468 + 0.756090i \(0.272893\pi\)
\(984\) 0 0
\(985\) 2.80776 0.0894628
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −10.2462 −0.325811
\(990\) 0 0
\(991\) 18.4384 0.585717 0.292858 0.956156i \(-0.405394\pi\)
0.292858 + 0.956156i \(0.405394\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.0000 0.570638
\(996\) 0 0
\(997\) −46.1080 −1.46025 −0.730127 0.683312i \(-0.760539\pi\)
−0.730127 + 0.683312i \(0.760539\pi\)
\(998\) 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 8280.2.a.bf.1.2 2
3.2 odd 2 920.2.a.e.1.1 2
12.11 even 2 1840.2.a.o.1.2 2
15.2 even 4 4600.2.e.n.4049.4 4
15.8 even 4 4600.2.e.n.4049.1 4
15.14 odd 2 4600.2.a.t.1.2 2
24.5 odd 2 7360.2.a.bp.1.2 2
24.11 even 2 7360.2.a.bl.1.1 2
60.59 even 2 9200.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.e.1.1 2 3.2 odd 2
1840.2.a.o.1.2 2 12.11 even 2
4600.2.a.t.1.2 2 15.14 odd 2
4600.2.e.n.4049.1 4 15.8 even 4
4600.2.e.n.4049.4 4 15.2 even 4
7360.2.a.bl.1.1 2 24.11 even 2
7360.2.a.bp.1.2 2 24.5 odd 2
8280.2.a.bf.1.2 2 1.1 even 1 trivial
9200.2.a.bq.1.1 2 60.59 even 2