Properties

Label 8280.2.a.be.1.1
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 2760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 8280.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} -2.56155 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -2.56155 q^{7} +5.12311 q^{11} +2.00000 q^{13} -4.56155 q^{17} +1.12311 q^{19} +1.00000 q^{23} +1.00000 q^{25} -8.56155 q^{29} -3.68466 q^{31} -2.56155 q^{35} +0.561553 q^{37} -3.43845 q^{41} -6.24621 q^{43} -8.00000 q^{47} -0.438447 q^{49} -0.561553 q^{53} +5.12311 q^{55} -6.56155 q^{59} +13.3693 q^{61} +2.00000 q^{65} -2.56155 q^{67} -11.6847 q^{71} +2.00000 q^{73} -13.1231 q^{77} +10.2462 q^{79} +0.315342 q^{83} -4.56155 q^{85} +3.12311 q^{89} -5.12311 q^{91} +1.12311 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - q^{7} + 2 q^{11} + 4 q^{13} - 5 q^{17} - 6 q^{19} + 2 q^{23} + 2 q^{25} - 13 q^{29} + 5 q^{31} - q^{35} - 3 q^{37} - 11 q^{41} + 4 q^{43} - 16 q^{47} - 5 q^{49} + 3 q^{53} + 2 q^{55} - 9 q^{59} + 2 q^{61} + 4 q^{65} - q^{67} - 11 q^{71} + 4 q^{73} - 18 q^{77} + 4 q^{79} + 13 q^{83} - 5 q^{85} - 2 q^{89} - 2 q^{91} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) 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
\(7\) −2.56155 −0.968176 −0.484088 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.12311 1.54467 0.772337 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.56155 −1.10634 −0.553170 0.833069i \(-0.686582\pi\)
−0.553170 + 0.833069i \(0.686582\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 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
\(29\) −8.56155 −1.58984 −0.794920 0.606714i \(-0.792487\pi\)
−0.794920 + 0.606714i \(0.792487\pi\)
\(30\) 0 0
\(31\) −3.68466 −0.661784 −0.330892 0.943669i \(-0.607350\pi\)
−0.330892 + 0.943669i \(0.607350\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.56155 −0.432981
\(36\) 0 0
\(37\) 0.561553 0.0923187 0.0461594 0.998934i \(-0.485302\pi\)
0.0461594 + 0.998934i \(0.485302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.43845 −0.536995 −0.268498 0.963280i \(-0.586527\pi\)
−0.268498 + 0.963280i \(0.586527\pi\)
\(42\) 0 0
\(43\) −6.24621 −0.952538 −0.476269 0.879300i \(-0.658011\pi\)
−0.476269 + 0.879300i \(0.658011\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.561553 −0.0771352 −0.0385676 0.999256i \(-0.512279\pi\)
−0.0385676 + 0.999256i \(0.512279\pi\)
\(54\) 0 0
\(55\) 5.12311 0.690799
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.56155 −0.854241 −0.427121 0.904195i \(-0.640472\pi\)
−0.427121 + 0.904195i \(0.640472\pi\)
\(60\) 0 0
\(61\) 13.3693 1.71177 0.855883 0.517170i \(-0.173014\pi\)
0.855883 + 0.517170i \(0.173014\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −2.56155 −0.312943 −0.156472 0.987682i \(-0.550012\pi\)
−0.156472 + 0.987682i \(0.550012\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.6847 −1.38671 −0.693357 0.720594i \(-0.743869\pi\)
−0.693357 + 0.720594i \(0.743869\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.1231 −1.49552
\(78\) 0 0
\(79\) 10.2462 1.15279 0.576394 0.817172i \(-0.304459\pi\)
0.576394 + 0.817172i \(0.304459\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.315342 0.0346132 0.0173066 0.999850i \(-0.494491\pi\)
0.0173066 + 0.999850i \(0.494491\pi\)
\(84\) 0 0
\(85\) −4.56155 −0.494770
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.12311 0.331049 0.165524 0.986206i \(-0.447068\pi\)
0.165524 + 0.986206i \(0.447068\pi\)
\(90\) 0 0
\(91\) −5.12311 −0.537047
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.12311 0.115228
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.31534 −0.628400 −0.314200 0.949357i \(-0.601736\pi\)
−0.314200 + 0.949357i \(0.601736\pi\)
\(102\) 0 0
\(103\) 14.2462 1.40372 0.701860 0.712314i \(-0.252353\pi\)
0.701860 + 0.712314i \(0.252353\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.68466 −0.742904 −0.371452 0.928452i \(-0.621140\pi\)
−0.371452 + 0.928452i \(0.621140\pi\)
\(108\) 0 0
\(109\) −4.24621 −0.406713 −0.203357 0.979105i \(-0.565185\pi\)
−0.203357 + 0.979105i \(0.565185\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.31534 −0.217809 −0.108905 0.994052i \(-0.534734\pi\)
−0.108905 + 0.994052i \(0.534734\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.6847 1.07113
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.2462 1.26415 0.632073 0.774909i \(-0.282204\pi\)
0.632073 + 0.774909i \(0.282204\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.2462 0.895216 0.447608 0.894230i \(-0.352276\pi\)
0.447608 + 0.894230i \(0.352276\pi\)
\(132\) 0 0
\(133\) −2.87689 −0.249458
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −12.8078 −1.08634 −0.543170 0.839623i \(-0.682776\pi\)
−0.543170 + 0.839623i \(0.682776\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.2462 0.856831
\(144\) 0 0
\(145\) −8.56155 −0.710998
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.3693 1.09526 0.547629 0.836722i \(-0.315531\pi\)
0.547629 + 0.836722i \(0.315531\pi\)
\(150\) 0 0
\(151\) 10.2462 0.833825 0.416912 0.908947i \(-0.363112\pi\)
0.416912 + 0.908947i \(0.363112\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.68466 −0.295959
\(156\) 0 0
\(157\) −17.0540 −1.36106 −0.680528 0.732722i \(-0.738249\pi\)
−0.680528 + 0.732722i \(0.738249\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.56155 −0.201879
\(162\) 0 0
\(163\) 19.3693 1.51712 0.758561 0.651602i \(-0.225903\pi\)
0.758561 + 0.651602i \(0.225903\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.4924 0.966693 0.483346 0.875429i \(-0.339421\pi\)
0.483346 + 0.875429i \(0.339421\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.1231 −1.14979 −0.574894 0.818228i \(-0.694957\pi\)
−0.574894 + 0.818228i \(0.694957\pi\)
\(174\) 0 0
\(175\) −2.56155 −0.193635
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.4924 −0.933727 −0.466864 0.884329i \(-0.654616\pi\)
−0.466864 + 0.884329i \(0.654616\pi\)
\(180\) 0 0
\(181\) 0.876894 0.0651790 0.0325895 0.999469i \(-0.489625\pi\)
0.0325895 + 0.999469i \(0.489625\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.561553 0.0412862
\(186\) 0 0
\(187\) −23.3693 −1.70893
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.4924 0.903920 0.451960 0.892038i \(-0.350725\pi\)
0.451960 + 0.892038i \(0.350725\pi\)
\(192\) 0 0
\(193\) −13.3693 −0.962344 −0.481172 0.876626i \(-0.659789\pi\)
−0.481172 + 0.876626i \(0.659789\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.2462 −1.44248 −0.721241 0.692684i \(-0.756428\pi\)
−0.721241 + 0.692684i \(0.756428\pi\)
\(198\) 0 0
\(199\) −2.87689 −0.203938 −0.101969 0.994788i \(-0.532514\pi\)
−0.101969 + 0.994788i \(0.532514\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 21.9309 1.53925
\(204\) 0 0
\(205\) −3.43845 −0.240152
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.75379 0.397998
\(210\) 0 0
\(211\) −13.4384 −0.925141 −0.462570 0.886583i \(-0.653073\pi\)
−0.462570 + 0.886583i \(0.653073\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.24621 −0.425988
\(216\) 0 0
\(217\) 9.43845 0.640724
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9.12311 −0.613686
\(222\) 0 0
\(223\) −19.3693 −1.29707 −0.648533 0.761187i \(-0.724617\pi\)
−0.648533 + 0.761187i \(0.724617\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.49242 −0.563662 −0.281831 0.959464i \(-0.590942\pi\)
−0.281831 + 0.959464i \(0.590942\pi\)
\(228\) 0 0
\(229\) −7.12311 −0.470708 −0.235354 0.971910i \(-0.575625\pi\)
−0.235354 + 0.971910i \(0.575625\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.3693 −0.875853 −0.437927 0.899011i \(-0.644287\pi\)
−0.437927 + 0.899011i \(0.644287\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.6847 1.27329 0.636647 0.771155i \(-0.280321\pi\)
0.636647 + 0.771155i \(0.280321\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.438447 −0.0280114
\(246\) 0 0
\(247\) 2.24621 0.142923
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 29.1231 1.83823 0.919117 0.393985i \(-0.128904\pi\)
0.919117 + 0.393985i \(0.128904\pi\)
\(252\) 0 0
\(253\) 5.12311 0.322087
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −23.6155 −1.47310 −0.736548 0.676385i \(-0.763545\pi\)
−0.736548 + 0.676385i \(0.763545\pi\)
\(258\) 0 0
\(259\) −1.43845 −0.0893808
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −30.5616 −1.88451 −0.942253 0.334902i \(-0.891297\pi\)
−0.942253 + 0.334902i \(0.891297\pi\)
\(264\) 0 0
\(265\) −0.561553 −0.0344959
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.4384 −0.697414 −0.348707 0.937232i \(-0.613379\pi\)
−0.348707 + 0.937232i \(0.613379\pi\)
\(270\) 0 0
\(271\) 11.6847 0.709792 0.354896 0.934906i \(-0.384516\pi\)
0.354896 + 0.934906i \(0.384516\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.12311 0.308935
\(276\) 0 0
\(277\) −8.24621 −0.495467 −0.247733 0.968828i \(-0.579686\pi\)
−0.247733 + 0.968828i \(0.579686\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 2.56155 0.152269 0.0761343 0.997098i \(-0.475742\pi\)
0.0761343 + 0.997098i \(0.475742\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.80776 0.519906
\(288\) 0 0
\(289\) 3.80776 0.223986
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.93087 −0.463326 −0.231663 0.972796i \(-0.574417\pi\)
−0.231663 + 0.972796i \(0.574417\pi\)
\(294\) 0 0
\(295\) −6.56155 −0.382028
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 13.3693 0.765525
\(306\) 0 0
\(307\) −29.6155 −1.69025 −0.845124 0.534571i \(-0.820473\pi\)
−0.845124 + 0.534571i \(0.820473\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 3.43845 0.194353 0.0971763 0.995267i \(-0.469019\pi\)
0.0971763 + 0.995267i \(0.469019\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.1231 0.624736 0.312368 0.949961i \(-0.398878\pi\)
0.312368 + 0.949961i \(0.398878\pi\)
\(318\) 0 0
\(319\) −43.8617 −2.45579
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.12311 −0.285057
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 20.4924 1.12978
\(330\) 0 0
\(331\) −3.19224 −0.175461 −0.0877306 0.996144i \(-0.527961\pi\)
−0.0877306 + 0.996144i \(0.527961\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.56155 −0.139953
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −18.8769 −1.02224
\(342\) 0 0
\(343\) 19.0540 1.02882
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.7386 −1.43541 −0.717703 0.696350i \(-0.754806\pi\)
−0.717703 + 0.696350i \(0.754806\pi\)
\(348\) 0 0
\(349\) 1.05398 0.0564180 0.0282090 0.999602i \(-0.491020\pi\)
0.0282090 + 0.999602i \(0.491020\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11.1231 −0.592023 −0.296012 0.955184i \(-0.595657\pi\)
−0.296012 + 0.955184i \(0.595657\pi\)
\(354\) 0 0
\(355\) −11.6847 −0.620157
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 17.3002 0.903062 0.451531 0.892255i \(-0.350878\pi\)
0.451531 + 0.892255i \(0.350878\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.43845 0.0746805
\(372\) 0 0
\(373\) −19.7538 −1.02281 −0.511406 0.859339i \(-0.670875\pi\)
−0.511406 + 0.859339i \(0.670875\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −17.1231 −0.881885
\(378\) 0 0
\(379\) −1.75379 −0.0900861 −0.0450430 0.998985i \(-0.514342\pi\)
−0.0450430 + 0.998985i \(0.514342\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.9309 1.12062 0.560308 0.828285i \(-0.310683\pi\)
0.560308 + 0.828285i \(0.310683\pi\)
\(384\) 0 0
\(385\) −13.1231 −0.668815
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.4924 −1.54603 −0.773014 0.634389i \(-0.781252\pi\)
−0.773014 + 0.634389i \(0.781252\pi\)
\(390\) 0 0
\(391\) −4.56155 −0.230688
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 10.2462 0.515543
\(396\) 0 0
\(397\) −23.6155 −1.18523 −0.592615 0.805486i \(-0.701904\pi\)
−0.592615 + 0.805486i \(0.701904\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −7.36932 −0.367092
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.87689 0.142602
\(408\) 0 0
\(409\) −2.31534 −0.114486 −0.0572431 0.998360i \(-0.518231\pi\)
−0.0572431 + 0.998360i \(0.518231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.8078 0.827056
\(414\) 0 0
\(415\) 0.315342 0.0154795
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.8769 1.31302 0.656511 0.754316i \(-0.272032\pi\)
0.656511 + 0.754316i \(0.272032\pi\)
\(420\) 0 0
\(421\) 26.4924 1.29116 0.645581 0.763692i \(-0.276615\pi\)
0.645581 + 0.763692i \(0.276615\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.56155 −0.221268
\(426\) 0 0
\(427\) −34.2462 −1.65729
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −20.4924 −0.987085 −0.493543 0.869722i \(-0.664298\pi\)
−0.493543 + 0.869722i \(0.664298\pi\)
\(432\) 0 0
\(433\) −7.43845 −0.357469 −0.178734 0.983897i \(-0.557200\pi\)
−0.178734 + 0.983897i \(0.557200\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.12311 0.0537254
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.4924 −0.783579 −0.391789 0.920055i \(-0.628144\pi\)
−0.391789 + 0.920055i \(0.628144\pi\)
\(444\) 0 0
\(445\) 3.12311 0.148049
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 40.4233 1.90769 0.953847 0.300294i \(-0.0970849\pi\)
0.953847 + 0.300294i \(0.0970849\pi\)
\(450\) 0 0
\(451\) −17.6155 −0.829483
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.12311 −0.240175
\(456\) 0 0
\(457\) −10.3153 −0.482531 −0.241266 0.970459i \(-0.577563\pi\)
−0.241266 + 0.970459i \(0.577563\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −20.2462 −0.942960 −0.471480 0.881877i \(-0.656280\pi\)
−0.471480 + 0.881877i \(0.656280\pi\)
\(462\) 0 0
\(463\) 4.63068 0.215206 0.107603 0.994194i \(-0.465682\pi\)
0.107603 + 0.994194i \(0.465682\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −31.6847 −1.46619 −0.733096 0.680126i \(-0.761925\pi\)
−0.733096 + 0.680126i \(0.761925\pi\)
\(468\) 0 0
\(469\) 6.56155 0.302984
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −32.0000 −1.47136
\(474\) 0 0
\(475\) 1.12311 0.0515316
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −22.7386 −1.03895 −0.519477 0.854484i \(-0.673873\pi\)
−0.519477 + 0.854484i \(0.673873\pi\)
\(480\) 0 0
\(481\) 1.12311 0.0512092
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) 18.7386 0.849129 0.424564 0.905398i \(-0.360427\pi\)
0.424564 + 0.905398i \(0.360427\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.31534 0.194749 0.0973743 0.995248i \(-0.468956\pi\)
0.0973743 + 0.995248i \(0.468956\pi\)
\(492\) 0 0
\(493\) 39.0540 1.75890
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 29.9309 1.34258
\(498\) 0 0
\(499\) 22.4233 1.00380 0.501902 0.864924i \(-0.332634\pi\)
0.501902 + 0.864924i \(0.332634\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.3153 −0.549114 −0.274557 0.961571i \(-0.588531\pi\)
−0.274557 + 0.961571i \(0.588531\pi\)
\(504\) 0 0
\(505\) −6.31534 −0.281029
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.49242 −0.287772 −0.143886 0.989594i \(-0.545960\pi\)
−0.143886 + 0.989594i \(0.545960\pi\)
\(510\) 0 0
\(511\) −5.12311 −0.226633
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.2462 0.627763
\(516\) 0 0
\(517\) −40.9848 −1.80251
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.50758 −0.241291 −0.120646 0.992696i \(-0.538496\pi\)
−0.120646 + 0.992696i \(0.538496\pi\)
\(522\) 0 0
\(523\) 0.492423 0.0215321 0.0107661 0.999942i \(-0.496573\pi\)
0.0107661 + 0.999942i \(0.496573\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.8078 0.732158
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.87689 −0.297871
\(534\) 0 0
\(535\) −7.68466 −0.332237
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.24621 −0.0967512
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.24621 −0.181888
\(546\) 0 0
\(547\) 9.75379 0.417042 0.208521 0.978018i \(-0.433135\pi\)
0.208521 + 0.978018i \(0.433135\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.61553 −0.409635
\(552\) 0 0
\(553\) −26.2462 −1.11610
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.9309 0.505527 0.252764 0.967528i \(-0.418661\pi\)
0.252764 + 0.967528i \(0.418661\pi\)
\(558\) 0 0
\(559\) −12.4924 −0.528373
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.6847 0.661030 0.330515 0.943801i \(-0.392778\pi\)
0.330515 + 0.943801i \(0.392778\pi\)
\(564\) 0 0
\(565\) −2.31534 −0.0974072
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 45.8617 1.90925 0.954625 0.297812i \(-0.0962568\pi\)
0.954625 + 0.297812i \(0.0962568\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.807764 −0.0335117
\(582\) 0 0
\(583\) −2.87689 −0.119149
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 39.8617 1.64527 0.822635 0.568570i \(-0.192503\pi\)
0.822635 + 0.568570i \(0.192503\pi\)
\(588\) 0 0
\(589\) −4.13826 −0.170514
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.8769 0.528791 0.264395 0.964414i \(-0.414828\pi\)
0.264395 + 0.964414i \(0.414828\pi\)
\(594\) 0 0
\(595\) 11.6847 0.479024
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) 0.561553 0.0229062 0.0114531 0.999934i \(-0.496354\pi\)
0.0114531 + 0.999934i \(0.496354\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.2462 0.619847
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 20.2462 0.817737 0.408868 0.912593i \(-0.365924\pi\)
0.408868 + 0.912593i \(0.365924\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.6847 −1.03403 −0.517013 0.855978i \(-0.672956\pi\)
−0.517013 + 0.855978i \(0.672956\pi\)
\(618\) 0 0
\(619\) −47.2311 −1.89838 −0.949188 0.314709i \(-0.898093\pi\)
−0.949188 + 0.314709i \(0.898093\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.56155 −0.102136
\(630\) 0 0
\(631\) −18.2462 −0.726370 −0.363185 0.931717i \(-0.618311\pi\)
−0.363185 + 0.931717i \(0.618311\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 14.2462 0.565344
\(636\) 0 0
\(637\) −0.876894 −0.0347438
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 32.2462 1.27365 0.636824 0.771009i \(-0.280248\pi\)
0.636824 + 0.771009i \(0.280248\pi\)
\(642\) 0 0
\(643\) −33.3002 −1.31323 −0.656616 0.754225i \(-0.728013\pi\)
−0.656616 + 0.754225i \(0.728013\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −34.8769 −1.37115 −0.685576 0.728001i \(-0.740450\pi\)
−0.685576 + 0.728001i \(0.740450\pi\)
\(648\) 0 0
\(649\) −33.6155 −1.31952
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.4924 1.34979 0.674896 0.737912i \(-0.264188\pi\)
0.674896 + 0.737912i \(0.264188\pi\)
\(654\) 0 0
\(655\) 10.2462 0.400353
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 15.3693 0.598704 0.299352 0.954143i \(-0.403230\pi\)
0.299352 + 0.954143i \(0.403230\pi\)
\(660\) 0 0
\(661\) −16.7386 −0.651057 −0.325529 0.945532i \(-0.605542\pi\)
−0.325529 + 0.945532i \(0.605542\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.87689 −0.111561
\(666\) 0 0
\(667\) −8.56155 −0.331505
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 68.4924 2.64412
\(672\) 0 0
\(673\) 18.6307 0.718160 0.359080 0.933307i \(-0.383091\pi\)
0.359080 + 0.933307i \(0.383091\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.3002 0.434301 0.217151 0.976138i \(-0.430324\pi\)
0.217151 + 0.976138i \(0.430324\pi\)
\(678\) 0 0
\(679\) 15.3693 0.589820
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.24621 0.239005 0.119502 0.992834i \(-0.461870\pi\)
0.119502 + 0.992834i \(0.461870\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.12311 −0.0427869
\(690\) 0 0
\(691\) −7.50758 −0.285602 −0.142801 0.989751i \(-0.545611\pi\)
−0.142801 + 0.989751i \(0.545611\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.8078 −0.485826
\(696\) 0 0
\(697\) 15.6847 0.594099
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −37.8617 −1.43002 −0.715009 0.699115i \(-0.753577\pi\)
−0.715009 + 0.699115i \(0.753577\pi\)
\(702\) 0 0
\(703\) 0.630683 0.0237867
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.1771 0.608402
\(708\) 0 0
\(709\) −28.8769 −1.08449 −0.542247 0.840219i \(-0.682426\pi\)
−0.542247 + 0.840219i \(0.682426\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.68466 −0.137992
\(714\) 0 0
\(715\) 10.2462 0.383187
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.3002 −0.496013 −0.248007 0.968758i \(-0.579775\pi\)
−0.248007 + 0.968758i \(0.579775\pi\)
\(720\) 0 0
\(721\) −36.4924 −1.35905
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.56155 −0.317968
\(726\) 0 0
\(727\) 12.8078 0.475014 0.237507 0.971386i \(-0.423670\pi\)
0.237507 + 0.971386i \(0.423670\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 28.4924 1.05383
\(732\) 0 0
\(733\) −27.9309 −1.03165 −0.515825 0.856694i \(-0.672515\pi\)
−0.515825 + 0.856694i \(0.672515\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.1231 −0.483396
\(738\) 0 0
\(739\) −25.9309 −0.953882 −0.476941 0.878935i \(-0.658255\pi\)
−0.476941 + 0.878935i \(0.658255\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −52.4924 −1.92576 −0.962880 0.269929i \(-0.913000\pi\)
−0.962880 + 0.269929i \(0.913000\pi\)
\(744\) 0 0
\(745\) 13.3693 0.489814
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.6847 0.719262
\(750\) 0 0
\(751\) 1.61553 0.0589515 0.0294757 0.999565i \(-0.490616\pi\)
0.0294757 + 0.999565i \(0.490616\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.2462 0.372898
\(756\) 0 0
\(757\) 2.17708 0.0791274 0.0395637 0.999217i \(-0.487403\pi\)
0.0395637 + 0.999217i \(0.487403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.19224 −0.0432185 −0.0216093 0.999766i \(-0.506879\pi\)
−0.0216093 + 0.999766i \(0.506879\pi\)
\(762\) 0 0
\(763\) 10.8769 0.393770
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.1231 −0.473848
\(768\) 0 0
\(769\) 25.3693 0.914841 0.457420 0.889251i \(-0.348773\pi\)
0.457420 + 0.889251i \(0.348773\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 52.7386 1.89688 0.948438 0.316961i \(-0.102663\pi\)
0.948438 + 0.316961i \(0.102663\pi\)
\(774\) 0 0
\(775\) −3.68466 −0.132357
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.86174 −0.138361
\(780\) 0 0
\(781\) −59.8617 −2.14202
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −17.0540 −0.608682
\(786\) 0 0
\(787\) 32.6695 1.16454 0.582271 0.812995i \(-0.302164\pi\)
0.582271 + 0.812995i \(0.302164\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.93087 0.210877
\(792\) 0 0
\(793\) 26.7386 0.949517
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 55.7926 1.97628 0.988138 0.153570i \(-0.0490770\pi\)
0.988138 + 0.153570i \(0.0490770\pi\)
\(798\) 0 0
\(799\) 36.4924 1.29101
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.2462 0.361581
\(804\) 0 0
\(805\) −2.56155 −0.0902829
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.1922 0.745079 0.372540 0.928016i \(-0.378487\pi\)
0.372540 + 0.928016i \(0.378487\pi\)
\(810\) 0 0
\(811\) 19.5464 0.686367 0.343183 0.939268i \(-0.388495\pi\)
0.343183 + 0.939268i \(0.388495\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 19.3693 0.678478
\(816\) 0 0
\(817\) −7.01515 −0.245429
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) 52.0000 1.81261 0.906303 0.422628i \(-0.138892\pi\)
0.906303 + 0.422628i \(0.138892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.94602 0.311084 0.155542 0.987829i \(-0.450288\pi\)
0.155542 + 0.987829i \(0.450288\pi\)
\(828\) 0 0
\(829\) 5.19224 0.180334 0.0901669 0.995927i \(-0.471260\pi\)
0.0901669 + 0.995927i \(0.471260\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 12.4924 0.432318
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.12311 0.176869 0.0884346 0.996082i \(-0.471814\pi\)
0.0884346 + 0.996082i \(0.471814\pi\)
\(840\) 0 0
\(841\) 44.3002 1.52759
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −39.0540 −1.34191
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.561553 0.0192498
\(852\) 0 0
\(853\) −49.2311 −1.68564 −0.842820 0.538196i \(-0.819106\pi\)
−0.842820 + 0.538196i \(0.819106\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.4924 −1.45151 −0.725757 0.687951i \(-0.758510\pi\)
−0.725757 + 0.687951i \(0.758510\pi\)
\(858\) 0 0
\(859\) −20.1771 −0.688433 −0.344217 0.938890i \(-0.611855\pi\)
−0.344217 + 0.938890i \(0.611855\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11.8617 0.403778 0.201889 0.979408i \(-0.435292\pi\)
0.201889 + 0.979408i \(0.435292\pi\)
\(864\) 0 0
\(865\) −15.1231 −0.514201
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 52.4924 1.78068
\(870\) 0 0
\(871\) −5.12311 −0.173590
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.56155 −0.0865963
\(876\) 0 0
\(877\) 53.2311 1.79749 0.898743 0.438477i \(-0.144482\pi\)
0.898743 + 0.438477i \(0.144482\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −26.6307 −0.897211 −0.448605 0.893730i \(-0.648079\pi\)
−0.448605 + 0.893730i \(0.648079\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.1231 0.709244 0.354622 0.935010i \(-0.384609\pi\)
0.354622 + 0.935010i \(0.384609\pi\)
\(888\) 0 0
\(889\) −36.4924 −1.22392
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.98485 −0.300666
\(894\) 0 0
\(895\) −12.4924 −0.417576
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 31.5464 1.05213
\(900\) 0 0
\(901\) 2.56155 0.0853377
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.876894 0.0291490
\(906\) 0 0
\(907\) −14.0691 −0.467158 −0.233579 0.972338i \(-0.575044\pi\)
−0.233579 + 0.972338i \(0.575044\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −30.7386 −1.01842 −0.509208 0.860643i \(-0.670062\pi\)
−0.509208 + 0.860643i \(0.670062\pi\)
\(912\) 0 0
\(913\) 1.61553 0.0534662
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −26.2462 −0.866726
\(918\) 0 0
\(919\) −48.9848 −1.61586 −0.807930 0.589278i \(-0.799412\pi\)
−0.807930 + 0.589278i \(0.799412\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −23.3693 −0.769210
\(924\) 0 0
\(925\) 0.561553 0.0184637
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −21.0540 −0.690759 −0.345379 0.938463i \(-0.612250\pi\)
−0.345379 + 0.938463i \(0.612250\pi\)
\(930\) 0 0
\(931\) −0.492423 −0.0161385
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −23.3693 −0.764258
\(936\) 0 0
\(937\) −34.4924 −1.12682 −0.563409 0.826178i \(-0.690511\pi\)
−0.563409 + 0.826178i \(0.690511\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.3693 0.696620 0.348310 0.937379i \(-0.386756\pi\)
0.348310 + 0.937379i \(0.386756\pi\)
\(942\) 0 0
\(943\) −3.43845 −0.111971
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.7538 −0.639888 −0.319944 0.947436i \(-0.603664\pi\)
−0.319944 + 0.947436i \(0.603664\pi\)
\(954\) 0 0
\(955\) 12.4924 0.404245
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 35.8617 1.15804
\(960\) 0 0
\(961\) −17.4233 −0.562042
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −13.3693 −0.430374
\(966\) 0 0
\(967\) −12.9848 −0.417564 −0.208782 0.977962i \(-0.566950\pi\)
−0.208782 + 0.977962i \(0.566950\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −46.7386 −1.49991 −0.749957 0.661487i \(-0.769926\pi\)
−0.749957 + 0.661487i \(0.769926\pi\)
\(972\) 0 0
\(973\) 32.8078 1.05177
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.4233 −1.54920 −0.774599 0.632452i \(-0.782048\pi\)
−0.774599 + 0.632452i \(0.782048\pi\)
\(978\) 0 0
\(979\) 16.0000 0.511362
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −24.1771 −0.771129 −0.385565 0.922681i \(-0.625993\pi\)
−0.385565 + 0.922681i \(0.625993\pi\)
\(984\) 0 0
\(985\) −20.2462 −0.645098
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.24621 −0.198618
\(990\) 0 0
\(991\) −12.9460 −0.411244 −0.205622 0.978631i \(-0.565922\pi\)
−0.205622 + 0.978631i \(0.565922\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.87689 −0.0912037
\(996\) 0 0
\(997\) 37.8617 1.19909 0.599547 0.800340i \(-0.295348\pi\)
0.599547 + 0.800340i \(0.295348\pi\)
\(998\) 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.be.1.1 2
3.2 odd 2 2760.2.a.p.1.1 2
12.11 even 2 5520.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.p.1.1 2 3.2 odd 2
5520.2.a.bh.1.2 2 12.11 even 2
8280.2.a.be.1.1 2 1.1 even 1 trivial