Properties

Label 5625.2.a.z.1.3
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.23762\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.23762 q^{2} -0.468306 q^{4} +2.08634 q^{7} +3.05482 q^{8} +O(q^{10})\) \(q-1.23762 q^{2} -0.468306 q^{4} +2.08634 q^{7} +3.05482 q^{8} -2.65287 q^{11} -0.149728 q^{13} -2.58209 q^{14} -2.84408 q^{16} +5.12810 q^{17} -3.08634 q^{19} +3.28323 q^{22} -2.76265 q^{23} +0.185305 q^{26} -0.977047 q^{28} +6.94530 q^{29} +1.52550 q^{31} -2.58976 q^{32} -6.34662 q^{34} -8.52550 q^{37} +3.81970 q^{38} -9.82722 q^{41} -3.43296 q^{43} +1.24236 q^{44} +3.41911 q^{46} +4.00501 q^{47} -2.64718 q^{49} +0.0701184 q^{52} +0.945455 q^{53} +6.37339 q^{56} -8.59562 q^{58} +8.18766 q^{59} -4.47450 q^{61} -1.88798 q^{62} +8.89328 q^{64} +3.34042 q^{67} -2.40152 q^{68} -12.4801 q^{71} -11.6018 q^{73} +10.5513 q^{74} +1.44535 q^{76} -5.53479 q^{77} +11.3696 q^{79} +12.1623 q^{82} -3.45676 q^{83} +4.24869 q^{86} -8.10403 q^{88} +14.8875 q^{89} -0.312383 q^{91} +1.29377 q^{92} -4.95666 q^{94} +0.0571908 q^{97} +3.27620 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73} + 56 q^{76} + 28 q^{79} - 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23762 −0.875127 −0.437563 0.899188i \(-0.644158\pi\)
−0.437563 + 0.899188i \(0.644158\pi\)
\(3\) 0 0
\(4\) −0.468306 −0.234153
\(5\) 0 0
\(6\) 0 0
\(7\) 2.08634 0.788563 0.394281 0.918990i \(-0.370994\pi\)
0.394281 + 0.918990i \(0.370994\pi\)
\(8\) 3.05482 1.08004
\(9\) 0 0
\(10\) 0 0
\(11\) −2.65287 −0.799870 −0.399935 0.916543i \(-0.630967\pi\)
−0.399935 + 0.916543i \(0.630967\pi\)
\(12\) 0 0
\(13\) −0.149728 −0.0415269 −0.0207635 0.999784i \(-0.506610\pi\)
−0.0207635 + 0.999784i \(0.506610\pi\)
\(14\) −2.58209 −0.690092
\(15\) 0 0
\(16\) −2.84408 −0.711019
\(17\) 5.12810 1.24375 0.621874 0.783118i \(-0.286372\pi\)
0.621874 + 0.783118i \(0.286372\pi\)
\(18\) 0 0
\(19\) −3.08634 −0.708055 −0.354028 0.935235i \(-0.615188\pi\)
−0.354028 + 0.935235i \(0.615188\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.28323 0.699988
\(23\) −2.76265 −0.576053 −0.288027 0.957622i \(-0.592999\pi\)
−0.288027 + 0.957622i \(0.592999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.185305 0.0363413
\(27\) 0 0
\(28\) −0.977047 −0.184644
\(29\) 6.94530 1.28971 0.644855 0.764305i \(-0.276918\pi\)
0.644855 + 0.764305i \(0.276918\pi\)
\(30\) 0 0
\(31\) 1.52550 0.273987 0.136994 0.990572i \(-0.456256\pi\)
0.136994 + 0.990572i \(0.456256\pi\)
\(32\) −2.58976 −0.457809
\(33\) 0 0
\(34\) −6.34662 −1.08844
\(35\) 0 0
\(36\) 0 0
\(37\) −8.52550 −1.40158 −0.700792 0.713366i \(-0.747170\pi\)
−0.700792 + 0.713366i \(0.747170\pi\)
\(38\) 3.81970 0.619638
\(39\) 0 0
\(40\) 0 0
\(41\) −9.82722 −1.53475 −0.767377 0.641196i \(-0.778438\pi\)
−0.767377 + 0.641196i \(0.778438\pi\)
\(42\) 0 0
\(43\) −3.43296 −0.523522 −0.261761 0.965133i \(-0.584303\pi\)
−0.261761 + 0.965133i \(0.584303\pi\)
\(44\) 1.24236 0.187292
\(45\) 0 0
\(46\) 3.41911 0.504120
\(47\) 4.00501 0.584191 0.292095 0.956389i \(-0.405648\pi\)
0.292095 + 0.956389i \(0.405648\pi\)
\(48\) 0 0
\(49\) −2.64718 −0.378169
\(50\) 0 0
\(51\) 0 0
\(52\) 0.0701184 0.00972367
\(53\) 0.945455 0.129868 0.0649341 0.997890i \(-0.479316\pi\)
0.0649341 + 0.997890i \(0.479316\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.37339 0.851679
\(57\) 0 0
\(58\) −8.59562 −1.12866
\(59\) 8.18766 1.06594 0.532971 0.846133i \(-0.321075\pi\)
0.532971 + 0.846133i \(0.321075\pi\)
\(60\) 0 0
\(61\) −4.47450 −0.572901 −0.286451 0.958095i \(-0.592475\pi\)
−0.286451 + 0.958095i \(0.592475\pi\)
\(62\) −1.88798 −0.239774
\(63\) 0 0
\(64\) 8.89328 1.11166
\(65\) 0 0
\(66\) 0 0
\(67\) 3.34042 0.408098 0.204049 0.978961i \(-0.434590\pi\)
0.204049 + 0.978961i \(0.434590\pi\)
\(68\) −2.40152 −0.291227
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4801 −1.48111 −0.740557 0.671994i \(-0.765438\pi\)
−0.740557 + 0.671994i \(0.765438\pi\)
\(72\) 0 0
\(73\) −11.6018 −1.35789 −0.678945 0.734189i \(-0.737562\pi\)
−0.678945 + 0.734189i \(0.737562\pi\)
\(74\) 10.5513 1.22656
\(75\) 0 0
\(76\) 1.44535 0.165793
\(77\) −5.53479 −0.630748
\(78\) 0 0
\(79\) 11.3696 1.27918 0.639588 0.768717i \(-0.279105\pi\)
0.639588 + 0.768717i \(0.279105\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 12.1623 1.34310
\(83\) −3.45676 −0.379429 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.24869 0.458148
\(87\) 0 0
\(88\) −8.10403 −0.863892
\(89\) 14.8875 1.57807 0.789034 0.614349i \(-0.210581\pi\)
0.789034 + 0.614349i \(0.210581\pi\)
\(90\) 0 0
\(91\) −0.312383 −0.0327466
\(92\) 1.29377 0.134885
\(93\) 0 0
\(94\) −4.95666 −0.511241
\(95\) 0 0
\(96\) 0 0
\(97\) 0.0571908 0.00580685 0.00290342 0.999996i \(-0.499076\pi\)
0.00290342 + 0.999996i \(0.499076\pi\)
\(98\) 3.27620 0.330946
\(99\) 0 0
\(100\) 0 0
\(101\) 16.2980 1.62171 0.810855 0.585247i \(-0.199003\pi\)
0.810855 + 0.585247i \(0.199003\pi\)
\(102\) 0 0
\(103\) −5.37814 −0.529924 −0.264962 0.964259i \(-0.585359\pi\)
−0.264962 + 0.964259i \(0.585359\pi\)
\(104\) −0.457390 −0.0448508
\(105\) 0 0
\(106\) −1.17011 −0.113651
\(107\) −20.2611 −1.95871 −0.979355 0.202147i \(-0.935208\pi\)
−0.979355 + 0.202147i \(0.935208\pi\)
\(108\) 0 0
\(109\) −16.0778 −1.53997 −0.769986 0.638061i \(-0.779737\pi\)
−0.769986 + 0.638061i \(0.779737\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −5.93371 −0.560683
\(113\) −8.40723 −0.790885 −0.395443 0.918491i \(-0.629409\pi\)
−0.395443 + 0.918491i \(0.629409\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.25253 −0.301990
\(117\) 0 0
\(118\) −10.1332 −0.932834
\(119\) 10.6990 0.980772
\(120\) 0 0
\(121\) −3.96229 −0.360208
\(122\) 5.53772 0.501361
\(123\) 0 0
\(124\) −0.714400 −0.0641550
\(125\) 0 0
\(126\) 0 0
\(127\) −21.8709 −1.94072 −0.970362 0.241654i \(-0.922310\pi\)
−0.970362 + 0.241654i \(0.922310\pi\)
\(128\) −5.82695 −0.515034
\(129\) 0 0
\(130\) 0 0
\(131\) 14.1197 1.23364 0.616820 0.787104i \(-0.288421\pi\)
0.616820 + 0.787104i \(0.288421\pi\)
\(132\) 0 0
\(133\) −6.43916 −0.558346
\(134\) −4.13416 −0.357137
\(135\) 0 0
\(136\) 15.6654 1.34330
\(137\) 9.59817 0.820027 0.410013 0.912079i \(-0.365524\pi\)
0.410013 + 0.912079i \(0.365524\pi\)
\(138\) 0 0
\(139\) 17.7945 1.50931 0.754657 0.656120i \(-0.227803\pi\)
0.754657 + 0.656120i \(0.227803\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.4456 1.29616
\(143\) 0.397208 0.0332162
\(144\) 0 0
\(145\) 0 0
\(146\) 14.3586 1.18833
\(147\) 0 0
\(148\) 3.99255 0.328185
\(149\) −1.01331 −0.0830132 −0.0415066 0.999138i \(-0.513216\pi\)
−0.0415066 + 0.999138i \(0.513216\pi\)
\(150\) 0 0
\(151\) −13.2809 −1.08078 −0.540391 0.841414i \(-0.681724\pi\)
−0.540391 + 0.841414i \(0.681724\pi\)
\(152\) −9.42820 −0.764728
\(153\) 0 0
\(154\) 6.84994 0.551984
\(155\) 0 0
\(156\) 0 0
\(157\) 21.5579 1.72051 0.860254 0.509866i \(-0.170305\pi\)
0.860254 + 0.509866i \(0.170305\pi\)
\(158\) −14.0712 −1.11944
\(159\) 0 0
\(160\) 0 0
\(161\) −5.76384 −0.454254
\(162\) 0 0
\(163\) −4.99144 −0.390960 −0.195480 0.980708i \(-0.562626\pi\)
−0.195480 + 0.980708i \(0.562626\pi\)
\(164\) 4.60215 0.359368
\(165\) 0 0
\(166\) 4.27815 0.332048
\(167\) 22.9559 1.77638 0.888189 0.459478i \(-0.151964\pi\)
0.888189 + 0.459478i \(0.151964\pi\)
\(168\) 0 0
\(169\) −12.9776 −0.998276
\(170\) 0 0
\(171\) 0 0
\(172\) 1.60768 0.122584
\(173\) 6.17748 0.469665 0.234833 0.972036i \(-0.424546\pi\)
0.234833 + 0.972036i \(0.424546\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 7.54496 0.568723
\(177\) 0 0
\(178\) −18.4250 −1.38101
\(179\) −14.6584 −1.09562 −0.547811 0.836602i \(-0.684539\pi\)
−0.547811 + 0.836602i \(0.684539\pi\)
\(180\) 0 0
\(181\) −2.87559 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(182\) 0.386610 0.0286574
\(183\) 0 0
\(184\) −8.43940 −0.622161
\(185\) 0 0
\(186\) 0 0
\(187\) −13.6042 −0.994836
\(188\) −1.87557 −0.136790
\(189\) 0 0
\(190\) 0 0
\(191\) −12.6216 −0.913270 −0.456635 0.889654i \(-0.650946\pi\)
−0.456635 + 0.889654i \(0.650946\pi\)
\(192\) 0 0
\(193\) −7.02059 −0.505353 −0.252676 0.967551i \(-0.581311\pi\)
−0.252676 + 0.967551i \(0.581311\pi\)
\(194\) −0.0707803 −0.00508173
\(195\) 0 0
\(196\) 1.23969 0.0885495
\(197\) 11.3056 0.805489 0.402745 0.915312i \(-0.368056\pi\)
0.402745 + 0.915312i \(0.368056\pi\)
\(198\) 0 0
\(199\) 13.4827 0.955763 0.477882 0.878424i \(-0.341405\pi\)
0.477882 + 0.878424i \(0.341405\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −20.1706 −1.41920
\(203\) 14.4903 1.01702
\(204\) 0 0
\(205\) 0 0
\(206\) 6.65607 0.463750
\(207\) 0 0
\(208\) 0.425837 0.0295265
\(209\) 8.18766 0.566352
\(210\) 0 0
\(211\) −10.1297 −0.697356 −0.348678 0.937243i \(-0.613369\pi\)
−0.348678 + 0.937243i \(0.613369\pi\)
\(212\) −0.442762 −0.0304091
\(213\) 0 0
\(214\) 25.0754 1.71412
\(215\) 0 0
\(216\) 0 0
\(217\) 3.18271 0.216056
\(218\) 19.8981 1.34767
\(219\) 0 0
\(220\) 0 0
\(221\) −0.767818 −0.0516490
\(222\) 0 0
\(223\) −22.6330 −1.51562 −0.757809 0.652477i \(-0.773730\pi\)
−0.757809 + 0.652477i \(0.773730\pi\)
\(224\) −5.40311 −0.361011
\(225\) 0 0
\(226\) 10.4049 0.692125
\(227\) −5.12810 −0.340364 −0.170182 0.985413i \(-0.554436\pi\)
−0.170182 + 0.985413i \(0.554436\pi\)
\(228\) 0 0
\(229\) 20.8709 1.37919 0.689593 0.724198i \(-0.257790\pi\)
0.689593 + 0.724198i \(0.257790\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 21.2166 1.39294
\(233\) 17.3988 1.13983 0.569916 0.821703i \(-0.306976\pi\)
0.569916 + 0.821703i \(0.306976\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.83433 −0.249594
\(237\) 0 0
\(238\) −13.2412 −0.858300
\(239\) 4.66304 0.301627 0.150814 0.988562i \(-0.451811\pi\)
0.150814 + 0.988562i \(0.451811\pi\)
\(240\) 0 0
\(241\) 13.3242 0.858287 0.429144 0.903236i \(-0.358815\pi\)
0.429144 + 0.903236i \(0.358815\pi\)
\(242\) 4.90379 0.315228
\(243\) 0 0
\(244\) 2.09544 0.134147
\(245\) 0 0
\(246\) 0 0
\(247\) 0.462110 0.0294034
\(248\) 4.66011 0.295917
\(249\) 0 0
\(250\) 0 0
\(251\) −13.1063 −0.827265 −0.413633 0.910444i \(-0.635740\pi\)
−0.413633 + 0.910444i \(0.635740\pi\)
\(252\) 0 0
\(253\) 7.32896 0.460768
\(254\) 27.0677 1.69838
\(255\) 0 0
\(256\) −10.5750 −0.660939
\(257\) 5.19595 0.324115 0.162057 0.986781i \(-0.448187\pi\)
0.162057 + 0.986781i \(0.448187\pi\)
\(258\) 0 0
\(259\) −17.7871 −1.10524
\(260\) 0 0
\(261\) 0 0
\(262\) −17.4747 −1.07959
\(263\) 0.145861 0.00899420 0.00449710 0.999990i \(-0.498569\pi\)
0.00449710 + 0.999990i \(0.498569\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.96920 0.488623
\(267\) 0 0
\(268\) −1.56434 −0.0955574
\(269\) 13.8640 0.845303 0.422652 0.906292i \(-0.361099\pi\)
0.422652 + 0.906292i \(0.361099\pi\)
\(270\) 0 0
\(271\) −3.09196 −0.187823 −0.0939117 0.995581i \(-0.529937\pi\)
−0.0939117 + 0.995581i \(0.529937\pi\)
\(272\) −14.5847 −0.884328
\(273\) 0 0
\(274\) −11.8788 −0.717627
\(275\) 0 0
\(276\) 0 0
\(277\) −11.3301 −0.680758 −0.340379 0.940288i \(-0.610555\pi\)
−0.340379 + 0.940288i \(0.610555\pi\)
\(278\) −22.0228 −1.32084
\(279\) 0 0
\(280\) 0 0
\(281\) −3.42069 −0.204061 −0.102031 0.994781i \(-0.532534\pi\)
−0.102031 + 0.994781i \(0.532534\pi\)
\(282\) 0 0
\(283\) −27.4747 −1.63320 −0.816601 0.577203i \(-0.804144\pi\)
−0.816601 + 0.577203i \(0.804144\pi\)
\(284\) 5.84451 0.346808
\(285\) 0 0
\(286\) −0.491590 −0.0290684
\(287\) −20.5029 −1.21025
\(288\) 0 0
\(289\) 9.29742 0.546907
\(290\) 0 0
\(291\) 0 0
\(292\) 5.43320 0.317954
\(293\) −23.5916 −1.37824 −0.689118 0.724649i \(-0.742002\pi\)
−0.689118 + 0.724649i \(0.742002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −26.0438 −1.51377
\(297\) 0 0
\(298\) 1.25408 0.0726471
\(299\) 0.413645 0.0239217
\(300\) 0 0
\(301\) −7.16232 −0.412830
\(302\) 16.4366 0.945821
\(303\) 0 0
\(304\) 8.77779 0.503441
\(305\) 0 0
\(306\) 0 0
\(307\) −4.34390 −0.247919 −0.123960 0.992287i \(-0.539559\pi\)
−0.123960 + 0.992287i \(0.539559\pi\)
\(308\) 2.59198 0.147692
\(309\) 0 0
\(310\) 0 0
\(311\) 6.69981 0.379912 0.189956 0.981793i \(-0.439166\pi\)
0.189956 + 0.981793i \(0.439166\pi\)
\(312\) 0 0
\(313\) −16.0033 −0.904558 −0.452279 0.891877i \(-0.649389\pi\)
−0.452279 + 0.891877i \(0.649389\pi\)
\(314\) −26.6804 −1.50566
\(315\) 0 0
\(316\) −5.32444 −0.299523
\(317\) −21.0650 −1.18313 −0.591563 0.806259i \(-0.701489\pi\)
−0.591563 + 0.806259i \(0.701489\pi\)
\(318\) 0 0
\(319\) −18.4250 −1.03160
\(320\) 0 0
\(321\) 0 0
\(322\) 7.13342 0.397530
\(323\) −15.8271 −0.880641
\(324\) 0 0
\(325\) 0 0
\(326\) 6.17748 0.342139
\(327\) 0 0
\(328\) −30.0203 −1.65760
\(329\) 8.35581 0.460671
\(330\) 0 0
\(331\) 30.4432 1.67331 0.836655 0.547731i \(-0.184508\pi\)
0.836655 + 0.547731i \(0.184508\pi\)
\(332\) 1.61882 0.0888445
\(333\) 0 0
\(334\) −28.4105 −1.55456
\(335\) 0 0
\(336\) 0 0
\(337\) −28.9925 −1.57932 −0.789662 0.613542i \(-0.789744\pi\)
−0.789662 + 0.613542i \(0.789744\pi\)
\(338\) 16.0613 0.873618
\(339\) 0 0
\(340\) 0 0
\(341\) −4.04694 −0.219154
\(342\) 0 0
\(343\) −20.1273 −1.08677
\(344\) −10.4871 −0.565424
\(345\) 0 0
\(346\) −7.64535 −0.411017
\(347\) −5.55708 −0.298320 −0.149160 0.988813i \(-0.547657\pi\)
−0.149160 + 0.988813i \(0.547657\pi\)
\(348\) 0 0
\(349\) −13.9703 −0.747812 −0.373906 0.927467i \(-0.621982\pi\)
−0.373906 + 0.927467i \(0.621982\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.87029 0.366187
\(353\) 19.7642 1.05194 0.525972 0.850502i \(-0.323702\pi\)
0.525972 + 0.850502i \(0.323702\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.97190 −0.369510
\(357\) 0 0
\(358\) 18.1415 0.958808
\(359\) −11.6083 −0.612665 −0.306332 0.951925i \(-0.599102\pi\)
−0.306332 + 0.951925i \(0.599102\pi\)
\(360\) 0 0
\(361\) −9.47450 −0.498658
\(362\) 3.55888 0.187051
\(363\) 0 0
\(364\) 0.146291 0.00766772
\(365\) 0 0
\(366\) 0 0
\(367\) 0.0182198 0.000951065 0 0.000475533 1.00000i \(-0.499849\pi\)
0.000475533 1.00000i \(0.499849\pi\)
\(368\) 7.85720 0.409585
\(369\) 0 0
\(370\) 0 0
\(371\) 1.97254 0.102409
\(372\) 0 0
\(373\) −16.5665 −0.857779 −0.428890 0.903357i \(-0.641095\pi\)
−0.428890 + 0.903357i \(0.641095\pi\)
\(374\) 16.8368 0.870608
\(375\) 0 0
\(376\) 12.2346 0.630950
\(377\) −1.03990 −0.0535577
\(378\) 0 0
\(379\) −2.52370 −0.129634 −0.0648170 0.997897i \(-0.520646\pi\)
−0.0648170 + 0.997897i \(0.520646\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 15.6208 0.799227
\(383\) 15.7910 0.806882 0.403441 0.915006i \(-0.367814\pi\)
0.403441 + 0.915006i \(0.367814\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.68879 0.442248
\(387\) 0 0
\(388\) −0.0267828 −0.00135969
\(389\) 4.36976 0.221556 0.110778 0.993845i \(-0.464666\pi\)
0.110778 + 0.993845i \(0.464666\pi\)
\(390\) 0 0
\(391\) −14.1672 −0.716465
\(392\) −8.08666 −0.408438
\(393\) 0 0
\(394\) −13.9920 −0.704905
\(395\) 0 0
\(396\) 0 0
\(397\) −11.6729 −0.585844 −0.292922 0.956136i \(-0.594628\pi\)
−0.292922 + 0.956136i \(0.594628\pi\)
\(398\) −16.6864 −0.836414
\(399\) 0 0
\(400\) 0 0
\(401\) −19.8999 −0.993755 −0.496877 0.867821i \(-0.665520\pi\)
−0.496877 + 0.867821i \(0.665520\pi\)
\(402\) 0 0
\(403\) −0.228409 −0.0113779
\(404\) −7.63245 −0.379729
\(405\) 0 0
\(406\) −17.9334 −0.890019
\(407\) 22.6170 1.12108
\(408\) 0 0
\(409\) −10.2084 −0.504772 −0.252386 0.967627i \(-0.581215\pi\)
−0.252386 + 0.967627i \(0.581215\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.51862 0.124083
\(413\) 17.0822 0.840562
\(414\) 0 0
\(415\) 0 0
\(416\) 0.387758 0.0190114
\(417\) 0 0
\(418\) −10.1332 −0.495630
\(419\) −37.8375 −1.84848 −0.924241 0.381810i \(-0.875301\pi\)
−0.924241 + 0.381810i \(0.875301\pi\)
\(420\) 0 0
\(421\) 10.1730 0.495802 0.247901 0.968785i \(-0.420259\pi\)
0.247901 + 0.968785i \(0.420259\pi\)
\(422\) 12.5366 0.610275
\(423\) 0 0
\(424\) 2.88819 0.140263
\(425\) 0 0
\(426\) 0 0
\(427\) −9.33534 −0.451769
\(428\) 9.48838 0.458638
\(429\) 0 0
\(430\) 0 0
\(431\) −37.9626 −1.82859 −0.914297 0.405045i \(-0.867256\pi\)
−0.914297 + 0.405045i \(0.867256\pi\)
\(432\) 0 0
\(433\) −16.4745 −0.791714 −0.395857 0.918312i \(-0.629552\pi\)
−0.395857 + 0.918312i \(0.629552\pi\)
\(434\) −3.93897 −0.189077
\(435\) 0 0
\(436\) 7.52933 0.360589
\(437\) 8.52649 0.407877
\(438\) 0 0
\(439\) 16.4231 0.783834 0.391917 0.920001i \(-0.371812\pi\)
0.391917 + 0.920001i \(0.371812\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.950264 0.0451994
\(443\) −34.6321 −1.64542 −0.822709 0.568462i \(-0.807539\pi\)
−0.822709 + 0.568462i \(0.807539\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 28.0110 1.32636
\(447\) 0 0
\(448\) 18.5544 0.876613
\(449\) −14.7835 −0.697678 −0.348839 0.937183i \(-0.613424\pi\)
−0.348839 + 0.937183i \(0.613424\pi\)
\(450\) 0 0
\(451\) 26.0703 1.22760
\(452\) 3.93716 0.185188
\(453\) 0 0
\(454\) 6.34662 0.297862
\(455\) 0 0
\(456\) 0 0
\(457\) 23.8075 1.11367 0.556833 0.830624i \(-0.312016\pi\)
0.556833 + 0.830624i \(0.312016\pi\)
\(458\) −25.8301 −1.20696
\(459\) 0 0
\(460\) 0 0
\(461\) −23.1017 −1.07595 −0.537977 0.842959i \(-0.680811\pi\)
−0.537977 + 0.842959i \(0.680811\pi\)
\(462\) 0 0
\(463\) 0.884510 0.0411067 0.0205533 0.999789i \(-0.493457\pi\)
0.0205533 + 0.999789i \(0.493457\pi\)
\(464\) −19.7530 −0.917008
\(465\) 0 0
\(466\) −21.5330 −0.997497
\(467\) −19.3835 −0.896959 −0.448480 0.893793i \(-0.648034\pi\)
−0.448480 + 0.893793i \(0.648034\pi\)
\(468\) 0 0
\(469\) 6.96926 0.321811
\(470\) 0 0
\(471\) 0 0
\(472\) 25.0118 1.15126
\(473\) 9.10719 0.418749
\(474\) 0 0
\(475\) 0 0
\(476\) −5.01039 −0.229651
\(477\) 0 0
\(478\) −5.77106 −0.263962
\(479\) −32.8149 −1.49935 −0.749674 0.661807i \(-0.769790\pi\)
−0.749674 + 0.661807i \(0.769790\pi\)
\(480\) 0 0
\(481\) 1.27650 0.0582035
\(482\) −16.4902 −0.751110
\(483\) 0 0
\(484\) 1.85556 0.0843439
\(485\) 0 0
\(486\) 0 0
\(487\) −18.5956 −0.842648 −0.421324 0.906910i \(-0.638434\pi\)
−0.421324 + 0.906910i \(0.638434\pi\)
\(488\) −13.6688 −0.618757
\(489\) 0 0
\(490\) 0 0
\(491\) 27.3676 1.23508 0.617540 0.786539i \(-0.288129\pi\)
0.617540 + 0.786539i \(0.288129\pi\)
\(492\) 0 0
\(493\) 35.6162 1.60407
\(494\) −0.571915 −0.0257317
\(495\) 0 0
\(496\) −4.33863 −0.194810
\(497\) −26.0377 −1.16795
\(498\) 0 0
\(499\) −0.852639 −0.0381694 −0.0190847 0.999818i \(-0.506075\pi\)
−0.0190847 + 0.999818i \(0.506075\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 16.2206 0.723962
\(503\) −30.4176 −1.35626 −0.678128 0.734944i \(-0.737208\pi\)
−0.678128 + 0.734944i \(0.737208\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −9.07044 −0.403230
\(507\) 0 0
\(508\) 10.2423 0.454427
\(509\) −4.30887 −0.190987 −0.0954936 0.995430i \(-0.530443\pi\)
−0.0954936 + 0.995430i \(0.530443\pi\)
\(510\) 0 0
\(511\) −24.2053 −1.07078
\(512\) 24.7417 1.09344
\(513\) 0 0
\(514\) −6.43059 −0.283641
\(515\) 0 0
\(516\) 0 0
\(517\) −10.6248 −0.467277
\(518\) 22.0136 0.967222
\(519\) 0 0
\(520\) 0 0
\(521\) 9.99538 0.437905 0.218953 0.975735i \(-0.429736\pi\)
0.218953 + 0.975735i \(0.429736\pi\)
\(522\) 0 0
\(523\) −13.1913 −0.576814 −0.288407 0.957508i \(-0.593126\pi\)
−0.288407 + 0.957508i \(0.593126\pi\)
\(524\) −6.61232 −0.288861
\(525\) 0 0
\(526\) −0.180521 −0.00787107
\(527\) 7.82290 0.340771
\(528\) 0 0
\(529\) −15.3677 −0.668163
\(530\) 0 0
\(531\) 0 0
\(532\) 3.01550 0.130738
\(533\) 1.47141 0.0637336
\(534\) 0 0
\(535\) 0 0
\(536\) 10.2044 0.440762
\(537\) 0 0
\(538\) −17.1583 −0.739747
\(539\) 7.02263 0.302486
\(540\) 0 0
\(541\) 35.6710 1.53362 0.766809 0.641876i \(-0.221844\pi\)
0.766809 + 0.641876i \(0.221844\pi\)
\(542\) 3.82666 0.164369
\(543\) 0 0
\(544\) −13.2805 −0.569398
\(545\) 0 0
\(546\) 0 0
\(547\) 16.6018 0.709842 0.354921 0.934896i \(-0.384508\pi\)
0.354921 + 0.934896i \(0.384508\pi\)
\(548\) −4.49488 −0.192012
\(549\) 0 0
\(550\) 0 0
\(551\) −21.4356 −0.913186
\(552\) 0 0
\(553\) 23.7208 1.00871
\(554\) 14.0223 0.595749
\(555\) 0 0
\(556\) −8.33330 −0.353411
\(557\) 14.8875 0.630802 0.315401 0.948958i \(-0.397861\pi\)
0.315401 + 0.948958i \(0.397861\pi\)
\(558\) 0 0
\(559\) 0.514009 0.0217403
\(560\) 0 0
\(561\) 0 0
\(562\) 4.23350 0.178579
\(563\) −46.4541 −1.95781 −0.978904 0.204322i \(-0.934501\pi\)
−0.978904 + 0.204322i \(0.934501\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 34.0031 1.42926
\(567\) 0 0
\(568\) −38.1244 −1.59966
\(569\) −9.57157 −0.401261 −0.200631 0.979667i \(-0.564299\pi\)
−0.200631 + 0.979667i \(0.564299\pi\)
\(570\) 0 0
\(571\) 0.921854 0.0385784 0.0192892 0.999814i \(-0.493860\pi\)
0.0192892 + 0.999814i \(0.493860\pi\)
\(572\) −0.186015 −0.00777767
\(573\) 0 0
\(574\) 25.3747 1.05912
\(575\) 0 0
\(576\) 0 0
\(577\) −20.5519 −0.855589 −0.427794 0.903876i \(-0.640709\pi\)
−0.427794 + 0.903876i \(0.640709\pi\)
\(578\) −11.5066 −0.478613
\(579\) 0 0
\(580\) 0 0
\(581\) −7.21198 −0.299204
\(582\) 0 0
\(583\) −2.50817 −0.103878
\(584\) −35.4414 −1.46658
\(585\) 0 0
\(586\) 29.1973 1.20613
\(587\) 0.487355 0.0201153 0.0100576 0.999949i \(-0.496798\pi\)
0.0100576 + 0.999949i \(0.496798\pi\)
\(588\) 0 0
\(589\) −4.70820 −0.193998
\(590\) 0 0
\(591\) 0 0
\(592\) 24.2472 0.996552
\(593\) 34.3654 1.41122 0.705608 0.708602i \(-0.250674\pi\)
0.705608 + 0.708602i \(0.250674\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.474538 0.0194378
\(597\) 0 0
\(598\) −0.511934 −0.0209345
\(599\) −39.2856 −1.60517 −0.802583 0.596540i \(-0.796542\pi\)
−0.802583 + 0.596540i \(0.796542\pi\)
\(600\) 0 0
\(601\) −48.6333 −1.98379 −0.991897 0.127044i \(-0.959451\pi\)
−0.991897 + 0.127044i \(0.959451\pi\)
\(602\) 8.86421 0.361278
\(603\) 0 0
\(604\) 6.21952 0.253068
\(605\) 0 0
\(606\) 0 0
\(607\) −37.4916 −1.52174 −0.760869 0.648906i \(-0.775227\pi\)
−0.760869 + 0.648906i \(0.775227\pi\)
\(608\) 7.99287 0.324154
\(609\) 0 0
\(610\) 0 0
\(611\) −0.599660 −0.0242597
\(612\) 0 0
\(613\) 37.7346 1.52409 0.762043 0.647526i \(-0.224196\pi\)
0.762043 + 0.647526i \(0.224196\pi\)
\(614\) 5.37608 0.216961
\(615\) 0 0
\(616\) −16.9078 −0.681233
\(617\) −41.3260 −1.66372 −0.831861 0.554984i \(-0.812724\pi\)
−0.831861 + 0.554984i \(0.812724\pi\)
\(618\) 0 0
\(619\) 11.7256 0.471294 0.235647 0.971839i \(-0.424279\pi\)
0.235647 + 0.971839i \(0.424279\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.29180 −0.332471
\(623\) 31.0603 1.24441
\(624\) 0 0
\(625\) 0 0
\(626\) 19.8059 0.791602
\(627\) 0 0
\(628\) −10.0957 −0.402862
\(629\) −43.7196 −1.74322
\(630\) 0 0
\(631\) 24.4523 0.973430 0.486715 0.873561i \(-0.338195\pi\)
0.486715 + 0.873561i \(0.338195\pi\)
\(632\) 34.7320 1.38156
\(633\) 0 0
\(634\) 26.0703 1.03538
\(635\) 0 0
\(636\) 0 0
\(637\) 0.396356 0.0157042
\(638\) 22.8030 0.902781
\(639\) 0 0
\(640\) 0 0
\(641\) 20.8359 0.822969 0.411484 0.911417i \(-0.365010\pi\)
0.411484 + 0.911417i \(0.365010\pi\)
\(642\) 0 0
\(643\) 20.6793 0.815510 0.407755 0.913091i \(-0.366312\pi\)
0.407755 + 0.913091i \(0.366312\pi\)
\(644\) 2.69924 0.106365
\(645\) 0 0
\(646\) 19.5878 0.770673
\(647\) −15.5302 −0.610554 −0.305277 0.952264i \(-0.598749\pi\)
−0.305277 + 0.952264i \(0.598749\pi\)
\(648\) 0 0
\(649\) −21.7208 −0.852615
\(650\) 0 0
\(651\) 0 0
\(652\) 2.33752 0.0915444
\(653\) 28.4323 1.11264 0.556320 0.830968i \(-0.312213\pi\)
0.556320 + 0.830968i \(0.312213\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 27.9494 1.09124
\(657\) 0 0
\(658\) −10.3413 −0.403145
\(659\) 17.9375 0.698748 0.349374 0.936983i \(-0.386394\pi\)
0.349374 + 0.936983i \(0.386394\pi\)
\(660\) 0 0
\(661\) −5.32444 −0.207097 −0.103548 0.994624i \(-0.533020\pi\)
−0.103548 + 0.994624i \(0.533020\pi\)
\(662\) −37.6770 −1.46436
\(663\) 0 0
\(664\) −10.5598 −0.409799
\(665\) 0 0
\(666\) 0 0
\(667\) −19.1875 −0.742942
\(668\) −10.7504 −0.415945
\(669\) 0 0
\(670\) 0 0
\(671\) 11.8703 0.458247
\(672\) 0 0
\(673\) 7.66248 0.295367 0.147683 0.989035i \(-0.452818\pi\)
0.147683 + 0.989035i \(0.452818\pi\)
\(674\) 35.8816 1.38211
\(675\) 0 0
\(676\) 6.07749 0.233749
\(677\) 37.4111 1.43783 0.718914 0.695099i \(-0.244640\pi\)
0.718914 + 0.695099i \(0.244640\pi\)
\(678\) 0 0
\(679\) 0.119320 0.00457906
\(680\) 0 0
\(681\) 0 0
\(682\) 5.00856 0.191788
\(683\) −18.3023 −0.700318 −0.350159 0.936690i \(-0.613872\pi\)
−0.350159 + 0.936690i \(0.613872\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 24.9099 0.951064
\(687\) 0 0
\(688\) 9.76360 0.372234
\(689\) −0.141561 −0.00539303
\(690\) 0 0
\(691\) 19.9505 0.758952 0.379476 0.925202i \(-0.376104\pi\)
0.379476 + 0.925202i \(0.376104\pi\)
\(692\) −2.89295 −0.109974
\(693\) 0 0
\(694\) 6.87754 0.261068
\(695\) 0 0
\(696\) 0 0
\(697\) −50.3950 −1.90885
\(698\) 17.2898 0.654430
\(699\) 0 0
\(700\) 0 0
\(701\) −41.9220 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(702\) 0 0
\(703\) 26.3126 0.992398
\(704\) −23.5927 −0.889183
\(705\) 0 0
\(706\) −24.4605 −0.920584
\(707\) 34.0031 1.27882
\(708\) 0 0
\(709\) 6.12755 0.230125 0.115063 0.993358i \(-0.463293\pi\)
0.115063 + 0.993358i \(0.463293\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 45.4785 1.70438
\(713\) −4.21442 −0.157831
\(714\) 0 0
\(715\) 0 0
\(716\) 6.86463 0.256543
\(717\) 0 0
\(718\) 14.3667 0.536159
\(719\) 40.2449 1.50088 0.750440 0.660939i \(-0.229842\pi\)
0.750440 + 0.660939i \(0.229842\pi\)
\(720\) 0 0
\(721\) −11.2206 −0.417878
\(722\) 11.7258 0.436389
\(723\) 0 0
\(724\) 1.34666 0.0500482
\(725\) 0 0
\(726\) 0 0
\(727\) 9.61564 0.356624 0.178312 0.983974i \(-0.442936\pi\)
0.178312 + 0.983974i \(0.442936\pi\)
\(728\) −0.954271 −0.0353676
\(729\) 0 0
\(730\) 0 0
\(731\) −17.6046 −0.651128
\(732\) 0 0
\(733\) −9.23751 −0.341195 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(734\) −0.0225491 −0.000832303 0
\(735\) 0 0
\(736\) 7.15460 0.263722
\(737\) −8.86171 −0.326425
\(738\) 0 0
\(739\) 11.7962 0.433931 0.216966 0.976179i \(-0.430384\pi\)
0.216966 + 0.976179i \(0.430384\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.44125 −0.0896210
\(743\) −7.23858 −0.265558 −0.132779 0.991146i \(-0.542390\pi\)
−0.132779 + 0.991146i \(0.542390\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 20.5029 0.750665
\(747\) 0 0
\(748\) 6.37092 0.232944
\(749\) −42.2715 −1.54457
\(750\) 0 0
\(751\) 4.76976 0.174051 0.0870255 0.996206i \(-0.472264\pi\)
0.0870255 + 0.996206i \(0.472264\pi\)
\(752\) −11.3906 −0.415371
\(753\) 0 0
\(754\) 1.28700 0.0468698
\(755\) 0 0
\(756\) 0 0
\(757\) −15.7834 −0.573658 −0.286829 0.957982i \(-0.592601\pi\)
−0.286829 + 0.957982i \(0.592601\pi\)
\(758\) 3.12338 0.113446
\(759\) 0 0
\(760\) 0 0
\(761\) 28.6365 1.03807 0.519036 0.854752i \(-0.326291\pi\)
0.519036 + 0.854752i \(0.326291\pi\)
\(762\) 0 0
\(763\) −33.5437 −1.21436
\(764\) 5.91080 0.213845
\(765\) 0 0
\(766\) −19.5432 −0.706124
\(767\) −1.22592 −0.0442653
\(768\) 0 0
\(769\) 44.2898 1.59713 0.798566 0.601907i \(-0.205592\pi\)
0.798566 + 0.601907i \(0.205592\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.28779 0.118330
\(773\) −14.3392 −0.515746 −0.257873 0.966179i \(-0.583022\pi\)
−0.257873 + 0.966179i \(0.583022\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.174707 0.00627163
\(777\) 0 0
\(778\) −5.40809 −0.193889
\(779\) 30.3301 1.08669
\(780\) 0 0
\(781\) 33.1080 1.18470
\(782\) 17.5335 0.626997
\(783\) 0 0
\(784\) 7.52879 0.268885
\(785\) 0 0
\(786\) 0 0
\(787\) 49.6574 1.77009 0.885047 0.465501i \(-0.154126\pi\)
0.885047 + 0.465501i \(0.154126\pi\)
\(788\) −5.29448 −0.188608
\(789\) 0 0
\(790\) 0 0
\(791\) −17.5403 −0.623663
\(792\) 0 0
\(793\) 0.669956 0.0237908
\(794\) 14.4465 0.512688
\(795\) 0 0
\(796\) −6.31403 −0.223795
\(797\) −3.33054 −0.117974 −0.0589869 0.998259i \(-0.518787\pi\)
−0.0589869 + 0.998259i \(0.518787\pi\)
\(798\) 0 0
\(799\) 20.5381 0.726586
\(800\) 0 0
\(801\) 0 0
\(802\) 24.6285 0.869661
\(803\) 30.7781 1.08614
\(804\) 0 0
\(805\) 0 0
\(806\) 0.282683 0.00995707
\(807\) 0 0
\(808\) 49.7873 1.75151
\(809\) −22.9978 −0.808559 −0.404280 0.914635i \(-0.632478\pi\)
−0.404280 + 0.914635i \(0.632478\pi\)
\(810\) 0 0
\(811\) 1.70528 0.0598804 0.0299402 0.999552i \(-0.490468\pi\)
0.0299402 + 0.999552i \(0.490468\pi\)
\(812\) −6.78588 −0.238138
\(813\) 0 0
\(814\) −27.9912 −0.981091
\(815\) 0 0
\(816\) 0 0
\(817\) 10.5953 0.370682
\(818\) 12.6341 0.441739
\(819\) 0 0
\(820\) 0 0
\(821\) −19.5935 −0.683819 −0.341910 0.939733i \(-0.611074\pi\)
−0.341910 + 0.939733i \(0.611074\pi\)
\(822\) 0 0
\(823\) −6.41237 −0.223521 −0.111761 0.993735i \(-0.535649\pi\)
−0.111761 + 0.993735i \(0.535649\pi\)
\(824\) −16.4292 −0.572339
\(825\) 0 0
\(826\) −21.1413 −0.735598
\(827\) 31.6309 1.09991 0.549957 0.835193i \(-0.314644\pi\)
0.549957 + 0.835193i \(0.314644\pi\)
\(828\) 0 0
\(829\) −3.08634 −0.107193 −0.0535965 0.998563i \(-0.517068\pi\)
−0.0535965 + 0.998563i \(0.517068\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.33157 −0.0461638
\(833\) −13.5750 −0.470347
\(834\) 0 0
\(835\) 0 0
\(836\) −3.83433 −0.132613
\(837\) 0 0
\(838\) 46.8283 1.61766
\(839\) −15.2136 −0.525233 −0.262616 0.964900i \(-0.584585\pi\)
−0.262616 + 0.964900i \(0.584585\pi\)
\(840\) 0 0
\(841\) 19.2372 0.663352
\(842\) −12.5903 −0.433890
\(843\) 0 0
\(844\) 4.74379 0.163288
\(845\) 0 0
\(846\) 0 0
\(847\) −8.26668 −0.284046
\(848\) −2.68894 −0.0923387
\(849\) 0 0
\(850\) 0 0
\(851\) 23.5530 0.807386
\(852\) 0 0
\(853\) 25.1005 0.859426 0.429713 0.902966i \(-0.358615\pi\)
0.429713 + 0.902966i \(0.358615\pi\)
\(854\) 11.5536 0.395355
\(855\) 0 0
\(856\) −61.8938 −2.11549
\(857\) −34.7042 −1.18547 −0.592737 0.805396i \(-0.701952\pi\)
−0.592737 + 0.805396i \(0.701952\pi\)
\(858\) 0 0
\(859\) 38.5260 1.31449 0.657246 0.753676i \(-0.271721\pi\)
0.657246 + 0.753676i \(0.271721\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 46.9831 1.60025
\(863\) 45.6656 1.55447 0.777237 0.629208i \(-0.216621\pi\)
0.777237 + 0.629208i \(0.216621\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 20.3891 0.692850
\(867\) 0 0
\(868\) −1.49048 −0.0505903
\(869\) −30.1620 −1.02318
\(870\) 0 0
\(871\) −0.500153 −0.0169471
\(872\) −49.1146 −1.66323
\(873\) 0 0
\(874\) −10.5525 −0.356944
\(875\) 0 0
\(876\) 0 0
\(877\) 6.58359 0.222312 0.111156 0.993803i \(-0.464545\pi\)
0.111156 + 0.993803i \(0.464545\pi\)
\(878\) −20.3255 −0.685954
\(879\) 0 0
\(880\) 0 0
\(881\) −42.9142 −1.44581 −0.722907 0.690945i \(-0.757195\pi\)
−0.722907 + 0.690945i \(0.757195\pi\)
\(882\) 0 0
\(883\) −35.4942 −1.19447 −0.597237 0.802065i \(-0.703735\pi\)
−0.597237 + 0.802065i \(0.703735\pi\)
\(884\) 0.359574 0.0120938
\(885\) 0 0
\(886\) 42.8612 1.43995
\(887\) 24.5790 0.825282 0.412641 0.910894i \(-0.364606\pi\)
0.412641 + 0.910894i \(0.364606\pi\)
\(888\) 0 0
\(889\) −45.6301 −1.53038
\(890\) 0 0
\(891\) 0 0
\(892\) 10.5992 0.354887
\(893\) −12.3608 −0.413639
\(894\) 0 0
\(895\) 0 0
\(896\) −12.1570 −0.406137
\(897\) 0 0
\(898\) 18.2963 0.610557
\(899\) 10.5950 0.353364
\(900\) 0 0
\(901\) 4.84839 0.161523
\(902\) −32.2651 −1.07431
\(903\) 0 0
\(904\) −25.6825 −0.854188
\(905\) 0 0
\(906\) 0 0
\(907\) −46.5140 −1.54447 −0.772236 0.635336i \(-0.780862\pi\)
−0.772236 + 0.635336i \(0.780862\pi\)
\(908\) 2.40152 0.0796973
\(909\) 0 0
\(910\) 0 0
\(911\) −15.2581 −0.505523 −0.252761 0.967529i \(-0.581339\pi\)
−0.252761 + 0.967529i \(0.581339\pi\)
\(912\) 0 0
\(913\) 9.17034 0.303494
\(914\) −29.4645 −0.974599
\(915\) 0 0
\(916\) −9.77396 −0.322941
\(917\) 29.4584 0.972802
\(918\) 0 0
\(919\) −12.0177 −0.396427 −0.198213 0.980159i \(-0.563514\pi\)
−0.198213 + 0.980159i \(0.563514\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 28.5911 0.941597
\(923\) 1.86861 0.0615061
\(924\) 0 0
\(925\) 0 0
\(926\) −1.09468 −0.0359735
\(927\) 0 0
\(928\) −17.9866 −0.590440
\(929\) −1.48156 −0.0486086 −0.0243043 0.999705i \(-0.507737\pi\)
−0.0243043 + 0.999705i \(0.507737\pi\)
\(930\) 0 0
\(931\) 8.17011 0.267765
\(932\) −8.14796 −0.266895
\(933\) 0 0
\(934\) 23.9893 0.784953
\(935\) 0 0
\(936\) 0 0
\(937\) −38.5057 −1.25793 −0.628963 0.777435i \(-0.716520\pi\)
−0.628963 + 0.777435i \(0.716520\pi\)
\(938\) −8.62527 −0.281625
\(939\) 0 0
\(940\) 0 0
\(941\) 24.8562 0.810291 0.405145 0.914252i \(-0.367221\pi\)
0.405145 + 0.914252i \(0.367221\pi\)
\(942\) 0 0
\(943\) 27.1492 0.884100
\(944\) −23.2863 −0.757905
\(945\) 0 0
\(946\) −11.2712 −0.366459
\(947\) 10.6476 0.345999 0.172999 0.984922i \(-0.444654\pi\)
0.172999 + 0.984922i \(0.444654\pi\)
\(948\) 0 0
\(949\) 1.73711 0.0563890
\(950\) 0 0
\(951\) 0 0
\(952\) 32.6834 1.05927
\(953\) −27.2483 −0.882659 −0.441329 0.897345i \(-0.645493\pi\)
−0.441329 + 0.897345i \(0.645493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2.18373 −0.0706270
\(957\) 0 0
\(958\) 40.6122 1.31212
\(959\) 20.0250 0.646643
\(960\) 0 0
\(961\) −28.6729 −0.924931
\(962\) −1.57982 −0.0509354
\(963\) 0 0
\(964\) −6.23981 −0.200971
\(965\) 0 0
\(966\) 0 0
\(967\) −37.9832 −1.22146 −0.610729 0.791840i \(-0.709123\pi\)
−0.610729 + 0.791840i \(0.709123\pi\)
\(968\) −12.1041 −0.389039
\(969\) 0 0
\(970\) 0 0
\(971\) −4.14071 −0.132882 −0.0664409 0.997790i \(-0.521164\pi\)
−0.0664409 + 0.997790i \(0.521164\pi\)
\(972\) 0 0
\(973\) 37.1255 1.19019
\(974\) 23.0142 0.737424
\(975\) 0 0
\(976\) 12.7258 0.407344
\(977\) −50.3811 −1.61183 −0.805917 0.592028i \(-0.798327\pi\)
−0.805917 + 0.592028i \(0.798327\pi\)
\(978\) 0 0
\(979\) −39.4945 −1.26225
\(980\) 0 0
\(981\) 0 0
\(982\) −33.8705 −1.08085
\(983\) 30.0565 0.958654 0.479327 0.877637i \(-0.340881\pi\)
0.479327 + 0.877637i \(0.340881\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −44.0792 −1.40377
\(987\) 0 0
\(988\) −0.216409 −0.00688489
\(989\) 9.48408 0.301576
\(990\) 0 0
\(991\) 4.37287 0.138909 0.0694544 0.997585i \(-0.477874\pi\)
0.0694544 + 0.997585i \(0.477874\pi\)
\(992\) −3.95067 −0.125434
\(993\) 0 0
\(994\) 32.2247 1.02211
\(995\) 0 0
\(996\) 0 0
\(997\) 2.57051 0.0814090 0.0407045 0.999171i \(-0.487040\pi\)
0.0407045 + 0.999171i \(0.487040\pi\)
\(998\) 1.05524 0.0334030
\(999\) 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 5625.2.a.z.1.3 8
3.2 odd 2 inner 5625.2.a.z.1.6 yes 8
5.4 even 2 5625.2.a.bb.1.6 yes 8
15.14 odd 2 5625.2.a.bb.1.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.3 8 1.1 even 1 trivial
5625.2.a.z.1.6 yes 8 3.2 odd 2 inner
5625.2.a.bb.1.3 yes 8 15.14 odd 2
5625.2.a.bb.1.6 yes 8 5.4 even 2