Properties

Label 7800.2.a.bw.1.1
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.10304.1
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.692297\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} -2.97906 q^{11} -1.00000 q^{13} +1.44383 q^{17} +4.17113 q^{19} -2.82843 q^{21} +6.21302 q^{23} +1.00000 q^{27} -0.828427 q^{29} +1.65685 q^{31} -2.97906 q^{33} -0.0418875 q^{37} -1.00000 q^{39} -4.36365 q^{41} -8.99956 q^{43} -4.40554 q^{47} +1.00000 q^{49} +1.44383 q^{51} -6.72730 q^{53} +4.17113 q^{57} -2.97906 q^{59} +6.27226 q^{61} -2.82843 q^{63} +0.727302 q^{67} +6.21302 q^{69} -8.32176 q^{71} -6.87031 q^{73} +8.42604 q^{77} +6.11190 q^{79} +1.00000 q^{81} +14.7059 q^{83} -0.828427 q^{87} +14.7478 q^{89} +2.82843 q^{91} +1.65685 q^{93} -6.78654 q^{97} -2.97906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} - 12 q^{19} + 4 q^{23} + 4 q^{27} + 8 q^{29} - 16 q^{31} - 8 q^{33} - 8 q^{37} - 4 q^{39} - 4 q^{41} + 4 q^{43} - 12 q^{47} + 4 q^{49} + 4 q^{51} - 12 q^{57} - 8 q^{59} + 12 q^{61} - 24 q^{67} + 4 q^{69} - 12 q^{71} - 24 q^{73} - 8 q^{77} - 12 q^{79} + 4 q^{81} - 12 q^{83} + 8 q^{87} - 4 q^{89} - 16 q^{93} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.97906 −0.898219 −0.449110 0.893477i \(-0.648259\pi\)
−0.449110 + 0.893477i \(0.648259\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.44383 0.350181 0.175090 0.984552i \(-0.443978\pi\)
0.175090 + 0.984552i \(0.443978\pi\)
\(18\) 0 0
\(19\) 4.17113 0.956924 0.478462 0.878108i \(-0.341194\pi\)
0.478462 + 0.878108i \(0.341194\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 0 0
\(23\) 6.21302 1.29550 0.647752 0.761851i \(-0.275709\pi\)
0.647752 + 0.761851i \(0.275709\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 1.65685 0.297580 0.148790 0.988869i \(-0.452462\pi\)
0.148790 + 0.988869i \(0.452462\pi\)
\(32\) 0 0
\(33\) −2.97906 −0.518587
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.0418875 −0.00688626 −0.00344313 0.999994i \(-0.501096\pi\)
−0.00344313 + 0.999994i \(0.501096\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −4.36365 −0.681488 −0.340744 0.940156i \(-0.610679\pi\)
−0.340744 + 0.940156i \(0.610679\pi\)
\(42\) 0 0
\(43\) −8.99956 −1.37242 −0.686210 0.727403i \(-0.740727\pi\)
−0.686210 + 0.727403i \(0.740727\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.40554 −0.642614 −0.321307 0.946975i \(-0.604122\pi\)
−0.321307 + 0.946975i \(0.604122\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.44383 0.202177
\(52\) 0 0
\(53\) −6.72730 −0.924066 −0.462033 0.886863i \(-0.652880\pi\)
−0.462033 + 0.886863i \(0.652880\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.17113 0.552480
\(58\) 0 0
\(59\) −2.97906 −0.387840 −0.193920 0.981017i \(-0.562120\pi\)
−0.193920 + 0.981017i \(0.562120\pi\)
\(60\) 0 0
\(61\) 6.27226 0.803081 0.401540 0.915841i \(-0.368475\pi\)
0.401540 + 0.915841i \(0.368475\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.727302 0.0888540 0.0444270 0.999013i \(-0.485854\pi\)
0.0444270 + 0.999013i \(0.485854\pi\)
\(68\) 0 0
\(69\) 6.21302 0.747960
\(70\) 0 0
\(71\) −8.32176 −0.987612 −0.493806 0.869572i \(-0.664395\pi\)
−0.493806 + 0.869572i \(0.664395\pi\)
\(72\) 0 0
\(73\) −6.87031 −0.804110 −0.402055 0.915616i \(-0.631704\pi\)
−0.402055 + 0.915616i \(0.631704\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.42604 0.960237
\(78\) 0 0
\(79\) 6.11190 0.687642 0.343821 0.939035i \(-0.388279\pi\)
0.343821 + 0.939035i \(0.388279\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.7059 1.61418 0.807092 0.590425i \(-0.201040\pi\)
0.807092 + 0.590425i \(0.201040\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.828427 −0.0888167
\(88\) 0 0
\(89\) 14.7478 1.56326 0.781632 0.623740i \(-0.214387\pi\)
0.781632 + 0.623740i \(0.214387\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) 1.65685 0.171808
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.78654 −0.689069 −0.344534 0.938774i \(-0.611963\pi\)
−0.344534 + 0.938774i \(0.611963\pi\)
\(98\) 0 0
\(99\) −2.97906 −0.299406
\(100\) 0 0
\(101\) 12.1421 1.20819 0.604094 0.796913i \(-0.293535\pi\)
0.604094 + 0.796913i \(0.293535\pi\)
\(102\) 0 0
\(103\) 2.81108 0.276984 0.138492 0.990364i \(-0.455775\pi\)
0.138492 + 0.990364i \(0.455775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.3842 −1.77726 −0.888632 0.458621i \(-0.848343\pi\)
−0.888632 + 0.458621i \(0.848343\pi\)
\(108\) 0 0
\(109\) 1.44383 0.138294 0.0691470 0.997606i \(-0.477972\pi\)
0.0691470 + 0.997606i \(0.477972\pi\)
\(110\) 0 0
\(111\) −0.0418875 −0.00397579
\(112\) 0 0
\(113\) −3.40194 −0.320028 −0.160014 0.987115i \(-0.551154\pi\)
−0.160014 + 0.987115i \(0.551154\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −4.08378 −0.374359
\(120\) 0 0
\(121\) −2.12522 −0.193202
\(122\) 0 0
\(123\) −4.36365 −0.393457
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −15.8106 −1.40297 −0.701484 0.712686i \(-0.747479\pi\)
−0.701484 + 0.712686i \(0.747479\pi\)
\(128\) 0 0
\(129\) −8.99956 −0.792367
\(130\) 0 0
\(131\) −19.1707 −1.67495 −0.837476 0.546475i \(-0.815970\pi\)
−0.837476 + 0.546475i \(0.815970\pi\)
\(132\) 0 0
\(133\) −11.7977 −1.02299
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.67824 −0.143381 −0.0716907 0.997427i \(-0.522839\pi\)
−0.0716907 + 0.997427i \(0.522839\pi\)
\(138\) 0 0
\(139\) −12.3423 −1.04686 −0.523429 0.852069i \(-0.675347\pi\)
−0.523429 + 0.852069i \(0.675347\pi\)
\(140\) 0 0
\(141\) −4.40554 −0.371013
\(142\) 0 0
\(143\) 2.97906 0.249121
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 0.677798 0.0555274 0.0277637 0.999615i \(-0.491161\pi\)
0.0277637 + 0.999615i \(0.491161\pi\)
\(150\) 0 0
\(151\) −17.9991 −1.46475 −0.732374 0.680903i \(-0.761588\pi\)
−0.732374 + 0.680903i \(0.761588\pi\)
\(152\) 0 0
\(153\) 1.44383 0.116727
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.8392 1.02468 0.512340 0.858783i \(-0.328779\pi\)
0.512340 + 0.858783i \(0.328779\pi\)
\(158\) 0 0
\(159\) −6.72730 −0.533510
\(160\) 0 0
\(161\) −17.5731 −1.38495
\(162\) 0 0
\(163\) −15.3137 −1.19946 −0.599731 0.800202i \(-0.704726\pi\)
−0.599731 + 0.800202i \(0.704726\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.3922 −1.19109 −0.595543 0.803324i \(-0.703063\pi\)
−0.595543 + 0.803324i \(0.703063\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 4.17113 0.318975
\(172\) 0 0
\(173\) 14.9706 1.13819 0.569095 0.822272i \(-0.307294\pi\)
0.569095 + 0.822272i \(0.307294\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.97906 −0.223920
\(178\) 0 0
\(179\) −7.63950 −0.571003 −0.285502 0.958378i \(-0.592160\pi\)
−0.285502 + 0.958378i \(0.592160\pi\)
\(180\) 0 0
\(181\) 15.0286 1.11706 0.558532 0.829483i \(-0.311365\pi\)
0.558532 + 0.829483i \(0.311365\pi\)
\(182\) 0 0
\(183\) 6.27226 0.463659
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.30126 −0.314539
\(188\) 0 0
\(189\) −2.82843 −0.205738
\(190\) 0 0
\(191\) −10.8877 −0.787804 −0.393902 0.919152i \(-0.628875\pi\)
−0.393902 + 0.919152i \(0.628875\pi\)
\(192\) 0 0
\(193\) −4.94690 −0.356086 −0.178043 0.984023i \(-0.556977\pi\)
−0.178043 + 0.984023i \(0.556977\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.3628 −1.87827 −0.939135 0.343549i \(-0.888371\pi\)
−0.939135 + 0.343549i \(0.888371\pi\)
\(198\) 0 0
\(199\) 6.11190 0.433261 0.216630 0.976254i \(-0.430493\pi\)
0.216630 + 0.976254i \(0.430493\pi\)
\(200\) 0 0
\(201\) 0.727302 0.0512999
\(202\) 0 0
\(203\) 2.34315 0.164457
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.21302 0.431835
\(208\) 0 0
\(209\) −12.4260 −0.859527
\(210\) 0 0
\(211\) 9.68585 0.666802 0.333401 0.942785i \(-0.391804\pi\)
0.333401 + 0.942785i \(0.391804\pi\)
\(212\) 0 0
\(213\) −8.32176 −0.570198
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.68629 −0.318126
\(218\) 0 0
\(219\) −6.87031 −0.464253
\(220\) 0 0
\(221\) −1.44383 −0.0971227
\(222\) 0 0
\(223\) −9.21258 −0.616920 −0.308460 0.951237i \(-0.599814\pi\)
−0.308460 + 0.951237i \(0.599814\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.3218 −1.34880 −0.674401 0.738365i \(-0.735598\pi\)
−0.674401 + 0.738365i \(0.735598\pi\)
\(228\) 0 0
\(229\) −16.4715 −1.08847 −0.544234 0.838933i \(-0.683180\pi\)
−0.544234 + 0.838933i \(0.683180\pi\)
\(230\) 0 0
\(231\) 8.42604 0.554393
\(232\) 0 0
\(233\) −2.37339 −0.155486 −0.0777428 0.996973i \(-0.524771\pi\)
−0.0777428 + 0.996973i \(0.524771\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.11190 0.397010
\(238\) 0 0
\(239\) −13.6363 −0.882062 −0.441031 0.897492i \(-0.645387\pi\)
−0.441031 + 0.897492i \(0.645387\pi\)
\(240\) 0 0
\(241\) 1.07045 0.0689536 0.0344768 0.999405i \(-0.489024\pi\)
0.0344768 + 0.999405i \(0.489024\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.17113 −0.265403
\(248\) 0 0
\(249\) 14.7059 0.931950
\(250\) 0 0
\(251\) 14.6262 0.923196 0.461598 0.887089i \(-0.347276\pi\)
0.461598 + 0.887089i \(0.347276\pi\)
\(252\) 0 0
\(253\) −18.5089 −1.16365
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.9403 −1.55573 −0.777867 0.628429i \(-0.783698\pi\)
−0.777867 + 0.628429i \(0.783698\pi\)
\(258\) 0 0
\(259\) 0.118476 0.00736173
\(260\) 0 0
\(261\) −0.828427 −0.0512784
\(262\) 0 0
\(263\) 9.01691 0.556007 0.278003 0.960580i \(-0.410327\pi\)
0.278003 + 0.960580i \(0.410327\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 14.7478 0.902551
\(268\) 0 0
\(269\) 8.28391 0.505079 0.252539 0.967587i \(-0.418734\pi\)
0.252539 + 0.967587i \(0.418734\pi\)
\(270\) 0 0
\(271\) −19.9572 −1.21232 −0.606158 0.795344i \(-0.707290\pi\)
−0.606158 + 0.795344i \(0.707290\pi\)
\(272\) 0 0
\(273\) 2.82843 0.171184
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.6145 1.35878 0.679388 0.733780i \(-0.262245\pi\)
0.679388 + 0.733780i \(0.262245\pi\)
\(278\) 0 0
\(279\) 1.65685 0.0991933
\(280\) 0 0
\(281\) −1.89572 −0.113089 −0.0565446 0.998400i \(-0.518008\pi\)
−0.0565446 + 0.998400i \(0.518008\pi\)
\(282\) 0 0
\(283\) 11.2299 0.667550 0.333775 0.942653i \(-0.391677\pi\)
0.333775 + 0.942653i \(0.391677\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.3423 0.728541
\(288\) 0 0
\(289\) −14.9153 −0.877373
\(290\) 0 0
\(291\) −6.78654 −0.397834
\(292\) 0 0
\(293\) −27.6774 −1.61693 −0.808464 0.588545i \(-0.799701\pi\)
−0.808464 + 0.588545i \(0.799701\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.97906 −0.172862
\(298\) 0 0
\(299\) −6.21302 −0.359308
\(300\) 0 0
\(301\) 25.4546 1.46718
\(302\) 0 0
\(303\) 12.1421 0.697547
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −24.9858 −1.42601 −0.713007 0.701157i \(-0.752667\pi\)
−0.713007 + 0.701157i \(0.752667\pi\)
\(308\) 0 0
\(309\) 2.81108 0.159917
\(310\) 0 0
\(311\) 18.3842 1.04247 0.521235 0.853413i \(-0.325472\pi\)
0.521235 + 0.853413i \(0.325472\pi\)
\(312\) 0 0
\(313\) 8.61453 0.486922 0.243461 0.969911i \(-0.421717\pi\)
0.243461 + 0.969911i \(0.421717\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.0491 −0.845240 −0.422620 0.906307i \(-0.638889\pi\)
−0.422620 + 0.906307i \(0.638889\pi\)
\(318\) 0 0
\(319\) 2.46793 0.138178
\(320\) 0 0
\(321\) −18.3842 −1.02610
\(322\) 0 0
\(323\) 6.02242 0.335096
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.44383 0.0798441
\(328\) 0 0
\(329\) 12.4607 0.686983
\(330\) 0 0
\(331\) 22.6738 1.24626 0.623131 0.782117i \(-0.285860\pi\)
0.623131 + 0.782117i \(0.285860\pi\)
\(332\) 0 0
\(333\) −0.0418875 −0.00229542
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −27.9282 −1.52135 −0.760674 0.649134i \(-0.775131\pi\)
−0.760674 + 0.649134i \(0.775131\pi\)
\(338\) 0 0
\(339\) −3.40194 −0.184768
\(340\) 0 0
\(341\) −4.93586 −0.267292
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −19.7398 −1.05969 −0.529843 0.848096i \(-0.677749\pi\)
−0.529843 + 0.848096i \(0.677749\pi\)
\(348\) 0 0
\(349\) 30.7157 1.64417 0.822086 0.569364i \(-0.192810\pi\)
0.822086 + 0.569364i \(0.192810\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 17.0072 0.905201 0.452600 0.891713i \(-0.350496\pi\)
0.452600 + 0.891713i \(0.350496\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.08378 −0.216136
\(358\) 0 0
\(359\) −7.90203 −0.417053 −0.208527 0.978017i \(-0.566867\pi\)
−0.208527 + 0.978017i \(0.566867\pi\)
\(360\) 0 0
\(361\) −1.60164 −0.0842968
\(362\) 0 0
\(363\) −2.12522 −0.111545
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 34.1810 1.78424 0.892118 0.451803i \(-0.149219\pi\)
0.892118 + 0.451803i \(0.149219\pi\)
\(368\) 0 0
\(369\) −4.36365 −0.227163
\(370\) 0 0
\(371\) 19.0277 0.987868
\(372\) 0 0
\(373\) −18.5098 −0.958402 −0.479201 0.877705i \(-0.659074\pi\)
−0.479201 + 0.877705i \(0.659074\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.828427 0.0426662
\(378\) 0 0
\(379\) −12.7994 −0.657462 −0.328731 0.944424i \(-0.606621\pi\)
−0.328731 + 0.944424i \(0.606621\pi\)
\(380\) 0 0
\(381\) −15.8106 −0.810004
\(382\) 0 0
\(383\) 0.664032 0.0339304 0.0169652 0.999856i \(-0.494600\pi\)
0.0169652 + 0.999856i \(0.494600\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.99956 −0.457473
\(388\) 0 0
\(389\) 2.32580 0.117923 0.0589613 0.998260i \(-0.481221\pi\)
0.0589613 + 0.998260i \(0.481221\pi\)
\(390\) 0 0
\(391\) 8.97056 0.453661
\(392\) 0 0
\(393\) −19.1707 −0.967034
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −18.2433 −0.915603 −0.457802 0.889054i \(-0.651363\pi\)
−0.457802 + 0.889054i \(0.651363\pi\)
\(398\) 0 0
\(399\) −11.7977 −0.590626
\(400\) 0 0
\(401\) 16.9919 0.848537 0.424269 0.905536i \(-0.360531\pi\)
0.424269 + 0.905536i \(0.360531\pi\)
\(402\) 0 0
\(403\) −1.65685 −0.0825338
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.124785 0.00618538
\(408\) 0 0
\(409\) −10.8111 −0.534573 −0.267287 0.963617i \(-0.586127\pi\)
−0.267287 + 0.963617i \(0.586127\pi\)
\(410\) 0 0
\(411\) −1.67824 −0.0827813
\(412\) 0 0
\(413\) 8.42604 0.414618
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.3423 −0.604403
\(418\) 0 0
\(419\) −5.08780 −0.248555 −0.124278 0.992247i \(-0.539661\pi\)
−0.124278 + 0.992247i \(0.539661\pi\)
\(420\) 0 0
\(421\) 12.0945 0.589452 0.294726 0.955582i \(-0.404772\pi\)
0.294726 + 0.955582i \(0.404772\pi\)
\(422\) 0 0
\(423\) −4.40554 −0.214205
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −17.7406 −0.858529
\(428\) 0 0
\(429\) 2.97906 0.143830
\(430\) 0 0
\(431\) −8.48931 −0.408916 −0.204458 0.978875i \(-0.565543\pi\)
−0.204458 + 0.978875i \(0.565543\pi\)
\(432\) 0 0
\(433\) 13.8453 0.665365 0.332682 0.943039i \(-0.392046\pi\)
0.332682 + 0.943039i \(0.392046\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.9153 1.23970
\(438\) 0 0
\(439\) 0.602516 0.0287565 0.0143783 0.999897i \(-0.495423\pi\)
0.0143783 + 0.999897i \(0.495423\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −17.6721 −0.839626 −0.419813 0.907611i \(-0.637904\pi\)
−0.419813 + 0.907611i \(0.637904\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.677798 0.0320587
\(448\) 0 0
\(449\) 3.33509 0.157393 0.0786963 0.996899i \(-0.474924\pi\)
0.0786963 + 0.996899i \(0.474924\pi\)
\(450\) 0 0
\(451\) 12.9996 0.612125
\(452\) 0 0
\(453\) −17.9991 −0.845673
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −38.9266 −1.82091 −0.910454 0.413611i \(-0.864267\pi\)
−0.910454 + 0.413611i \(0.864267\pi\)
\(458\) 0 0
\(459\) 1.44383 0.0673923
\(460\) 0 0
\(461\) 28.3918 1.32234 0.661168 0.750238i \(-0.270061\pi\)
0.661168 + 0.750238i \(0.270061\pi\)
\(462\) 0 0
\(463\) 10.2259 0.475238 0.237619 0.971358i \(-0.423633\pi\)
0.237619 + 0.971358i \(0.423633\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.61584 −0.259870 −0.129935 0.991522i \(-0.541477\pi\)
−0.129935 + 0.991522i \(0.541477\pi\)
\(468\) 0 0
\(469\) −2.05712 −0.0949890
\(470\) 0 0
\(471\) 12.8392 0.591599
\(472\) 0 0
\(473\) 26.8102 1.23273
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.72730 −0.308022
\(478\) 0 0
\(479\) −8.95006 −0.408939 −0.204469 0.978873i \(-0.565547\pi\)
−0.204469 + 0.978873i \(0.565547\pi\)
\(480\) 0 0
\(481\) 0.0418875 0.00190991
\(482\) 0 0
\(483\) −17.5731 −0.799603
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 18.3596 0.831954 0.415977 0.909375i \(-0.363440\pi\)
0.415977 + 0.909375i \(0.363440\pi\)
\(488\) 0 0
\(489\) −15.3137 −0.692510
\(490\) 0 0
\(491\) −33.7981 −1.52529 −0.762644 0.646819i \(-0.776099\pi\)
−0.762644 + 0.646819i \(0.776099\pi\)
\(492\) 0 0
\(493\) −1.19611 −0.0538701
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 23.5375 1.05580
\(498\) 0 0
\(499\) 5.72898 0.256464 0.128232 0.991744i \(-0.459070\pi\)
0.128232 + 0.991744i \(0.459070\pi\)
\(500\) 0 0
\(501\) −15.3922 −0.687673
\(502\) 0 0
\(503\) −20.2121 −0.901215 −0.450607 0.892722i \(-0.648793\pi\)
−0.450607 + 0.892722i \(0.648793\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 6.02708 0.267146 0.133573 0.991039i \(-0.457355\pi\)
0.133573 + 0.991039i \(0.457355\pi\)
\(510\) 0 0
\(511\) 19.4322 0.859629
\(512\) 0 0
\(513\) 4.17113 0.184160
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13.1243 0.577208
\(518\) 0 0
\(519\) 14.9706 0.657135
\(520\) 0 0
\(521\) −1.95899 −0.0858249 −0.0429125 0.999079i \(-0.513664\pi\)
−0.0429125 + 0.999079i \(0.513664\pi\)
\(522\) 0 0
\(523\) −21.6859 −0.948256 −0.474128 0.880456i \(-0.657237\pi\)
−0.474128 + 0.880456i \(0.657237\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.39222 0.104207
\(528\) 0 0
\(529\) 15.6016 0.678332
\(530\) 0 0
\(531\) −2.97906 −0.129280
\(532\) 0 0
\(533\) 4.36365 0.189011
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −7.63950 −0.329669
\(538\) 0 0
\(539\) −2.97906 −0.128317
\(540\) 0 0
\(541\) 39.3245 1.69069 0.845346 0.534220i \(-0.179394\pi\)
0.845346 + 0.534220i \(0.179394\pi\)
\(542\) 0 0
\(543\) 15.0286 0.644937
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.1466 0.604865 0.302432 0.953171i \(-0.402201\pi\)
0.302432 + 0.953171i \(0.402201\pi\)
\(548\) 0 0
\(549\) 6.27226 0.267694
\(550\) 0 0
\(551\) −3.45548 −0.147208
\(552\) 0 0
\(553\) −17.2871 −0.735120
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −29.7603 −1.26098 −0.630491 0.776196i \(-0.717147\pi\)
−0.630491 + 0.776196i \(0.717147\pi\)
\(558\) 0 0
\(559\) 8.99956 0.380641
\(560\) 0 0
\(561\) −4.30126 −0.181599
\(562\) 0 0
\(563\) −28.6498 −1.20745 −0.603723 0.797194i \(-0.706317\pi\)
−0.603723 + 0.797194i \(0.706317\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) 39.7326 1.66568 0.832838 0.553517i \(-0.186715\pi\)
0.832838 + 0.553517i \(0.186715\pi\)
\(570\) 0 0
\(571\) 30.5726 1.27943 0.639713 0.768614i \(-0.279053\pi\)
0.639713 + 0.768614i \(0.279053\pi\)
\(572\) 0 0
\(573\) −10.8877 −0.454839
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −35.3116 −1.47004 −0.735020 0.678045i \(-0.762827\pi\)
−0.735020 + 0.678045i \(0.762827\pi\)
\(578\) 0 0
\(579\) −4.94690 −0.205586
\(580\) 0 0
\(581\) −41.5946 −1.72564
\(582\) 0 0
\(583\) 20.0410 0.830014
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.5517 0.806985 0.403492 0.914983i \(-0.367796\pi\)
0.403492 + 0.914983i \(0.367796\pi\)
\(588\) 0 0
\(589\) 6.91096 0.284761
\(590\) 0 0
\(591\) −26.3628 −1.08442
\(592\) 0 0
\(593\) 4.66491 0.191565 0.0957824 0.995402i \(-0.469465\pi\)
0.0957824 + 0.995402i \(0.469465\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.11190 0.250143
\(598\) 0 0
\(599\) −19.6497 −0.802864 −0.401432 0.915889i \(-0.631487\pi\)
−0.401432 + 0.915889i \(0.631487\pi\)
\(600\) 0 0
\(601\) 44.2714 1.80587 0.902934 0.429780i \(-0.141409\pi\)
0.902934 + 0.429780i \(0.141409\pi\)
\(602\) 0 0
\(603\) 0.727302 0.0296180
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 34.2648 1.39077 0.695383 0.718640i \(-0.255235\pi\)
0.695383 + 0.718640i \(0.255235\pi\)
\(608\) 0 0
\(609\) 2.34315 0.0949491
\(610\) 0 0
\(611\) 4.40554 0.178229
\(612\) 0 0
\(613\) 33.7203 1.36195 0.680975 0.732307i \(-0.261556\pi\)
0.680975 + 0.732307i \(0.261556\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.0901 −1.41267 −0.706337 0.707876i \(-0.749653\pi\)
−0.706337 + 0.707876i \(0.749653\pi\)
\(618\) 0 0
\(619\) 2.63907 0.106073 0.0530365 0.998593i \(-0.483110\pi\)
0.0530365 + 0.998593i \(0.483110\pi\)
\(620\) 0 0
\(621\) 6.21302 0.249320
\(622\) 0 0
\(623\) −41.7131 −1.67120
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −12.4260 −0.496248
\(628\) 0 0
\(629\) −0.0604785 −0.00241144
\(630\) 0 0
\(631\) −28.7683 −1.14525 −0.572624 0.819818i \(-0.694075\pi\)
−0.572624 + 0.819818i \(0.694075\pi\)
\(632\) 0 0
\(633\) 9.68585 0.384978
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −8.32176 −0.329204
\(640\) 0 0
\(641\) 35.1944 1.39009 0.695047 0.718965i \(-0.255384\pi\)
0.695047 + 0.718965i \(0.255384\pi\)
\(642\) 0 0
\(643\) −27.2709 −1.07546 −0.537731 0.843117i \(-0.680718\pi\)
−0.537731 + 0.843117i \(0.680718\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.22913 −0.166264 −0.0831322 0.996539i \(-0.526492\pi\)
−0.0831322 + 0.996539i \(0.526492\pi\)
\(648\) 0 0
\(649\) 8.87478 0.348365
\(650\) 0 0
\(651\) −4.68629 −0.183670
\(652\) 0 0
\(653\) −22.1400 −0.866406 −0.433203 0.901296i \(-0.642617\pi\)
−0.433203 + 0.901296i \(0.642617\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.87031 −0.268037
\(658\) 0 0
\(659\) 27.0932 1.05540 0.527701 0.849430i \(-0.323054\pi\)
0.527701 + 0.849430i \(0.323054\pi\)
\(660\) 0 0
\(661\) 19.0597 0.741335 0.370668 0.928766i \(-0.379129\pi\)
0.370668 + 0.928766i \(0.379129\pi\)
\(662\) 0 0
\(663\) −1.44383 −0.0560738
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.14704 −0.199294
\(668\) 0 0
\(669\) −9.21258 −0.356179
\(670\) 0 0
\(671\) −18.6854 −0.721342
\(672\) 0 0
\(673\) 23.6213 0.910533 0.455267 0.890355i \(-0.349544\pi\)
0.455267 + 0.890355i \(0.349544\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.8111 0.722968 0.361484 0.932378i \(-0.382270\pi\)
0.361484 + 0.932378i \(0.382270\pi\)
\(678\) 0 0
\(679\) 19.1952 0.736645
\(680\) 0 0
\(681\) −20.3218 −0.778732
\(682\) 0 0
\(683\) −33.7827 −1.29266 −0.646329 0.763059i \(-0.723697\pi\)
−0.646329 + 0.763059i \(0.723697\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −16.4715 −0.628428
\(688\) 0 0
\(689\) 6.72730 0.256290
\(690\) 0 0
\(691\) −47.3236 −1.80027 −0.900137 0.435606i \(-0.856534\pi\)
−0.900137 + 0.435606i \(0.856534\pi\)
\(692\) 0 0
\(693\) 8.42604 0.320079
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −6.30038 −0.238644
\(698\) 0 0
\(699\) −2.37339 −0.0897697
\(700\) 0 0
\(701\) −3.29636 −0.124502 −0.0622509 0.998061i \(-0.519828\pi\)
−0.0622509 + 0.998061i \(0.519828\pi\)
\(702\) 0 0
\(703\) −0.174718 −0.00658963
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −34.3431 −1.29161
\(708\) 0 0
\(709\) 46.8404 1.75913 0.879565 0.475779i \(-0.157834\pi\)
0.879565 + 0.475779i \(0.157834\pi\)
\(710\) 0 0
\(711\) 6.11190 0.229214
\(712\) 0 0
\(713\) 10.2941 0.385516
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −13.6363 −0.509259
\(718\) 0 0
\(719\) −20.0152 −0.746442 −0.373221 0.927742i \(-0.621747\pi\)
−0.373221 + 0.927742i \(0.621747\pi\)
\(720\) 0 0
\(721\) −7.95093 −0.296108
\(722\) 0 0
\(723\) 1.07045 0.0398104
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −10.8383 −0.401971 −0.200986 0.979594i \(-0.564414\pi\)
−0.200986 + 0.979594i \(0.564414\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.9939 −0.480595
\(732\) 0 0
\(733\) 21.1952 0.782864 0.391432 0.920207i \(-0.371980\pi\)
0.391432 + 0.920207i \(0.371980\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.16667 −0.0798104
\(738\) 0 0
\(739\) 7.07403 0.260222 0.130111 0.991499i \(-0.458467\pi\)
0.130111 + 0.991499i \(0.458467\pi\)
\(740\) 0 0
\(741\) −4.17113 −0.153230
\(742\) 0 0
\(743\) 21.3913 0.784772 0.392386 0.919801i \(-0.371650\pi\)
0.392386 + 0.919801i \(0.371650\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 14.7059 0.538061
\(748\) 0 0
\(749\) 51.9982 1.89997
\(750\) 0 0
\(751\) 18.2247 0.665028 0.332514 0.943098i \(-0.392103\pi\)
0.332514 + 0.943098i \(0.392103\pi\)
\(752\) 0 0
\(753\) 14.6262 0.533007
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −39.3708 −1.43096 −0.715479 0.698635i \(-0.753791\pi\)
−0.715479 + 0.698635i \(0.753791\pi\)
\(758\) 0 0
\(759\) −18.5089 −0.671832
\(760\) 0 0
\(761\) −13.8610 −0.502462 −0.251231 0.967927i \(-0.580835\pi\)
−0.251231 + 0.967927i \(0.580835\pi\)
\(762\) 0 0
\(763\) −4.08378 −0.147843
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.97906 0.107567
\(768\) 0 0
\(769\) −12.6274 −0.455356 −0.227678 0.973736i \(-0.573113\pi\)
−0.227678 + 0.973736i \(0.573113\pi\)
\(770\) 0 0
\(771\) −24.9403 −0.898204
\(772\) 0 0
\(773\) −34.7825 −1.25104 −0.625520 0.780208i \(-0.715113\pi\)
−0.625520 + 0.780208i \(0.715113\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.118476 0.00425029
\(778\) 0 0
\(779\) −18.2014 −0.652132
\(780\) 0 0
\(781\) 24.7910 0.887092
\(782\) 0 0
\(783\) −0.828427 −0.0296056
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 36.7835 1.31119 0.655596 0.755112i \(-0.272418\pi\)
0.655596 + 0.755112i \(0.272418\pi\)
\(788\) 0 0
\(789\) 9.01691 0.321011
\(790\) 0 0
\(791\) 9.62215 0.342124
\(792\) 0 0
\(793\) −6.27226 −0.222734
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.4413 1.50335 0.751674 0.659535i \(-0.229247\pi\)
0.751674 + 0.659535i \(0.229247\pi\)
\(798\) 0 0
\(799\) −6.36086 −0.225031
\(800\) 0 0
\(801\) 14.7478 0.521088
\(802\) 0 0
\(803\) 20.4671 0.722267
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.28391 0.291607
\(808\) 0 0
\(809\) 55.3771 1.94696 0.973478 0.228780i \(-0.0734735\pi\)
0.973478 + 0.228780i \(0.0734735\pi\)
\(810\) 0 0
\(811\) −17.1159 −0.601021 −0.300511 0.953778i \(-0.597157\pi\)
−0.300511 + 0.953778i \(0.597157\pi\)
\(812\) 0 0
\(813\) −19.9572 −0.699931
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −37.5384 −1.31330
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) 15.5460 0.542559 0.271279 0.962501i \(-0.412553\pi\)
0.271279 + 0.962501i \(0.412553\pi\)
\(822\) 0 0
\(823\) −0.0998847 −0.00348176 −0.00174088 0.999998i \(-0.500554\pi\)
−0.00174088 + 0.999998i \(0.500554\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.53311 0.331499 0.165749 0.986168i \(-0.446996\pi\)
0.165749 + 0.986168i \(0.446996\pi\)
\(828\) 0 0
\(829\) 50.6361 1.75866 0.879332 0.476210i \(-0.157990\pi\)
0.879332 + 0.476210i \(0.157990\pi\)
\(830\) 0 0
\(831\) 22.6145 0.784489
\(832\) 0 0
\(833\) 1.44383 0.0500258
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.65685 0.0572693
\(838\) 0 0
\(839\) −43.2310 −1.49250 −0.746249 0.665666i \(-0.768147\pi\)
−0.746249 + 0.665666i \(0.768147\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 0 0
\(843\) −1.89572 −0.0652921
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.01104 0.206542
\(848\) 0 0
\(849\) 11.2299 0.385410
\(850\) 0 0
\(851\) −0.260248 −0.00892119
\(852\) 0 0
\(853\) 31.6979 1.08531 0.542657 0.839954i \(-0.317418\pi\)
0.542657 + 0.839954i \(0.317418\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.8912 0.986906 0.493453 0.869772i \(-0.335734\pi\)
0.493453 + 0.869772i \(0.335734\pi\)
\(858\) 0 0
\(859\) 49.7099 1.69608 0.848040 0.529933i \(-0.177783\pi\)
0.848040 + 0.529933i \(0.177783\pi\)
\(860\) 0 0
\(861\) 12.3423 0.420623
\(862\) 0 0
\(863\) 4.56502 0.155395 0.0776976 0.996977i \(-0.475243\pi\)
0.0776976 + 0.996977i \(0.475243\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −14.9153 −0.506552
\(868\) 0 0
\(869\) −18.2077 −0.617653
\(870\) 0 0
\(871\) −0.727302 −0.0246437
\(872\) 0 0
\(873\) −6.78654 −0.229690
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −13.5312 −0.456916 −0.228458 0.973554i \(-0.573368\pi\)
−0.228458 + 0.973554i \(0.573368\pi\)
\(878\) 0 0
\(879\) −27.6774 −0.933534
\(880\) 0 0
\(881\) −28.0634 −0.945481 −0.472740 0.881202i \(-0.656735\pi\)
−0.472740 + 0.881202i \(0.656735\pi\)
\(882\) 0 0
\(883\) 47.3700 1.59413 0.797063 0.603896i \(-0.206386\pi\)
0.797063 + 0.603896i \(0.206386\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.606101 0.0203509 0.0101754 0.999948i \(-0.496761\pi\)
0.0101754 + 0.999948i \(0.496761\pi\)
\(888\) 0 0
\(889\) 44.7192 1.49984
\(890\) 0 0
\(891\) −2.97906 −0.0998021
\(892\) 0 0
\(893\) −18.3761 −0.614932
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −6.21302 −0.207447
\(898\) 0 0
\(899\) −1.37258 −0.0457782
\(900\) 0 0
\(901\) −9.71310 −0.323590
\(902\) 0 0
\(903\) 25.4546 0.847076
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 11.8582 0.393746 0.196873 0.980429i \(-0.436921\pi\)
0.196873 + 0.980429i \(0.436921\pi\)
\(908\) 0 0
\(909\) 12.1421 0.402729
\(910\) 0 0
\(911\) −33.5642 −1.11203 −0.556015 0.831172i \(-0.687670\pi\)
−0.556015 + 0.831172i \(0.687670\pi\)
\(912\) 0 0
\(913\) −43.8098 −1.44989
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 54.2229 1.79060
\(918\) 0 0
\(919\) −19.2242 −0.634149 −0.317074 0.948401i \(-0.602701\pi\)
−0.317074 + 0.948401i \(0.602701\pi\)
\(920\) 0 0
\(921\) −24.9858 −0.823310
\(922\) 0 0
\(923\) 8.32176 0.273914
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.81108 0.0923279
\(928\) 0 0
\(929\) −36.0349 −1.18227 −0.591133 0.806574i \(-0.701319\pi\)
−0.591133 + 0.806574i \(0.701319\pi\)
\(930\) 0 0
\(931\) 4.17113 0.136703
\(932\) 0 0
\(933\) 18.3842 0.601870
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 41.5851 1.35853 0.679263 0.733895i \(-0.262300\pi\)
0.679263 + 0.733895i \(0.262300\pi\)
\(938\) 0 0
\(939\) 8.61453 0.281124
\(940\) 0 0
\(941\) −56.4933 −1.84163 −0.920814 0.390002i \(-0.872474\pi\)
−0.920814 + 0.390002i \(0.872474\pi\)
\(942\) 0 0
\(943\) −27.1115 −0.882871
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.3922 −0.370197 −0.185099 0.982720i \(-0.559260\pi\)
−0.185099 + 0.982720i \(0.559260\pi\)
\(948\) 0 0
\(949\) 6.87031 0.223020
\(950\) 0 0
\(951\) −15.0491 −0.487999
\(952\) 0 0
\(953\) 56.7986 1.83989 0.919943 0.392053i \(-0.128235\pi\)
0.919943 + 0.392053i \(0.128235\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.46793 0.0797769
\(958\) 0 0
\(959\) 4.74677 0.153281
\(960\) 0 0
\(961\) −28.2548 −0.911446
\(962\) 0 0
\(963\) −18.3842 −0.592421
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −25.7224 −0.827177 −0.413588 0.910464i \(-0.635725\pi\)
−0.413588 + 0.910464i \(0.635725\pi\)
\(968\) 0 0
\(969\) 6.02242 0.193468
\(970\) 0 0
\(971\) −32.8703 −1.05486 −0.527429 0.849599i \(-0.676844\pi\)
−0.527429 + 0.849599i \(0.676844\pi\)
\(972\) 0 0
\(973\) 34.9092 1.11914
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.1453 0.964433 0.482217 0.876052i \(-0.339832\pi\)
0.482217 + 0.876052i \(0.339832\pi\)
\(978\) 0 0
\(979\) −43.9345 −1.40415
\(980\) 0 0
\(981\) 1.44383 0.0460980
\(982\) 0 0
\(983\) −32.4965 −1.03648 −0.518238 0.855236i \(-0.673412\pi\)
−0.518238 + 0.855236i \(0.673412\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 12.4607 0.396630
\(988\) 0 0
\(989\) −55.9145 −1.77798
\(990\) 0 0
\(991\) 37.4893 1.19089 0.595443 0.803397i \(-0.296976\pi\)
0.595443 + 0.803397i \(0.296976\pi\)
\(992\) 0 0
\(993\) 22.6738 0.719530
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −18.8659 −0.597488 −0.298744 0.954333i \(-0.596568\pi\)
−0.298744 + 0.954333i \(0.596568\pi\)
\(998\) 0 0
\(999\) −0.0418875 −0.00132526
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 7800.2.a.bw.1.1 4
5.2 odd 4 1560.2.l.e.1249.3 8
5.3 odd 4 1560.2.l.e.1249.7 yes 8
5.4 even 2 7800.2.a.bv.1.3 4
15.2 even 4 4680.2.l.f.2809.4 8
15.8 even 4 4680.2.l.f.2809.3 8
20.3 even 4 3120.2.l.o.1249.3 8
20.7 even 4 3120.2.l.o.1249.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.e.1249.3 8 5.2 odd 4
1560.2.l.e.1249.7 yes 8 5.3 odd 4
3120.2.l.o.1249.3 8 20.3 even 4
3120.2.l.o.1249.7 8 20.7 even 4
4680.2.l.f.2809.3 8 15.8 even 4
4680.2.l.f.2809.4 8 15.2 even 4
7800.2.a.bv.1.3 4 5.4 even 2
7800.2.a.bw.1.1 4 1.1 even 1 trivial