Properties

Label 8004.2.a.g.1.1
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.15588\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.15588 q^{5} -1.11441 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.15588 q^{5} -1.11441 q^{7} +1.00000 q^{9} -5.32243 q^{11} +0.315594 q^{13} +4.15588 q^{15} -1.36640 q^{17} +5.63415 q^{19} +1.11441 q^{21} +1.00000 q^{23} +12.2713 q^{25} -1.00000 q^{27} -1.00000 q^{29} -7.05741 q^{31} +5.32243 q^{33} +4.63136 q^{35} -5.05783 q^{37} -0.315594 q^{39} +2.34509 q^{41} +4.48164 q^{43} -4.15588 q^{45} +3.25596 q^{47} -5.75809 q^{49} +1.36640 q^{51} +7.64907 q^{53} +22.1194 q^{55} -5.63415 q^{57} -5.11563 q^{59} +1.44172 q^{61} -1.11441 q^{63} -1.31157 q^{65} +1.32653 q^{67} -1.00000 q^{69} +5.18601 q^{71} +2.17074 q^{73} -12.2713 q^{75} +5.93137 q^{77} +16.7829 q^{79} +1.00000 q^{81} +10.5447 q^{83} +5.67859 q^{85} +1.00000 q^{87} +7.27517 q^{89} -0.351701 q^{91} +7.05741 q^{93} -23.4149 q^{95} +17.8471 q^{97} -5.32243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −4.15588 −1.85857 −0.929283 0.369368i \(-0.879574\pi\)
−0.929283 + 0.369368i \(0.879574\pi\)
\(6\) 0 0
\(7\) −1.11441 −0.421208 −0.210604 0.977572i \(-0.567543\pi\)
−0.210604 + 0.977572i \(0.567543\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.32243 −1.60477 −0.802386 0.596805i \(-0.796437\pi\)
−0.802386 + 0.596805i \(0.796437\pi\)
\(12\) 0 0
\(13\) 0.315594 0.0875301 0.0437650 0.999042i \(-0.486065\pi\)
0.0437650 + 0.999042i \(0.486065\pi\)
\(14\) 0 0
\(15\) 4.15588 1.07304
\(16\) 0 0
\(17\) −1.36640 −0.331400 −0.165700 0.986176i \(-0.552988\pi\)
−0.165700 + 0.986176i \(0.552988\pi\)
\(18\) 0 0
\(19\) 5.63415 1.29256 0.646282 0.763099i \(-0.276323\pi\)
0.646282 + 0.763099i \(0.276323\pi\)
\(20\) 0 0
\(21\) 1.11441 0.243184
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 12.2713 2.45427
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −7.05741 −1.26755 −0.633774 0.773518i \(-0.718495\pi\)
−0.633774 + 0.773518i \(0.718495\pi\)
\(32\) 0 0
\(33\) 5.32243 0.926516
\(34\) 0 0
\(35\) 4.63136 0.782842
\(36\) 0 0
\(37\) −5.05783 −0.831502 −0.415751 0.909478i \(-0.636481\pi\)
−0.415751 + 0.909478i \(0.636481\pi\)
\(38\) 0 0
\(39\) −0.315594 −0.0505355
\(40\) 0 0
\(41\) 2.34509 0.366241 0.183120 0.983090i \(-0.441380\pi\)
0.183120 + 0.983090i \(0.441380\pi\)
\(42\) 0 0
\(43\) 4.48164 0.683444 0.341722 0.939801i \(-0.388990\pi\)
0.341722 + 0.939801i \(0.388990\pi\)
\(44\) 0 0
\(45\) −4.15588 −0.619522
\(46\) 0 0
\(47\) 3.25596 0.474930 0.237465 0.971396i \(-0.423683\pi\)
0.237465 + 0.971396i \(0.423683\pi\)
\(48\) 0 0
\(49\) −5.75809 −0.822584
\(50\) 0 0
\(51\) 1.36640 0.191334
\(52\) 0 0
\(53\) 7.64907 1.05068 0.525341 0.850892i \(-0.323938\pi\)
0.525341 + 0.850892i \(0.323938\pi\)
\(54\) 0 0
\(55\) 22.1194 2.98258
\(56\) 0 0
\(57\) −5.63415 −0.746262
\(58\) 0 0
\(59\) −5.11563 −0.665998 −0.332999 0.942927i \(-0.608061\pi\)
−0.332999 + 0.942927i \(0.608061\pi\)
\(60\) 0 0
\(61\) 1.44172 0.184593 0.0922965 0.995732i \(-0.470579\pi\)
0.0922965 + 0.995732i \(0.470579\pi\)
\(62\) 0 0
\(63\) −1.11441 −0.140403
\(64\) 0 0
\(65\) −1.31157 −0.162680
\(66\) 0 0
\(67\) 1.32653 0.162062 0.0810310 0.996712i \(-0.474179\pi\)
0.0810310 + 0.996712i \(0.474179\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 5.18601 0.615467 0.307733 0.951473i \(-0.400430\pi\)
0.307733 + 0.951473i \(0.400430\pi\)
\(72\) 0 0
\(73\) 2.17074 0.254066 0.127033 0.991898i \(-0.459455\pi\)
0.127033 + 0.991898i \(0.459455\pi\)
\(74\) 0 0
\(75\) −12.2713 −1.41697
\(76\) 0 0
\(77\) 5.93137 0.675942
\(78\) 0 0
\(79\) 16.7829 1.88822 0.944111 0.329629i \(-0.106924\pi\)
0.944111 + 0.329629i \(0.106924\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.5447 1.15743 0.578714 0.815531i \(-0.303555\pi\)
0.578714 + 0.815531i \(0.303555\pi\)
\(84\) 0 0
\(85\) 5.67859 0.615930
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 7.27517 0.771166 0.385583 0.922673i \(-0.374000\pi\)
0.385583 + 0.922673i \(0.374000\pi\)
\(90\) 0 0
\(91\) −0.351701 −0.0368683
\(92\) 0 0
\(93\) 7.05741 0.731819
\(94\) 0 0
\(95\) −23.4149 −2.40231
\(96\) 0 0
\(97\) 17.8471 1.81210 0.906051 0.423168i \(-0.139082\pi\)
0.906051 + 0.423168i \(0.139082\pi\)
\(98\) 0 0
\(99\) −5.32243 −0.534924
\(100\) 0 0
\(101\) 2.52471 0.251218 0.125609 0.992080i \(-0.459912\pi\)
0.125609 + 0.992080i \(0.459912\pi\)
\(102\) 0 0
\(103\) 7.55977 0.744886 0.372443 0.928055i \(-0.378520\pi\)
0.372443 + 0.928055i \(0.378520\pi\)
\(104\) 0 0
\(105\) −4.63136 −0.451974
\(106\) 0 0
\(107\) −6.96602 −0.673431 −0.336715 0.941606i \(-0.609316\pi\)
−0.336715 + 0.941606i \(0.609316\pi\)
\(108\) 0 0
\(109\) 15.0178 1.43844 0.719221 0.694781i \(-0.244499\pi\)
0.719221 + 0.694781i \(0.244499\pi\)
\(110\) 0 0
\(111\) 5.05783 0.480068
\(112\) 0 0
\(113\) −16.7693 −1.57752 −0.788759 0.614702i \(-0.789276\pi\)
−0.788759 + 0.614702i \(0.789276\pi\)
\(114\) 0 0
\(115\) −4.15588 −0.387538
\(116\) 0 0
\(117\) 0.315594 0.0291767
\(118\) 0 0
\(119\) 1.52273 0.139588
\(120\) 0 0
\(121\) 17.3282 1.57530
\(122\) 0 0
\(123\) −2.34509 −0.211449
\(124\) 0 0
\(125\) −30.2189 −2.70286
\(126\) 0 0
\(127\) 10.4769 0.929676 0.464838 0.885396i \(-0.346113\pi\)
0.464838 + 0.885396i \(0.346113\pi\)
\(128\) 0 0
\(129\) −4.48164 −0.394587
\(130\) 0 0
\(131\) 1.45472 0.127099 0.0635496 0.997979i \(-0.479758\pi\)
0.0635496 + 0.997979i \(0.479758\pi\)
\(132\) 0 0
\(133\) −6.27876 −0.544437
\(134\) 0 0
\(135\) 4.15588 0.357681
\(136\) 0 0
\(137\) −17.7948 −1.52031 −0.760155 0.649742i \(-0.774877\pi\)
−0.760155 + 0.649742i \(0.774877\pi\)
\(138\) 0 0
\(139\) 0.750874 0.0636883 0.0318442 0.999493i \(-0.489862\pi\)
0.0318442 + 0.999493i \(0.489862\pi\)
\(140\) 0 0
\(141\) −3.25596 −0.274201
\(142\) 0 0
\(143\) −1.67973 −0.140466
\(144\) 0 0
\(145\) 4.15588 0.345127
\(146\) 0 0
\(147\) 5.75809 0.474919
\(148\) 0 0
\(149\) −3.25094 −0.266327 −0.133164 0.991094i \(-0.542514\pi\)
−0.133164 + 0.991094i \(0.542514\pi\)
\(150\) 0 0
\(151\) 10.2670 0.835513 0.417756 0.908559i \(-0.362817\pi\)
0.417756 + 0.908559i \(0.362817\pi\)
\(152\) 0 0
\(153\) −1.36640 −0.110467
\(154\) 0 0
\(155\) 29.3298 2.35582
\(156\) 0 0
\(157\) −5.19176 −0.414347 −0.207174 0.978304i \(-0.566426\pi\)
−0.207174 + 0.978304i \(0.566426\pi\)
\(158\) 0 0
\(159\) −7.64907 −0.606611
\(160\) 0 0
\(161\) −1.11441 −0.0878278
\(162\) 0 0
\(163\) −21.8048 −1.70788 −0.853942 0.520368i \(-0.825795\pi\)
−0.853942 + 0.520368i \(0.825795\pi\)
\(164\) 0 0
\(165\) −22.1194 −1.72199
\(166\) 0 0
\(167\) −13.6574 −1.05684 −0.528420 0.848983i \(-0.677215\pi\)
−0.528420 + 0.848983i \(0.677215\pi\)
\(168\) 0 0
\(169\) −12.9004 −0.992338
\(170\) 0 0
\(171\) 5.63415 0.430854
\(172\) 0 0
\(173\) −16.9846 −1.29131 −0.645657 0.763627i \(-0.723417\pi\)
−0.645657 + 0.763627i \(0.723417\pi\)
\(174\) 0 0
\(175\) −13.6753 −1.03376
\(176\) 0 0
\(177\) 5.11563 0.384514
\(178\) 0 0
\(179\) −24.7924 −1.85307 −0.926537 0.376203i \(-0.877229\pi\)
−0.926537 + 0.376203i \(0.877229\pi\)
\(180\) 0 0
\(181\) 1.46156 0.108637 0.0543184 0.998524i \(-0.482701\pi\)
0.0543184 + 0.998524i \(0.482701\pi\)
\(182\) 0 0
\(183\) −1.44172 −0.106575
\(184\) 0 0
\(185\) 21.0197 1.54540
\(186\) 0 0
\(187\) 7.27256 0.531822
\(188\) 0 0
\(189\) 1.11441 0.0810614
\(190\) 0 0
\(191\) 23.7095 1.71556 0.857780 0.514018i \(-0.171843\pi\)
0.857780 + 0.514018i \(0.171843\pi\)
\(192\) 0 0
\(193\) −10.7989 −0.777325 −0.388662 0.921380i \(-0.627063\pi\)
−0.388662 + 0.921380i \(0.627063\pi\)
\(194\) 0 0
\(195\) 1.31157 0.0939236
\(196\) 0 0
\(197\) 4.91211 0.349973 0.174987 0.984571i \(-0.444012\pi\)
0.174987 + 0.984571i \(0.444012\pi\)
\(198\) 0 0
\(199\) −15.6593 −1.11006 −0.555030 0.831830i \(-0.687293\pi\)
−0.555030 + 0.831830i \(0.687293\pi\)
\(200\) 0 0
\(201\) −1.32653 −0.0935665
\(202\) 0 0
\(203\) 1.11441 0.0782163
\(204\) 0 0
\(205\) −9.74590 −0.680683
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −29.9874 −2.07427
\(210\) 0 0
\(211\) −16.1130 −1.10927 −0.554633 0.832095i \(-0.687141\pi\)
−0.554633 + 0.832095i \(0.687141\pi\)
\(212\) 0 0
\(213\) −5.18601 −0.355340
\(214\) 0 0
\(215\) −18.6252 −1.27023
\(216\) 0 0
\(217\) 7.86485 0.533901
\(218\) 0 0
\(219\) −2.17074 −0.146685
\(220\) 0 0
\(221\) −0.431227 −0.0290075
\(222\) 0 0
\(223\) 15.7109 1.05208 0.526038 0.850461i \(-0.323677\pi\)
0.526038 + 0.850461i \(0.323677\pi\)
\(224\) 0 0
\(225\) 12.2713 0.818090
\(226\) 0 0
\(227\) 6.06788 0.402739 0.201370 0.979515i \(-0.435461\pi\)
0.201370 + 0.979515i \(0.435461\pi\)
\(228\) 0 0
\(229\) −17.2636 −1.14081 −0.570405 0.821364i \(-0.693214\pi\)
−0.570405 + 0.821364i \(0.693214\pi\)
\(230\) 0 0
\(231\) −5.93137 −0.390255
\(232\) 0 0
\(233\) −13.0603 −0.855608 −0.427804 0.903871i \(-0.640713\pi\)
−0.427804 + 0.903871i \(0.640713\pi\)
\(234\) 0 0
\(235\) −13.5314 −0.882689
\(236\) 0 0
\(237\) −16.7829 −1.09017
\(238\) 0 0
\(239\) 5.18526 0.335406 0.167703 0.985838i \(-0.446365\pi\)
0.167703 + 0.985838i \(0.446365\pi\)
\(240\) 0 0
\(241\) 1.56824 0.101019 0.0505097 0.998724i \(-0.483915\pi\)
0.0505097 + 0.998724i \(0.483915\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 23.9299 1.52883
\(246\) 0 0
\(247\) 1.77811 0.113138
\(248\) 0 0
\(249\) −10.5447 −0.668241
\(250\) 0 0
\(251\) −10.7456 −0.678257 −0.339128 0.940740i \(-0.610132\pi\)
−0.339128 + 0.940740i \(0.610132\pi\)
\(252\) 0 0
\(253\) −5.32243 −0.334618
\(254\) 0 0
\(255\) −5.67859 −0.355607
\(256\) 0 0
\(257\) 15.6578 0.976705 0.488352 0.872647i \(-0.337598\pi\)
0.488352 + 0.872647i \(0.337598\pi\)
\(258\) 0 0
\(259\) 5.63650 0.350235
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −26.4212 −1.62920 −0.814602 0.580020i \(-0.803045\pi\)
−0.814602 + 0.580020i \(0.803045\pi\)
\(264\) 0 0
\(265\) −31.7886 −1.95276
\(266\) 0 0
\(267\) −7.27517 −0.445233
\(268\) 0 0
\(269\) −4.75677 −0.290025 −0.145013 0.989430i \(-0.546322\pi\)
−0.145013 + 0.989430i \(0.546322\pi\)
\(270\) 0 0
\(271\) 28.0807 1.70578 0.852889 0.522092i \(-0.174848\pi\)
0.852889 + 0.522092i \(0.174848\pi\)
\(272\) 0 0
\(273\) 0.351701 0.0212859
\(274\) 0 0
\(275\) −65.3134 −3.93855
\(276\) 0 0
\(277\) −4.76867 −0.286522 −0.143261 0.989685i \(-0.545759\pi\)
−0.143261 + 0.989685i \(0.545759\pi\)
\(278\) 0 0
\(279\) −7.05741 −0.422516
\(280\) 0 0
\(281\) 9.24418 0.551462 0.275731 0.961235i \(-0.411080\pi\)
0.275731 + 0.961235i \(0.411080\pi\)
\(282\) 0 0
\(283\) 1.38881 0.0825564 0.0412782 0.999148i \(-0.486857\pi\)
0.0412782 + 0.999148i \(0.486857\pi\)
\(284\) 0 0
\(285\) 23.4149 1.38698
\(286\) 0 0
\(287\) −2.61339 −0.154263
\(288\) 0 0
\(289\) −15.1330 −0.890174
\(290\) 0 0
\(291\) −17.8471 −1.04622
\(292\) 0 0
\(293\) 16.1908 0.945874 0.472937 0.881096i \(-0.343194\pi\)
0.472937 + 0.881096i \(0.343194\pi\)
\(294\) 0 0
\(295\) 21.2600 1.23780
\(296\) 0 0
\(297\) 5.32243 0.308839
\(298\) 0 0
\(299\) 0.315594 0.0182513
\(300\) 0 0
\(301\) −4.99439 −0.287872
\(302\) 0 0
\(303\) −2.52471 −0.145041
\(304\) 0 0
\(305\) −5.99161 −0.343078
\(306\) 0 0
\(307\) −27.3469 −1.56077 −0.780386 0.625298i \(-0.784977\pi\)
−0.780386 + 0.625298i \(0.784977\pi\)
\(308\) 0 0
\(309\) −7.55977 −0.430060
\(310\) 0 0
\(311\) 3.23554 0.183471 0.0917353 0.995783i \(-0.470759\pi\)
0.0917353 + 0.995783i \(0.470759\pi\)
\(312\) 0 0
\(313\) 15.5299 0.877804 0.438902 0.898535i \(-0.355368\pi\)
0.438902 + 0.898535i \(0.355368\pi\)
\(314\) 0 0
\(315\) 4.63136 0.260947
\(316\) 0 0
\(317\) 19.2374 1.08048 0.540241 0.841510i \(-0.318333\pi\)
0.540241 + 0.841510i \(0.318333\pi\)
\(318\) 0 0
\(319\) 5.32243 0.297999
\(320\) 0 0
\(321\) 6.96602 0.388805
\(322\) 0 0
\(323\) −7.69850 −0.428356
\(324\) 0 0
\(325\) 3.87277 0.214822
\(326\) 0 0
\(327\) −15.0178 −0.830485
\(328\) 0 0
\(329\) −3.62847 −0.200044
\(330\) 0 0
\(331\) 11.7753 0.647228 0.323614 0.946189i \(-0.395102\pi\)
0.323614 + 0.946189i \(0.395102\pi\)
\(332\) 0 0
\(333\) −5.05783 −0.277167
\(334\) 0 0
\(335\) −5.51292 −0.301203
\(336\) 0 0
\(337\) −8.90018 −0.484824 −0.242412 0.970173i \(-0.577939\pi\)
−0.242412 + 0.970173i \(0.577939\pi\)
\(338\) 0 0
\(339\) 16.7693 0.910781
\(340\) 0 0
\(341\) 37.5626 2.03413
\(342\) 0 0
\(343\) 14.2177 0.767686
\(344\) 0 0
\(345\) 4.15588 0.223745
\(346\) 0 0
\(347\) −21.1007 −1.13274 −0.566371 0.824150i \(-0.691653\pi\)
−0.566371 + 0.824150i \(0.691653\pi\)
\(348\) 0 0
\(349\) 0.752308 0.0402701 0.0201351 0.999797i \(-0.493590\pi\)
0.0201351 + 0.999797i \(0.493590\pi\)
\(350\) 0 0
\(351\) −0.315594 −0.0168452
\(352\) 0 0
\(353\) −1.23784 −0.0658834 −0.0329417 0.999457i \(-0.510488\pi\)
−0.0329417 + 0.999457i \(0.510488\pi\)
\(354\) 0 0
\(355\) −21.5525 −1.14389
\(356\) 0 0
\(357\) −1.52273 −0.0805913
\(358\) 0 0
\(359\) −10.5446 −0.556525 −0.278263 0.960505i \(-0.589759\pi\)
−0.278263 + 0.960505i \(0.589759\pi\)
\(360\) 0 0
\(361\) 12.7437 0.670719
\(362\) 0 0
\(363\) −17.3282 −0.909497
\(364\) 0 0
\(365\) −9.02135 −0.472199
\(366\) 0 0
\(367\) −16.2585 −0.848685 −0.424343 0.905502i \(-0.639495\pi\)
−0.424343 + 0.905502i \(0.639495\pi\)
\(368\) 0 0
\(369\) 2.34509 0.122080
\(370\) 0 0
\(371\) −8.52421 −0.442555
\(372\) 0 0
\(373\) 11.3203 0.586144 0.293072 0.956090i \(-0.405322\pi\)
0.293072 + 0.956090i \(0.405322\pi\)
\(374\) 0 0
\(375\) 30.2189 1.56050
\(376\) 0 0
\(377\) −0.315594 −0.0162539
\(378\) 0 0
\(379\) 32.7924 1.68443 0.842217 0.539138i \(-0.181250\pi\)
0.842217 + 0.539138i \(0.181250\pi\)
\(380\) 0 0
\(381\) −10.4769 −0.536749
\(382\) 0 0
\(383\) 25.3286 1.29423 0.647114 0.762393i \(-0.275976\pi\)
0.647114 + 0.762393i \(0.275976\pi\)
\(384\) 0 0
\(385\) −24.6501 −1.25628
\(386\) 0 0
\(387\) 4.48164 0.227815
\(388\) 0 0
\(389\) 3.40417 0.172598 0.0862991 0.996269i \(-0.472496\pi\)
0.0862991 + 0.996269i \(0.472496\pi\)
\(390\) 0 0
\(391\) −1.36640 −0.0691017
\(392\) 0 0
\(393\) −1.45472 −0.0733807
\(394\) 0 0
\(395\) −69.7476 −3.50939
\(396\) 0 0
\(397\) 14.9864 0.752145 0.376073 0.926590i \(-0.377274\pi\)
0.376073 + 0.926590i \(0.377274\pi\)
\(398\) 0 0
\(399\) 6.27876 0.314331
\(400\) 0 0
\(401\) 30.3023 1.51323 0.756613 0.653863i \(-0.226853\pi\)
0.756613 + 0.653863i \(0.226853\pi\)
\(402\) 0 0
\(403\) −2.22728 −0.110949
\(404\) 0 0
\(405\) −4.15588 −0.206507
\(406\) 0 0
\(407\) 26.9199 1.33437
\(408\) 0 0
\(409\) −16.9216 −0.836717 −0.418359 0.908282i \(-0.637394\pi\)
−0.418359 + 0.908282i \(0.637394\pi\)
\(410\) 0 0
\(411\) 17.7948 0.877752
\(412\) 0 0
\(413\) 5.70091 0.280524
\(414\) 0 0
\(415\) −43.8224 −2.15116
\(416\) 0 0
\(417\) −0.750874 −0.0367705
\(418\) 0 0
\(419\) −18.5534 −0.906394 −0.453197 0.891410i \(-0.649717\pi\)
−0.453197 + 0.891410i \(0.649717\pi\)
\(420\) 0 0
\(421\) −30.7341 −1.49789 −0.748944 0.662634i \(-0.769439\pi\)
−0.748944 + 0.662634i \(0.769439\pi\)
\(422\) 0 0
\(423\) 3.25596 0.158310
\(424\) 0 0
\(425\) −16.7676 −0.813346
\(426\) 0 0
\(427\) −1.60666 −0.0777519
\(428\) 0 0
\(429\) 1.67973 0.0810980
\(430\) 0 0
\(431\) 35.0770 1.68960 0.844799 0.535083i \(-0.179720\pi\)
0.844799 + 0.535083i \(0.179720\pi\)
\(432\) 0 0
\(433\) −27.1107 −1.30286 −0.651428 0.758711i \(-0.725830\pi\)
−0.651428 + 0.758711i \(0.725830\pi\)
\(434\) 0 0
\(435\) −4.15588 −0.199259
\(436\) 0 0
\(437\) 5.63415 0.269518
\(438\) 0 0
\(439\) −17.4657 −0.833594 −0.416797 0.909000i \(-0.636847\pi\)
−0.416797 + 0.909000i \(0.636847\pi\)
\(440\) 0 0
\(441\) −5.75809 −0.274195
\(442\) 0 0
\(443\) −7.96682 −0.378515 −0.189258 0.981927i \(-0.560608\pi\)
−0.189258 + 0.981927i \(0.560608\pi\)
\(444\) 0 0
\(445\) −30.2347 −1.43326
\(446\) 0 0
\(447\) 3.25094 0.153764
\(448\) 0 0
\(449\) 3.54966 0.167519 0.0837594 0.996486i \(-0.473307\pi\)
0.0837594 + 0.996486i \(0.473307\pi\)
\(450\) 0 0
\(451\) −12.4816 −0.587733
\(452\) 0 0
\(453\) −10.2670 −0.482383
\(454\) 0 0
\(455\) 1.46163 0.0685222
\(456\) 0 0
\(457\) 0.905773 0.0423703 0.0211851 0.999776i \(-0.493256\pi\)
0.0211851 + 0.999776i \(0.493256\pi\)
\(458\) 0 0
\(459\) 1.36640 0.0637780
\(460\) 0 0
\(461\) −7.28459 −0.339277 −0.169639 0.985506i \(-0.554260\pi\)
−0.169639 + 0.985506i \(0.554260\pi\)
\(462\) 0 0
\(463\) −5.99214 −0.278478 −0.139239 0.990259i \(-0.544466\pi\)
−0.139239 + 0.990259i \(0.544466\pi\)
\(464\) 0 0
\(465\) −29.3298 −1.36014
\(466\) 0 0
\(467\) −28.8181 −1.33354 −0.666772 0.745262i \(-0.732324\pi\)
−0.666772 + 0.745262i \(0.732324\pi\)
\(468\) 0 0
\(469\) −1.47830 −0.0682617
\(470\) 0 0
\(471\) 5.19176 0.239223
\(472\) 0 0
\(473\) −23.8532 −1.09677
\(474\) 0 0
\(475\) 69.1386 3.17230
\(476\) 0 0
\(477\) 7.64907 0.350227
\(478\) 0 0
\(479\) −1.47774 −0.0675197 −0.0337598 0.999430i \(-0.510748\pi\)
−0.0337598 + 0.999430i \(0.510748\pi\)
\(480\) 0 0
\(481\) −1.59622 −0.0727814
\(482\) 0 0
\(483\) 1.11441 0.0507074
\(484\) 0 0
\(485\) −74.1706 −3.36791
\(486\) 0 0
\(487\) 31.2807 1.41746 0.708732 0.705478i \(-0.249268\pi\)
0.708732 + 0.705478i \(0.249268\pi\)
\(488\) 0 0
\(489\) 21.8048 0.986047
\(490\) 0 0
\(491\) 16.5707 0.747827 0.373913 0.927464i \(-0.378016\pi\)
0.373913 + 0.927464i \(0.378016\pi\)
\(492\) 0 0
\(493\) 1.36640 0.0615395
\(494\) 0 0
\(495\) 22.1194 0.994192
\(496\) 0 0
\(497\) −5.77935 −0.259239
\(498\) 0 0
\(499\) −36.5634 −1.63680 −0.818401 0.574648i \(-0.805139\pi\)
−0.818401 + 0.574648i \(0.805139\pi\)
\(500\) 0 0
\(501\) 13.6574 0.610167
\(502\) 0 0
\(503\) −11.0931 −0.494618 −0.247309 0.968937i \(-0.579546\pi\)
−0.247309 + 0.968937i \(0.579546\pi\)
\(504\) 0 0
\(505\) −10.4924 −0.466906
\(506\) 0 0
\(507\) 12.9004 0.572927
\(508\) 0 0
\(509\) 17.2458 0.764406 0.382203 0.924078i \(-0.375166\pi\)
0.382203 + 0.924078i \(0.375166\pi\)
\(510\) 0 0
\(511\) −2.41910 −0.107015
\(512\) 0 0
\(513\) −5.63415 −0.248754
\(514\) 0 0
\(515\) −31.4175 −1.38442
\(516\) 0 0
\(517\) −17.3296 −0.762155
\(518\) 0 0
\(519\) 16.9846 0.745541
\(520\) 0 0
\(521\) −34.7972 −1.52449 −0.762247 0.647286i \(-0.775904\pi\)
−0.762247 + 0.647286i \(0.775904\pi\)
\(522\) 0 0
\(523\) 35.5395 1.55404 0.777018 0.629479i \(-0.216732\pi\)
0.777018 + 0.629479i \(0.216732\pi\)
\(524\) 0 0
\(525\) 13.6753 0.596840
\(526\) 0 0
\(527\) 9.64324 0.420066
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.11563 −0.221999
\(532\) 0 0
\(533\) 0.740095 0.0320571
\(534\) 0 0
\(535\) 28.9500 1.25162
\(536\) 0 0
\(537\) 24.7924 1.06987
\(538\) 0 0
\(539\) 30.6470 1.32006
\(540\) 0 0
\(541\) −37.3517 −1.60587 −0.802936 0.596065i \(-0.796730\pi\)
−0.802936 + 0.596065i \(0.796730\pi\)
\(542\) 0 0
\(543\) −1.46156 −0.0627215
\(544\) 0 0
\(545\) −62.4121 −2.67344
\(546\) 0 0
\(547\) −37.1011 −1.58633 −0.793165 0.609007i \(-0.791568\pi\)
−0.793165 + 0.609007i \(0.791568\pi\)
\(548\) 0 0
\(549\) 1.44172 0.0615310
\(550\) 0 0
\(551\) −5.63415 −0.240023
\(552\) 0 0
\(553\) −18.7030 −0.795333
\(554\) 0 0
\(555\) −21.0197 −0.892238
\(556\) 0 0
\(557\) −37.2436 −1.57806 −0.789030 0.614355i \(-0.789416\pi\)
−0.789030 + 0.614355i \(0.789416\pi\)
\(558\) 0 0
\(559\) 1.41438 0.0598219
\(560\) 0 0
\(561\) −7.27256 −0.307048
\(562\) 0 0
\(563\) 21.7473 0.916540 0.458270 0.888813i \(-0.348469\pi\)
0.458270 + 0.888813i \(0.348469\pi\)
\(564\) 0 0
\(565\) 69.6910 2.93192
\(566\) 0 0
\(567\) −1.11441 −0.0468008
\(568\) 0 0
\(569\) 2.87553 0.120548 0.0602741 0.998182i \(-0.480803\pi\)
0.0602741 + 0.998182i \(0.480803\pi\)
\(570\) 0 0
\(571\) 11.8603 0.496339 0.248170 0.968717i \(-0.420171\pi\)
0.248170 + 0.968717i \(0.420171\pi\)
\(572\) 0 0
\(573\) −23.7095 −0.990479
\(574\) 0 0
\(575\) 12.2713 0.511751
\(576\) 0 0
\(577\) 13.2791 0.552818 0.276409 0.961040i \(-0.410856\pi\)
0.276409 + 0.961040i \(0.410856\pi\)
\(578\) 0 0
\(579\) 10.7989 0.448789
\(580\) 0 0
\(581\) −11.7511 −0.487517
\(582\) 0 0
\(583\) −40.7116 −1.68610
\(584\) 0 0
\(585\) −1.31157 −0.0542268
\(586\) 0 0
\(587\) −13.3679 −0.551753 −0.275877 0.961193i \(-0.588968\pi\)
−0.275877 + 0.961193i \(0.588968\pi\)
\(588\) 0 0
\(589\) −39.7625 −1.63839
\(590\) 0 0
\(591\) −4.91211 −0.202057
\(592\) 0 0
\(593\) 4.50293 0.184913 0.0924566 0.995717i \(-0.470528\pi\)
0.0924566 + 0.995717i \(0.470528\pi\)
\(594\) 0 0
\(595\) −6.32828 −0.259434
\(596\) 0 0
\(597\) 15.6593 0.640893
\(598\) 0 0
\(599\) 7.84271 0.320445 0.160222 0.987081i \(-0.448779\pi\)
0.160222 + 0.987081i \(0.448779\pi\)
\(600\) 0 0
\(601\) −8.22447 −0.335483 −0.167742 0.985831i \(-0.553647\pi\)
−0.167742 + 0.985831i \(0.553647\pi\)
\(602\) 0 0
\(603\) 1.32653 0.0540207
\(604\) 0 0
\(605\) −72.0141 −2.92779
\(606\) 0 0
\(607\) 38.0418 1.54407 0.772035 0.635580i \(-0.219239\pi\)
0.772035 + 0.635580i \(0.219239\pi\)
\(608\) 0 0
\(609\) −1.11441 −0.0451582
\(610\) 0 0
\(611\) 1.02756 0.0415707
\(612\) 0 0
\(613\) −9.57247 −0.386628 −0.193314 0.981137i \(-0.561924\pi\)
−0.193314 + 0.981137i \(0.561924\pi\)
\(614\) 0 0
\(615\) 9.74590 0.392993
\(616\) 0 0
\(617\) 21.5698 0.868366 0.434183 0.900825i \(-0.357037\pi\)
0.434183 + 0.900825i \(0.357037\pi\)
\(618\) 0 0
\(619\) 30.3002 1.21787 0.608934 0.793221i \(-0.291598\pi\)
0.608934 + 0.793221i \(0.291598\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −8.10752 −0.324821
\(624\) 0 0
\(625\) 64.2293 2.56917
\(626\) 0 0
\(627\) 29.9874 1.19758
\(628\) 0 0
\(629\) 6.91101 0.275560
\(630\) 0 0
\(631\) −1.11720 −0.0444749 −0.0222374 0.999753i \(-0.507079\pi\)
−0.0222374 + 0.999753i \(0.507079\pi\)
\(632\) 0 0
\(633\) 16.1130 0.640435
\(634\) 0 0
\(635\) −43.5408 −1.72786
\(636\) 0 0
\(637\) −1.81722 −0.0720009
\(638\) 0 0
\(639\) 5.18601 0.205156
\(640\) 0 0
\(641\) −7.79220 −0.307774 −0.153887 0.988088i \(-0.549179\pi\)
−0.153887 + 0.988088i \(0.549179\pi\)
\(642\) 0 0
\(643\) 35.8453 1.41360 0.706800 0.707413i \(-0.250138\pi\)
0.706800 + 0.707413i \(0.250138\pi\)
\(644\) 0 0
\(645\) 18.6252 0.733365
\(646\) 0 0
\(647\) 13.9579 0.548742 0.274371 0.961624i \(-0.411530\pi\)
0.274371 + 0.961624i \(0.411530\pi\)
\(648\) 0 0
\(649\) 27.2276 1.06878
\(650\) 0 0
\(651\) −7.86485 −0.308248
\(652\) 0 0
\(653\) −9.11600 −0.356737 −0.178368 0.983964i \(-0.557082\pi\)
−0.178368 + 0.983964i \(0.557082\pi\)
\(654\) 0 0
\(655\) −6.04563 −0.236222
\(656\) 0 0
\(657\) 2.17074 0.0846887
\(658\) 0 0
\(659\) −33.7781 −1.31581 −0.657904 0.753101i \(-0.728557\pi\)
−0.657904 + 0.753101i \(0.728557\pi\)
\(660\) 0 0
\(661\) −28.6418 −1.11404 −0.557018 0.830500i \(-0.688055\pi\)
−0.557018 + 0.830500i \(0.688055\pi\)
\(662\) 0 0
\(663\) 0.431227 0.0167475
\(664\) 0 0
\(665\) 26.0938 1.01187
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −15.7109 −0.607417
\(670\) 0 0
\(671\) −7.67344 −0.296230
\(672\) 0 0
\(673\) 20.2837 0.781879 0.390939 0.920416i \(-0.372150\pi\)
0.390939 + 0.920416i \(0.372150\pi\)
\(674\) 0 0
\(675\) −12.2713 −0.472324
\(676\) 0 0
\(677\) 2.54900 0.0979659 0.0489830 0.998800i \(-0.484402\pi\)
0.0489830 + 0.998800i \(0.484402\pi\)
\(678\) 0 0
\(679\) −19.8890 −0.763271
\(680\) 0 0
\(681\) −6.06788 −0.232522
\(682\) 0 0
\(683\) 9.01416 0.344917 0.172459 0.985017i \(-0.444829\pi\)
0.172459 + 0.985017i \(0.444829\pi\)
\(684\) 0 0
\(685\) 73.9530 2.82560
\(686\) 0 0
\(687\) 17.2636 0.658646
\(688\) 0 0
\(689\) 2.41400 0.0919662
\(690\) 0 0
\(691\) −4.03353 −0.153443 −0.0767214 0.997053i \(-0.524445\pi\)
−0.0767214 + 0.997053i \(0.524445\pi\)
\(692\) 0 0
\(693\) 5.93137 0.225314
\(694\) 0 0
\(695\) −3.12054 −0.118369
\(696\) 0 0
\(697\) −3.20432 −0.121372
\(698\) 0 0
\(699\) 13.0603 0.493986
\(700\) 0 0
\(701\) −25.9923 −0.981717 −0.490859 0.871239i \(-0.663317\pi\)
−0.490859 + 0.871239i \(0.663317\pi\)
\(702\) 0 0
\(703\) −28.4966 −1.07477
\(704\) 0 0
\(705\) 13.5314 0.509621
\(706\) 0 0
\(707\) −2.81356 −0.105815
\(708\) 0 0
\(709\) −18.6909 −0.701951 −0.350975 0.936385i \(-0.614150\pi\)
−0.350975 + 0.936385i \(0.614150\pi\)
\(710\) 0 0
\(711\) 16.7829 0.629407
\(712\) 0 0
\(713\) −7.05741 −0.264302
\(714\) 0 0
\(715\) 6.98075 0.261065
\(716\) 0 0
\(717\) −5.18526 −0.193647
\(718\) 0 0
\(719\) −44.0786 −1.64386 −0.821928 0.569591i \(-0.807102\pi\)
−0.821928 + 0.569591i \(0.807102\pi\)
\(720\) 0 0
\(721\) −8.42468 −0.313752
\(722\) 0 0
\(723\) −1.56824 −0.0583236
\(724\) 0 0
\(725\) −12.2713 −0.455746
\(726\) 0 0
\(727\) −27.0431 −1.00297 −0.501487 0.865165i \(-0.667213\pi\)
−0.501487 + 0.865165i \(0.667213\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.12371 −0.226494
\(732\) 0 0
\(733\) −11.0106 −0.406685 −0.203343 0.979108i \(-0.565181\pi\)
−0.203343 + 0.979108i \(0.565181\pi\)
\(734\) 0 0
\(735\) −23.9299 −0.882669
\(736\) 0 0
\(737\) −7.06039 −0.260073
\(738\) 0 0
\(739\) −4.77272 −0.175567 −0.0877836 0.996140i \(-0.527978\pi\)
−0.0877836 + 0.996140i \(0.527978\pi\)
\(740\) 0 0
\(741\) −1.77811 −0.0653203
\(742\) 0 0
\(743\) 25.1948 0.924307 0.462153 0.886800i \(-0.347077\pi\)
0.462153 + 0.886800i \(0.347077\pi\)
\(744\) 0 0
\(745\) 13.5105 0.494987
\(746\) 0 0
\(747\) 10.5447 0.385809
\(748\) 0 0
\(749\) 7.76301 0.283654
\(750\) 0 0
\(751\) 20.4836 0.747457 0.373729 0.927538i \(-0.378079\pi\)
0.373729 + 0.927538i \(0.378079\pi\)
\(752\) 0 0
\(753\) 10.7456 0.391592
\(754\) 0 0
\(755\) −42.6682 −1.55286
\(756\) 0 0
\(757\) −28.9761 −1.05315 −0.526577 0.850127i \(-0.676525\pi\)
−0.526577 + 0.850127i \(0.676525\pi\)
\(758\) 0 0
\(759\) 5.32243 0.193192
\(760\) 0 0
\(761\) −34.4284 −1.24803 −0.624015 0.781413i \(-0.714499\pi\)
−0.624015 + 0.781413i \(0.714499\pi\)
\(762\) 0 0
\(763\) −16.7360 −0.605883
\(764\) 0 0
\(765\) 5.67859 0.205310
\(766\) 0 0
\(767\) −1.61446 −0.0582949
\(768\) 0 0
\(769\) −14.7722 −0.532699 −0.266350 0.963876i \(-0.585818\pi\)
−0.266350 + 0.963876i \(0.585818\pi\)
\(770\) 0 0
\(771\) −15.6578 −0.563901
\(772\) 0 0
\(773\) 32.7055 1.17634 0.588168 0.808739i \(-0.299849\pi\)
0.588168 + 0.808739i \(0.299849\pi\)
\(774\) 0 0
\(775\) −86.6040 −3.11091
\(776\) 0 0
\(777\) −5.63650 −0.202208
\(778\) 0 0
\(779\) 13.2126 0.473389
\(780\) 0 0
\(781\) −27.6022 −0.987684
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 21.5763 0.770092
\(786\) 0 0
\(787\) 26.9355 0.960147 0.480074 0.877228i \(-0.340610\pi\)
0.480074 + 0.877228i \(0.340610\pi\)
\(788\) 0 0
\(789\) 26.4212 0.940621
\(790\) 0 0
\(791\) 18.6878 0.664463
\(792\) 0 0
\(793\) 0.454998 0.0161574
\(794\) 0 0
\(795\) 31.7886 1.12743
\(796\) 0 0
\(797\) 24.6641 0.873648 0.436824 0.899547i \(-0.356103\pi\)
0.436824 + 0.899547i \(0.356103\pi\)
\(798\) 0 0
\(799\) −4.44894 −0.157392
\(800\) 0 0
\(801\) 7.27517 0.257055
\(802\) 0 0
\(803\) −11.5536 −0.407718
\(804\) 0 0
\(805\) 4.63136 0.163234
\(806\) 0 0
\(807\) 4.75677 0.167446
\(808\) 0 0
\(809\) −1.47151 −0.0517354 −0.0258677 0.999665i \(-0.508235\pi\)
−0.0258677 + 0.999665i \(0.508235\pi\)
\(810\) 0 0
\(811\) −1.29574 −0.0454997 −0.0227499 0.999741i \(-0.507242\pi\)
−0.0227499 + 0.999741i \(0.507242\pi\)
\(812\) 0 0
\(813\) −28.0807 −0.984832
\(814\) 0 0
\(815\) 90.6182 3.17422
\(816\) 0 0
\(817\) 25.2502 0.883394
\(818\) 0 0
\(819\) −0.351701 −0.0122894
\(820\) 0 0
\(821\) 21.4366 0.748142 0.374071 0.927400i \(-0.377962\pi\)
0.374071 + 0.927400i \(0.377962\pi\)
\(822\) 0 0
\(823\) 19.9990 0.697120 0.348560 0.937286i \(-0.386671\pi\)
0.348560 + 0.937286i \(0.386671\pi\)
\(824\) 0 0
\(825\) 65.3134 2.27392
\(826\) 0 0
\(827\) 22.0401 0.766409 0.383205 0.923663i \(-0.374820\pi\)
0.383205 + 0.923663i \(0.374820\pi\)
\(828\) 0 0
\(829\) −38.6535 −1.34249 −0.671245 0.741235i \(-0.734240\pi\)
−0.671245 + 0.741235i \(0.734240\pi\)
\(830\) 0 0
\(831\) 4.76867 0.165423
\(832\) 0 0
\(833\) 7.86784 0.272605
\(834\) 0 0
\(835\) 56.7585 1.96421
\(836\) 0 0
\(837\) 7.05741 0.243940
\(838\) 0 0
\(839\) −4.43195 −0.153008 −0.0765039 0.997069i \(-0.524376\pi\)
−0.0765039 + 0.997069i \(0.524376\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −9.24418 −0.318387
\(844\) 0 0
\(845\) 53.6125 1.84433
\(846\) 0 0
\(847\) −19.3108 −0.663526
\(848\) 0 0
\(849\) −1.38881 −0.0476640
\(850\) 0 0
\(851\) −5.05783 −0.173380
\(852\) 0 0
\(853\) −16.3932 −0.561291 −0.280646 0.959811i \(-0.590549\pi\)
−0.280646 + 0.959811i \(0.590549\pi\)
\(854\) 0 0
\(855\) −23.4149 −0.800772
\(856\) 0 0
\(857\) 4.15179 0.141822 0.0709112 0.997483i \(-0.477409\pi\)
0.0709112 + 0.997483i \(0.477409\pi\)
\(858\) 0 0
\(859\) 30.0383 1.02489 0.512447 0.858719i \(-0.328739\pi\)
0.512447 + 0.858719i \(0.328739\pi\)
\(860\) 0 0
\(861\) 2.61339 0.0890640
\(862\) 0 0
\(863\) 24.1245 0.821208 0.410604 0.911814i \(-0.365318\pi\)
0.410604 + 0.911814i \(0.365318\pi\)
\(864\) 0 0
\(865\) 70.5860 2.39999
\(866\) 0 0
\(867\) 15.1330 0.513942
\(868\) 0 0
\(869\) −89.3257 −3.03017
\(870\) 0 0
\(871\) 0.418647 0.0141853
\(872\) 0 0
\(873\) 17.8471 0.604034
\(874\) 0 0
\(875\) 33.6762 1.13846
\(876\) 0 0
\(877\) 43.4529 1.46730 0.733650 0.679528i \(-0.237815\pi\)
0.733650 + 0.679528i \(0.237815\pi\)
\(878\) 0 0
\(879\) −16.1908 −0.546100
\(880\) 0 0
\(881\) 10.3229 0.347786 0.173893 0.984765i \(-0.444365\pi\)
0.173893 + 0.984765i \(0.444365\pi\)
\(882\) 0 0
\(883\) 30.1615 1.01502 0.507508 0.861647i \(-0.330567\pi\)
0.507508 + 0.861647i \(0.330567\pi\)
\(884\) 0 0
\(885\) −21.2600 −0.714646
\(886\) 0 0
\(887\) −35.0652 −1.17737 −0.588687 0.808361i \(-0.700355\pi\)
−0.588687 + 0.808361i \(0.700355\pi\)
\(888\) 0 0
\(889\) −11.6756 −0.391587
\(890\) 0 0
\(891\) −5.32243 −0.178308
\(892\) 0 0
\(893\) 18.3446 0.613877
\(894\) 0 0
\(895\) 103.034 3.44406
\(896\) 0 0
\(897\) −0.315594 −0.0105374
\(898\) 0 0
\(899\) 7.05741 0.235378
\(900\) 0 0
\(901\) −10.4517 −0.348196
\(902\) 0 0
\(903\) 4.99439 0.166203
\(904\) 0 0
\(905\) −6.07406 −0.201909
\(906\) 0 0
\(907\) 18.5729 0.616704 0.308352 0.951272i \(-0.400223\pi\)
0.308352 + 0.951272i \(0.400223\pi\)
\(908\) 0 0
\(909\) 2.52471 0.0837394
\(910\) 0 0
\(911\) 25.8275 0.855704 0.427852 0.903849i \(-0.359270\pi\)
0.427852 + 0.903849i \(0.359270\pi\)
\(912\) 0 0
\(913\) −56.1232 −1.85741
\(914\) 0 0
\(915\) 5.99161 0.198076
\(916\) 0 0
\(917\) −1.62115 −0.0535351
\(918\) 0 0
\(919\) −25.4838 −0.840632 −0.420316 0.907378i \(-0.638081\pi\)
−0.420316 + 0.907378i \(0.638081\pi\)
\(920\) 0 0
\(921\) 27.3469 0.901112
\(922\) 0 0
\(923\) 1.63668 0.0538718
\(924\) 0 0
\(925\) −62.0664 −2.04073
\(926\) 0 0
\(927\) 7.55977 0.248295
\(928\) 0 0
\(929\) 9.01630 0.295815 0.147908 0.989001i \(-0.452746\pi\)
0.147908 + 0.989001i \(0.452746\pi\)
\(930\) 0 0
\(931\) −32.4419 −1.06324
\(932\) 0 0
\(933\) −3.23554 −0.105927
\(934\) 0 0
\(935\) −30.2239 −0.988427
\(936\) 0 0
\(937\) 30.2582 0.988491 0.494245 0.869322i \(-0.335444\pi\)
0.494245 + 0.869322i \(0.335444\pi\)
\(938\) 0 0
\(939\) −15.5299 −0.506800
\(940\) 0 0
\(941\) −22.4808 −0.732852 −0.366426 0.930447i \(-0.619419\pi\)
−0.366426 + 0.930447i \(0.619419\pi\)
\(942\) 0 0
\(943\) 2.34509 0.0763665
\(944\) 0 0
\(945\) −4.63136 −0.150658
\(946\) 0 0
\(947\) −22.8549 −0.742684 −0.371342 0.928496i \(-0.621102\pi\)
−0.371342 + 0.928496i \(0.621102\pi\)
\(948\) 0 0
\(949\) 0.685073 0.0222384
\(950\) 0 0
\(951\) −19.2374 −0.623817
\(952\) 0 0
\(953\) −27.3766 −0.886814 −0.443407 0.896320i \(-0.646230\pi\)
−0.443407 + 0.896320i \(0.646230\pi\)
\(954\) 0 0
\(955\) −98.5339 −3.18848
\(956\) 0 0
\(957\) −5.32243 −0.172050
\(958\) 0 0
\(959\) 19.8307 0.640366
\(960\) 0 0
\(961\) 18.8071 0.606679
\(962\) 0 0
\(963\) −6.96602 −0.224477
\(964\) 0 0
\(965\) 44.8791 1.44471
\(966\) 0 0
\(967\) 51.4022 1.65298 0.826492 0.562949i \(-0.190333\pi\)
0.826492 + 0.562949i \(0.190333\pi\)
\(968\) 0 0
\(969\) 7.69850 0.247311
\(970\) 0 0
\(971\) 16.1002 0.516679 0.258340 0.966054i \(-0.416825\pi\)
0.258340 + 0.966054i \(0.416825\pi\)
\(972\) 0 0
\(973\) −0.836782 −0.0268260
\(974\) 0 0
\(975\) −3.87277 −0.124028
\(976\) 0 0
\(977\) 23.1830 0.741689 0.370845 0.928695i \(-0.379068\pi\)
0.370845 + 0.928695i \(0.379068\pi\)
\(978\) 0 0
\(979\) −38.7216 −1.23755
\(980\) 0 0
\(981\) 15.0178 0.479481
\(982\) 0 0
\(983\) 20.7740 0.662588 0.331294 0.943528i \(-0.392515\pi\)
0.331294 + 0.943528i \(0.392515\pi\)
\(984\) 0 0
\(985\) −20.4142 −0.650449
\(986\) 0 0
\(987\) 3.62847 0.115496
\(988\) 0 0
\(989\) 4.48164 0.142508
\(990\) 0 0
\(991\) −53.1822 −1.68939 −0.844694 0.535250i \(-0.820217\pi\)
−0.844694 + 0.535250i \(0.820217\pi\)
\(992\) 0 0
\(993\) −11.7753 −0.373677
\(994\) 0 0
\(995\) 65.0783 2.06312
\(996\) 0 0
\(997\) 39.0853 1.23785 0.618923 0.785452i \(-0.287569\pi\)
0.618923 + 0.785452i \(0.287569\pi\)
\(998\) 0 0
\(999\) 5.05783 0.160023
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 8004.2.a.g.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.1 12 1.1 even 1 trivial