Properties

Label 8004.2.a.g.1.2
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 1.2
Root \(3.12492\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.12492 q^{5} +4.33538 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.12492 q^{5} +4.33538 q^{7} +1.00000 q^{9} -3.88477 q^{11} -5.76924 q^{13} +3.12492 q^{15} +1.77786 q^{17} +1.14116 q^{19} -4.33538 q^{21} +1.00000 q^{23} +4.76514 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.76454 q^{31} +3.88477 q^{33} -13.5477 q^{35} +9.00519 q^{37} +5.76924 q^{39} -1.21895 q^{41} +6.15921 q^{43} -3.12492 q^{45} -1.33639 q^{47} +11.7955 q^{49} -1.77786 q^{51} -5.98281 q^{53} +12.1396 q^{55} -1.14116 q^{57} -7.53101 q^{59} +2.35532 q^{61} +4.33538 q^{63} +18.0284 q^{65} -3.03611 q^{67} -1.00000 q^{69} +6.64492 q^{71} +2.08303 q^{73} -4.76514 q^{75} -16.8420 q^{77} -13.0123 q^{79} +1.00000 q^{81} -14.3584 q^{83} -5.55566 q^{85} +1.00000 q^{87} -4.89965 q^{89} -25.0119 q^{91} -5.76454 q^{93} -3.56603 q^{95} +7.62077 q^{97} -3.88477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) −3.12492 −1.39751 −0.698754 0.715362i \(-0.746262\pi\)
−0.698754 + 0.715362i \(0.746262\pi\)
\(6\) 0 0
\(7\) 4.33538 1.63862 0.819310 0.573351i \(-0.194357\pi\)
0.819310 + 0.573351i \(0.194357\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.88477 −1.17130 −0.585652 0.810563i \(-0.699161\pi\)
−0.585652 + 0.810563i \(0.699161\pi\)
\(12\) 0 0
\(13\) −5.76924 −1.60010 −0.800050 0.599933i \(-0.795194\pi\)
−0.800050 + 0.599933i \(0.795194\pi\)
\(14\) 0 0
\(15\) 3.12492 0.806851
\(16\) 0 0
\(17\) 1.77786 0.431194 0.215597 0.976482i \(-0.430830\pi\)
0.215597 + 0.976482i \(0.430830\pi\)
\(18\) 0 0
\(19\) 1.14116 0.261799 0.130900 0.991396i \(-0.458213\pi\)
0.130900 + 0.991396i \(0.458213\pi\)
\(20\) 0 0
\(21\) −4.33538 −0.946058
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.76514 0.953028
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 5.76454 1.03534 0.517671 0.855580i \(-0.326799\pi\)
0.517671 + 0.855580i \(0.326799\pi\)
\(32\) 0 0
\(33\) 3.88477 0.676252
\(34\) 0 0
\(35\) −13.5477 −2.28998
\(36\) 0 0
\(37\) 9.00519 1.48044 0.740222 0.672362i \(-0.234720\pi\)
0.740222 + 0.672362i \(0.234720\pi\)
\(38\) 0 0
\(39\) 5.76924 0.923818
\(40\) 0 0
\(41\) −1.21895 −0.190368 −0.0951839 0.995460i \(-0.530344\pi\)
−0.0951839 + 0.995460i \(0.530344\pi\)
\(42\) 0 0
\(43\) 6.15921 0.939271 0.469636 0.882860i \(-0.344385\pi\)
0.469636 + 0.882860i \(0.344385\pi\)
\(44\) 0 0
\(45\) −3.12492 −0.465836
\(46\) 0 0
\(47\) −1.33639 −0.194932 −0.0974661 0.995239i \(-0.531074\pi\)
−0.0974661 + 0.995239i \(0.531074\pi\)
\(48\) 0 0
\(49\) 11.7955 1.68508
\(50\) 0 0
\(51\) −1.77786 −0.248950
\(52\) 0 0
\(53\) −5.98281 −0.821802 −0.410901 0.911680i \(-0.634786\pi\)
−0.410901 + 0.911680i \(0.634786\pi\)
\(54\) 0 0
\(55\) 12.1396 1.63691
\(56\) 0 0
\(57\) −1.14116 −0.151150
\(58\) 0 0
\(59\) −7.53101 −0.980454 −0.490227 0.871595i \(-0.663086\pi\)
−0.490227 + 0.871595i \(0.663086\pi\)
\(60\) 0 0
\(61\) 2.35532 0.301568 0.150784 0.988567i \(-0.451820\pi\)
0.150784 + 0.988567i \(0.451820\pi\)
\(62\) 0 0
\(63\) 4.33538 0.546207
\(64\) 0 0
\(65\) 18.0284 2.23615
\(66\) 0 0
\(67\) −3.03611 −0.370920 −0.185460 0.982652i \(-0.559377\pi\)
−0.185460 + 0.982652i \(0.559377\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 6.64492 0.788607 0.394304 0.918980i \(-0.370986\pi\)
0.394304 + 0.918980i \(0.370986\pi\)
\(72\) 0 0
\(73\) 2.08303 0.243800 0.121900 0.992542i \(-0.461101\pi\)
0.121900 + 0.992542i \(0.461101\pi\)
\(74\) 0 0
\(75\) −4.76514 −0.550231
\(76\) 0 0
\(77\) −16.8420 −1.91932
\(78\) 0 0
\(79\) −13.0123 −1.46400 −0.732001 0.681304i \(-0.761413\pi\)
−0.732001 + 0.681304i \(0.761413\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.3584 −1.57604 −0.788019 0.615651i \(-0.788893\pi\)
−0.788019 + 0.615651i \(0.788893\pi\)
\(84\) 0 0
\(85\) −5.55566 −0.602596
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −4.89965 −0.519362 −0.259681 0.965694i \(-0.583617\pi\)
−0.259681 + 0.965694i \(0.583617\pi\)
\(90\) 0 0
\(91\) −25.0119 −2.62196
\(92\) 0 0
\(93\) −5.76454 −0.597755
\(94\) 0 0
\(95\) −3.56603 −0.365866
\(96\) 0 0
\(97\) 7.62077 0.773772 0.386886 0.922128i \(-0.373551\pi\)
0.386886 + 0.922128i \(0.373551\pi\)
\(98\) 0 0
\(99\) −3.88477 −0.390435
\(100\) 0 0
\(101\) 16.8588 1.67752 0.838758 0.544504i \(-0.183282\pi\)
0.838758 + 0.544504i \(0.183282\pi\)
\(102\) 0 0
\(103\) −1.87256 −0.184509 −0.0922543 0.995735i \(-0.529407\pi\)
−0.0922543 + 0.995735i \(0.529407\pi\)
\(104\) 0 0
\(105\) 13.5477 1.32212
\(106\) 0 0
\(107\) 12.8890 1.24603 0.623015 0.782210i \(-0.285907\pi\)
0.623015 + 0.782210i \(0.285907\pi\)
\(108\) 0 0
\(109\) 0.163721 0.0156817 0.00784083 0.999969i \(-0.497504\pi\)
0.00784083 + 0.999969i \(0.497504\pi\)
\(110\) 0 0
\(111\) −9.00519 −0.854735
\(112\) 0 0
\(113\) −1.58635 −0.149231 −0.0746155 0.997212i \(-0.523773\pi\)
−0.0746155 + 0.997212i \(0.523773\pi\)
\(114\) 0 0
\(115\) −3.12492 −0.291401
\(116\) 0 0
\(117\) −5.76924 −0.533367
\(118\) 0 0
\(119\) 7.70769 0.706563
\(120\) 0 0
\(121\) 4.09147 0.371952
\(122\) 0 0
\(123\) 1.21895 0.109909
\(124\) 0 0
\(125\) 0.733922 0.0656440
\(126\) 0 0
\(127\) −19.0449 −1.68996 −0.844981 0.534796i \(-0.820388\pi\)
−0.844981 + 0.534796i \(0.820388\pi\)
\(128\) 0 0
\(129\) −6.15921 −0.542289
\(130\) 0 0
\(131\) 13.9702 1.22058 0.610292 0.792176i \(-0.291052\pi\)
0.610292 + 0.792176i \(0.291052\pi\)
\(132\) 0 0
\(133\) 4.94735 0.428990
\(134\) 0 0
\(135\) 3.12492 0.268950
\(136\) 0 0
\(137\) 6.51252 0.556402 0.278201 0.960523i \(-0.410262\pi\)
0.278201 + 0.960523i \(0.410262\pi\)
\(138\) 0 0
\(139\) 7.91579 0.671409 0.335704 0.941967i \(-0.391026\pi\)
0.335704 + 0.941967i \(0.391026\pi\)
\(140\) 0 0
\(141\) 1.33639 0.112544
\(142\) 0 0
\(143\) 22.4122 1.87420
\(144\) 0 0
\(145\) 3.12492 0.259511
\(146\) 0 0
\(147\) −11.7955 −0.972879
\(148\) 0 0
\(149\) −17.3544 −1.42173 −0.710865 0.703328i \(-0.751696\pi\)
−0.710865 + 0.703328i \(0.751696\pi\)
\(150\) 0 0
\(151\) −20.6742 −1.68244 −0.841220 0.540693i \(-0.818162\pi\)
−0.841220 + 0.540693i \(0.818162\pi\)
\(152\) 0 0
\(153\) 1.77786 0.143731
\(154\) 0 0
\(155\) −18.0137 −1.44690
\(156\) 0 0
\(157\) 1.53395 0.122422 0.0612112 0.998125i \(-0.480504\pi\)
0.0612112 + 0.998125i \(0.480504\pi\)
\(158\) 0 0
\(159\) 5.98281 0.474468
\(160\) 0 0
\(161\) 4.33538 0.341676
\(162\) 0 0
\(163\) 3.92716 0.307599 0.153800 0.988102i \(-0.450849\pi\)
0.153800 + 0.988102i \(0.450849\pi\)
\(164\) 0 0
\(165\) −12.1396 −0.945068
\(166\) 0 0
\(167\) 19.2364 1.48855 0.744277 0.667871i \(-0.232794\pi\)
0.744277 + 0.667871i \(0.232794\pi\)
\(168\) 0 0
\(169\) 20.2842 1.56032
\(170\) 0 0
\(171\) 1.14116 0.0872664
\(172\) 0 0
\(173\) −16.6806 −1.26820 −0.634102 0.773249i \(-0.718630\pi\)
−0.634102 + 0.773249i \(0.718630\pi\)
\(174\) 0 0
\(175\) 20.6587 1.56165
\(176\) 0 0
\(177\) 7.53101 0.566066
\(178\) 0 0
\(179\) −9.82298 −0.734204 −0.367102 0.930181i \(-0.619650\pi\)
−0.367102 + 0.930181i \(0.619650\pi\)
\(180\) 0 0
\(181\) 15.3057 1.13766 0.568831 0.822454i \(-0.307396\pi\)
0.568831 + 0.822454i \(0.307396\pi\)
\(182\) 0 0
\(183\) −2.35532 −0.174110
\(184\) 0 0
\(185\) −28.1405 −2.06893
\(186\) 0 0
\(187\) −6.90657 −0.505059
\(188\) 0 0
\(189\) −4.33538 −0.315353
\(190\) 0 0
\(191\) −15.6137 −1.12977 −0.564884 0.825171i \(-0.691079\pi\)
−0.564884 + 0.825171i \(0.691079\pi\)
\(192\) 0 0
\(193\) −4.76207 −0.342781 −0.171391 0.985203i \(-0.554826\pi\)
−0.171391 + 0.985203i \(0.554826\pi\)
\(194\) 0 0
\(195\) −18.0284 −1.29104
\(196\) 0 0
\(197\) −7.66400 −0.546037 −0.273019 0.962009i \(-0.588022\pi\)
−0.273019 + 0.962009i \(0.588022\pi\)
\(198\) 0 0
\(199\) 26.8524 1.90351 0.951757 0.306853i \(-0.0992761\pi\)
0.951757 + 0.306853i \(0.0992761\pi\)
\(200\) 0 0
\(201\) 3.03611 0.214151
\(202\) 0 0
\(203\) −4.33538 −0.304284
\(204\) 0 0
\(205\) 3.80912 0.266040
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −4.43314 −0.306646
\(210\) 0 0
\(211\) −14.9556 −1.02959 −0.514793 0.857314i \(-0.672131\pi\)
−0.514793 + 0.857314i \(0.672131\pi\)
\(212\) 0 0
\(213\) −6.64492 −0.455303
\(214\) 0 0
\(215\) −19.2471 −1.31264
\(216\) 0 0
\(217\) 24.9915 1.69653
\(218\) 0 0
\(219\) −2.08303 −0.140758
\(220\) 0 0
\(221\) −10.2569 −0.689953
\(222\) 0 0
\(223\) −25.6301 −1.71632 −0.858160 0.513383i \(-0.828392\pi\)
−0.858160 + 0.513383i \(0.828392\pi\)
\(224\) 0 0
\(225\) 4.76514 0.317676
\(226\) 0 0
\(227\) −21.3625 −1.41788 −0.708939 0.705270i \(-0.750826\pi\)
−0.708939 + 0.705270i \(0.750826\pi\)
\(228\) 0 0
\(229\) −25.1208 −1.66003 −0.830015 0.557741i \(-0.811668\pi\)
−0.830015 + 0.557741i \(0.811668\pi\)
\(230\) 0 0
\(231\) 16.8420 1.10812
\(232\) 0 0
\(233\) −5.74498 −0.376366 −0.188183 0.982134i \(-0.560260\pi\)
−0.188183 + 0.982134i \(0.560260\pi\)
\(234\) 0 0
\(235\) 4.17611 0.272419
\(236\) 0 0
\(237\) 13.0123 0.845241
\(238\) 0 0
\(239\) 9.74972 0.630657 0.315328 0.948983i \(-0.397885\pi\)
0.315328 + 0.948983i \(0.397885\pi\)
\(240\) 0 0
\(241\) −0.660892 −0.0425718 −0.0212859 0.999773i \(-0.506776\pi\)
−0.0212859 + 0.999773i \(0.506776\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −36.8601 −2.35491
\(246\) 0 0
\(247\) −6.58361 −0.418905
\(248\) 0 0
\(249\) 14.3584 0.909926
\(250\) 0 0
\(251\) 3.07690 0.194212 0.0971062 0.995274i \(-0.469041\pi\)
0.0971062 + 0.995274i \(0.469041\pi\)
\(252\) 0 0
\(253\) −3.88477 −0.244234
\(254\) 0 0
\(255\) 5.55566 0.347909
\(256\) 0 0
\(257\) 1.30458 0.0813772 0.0406886 0.999172i \(-0.487045\pi\)
0.0406886 + 0.999172i \(0.487045\pi\)
\(258\) 0 0
\(259\) 39.0409 2.42589
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 14.7719 0.910874 0.455437 0.890268i \(-0.349483\pi\)
0.455437 + 0.890268i \(0.349483\pi\)
\(264\) 0 0
\(265\) 18.6958 1.14847
\(266\) 0 0
\(267\) 4.89965 0.299854
\(268\) 0 0
\(269\) 13.6664 0.833253 0.416627 0.909078i \(-0.363212\pi\)
0.416627 + 0.909078i \(0.363212\pi\)
\(270\) 0 0
\(271\) 3.00165 0.182337 0.0911687 0.995835i \(-0.470940\pi\)
0.0911687 + 0.995835i \(0.470940\pi\)
\(272\) 0 0
\(273\) 25.0119 1.51379
\(274\) 0 0
\(275\) −18.5115 −1.11628
\(276\) 0 0
\(277\) 18.1613 1.09121 0.545603 0.838044i \(-0.316301\pi\)
0.545603 + 0.838044i \(0.316301\pi\)
\(278\) 0 0
\(279\) 5.76454 0.345114
\(280\) 0 0
\(281\) 3.73451 0.222782 0.111391 0.993777i \(-0.464469\pi\)
0.111391 + 0.993777i \(0.464469\pi\)
\(282\) 0 0
\(283\) −23.6268 −1.40447 −0.702235 0.711945i \(-0.747814\pi\)
−0.702235 + 0.711945i \(0.747814\pi\)
\(284\) 0 0
\(285\) 3.56603 0.211233
\(286\) 0 0
\(287\) −5.28461 −0.311941
\(288\) 0 0
\(289\) −13.8392 −0.814072
\(290\) 0 0
\(291\) −7.62077 −0.446737
\(292\) 0 0
\(293\) −7.17862 −0.419379 −0.209690 0.977768i \(-0.567245\pi\)
−0.209690 + 0.977768i \(0.567245\pi\)
\(294\) 0 0
\(295\) 23.5338 1.37019
\(296\) 0 0
\(297\) 3.88477 0.225417
\(298\) 0 0
\(299\) −5.76924 −0.333644
\(300\) 0 0
\(301\) 26.7025 1.53911
\(302\) 0 0
\(303\) −16.8588 −0.968514
\(304\) 0 0
\(305\) −7.36019 −0.421443
\(306\) 0 0
\(307\) 10.3673 0.591690 0.295845 0.955236i \(-0.404399\pi\)
0.295845 + 0.955236i \(0.404399\pi\)
\(308\) 0 0
\(309\) 1.87256 0.106526
\(310\) 0 0
\(311\) −29.7060 −1.68447 −0.842236 0.539109i \(-0.818761\pi\)
−0.842236 + 0.539109i \(0.818761\pi\)
\(312\) 0 0
\(313\) 10.6960 0.604571 0.302286 0.953217i \(-0.402250\pi\)
0.302286 + 0.953217i \(0.402250\pi\)
\(314\) 0 0
\(315\) −13.5477 −0.763328
\(316\) 0 0
\(317\) 11.5388 0.648083 0.324041 0.946043i \(-0.394958\pi\)
0.324041 + 0.946043i \(0.394958\pi\)
\(318\) 0 0
\(319\) 3.88477 0.217506
\(320\) 0 0
\(321\) −12.8890 −0.719396
\(322\) 0 0
\(323\) 2.02881 0.112886
\(324\) 0 0
\(325\) −27.4913 −1.52494
\(326\) 0 0
\(327\) −0.163721 −0.00905381
\(328\) 0 0
\(329\) −5.79375 −0.319420
\(330\) 0 0
\(331\) 12.4593 0.684827 0.342414 0.939549i \(-0.388756\pi\)
0.342414 + 0.939549i \(0.388756\pi\)
\(332\) 0 0
\(333\) 9.00519 0.493481
\(334\) 0 0
\(335\) 9.48761 0.518363
\(336\) 0 0
\(337\) −17.0473 −0.928623 −0.464312 0.885672i \(-0.653698\pi\)
−0.464312 + 0.885672i \(0.653698\pi\)
\(338\) 0 0
\(339\) 1.58635 0.0861585
\(340\) 0 0
\(341\) −22.3939 −1.21270
\(342\) 0 0
\(343\) 20.7905 1.12258
\(344\) 0 0
\(345\) 3.12492 0.168240
\(346\) 0 0
\(347\) −22.8400 −1.22612 −0.613059 0.790037i \(-0.710061\pi\)
−0.613059 + 0.790037i \(0.710061\pi\)
\(348\) 0 0
\(349\) −1.40175 −0.0750337 −0.0375169 0.999296i \(-0.511945\pi\)
−0.0375169 + 0.999296i \(0.511945\pi\)
\(350\) 0 0
\(351\) 5.76924 0.307939
\(352\) 0 0
\(353\) 1.34698 0.0716923 0.0358462 0.999357i \(-0.488587\pi\)
0.0358462 + 0.999357i \(0.488587\pi\)
\(354\) 0 0
\(355\) −20.7649 −1.10208
\(356\) 0 0
\(357\) −7.70769 −0.407934
\(358\) 0 0
\(359\) −10.1026 −0.533195 −0.266597 0.963808i \(-0.585899\pi\)
−0.266597 + 0.963808i \(0.585899\pi\)
\(360\) 0 0
\(361\) −17.6978 −0.931461
\(362\) 0 0
\(363\) −4.09147 −0.214747
\(364\) 0 0
\(365\) −6.50930 −0.340712
\(366\) 0 0
\(367\) 2.38990 0.124752 0.0623758 0.998053i \(-0.480132\pi\)
0.0623758 + 0.998053i \(0.480132\pi\)
\(368\) 0 0
\(369\) −1.21895 −0.0634559
\(370\) 0 0
\(371\) −25.9378 −1.34662
\(372\) 0 0
\(373\) −35.2159 −1.82341 −0.911705 0.410845i \(-0.865234\pi\)
−0.911705 + 0.410845i \(0.865234\pi\)
\(374\) 0 0
\(375\) −0.733922 −0.0378996
\(376\) 0 0
\(377\) 5.76924 0.297131
\(378\) 0 0
\(379\) 9.92302 0.509711 0.254856 0.966979i \(-0.417972\pi\)
0.254856 + 0.966979i \(0.417972\pi\)
\(380\) 0 0
\(381\) 19.0449 0.975700
\(382\) 0 0
\(383\) −23.2260 −1.18680 −0.593398 0.804909i \(-0.702214\pi\)
−0.593398 + 0.804909i \(0.702214\pi\)
\(384\) 0 0
\(385\) 52.6299 2.68227
\(386\) 0 0
\(387\) 6.15921 0.313090
\(388\) 0 0
\(389\) −37.3840 −1.89545 −0.947723 0.319094i \(-0.896622\pi\)
−0.947723 + 0.319094i \(0.896622\pi\)
\(390\) 0 0
\(391\) 1.77786 0.0899101
\(392\) 0 0
\(393\) −13.9702 −0.704705
\(394\) 0 0
\(395\) 40.6625 2.04595
\(396\) 0 0
\(397\) 18.5902 0.933016 0.466508 0.884517i \(-0.345512\pi\)
0.466508 + 0.884517i \(0.345512\pi\)
\(398\) 0 0
\(399\) −4.94735 −0.247677
\(400\) 0 0
\(401\) −12.2176 −0.610119 −0.305059 0.952333i \(-0.598676\pi\)
−0.305059 + 0.952333i \(0.598676\pi\)
\(402\) 0 0
\(403\) −33.2570 −1.65665
\(404\) 0 0
\(405\) −3.12492 −0.155279
\(406\) 0 0
\(407\) −34.9831 −1.73405
\(408\) 0 0
\(409\) −1.97213 −0.0975158 −0.0487579 0.998811i \(-0.515526\pi\)
−0.0487579 + 0.998811i \(0.515526\pi\)
\(410\) 0 0
\(411\) −6.51252 −0.321239
\(412\) 0 0
\(413\) −32.6498 −1.60659
\(414\) 0 0
\(415\) 44.8688 2.20252
\(416\) 0 0
\(417\) −7.91579 −0.387638
\(418\) 0 0
\(419\) 13.3906 0.654174 0.327087 0.944994i \(-0.393933\pi\)
0.327087 + 0.944994i \(0.393933\pi\)
\(420\) 0 0
\(421\) −25.8993 −1.26226 −0.631128 0.775678i \(-0.717408\pi\)
−0.631128 + 0.775678i \(0.717408\pi\)
\(422\) 0 0
\(423\) −1.33639 −0.0649774
\(424\) 0 0
\(425\) 8.47173 0.410939
\(426\) 0 0
\(427\) 10.2112 0.494155
\(428\) 0 0
\(429\) −22.4122 −1.08207
\(430\) 0 0
\(431\) −17.1828 −0.827667 −0.413834 0.910353i \(-0.635810\pi\)
−0.413834 + 0.910353i \(0.635810\pi\)
\(432\) 0 0
\(433\) −0.495336 −0.0238043 −0.0119022 0.999929i \(-0.503789\pi\)
−0.0119022 + 0.999929i \(0.503789\pi\)
\(434\) 0 0
\(435\) −3.12492 −0.149829
\(436\) 0 0
\(437\) 1.14116 0.0545889
\(438\) 0 0
\(439\) −22.2342 −1.06118 −0.530589 0.847629i \(-0.678029\pi\)
−0.530589 + 0.847629i \(0.678029\pi\)
\(440\) 0 0
\(441\) 11.7955 0.561692
\(442\) 0 0
\(443\) −17.4641 −0.829745 −0.414873 0.909880i \(-0.636174\pi\)
−0.414873 + 0.909880i \(0.636174\pi\)
\(444\) 0 0
\(445\) 15.3110 0.725813
\(446\) 0 0
\(447\) 17.3544 0.820836
\(448\) 0 0
\(449\) −32.9362 −1.55436 −0.777178 0.629280i \(-0.783350\pi\)
−0.777178 + 0.629280i \(0.783350\pi\)
\(450\) 0 0
\(451\) 4.73534 0.222978
\(452\) 0 0
\(453\) 20.6742 0.971357
\(454\) 0 0
\(455\) 78.1602 3.66421
\(456\) 0 0
\(457\) 12.8131 0.599370 0.299685 0.954038i \(-0.403118\pi\)
0.299685 + 0.954038i \(0.403118\pi\)
\(458\) 0 0
\(459\) −1.77786 −0.0829832
\(460\) 0 0
\(461\) 10.7073 0.498688 0.249344 0.968415i \(-0.419785\pi\)
0.249344 + 0.968415i \(0.419785\pi\)
\(462\) 0 0
\(463\) −33.9367 −1.57717 −0.788587 0.614923i \(-0.789187\pi\)
−0.788587 + 0.614923i \(0.789187\pi\)
\(464\) 0 0
\(465\) 18.0137 0.835367
\(466\) 0 0
\(467\) −21.4185 −0.991129 −0.495565 0.868571i \(-0.665039\pi\)
−0.495565 + 0.868571i \(0.665039\pi\)
\(468\) 0 0
\(469\) −13.1627 −0.607797
\(470\) 0 0
\(471\) −1.53395 −0.0706807
\(472\) 0 0
\(473\) −23.9272 −1.10017
\(474\) 0 0
\(475\) 5.43777 0.249502
\(476\) 0 0
\(477\) −5.98281 −0.273934
\(478\) 0 0
\(479\) 18.9177 0.864373 0.432186 0.901784i \(-0.357742\pi\)
0.432186 + 0.901784i \(0.357742\pi\)
\(480\) 0 0
\(481\) −51.9531 −2.36886
\(482\) 0 0
\(483\) −4.33538 −0.197267
\(484\) 0 0
\(485\) −23.8143 −1.08135
\(486\) 0 0
\(487\) 29.4226 1.33327 0.666634 0.745386i \(-0.267735\pi\)
0.666634 + 0.745386i \(0.267735\pi\)
\(488\) 0 0
\(489\) −3.92716 −0.177592
\(490\) 0 0
\(491\) 36.0670 1.62768 0.813841 0.581088i \(-0.197373\pi\)
0.813841 + 0.581088i \(0.197373\pi\)
\(492\) 0 0
\(493\) −1.77786 −0.0800706
\(494\) 0 0
\(495\) 12.1396 0.545635
\(496\) 0 0
\(497\) 28.8083 1.29223
\(498\) 0 0
\(499\) 16.9469 0.758649 0.379325 0.925264i \(-0.376156\pi\)
0.379325 + 0.925264i \(0.376156\pi\)
\(500\) 0 0
\(501\) −19.2364 −0.859417
\(502\) 0 0
\(503\) 8.80429 0.392564 0.196282 0.980547i \(-0.437113\pi\)
0.196282 + 0.980547i \(0.437113\pi\)
\(504\) 0 0
\(505\) −52.6825 −2.34434
\(506\) 0 0
\(507\) −20.2842 −0.900852
\(508\) 0 0
\(509\) 26.2927 1.16540 0.582701 0.812687i \(-0.301996\pi\)
0.582701 + 0.812687i \(0.301996\pi\)
\(510\) 0 0
\(511\) 9.03072 0.399495
\(512\) 0 0
\(513\) −1.14116 −0.0503833
\(514\) 0 0
\(515\) 5.85160 0.257852
\(516\) 0 0
\(517\) 5.19157 0.228325
\(518\) 0 0
\(519\) 16.6806 0.732198
\(520\) 0 0
\(521\) −8.01412 −0.351105 −0.175552 0.984470i \(-0.556171\pi\)
−0.175552 + 0.984470i \(0.556171\pi\)
\(522\) 0 0
\(523\) −37.7616 −1.65120 −0.825599 0.564258i \(-0.809162\pi\)
−0.825599 + 0.564258i \(0.809162\pi\)
\(524\) 0 0
\(525\) −20.6587 −0.901619
\(526\) 0 0
\(527\) 10.2485 0.446433
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −7.53101 −0.326818
\(532\) 0 0
\(533\) 7.03241 0.304608
\(534\) 0 0
\(535\) −40.2772 −1.74134
\(536\) 0 0
\(537\) 9.82298 0.423893
\(538\) 0 0
\(539\) −45.8230 −1.97374
\(540\) 0 0
\(541\) −32.1345 −1.38157 −0.690785 0.723060i \(-0.742735\pi\)
−0.690785 + 0.723060i \(0.742735\pi\)
\(542\) 0 0
\(543\) −15.3057 −0.656830
\(544\) 0 0
\(545\) −0.511616 −0.0219152
\(546\) 0 0
\(547\) 16.9345 0.724069 0.362034 0.932165i \(-0.382082\pi\)
0.362034 + 0.932165i \(0.382082\pi\)
\(548\) 0 0
\(549\) 2.35532 0.100523
\(550\) 0 0
\(551\) −1.14116 −0.0486149
\(552\) 0 0
\(553\) −56.4134 −2.39894
\(554\) 0 0
\(555\) 28.1405 1.19450
\(556\) 0 0
\(557\) 3.86058 0.163578 0.0817890 0.996650i \(-0.473937\pi\)
0.0817890 + 0.996650i \(0.473937\pi\)
\(558\) 0 0
\(559\) −35.5340 −1.50293
\(560\) 0 0
\(561\) 6.90657 0.291596
\(562\) 0 0
\(563\) −32.7287 −1.37935 −0.689676 0.724118i \(-0.742247\pi\)
−0.689676 + 0.724118i \(0.742247\pi\)
\(564\) 0 0
\(565\) 4.95721 0.208551
\(566\) 0 0
\(567\) 4.33538 0.182069
\(568\) 0 0
\(569\) −27.1935 −1.14001 −0.570005 0.821641i \(-0.693059\pi\)
−0.570005 + 0.821641i \(0.693059\pi\)
\(570\) 0 0
\(571\) −9.01198 −0.377140 −0.188570 0.982060i \(-0.560385\pi\)
−0.188570 + 0.982060i \(0.560385\pi\)
\(572\) 0 0
\(573\) 15.6137 0.652271
\(574\) 0 0
\(575\) 4.76514 0.198720
\(576\) 0 0
\(577\) −16.5121 −0.687408 −0.343704 0.939078i \(-0.611682\pi\)
−0.343704 + 0.939078i \(0.611682\pi\)
\(578\) 0 0
\(579\) 4.76207 0.197905
\(580\) 0 0
\(581\) −62.2491 −2.58253
\(582\) 0 0
\(583\) 23.2419 0.962579
\(584\) 0 0
\(585\) 18.0284 0.745384
\(586\) 0 0
\(587\) 1.61443 0.0666346 0.0333173 0.999445i \(-0.489393\pi\)
0.0333173 + 0.999445i \(0.489393\pi\)
\(588\) 0 0
\(589\) 6.57824 0.271052
\(590\) 0 0
\(591\) 7.66400 0.315255
\(592\) 0 0
\(593\) −11.2398 −0.461562 −0.230781 0.973006i \(-0.574128\pi\)
−0.230781 + 0.973006i \(0.574128\pi\)
\(594\) 0 0
\(595\) −24.0859 −0.987427
\(596\) 0 0
\(597\) −26.8524 −1.09899
\(598\) 0 0
\(599\) 13.3436 0.545204 0.272602 0.962127i \(-0.412116\pi\)
0.272602 + 0.962127i \(0.412116\pi\)
\(600\) 0 0
\(601\) −31.6297 −1.29020 −0.645101 0.764097i \(-0.723185\pi\)
−0.645101 + 0.764097i \(0.723185\pi\)
\(602\) 0 0
\(603\) −3.03611 −0.123640
\(604\) 0 0
\(605\) −12.7855 −0.519806
\(606\) 0 0
\(607\) −28.1421 −1.14225 −0.571127 0.820862i \(-0.693493\pi\)
−0.571127 + 0.820862i \(0.693493\pi\)
\(608\) 0 0
\(609\) 4.33538 0.175679
\(610\) 0 0
\(611\) 7.70995 0.311911
\(612\) 0 0
\(613\) 27.5963 1.11460 0.557301 0.830310i \(-0.311837\pi\)
0.557301 + 0.830310i \(0.311837\pi\)
\(614\) 0 0
\(615\) −3.80912 −0.153599
\(616\) 0 0
\(617\) −29.9733 −1.20668 −0.603341 0.797484i \(-0.706164\pi\)
−0.603341 + 0.797484i \(0.706164\pi\)
\(618\) 0 0
\(619\) 23.8123 0.957097 0.478548 0.878061i \(-0.341163\pi\)
0.478548 + 0.878061i \(0.341163\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −21.2419 −0.851038
\(624\) 0 0
\(625\) −26.1191 −1.04477
\(626\) 0 0
\(627\) 4.43314 0.177042
\(628\) 0 0
\(629\) 16.0099 0.638358
\(630\) 0 0
\(631\) 10.7203 0.426769 0.213384 0.976968i \(-0.431551\pi\)
0.213384 + 0.976968i \(0.431551\pi\)
\(632\) 0 0
\(633\) 14.9556 0.594432
\(634\) 0 0
\(635\) 59.5138 2.36173
\(636\) 0 0
\(637\) −68.0513 −2.69629
\(638\) 0 0
\(639\) 6.64492 0.262869
\(640\) 0 0
\(641\) −27.7824 −1.09734 −0.548670 0.836039i \(-0.684866\pi\)
−0.548670 + 0.836039i \(0.684866\pi\)
\(642\) 0 0
\(643\) −14.7862 −0.583109 −0.291555 0.956554i \(-0.594173\pi\)
−0.291555 + 0.956554i \(0.594173\pi\)
\(644\) 0 0
\(645\) 19.2471 0.757853
\(646\) 0 0
\(647\) −21.4419 −0.842969 −0.421485 0.906836i \(-0.638491\pi\)
−0.421485 + 0.906836i \(0.638491\pi\)
\(648\) 0 0
\(649\) 29.2563 1.14841
\(650\) 0 0
\(651\) −24.9915 −0.979493
\(652\) 0 0
\(653\) 1.45978 0.0571255 0.0285627 0.999592i \(-0.490907\pi\)
0.0285627 + 0.999592i \(0.490907\pi\)
\(654\) 0 0
\(655\) −43.6559 −1.70578
\(656\) 0 0
\(657\) 2.08303 0.0812666
\(658\) 0 0
\(659\) −46.9684 −1.82963 −0.914815 0.403872i \(-0.867664\pi\)
−0.914815 + 0.403872i \(0.867664\pi\)
\(660\) 0 0
\(661\) −0.829538 −0.0322653 −0.0161326 0.999870i \(-0.505135\pi\)
−0.0161326 + 0.999870i \(0.505135\pi\)
\(662\) 0 0
\(663\) 10.2569 0.398345
\(664\) 0 0
\(665\) −15.4601 −0.599516
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 25.6301 0.990917
\(670\) 0 0
\(671\) −9.14988 −0.353227
\(672\) 0 0
\(673\) −3.40741 −0.131346 −0.0656729 0.997841i \(-0.520919\pi\)
−0.0656729 + 0.997841i \(0.520919\pi\)
\(674\) 0 0
\(675\) −4.76514 −0.183410
\(676\) 0 0
\(677\) 48.8238 1.87645 0.938225 0.346027i \(-0.112469\pi\)
0.938225 + 0.346027i \(0.112469\pi\)
\(678\) 0 0
\(679\) 33.0389 1.26792
\(680\) 0 0
\(681\) 21.3625 0.818612
\(682\) 0 0
\(683\) −41.0929 −1.57238 −0.786188 0.617987i \(-0.787948\pi\)
−0.786188 + 0.617987i \(0.787948\pi\)
\(684\) 0 0
\(685\) −20.3511 −0.777577
\(686\) 0 0
\(687\) 25.1208 0.958419
\(688\) 0 0
\(689\) 34.5163 1.31497
\(690\) 0 0
\(691\) 34.2118 1.30148 0.650739 0.759301i \(-0.274459\pi\)
0.650739 + 0.759301i \(0.274459\pi\)
\(692\) 0 0
\(693\) −16.8420 −0.639774
\(694\) 0 0
\(695\) −24.7362 −0.938299
\(696\) 0 0
\(697\) −2.16712 −0.0820854
\(698\) 0 0
\(699\) 5.74498 0.217295
\(700\) 0 0
\(701\) −3.58511 −0.135408 −0.0677038 0.997705i \(-0.521567\pi\)
−0.0677038 + 0.997705i \(0.521567\pi\)
\(702\) 0 0
\(703\) 10.2763 0.387579
\(704\) 0 0
\(705\) −4.17611 −0.157281
\(706\) 0 0
\(707\) 73.0894 2.74881
\(708\) 0 0
\(709\) 8.14577 0.305921 0.152960 0.988232i \(-0.451119\pi\)
0.152960 + 0.988232i \(0.451119\pi\)
\(710\) 0 0
\(711\) −13.0123 −0.488000
\(712\) 0 0
\(713\) 5.76454 0.215884
\(714\) 0 0
\(715\) −70.0364 −2.61921
\(716\) 0 0
\(717\) −9.74972 −0.364110
\(718\) 0 0
\(719\) 1.60257 0.0597658 0.0298829 0.999553i \(-0.490487\pi\)
0.0298829 + 0.999553i \(0.490487\pi\)
\(720\) 0 0
\(721\) −8.11825 −0.302339
\(722\) 0 0
\(723\) 0.660892 0.0245788
\(724\) 0 0
\(725\) −4.76514 −0.176973
\(726\) 0 0
\(727\) 8.45074 0.313421 0.156710 0.987645i \(-0.449911\pi\)
0.156710 + 0.987645i \(0.449911\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.9502 0.405008
\(732\) 0 0
\(733\) 20.1821 0.745442 0.372721 0.927943i \(-0.378425\pi\)
0.372721 + 0.927943i \(0.378425\pi\)
\(734\) 0 0
\(735\) 36.8601 1.35961
\(736\) 0 0
\(737\) 11.7946 0.434460
\(738\) 0 0
\(739\) 12.0703 0.444015 0.222007 0.975045i \(-0.428739\pi\)
0.222007 + 0.975045i \(0.428739\pi\)
\(740\) 0 0
\(741\) 6.58361 0.241855
\(742\) 0 0
\(743\) −27.2302 −0.998980 −0.499490 0.866320i \(-0.666479\pi\)
−0.499490 + 0.866320i \(0.666479\pi\)
\(744\) 0 0
\(745\) 54.2312 1.98688
\(746\) 0 0
\(747\) −14.3584 −0.525346
\(748\) 0 0
\(749\) 55.8789 2.04177
\(750\) 0 0
\(751\) 35.2040 1.28461 0.642306 0.766448i \(-0.277978\pi\)
0.642306 + 0.766448i \(0.277978\pi\)
\(752\) 0 0
\(753\) −3.07690 −0.112129
\(754\) 0 0
\(755\) 64.6052 2.35122
\(756\) 0 0
\(757\) −28.0464 −1.01936 −0.509682 0.860363i \(-0.670237\pi\)
−0.509682 + 0.860363i \(0.670237\pi\)
\(758\) 0 0
\(759\) 3.88477 0.141008
\(760\) 0 0
\(761\) 21.7067 0.786866 0.393433 0.919353i \(-0.371287\pi\)
0.393433 + 0.919353i \(0.371287\pi\)
\(762\) 0 0
\(763\) 0.709794 0.0256963
\(764\) 0 0
\(765\) −5.55566 −0.200865
\(766\) 0 0
\(767\) 43.4483 1.56883
\(768\) 0 0
\(769\) 23.4309 0.844941 0.422471 0.906377i \(-0.361163\pi\)
0.422471 + 0.906377i \(0.361163\pi\)
\(770\) 0 0
\(771\) −1.30458 −0.0469831
\(772\) 0 0
\(773\) −25.0981 −0.902717 −0.451358 0.892343i \(-0.649060\pi\)
−0.451358 + 0.892343i \(0.649060\pi\)
\(774\) 0 0
\(775\) 27.4688 0.986710
\(776\) 0 0
\(777\) −39.0409 −1.40059
\(778\) 0 0
\(779\) −1.39101 −0.0498381
\(780\) 0 0
\(781\) −25.8140 −0.923698
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −4.79347 −0.171086
\(786\) 0 0
\(787\) −44.9599 −1.60265 −0.801325 0.598230i \(-0.795871\pi\)
−0.801325 + 0.598230i \(0.795871\pi\)
\(788\) 0 0
\(789\) −14.7719 −0.525894
\(790\) 0 0
\(791\) −6.87742 −0.244533
\(792\) 0 0
\(793\) −13.5884 −0.482538
\(794\) 0 0
\(795\) −18.6958 −0.663072
\(796\) 0 0
\(797\) −24.0899 −0.853310 −0.426655 0.904415i \(-0.640308\pi\)
−0.426655 + 0.904415i \(0.640308\pi\)
\(798\) 0 0
\(799\) −2.37591 −0.0840535
\(800\) 0 0
\(801\) −4.89965 −0.173121
\(802\) 0 0
\(803\) −8.09209 −0.285564
\(804\) 0 0
\(805\) −13.5477 −0.477495
\(806\) 0 0
\(807\) −13.6664 −0.481079
\(808\) 0 0
\(809\) −49.4644 −1.73908 −0.869538 0.493866i \(-0.835583\pi\)
−0.869538 + 0.493866i \(0.835583\pi\)
\(810\) 0 0
\(811\) 9.55785 0.335622 0.167811 0.985819i \(-0.446330\pi\)
0.167811 + 0.985819i \(0.446330\pi\)
\(812\) 0 0
\(813\) −3.00165 −0.105273
\(814\) 0 0
\(815\) −12.2721 −0.429872
\(816\) 0 0
\(817\) 7.02863 0.245901
\(818\) 0 0
\(819\) −25.0119 −0.873986
\(820\) 0 0
\(821\) −18.1610 −0.633824 −0.316912 0.948455i \(-0.602646\pi\)
−0.316912 + 0.948455i \(0.602646\pi\)
\(822\) 0 0
\(823\) 43.5039 1.51645 0.758226 0.651992i \(-0.226067\pi\)
0.758226 + 0.651992i \(0.226067\pi\)
\(824\) 0 0
\(825\) 18.5115 0.644487
\(826\) 0 0
\(827\) −3.09541 −0.107638 −0.0538190 0.998551i \(-0.517139\pi\)
−0.0538190 + 0.998551i \(0.517139\pi\)
\(828\) 0 0
\(829\) −39.8485 −1.38399 −0.691997 0.721900i \(-0.743269\pi\)
−0.691997 + 0.721900i \(0.743269\pi\)
\(830\) 0 0
\(831\) −18.1613 −0.630009
\(832\) 0 0
\(833\) 20.9708 0.726594
\(834\) 0 0
\(835\) −60.1121 −2.08027
\(836\) 0 0
\(837\) −5.76454 −0.199252
\(838\) 0 0
\(839\) 27.0042 0.932289 0.466144 0.884709i \(-0.345643\pi\)
0.466144 + 0.884709i \(0.345643\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −3.73451 −0.128623
\(844\) 0 0
\(845\) −63.3865 −2.18056
\(846\) 0 0
\(847\) 17.7381 0.609488
\(848\) 0 0
\(849\) 23.6268 0.810871
\(850\) 0 0
\(851\) 9.00519 0.308694
\(852\) 0 0
\(853\) 19.2754 0.659978 0.329989 0.943985i \(-0.392955\pi\)
0.329989 + 0.943985i \(0.392955\pi\)
\(854\) 0 0
\(855\) −3.56603 −0.121955
\(856\) 0 0
\(857\) 21.5762 0.737030 0.368515 0.929622i \(-0.379866\pi\)
0.368515 + 0.929622i \(0.379866\pi\)
\(858\) 0 0
\(859\) −18.2042 −0.621121 −0.310560 0.950554i \(-0.600517\pi\)
−0.310560 + 0.950554i \(0.600517\pi\)
\(860\) 0 0
\(861\) 5.28461 0.180099
\(862\) 0 0
\(863\) 26.0426 0.886500 0.443250 0.896398i \(-0.353825\pi\)
0.443250 + 0.896398i \(0.353825\pi\)
\(864\) 0 0
\(865\) 52.1256 1.77233
\(866\) 0 0
\(867\) 13.8392 0.470005
\(868\) 0 0
\(869\) 50.5500 1.71479
\(870\) 0 0
\(871\) 17.5161 0.593509
\(872\) 0 0
\(873\) 7.62077 0.257924
\(874\) 0 0
\(875\) 3.18183 0.107566
\(876\) 0 0
\(877\) −32.1393 −1.08527 −0.542633 0.839970i \(-0.682573\pi\)
−0.542633 + 0.839970i \(0.682573\pi\)
\(878\) 0 0
\(879\) 7.17862 0.242129
\(880\) 0 0
\(881\) 1.44207 0.0485846 0.0242923 0.999705i \(-0.492267\pi\)
0.0242923 + 0.999705i \(0.492267\pi\)
\(882\) 0 0
\(883\) −6.05643 −0.203815 −0.101908 0.994794i \(-0.532495\pi\)
−0.101908 + 0.994794i \(0.532495\pi\)
\(884\) 0 0
\(885\) −23.5338 −0.791081
\(886\) 0 0
\(887\) 53.2715 1.78868 0.894340 0.447388i \(-0.147646\pi\)
0.894340 + 0.447388i \(0.147646\pi\)
\(888\) 0 0
\(889\) −82.5669 −2.76921
\(890\) 0 0
\(891\) −3.88477 −0.130145
\(892\) 0 0
\(893\) −1.52503 −0.0510331
\(894\) 0 0
\(895\) 30.6961 1.02606
\(896\) 0 0
\(897\) 5.76924 0.192629
\(898\) 0 0
\(899\) −5.76454 −0.192258
\(900\) 0 0
\(901\) −10.6366 −0.354356
\(902\) 0 0
\(903\) −26.7025 −0.888605
\(904\) 0 0
\(905\) −47.8291 −1.58989
\(906\) 0 0
\(907\) 26.7289 0.887517 0.443759 0.896146i \(-0.353645\pi\)
0.443759 + 0.896146i \(0.353645\pi\)
\(908\) 0 0
\(909\) 16.8588 0.559172
\(910\) 0 0
\(911\) 9.93260 0.329082 0.164541 0.986370i \(-0.447386\pi\)
0.164541 + 0.986370i \(0.447386\pi\)
\(912\) 0 0
\(913\) 55.7791 1.84602
\(914\) 0 0
\(915\) 7.36019 0.243320
\(916\) 0 0
\(917\) 60.5663 2.00007
\(918\) 0 0
\(919\) 42.8801 1.41448 0.707242 0.706971i \(-0.249939\pi\)
0.707242 + 0.706971i \(0.249939\pi\)
\(920\) 0 0
\(921\) −10.3673 −0.341613
\(922\) 0 0
\(923\) −38.3362 −1.26185
\(924\) 0 0
\(925\) 42.9110 1.41090
\(926\) 0 0
\(927\) −1.87256 −0.0615028
\(928\) 0 0
\(929\) 55.6843 1.82694 0.913472 0.406903i \(-0.133391\pi\)
0.913472 + 0.406903i \(0.133391\pi\)
\(930\) 0 0
\(931\) 13.4606 0.441152
\(932\) 0 0
\(933\) 29.7060 0.972530
\(934\) 0 0
\(935\) 21.5825 0.705823
\(936\) 0 0
\(937\) 55.2344 1.80443 0.902215 0.431286i \(-0.141940\pi\)
0.902215 + 0.431286i \(0.141940\pi\)
\(938\) 0 0
\(939\) −10.6960 −0.349049
\(940\) 0 0
\(941\) 13.1855 0.429835 0.214918 0.976632i \(-0.431052\pi\)
0.214918 + 0.976632i \(0.431052\pi\)
\(942\) 0 0
\(943\) −1.21895 −0.0396944
\(944\) 0 0
\(945\) 13.5477 0.440708
\(946\) 0 0
\(947\) 12.4167 0.403488 0.201744 0.979438i \(-0.435339\pi\)
0.201744 + 0.979438i \(0.435339\pi\)
\(948\) 0 0
\(949\) −12.0175 −0.390104
\(950\) 0 0
\(951\) −11.5388 −0.374171
\(952\) 0 0
\(953\) −9.96101 −0.322669 −0.161334 0.986900i \(-0.551580\pi\)
−0.161334 + 0.986900i \(0.551580\pi\)
\(954\) 0 0
\(955\) 48.7916 1.57886
\(956\) 0 0
\(957\) −3.88477 −0.125577
\(958\) 0 0
\(959\) 28.2343 0.911732
\(960\) 0 0
\(961\) 2.22992 0.0719328
\(962\) 0 0
\(963\) 12.8890 0.415343
\(964\) 0 0
\(965\) 14.8811 0.479040
\(966\) 0 0
\(967\) −42.7790 −1.37568 −0.687839 0.725863i \(-0.741441\pi\)
−0.687839 + 0.725863i \(0.741441\pi\)
\(968\) 0 0
\(969\) −2.02881 −0.0651748
\(970\) 0 0
\(971\) −8.09189 −0.259681 −0.129841 0.991535i \(-0.541447\pi\)
−0.129841 + 0.991535i \(0.541447\pi\)
\(972\) 0 0
\(973\) 34.3180 1.10018
\(974\) 0 0
\(975\) 27.4913 0.880425
\(976\) 0 0
\(977\) 14.6916 0.470025 0.235013 0.971992i \(-0.424487\pi\)
0.235013 + 0.971992i \(0.424487\pi\)
\(978\) 0 0
\(979\) 19.0341 0.608331
\(980\) 0 0
\(981\) 0.163721 0.00522722
\(982\) 0 0
\(983\) 45.8602 1.46271 0.731357 0.681995i \(-0.238887\pi\)
0.731357 + 0.681995i \(0.238887\pi\)
\(984\) 0 0
\(985\) 23.9494 0.763091
\(986\) 0 0
\(987\) 5.79375 0.184417
\(988\) 0 0
\(989\) 6.15921 0.195852
\(990\) 0 0
\(991\) −13.0647 −0.415012 −0.207506 0.978234i \(-0.566535\pi\)
−0.207506 + 0.978234i \(0.566535\pi\)
\(992\) 0 0
\(993\) −12.4593 −0.395385
\(994\) 0 0
\(995\) −83.9116 −2.66018
\(996\) 0 0
\(997\) −8.55526 −0.270948 −0.135474 0.990781i \(-0.543256\pi\)
−0.135474 + 0.990781i \(0.543256\pi\)
\(998\) 0 0
\(999\) −9.00519 −0.284912
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.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.2 12 1.1 even 1 trivial