Properties

Label 1216.2.a.t.1.2
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{3} -1.56155 q^{5} -3.00000 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q+2.56155 q^{3} -1.56155 q^{5} -3.00000 q^{7} +3.56155 q^{9} -3.56155 q^{11} -2.56155 q^{13} -4.00000 q^{15} -8.12311 q^{17} +1.00000 q^{19} -7.68466 q^{21} -1.43845 q^{23} -2.56155 q^{25} +1.43845 q^{27} +7.68466 q^{29} -0.876894 q^{31} -9.12311 q^{33} +4.68466 q^{35} +1.12311 q^{37} -6.56155 q^{39} +4.00000 q^{41} +9.56155 q^{43} -5.56155 q^{45} -8.68466 q^{47} +2.00000 q^{49} -20.8078 q^{51} +8.56155 q^{53} +5.56155 q^{55} +2.56155 q^{57} -8.56155 q^{59} +5.80776 q^{61} -10.6847 q^{63} +4.00000 q^{65} -4.56155 q^{67} -3.68466 q^{69} -12.2462 q^{71} +7.24621 q^{73} -6.56155 q^{75} +10.6847 q^{77} -10.0000 q^{79} -7.00000 q^{81} +7.36932 q^{83} +12.6847 q^{85} +19.6847 q^{87} +9.36932 q^{89} +7.68466 q^{91} -2.24621 q^{93} -1.56155 q^{95} -1.12311 q^{97} -12.6847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{5} - 6q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} + q^{5} - 6q^{7} + 3q^{9} - 3q^{11} - q^{13} - 8q^{15} - 8q^{17} + 2q^{19} - 3q^{21} - 7q^{23} - q^{25} + 7q^{27} + 3q^{29} - 10q^{31} - 10q^{33} - 3q^{35} - 6q^{37} - 9q^{39} + 8q^{41} + 15q^{43} - 7q^{45} - 5q^{47} + 4q^{49} - 21q^{51} + 13q^{53} + 7q^{55} + q^{57} - 13q^{59} - 9q^{61} - 9q^{63} + 8q^{65} - 5q^{67} + 5q^{69} - 8q^{71} - 2q^{73} - 9q^{75} + 9q^{77} - 20q^{79} - 14q^{81} - 10q^{83} + 13q^{85} + 27q^{87} - 6q^{89} + 3q^{91} + 12q^{93} + q^{95} + 6q^{97} - 13q^{99} + O(q^{100}) \)

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
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) −1.56155 −0.698348 −0.349174 0.937058i \(-0.613538\pi\)
−0.349174 + 0.937058i \(0.613538\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −3.56155 −1.07385 −0.536924 0.843630i \(-0.680414\pi\)
−0.536924 + 0.843630i \(0.680414\pi\)
\(12\) 0 0
\(13\) −2.56155 −0.710447 −0.355223 0.934781i \(-0.615595\pi\)
−0.355223 + 0.934781i \(0.615595\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −8.12311 −1.97014 −0.985071 0.172147i \(-0.944930\pi\)
−0.985071 + 0.172147i \(0.944930\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −7.68466 −1.67693
\(22\) 0 0
\(23\) −1.43845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) 7.68466 1.42701 0.713503 0.700653i \(-0.247108\pi\)
0.713503 + 0.700653i \(0.247108\pi\)
\(30\) 0 0
\(31\) −0.876894 −0.157495 −0.0787474 0.996895i \(-0.525092\pi\)
−0.0787474 + 0.996895i \(0.525092\pi\)
\(32\) 0 0
\(33\) −9.12311 −1.58813
\(34\) 0 0
\(35\) 4.68466 0.791852
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) −6.56155 −1.05069
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 9.56155 1.45812 0.729062 0.684448i \(-0.239957\pi\)
0.729062 + 0.684448i \(0.239957\pi\)
\(44\) 0 0
\(45\) −5.56155 −0.829067
\(46\) 0 0
\(47\) −8.68466 −1.26679 −0.633394 0.773830i \(-0.718339\pi\)
−0.633394 + 0.773830i \(0.718339\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −20.8078 −2.91367
\(52\) 0 0
\(53\) 8.56155 1.17602 0.588010 0.808854i \(-0.299912\pi\)
0.588010 + 0.808854i \(0.299912\pi\)
\(54\) 0 0
\(55\) 5.56155 0.749920
\(56\) 0 0
\(57\) 2.56155 0.339286
\(58\) 0 0
\(59\) −8.56155 −1.11462 −0.557310 0.830305i \(-0.688166\pi\)
−0.557310 + 0.830305i \(0.688166\pi\)
\(60\) 0 0
\(61\) 5.80776 0.743608 0.371804 0.928311i \(-0.378739\pi\)
0.371804 + 0.928311i \(0.378739\pi\)
\(62\) 0 0
\(63\) −10.6847 −1.34614
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −4.56155 −0.557282 −0.278641 0.960395i \(-0.589884\pi\)
−0.278641 + 0.960395i \(0.589884\pi\)
\(68\) 0 0
\(69\) −3.68466 −0.443581
\(70\) 0 0
\(71\) −12.2462 −1.45336 −0.726679 0.686977i \(-0.758937\pi\)
−0.726679 + 0.686977i \(0.758937\pi\)
\(72\) 0 0
\(73\) 7.24621 0.848105 0.424052 0.905638i \(-0.360607\pi\)
0.424052 + 0.905638i \(0.360607\pi\)
\(74\) 0 0
\(75\) −6.56155 −0.757663
\(76\) 0 0
\(77\) 10.6847 1.21763
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 7.36932 0.808888 0.404444 0.914563i \(-0.367465\pi\)
0.404444 + 0.914563i \(0.367465\pi\)
\(84\) 0 0
\(85\) 12.6847 1.37584
\(86\) 0 0
\(87\) 19.6847 2.11042
\(88\) 0 0
\(89\) 9.36932 0.993146 0.496573 0.867995i \(-0.334592\pi\)
0.496573 + 0.867995i \(0.334592\pi\)
\(90\) 0 0
\(91\) 7.68466 0.805571
\(92\) 0 0
\(93\) −2.24621 −0.232921
\(94\) 0 0
\(95\) −1.56155 −0.160212
\(96\) 0 0
\(97\) −1.12311 −0.114034 −0.0570170 0.998373i \(-0.518159\pi\)
−0.0570170 + 0.998373i \(0.518159\pi\)
\(98\) 0 0
\(99\) −12.6847 −1.27486
\(100\) 0 0
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 0 0
\(103\) 18.4924 1.82211 0.911056 0.412282i \(-0.135268\pi\)
0.911056 + 0.412282i \(0.135268\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) −11.9309 −1.15340 −0.576700 0.816956i \(-0.695660\pi\)
−0.576700 + 0.816956i \(0.695660\pi\)
\(108\) 0 0
\(109\) −0.561553 −0.0537870 −0.0268935 0.999638i \(-0.508561\pi\)
−0.0268935 + 0.999638i \(0.508561\pi\)
\(110\) 0 0
\(111\) 2.87689 0.273063
\(112\) 0 0
\(113\) 13.1231 1.23452 0.617259 0.786760i \(-0.288243\pi\)
0.617259 + 0.786760i \(0.288243\pi\)
\(114\) 0 0
\(115\) 2.24621 0.209460
\(116\) 0 0
\(117\) −9.12311 −0.843431
\(118\) 0 0
\(119\) 24.3693 2.23393
\(120\) 0 0
\(121\) 1.68466 0.153151
\(122\) 0 0
\(123\) 10.2462 0.923870
\(124\) 0 0
\(125\) 11.8078 1.05612
\(126\) 0 0
\(127\) −6.87689 −0.610226 −0.305113 0.952316i \(-0.598694\pi\)
−0.305113 + 0.952316i \(0.598694\pi\)
\(128\) 0 0
\(129\) 24.4924 2.15644
\(130\) 0 0
\(131\) −17.5616 −1.53436 −0.767180 0.641432i \(-0.778341\pi\)
−0.767180 + 0.641432i \(0.778341\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) −2.24621 −0.193323
\(136\) 0 0
\(137\) −10.3693 −0.885911 −0.442955 0.896544i \(-0.646070\pi\)
−0.442955 + 0.896544i \(0.646070\pi\)
\(138\) 0 0
\(139\) 9.56155 0.811000 0.405500 0.914095i \(-0.367097\pi\)
0.405500 + 0.914095i \(0.367097\pi\)
\(140\) 0 0
\(141\) −22.2462 −1.87347
\(142\) 0 0
\(143\) 9.12311 0.762912
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 5.12311 0.422547
\(148\) 0 0
\(149\) −6.43845 −0.527458 −0.263729 0.964597i \(-0.584952\pi\)
−0.263729 + 0.964597i \(0.584952\pi\)
\(150\) 0 0
\(151\) −10.4924 −0.853861 −0.426931 0.904284i \(-0.640405\pi\)
−0.426931 + 0.904284i \(0.640405\pi\)
\(152\) 0 0
\(153\) −28.9309 −2.33892
\(154\) 0 0
\(155\) 1.36932 0.109986
\(156\) 0 0
\(157\) 24.2462 1.93506 0.967529 0.252759i \(-0.0813381\pi\)
0.967529 + 0.252759i \(0.0813381\pi\)
\(158\) 0 0
\(159\) 21.9309 1.73923
\(160\) 0 0
\(161\) 4.31534 0.340097
\(162\) 0 0
\(163\) −16.4924 −1.29179 −0.645893 0.763428i \(-0.723515\pi\)
−0.645893 + 0.763428i \(0.723515\pi\)
\(164\) 0 0
\(165\) 14.2462 1.10907
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −6.43845 −0.495265
\(170\) 0 0
\(171\) 3.56155 0.272359
\(172\) 0 0
\(173\) 3.75379 0.285395 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(174\) 0 0
\(175\) 7.68466 0.580906
\(176\) 0 0
\(177\) −21.9309 −1.64843
\(178\) 0 0
\(179\) −14.2462 −1.06481 −0.532406 0.846489i \(-0.678712\pi\)
−0.532406 + 0.846489i \(0.678712\pi\)
\(180\) 0 0
\(181\) −16.2462 −1.20757 −0.603786 0.797147i \(-0.706342\pi\)
−0.603786 + 0.797147i \(0.706342\pi\)
\(182\) 0 0
\(183\) 14.8769 1.09973
\(184\) 0 0
\(185\) −1.75379 −0.128941
\(186\) 0 0
\(187\) 28.9309 2.11563
\(188\) 0 0
\(189\) −4.31534 −0.313895
\(190\) 0 0
\(191\) 1.00000 0.0723575 0.0361787 0.999345i \(-0.488481\pi\)
0.0361787 + 0.999345i \(0.488481\pi\)
\(192\) 0 0
\(193\) −14.4924 −1.04319 −0.521594 0.853194i \(-0.674662\pi\)
−0.521594 + 0.853194i \(0.674662\pi\)
\(194\) 0 0
\(195\) 10.2462 0.733746
\(196\) 0 0
\(197\) 13.3693 0.952524 0.476262 0.879303i \(-0.341991\pi\)
0.476262 + 0.879303i \(0.341991\pi\)
\(198\) 0 0
\(199\) −25.7386 −1.82456 −0.912282 0.409563i \(-0.865681\pi\)
−0.912282 + 0.409563i \(0.865681\pi\)
\(200\) 0 0
\(201\) −11.6847 −0.824172
\(202\) 0 0
\(203\) −23.0540 −1.61807
\(204\) 0 0
\(205\) −6.24621 −0.436254
\(206\) 0 0
\(207\) −5.12311 −0.356080
\(208\) 0 0
\(209\) −3.56155 −0.246358
\(210\) 0 0
\(211\) 11.4384 0.787455 0.393728 0.919227i \(-0.371185\pi\)
0.393728 + 0.919227i \(0.371185\pi\)
\(212\) 0 0
\(213\) −31.3693 −2.14939
\(214\) 0 0
\(215\) −14.9309 −1.01828
\(216\) 0 0
\(217\) 2.63068 0.178582
\(218\) 0 0
\(219\) 18.5616 1.25427
\(220\) 0 0
\(221\) 20.8078 1.39968
\(222\) 0 0
\(223\) −23.3693 −1.56493 −0.782463 0.622698i \(-0.786037\pi\)
−0.782463 + 0.622698i \(0.786037\pi\)
\(224\) 0 0
\(225\) −9.12311 −0.608207
\(226\) 0 0
\(227\) 4.31534 0.286419 0.143210 0.989692i \(-0.454258\pi\)
0.143210 + 0.989692i \(0.454258\pi\)
\(228\) 0 0
\(229\) 8.43845 0.557628 0.278814 0.960345i \(-0.410059\pi\)
0.278814 + 0.960345i \(0.410059\pi\)
\(230\) 0 0
\(231\) 27.3693 1.80077
\(232\) 0 0
\(233\) 8.93087 0.585081 0.292540 0.956253i \(-0.405499\pi\)
0.292540 + 0.956253i \(0.405499\pi\)
\(234\) 0 0
\(235\) 13.5616 0.884658
\(236\) 0 0
\(237\) −25.6155 −1.66391
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) 2.24621 0.144691 0.0723456 0.997380i \(-0.476952\pi\)
0.0723456 + 0.997380i \(0.476952\pi\)
\(242\) 0 0
\(243\) −22.2462 −1.42710
\(244\) 0 0
\(245\) −3.12311 −0.199528
\(246\) 0 0
\(247\) −2.56155 −0.162988
\(248\) 0 0
\(249\) 18.8769 1.19627
\(250\) 0 0
\(251\) −6.68466 −0.421932 −0.210966 0.977493i \(-0.567661\pi\)
−0.210966 + 0.977493i \(0.567661\pi\)
\(252\) 0 0
\(253\) 5.12311 0.322087
\(254\) 0 0
\(255\) 32.4924 2.03475
\(256\) 0 0
\(257\) 9.75379 0.608425 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(258\) 0 0
\(259\) −3.36932 −0.209359
\(260\) 0 0
\(261\) 27.3693 1.69412
\(262\) 0 0
\(263\) 26.0540 1.60656 0.803278 0.595604i \(-0.203087\pi\)
0.803278 + 0.595604i \(0.203087\pi\)
\(264\) 0 0
\(265\) −13.3693 −0.821271
\(266\) 0 0
\(267\) 24.0000 1.46878
\(268\) 0 0
\(269\) −14.4924 −0.883619 −0.441809 0.897109i \(-0.645663\pi\)
−0.441809 + 0.897109i \(0.645663\pi\)
\(270\) 0 0
\(271\) 27.0540 1.64341 0.821706 0.569912i \(-0.193023\pi\)
0.821706 + 0.569912i \(0.193023\pi\)
\(272\) 0 0
\(273\) 19.6847 1.19137
\(274\) 0 0
\(275\) 9.12311 0.550144
\(276\) 0 0
\(277\) −0.684658 −0.0411371 −0.0205686 0.999788i \(-0.506548\pi\)
−0.0205686 + 0.999788i \(0.506548\pi\)
\(278\) 0 0
\(279\) −3.12311 −0.186975
\(280\) 0 0
\(281\) −25.3693 −1.51341 −0.756703 0.653758i \(-0.773191\pi\)
−0.756703 + 0.653758i \(0.773191\pi\)
\(282\) 0 0
\(283\) 24.4384 1.45271 0.726357 0.687317i \(-0.241212\pi\)
0.726357 + 0.687317i \(0.241212\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 48.9848 2.88146
\(290\) 0 0
\(291\) −2.87689 −0.168647
\(292\) 0 0
\(293\) −27.6847 −1.61736 −0.808678 0.588252i \(-0.799816\pi\)
−0.808678 + 0.588252i \(0.799816\pi\)
\(294\) 0 0
\(295\) 13.3693 0.778392
\(296\) 0 0
\(297\) −5.12311 −0.297273
\(298\) 0 0
\(299\) 3.68466 0.213089
\(300\) 0 0
\(301\) −28.6847 −1.65336
\(302\) 0 0
\(303\) −41.6155 −2.39075
\(304\) 0 0
\(305\) −9.06913 −0.519297
\(306\) 0 0
\(307\) 29.3693 1.67620 0.838098 0.545520i \(-0.183668\pi\)
0.838098 + 0.545520i \(0.183668\pi\)
\(308\) 0 0
\(309\) 47.3693 2.69475
\(310\) 0 0
\(311\) −2.12311 −0.120390 −0.0601951 0.998187i \(-0.519172\pi\)
−0.0601951 + 0.998187i \(0.519172\pi\)
\(312\) 0 0
\(313\) 27.9309 1.57875 0.789373 0.613914i \(-0.210406\pi\)
0.789373 + 0.613914i \(0.210406\pi\)
\(314\) 0 0
\(315\) 16.6847 0.940074
\(316\) 0 0
\(317\) −20.5616 −1.15485 −0.577426 0.816443i \(-0.695943\pi\)
−0.577426 + 0.816443i \(0.695943\pi\)
\(318\) 0 0
\(319\) −27.3693 −1.53239
\(320\) 0 0
\(321\) −30.5616 −1.70578
\(322\) 0 0
\(323\) −8.12311 −0.451982
\(324\) 0 0
\(325\) 6.56155 0.363969
\(326\) 0 0
\(327\) −1.43845 −0.0795463
\(328\) 0 0
\(329\) 26.0540 1.43640
\(330\) 0 0
\(331\) 8.56155 0.470586 0.235293 0.971925i \(-0.424395\pi\)
0.235293 + 0.971925i \(0.424395\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 7.12311 0.389177
\(336\) 0 0
\(337\) 12.4924 0.680506 0.340253 0.940334i \(-0.389487\pi\)
0.340253 + 0.940334i \(0.389487\pi\)
\(338\) 0 0
\(339\) 33.6155 1.82574
\(340\) 0 0
\(341\) 3.12311 0.169126
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 5.75379 0.309774
\(346\) 0 0
\(347\) −0.438447 −0.0235371 −0.0117685 0.999931i \(-0.503746\pi\)
−0.0117685 + 0.999931i \(0.503746\pi\)
\(348\) 0 0
\(349\) −1.31534 −0.0704086 −0.0352043 0.999380i \(-0.511208\pi\)
−0.0352043 + 0.999380i \(0.511208\pi\)
\(350\) 0 0
\(351\) −3.68466 −0.196673
\(352\) 0 0
\(353\) −0.0691303 −0.00367944 −0.00183972 0.999998i \(-0.500586\pi\)
−0.00183972 + 0.999998i \(0.500586\pi\)
\(354\) 0 0
\(355\) 19.1231 1.01495
\(356\) 0 0
\(357\) 62.4233 3.30379
\(358\) 0 0
\(359\) −36.1231 −1.90650 −0.953252 0.302176i \(-0.902287\pi\)
−0.953252 + 0.302176i \(0.902287\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 4.31534 0.226497
\(364\) 0 0
\(365\) −11.3153 −0.592272
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) 14.2462 0.741628
\(370\) 0 0
\(371\) −25.6847 −1.33348
\(372\) 0 0
\(373\) −13.9309 −0.721313 −0.360657 0.932699i \(-0.617447\pi\)
−0.360657 + 0.932699i \(0.617447\pi\)
\(374\) 0 0
\(375\) 30.2462 1.56191
\(376\) 0 0
\(377\) −19.6847 −1.01381
\(378\) 0 0
\(379\) −21.0540 −1.08147 −0.540735 0.841193i \(-0.681854\pi\)
−0.540735 + 0.841193i \(0.681854\pi\)
\(380\) 0 0
\(381\) −17.6155 −0.902471
\(382\) 0 0
\(383\) −7.75379 −0.396200 −0.198100 0.980182i \(-0.563477\pi\)
−0.198100 + 0.980182i \(0.563477\pi\)
\(384\) 0 0
\(385\) −16.6847 −0.850329
\(386\) 0 0
\(387\) 34.0540 1.73106
\(388\) 0 0
\(389\) −6.19224 −0.313959 −0.156979 0.987602i \(-0.550176\pi\)
−0.156979 + 0.987602i \(0.550176\pi\)
\(390\) 0 0
\(391\) 11.6847 0.590919
\(392\) 0 0
\(393\) −44.9848 −2.26919
\(394\) 0 0
\(395\) 15.6155 0.785702
\(396\) 0 0
\(397\) 9.56155 0.479881 0.239940 0.970788i \(-0.422872\pi\)
0.239940 + 0.970788i \(0.422872\pi\)
\(398\) 0 0
\(399\) −7.68466 −0.384714
\(400\) 0 0
\(401\) −1.36932 −0.0683804 −0.0341902 0.999415i \(-0.510885\pi\)
−0.0341902 + 0.999415i \(0.510885\pi\)
\(402\) 0 0
\(403\) 2.24621 0.111892
\(404\) 0 0
\(405\) 10.9309 0.543159
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 27.1231 1.34115 0.670576 0.741841i \(-0.266047\pi\)
0.670576 + 0.741841i \(0.266047\pi\)
\(410\) 0 0
\(411\) −26.5616 −1.31018
\(412\) 0 0
\(413\) 25.6847 1.26386
\(414\) 0 0
\(415\) −11.5076 −0.564885
\(416\) 0 0
\(417\) 24.4924 1.19940
\(418\) 0 0
\(419\) −9.75379 −0.476504 −0.238252 0.971203i \(-0.576574\pi\)
−0.238252 + 0.971203i \(0.576574\pi\)
\(420\) 0 0
\(421\) 17.0540 0.831160 0.415580 0.909557i \(-0.363579\pi\)
0.415580 + 0.909557i \(0.363579\pi\)
\(422\) 0 0
\(423\) −30.9309 −1.50391
\(424\) 0 0
\(425\) 20.8078 1.00932
\(426\) 0 0
\(427\) −17.4233 −0.843172
\(428\) 0 0
\(429\) 23.3693 1.12828
\(430\) 0 0
\(431\) 22.8769 1.10194 0.550971 0.834525i \(-0.314258\pi\)
0.550971 + 0.834525i \(0.314258\pi\)
\(432\) 0 0
\(433\) −1.61553 −0.0776373 −0.0388187 0.999246i \(-0.512359\pi\)
−0.0388187 + 0.999246i \(0.512359\pi\)
\(434\) 0 0
\(435\) −30.7386 −1.47380
\(436\) 0 0
\(437\) −1.43845 −0.0688103
\(438\) 0 0
\(439\) −18.2462 −0.870844 −0.435422 0.900226i \(-0.643401\pi\)
−0.435422 + 0.900226i \(0.643401\pi\)
\(440\) 0 0
\(441\) 7.12311 0.339196
\(442\) 0 0
\(443\) −10.6847 −0.507643 −0.253822 0.967251i \(-0.581688\pi\)
−0.253822 + 0.967251i \(0.581688\pi\)
\(444\) 0 0
\(445\) −14.6307 −0.693561
\(446\) 0 0
\(447\) −16.4924 −0.780065
\(448\) 0 0
\(449\) −6.87689 −0.324541 −0.162270 0.986746i \(-0.551882\pi\)
−0.162270 + 0.986746i \(0.551882\pi\)
\(450\) 0 0
\(451\) −14.2462 −0.670828
\(452\) 0 0
\(453\) −26.8769 −1.26279
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −27.8769 −1.30403 −0.652013 0.758208i \(-0.726075\pi\)
−0.652013 + 0.758208i \(0.726075\pi\)
\(458\) 0 0
\(459\) −11.6847 −0.545393
\(460\) 0 0
\(461\) 5.06913 0.236093 0.118046 0.993008i \(-0.462337\pi\)
0.118046 + 0.993008i \(0.462337\pi\)
\(462\) 0 0
\(463\) 22.0540 1.02494 0.512468 0.858707i \(-0.328731\pi\)
0.512468 + 0.858707i \(0.328731\pi\)
\(464\) 0 0
\(465\) 3.50758 0.162660
\(466\) 0 0
\(467\) 5.31534 0.245965 0.122982 0.992409i \(-0.460754\pi\)
0.122982 + 0.992409i \(0.460754\pi\)
\(468\) 0 0
\(469\) 13.6847 0.631899
\(470\) 0 0
\(471\) 62.1080 2.86178
\(472\) 0 0
\(473\) −34.0540 −1.56580
\(474\) 0 0
\(475\) −2.56155 −0.117532
\(476\) 0 0
\(477\) 30.4924 1.39615
\(478\) 0 0
\(479\) −2.24621 −0.102632 −0.0513160 0.998682i \(-0.516342\pi\)
−0.0513160 + 0.998682i \(0.516342\pi\)
\(480\) 0 0
\(481\) −2.87689 −0.131175
\(482\) 0 0
\(483\) 11.0540 0.502973
\(484\) 0 0
\(485\) 1.75379 0.0796354
\(486\) 0 0
\(487\) −26.4924 −1.20049 −0.600243 0.799818i \(-0.704930\pi\)
−0.600243 + 0.799818i \(0.704930\pi\)
\(488\) 0 0
\(489\) −42.2462 −1.91044
\(490\) 0 0
\(491\) 0.492423 0.0222227 0.0111114 0.999938i \(-0.496463\pi\)
0.0111114 + 0.999938i \(0.496463\pi\)
\(492\) 0 0
\(493\) −62.4233 −2.81140
\(494\) 0 0
\(495\) 19.8078 0.890293
\(496\) 0 0
\(497\) 36.7386 1.64795
\(498\) 0 0
\(499\) 8.68466 0.388779 0.194389 0.980924i \(-0.437727\pi\)
0.194389 + 0.980924i \(0.437727\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.56155 0.292565 0.146283 0.989243i \(-0.453269\pi\)
0.146283 + 0.989243i \(0.453269\pi\)
\(504\) 0 0
\(505\) 25.3693 1.12892
\(506\) 0 0
\(507\) −16.4924 −0.732454
\(508\) 0 0
\(509\) −21.6155 −0.958091 −0.479046 0.877790i \(-0.659017\pi\)
−0.479046 + 0.877790i \(0.659017\pi\)
\(510\) 0 0
\(511\) −21.7386 −0.961661
\(512\) 0 0
\(513\) 1.43845 0.0635090
\(514\) 0 0
\(515\) −28.8769 −1.27247
\(516\) 0 0
\(517\) 30.9309 1.36034
\(518\) 0 0
\(519\) 9.61553 0.422075
\(520\) 0 0
\(521\) −1.36932 −0.0599909 −0.0299954 0.999550i \(-0.509549\pi\)
−0.0299954 + 0.999550i \(0.509549\pi\)
\(522\) 0 0
\(523\) 37.5464 1.64179 0.820895 0.571080i \(-0.193475\pi\)
0.820895 + 0.571080i \(0.193475\pi\)
\(524\) 0 0
\(525\) 19.6847 0.859109
\(526\) 0 0
\(527\) 7.12311 0.310287
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) 0 0
\(531\) −30.4924 −1.32326
\(532\) 0 0
\(533\) −10.2462 −0.443813
\(534\) 0 0
\(535\) 18.6307 0.805475
\(536\) 0 0
\(537\) −36.4924 −1.57476
\(538\) 0 0
\(539\) −7.12311 −0.306814
\(540\) 0 0
\(541\) 39.5616 1.70088 0.850442 0.526069i \(-0.176335\pi\)
0.850442 + 0.526069i \(0.176335\pi\)
\(542\) 0 0
\(543\) −41.6155 −1.78589
\(544\) 0 0
\(545\) 0.876894 0.0375620
\(546\) 0 0
\(547\) −8.49242 −0.363110 −0.181555 0.983381i \(-0.558113\pi\)
−0.181555 + 0.983381i \(0.558113\pi\)
\(548\) 0 0
\(549\) 20.6847 0.882800
\(550\) 0 0
\(551\) 7.68466 0.327377
\(552\) 0 0
\(553\) 30.0000 1.27573
\(554\) 0 0
\(555\) −4.49242 −0.190693
\(556\) 0 0
\(557\) −10.9309 −0.463156 −0.231578 0.972816i \(-0.574389\pi\)
−0.231578 + 0.972816i \(0.574389\pi\)
\(558\) 0 0
\(559\) −24.4924 −1.03592
\(560\) 0 0
\(561\) 74.1080 3.12884
\(562\) 0 0
\(563\) −13.7538 −0.579653 −0.289827 0.957079i \(-0.593598\pi\)
−0.289827 + 0.957079i \(0.593598\pi\)
\(564\) 0 0
\(565\) −20.4924 −0.862123
\(566\) 0 0
\(567\) 21.0000 0.881917
\(568\) 0 0
\(569\) 22.7386 0.953253 0.476627 0.879106i \(-0.341859\pi\)
0.476627 + 0.879106i \(0.341859\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 2.56155 0.107010
\(574\) 0 0
\(575\) 3.68466 0.153661
\(576\) 0 0
\(577\) −11.2462 −0.468186 −0.234093 0.972214i \(-0.575212\pi\)
−0.234093 + 0.972214i \(0.575212\pi\)
\(578\) 0 0
\(579\) −37.1231 −1.54278
\(580\) 0 0
\(581\) −22.1080 −0.917192
\(582\) 0 0
\(583\) −30.4924 −1.26287
\(584\) 0 0
\(585\) 14.2462 0.589008
\(586\) 0 0
\(587\) −5.06913 −0.209225 −0.104613 0.994513i \(-0.533360\pi\)
−0.104613 + 0.994513i \(0.533360\pi\)
\(588\) 0 0
\(589\) −0.876894 −0.0361318
\(590\) 0 0
\(591\) 34.2462 1.40870
\(592\) 0 0
\(593\) −16.7386 −0.687373 −0.343687 0.939084i \(-0.611676\pi\)
−0.343687 + 0.939084i \(0.611676\pi\)
\(594\) 0 0
\(595\) −38.0540 −1.56006
\(596\) 0 0
\(597\) −65.9309 −2.69837
\(598\) 0 0
\(599\) 13.8617 0.566375 0.283188 0.959065i \(-0.408608\pi\)
0.283188 + 0.959065i \(0.408608\pi\)
\(600\) 0 0
\(601\) −29.3693 −1.19800 −0.599000 0.800749i \(-0.704435\pi\)
−0.599000 + 0.800749i \(0.704435\pi\)
\(602\) 0 0
\(603\) −16.2462 −0.661597
\(604\) 0 0
\(605\) −2.63068 −0.106952
\(606\) 0 0
\(607\) 21.1231 0.857360 0.428680 0.903456i \(-0.358979\pi\)
0.428680 + 0.903456i \(0.358979\pi\)
\(608\) 0 0
\(609\) −59.0540 −2.39299
\(610\) 0 0
\(611\) 22.2462 0.899985
\(612\) 0 0
\(613\) −28.5464 −1.15298 −0.576489 0.817105i \(-0.695578\pi\)
−0.576489 + 0.817105i \(0.695578\pi\)
\(614\) 0 0
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) 9.31534 0.375022 0.187511 0.982263i \(-0.439958\pi\)
0.187511 + 0.982263i \(0.439958\pi\)
\(618\) 0 0
\(619\) −38.8769 −1.56259 −0.781297 0.624159i \(-0.785442\pi\)
−0.781297 + 0.624159i \(0.785442\pi\)
\(620\) 0 0
\(621\) −2.06913 −0.0830313
\(622\) 0 0
\(623\) −28.1080 −1.12612
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 0 0
\(627\) −9.12311 −0.364342
\(628\) 0 0
\(629\) −9.12311 −0.363762
\(630\) 0 0
\(631\) −8.30019 −0.330425 −0.165213 0.986258i \(-0.552831\pi\)
−0.165213 + 0.986258i \(0.552831\pi\)
\(632\) 0 0
\(633\) 29.3002 1.16458
\(634\) 0 0
\(635\) 10.7386 0.426150
\(636\) 0 0
\(637\) −5.12311 −0.202985
\(638\) 0 0
\(639\) −43.6155 −1.72540
\(640\) 0 0
\(641\) −50.1080 −1.97915 −0.989573 0.144035i \(-0.953992\pi\)
−0.989573 + 0.144035i \(0.953992\pi\)
\(642\) 0 0
\(643\) −26.3002 −1.03718 −0.518589 0.855024i \(-0.673543\pi\)
−0.518589 + 0.855024i \(0.673543\pi\)
\(644\) 0 0
\(645\) −38.2462 −1.50594
\(646\) 0 0
\(647\) −47.0000 −1.84776 −0.923880 0.382682i \(-0.875001\pi\)
−0.923880 + 0.382682i \(0.875001\pi\)
\(648\) 0 0
\(649\) 30.4924 1.19693
\(650\) 0 0
\(651\) 6.73863 0.264108
\(652\) 0 0
\(653\) 11.8078 0.462074 0.231037 0.972945i \(-0.425788\pi\)
0.231037 + 0.972945i \(0.425788\pi\)
\(654\) 0 0
\(655\) 27.4233 1.07152
\(656\) 0 0
\(657\) 25.8078 1.00686
\(658\) 0 0
\(659\) 37.5464 1.46260 0.731300 0.682056i \(-0.238914\pi\)
0.731300 + 0.682056i \(0.238914\pi\)
\(660\) 0 0
\(661\) −24.1771 −0.940379 −0.470190 0.882565i \(-0.655815\pi\)
−0.470190 + 0.882565i \(0.655815\pi\)
\(662\) 0 0
\(663\) 53.3002 2.07001
\(664\) 0 0
\(665\) 4.68466 0.181663
\(666\) 0 0
\(667\) −11.0540 −0.428012
\(668\) 0 0
\(669\) −59.8617 −2.31439
\(670\) 0 0
\(671\) −20.6847 −0.798522
\(672\) 0 0
\(673\) −12.8769 −0.496368 −0.248184 0.968713i \(-0.579834\pi\)
−0.248184 + 0.968713i \(0.579834\pi\)
\(674\) 0 0
\(675\) −3.68466 −0.141823
\(676\) 0 0
\(677\) 6.56155 0.252181 0.126090 0.992019i \(-0.459757\pi\)
0.126090 + 0.992019i \(0.459757\pi\)
\(678\) 0 0
\(679\) 3.36932 0.129303
\(680\) 0 0
\(681\) 11.0540 0.423589
\(682\) 0 0
\(683\) −34.2462 −1.31039 −0.655197 0.755458i \(-0.727415\pi\)
−0.655197 + 0.755458i \(0.727415\pi\)
\(684\) 0 0
\(685\) 16.1922 0.618674
\(686\) 0 0
\(687\) 21.6155 0.824684
\(688\) 0 0
\(689\) −21.9309 −0.835500
\(690\) 0 0
\(691\) −7.94602 −0.302281 −0.151141 0.988512i \(-0.548295\pi\)
−0.151141 + 0.988512i \(0.548295\pi\)
\(692\) 0 0
\(693\) 38.0540 1.44555
\(694\) 0 0
\(695\) −14.9309 −0.566360
\(696\) 0 0
\(697\) −32.4924 −1.23074
\(698\) 0 0
\(699\) 22.8769 0.865284
\(700\) 0 0
\(701\) 27.6155 1.04302 0.521512 0.853244i \(-0.325368\pi\)
0.521512 + 0.853244i \(0.325368\pi\)
\(702\) 0 0
\(703\) 1.12311 0.0423587
\(704\) 0 0
\(705\) 34.7386 1.30833
\(706\) 0 0
\(707\) 48.7386 1.83300
\(708\) 0 0
\(709\) −17.5076 −0.657511 −0.328755 0.944415i \(-0.606629\pi\)
−0.328755 + 0.944415i \(0.606629\pi\)
\(710\) 0 0
\(711\) −35.6155 −1.33569
\(712\) 0 0
\(713\) 1.26137 0.0472385
\(714\) 0 0
\(715\) −14.2462 −0.532778
\(716\) 0 0
\(717\) −38.4233 −1.43494
\(718\) 0 0
\(719\) 9.49242 0.354008 0.177004 0.984210i \(-0.443360\pi\)
0.177004 + 0.984210i \(0.443360\pi\)
\(720\) 0 0
\(721\) −55.4773 −2.06608
\(722\) 0 0
\(723\) 5.75379 0.213986
\(724\) 0 0
\(725\) −19.6847 −0.731070
\(726\) 0 0
\(727\) 2.36932 0.0878731 0.0439365 0.999034i \(-0.486010\pi\)
0.0439365 + 0.999034i \(0.486010\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −77.6695 −2.87271
\(732\) 0 0
\(733\) 21.3693 0.789294 0.394647 0.918833i \(-0.370867\pi\)
0.394647 + 0.918833i \(0.370867\pi\)
\(734\) 0 0
\(735\) −8.00000 −0.295084
\(736\) 0 0
\(737\) 16.2462 0.598437
\(738\) 0 0
\(739\) 5.56155 0.204585 0.102293 0.994754i \(-0.467382\pi\)
0.102293 + 0.994754i \(0.467382\pi\)
\(740\) 0 0
\(741\) −6.56155 −0.241045
\(742\) 0 0
\(743\) −31.1231 −1.14180 −0.570898 0.821021i \(-0.693405\pi\)
−0.570898 + 0.821021i \(0.693405\pi\)
\(744\) 0 0
\(745\) 10.0540 0.368349
\(746\) 0 0
\(747\) 26.2462 0.960299
\(748\) 0 0
\(749\) 35.7926 1.30783
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 0 0
\(753\) −17.1231 −0.624001
\(754\) 0 0
\(755\) 16.3845 0.596292
\(756\) 0 0
\(757\) 16.0540 0.583492 0.291746 0.956496i \(-0.405764\pi\)
0.291746 + 0.956496i \(0.405764\pi\)
\(758\) 0 0
\(759\) 13.1231 0.476339
\(760\) 0 0
\(761\) 16.8617 0.611238 0.305619 0.952154i \(-0.401137\pi\)
0.305619 + 0.952154i \(0.401137\pi\)
\(762\) 0 0
\(763\) 1.68466 0.0609887
\(764\) 0 0
\(765\) 45.1771 1.63338
\(766\) 0 0
\(767\) 21.9309 0.791878
\(768\) 0 0
\(769\) −12.3693 −0.446049 −0.223024 0.974813i \(-0.571593\pi\)
−0.223024 + 0.974813i \(0.571593\pi\)
\(770\) 0 0
\(771\) 24.9848 0.899807
\(772\) 0 0
\(773\) −5.05398 −0.181779 −0.0908894 0.995861i \(-0.528971\pi\)
−0.0908894 + 0.995861i \(0.528971\pi\)
\(774\) 0 0
\(775\) 2.24621 0.0806863
\(776\) 0 0
\(777\) −8.63068 −0.309624
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 43.6155 1.56069
\(782\) 0 0
\(783\) 11.0540 0.395037
\(784\) 0 0
\(785\) −37.8617 −1.35134
\(786\) 0 0
\(787\) 26.5616 0.946817 0.473409 0.880843i \(-0.343023\pi\)
0.473409 + 0.880843i \(0.343023\pi\)
\(788\) 0 0
\(789\) 66.7386 2.37596
\(790\) 0 0
\(791\) −39.3693 −1.39981
\(792\) 0 0
\(793\) −14.8769 −0.528294
\(794\) 0 0
\(795\) −34.2462 −1.21459
\(796\) 0 0
\(797\) −9.68466 −0.343048 −0.171524 0.985180i \(-0.554869\pi\)
−0.171524 + 0.985180i \(0.554869\pi\)
\(798\) 0 0
\(799\) 70.5464 2.49575
\(800\) 0 0
\(801\) 33.3693 1.17905
\(802\) 0 0
\(803\) −25.8078 −0.910736
\(804\) 0 0
\(805\) −6.73863 −0.237506
\(806\) 0 0
\(807\) −37.1231 −1.30680
\(808\) 0 0
\(809\) −2.12311 −0.0746444 −0.0373222 0.999303i \(-0.511883\pi\)
−0.0373222 + 0.999303i \(0.511883\pi\)
\(810\) 0 0
\(811\) 50.4233 1.77060 0.885301 0.465019i \(-0.153953\pi\)
0.885301 + 0.465019i \(0.153953\pi\)
\(812\) 0 0
\(813\) 69.3002 2.43046
\(814\) 0 0
\(815\) 25.7538 0.902116
\(816\) 0 0
\(817\) 9.56155 0.334516
\(818\) 0 0
\(819\) 27.3693 0.956361
\(820\) 0 0
\(821\) −16.6847 −0.582299 −0.291149 0.956678i \(-0.594038\pi\)
−0.291149 + 0.956678i \(0.594038\pi\)
\(822\) 0 0
\(823\) 4.36932 0.152305 0.0761524 0.997096i \(-0.475736\pi\)
0.0761524 + 0.997096i \(0.475736\pi\)
\(824\) 0 0
\(825\) 23.3693 0.813615
\(826\) 0 0
\(827\) 15.4384 0.536847 0.268424 0.963301i \(-0.413497\pi\)
0.268424 + 0.963301i \(0.413497\pi\)
\(828\) 0 0
\(829\) 51.5464 1.79028 0.895140 0.445785i \(-0.147075\pi\)
0.895140 + 0.445785i \(0.147075\pi\)
\(830\) 0 0
\(831\) −1.75379 −0.0608383
\(832\) 0 0
\(833\) −16.2462 −0.562898
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.26137 −0.0435992
\(838\) 0 0
\(839\) 11.6155 0.401013 0.200506 0.979692i \(-0.435741\pi\)
0.200506 + 0.979692i \(0.435741\pi\)
\(840\) 0 0
\(841\) 30.0540 1.03634
\(842\) 0 0
\(843\) −64.9848 −2.23820
\(844\) 0 0
\(845\) 10.0540 0.345867
\(846\) 0 0
\(847\) −5.05398 −0.173657
\(848\) 0 0
\(849\) 62.6004 2.14844
\(850\) 0 0
\(851\) −1.61553 −0.0553796
\(852\) 0 0
\(853\) −15.7538 −0.539399 −0.269700 0.962944i \(-0.586924\pi\)
−0.269700 + 0.962944i \(0.586924\pi\)
\(854\) 0 0
\(855\) −5.56155 −0.190201
\(856\) 0 0
\(857\) 37.3693 1.27651 0.638256 0.769824i \(-0.279656\pi\)
0.638256 + 0.769824i \(0.279656\pi\)
\(858\) 0 0
\(859\) −7.06913 −0.241196 −0.120598 0.992701i \(-0.538481\pi\)
−0.120598 + 0.992701i \(0.538481\pi\)
\(860\) 0 0
\(861\) −30.7386 −1.04757
\(862\) 0 0
\(863\) −47.3693 −1.61247 −0.806235 0.591595i \(-0.798498\pi\)
−0.806235 + 0.591595i \(0.798498\pi\)
\(864\) 0 0
\(865\) −5.86174 −0.199305
\(866\) 0 0
\(867\) 125.477 4.26143
\(868\) 0 0
\(869\) 35.6155 1.20817
\(870\) 0 0
\(871\) 11.6847 0.395920
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) −35.4233 −1.19753
\(876\) 0 0
\(877\) 9.68466 0.327028 0.163514 0.986541i \(-0.447717\pi\)
0.163514 + 0.986541i \(0.447717\pi\)
\(878\) 0 0
\(879\) −70.9157 −2.39193
\(880\) 0 0
\(881\) 37.3153 1.25719 0.628593 0.777735i \(-0.283631\pi\)
0.628593 + 0.777735i \(0.283631\pi\)
\(882\) 0 0
\(883\) 28.9309 0.973601 0.486801 0.873513i \(-0.338164\pi\)
0.486801 + 0.873513i \(0.338164\pi\)
\(884\) 0 0
\(885\) 34.2462 1.15117
\(886\) 0 0
\(887\) −45.2311 −1.51871 −0.759355 0.650676i \(-0.774485\pi\)
−0.759355 + 0.650676i \(0.774485\pi\)
\(888\) 0 0
\(889\) 20.6307 0.691931
\(890\) 0 0
\(891\) 24.9309 0.835216
\(892\) 0 0
\(893\) −8.68466 −0.290621
\(894\) 0 0
\(895\) 22.2462 0.743609
\(896\) 0 0
\(897\) 9.43845 0.315141
\(898\) 0 0
\(899\) −6.73863 −0.224746
\(900\) 0 0
\(901\) −69.5464 −2.31693
\(902\) 0 0
\(903\) −73.4773 −2.44517
\(904\) 0 0
\(905\) 25.3693 0.843305
\(906\) 0 0
\(907\) −11.4384 −0.379807 −0.189904 0.981803i \(-0.560818\pi\)
−0.189904 + 0.981803i \(0.560818\pi\)
\(908\) 0 0
\(909\) −57.8617 −1.91915
\(910\) 0 0
\(911\) 0.876894 0.0290528 0.0145264 0.999894i \(-0.495376\pi\)
0.0145264 + 0.999894i \(0.495376\pi\)
\(912\) 0 0
\(913\) −26.2462 −0.868623
\(914\) 0 0
\(915\) −23.2311 −0.767995
\(916\) 0 0
\(917\) 52.6847 1.73980
\(918\) 0 0
\(919\) −29.3002 −0.966524 −0.483262 0.875476i \(-0.660548\pi\)
−0.483262 + 0.875476i \(0.660548\pi\)
\(920\) 0 0
\(921\) 75.2311 2.47895
\(922\) 0 0
\(923\) 31.3693 1.03253
\(924\) 0 0
\(925\) −2.87689 −0.0945917
\(926\) 0 0
\(927\) 65.8617 2.16318
\(928\) 0 0
\(929\) −9.82292 −0.322280 −0.161140 0.986932i \(-0.551517\pi\)
−0.161140 + 0.986932i \(0.551517\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) −5.43845 −0.178047
\(934\) 0 0
\(935\) −45.1771 −1.47745
\(936\) 0 0
\(937\) −29.2462 −0.955432 −0.477716 0.878514i \(-0.658535\pi\)
−0.477716 + 0.878514i \(0.658535\pi\)
\(938\) 0 0
\(939\) 71.5464 2.33483
\(940\) 0 0
\(941\) 35.3002 1.15075 0.575377 0.817889i \(-0.304856\pi\)
0.575377 + 0.817889i \(0.304856\pi\)
\(942\) 0 0
\(943\) −5.75379 −0.187369
\(944\) 0 0
\(945\) 6.73863 0.219208
\(946\) 0 0
\(947\) −28.9848 −0.941881 −0.470940 0.882165i \(-0.656085\pi\)
−0.470940 + 0.882165i \(0.656085\pi\)
\(948\) 0 0
\(949\) −18.5616 −0.602534
\(950\) 0 0
\(951\) −52.6695 −1.70793
\(952\) 0 0
\(953\) 8.24621 0.267121 0.133560 0.991041i \(-0.457359\pi\)
0.133560 + 0.991041i \(0.457359\pi\)
\(954\) 0 0
\(955\) −1.56155 −0.0505307
\(956\) 0 0
\(957\) −70.1080 −2.26627
\(958\) 0 0
\(959\) 31.1080 1.00453
\(960\) 0 0
\(961\) −30.2311 −0.975195
\(962\) 0 0
\(963\) −42.4924 −1.36930
\(964\) 0 0
\(965\) 22.6307 0.728507
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 0 0
\(969\) −20.8078 −0.668442
\(970\) 0 0
\(971\) −56.4924 −1.81293 −0.906464 0.422283i \(-0.861229\pi\)
−0.906464 + 0.422283i \(0.861229\pi\)
\(972\) 0 0
\(973\) −28.6847 −0.919588
\(974\) 0 0
\(975\) 16.8078 0.538279
\(976\) 0 0
\(977\) −12.3845 −0.396214 −0.198107 0.980180i \(-0.563479\pi\)
−0.198107 + 0.980180i \(0.563479\pi\)
\(978\) 0 0
\(979\) −33.3693 −1.06649
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) 29.6155 0.944589 0.472294 0.881441i \(-0.343426\pi\)
0.472294 + 0.881441i \(0.343426\pi\)
\(984\) 0 0
\(985\) −20.8769 −0.665193
\(986\) 0 0
\(987\) 66.7386 2.12431
\(988\) 0 0
\(989\) −13.7538 −0.437345
\(990\) 0 0
\(991\) 1.75379 0.0557109 0.0278555 0.999612i \(-0.491132\pi\)
0.0278555 + 0.999612i \(0.491132\pi\)
\(992\) 0 0
\(993\) 21.9309 0.695955
\(994\) 0 0
\(995\) 40.1922 1.27418
\(996\) 0 0
\(997\) −47.9157 −1.51751 −0.758753 0.651379i \(-0.774191\pi\)
−0.758753 + 0.651379i \(0.774191\pi\)
\(998\) 0 0
\(999\) 1.61553 0.0511130
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 1216.2.a.t.1.2 2
4.3 odd 2 1216.2.a.s.1.1 2
8.3 odd 2 608.2.a.h.1.2 yes 2
8.5 even 2 608.2.a.g.1.1 2
24.5 odd 2 5472.2.a.bc.1.1 2
24.11 even 2 5472.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.g.1.1 2 8.5 even 2
608.2.a.h.1.2 yes 2 8.3 odd 2
1216.2.a.s.1.1 2 4.3 odd 2
1216.2.a.t.1.2 2 1.1 even 1 trivial
5472.2.a.bc.1.1 2 24.5 odd 2
5472.2.a.bf.1.1 2 24.11 even 2