Properties

Label 1521.2.a.f.1.2
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.82843 q^{4} -2.82843 q^{5} -2.82843 q^{7} -1.58579 q^{8} +O(q^{10})\) \(q+0.414214 q^{2} -1.82843 q^{4} -2.82843 q^{5} -2.82843 q^{7} -1.58579 q^{8} -1.17157 q^{10} -2.00000 q^{11} -1.17157 q^{14} +3.00000 q^{16} -7.65685 q^{17} +2.82843 q^{19} +5.17157 q^{20} -0.828427 q^{22} +4.00000 q^{23} +3.00000 q^{25} +5.17157 q^{28} -2.00000 q^{29} +1.17157 q^{31} +4.41421 q^{32} -3.17157 q^{34} +8.00000 q^{35} +7.65685 q^{37} +1.17157 q^{38} +4.48528 q^{40} +5.17157 q^{41} -1.65685 q^{43} +3.65685 q^{44} +1.65685 q^{46} -11.6569 q^{47} +1.00000 q^{49} +1.24264 q^{50} +2.00000 q^{53} +5.65685 q^{55} +4.48528 q^{56} -0.828427 q^{58} +7.65685 q^{59} +13.3137 q^{61} +0.485281 q^{62} -4.17157 q^{64} -6.82843 q^{67} +14.0000 q^{68} +3.31371 q^{70} +2.00000 q^{71} -0.343146 q^{73} +3.17157 q^{74} -5.17157 q^{76} +5.65685 q^{77} -11.3137 q^{79} -8.48528 q^{80} +2.14214 q^{82} +3.65685 q^{83} +21.6569 q^{85} -0.686292 q^{86} +3.17157 q^{88} +14.8284 q^{89} -7.31371 q^{92} -4.82843 q^{94} -8.00000 q^{95} -3.65685 q^{97} +0.414214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{8} - 8 q^{10} - 4 q^{11} - 8 q^{14} + 6 q^{16} - 4 q^{17} + 16 q^{20} + 4 q^{22} + 8 q^{23} + 6 q^{25} + 16 q^{28} - 4 q^{29} + 8 q^{31} + 6 q^{32} - 12 q^{34} + 16 q^{35} + 4 q^{37} + 8 q^{38} - 8 q^{40} + 16 q^{41} + 8 q^{43} - 4 q^{44} - 8 q^{46} - 12 q^{47} + 2 q^{49} - 6 q^{50} + 4 q^{53} - 8 q^{56} + 4 q^{58} + 4 q^{59} + 4 q^{61} - 16 q^{62} - 14 q^{64} - 8 q^{67} + 28 q^{68} - 16 q^{70} + 4 q^{71} - 12 q^{73} + 12 q^{74} - 16 q^{76} - 24 q^{82} - 4 q^{83} + 32 q^{85} - 24 q^{86} + 12 q^{88} + 24 q^{89} + 8 q^{92} - 4 q^{94} - 16 q^{95} + 4 q^{97} - 2 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) −1.58579 −0.560660
\(9\) 0 0
\(10\) −1.17157 −0.370484
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −1.17157 −0.313116
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 5.17157 1.15640
\(21\) 0 0
\(22\) −0.828427 −0.176621
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 5.17157 0.977335
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 4.41421 0.780330
\(33\) 0 0
\(34\) −3.17157 −0.543920
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) 7.65685 1.25878 0.629390 0.777090i \(-0.283305\pi\)
0.629390 + 0.777090i \(0.283305\pi\)
\(38\) 1.17157 0.190054
\(39\) 0 0
\(40\) 4.48528 0.709185
\(41\) 5.17157 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(42\) 0 0
\(43\) −1.65685 −0.252668 −0.126334 0.991988i \(-0.540321\pi\)
−0.126334 + 0.991988i \(0.540321\pi\)
\(44\) 3.65685 0.551292
\(45\) 0 0
\(46\) 1.65685 0.244290
\(47\) −11.6569 −1.70033 −0.850163 0.526519i \(-0.823497\pi\)
−0.850163 + 0.526519i \(0.823497\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.24264 0.175736
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 4.48528 0.599371
\(57\) 0 0
\(58\) −0.828427 −0.108778
\(59\) 7.65685 0.996838 0.498419 0.866936i \(-0.333914\pi\)
0.498419 + 0.866936i \(0.333914\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0.485281 0.0616308
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) −6.82843 −0.834225 −0.417113 0.908855i \(-0.636958\pi\)
−0.417113 + 0.908855i \(0.636958\pi\)
\(68\) 14.0000 1.69775
\(69\) 0 0
\(70\) 3.31371 0.396064
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −0.343146 −0.0401622 −0.0200811 0.999798i \(-0.506392\pi\)
−0.0200811 + 0.999798i \(0.506392\pi\)
\(74\) 3.17157 0.368688
\(75\) 0 0
\(76\) −5.17157 −0.593220
\(77\) 5.65685 0.644658
\(78\) 0 0
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) −8.48528 −0.948683
\(81\) 0 0
\(82\) 2.14214 0.236559
\(83\) 3.65685 0.401392 0.200696 0.979654i \(-0.435680\pi\)
0.200696 + 0.979654i \(0.435680\pi\)
\(84\) 0 0
\(85\) 21.6569 2.34902
\(86\) −0.686292 −0.0740047
\(87\) 0 0
\(88\) 3.17157 0.338091
\(89\) 14.8284 1.57181 0.785905 0.618347i \(-0.212197\pi\)
0.785905 + 0.618347i \(0.212197\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.31371 −0.762507
\(93\) 0 0
\(94\) −4.82843 −0.498014
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) 0.414214 0.0418419
\(99\) 0 0
\(100\) −5.48528 −0.548528
\(101\) −7.65685 −0.761885 −0.380943 0.924599i \(-0.624401\pi\)
−0.380943 + 0.924599i \(0.624401\pi\)
\(102\) 0 0
\(103\) 2.34315 0.230877 0.115439 0.993315i \(-0.463173\pi\)
0.115439 + 0.993315i \(0.463173\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.828427 0.0804640
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 0 0
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 2.34315 0.223410
\(111\) 0 0
\(112\) −8.48528 −0.801784
\(113\) 5.31371 0.499872 0.249936 0.968262i \(-0.419590\pi\)
0.249936 + 0.968262i \(0.419590\pi\)
\(114\) 0 0
\(115\) −11.3137 −1.05501
\(116\) 3.65685 0.339530
\(117\) 0 0
\(118\) 3.17157 0.291967
\(119\) 21.6569 1.98528
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.51472 0.499279
\(123\) 0 0
\(124\) −2.14214 −0.192369
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) −10.5563 −0.933058
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −2.82843 −0.244339
\(135\) 0 0
\(136\) 12.1421 1.04118
\(137\) −10.8284 −0.925135 −0.462567 0.886584i \(-0.653072\pi\)
−0.462567 + 0.886584i \(0.653072\pi\)
\(138\) 0 0
\(139\) −7.31371 −0.620341 −0.310170 0.950681i \(-0.600386\pi\)
−0.310170 + 0.950681i \(0.600386\pi\)
\(140\) −14.6274 −1.23624
\(141\) 0 0
\(142\) 0.828427 0.0695201
\(143\) 0 0
\(144\) 0 0
\(145\) 5.65685 0.469776
\(146\) −0.142136 −0.0117632
\(147\) 0 0
\(148\) −14.0000 −1.15079
\(149\) −9.17157 −0.751365 −0.375682 0.926749i \(-0.622592\pi\)
−0.375682 + 0.926749i \(0.622592\pi\)
\(150\) 0 0
\(151\) 3.51472 0.286024 0.143012 0.989721i \(-0.454321\pi\)
0.143012 + 0.989721i \(0.454321\pi\)
\(152\) −4.48528 −0.363804
\(153\) 0 0
\(154\) 2.34315 0.188816
\(155\) −3.31371 −0.266163
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −4.68629 −0.372821
\(159\) 0 0
\(160\) −12.4853 −0.987048
\(161\) −11.3137 −0.891645
\(162\) 0 0
\(163\) −18.8284 −1.47476 −0.737378 0.675480i \(-0.763936\pi\)
−0.737378 + 0.675480i \(0.763936\pi\)
\(164\) −9.45584 −0.738377
\(165\) 0 0
\(166\) 1.51472 0.117565
\(167\) −3.65685 −0.282976 −0.141488 0.989940i \(-0.545189\pi\)
−0.141488 + 0.989940i \(0.545189\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 8.97056 0.688011
\(171\) 0 0
\(172\) 3.02944 0.230992
\(173\) 11.6569 0.886254 0.443127 0.896459i \(-0.353869\pi\)
0.443127 + 0.896459i \(0.353869\pi\)
\(174\) 0 0
\(175\) −8.48528 −0.641427
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) 6.14214 0.460373
\(179\) 23.3137 1.74255 0.871274 0.490797i \(-0.163294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.34315 −0.467623
\(185\) −21.6569 −1.59224
\(186\) 0 0
\(187\) 15.3137 1.11985
\(188\) 21.3137 1.55446
\(189\) 0 0
\(190\) −3.31371 −0.240402
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) −5.31371 −0.382489 −0.191245 0.981542i \(-0.561252\pi\)
−0.191245 + 0.981542i \(0.561252\pi\)
\(194\) −1.51472 −0.108750
\(195\) 0 0
\(196\) −1.82843 −0.130602
\(197\) 0.485281 0.0345749 0.0172874 0.999851i \(-0.494497\pi\)
0.0172874 + 0.999851i \(0.494497\pi\)
\(198\) 0 0
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) −4.75736 −0.336396
\(201\) 0 0
\(202\) −3.17157 −0.223151
\(203\) 5.65685 0.397033
\(204\) 0 0
\(205\) −14.6274 −1.02162
\(206\) 0.970563 0.0676223
\(207\) 0 0
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −3.65685 −0.251154
\(213\) 0 0
\(214\) 4.68629 0.320348
\(215\) 4.68629 0.319602
\(216\) 0 0
\(217\) −3.31371 −0.224949
\(218\) −2.20101 −0.149071
\(219\) 0 0
\(220\) −10.3431 −0.697335
\(221\) 0 0
\(222\) 0 0
\(223\) 12.4853 0.836076 0.418038 0.908429i \(-0.362718\pi\)
0.418038 + 0.908429i \(0.362718\pi\)
\(224\) −12.4853 −0.834208
\(225\) 0 0
\(226\) 2.20101 0.146409
\(227\) −17.3137 −1.14915 −0.574576 0.818452i \(-0.694833\pi\)
−0.574576 + 0.818452i \(0.694833\pi\)
\(228\) 0 0
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) −4.68629 −0.309005
\(231\) 0 0
\(232\) 3.17157 0.208224
\(233\) −6.97056 −0.456657 −0.228328 0.973584i \(-0.573326\pi\)
−0.228328 + 0.973584i \(0.573326\pi\)
\(234\) 0 0
\(235\) 32.9706 2.15076
\(236\) −14.0000 −0.911322
\(237\) 0 0
\(238\) 8.97056 0.581475
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) −0.343146 −0.0221040 −0.0110520 0.999939i \(-0.503518\pi\)
−0.0110520 + 0.999939i \(0.503518\pi\)
\(242\) −2.89949 −0.186387
\(243\) 0 0
\(244\) −24.3431 −1.55841
\(245\) −2.82843 −0.180702
\(246\) 0 0
\(247\) 0 0
\(248\) −1.85786 −0.117975
\(249\) 0 0
\(250\) 2.34315 0.148194
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 2.34315 0.147022
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 4.34315 0.270918 0.135459 0.990783i \(-0.456749\pi\)
0.135459 + 0.990783i \(0.456749\pi\)
\(258\) 0 0
\(259\) −21.6569 −1.34569
\(260\) 0 0
\(261\) 0 0
\(262\) 3.31371 0.204722
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −5.65685 −0.347498
\(266\) −3.31371 −0.203177
\(267\) 0 0
\(268\) 12.4853 0.762660
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 27.7990 1.68867 0.844334 0.535817i \(-0.179996\pi\)
0.844334 + 0.535817i \(0.179996\pi\)
\(272\) −22.9706 −1.39279
\(273\) 0 0
\(274\) −4.48528 −0.270966
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −3.02944 −0.181694
\(279\) 0 0
\(280\) −12.6863 −0.758151
\(281\) 21.1716 1.26299 0.631495 0.775380i \(-0.282442\pi\)
0.631495 + 0.775380i \(0.282442\pi\)
\(282\) 0 0
\(283\) 28.9706 1.72212 0.861061 0.508502i \(-0.169801\pi\)
0.861061 + 0.508502i \(0.169801\pi\)
\(284\) −3.65685 −0.216994
\(285\) 0 0
\(286\) 0 0
\(287\) −14.6274 −0.863429
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 2.34315 0.137594
\(291\) 0 0
\(292\) 0.627417 0.0367168
\(293\) 2.14214 0.125145 0.0625724 0.998040i \(-0.480070\pi\)
0.0625724 + 0.998040i \(0.480070\pi\)
\(294\) 0 0
\(295\) −21.6569 −1.26091
\(296\) −12.1421 −0.705747
\(297\) 0 0
\(298\) −3.79899 −0.220070
\(299\) 0 0
\(300\) 0 0
\(301\) 4.68629 0.270113
\(302\) 1.45584 0.0837744
\(303\) 0 0
\(304\) 8.48528 0.486664
\(305\) −37.6569 −2.15623
\(306\) 0 0
\(307\) 22.8284 1.30289 0.651444 0.758697i \(-0.274164\pi\)
0.651444 + 0.758697i \(0.274164\pi\)
\(308\) −10.3431 −0.589355
\(309\) 0 0
\(310\) −1.37258 −0.0779575
\(311\) 10.6274 0.602626 0.301313 0.953525i \(-0.402575\pi\)
0.301313 + 0.953525i \(0.402575\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −4.14214 −0.233754
\(315\) 0 0
\(316\) 20.6863 1.16369
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 11.7990 0.659584
\(321\) 0 0
\(322\) −4.68629 −0.261157
\(323\) −21.6569 −1.20502
\(324\) 0 0
\(325\) 0 0
\(326\) −7.79899 −0.431946
\(327\) 0 0
\(328\) −8.20101 −0.452825
\(329\) 32.9706 1.81773
\(330\) 0 0
\(331\) −26.1421 −1.43690 −0.718451 0.695578i \(-0.755148\pi\)
−0.718451 + 0.695578i \(0.755148\pi\)
\(332\) −6.68629 −0.366958
\(333\) 0 0
\(334\) −1.51472 −0.0828817
\(335\) 19.3137 1.05522
\(336\) 0 0
\(337\) 9.31371 0.507350 0.253675 0.967290i \(-0.418361\pi\)
0.253675 + 0.967290i \(0.418361\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −39.5980 −2.14750
\(341\) −2.34315 −0.126888
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 2.62742 0.141661
\(345\) 0 0
\(346\) 4.82843 0.259578
\(347\) 8.68629 0.466305 0.233152 0.972440i \(-0.425096\pi\)
0.233152 + 0.972440i \(0.425096\pi\)
\(348\) 0 0
\(349\) −3.65685 −0.195747 −0.0978735 0.995199i \(-0.531204\pi\)
−0.0978735 + 0.995199i \(0.531204\pi\)
\(350\) −3.51472 −0.187870
\(351\) 0 0
\(352\) −8.82843 −0.470557
\(353\) −33.4558 −1.78067 −0.890337 0.455301i \(-0.849532\pi\)
−0.890337 + 0.455301i \(0.849532\pi\)
\(354\) 0 0
\(355\) −5.65685 −0.300235
\(356\) −27.1127 −1.43697
\(357\) 0 0
\(358\) 9.65685 0.510381
\(359\) 34.9706 1.84568 0.922838 0.385189i \(-0.125864\pi\)
0.922838 + 0.385189i \(0.125864\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 5.79899 0.304788
\(363\) 0 0
\(364\) 0 0
\(365\) 0.970563 0.0508016
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 12.0000 0.625543
\(369\) 0 0
\(370\) −8.97056 −0.466357
\(371\) −5.65685 −0.293689
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 6.34315 0.327996
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) 0 0
\(378\) 0 0
\(379\) −0.485281 −0.0249272 −0.0124636 0.999922i \(-0.503967\pi\)
−0.0124636 + 0.999922i \(0.503967\pi\)
\(380\) 14.6274 0.750371
\(381\) 0 0
\(382\) −1.37258 −0.0702275
\(383\) 30.9706 1.58252 0.791261 0.611479i \(-0.209425\pi\)
0.791261 + 0.611479i \(0.209425\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) −2.20101 −0.112028
\(387\) 0 0
\(388\) 6.68629 0.339445
\(389\) 26.9706 1.36746 0.683731 0.729734i \(-0.260356\pi\)
0.683731 + 0.729734i \(0.260356\pi\)
\(390\) 0 0
\(391\) −30.6274 −1.54890
\(392\) −1.58579 −0.0800943
\(393\) 0 0
\(394\) 0.201010 0.0101267
\(395\) 32.0000 1.61009
\(396\) 0 0
\(397\) −30.9706 −1.55437 −0.777184 0.629273i \(-0.783353\pi\)
−0.777184 + 0.629273i \(0.783353\pi\)
\(398\) 8.97056 0.449654
\(399\) 0 0
\(400\) 9.00000 0.450000
\(401\) 26.1421 1.30548 0.652738 0.757584i \(-0.273620\pi\)
0.652738 + 0.757584i \(0.273620\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 2.34315 0.116288
\(407\) −15.3137 −0.759072
\(408\) 0 0
\(409\) 34.9706 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(410\) −6.05887 −0.299226
\(411\) 0 0
\(412\) −4.28427 −0.211071
\(413\) −21.6569 −1.06566
\(414\) 0 0
\(415\) −10.3431 −0.507725
\(416\) 0 0
\(417\) 0 0
\(418\) −2.34315 −0.114607
\(419\) −14.6274 −0.714596 −0.357298 0.933990i \(-0.616302\pi\)
−0.357298 + 0.933990i \(0.616302\pi\)
\(420\) 0 0
\(421\) −37.3137 −1.81856 −0.909279 0.416186i \(-0.863366\pi\)
−0.909279 + 0.416186i \(0.863366\pi\)
\(422\) −4.97056 −0.241963
\(423\) 0 0
\(424\) −3.17157 −0.154025
\(425\) −22.9706 −1.11424
\(426\) 0 0
\(427\) −37.6569 −1.82234
\(428\) −20.6863 −0.999910
\(429\) 0 0
\(430\) 1.94113 0.0936094
\(431\) 8.34315 0.401875 0.200938 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(432\) 0 0
\(433\) −21.3137 −1.02427 −0.512136 0.858905i \(-0.671146\pi\)
−0.512136 + 0.858905i \(0.671146\pi\)
\(434\) −1.37258 −0.0658861
\(435\) 0 0
\(436\) 9.71573 0.465299
\(437\) 11.3137 0.541208
\(438\) 0 0
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) −8.97056 −0.427655
\(441\) 0 0
\(442\) 0 0
\(443\) 25.9411 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(444\) 0 0
\(445\) −41.9411 −1.98820
\(446\) 5.17157 0.244881
\(447\) 0 0
\(448\) 11.7990 0.557450
\(449\) −31.7990 −1.50069 −0.750344 0.661048i \(-0.770112\pi\)
−0.750344 + 0.661048i \(0.770112\pi\)
\(450\) 0 0
\(451\) −10.3431 −0.487040
\(452\) −9.71573 −0.456989
\(453\) 0 0
\(454\) −7.17157 −0.336579
\(455\) 0 0
\(456\) 0 0
\(457\) 7.65685 0.358173 0.179086 0.983833i \(-0.442686\pi\)
0.179086 + 0.983833i \(0.442686\pi\)
\(458\) 0.544156 0.0254267
\(459\) 0 0
\(460\) 20.6863 0.964503
\(461\) 5.17157 0.240864 0.120432 0.992722i \(-0.461572\pi\)
0.120432 + 0.992722i \(0.461572\pi\)
\(462\) 0 0
\(463\) 24.4853 1.13793 0.568964 0.822363i \(-0.307344\pi\)
0.568964 + 0.822363i \(0.307344\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −2.88730 −0.133752
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 19.3137 0.891824
\(470\) 13.6569 0.629944
\(471\) 0 0
\(472\) −12.1421 −0.558887
\(473\) 3.31371 0.152364
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) −39.5980 −1.81497
\(477\) 0 0
\(478\) 0.828427 0.0378914
\(479\) 25.3137 1.15661 0.578306 0.815820i \(-0.303714\pi\)
0.578306 + 0.815820i \(0.303714\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.142136 −0.00647410
\(483\) 0 0
\(484\) 12.7990 0.581772
\(485\) 10.3431 0.469658
\(486\) 0 0
\(487\) 7.79899 0.353406 0.176703 0.984264i \(-0.443457\pi\)
0.176703 + 0.984264i \(0.443457\pi\)
\(488\) −21.1127 −0.955727
\(489\) 0 0
\(490\) −1.17157 −0.0529263
\(491\) −30.6274 −1.38220 −0.691098 0.722761i \(-0.742873\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(492\) 0 0
\(493\) 15.3137 0.689695
\(494\) 0 0
\(495\) 0 0
\(496\) 3.51472 0.157816
\(497\) −5.65685 −0.253745
\(498\) 0 0
\(499\) −26.1421 −1.17028 −0.585141 0.810931i \(-0.698961\pi\)
−0.585141 + 0.810931i \(0.698961\pi\)
\(500\) −10.3431 −0.462560
\(501\) 0 0
\(502\) 0 0
\(503\) 7.31371 0.326102 0.163051 0.986618i \(-0.447866\pi\)
0.163051 + 0.986618i \(0.447866\pi\)
\(504\) 0 0
\(505\) 21.6569 0.963717
\(506\) −3.31371 −0.147312
\(507\) 0 0
\(508\) −10.3431 −0.458903
\(509\) 11.7990 0.522981 0.261491 0.965206i \(-0.415786\pi\)
0.261491 + 0.965206i \(0.415786\pi\)
\(510\) 0 0
\(511\) 0.970563 0.0429352
\(512\) 22.7574 1.00574
\(513\) 0 0
\(514\) 1.79899 0.0793500
\(515\) −6.62742 −0.292039
\(516\) 0 0
\(517\) 23.3137 1.02534
\(518\) −8.97056 −0.394144
\(519\) 0 0
\(520\) 0 0
\(521\) −25.3137 −1.10901 −0.554507 0.832179i \(-0.687093\pi\)
−0.554507 + 0.832179i \(0.687093\pi\)
\(522\) 0 0
\(523\) −15.3137 −0.669622 −0.334811 0.942285i \(-0.608672\pi\)
−0.334811 + 0.942285i \(0.608672\pi\)
\(524\) −14.6274 −0.639002
\(525\) 0 0
\(526\) −4.97056 −0.216727
\(527\) −8.97056 −0.390764
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −2.34315 −0.101780
\(531\) 0 0
\(532\) 14.6274 0.634179
\(533\) 0 0
\(534\) 0 0
\(535\) −32.0000 −1.38348
\(536\) 10.8284 0.467717
\(537\) 0 0
\(538\) −7.45584 −0.321444
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 11.5147 0.494600
\(543\) 0 0
\(544\) −33.7990 −1.44912
\(545\) 15.0294 0.643790
\(546\) 0 0
\(547\) 23.3137 0.996822 0.498411 0.866941i \(-0.333917\pi\)
0.498411 + 0.866941i \(0.333917\pi\)
\(548\) 19.7990 0.845771
\(549\) 0 0
\(550\) −2.48528 −0.105973
\(551\) −5.65685 −0.240990
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) −0.828427 −0.0351965
\(555\) 0 0
\(556\) 13.3726 0.567124
\(557\) −7.79899 −0.330454 −0.165227 0.986256i \(-0.552836\pi\)
−0.165227 + 0.986256i \(0.552836\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 24.0000 1.01419
\(561\) 0 0
\(562\) 8.76955 0.369921
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −15.0294 −0.632293
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) −3.17157 −0.133076
\(569\) 42.9706 1.80142 0.900710 0.434421i \(-0.143047\pi\)
0.900710 + 0.434421i \(0.143047\pi\)
\(570\) 0 0
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.05887 −0.252893
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 31.9411 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(578\) 17.2426 0.717199
\(579\) 0 0
\(580\) −10.3431 −0.429476
\(581\) −10.3431 −0.429106
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0.544156 0.0225173
\(585\) 0 0
\(586\) 0.887302 0.0366541
\(587\) −10.9706 −0.452804 −0.226402 0.974034i \(-0.572696\pi\)
−0.226402 + 0.974034i \(0.572696\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) −8.97056 −0.369312
\(591\) 0 0
\(592\) 22.9706 0.944084
\(593\) −20.4853 −0.841230 −0.420615 0.907239i \(-0.638186\pi\)
−0.420615 + 0.907239i \(0.638186\pi\)
\(594\) 0 0
\(595\) −61.2548 −2.51120
\(596\) 16.7696 0.686908
\(597\) 0 0
\(598\) 0 0
\(599\) 23.3137 0.952572 0.476286 0.879290i \(-0.341983\pi\)
0.476286 + 0.879290i \(0.341983\pi\)
\(600\) 0 0
\(601\) −0.627417 −0.0255929 −0.0127964 0.999918i \(-0.504073\pi\)
−0.0127964 + 0.999918i \(0.504073\pi\)
\(602\) 1.94113 0.0791144
\(603\) 0 0
\(604\) −6.42641 −0.261487
\(605\) 19.7990 0.804943
\(606\) 0 0
\(607\) 41.9411 1.70234 0.851169 0.524892i \(-0.175894\pi\)
0.851169 + 0.524892i \(0.175894\pi\)
\(608\) 12.4853 0.506345
\(609\) 0 0
\(610\) −15.5980 −0.631544
\(611\) 0 0
\(612\) 0 0
\(613\) 47.6569 1.92484 0.962421 0.271561i \(-0.0875400\pi\)
0.962421 + 0.271561i \(0.0875400\pi\)
\(614\) 9.45584 0.381607
\(615\) 0 0
\(616\) −8.97056 −0.361434
\(617\) −34.8284 −1.40214 −0.701070 0.713093i \(-0.747294\pi\)
−0.701070 + 0.713093i \(0.747294\pi\)
\(618\) 0 0
\(619\) −23.7990 −0.956562 −0.478281 0.878207i \(-0.658740\pi\)
−0.478281 + 0.878207i \(0.658740\pi\)
\(620\) 6.05887 0.243330
\(621\) 0 0
\(622\) 4.40202 0.176505
\(623\) −41.9411 −1.68034
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 2.48528 0.0993318
\(627\) 0 0
\(628\) 18.2843 0.729622
\(629\) −58.6274 −2.33763
\(630\) 0 0
\(631\) −43.1127 −1.71629 −0.858145 0.513408i \(-0.828383\pi\)
−0.858145 + 0.513408i \(0.828383\pi\)
\(632\) 17.9411 0.713660
\(633\) 0 0
\(634\) 3.51472 0.139587
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 1.65685 0.0655955
\(639\) 0 0
\(640\) 29.8579 1.18024
\(641\) −30.2843 −1.19616 −0.598078 0.801438i \(-0.704069\pi\)
−0.598078 + 0.801438i \(0.704069\pi\)
\(642\) 0 0
\(643\) −22.8284 −0.900265 −0.450133 0.892962i \(-0.648623\pi\)
−0.450133 + 0.892962i \(0.648623\pi\)
\(644\) 20.6863 0.815154
\(645\) 0 0
\(646\) −8.97056 −0.352942
\(647\) −11.3137 −0.444788 −0.222394 0.974957i \(-0.571387\pi\)
−0.222394 + 0.974957i \(0.571387\pi\)
\(648\) 0 0
\(649\) −15.3137 −0.601116
\(650\) 0 0
\(651\) 0 0
\(652\) 34.4264 1.34824
\(653\) 25.3137 0.990602 0.495301 0.868721i \(-0.335058\pi\)
0.495301 + 0.868721i \(0.335058\pi\)
\(654\) 0 0
\(655\) −22.6274 −0.884126
\(656\) 15.5147 0.605748
\(657\) 0 0
\(658\) 13.6569 0.532400
\(659\) 47.3137 1.84308 0.921540 0.388283i \(-0.126932\pi\)
0.921540 + 0.388283i \(0.126932\pi\)
\(660\) 0 0
\(661\) 34.9706 1.36020 0.680099 0.733121i \(-0.261937\pi\)
0.680099 + 0.733121i \(0.261937\pi\)
\(662\) −10.8284 −0.420859
\(663\) 0 0
\(664\) −5.79899 −0.225044
\(665\) 22.6274 0.877454
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 6.68629 0.258700
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) −26.6274 −1.02794
\(672\) 0 0
\(673\) 16.6274 0.640940 0.320470 0.947259i \(-0.396159\pi\)
0.320470 + 0.947259i \(0.396159\pi\)
\(674\) 3.85786 0.148599
\(675\) 0 0
\(676\) 0 0
\(677\) −26.6863 −1.02564 −0.512819 0.858497i \(-0.671399\pi\)
−0.512819 + 0.858497i \(0.671399\pi\)
\(678\) 0 0
\(679\) 10.3431 0.396934
\(680\) −34.3431 −1.31700
\(681\) 0 0
\(682\) −0.970563 −0.0371648
\(683\) 47.9411 1.83442 0.917208 0.398408i \(-0.130437\pi\)
0.917208 + 0.398408i \(0.130437\pi\)
\(684\) 0 0
\(685\) 30.6274 1.17021
\(686\) 7.02944 0.268385
\(687\) 0 0
\(688\) −4.97056 −0.189501
\(689\) 0 0
\(690\) 0 0
\(691\) 5.85786 0.222844 0.111422 0.993773i \(-0.464460\pi\)
0.111422 + 0.993773i \(0.464460\pi\)
\(692\) −21.3137 −0.810226
\(693\) 0 0
\(694\) 3.59798 0.136577
\(695\) 20.6863 0.784676
\(696\) 0 0
\(697\) −39.5980 −1.49988
\(698\) −1.51472 −0.0573329
\(699\) 0 0
\(700\) 15.5147 0.586401
\(701\) −5.02944 −0.189959 −0.0949796 0.995479i \(-0.530279\pi\)
−0.0949796 + 0.995479i \(0.530279\pi\)
\(702\) 0 0
\(703\) 21.6569 0.816804
\(704\) 8.34315 0.314444
\(705\) 0 0
\(706\) −13.8579 −0.521548
\(707\) 21.6569 0.814490
\(708\) 0 0
\(709\) 4.62742 0.173786 0.0868931 0.996218i \(-0.472306\pi\)
0.0868931 + 0.996218i \(0.472306\pi\)
\(710\) −2.34315 −0.0879367
\(711\) 0 0
\(712\) −23.5147 −0.881251
\(713\) 4.68629 0.175503
\(714\) 0 0
\(715\) 0 0
\(716\) −42.6274 −1.59306
\(717\) 0 0
\(718\) 14.4853 0.540586
\(719\) 29.9411 1.11662 0.558308 0.829634i \(-0.311451\pi\)
0.558308 + 0.829634i \(0.311451\pi\)
\(720\) 0 0
\(721\) −6.62742 −0.246818
\(722\) −4.55635 −0.169570
\(723\) 0 0
\(724\) −25.5980 −0.951341
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −10.3431 −0.383606 −0.191803 0.981433i \(-0.561433\pi\)
−0.191803 + 0.981433i \(0.561433\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0.402020 0.0148794
\(731\) 12.6863 0.469219
\(732\) 0 0
\(733\) 36.6274 1.35286 0.676432 0.736505i \(-0.263525\pi\)
0.676432 + 0.736505i \(0.263525\pi\)
\(734\) −9.94113 −0.366934
\(735\) 0 0
\(736\) 17.6569 0.650840
\(737\) 13.6569 0.503057
\(738\) 0 0
\(739\) 18.1421 0.667369 0.333685 0.942685i \(-0.391708\pi\)
0.333685 + 0.942685i \(0.391708\pi\)
\(740\) 39.5980 1.45565
\(741\) 0 0
\(742\) −2.34315 −0.0860196
\(743\) 2.00000 0.0733729 0.0366864 0.999327i \(-0.488320\pi\)
0.0366864 + 0.999327i \(0.488320\pi\)
\(744\) 0 0
\(745\) 25.9411 0.950409
\(746\) 4.14214 0.151654
\(747\) 0 0
\(748\) −28.0000 −1.02378
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −0.970563 −0.0354163 −0.0177082 0.999843i \(-0.505637\pi\)
−0.0177082 + 0.999843i \(0.505637\pi\)
\(752\) −34.9706 −1.27525
\(753\) 0 0
\(754\) 0 0
\(755\) −9.94113 −0.361795
\(756\) 0 0
\(757\) 51.9411 1.88783 0.943916 0.330185i \(-0.107111\pi\)
0.943916 + 0.330185i \(0.107111\pi\)
\(758\) −0.201010 −0.00730102
\(759\) 0 0
\(760\) 12.6863 0.460180
\(761\) 32.4853 1.17759 0.588795 0.808282i \(-0.299602\pi\)
0.588795 + 0.808282i \(0.299602\pi\)
\(762\) 0 0
\(763\) 15.0294 0.544102
\(764\) 6.05887 0.219202
\(765\) 0 0
\(766\) 12.8284 0.463510
\(767\) 0 0
\(768\) 0 0
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) −6.62742 −0.238836
\(771\) 0 0
\(772\) 9.71573 0.349677
\(773\) 34.1421 1.22801 0.614004 0.789303i \(-0.289558\pi\)
0.614004 + 0.789303i \(0.289558\pi\)
\(774\) 0 0
\(775\) 3.51472 0.126252
\(776\) 5.79899 0.208172
\(777\) 0 0
\(778\) 11.1716 0.400520
\(779\) 14.6274 0.524082
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −12.6863 −0.453661
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 28.2843 1.00951
\(786\) 0 0
\(787\) 40.7696 1.45328 0.726639 0.687020i \(-0.241081\pi\)
0.726639 + 0.687020i \(0.241081\pi\)
\(788\) −0.887302 −0.0316088
\(789\) 0 0
\(790\) 13.2548 0.471586
\(791\) −15.0294 −0.534385
\(792\) 0 0
\(793\) 0 0
\(794\) −12.8284 −0.455264
\(795\) 0 0
\(796\) −39.5980 −1.40351
\(797\) −24.3431 −0.862278 −0.431139 0.902285i \(-0.641888\pi\)
−0.431139 + 0.902285i \(0.641888\pi\)
\(798\) 0 0
\(799\) 89.2548 3.15761
\(800\) 13.2426 0.468198
\(801\) 0 0
\(802\) 10.8284 0.382365
\(803\) 0.686292 0.0242187
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) 0 0
\(808\) 12.1421 0.427159
\(809\) −18.6863 −0.656975 −0.328488 0.944508i \(-0.606539\pi\)
−0.328488 + 0.944508i \(0.606539\pi\)
\(810\) 0 0
\(811\) 30.1421 1.05843 0.529217 0.848487i \(-0.322486\pi\)
0.529217 + 0.848487i \(0.322486\pi\)
\(812\) −10.3431 −0.362973
\(813\) 0 0
\(814\) −6.34315 −0.222327
\(815\) 53.2548 1.86544
\(816\) 0 0
\(817\) −4.68629 −0.163953
\(818\) 14.4853 0.506466
\(819\) 0 0
\(820\) 26.7452 0.933982
\(821\) −23.7990 −0.830590 −0.415295 0.909687i \(-0.636322\pi\)
−0.415295 + 0.909687i \(0.636322\pi\)
\(822\) 0 0
\(823\) 15.0294 0.523893 0.261947 0.965082i \(-0.415636\pi\)
0.261947 + 0.965082i \(0.415636\pi\)
\(824\) −3.71573 −0.129444
\(825\) 0 0
\(826\) −8.97056 −0.312126
\(827\) −26.0000 −0.904109 −0.452054 0.891990i \(-0.649309\pi\)
−0.452054 + 0.891990i \(0.649309\pi\)
\(828\) 0 0
\(829\) −17.3137 −0.601330 −0.300665 0.953730i \(-0.597209\pi\)
−0.300665 + 0.953730i \(0.597209\pi\)
\(830\) −4.28427 −0.148709
\(831\) 0 0
\(832\) 0 0
\(833\) −7.65685 −0.265294
\(834\) 0 0
\(835\) 10.3431 0.357939
\(836\) 10.3431 0.357725
\(837\) 0 0
\(838\) −6.05887 −0.209300
\(839\) 43.2548 1.49332 0.746661 0.665204i \(-0.231656\pi\)
0.746661 + 0.665204i \(0.231656\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −15.4558 −0.532644
\(843\) 0 0
\(844\) 21.9411 0.755245
\(845\) 0 0
\(846\) 0 0
\(847\) 19.7990 0.680301
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −9.51472 −0.326352
\(851\) 30.6274 1.04989
\(852\) 0 0
\(853\) −3.65685 −0.125208 −0.0626042 0.998038i \(-0.519941\pi\)
−0.0626042 + 0.998038i \(0.519941\pi\)
\(854\) −15.5980 −0.533752
\(855\) 0 0
\(856\) −17.9411 −0.613215
\(857\) 49.5980 1.69423 0.847117 0.531406i \(-0.178336\pi\)
0.847117 + 0.531406i \(0.178336\pi\)
\(858\) 0 0
\(859\) −0.686292 −0.0234160 −0.0117080 0.999931i \(-0.503727\pi\)
−0.0117080 + 0.999931i \(0.503727\pi\)
\(860\) −8.56854 −0.292185
\(861\) 0 0
\(862\) 3.45584 0.117707
\(863\) −28.3431 −0.964812 −0.482406 0.875948i \(-0.660237\pi\)
−0.482406 + 0.875948i \(0.660237\pi\)
\(864\) 0 0
\(865\) −32.9706 −1.12103
\(866\) −8.82843 −0.300002
\(867\) 0 0
\(868\) 6.05887 0.205652
\(869\) 22.6274 0.767583
\(870\) 0 0
\(871\) 0 0
\(872\) 8.42641 0.285354
\(873\) 0 0
\(874\) 4.68629 0.158516
\(875\) −16.0000 −0.540899
\(876\) 0 0
\(877\) −42.2843 −1.42784 −0.713919 0.700228i \(-0.753082\pi\)
−0.713919 + 0.700228i \(0.753082\pi\)
\(878\) 7.02944 0.237232
\(879\) 0 0
\(880\) 16.9706 0.572078
\(881\) 25.5980 0.862418 0.431209 0.902252i \(-0.358087\pi\)
0.431209 + 0.902252i \(0.358087\pi\)
\(882\) 0 0
\(883\) 27.5980 0.928746 0.464373 0.885640i \(-0.346280\pi\)
0.464373 + 0.885640i \(0.346280\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 10.7452 0.360991
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −17.3726 −0.582330
\(891\) 0 0
\(892\) −22.8284 −0.764352
\(893\) −32.9706 −1.10332
\(894\) 0 0
\(895\) −65.9411 −2.20417
\(896\) 29.8579 0.997481
\(897\) 0 0
\(898\) −13.1716 −0.439541
\(899\) −2.34315 −0.0781483
\(900\) 0 0
\(901\) −15.3137 −0.510174
\(902\) −4.28427 −0.142651
\(903\) 0 0
\(904\) −8.42641 −0.280258
\(905\) −39.5980 −1.31628
\(906\) 0 0
\(907\) −12.9706 −0.430680 −0.215340 0.976539i \(-0.569086\pi\)
−0.215340 + 0.976539i \(0.569086\pi\)
\(908\) 31.6569 1.05057
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −7.31371 −0.242048
\(914\) 3.17157 0.104906
\(915\) 0 0
\(916\) −2.40202 −0.0793650
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) 3.31371 0.109309 0.0546546 0.998505i \(-0.482594\pi\)
0.0546546 + 0.998505i \(0.482594\pi\)
\(920\) 17.9411 0.591501
\(921\) 0 0
\(922\) 2.14214 0.0705475
\(923\) 0 0
\(924\) 0 0
\(925\) 22.9706 0.755267
\(926\) 10.1421 0.333291
\(927\) 0 0
\(928\) −8.82843 −0.289807
\(929\) 11.7990 0.387112 0.193556 0.981089i \(-0.437998\pi\)
0.193556 + 0.981089i \(0.437998\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) 12.7452 0.417482
\(933\) 0 0
\(934\) 3.31371 0.108428
\(935\) −43.3137 −1.41651
\(936\) 0 0
\(937\) −21.3137 −0.696289 −0.348144 0.937441i \(-0.613188\pi\)
−0.348144 + 0.937441i \(0.613188\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) −60.2843 −1.96626
\(941\) 34.1421 1.11300 0.556501 0.830847i \(-0.312144\pi\)
0.556501 + 0.830847i \(0.312144\pi\)
\(942\) 0 0
\(943\) 20.6863 0.673638
\(944\) 22.9706 0.747628
\(945\) 0 0
\(946\) 1.37258 0.0446265
\(947\) −21.0294 −0.683365 −0.341682 0.939815i \(-0.610997\pi\)
−0.341682 + 0.939815i \(0.610997\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 3.51472 0.114033
\(951\) 0 0
\(952\) −34.3431 −1.11307
\(953\) −40.3431 −1.30684 −0.653421 0.756994i \(-0.726667\pi\)
−0.653421 + 0.756994i \(0.726667\pi\)
\(954\) 0 0
\(955\) 9.37258 0.303290
\(956\) −3.65685 −0.118271
\(957\) 0 0
\(958\) 10.4853 0.338764
\(959\) 30.6274 0.989011
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) 0 0
\(963\) 0 0
\(964\) 0.627417 0.0202077
\(965\) 15.0294 0.483815
\(966\) 0 0
\(967\) −18.1421 −0.583412 −0.291706 0.956508i \(-0.594223\pi\)
−0.291706 + 0.956508i \(0.594223\pi\)
\(968\) 11.1005 0.356784
\(969\) 0 0
\(970\) 4.28427 0.137560
\(971\) 15.3137 0.491440 0.245720 0.969341i \(-0.420976\pi\)
0.245720 + 0.969341i \(0.420976\pi\)
\(972\) 0 0
\(973\) 20.6863 0.663172
\(974\) 3.23045 0.103510
\(975\) 0 0
\(976\) 39.9411 1.27848
\(977\) 42.1421 1.34825 0.674123 0.738619i \(-0.264522\pi\)
0.674123 + 0.738619i \(0.264522\pi\)
\(978\) 0 0
\(979\) −29.6569 −0.947837
\(980\) 5.17157 0.165200
\(981\) 0 0
\(982\) −12.6863 −0.404836
\(983\) 25.3137 0.807382 0.403691 0.914895i \(-0.367727\pi\)
0.403691 + 0.914895i \(0.367727\pi\)
\(984\) 0 0
\(985\) −1.37258 −0.0437341
\(986\) 6.34315 0.202007
\(987\) 0 0
\(988\) 0 0
\(989\) −6.62742 −0.210740
\(990\) 0 0
\(991\) 4.68629 0.148865 0.0744325 0.997226i \(-0.476285\pi\)
0.0744325 + 0.997226i \(0.476285\pi\)
\(992\) 5.17157 0.164198
\(993\) 0 0
\(994\) −2.34315 −0.0743201
\(995\) −61.2548 −1.94191
\(996\) 0 0
\(997\) −39.2548 −1.24321 −0.621607 0.783330i \(-0.713520\pi\)
−0.621607 + 0.783330i \(0.713520\pi\)
\(998\) −10.8284 −0.342768
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.2.a.f.1.2 2
3.2 odd 2 507.2.a.h.1.1 2
12.11 even 2 8112.2.a.bm.1.2 2
13.5 odd 4 1521.2.b.j.1351.2 4
13.8 odd 4 1521.2.b.j.1351.3 4
13.12 even 2 117.2.a.c.1.1 2
39.2 even 12 507.2.j.f.316.2 8
39.5 even 4 507.2.b.e.337.3 4
39.8 even 4 507.2.b.e.337.2 4
39.11 even 12 507.2.j.f.316.3 8
39.17 odd 6 507.2.e.h.484.1 4
39.20 even 12 507.2.j.f.361.2 8
39.23 odd 6 507.2.e.h.22.1 4
39.29 odd 6 507.2.e.d.22.2 4
39.32 even 12 507.2.j.f.361.3 8
39.35 odd 6 507.2.e.d.484.2 4
39.38 odd 2 39.2.a.b.1.2 2
52.51 odd 2 1872.2.a.w.1.2 2
65.12 odd 4 2925.2.c.u.2224.2 4
65.38 odd 4 2925.2.c.u.2224.3 4
65.64 even 2 2925.2.a.v.1.2 2
91.90 odd 2 5733.2.a.u.1.1 2
104.51 odd 2 7488.2.a.co.1.1 2
104.77 even 2 7488.2.a.cl.1.1 2
117.25 even 6 1053.2.e.e.352.2 4
117.38 odd 6 1053.2.e.m.352.1 4
117.77 odd 6 1053.2.e.m.703.1 4
117.103 even 6 1053.2.e.e.703.2 4
156.155 even 2 624.2.a.k.1.1 2
195.38 even 4 975.2.c.h.274.2 4
195.77 even 4 975.2.c.h.274.3 4
195.194 odd 2 975.2.a.l.1.1 2
273.272 even 2 1911.2.a.h.1.2 2
312.77 odd 2 2496.2.a.bf.1.2 2
312.155 even 2 2496.2.a.bi.1.2 2
429.428 even 2 4719.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.2 2 39.38 odd 2
117.2.a.c.1.1 2 13.12 even 2
507.2.a.h.1.1 2 3.2 odd 2
507.2.b.e.337.2 4 39.8 even 4
507.2.b.e.337.3 4 39.5 even 4
507.2.e.d.22.2 4 39.29 odd 6
507.2.e.d.484.2 4 39.35 odd 6
507.2.e.h.22.1 4 39.23 odd 6
507.2.e.h.484.1 4 39.17 odd 6
507.2.j.f.316.2 8 39.2 even 12
507.2.j.f.316.3 8 39.11 even 12
507.2.j.f.361.2 8 39.20 even 12
507.2.j.f.361.3 8 39.32 even 12
624.2.a.k.1.1 2 156.155 even 2
975.2.a.l.1.1 2 195.194 odd 2
975.2.c.h.274.2 4 195.38 even 4
975.2.c.h.274.3 4 195.77 even 4
1053.2.e.e.352.2 4 117.25 even 6
1053.2.e.e.703.2 4 117.103 even 6
1053.2.e.m.352.1 4 117.38 odd 6
1053.2.e.m.703.1 4 117.77 odd 6
1521.2.a.f.1.2 2 1.1 even 1 trivial
1521.2.b.j.1351.2 4 13.5 odd 4
1521.2.b.j.1351.3 4 13.8 odd 4
1872.2.a.w.1.2 2 52.51 odd 2
1911.2.a.h.1.2 2 273.272 even 2
2496.2.a.bf.1.2 2 312.77 odd 2
2496.2.a.bi.1.2 2 312.155 even 2
2925.2.a.v.1.2 2 65.64 even 2
2925.2.c.u.2224.2 4 65.12 odd 4
2925.2.c.u.2224.3 4 65.38 odd 4
4719.2.a.p.1.1 2 429.428 even 2
5733.2.a.u.1.1 2 91.90 odd 2
7488.2.a.cl.1.1 2 104.77 even 2
7488.2.a.co.1.1 2 104.51 odd 2
8112.2.a.bm.1.2 2 12.11 even 2