Properties

Label 1911.2.a.h.1.2
Level $1911$
Weight $2$
Character 1911.1
Self dual yes
Analytic conductor $15.259$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.2594118263\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1911.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} +2.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} +2.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +1.17157 q^{10} -2.00000 q^{11} +1.82843 q^{12} +1.00000 q^{13} -2.82843 q^{15} +3.00000 q^{16} -7.65685 q^{17} +0.414214 q^{18} +2.82843 q^{19} -5.17157 q^{20} -0.828427 q^{22} -4.00000 q^{23} +1.58579 q^{24} +3.00000 q^{25} +0.414214 q^{26} -1.00000 q^{27} +2.00000 q^{29} -1.17157 q^{30} +1.17157 q^{31} +4.41421 q^{32} +2.00000 q^{33} -3.17157 q^{34} -1.82843 q^{36} -7.65685 q^{37} +1.17157 q^{38} -1.00000 q^{39} -4.48528 q^{40} -5.17157 q^{41} -1.65685 q^{43} +3.65685 q^{44} +2.82843 q^{45} -1.65685 q^{46} +11.6569 q^{47} -3.00000 q^{48} +1.24264 q^{50} +7.65685 q^{51} -1.82843 q^{52} -2.00000 q^{53} -0.414214 q^{54} -5.65685 q^{55} -2.82843 q^{57} +0.828427 q^{58} -7.65685 q^{59} +5.17157 q^{60} -13.3137 q^{61} +0.485281 q^{62} -4.17157 q^{64} +2.82843 q^{65} +0.828427 q^{66} +6.82843 q^{67} +14.0000 q^{68} +4.00000 q^{69} +2.00000 q^{71} -1.58579 q^{72} -0.343146 q^{73} -3.17157 q^{74} -3.00000 q^{75} -5.17157 q^{76} -0.414214 q^{78} -11.3137 q^{79} +8.48528 q^{80} +1.00000 q^{81} -2.14214 q^{82} -3.65685 q^{83} -21.6569 q^{85} -0.686292 q^{86} -2.00000 q^{87} +3.17157 q^{88} -14.8284 q^{89} +1.17157 q^{90} +7.31371 q^{92} -1.17157 q^{93} +4.82843 q^{94} +8.00000 q^{95} -4.41421 q^{96} -3.65685 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 6q^{8} + 2q^{9} + 8q^{10} - 4q^{11} - 2q^{12} + 2q^{13} + 6q^{16} - 4q^{17} - 2q^{18} - 16q^{20} + 4q^{22} - 8q^{23} + 6q^{24} + 6q^{25} - 2q^{26} - 2q^{27} + 4q^{29} - 8q^{30} + 8q^{31} + 6q^{32} + 4q^{33} - 12q^{34} + 2q^{36} - 4q^{37} + 8q^{38} - 2q^{39} + 8q^{40} - 16q^{41} + 8q^{43} - 4q^{44} + 8q^{46} + 12q^{47} - 6q^{48} - 6q^{50} + 4q^{51} + 2q^{52} - 4q^{53} + 2q^{54} - 4q^{58} - 4q^{59} + 16q^{60} - 4q^{61} - 16q^{62} - 14q^{64} - 4q^{66} + 8q^{67} + 28q^{68} + 8q^{69} + 4q^{71} - 6q^{72} - 12q^{73} - 12q^{74} - 6q^{75} - 16q^{76} + 2q^{78} + 2q^{81} + 24q^{82} + 4q^{83} - 32q^{85} - 24q^{86} - 4q^{87} + 12q^{88} - 24q^{89} + 8q^{90} - 8q^{92} - 8q^{93} + 4q^{94} + 16q^{95} - 6q^{96} + 4q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.82843 −0.914214
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) −0.414214 −0.169102
\(7\) 0 0
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) 1.17157 0.370484
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.82843 0.527821
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −2.82843 −0.730297
\(16\) 3.00000 0.750000
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0.414214 0.0976311
\(19\) 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 \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.58579 0.323697
\(25\) 3.00000 0.600000
\(26\) 0.414214 0.0812340
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.17157 −0.213899
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 4.41421 0.780330
\(33\) 2.00000 0.348155
\(34\) −3.17157 −0.543920
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 1.17157 0.190054
\(39\) −1.00000 −0.160128
\(40\) −4.48528 −0.709185
\(41\) −5.17157 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(42\) 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
\(45\) 2.82843 0.421637
\(46\) −1.65685 −0.244290
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) −3.00000 −0.433013
\(49\) 0 0
\(50\) 1.24264 0.175736
\(51\) 7.65685 1.07217
\(52\) −1.82843 −0.253557
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −0.414214 −0.0563673
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0.828427 0.108778
\(59\) −7.65685 −0.996838 −0.498419 0.866936i \(-0.666086\pi\)
−0.498419 + 0.866936i \(0.666086\pi\)
\(60\) 5.17157 0.667647
\(61\) −13.3137 −1.70465 −0.852323 0.523016i \(-0.824807\pi\)
−0.852323 + 0.523016i \(0.824807\pi\)
\(62\) 0.485281 0.0616308
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 2.82843 0.350823
\(66\) 0.828427 0.101972
\(67\) 6.82843 0.834225 0.417113 0.908855i \(-0.363042\pi\)
0.417113 + 0.908855i \(0.363042\pi\)
\(68\) 14.0000 1.69775
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −1.58579 −0.186887
\(73\) −0.343146 −0.0401622 −0.0200811 0.999798i \(-0.506392\pi\)
−0.0200811 + 0.999798i \(0.506392\pi\)
\(74\) −3.17157 −0.368688
\(75\) −3.00000 −0.346410
\(76\) −5.17157 −0.593220
\(77\) 0 0
\(78\) −0.414214 −0.0469005
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 8.48528 0.948683
\(81\) 1.00000 0.111111
\(82\) −2.14214 −0.236559
\(83\) −3.65685 −0.401392 −0.200696 0.979654i \(-0.564320\pi\)
−0.200696 + 0.979654i \(0.564320\pi\)
\(84\) 0 0
\(85\) −21.6569 −2.34902
\(86\) −0.686292 −0.0740047
\(87\) −2.00000 −0.214423
\(88\) 3.17157 0.338091
\(89\) −14.8284 −1.57181 −0.785905 0.618347i \(-0.787803\pi\)
−0.785905 + 0.618347i \(0.787803\pi\)
\(90\) 1.17157 0.123495
\(91\) 0 0
\(92\) 7.31371 0.762507
\(93\) −1.17157 −0.121486
\(94\) 4.82843 0.498014
\(95\) 8.00000 0.820783
\(96\) −4.41421 −0.450524
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −5.48528 −0.548528
\(101\) −7.65685 −0.761885 −0.380943 0.924599i \(-0.624401\pi\)
−0.380943 + 0.924599i \(0.624401\pi\)
\(102\) 3.17157 0.314033
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) −1.58579 −0.155499
\(105\) 0 0
\(106\) −0.828427 −0.0804640
\(107\) −11.3137 −1.09374 −0.546869 0.837218i \(-0.684180\pi\)
−0.546869 + 0.837218i \(0.684180\pi\)
\(108\) 1.82843 0.175940
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) −2.34315 −0.223410
\(111\) 7.65685 0.726756
\(112\) 0 0
\(113\) −5.31371 −0.499872 −0.249936 0.968262i \(-0.580410\pi\)
−0.249936 + 0.968262i \(0.580410\pi\)
\(114\) −1.17157 −0.109728
\(115\) −11.3137 −1.05501
\(116\) −3.65685 −0.339530
\(117\) 1.00000 0.0924500
\(118\) −3.17157 −0.291967
\(119\) 0 0
\(120\) 4.48528 0.409448
\(121\) −7.00000 −0.636364
\(122\) −5.51472 −0.499279
\(123\) 5.17157 0.466305
\(124\) −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
\(129\) 1.65685 0.145878
\(130\) 1.17157 0.102754
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −3.65685 −0.318288
\(133\) 0 0
\(134\) 2.82843 0.244339
\(135\) −2.82843 −0.243432
\(136\) 12.1421 1.04118
\(137\) −10.8284 −0.925135 −0.462567 0.886584i \(-0.653072\pi\)
−0.462567 + 0.886584i \(0.653072\pi\)
\(138\) 1.65685 0.141041
\(139\) 7.31371 0.620341 0.310170 0.950681i \(-0.399614\pi\)
0.310170 + 0.950681i \(0.399614\pi\)
\(140\) 0 0
\(141\) −11.6569 −0.981684
\(142\) 0.828427 0.0695201
\(143\) −2.00000 −0.167248
\(144\) 3.00000 0.250000
\(145\) 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 \(0.622592\pi\)
\(150\) −1.24264 −0.101461
\(151\) −3.51472 −0.286024 −0.143012 0.989721i \(-0.545679\pi\)
−0.143012 + 0.989721i \(0.545679\pi\)
\(152\) −4.48528 −0.363804
\(153\) −7.65685 −0.619020
\(154\) 0 0
\(155\) 3.31371 0.266163
\(156\) 1.82843 0.146391
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −4.68629 −0.372821
\(159\) 2.00000 0.158610
\(160\) 12.4853 0.987048
\(161\) 0 0
\(162\) 0.414214 0.0325437
\(163\) 18.8284 1.47476 0.737378 0.675480i \(-0.236064\pi\)
0.737378 + 0.675480i \(0.236064\pi\)
\(164\) 9.45584 0.738377
\(165\) 5.65685 0.440386
\(166\) −1.51472 −0.117565
\(167\) 3.65685 0.282976 0.141488 0.989940i \(-0.454811\pi\)
0.141488 + 0.989940i \(0.454811\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −8.97056 −0.688011
\(171\) 2.82843 0.216295
\(172\) 3.02944 0.230992
\(173\) 11.6569 0.886254 0.443127 0.896459i \(-0.353869\pi\)
0.443127 + 0.896459i \(0.353869\pi\)
\(174\) −0.828427 −0.0628029
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 7.65685 0.575524
\(178\) −6.14214 −0.460373
\(179\) −23.3137 −1.74255 −0.871274 0.490797i \(-0.836706\pi\)
−0.871274 + 0.490797i \(0.836706\pi\)
\(180\) −5.17157 −0.385466
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 13.3137 0.984178
\(184\) 6.34315 0.467623
\(185\) −21.6569 −1.59224
\(186\) −0.485281 −0.0355826
\(187\) 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 \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) 4.17157 0.301057
\(193\) 5.31371 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(194\) −1.51472 −0.108750
\(195\) −2.82843 −0.202548
\(196\) 0 0
\(197\) 0.485281 0.0345749 0.0172874 0.999851i \(-0.494497\pi\)
0.0172874 + 0.999851i \(0.494497\pi\)
\(198\) −0.828427 −0.0588738
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) −4.75736 −0.336396
\(201\) −6.82843 −0.481640
\(202\) −3.17157 −0.223151
\(203\) 0 0
\(204\) −14.0000 −0.980196
\(205\) −14.6274 −1.02162
\(206\) −0.970563 −0.0676223
\(207\) −4.00000 −0.278019
\(208\) 3.00000 0.208013
\(209\) −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
\(213\) −2.00000 −0.137038
\(214\) −4.68629 −0.320348
\(215\) −4.68629 −0.319602
\(216\) 1.58579 0.107899
\(217\) 0 0
\(218\) 2.20101 0.149071
\(219\) 0.343146 0.0231876
\(220\) 10.3431 0.697335
\(221\) −7.65685 −0.515056
\(222\) 3.17157 0.212862
\(223\) 12.4853 0.836076 0.418038 0.908429i \(-0.362718\pi\)
0.418038 + 0.908429i \(0.362718\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) −2.20101 −0.146409
\(227\) 17.3137 1.14915 0.574576 0.818452i \(-0.305167\pi\)
0.574576 + 0.818452i \(0.305167\pi\)
\(228\) 5.17157 0.342496
\(229\) 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 \(-0.426674\pi\)
0.228328 + 0.973584i \(0.426674\pi\)
\(234\) 0.414214 0.0270780
\(235\) 32.9706 2.15076
\(236\) 14.0000 0.911322
\(237\) 11.3137 0.734904
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) −8.48528 −0.547723
\(241\) −0.343146 −0.0221040 −0.0110520 0.999939i \(-0.503518\pi\)
−0.0110520 + 0.999939i \(0.503518\pi\)
\(242\) −2.89949 −0.186387
\(243\) −1.00000 −0.0641500
\(244\) 24.3431 1.55841
\(245\) 0 0
\(246\) 2.14214 0.136578
\(247\) 2.82843 0.179969
\(248\) −1.85786 −0.117975
\(249\) 3.65685 0.231744
\(250\) −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
\(255\) 21.6569 1.35620
\(256\) 3.97056 0.248160
\(257\) 4.34315 0.270918 0.135459 0.990783i \(-0.456749\pi\)
0.135459 + 0.990783i \(0.456749\pi\)
\(258\) 0.686292 0.0427266
\(259\) 0 0
\(260\) −5.17157 −0.320727
\(261\) 2.00000 0.123797
\(262\) 3.31371 0.204722
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −3.17157 −0.195197
\(265\) −5.65685 −0.347498
\(266\) 0 0
\(267\) 14.8284 0.907485
\(268\) −12.4853 −0.762660
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.17157 −0.0712997
\(271\) 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
\(276\) −7.31371 −0.440234
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 3.02944 0.181694
\(279\) 1.17157 0.0701402
\(280\) 0 0
\(281\) 21.1716 1.26299 0.631495 0.775380i \(-0.282442\pi\)
0.631495 + 0.775380i \(0.282442\pi\)
\(282\) −4.82843 −0.287529
\(283\) −28.9706 −1.72212 −0.861061 0.508502i \(-0.830199\pi\)
−0.861061 + 0.508502i \(0.830199\pi\)
\(284\) −3.65685 −0.216994
\(285\) −8.00000 −0.473879
\(286\) −0.828427 −0.0489859
\(287\) 0 0
\(288\) 4.41421 0.260110
\(289\) 41.6274 2.44867
\(290\) 2.34315 0.137594
\(291\) 3.65685 0.214369
\(292\) 0.627417 0.0367168
\(293\) −2.14214 −0.125145 −0.0625724 0.998040i \(-0.519930\pi\)
−0.0625724 + 0.998040i \(0.519930\pi\)
\(294\) 0 0
\(295\) −21.6569 −1.26091
\(296\) 12.1421 0.705747
\(297\) 2.00000 0.116052
\(298\) −3.79899 −0.220070
\(299\) −4.00000 −0.231326
\(300\) 5.48528 0.316693
\(301\) 0 0
\(302\) −1.45584 −0.0837744
\(303\) 7.65685 0.439875
\(304\) 8.48528 0.486664
\(305\) −37.6569 −2.15623
\(306\) −3.17157 −0.181307
\(307\) 22.8284 1.30289 0.651444 0.758697i \(-0.274164\pi\)
0.651444 + 0.758697i \(0.274164\pi\)
\(308\) 0 0
\(309\) 2.34315 0.133297
\(310\) 1.37258 0.0779575
\(311\) 10.6274 0.602626 0.301313 0.953525i \(-0.402575\pi\)
0.301313 + 0.953525i \(0.402575\pi\)
\(312\) 1.58579 0.0897775
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 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 \(0.423413\pi\)
\(318\) 0.828427 0.0464559
\(319\) −4.00000 −0.223957
\(320\) −11.7990 −0.659584
\(321\) 11.3137 0.631470
\(322\) 0 0
\(323\) −21.6569 −1.20502
\(324\) −1.82843 −0.101579
\(325\) 3.00000 0.166410
\(326\) 7.79899 0.431946
\(327\) −5.31371 −0.293849
\(328\) 8.20101 0.452825
\(329\) 0 0
\(330\) 2.34315 0.128986
\(331\) 26.1421 1.43690 0.718451 0.695578i \(-0.244852\pi\)
0.718451 + 0.695578i \(0.244852\pi\)
\(332\) 6.68629 0.366958
\(333\) −7.65685 −0.419593
\(334\) 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 \(0.418361\pi\)
\(338\) 0.414214 0.0225302
\(339\) 5.31371 0.288601
\(340\) 39.5980 2.14750
\(341\) −2.34315 −0.126888
\(342\) 1.17157 0.0633514
\(343\) 0 0
\(344\) 2.62742 0.141661
\(345\) 11.3137 0.609110
\(346\) 4.82843 0.259578
\(347\) −8.68629 −0.466305 −0.233152 0.972440i \(-0.574904\pi\)
−0.233152 + 0.972440i \(0.574904\pi\)
\(348\) 3.65685 0.196028
\(349\) −3.65685 −0.195747 −0.0978735 0.995199i \(-0.531204\pi\)
−0.0978735 + 0.995199i \(0.531204\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −8.82843 −0.470557
\(353\) 33.4558 1.78067 0.890337 0.455301i \(-0.150468\pi\)
0.890337 + 0.455301i \(0.150468\pi\)
\(354\) 3.17157 0.168567
\(355\) 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 \(0.125864\pi\)
\(360\) −4.48528 −0.236395
\(361\) −11.0000 −0.578947
\(362\) −5.79899 −0.304788
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −0.970563 −0.0508016
\(366\) 5.51472 0.288259
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) −12.0000 −0.625543
\(369\) −5.17157 −0.269221
\(370\) −8.97056 −0.466357
\(371\) 0 0
\(372\) 2.14214 0.111065
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 6.34315 0.327996
\(375\) 5.65685 0.292119
\(376\) −18.4853 −0.953306
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 0.485281 0.0249272 0.0124636 0.999922i \(-0.496033\pi\)
0.0124636 + 0.999922i \(0.496033\pi\)
\(380\) −14.6274 −0.750371
\(381\) −5.65685 −0.289809
\(382\) 1.37258 0.0702275
\(383\) −30.9706 −1.58252 −0.791261 0.611479i \(-0.790575\pi\)
−0.791261 + 0.611479i \(0.790575\pi\)
\(384\) 10.5563 0.538701
\(385\) 0 0
\(386\) 2.20101 0.112028
\(387\) −1.65685 −0.0842226
\(388\) 6.68629 0.339445
\(389\) −26.9706 −1.36746 −0.683731 0.729734i \(-0.739644\pi\)
−0.683731 + 0.729734i \(0.739644\pi\)
\(390\) −1.17157 −0.0593249
\(391\) 30.6274 1.54890
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0.201010 0.0101267
\(395\) −32.0000 −1.61009
\(396\) 3.65685 0.183764
\(397\) −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 \(0.273620\pi\)
\(402\) −2.82843 −0.141069
\(403\) 1.17157 0.0583602
\(404\) 14.0000 0.696526
\(405\) 2.82843 0.140546
\(406\) 0 0
\(407\) 15.3137 0.759072
\(408\) −12.1421 −0.601125
\(409\) 34.9706 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(410\) −6.05887 −0.299226
\(411\) 10.8284 0.534127
\(412\) 4.28427 0.211071
\(413\) 0 0
\(414\) −1.65685 −0.0814299
\(415\) −10.3431 −0.507725
\(416\) 4.41421 0.216425
\(417\) −7.31371 −0.358154
\(418\) −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 \(-0.136634\pi\)
0.909279 + 0.416186i \(0.136634\pi\)
\(422\) −4.97056 −0.241963
\(423\) 11.6569 0.566776
\(424\) 3.17157 0.154025
\(425\) −22.9706 −1.11424
\(426\) −0.828427 −0.0401374
\(427\) 0 0
\(428\) 20.6863 0.999910
\(429\) 2.00000 0.0965609
\(430\) −1.94113 −0.0936094
\(431\) 8.34315 0.401875 0.200938 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(432\) −3.00000 −0.144338
\(433\) 21.3137 1.02427 0.512136 0.858905i \(-0.328854\pi\)
0.512136 + 0.858905i \(0.328854\pi\)
\(434\) 0 0
\(435\) −5.65685 −0.271225
\(436\) −9.71573 −0.465299
\(437\) −11.3137 −0.541208
\(438\) 0.142136 0.00679150
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 8.97056 0.427655
\(441\) 0 0
\(442\) −3.17157 −0.150856
\(443\) −25.9411 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(444\) −14.0000 −0.664411
\(445\) −41.9411 −1.98820
\(446\) 5.17157 0.244881
\(447\) 9.17157 0.433801
\(448\) 0 0
\(449\) −31.7990 −1.50069 −0.750344 0.661048i \(-0.770112\pi\)
−0.750344 + 0.661048i \(0.770112\pi\)
\(450\) 1.24264 0.0585786
\(451\) 10.3431 0.487040
\(452\) 9.71573 0.456989
\(453\) 3.51472 0.165136
\(454\) 7.17157 0.336579
\(455\) 0 0
\(456\) 4.48528 0.210043
\(457\) −7.65685 −0.358173 −0.179086 0.983833i \(-0.557314\pi\)
−0.179086 + 0.983833i \(0.557314\pi\)
\(458\) 0.544156 0.0254267
\(459\) 7.65685 0.357391
\(460\) 20.6863 0.964503
\(461\) −5.17157 −0.240864 −0.120432 0.992722i \(-0.538428\pi\)
−0.120432 + 0.992722i \(0.538428\pi\)
\(462\) 0 0
\(463\) −24.4853 −1.13793 −0.568964 0.822363i \(-0.692656\pi\)
−0.568964 + 0.822363i \(0.692656\pi\)
\(464\) 6.00000 0.278543
\(465\) −3.31371 −0.153670
\(466\) 2.88730 0.133752
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −1.82843 −0.0845191
\(469\) 0 0
\(470\) 13.6569 0.629944
\(471\) −10.0000 −0.460776
\(472\) 12.1421 0.558887
\(473\) 3.31371 0.152364
\(474\) 4.68629 0.215248
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0.828427 0.0378914
\(479\) −25.3137 −1.15661 −0.578306 0.815820i \(-0.696286\pi\)
−0.578306 + 0.815820i \(0.696286\pi\)
\(480\) −12.4853 −0.569873
\(481\) −7.65685 −0.349123
\(482\) −0.142136 −0.00647410
\(483\) 0 0
\(484\) 12.7990 0.581772
\(485\) −10.3431 −0.469658
\(486\) −0.414214 −0.0187891
\(487\) −7.79899 −0.353406 −0.176703 0.984264i \(-0.556543\pi\)
−0.176703 + 0.984264i \(0.556543\pi\)
\(488\) 21.1127 0.955727
\(489\) −18.8284 −0.851451
\(490\) 0 0
\(491\) 30.6274 1.38220 0.691098 0.722761i \(-0.257127\pi\)
0.691098 + 0.722761i \(0.257127\pi\)
\(492\) −9.45584 −0.426302
\(493\) −15.3137 −0.689695
\(494\) 1.17157 0.0527116
\(495\) −5.65685 −0.254257
\(496\) 3.51472 0.157816
\(497\) 0 0
\(498\) 1.51472 0.0678762
\(499\) 26.1421 1.17028 0.585141 0.810931i \(-0.301039\pi\)
0.585141 + 0.810931i \(0.301039\pi\)
\(500\) 10.3431 0.462560
\(501\) −3.65685 −0.163376
\(502\) 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
\(507\) −1.00000 −0.0444116
\(508\) −10.3431 −0.458903
\(509\) −11.7990 −0.522981 −0.261491 0.965206i \(-0.584214\pi\)
−0.261491 + 0.965206i \(0.584214\pi\)
\(510\) 8.97056 0.397223
\(511\) 0 0
\(512\) 22.7574 1.00574
\(513\) −2.82843 −0.124878
\(514\) 1.79899 0.0793500
\(515\) −6.62742 −0.292039
\(516\) −3.02944 −0.133364
\(517\) −23.3137 −1.02534
\(518\) 0 0
\(519\) −11.6569 −0.511679
\(520\) −4.48528 −0.196693
\(521\) −25.3137 −1.10901 −0.554507 0.832179i \(-0.687093\pi\)
−0.554507 + 0.832179i \(0.687093\pi\)
\(522\) 0.828427 0.0362593
\(523\) 15.3137 0.669622 0.334811 0.942285i \(-0.391328\pi\)
0.334811 + 0.942285i \(0.391328\pi\)
\(524\) −14.6274 −0.639002
\(525\) 0 0
\(526\) 4.97056 0.216727
\(527\) −8.97056 −0.390764
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) −2.34315 −0.101780
\(531\) −7.65685 −0.332279
\(532\) 0 0
\(533\) −5.17157 −0.224006
\(534\) 6.14214 0.265796
\(535\) −32.0000 −1.38348
\(536\) −10.8284 −0.467717
\(537\) 23.3137 1.00606
\(538\) −7.45584 −0.321444
\(539\) 0 0
\(540\) 5.17157 0.222549
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 11.5147 0.494600
\(543\) 14.0000 0.600798
\(544\) −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
\(549\) −13.3137 −0.568215
\(550\) −2.48528 −0.105973
\(551\) 5.65685 0.240990
\(552\) −6.34315 −0.269982
\(553\) 0 0
\(554\) −0.828427 −0.0351965
\(555\) 21.6569 0.919282
\(556\) −13.3726 −0.567124
\(557\) −7.79899 −0.330454 −0.165227 0.986256i \(-0.552836\pi\)
−0.165227 + 0.986256i \(0.552836\pi\)
\(558\) 0.485281 0.0205436
\(559\) −1.65685 −0.0700775
\(560\) 0 0
\(561\) −15.3137 −0.646545
\(562\) 8.76955 0.369921
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 21.3137 0.897469
\(565\) −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 \(-0.856953\pi\)
−0.900710 + 0.434421i \(0.856953\pi\)
\(570\) −3.31371 −0.138796
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 3.65685 0.152901
\(573\) −3.31371 −0.138432
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) −4.17157 −0.173816
\(577\) 31.9411 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(578\) 17.2426 0.717199
\(579\) −5.31371 −0.220830
\(580\) −10.3431 −0.429476
\(581\) 0 0
\(582\) 1.51472 0.0627871
\(583\) 4.00000 0.165663
\(584\) 0.544156 0.0225173
\(585\) 2.82843 0.116941
\(586\) −0.887302 −0.0366541
\(587\) 10.9706 0.452804 0.226402 0.974034i \(-0.427304\pi\)
0.226402 + 0.974034i \(0.427304\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) −8.97056 −0.369312
\(591\) −0.485281 −0.0199618
\(592\) −22.9706 −0.944084
\(593\) 20.4853 0.841230 0.420615 0.907239i \(-0.361814\pi\)
0.420615 + 0.907239i \(0.361814\pi\)
\(594\) 0.828427 0.0339908
\(595\) 0 0
\(596\) 16.7696 0.686908
\(597\) 21.6569 0.886356
\(598\) −1.65685 −0.0677538
\(599\) −23.3137 −0.952572 −0.476286 0.879290i \(-0.658017\pi\)
−0.476286 + 0.879290i \(0.658017\pi\)
\(600\) 4.75736 0.194218
\(601\) 0.627417 0.0255929 0.0127964 0.999918i \(-0.495927\pi\)
0.0127964 + 0.999918i \(0.495927\pi\)
\(602\) 0 0
\(603\) 6.82843 0.278075
\(604\) 6.42641 0.261487
\(605\) −19.7990 −0.804943
\(606\) 3.17157 0.128836
\(607\) −41.9411 −1.70234 −0.851169 0.524892i \(-0.824106\pi\)
−0.851169 + 0.524892i \(0.824106\pi\)
\(608\) 12.4853 0.506345
\(609\) 0 0
\(610\) −15.5980 −0.631544
\(611\) 11.6569 0.471586
\(612\) 14.0000 0.565916
\(613\) −47.6569 −1.92484 −0.962421 0.271561i \(-0.912460\pi\)
−0.962421 + 0.271561i \(0.912460\pi\)
\(614\) 9.45584 0.381607
\(615\) 14.6274 0.589834
\(616\) 0 0
\(617\) −34.8284 −1.40214 −0.701070 0.713093i \(-0.747294\pi\)
−0.701070 + 0.713093i \(0.747294\pi\)
\(618\) 0.970563 0.0390418
\(619\) −23.7990 −0.956562 −0.478281 0.878207i \(-0.658740\pi\)
−0.478281 + 0.878207i \(0.658740\pi\)
\(620\) −6.05887 −0.243330
\(621\) 4.00000 0.160514
\(622\) 4.40202 0.176505
\(623\) 0 0
\(624\) −3.00000 −0.120096
\(625\) −31.0000 −1.24000
\(626\) −2.48528 −0.0993318
\(627\) 5.65685 0.225913
\(628\) −18.2843 −0.729622
\(629\) 58.6274 2.33763
\(630\) 0 0
\(631\) 43.1127 1.71629 0.858145 0.513408i \(-0.171617\pi\)
0.858145 + 0.513408i \(0.171617\pi\)
\(632\) 17.9411 0.713660
\(633\) 12.0000 0.476957
\(634\) 3.51472 0.139587
\(635\) 16.0000 0.634941
\(636\) −3.65685 −0.145004
\(637\) 0 0
\(638\) −1.65685 −0.0655955
\(639\) 2.00000 0.0791188
\(640\) −29.8579 −1.18024
\(641\) 30.2843 1.19616 0.598078 0.801438i \(-0.295931\pi\)
0.598078 + 0.801438i \(0.295931\pi\)
\(642\) 4.68629 0.184953
\(643\) −22.8284 −0.900265 −0.450133 0.892962i \(-0.648623\pi\)
−0.450133 + 0.892962i \(0.648623\pi\)
\(644\) 0 0
\(645\) 4.68629 0.184523
\(646\) −8.97056 −0.352942
\(647\) −11.3137 −0.444788 −0.222394 0.974957i \(-0.571387\pi\)
−0.222394 + 0.974957i \(0.571387\pi\)
\(648\) −1.58579 −0.0622956
\(649\) 15.3137 0.601116
\(650\) 1.24264 0.0487404
\(651\) 0 0
\(652\) −34.4264 −1.34824
\(653\) −25.3137 −0.990602 −0.495301 0.868721i \(-0.664942\pi\)
−0.495301 + 0.868721i \(0.664942\pi\)
\(654\) −2.20101 −0.0860663
\(655\) 22.6274 0.884126
\(656\) −15.5147 −0.605748
\(657\) −0.343146 −0.0133874
\(658\) 0 0
\(659\) −47.3137 −1.84308 −0.921540 0.388283i \(-0.873068\pi\)
−0.921540 + 0.388283i \(0.873068\pi\)
\(660\) −10.3431 −0.402606
\(661\) 34.9706 1.36020 0.680099 0.733121i \(-0.261937\pi\)
0.680099 + 0.733121i \(0.261937\pi\)
\(662\) 10.8284 0.420859
\(663\) 7.65685 0.297368
\(664\) 5.79899 0.225044
\(665\) 0 0
\(666\) −3.17157 −0.122896
\(667\) −8.00000 −0.309761
\(668\) −6.68629 −0.258700
\(669\) −12.4853 −0.482709
\(670\) 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
\(675\) −3.00000 −0.115470
\(676\) −1.82843 −0.0703241
\(677\) −26.6863 −1.02564 −0.512819 0.858497i \(-0.671399\pi\)
−0.512819 + 0.858497i \(0.671399\pi\)
\(678\) 2.20101 0.0845293
\(679\) 0 0
\(680\) 34.3431 1.31700
\(681\) −17.3137 −0.663463
\(682\) −0.970563 −0.0371648
\(683\) 47.9411 1.83442 0.917208 0.398408i \(-0.130437\pi\)
0.917208 + 0.398408i \(0.130437\pi\)
\(684\) −5.17157 −0.197740
\(685\) −30.6274 −1.17021
\(686\) 0 0
\(687\) −1.31371 −0.0501211
\(688\) −4.97056 −0.189501
\(689\) −2.00000 −0.0761939
\(690\) 4.68629 0.178404
\(691\) 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
\(696\) 3.17157 0.120218
\(697\) 39.5980 1.49988
\(698\) −1.51472 −0.0573329
\(699\) −6.97056 −0.263651
\(700\) 0 0
\(701\) 5.02944 0.189959 0.0949796 0.995479i \(-0.469721\pi\)
0.0949796 + 0.995479i \(0.469721\pi\)
\(702\) −0.414214 −0.0156335
\(703\) −21.6569 −0.816804
\(704\) 8.34315 0.314444
\(705\) −32.9706 −1.24174
\(706\) 13.8579 0.521548
\(707\) 0 0
\(708\) −14.0000 −0.526152
\(709\) −4.62742 −0.173786 −0.0868931 0.996218i \(-0.527694\pi\)
−0.0868931 + 0.996218i \(0.527694\pi\)
\(710\) 2.34315 0.0879367
\(711\) −11.3137 −0.424297
\(712\) 23.5147 0.881251
\(713\) −4.68629 −0.175503
\(714\) 0 0
\(715\) −5.65685 −0.211554
\(716\) 42.6274 1.59306
\(717\) −2.00000 −0.0746914
\(718\) 14.4853 0.540586
\(719\) 29.9411 1.11662 0.558308 0.829634i \(-0.311451\pi\)
0.558308 + 0.829634i \(0.311451\pi\)
\(720\) 8.48528 0.316228
\(721\) 0 0
\(722\) −4.55635 −0.169570
\(723\) 0.343146 0.0127617
\(724\) 25.5980 0.951341
\(725\) 6.00000 0.222834
\(726\) 2.89949 0.107610
\(727\) 10.3431 0.383606 0.191803 0.981433i \(-0.438567\pi\)
0.191803 + 0.981433i \(0.438567\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −0.402020 −0.0148794
\(731\) 12.6863 0.469219
\(732\) −24.3431 −0.899749
\(733\) 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
\(738\) −2.14214 −0.0788531
\(739\) −18.1421 −0.667369 −0.333685 0.942685i \(-0.608292\pi\)
−0.333685 + 0.942685i \(0.608292\pi\)
\(740\) 39.5980 1.45565
\(741\) −2.82843 −0.103905
\(742\) 0 0
\(743\) 2.00000 0.0733729 0.0366864 0.999327i \(-0.488320\pi\)
0.0366864 + 0.999327i \(0.488320\pi\)
\(744\) 1.85786 0.0681126
\(745\) −25.9411 −0.950409
\(746\) 4.14214 0.151654
\(747\) −3.65685 −0.133797
\(748\) −28.0000 −1.02378
\(749\) 0 0
\(750\) 2.34315 0.0855596
\(751\) −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
\(754\) 0.828427 0.0301695
\(755\) −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
\(759\) −8.00000 −0.290382
\(760\) −12.6863 −0.460180
\(761\) −32.4853 −1.17759 −0.588795 0.808282i \(-0.700398\pi\)
−0.588795 + 0.808282i \(0.700398\pi\)
\(762\) −2.34315 −0.0848832
\(763\) 0 0
\(764\) −6.05887 −0.219202
\(765\) −21.6569 −0.783005
\(766\) −12.8284 −0.463510
\(767\) −7.65685 −0.276473
\(768\) −3.97056 −0.143275
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) −4.34315 −0.156415
\(772\) −9.71573 −0.349677
\(773\) −34.1421 −1.22801 −0.614004 0.789303i \(-0.710442\pi\)
−0.614004 + 0.789303i \(0.710442\pi\)
\(774\) −0.686292 −0.0246682
\(775\) 3.51472 0.126252
\(776\) 5.79899 0.208172
\(777\) 0 0
\(778\) −11.1716 −0.400520
\(779\) −14.6274 −0.524082
\(780\) 5.17157 0.185172
\(781\) −4.00000 −0.143131
\(782\) 12.6863 0.453661
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 28.2843 1.00951
\(786\) −3.31371 −0.118196
\(787\) 40.7696 1.45328 0.726639 0.687020i \(-0.241081\pi\)
0.726639 + 0.687020i \(0.241081\pi\)
\(788\) −0.887302 −0.0316088
\(789\) −12.0000 −0.427211
\(790\) −13.2548 −0.471586
\(791\) 0 0
\(792\) 3.17157 0.112697
\(793\) −13.3137 −0.472784
\(794\) −12.8284 −0.455264
\(795\) 5.65685 0.200628
\(796\) 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
\(801\) −14.8284 −0.523937
\(802\) 10.8284 0.382365
\(803\) 0.686292 0.0242187
\(804\) 12.4853 0.440322
\(805\) 0 0
\(806\) 0.485281 0.0170933
\(807\) 18.0000 0.633630
\(808\) 12.1421 0.427159
\(809\) 18.6863 0.656975 0.328488 0.944508i \(-0.393461\pi\)
0.328488 + 0.944508i \(0.393461\pi\)
\(810\) 1.17157 0.0411649
\(811\) 30.1421 1.05843 0.529217 0.848487i \(-0.322486\pi\)
0.529217 + 0.848487i \(0.322486\pi\)
\(812\) 0 0
\(813\) −27.7990 −0.974953
\(814\) 6.34315 0.222327
\(815\) 53.2548 1.86544
\(816\) 22.9706 0.804131
\(817\) −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 \(0.636322\pi\)
\(822\) 4.48528 0.156442
\(823\) 15.0294 0.523893 0.261947 0.965082i \(-0.415636\pi\)
0.261947 + 0.965082i \(0.415636\pi\)
\(824\) 3.71573 0.129444
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −26.0000 −0.904109 −0.452054 0.891990i \(-0.649309\pi\)
−0.452054 + 0.891990i \(0.649309\pi\)
\(828\) 7.31371 0.254169
\(829\) 17.3137 0.601330 0.300665 0.953730i \(-0.402791\pi\)
0.300665 + 0.953730i \(0.402791\pi\)
\(830\) −4.28427 −0.148709
\(831\) 2.00000 0.0693792
\(832\) −4.17157 −0.144623
\(833\) 0 0
\(834\) −3.02944 −0.104901
\(835\) 10.3431 0.357939
\(836\) 10.3431 0.357725
\(837\) −1.17157 −0.0404955
\(838\) −6.05887 −0.209300
\(839\) −43.2548 −1.49332 −0.746661 0.665204i \(-0.768344\pi\)
−0.746661 + 0.665204i \(0.768344\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 15.4558 0.532644
\(843\) −21.1716 −0.729188
\(844\) 21.9411 0.755245
\(845\) 2.82843 0.0973009
\(846\) 4.82843 0.166005
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 28.9706 0.994267
\(850\) −9.51472 −0.326352
\(851\) 30.6274 1.04989
\(852\) 3.65685 0.125282
\(853\) −3.65685 −0.125208 −0.0626042 0.998038i \(-0.519941\pi\)
−0.0626042 + 0.998038i \(0.519941\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 17.9411 0.613215
\(857\) 49.5980 1.69423 0.847117 0.531406i \(-0.178336\pi\)
0.847117 + 0.531406i \(0.178336\pi\)
\(858\) 0.828427 0.0282820
\(859\) 0.686292 0.0234160 0.0117080 0.999931i \(-0.496273\pi\)
0.0117080 + 0.999931i \(0.496273\pi\)
\(860\) 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 \(0.660237\pi\)
\(864\) −4.41421 −0.150175
\(865\) 32.9706 1.12103
\(866\) 8.82843 0.300002
\(867\) −41.6274 −1.41374
\(868\) 0 0
\(869\) 22.6274 0.767583
\(870\) −2.34315 −0.0794401
\(871\) 6.82843 0.231372
\(872\) −8.42641 −0.285354
\(873\) −3.65685 −0.123766
\(874\) −4.68629 −0.158516
\(875\) 0 0
\(876\) −0.627417 −0.0211985
\(877\) 42.2843 1.42784 0.713919 0.700228i \(-0.246918\pi\)
0.713919 + 0.700228i \(0.246918\pi\)
\(878\) −7.02944 −0.237232
\(879\) 2.14214 0.0722524
\(880\) −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 \(0.346280\pi\)
\(884\) 14.0000 0.470871
\(885\) 21.6569 0.727987
\(886\) −10.7452 −0.360991
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −12.1421 −0.407463
\(889\) 0 0
\(890\) −17.3726 −0.582330
\(891\) −2.00000 −0.0670025
\(892\) −22.8284 −0.764352
\(893\) 32.9706 1.10332
\(894\) 3.79899 0.127057
\(895\) −65.9411 −2.20417
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) −13.1716 −0.439541
\(899\) 2.34315 0.0781483
\(900\) −5.48528 −0.182843
\(901\) 15.3137 0.510174
\(902\) 4.28427 0.142651
\(903\) 0 0
\(904\) 8.42641 0.280258
\(905\) −39.5980 −1.31628
\(906\) 1.45584 0.0483672
\(907\) −12.9706 −0.430680 −0.215340 0.976539i \(-0.569086\pi\)
−0.215340 + 0.976539i \(0.569086\pi\)
\(908\) −31.6569 −1.05057
\(909\) −7.65685 −0.253962
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −8.48528 −0.280976
\(913\) 7.31371 0.242048
\(914\) −3.17157 −0.104906
\(915\) 37.6569 1.24490
\(916\) −2.40202 −0.0793650
\(917\) 0 0
\(918\) 3.17157 0.104678
\(919\) 3.31371 0.109309 0.0546546 0.998505i \(-0.482594\pi\)
0.0546546 + 0.998505i \(0.482594\pi\)
\(920\) 17.9411 0.591501
\(921\) −22.8284 −0.752222
\(922\) −2.14214 −0.0705475
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) −22.9706 −0.755267
\(926\) −10.1421 −0.333291
\(927\) −2.34315 −0.0769590
\(928\) 8.82843 0.289807
\(929\) −11.7990 −0.387112 −0.193556 0.981089i \(-0.562002\pi\)
−0.193556 + 0.981089i \(0.562002\pi\)
\(930\) −1.37258 −0.0450088
\(931\) 0 0
\(932\) −12.7452 −0.417482
\(933\) −10.6274 −0.347926
\(934\) 3.31371 0.108428
\(935\) 43.3137 1.41651
\(936\) −1.58579 −0.0518331
\(937\) 21.3137 0.696289 0.348144 0.937441i \(-0.386812\pi\)
0.348144 + 0.937441i \(0.386812\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) −60.2843 −1.96626
\(941\) −34.1421 −1.11300 −0.556501 0.830847i \(-0.687856\pi\)
−0.556501 + 0.830847i \(0.687856\pi\)
\(942\) −4.14214 −0.134958
\(943\) 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 \(0.610997\pi\)
\(948\) −20.6863 −0.671860
\(949\) −0.343146 −0.0111390
\(950\) 3.51472 0.114033
\(951\) −8.48528 −0.275154
\(952\) 0 0
\(953\) 40.3431 1.30684 0.653421 0.756994i \(-0.273333\pi\)
0.653421 + 0.756994i \(0.273333\pi\)
\(954\) −0.828427 −0.0268213
\(955\) 9.37258 0.303290
\(956\) −3.65685 −0.118271
\(957\) 4.00000 0.129302
\(958\) −10.4853 −0.338764
\(959\) 0 0
\(960\) 11.7990 0.380811
\(961\) −29.6274 −0.955723
\(962\) −3.17157 −0.102256
\(963\) −11.3137 −0.364579
\(964\) 0.627417 0.0202077
\(965\) 15.0294 0.483815
\(966\) 0 0
\(967\) 18.1421 0.583412 0.291706 0.956508i \(-0.405777\pi\)
0.291706 + 0.956508i \(0.405777\pi\)
\(968\) 11.1005 0.356784
\(969\) 21.6569 0.695718
\(970\) −4.28427 −0.137560
\(971\) 15.3137 0.491440 0.245720 0.969341i \(-0.420976\pi\)
0.245720 + 0.969341i \(0.420976\pi\)
\(972\) 1.82843 0.0586468
\(973\) 0 0
\(974\) −3.23045 −0.103510
\(975\) −3.00000 −0.0960769
\(976\) −39.9411 −1.27848
\(977\) 42.1421 1.34825 0.674123 0.738619i \(-0.264522\pi\)
0.674123 + 0.738619i \(0.264522\pi\)
\(978\) −7.79899 −0.249384
\(979\) 29.6569 0.947837
\(980\) 0 0
\(981\) 5.31371 0.169654
\(982\) 12.6863 0.404836
\(983\) −25.3137 −0.807382 −0.403691 0.914895i \(-0.632273\pi\)
−0.403691 + 0.914895i \(0.632273\pi\)
\(984\) −8.20101 −0.261439
\(985\) 1.37258 0.0437341
\(986\) −6.34315 −0.202007
\(987\) 0 0
\(988\) −5.17157 −0.164530
\(989\) 6.62742 0.210740
\(990\) −2.34315 −0.0744701
\(991\) 4.68629 0.148865 0.0744325 0.997226i \(-0.476285\pi\)
0.0744325 + 0.997226i \(0.476285\pi\)
\(992\) 5.17157 0.164198
\(993\) −26.1421 −0.829596
\(994\) 0 0
\(995\) −61.2548 −1.94191
\(996\) −6.68629 −0.211863
\(997\) 39.2548 1.24321 0.621607 0.783330i \(-0.286480\pi\)
0.621607 + 0.783330i \(0.286480\pi\)
\(998\) 10.8284 0.342768
\(999\) 7.65685 0.242252
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 1911.2.a.h.1.2 2
3.2 odd 2 5733.2.a.u.1.1 2
7.6 odd 2 39.2.a.b.1.2 2
21.20 even 2 117.2.a.c.1.1 2
28.27 even 2 624.2.a.k.1.1 2
35.13 even 4 975.2.c.h.274.2 4
35.27 even 4 975.2.c.h.274.3 4
35.34 odd 2 975.2.a.l.1.1 2
56.13 odd 2 2496.2.a.bf.1.2 2
56.27 even 2 2496.2.a.bi.1.2 2
63.13 odd 6 1053.2.e.m.703.1 4
63.20 even 6 1053.2.e.e.352.2 4
63.34 odd 6 1053.2.e.m.352.1 4
63.41 even 6 1053.2.e.e.703.2 4
77.76 even 2 4719.2.a.p.1.1 2
84.83 odd 2 1872.2.a.w.1.2 2
91.6 even 12 507.2.j.f.361.2 8
91.20 even 12 507.2.j.f.361.3 8
91.34 even 4 507.2.b.e.337.3 4
91.41 even 12 507.2.j.f.316.3 8
91.48 odd 6 507.2.e.h.484.1 4
91.55 odd 6 507.2.e.h.22.1 4
91.62 odd 6 507.2.e.d.22.2 4
91.69 odd 6 507.2.e.d.484.2 4
91.76 even 12 507.2.j.f.316.2 8
91.83 even 4 507.2.b.e.337.2 4
91.90 odd 2 507.2.a.h.1.1 2
105.62 odd 4 2925.2.c.u.2224.2 4
105.83 odd 4 2925.2.c.u.2224.3 4
105.104 even 2 2925.2.a.v.1.2 2
168.83 odd 2 7488.2.a.co.1.1 2
168.125 even 2 7488.2.a.cl.1.1 2
273.83 odd 4 1521.2.b.j.1351.3 4
273.125 odd 4 1521.2.b.j.1351.2 4
273.272 even 2 1521.2.a.f.1.2 2
364.363 even 2 8112.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.2 2 7.6 odd 2
117.2.a.c.1.1 2 21.20 even 2
507.2.a.h.1.1 2 91.90 odd 2
507.2.b.e.337.2 4 91.83 even 4
507.2.b.e.337.3 4 91.34 even 4
507.2.e.d.22.2 4 91.62 odd 6
507.2.e.d.484.2 4 91.69 odd 6
507.2.e.h.22.1 4 91.55 odd 6
507.2.e.h.484.1 4 91.48 odd 6
507.2.j.f.316.2 8 91.76 even 12
507.2.j.f.316.3 8 91.41 even 12
507.2.j.f.361.2 8 91.6 even 12
507.2.j.f.361.3 8 91.20 even 12
624.2.a.k.1.1 2 28.27 even 2
975.2.a.l.1.1 2 35.34 odd 2
975.2.c.h.274.2 4 35.13 even 4
975.2.c.h.274.3 4 35.27 even 4
1053.2.e.e.352.2 4 63.20 even 6
1053.2.e.e.703.2 4 63.41 even 6
1053.2.e.m.352.1 4 63.34 odd 6
1053.2.e.m.703.1 4 63.13 odd 6
1521.2.a.f.1.2 2 273.272 even 2
1521.2.b.j.1351.2 4 273.125 odd 4
1521.2.b.j.1351.3 4 273.83 odd 4
1872.2.a.w.1.2 2 84.83 odd 2
1911.2.a.h.1.2 2 1.1 even 1 trivial
2496.2.a.bf.1.2 2 56.13 odd 2
2496.2.a.bi.1.2 2 56.27 even 2
2925.2.a.v.1.2 2 105.104 even 2
2925.2.c.u.2224.2 4 105.62 odd 4
2925.2.c.u.2224.3 4 105.83 odd 4
4719.2.a.p.1.1 2 77.76 even 2
5733.2.a.u.1.1 2 3.2 odd 2
7488.2.a.cl.1.1 2 168.125 even 2
7488.2.a.co.1.1 2 168.83 odd 2
8112.2.a.bm.1.2 2 364.363 even 2