Properties

Label 8034.2.a.bc.1.13
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-3.01123\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.01123 q^{5} -1.00000 q^{6} -0.871929 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.01123 q^{5} -1.00000 q^{6} -0.871929 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.01123 q^{10} -3.54795 q^{11} -1.00000 q^{12} -1.00000 q^{13} -0.871929 q^{14} -3.01123 q^{15} +1.00000 q^{16} -1.08800 q^{17} +1.00000 q^{18} -2.36288 q^{19} +3.01123 q^{20} +0.871929 q^{21} -3.54795 q^{22} +6.14753 q^{23} -1.00000 q^{24} +4.06751 q^{25} -1.00000 q^{26} -1.00000 q^{27} -0.871929 q^{28} -2.17058 q^{29} -3.01123 q^{30} +0.257032 q^{31} +1.00000 q^{32} +3.54795 q^{33} -1.08800 q^{34} -2.62558 q^{35} +1.00000 q^{36} -2.01359 q^{37} -2.36288 q^{38} +1.00000 q^{39} +3.01123 q^{40} +8.60323 q^{41} +0.871929 q^{42} -1.44174 q^{43} -3.54795 q^{44} +3.01123 q^{45} +6.14753 q^{46} +8.61755 q^{47} -1.00000 q^{48} -6.23974 q^{49} +4.06751 q^{50} +1.08800 q^{51} -1.00000 q^{52} +8.32354 q^{53} -1.00000 q^{54} -10.6837 q^{55} -0.871929 q^{56} +2.36288 q^{57} -2.17058 q^{58} +6.35243 q^{59} -3.01123 q^{60} +15.0315 q^{61} +0.257032 q^{62} -0.871929 q^{63} +1.00000 q^{64} -3.01123 q^{65} +3.54795 q^{66} -4.89108 q^{67} -1.08800 q^{68} -6.14753 q^{69} -2.62558 q^{70} -6.82395 q^{71} +1.00000 q^{72} +15.2479 q^{73} -2.01359 q^{74} -4.06751 q^{75} -2.36288 q^{76} +3.09356 q^{77} +1.00000 q^{78} -5.43391 q^{79} +3.01123 q^{80} +1.00000 q^{81} +8.60323 q^{82} -6.99737 q^{83} +0.871929 q^{84} -3.27622 q^{85} -1.44174 q^{86} +2.17058 q^{87} -3.54795 q^{88} +13.7319 q^{89} +3.01123 q^{90} +0.871929 q^{91} +6.14753 q^{92} -0.257032 q^{93} +8.61755 q^{94} -7.11519 q^{95} -1.00000 q^{96} +14.3069 q^{97} -6.23974 q^{98} -3.54795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.01123 1.34666 0.673332 0.739341i \(-0.264863\pi\)
0.673332 + 0.739341i \(0.264863\pi\)
\(6\) −1.00000 −0.408248
\(7\) −0.871929 −0.329558 −0.164779 0.986331i \(-0.552691\pi\)
−0.164779 + 0.986331i \(0.552691\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.01123 0.952235
\(11\) −3.54795 −1.06975 −0.534874 0.844932i \(-0.679641\pi\)
−0.534874 + 0.844932i \(0.679641\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −0.871929 −0.233033
\(15\) −3.01123 −0.777496
\(16\) 1.00000 0.250000
\(17\) −1.08800 −0.263879 −0.131939 0.991258i \(-0.542120\pi\)
−0.131939 + 0.991258i \(0.542120\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.36288 −0.542083 −0.271041 0.962568i \(-0.587368\pi\)
−0.271041 + 0.962568i \(0.587368\pi\)
\(20\) 3.01123 0.673332
\(21\) 0.871929 0.190270
\(22\) −3.54795 −0.756426
\(23\) 6.14753 1.28185 0.640924 0.767604i \(-0.278551\pi\)
0.640924 + 0.767604i \(0.278551\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.06751 0.813501
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −0.871929 −0.164779
\(29\) −2.17058 −0.403066 −0.201533 0.979482i \(-0.564592\pi\)
−0.201533 + 0.979482i \(0.564592\pi\)
\(30\) −3.01123 −0.549773
\(31\) 0.257032 0.0461643 0.0230822 0.999734i \(-0.492652\pi\)
0.0230822 + 0.999734i \(0.492652\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.54795 0.617619
\(34\) −1.08800 −0.186591
\(35\) −2.62558 −0.443804
\(36\) 1.00000 0.166667
\(37\) −2.01359 −0.331032 −0.165516 0.986207i \(-0.552929\pi\)
−0.165516 + 0.986207i \(0.552929\pi\)
\(38\) −2.36288 −0.383311
\(39\) 1.00000 0.160128
\(40\) 3.01123 0.476117
\(41\) 8.60323 1.34360 0.671800 0.740733i \(-0.265522\pi\)
0.671800 + 0.740733i \(0.265522\pi\)
\(42\) 0.871929 0.134542
\(43\) −1.44174 −0.219863 −0.109932 0.993939i \(-0.535063\pi\)
−0.109932 + 0.993939i \(0.535063\pi\)
\(44\) −3.54795 −0.534874
\(45\) 3.01123 0.448888
\(46\) 6.14753 0.906404
\(47\) 8.61755 1.25700 0.628500 0.777810i \(-0.283669\pi\)
0.628500 + 0.777810i \(0.283669\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.23974 −0.891391
\(50\) 4.06751 0.575232
\(51\) 1.08800 0.152351
\(52\) −1.00000 −0.138675
\(53\) 8.32354 1.14333 0.571663 0.820488i \(-0.306298\pi\)
0.571663 + 0.820488i \(0.306298\pi\)
\(54\) −1.00000 −0.136083
\(55\) −10.6837 −1.44059
\(56\) −0.871929 −0.116516
\(57\) 2.36288 0.312972
\(58\) −2.17058 −0.285011
\(59\) 6.35243 0.827016 0.413508 0.910501i \(-0.364303\pi\)
0.413508 + 0.910501i \(0.364303\pi\)
\(60\) −3.01123 −0.388748
\(61\) 15.0315 1.92458 0.962290 0.272026i \(-0.0876935\pi\)
0.962290 + 0.272026i \(0.0876935\pi\)
\(62\) 0.257032 0.0326431
\(63\) −0.871929 −0.109853
\(64\) 1.00000 0.125000
\(65\) −3.01123 −0.373497
\(66\) 3.54795 0.436723
\(67\) −4.89108 −0.597541 −0.298770 0.954325i \(-0.596576\pi\)
−0.298770 + 0.954325i \(0.596576\pi\)
\(68\) −1.08800 −0.131939
\(69\) −6.14753 −0.740075
\(70\) −2.62558 −0.313817
\(71\) −6.82395 −0.809853 −0.404927 0.914349i \(-0.632703\pi\)
−0.404927 + 0.914349i \(0.632703\pi\)
\(72\) 1.00000 0.117851
\(73\) 15.2479 1.78463 0.892315 0.451414i \(-0.149080\pi\)
0.892315 + 0.451414i \(0.149080\pi\)
\(74\) −2.01359 −0.234075
\(75\) −4.06751 −0.469675
\(76\) −2.36288 −0.271041
\(77\) 3.09356 0.352544
\(78\) 1.00000 0.113228
\(79\) −5.43391 −0.611362 −0.305681 0.952134i \(-0.598884\pi\)
−0.305681 + 0.952134i \(0.598884\pi\)
\(80\) 3.01123 0.336666
\(81\) 1.00000 0.111111
\(82\) 8.60323 0.950068
\(83\) −6.99737 −0.768062 −0.384031 0.923320i \(-0.625464\pi\)
−0.384031 + 0.923320i \(0.625464\pi\)
\(84\) 0.871929 0.0951352
\(85\) −3.27622 −0.355356
\(86\) −1.44174 −0.155467
\(87\) 2.17058 0.232710
\(88\) −3.54795 −0.378213
\(89\) 13.7319 1.45557 0.727787 0.685803i \(-0.240549\pi\)
0.727787 + 0.685803i \(0.240549\pi\)
\(90\) 3.01123 0.317412
\(91\) 0.871929 0.0914030
\(92\) 6.14753 0.640924
\(93\) −0.257032 −0.0266530
\(94\) 8.61755 0.888833
\(95\) −7.11519 −0.730003
\(96\) −1.00000 −0.102062
\(97\) 14.3069 1.45264 0.726322 0.687355i \(-0.241228\pi\)
0.726322 + 0.687355i \(0.241228\pi\)
\(98\) −6.23974 −0.630309
\(99\) −3.54795 −0.356583
\(100\) 4.06751 0.406751
\(101\) 12.5408 1.24786 0.623929 0.781481i \(-0.285535\pi\)
0.623929 + 0.781481i \(0.285535\pi\)
\(102\) 1.08800 0.107728
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 2.62558 0.256230
\(106\) 8.32354 0.808454
\(107\) 6.50708 0.629063 0.314531 0.949247i \(-0.398153\pi\)
0.314531 + 0.949247i \(0.398153\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.0157 1.05511 0.527557 0.849520i \(-0.323108\pi\)
0.527557 + 0.849520i \(0.323108\pi\)
\(110\) −10.6837 −1.01865
\(111\) 2.01359 0.191121
\(112\) −0.871929 −0.0823895
\(113\) 8.84167 0.831754 0.415877 0.909421i \(-0.363475\pi\)
0.415877 + 0.909421i \(0.363475\pi\)
\(114\) 2.36288 0.221304
\(115\) 18.5116 1.72622
\(116\) −2.17058 −0.201533
\(117\) −1.00000 −0.0924500
\(118\) 6.35243 0.584789
\(119\) 0.948659 0.0869634
\(120\) −3.01123 −0.274886
\(121\) 1.58796 0.144360
\(122\) 15.0315 1.36088
\(123\) −8.60323 −0.775728
\(124\) 0.257032 0.0230822
\(125\) −2.80795 −0.251151
\(126\) −0.871929 −0.0776776
\(127\) 6.76430 0.600234 0.300117 0.953902i \(-0.402974\pi\)
0.300117 + 0.953902i \(0.402974\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.44174 0.126938
\(130\) −3.01123 −0.264102
\(131\) 19.9488 1.74293 0.871466 0.490456i \(-0.163170\pi\)
0.871466 + 0.490456i \(0.163170\pi\)
\(132\) 3.54795 0.308810
\(133\) 2.06027 0.178648
\(134\) −4.89108 −0.422525
\(135\) −3.01123 −0.259165
\(136\) −1.08800 −0.0932953
\(137\) −21.0698 −1.80011 −0.900055 0.435775i \(-0.856474\pi\)
−0.900055 + 0.435775i \(0.856474\pi\)
\(138\) −6.14753 −0.523312
\(139\) 11.4453 0.970777 0.485389 0.874299i \(-0.338678\pi\)
0.485389 + 0.874299i \(0.338678\pi\)
\(140\) −2.62558 −0.221902
\(141\) −8.61755 −0.725729
\(142\) −6.82395 −0.572653
\(143\) 3.54795 0.296695
\(144\) 1.00000 0.0833333
\(145\) −6.53610 −0.542794
\(146\) 15.2479 1.26192
\(147\) 6.23974 0.514645
\(148\) −2.01359 −0.165516
\(149\) −16.5724 −1.35766 −0.678830 0.734295i \(-0.737513\pi\)
−0.678830 + 0.734295i \(0.737513\pi\)
\(150\) −4.06751 −0.332111
\(151\) 1.15504 0.0939960 0.0469980 0.998895i \(-0.485035\pi\)
0.0469980 + 0.998895i \(0.485035\pi\)
\(152\) −2.36288 −0.191655
\(153\) −1.08800 −0.0879596
\(154\) 3.09356 0.249286
\(155\) 0.773983 0.0621678
\(156\) 1.00000 0.0800641
\(157\) 3.32560 0.265412 0.132706 0.991155i \(-0.457633\pi\)
0.132706 + 0.991155i \(0.457633\pi\)
\(158\) −5.43391 −0.432298
\(159\) −8.32354 −0.660100
\(160\) 3.01123 0.238059
\(161\) −5.36021 −0.422443
\(162\) 1.00000 0.0785674
\(163\) −10.0530 −0.787409 −0.393704 0.919237i \(-0.628807\pi\)
−0.393704 + 0.919237i \(0.628807\pi\)
\(164\) 8.60323 0.671800
\(165\) 10.6837 0.831725
\(166\) −6.99737 −0.543102
\(167\) 9.56882 0.740458 0.370229 0.928941i \(-0.379279\pi\)
0.370229 + 0.928941i \(0.379279\pi\)
\(168\) 0.871929 0.0672708
\(169\) 1.00000 0.0769231
\(170\) −3.27622 −0.251275
\(171\) −2.36288 −0.180694
\(172\) −1.44174 −0.109932
\(173\) −7.30957 −0.555737 −0.277868 0.960619i \(-0.589628\pi\)
−0.277868 + 0.960619i \(0.589628\pi\)
\(174\) 2.17058 0.164551
\(175\) −3.54658 −0.268096
\(176\) −3.54795 −0.267437
\(177\) −6.35243 −0.477478
\(178\) 13.7319 1.02925
\(179\) 11.1967 0.836882 0.418441 0.908244i \(-0.362577\pi\)
0.418441 + 0.908244i \(0.362577\pi\)
\(180\) 3.01123 0.224444
\(181\) 1.94900 0.144868 0.0724340 0.997373i \(-0.476923\pi\)
0.0724340 + 0.997373i \(0.476923\pi\)
\(182\) 0.871929 0.0646317
\(183\) −15.0315 −1.11116
\(184\) 6.14753 0.453202
\(185\) −6.06337 −0.445788
\(186\) −0.257032 −0.0188465
\(187\) 3.86017 0.282284
\(188\) 8.61755 0.628500
\(189\) 0.871929 0.0634235
\(190\) −7.11519 −0.516190
\(191\) −10.9502 −0.792330 −0.396165 0.918179i \(-0.629659\pi\)
−0.396165 + 0.918179i \(0.629659\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −5.43921 −0.391523 −0.195761 0.980652i \(-0.562718\pi\)
−0.195761 + 0.980652i \(0.562718\pi\)
\(194\) 14.3069 1.02717
\(195\) 3.01123 0.215639
\(196\) −6.23974 −0.445696
\(197\) 21.7761 1.55148 0.775740 0.631052i \(-0.217377\pi\)
0.775740 + 0.631052i \(0.217377\pi\)
\(198\) −3.54795 −0.252142
\(199\) −13.6495 −0.967586 −0.483793 0.875182i \(-0.660741\pi\)
−0.483793 + 0.875182i \(0.660741\pi\)
\(200\) 4.06751 0.287616
\(201\) 4.89108 0.344990
\(202\) 12.5408 0.882369
\(203\) 1.89259 0.132834
\(204\) 1.08800 0.0761753
\(205\) 25.9063 1.80938
\(206\) −1.00000 −0.0696733
\(207\) 6.14753 0.427283
\(208\) −1.00000 −0.0693375
\(209\) 8.38340 0.579892
\(210\) 2.62558 0.181182
\(211\) −19.2055 −1.32216 −0.661081 0.750315i \(-0.729902\pi\)
−0.661081 + 0.750315i \(0.729902\pi\)
\(212\) 8.32354 0.571663
\(213\) 6.82395 0.467569
\(214\) 6.50708 0.444815
\(215\) −4.34141 −0.296081
\(216\) −1.00000 −0.0680414
\(217\) −0.224114 −0.0152138
\(218\) 11.0157 0.746078
\(219\) −15.2479 −1.03036
\(220\) −10.6837 −0.720295
\(221\) 1.08800 0.0731868
\(222\) 2.01359 0.135143
\(223\) 7.76423 0.519931 0.259966 0.965618i \(-0.416289\pi\)
0.259966 + 0.965618i \(0.416289\pi\)
\(224\) −0.871929 −0.0582582
\(225\) 4.06751 0.271167
\(226\) 8.84167 0.588139
\(227\) −9.20208 −0.610764 −0.305382 0.952230i \(-0.598784\pi\)
−0.305382 + 0.952230i \(0.598784\pi\)
\(228\) 2.36288 0.156486
\(229\) −22.1657 −1.46475 −0.732375 0.680902i \(-0.761588\pi\)
−0.732375 + 0.680902i \(0.761588\pi\)
\(230\) 18.5116 1.22062
\(231\) −3.09356 −0.203541
\(232\) −2.17058 −0.142505
\(233\) −16.5704 −1.08556 −0.542780 0.839875i \(-0.682628\pi\)
−0.542780 + 0.839875i \(0.682628\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 25.9494 1.69275
\(236\) 6.35243 0.413508
\(237\) 5.43391 0.352970
\(238\) 0.948659 0.0614924
\(239\) 21.6746 1.40201 0.701007 0.713155i \(-0.252734\pi\)
0.701007 + 0.713155i \(0.252734\pi\)
\(240\) −3.01123 −0.194374
\(241\) −16.7333 −1.07789 −0.538943 0.842342i \(-0.681176\pi\)
−0.538943 + 0.842342i \(0.681176\pi\)
\(242\) 1.58796 0.102078
\(243\) −1.00000 −0.0641500
\(244\) 15.0315 0.962290
\(245\) −18.7893 −1.20040
\(246\) −8.60323 −0.548522
\(247\) 2.36288 0.150347
\(248\) 0.257032 0.0163216
\(249\) 6.99737 0.443441
\(250\) −2.80795 −0.177590
\(251\) 23.0103 1.45240 0.726200 0.687483i \(-0.241285\pi\)
0.726200 + 0.687483i \(0.241285\pi\)
\(252\) −0.871929 −0.0549263
\(253\) −21.8111 −1.37125
\(254\) 6.76430 0.424430
\(255\) 3.27622 0.205165
\(256\) 1.00000 0.0625000
\(257\) −4.89130 −0.305111 −0.152556 0.988295i \(-0.548750\pi\)
−0.152556 + 0.988295i \(0.548750\pi\)
\(258\) 1.44174 0.0897587
\(259\) 1.75570 0.109094
\(260\) −3.01123 −0.186749
\(261\) −2.17058 −0.134355
\(262\) 19.9488 1.23244
\(263\) −5.62766 −0.347016 −0.173508 0.984832i \(-0.555510\pi\)
−0.173508 + 0.984832i \(0.555510\pi\)
\(264\) 3.54795 0.218361
\(265\) 25.0641 1.53968
\(266\) 2.06027 0.126323
\(267\) −13.7319 −0.840377
\(268\) −4.89108 −0.298770
\(269\) −9.72299 −0.592821 −0.296411 0.955061i \(-0.595790\pi\)
−0.296411 + 0.955061i \(0.595790\pi\)
\(270\) −3.01123 −0.183258
\(271\) −0.275577 −0.0167401 −0.00837007 0.999965i \(-0.502664\pi\)
−0.00837007 + 0.999965i \(0.502664\pi\)
\(272\) −1.08800 −0.0659697
\(273\) −0.871929 −0.0527715
\(274\) −21.0698 −1.27287
\(275\) −14.4313 −0.870241
\(276\) −6.14753 −0.370038
\(277\) 9.98438 0.599903 0.299951 0.953954i \(-0.403030\pi\)
0.299951 + 0.953954i \(0.403030\pi\)
\(278\) 11.4453 0.686443
\(279\) 0.257032 0.0153881
\(280\) −2.62558 −0.156908
\(281\) 10.8011 0.644338 0.322169 0.946682i \(-0.395588\pi\)
0.322169 + 0.946682i \(0.395588\pi\)
\(282\) −8.61755 −0.513168
\(283\) −6.93304 −0.412126 −0.206063 0.978539i \(-0.566065\pi\)
−0.206063 + 0.978539i \(0.566065\pi\)
\(284\) −6.82395 −0.404927
\(285\) 7.11519 0.421467
\(286\) 3.54795 0.209795
\(287\) −7.50141 −0.442794
\(288\) 1.00000 0.0589256
\(289\) −15.8163 −0.930368
\(290\) −6.53610 −0.383813
\(291\) −14.3069 −0.838684
\(292\) 15.2479 0.892315
\(293\) −0.802976 −0.0469104 −0.0234552 0.999725i \(-0.507467\pi\)
−0.0234552 + 0.999725i \(0.507467\pi\)
\(294\) 6.23974 0.363909
\(295\) 19.1286 1.11371
\(296\) −2.01359 −0.117037
\(297\) 3.54795 0.205873
\(298\) −16.5724 −0.960011
\(299\) −6.14753 −0.355521
\(300\) −4.06751 −0.234838
\(301\) 1.25709 0.0724577
\(302\) 1.15504 0.0664652
\(303\) −12.5408 −0.720451
\(304\) −2.36288 −0.135521
\(305\) 45.2632 2.59176
\(306\) −1.08800 −0.0621968
\(307\) 0.918828 0.0524403 0.0262201 0.999656i \(-0.491653\pi\)
0.0262201 + 0.999656i \(0.491653\pi\)
\(308\) 3.09356 0.176272
\(309\) 1.00000 0.0568880
\(310\) 0.773983 0.0439593
\(311\) −22.0719 −1.25158 −0.625792 0.779990i \(-0.715224\pi\)
−0.625792 + 0.779990i \(0.715224\pi\)
\(312\) 1.00000 0.0566139
\(313\) −15.5057 −0.876436 −0.438218 0.898869i \(-0.644390\pi\)
−0.438218 + 0.898869i \(0.644390\pi\)
\(314\) 3.32560 0.187674
\(315\) −2.62558 −0.147935
\(316\) −5.43391 −0.305681
\(317\) 26.8110 1.50585 0.752927 0.658104i \(-0.228641\pi\)
0.752927 + 0.658104i \(0.228641\pi\)
\(318\) −8.32354 −0.466761
\(319\) 7.70110 0.431179
\(320\) 3.01123 0.168333
\(321\) −6.50708 −0.363190
\(322\) −5.36021 −0.298713
\(323\) 2.57082 0.143044
\(324\) 1.00000 0.0555556
\(325\) −4.06751 −0.225625
\(326\) −10.0530 −0.556782
\(327\) −11.0157 −0.609170
\(328\) 8.60323 0.475034
\(329\) −7.51389 −0.414254
\(330\) 10.6837 0.588118
\(331\) 3.71332 0.204102 0.102051 0.994779i \(-0.467459\pi\)
0.102051 + 0.994779i \(0.467459\pi\)
\(332\) −6.99737 −0.384031
\(333\) −2.01359 −0.110344
\(334\) 9.56882 0.523583
\(335\) −14.7282 −0.804686
\(336\) 0.871929 0.0475676
\(337\) 8.97722 0.489021 0.244510 0.969647i \(-0.421373\pi\)
0.244510 + 0.969647i \(0.421373\pi\)
\(338\) 1.00000 0.0543928
\(339\) −8.84167 −0.480214
\(340\) −3.27622 −0.177678
\(341\) −0.911938 −0.0493842
\(342\) −2.36288 −0.127770
\(343\) 11.5441 0.623323
\(344\) −1.44174 −0.0777333
\(345\) −18.5116 −0.996632
\(346\) −7.30957 −0.392965
\(347\) −11.5432 −0.619672 −0.309836 0.950790i \(-0.600274\pi\)
−0.309836 + 0.950790i \(0.600274\pi\)
\(348\) 2.17058 0.116355
\(349\) 0.844753 0.0452186 0.0226093 0.999744i \(-0.492803\pi\)
0.0226093 + 0.999744i \(0.492803\pi\)
\(350\) −3.54658 −0.189572
\(351\) 1.00000 0.0533761
\(352\) −3.54795 −0.189106
\(353\) 19.3423 1.02949 0.514744 0.857344i \(-0.327887\pi\)
0.514744 + 0.857344i \(0.327887\pi\)
\(354\) −6.35243 −0.337628
\(355\) −20.5485 −1.09060
\(356\) 13.7319 0.727787
\(357\) −0.948659 −0.0502083
\(358\) 11.1967 0.591765
\(359\) 10.1494 0.535663 0.267831 0.963466i \(-0.413693\pi\)
0.267831 + 0.963466i \(0.413693\pi\)
\(360\) 3.01123 0.158706
\(361\) −13.4168 −0.706146
\(362\) 1.94900 0.102437
\(363\) −1.58796 −0.0833463
\(364\) 0.871929 0.0457015
\(365\) 45.9149 2.40330
\(366\) −15.0315 −0.785707
\(367\) −7.35480 −0.383917 −0.191959 0.981403i \(-0.561484\pi\)
−0.191959 + 0.981403i \(0.561484\pi\)
\(368\) 6.14753 0.320462
\(369\) 8.60323 0.447867
\(370\) −6.06337 −0.315220
\(371\) −7.25753 −0.376792
\(372\) −0.257032 −0.0133265
\(373\) −0.00733682 −0.000379886 0 −0.000189943 1.00000i \(-0.500060\pi\)
−0.000189943 1.00000i \(0.500060\pi\)
\(374\) 3.86017 0.199605
\(375\) 2.80795 0.145002
\(376\) 8.61755 0.444416
\(377\) 2.17058 0.111790
\(378\) 0.871929 0.0448472
\(379\) −17.2292 −0.885004 −0.442502 0.896768i \(-0.645909\pi\)
−0.442502 + 0.896768i \(0.645909\pi\)
\(380\) −7.11519 −0.365002
\(381\) −6.76430 −0.346545
\(382\) −10.9502 −0.560262
\(383\) −37.0826 −1.89483 −0.947417 0.320003i \(-0.896316\pi\)
−0.947417 + 0.320003i \(0.896316\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 9.31542 0.474758
\(386\) −5.43921 −0.276849
\(387\) −1.44174 −0.0732877
\(388\) 14.3069 0.726322
\(389\) 16.4678 0.834949 0.417474 0.908689i \(-0.362915\pi\)
0.417474 + 0.908689i \(0.362915\pi\)
\(390\) 3.01123 0.152480
\(391\) −6.68851 −0.338253
\(392\) −6.23974 −0.315154
\(393\) −19.9488 −1.00628
\(394\) 21.7761 1.09706
\(395\) −16.3627 −0.823299
\(396\) −3.54795 −0.178291
\(397\) 33.6723 1.68996 0.844982 0.534795i \(-0.179611\pi\)
0.844982 + 0.534795i \(0.179611\pi\)
\(398\) −13.6495 −0.684187
\(399\) −2.06027 −0.103142
\(400\) 4.06751 0.203375
\(401\) −18.2404 −0.910882 −0.455441 0.890266i \(-0.650518\pi\)
−0.455441 + 0.890266i \(0.650518\pi\)
\(402\) 4.89108 0.243945
\(403\) −0.257032 −0.0128037
\(404\) 12.5408 0.623929
\(405\) 3.01123 0.149629
\(406\) 1.89259 0.0939275
\(407\) 7.14411 0.354120
\(408\) 1.08800 0.0538640
\(409\) −24.8804 −1.23025 −0.615127 0.788428i \(-0.710895\pi\)
−0.615127 + 0.788428i \(0.710895\pi\)
\(410\) 25.9063 1.27942
\(411\) 21.0698 1.03929
\(412\) −1.00000 −0.0492665
\(413\) −5.53887 −0.272550
\(414\) 6.14753 0.302135
\(415\) −21.0707 −1.03432
\(416\) −1.00000 −0.0490290
\(417\) −11.4453 −0.560478
\(418\) 8.38340 0.410046
\(419\) 1.72810 0.0844232 0.0422116 0.999109i \(-0.486560\pi\)
0.0422116 + 0.999109i \(0.486560\pi\)
\(420\) 2.62558 0.128115
\(421\) 40.5952 1.97849 0.989245 0.146266i \(-0.0467255\pi\)
0.989245 + 0.146266i \(0.0467255\pi\)
\(422\) −19.2055 −0.934909
\(423\) 8.61755 0.419000
\(424\) 8.32354 0.404227
\(425\) −4.42545 −0.214666
\(426\) 6.82395 0.330621
\(427\) −13.1064 −0.634261
\(428\) 6.50708 0.314531
\(429\) −3.54795 −0.171297
\(430\) −4.34141 −0.209361
\(431\) −21.6324 −1.04199 −0.520997 0.853559i \(-0.674440\pi\)
−0.520997 + 0.853559i \(0.674440\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.23507 0.107411 0.0537054 0.998557i \(-0.482897\pi\)
0.0537054 + 0.998557i \(0.482897\pi\)
\(434\) −0.224114 −0.0107578
\(435\) 6.53610 0.313382
\(436\) 11.0157 0.527557
\(437\) −14.5259 −0.694868
\(438\) −15.2479 −0.728572
\(439\) −28.8935 −1.37901 −0.689505 0.724281i \(-0.742172\pi\)
−0.689505 + 0.724281i \(0.742172\pi\)
\(440\) −10.6837 −0.509325
\(441\) −6.23974 −0.297130
\(442\) 1.08800 0.0517509
\(443\) 18.5915 0.883308 0.441654 0.897186i \(-0.354392\pi\)
0.441654 + 0.897186i \(0.354392\pi\)
\(444\) 2.01359 0.0955606
\(445\) 41.3498 1.96017
\(446\) 7.76423 0.367647
\(447\) 16.5724 0.783846
\(448\) −0.871929 −0.0411948
\(449\) −0.663272 −0.0313018 −0.0156509 0.999878i \(-0.504982\pi\)
−0.0156509 + 0.999878i \(0.504982\pi\)
\(450\) 4.06751 0.191744
\(451\) −30.5239 −1.43731
\(452\) 8.84167 0.415877
\(453\) −1.15504 −0.0542686
\(454\) −9.20208 −0.431875
\(455\) 2.62558 0.123089
\(456\) 2.36288 0.110652
\(457\) 5.01979 0.234816 0.117408 0.993084i \(-0.462542\pi\)
0.117408 + 0.993084i \(0.462542\pi\)
\(458\) −22.1657 −1.03573
\(459\) 1.08800 0.0507835
\(460\) 18.5116 0.863109
\(461\) 38.2064 1.77945 0.889725 0.456497i \(-0.150896\pi\)
0.889725 + 0.456497i \(0.150896\pi\)
\(462\) −3.09356 −0.143925
\(463\) 14.6836 0.682403 0.341201 0.939990i \(-0.389166\pi\)
0.341201 + 0.939990i \(0.389166\pi\)
\(464\) −2.17058 −0.100766
\(465\) −0.773983 −0.0358926
\(466\) −16.5704 −0.767607
\(467\) −0.537681 −0.0248809 −0.0124405 0.999923i \(-0.503960\pi\)
−0.0124405 + 0.999923i \(0.503960\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 4.26468 0.196924
\(470\) 25.9494 1.19696
\(471\) −3.32560 −0.153235
\(472\) 6.35243 0.292394
\(473\) 5.11522 0.235198
\(474\) 5.43391 0.249588
\(475\) −9.61105 −0.440985
\(476\) 0.948659 0.0434817
\(477\) 8.32354 0.381109
\(478\) 21.6746 0.991373
\(479\) −35.3522 −1.61528 −0.807642 0.589673i \(-0.799257\pi\)
−0.807642 + 0.589673i \(0.799257\pi\)
\(480\) −3.01123 −0.137443
\(481\) 2.01359 0.0918116
\(482\) −16.7333 −0.762181
\(483\) 5.36021 0.243898
\(484\) 1.58796 0.0721800
\(485\) 43.0813 1.95622
\(486\) −1.00000 −0.0453609
\(487\) 5.58605 0.253128 0.126564 0.991958i \(-0.459605\pi\)
0.126564 + 0.991958i \(0.459605\pi\)
\(488\) 15.0315 0.680442
\(489\) 10.0530 0.454611
\(490\) −18.7893 −0.848814
\(491\) 8.79074 0.396721 0.198360 0.980129i \(-0.436438\pi\)
0.198360 + 0.980129i \(0.436438\pi\)
\(492\) −8.60323 −0.387864
\(493\) 2.36159 0.106361
\(494\) 2.36288 0.106311
\(495\) −10.6837 −0.480197
\(496\) 0.257032 0.0115411
\(497\) 5.94999 0.266894
\(498\) 6.99737 0.313560
\(499\) −5.15866 −0.230933 −0.115467 0.993311i \(-0.536836\pi\)
−0.115467 + 0.993311i \(0.536836\pi\)
\(500\) −2.80795 −0.125575
\(501\) −9.56882 −0.427503
\(502\) 23.0103 1.02700
\(503\) −5.12429 −0.228481 −0.114240 0.993453i \(-0.536443\pi\)
−0.114240 + 0.993453i \(0.536443\pi\)
\(504\) −0.871929 −0.0388388
\(505\) 37.7633 1.68044
\(506\) −21.8111 −0.969623
\(507\) −1.00000 −0.0444116
\(508\) 6.76430 0.300117
\(509\) 10.2395 0.453858 0.226929 0.973911i \(-0.427132\pi\)
0.226929 + 0.973911i \(0.427132\pi\)
\(510\) 3.27622 0.145073
\(511\) −13.2951 −0.588139
\(512\) 1.00000 0.0441942
\(513\) 2.36288 0.104324
\(514\) −4.89130 −0.215746
\(515\) −3.01123 −0.132691
\(516\) 1.44174 0.0634690
\(517\) −30.5747 −1.34467
\(518\) 1.75570 0.0771412
\(519\) 7.30957 0.320855
\(520\) −3.01123 −0.132051
\(521\) −28.3680 −1.24282 −0.621412 0.783484i \(-0.713441\pi\)
−0.621412 + 0.783484i \(0.713441\pi\)
\(522\) −2.17058 −0.0950035
\(523\) 7.52975 0.329253 0.164627 0.986356i \(-0.447358\pi\)
0.164627 + 0.986356i \(0.447358\pi\)
\(524\) 19.9488 0.871466
\(525\) 3.54658 0.154785
\(526\) −5.62766 −0.245377
\(527\) −0.279651 −0.0121818
\(528\) 3.54795 0.154405
\(529\) 14.7921 0.643135
\(530\) 25.0641 1.08871
\(531\) 6.35243 0.275672
\(532\) 2.06027 0.0893239
\(533\) −8.60323 −0.372647
\(534\) −13.7319 −0.594236
\(535\) 19.5943 0.847136
\(536\) −4.89108 −0.211263
\(537\) −11.1967 −0.483174
\(538\) −9.72299 −0.419188
\(539\) 22.1383 0.953564
\(540\) −3.01123 −0.129583
\(541\) 34.4659 1.48181 0.740903 0.671613i \(-0.234398\pi\)
0.740903 + 0.671613i \(0.234398\pi\)
\(542\) −0.275577 −0.0118371
\(543\) −1.94900 −0.0836396
\(544\) −1.08800 −0.0466476
\(545\) 33.1708 1.42088
\(546\) −0.871929 −0.0373151
\(547\) −18.3243 −0.783490 −0.391745 0.920074i \(-0.628128\pi\)
−0.391745 + 0.920074i \(0.628128\pi\)
\(548\) −21.0698 −0.900055
\(549\) 15.0315 0.641527
\(550\) −14.4313 −0.615354
\(551\) 5.12882 0.218495
\(552\) −6.14753 −0.261656
\(553\) 4.73798 0.201479
\(554\) 9.98438 0.424195
\(555\) 6.06337 0.257376
\(556\) 11.4453 0.485389
\(557\) 6.68277 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(558\) 0.257032 0.0108810
\(559\) 1.44174 0.0609790
\(560\) −2.62558 −0.110951
\(561\) −3.86017 −0.162977
\(562\) 10.8011 0.455615
\(563\) 24.3222 1.02506 0.512528 0.858670i \(-0.328709\pi\)
0.512528 + 0.858670i \(0.328709\pi\)
\(564\) −8.61755 −0.362864
\(565\) 26.6243 1.12009
\(566\) −6.93304 −0.291417
\(567\) −0.871929 −0.0366176
\(568\) −6.82395 −0.286326
\(569\) 12.3646 0.518353 0.259176 0.965830i \(-0.416549\pi\)
0.259176 + 0.965830i \(0.416549\pi\)
\(570\) 7.11519 0.298023
\(571\) −32.7042 −1.36863 −0.684314 0.729187i \(-0.739898\pi\)
−0.684314 + 0.729187i \(0.739898\pi\)
\(572\) 3.54795 0.148347
\(573\) 10.9502 0.457452
\(574\) −7.50141 −0.313103
\(575\) 25.0051 1.04279
\(576\) 1.00000 0.0416667
\(577\) 40.8430 1.70032 0.850159 0.526527i \(-0.176506\pi\)
0.850159 + 0.526527i \(0.176506\pi\)
\(578\) −15.8163 −0.657869
\(579\) 5.43921 0.226046
\(580\) −6.53610 −0.271397
\(581\) 6.10121 0.253121
\(582\) −14.3069 −0.593039
\(583\) −29.5315 −1.22307
\(584\) 15.2479 0.630962
\(585\) −3.01123 −0.124499
\(586\) −0.802976 −0.0331706
\(587\) −4.37148 −0.180430 −0.0902151 0.995922i \(-0.528755\pi\)
−0.0902151 + 0.995922i \(0.528755\pi\)
\(588\) 6.23974 0.257323
\(589\) −0.607337 −0.0250249
\(590\) 19.1286 0.787513
\(591\) −21.7761 −0.895748
\(592\) −2.01359 −0.0827579
\(593\) −1.68463 −0.0691793 −0.0345897 0.999402i \(-0.511012\pi\)
−0.0345897 + 0.999402i \(0.511012\pi\)
\(594\) 3.54795 0.145574
\(595\) 2.85663 0.117110
\(596\) −16.5724 −0.678830
\(597\) 13.6495 0.558636
\(598\) −6.14753 −0.251391
\(599\) 23.8658 0.975132 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(600\) −4.06751 −0.166055
\(601\) −5.07650 −0.207075 −0.103537 0.994626i \(-0.533016\pi\)
−0.103537 + 0.994626i \(0.533016\pi\)
\(602\) 1.25709 0.0512353
\(603\) −4.89108 −0.199180
\(604\) 1.15504 0.0469980
\(605\) 4.78172 0.194404
\(606\) −12.5408 −0.509436
\(607\) −9.89991 −0.401825 −0.200912 0.979609i \(-0.564391\pi\)
−0.200912 + 0.979609i \(0.564391\pi\)
\(608\) −2.36288 −0.0958276
\(609\) −1.89259 −0.0766915
\(610\) 45.2632 1.83265
\(611\) −8.61755 −0.348629
\(612\) −1.08800 −0.0439798
\(613\) 9.11675 0.368222 0.184111 0.982905i \(-0.441059\pi\)
0.184111 + 0.982905i \(0.441059\pi\)
\(614\) 0.918828 0.0370809
\(615\) −25.9063 −1.04464
\(616\) 3.09356 0.124643
\(617\) −23.4627 −0.944575 −0.472287 0.881445i \(-0.656572\pi\)
−0.472287 + 0.881445i \(0.656572\pi\)
\(618\) 1.00000 0.0402259
\(619\) 10.6859 0.429503 0.214751 0.976669i \(-0.431106\pi\)
0.214751 + 0.976669i \(0.431106\pi\)
\(620\) 0.773983 0.0310839
\(621\) −6.14753 −0.246692
\(622\) −22.0719 −0.885004
\(623\) −11.9732 −0.479696
\(624\) 1.00000 0.0400320
\(625\) −28.7929 −1.15172
\(626\) −15.5057 −0.619734
\(627\) −8.38340 −0.334801
\(628\) 3.32560 0.132706
\(629\) 2.19078 0.0873522
\(630\) −2.62558 −0.104606
\(631\) 2.34265 0.0932596 0.0466298 0.998912i \(-0.485152\pi\)
0.0466298 + 0.998912i \(0.485152\pi\)
\(632\) −5.43391 −0.216149
\(633\) 19.2055 0.763350
\(634\) 26.8110 1.06480
\(635\) 20.3689 0.808313
\(636\) −8.32354 −0.330050
\(637\) 6.23974 0.247228
\(638\) 7.70110 0.304889
\(639\) −6.82395 −0.269951
\(640\) 3.01123 0.119029
\(641\) 8.44115 0.333405 0.166703 0.986007i \(-0.446688\pi\)
0.166703 + 0.986007i \(0.446688\pi\)
\(642\) −6.50708 −0.256814
\(643\) 29.9959 1.18292 0.591462 0.806333i \(-0.298551\pi\)
0.591462 + 0.806333i \(0.298551\pi\)
\(644\) −5.36021 −0.211222
\(645\) 4.34141 0.170943
\(646\) 2.57082 0.101148
\(647\) 19.8516 0.780446 0.390223 0.920720i \(-0.372398\pi\)
0.390223 + 0.920720i \(0.372398\pi\)
\(648\) 1.00000 0.0392837
\(649\) −22.5381 −0.884698
\(650\) −4.06751 −0.159541
\(651\) 0.224114 0.00878371
\(652\) −10.0530 −0.393704
\(653\) −12.4747 −0.488173 −0.244087 0.969753i \(-0.578488\pi\)
−0.244087 + 0.969753i \(0.578488\pi\)
\(654\) −11.0157 −0.430748
\(655\) 60.0703 2.34714
\(656\) 8.60323 0.335900
\(657\) 15.2479 0.594877
\(658\) −7.51389 −0.292922
\(659\) 28.5645 1.11271 0.556357 0.830944i \(-0.312199\pi\)
0.556357 + 0.830944i \(0.312199\pi\)
\(660\) 10.6837 0.415862
\(661\) 1.94352 0.0755942 0.0377971 0.999285i \(-0.487966\pi\)
0.0377971 + 0.999285i \(0.487966\pi\)
\(662\) 3.71332 0.144322
\(663\) −1.08800 −0.0422544
\(664\) −6.99737 −0.271551
\(665\) 6.20394 0.240578
\(666\) −2.01359 −0.0780249
\(667\) −13.3437 −0.516669
\(668\) 9.56882 0.370229
\(669\) −7.76423 −0.300183
\(670\) −14.7282 −0.568999
\(671\) −53.3309 −2.05881
\(672\) 0.871929 0.0336354
\(673\) 29.4988 1.13709 0.568547 0.822650i \(-0.307506\pi\)
0.568547 + 0.822650i \(0.307506\pi\)
\(674\) 8.97722 0.345790
\(675\) −4.06751 −0.156558
\(676\) 1.00000 0.0384615
\(677\) −35.1531 −1.35104 −0.675522 0.737340i \(-0.736082\pi\)
−0.675522 + 0.737340i \(0.736082\pi\)
\(678\) −8.84167 −0.339562
\(679\) −12.4746 −0.478730
\(680\) −3.27622 −0.125637
\(681\) 9.20208 0.352625
\(682\) −0.911938 −0.0349199
\(683\) 11.0341 0.422207 0.211103 0.977464i \(-0.432294\pi\)
0.211103 + 0.977464i \(0.432294\pi\)
\(684\) −2.36288 −0.0903472
\(685\) −63.4459 −2.42414
\(686\) 11.5441 0.440756
\(687\) 22.1657 0.845674
\(688\) −1.44174 −0.0549658
\(689\) −8.32354 −0.317102
\(690\) −18.5116 −0.704725
\(691\) 16.5485 0.629536 0.314768 0.949169i \(-0.398073\pi\)
0.314768 + 0.949169i \(0.398073\pi\)
\(692\) −7.30957 −0.277868
\(693\) 3.09356 0.117515
\(694\) −11.5432 −0.438174
\(695\) 34.4644 1.30731
\(696\) 2.17058 0.0822755
\(697\) −9.36032 −0.354547
\(698\) 0.844753 0.0319744
\(699\) 16.5704 0.626748
\(700\) −3.54658 −0.134048
\(701\) 43.0980 1.62779 0.813894 0.581013i \(-0.197343\pi\)
0.813894 + 0.581013i \(0.197343\pi\)
\(702\) 1.00000 0.0377426
\(703\) 4.75787 0.179447
\(704\) −3.54795 −0.133718
\(705\) −25.9494 −0.977312
\(706\) 19.3423 0.727958
\(707\) −10.9347 −0.411242
\(708\) −6.35243 −0.238739
\(709\) −14.8018 −0.555892 −0.277946 0.960597i \(-0.589654\pi\)
−0.277946 + 0.960597i \(0.589654\pi\)
\(710\) −20.5485 −0.771170
\(711\) −5.43391 −0.203787
\(712\) 13.7319 0.514623
\(713\) 1.58011 0.0591757
\(714\) −0.948659 −0.0355027
\(715\) 10.6837 0.399548
\(716\) 11.1967 0.418441
\(717\) −21.6746 −0.809453
\(718\) 10.1494 0.378771
\(719\) 6.56249 0.244740 0.122370 0.992485i \(-0.460951\pi\)
0.122370 + 0.992485i \(0.460951\pi\)
\(720\) 3.01123 0.112222
\(721\) 0.871929 0.0324723
\(722\) −13.4168 −0.499321
\(723\) 16.7333 0.622318
\(724\) 1.94900 0.0724340
\(725\) −8.82883 −0.327895
\(726\) −1.58796 −0.0589348
\(727\) 16.4538 0.610239 0.305120 0.952314i \(-0.401304\pi\)
0.305120 + 0.952314i \(0.401304\pi\)
\(728\) 0.871929 0.0323158
\(729\) 1.00000 0.0370370
\(730\) 45.9149 1.69939
\(731\) 1.56861 0.0580172
\(732\) −15.0315 −0.555578
\(733\) −24.6565 −0.910707 −0.455354 0.890311i \(-0.650487\pi\)
−0.455354 + 0.890311i \(0.650487\pi\)
\(734\) −7.35480 −0.271471
\(735\) 18.7893 0.693054
\(736\) 6.14753 0.226601
\(737\) 17.3533 0.639218
\(738\) 8.60323 0.316689
\(739\) −37.4776 −1.37864 −0.689318 0.724459i \(-0.742090\pi\)
−0.689318 + 0.724459i \(0.742090\pi\)
\(740\) −6.06337 −0.222894
\(741\) −2.36288 −0.0868027
\(742\) −7.25753 −0.266432
\(743\) 18.1213 0.664807 0.332404 0.943137i \(-0.392140\pi\)
0.332404 + 0.943137i \(0.392140\pi\)
\(744\) −0.257032 −0.00942325
\(745\) −49.9032 −1.82831
\(746\) −0.00733682 −0.000268620 0
\(747\) −6.99737 −0.256021
\(748\) 3.86017 0.141142
\(749\) −5.67371 −0.207313
\(750\) 2.80795 0.102532
\(751\) 27.8326 1.01563 0.507813 0.861468i \(-0.330454\pi\)
0.507813 + 0.861468i \(0.330454\pi\)
\(752\) 8.61755 0.314250
\(753\) −23.0103 −0.838544
\(754\) 2.17058 0.0790477
\(755\) 3.47810 0.126581
\(756\) 0.871929 0.0317117
\(757\) 14.3010 0.519778 0.259889 0.965638i \(-0.416314\pi\)
0.259889 + 0.965638i \(0.416314\pi\)
\(758\) −17.2292 −0.625792
\(759\) 21.8111 0.791694
\(760\) −7.11519 −0.258095
\(761\) −12.3610 −0.448085 −0.224042 0.974579i \(-0.571925\pi\)
−0.224042 + 0.974579i \(0.571925\pi\)
\(762\) −6.76430 −0.245045
\(763\) −9.60491 −0.347721
\(764\) −10.9502 −0.396165
\(765\) −3.27622 −0.118452
\(766\) −37.0826 −1.33985
\(767\) −6.35243 −0.229373
\(768\) −1.00000 −0.0360844
\(769\) −9.39885 −0.338931 −0.169466 0.985536i \(-0.554204\pi\)
−0.169466 + 0.985536i \(0.554204\pi\)
\(770\) 9.31542 0.335705
\(771\) 4.89130 0.176156
\(772\) −5.43921 −0.195761
\(773\) −31.8392 −1.14518 −0.572589 0.819843i \(-0.694061\pi\)
−0.572589 + 0.819843i \(0.694061\pi\)
\(774\) −1.44174 −0.0518222
\(775\) 1.04548 0.0375548
\(776\) 14.3069 0.513587
\(777\) −1.75570 −0.0629855
\(778\) 16.4678 0.590398
\(779\) −20.3285 −0.728342
\(780\) 3.01123 0.107819
\(781\) 24.2110 0.866339
\(782\) −6.68851 −0.239181
\(783\) 2.17058 0.0775700
\(784\) −6.23974 −0.222848
\(785\) 10.0141 0.357420
\(786\) −19.9488 −0.711549
\(787\) −0.192285 −0.00685421 −0.00342711 0.999994i \(-0.501091\pi\)
−0.00342711 + 0.999994i \(0.501091\pi\)
\(788\) 21.7761 0.775740
\(789\) 5.62766 0.200350
\(790\) −16.3627 −0.582160
\(791\) −7.70931 −0.274111
\(792\) −3.54795 −0.126071
\(793\) −15.0315 −0.533782
\(794\) 33.6723 1.19498
\(795\) −25.0641 −0.888932
\(796\) −13.6495 −0.483793
\(797\) −43.0459 −1.52477 −0.762383 0.647126i \(-0.775971\pi\)
−0.762383 + 0.647126i \(0.775971\pi\)
\(798\) −2.06027 −0.0729327
\(799\) −9.37590 −0.331696
\(800\) 4.06751 0.143808
\(801\) 13.7319 0.485192
\(802\) −18.2404 −0.644091
\(803\) −54.0988 −1.90910
\(804\) 4.89108 0.172495
\(805\) −16.1408 −0.568889
\(806\) −0.257032 −0.00905357
\(807\) 9.72299 0.342265
\(808\) 12.5408 0.441184
\(809\) −18.9263 −0.665415 −0.332707 0.943030i \(-0.607962\pi\)
−0.332707 + 0.943030i \(0.607962\pi\)
\(810\) 3.01123 0.105804
\(811\) −4.82654 −0.169483 −0.0847414 0.996403i \(-0.527006\pi\)
−0.0847414 + 0.996403i \(0.527006\pi\)
\(812\) 1.89259 0.0664168
\(813\) 0.275577 0.00966492
\(814\) 7.14411 0.250401
\(815\) −30.2718 −1.06037
\(816\) 1.08800 0.0380876
\(817\) 3.40666 0.119184
\(818\) −24.8804 −0.869921
\(819\) 0.871929 0.0304677
\(820\) 25.9063 0.904688
\(821\) 5.60454 0.195600 0.0977999 0.995206i \(-0.468819\pi\)
0.0977999 + 0.995206i \(0.468819\pi\)
\(822\) 21.0698 0.734892
\(823\) −52.8892 −1.84360 −0.921801 0.387664i \(-0.873282\pi\)
−0.921801 + 0.387664i \(0.873282\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 14.4313 0.502434
\(826\) −5.53887 −0.192722
\(827\) −55.8331 −1.94151 −0.970753 0.240079i \(-0.922827\pi\)
−0.970753 + 0.240079i \(0.922827\pi\)
\(828\) 6.14753 0.213641
\(829\) 53.0679 1.84313 0.921563 0.388229i \(-0.126913\pi\)
0.921563 + 0.388229i \(0.126913\pi\)
\(830\) −21.0707 −0.731375
\(831\) −9.98438 −0.346354
\(832\) −1.00000 −0.0346688
\(833\) 6.78884 0.235219
\(834\) −11.4453 −0.396318
\(835\) 28.8139 0.997147
\(836\) 8.38340 0.289946
\(837\) −0.257032 −0.00888433
\(838\) 1.72810 0.0596962
\(839\) 31.5211 1.08823 0.544114 0.839011i \(-0.316866\pi\)
0.544114 + 0.839011i \(0.316866\pi\)
\(840\) 2.62558 0.0905911
\(841\) −24.2886 −0.837538
\(842\) 40.5952 1.39900
\(843\) −10.8011 −0.372008
\(844\) −19.2055 −0.661081
\(845\) 3.01123 0.103589
\(846\) 8.61755 0.296278
\(847\) −1.38459 −0.0475750
\(848\) 8.32354 0.285832
\(849\) 6.93304 0.237941
\(850\) −4.42545 −0.151792
\(851\) −12.3786 −0.424332
\(852\) 6.82395 0.233785
\(853\) 31.3549 1.07357 0.536787 0.843718i \(-0.319638\pi\)
0.536787 + 0.843718i \(0.319638\pi\)
\(854\) −13.1064 −0.448490
\(855\) −7.11519 −0.243334
\(856\) 6.50708 0.222407
\(857\) 39.4533 1.34770 0.673849 0.738869i \(-0.264640\pi\)
0.673849 + 0.738869i \(0.264640\pi\)
\(858\) −3.54795 −0.121125
\(859\) 17.7902 0.606994 0.303497 0.952832i \(-0.401846\pi\)
0.303497 + 0.952832i \(0.401846\pi\)
\(860\) −4.34141 −0.148041
\(861\) 7.50141 0.255647
\(862\) −21.6324 −0.736801
\(863\) −10.7348 −0.365418 −0.182709 0.983167i \(-0.558487\pi\)
−0.182709 + 0.983167i \(0.558487\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −22.0108 −0.748390
\(866\) 2.23507 0.0759509
\(867\) 15.8163 0.537148
\(868\) −0.224114 −0.00760691
\(869\) 19.2792 0.654003
\(870\) 6.53610 0.221595
\(871\) 4.89108 0.165728
\(872\) 11.0157 0.373039
\(873\) 14.3069 0.484215
\(874\) −14.5259 −0.491346
\(875\) 2.44833 0.0827687
\(876\) −15.2479 −0.515178
\(877\) 26.3710 0.890486 0.445243 0.895410i \(-0.353117\pi\)
0.445243 + 0.895410i \(0.353117\pi\)
\(878\) −28.8935 −0.975107
\(879\) 0.802976 0.0270837
\(880\) −10.6837 −0.360147
\(881\) −7.45488 −0.251161 −0.125581 0.992083i \(-0.540079\pi\)
−0.125581 + 0.992083i \(0.540079\pi\)
\(882\) −6.23974 −0.210103
\(883\) 58.9504 1.98384 0.991919 0.126871i \(-0.0404935\pi\)
0.991919 + 0.126871i \(0.0404935\pi\)
\(884\) 1.08800 0.0365934
\(885\) −19.1286 −0.643002
\(886\) 18.5915 0.624593
\(887\) −43.4323 −1.45831 −0.729157 0.684347i \(-0.760087\pi\)
−0.729157 + 0.684347i \(0.760087\pi\)
\(888\) 2.01359 0.0675715
\(889\) −5.89798 −0.197812
\(890\) 41.3498 1.38605
\(891\) −3.54795 −0.118861
\(892\) 7.76423 0.259966
\(893\) −20.3623 −0.681398
\(894\) 16.5724 0.554263
\(895\) 33.7159 1.12700
\(896\) −0.871929 −0.0291291
\(897\) 6.14753 0.205260
\(898\) −0.663272 −0.0221337
\(899\) −0.557908 −0.0186073
\(900\) 4.06751 0.135584
\(901\) −9.05601 −0.301700
\(902\) −30.5239 −1.01633
\(903\) −1.25709 −0.0418334
\(904\) 8.84167 0.294070
\(905\) 5.86888 0.195088
\(906\) −1.15504 −0.0383737
\(907\) −31.4571 −1.04452 −0.522258 0.852787i \(-0.674910\pi\)
−0.522258 + 0.852787i \(0.674910\pi\)
\(908\) −9.20208 −0.305382
\(909\) 12.5408 0.415953
\(910\) 2.62558 0.0870371
\(911\) −23.6402 −0.783236 −0.391618 0.920128i \(-0.628084\pi\)
−0.391618 + 0.920128i \(0.628084\pi\)
\(912\) 2.36288 0.0782429
\(913\) 24.8263 0.821632
\(914\) 5.01979 0.166040
\(915\) −45.2632 −1.49635
\(916\) −22.1657 −0.732375
\(917\) −17.3939 −0.574397
\(918\) 1.08800 0.0359094
\(919\) 20.1374 0.664273 0.332136 0.943231i \(-0.392231\pi\)
0.332136 + 0.943231i \(0.392231\pi\)
\(920\) 18.5116 0.610310
\(921\) −0.918828 −0.0302764
\(922\) 38.2064 1.25826
\(923\) 6.82395 0.224613
\(924\) −3.09356 −0.101771
\(925\) −8.19028 −0.269295
\(926\) 14.6836 0.482532
\(927\) −1.00000 −0.0328443
\(928\) −2.17058 −0.0712526
\(929\) −30.8944 −1.01361 −0.506807 0.862060i \(-0.669174\pi\)
−0.506807 + 0.862060i \(0.669174\pi\)
\(930\) −0.773983 −0.0253799
\(931\) 14.7438 0.483208
\(932\) −16.5704 −0.542780
\(933\) 22.0719 0.722603
\(934\) −0.537681 −0.0175935
\(935\) 11.6239 0.380141
\(936\) −1.00000 −0.0326860
\(937\) −41.1730 −1.34506 −0.672532 0.740068i \(-0.734793\pi\)
−0.672532 + 0.740068i \(0.734793\pi\)
\(938\) 4.26468 0.139247
\(939\) 15.5057 0.506011
\(940\) 25.9494 0.846377
\(941\) 23.8092 0.776158 0.388079 0.921626i \(-0.373139\pi\)
0.388079 + 0.921626i \(0.373139\pi\)
\(942\) −3.32560 −0.108354
\(943\) 52.8886 1.72229
\(944\) 6.35243 0.206754
\(945\) 2.62558 0.0854101
\(946\) 5.11522 0.166310
\(947\) −43.1387 −1.40182 −0.700910 0.713250i \(-0.747223\pi\)
−0.700910 + 0.713250i \(0.747223\pi\)
\(948\) 5.43391 0.176485
\(949\) −15.2479 −0.494967
\(950\) −9.61105 −0.311824
\(951\) −26.8110 −0.869405
\(952\) 0.948659 0.0307462
\(953\) 8.71067 0.282166 0.141083 0.989998i \(-0.454942\pi\)
0.141083 + 0.989998i \(0.454942\pi\)
\(954\) 8.32354 0.269485
\(955\) −32.9736 −1.06700
\(956\) 21.6746 0.701007
\(957\) −7.70110 −0.248941
\(958\) −35.3522 −1.14218
\(959\) 18.3713 0.593241
\(960\) −3.01123 −0.0971870
\(961\) −30.9339 −0.997869
\(962\) 2.01359 0.0649206
\(963\) 6.50708 0.209688
\(964\) −16.7333 −0.538943
\(965\) −16.3787 −0.527249
\(966\) 5.36021 0.172462
\(967\) −5.99975 −0.192939 −0.0964694 0.995336i \(-0.530755\pi\)
−0.0964694 + 0.995336i \(0.530755\pi\)
\(968\) 1.58796 0.0510390
\(969\) −2.57082 −0.0825866
\(970\) 43.0813 1.38326
\(971\) −56.2319 −1.80457 −0.902283 0.431144i \(-0.858110\pi\)
−0.902283 + 0.431144i \(0.858110\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −9.97948 −0.319927
\(974\) 5.58605 0.178989
\(975\) 4.06751 0.130264
\(976\) 15.0315 0.481145
\(977\) 15.8848 0.508198 0.254099 0.967178i \(-0.418221\pi\)
0.254099 + 0.967178i \(0.418221\pi\)
\(978\) 10.0530 0.321458
\(979\) −48.7200 −1.55710
\(980\) −18.7893 −0.600202
\(981\) 11.0157 0.351704
\(982\) 8.79074 0.280524
\(983\) −33.7501 −1.07646 −0.538230 0.842798i \(-0.680907\pi\)
−0.538230 + 0.842798i \(0.680907\pi\)
\(984\) −8.60323 −0.274261
\(985\) 65.5727 2.08932
\(986\) 2.36159 0.0752082
\(987\) 7.51389 0.239170
\(988\) 2.36288 0.0751734
\(989\) −8.86313 −0.281831
\(990\) −10.6837 −0.339550
\(991\) 50.6049 1.60752 0.803759 0.594954i \(-0.202830\pi\)
0.803759 + 0.594954i \(0.202830\pi\)
\(992\) 0.257032 0.00816078
\(993\) −3.71332 −0.117839
\(994\) 5.94999 0.188722
\(995\) −41.1017 −1.30301
\(996\) 6.99737 0.221720
\(997\) 53.0623 1.68050 0.840250 0.542199i \(-0.182408\pi\)
0.840250 + 0.542199i \(0.182408\pi\)
\(998\) −5.15866 −0.163294
\(999\) 2.01359 0.0637070
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 8034.2.a.bc.1.13 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.13 15 1.1 even 1 trivial