Properties

Label 8034.2.a.bc.1.9
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 1.9
Root \(-0.656329\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.656329 q^{5} -1.00000 q^{6} +2.89208 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.656329 q^{5} -1.00000 q^{6} +2.89208 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.656329 q^{10} +2.55119 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.89208 q^{14} -0.656329 q^{15} +1.00000 q^{16} -3.68086 q^{17} +1.00000 q^{18} +2.16301 q^{19} +0.656329 q^{20} -2.89208 q^{21} +2.55119 q^{22} +6.84289 q^{23} -1.00000 q^{24} -4.56923 q^{25} -1.00000 q^{26} -1.00000 q^{27} +2.89208 q^{28} -8.30733 q^{29} -0.656329 q^{30} +8.50156 q^{31} +1.00000 q^{32} -2.55119 q^{33} -3.68086 q^{34} +1.89816 q^{35} +1.00000 q^{36} -2.04906 q^{37} +2.16301 q^{38} +1.00000 q^{39} +0.656329 q^{40} +10.0847 q^{41} -2.89208 q^{42} -0.666049 q^{43} +2.55119 q^{44} +0.656329 q^{45} +6.84289 q^{46} -5.56855 q^{47} -1.00000 q^{48} +1.36415 q^{49} -4.56923 q^{50} +3.68086 q^{51} -1.00000 q^{52} +8.88913 q^{53} -1.00000 q^{54} +1.67442 q^{55} +2.89208 q^{56} -2.16301 q^{57} -8.30733 q^{58} +6.49563 q^{59} -0.656329 q^{60} +3.88458 q^{61} +8.50156 q^{62} +2.89208 q^{63} +1.00000 q^{64} -0.656329 q^{65} -2.55119 q^{66} +9.97687 q^{67} -3.68086 q^{68} -6.84289 q^{69} +1.89816 q^{70} +10.0087 q^{71} +1.00000 q^{72} -2.34621 q^{73} -2.04906 q^{74} +4.56923 q^{75} +2.16301 q^{76} +7.37825 q^{77} +1.00000 q^{78} +14.6755 q^{79} +0.656329 q^{80} +1.00000 q^{81} +10.0847 q^{82} -0.205035 q^{83} -2.89208 q^{84} -2.41586 q^{85} -0.666049 q^{86} +8.30733 q^{87} +2.55119 q^{88} -15.8023 q^{89} +0.656329 q^{90} -2.89208 q^{91} +6.84289 q^{92} -8.50156 q^{93} -5.56855 q^{94} +1.41965 q^{95} -1.00000 q^{96} -11.3618 q^{97} +1.36415 q^{98} +2.55119 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) 0.656329 0.293519 0.146760 0.989172i \(-0.453116\pi\)
0.146760 + 0.989172i \(0.453116\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.89208 1.09310 0.546552 0.837425i \(-0.315940\pi\)
0.546552 + 0.837425i \(0.315940\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.656329 0.207550
\(11\) 2.55119 0.769212 0.384606 0.923081i \(-0.374337\pi\)
0.384606 + 0.923081i \(0.374337\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.89208 0.772942
\(15\) −0.656329 −0.169464
\(16\) 1.00000 0.250000
\(17\) −3.68086 −0.892740 −0.446370 0.894848i \(-0.647284\pi\)
−0.446370 + 0.894848i \(0.647284\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.16301 0.496228 0.248114 0.968731i \(-0.420189\pi\)
0.248114 + 0.968731i \(0.420189\pi\)
\(20\) 0.656329 0.146760
\(21\) −2.89208 −0.631104
\(22\) 2.55119 0.543915
\(23\) 6.84289 1.42684 0.713421 0.700736i \(-0.247145\pi\)
0.713421 + 0.700736i \(0.247145\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.56923 −0.913846
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 2.89208 0.546552
\(29\) −8.30733 −1.54263 −0.771316 0.636452i \(-0.780401\pi\)
−0.771316 + 0.636452i \(0.780401\pi\)
\(30\) −0.656329 −0.119829
\(31\) 8.50156 1.52693 0.763463 0.645852i \(-0.223498\pi\)
0.763463 + 0.645852i \(0.223498\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.55119 −0.444105
\(34\) −3.68086 −0.631263
\(35\) 1.89816 0.320847
\(36\) 1.00000 0.166667
\(37\) −2.04906 −0.336863 −0.168432 0.985713i \(-0.553870\pi\)
−0.168432 + 0.985713i \(0.553870\pi\)
\(38\) 2.16301 0.350886
\(39\) 1.00000 0.160128
\(40\) 0.656329 0.103775
\(41\) 10.0847 1.57497 0.787483 0.616336i \(-0.211384\pi\)
0.787483 + 0.616336i \(0.211384\pi\)
\(42\) −2.89208 −0.446258
\(43\) −0.666049 −0.101571 −0.0507857 0.998710i \(-0.516173\pi\)
−0.0507857 + 0.998710i \(0.516173\pi\)
\(44\) 2.55119 0.384606
\(45\) 0.656329 0.0978398
\(46\) 6.84289 1.00893
\(47\) −5.56855 −0.812256 −0.406128 0.913816i \(-0.633121\pi\)
−0.406128 + 0.913816i \(0.633121\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.36415 0.194878
\(50\) −4.56923 −0.646187
\(51\) 3.68086 0.515424
\(52\) −1.00000 −0.138675
\(53\) 8.88913 1.22102 0.610508 0.792010i \(-0.290965\pi\)
0.610508 + 0.792010i \(0.290965\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.67442 0.225779
\(56\) 2.89208 0.386471
\(57\) −2.16301 −0.286497
\(58\) −8.30733 −1.09081
\(59\) 6.49563 0.845659 0.422829 0.906209i \(-0.361037\pi\)
0.422829 + 0.906209i \(0.361037\pi\)
\(60\) −0.656329 −0.0847318
\(61\) 3.88458 0.497370 0.248685 0.968584i \(-0.420002\pi\)
0.248685 + 0.968584i \(0.420002\pi\)
\(62\) 8.50156 1.07970
\(63\) 2.89208 0.364368
\(64\) 1.00000 0.125000
\(65\) −0.656329 −0.0814076
\(66\) −2.55119 −0.314030
\(67\) 9.97687 1.21887 0.609434 0.792837i \(-0.291397\pi\)
0.609434 + 0.792837i \(0.291397\pi\)
\(68\) −3.68086 −0.446370
\(69\) −6.84289 −0.823788
\(70\) 1.89816 0.226873
\(71\) 10.0087 1.18781 0.593905 0.804535i \(-0.297585\pi\)
0.593905 + 0.804535i \(0.297585\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.34621 −0.274604 −0.137302 0.990529i \(-0.543843\pi\)
−0.137302 + 0.990529i \(0.543843\pi\)
\(74\) −2.04906 −0.238198
\(75\) 4.56923 0.527609
\(76\) 2.16301 0.248114
\(77\) 7.37825 0.840830
\(78\) 1.00000 0.113228
\(79\) 14.6755 1.65112 0.825562 0.564311i \(-0.190858\pi\)
0.825562 + 0.564311i \(0.190858\pi\)
\(80\) 0.656329 0.0733799
\(81\) 1.00000 0.111111
\(82\) 10.0847 1.11367
\(83\) −0.205035 −0.0225056 −0.0112528 0.999937i \(-0.503582\pi\)
−0.0112528 + 0.999937i \(0.503582\pi\)
\(84\) −2.89208 −0.315552
\(85\) −2.41586 −0.262037
\(86\) −0.666049 −0.0718219
\(87\) 8.30733 0.890640
\(88\) 2.55119 0.271958
\(89\) −15.8023 −1.67504 −0.837520 0.546407i \(-0.815995\pi\)
−0.837520 + 0.546407i \(0.815995\pi\)
\(90\) 0.656329 0.0691832
\(91\) −2.89208 −0.303173
\(92\) 6.84289 0.713421
\(93\) −8.50156 −0.881571
\(94\) −5.56855 −0.574352
\(95\) 1.41965 0.145653
\(96\) −1.00000 −0.102062
\(97\) −11.3618 −1.15362 −0.576810 0.816878i \(-0.695703\pi\)
−0.576810 + 0.816878i \(0.695703\pi\)
\(98\) 1.36415 0.137800
\(99\) 2.55119 0.256404
\(100\) −4.56923 −0.456923
\(101\) 6.72895 0.669555 0.334778 0.942297i \(-0.391339\pi\)
0.334778 + 0.942297i \(0.391339\pi\)
\(102\) 3.68086 0.364460
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −1.89816 −0.185241
\(106\) 8.88913 0.863389
\(107\) −7.54576 −0.729476 −0.364738 0.931110i \(-0.618841\pi\)
−0.364738 + 0.931110i \(0.618841\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.71913 0.452010 0.226005 0.974126i \(-0.427433\pi\)
0.226005 + 0.974126i \(0.427433\pi\)
\(110\) 1.67442 0.159650
\(111\) 2.04906 0.194488
\(112\) 2.89208 0.273276
\(113\) −14.2188 −1.33759 −0.668794 0.743447i \(-0.733189\pi\)
−0.668794 + 0.743447i \(0.733189\pi\)
\(114\) −2.16301 −0.202584
\(115\) 4.49119 0.418806
\(116\) −8.30733 −0.771316
\(117\) −1.00000 −0.0924500
\(118\) 6.49563 0.597971
\(119\) −10.6454 −0.975859
\(120\) −0.656329 −0.0599144
\(121\) −4.49143 −0.408312
\(122\) 3.88458 0.351694
\(123\) −10.0847 −0.909308
\(124\) 8.50156 0.763463
\(125\) −6.28057 −0.561751
\(126\) 2.89208 0.257647
\(127\) 3.34533 0.296850 0.148425 0.988924i \(-0.452580\pi\)
0.148425 + 0.988924i \(0.452580\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.666049 0.0586423
\(130\) −0.656329 −0.0575639
\(131\) −14.1498 −1.23627 −0.618136 0.786071i \(-0.712112\pi\)
−0.618136 + 0.786071i \(0.712112\pi\)
\(132\) −2.55119 −0.222052
\(133\) 6.25560 0.542429
\(134\) 9.97687 0.861870
\(135\) −0.656329 −0.0564878
\(136\) −3.68086 −0.315631
\(137\) −4.05511 −0.346451 −0.173226 0.984882i \(-0.555419\pi\)
−0.173226 + 0.984882i \(0.555419\pi\)
\(138\) −6.84289 −0.582506
\(139\) 0.498690 0.0422983 0.0211491 0.999776i \(-0.493268\pi\)
0.0211491 + 0.999776i \(0.493268\pi\)
\(140\) 1.89816 0.160424
\(141\) 5.56855 0.468956
\(142\) 10.0087 0.839909
\(143\) −2.55119 −0.213341
\(144\) 1.00000 0.0833333
\(145\) −5.45235 −0.452793
\(146\) −2.34621 −0.194174
\(147\) −1.36415 −0.112513
\(148\) −2.04906 −0.168432
\(149\) 20.6355 1.69052 0.845262 0.534353i \(-0.179445\pi\)
0.845262 + 0.534353i \(0.179445\pi\)
\(150\) 4.56923 0.373076
\(151\) 10.7708 0.876516 0.438258 0.898849i \(-0.355596\pi\)
0.438258 + 0.898849i \(0.355596\pi\)
\(152\) 2.16301 0.175443
\(153\) −3.68086 −0.297580
\(154\) 7.37825 0.594556
\(155\) 5.57983 0.448182
\(156\) 1.00000 0.0800641
\(157\) −11.2398 −0.897030 −0.448515 0.893775i \(-0.648047\pi\)
−0.448515 + 0.893775i \(0.648047\pi\)
\(158\) 14.6755 1.16752
\(159\) −8.88913 −0.704954
\(160\) 0.656329 0.0518874
\(161\) 19.7902 1.55969
\(162\) 1.00000 0.0785674
\(163\) 10.5560 0.826808 0.413404 0.910548i \(-0.364340\pi\)
0.413404 + 0.910548i \(0.364340\pi\)
\(164\) 10.0847 0.787483
\(165\) −1.67442 −0.130353
\(166\) −0.205035 −0.0159138
\(167\) −8.41456 −0.651138 −0.325569 0.945518i \(-0.605556\pi\)
−0.325569 + 0.945518i \(0.605556\pi\)
\(168\) −2.89208 −0.223129
\(169\) 1.00000 0.0769231
\(170\) −2.41586 −0.185288
\(171\) 2.16301 0.165409
\(172\) −0.666049 −0.0507857
\(173\) −6.31337 −0.479996 −0.239998 0.970773i \(-0.577147\pi\)
−0.239998 + 0.970773i \(0.577147\pi\)
\(174\) 8.30733 0.629777
\(175\) −13.2146 −0.998930
\(176\) 2.55119 0.192303
\(177\) −6.49563 −0.488241
\(178\) −15.8023 −1.18443
\(179\) 11.2133 0.838122 0.419061 0.907958i \(-0.362359\pi\)
0.419061 + 0.907958i \(0.362359\pi\)
\(180\) 0.656329 0.0489199
\(181\) 4.89458 0.363811 0.181906 0.983316i \(-0.441773\pi\)
0.181906 + 0.983316i \(0.441773\pi\)
\(182\) −2.89208 −0.214375
\(183\) −3.88458 −0.287157
\(184\) 6.84289 0.504465
\(185\) −1.34486 −0.0988759
\(186\) −8.50156 −0.623365
\(187\) −9.39058 −0.686707
\(188\) −5.56855 −0.406128
\(189\) −2.89208 −0.210368
\(190\) 1.41965 0.102992
\(191\) 17.2636 1.24915 0.624574 0.780966i \(-0.285273\pi\)
0.624574 + 0.780966i \(0.285273\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −17.8750 −1.28667 −0.643335 0.765585i \(-0.722450\pi\)
−0.643335 + 0.765585i \(0.722450\pi\)
\(194\) −11.3618 −0.815733
\(195\) 0.656329 0.0470007
\(196\) 1.36415 0.0974390
\(197\) 15.6986 1.11848 0.559238 0.829007i \(-0.311094\pi\)
0.559238 + 0.829007i \(0.311094\pi\)
\(198\) 2.55119 0.181305
\(199\) 10.1854 0.722027 0.361013 0.932561i \(-0.382431\pi\)
0.361013 + 0.932561i \(0.382431\pi\)
\(200\) −4.56923 −0.323093
\(201\) −9.97687 −0.703714
\(202\) 6.72895 0.473447
\(203\) −24.0255 −1.68626
\(204\) 3.68086 0.257712
\(205\) 6.61889 0.462283
\(206\) −1.00000 −0.0696733
\(207\) 6.84289 0.475614
\(208\) −1.00000 −0.0693375
\(209\) 5.51824 0.381705
\(210\) −1.89816 −0.130985
\(211\) −11.2856 −0.776930 −0.388465 0.921464i \(-0.626995\pi\)
−0.388465 + 0.921464i \(0.626995\pi\)
\(212\) 8.88913 0.610508
\(213\) −10.0087 −0.685782
\(214\) −7.54576 −0.515818
\(215\) −0.437147 −0.0298132
\(216\) −1.00000 −0.0680414
\(217\) 24.5872 1.66909
\(218\) 4.71913 0.319620
\(219\) 2.34621 0.158542
\(220\) 1.67442 0.112889
\(221\) 3.68086 0.247602
\(222\) 2.04906 0.137524
\(223\) 12.1915 0.816406 0.408203 0.912891i \(-0.366156\pi\)
0.408203 + 0.912891i \(0.366156\pi\)
\(224\) 2.89208 0.193235
\(225\) −4.56923 −0.304615
\(226\) −14.2188 −0.945818
\(227\) 2.61611 0.173638 0.0868188 0.996224i \(-0.472330\pi\)
0.0868188 + 0.996224i \(0.472330\pi\)
\(228\) −2.16301 −0.143249
\(229\) −3.08691 −0.203989 −0.101994 0.994785i \(-0.532522\pi\)
−0.101994 + 0.994785i \(0.532522\pi\)
\(230\) 4.49119 0.296140
\(231\) −7.37825 −0.485453
\(232\) −8.30733 −0.545403
\(233\) 0.417209 0.0273323 0.0136661 0.999907i \(-0.495650\pi\)
0.0136661 + 0.999907i \(0.495650\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −3.65480 −0.238413
\(236\) 6.49563 0.422829
\(237\) −14.6755 −0.953277
\(238\) −10.6454 −0.690036
\(239\) −21.8617 −1.41412 −0.707058 0.707155i \(-0.749978\pi\)
−0.707058 + 0.707155i \(0.749978\pi\)
\(240\) −0.656329 −0.0423659
\(241\) 26.3640 1.69826 0.849128 0.528187i \(-0.177128\pi\)
0.849128 + 0.528187i \(0.177128\pi\)
\(242\) −4.49143 −0.288720
\(243\) −1.00000 −0.0641500
\(244\) 3.88458 0.248685
\(245\) 0.895329 0.0572005
\(246\) −10.0847 −0.642978
\(247\) −2.16301 −0.137629
\(248\) 8.50156 0.539850
\(249\) 0.205035 0.0129936
\(250\) −6.28057 −0.397218
\(251\) 21.5806 1.36215 0.681077 0.732212i \(-0.261512\pi\)
0.681077 + 0.732212i \(0.261512\pi\)
\(252\) 2.89208 0.182184
\(253\) 17.4575 1.09754
\(254\) 3.34533 0.209905
\(255\) 2.41586 0.151287
\(256\) 1.00000 0.0625000
\(257\) 14.0027 0.873465 0.436733 0.899591i \(-0.356136\pi\)
0.436733 + 0.899591i \(0.356136\pi\)
\(258\) 0.666049 0.0414664
\(259\) −5.92605 −0.368227
\(260\) −0.656329 −0.0407038
\(261\) −8.30733 −0.514211
\(262\) −14.1498 −0.874176
\(263\) 13.3638 0.824049 0.412024 0.911173i \(-0.364822\pi\)
0.412024 + 0.911173i \(0.364822\pi\)
\(264\) −2.55119 −0.157015
\(265\) 5.83420 0.358392
\(266\) 6.25560 0.383556
\(267\) 15.8023 0.967084
\(268\) 9.97687 0.609434
\(269\) −14.9933 −0.914159 −0.457079 0.889426i \(-0.651105\pi\)
−0.457079 + 0.889426i \(0.651105\pi\)
\(270\) −0.656329 −0.0399429
\(271\) −8.95971 −0.544263 −0.272132 0.962260i \(-0.587729\pi\)
−0.272132 + 0.962260i \(0.587729\pi\)
\(272\) −3.68086 −0.223185
\(273\) 2.89208 0.175037
\(274\) −4.05511 −0.244978
\(275\) −11.6570 −0.702942
\(276\) −6.84289 −0.411894
\(277\) 15.9036 0.955556 0.477778 0.878480i \(-0.341442\pi\)
0.477778 + 0.878480i \(0.341442\pi\)
\(278\) 0.498690 0.0299094
\(279\) 8.50156 0.508975
\(280\) 1.89816 0.113437
\(281\) −11.1504 −0.665177 −0.332589 0.943072i \(-0.607922\pi\)
−0.332589 + 0.943072i \(0.607922\pi\)
\(282\) 5.56855 0.331602
\(283\) 27.1978 1.61674 0.808370 0.588674i \(-0.200350\pi\)
0.808370 + 0.588674i \(0.200350\pi\)
\(284\) 10.0087 0.593905
\(285\) −1.41965 −0.0840926
\(286\) −2.55119 −0.150855
\(287\) 29.1658 1.72160
\(288\) 1.00000 0.0589256
\(289\) −3.45125 −0.203014
\(290\) −5.45235 −0.320173
\(291\) 11.3618 0.666043
\(292\) −2.34621 −0.137302
\(293\) −26.2040 −1.53086 −0.765428 0.643521i \(-0.777473\pi\)
−0.765428 + 0.643521i \(0.777473\pi\)
\(294\) −1.36415 −0.0795586
\(295\) 4.26327 0.248217
\(296\) −2.04906 −0.119099
\(297\) −2.55119 −0.148035
\(298\) 20.6355 1.19538
\(299\) −6.84289 −0.395735
\(300\) 4.56923 0.263805
\(301\) −1.92627 −0.111028
\(302\) 10.7708 0.619790
\(303\) −6.72895 −0.386568
\(304\) 2.16301 0.124057
\(305\) 2.54957 0.145988
\(306\) −3.68086 −0.210421
\(307\) 7.42744 0.423906 0.211953 0.977280i \(-0.432018\pi\)
0.211953 + 0.977280i \(0.432018\pi\)
\(308\) 7.37825 0.420415
\(309\) 1.00000 0.0568880
\(310\) 5.57983 0.316913
\(311\) −2.89222 −0.164003 −0.0820014 0.996632i \(-0.526131\pi\)
−0.0820014 + 0.996632i \(0.526131\pi\)
\(312\) 1.00000 0.0566139
\(313\) −0.277979 −0.0157123 −0.00785615 0.999969i \(-0.502501\pi\)
−0.00785615 + 0.999969i \(0.502501\pi\)
\(314\) −11.2398 −0.634296
\(315\) 1.89816 0.106949
\(316\) 14.6755 0.825562
\(317\) −8.02479 −0.450717 −0.225359 0.974276i \(-0.572355\pi\)
−0.225359 + 0.974276i \(0.572355\pi\)
\(318\) −8.88913 −0.498478
\(319\) −21.1936 −1.18661
\(320\) 0.656329 0.0366899
\(321\) 7.54576 0.421163
\(322\) 19.7902 1.10287
\(323\) −7.96174 −0.443003
\(324\) 1.00000 0.0555556
\(325\) 4.56923 0.253455
\(326\) 10.5560 0.584641
\(327\) −4.71913 −0.260968
\(328\) 10.0847 0.556835
\(329\) −16.1047 −0.887881
\(330\) −1.67442 −0.0921738
\(331\) −24.2699 −1.33400 −0.666998 0.745059i \(-0.732421\pi\)
−0.666998 + 0.745059i \(0.732421\pi\)
\(332\) −0.205035 −0.0112528
\(333\) −2.04906 −0.112288
\(334\) −8.41456 −0.460424
\(335\) 6.54811 0.357762
\(336\) −2.89208 −0.157776
\(337\) 14.8739 0.810235 0.405118 0.914265i \(-0.367231\pi\)
0.405118 + 0.914265i \(0.367231\pi\)
\(338\) 1.00000 0.0543928
\(339\) 14.2188 0.772257
\(340\) −2.41586 −0.131018
\(341\) 21.6891 1.17453
\(342\) 2.16301 0.116962
\(343\) −16.2994 −0.880083
\(344\) −0.666049 −0.0359109
\(345\) −4.49119 −0.241798
\(346\) −6.31337 −0.339409
\(347\) −14.8380 −0.796548 −0.398274 0.917267i \(-0.630391\pi\)
−0.398274 + 0.917267i \(0.630391\pi\)
\(348\) 8.30733 0.445320
\(349\) 4.95172 0.265059 0.132530 0.991179i \(-0.457690\pi\)
0.132530 + 0.991179i \(0.457690\pi\)
\(350\) −13.2146 −0.706350
\(351\) 1.00000 0.0533761
\(352\) 2.55119 0.135979
\(353\) −18.8072 −1.00101 −0.500504 0.865734i \(-0.666852\pi\)
−0.500504 + 0.865734i \(0.666852\pi\)
\(354\) −6.49563 −0.345239
\(355\) 6.56898 0.348645
\(356\) −15.8023 −0.837520
\(357\) 10.6454 0.563412
\(358\) 11.2133 0.592641
\(359\) 23.9366 1.26333 0.631663 0.775243i \(-0.282372\pi\)
0.631663 + 0.775243i \(0.282372\pi\)
\(360\) 0.656329 0.0345916
\(361\) −14.3214 −0.753758
\(362\) 4.89458 0.257254
\(363\) 4.49143 0.235739
\(364\) −2.89208 −0.151586
\(365\) −1.53989 −0.0806015
\(366\) −3.88458 −0.203050
\(367\) 37.8746 1.97704 0.988519 0.151095i \(-0.0482799\pi\)
0.988519 + 0.151095i \(0.0482799\pi\)
\(368\) 6.84289 0.356710
\(369\) 10.0847 0.524989
\(370\) −1.34486 −0.0699158
\(371\) 25.7081 1.33470
\(372\) −8.50156 −0.440785
\(373\) 11.4552 0.593127 0.296563 0.955013i \(-0.404159\pi\)
0.296563 + 0.955013i \(0.404159\pi\)
\(374\) −9.39058 −0.485575
\(375\) 6.28057 0.324327
\(376\) −5.56855 −0.287176
\(377\) 8.30733 0.427849
\(378\) −2.89208 −0.148753
\(379\) 27.4077 1.40784 0.703919 0.710280i \(-0.251432\pi\)
0.703919 + 0.710280i \(0.251432\pi\)
\(380\) 1.41965 0.0728263
\(381\) −3.34533 −0.171386
\(382\) 17.2636 0.883281
\(383\) −9.69866 −0.495578 −0.247789 0.968814i \(-0.579704\pi\)
−0.247789 + 0.968814i \(0.579704\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.84256 0.246800
\(386\) −17.8750 −0.909812
\(387\) −0.666049 −0.0338571
\(388\) −11.3618 −0.576810
\(389\) −13.6955 −0.694390 −0.347195 0.937793i \(-0.612866\pi\)
−0.347195 + 0.937793i \(0.612866\pi\)
\(390\) 0.656329 0.0332345
\(391\) −25.1878 −1.27380
\(392\) 1.36415 0.0688998
\(393\) 14.1498 0.713762
\(394\) 15.6986 0.790883
\(395\) 9.63197 0.484637
\(396\) 2.55119 0.128202
\(397\) 0.618503 0.0310418 0.0155209 0.999880i \(-0.495059\pi\)
0.0155209 + 0.999880i \(0.495059\pi\)
\(398\) 10.1854 0.510550
\(399\) −6.25560 −0.313172
\(400\) −4.56923 −0.228462
\(401\) −16.0957 −0.803781 −0.401890 0.915688i \(-0.631647\pi\)
−0.401890 + 0.915688i \(0.631647\pi\)
\(402\) −9.97687 −0.497601
\(403\) −8.50156 −0.423493
\(404\) 6.72895 0.334778
\(405\) 0.656329 0.0326133
\(406\) −24.0255 −1.19237
\(407\) −5.22754 −0.259119
\(408\) 3.68086 0.182230
\(409\) −23.0712 −1.14080 −0.570399 0.821367i \(-0.693212\pi\)
−0.570399 + 0.821367i \(0.693212\pi\)
\(410\) 6.61889 0.326884
\(411\) 4.05511 0.200024
\(412\) −1.00000 −0.0492665
\(413\) 18.7859 0.924393
\(414\) 6.84289 0.336310
\(415\) −0.134571 −0.00660582
\(416\) −1.00000 −0.0490290
\(417\) −0.498690 −0.0244209
\(418\) 5.51824 0.269906
\(419\) 21.0454 1.02813 0.514067 0.857750i \(-0.328138\pi\)
0.514067 + 0.857750i \(0.328138\pi\)
\(420\) −1.89816 −0.0926207
\(421\) 28.0643 1.36777 0.683884 0.729591i \(-0.260289\pi\)
0.683884 + 0.729591i \(0.260289\pi\)
\(422\) −11.2856 −0.549372
\(423\) −5.56855 −0.270752
\(424\) 8.88913 0.431694
\(425\) 16.8187 0.815828
\(426\) −10.0087 −0.484921
\(427\) 11.2345 0.543677
\(428\) −7.54576 −0.364738
\(429\) 2.55119 0.123173
\(430\) −0.437147 −0.0210811
\(431\) −21.0838 −1.01557 −0.507785 0.861484i \(-0.669536\pi\)
−0.507785 + 0.861484i \(0.669536\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.3609 0.786255 0.393127 0.919484i \(-0.371393\pi\)
0.393127 + 0.919484i \(0.371393\pi\)
\(434\) 24.5872 1.18022
\(435\) 5.45235 0.261420
\(436\) 4.71913 0.226005
\(437\) 14.8012 0.708039
\(438\) 2.34621 0.112106
\(439\) −29.0513 −1.38654 −0.693272 0.720676i \(-0.743832\pi\)
−0.693272 + 0.720676i \(0.743832\pi\)
\(440\) 1.67442 0.0798249
\(441\) 1.36415 0.0649593
\(442\) 3.68086 0.175081
\(443\) −38.5298 −1.83060 −0.915302 0.402768i \(-0.868048\pi\)
−0.915302 + 0.402768i \(0.868048\pi\)
\(444\) 2.04906 0.0972441
\(445\) −10.3715 −0.491657
\(446\) 12.1915 0.577286
\(447\) −20.6355 −0.976024
\(448\) 2.89208 0.136638
\(449\) 0.892366 0.0421134 0.0210567 0.999778i \(-0.493297\pi\)
0.0210567 + 0.999778i \(0.493297\pi\)
\(450\) −4.56923 −0.215396
\(451\) 25.7280 1.21148
\(452\) −14.2188 −0.668794
\(453\) −10.7708 −0.506057
\(454\) 2.61611 0.122780
\(455\) −1.89816 −0.0889871
\(456\) −2.16301 −0.101292
\(457\) −22.3844 −1.04710 −0.523549 0.851995i \(-0.675392\pi\)
−0.523549 + 0.851995i \(0.675392\pi\)
\(458\) −3.08691 −0.144242
\(459\) 3.68086 0.171808
\(460\) 4.49119 0.209403
\(461\) −12.2927 −0.572530 −0.286265 0.958150i \(-0.592414\pi\)
−0.286265 + 0.958150i \(0.592414\pi\)
\(462\) −7.37825 −0.343267
\(463\) −22.6408 −1.05221 −0.526103 0.850421i \(-0.676347\pi\)
−0.526103 + 0.850421i \(0.676347\pi\)
\(464\) −8.30733 −0.385658
\(465\) −5.57983 −0.258758
\(466\) 0.417209 0.0193268
\(467\) −2.86491 −0.132572 −0.0662861 0.997801i \(-0.521115\pi\)
−0.0662861 + 0.997801i \(0.521115\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 28.8539 1.33235
\(470\) −3.65480 −0.168583
\(471\) 11.2398 0.517901
\(472\) 6.49563 0.298985
\(473\) −1.69922 −0.0781300
\(474\) −14.6755 −0.674069
\(475\) −9.88329 −0.453476
\(476\) −10.6454 −0.487929
\(477\) 8.88913 0.407005
\(478\) −21.8617 −0.999932
\(479\) 32.3676 1.47891 0.739456 0.673205i \(-0.235083\pi\)
0.739456 + 0.673205i \(0.235083\pi\)
\(480\) −0.656329 −0.0299572
\(481\) 2.04906 0.0934291
\(482\) 26.3640 1.20085
\(483\) −19.7902 −0.900486
\(484\) −4.49143 −0.204156
\(485\) −7.45711 −0.338610
\(486\) −1.00000 −0.0453609
\(487\) 5.28092 0.239301 0.119651 0.992816i \(-0.461823\pi\)
0.119651 + 0.992816i \(0.461823\pi\)
\(488\) 3.88458 0.175847
\(489\) −10.5560 −0.477358
\(490\) 0.895329 0.0404468
\(491\) 2.25221 0.101641 0.0508205 0.998708i \(-0.483816\pi\)
0.0508205 + 0.998708i \(0.483816\pi\)
\(492\) −10.0847 −0.454654
\(493\) 30.5782 1.37717
\(494\) −2.16301 −0.0973184
\(495\) 1.67442 0.0752596
\(496\) 8.50156 0.381731
\(497\) 28.9459 1.29840
\(498\) 0.205035 0.00918786
\(499\) 7.45562 0.333759 0.166880 0.985977i \(-0.446631\pi\)
0.166880 + 0.985977i \(0.446631\pi\)
\(500\) −6.28057 −0.280876
\(501\) 8.41456 0.375935
\(502\) 21.5806 0.963188
\(503\) 23.1612 1.03271 0.516353 0.856376i \(-0.327289\pi\)
0.516353 + 0.856376i \(0.327289\pi\)
\(504\) 2.89208 0.128824
\(505\) 4.41641 0.196528
\(506\) 17.4575 0.776081
\(507\) −1.00000 −0.0444116
\(508\) 3.34533 0.148425
\(509\) 25.8002 1.14358 0.571788 0.820402i \(-0.306250\pi\)
0.571788 + 0.820402i \(0.306250\pi\)
\(510\) 2.41586 0.106976
\(511\) −6.78545 −0.300170
\(512\) 1.00000 0.0441942
\(513\) −2.16301 −0.0954992
\(514\) 14.0027 0.617633
\(515\) −0.656329 −0.0289213
\(516\) 0.666049 0.0293212
\(517\) −14.2064 −0.624798
\(518\) −5.92605 −0.260376
\(519\) 6.31337 0.277126
\(520\) −0.656329 −0.0287819
\(521\) 17.1592 0.751758 0.375879 0.926669i \(-0.377341\pi\)
0.375879 + 0.926669i \(0.377341\pi\)
\(522\) −8.30733 −0.363602
\(523\) 25.6104 1.11986 0.559932 0.828538i \(-0.310827\pi\)
0.559932 + 0.828538i \(0.310827\pi\)
\(524\) −14.1498 −0.618136
\(525\) 13.2146 0.576732
\(526\) 13.3638 0.582691
\(527\) −31.2931 −1.36315
\(528\) −2.55119 −0.111026
\(529\) 23.8252 1.03588
\(530\) 5.83420 0.253421
\(531\) 6.49563 0.281886
\(532\) 6.25560 0.271215
\(533\) −10.0847 −0.436817
\(534\) 15.8023 0.683832
\(535\) −4.95251 −0.214115
\(536\) 9.97687 0.430935
\(537\) −11.2133 −0.483890
\(538\) −14.9933 −0.646408
\(539\) 3.48019 0.149903
\(540\) −0.656329 −0.0282439
\(541\) 15.0116 0.645397 0.322698 0.946502i \(-0.395410\pi\)
0.322698 + 0.946502i \(0.395410\pi\)
\(542\) −8.95971 −0.384852
\(543\) −4.89458 −0.210047
\(544\) −3.68086 −0.157816
\(545\) 3.09730 0.132674
\(546\) 2.89208 0.123770
\(547\) 27.7252 1.18545 0.592723 0.805406i \(-0.298053\pi\)
0.592723 + 0.805406i \(0.298053\pi\)
\(548\) −4.05511 −0.173226
\(549\) 3.88458 0.165790
\(550\) −11.6570 −0.497055
\(551\) −17.9688 −0.765498
\(552\) −6.84289 −0.291253
\(553\) 42.4428 1.80485
\(554\) 15.9036 0.675680
\(555\) 1.34486 0.0570860
\(556\) 0.498690 0.0211491
\(557\) 3.21139 0.136071 0.0680354 0.997683i \(-0.478327\pi\)
0.0680354 + 0.997683i \(0.478327\pi\)
\(558\) 8.50156 0.359900
\(559\) 0.666049 0.0281709
\(560\) 1.89816 0.0802119
\(561\) 9.39058 0.396470
\(562\) −11.1504 −0.470351
\(563\) −39.7260 −1.67425 −0.837125 0.547011i \(-0.815765\pi\)
−0.837125 + 0.547011i \(0.815765\pi\)
\(564\) 5.56855 0.234478
\(565\) −9.33219 −0.392608
\(566\) 27.1978 1.14321
\(567\) 2.89208 0.121456
\(568\) 10.0087 0.419954
\(569\) −7.40803 −0.310561 −0.155280 0.987870i \(-0.549628\pi\)
−0.155280 + 0.987870i \(0.549628\pi\)
\(570\) −1.41965 −0.0594624
\(571\) 28.8822 1.20868 0.604341 0.796726i \(-0.293437\pi\)
0.604341 + 0.796726i \(0.293437\pi\)
\(572\) −2.55119 −0.106671
\(573\) −17.2636 −0.721196
\(574\) 29.1658 1.21736
\(575\) −31.2668 −1.30391
\(576\) 1.00000 0.0416667
\(577\) −15.8961 −0.661764 −0.330882 0.943672i \(-0.607346\pi\)
−0.330882 + 0.943672i \(0.607346\pi\)
\(578\) −3.45125 −0.143553
\(579\) 17.8750 0.742859
\(580\) −5.45235 −0.226396
\(581\) −0.592979 −0.0246009
\(582\) 11.3618 0.470964
\(583\) 22.6779 0.939221
\(584\) −2.34621 −0.0970870
\(585\) −0.656329 −0.0271359
\(586\) −26.2040 −1.08248
\(587\) 10.1734 0.419900 0.209950 0.977712i \(-0.432670\pi\)
0.209950 + 0.977712i \(0.432670\pi\)
\(588\) −1.36415 −0.0562564
\(589\) 18.3890 0.757704
\(590\) 4.26327 0.175516
\(591\) −15.6986 −0.645753
\(592\) −2.04906 −0.0842158
\(593\) 34.3987 1.41259 0.706293 0.707920i \(-0.250366\pi\)
0.706293 + 0.707920i \(0.250366\pi\)
\(594\) −2.55119 −0.104677
\(595\) −6.98686 −0.286434
\(596\) 20.6355 0.845262
\(597\) −10.1854 −0.416862
\(598\) −6.84289 −0.279827
\(599\) −11.9637 −0.488823 −0.244411 0.969672i \(-0.578595\pi\)
−0.244411 + 0.969672i \(0.578595\pi\)
\(600\) 4.56923 0.186538
\(601\) −5.59948 −0.228408 −0.114204 0.993457i \(-0.536432\pi\)
−0.114204 + 0.993457i \(0.536432\pi\)
\(602\) −1.92627 −0.0785088
\(603\) 9.97687 0.406289
\(604\) 10.7708 0.438258
\(605\) −2.94786 −0.119848
\(606\) −6.72895 −0.273345
\(607\) −39.1365 −1.58850 −0.794251 0.607590i \(-0.792136\pi\)
−0.794251 + 0.607590i \(0.792136\pi\)
\(608\) 2.16301 0.0877216
\(609\) 24.0255 0.973562
\(610\) 2.54957 0.103229
\(611\) 5.56855 0.225279
\(612\) −3.68086 −0.148790
\(613\) −10.9031 −0.440372 −0.220186 0.975458i \(-0.570666\pi\)
−0.220186 + 0.975458i \(0.570666\pi\)
\(614\) 7.42744 0.299747
\(615\) −6.61889 −0.266899
\(616\) 7.37825 0.297278
\(617\) −37.6860 −1.51718 −0.758590 0.651569i \(-0.774111\pi\)
−0.758590 + 0.651569i \(0.774111\pi\)
\(618\) 1.00000 0.0402259
\(619\) −10.7806 −0.433307 −0.216653 0.976249i \(-0.569514\pi\)
−0.216653 + 0.976249i \(0.569514\pi\)
\(620\) 5.57983 0.224091
\(621\) −6.84289 −0.274596
\(622\) −2.89222 −0.115967
\(623\) −45.7015 −1.83099
\(624\) 1.00000 0.0400320
\(625\) 18.7240 0.748961
\(626\) −0.277979 −0.0111103
\(627\) −5.51824 −0.220377
\(628\) −11.2398 −0.448515
\(629\) 7.54231 0.300732
\(630\) 1.89816 0.0756245
\(631\) 15.7530 0.627116 0.313558 0.949569i \(-0.398479\pi\)
0.313558 + 0.949569i \(0.398479\pi\)
\(632\) 14.6755 0.583761
\(633\) 11.2856 0.448561
\(634\) −8.02479 −0.318705
\(635\) 2.19564 0.0871312
\(636\) −8.88913 −0.352477
\(637\) −1.36415 −0.0540494
\(638\) −21.1936 −0.839062
\(639\) 10.0087 0.395937
\(640\) 0.656329 0.0259437
\(641\) −23.3905 −0.923870 −0.461935 0.886914i \(-0.652845\pi\)
−0.461935 + 0.886914i \(0.652845\pi\)
\(642\) 7.54576 0.297807
\(643\) 23.0205 0.907840 0.453920 0.891043i \(-0.350025\pi\)
0.453920 + 0.891043i \(0.350025\pi\)
\(644\) 19.7902 0.779844
\(645\) 0.437147 0.0172127
\(646\) −7.96174 −0.313250
\(647\) 12.2922 0.483255 0.241628 0.970369i \(-0.422319\pi\)
0.241628 + 0.970369i \(0.422319\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.5716 0.650491
\(650\) 4.56923 0.179220
\(651\) −24.5872 −0.963649
\(652\) 10.5560 0.413404
\(653\) 38.7002 1.51446 0.757228 0.653151i \(-0.226553\pi\)
0.757228 + 0.653151i \(0.226553\pi\)
\(654\) −4.71913 −0.184533
\(655\) −9.28691 −0.362870
\(656\) 10.0847 0.393742
\(657\) −2.34621 −0.0915345
\(658\) −16.1047 −0.627827
\(659\) −40.2741 −1.56886 −0.784429 0.620219i \(-0.787044\pi\)
−0.784429 + 0.620219i \(0.787044\pi\)
\(660\) −1.67442 −0.0651767
\(661\) 13.5937 0.528732 0.264366 0.964422i \(-0.414837\pi\)
0.264366 + 0.964422i \(0.414837\pi\)
\(662\) −24.2699 −0.943278
\(663\) −3.68086 −0.142953
\(664\) −0.205035 −0.00795692
\(665\) 4.10573 0.159214
\(666\) −2.04906 −0.0793994
\(667\) −56.8462 −2.20109
\(668\) −8.41456 −0.325569
\(669\) −12.1915 −0.471352
\(670\) 6.54811 0.252976
\(671\) 9.91030 0.382583
\(672\) −2.89208 −0.111565
\(673\) 41.0267 1.58146 0.790732 0.612163i \(-0.209700\pi\)
0.790732 + 0.612163i \(0.209700\pi\)
\(674\) 14.8739 0.572923
\(675\) 4.56923 0.175870
\(676\) 1.00000 0.0384615
\(677\) −11.2949 −0.434099 −0.217049 0.976161i \(-0.569643\pi\)
−0.217049 + 0.976161i \(0.569643\pi\)
\(678\) 14.2188 0.546068
\(679\) −32.8594 −1.26103
\(680\) −2.41586 −0.0926440
\(681\) −2.61611 −0.100250
\(682\) 21.6891 0.830518
\(683\) −8.47663 −0.324349 −0.162175 0.986762i \(-0.551851\pi\)
−0.162175 + 0.986762i \(0.551851\pi\)
\(684\) 2.16301 0.0827047
\(685\) −2.66149 −0.101690
\(686\) −16.2994 −0.622312
\(687\) 3.08691 0.117773
\(688\) −0.666049 −0.0253929
\(689\) −8.88913 −0.338649
\(690\) −4.49119 −0.170977
\(691\) −26.4454 −1.00603 −0.503015 0.864277i \(-0.667776\pi\)
−0.503015 + 0.864277i \(0.667776\pi\)
\(692\) −6.31337 −0.239998
\(693\) 7.37825 0.280277
\(694\) −14.8380 −0.563244
\(695\) 0.327305 0.0124154
\(696\) 8.30733 0.314889
\(697\) −37.1204 −1.40604
\(698\) 4.95172 0.187425
\(699\) −0.417209 −0.0157803
\(700\) −13.2146 −0.499465
\(701\) 44.4712 1.67965 0.839826 0.542855i \(-0.182657\pi\)
0.839826 + 0.542855i \(0.182657\pi\)
\(702\) 1.00000 0.0377426
\(703\) −4.43213 −0.167161
\(704\) 2.55119 0.0961516
\(705\) 3.65480 0.137648
\(706\) −18.8072 −0.707820
\(707\) 19.4607 0.731894
\(708\) −6.49563 −0.244121
\(709\) 39.4804 1.48272 0.741360 0.671108i \(-0.234181\pi\)
0.741360 + 0.671108i \(0.234181\pi\)
\(710\) 6.56898 0.246529
\(711\) 14.6755 0.550375
\(712\) −15.8023 −0.592216
\(713\) 58.1753 2.17868
\(714\) 10.6454 0.398393
\(715\) −1.67442 −0.0626198
\(716\) 11.2133 0.419061
\(717\) 21.8617 0.816441
\(718\) 23.9366 0.893307
\(719\) −24.2062 −0.902738 −0.451369 0.892337i \(-0.649064\pi\)
−0.451369 + 0.892337i \(0.649064\pi\)
\(720\) 0.656329 0.0244600
\(721\) −2.89208 −0.107707
\(722\) −14.3214 −0.532987
\(723\) −26.3640 −0.980489
\(724\) 4.89458 0.181906
\(725\) 37.9581 1.40973
\(726\) 4.49143 0.166693
\(727\) −35.5726 −1.31932 −0.659658 0.751566i \(-0.729299\pi\)
−0.659658 + 0.751566i \(0.729299\pi\)
\(728\) −2.89208 −0.107188
\(729\) 1.00000 0.0370370
\(730\) −1.53989 −0.0569939
\(731\) 2.45163 0.0906769
\(732\) −3.88458 −0.143578
\(733\) 11.7305 0.433276 0.216638 0.976252i \(-0.430491\pi\)
0.216638 + 0.976252i \(0.430491\pi\)
\(734\) 37.8746 1.39798
\(735\) −0.895329 −0.0330247
\(736\) 6.84289 0.252232
\(737\) 25.4529 0.937569
\(738\) 10.0847 0.371223
\(739\) 18.7704 0.690480 0.345240 0.938514i \(-0.387798\pi\)
0.345240 + 0.938514i \(0.387798\pi\)
\(740\) −1.34486 −0.0494380
\(741\) 2.16301 0.0794601
\(742\) 25.7081 0.943774
\(743\) −36.6647 −1.34510 −0.672549 0.740053i \(-0.734801\pi\)
−0.672549 + 0.740053i \(0.734801\pi\)
\(744\) −8.50156 −0.311682
\(745\) 13.5437 0.496201
\(746\) 11.4552 0.419404
\(747\) −0.205035 −0.00750185
\(748\) −9.39058 −0.343354
\(749\) −21.8230 −0.797394
\(750\) 6.28057 0.229334
\(751\) −29.2857 −1.06865 −0.534325 0.845279i \(-0.679434\pi\)
−0.534325 + 0.845279i \(0.679434\pi\)
\(752\) −5.56855 −0.203064
\(753\) −21.5806 −0.786440
\(754\) 8.30733 0.302535
\(755\) 7.06920 0.257274
\(756\) −2.89208 −0.105184
\(757\) −42.7918 −1.55529 −0.777647 0.628702i \(-0.783587\pi\)
−0.777647 + 0.628702i \(0.783587\pi\)
\(758\) 27.4077 0.995492
\(759\) −17.4575 −0.633668
\(760\) 1.41965 0.0514960
\(761\) −10.8233 −0.392345 −0.196172 0.980569i \(-0.562851\pi\)
−0.196172 + 0.980569i \(0.562851\pi\)
\(762\) −3.34533 −0.121188
\(763\) 13.6481 0.494095
\(764\) 17.2636 0.624574
\(765\) −2.41586 −0.0873456
\(766\) −9.69866 −0.350427
\(767\) −6.49563 −0.234543
\(768\) −1.00000 −0.0360844
\(769\) −49.8448 −1.79745 −0.898725 0.438513i \(-0.855505\pi\)
−0.898725 + 0.438513i \(0.855505\pi\)
\(770\) 4.84256 0.174514
\(771\) −14.0027 −0.504295
\(772\) −17.8750 −0.643335
\(773\) −3.40980 −0.122642 −0.0613211 0.998118i \(-0.519531\pi\)
−0.0613211 + 0.998118i \(0.519531\pi\)
\(774\) −0.666049 −0.0239406
\(775\) −38.8456 −1.39538
\(776\) −11.3618 −0.407866
\(777\) 5.92605 0.212596
\(778\) −13.6955 −0.491008
\(779\) 21.8133 0.781543
\(780\) 0.656329 0.0235004
\(781\) 25.5340 0.913678
\(782\) −25.1878 −0.900712
\(783\) 8.30733 0.296880
\(784\) 1.36415 0.0487195
\(785\) −7.37698 −0.263296
\(786\) 14.1498 0.504706
\(787\) −10.6139 −0.378345 −0.189173 0.981944i \(-0.560581\pi\)
−0.189173 + 0.981944i \(0.560581\pi\)
\(788\) 15.6986 0.559238
\(789\) −13.3638 −0.475765
\(790\) 9.63197 0.342690
\(791\) −41.1219 −1.46212
\(792\) 2.55119 0.0906526
\(793\) −3.88458 −0.137946
\(794\) 0.618503 0.0219498
\(795\) −5.83420 −0.206918
\(796\) 10.1854 0.361013
\(797\) 12.9337 0.458134 0.229067 0.973411i \(-0.426432\pi\)
0.229067 + 0.973411i \(0.426432\pi\)
\(798\) −6.25560 −0.221446
\(799\) 20.4971 0.725134
\(800\) −4.56923 −0.161547
\(801\) −15.8023 −0.558346
\(802\) −16.0957 −0.568359
\(803\) −5.98563 −0.211228
\(804\) −9.97687 −0.351857
\(805\) 12.9889 0.457799
\(806\) −8.50156 −0.299455
\(807\) 14.9933 0.527790
\(808\) 6.72895 0.236724
\(809\) −28.4844 −1.00146 −0.500730 0.865604i \(-0.666935\pi\)
−0.500730 + 0.865604i \(0.666935\pi\)
\(810\) 0.656329 0.0230611
\(811\) −42.7330 −1.50056 −0.750279 0.661122i \(-0.770081\pi\)
−0.750279 + 0.661122i \(0.770081\pi\)
\(812\) −24.0255 −0.843130
\(813\) 8.95971 0.314231
\(814\) −5.22754 −0.183225
\(815\) 6.92820 0.242684
\(816\) 3.68086 0.128856
\(817\) −1.44067 −0.0504026
\(818\) −23.0712 −0.806667
\(819\) −2.89208 −0.101058
\(820\) 6.61889 0.231142
\(821\) 0.941138 0.0328459 0.0164230 0.999865i \(-0.494772\pi\)
0.0164230 + 0.999865i \(0.494772\pi\)
\(822\) 4.05511 0.141438
\(823\) 14.2890 0.498082 0.249041 0.968493i \(-0.419885\pi\)
0.249041 + 0.968493i \(0.419885\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 11.6570 0.405844
\(826\) 18.7859 0.653645
\(827\) −5.91860 −0.205810 −0.102905 0.994691i \(-0.532814\pi\)
−0.102905 + 0.994691i \(0.532814\pi\)
\(828\) 6.84289 0.237807
\(829\) 22.6907 0.788080 0.394040 0.919093i \(-0.371077\pi\)
0.394040 + 0.919093i \(0.371077\pi\)
\(830\) −0.134571 −0.00467102
\(831\) −15.9036 −0.551691
\(832\) −1.00000 −0.0346688
\(833\) −5.02123 −0.173975
\(834\) −0.498690 −0.0172682
\(835\) −5.52272 −0.191122
\(836\) 5.51824 0.190852
\(837\) −8.50156 −0.293857
\(838\) 21.0454 0.727001
\(839\) −22.8837 −0.790035 −0.395017 0.918674i \(-0.629261\pi\)
−0.395017 + 0.918674i \(0.629261\pi\)
\(840\) −1.89816 −0.0654927
\(841\) 40.0118 1.37972
\(842\) 28.0643 0.967158
\(843\) 11.1504 0.384040
\(844\) −11.2856 −0.388465
\(845\) 0.656329 0.0225784
\(846\) −5.56855 −0.191451
\(847\) −12.9896 −0.446328
\(848\) 8.88913 0.305254
\(849\) −27.1978 −0.933426
\(850\) 16.8187 0.576877
\(851\) −14.0215 −0.480651
\(852\) −10.0087 −0.342891
\(853\) 0.575726 0.0197125 0.00985625 0.999951i \(-0.496863\pi\)
0.00985625 + 0.999951i \(0.496863\pi\)
\(854\) 11.2345 0.384438
\(855\) 1.41965 0.0485509
\(856\) −7.54576 −0.257909
\(857\) 29.1028 0.994133 0.497066 0.867713i \(-0.334411\pi\)
0.497066 + 0.867713i \(0.334411\pi\)
\(858\) 2.55119 0.0870962
\(859\) −18.1428 −0.619025 −0.309513 0.950895i \(-0.600166\pi\)
−0.309513 + 0.950895i \(0.600166\pi\)
\(860\) −0.437147 −0.0149066
\(861\) −29.1658 −0.993968
\(862\) −21.0838 −0.718117
\(863\) 11.7829 0.401094 0.200547 0.979684i \(-0.435728\pi\)
0.200547 + 0.979684i \(0.435728\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −4.14365 −0.140888
\(866\) 16.3609 0.555966
\(867\) 3.45125 0.117210
\(868\) 24.5872 0.834545
\(869\) 37.4400 1.27007
\(870\) 5.45235 0.184852
\(871\) −9.97687 −0.338053
\(872\) 4.71913 0.159810
\(873\) −11.3618 −0.384540
\(874\) 14.8012 0.500659
\(875\) −18.1639 −0.614053
\(876\) 2.34621 0.0792712
\(877\) 1.75567 0.0592847 0.0296424 0.999561i \(-0.490563\pi\)
0.0296424 + 0.999561i \(0.490563\pi\)
\(878\) −29.0513 −0.980435
\(879\) 26.2040 0.883840
\(880\) 1.67442 0.0564447
\(881\) −26.1340 −0.880477 −0.440239 0.897881i \(-0.645106\pi\)
−0.440239 + 0.897881i \(0.645106\pi\)
\(882\) 1.36415 0.0459332
\(883\) −3.28660 −0.110603 −0.0553014 0.998470i \(-0.517612\pi\)
−0.0553014 + 0.998470i \(0.517612\pi\)
\(884\) 3.68086 0.123801
\(885\) −4.26327 −0.143308
\(886\) −38.5298 −1.29443
\(887\) −25.5225 −0.856960 −0.428480 0.903551i \(-0.640951\pi\)
−0.428480 + 0.903551i \(0.640951\pi\)
\(888\) 2.04906 0.0687619
\(889\) 9.67497 0.324488
\(890\) −10.3715 −0.347654
\(891\) 2.55119 0.0854680
\(892\) 12.1915 0.408203
\(893\) −12.0448 −0.403065
\(894\) −20.6355 −0.690153
\(895\) 7.35962 0.246005
\(896\) 2.89208 0.0966177
\(897\) 6.84289 0.228478
\(898\) 0.892366 0.0297786
\(899\) −70.6253 −2.35549
\(900\) −4.56923 −0.152308
\(901\) −32.7197 −1.09005
\(902\) 25.7280 0.856649
\(903\) 1.92627 0.0641022
\(904\) −14.2188 −0.472909
\(905\) 3.21246 0.106786
\(906\) −10.7708 −0.357836
\(907\) −54.8934 −1.82270 −0.911352 0.411628i \(-0.864960\pi\)
−0.911352 + 0.411628i \(0.864960\pi\)
\(908\) 2.61611 0.0868188
\(909\) 6.72895 0.223185
\(910\) −1.89816 −0.0629234
\(911\) 53.0254 1.75681 0.878404 0.477918i \(-0.158609\pi\)
0.878404 + 0.477918i \(0.158609\pi\)
\(912\) −2.16301 −0.0716244
\(913\) −0.523084 −0.0173116
\(914\) −22.3844 −0.740411
\(915\) −2.54957 −0.0842861
\(916\) −3.08691 −0.101994
\(917\) −40.9223 −1.35137
\(918\) 3.68086 0.121487
\(919\) −21.2737 −0.701754 −0.350877 0.936422i \(-0.614116\pi\)
−0.350877 + 0.936422i \(0.614116\pi\)
\(920\) 4.49119 0.148070
\(921\) −7.42744 −0.244742
\(922\) −12.2927 −0.404840
\(923\) −10.0087 −0.329439
\(924\) −7.37825 −0.242727
\(925\) 9.36263 0.307841
\(926\) −22.6408 −0.744022
\(927\) −1.00000 −0.0328443
\(928\) −8.30733 −0.272702
\(929\) −24.6312 −0.808122 −0.404061 0.914732i \(-0.632402\pi\)
−0.404061 + 0.914732i \(0.632402\pi\)
\(930\) −5.57983 −0.182970
\(931\) 2.95066 0.0967040
\(932\) 0.417209 0.0136661
\(933\) 2.89222 0.0946870
\(934\) −2.86491 −0.0937427
\(935\) −6.16331 −0.201562
\(936\) −1.00000 −0.0326860
\(937\) 41.3911 1.35219 0.676093 0.736816i \(-0.263672\pi\)
0.676093 + 0.736816i \(0.263672\pi\)
\(938\) 28.8539 0.942114
\(939\) 0.277979 0.00907150
\(940\) −3.65480 −0.119207
\(941\) −18.1667 −0.592217 −0.296108 0.955154i \(-0.595689\pi\)
−0.296108 + 0.955154i \(0.595689\pi\)
\(942\) 11.2398 0.366211
\(943\) 69.0086 2.24723
\(944\) 6.49563 0.211415
\(945\) −1.89816 −0.0617471
\(946\) −1.69922 −0.0552463
\(947\) 12.1010 0.393230 0.196615 0.980481i \(-0.437005\pi\)
0.196615 + 0.980481i \(0.437005\pi\)
\(948\) −14.6755 −0.476638
\(949\) 2.34621 0.0761613
\(950\) −9.88329 −0.320656
\(951\) 8.02479 0.260222
\(952\) −10.6454 −0.345018
\(953\) −17.9947 −0.582905 −0.291453 0.956585i \(-0.594139\pi\)
−0.291453 + 0.956585i \(0.594139\pi\)
\(954\) 8.88913 0.287796
\(955\) 11.3306 0.366649
\(956\) −21.8617 −0.707058
\(957\) 21.1936 0.685091
\(958\) 32.3676 1.04575
\(959\) −11.7277 −0.378707
\(960\) −0.656329 −0.0211829
\(961\) 41.2766 1.33150
\(962\) 2.04906 0.0660643
\(963\) −7.54576 −0.243159
\(964\) 26.3640 0.849128
\(965\) −11.7319 −0.377662
\(966\) −19.7902 −0.636740
\(967\) −11.3422 −0.364741 −0.182371 0.983230i \(-0.558377\pi\)
−0.182371 + 0.983230i \(0.558377\pi\)
\(968\) −4.49143 −0.144360
\(969\) 7.96174 0.255768
\(970\) −7.45711 −0.239433
\(971\) −22.8430 −0.733067 −0.366534 0.930405i \(-0.619456\pi\)
−0.366534 + 0.930405i \(0.619456\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 1.44225 0.0462365
\(974\) 5.28092 0.169212
\(975\) −4.56923 −0.146333
\(976\) 3.88458 0.124342
\(977\) 39.8336 1.27439 0.637195 0.770703i \(-0.280095\pi\)
0.637195 + 0.770703i \(0.280095\pi\)
\(978\) −10.5560 −0.337543
\(979\) −40.3146 −1.28846
\(980\) 0.895329 0.0286002
\(981\) 4.71913 0.150670
\(982\) 2.25221 0.0718710
\(983\) −27.4531 −0.875617 −0.437809 0.899068i \(-0.644245\pi\)
−0.437809 + 0.899068i \(0.644245\pi\)
\(984\) −10.0847 −0.321489
\(985\) 10.3034 0.328295
\(986\) 30.5782 0.973807
\(987\) 16.1047 0.512619
\(988\) −2.16301 −0.0688145
\(989\) −4.55770 −0.144926
\(990\) 1.67442 0.0532166
\(991\) 4.04537 0.128506 0.0642528 0.997934i \(-0.479534\pi\)
0.0642528 + 0.997934i \(0.479534\pi\)
\(992\) 8.50156 0.269925
\(993\) 24.2699 0.770183
\(994\) 28.9459 0.918108
\(995\) 6.68501 0.211929
\(996\) 0.205035 0.00649680
\(997\) −28.9642 −0.917305 −0.458652 0.888616i \(-0.651668\pi\)
−0.458652 + 0.888616i \(0.651668\pi\)
\(998\) 7.45562 0.236004
\(999\) 2.04906 0.0648294
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.1.9 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.9 15 1.1 even 1 trivial