Properties

Label 5054.2.a.bm.1.12
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} - 2326 x^{3} + 570 x^{2} + 105 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(3.20919\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.20919 q^{3} +1.00000 q^{4} +2.74348 q^{5} +3.20919 q^{6} +1.00000 q^{7} +1.00000 q^{8} +7.29887 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.20919 q^{3} +1.00000 q^{4} +2.74348 q^{5} +3.20919 q^{6} +1.00000 q^{7} +1.00000 q^{8} +7.29887 q^{9} +2.74348 q^{10} -3.98639 q^{11} +3.20919 q^{12} -4.64202 q^{13} +1.00000 q^{14} +8.80432 q^{15} +1.00000 q^{16} +5.22474 q^{17} +7.29887 q^{18} +2.74348 q^{20} +3.20919 q^{21} -3.98639 q^{22} +0.520464 q^{23} +3.20919 q^{24} +2.52666 q^{25} -4.64202 q^{26} +13.7959 q^{27} +1.00000 q^{28} +5.83623 q^{29} +8.80432 q^{30} -1.38498 q^{31} +1.00000 q^{32} -12.7931 q^{33} +5.22474 q^{34} +2.74348 q^{35} +7.29887 q^{36} -9.44185 q^{37} -14.8971 q^{39} +2.74348 q^{40} -9.40284 q^{41} +3.20919 q^{42} +2.55638 q^{43} -3.98639 q^{44} +20.0243 q^{45} +0.520464 q^{46} +4.77001 q^{47} +3.20919 q^{48} +1.00000 q^{49} +2.52666 q^{50} +16.7672 q^{51} -4.64202 q^{52} +9.11202 q^{53} +13.7959 q^{54} -10.9366 q^{55} +1.00000 q^{56} +5.83623 q^{58} +1.07661 q^{59} +8.80432 q^{60} -1.53726 q^{61} -1.38498 q^{62} +7.29887 q^{63} +1.00000 q^{64} -12.7353 q^{65} -12.7931 q^{66} -12.3091 q^{67} +5.22474 q^{68} +1.67027 q^{69} +2.74348 q^{70} -4.00739 q^{71} +7.29887 q^{72} -0.422816 q^{73} -9.44185 q^{74} +8.10851 q^{75} -3.98639 q^{77} -14.8971 q^{78} -12.7880 q^{79} +2.74348 q^{80} +22.3769 q^{81} -9.40284 q^{82} -1.49938 q^{83} +3.20919 q^{84} +14.3339 q^{85} +2.55638 q^{86} +18.7295 q^{87} -3.98639 q^{88} -8.77859 q^{89} +20.0243 q^{90} -4.64202 q^{91} +0.520464 q^{92} -4.44464 q^{93} +4.77001 q^{94} +3.20919 q^{96} -4.61801 q^{97} +1.00000 q^{98} -29.0962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} + 12 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} + 12 q^{8} + 21 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} - 6 q^{13} + 12 q^{14} - 6 q^{15} + 12 q^{16} + 3 q^{17} + 21 q^{18} + 3 q^{20} + 3 q^{21} + 3 q^{22} + 12 q^{23} + 3 q^{24} + 33 q^{25} - 6 q^{26} + 21 q^{27} + 12 q^{28} + 6 q^{29} - 6 q^{30} + 3 q^{31} + 12 q^{32} - 6 q^{33} + 3 q^{34} + 3 q^{35} + 21 q^{36} + 27 q^{37} + 27 q^{39} + 3 q^{40} - 27 q^{41} + 3 q^{42} + 30 q^{43} + 3 q^{44} + 30 q^{45} + 12 q^{46} + 21 q^{47} + 3 q^{48} + 12 q^{49} + 33 q^{50} + 3 q^{51} - 6 q^{52} + 6 q^{53} + 21 q^{54} + 48 q^{55} + 12 q^{56} + 6 q^{58} - 27 q^{59} - 6 q^{60} + 18 q^{61} + 3 q^{62} + 21 q^{63} + 12 q^{64} - 9 q^{65} - 6 q^{66} + 9 q^{67} + 3 q^{68} + 18 q^{69} + 3 q^{70} + 27 q^{71} + 21 q^{72} - 9 q^{73} + 27 q^{74} + 33 q^{75} + 3 q^{77} + 27 q^{78} - 12 q^{79} + 3 q^{80} + 24 q^{81} - 27 q^{82} + 9 q^{83} + 3 q^{84} + 78 q^{85} + 30 q^{86} - 45 q^{87} + 3 q^{88} - 24 q^{89} + 30 q^{90} - 6 q^{91} + 12 q^{92} + 3 q^{93} + 21 q^{94} + 3 q^{96} - 18 q^{97} + 12 q^{98} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 3.20919 1.85282 0.926412 0.376511i \(-0.122876\pi\)
0.926412 + 0.376511i \(0.122876\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.74348 1.22692 0.613460 0.789726i \(-0.289777\pi\)
0.613460 + 0.789726i \(0.289777\pi\)
\(6\) 3.20919 1.31014
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 7.29887 2.43296
\(10\) 2.74348 0.867563
\(11\) −3.98639 −1.20194 −0.600971 0.799271i \(-0.705219\pi\)
−0.600971 + 0.799271i \(0.705219\pi\)
\(12\) 3.20919 0.926412
\(13\) −4.64202 −1.28746 −0.643732 0.765251i \(-0.722615\pi\)
−0.643732 + 0.765251i \(0.722615\pi\)
\(14\) 1.00000 0.267261
\(15\) 8.80432 2.27327
\(16\) 1.00000 0.250000
\(17\) 5.22474 1.26719 0.633593 0.773667i \(-0.281580\pi\)
0.633593 + 0.773667i \(0.281580\pi\)
\(18\) 7.29887 1.72036
\(19\) 0 0
\(20\) 2.74348 0.613460
\(21\) 3.20919 0.700302
\(22\) −3.98639 −0.849901
\(23\) 0.520464 0.108524 0.0542621 0.998527i \(-0.482719\pi\)
0.0542621 + 0.998527i \(0.482719\pi\)
\(24\) 3.20919 0.655072
\(25\) 2.52666 0.505331
\(26\) −4.64202 −0.910374
\(27\) 13.7959 2.65502
\(28\) 1.00000 0.188982
\(29\) 5.83623 1.08376 0.541880 0.840456i \(-0.317713\pi\)
0.541880 + 0.840456i \(0.317713\pi\)
\(30\) 8.80432 1.60744
\(31\) −1.38498 −0.248749 −0.124374 0.992235i \(-0.539692\pi\)
−0.124374 + 0.992235i \(0.539692\pi\)
\(32\) 1.00000 0.176777
\(33\) −12.7931 −2.22699
\(34\) 5.22474 0.896035
\(35\) 2.74348 0.463732
\(36\) 7.29887 1.21648
\(37\) −9.44185 −1.55223 −0.776115 0.630591i \(-0.782812\pi\)
−0.776115 + 0.630591i \(0.782812\pi\)
\(38\) 0 0
\(39\) −14.8971 −2.38544
\(40\) 2.74348 0.433781
\(41\) −9.40284 −1.46848 −0.734239 0.678891i \(-0.762461\pi\)
−0.734239 + 0.678891i \(0.762461\pi\)
\(42\) 3.20919 0.495188
\(43\) 2.55638 0.389845 0.194922 0.980819i \(-0.437555\pi\)
0.194922 + 0.980819i \(0.437555\pi\)
\(44\) −3.98639 −0.600971
\(45\) 20.0243 2.98504
\(46\) 0.520464 0.0767382
\(47\) 4.77001 0.695777 0.347889 0.937536i \(-0.386899\pi\)
0.347889 + 0.937536i \(0.386899\pi\)
\(48\) 3.20919 0.463206
\(49\) 1.00000 0.142857
\(50\) 2.52666 0.357323
\(51\) 16.7672 2.34787
\(52\) −4.64202 −0.643732
\(53\) 9.11202 1.25163 0.625816 0.779971i \(-0.284766\pi\)
0.625816 + 0.779971i \(0.284766\pi\)
\(54\) 13.7959 1.87738
\(55\) −10.9366 −1.47469
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 5.83623 0.766334
\(59\) 1.07661 0.140163 0.0700815 0.997541i \(-0.477674\pi\)
0.0700815 + 0.997541i \(0.477674\pi\)
\(60\) 8.80432 1.13663
\(61\) −1.53726 −0.196826 −0.0984132 0.995146i \(-0.531377\pi\)
−0.0984132 + 0.995146i \(0.531377\pi\)
\(62\) −1.38498 −0.175892
\(63\) 7.29887 0.919571
\(64\) 1.00000 0.125000
\(65\) −12.7353 −1.57961
\(66\) −12.7931 −1.57472
\(67\) −12.3091 −1.50380 −0.751898 0.659280i \(-0.770861\pi\)
−0.751898 + 0.659280i \(0.770861\pi\)
\(68\) 5.22474 0.633593
\(69\) 1.67027 0.201076
\(70\) 2.74348 0.327908
\(71\) −4.00739 −0.475589 −0.237795 0.971315i \(-0.576425\pi\)
−0.237795 + 0.971315i \(0.576425\pi\)
\(72\) 7.29887 0.860180
\(73\) −0.422816 −0.0494868 −0.0247434 0.999694i \(-0.507877\pi\)
−0.0247434 + 0.999694i \(0.507877\pi\)
\(74\) −9.44185 −1.09759
\(75\) 8.10851 0.936290
\(76\) 0 0
\(77\) −3.98639 −0.454291
\(78\) −14.8971 −1.68676
\(79\) −12.7880 −1.43876 −0.719381 0.694616i \(-0.755574\pi\)
−0.719381 + 0.694616i \(0.755574\pi\)
\(80\) 2.74348 0.306730
\(81\) 22.3769 2.48632
\(82\) −9.40284 −1.03837
\(83\) −1.49938 −0.164578 −0.0822892 0.996608i \(-0.526223\pi\)
−0.0822892 + 0.996608i \(0.526223\pi\)
\(84\) 3.20919 0.350151
\(85\) 14.3339 1.55473
\(86\) 2.55638 0.275662
\(87\) 18.7295 2.00802
\(88\) −3.98639 −0.424951
\(89\) −8.77859 −0.930529 −0.465265 0.885172i \(-0.654041\pi\)
−0.465265 + 0.885172i \(0.654041\pi\)
\(90\) 20.0243 2.11074
\(91\) −4.64202 −0.486615
\(92\) 0.520464 0.0542621
\(93\) −4.44464 −0.460888
\(94\) 4.77001 0.491989
\(95\) 0 0
\(96\) 3.20919 0.327536
\(97\) −4.61801 −0.468888 −0.234444 0.972130i \(-0.575327\pi\)
−0.234444 + 0.972130i \(0.575327\pi\)
\(98\) 1.00000 0.101015
\(99\) −29.0962 −2.92427
\(100\) 2.52666 0.252666
\(101\) 5.07336 0.504818 0.252409 0.967621i \(-0.418777\pi\)
0.252409 + 0.967621i \(0.418777\pi\)
\(102\) 16.7672 1.66020
\(103\) 1.40012 0.137958 0.0689790 0.997618i \(-0.478026\pi\)
0.0689790 + 0.997618i \(0.478026\pi\)
\(104\) −4.64202 −0.455187
\(105\) 8.80432 0.859214
\(106\) 9.11202 0.885038
\(107\) 18.3546 1.77441 0.887205 0.461375i \(-0.152644\pi\)
0.887205 + 0.461375i \(0.152644\pi\)
\(108\) 13.7959 1.32751
\(109\) 8.49097 0.813287 0.406643 0.913587i \(-0.366699\pi\)
0.406643 + 0.913587i \(0.366699\pi\)
\(110\) −10.9366 −1.04276
\(111\) −30.3006 −2.87601
\(112\) 1.00000 0.0944911
\(113\) 8.06367 0.758566 0.379283 0.925281i \(-0.376171\pi\)
0.379283 + 0.925281i \(0.376171\pi\)
\(114\) 0 0
\(115\) 1.42788 0.133150
\(116\) 5.83623 0.541880
\(117\) −33.8815 −3.13234
\(118\) 1.07661 0.0991102
\(119\) 5.22474 0.478951
\(120\) 8.80432 0.803721
\(121\) 4.89131 0.444665
\(122\) −1.53726 −0.139177
\(123\) −30.1755 −2.72083
\(124\) −1.38498 −0.124374
\(125\) −6.78556 −0.606919
\(126\) 7.29887 0.650235
\(127\) 13.1420 1.16617 0.583084 0.812412i \(-0.301846\pi\)
0.583084 + 0.812412i \(0.301846\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.20390 0.722313
\(130\) −12.7353 −1.11696
\(131\) 18.3208 1.60069 0.800347 0.599537i \(-0.204649\pi\)
0.800347 + 0.599537i \(0.204649\pi\)
\(132\) −12.7931 −1.11349
\(133\) 0 0
\(134\) −12.3091 −1.06334
\(135\) 37.8486 3.25749
\(136\) 5.22474 0.448018
\(137\) −4.18816 −0.357818 −0.178909 0.983866i \(-0.557257\pi\)
−0.178909 + 0.983866i \(0.557257\pi\)
\(138\) 1.67027 0.142182
\(139\) −14.9473 −1.26781 −0.633905 0.773411i \(-0.718549\pi\)
−0.633905 + 0.773411i \(0.718549\pi\)
\(140\) 2.74348 0.231866
\(141\) 15.3078 1.28915
\(142\) −4.00739 −0.336293
\(143\) 18.5049 1.54746
\(144\) 7.29887 0.608239
\(145\) 16.0115 1.32969
\(146\) −0.422816 −0.0349925
\(147\) 3.20919 0.264689
\(148\) −9.44185 −0.776115
\(149\) −0.0172226 −0.00141093 −0.000705467 1.00000i \(-0.500225\pi\)
−0.000705467 1.00000i \(0.500225\pi\)
\(150\) 8.10851 0.662057
\(151\) −6.94979 −0.565566 −0.282783 0.959184i \(-0.591258\pi\)
−0.282783 + 0.959184i \(0.591258\pi\)
\(152\) 0 0
\(153\) 38.1347 3.08301
\(154\) −3.98639 −0.321233
\(155\) −3.79964 −0.305195
\(156\) −14.8971 −1.19272
\(157\) −20.9105 −1.66884 −0.834419 0.551130i \(-0.814197\pi\)
−0.834419 + 0.551130i \(0.814197\pi\)
\(158\) −12.7880 −1.01736
\(159\) 29.2422 2.31905
\(160\) 2.74348 0.216891
\(161\) 0.520464 0.0410183
\(162\) 22.3769 1.75810
\(163\) 8.93937 0.700186 0.350093 0.936715i \(-0.386150\pi\)
0.350093 + 0.936715i \(0.386150\pi\)
\(164\) −9.40284 −0.734239
\(165\) −35.0975 −2.73233
\(166\) −1.49938 −0.116375
\(167\) 18.1411 1.40380 0.701901 0.712275i \(-0.252335\pi\)
0.701901 + 0.712275i \(0.252335\pi\)
\(168\) 3.20919 0.247594
\(169\) 8.54831 0.657562
\(170\) 14.3339 1.09936
\(171\) 0 0
\(172\) 2.55638 0.194922
\(173\) −1.97709 −0.150315 −0.0751577 0.997172i \(-0.523946\pi\)
−0.0751577 + 0.997172i \(0.523946\pi\)
\(174\) 18.7295 1.41988
\(175\) 2.52666 0.190997
\(176\) −3.98639 −0.300486
\(177\) 3.45505 0.259697
\(178\) −8.77859 −0.657983
\(179\) −19.4213 −1.45161 −0.725807 0.687899i \(-0.758533\pi\)
−0.725807 + 0.687899i \(0.758533\pi\)
\(180\) 20.0243 1.49252
\(181\) −2.54502 −0.189170 −0.0945848 0.995517i \(-0.530152\pi\)
−0.0945848 + 0.995517i \(0.530152\pi\)
\(182\) −4.64202 −0.344089
\(183\) −4.93336 −0.364685
\(184\) 0.520464 0.0383691
\(185\) −25.9035 −1.90446
\(186\) −4.44464 −0.325897
\(187\) −20.8278 −1.52308
\(188\) 4.77001 0.347889
\(189\) 13.7959 1.00350
\(190\) 0 0
\(191\) −18.1665 −1.31448 −0.657241 0.753681i \(-0.728276\pi\)
−0.657241 + 0.753681i \(0.728276\pi\)
\(192\) 3.20919 0.231603
\(193\) −12.5306 −0.901971 −0.450986 0.892531i \(-0.648927\pi\)
−0.450986 + 0.892531i \(0.648927\pi\)
\(194\) −4.61801 −0.331554
\(195\) −40.8698 −2.92675
\(196\) 1.00000 0.0714286
\(197\) −11.0525 −0.787456 −0.393728 0.919227i \(-0.628815\pi\)
−0.393728 + 0.919227i \(0.628815\pi\)
\(198\) −29.0962 −2.06777
\(199\) −2.42328 −0.171782 −0.0858910 0.996305i \(-0.527374\pi\)
−0.0858910 + 0.996305i \(0.527374\pi\)
\(200\) 2.52666 0.178662
\(201\) −39.5022 −2.78627
\(202\) 5.07336 0.356960
\(203\) 5.83623 0.409623
\(204\) 16.7672 1.17394
\(205\) −25.7965 −1.80170
\(206\) 1.40012 0.0975510
\(207\) 3.79880 0.264035
\(208\) −4.64202 −0.321866
\(209\) 0 0
\(210\) 8.80432 0.607556
\(211\) −3.19423 −0.219900 −0.109950 0.993937i \(-0.535069\pi\)
−0.109950 + 0.993937i \(0.535069\pi\)
\(212\) 9.11202 0.625816
\(213\) −12.8605 −0.881184
\(214\) 18.3546 1.25470
\(215\) 7.01337 0.478308
\(216\) 13.7959 0.938691
\(217\) −1.38498 −0.0940182
\(218\) 8.49097 0.575081
\(219\) −1.35689 −0.0916904
\(220\) −10.9366 −0.737343
\(221\) −24.2533 −1.63145
\(222\) −30.3006 −2.03365
\(223\) 7.11759 0.476629 0.238314 0.971188i \(-0.423405\pi\)
0.238314 + 0.971188i \(0.423405\pi\)
\(224\) 1.00000 0.0668153
\(225\) 18.4417 1.22945
\(226\) 8.06367 0.536388
\(227\) −15.0860 −1.00130 −0.500648 0.865651i \(-0.666905\pi\)
−0.500648 + 0.865651i \(0.666905\pi\)
\(228\) 0 0
\(229\) 0.806298 0.0532817 0.0266409 0.999645i \(-0.491519\pi\)
0.0266409 + 0.999645i \(0.491519\pi\)
\(230\) 1.42788 0.0941516
\(231\) −12.7931 −0.841722
\(232\) 5.83623 0.383167
\(233\) −9.43769 −0.618284 −0.309142 0.951016i \(-0.600042\pi\)
−0.309142 + 0.951016i \(0.600042\pi\)
\(234\) −33.8815 −2.21490
\(235\) 13.0864 0.853662
\(236\) 1.07661 0.0700815
\(237\) −41.0390 −2.66577
\(238\) 5.22474 0.338669
\(239\) −1.94285 −0.125672 −0.0628362 0.998024i \(-0.520015\pi\)
−0.0628362 + 0.998024i \(0.520015\pi\)
\(240\) 8.80432 0.568316
\(241\) −23.4255 −1.50897 −0.754483 0.656319i \(-0.772112\pi\)
−0.754483 + 0.656319i \(0.772112\pi\)
\(242\) 4.89131 0.314425
\(243\) 30.4241 1.95170
\(244\) −1.53726 −0.0984132
\(245\) 2.74348 0.175274
\(246\) −30.1755 −1.92392
\(247\) 0 0
\(248\) −1.38498 −0.0879460
\(249\) −4.81179 −0.304935
\(250\) −6.78556 −0.429156
\(251\) 1.46460 0.0924449 0.0462224 0.998931i \(-0.485282\pi\)
0.0462224 + 0.998931i \(0.485282\pi\)
\(252\) 7.29887 0.459786
\(253\) −2.07477 −0.130440
\(254\) 13.1420 0.824605
\(255\) 46.0003 2.88065
\(256\) 1.00000 0.0625000
\(257\) 20.5292 1.28058 0.640289 0.768134i \(-0.278815\pi\)
0.640289 + 0.768134i \(0.278815\pi\)
\(258\) 8.20390 0.510753
\(259\) −9.44185 −0.586688
\(260\) −12.7353 −0.789807
\(261\) 42.5979 2.63674
\(262\) 18.3208 1.13186
\(263\) 17.3213 1.06808 0.534039 0.845460i \(-0.320673\pi\)
0.534039 + 0.845460i \(0.320673\pi\)
\(264\) −12.7931 −0.787359
\(265\) 24.9986 1.53565
\(266\) 0 0
\(267\) −28.1721 −1.72411
\(268\) −12.3091 −0.751898
\(269\) 11.0472 0.673562 0.336781 0.941583i \(-0.390662\pi\)
0.336781 + 0.941583i \(0.390662\pi\)
\(270\) 37.8486 2.30340
\(271\) 26.8694 1.63220 0.816101 0.577910i \(-0.196131\pi\)
0.816101 + 0.577910i \(0.196131\pi\)
\(272\) 5.22474 0.316796
\(273\) −14.8971 −0.901613
\(274\) −4.18816 −0.253016
\(275\) −10.0722 −0.607379
\(276\) 1.67027 0.100538
\(277\) −17.9579 −1.07899 −0.539493 0.841990i \(-0.681384\pi\)
−0.539493 + 0.841990i \(0.681384\pi\)
\(278\) −14.9473 −0.896478
\(279\) −10.1088 −0.605195
\(280\) 2.74348 0.163954
\(281\) −8.35573 −0.498461 −0.249231 0.968444i \(-0.580178\pi\)
−0.249231 + 0.968444i \(0.580178\pi\)
\(282\) 15.3078 0.911569
\(283\) 9.89060 0.587935 0.293968 0.955815i \(-0.405024\pi\)
0.293968 + 0.955815i \(0.405024\pi\)
\(284\) −4.00739 −0.237795
\(285\) 0 0
\(286\) 18.5049 1.09422
\(287\) −9.40284 −0.555032
\(288\) 7.29887 0.430090
\(289\) 10.2979 0.605758
\(290\) 16.0115 0.940230
\(291\) −14.8201 −0.868767
\(292\) −0.422816 −0.0247434
\(293\) 25.5978 1.49544 0.747719 0.664015i \(-0.231149\pi\)
0.747719 + 0.664015i \(0.231149\pi\)
\(294\) 3.20919 0.187164
\(295\) 2.95366 0.171969
\(296\) −9.44185 −0.548796
\(297\) −54.9958 −3.19118
\(298\) −0.0172226 −0.000997681 0
\(299\) −2.41600 −0.139721
\(300\) 8.10851 0.468145
\(301\) 2.55638 0.147347
\(302\) −6.94979 −0.399915
\(303\) 16.2813 0.935339
\(304\) 0 0
\(305\) −4.21744 −0.241490
\(306\) 38.1347 2.18002
\(307\) −9.29373 −0.530421 −0.265211 0.964190i \(-0.585442\pi\)
−0.265211 + 0.964190i \(0.585442\pi\)
\(308\) −3.98639 −0.227146
\(309\) 4.49324 0.255612
\(310\) −3.79964 −0.215805
\(311\) −7.03492 −0.398914 −0.199457 0.979907i \(-0.563918\pi\)
−0.199457 + 0.979907i \(0.563918\pi\)
\(312\) −14.8971 −0.843382
\(313\) 29.7865 1.68363 0.841817 0.539762i \(-0.181486\pi\)
0.841817 + 0.539762i \(0.181486\pi\)
\(314\) −20.9105 −1.18005
\(315\) 20.0243 1.12824
\(316\) −12.7880 −0.719381
\(317\) −18.9853 −1.06632 −0.533160 0.846015i \(-0.678995\pi\)
−0.533160 + 0.846015i \(0.678995\pi\)
\(318\) 29.2422 1.63982
\(319\) −23.2655 −1.30262
\(320\) 2.74348 0.153365
\(321\) 58.9035 3.28767
\(322\) 0.520464 0.0290043
\(323\) 0 0
\(324\) 22.3769 1.24316
\(325\) −11.7288 −0.650595
\(326\) 8.93937 0.495106
\(327\) 27.2491 1.50688
\(328\) −9.40284 −0.519185
\(329\) 4.77001 0.262979
\(330\) −35.0975 −1.93205
\(331\) 8.68202 0.477207 0.238603 0.971117i \(-0.423310\pi\)
0.238603 + 0.971117i \(0.423310\pi\)
\(332\) −1.49938 −0.0822892
\(333\) −68.9148 −3.77651
\(334\) 18.1411 0.992637
\(335\) −33.7697 −1.84504
\(336\) 3.20919 0.175075
\(337\) −16.7894 −0.914575 −0.457287 0.889319i \(-0.651179\pi\)
−0.457287 + 0.889319i \(0.651179\pi\)
\(338\) 8.54831 0.464967
\(339\) 25.8778 1.40549
\(340\) 14.3339 0.777367
\(341\) 5.52105 0.298982
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 2.55638 0.137831
\(345\) 4.58233 0.246704
\(346\) −1.97709 −0.106289
\(347\) 13.9880 0.750918 0.375459 0.926839i \(-0.377485\pi\)
0.375459 + 0.926839i \(0.377485\pi\)
\(348\) 18.7295 1.00401
\(349\) −17.7151 −0.948269 −0.474134 0.880453i \(-0.657239\pi\)
−0.474134 + 0.880453i \(0.657239\pi\)
\(350\) 2.52666 0.135055
\(351\) −64.0407 −3.41824
\(352\) −3.98639 −0.212475
\(353\) −7.84139 −0.417355 −0.208678 0.977984i \(-0.566916\pi\)
−0.208678 + 0.977984i \(0.566916\pi\)
\(354\) 3.45505 0.183634
\(355\) −10.9942 −0.583510
\(356\) −8.77859 −0.465265
\(357\) 16.7672 0.887412
\(358\) −19.4213 −1.02645
\(359\) 14.7718 0.779625 0.389813 0.920894i \(-0.372540\pi\)
0.389813 + 0.920894i \(0.372540\pi\)
\(360\) 20.0243 1.05537
\(361\) 0 0
\(362\) −2.54502 −0.133763
\(363\) 15.6971 0.823885
\(364\) −4.64202 −0.243308
\(365\) −1.15998 −0.0607164
\(366\) −4.93336 −0.257871
\(367\) 5.42648 0.283260 0.141630 0.989920i \(-0.454766\pi\)
0.141630 + 0.989920i \(0.454766\pi\)
\(368\) 0.520464 0.0271311
\(369\) −68.6301 −3.57274
\(370\) −25.9035 −1.34666
\(371\) 9.11202 0.473073
\(372\) −4.44464 −0.230444
\(373\) 30.1582 1.56153 0.780767 0.624822i \(-0.214828\pi\)
0.780767 + 0.624822i \(0.214828\pi\)
\(374\) −20.8278 −1.07698
\(375\) −21.7761 −1.12451
\(376\) 4.77001 0.245994
\(377\) −27.0919 −1.39530
\(378\) 13.7959 0.709583
\(379\) −22.1827 −1.13945 −0.569726 0.821835i \(-0.692951\pi\)
−0.569726 + 0.821835i \(0.692951\pi\)
\(380\) 0 0
\(381\) 42.1752 2.16070
\(382\) −18.1665 −0.929479
\(383\) 13.5571 0.692734 0.346367 0.938099i \(-0.387415\pi\)
0.346367 + 0.938099i \(0.387415\pi\)
\(384\) 3.20919 0.163768
\(385\) −10.9366 −0.557379
\(386\) −12.5306 −0.637790
\(387\) 18.6587 0.948475
\(388\) −4.61801 −0.234444
\(389\) 30.2938 1.53596 0.767979 0.640475i \(-0.221262\pi\)
0.767979 + 0.640475i \(0.221262\pi\)
\(390\) −40.8698 −2.06952
\(391\) 2.71929 0.137520
\(392\) 1.00000 0.0505076
\(393\) 58.7948 2.96580
\(394\) −11.0525 −0.556816
\(395\) −35.0835 −1.76524
\(396\) −29.0962 −1.46214
\(397\) −21.9430 −1.10129 −0.550644 0.834740i \(-0.685618\pi\)
−0.550644 + 0.834740i \(0.685618\pi\)
\(398\) −2.42328 −0.121468
\(399\) 0 0
\(400\) 2.52666 0.126333
\(401\) −31.7184 −1.58394 −0.791970 0.610560i \(-0.790945\pi\)
−0.791970 + 0.610560i \(0.790945\pi\)
\(402\) −39.5022 −1.97019
\(403\) 6.42908 0.320255
\(404\) 5.07336 0.252409
\(405\) 61.3905 3.05052
\(406\) 5.83623 0.289647
\(407\) 37.6389 1.86569
\(408\) 16.7672 0.830098
\(409\) 12.3597 0.611146 0.305573 0.952169i \(-0.401152\pi\)
0.305573 + 0.952169i \(0.401152\pi\)
\(410\) −25.7965 −1.27400
\(411\) −13.4406 −0.662974
\(412\) 1.40012 0.0689790
\(413\) 1.07661 0.0529766
\(414\) 3.79880 0.186701
\(415\) −4.11352 −0.201925
\(416\) −4.64202 −0.227594
\(417\) −47.9686 −2.34903
\(418\) 0 0
\(419\) 19.6277 0.958875 0.479438 0.877576i \(-0.340841\pi\)
0.479438 + 0.877576i \(0.340841\pi\)
\(420\) 8.80432 0.429607
\(421\) −17.1735 −0.836987 −0.418493 0.908220i \(-0.637442\pi\)
−0.418493 + 0.908220i \(0.637442\pi\)
\(422\) −3.19423 −0.155493
\(423\) 34.8157 1.69280
\(424\) 9.11202 0.442519
\(425\) 13.2011 0.640348
\(426\) −12.8605 −0.623091
\(427\) −1.53726 −0.0743934
\(428\) 18.3546 0.887205
\(429\) 59.3856 2.86716
\(430\) 7.01337 0.338215
\(431\) −32.3852 −1.55994 −0.779969 0.625818i \(-0.784765\pi\)
−0.779969 + 0.625818i \(0.784765\pi\)
\(432\) 13.7959 0.663755
\(433\) −36.9496 −1.77568 −0.887842 0.460148i \(-0.847797\pi\)
−0.887842 + 0.460148i \(0.847797\pi\)
\(434\) −1.38498 −0.0664809
\(435\) 51.3840 2.46367
\(436\) 8.49097 0.406643
\(437\) 0 0
\(438\) −1.35689 −0.0648349
\(439\) 17.0885 0.815588 0.407794 0.913074i \(-0.366298\pi\)
0.407794 + 0.913074i \(0.366298\pi\)
\(440\) −10.9366 −0.521380
\(441\) 7.29887 0.347565
\(442\) −24.2533 −1.15361
\(443\) 12.9797 0.616682 0.308341 0.951276i \(-0.400226\pi\)
0.308341 + 0.951276i \(0.400226\pi\)
\(444\) −30.3006 −1.43800
\(445\) −24.0839 −1.14168
\(446\) 7.11759 0.337027
\(447\) −0.0552706 −0.00261421
\(448\) 1.00000 0.0472456
\(449\) −4.13018 −0.194915 −0.0974575 0.995240i \(-0.531071\pi\)
−0.0974575 + 0.995240i \(0.531071\pi\)
\(450\) 18.4417 0.869352
\(451\) 37.4834 1.76502
\(452\) 8.06367 0.379283
\(453\) −22.3032 −1.04789
\(454\) −15.0860 −0.708023
\(455\) −12.7353 −0.597038
\(456\) 0 0
\(457\) 32.4479 1.51785 0.758925 0.651178i \(-0.225725\pi\)
0.758925 + 0.651178i \(0.225725\pi\)
\(458\) 0.806298 0.0376759
\(459\) 72.0799 3.36440
\(460\) 1.42788 0.0665752
\(461\) 19.0076 0.885272 0.442636 0.896701i \(-0.354043\pi\)
0.442636 + 0.896701i \(0.354043\pi\)
\(462\) −12.7931 −0.595187
\(463\) 11.7900 0.547930 0.273965 0.961740i \(-0.411665\pi\)
0.273965 + 0.961740i \(0.411665\pi\)
\(464\) 5.83623 0.270940
\(465\) −12.1938 −0.565472
\(466\) −9.43769 −0.437193
\(467\) −31.0653 −1.43753 −0.718766 0.695252i \(-0.755293\pi\)
−0.718766 + 0.695252i \(0.755293\pi\)
\(468\) −33.8815 −1.56617
\(469\) −12.3091 −0.568381
\(470\) 13.0864 0.603631
\(471\) −67.1056 −3.09206
\(472\) 1.07661 0.0495551
\(473\) −10.1907 −0.468571
\(474\) −41.0390 −1.88499
\(475\) 0 0
\(476\) 5.22474 0.239475
\(477\) 66.5075 3.04517
\(478\) −1.94285 −0.0888638
\(479\) −14.0514 −0.642026 −0.321013 0.947075i \(-0.604023\pi\)
−0.321013 + 0.947075i \(0.604023\pi\)
\(480\) 8.80432 0.401860
\(481\) 43.8292 1.99844
\(482\) −23.4255 −1.06700
\(483\) 1.67027 0.0759997
\(484\) 4.89131 0.222332
\(485\) −12.6694 −0.575288
\(486\) 30.4241 1.38006
\(487\) 10.4919 0.475431 0.237716 0.971335i \(-0.423601\pi\)
0.237716 + 0.971335i \(0.423601\pi\)
\(488\) −1.53726 −0.0695886
\(489\) 28.6881 1.29732
\(490\) 2.74348 0.123938
\(491\) −1.39469 −0.0629415 −0.0314707 0.999505i \(-0.510019\pi\)
−0.0314707 + 0.999505i \(0.510019\pi\)
\(492\) −30.1755 −1.36042
\(493\) 30.4928 1.37332
\(494\) 0 0
\(495\) −79.8246 −3.58785
\(496\) −1.38498 −0.0621872
\(497\) −4.00739 −0.179756
\(498\) −4.81179 −0.215622
\(499\) 20.9626 0.938417 0.469209 0.883087i \(-0.344539\pi\)
0.469209 + 0.883087i \(0.344539\pi\)
\(500\) −6.78556 −0.303459
\(501\) 58.2182 2.60100
\(502\) 1.46460 0.0653684
\(503\) 24.5284 1.09367 0.546833 0.837242i \(-0.315833\pi\)
0.546833 + 0.837242i \(0.315833\pi\)
\(504\) 7.29887 0.325118
\(505\) 13.9186 0.619371
\(506\) −2.07477 −0.0922349
\(507\) 27.4331 1.21835
\(508\) 13.1420 0.583084
\(509\) 9.65017 0.427736 0.213868 0.976863i \(-0.431394\pi\)
0.213868 + 0.976863i \(0.431394\pi\)
\(510\) 46.0003 2.03693
\(511\) −0.422816 −0.0187043
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 20.5292 0.905505
\(515\) 3.84119 0.169263
\(516\) 8.20390 0.361157
\(517\) −19.0151 −0.836284
\(518\) −9.44185 −0.414851
\(519\) −6.34485 −0.278508
\(520\) −12.7353 −0.558478
\(521\) 22.4183 0.982164 0.491082 0.871113i \(-0.336602\pi\)
0.491082 + 0.871113i \(0.336602\pi\)
\(522\) 42.5979 1.86446
\(523\) −38.7427 −1.69410 −0.847050 0.531514i \(-0.821623\pi\)
−0.847050 + 0.531514i \(0.821623\pi\)
\(524\) 18.3208 0.800347
\(525\) 8.10851 0.353884
\(526\) 17.3213 0.755246
\(527\) −7.23613 −0.315211
\(528\) −12.7931 −0.556747
\(529\) −22.7291 −0.988222
\(530\) 24.9986 1.08587
\(531\) 7.85806 0.341011
\(532\) 0 0
\(533\) 43.6481 1.89061
\(534\) −28.1721 −1.21913
\(535\) 50.3555 2.17706
\(536\) −12.3091 −0.531672
\(537\) −62.3264 −2.68958
\(538\) 11.0472 0.476280
\(539\) −3.98639 −0.171706
\(540\) 37.8486 1.62875
\(541\) −11.0477 −0.474979 −0.237489 0.971390i \(-0.576324\pi\)
−0.237489 + 0.971390i \(0.576324\pi\)
\(542\) 26.8694 1.15414
\(543\) −8.16743 −0.350498
\(544\) 5.22474 0.224009
\(545\) 23.2948 0.997837
\(546\) −14.8971 −0.637537
\(547\) 32.3458 1.38301 0.691503 0.722374i \(-0.256949\pi\)
0.691503 + 0.722374i \(0.256949\pi\)
\(548\) −4.18816 −0.178909
\(549\) −11.2203 −0.478870
\(550\) −10.0722 −0.429482
\(551\) 0 0
\(552\) 1.67027 0.0710912
\(553\) −12.7880 −0.543801
\(554\) −17.9579 −0.762958
\(555\) −83.1290 −3.52863
\(556\) −14.9473 −0.633905
\(557\) −3.95763 −0.167690 −0.0838451 0.996479i \(-0.526720\pi\)
−0.0838451 + 0.996479i \(0.526720\pi\)
\(558\) −10.1088 −0.427938
\(559\) −11.8668 −0.501911
\(560\) 2.74348 0.115933
\(561\) −66.8404 −2.82201
\(562\) −8.35573 −0.352465
\(563\) 10.8437 0.457006 0.228503 0.973543i \(-0.426617\pi\)
0.228503 + 0.973543i \(0.426617\pi\)
\(564\) 15.3078 0.644576
\(565\) 22.1225 0.930700
\(566\) 9.89060 0.415733
\(567\) 22.3769 0.939742
\(568\) −4.00739 −0.168146
\(569\) −35.0709 −1.47025 −0.735123 0.677933i \(-0.762876\pi\)
−0.735123 + 0.677933i \(0.762876\pi\)
\(570\) 0 0
\(571\) 4.49506 0.188112 0.0940561 0.995567i \(-0.470017\pi\)
0.0940561 + 0.995567i \(0.470017\pi\)
\(572\) 18.5049 0.773728
\(573\) −58.2996 −2.43550
\(574\) −9.40284 −0.392467
\(575\) 1.31503 0.0548407
\(576\) 7.29887 0.304120
\(577\) −33.3015 −1.38636 −0.693179 0.720765i \(-0.743791\pi\)
−0.693179 + 0.720765i \(0.743791\pi\)
\(578\) 10.2979 0.428336
\(579\) −40.2130 −1.67119
\(580\) 16.0115 0.664843
\(581\) −1.49938 −0.0622048
\(582\) −14.8201 −0.614311
\(583\) −36.3241 −1.50439
\(584\) −0.422816 −0.0174962
\(585\) −92.9530 −3.84313
\(586\) 25.5978 1.05743
\(587\) 3.00415 0.123995 0.0619973 0.998076i \(-0.480253\pi\)
0.0619973 + 0.998076i \(0.480253\pi\)
\(588\) 3.20919 0.132345
\(589\) 0 0
\(590\) 2.95366 0.121600
\(591\) −35.4694 −1.45902
\(592\) −9.44185 −0.388057
\(593\) 12.4444 0.511030 0.255515 0.966805i \(-0.417755\pi\)
0.255515 + 0.966805i \(0.417755\pi\)
\(594\) −54.9958 −2.25650
\(595\) 14.3339 0.587634
\(596\) −0.0172226 −0.000705467 0
\(597\) −7.77677 −0.318282
\(598\) −2.41600 −0.0987976
\(599\) 34.0809 1.39251 0.696255 0.717795i \(-0.254848\pi\)
0.696255 + 0.717795i \(0.254848\pi\)
\(600\) 8.10851 0.331028
\(601\) 13.3248 0.543529 0.271764 0.962364i \(-0.412393\pi\)
0.271764 + 0.962364i \(0.412393\pi\)
\(602\) 2.55638 0.104190
\(603\) −89.8425 −3.65867
\(604\) −6.94979 −0.282783
\(605\) 13.4192 0.545568
\(606\) 16.2813 0.661385
\(607\) 42.2842 1.71626 0.858132 0.513430i \(-0.171625\pi\)
0.858132 + 0.513430i \(0.171625\pi\)
\(608\) 0 0
\(609\) 18.7295 0.758959
\(610\) −4.21744 −0.170759
\(611\) −22.1425 −0.895788
\(612\) 38.1347 1.54150
\(613\) 23.4473 0.947027 0.473513 0.880787i \(-0.342986\pi\)
0.473513 + 0.880787i \(0.342986\pi\)
\(614\) −9.29373 −0.375065
\(615\) −82.7856 −3.33824
\(616\) −3.98639 −0.160616
\(617\) 21.3865 0.860989 0.430495 0.902593i \(-0.358339\pi\)
0.430495 + 0.902593i \(0.358339\pi\)
\(618\) 4.49324 0.180745
\(619\) 21.3208 0.856954 0.428477 0.903553i \(-0.359050\pi\)
0.428477 + 0.903553i \(0.359050\pi\)
\(620\) −3.79964 −0.152597
\(621\) 7.18026 0.288134
\(622\) −7.03492 −0.282075
\(623\) −8.77859 −0.351707
\(624\) −14.8971 −0.596361
\(625\) −31.2493 −1.24997
\(626\) 29.7865 1.19051
\(627\) 0 0
\(628\) −20.9105 −0.834419
\(629\) −49.3312 −1.96696
\(630\) 20.0243 0.797786
\(631\) 14.5419 0.578903 0.289452 0.957193i \(-0.406527\pi\)
0.289452 + 0.957193i \(0.406527\pi\)
\(632\) −12.7880 −0.508679
\(633\) −10.2509 −0.407436
\(634\) −18.9853 −0.754002
\(635\) 36.0549 1.43079
\(636\) 29.2422 1.15953
\(637\) −4.64202 −0.183923
\(638\) −23.2655 −0.921089
\(639\) −29.2494 −1.15709
\(640\) 2.74348 0.108445
\(641\) 6.21640 0.245533 0.122767 0.992436i \(-0.460823\pi\)
0.122767 + 0.992436i \(0.460823\pi\)
\(642\) 58.9035 2.32473
\(643\) 38.8654 1.53270 0.766350 0.642423i \(-0.222071\pi\)
0.766350 + 0.642423i \(0.222071\pi\)
\(644\) 0.520464 0.0205091
\(645\) 22.5072 0.886220
\(646\) 0 0
\(647\) −26.3165 −1.03461 −0.517303 0.855802i \(-0.673064\pi\)
−0.517303 + 0.855802i \(0.673064\pi\)
\(648\) 22.3769 0.879048
\(649\) −4.29180 −0.168468
\(650\) −11.7288 −0.460040
\(651\) −4.44464 −0.174199
\(652\) 8.93937 0.350093
\(653\) 21.6104 0.845679 0.422840 0.906205i \(-0.361033\pi\)
0.422840 + 0.906205i \(0.361033\pi\)
\(654\) 27.2491 1.06552
\(655\) 50.2626 1.96392
\(656\) −9.40284 −0.367119
\(657\) −3.08608 −0.120399
\(658\) 4.77001 0.185954
\(659\) −6.45405 −0.251414 −0.125707 0.992067i \(-0.540120\pi\)
−0.125707 + 0.992067i \(0.540120\pi\)
\(660\) −35.0975 −1.36617
\(661\) −6.13106 −0.238470 −0.119235 0.992866i \(-0.538044\pi\)
−0.119235 + 0.992866i \(0.538044\pi\)
\(662\) 8.68202 0.337436
\(663\) −77.8334 −3.02280
\(664\) −1.49938 −0.0581873
\(665\) 0 0
\(666\) −68.9148 −2.67040
\(667\) 3.03755 0.117614
\(668\) 18.1411 0.701901
\(669\) 22.8417 0.883109
\(670\) −33.7697 −1.30464
\(671\) 6.12813 0.236574
\(672\) 3.20919 0.123797
\(673\) −9.63805 −0.371520 −0.185760 0.982595i \(-0.559475\pi\)
−0.185760 + 0.982595i \(0.559475\pi\)
\(674\) −16.7894 −0.646702
\(675\) 34.8574 1.34166
\(676\) 8.54831 0.328781
\(677\) −1.71964 −0.0660909 −0.0330455 0.999454i \(-0.510521\pi\)
−0.0330455 + 0.999454i \(0.510521\pi\)
\(678\) 25.8778 0.993832
\(679\) −4.61801 −0.177223
\(680\) 14.3339 0.549681
\(681\) −48.4139 −1.85523
\(682\) 5.52105 0.211412
\(683\) −9.63395 −0.368633 −0.184317 0.982867i \(-0.559007\pi\)
−0.184317 + 0.982867i \(0.559007\pi\)
\(684\) 0 0
\(685\) −11.4901 −0.439014
\(686\) 1.00000 0.0381802
\(687\) 2.58756 0.0987216
\(688\) 2.55638 0.0974611
\(689\) −42.2981 −1.61143
\(690\) 4.58233 0.174446
\(691\) 38.5682 1.46720 0.733602 0.679579i \(-0.237838\pi\)
0.733602 + 0.679579i \(0.237838\pi\)
\(692\) −1.97709 −0.0751577
\(693\) −29.0962 −1.10527
\(694\) 13.9880 0.530979
\(695\) −41.0075 −1.55550
\(696\) 18.7295 0.709941
\(697\) −49.1274 −1.86083
\(698\) −17.7151 −0.670527
\(699\) −30.2873 −1.14557
\(700\) 2.52666 0.0954986
\(701\) −23.8793 −0.901910 −0.450955 0.892547i \(-0.648916\pi\)
−0.450955 + 0.892547i \(0.648916\pi\)
\(702\) −64.0407 −2.41706
\(703\) 0 0
\(704\) −3.98639 −0.150243
\(705\) 41.9967 1.58169
\(706\) −7.84139 −0.295115
\(707\) 5.07336 0.190803
\(708\) 3.45505 0.129849
\(709\) −17.7638 −0.667134 −0.333567 0.942726i \(-0.608252\pi\)
−0.333567 + 0.942726i \(0.608252\pi\)
\(710\) −10.9942 −0.412604
\(711\) −93.3379 −3.50045
\(712\) −8.77859 −0.328992
\(713\) −0.720830 −0.0269953
\(714\) 16.7672 0.627495
\(715\) 50.7677 1.89860
\(716\) −19.4213 −0.725807
\(717\) −6.23496 −0.232849
\(718\) 14.7718 0.551278
\(719\) −6.30664 −0.235198 −0.117599 0.993061i \(-0.537520\pi\)
−0.117599 + 0.993061i \(0.537520\pi\)
\(720\) 20.0243 0.746261
\(721\) 1.40012 0.0521432
\(722\) 0 0
\(723\) −75.1767 −2.79585
\(724\) −2.54502 −0.0945848
\(725\) 14.7461 0.547658
\(726\) 15.6971 0.582575
\(727\) −11.7046 −0.434101 −0.217050 0.976160i \(-0.569644\pi\)
−0.217050 + 0.976160i \(0.569644\pi\)
\(728\) −4.64202 −0.172045
\(729\) 30.5057 1.12984
\(730\) −1.15998 −0.0429330
\(731\) 13.3564 0.494005
\(732\) −4.93336 −0.182342
\(733\) 16.8530 0.622479 0.311239 0.950332i \(-0.399256\pi\)
0.311239 + 0.950332i \(0.399256\pi\)
\(734\) 5.42648 0.200295
\(735\) 8.80432 0.324752
\(736\) 0.520464 0.0191846
\(737\) 49.0689 1.80747
\(738\) −68.6301 −2.52631
\(739\) 33.5157 1.23289 0.616447 0.787396i \(-0.288571\pi\)
0.616447 + 0.787396i \(0.288571\pi\)
\(740\) −25.9035 −0.952230
\(741\) 0 0
\(742\) 9.11202 0.334513
\(743\) −21.1455 −0.775754 −0.387877 0.921711i \(-0.626792\pi\)
−0.387877 + 0.921711i \(0.626792\pi\)
\(744\) −4.44464 −0.162948
\(745\) −0.0472499 −0.00173110
\(746\) 30.1582 1.10417
\(747\) −10.9438 −0.400412
\(748\) −20.8278 −0.761542
\(749\) 18.3546 0.670664
\(750\) −21.7761 −0.795151
\(751\) 9.61118 0.350717 0.175359 0.984505i \(-0.443892\pi\)
0.175359 + 0.984505i \(0.443892\pi\)
\(752\) 4.77001 0.173944
\(753\) 4.70018 0.171284
\(754\) −27.0919 −0.986627
\(755\) −19.0666 −0.693903
\(756\) 13.7959 0.501751
\(757\) 33.2945 1.21011 0.605054 0.796185i \(-0.293152\pi\)
0.605054 + 0.796185i \(0.293152\pi\)
\(758\) −22.1827 −0.805714
\(759\) −6.65833 −0.241682
\(760\) 0 0
\(761\) −26.5600 −0.962801 −0.481400 0.876501i \(-0.659872\pi\)
−0.481400 + 0.876501i \(0.659872\pi\)
\(762\) 42.1752 1.52785
\(763\) 8.49097 0.307394
\(764\) −18.1665 −0.657241
\(765\) 104.622 3.78260
\(766\) 13.5571 0.489837
\(767\) −4.99765 −0.180455
\(768\) 3.20919 0.115802
\(769\) 6.41593 0.231364 0.115682 0.993286i \(-0.463095\pi\)
0.115682 + 0.993286i \(0.463095\pi\)
\(770\) −10.9366 −0.394126
\(771\) 65.8821 2.37268
\(772\) −12.5306 −0.450986
\(773\) 32.9261 1.18427 0.592134 0.805839i \(-0.298286\pi\)
0.592134 + 0.805839i \(0.298286\pi\)
\(774\) 18.6587 0.670673
\(775\) −3.49935 −0.125701
\(776\) −4.61801 −0.165777
\(777\) −30.3006 −1.08703
\(778\) 30.2938 1.08609
\(779\) 0 0
\(780\) −40.8698 −1.46337
\(781\) 15.9750 0.571631
\(782\) 2.71929 0.0972415
\(783\) 80.5159 2.87740
\(784\) 1.00000 0.0357143
\(785\) −57.3674 −2.04753
\(786\) 58.7948 2.09714
\(787\) −46.7761 −1.66739 −0.833694 0.552227i \(-0.813778\pi\)
−0.833694 + 0.552227i \(0.813778\pi\)
\(788\) −11.0525 −0.393728
\(789\) 55.5873 1.97896
\(790\) −35.0835 −1.24822
\(791\) 8.06367 0.286711
\(792\) −29.0962 −1.03389
\(793\) 7.13600 0.253407
\(794\) −21.9430 −0.778728
\(795\) 80.2252 2.84529
\(796\) −2.42328 −0.0858910
\(797\) 52.2887 1.85216 0.926080 0.377327i \(-0.123157\pi\)
0.926080 + 0.377327i \(0.123157\pi\)
\(798\) 0 0
\(799\) 24.9220 0.881678
\(800\) 2.52666 0.0893308
\(801\) −64.0738 −2.26394
\(802\) −31.7184 −1.12002
\(803\) 1.68551 0.0594803
\(804\) −39.5022 −1.39313
\(805\) 1.42788 0.0503261
\(806\) 6.42908 0.226455
\(807\) 35.4527 1.24799
\(808\) 5.07336 0.178480
\(809\) 42.2566 1.48566 0.742832 0.669478i \(-0.233482\pi\)
0.742832 + 0.669478i \(0.233482\pi\)
\(810\) 61.3905 2.15704
\(811\) 14.5252 0.510050 0.255025 0.966934i \(-0.417916\pi\)
0.255025 + 0.966934i \(0.417916\pi\)
\(812\) 5.83623 0.204811
\(813\) 86.2290 3.02418
\(814\) 37.6389 1.31924
\(815\) 24.5249 0.859071
\(816\) 16.7672 0.586968
\(817\) 0 0
\(818\) 12.3597 0.432145
\(819\) −33.8815 −1.18391
\(820\) −25.7965 −0.900852
\(821\) 24.1732 0.843651 0.421826 0.906677i \(-0.361389\pi\)
0.421826 + 0.906677i \(0.361389\pi\)
\(822\) −13.4406 −0.468794
\(823\) −48.3154 −1.68417 −0.842084 0.539346i \(-0.818671\pi\)
−0.842084 + 0.539346i \(0.818671\pi\)
\(824\) 1.40012 0.0487755
\(825\) −32.3237 −1.12537
\(826\) 1.07661 0.0374601
\(827\) 17.5537 0.610404 0.305202 0.952288i \(-0.401276\pi\)
0.305202 + 0.952288i \(0.401276\pi\)
\(828\) 3.79880 0.132017
\(829\) 7.73032 0.268485 0.134242 0.990949i \(-0.457140\pi\)
0.134242 + 0.990949i \(0.457140\pi\)
\(830\) −4.11352 −0.142782
\(831\) −57.6303 −1.99917
\(832\) −4.64202 −0.160933
\(833\) 5.22474 0.181026
\(834\) −47.9686 −1.66102
\(835\) 49.7697 1.72235
\(836\) 0 0
\(837\) −19.1069 −0.660433
\(838\) 19.6277 0.678027
\(839\) 23.1192 0.798163 0.399081 0.916916i \(-0.369329\pi\)
0.399081 + 0.916916i \(0.369329\pi\)
\(840\) 8.80432 0.303778
\(841\) 5.06154 0.174536
\(842\) −17.1735 −0.591839
\(843\) −26.8151 −0.923561
\(844\) −3.19423 −0.109950
\(845\) 23.4521 0.806776
\(846\) 34.8157 1.19699
\(847\) 4.89131 0.168067
\(848\) 9.11202 0.312908
\(849\) 31.7408 1.08934
\(850\) 13.2011 0.452794
\(851\) −4.91414 −0.168455
\(852\) −12.8605 −0.440592
\(853\) −27.0971 −0.927787 −0.463894 0.885891i \(-0.653548\pi\)
−0.463894 + 0.885891i \(0.653548\pi\)
\(854\) −1.53726 −0.0526041
\(855\) 0 0
\(856\) 18.3546 0.627349
\(857\) 40.3472 1.37823 0.689117 0.724650i \(-0.257999\pi\)
0.689117 + 0.724650i \(0.257999\pi\)
\(858\) 59.3856 2.02739
\(859\) −35.7644 −1.22027 −0.610133 0.792299i \(-0.708884\pi\)
−0.610133 + 0.792299i \(0.708884\pi\)
\(860\) 7.01337 0.239154
\(861\) −30.1755 −1.02838
\(862\) −32.3852 −1.10304
\(863\) 11.3420 0.386085 0.193042 0.981190i \(-0.438165\pi\)
0.193042 + 0.981190i \(0.438165\pi\)
\(864\) 13.7959 0.469345
\(865\) −5.42410 −0.184425
\(866\) −36.9496 −1.25560
\(867\) 33.0478 1.12236
\(868\) −1.38498 −0.0470091
\(869\) 50.9779 1.72931
\(870\) 51.3840 1.74208
\(871\) 57.1390 1.93608
\(872\) 8.49097 0.287540
\(873\) −33.7063 −1.14078
\(874\) 0 0
\(875\) −6.78556 −0.229394
\(876\) −1.35689 −0.0458452
\(877\) 21.1454 0.714028 0.357014 0.934099i \(-0.383795\pi\)
0.357014 + 0.934099i \(0.383795\pi\)
\(878\) 17.0885 0.576708
\(879\) 82.1480 2.77078
\(880\) −10.9366 −0.368671
\(881\) 15.5245 0.523034 0.261517 0.965199i \(-0.415777\pi\)
0.261517 + 0.965199i \(0.415777\pi\)
\(882\) 7.29887 0.245766
\(883\) −5.25391 −0.176808 −0.0884040 0.996085i \(-0.528177\pi\)
−0.0884040 + 0.996085i \(0.528177\pi\)
\(884\) −24.2533 −0.815727
\(885\) 9.47884 0.318628
\(886\) 12.9797 0.436060
\(887\) −25.4039 −0.852979 −0.426489 0.904493i \(-0.640250\pi\)
−0.426489 + 0.904493i \(0.640250\pi\)
\(888\) −30.3006 −1.01682
\(889\) 13.1420 0.440770
\(890\) −24.0839 −0.807293
\(891\) −89.2031 −2.98842
\(892\) 7.11759 0.238314
\(893\) 0 0
\(894\) −0.0552706 −0.00184853
\(895\) −53.2818 −1.78101
\(896\) 1.00000 0.0334077
\(897\) −7.75340 −0.258878
\(898\) −4.13018 −0.137826
\(899\) −8.08303 −0.269584
\(900\) 18.4417 0.614724
\(901\) 47.6079 1.58605
\(902\) 37.4834 1.24806
\(903\) 8.20390 0.273009
\(904\) 8.06367 0.268194
\(905\) −6.98219 −0.232096
\(906\) −22.3032 −0.740973
\(907\) 9.07682 0.301391 0.150695 0.988580i \(-0.451849\pi\)
0.150695 + 0.988580i \(0.451849\pi\)
\(908\) −15.0860 −0.500648
\(909\) 37.0298 1.22820
\(910\) −12.7353 −0.422170
\(911\) −46.2125 −1.53109 −0.765544 0.643383i \(-0.777530\pi\)
−0.765544 + 0.643383i \(0.777530\pi\)
\(912\) 0 0
\(913\) 5.97712 0.197814
\(914\) 32.4479 1.07328
\(915\) −13.5346 −0.447439
\(916\) 0.806298 0.0266409
\(917\) 18.3208 0.605005
\(918\) 72.0799 2.37899
\(919\) 12.9003 0.425540 0.212770 0.977102i \(-0.431751\pi\)
0.212770 + 0.977102i \(0.431751\pi\)
\(920\) 1.42788 0.0470758
\(921\) −29.8253 −0.982778
\(922\) 19.0076 0.625982
\(923\) 18.6024 0.612304
\(924\) −12.7931 −0.420861
\(925\) −23.8563 −0.784390
\(926\) 11.7900 0.387445
\(927\) 10.2193 0.335646
\(928\) 5.83623 0.191584
\(929\) −2.93205 −0.0961975 −0.0480987 0.998843i \(-0.515316\pi\)
−0.0480987 + 0.998843i \(0.515316\pi\)
\(930\) −12.1938 −0.399849
\(931\) 0 0
\(932\) −9.43769 −0.309142
\(933\) −22.5764 −0.739117
\(934\) −31.0653 −1.01649
\(935\) −57.1407 −1.86870
\(936\) −33.8815 −1.10745
\(937\) 38.6251 1.26183 0.630914 0.775853i \(-0.282680\pi\)
0.630914 + 0.775853i \(0.282680\pi\)
\(938\) −12.3091 −0.401906
\(939\) 95.5905 3.11948
\(940\) 13.0864 0.426831
\(941\) 23.1348 0.754172 0.377086 0.926178i \(-0.376926\pi\)
0.377086 + 0.926178i \(0.376926\pi\)
\(942\) −67.1056 −2.18642
\(943\) −4.89384 −0.159365
\(944\) 1.07661 0.0350408
\(945\) 37.8486 1.23122
\(946\) −10.1907 −0.331329
\(947\) −43.9090 −1.42685 −0.713425 0.700732i \(-0.752857\pi\)
−0.713425 + 0.700732i \(0.752857\pi\)
\(948\) −41.0390 −1.33289
\(949\) 1.96272 0.0637125
\(950\) 0 0
\(951\) −60.9273 −1.97570
\(952\) 5.22474 0.169335
\(953\) 59.8811 1.93974 0.969869 0.243625i \(-0.0783366\pi\)
0.969869 + 0.243625i \(0.0783366\pi\)
\(954\) 66.5075 2.15326
\(955\) −49.8393 −1.61276
\(956\) −1.94285 −0.0628362
\(957\) −74.6632 −2.41352
\(958\) −14.0514 −0.453981
\(959\) −4.18816 −0.135243
\(960\) 8.80432 0.284158
\(961\) −29.0818 −0.938124
\(962\) 43.8292 1.41311
\(963\) 133.968 4.31706
\(964\) −23.4255 −0.754483
\(965\) −34.3773 −1.10665
\(966\) 1.67027 0.0537399
\(967\) 2.27299 0.0730943 0.0365472 0.999332i \(-0.488364\pi\)
0.0365472 + 0.999332i \(0.488364\pi\)
\(968\) 4.89131 0.157213
\(969\) 0 0
\(970\) −12.6694 −0.406790
\(971\) −37.9775 −1.21876 −0.609378 0.792880i \(-0.708581\pi\)
−0.609378 + 0.792880i \(0.708581\pi\)
\(972\) 30.4241 0.975852
\(973\) −14.9473 −0.479187
\(974\) 10.4919 0.336181
\(975\) −37.6398 −1.20544
\(976\) −1.53726 −0.0492066
\(977\) −7.14080 −0.228455 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(978\) 28.6881 0.917345
\(979\) 34.9949 1.11844
\(980\) 2.74348 0.0876371
\(981\) 61.9745 1.97869
\(982\) −1.39469 −0.0445064
\(983\) −0.297396 −0.00948547 −0.00474273 0.999989i \(-0.501510\pi\)
−0.00474273 + 0.999989i \(0.501510\pi\)
\(984\) −30.1755 −0.961959
\(985\) −30.3222 −0.966145
\(986\) 30.4928 0.971087
\(987\) 15.3078 0.487254
\(988\) 0 0
\(989\) 1.33050 0.0423076
\(990\) −79.8246 −2.53699
\(991\) −24.7749 −0.786999 −0.393500 0.919325i \(-0.628736\pi\)
−0.393500 + 0.919325i \(0.628736\pi\)
\(992\) −1.38498 −0.0439730
\(993\) 27.8622 0.884180
\(994\) −4.00739 −0.127107
\(995\) −6.64822 −0.210763
\(996\) −4.81179 −0.152467
\(997\) −29.9031 −0.947040 −0.473520 0.880783i \(-0.657017\pi\)
−0.473520 + 0.880783i \(0.657017\pi\)
\(998\) 20.9626 0.663561
\(999\) −130.259 −4.12120
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 5054.2.a.bm.1.12 12
19.2 odd 18 266.2.u.d.99.1 yes 24
19.10 odd 18 266.2.u.d.43.1 24
19.18 odd 2 5054.2.a.bl.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.u.d.43.1 24 19.10 odd 18
266.2.u.d.99.1 yes 24 19.2 odd 18
5054.2.a.bl.1.1 12 19.18 odd 2
5054.2.a.bm.1.12 12 1.1 even 1 trivial