Properties

Label 5915.2.a.l.1.2
Level $5915$
Weight $2$
Character 5915.1
Self dual yes
Analytic conductor $47.232$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5915.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.2315127956\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 5915.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} -1.00000 q^{5} +4.00000 q^{6} +1.00000 q^{7} +6.56155 q^{8} -0.561553 q^{9} +O(q^{10})\) \(q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} -1.00000 q^{5} +4.00000 q^{6} +1.00000 q^{7} +6.56155 q^{8} -0.561553 q^{9} -2.56155 q^{10} +1.56155 q^{11} +7.12311 q^{12} +2.56155 q^{14} -1.56155 q^{15} +7.68466 q^{16} -0.438447 q^{17} -1.43845 q^{18} +7.12311 q^{19} -4.56155 q^{20} +1.56155 q^{21} +4.00000 q^{22} +3.12311 q^{23} +10.2462 q^{24} +1.00000 q^{25} -5.56155 q^{27} +4.56155 q^{28} +6.68466 q^{29} -4.00000 q^{30} +6.56155 q^{32} +2.43845 q^{33} -1.12311 q^{34} -1.00000 q^{35} -2.56155 q^{36} -6.00000 q^{37} +18.2462 q^{38} -6.56155 q^{40} -5.12311 q^{41} +4.00000 q^{42} +0.876894 q^{43} +7.12311 q^{44} +0.561553 q^{45} +8.00000 q^{46} +8.68466 q^{47} +12.0000 q^{48} +1.00000 q^{49} +2.56155 q^{50} -0.684658 q^{51} -5.12311 q^{53} -14.2462 q^{54} -1.56155 q^{55} +6.56155 q^{56} +11.1231 q^{57} +17.1231 q^{58} +4.00000 q^{59} -7.12311 q^{60} +15.3693 q^{61} -0.561553 q^{63} +1.43845 q^{64} +6.24621 q^{66} -10.2462 q^{67} -2.00000 q^{68} +4.87689 q^{69} -2.56155 q^{70} -8.00000 q^{71} -3.68466 q^{72} +12.2462 q^{73} -15.3693 q^{74} +1.56155 q^{75} +32.4924 q^{76} +1.56155 q^{77} -2.43845 q^{79} -7.68466 q^{80} -7.00000 q^{81} -13.1231 q^{82} -4.00000 q^{83} +7.12311 q^{84} +0.438447 q^{85} +2.24621 q^{86} +10.4384 q^{87} +10.2462 q^{88} +1.12311 q^{89} +1.43845 q^{90} +14.2462 q^{92} +22.2462 q^{94} -7.12311 q^{95} +10.2462 q^{96} -5.80776 q^{97} +2.56155 q^{98} -0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} + 5q^{4} - 2q^{5} + 8q^{6} + 2q^{7} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} + 5q^{4} - 2q^{5} + 8q^{6} + 2q^{7} + 9q^{8} + 3q^{9} - q^{10} - q^{11} + 6q^{12} + q^{14} + q^{15} + 3q^{16} - 5q^{17} - 7q^{18} + 6q^{19} - 5q^{20} - q^{21} + 8q^{22} - 2q^{23} + 4q^{24} + 2q^{25} - 7q^{27} + 5q^{28} + q^{29} - 8q^{30} + 9q^{32} + 9q^{33} + 6q^{34} - 2q^{35} - q^{36} - 12q^{37} + 20q^{38} - 9q^{40} - 2q^{41} + 8q^{42} + 10q^{43} + 6q^{44} - 3q^{45} + 16q^{46} + 5q^{47} + 24q^{48} + 2q^{49} + q^{50} + 11q^{51} - 2q^{53} - 12q^{54} + q^{55} + 9q^{56} + 14q^{57} + 26q^{58} + 8q^{59} - 6q^{60} + 6q^{61} + 3q^{63} + 7q^{64} - 4q^{66} - 4q^{67} - 4q^{68} + 18q^{69} - q^{70} - 16q^{71} + 5q^{72} + 8q^{73} - 6q^{74} - q^{75} + 32q^{76} - q^{77} - 9q^{79} - 3q^{80} - 14q^{81} - 18q^{82} - 8q^{83} + 6q^{84} + 5q^{85} - 12q^{86} + 25q^{87} + 4q^{88} - 6q^{89} + 7q^{90} + 12q^{92} + 28q^{94} - 6q^{95} + 4q^{96} + 9q^{97} + q^{98} - 10q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 4.56155 2.28078
\(5\) −1.00000 −0.447214
\(6\) 4.00000 1.63299
\(7\) 1.00000 0.377964
\(8\) 6.56155 2.31986
\(9\) −0.561553 −0.187184
\(10\) −2.56155 −0.810034
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 7.12311 2.05626
\(13\) 0 0
\(14\) 2.56155 0.684604
\(15\) −1.56155 −0.403191
\(16\) 7.68466 1.92116
\(17\) −0.438447 −0.106339 −0.0531695 0.998586i \(-0.516932\pi\)
−0.0531695 + 0.998586i \(0.516932\pi\)
\(18\) −1.43845 −0.339045
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) −4.56155 −1.01999
\(21\) 1.56155 0.340759
\(22\) 4.00000 0.852803
\(23\) 3.12311 0.651213 0.325606 0.945505i \(-0.394432\pi\)
0.325606 + 0.945505i \(0.394432\pi\)
\(24\) 10.2462 2.09150
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 4.56155 0.862052
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) −4.00000 −0.730297
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 6.56155 1.15993
\(33\) 2.43845 0.424479
\(34\) −1.12311 −0.192611
\(35\) −1.00000 −0.169031
\(36\) −2.56155 −0.426925
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 18.2462 2.95993
\(39\) 0 0
\(40\) −6.56155 −1.03747
\(41\) −5.12311 −0.800095 −0.400047 0.916494i \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 4.00000 0.617213
\(43\) 0.876894 0.133725 0.0668626 0.997762i \(-0.478701\pi\)
0.0668626 + 0.997762i \(0.478701\pi\)
\(44\) 7.12311 1.07385
\(45\) 0.561553 0.0837114
\(46\) 8.00000 1.17954
\(47\) 8.68466 1.26679 0.633394 0.773830i \(-0.281661\pi\)
0.633394 + 0.773830i \(0.281661\pi\)
\(48\) 12.0000 1.73205
\(49\) 1.00000 0.142857
\(50\) 2.56155 0.362258
\(51\) −0.684658 −0.0958714
\(52\) 0 0
\(53\) −5.12311 −0.703713 −0.351856 0.936054i \(-0.614449\pi\)
−0.351856 + 0.936054i \(0.614449\pi\)
\(54\) −14.2462 −1.93866
\(55\) −1.56155 −0.210560
\(56\) 6.56155 0.876824
\(57\) 11.1231 1.47329
\(58\) 17.1231 2.24837
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −7.12311 −0.919589
\(61\) 15.3693 1.96784 0.983920 0.178611i \(-0.0571605\pi\)
0.983920 + 0.178611i \(0.0571605\pi\)
\(62\) 0 0
\(63\) −0.561553 −0.0707490
\(64\) 1.43845 0.179806
\(65\) 0 0
\(66\) 6.24621 0.768855
\(67\) −10.2462 −1.25177 −0.625887 0.779914i \(-0.715263\pi\)
−0.625887 + 0.779914i \(0.715263\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.87689 0.587109
\(70\) −2.56155 −0.306164
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −3.68466 −0.434241
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) −15.3693 −1.78665
\(75\) 1.56155 0.180313
\(76\) 32.4924 3.72714
\(77\) 1.56155 0.177955
\(78\) 0 0
\(79\) −2.43845 −0.274347 −0.137173 0.990547i \(-0.543802\pi\)
−0.137173 + 0.990547i \(0.543802\pi\)
\(80\) −7.68466 −0.859171
\(81\) −7.00000 −0.777778
\(82\) −13.1231 −1.44920
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 7.12311 0.777195
\(85\) 0.438447 0.0475563
\(86\) 2.24621 0.242215
\(87\) 10.4384 1.11912
\(88\) 10.2462 1.09225
\(89\) 1.12311 0.119049 0.0595245 0.998227i \(-0.481042\pi\)
0.0595245 + 0.998227i \(0.481042\pi\)
\(90\) 1.43845 0.151626
\(91\) 0 0
\(92\) 14.2462 1.48527
\(93\) 0 0
\(94\) 22.2462 2.29452
\(95\) −7.12311 −0.730815
\(96\) 10.2462 1.04575
\(97\) −5.80776 −0.589689 −0.294845 0.955545i \(-0.595268\pi\)
−0.294845 + 0.955545i \(0.595268\pi\)
\(98\) 2.56155 0.258756
\(99\) −0.876894 −0.0881312
\(100\) 4.56155 0.456155
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) −1.75379 −0.173651
\(103\) 5.56155 0.547996 0.273998 0.961730i \(-0.411654\pi\)
0.273998 + 0.961730i \(0.411654\pi\)
\(104\) 0 0
\(105\) −1.56155 −0.152392
\(106\) −13.1231 −1.27463
\(107\) 13.3693 1.29246 0.646230 0.763142i \(-0.276345\pi\)
0.646230 + 0.763142i \(0.276345\pi\)
\(108\) −25.3693 −2.44116
\(109\) −5.31534 −0.509117 −0.254559 0.967057i \(-0.581930\pi\)
−0.254559 + 0.967057i \(0.581930\pi\)
\(110\) −4.00000 −0.381385
\(111\) −9.36932 −0.889296
\(112\) 7.68466 0.726132
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 28.4924 2.66856
\(115\) −3.12311 −0.291231
\(116\) 30.4924 2.83115
\(117\) 0 0
\(118\) 10.2462 0.943240
\(119\) −0.438447 −0.0401924
\(120\) −10.2462 −0.935347
\(121\) −8.56155 −0.778323
\(122\) 39.3693 3.56433
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −1.43845 −0.128147
\(127\) −6.24621 −0.554262 −0.277131 0.960832i \(-0.589384\pi\)
−0.277131 + 0.960832i \(0.589384\pi\)
\(128\) −9.43845 −0.834249
\(129\) 1.36932 0.120562
\(130\) 0 0
\(131\) −0.876894 −0.0766146 −0.0383073 0.999266i \(-0.512197\pi\)
−0.0383073 + 0.999266i \(0.512197\pi\)
\(132\) 11.1231 0.968142
\(133\) 7.12311 0.617652
\(134\) −26.2462 −2.26733
\(135\) 5.56155 0.478662
\(136\) −2.87689 −0.246692
\(137\) 17.1231 1.46293 0.731463 0.681881i \(-0.238838\pi\)
0.731463 + 0.681881i \(0.238838\pi\)
\(138\) 12.4924 1.06343
\(139\) −15.1231 −1.28273 −0.641363 0.767238i \(-0.721631\pi\)
−0.641363 + 0.767238i \(0.721631\pi\)
\(140\) −4.56155 −0.385522
\(141\) 13.5616 1.14209
\(142\) −20.4924 −1.71969
\(143\) 0 0
\(144\) −4.31534 −0.359612
\(145\) −6.68466 −0.555131
\(146\) 31.3693 2.59614
\(147\) 1.56155 0.128795
\(148\) −27.3693 −2.24974
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 4.00000 0.326599
\(151\) 6.93087 0.564026 0.282013 0.959411i \(-0.408998\pi\)
0.282013 + 0.959411i \(0.408998\pi\)
\(152\) 46.7386 3.79100
\(153\) 0.246211 0.0199050
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 20.2462 1.61582 0.807912 0.589303i \(-0.200598\pi\)
0.807912 + 0.589303i \(0.200598\pi\)
\(158\) −6.24621 −0.496922
\(159\) −8.00000 −0.634441
\(160\) −6.56155 −0.518736
\(161\) 3.12311 0.246135
\(162\) −17.9309 −1.40878
\(163\) 7.12311 0.557925 0.278962 0.960302i \(-0.410010\pi\)
0.278962 + 0.960302i \(0.410010\pi\)
\(164\) −23.3693 −1.82484
\(165\) −2.43845 −0.189833
\(166\) −10.2462 −0.795260
\(167\) 6.93087 0.536327 0.268163 0.963373i \(-0.413583\pi\)
0.268163 + 0.963373i \(0.413583\pi\)
\(168\) 10.2462 0.790512
\(169\) 0 0
\(170\) 1.12311 0.0861383
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) −4.43845 −0.337449 −0.168724 0.985663i \(-0.553965\pi\)
−0.168724 + 0.985663i \(0.553965\pi\)
\(174\) 26.7386 2.02705
\(175\) 1.00000 0.0755929
\(176\) 12.0000 0.904534
\(177\) 6.24621 0.469494
\(178\) 2.87689 0.215632
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 2.56155 0.190927
\(181\) −17.6155 −1.30935 −0.654676 0.755910i \(-0.727195\pi\)
−0.654676 + 0.755910i \(0.727195\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 20.4924 1.51072
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) −0.684658 −0.0500672
\(188\) 39.6155 2.88926
\(189\) −5.56155 −0.404543
\(190\) −18.2462 −1.32372
\(191\) −13.5616 −0.981280 −0.490640 0.871363i \(-0.663237\pi\)
−0.490640 + 0.871363i \(0.663237\pi\)
\(192\) 2.24621 0.162106
\(193\) −19.3693 −1.39423 −0.697117 0.716957i \(-0.745534\pi\)
−0.697117 + 0.716957i \(0.745534\pi\)
\(194\) −14.8769 −1.06810
\(195\) 0 0
\(196\) 4.56155 0.325825
\(197\) −1.12311 −0.0800180 −0.0400090 0.999199i \(-0.512739\pi\)
−0.0400090 + 0.999199i \(0.512739\pi\)
\(198\) −2.24621 −0.159631
\(199\) −1.75379 −0.124323 −0.0621614 0.998066i \(-0.519799\pi\)
−0.0621614 + 0.998066i \(0.519799\pi\)
\(200\) 6.56155 0.463972
\(201\) −16.0000 −1.12855
\(202\) −41.6155 −2.92806
\(203\) 6.68466 0.469171
\(204\) −3.12311 −0.218661
\(205\) 5.12311 0.357813
\(206\) 14.2462 0.992581
\(207\) −1.75379 −0.121897
\(208\) 0 0
\(209\) 11.1231 0.769401
\(210\) −4.00000 −0.276026
\(211\) 14.0540 0.967516 0.483758 0.875202i \(-0.339272\pi\)
0.483758 + 0.875202i \(0.339272\pi\)
\(212\) −23.3693 −1.60501
\(213\) −12.4924 −0.855967
\(214\) 34.2462 2.34102
\(215\) −0.876894 −0.0598037
\(216\) −36.4924 −2.48299
\(217\) 0 0
\(218\) −13.6155 −0.922160
\(219\) 19.1231 1.29222
\(220\) −7.12311 −0.480240
\(221\) 0 0
\(222\) −24.0000 −1.61077
\(223\) 2.43845 0.163291 0.0816453 0.996661i \(-0.473983\pi\)
0.0816453 + 0.996661i \(0.473983\pi\)
\(224\) 6.56155 0.438412
\(225\) −0.561553 −0.0374369
\(226\) −35.8617 −2.38549
\(227\) −11.3153 −0.751026 −0.375513 0.926817i \(-0.622533\pi\)
−0.375513 + 0.926817i \(0.622533\pi\)
\(228\) 50.7386 3.36025
\(229\) −10.8769 −0.718765 −0.359383 0.933190i \(-0.617013\pi\)
−0.359383 + 0.933190i \(0.617013\pi\)
\(230\) −8.00000 −0.527504
\(231\) 2.43845 0.160438
\(232\) 43.8617 2.87966
\(233\) 5.12311 0.335626 0.167813 0.985819i \(-0.446330\pi\)
0.167813 + 0.985819i \(0.446330\pi\)
\(234\) 0 0
\(235\) −8.68466 −0.566525
\(236\) 18.2462 1.18773
\(237\) −3.80776 −0.247341
\(238\) −1.12311 −0.0728001
\(239\) −19.8078 −1.28126 −0.640629 0.767851i \(-0.721326\pi\)
−0.640629 + 0.767851i \(0.721326\pi\)
\(240\) −12.0000 −0.774597
\(241\) 4.24621 0.273523 0.136761 0.990604i \(-0.456331\pi\)
0.136761 + 0.990604i \(0.456331\pi\)
\(242\) −21.9309 −1.40977
\(243\) 5.75379 0.369106
\(244\) 70.1080 4.48820
\(245\) −1.00000 −0.0638877
\(246\) −20.4924 −1.30655
\(247\) 0 0
\(248\) 0 0
\(249\) −6.24621 −0.395838
\(250\) −2.56155 −0.162007
\(251\) −8.87689 −0.560305 −0.280152 0.959956i \(-0.590385\pi\)
−0.280152 + 0.959956i \(0.590385\pi\)
\(252\) −2.56155 −0.161363
\(253\) 4.87689 0.306608
\(254\) −16.0000 −1.00393
\(255\) 0.684658 0.0428750
\(256\) −27.0540 −1.69087
\(257\) −10.4924 −0.654499 −0.327250 0.944938i \(-0.606122\pi\)
−0.327250 + 0.944938i \(0.606122\pi\)
\(258\) 3.50758 0.218372
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −3.75379 −0.232354
\(262\) −2.24621 −0.138771
\(263\) −12.8769 −0.794023 −0.397012 0.917814i \(-0.629953\pi\)
−0.397012 + 0.917814i \(0.629953\pi\)
\(264\) 16.0000 0.984732
\(265\) 5.12311 0.314710
\(266\) 18.2462 1.11875
\(267\) 1.75379 0.107330
\(268\) −46.7386 −2.85502
\(269\) −20.7386 −1.26446 −0.632228 0.774782i \(-0.717860\pi\)
−0.632228 + 0.774782i \(0.717860\pi\)
\(270\) 14.2462 0.866997
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −3.36932 −0.204295
\(273\) 0 0
\(274\) 43.8617 2.64978
\(275\) 1.56155 0.0941652
\(276\) 22.2462 1.33906
\(277\) −0.246211 −0.0147934 −0.00739670 0.999973i \(-0.502354\pi\)
−0.00739670 + 0.999973i \(0.502354\pi\)
\(278\) −38.7386 −2.32339
\(279\) 0 0
\(280\) −6.56155 −0.392128
\(281\) −12.4384 −0.742016 −0.371008 0.928630i \(-0.620988\pi\)
−0.371008 + 0.928630i \(0.620988\pi\)
\(282\) 34.7386 2.06866
\(283\) −11.3153 −0.672627 −0.336314 0.941750i \(-0.609180\pi\)
−0.336314 + 0.941750i \(0.609180\pi\)
\(284\) −36.4924 −2.16543
\(285\) −11.1231 −0.658876
\(286\) 0 0
\(287\) −5.12311 −0.302407
\(288\) −3.68466 −0.217121
\(289\) −16.8078 −0.988692
\(290\) −17.1231 −1.00550
\(291\) −9.06913 −0.531642
\(292\) 55.8617 3.26906
\(293\) 2.68466 0.156839 0.0784197 0.996920i \(-0.475013\pi\)
0.0784197 + 0.996920i \(0.475013\pi\)
\(294\) 4.00000 0.233285
\(295\) −4.00000 −0.232889
\(296\) −39.3693 −2.28830
\(297\) −8.68466 −0.503935
\(298\) −31.3693 −1.81718
\(299\) 0 0
\(300\) 7.12311 0.411253
\(301\) 0.876894 0.0505434
\(302\) 17.7538 1.02162
\(303\) −25.3693 −1.45743
\(304\) 54.7386 3.13948
\(305\) −15.3693 −0.880045
\(306\) 0.630683 0.0360538
\(307\) 19.3153 1.10238 0.551192 0.834378i \(-0.314173\pi\)
0.551192 + 0.834378i \(0.314173\pi\)
\(308\) 7.12311 0.405877
\(309\) 8.68466 0.494053
\(310\) 0 0
\(311\) 31.6155 1.79275 0.896376 0.443294i \(-0.146190\pi\)
0.896376 + 0.443294i \(0.146190\pi\)
\(312\) 0 0
\(313\) −22.3002 −1.26048 −0.630241 0.776400i \(-0.717044\pi\)
−0.630241 + 0.776400i \(0.717044\pi\)
\(314\) 51.8617 2.92673
\(315\) 0.561553 0.0316399
\(316\) −11.1231 −0.625724
\(317\) −10.4924 −0.589313 −0.294657 0.955603i \(-0.595205\pi\)
−0.294657 + 0.955603i \(0.595205\pi\)
\(318\) −20.4924 −1.14916
\(319\) 10.4384 0.584441
\(320\) −1.43845 −0.0804116
\(321\) 20.8769 1.16523
\(322\) 8.00000 0.445823
\(323\) −3.12311 −0.173774
\(324\) −31.9309 −1.77394
\(325\) 0 0
\(326\) 18.2462 1.01056
\(327\) −8.30019 −0.459001
\(328\) −33.6155 −1.85611
\(329\) 8.68466 0.478801
\(330\) −6.24621 −0.343843
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −18.2462 −1.00139
\(333\) 3.36932 0.184637
\(334\) 17.7538 0.971444
\(335\) 10.2462 0.559810
\(336\) 12.0000 0.654654
\(337\) −1.50758 −0.0821230 −0.0410615 0.999157i \(-0.513074\pi\)
−0.0410615 + 0.999157i \(0.513074\pi\)
\(338\) 0 0
\(339\) −21.8617 −1.18737
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) −10.2462 −0.554052
\(343\) 1.00000 0.0539949
\(344\) 5.75379 0.310223
\(345\) −4.87689 −0.262563
\(346\) −11.3693 −0.611218
\(347\) 7.12311 0.382388 0.191194 0.981552i \(-0.438764\pi\)
0.191194 + 0.981552i \(0.438764\pi\)
\(348\) 47.6155 2.55246
\(349\) −10.4924 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(350\) 2.56155 0.136921
\(351\) 0 0
\(352\) 10.2462 0.546125
\(353\) −5.80776 −0.309116 −0.154558 0.987984i \(-0.549395\pi\)
−0.154558 + 0.987984i \(0.549395\pi\)
\(354\) 16.0000 0.850390
\(355\) 8.00000 0.424596
\(356\) 5.12311 0.271524
\(357\) −0.684658 −0.0362360
\(358\) 51.2311 2.70765
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 3.68466 0.194199
\(361\) 31.7386 1.67045
\(362\) −45.1231 −2.37162
\(363\) −13.3693 −0.701707
\(364\) 0 0
\(365\) −12.2462 −0.640996
\(366\) 61.4773 3.21347
\(367\) −8.68466 −0.453335 −0.226668 0.973972i \(-0.572783\pi\)
−0.226668 + 0.973972i \(0.572783\pi\)
\(368\) 24.0000 1.25109
\(369\) 2.87689 0.149765
\(370\) 15.3693 0.799013
\(371\) −5.12311 −0.265978
\(372\) 0 0
\(373\) 4.63068 0.239768 0.119884 0.992788i \(-0.461748\pi\)
0.119884 + 0.992788i \(0.461748\pi\)
\(374\) −1.75379 −0.0906863
\(375\) −1.56155 −0.0806382
\(376\) 56.9848 2.93877
\(377\) 0 0
\(378\) −14.2462 −0.732746
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) −32.4924 −1.66683
\(381\) −9.75379 −0.499702
\(382\) −34.7386 −1.77738
\(383\) −6.24621 −0.319166 −0.159583 0.987184i \(-0.551015\pi\)
−0.159583 + 0.987184i \(0.551015\pi\)
\(384\) −14.7386 −0.752128
\(385\) −1.56155 −0.0795841
\(386\) −49.6155 −2.52536
\(387\) −0.492423 −0.0250312
\(388\) −26.4924 −1.34495
\(389\) −24.9309 −1.26405 −0.632023 0.774950i \(-0.717775\pi\)
−0.632023 + 0.774950i \(0.717775\pi\)
\(390\) 0 0
\(391\) −1.36932 −0.0692493
\(392\) 6.56155 0.331408
\(393\) −1.36932 −0.0690729
\(394\) −2.87689 −0.144936
\(395\) 2.43845 0.122692
\(396\) −4.00000 −0.201008
\(397\) −27.5616 −1.38327 −0.691637 0.722245i \(-0.743110\pi\)
−0.691637 + 0.722245i \(0.743110\pi\)
\(398\) −4.49242 −0.225185
\(399\) 11.1231 0.556852
\(400\) 7.68466 0.384233
\(401\) −31.5616 −1.57611 −0.788054 0.615606i \(-0.788911\pi\)
−0.788054 + 0.615606i \(0.788911\pi\)
\(402\) −40.9848 −2.04414
\(403\) 0 0
\(404\) −74.1080 −3.68701
\(405\) 7.00000 0.347833
\(406\) 17.1231 0.849805
\(407\) −9.36932 −0.464420
\(408\) −4.49242 −0.222408
\(409\) −6.49242 −0.321030 −0.160515 0.987033i \(-0.551315\pi\)
−0.160515 + 0.987033i \(0.551315\pi\)
\(410\) 13.1231 0.648104
\(411\) 26.7386 1.31892
\(412\) 25.3693 1.24986
\(413\) 4.00000 0.196827
\(414\) −4.49242 −0.220791
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −23.6155 −1.15646
\(418\) 28.4924 1.39361
\(419\) 26.2462 1.28221 0.641106 0.767453i \(-0.278476\pi\)
0.641106 + 0.767453i \(0.278476\pi\)
\(420\) −7.12311 −0.347572
\(421\) 2.68466 0.130842 0.0654211 0.997858i \(-0.479161\pi\)
0.0654211 + 0.997858i \(0.479161\pi\)
\(422\) 36.0000 1.75245
\(423\) −4.87689 −0.237123
\(424\) −33.6155 −1.63251
\(425\) −0.438447 −0.0212678
\(426\) −32.0000 −1.55041
\(427\) 15.3693 0.743773
\(428\) 60.9848 2.94781
\(429\) 0 0
\(430\) −2.24621 −0.108322
\(431\) 19.8078 0.954106 0.477053 0.878874i \(-0.341705\pi\)
0.477053 + 0.878874i \(0.341705\pi\)
\(432\) −42.7386 −2.05626
\(433\) 8.24621 0.396288 0.198144 0.980173i \(-0.436509\pi\)
0.198144 + 0.980173i \(0.436509\pi\)
\(434\) 0 0
\(435\) −10.4384 −0.500485
\(436\) −24.2462 −1.16118
\(437\) 22.2462 1.06418
\(438\) 48.9848 2.34059
\(439\) 9.36932 0.447173 0.223587 0.974684i \(-0.428223\pi\)
0.223587 + 0.974684i \(0.428223\pi\)
\(440\) −10.2462 −0.488469
\(441\) −0.561553 −0.0267406
\(442\) 0 0
\(443\) −2.63068 −0.124988 −0.0624938 0.998045i \(-0.519905\pi\)
−0.0624938 + 0.998045i \(0.519905\pi\)
\(444\) −42.7386 −2.02829
\(445\) −1.12311 −0.0532403
\(446\) 6.24621 0.295767
\(447\) −19.1231 −0.904492
\(448\) 1.43845 0.0679602
\(449\) 1.80776 0.0853137 0.0426568 0.999090i \(-0.486418\pi\)
0.0426568 + 0.999090i \(0.486418\pi\)
\(450\) −1.43845 −0.0678091
\(451\) −8.00000 −0.376705
\(452\) −63.8617 −3.00380
\(453\) 10.8229 0.508505
\(454\) −28.9848 −1.36033
\(455\) 0 0
\(456\) 72.9848 3.41783
\(457\) 17.1231 0.800985 0.400493 0.916300i \(-0.368839\pi\)
0.400493 + 0.916300i \(0.368839\pi\)
\(458\) −27.8617 −1.30189
\(459\) 2.43845 0.113817
\(460\) −14.2462 −0.664233
\(461\) 13.1231 0.611204 0.305602 0.952159i \(-0.401142\pi\)
0.305602 + 0.952159i \(0.401142\pi\)
\(462\) 6.24621 0.290600
\(463\) −12.4924 −0.580572 −0.290286 0.956940i \(-0.593750\pi\)
−0.290286 + 0.956940i \(0.593750\pi\)
\(464\) 51.3693 2.38476
\(465\) 0 0
\(466\) 13.1231 0.607916
\(467\) 22.4384 1.03833 0.519164 0.854675i \(-0.326243\pi\)
0.519164 + 0.854675i \(0.326243\pi\)
\(468\) 0 0
\(469\) −10.2462 −0.473126
\(470\) −22.2462 −1.02614
\(471\) 31.6155 1.45677
\(472\) 26.2462 1.20808
\(473\) 1.36932 0.0629613
\(474\) −9.75379 −0.448006
\(475\) 7.12311 0.326831
\(476\) −2.00000 −0.0916698
\(477\) 2.87689 0.131724
\(478\) −50.7386 −2.32073
\(479\) −4.87689 −0.222831 −0.111415 0.993774i \(-0.535538\pi\)
−0.111415 + 0.993774i \(0.535538\pi\)
\(480\) −10.2462 −0.467673
\(481\) 0 0
\(482\) 10.8769 0.495429
\(483\) 4.87689 0.221906
\(484\) −39.0540 −1.77518
\(485\) 5.80776 0.263717
\(486\) 14.7386 0.668558
\(487\) 3.12311 0.141521 0.0707607 0.997493i \(-0.477457\pi\)
0.0707607 + 0.997493i \(0.477457\pi\)
\(488\) 100.847 4.56511
\(489\) 11.1231 0.503004
\(490\) −2.56155 −0.115719
\(491\) −41.1771 −1.85830 −0.929148 0.369708i \(-0.879458\pi\)
−0.929148 + 0.369708i \(0.879458\pi\)
\(492\) −36.4924 −1.64521
\(493\) −2.93087 −0.132000
\(494\) 0 0
\(495\) 0.876894 0.0394135
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) −16.0000 −0.716977
\(499\) −41.1771 −1.84334 −0.921670 0.387976i \(-0.873174\pi\)
−0.921670 + 0.387976i \(0.873174\pi\)
\(500\) −4.56155 −0.203999
\(501\) 10.8229 0.483532
\(502\) −22.7386 −1.01487
\(503\) 38.9309 1.73584 0.867921 0.496703i \(-0.165456\pi\)
0.867921 + 0.496703i \(0.165456\pi\)
\(504\) −3.68466 −0.164128
\(505\) 16.2462 0.722947
\(506\) 12.4924 0.555356
\(507\) 0 0
\(508\) −28.4924 −1.26415
\(509\) 11.7538 0.520978 0.260489 0.965477i \(-0.416116\pi\)
0.260489 + 0.965477i \(0.416116\pi\)
\(510\) 1.75379 0.0776591
\(511\) 12.2462 0.541740
\(512\) −50.4233 −2.22842
\(513\) −39.6155 −1.74907
\(514\) −26.8769 −1.18549
\(515\) −5.56155 −0.245071
\(516\) 6.24621 0.274974
\(517\) 13.5616 0.596436
\(518\) −15.3693 −0.675289
\(519\) −6.93087 −0.304231
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −9.61553 −0.420860
\(523\) 40.4924 1.77061 0.885305 0.465011i \(-0.153950\pi\)
0.885305 + 0.465011i \(0.153950\pi\)
\(524\) −4.00000 −0.174741
\(525\) 1.56155 0.0681518
\(526\) −32.9848 −1.43821
\(527\) 0 0
\(528\) 18.7386 0.815494
\(529\) −13.2462 −0.575922
\(530\) 13.1231 0.570031
\(531\) −2.24621 −0.0974773
\(532\) 32.4924 1.40873
\(533\) 0 0
\(534\) 4.49242 0.194406
\(535\) −13.3693 −0.578006
\(536\) −67.2311 −2.90394
\(537\) 31.2311 1.34772
\(538\) −53.1231 −2.29030
\(539\) 1.56155 0.0672608
\(540\) 25.3693 1.09172
\(541\) 37.8078 1.62548 0.812741 0.582625i \(-0.197974\pi\)
0.812741 + 0.582625i \(0.197974\pi\)
\(542\) 40.9848 1.76045
\(543\) −27.5076 −1.18046
\(544\) −2.87689 −0.123346
\(545\) 5.31534 0.227684
\(546\) 0 0
\(547\) −2.24621 −0.0960411 −0.0480205 0.998846i \(-0.515291\pi\)
−0.0480205 + 0.998846i \(0.515291\pi\)
\(548\) 78.1080 3.33661
\(549\) −8.63068 −0.368349
\(550\) 4.00000 0.170561
\(551\) 47.6155 2.02849
\(552\) 32.0000 1.36201
\(553\) −2.43845 −0.103693
\(554\) −0.630683 −0.0267952
\(555\) 9.36932 0.397705
\(556\) −68.9848 −2.92561
\(557\) 13.1231 0.556044 0.278022 0.960575i \(-0.410321\pi\)
0.278022 + 0.960575i \(0.410321\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −7.68466 −0.324736
\(561\) −1.06913 −0.0451387
\(562\) −31.8617 −1.34401
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 61.8617 2.60485
\(565\) 14.0000 0.588984
\(566\) −28.9848 −1.21832
\(567\) −7.00000 −0.293972
\(568\) −52.4924 −2.20253
\(569\) −30.9848 −1.29895 −0.649476 0.760382i \(-0.725012\pi\)
−0.649476 + 0.760382i \(0.725012\pi\)
\(570\) −28.4924 −1.19342
\(571\) 40.4924 1.69456 0.847278 0.531150i \(-0.178240\pi\)
0.847278 + 0.531150i \(0.178240\pi\)
\(572\) 0 0
\(573\) −21.1771 −0.884685
\(574\) −13.1231 −0.547748
\(575\) 3.12311 0.130243
\(576\) −0.807764 −0.0336568
\(577\) 24.0540 1.00138 0.500690 0.865627i \(-0.333080\pi\)
0.500690 + 0.865627i \(0.333080\pi\)
\(578\) −43.0540 −1.79081
\(579\) −30.2462 −1.25699
\(580\) −30.4924 −1.26613
\(581\) −4.00000 −0.165948
\(582\) −23.2311 −0.962958
\(583\) −8.00000 −0.331326
\(584\) 80.3542 3.32508
\(585\) 0 0
\(586\) 6.87689 0.284082
\(587\) 26.2462 1.08330 0.541649 0.840605i \(-0.317800\pi\)
0.541649 + 0.840605i \(0.317800\pi\)
\(588\) 7.12311 0.293752
\(589\) 0 0
\(590\) −10.2462 −0.421830
\(591\) −1.75379 −0.0721412
\(592\) −46.1080 −1.89503
\(593\) 27.5616 1.13182 0.565909 0.824468i \(-0.308525\pi\)
0.565909 + 0.824468i \(0.308525\pi\)
\(594\) −22.2462 −0.912773
\(595\) 0.438447 0.0179746
\(596\) −55.8617 −2.28819
\(597\) −2.73863 −0.112085
\(598\) 0 0
\(599\) −11.8078 −0.482452 −0.241226 0.970469i \(-0.577550\pi\)
−0.241226 + 0.970469i \(0.577550\pi\)
\(600\) 10.2462 0.418300
\(601\) 6.49242 0.264831 0.132416 0.991194i \(-0.457727\pi\)
0.132416 + 0.991194i \(0.457727\pi\)
\(602\) 2.24621 0.0915487
\(603\) 5.75379 0.234312
\(604\) 31.6155 1.28642
\(605\) 8.56155 0.348077
\(606\) −64.9848 −2.63983
\(607\) −42.0540 −1.70692 −0.853459 0.521160i \(-0.825500\pi\)
−0.853459 + 0.521160i \(0.825500\pi\)
\(608\) 46.7386 1.89550
\(609\) 10.4384 0.422987
\(610\) −39.3693 −1.59402
\(611\) 0 0
\(612\) 1.12311 0.0453989
\(613\) −40.7386 −1.64542 −0.822709 0.568463i \(-0.807538\pi\)
−0.822709 + 0.568463i \(0.807538\pi\)
\(614\) 49.4773 1.99674
\(615\) 8.00000 0.322591
\(616\) 10.2462 0.412832
\(617\) −32.2462 −1.29818 −0.649092 0.760710i \(-0.724851\pi\)
−0.649092 + 0.760710i \(0.724851\pi\)
\(618\) 22.2462 0.894874
\(619\) −32.1080 −1.29053 −0.645264 0.763960i \(-0.723253\pi\)
−0.645264 + 0.763960i \(0.723253\pi\)
\(620\) 0 0
\(621\) −17.3693 −0.697007
\(622\) 80.9848 3.24720
\(623\) 1.12311 0.0449963
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −57.1231 −2.28310
\(627\) 17.3693 0.693664
\(628\) 92.3542 3.68533
\(629\) 2.63068 0.104892
\(630\) 1.43845 0.0573091
\(631\) 11.8078 0.470060 0.235030 0.971988i \(-0.424481\pi\)
0.235030 + 0.971988i \(0.424481\pi\)
\(632\) −16.0000 −0.636446
\(633\) 21.9460 0.872276
\(634\) −26.8769 −1.06742
\(635\) 6.24621 0.247873
\(636\) −36.4924 −1.44702
\(637\) 0 0
\(638\) 26.7386 1.05859
\(639\) 4.49242 0.177717
\(640\) 9.43845 0.373087
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 53.4773 2.11058
\(643\) −1.56155 −0.0615816 −0.0307908 0.999526i \(-0.509803\pi\)
−0.0307908 + 0.999526i \(0.509803\pi\)
\(644\) 14.2462 0.561379
\(645\) −1.36932 −0.0539168
\(646\) −8.00000 −0.314756
\(647\) 36.4924 1.43467 0.717333 0.696731i \(-0.245363\pi\)
0.717333 + 0.696731i \(0.245363\pi\)
\(648\) −45.9309 −1.80433
\(649\) 6.24621 0.245185
\(650\) 0 0
\(651\) 0 0
\(652\) 32.4924 1.27250
\(653\) −33.2311 −1.30043 −0.650216 0.759750i \(-0.725322\pi\)
−0.650216 + 0.759750i \(0.725322\pi\)
\(654\) −21.2614 −0.831385
\(655\) 0.876894 0.0342631
\(656\) −39.3693 −1.53711
\(657\) −6.87689 −0.268293
\(658\) 22.2462 0.867248
\(659\) 9.17708 0.357488 0.178744 0.983896i \(-0.442797\pi\)
0.178744 + 0.983896i \(0.442797\pi\)
\(660\) −11.1231 −0.432966
\(661\) 5.12311 0.199266 0.0996329 0.995024i \(-0.468233\pi\)
0.0996329 + 0.995024i \(0.468233\pi\)
\(662\) −30.7386 −1.19469
\(663\) 0 0
\(664\) −26.2462 −1.01855
\(665\) −7.12311 −0.276222
\(666\) 8.63068 0.334432
\(667\) 20.8769 0.808357
\(668\) 31.6155 1.22324
\(669\) 3.80776 0.147217
\(670\) 26.2462 1.01398
\(671\) 24.0000 0.926510
\(672\) 10.2462 0.395256
\(673\) 31.8617 1.22818 0.614090 0.789236i \(-0.289523\pi\)
0.614090 + 0.789236i \(0.289523\pi\)
\(674\) −3.86174 −0.148749
\(675\) −5.56155 −0.214064
\(676\) 0 0
\(677\) 4.93087 0.189509 0.0947544 0.995501i \(-0.469793\pi\)
0.0947544 + 0.995501i \(0.469793\pi\)
\(678\) −56.0000 −2.15067
\(679\) −5.80776 −0.222882
\(680\) 2.87689 0.110324
\(681\) −17.6695 −0.677097
\(682\) 0 0
\(683\) 6.73863 0.257847 0.128923 0.991655i \(-0.458848\pi\)
0.128923 + 0.991655i \(0.458848\pi\)
\(684\) −18.2462 −0.697661
\(685\) −17.1231 −0.654240
\(686\) 2.56155 0.0978005
\(687\) −16.9848 −0.648012
\(688\) 6.73863 0.256908
\(689\) 0 0
\(690\) −12.4924 −0.475578
\(691\) 24.4924 0.931736 0.465868 0.884854i \(-0.345742\pi\)
0.465868 + 0.884854i \(0.345742\pi\)
\(692\) −20.2462 −0.769645
\(693\) −0.876894 −0.0333105
\(694\) 18.2462 0.692617
\(695\) 15.1231 0.573652
\(696\) 68.4924 2.59620
\(697\) 2.24621 0.0850813
\(698\) −26.8769 −1.01731
\(699\) 8.00000 0.302588
\(700\) 4.56155 0.172410
\(701\) 28.9309 1.09270 0.546352 0.837556i \(-0.316016\pi\)
0.546352 + 0.837556i \(0.316016\pi\)
\(702\) 0 0
\(703\) −42.7386 −1.61192
\(704\) 2.24621 0.0846573
\(705\) −13.5616 −0.510758
\(706\) −14.8769 −0.559899
\(707\) −16.2462 −0.611002
\(708\) 28.4924 1.07081
\(709\) −27.1771 −1.02066 −0.510328 0.859980i \(-0.670476\pi\)
−0.510328 + 0.859980i \(0.670476\pi\)
\(710\) 20.4924 0.769067
\(711\) 1.36932 0.0513534
\(712\) 7.36932 0.276177
\(713\) 0 0
\(714\) −1.75379 −0.0656339
\(715\) 0 0
\(716\) 91.2311 3.40946
\(717\) −30.9309 −1.15513
\(718\) −20.4924 −0.764770
\(719\) 8.38447 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(720\) 4.31534 0.160823
\(721\) 5.56155 0.207123
\(722\) 81.3002 3.02568
\(723\) 6.63068 0.246598
\(724\) −80.3542 −2.98634
\(725\) 6.68466 0.248262
\(726\) −34.2462 −1.27100
\(727\) 52.4924 1.94684 0.973418 0.229035i \(-0.0735572\pi\)
0.973418 + 0.229035i \(0.0735572\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) −31.3693 −1.16103
\(731\) −0.384472 −0.0142202
\(732\) 109.477 4.04640
\(733\) −6.68466 −0.246903 −0.123452 0.992351i \(-0.539396\pi\)
−0.123452 + 0.992351i \(0.539396\pi\)
\(734\) −22.2462 −0.821123
\(735\) −1.56155 −0.0575987
\(736\) 20.4924 0.755361
\(737\) −16.0000 −0.589368
\(738\) 7.36932 0.271268
\(739\) −34.9309 −1.28495 −0.642476 0.766305i \(-0.722093\pi\)
−0.642476 + 0.766305i \(0.722093\pi\)
\(740\) 27.3693 1.00612
\(741\) 0 0
\(742\) −13.1231 −0.481764
\(743\) 32.9848 1.21010 0.605048 0.796189i \(-0.293154\pi\)
0.605048 + 0.796189i \(0.293154\pi\)
\(744\) 0 0
\(745\) 12.2462 0.448666
\(746\) 11.8617 0.434289
\(747\) 2.24621 0.0821846
\(748\) −3.12311 −0.114192
\(749\) 13.3693 0.488504
\(750\) −4.00000 −0.146059
\(751\) 17.0691 0.622861 0.311431 0.950269i \(-0.399192\pi\)
0.311431 + 0.950269i \(0.399192\pi\)
\(752\) 66.7386 2.43371
\(753\) −13.8617 −0.505150
\(754\) 0 0
\(755\) −6.93087 −0.252240
\(756\) −25.3693 −0.922673
\(757\) 39.3693 1.43090 0.715451 0.698663i \(-0.246221\pi\)
0.715451 + 0.698663i \(0.246221\pi\)
\(758\) 42.2462 1.53445
\(759\) 7.61553 0.276426
\(760\) −46.7386 −1.69539
\(761\) −48.2462 −1.74892 −0.874462 0.485094i \(-0.838785\pi\)
−0.874462 + 0.485094i \(0.838785\pi\)
\(762\) −24.9848 −0.905105
\(763\) −5.31534 −0.192428
\(764\) −61.8617 −2.23808
\(765\) −0.246211 −0.00890179
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) −42.2462 −1.52443
\(769\) 42.4924 1.53232 0.766158 0.642652i \(-0.222166\pi\)
0.766158 + 0.642652i \(0.222166\pi\)
\(770\) −4.00000 −0.144150
\(771\) −16.3845 −0.590072
\(772\) −88.3542 −3.17994
\(773\) −36.9309 −1.32831 −0.664156 0.747594i \(-0.731209\pi\)
−0.664156 + 0.747594i \(0.731209\pi\)
\(774\) −1.26137 −0.0453389
\(775\) 0 0
\(776\) −38.1080 −1.36800
\(777\) −9.36932 −0.336122
\(778\) −63.8617 −2.28955
\(779\) −36.4924 −1.30748
\(780\) 0 0
\(781\) −12.4924 −0.447014
\(782\) −3.50758 −0.125431
\(783\) −37.1771 −1.32860
\(784\) 7.68466 0.274452
\(785\) −20.2462 −0.722618
\(786\) −3.50758 −0.125111
\(787\) 49.1771 1.75297 0.876487 0.481426i \(-0.159881\pi\)
0.876487 + 0.481426i \(0.159881\pi\)
\(788\) −5.12311 −0.182503
\(789\) −20.1080 −0.715862
\(790\) 6.24621 0.222230
\(791\) −14.0000 −0.497783
\(792\) −5.75379 −0.204452
\(793\) 0 0
\(794\) −70.6004 −2.50551
\(795\) 8.00000 0.283731
\(796\) −8.00000 −0.283552
\(797\) 24.0540 0.852036 0.426018 0.904715i \(-0.359916\pi\)
0.426018 + 0.904715i \(0.359916\pi\)
\(798\) 28.4924 1.00862
\(799\) −3.80776 −0.134709
\(800\) 6.56155 0.231986
\(801\) −0.630683 −0.0222841
\(802\) −80.8466 −2.85479
\(803\) 19.1231 0.674840
\(804\) −72.9848 −2.57398
\(805\) −3.12311 −0.110075
\(806\) 0 0
\(807\) −32.3845 −1.13999
\(808\) −106.600 −3.75019
\(809\) 16.5464 0.581740 0.290870 0.956763i \(-0.406055\pi\)
0.290870 + 0.956763i \(0.406055\pi\)
\(810\) 17.9309 0.630027
\(811\) −19.6155 −0.688794 −0.344397 0.938824i \(-0.611917\pi\)
−0.344397 + 0.938824i \(0.611917\pi\)
\(812\) 30.4924 1.07007
\(813\) 24.9848 0.876257
\(814\) −24.0000 −0.841200
\(815\) −7.12311 −0.249512
\(816\) −5.26137 −0.184185
\(817\) 6.24621 0.218527
\(818\) −16.6307 −0.581478
\(819\) 0 0
\(820\) 23.3693 0.816092
\(821\) 21.4233 0.747678 0.373839 0.927494i \(-0.378041\pi\)
0.373839 + 0.927494i \(0.378041\pi\)
\(822\) 68.4924 2.38895
\(823\) −36.4924 −1.27205 −0.636023 0.771670i \(-0.719422\pi\)
−0.636023 + 0.771670i \(0.719422\pi\)
\(824\) 36.4924 1.27127
\(825\) 2.43845 0.0848958
\(826\) 10.2462 0.356511
\(827\) 5.36932 0.186709 0.0933547 0.995633i \(-0.470241\pi\)
0.0933547 + 0.995633i \(0.470241\pi\)
\(828\) −8.00000 −0.278019
\(829\) 34.8769 1.21132 0.605662 0.795722i \(-0.292908\pi\)
0.605662 + 0.795722i \(0.292908\pi\)
\(830\) 10.2462 0.355651
\(831\) −0.384472 −0.0133372
\(832\) 0 0
\(833\) −0.438447 −0.0151913
\(834\) −60.4924 −2.09468
\(835\) −6.93087 −0.239853
\(836\) 50.7386 1.75483
\(837\) 0 0
\(838\) 67.2311 2.32246
\(839\) 28.8769 0.996941 0.498471 0.866907i \(-0.333895\pi\)
0.498471 + 0.866907i \(0.333895\pi\)
\(840\) −10.2462 −0.353528
\(841\) 15.6847 0.540850
\(842\) 6.87689 0.236993
\(843\) −19.4233 −0.668974
\(844\) 64.1080 2.20669
\(845\) 0 0
\(846\) −12.4924 −0.429498
\(847\) −8.56155 −0.294178
\(848\) −39.3693 −1.35195
\(849\) −17.6695 −0.606416
\(850\) −1.12311 −0.0385222
\(851\) −18.7386 −0.642352
\(852\) −56.9848 −1.95227
\(853\) 7.26137 0.248624 0.124312 0.992243i \(-0.460328\pi\)
0.124312 + 0.992243i \(0.460328\pi\)
\(854\) 39.3693 1.34719
\(855\) 4.00000 0.136797
\(856\) 87.7235 2.99833
\(857\) −15.7538 −0.538139 −0.269070 0.963121i \(-0.586716\pi\)
−0.269070 + 0.963121i \(0.586716\pi\)
\(858\) 0 0
\(859\) −16.4924 −0.562714 −0.281357 0.959603i \(-0.590785\pi\)
−0.281357 + 0.959603i \(0.590785\pi\)
\(860\) −4.00000 −0.136399
\(861\) −8.00000 −0.272639
\(862\) 50.7386 1.72816
\(863\) 25.7538 0.876669 0.438335 0.898812i \(-0.355569\pi\)
0.438335 + 0.898812i \(0.355569\pi\)
\(864\) −36.4924 −1.24150
\(865\) 4.43845 0.150912
\(866\) 21.1231 0.717792
\(867\) −26.2462 −0.891368
\(868\) 0 0
\(869\) −3.80776 −0.129170
\(870\) −26.7386 −0.906525
\(871\) 0 0
\(872\) −34.8769 −1.18108
\(873\) 3.26137 0.110381
\(874\) 56.9848 1.92754
\(875\) −1.00000 −0.0338062
\(876\) 87.2311 2.94726
\(877\) 40.2462 1.35902 0.679509 0.733667i \(-0.262193\pi\)
0.679509 + 0.733667i \(0.262193\pi\)
\(878\) 24.0000 0.809961
\(879\) 4.19224 0.141401
\(880\) −12.0000 −0.404520
\(881\) −11.8617 −0.399632 −0.199816 0.979833i \(-0.564034\pi\)
−0.199816 + 0.979833i \(0.564034\pi\)
\(882\) −1.43845 −0.0484350
\(883\) −8.49242 −0.285793 −0.142896 0.989738i \(-0.545642\pi\)
−0.142896 + 0.989738i \(0.545642\pi\)
\(884\) 0 0
\(885\) −6.24621 −0.209964
\(886\) −6.73863 −0.226389
\(887\) 20.4924 0.688068 0.344034 0.938957i \(-0.388206\pi\)
0.344034 + 0.938957i \(0.388206\pi\)
\(888\) −61.4773 −2.06304
\(889\) −6.24621 −0.209491
\(890\) −2.87689 −0.0964337
\(891\) −10.9309 −0.366198
\(892\) 11.1231 0.372429
\(893\) 61.8617 2.07012
\(894\) −48.9848 −1.63830
\(895\) −20.0000 −0.668526
\(896\) −9.43845 −0.315316
\(897\) 0 0
\(898\) 4.63068 0.154528
\(899\) 0 0
\(900\) −2.56155 −0.0853851
\(901\) 2.24621 0.0748321
\(902\) −20.4924 −0.682323
\(903\) 1.36932 0.0455680
\(904\) −91.8617 −3.05528
\(905\) 17.6155 0.585560
\(906\) 27.7235 0.921051
\(907\) −24.1080 −0.800491 −0.400246 0.916408i \(-0.631075\pi\)
−0.400246 + 0.916408i \(0.631075\pi\)
\(908\) −51.6155 −1.71292
\(909\) 9.12311 0.302594
\(910\) 0 0
\(911\) −28.4924 −0.943996 −0.471998 0.881600i \(-0.656467\pi\)
−0.471998 + 0.881600i \(0.656467\pi\)
\(912\) 85.4773 2.83044
\(913\) −6.24621 −0.206719
\(914\) 43.8617 1.45082
\(915\) −24.0000 −0.793416
\(916\) −49.6155 −1.63934
\(917\) −0.876894 −0.0289576
\(918\) 6.24621 0.206156
\(919\) 40.3002 1.32938 0.664690 0.747119i \(-0.268564\pi\)
0.664690 + 0.747119i \(0.268564\pi\)
\(920\) −20.4924 −0.675615
\(921\) 30.1619 0.993869
\(922\) 33.6155 1.10707
\(923\) 0 0
\(924\) 11.1231 0.365923
\(925\) −6.00000 −0.197279
\(926\) −32.0000 −1.05159
\(927\) −3.12311 −0.102576
\(928\) 43.8617 1.43983
\(929\) −22.1080 −0.725338 −0.362669 0.931918i \(-0.618134\pi\)
−0.362669 + 0.931918i \(0.618134\pi\)
\(930\) 0 0
\(931\) 7.12311 0.233450
\(932\) 23.3693 0.765487
\(933\) 49.3693 1.61628
\(934\) 57.4773 1.88071
\(935\) 0.684658 0.0223907
\(936\) 0 0
\(937\) −55.6695 −1.81864 −0.909322 0.416094i \(-0.863399\pi\)
−0.909322 + 0.416094i \(0.863399\pi\)
\(938\) −26.2462 −0.856969
\(939\) −34.8229 −1.13640
\(940\) −39.6155 −1.29212
\(941\) −43.8617 −1.42985 −0.714926 0.699200i \(-0.753540\pi\)
−0.714926 + 0.699200i \(0.753540\pi\)
\(942\) 80.9848 2.63863
\(943\) −16.0000 −0.521032
\(944\) 30.7386 1.00046
\(945\) 5.56155 0.180917
\(946\) 3.50758 0.114041
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −17.3693 −0.564129
\(949\) 0 0
\(950\) 18.2462 0.591985
\(951\) −16.3845 −0.531303
\(952\) −2.87689 −0.0932407
\(953\) −33.1231 −1.07296 −0.536481 0.843912i \(-0.680247\pi\)
−0.536481 + 0.843912i \(0.680247\pi\)
\(954\) 7.36932 0.238590
\(955\) 13.5616 0.438842
\(956\) −90.3542 −2.92226
\(957\) 16.3002 0.526910
\(958\) −12.4924 −0.403612
\(959\) 17.1231 0.552934
\(960\) −2.24621 −0.0724962
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −7.50758 −0.241928
\(964\) 19.3693 0.623844
\(965\) 19.3693 0.623520
\(966\) 12.4924 0.401937
\(967\) 35.1231 1.12948 0.564741 0.825268i \(-0.308976\pi\)
0.564741 + 0.825268i \(0.308976\pi\)
\(968\) −56.1771 −1.80560
\(969\) −4.87689 −0.156668
\(970\) 14.8769 0.477668
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 26.2462 0.841848
\(973\) −15.1231 −0.484825
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 118.108 3.78054
\(977\) −33.2311 −1.06316 −0.531578 0.847009i \(-0.678401\pi\)
−0.531578 + 0.847009i \(0.678401\pi\)
\(978\) 28.4924 0.911087
\(979\) 1.75379 0.0560513
\(980\) −4.56155 −0.145713
\(981\) 2.98485 0.0952988
\(982\) −105.477 −3.36591
\(983\) −51.4233 −1.64015 −0.820074 0.572257i \(-0.806068\pi\)
−0.820074 + 0.572257i \(0.806068\pi\)
\(984\) −52.4924 −1.67340
\(985\) 1.12311 0.0357851
\(986\) −7.50758 −0.239090
\(987\) 13.5616 0.431669
\(988\) 0 0
\(989\) 2.73863 0.0870835
\(990\) 2.24621 0.0713893
\(991\) 12.4924 0.396835 0.198417 0.980118i \(-0.436420\pi\)
0.198417 + 0.980118i \(0.436420\pi\)
\(992\) 0 0
\(993\) −18.7386 −0.594653
\(994\) −20.4924 −0.649980
\(995\) 1.75379 0.0555988
\(996\) −28.4924 −0.902817
\(997\) −2.68466 −0.0850240 −0.0425120 0.999096i \(-0.513536\pi\)
−0.0425120 + 0.999096i \(0.513536\pi\)
\(998\) −105.477 −3.33882
\(999\) 33.3693 1.05576
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 5915.2.a.l.1.2 2
13.12 even 2 35.2.a.b.1.1 2
39.38 odd 2 315.2.a.e.1.2 2
52.51 odd 2 560.2.a.i.1.1 2
65.12 odd 4 175.2.b.b.99.1 4
65.38 odd 4 175.2.b.b.99.4 4
65.64 even 2 175.2.a.f.1.2 2
91.12 odd 6 245.2.e.h.116.2 4
91.25 even 6 245.2.e.i.226.2 4
91.38 odd 6 245.2.e.h.226.2 4
91.51 even 6 245.2.e.i.116.2 4
91.90 odd 2 245.2.a.d.1.1 2
104.51 odd 2 2240.2.a.bd.1.2 2
104.77 even 2 2240.2.a.bh.1.1 2
143.142 odd 2 4235.2.a.m.1.2 2
156.155 even 2 5040.2.a.bt.1.1 2
195.38 even 4 1575.2.d.e.1324.1 4
195.77 even 4 1575.2.d.e.1324.4 4
195.194 odd 2 1575.2.a.p.1.1 2
260.103 even 4 2800.2.g.t.449.2 4
260.207 even 4 2800.2.g.t.449.3 4
260.259 odd 2 2800.2.a.bi.1.2 2
273.272 even 2 2205.2.a.x.1.2 2
364.363 even 2 3920.2.a.bs.1.2 2
455.272 even 4 1225.2.b.f.99.1 4
455.363 even 4 1225.2.b.f.99.4 4
455.454 odd 2 1225.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 13.12 even 2
175.2.a.f.1.2 2 65.64 even 2
175.2.b.b.99.1 4 65.12 odd 4
175.2.b.b.99.4 4 65.38 odd 4
245.2.a.d.1.1 2 91.90 odd 2
245.2.e.h.116.2 4 91.12 odd 6
245.2.e.h.226.2 4 91.38 odd 6
245.2.e.i.116.2 4 91.51 even 6
245.2.e.i.226.2 4 91.25 even 6
315.2.a.e.1.2 2 39.38 odd 2
560.2.a.i.1.1 2 52.51 odd 2
1225.2.a.s.1.2 2 455.454 odd 2
1225.2.b.f.99.1 4 455.272 even 4
1225.2.b.f.99.4 4 455.363 even 4
1575.2.a.p.1.1 2 195.194 odd 2
1575.2.d.e.1324.1 4 195.38 even 4
1575.2.d.e.1324.4 4 195.77 even 4
2205.2.a.x.1.2 2 273.272 even 2
2240.2.a.bd.1.2 2 104.51 odd 2
2240.2.a.bh.1.1 2 104.77 even 2
2800.2.a.bi.1.2 2 260.259 odd 2
2800.2.g.t.449.2 4 260.103 even 4
2800.2.g.t.449.3 4 260.207 even 4
3920.2.a.bs.1.2 2 364.363 even 2
4235.2.a.m.1.2 2 143.142 odd 2
5040.2.a.bt.1.1 2 156.155 even 2
5915.2.a.l.1.2 2 1.1 even 1 trivial