Properties

Label 3549.2.a.bh.1.11
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + 4553 x^{7} - 6393 x^{6} - 6785 x^{5} + 7806 x^{4} + 4632 x^{3} - 3811 x^{2} - 1041 x + 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(1.86754\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.86754 q^{2} +1.00000 q^{3} +1.48770 q^{4} -4.30941 q^{5} +1.86754 q^{6} -1.00000 q^{7} -0.956744 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.86754 q^{2} +1.00000 q^{3} +1.48770 q^{4} -4.30941 q^{5} +1.86754 q^{6} -1.00000 q^{7} -0.956744 q^{8} +1.00000 q^{9} -8.04798 q^{10} -3.88982 q^{11} +1.48770 q^{12} -1.86754 q^{14} -4.30941 q^{15} -4.76215 q^{16} +6.82373 q^{17} +1.86754 q^{18} +5.88962 q^{19} -6.41110 q^{20} -1.00000 q^{21} -7.26438 q^{22} -1.28080 q^{23} -0.956744 q^{24} +13.5710 q^{25} +1.00000 q^{27} -1.48770 q^{28} +1.83066 q^{29} -8.04798 q^{30} +1.37027 q^{31} -6.98001 q^{32} -3.88982 q^{33} +12.7436 q^{34} +4.30941 q^{35} +1.48770 q^{36} +10.7371 q^{37} +10.9991 q^{38} +4.12300 q^{40} +6.71425 q^{41} -1.86754 q^{42} +9.00507 q^{43} -5.78687 q^{44} -4.30941 q^{45} -2.39195 q^{46} -9.83977 q^{47} -4.76215 q^{48} +1.00000 q^{49} +25.3444 q^{50} +6.82373 q^{51} +2.63020 q^{53} +1.86754 q^{54} +16.7628 q^{55} +0.956744 q^{56} +5.88962 q^{57} +3.41883 q^{58} +3.34355 q^{59} -6.41110 q^{60} -10.4539 q^{61} +2.55904 q^{62} -1.00000 q^{63} -3.51113 q^{64} -7.26438 q^{66} -7.48179 q^{67} +10.1516 q^{68} -1.28080 q^{69} +8.04798 q^{70} +4.78383 q^{71} -0.956744 q^{72} -7.13412 q^{73} +20.0519 q^{74} +13.5710 q^{75} +8.76197 q^{76} +3.88982 q^{77} +3.77176 q^{79} +20.5221 q^{80} +1.00000 q^{81} +12.5391 q^{82} +3.87193 q^{83} -1.48770 q^{84} -29.4062 q^{85} +16.8173 q^{86} +1.83066 q^{87} +3.72156 q^{88} -9.71508 q^{89} -8.04798 q^{90} -1.90545 q^{92} +1.37027 q^{93} -18.3761 q^{94} -25.3808 q^{95} -6.98001 q^{96} +1.63979 q^{97} +1.86754 q^{98} -3.88982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + 21q^{10} + 5q^{11} + 28q^{12} - 2q^{14} - 9q^{15} + 50q^{16} - q^{17} + 2q^{18} + 3q^{19} - 23q^{20} - 15q^{21} + 21q^{22} + 4q^{23} + 9q^{24} + 50q^{25} + 15q^{27} - 28q^{28} + 9q^{29} + 21q^{30} + 7q^{31} + 35q^{32} + 5q^{33} - 2q^{34} + 9q^{35} + 28q^{36} + 17q^{37} - 12q^{38} + 46q^{40} - 22q^{41} - 2q^{42} + 36q^{43} + 29q^{44} - 9q^{45} - q^{46} - 12q^{47} + 50q^{48} + 15q^{49} + 53q^{50} - q^{51} - 5q^{53} + 2q^{54} + 43q^{55} - 9q^{56} + 3q^{57} + 29q^{58} - 29q^{59} - 23q^{60} + 12q^{61} + 14q^{62} - 15q^{63} + 95q^{64} + 21q^{66} - 12q^{67} - 16q^{68} + 4q^{69} - 21q^{70} + 36q^{71} + 9q^{72} - 29q^{73} - 5q^{74} + 50q^{75} + 25q^{76} - 5q^{77} + 35q^{79} - 89q^{80} + 15q^{81} + 51q^{82} - 10q^{83} - 28q^{84} + 23q^{85} + 19q^{86} + 9q^{87} + 73q^{88} + 25q^{89} + 21q^{90} - 31q^{92} + 7q^{93} - 19q^{94} - 7q^{95} + 35q^{96} - 26q^{97} + 2q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.86754 1.32055 0.660274 0.751025i \(-0.270440\pi\)
0.660274 + 0.751025i \(0.270440\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.48770 0.743849
\(5\) −4.30941 −1.92723 −0.963613 0.267301i \(-0.913868\pi\)
−0.963613 + 0.267301i \(0.913868\pi\)
\(6\) 1.86754 0.762419
\(7\) −1.00000 −0.377964
\(8\) −0.956744 −0.338260
\(9\) 1.00000 0.333333
\(10\) −8.04798 −2.54500
\(11\) −3.88982 −1.17282 −0.586412 0.810013i \(-0.699460\pi\)
−0.586412 + 0.810013i \(0.699460\pi\)
\(12\) 1.48770 0.429461
\(13\) 0 0
\(14\) −1.86754 −0.499120
\(15\) −4.30941 −1.11268
\(16\) −4.76215 −1.19054
\(17\) 6.82373 1.65500 0.827498 0.561468i \(-0.189763\pi\)
0.827498 + 0.561468i \(0.189763\pi\)
\(18\) 1.86754 0.440183
\(19\) 5.88962 1.35117 0.675586 0.737282i \(-0.263891\pi\)
0.675586 + 0.737282i \(0.263891\pi\)
\(20\) −6.41110 −1.43356
\(21\) −1.00000 −0.218218
\(22\) −7.26438 −1.54877
\(23\) −1.28080 −0.267066 −0.133533 0.991044i \(-0.542632\pi\)
−0.133533 + 0.991044i \(0.542632\pi\)
\(24\) −0.956744 −0.195295
\(25\) 13.5710 2.71420
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.48770 −0.281148
\(29\) 1.83066 0.339945 0.169973 0.985449i \(-0.445632\pi\)
0.169973 + 0.985449i \(0.445632\pi\)
\(30\) −8.04798 −1.46935
\(31\) 1.37027 0.246108 0.123054 0.992400i \(-0.460731\pi\)
0.123054 + 0.992400i \(0.460731\pi\)
\(32\) −6.98001 −1.23390
\(33\) −3.88982 −0.677130
\(34\) 12.7436 2.18550
\(35\) 4.30941 0.728423
\(36\) 1.48770 0.247950
\(37\) 10.7371 1.76516 0.882580 0.470162i \(-0.155804\pi\)
0.882580 + 0.470162i \(0.155804\pi\)
\(38\) 10.9991 1.78429
\(39\) 0 0
\(40\) 4.12300 0.651904
\(41\) 6.71425 1.04859 0.524295 0.851537i \(-0.324329\pi\)
0.524295 + 0.851537i \(0.324329\pi\)
\(42\) −1.86754 −0.288167
\(43\) 9.00507 1.37326 0.686630 0.727007i \(-0.259089\pi\)
0.686630 + 0.727007i \(0.259089\pi\)
\(44\) −5.78687 −0.872404
\(45\) −4.30941 −0.642409
\(46\) −2.39195 −0.352674
\(47\) −9.83977 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(48\) −4.76215 −0.687357
\(49\) 1.00000 0.142857
\(50\) 25.3444 3.58423
\(51\) 6.82373 0.955513
\(52\) 0 0
\(53\) 2.63020 0.361285 0.180643 0.983549i \(-0.442182\pi\)
0.180643 + 0.983549i \(0.442182\pi\)
\(54\) 1.86754 0.254140
\(55\) 16.7628 2.26030
\(56\) 0.956744 0.127850
\(57\) 5.88962 0.780099
\(58\) 3.41883 0.448914
\(59\) 3.34355 0.435293 0.217647 0.976028i \(-0.430162\pi\)
0.217647 + 0.976028i \(0.430162\pi\)
\(60\) −6.41110 −0.827669
\(61\) −10.4539 −1.33848 −0.669241 0.743045i \(-0.733381\pi\)
−0.669241 + 0.743045i \(0.733381\pi\)
\(62\) 2.55904 0.324998
\(63\) −1.00000 −0.125988
\(64\) −3.51113 −0.438891
\(65\) 0 0
\(66\) −7.26438 −0.894184
\(67\) −7.48179 −0.914046 −0.457023 0.889455i \(-0.651084\pi\)
−0.457023 + 0.889455i \(0.651084\pi\)
\(68\) 10.1516 1.23107
\(69\) −1.28080 −0.154191
\(70\) 8.04798 0.961918
\(71\) 4.78383 0.567736 0.283868 0.958863i \(-0.408382\pi\)
0.283868 + 0.958863i \(0.408382\pi\)
\(72\) −0.956744 −0.112753
\(73\) −7.13412 −0.834986 −0.417493 0.908680i \(-0.637091\pi\)
−0.417493 + 0.908680i \(0.637091\pi\)
\(74\) 20.0519 2.33098
\(75\) 13.5710 1.56704
\(76\) 8.76197 1.00507
\(77\) 3.88982 0.443286
\(78\) 0 0
\(79\) 3.77176 0.424356 0.212178 0.977231i \(-0.431944\pi\)
0.212178 + 0.977231i \(0.431944\pi\)
\(80\) 20.5221 2.29444
\(81\) 1.00000 0.111111
\(82\) 12.5391 1.38471
\(83\) 3.87193 0.425000 0.212500 0.977161i \(-0.431840\pi\)
0.212500 + 0.977161i \(0.431840\pi\)
\(84\) −1.48770 −0.162321
\(85\) −29.4062 −3.18955
\(86\) 16.8173 1.81346
\(87\) 1.83066 0.196268
\(88\) 3.72156 0.396720
\(89\) −9.71508 −1.02980 −0.514898 0.857251i \(-0.672170\pi\)
−0.514898 + 0.857251i \(0.672170\pi\)
\(90\) −8.04798 −0.848332
\(91\) 0 0
\(92\) −1.90545 −0.198657
\(93\) 1.37027 0.142091
\(94\) −18.3761 −1.89535
\(95\) −25.3808 −2.60401
\(96\) −6.98001 −0.712394
\(97\) 1.63979 0.166495 0.0832475 0.996529i \(-0.473471\pi\)
0.0832475 + 0.996529i \(0.473471\pi\)
\(98\) 1.86754 0.188650
\(99\) −3.88982 −0.390941
\(100\) 20.1895 2.01895
\(101\) −4.24015 −0.421910 −0.210955 0.977496i \(-0.567657\pi\)
−0.210955 + 0.977496i \(0.567657\pi\)
\(102\) 12.7436 1.26180
\(103\) 6.12865 0.603874 0.301937 0.953328i \(-0.402367\pi\)
0.301937 + 0.953328i \(0.402367\pi\)
\(104\) 0 0
\(105\) 4.30941 0.420555
\(106\) 4.91199 0.477095
\(107\) 3.75644 0.363149 0.181574 0.983377i \(-0.441881\pi\)
0.181574 + 0.983377i \(0.441881\pi\)
\(108\) 1.48770 0.143154
\(109\) 18.2430 1.74736 0.873681 0.486499i \(-0.161726\pi\)
0.873681 + 0.486499i \(0.161726\pi\)
\(110\) 31.3052 2.98483
\(111\) 10.7371 1.01912
\(112\) 4.76215 0.449981
\(113\) −10.1730 −0.957000 −0.478500 0.878088i \(-0.658819\pi\)
−0.478500 + 0.878088i \(0.658819\pi\)
\(114\) 10.9991 1.03016
\(115\) 5.51951 0.514697
\(116\) 2.72347 0.252868
\(117\) 0 0
\(118\) 6.24421 0.574826
\(119\) −6.82373 −0.625530
\(120\) 4.12300 0.376377
\(121\) 4.13068 0.375517
\(122\) −19.5230 −1.76753
\(123\) 6.71425 0.605404
\(124\) 2.03855 0.183067
\(125\) −36.9359 −3.30365
\(126\) −1.86754 −0.166373
\(127\) 0.713456 0.0633090 0.0316545 0.999499i \(-0.489922\pi\)
0.0316545 + 0.999499i \(0.489922\pi\)
\(128\) 7.40285 0.654326
\(129\) 9.00507 0.792852
\(130\) 0 0
\(131\) −3.47549 −0.303655 −0.151828 0.988407i \(-0.548516\pi\)
−0.151828 + 0.988407i \(0.548516\pi\)
\(132\) −5.78687 −0.503683
\(133\) −5.88962 −0.510695
\(134\) −13.9725 −1.20704
\(135\) −4.30941 −0.370895
\(136\) −6.52856 −0.559819
\(137\) 6.49365 0.554790 0.277395 0.960756i \(-0.410529\pi\)
0.277395 + 0.960756i \(0.410529\pi\)
\(138\) −2.39195 −0.203616
\(139\) 9.71185 0.823749 0.411874 0.911241i \(-0.364874\pi\)
0.411874 + 0.911241i \(0.364874\pi\)
\(140\) 6.41110 0.541836
\(141\) −9.83977 −0.828658
\(142\) 8.93398 0.749723
\(143\) 0 0
\(144\) −4.76215 −0.396846
\(145\) −7.88907 −0.655151
\(146\) −13.3232 −1.10264
\(147\) 1.00000 0.0824786
\(148\) 15.9735 1.31301
\(149\) 7.63897 0.625808 0.312904 0.949785i \(-0.398698\pi\)
0.312904 + 0.949785i \(0.398698\pi\)
\(150\) 25.3444 2.06936
\(151\) 0.596432 0.0485369 0.0242685 0.999705i \(-0.492274\pi\)
0.0242685 + 0.999705i \(0.492274\pi\)
\(152\) −5.63486 −0.457047
\(153\) 6.82373 0.551665
\(154\) 7.26438 0.585381
\(155\) −5.90507 −0.474306
\(156\) 0 0
\(157\) −15.8594 −1.26572 −0.632858 0.774268i \(-0.718118\pi\)
−0.632858 + 0.774268i \(0.718118\pi\)
\(158\) 7.04390 0.560383
\(159\) 2.63020 0.208588
\(160\) 30.0797 2.37801
\(161\) 1.28080 0.100942
\(162\) 1.86754 0.146728
\(163\) 14.9533 1.17123 0.585617 0.810588i \(-0.300852\pi\)
0.585617 + 0.810588i \(0.300852\pi\)
\(164\) 9.98878 0.779992
\(165\) 16.7628 1.30498
\(166\) 7.23098 0.561233
\(167\) −2.49975 −0.193436 −0.0967182 0.995312i \(-0.530835\pi\)
−0.0967182 + 0.995312i \(0.530835\pi\)
\(168\) 0.956744 0.0738144
\(169\) 0 0
\(170\) −54.9172 −4.21196
\(171\) 5.88962 0.450390
\(172\) 13.3968 1.02150
\(173\) 2.05122 0.155952 0.0779758 0.996955i \(-0.475154\pi\)
0.0779758 + 0.996955i \(0.475154\pi\)
\(174\) 3.41883 0.259181
\(175\) −13.5710 −1.02587
\(176\) 18.5239 1.39629
\(177\) 3.34355 0.251317
\(178\) −18.1433 −1.35990
\(179\) 12.4319 0.929206 0.464603 0.885519i \(-0.346197\pi\)
0.464603 + 0.885519i \(0.346197\pi\)
\(180\) −6.41110 −0.477855
\(181\) −11.3497 −0.843617 −0.421808 0.906685i \(-0.638605\pi\)
−0.421808 + 0.906685i \(0.638605\pi\)
\(182\) 0 0
\(183\) −10.4539 −0.772773
\(184\) 1.22540 0.0903378
\(185\) −46.2703 −3.40186
\(186\) 2.55904 0.187638
\(187\) −26.5430 −1.94102
\(188\) −14.6386 −1.06763
\(189\) −1.00000 −0.0727393
\(190\) −47.3996 −3.43872
\(191\) −3.73903 −0.270547 −0.135273 0.990808i \(-0.543191\pi\)
−0.135273 + 0.990808i \(0.543191\pi\)
\(192\) −3.51113 −0.253394
\(193\) 9.51514 0.684915 0.342457 0.939533i \(-0.388741\pi\)
0.342457 + 0.939533i \(0.388741\pi\)
\(194\) 3.06236 0.219865
\(195\) 0 0
\(196\) 1.48770 0.106264
\(197\) 15.0989 1.07575 0.537877 0.843023i \(-0.319226\pi\)
0.537877 + 0.843023i \(0.319226\pi\)
\(198\) −7.26438 −0.516257
\(199\) 4.23261 0.300042 0.150021 0.988683i \(-0.452066\pi\)
0.150021 + 0.988683i \(0.452066\pi\)
\(200\) −12.9840 −0.918106
\(201\) −7.48179 −0.527725
\(202\) −7.91863 −0.557153
\(203\) −1.83066 −0.128487
\(204\) 10.1516 0.710757
\(205\) −28.9345 −2.02087
\(206\) 11.4455 0.797444
\(207\) −1.28080 −0.0890220
\(208\) 0 0
\(209\) −22.9095 −1.58469
\(210\) 8.04798 0.555364
\(211\) 12.5911 0.866805 0.433403 0.901200i \(-0.357313\pi\)
0.433403 + 0.901200i \(0.357313\pi\)
\(212\) 3.91294 0.268741
\(213\) 4.78383 0.327783
\(214\) 7.01529 0.479556
\(215\) −38.8065 −2.64658
\(216\) −0.956744 −0.0650982
\(217\) −1.37027 −0.0930202
\(218\) 34.0695 2.30748
\(219\) −7.13412 −0.482079
\(220\) 24.9380 1.68132
\(221\) 0 0
\(222\) 20.0519 1.34579
\(223\) 17.9666 1.20313 0.601565 0.798824i \(-0.294544\pi\)
0.601565 + 0.798824i \(0.294544\pi\)
\(224\) 6.98001 0.466371
\(225\) 13.5710 0.904733
\(226\) −18.9986 −1.26377
\(227\) 7.72887 0.512983 0.256491 0.966547i \(-0.417433\pi\)
0.256491 + 0.966547i \(0.417433\pi\)
\(228\) 8.76197 0.580276
\(229\) 21.7426 1.43679 0.718396 0.695635i \(-0.244877\pi\)
0.718396 + 0.695635i \(0.244877\pi\)
\(230\) 10.3079 0.679682
\(231\) 3.88982 0.255931
\(232\) −1.75148 −0.114990
\(233\) −16.1360 −1.05711 −0.528553 0.848900i \(-0.677265\pi\)
−0.528553 + 0.848900i \(0.677265\pi\)
\(234\) 0 0
\(235\) 42.4036 2.76610
\(236\) 4.97419 0.323792
\(237\) 3.77176 0.245002
\(238\) −12.7436 −0.826043
\(239\) 19.3434 1.25122 0.625611 0.780135i \(-0.284850\pi\)
0.625611 + 0.780135i \(0.284850\pi\)
\(240\) 20.5221 1.32469
\(241\) −10.2012 −0.657119 −0.328560 0.944483i \(-0.606563\pi\)
−0.328560 + 0.944483i \(0.606563\pi\)
\(242\) 7.71421 0.495888
\(243\) 1.00000 0.0641500
\(244\) −15.5522 −0.995628
\(245\) −4.30941 −0.275318
\(246\) 12.5391 0.799465
\(247\) 0 0
\(248\) −1.31100 −0.0832487
\(249\) 3.87193 0.245374
\(250\) −68.9792 −4.36263
\(251\) 18.1674 1.14671 0.573357 0.819306i \(-0.305641\pi\)
0.573357 + 0.819306i \(0.305641\pi\)
\(252\) −1.48770 −0.0937161
\(253\) 4.98209 0.313222
\(254\) 1.33241 0.0836026
\(255\) −29.4062 −1.84149
\(256\) 20.8474 1.30296
\(257\) −12.2798 −0.765991 −0.382995 0.923750i \(-0.625107\pi\)
−0.382995 + 0.923750i \(0.625107\pi\)
\(258\) 16.8173 1.04700
\(259\) −10.7371 −0.667168
\(260\) 0 0
\(261\) 1.83066 0.113315
\(262\) −6.49061 −0.400991
\(263\) 11.5962 0.715053 0.357527 0.933903i \(-0.383620\pi\)
0.357527 + 0.933903i \(0.383620\pi\)
\(264\) 3.72156 0.229046
\(265\) −11.3346 −0.696278
\(266\) −10.9991 −0.674397
\(267\) −9.71508 −0.594554
\(268\) −11.1306 −0.679912
\(269\) −6.89939 −0.420664 −0.210332 0.977630i \(-0.567454\pi\)
−0.210332 + 0.977630i \(0.567454\pi\)
\(270\) −8.04798 −0.489785
\(271\) −4.69868 −0.285424 −0.142712 0.989764i \(-0.545582\pi\)
−0.142712 + 0.989764i \(0.545582\pi\)
\(272\) −32.4956 −1.97034
\(273\) 0 0
\(274\) 12.1271 0.732627
\(275\) −52.7887 −3.18328
\(276\) −1.90545 −0.114695
\(277\) 12.9421 0.777618 0.388809 0.921318i \(-0.372887\pi\)
0.388809 + 0.921318i \(0.372887\pi\)
\(278\) 18.1373 1.08780
\(279\) 1.37027 0.0820361
\(280\) −4.12300 −0.246396
\(281\) 30.0052 1.78996 0.894981 0.446104i \(-0.147189\pi\)
0.894981 + 0.446104i \(0.147189\pi\)
\(282\) −18.3761 −1.09428
\(283\) 25.8570 1.53704 0.768521 0.639825i \(-0.220993\pi\)
0.768521 + 0.639825i \(0.220993\pi\)
\(284\) 7.11689 0.422310
\(285\) −25.3808 −1.50343
\(286\) 0 0
\(287\) −6.71425 −0.396330
\(288\) −6.98001 −0.411301
\(289\) 29.5632 1.73901
\(290\) −14.7331 −0.865159
\(291\) 1.63979 0.0961259
\(292\) −10.6134 −0.621103
\(293\) −30.7151 −1.79440 −0.897199 0.441627i \(-0.854401\pi\)
−0.897199 + 0.441627i \(0.854401\pi\)
\(294\) 1.86754 0.108917
\(295\) −14.4087 −0.838909
\(296\) −10.2726 −0.597084
\(297\) −3.88982 −0.225710
\(298\) 14.2661 0.826410
\(299\) 0 0
\(300\) 20.1895 1.16564
\(301\) −9.00507 −0.519044
\(302\) 1.11386 0.0640954
\(303\) −4.24015 −0.243590
\(304\) −28.0473 −1.60862
\(305\) 45.0500 2.57956
\(306\) 12.7436 0.728501
\(307\) 13.1810 0.752277 0.376138 0.926564i \(-0.377252\pi\)
0.376138 + 0.926564i \(0.377252\pi\)
\(308\) 5.78687 0.329738
\(309\) 6.12865 0.348647
\(310\) −11.0279 −0.626345
\(311\) −22.3404 −1.26681 −0.633404 0.773821i \(-0.718343\pi\)
−0.633404 + 0.773821i \(0.718343\pi\)
\(312\) 0 0
\(313\) 26.5372 1.49997 0.749985 0.661455i \(-0.230061\pi\)
0.749985 + 0.661455i \(0.230061\pi\)
\(314\) −29.6180 −1.67144
\(315\) 4.30941 0.242808
\(316\) 5.61124 0.315657
\(317\) −4.21944 −0.236987 −0.118494 0.992955i \(-0.537807\pi\)
−0.118494 + 0.992955i \(0.537807\pi\)
\(318\) 4.91199 0.275451
\(319\) −7.12094 −0.398696
\(320\) 15.1309 0.845842
\(321\) 3.75644 0.209664
\(322\) 2.39195 0.133298
\(323\) 40.1891 2.23618
\(324\) 1.48770 0.0826499
\(325\) 0 0
\(326\) 27.9259 1.54667
\(327\) 18.2430 1.00884
\(328\) −6.42382 −0.354696
\(329\) 9.83977 0.542484
\(330\) 31.3052 1.72329
\(331\) −16.1732 −0.888960 −0.444480 0.895789i \(-0.646612\pi\)
−0.444480 + 0.895789i \(0.646612\pi\)
\(332\) 5.76026 0.316135
\(333\) 10.7371 0.588387
\(334\) −4.66837 −0.255442
\(335\) 32.2421 1.76157
\(336\) 4.76215 0.259797
\(337\) −10.2533 −0.558534 −0.279267 0.960214i \(-0.590091\pi\)
−0.279267 + 0.960214i \(0.590091\pi\)
\(338\) 0 0
\(339\) −10.1730 −0.552524
\(340\) −43.7476 −2.37254
\(341\) −5.33011 −0.288642
\(342\) 10.9991 0.594762
\(343\) −1.00000 −0.0539949
\(344\) −8.61555 −0.464519
\(345\) 5.51951 0.297160
\(346\) 3.83073 0.205942
\(347\) 26.5008 1.42264 0.711318 0.702871i \(-0.248099\pi\)
0.711318 + 0.702871i \(0.248099\pi\)
\(348\) 2.72347 0.145993
\(349\) 17.4402 0.933555 0.466778 0.884375i \(-0.345415\pi\)
0.466778 + 0.884375i \(0.345415\pi\)
\(350\) −25.3444 −1.35471
\(351\) 0 0
\(352\) 27.1510 1.44715
\(353\) 5.00425 0.266349 0.133175 0.991093i \(-0.457483\pi\)
0.133175 + 0.991093i \(0.457483\pi\)
\(354\) 6.24421 0.331876
\(355\) −20.6155 −1.09416
\(356\) −14.4531 −0.766013
\(357\) −6.82373 −0.361150
\(358\) 23.2171 1.22706
\(359\) 4.95366 0.261444 0.130722 0.991419i \(-0.458270\pi\)
0.130722 + 0.991419i \(0.458270\pi\)
\(360\) 4.12300 0.217301
\(361\) 15.6876 0.825664
\(362\) −21.1960 −1.11404
\(363\) 4.13068 0.216805
\(364\) 0 0
\(365\) 30.7438 1.60921
\(366\) −19.5230 −1.02048
\(367\) 12.3603 0.645201 0.322601 0.946535i \(-0.395443\pi\)
0.322601 + 0.946535i \(0.395443\pi\)
\(368\) 6.09938 0.317952
\(369\) 6.71425 0.349530
\(370\) −86.4116 −4.49233
\(371\) −2.63020 −0.136553
\(372\) 2.03855 0.105694
\(373\) −16.5053 −0.854613 −0.427307 0.904107i \(-0.640537\pi\)
−0.427307 + 0.904107i \(0.640537\pi\)
\(374\) −49.5701 −2.56321
\(375\) −36.9359 −1.90736
\(376\) 9.41414 0.485497
\(377\) 0 0
\(378\) −1.86754 −0.0960558
\(379\) −26.1674 −1.34413 −0.672066 0.740492i \(-0.734593\pi\)
−0.672066 + 0.740492i \(0.734593\pi\)
\(380\) −37.7589 −1.93699
\(381\) 0.713456 0.0365515
\(382\) −6.98278 −0.357270
\(383\) −27.8933 −1.42528 −0.712640 0.701530i \(-0.752500\pi\)
−0.712640 + 0.701530i \(0.752500\pi\)
\(384\) 7.40285 0.377775
\(385\) −16.7628 −0.854312
\(386\) 17.7699 0.904463
\(387\) 9.00507 0.457753
\(388\) 2.43950 0.123847
\(389\) 0.413312 0.0209557 0.0104779 0.999945i \(-0.496665\pi\)
0.0104779 + 0.999945i \(0.496665\pi\)
\(390\) 0 0
\(391\) −8.73986 −0.441993
\(392\) −0.956744 −0.0483229
\(393\) −3.47549 −0.175315
\(394\) 28.1978 1.42059
\(395\) −16.2540 −0.817830
\(396\) −5.78687 −0.290801
\(397\) −10.3705 −0.520482 −0.260241 0.965544i \(-0.583802\pi\)
−0.260241 + 0.965544i \(0.583802\pi\)
\(398\) 7.90457 0.396220
\(399\) −5.88962 −0.294850
\(400\) −64.6271 −3.23136
\(401\) −16.6369 −0.830806 −0.415403 0.909638i \(-0.636359\pi\)
−0.415403 + 0.909638i \(0.636359\pi\)
\(402\) −13.9725 −0.696886
\(403\) 0 0
\(404\) −6.30806 −0.313837
\(405\) −4.30941 −0.214136
\(406\) −3.41883 −0.169674
\(407\) −41.7652 −2.07022
\(408\) −6.52856 −0.323212
\(409\) 7.05450 0.348822 0.174411 0.984673i \(-0.444198\pi\)
0.174411 + 0.984673i \(0.444198\pi\)
\(410\) −54.0362 −2.66866
\(411\) 6.49365 0.320308
\(412\) 9.11757 0.449191
\(413\) −3.34355 −0.164525
\(414\) −2.39195 −0.117558
\(415\) −16.6857 −0.819070
\(416\) 0 0
\(417\) 9.71185 0.475592
\(418\) −42.7844 −2.09266
\(419\) 35.8189 1.74987 0.874934 0.484242i \(-0.160904\pi\)
0.874934 + 0.484242i \(0.160904\pi\)
\(420\) 6.41110 0.312829
\(421\) −20.5380 −1.00096 −0.500479 0.865748i \(-0.666843\pi\)
−0.500479 + 0.865748i \(0.666843\pi\)
\(422\) 23.5143 1.14466
\(423\) −9.83977 −0.478426
\(424\) −2.51642 −0.122208
\(425\) 92.6048 4.49199
\(426\) 8.93398 0.432853
\(427\) 10.4539 0.505899
\(428\) 5.58845 0.270128
\(429\) 0 0
\(430\) −72.4727 −3.49494
\(431\) 13.1827 0.634987 0.317493 0.948261i \(-0.397159\pi\)
0.317493 + 0.948261i \(0.397159\pi\)
\(432\) −4.76215 −0.229119
\(433\) 5.34891 0.257052 0.128526 0.991706i \(-0.458975\pi\)
0.128526 + 0.991706i \(0.458975\pi\)
\(434\) −2.55904 −0.122838
\(435\) −7.88907 −0.378252
\(436\) 27.1401 1.29977
\(437\) −7.54345 −0.360852
\(438\) −13.3232 −0.636609
\(439\) −32.3209 −1.54259 −0.771296 0.636476i \(-0.780391\pi\)
−0.771296 + 0.636476i \(0.780391\pi\)
\(440\) −16.0377 −0.764569
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 19.3799 0.920769 0.460385 0.887720i \(-0.347712\pi\)
0.460385 + 0.887720i \(0.347712\pi\)
\(444\) 15.9735 0.758068
\(445\) 41.8663 1.98465
\(446\) 33.5533 1.58879
\(447\) 7.63897 0.361311
\(448\) 3.51113 0.165885
\(449\) −36.3017 −1.71318 −0.856592 0.515995i \(-0.827422\pi\)
−0.856592 + 0.515995i \(0.827422\pi\)
\(450\) 25.3444 1.19474
\(451\) −26.1172 −1.22981
\(452\) −15.1344 −0.711863
\(453\) 0.596432 0.0280228
\(454\) 14.4339 0.677419
\(455\) 0 0
\(456\) −5.63486 −0.263876
\(457\) 2.08528 0.0975454 0.0487727 0.998810i \(-0.484469\pi\)
0.0487727 + 0.998810i \(0.484469\pi\)
\(458\) 40.6051 1.89735
\(459\) 6.82373 0.318504
\(460\) 8.21136 0.382857
\(461\) 28.1146 1.30943 0.654714 0.755877i \(-0.272789\pi\)
0.654714 + 0.755877i \(0.272789\pi\)
\(462\) 7.26438 0.337970
\(463\) 10.3933 0.483017 0.241509 0.970399i \(-0.422358\pi\)
0.241509 + 0.970399i \(0.422358\pi\)
\(464\) −8.71789 −0.404718
\(465\) −5.90507 −0.273841
\(466\) −30.1346 −1.39596
\(467\) 4.78281 0.221322 0.110661 0.993858i \(-0.464703\pi\)
0.110661 + 0.993858i \(0.464703\pi\)
\(468\) 0 0
\(469\) 7.48179 0.345477
\(470\) 79.1903 3.65277
\(471\) −15.8594 −0.730761
\(472\) −3.19892 −0.147242
\(473\) −35.0281 −1.61059
\(474\) 7.04390 0.323537
\(475\) 79.9280 3.66735
\(476\) −10.1516 −0.465300
\(477\) 2.63020 0.120428
\(478\) 36.1246 1.65230
\(479\) −8.07411 −0.368916 −0.184458 0.982840i \(-0.559053\pi\)
−0.184458 + 0.982840i \(0.559053\pi\)
\(480\) 30.0797 1.37294
\(481\) 0 0
\(482\) −19.0512 −0.867758
\(483\) 1.28080 0.0582786
\(484\) 6.14521 0.279328
\(485\) −7.06650 −0.320873
\(486\) 1.86754 0.0847132
\(487\) −11.1535 −0.505412 −0.252706 0.967543i \(-0.581321\pi\)
−0.252706 + 0.967543i \(0.581321\pi\)
\(488\) 10.0017 0.452755
\(489\) 14.9533 0.676213
\(490\) −8.04798 −0.363571
\(491\) −5.11302 −0.230747 −0.115374 0.993322i \(-0.536807\pi\)
−0.115374 + 0.993322i \(0.536807\pi\)
\(492\) 9.98878 0.450329
\(493\) 12.4919 0.562608
\(494\) 0 0
\(495\) 16.7628 0.753432
\(496\) −6.52545 −0.293001
\(497\) −4.78383 −0.214584
\(498\) 7.23098 0.324028
\(499\) −38.6257 −1.72912 −0.864561 0.502528i \(-0.832403\pi\)
−0.864561 + 0.502528i \(0.832403\pi\)
\(500\) −54.9495 −2.45742
\(501\) −2.49975 −0.111681
\(502\) 33.9283 1.51429
\(503\) −43.5039 −1.93974 −0.969871 0.243620i \(-0.921665\pi\)
−0.969871 + 0.243620i \(0.921665\pi\)
\(504\) 0.956744 0.0426168
\(505\) 18.2725 0.813117
\(506\) 9.30425 0.413624
\(507\) 0 0
\(508\) 1.06141 0.0470923
\(509\) −19.3651 −0.858344 −0.429172 0.903223i \(-0.641195\pi\)
−0.429172 + 0.903223i \(0.641195\pi\)
\(510\) −54.9172 −2.43178
\(511\) 7.13412 0.315595
\(512\) 24.1275 1.06630
\(513\) 5.88962 0.260033
\(514\) −22.9329 −1.01153
\(515\) −26.4108 −1.16380
\(516\) 13.3968 0.589762
\(517\) 38.2749 1.68333
\(518\) −20.0519 −0.881028
\(519\) 2.05122 0.0900386
\(520\) 0 0
\(521\) 4.69912 0.205872 0.102936 0.994688i \(-0.467176\pi\)
0.102936 + 0.994688i \(0.467176\pi\)
\(522\) 3.41883 0.149638
\(523\) −11.7149 −0.512256 −0.256128 0.966643i \(-0.582447\pi\)
−0.256128 + 0.966643i \(0.582447\pi\)
\(524\) −5.17048 −0.225874
\(525\) −13.5710 −0.592287
\(526\) 21.6564 0.944263
\(527\) 9.35037 0.407309
\(528\) 18.5239 0.806149
\(529\) −21.3595 −0.928676
\(530\) −21.1678 −0.919469
\(531\) 3.34355 0.145098
\(532\) −8.76197 −0.379880
\(533\) 0 0
\(534\) −18.1433 −0.785137
\(535\) −16.1880 −0.699870
\(536\) 7.15816 0.309185
\(537\) 12.4319 0.536477
\(538\) −12.8849 −0.555507
\(539\) −3.88982 −0.167546
\(540\) −6.41110 −0.275890
\(541\) −12.0409 −0.517677 −0.258838 0.965921i \(-0.583340\pi\)
−0.258838 + 0.965921i \(0.583340\pi\)
\(542\) −8.77495 −0.376917
\(543\) −11.3497 −0.487062
\(544\) −47.6297 −2.04210
\(545\) −78.6165 −3.36756
\(546\) 0 0
\(547\) −41.9904 −1.79538 −0.897690 0.440628i \(-0.854756\pi\)
−0.897690 + 0.440628i \(0.854756\pi\)
\(548\) 9.66059 0.412680
\(549\) −10.4539 −0.446161
\(550\) −98.5849 −4.20368
\(551\) 10.7819 0.459324
\(552\) 1.22540 0.0521566
\(553\) −3.77176 −0.160392
\(554\) 24.1699 1.02688
\(555\) −46.2703 −1.96407
\(556\) 14.4483 0.612744
\(557\) −11.1177 −0.471072 −0.235536 0.971866i \(-0.575685\pi\)
−0.235536 + 0.971866i \(0.575685\pi\)
\(558\) 2.55904 0.108333
\(559\) 0 0
\(560\) −20.5221 −0.867215
\(561\) −26.5430 −1.12065
\(562\) 56.0359 2.36373
\(563\) 4.88687 0.205957 0.102979 0.994684i \(-0.467163\pi\)
0.102979 + 0.994684i \(0.467163\pi\)
\(564\) −14.6386 −0.616396
\(565\) 43.8398 1.84436
\(566\) 48.2890 2.02974
\(567\) −1.00000 −0.0419961
\(568\) −4.57690 −0.192043
\(569\) 23.5596 0.987669 0.493835 0.869556i \(-0.335595\pi\)
0.493835 + 0.869556i \(0.335595\pi\)
\(570\) −47.3996 −1.98535
\(571\) 17.6931 0.740432 0.370216 0.928946i \(-0.379284\pi\)
0.370216 + 0.928946i \(0.379284\pi\)
\(572\) 0 0
\(573\) −3.73903 −0.156200
\(574\) −12.5391 −0.523373
\(575\) −17.3818 −0.724871
\(576\) −3.51113 −0.146297
\(577\) 43.4059 1.80701 0.903507 0.428574i \(-0.140984\pi\)
0.903507 + 0.428574i \(0.140984\pi\)
\(578\) 55.2104 2.29645
\(579\) 9.51514 0.395436
\(580\) −11.7365 −0.487334
\(581\) −3.87193 −0.160635
\(582\) 3.06236 0.126939
\(583\) −10.2310 −0.423724
\(584\) 6.82553 0.282442
\(585\) 0 0
\(586\) −57.3617 −2.36959
\(587\) 20.6405 0.851923 0.425962 0.904741i \(-0.359936\pi\)
0.425962 + 0.904741i \(0.359936\pi\)
\(588\) 1.48770 0.0613516
\(589\) 8.07039 0.332535
\(590\) −26.9088 −1.10782
\(591\) 15.0989 0.621087
\(592\) −51.1315 −2.10149
\(593\) 36.5043 1.49905 0.749526 0.661974i \(-0.230281\pi\)
0.749526 + 0.661974i \(0.230281\pi\)
\(594\) −7.26438 −0.298061
\(595\) 29.4062 1.20554
\(596\) 11.3645 0.465507
\(597\) 4.23261 0.173229
\(598\) 0 0
\(599\) 18.7221 0.764962 0.382481 0.923963i \(-0.375070\pi\)
0.382481 + 0.923963i \(0.375070\pi\)
\(600\) −12.9840 −0.530069
\(601\) −38.4712 −1.56927 −0.784636 0.619956i \(-0.787150\pi\)
−0.784636 + 0.619956i \(0.787150\pi\)
\(602\) −16.8173 −0.685422
\(603\) −7.48179 −0.304682
\(604\) 0.887310 0.0361041
\(605\) −17.8008 −0.723705
\(606\) −7.91863 −0.321673
\(607\) 34.0718 1.38293 0.691466 0.722409i \(-0.256965\pi\)
0.691466 + 0.722409i \(0.256965\pi\)
\(608\) −41.1096 −1.66721
\(609\) −1.83066 −0.0741822
\(610\) 84.1327 3.40643
\(611\) 0 0
\(612\) 10.1516 0.410356
\(613\) −14.9448 −0.603616 −0.301808 0.953369i \(-0.597590\pi\)
−0.301808 + 0.953369i \(0.597590\pi\)
\(614\) 24.6159 0.993418
\(615\) −28.9345 −1.16675
\(616\) −3.72156 −0.149946
\(617\) −30.6354 −1.23334 −0.616668 0.787223i \(-0.711518\pi\)
−0.616668 + 0.787223i \(0.711518\pi\)
\(618\) 11.4455 0.460405
\(619\) −44.8329 −1.80198 −0.900992 0.433835i \(-0.857160\pi\)
−0.900992 + 0.433835i \(0.857160\pi\)
\(620\) −8.78495 −0.352812
\(621\) −1.28080 −0.0513969
\(622\) −41.7215 −1.67288
\(623\) 9.71508 0.389227
\(624\) 0 0
\(625\) 91.3170 3.65268
\(626\) 49.5592 1.98078
\(627\) −22.9095 −0.914919
\(628\) −23.5939 −0.941500
\(629\) 73.2667 2.92133
\(630\) 8.04798 0.320639
\(631\) 39.4759 1.57151 0.785756 0.618536i \(-0.212274\pi\)
0.785756 + 0.618536i \(0.212274\pi\)
\(632\) −3.60861 −0.143543
\(633\) 12.5911 0.500450
\(634\) −7.87997 −0.312953
\(635\) −3.07457 −0.122011
\(636\) 3.91294 0.155158
\(637\) 0 0
\(638\) −13.2986 −0.526498
\(639\) 4.78383 0.189245
\(640\) −31.9019 −1.26103
\(641\) −26.1610 −1.03330 −0.516649 0.856197i \(-0.672821\pi\)
−0.516649 + 0.856197i \(0.672821\pi\)
\(642\) 7.01529 0.276872
\(643\) −12.8679 −0.507460 −0.253730 0.967275i \(-0.581657\pi\)
−0.253730 + 0.967275i \(0.581657\pi\)
\(644\) 1.90545 0.0750852
\(645\) −38.8065 −1.52801
\(646\) 75.0547 2.95299
\(647\) 9.13149 0.358996 0.179498 0.983758i \(-0.442553\pi\)
0.179498 + 0.983758i \(0.442553\pi\)
\(648\) −0.956744 −0.0375845
\(649\) −13.0058 −0.510523
\(650\) 0 0
\(651\) −1.37027 −0.0537053
\(652\) 22.2460 0.871222
\(653\) 21.3744 0.836445 0.418222 0.908345i \(-0.362653\pi\)
0.418222 + 0.908345i \(0.362653\pi\)
\(654\) 34.0695 1.33222
\(655\) 14.9773 0.585212
\(656\) −31.9743 −1.24839
\(657\) −7.13412 −0.278329
\(658\) 18.3761 0.716376
\(659\) −31.1995 −1.21536 −0.607679 0.794183i \(-0.707899\pi\)
−0.607679 + 0.794183i \(0.707899\pi\)
\(660\) 24.9380 0.970710
\(661\) 13.1927 0.513135 0.256568 0.966526i \(-0.417408\pi\)
0.256568 + 0.966526i \(0.417408\pi\)
\(662\) −30.2041 −1.17392
\(663\) 0 0
\(664\) −3.70445 −0.143760
\(665\) 25.3808 0.984224
\(666\) 20.0519 0.776993
\(667\) −2.34472 −0.0907879
\(668\) −3.71887 −0.143887
\(669\) 17.9666 0.694628
\(670\) 60.2133 2.32624
\(671\) 40.6637 1.56980
\(672\) 6.98001 0.269260
\(673\) 24.0900 0.928600 0.464300 0.885678i \(-0.346306\pi\)
0.464300 + 0.885678i \(0.346306\pi\)
\(674\) −19.1485 −0.737571
\(675\) 13.5710 0.522348
\(676\) 0 0
\(677\) −29.3251 −1.12705 −0.563527 0.826097i \(-0.690556\pi\)
−0.563527 + 0.826097i \(0.690556\pi\)
\(678\) −18.9986 −0.729635
\(679\) −1.63979 −0.0629292
\(680\) 28.1342 1.07890
\(681\) 7.72887 0.296171
\(682\) −9.95419 −0.381166
\(683\) 48.4183 1.85267 0.926337 0.376697i \(-0.122940\pi\)
0.926337 + 0.376697i \(0.122940\pi\)
\(684\) 8.76197 0.335022
\(685\) −27.9838 −1.06921
\(686\) −1.86754 −0.0713029
\(687\) 21.7426 0.829532
\(688\) −42.8835 −1.63492
\(689\) 0 0
\(690\) 10.3079 0.392415
\(691\) 20.4556 0.778168 0.389084 0.921202i \(-0.372792\pi\)
0.389084 + 0.921202i \(0.372792\pi\)
\(692\) 3.05160 0.116004
\(693\) 3.88982 0.147762
\(694\) 49.4912 1.87866
\(695\) −41.8523 −1.58755
\(696\) −1.75148 −0.0663895
\(697\) 45.8162 1.73541
\(698\) 32.5703 1.23280
\(699\) −16.1360 −0.610320
\(700\) −20.1895 −0.763093
\(701\) 22.4924 0.849527 0.424763 0.905304i \(-0.360357\pi\)
0.424763 + 0.905304i \(0.360357\pi\)
\(702\) 0 0
\(703\) 63.2371 2.38503
\(704\) 13.6576 0.514742
\(705\) 42.4036 1.59701
\(706\) 9.34563 0.351727
\(707\) 4.24015 0.159467
\(708\) 4.97419 0.186942
\(709\) 9.36836 0.351836 0.175918 0.984405i \(-0.443711\pi\)
0.175918 + 0.984405i \(0.443711\pi\)
\(710\) −38.5002 −1.44489
\(711\) 3.77176 0.141452
\(712\) 9.29485 0.348339
\(713\) −1.75505 −0.0657272
\(714\) −12.7436 −0.476916
\(715\) 0 0
\(716\) 18.4950 0.691189
\(717\) 19.3434 0.722394
\(718\) 9.25115 0.345250
\(719\) −26.0078 −0.969929 −0.484965 0.874534i \(-0.661167\pi\)
−0.484965 + 0.874534i \(0.661167\pi\)
\(720\) 20.5221 0.764812
\(721\) −6.12865 −0.228243
\(722\) 29.2972 1.09033
\(723\) −10.2012 −0.379388
\(724\) −16.8849 −0.627523
\(725\) 24.8439 0.922680
\(726\) 7.71421 0.286301
\(727\) −45.4636 −1.68615 −0.843076 0.537795i \(-0.819258\pi\)
−0.843076 + 0.537795i \(0.819258\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 57.4153 2.12503
\(731\) 61.4481 2.27274
\(732\) −15.5522 −0.574826
\(733\) −37.7796 −1.39542 −0.697711 0.716380i \(-0.745798\pi\)
−0.697711 + 0.716380i \(0.745798\pi\)
\(734\) 23.0833 0.852020
\(735\) −4.30941 −0.158955
\(736\) 8.94002 0.329534
\(737\) 29.1028 1.07202
\(738\) 12.5391 0.461571
\(739\) 1.34453 0.0494595 0.0247297 0.999694i \(-0.492127\pi\)
0.0247297 + 0.999694i \(0.492127\pi\)
\(740\) −68.8363 −2.53047
\(741\) 0 0
\(742\) −4.91199 −0.180325
\(743\) 13.2274 0.485265 0.242633 0.970118i \(-0.421989\pi\)
0.242633 + 0.970118i \(0.421989\pi\)
\(744\) −1.31100 −0.0480636
\(745\) −32.9194 −1.20607
\(746\) −30.8243 −1.12856
\(747\) 3.87193 0.141667
\(748\) −39.4880 −1.44383
\(749\) −3.75644 −0.137257
\(750\) −68.9792 −2.51877
\(751\) −3.03878 −0.110887 −0.0554433 0.998462i \(-0.517657\pi\)
−0.0554433 + 0.998462i \(0.517657\pi\)
\(752\) 46.8585 1.70875
\(753\) 18.1674 0.662056
\(754\) 0 0
\(755\) −2.57027 −0.0935417
\(756\) −1.48770 −0.0541070
\(757\) 28.1535 1.02326 0.511628 0.859207i \(-0.329043\pi\)
0.511628 + 0.859207i \(0.329043\pi\)
\(758\) −48.8687 −1.77499
\(759\) 4.98209 0.180839
\(760\) 24.2829 0.880834
\(761\) −21.5087 −0.779691 −0.389845 0.920880i \(-0.627472\pi\)
−0.389845 + 0.920880i \(0.627472\pi\)
\(762\) 1.33241 0.0482680
\(763\) −18.2430 −0.660441
\(764\) −5.56255 −0.201246
\(765\) −29.4062 −1.06318
\(766\) −52.0917 −1.88215
\(767\) 0 0
\(768\) 20.8474 0.752264
\(769\) 28.7707 1.03750 0.518749 0.854927i \(-0.326398\pi\)
0.518749 + 0.854927i \(0.326398\pi\)
\(770\) −31.3052 −1.12816
\(771\) −12.2798 −0.442245
\(772\) 14.1556 0.509473
\(773\) −28.8915 −1.03916 −0.519578 0.854423i \(-0.673911\pi\)
−0.519578 + 0.854423i \(0.673911\pi\)
\(774\) 16.8173 0.604486
\(775\) 18.5960 0.667987
\(776\) −1.56886 −0.0563186
\(777\) −10.7371 −0.385190
\(778\) 0.771876 0.0276731
\(779\) 39.5444 1.41682
\(780\) 0 0
\(781\) −18.6082 −0.665855
\(782\) −16.3220 −0.583674
\(783\) 1.83066 0.0654225
\(784\) −4.76215 −0.170077
\(785\) 68.3445 2.43932
\(786\) −6.49061 −0.231512
\(787\) 46.3190 1.65109 0.825547 0.564333i \(-0.190866\pi\)
0.825547 + 0.564333i \(0.190866\pi\)
\(788\) 22.4626 0.800198
\(789\) 11.5962 0.412836
\(790\) −30.3551 −1.07998
\(791\) 10.1730 0.361712
\(792\) 3.72156 0.132240
\(793\) 0 0
\(794\) −19.3674 −0.687322
\(795\) −11.3346 −0.401996
\(796\) 6.29685 0.223186
\(797\) −7.09265 −0.251235 −0.125617 0.992079i \(-0.540091\pi\)
−0.125617 + 0.992079i \(0.540091\pi\)
\(798\) −10.9991 −0.389363
\(799\) −67.1439 −2.37538
\(800\) −94.7257 −3.34906
\(801\) −9.71508 −0.343266
\(802\) −31.0700 −1.09712
\(803\) 27.7504 0.979291
\(804\) −11.1306 −0.392547
\(805\) −5.51951 −0.194537
\(806\) 0 0
\(807\) −6.89939 −0.242870
\(808\) 4.05674 0.142715
\(809\) 26.8422 0.943721 0.471861 0.881673i \(-0.343583\pi\)
0.471861 + 0.881673i \(0.343583\pi\)
\(810\) −8.04798 −0.282777
\(811\) 6.59151 0.231459 0.115730 0.993281i \(-0.463079\pi\)
0.115730 + 0.993281i \(0.463079\pi\)
\(812\) −2.72347 −0.0955751
\(813\) −4.69868 −0.164790
\(814\) −77.9981 −2.73383
\(815\) −64.4400 −2.25723
\(816\) −32.4956 −1.13757
\(817\) 53.0364 1.85551
\(818\) 13.1745 0.460637
\(819\) 0 0
\(820\) −43.0457 −1.50322
\(821\) −16.9329 −0.590961 −0.295481 0.955349i \(-0.595480\pi\)
−0.295481 + 0.955349i \(0.595480\pi\)
\(822\) 12.1271 0.422983
\(823\) −15.7875 −0.550318 −0.275159 0.961399i \(-0.588730\pi\)
−0.275159 + 0.961399i \(0.588730\pi\)
\(824\) −5.86355 −0.204266
\(825\) −52.7887 −1.83787
\(826\) −6.24421 −0.217264
\(827\) −20.9917 −0.729954 −0.364977 0.931016i \(-0.618923\pi\)
−0.364977 + 0.931016i \(0.618923\pi\)
\(828\) −1.90545 −0.0662189
\(829\) 27.4341 0.952827 0.476414 0.879221i \(-0.341937\pi\)
0.476414 + 0.879221i \(0.341937\pi\)
\(830\) −31.1612 −1.08162
\(831\) 12.9421 0.448958
\(832\) 0 0
\(833\) 6.82373 0.236428
\(834\) 18.1373 0.628042
\(835\) 10.7724 0.372796
\(836\) −34.0825 −1.17877
\(837\) 1.37027 0.0473636
\(838\) 66.8932 2.31079
\(839\) −44.6389 −1.54111 −0.770553 0.637376i \(-0.780020\pi\)
−0.770553 + 0.637376i \(0.780020\pi\)
\(840\) −4.12300 −0.142257
\(841\) −25.6487 −0.884437
\(842\) −38.3554 −1.32181
\(843\) 30.0052 1.03344
\(844\) 18.7317 0.644772
\(845\) 0 0
\(846\) −18.3761 −0.631785
\(847\) −4.13068 −0.141932
\(848\) −12.5254 −0.430124
\(849\) 25.8570 0.887411
\(850\) 172.943 5.93189
\(851\) −13.7521 −0.471415
\(852\) 7.11689 0.243821
\(853\) −5.62125 −0.192468 −0.0962340 0.995359i \(-0.530680\pi\)
−0.0962340 + 0.995359i \(0.530680\pi\)
\(854\) 19.5230 0.668064
\(855\) −25.3808 −0.868004
\(856\) −3.59395 −0.122839
\(857\) −33.0059 −1.12746 −0.563729 0.825960i \(-0.690634\pi\)
−0.563729 + 0.825960i \(0.690634\pi\)
\(858\) 0 0
\(859\) −3.61943 −0.123493 −0.0617467 0.998092i \(-0.519667\pi\)
−0.0617467 + 0.998092i \(0.519667\pi\)
\(860\) −57.7324 −1.96866
\(861\) −6.71425 −0.228821
\(862\) 24.6191 0.838531
\(863\) 38.2126 1.30077 0.650386 0.759604i \(-0.274607\pi\)
0.650386 + 0.759604i \(0.274607\pi\)
\(864\) −6.98001 −0.237465
\(865\) −8.83955 −0.300554
\(866\) 9.98929 0.339450
\(867\) 29.5632 1.00402
\(868\) −2.03855 −0.0691930
\(869\) −14.6715 −0.497695
\(870\) −14.7331 −0.499500
\(871\) 0 0
\(872\) −17.4539 −0.591063
\(873\) 1.63979 0.0554983
\(874\) −14.0877 −0.476523
\(875\) 36.9359 1.24866
\(876\) −10.6134 −0.358594
\(877\) −3.37408 −0.113935 −0.0569673 0.998376i \(-0.518143\pi\)
−0.0569673 + 0.998376i \(0.518143\pi\)
\(878\) −60.3605 −2.03707
\(879\) −30.7151 −1.03600
\(880\) −79.8270 −2.69097
\(881\) 22.2190 0.748578 0.374289 0.927312i \(-0.377887\pi\)
0.374289 + 0.927312i \(0.377887\pi\)
\(882\) 1.86754 0.0628833
\(883\) 12.9182 0.434733 0.217367 0.976090i \(-0.430253\pi\)
0.217367 + 0.976090i \(0.430253\pi\)
\(884\) 0 0
\(885\) −14.4087 −0.484344
\(886\) 36.1928 1.21592
\(887\) 8.88838 0.298443 0.149221 0.988804i \(-0.452323\pi\)
0.149221 + 0.988804i \(0.452323\pi\)
\(888\) −10.2726 −0.344726
\(889\) −0.713456 −0.0239285
\(890\) 78.1868 2.62083
\(891\) −3.88982 −0.130314
\(892\) 26.7288 0.894947
\(893\) −57.9525 −1.93931
\(894\) 14.2661 0.477128
\(895\) −53.5743 −1.79079
\(896\) −7.40285 −0.247312
\(897\) 0 0
\(898\) −67.7948 −2.26234
\(899\) 2.50851 0.0836634
\(900\) 20.1895 0.672985
\(901\) 17.9477 0.597926
\(902\) −48.7749 −1.62403
\(903\) −9.00507 −0.299670
\(904\) 9.73301 0.323715
\(905\) 48.9105 1.62584
\(906\) 1.11386 0.0370055
\(907\) 2.04682 0.0679636 0.0339818 0.999422i \(-0.489181\pi\)
0.0339818 + 0.999422i \(0.489181\pi\)
\(908\) 11.4982 0.381582
\(909\) −4.24015 −0.140637
\(910\) 0 0
\(911\) −42.2742 −1.40061 −0.700303 0.713846i \(-0.746952\pi\)
−0.700303 + 0.713846i \(0.746952\pi\)
\(912\) −28.0473 −0.928737
\(913\) −15.0611 −0.498450
\(914\) 3.89435 0.128814
\(915\) 45.0500 1.48931
\(916\) 32.3464 1.06876
\(917\) 3.47549 0.114771
\(918\) 12.7436 0.420600
\(919\) 21.0914 0.695743 0.347871 0.937542i \(-0.386905\pi\)
0.347871 + 0.937542i \(0.386905\pi\)
\(920\) −5.28076 −0.174101
\(921\) 13.1810 0.434327
\(922\) 52.5051 1.72916
\(923\) 0 0
\(924\) 5.78687 0.190374
\(925\) 145.713 4.79100
\(926\) 19.4099 0.637847
\(927\) 6.12865 0.201291
\(928\) −12.7780 −0.419460
\(929\) −14.2296 −0.466859 −0.233430 0.972374i \(-0.574995\pi\)
−0.233430 + 0.972374i \(0.574995\pi\)
\(930\) −11.0279 −0.361620
\(931\) 5.88962 0.193024
\(932\) −24.0055 −0.786327
\(933\) −22.3404 −0.731392
\(934\) 8.93208 0.292267
\(935\) 114.385 3.74078
\(936\) 0 0
\(937\) 1.73924 0.0568186 0.0284093 0.999596i \(-0.490956\pi\)
0.0284093 + 0.999596i \(0.490956\pi\)
\(938\) 13.9725 0.456219
\(939\) 26.5372 0.866008
\(940\) 63.0837 2.05756
\(941\) −9.02109 −0.294079 −0.147040 0.989131i \(-0.546974\pi\)
−0.147040 + 0.989131i \(0.546974\pi\)
\(942\) −29.6180 −0.965005
\(943\) −8.59964 −0.280043
\(944\) −15.9225 −0.518233
\(945\) 4.30941 0.140185
\(946\) −65.4163 −2.12687
\(947\) −18.8346 −0.612041 −0.306020 0.952025i \(-0.598998\pi\)
−0.306020 + 0.952025i \(0.598998\pi\)
\(948\) 5.61124 0.182244
\(949\) 0 0
\(950\) 149.269 4.84291
\(951\) −4.21944 −0.136825
\(952\) 6.52856 0.211592
\(953\) −39.3935 −1.27608 −0.638040 0.770003i \(-0.720254\pi\)
−0.638040 + 0.770003i \(0.720254\pi\)
\(954\) 4.91199 0.159032
\(955\) 16.1130 0.521405
\(956\) 28.7772 0.930720
\(957\) −7.12094 −0.230187
\(958\) −15.0787 −0.487171
\(959\) −6.49365 −0.209691
\(960\) 15.1309 0.488347
\(961\) −29.1224 −0.939431
\(962\) 0 0
\(963\) 3.75644 0.121050
\(964\) −15.1763 −0.488797
\(965\) −41.0046 −1.31998
\(966\) 2.39195 0.0769597
\(967\) −20.0823 −0.645804 −0.322902 0.946432i \(-0.604658\pi\)
−0.322902 + 0.946432i \(0.604658\pi\)
\(968\) −3.95201 −0.127022
\(969\) 40.1891 1.29106
\(970\) −13.1970 −0.423729
\(971\) −28.4437 −0.912803 −0.456401 0.889774i \(-0.650862\pi\)
−0.456401 + 0.889774i \(0.650862\pi\)
\(972\) 1.48770 0.0477179
\(973\) −9.71185 −0.311348
\(974\) −20.8295 −0.667421
\(975\) 0 0
\(976\) 49.7830 1.59351
\(977\) −36.8889 −1.18018 −0.590091 0.807337i \(-0.700908\pi\)
−0.590091 + 0.807337i \(0.700908\pi\)
\(978\) 27.9259 0.892972
\(979\) 37.7899 1.20777
\(980\) −6.41110 −0.204795
\(981\) 18.2430 0.582454
\(982\) −9.54875 −0.304713
\(983\) −17.9973 −0.574026 −0.287013 0.957927i \(-0.592662\pi\)
−0.287013 + 0.957927i \(0.592662\pi\)
\(984\) −6.42382 −0.204784
\(985\) −65.0674 −2.07322
\(986\) 23.3292 0.742952
\(987\) 9.83977 0.313203
\(988\) 0 0
\(989\) −11.5337 −0.366751
\(990\) 31.3052 0.994944
\(991\) −46.5528 −1.47880 −0.739400 0.673267i \(-0.764891\pi\)
−0.739400 + 0.673267i \(0.764891\pi\)
\(992\) −9.56452 −0.303674
\(993\) −16.1732 −0.513241
\(994\) −8.93398 −0.283369
\(995\) −18.2401 −0.578249
\(996\) 5.76026 0.182521
\(997\) 41.4216 1.31183 0.655917 0.754833i \(-0.272282\pi\)
0.655917 + 0.754833i \(0.272282\pi\)
\(998\) −72.1349 −2.28339
\(999\) 10.7371 0.339705
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 3549.2.a.bh.1.11 yes 15
13.12 even 2 3549.2.a.bg.1.5 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.bg.1.5 15 13.12 even 2
3549.2.a.bh.1.11 yes 15 1.1 even 1 trivial