Properties

Label 3630.2.a.bs.1.3
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.52625.1
Defining polynomial: \(x^{4} - x^{3} - 24 x^{2} + 19 x + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.17549\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.19863 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.19863 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +6.75608 q^{13} +3.19863 q^{14} -1.00000 q^{15} +1.00000 q^{16} -0.557453 q^{17} +1.00000 q^{18} -6.41156 q^{19} -1.00000 q^{20} +3.19863 q^{21} +2.59489 q^{23} +1.00000 q^{24} +1.00000 q^{25} +6.75608 q^{26} +1.00000 q^{27} +3.19863 q^{28} +4.89154 q^{29} -1.00000 q^{30} +2.41941 q^{31} +1.00000 q^{32} -0.557453 q^{34} -3.19863 q^{35} +1.00000 q^{36} -5.73294 q^{37} -6.41156 q^{38} +6.75608 q^{39} -1.00000 q^{40} -1.31979 q^{41} +3.19863 q^{42} +1.32139 q^{43} -1.00000 q^{45} +2.59489 q^{46} -5.99215 q^{47} +1.00000 q^{48} +3.23122 q^{49} +1.00000 q^{50} -0.557453 q^{51} +6.75608 q^{52} -2.43470 q^{53} +1.00000 q^{54} +3.19863 q^{56} -6.41156 q^{57} +4.89154 q^{58} +10.3262 q^{59} -1.00000 q^{60} -3.92512 q^{61} +2.41941 q^{62} +3.19863 q^{63} +1.00000 q^{64} -6.75608 q^{65} +15.1676 q^{67} -0.557453 q^{68} +2.59489 q^{69} -3.19863 q^{70} -6.35097 q^{71} +1.00000 q^{72} +6.94427 q^{73} -5.73294 q^{74} +1.00000 q^{75} -6.41156 q^{76} +6.75608 q^{78} -11.6724 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.31979 q^{82} +8.00000 q^{83} +3.19863 q^{84} +0.557453 q^{85} +1.32139 q^{86} +4.89154 q^{87} -13.0775 q^{89} -1.00000 q^{90} +21.6102 q^{91} +2.59489 q^{92} +2.41941 q^{93} -5.99215 q^{94} +6.41156 q^{95} +1.00000 q^{96} +7.63332 q^{97} +3.23122 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 4q^{12} + 2q^{13} + q^{14} - 4q^{15} + 4q^{16} + 11q^{17} + 4q^{18} + q^{19} - 4q^{20} + q^{21} + 4q^{24} + 4q^{25} + 2q^{26} + 4q^{27} + q^{28} + 9q^{29} - 4q^{30} + 17q^{31} + 4q^{32} + 11q^{34} - q^{35} + 4q^{36} + 8q^{37} + q^{38} + 2q^{39} - 4q^{40} - 11q^{41} + q^{42} + q^{43} - 4q^{45} + 10q^{47} + 4q^{48} + 31q^{49} + 4q^{50} + 11q^{51} + 2q^{52} + 11q^{53} + 4q^{54} + q^{56} + q^{57} + 9q^{58} + 10q^{59} - 4q^{60} - 10q^{61} + 17q^{62} + q^{63} + 4q^{64} - 2q^{65} + 9q^{67} + 11q^{68} - q^{70} + 10q^{71} + 4q^{72} - 8q^{73} + 8q^{74} + 4q^{75} + q^{76} + 2q^{78} - 7q^{79} - 4q^{80} + 4q^{81} - 11q^{82} + 32q^{83} + q^{84} - 11q^{85} + q^{86} + 9q^{87} - 23q^{89} - 4q^{90} + 48q^{91} + 17q^{93} + 10q^{94} - q^{95} + 4q^{96} - 2q^{97} + 31q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 3.19863 1.20897 0.604484 0.796618i \(-0.293379\pi\)
0.604484 + 0.796618i \(0.293379\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 6.75608 1.87380 0.936900 0.349598i \(-0.113682\pi\)
0.936900 + 0.349598i \(0.113682\pi\)
\(14\) 3.19863 0.854869
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −0.557453 −0.135202 −0.0676012 0.997712i \(-0.521535\pi\)
−0.0676012 + 0.997712i \(0.521535\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.41156 −1.47091 −0.735456 0.677573i \(-0.763032\pi\)
−0.735456 + 0.677573i \(0.763032\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.19863 0.697998
\(22\) 0 0
\(23\) 2.59489 0.541073 0.270536 0.962710i \(-0.412799\pi\)
0.270536 + 0.962710i \(0.412799\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 6.75608 1.32498
\(27\) 1.00000 0.192450
\(28\) 3.19863 0.604484
\(29\) 4.89154 0.908337 0.454168 0.890916i \(-0.349936\pi\)
0.454168 + 0.890916i \(0.349936\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.41941 0.434538 0.217269 0.976112i \(-0.430285\pi\)
0.217269 + 0.976112i \(0.430285\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.557453 −0.0956025
\(35\) −3.19863 −0.540667
\(36\) 1.00000 0.166667
\(37\) −5.73294 −0.942490 −0.471245 0.882002i \(-0.656195\pi\)
−0.471245 + 0.882002i \(0.656195\pi\)
\(38\) −6.41156 −1.04009
\(39\) 6.75608 1.08184
\(40\) −1.00000 −0.158114
\(41\) −1.31979 −0.206116 −0.103058 0.994675i \(-0.532863\pi\)
−0.103058 + 0.994675i \(0.532863\pi\)
\(42\) 3.19863 0.493559
\(43\) 1.32139 0.201509 0.100755 0.994911i \(-0.467874\pi\)
0.100755 + 0.994911i \(0.467874\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 2.59489 0.382596
\(47\) −5.99215 −0.874045 −0.437022 0.899451i \(-0.643967\pi\)
−0.437022 + 0.899451i \(0.643967\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.23122 0.461602
\(50\) 1.00000 0.141421
\(51\) −0.557453 −0.0780591
\(52\) 6.75608 0.936900
\(53\) −2.43470 −0.334431 −0.167216 0.985920i \(-0.553478\pi\)
−0.167216 + 0.985920i \(0.553478\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.19863 0.427435
\(57\) −6.41156 −0.849231
\(58\) 4.89154 0.642291
\(59\) 10.3262 1.34436 0.672181 0.740387i \(-0.265358\pi\)
0.672181 + 0.740387i \(0.265358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −3.92512 −0.502560 −0.251280 0.967914i \(-0.580851\pi\)
−0.251280 + 0.967914i \(0.580851\pi\)
\(62\) 2.41941 0.307265
\(63\) 3.19863 0.402989
\(64\) 1.00000 0.125000
\(65\) −6.75608 −0.837989
\(66\) 0 0
\(67\) 15.1676 1.85302 0.926511 0.376268i \(-0.122793\pi\)
0.926511 + 0.376268i \(0.122793\pi\)
\(68\) −0.557453 −0.0676012
\(69\) 2.59489 0.312389
\(70\) −3.19863 −0.382309
\(71\) −6.35097 −0.753722 −0.376861 0.926270i \(-0.622997\pi\)
−0.376861 + 0.926270i \(0.622997\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.94427 0.812766 0.406383 0.913703i \(-0.366790\pi\)
0.406383 + 0.913703i \(0.366790\pi\)
\(74\) −5.73294 −0.666441
\(75\) 1.00000 0.115470
\(76\) −6.41156 −0.735456
\(77\) 0 0
\(78\) 6.75608 0.764975
\(79\) −11.6724 −1.31324 −0.656622 0.754220i \(-0.728015\pi\)
−0.656622 + 0.754220i \(0.728015\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −1.31979 −0.145746
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 3.19863 0.348999
\(85\) 0.557453 0.0604643
\(86\) 1.32139 0.142489
\(87\) 4.89154 0.524428
\(88\) 0 0
\(89\) −13.0775 −1.38621 −0.693104 0.720837i \(-0.743757\pi\)
−0.693104 + 0.720837i \(0.743757\pi\)
\(90\) −1.00000 −0.105409
\(91\) 21.6102 2.26536
\(92\) 2.59489 0.270536
\(93\) 2.41941 0.250881
\(94\) −5.99215 −0.618043
\(95\) 6.41156 0.657812
\(96\) 1.00000 0.102062
\(97\) 7.63332 0.775046 0.387523 0.921860i \(-0.373331\pi\)
0.387523 + 0.921860i \(0.373331\pi\)
\(98\) 3.23122 0.326402
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −15.9739 −1.58946 −0.794729 0.606964i \(-0.792387\pi\)
−0.794729 + 0.606964i \(0.792387\pi\)
\(102\) −0.557453 −0.0551961
\(103\) 18.2904 1.80221 0.901103 0.433605i \(-0.142759\pi\)
0.901103 + 0.433605i \(0.142759\pi\)
\(104\) 6.75608 0.662488
\(105\) −3.19863 −0.312154
\(106\) −2.43470 −0.236478
\(107\) −17.1149 −1.65456 −0.827280 0.561789i \(-0.810113\pi\)
−0.827280 + 0.561789i \(0.810113\pi\)
\(108\) 1.00000 0.0962250
\(109\) −8.94427 −0.856706 −0.428353 0.903612i \(-0.640906\pi\)
−0.428353 + 0.903612i \(0.640906\pi\)
\(110\) 0 0
\(111\) −5.73294 −0.544147
\(112\) 3.19863 0.302242
\(113\) 4.81666 0.453113 0.226557 0.973998i \(-0.427253\pi\)
0.226557 + 0.973998i \(0.427253\pi\)
\(114\) −6.41156 −0.600497
\(115\) −2.59489 −0.241975
\(116\) 4.89154 0.454168
\(117\) 6.75608 0.624600
\(118\) 10.3262 0.950607
\(119\) −1.78309 −0.163455
\(120\) −1.00000 −0.0912871
\(121\) 0 0
\(122\) −3.92512 −0.355364
\(123\) −1.31979 −0.119001
\(124\) 2.41941 0.217269
\(125\) −1.00000 −0.0894427
\(126\) 3.19863 0.284956
\(127\) 16.2155 1.43889 0.719447 0.694547i \(-0.244395\pi\)
0.719447 + 0.694547i \(0.244395\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.32139 0.116342
\(130\) −6.75608 −0.592547
\(131\) −17.2888 −1.51053 −0.755265 0.655420i \(-0.772492\pi\)
−0.755265 + 0.655420i \(0.772492\pi\)
\(132\) 0 0
\(133\) −20.5082 −1.77828
\(134\) 15.1676 1.31028
\(135\) −1.00000 −0.0860663
\(136\) −0.557453 −0.0478012
\(137\) 7.93157 0.677640 0.338820 0.940851i \(-0.389972\pi\)
0.338820 + 0.940851i \(0.389972\pi\)
\(138\) 2.59489 0.220892
\(139\) 8.21778 0.697023 0.348512 0.937304i \(-0.386687\pi\)
0.348512 + 0.937304i \(0.386687\pi\)
\(140\) −3.19863 −0.270333
\(141\) −5.99215 −0.504630
\(142\) −6.35097 −0.532962
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.89154 −0.406221
\(146\) 6.94427 0.574712
\(147\) 3.23122 0.266506
\(148\) −5.73294 −0.471245
\(149\) 6.08532 0.498529 0.249264 0.968435i \(-0.419811\pi\)
0.249264 + 0.968435i \(0.419811\pi\)
\(150\) 1.00000 0.0816497
\(151\) −18.2761 −1.48729 −0.743644 0.668576i \(-0.766904\pi\)
−0.743644 + 0.668576i \(0.766904\pi\)
\(152\) −6.41156 −0.520046
\(153\) −0.557453 −0.0450674
\(154\) 0 0
\(155\) −2.41941 −0.194331
\(156\) 6.75608 0.540919
\(157\) 18.4874 1.47546 0.737729 0.675097i \(-0.235898\pi\)
0.737729 + 0.675097i \(0.235898\pi\)
\(158\) −11.6724 −0.928603
\(159\) −2.43470 −0.193084
\(160\) −1.00000 −0.0790569
\(161\) 8.30010 0.654139
\(162\) 1.00000 0.0785674
\(163\) 11.5480 0.904510 0.452255 0.891889i \(-0.350620\pi\)
0.452255 + 0.891889i \(0.350620\pi\)
\(164\) −1.31979 −0.103058
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) −2.95212 −0.228442 −0.114221 0.993455i \(-0.536437\pi\)
−0.114221 + 0.993455i \(0.536437\pi\)
\(168\) 3.19863 0.246779
\(169\) 32.6446 2.51112
\(170\) 0.557453 0.0427547
\(171\) −6.41156 −0.490304
\(172\) 1.32139 0.100755
\(173\) −1.84140 −0.139999 −0.0699994 0.997547i \(-0.522300\pi\)
−0.0699994 + 0.997547i \(0.522300\pi\)
\(174\) 4.89154 0.370827
\(175\) 3.19863 0.241793
\(176\) 0 0
\(177\) 10.3262 0.776168
\(178\) −13.0775 −0.980198
\(179\) −18.5623 −1.38741 −0.693706 0.720258i \(-0.744023\pi\)
−0.693706 + 0.720258i \(0.744023\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 9.52786 0.708201 0.354100 0.935207i \(-0.384787\pi\)
0.354100 + 0.935207i \(0.384787\pi\)
\(182\) 21.6102 1.60185
\(183\) −3.92512 −0.290153
\(184\) 2.59489 0.191298
\(185\) 5.73294 0.421494
\(186\) 2.41941 0.177400
\(187\) 0 0
\(188\) −5.99215 −0.437022
\(189\) 3.19863 0.232666
\(190\) 6.41156 0.465143
\(191\) 0.913695 0.0661127 0.0330563 0.999453i \(-0.489476\pi\)
0.0330563 + 0.999453i \(0.489476\pi\)
\(192\) 1.00000 0.0721688
\(193\) −23.2952 −1.67683 −0.838414 0.545033i \(-0.816517\pi\)
−0.838414 + 0.545033i \(0.816517\pi\)
\(194\) 7.63332 0.548041
\(195\) −6.75608 −0.483813
\(196\) 3.23122 0.230801
\(197\) 24.5896 1.75194 0.875969 0.482367i \(-0.160223\pi\)
0.875969 + 0.482367i \(0.160223\pi\)
\(198\) 0 0
\(199\) 10.5112 0.745117 0.372559 0.928009i \(-0.378481\pi\)
0.372559 + 0.928009i \(0.378481\pi\)
\(200\) 1.00000 0.0707107
\(201\) 15.1676 1.06984
\(202\) −15.9739 −1.12392
\(203\) 15.6462 1.09815
\(204\) −0.557453 −0.0390295
\(205\) 1.31979 0.0921780
\(206\) 18.2904 1.27435
\(207\) 2.59489 0.180358
\(208\) 6.75608 0.468450
\(209\) 0 0
\(210\) −3.19863 −0.220726
\(211\) 7.06863 0.486624 0.243312 0.969948i \(-0.421766\pi\)
0.243312 + 0.969948i \(0.421766\pi\)
\(212\) −2.43470 −0.167216
\(213\) −6.35097 −0.435162
\(214\) −17.1149 −1.16995
\(215\) −1.32139 −0.0901177
\(216\) 1.00000 0.0680414
\(217\) 7.73878 0.525343
\(218\) −8.94427 −0.605783
\(219\) 6.94427 0.469250
\(220\) 0 0
\(221\) −3.76620 −0.253342
\(222\) −5.73294 −0.384770
\(223\) 0.801373 0.0536639 0.0268319 0.999640i \(-0.491458\pi\)
0.0268319 + 0.999640i \(0.491458\pi\)
\(224\) 3.19863 0.213717
\(225\) 1.00000 0.0666667
\(226\) 4.81666 0.320400
\(227\) 4.86939 0.323193 0.161596 0.986857i \(-0.448336\pi\)
0.161596 + 0.986857i \(0.448336\pi\)
\(228\) −6.41156 −0.424616
\(229\) −1.16119 −0.0767333 −0.0383667 0.999264i \(-0.512215\pi\)
−0.0383667 + 0.999264i \(0.512215\pi\)
\(230\) −2.59489 −0.171102
\(231\) 0 0
\(232\) 4.89154 0.321146
\(233\) −12.0527 −0.789601 −0.394800 0.918767i \(-0.629186\pi\)
−0.394800 + 0.918767i \(0.629186\pi\)
\(234\) 6.75608 0.441659
\(235\) 5.99215 0.390885
\(236\) 10.3262 0.672181
\(237\) −11.6724 −0.758201
\(238\) −1.78309 −0.115580
\(239\) 28.2604 1.82801 0.914006 0.405700i \(-0.132972\pi\)
0.914006 + 0.405700i \(0.132972\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −24.7857 −1.59659 −0.798293 0.602270i \(-0.794263\pi\)
−0.798293 + 0.602270i \(0.794263\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −3.92512 −0.251280
\(245\) −3.23122 −0.206435
\(246\) −1.31979 −0.0841466
\(247\) −43.3170 −2.75619
\(248\) 2.41941 0.153632
\(249\) 8.00000 0.506979
\(250\) −1.00000 −0.0632456
\(251\) −17.8615 −1.12741 −0.563705 0.825976i \(-0.690625\pi\)
−0.563705 + 0.825976i \(0.690625\pi\)
\(252\) 3.19863 0.201495
\(253\) 0 0
\(254\) 16.2155 1.01745
\(255\) 0.557453 0.0349091
\(256\) 1.00000 0.0625000
\(257\) 4.32237 0.269622 0.134811 0.990871i \(-0.456957\pi\)
0.134811 + 0.990871i \(0.456957\pi\)
\(258\) 1.32139 0.0822659
\(259\) −18.3375 −1.13944
\(260\) −6.75608 −0.418994
\(261\) 4.89154 0.302779
\(262\) −17.2888 −1.06811
\(263\) −23.0282 −1.41998 −0.709989 0.704212i \(-0.751300\pi\)
−0.709989 + 0.704212i \(0.751300\pi\)
\(264\) 0 0
\(265\) 2.43470 0.149562
\(266\) −20.5082 −1.25744
\(267\) −13.0775 −0.800328
\(268\) 15.1676 0.926511
\(269\) −11.8567 −0.722915 −0.361457 0.932389i \(-0.617721\pi\)
−0.361457 + 0.932389i \(0.617721\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −11.4268 −0.694131 −0.347066 0.937841i \(-0.612822\pi\)
−0.347066 + 0.937841i \(0.612822\pi\)
\(272\) −0.557453 −0.0338006
\(273\) 21.6102 1.30791
\(274\) 7.93157 0.479164
\(275\) 0 0
\(276\) 2.59489 0.156194
\(277\) −15.0384 −0.903572 −0.451786 0.892126i \(-0.649213\pi\)
−0.451786 + 0.892126i \(0.649213\pi\)
\(278\) 8.21778 0.492870
\(279\) 2.41941 0.144846
\(280\) −3.19863 −0.191155
\(281\) 13.2666 0.791422 0.395711 0.918375i \(-0.370498\pi\)
0.395711 + 0.918375i \(0.370498\pi\)
\(282\) −5.99215 −0.356827
\(283\) 15.1214 0.898871 0.449436 0.893313i \(-0.351625\pi\)
0.449436 + 0.893313i \(0.351625\pi\)
\(284\) −6.35097 −0.376861
\(285\) 6.41156 0.379788
\(286\) 0 0
\(287\) −4.22151 −0.249188
\(288\) 1.00000 0.0589256
\(289\) −16.6892 −0.981720
\(290\) −4.89154 −0.287241
\(291\) 7.63332 0.447473
\(292\) 6.94427 0.406383
\(293\) −13.9163 −0.812998 −0.406499 0.913651i \(-0.633250\pi\)
−0.406499 + 0.913651i \(0.633250\pi\)
\(294\) 3.23122 0.188448
\(295\) −10.3262 −0.601217
\(296\) −5.73294 −0.333220
\(297\) 0 0
\(298\) 6.08532 0.352513
\(299\) 17.5313 1.01386
\(300\) 1.00000 0.0577350
\(301\) 4.22662 0.243618
\(302\) −18.2761 −1.05167
\(303\) −15.9739 −0.917674
\(304\) −6.41156 −0.367728
\(305\) 3.92512 0.224752
\(306\) −0.557453 −0.0318675
\(307\) 24.0813 1.37439 0.687197 0.726471i \(-0.258841\pi\)
0.687197 + 0.726471i \(0.258841\pi\)
\(308\) 0 0
\(309\) 18.2904 1.04050
\(310\) −2.41941 −0.137413
\(311\) −30.3816 −1.72278 −0.861390 0.507944i \(-0.830406\pi\)
−0.861390 + 0.507944i \(0.830406\pi\)
\(312\) 6.75608 0.382488
\(313\) 16.2012 0.915747 0.457873 0.889017i \(-0.348611\pi\)
0.457873 + 0.889017i \(0.348611\pi\)
\(314\) 18.4874 1.04331
\(315\) −3.19863 −0.180222
\(316\) −11.6724 −0.656622
\(317\) −33.0640 −1.85706 −0.928531 0.371256i \(-0.878927\pi\)
−0.928531 + 0.371256i \(0.878927\pi\)
\(318\) −2.43470 −0.136531
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −17.1149 −0.955261
\(322\) 8.30010 0.462546
\(323\) 3.57414 0.198871
\(324\) 1.00000 0.0555556
\(325\) 6.75608 0.374760
\(326\) 11.5480 0.639585
\(327\) −8.94427 −0.494619
\(328\) −1.31979 −0.0728731
\(329\) −19.1666 −1.05669
\(330\) 0 0
\(331\) −0.733928 −0.0403403 −0.0201702 0.999797i \(-0.506421\pi\)
−0.0201702 + 0.999797i \(0.506421\pi\)
\(332\) 8.00000 0.439057
\(333\) −5.73294 −0.314163
\(334\) −2.95212 −0.161533
\(335\) −15.1676 −0.828696
\(336\) 3.19863 0.174499
\(337\) −16.3816 −0.892360 −0.446180 0.894943i \(-0.647216\pi\)
−0.446180 + 0.894943i \(0.647216\pi\)
\(338\) 32.6446 1.77563
\(339\) 4.81666 0.261605
\(340\) 0.557453 0.0302322
\(341\) 0 0
\(342\) −6.41156 −0.346697
\(343\) −12.0549 −0.650905
\(344\) 1.32139 0.0712443
\(345\) −2.59489 −0.139704
\(346\) −1.84140 −0.0989942
\(347\) −16.2012 −0.869727 −0.434863 0.900496i \(-0.643203\pi\)
−0.434863 + 0.900496i \(0.643203\pi\)
\(348\) 4.89154 0.262214
\(349\) 4.19604 0.224609 0.112305 0.993674i \(-0.464177\pi\)
0.112305 + 0.993674i \(0.464177\pi\)
\(350\) 3.19863 0.170974
\(351\) 6.75608 0.360613
\(352\) 0 0
\(353\) 3.32139 0.176780 0.0883898 0.996086i \(-0.471828\pi\)
0.0883898 + 0.996086i \(0.471828\pi\)
\(354\) 10.3262 0.548833
\(355\) 6.35097 0.337075
\(356\) −13.0775 −0.693104
\(357\) −1.78309 −0.0943709
\(358\) −18.5623 −0.981048
\(359\) −22.1803 −1.17063 −0.585317 0.810805i \(-0.699030\pi\)
−0.585317 + 0.810805i \(0.699030\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 22.1080 1.16358
\(362\) 9.52786 0.500773
\(363\) 0 0
\(364\) 21.6102 1.13268
\(365\) −6.94427 −0.363480
\(366\) −3.92512 −0.205169
\(367\) −17.0497 −0.889989 −0.444994 0.895533i \(-0.646794\pi\)
−0.444994 + 0.895533i \(0.646794\pi\)
\(368\) 2.59489 0.135268
\(369\) −1.31979 −0.0687054
\(370\) 5.73294 0.298041
\(371\) −7.78768 −0.404316
\(372\) 2.41941 0.125440
\(373\) −0.231215 −0.0119719 −0.00598594 0.999982i \(-0.501905\pi\)
−0.00598594 + 0.999982i \(0.501905\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −5.99215 −0.309021
\(377\) 33.0477 1.70204
\(378\) 3.19863 0.164520
\(379\) −11.9723 −0.614974 −0.307487 0.951552i \(-0.599488\pi\)
−0.307487 + 0.951552i \(0.599488\pi\)
\(380\) 6.41156 0.328906
\(381\) 16.2155 0.830746
\(382\) 0.913695 0.0467487
\(383\) 3.07102 0.156922 0.0784608 0.996917i \(-0.474999\pi\)
0.0784608 + 0.996917i \(0.474999\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −23.2952 −1.18570
\(387\) 1.32139 0.0671698
\(388\) 7.63332 0.387523
\(389\) 19.7569 1.00172 0.500858 0.865529i \(-0.333018\pi\)
0.500858 + 0.865529i \(0.333018\pi\)
\(390\) −6.75608 −0.342107
\(391\) −1.44653 −0.0731543
\(392\) 3.23122 0.163201
\(393\) −17.2888 −0.872104
\(394\) 24.5896 1.23881
\(395\) 11.6724 0.587300
\(396\) 0 0
\(397\) −11.4546 −0.574889 −0.287444 0.957797i \(-0.592806\pi\)
−0.287444 + 0.957797i \(0.592806\pi\)
\(398\) 10.5112 0.526878
\(399\) −20.5082 −1.02669
\(400\) 1.00000 0.0500000
\(401\) 22.3001 1.11361 0.556807 0.830642i \(-0.312026\pi\)
0.556807 + 0.830642i \(0.312026\pi\)
\(402\) 15.1676 0.756493
\(403\) 16.3457 0.814238
\(404\) −15.9739 −0.794729
\(405\) −1.00000 −0.0496904
\(406\) 15.6462 0.776509
\(407\) 0 0
\(408\) −0.557453 −0.0275981
\(409\) 36.2811 1.79399 0.896993 0.442044i \(-0.145746\pi\)
0.896993 + 0.442044i \(0.145746\pi\)
\(410\) 1.31979 0.0651797
\(411\) 7.93157 0.391235
\(412\) 18.2904 0.901103
\(413\) 33.0298 1.62529
\(414\) 2.59489 0.127532
\(415\) −8.00000 −0.392705
\(416\) 6.75608 0.331244
\(417\) 8.21778 0.402426
\(418\) 0 0
\(419\) 9.06157 0.442687 0.221343 0.975196i \(-0.428956\pi\)
0.221343 + 0.975196i \(0.428956\pi\)
\(420\) −3.19863 −0.156077
\(421\) −2.42586 −0.118229 −0.0591145 0.998251i \(-0.518828\pi\)
−0.0591145 + 0.998251i \(0.518828\pi\)
\(422\) 7.06863 0.344095
\(423\) −5.99215 −0.291348
\(424\) −2.43470 −0.118239
\(425\) −0.557453 −0.0270405
\(426\) −6.35097 −0.307706
\(427\) −12.5550 −0.607579
\(428\) −17.1149 −0.827280
\(429\) 0 0
\(430\) −1.32139 −0.0637229
\(431\) −2.65847 −0.128054 −0.0640271 0.997948i \(-0.520394\pi\)
−0.0640271 + 0.997948i \(0.520394\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.1918 −0.922297 −0.461149 0.887323i \(-0.652563\pi\)
−0.461149 + 0.887323i \(0.652563\pi\)
\(434\) 7.73878 0.371473
\(435\) −4.89154 −0.234532
\(436\) −8.94427 −0.428353
\(437\) −16.6373 −0.795870
\(438\) 6.94427 0.331810
\(439\) −12.6955 −0.605923 −0.302962 0.953003i \(-0.597975\pi\)
−0.302962 + 0.953003i \(0.597975\pi\)
\(440\) 0 0
\(441\) 3.23122 0.153867
\(442\) −3.76620 −0.179140
\(443\) 16.9189 0.803839 0.401920 0.915675i \(-0.368343\pi\)
0.401920 + 0.915675i \(0.368343\pi\)
\(444\) −5.73294 −0.272073
\(445\) 13.0775 0.619931
\(446\) 0.801373 0.0379461
\(447\) 6.08532 0.287826
\(448\) 3.19863 0.151121
\(449\) −19.2809 −0.909924 −0.454962 0.890511i \(-0.650347\pi\)
−0.454962 + 0.890511i \(0.650347\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 4.81666 0.226557
\(453\) −18.2761 −0.858686
\(454\) 4.86939 0.228532
\(455\) −21.6102 −1.01310
\(456\) −6.41156 −0.300249
\(457\) −39.5219 −1.84875 −0.924377 0.381479i \(-0.875415\pi\)
−0.924377 + 0.381479i \(0.875415\pi\)
\(458\) −1.16119 −0.0542587
\(459\) −0.557453 −0.0260197
\(460\) −2.59489 −0.120988
\(461\) −13.8398 −0.644584 −0.322292 0.946640i \(-0.604453\pi\)
−0.322292 + 0.946640i \(0.604453\pi\)
\(462\) 0 0
\(463\) 12.5919 0.585195 0.292598 0.956236i \(-0.405480\pi\)
0.292598 + 0.956236i \(0.405480\pi\)
\(464\) 4.89154 0.227084
\(465\) −2.41941 −0.112197
\(466\) −12.0527 −0.558332
\(467\) −11.7368 −0.543115 −0.271557 0.962422i \(-0.587539\pi\)
−0.271557 + 0.962422i \(0.587539\pi\)
\(468\) 6.75608 0.312300
\(469\) 48.5156 2.24024
\(470\) 5.99215 0.276397
\(471\) 18.4874 0.851856
\(472\) 10.3262 0.475304
\(473\) 0 0
\(474\) −11.6724 −0.536129
\(475\) −6.41156 −0.294182
\(476\) −1.78309 −0.0817276
\(477\) −2.43470 −0.111477
\(478\) 28.2604 1.29260
\(479\) −21.4007 −0.977823 −0.488912 0.872333i \(-0.662606\pi\)
−0.488912 + 0.872333i \(0.662606\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −38.7322 −1.76604
\(482\) −24.7857 −1.12896
\(483\) 8.30010 0.377668
\(484\) 0 0
\(485\) −7.63332 −0.346611
\(486\) 1.00000 0.0453609
\(487\) −15.8502 −0.718243 −0.359121 0.933291i \(-0.616924\pi\)
−0.359121 + 0.933291i \(0.616924\pi\)
\(488\) −3.92512 −0.177682
\(489\) 11.5480 0.522219
\(490\) −3.23122 −0.145971
\(491\) −27.6698 −1.24872 −0.624360 0.781137i \(-0.714640\pi\)
−0.624360 + 0.781137i \(0.714640\pi\)
\(492\) −1.31979 −0.0595007
\(493\) −2.72681 −0.122809
\(494\) −43.3170 −1.94892
\(495\) 0 0
\(496\) 2.41941 0.108635
\(497\) −20.3144 −0.911225
\(498\) 8.00000 0.358489
\(499\) −7.06176 −0.316128 −0.158064 0.987429i \(-0.550525\pi\)
−0.158064 + 0.987429i \(0.550525\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −2.95212 −0.131891
\(502\) −17.8615 −0.797199
\(503\) −1.15333 −0.0514247 −0.0257123 0.999669i \(-0.508185\pi\)
−0.0257123 + 0.999669i \(0.508185\pi\)
\(504\) 3.19863 0.142478
\(505\) 15.9739 0.710827
\(506\) 0 0
\(507\) 32.6446 1.44980
\(508\) 16.2155 0.719447
\(509\) −8.47539 −0.375665 −0.187833 0.982201i \(-0.560146\pi\)
−0.187833 + 0.982201i \(0.560146\pi\)
\(510\) 0.557453 0.0246845
\(511\) 22.2121 0.982607
\(512\) 1.00000 0.0441942
\(513\) −6.41156 −0.283077
\(514\) 4.32237 0.190652
\(515\) −18.2904 −0.805971
\(516\) 1.32139 0.0581708
\(517\) 0 0
\(518\) −18.3375 −0.805705
\(519\) −1.84140 −0.0808284
\(520\) −6.75608 −0.296274
\(521\) 31.0618 1.36084 0.680420 0.732822i \(-0.261797\pi\)
0.680420 + 0.732822i \(0.261797\pi\)
\(522\) 4.89154 0.214097
\(523\) −19.8326 −0.867217 −0.433609 0.901101i \(-0.642760\pi\)
−0.433609 + 0.901101i \(0.642760\pi\)
\(524\) −17.2888 −0.755265
\(525\) 3.19863 0.139600
\(526\) −23.0282 −1.00408
\(527\) −1.34871 −0.0587506
\(528\) 0 0
\(529\) −16.2665 −0.707240
\(530\) 2.43470 0.105756
\(531\) 10.3262 0.448121
\(532\) −20.5082 −0.889142
\(533\) −8.91660 −0.386221
\(534\) −13.0775 −0.565917
\(535\) 17.1149 0.739942
\(536\) 15.1676 0.655142
\(537\) −18.5623 −0.801023
\(538\) −11.8567 −0.511178
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −25.2333 −1.08486 −0.542431 0.840100i \(-0.682496\pi\)
−0.542431 + 0.840100i \(0.682496\pi\)
\(542\) −11.4268 −0.490825
\(543\) 9.52786 0.408880
\(544\) −0.557453 −0.0239006
\(545\) 8.94427 0.383131
\(546\) 21.6102 0.924830
\(547\) −13.4900 −0.576791 −0.288396 0.957511i \(-0.593122\pi\)
−0.288396 + 0.957511i \(0.593122\pi\)
\(548\) 7.93157 0.338820
\(549\) −3.92512 −0.167520
\(550\) 0 0
\(551\) −31.3624 −1.33608
\(552\) 2.59489 0.110446
\(553\) −37.3355 −1.58767
\(554\) −15.0384 −0.638922
\(555\) 5.73294 0.243350
\(556\) 8.21778 0.348512
\(557\) −38.4550 −1.62939 −0.814696 0.579888i \(-0.803096\pi\)
−0.814696 + 0.579888i \(0.803096\pi\)
\(558\) 2.41941 0.102422
\(559\) 8.92739 0.377588
\(560\) −3.19863 −0.135167
\(561\) 0 0
\(562\) 13.2666 0.559620
\(563\) 27.8423 1.17341 0.586706 0.809800i \(-0.300425\pi\)
0.586706 + 0.809800i \(0.300425\pi\)
\(564\) −5.99215 −0.252315
\(565\) −4.81666 −0.202638
\(566\) 15.1214 0.635598
\(567\) 3.19863 0.134330
\(568\) −6.35097 −0.266481
\(569\) 8.26209 0.346365 0.173182 0.984890i \(-0.444595\pi\)
0.173182 + 0.984890i \(0.444595\pi\)
\(570\) 6.41156 0.268551
\(571\) 6.71079 0.280838 0.140419 0.990092i \(-0.455155\pi\)
0.140419 + 0.990092i \(0.455155\pi\)
\(572\) 0 0
\(573\) 0.913695 0.0381702
\(574\) −4.22151 −0.176202
\(575\) 2.59489 0.108215
\(576\) 1.00000 0.0416667
\(577\) −39.8137 −1.65746 −0.828732 0.559645i \(-0.810937\pi\)
−0.828732 + 0.559645i \(0.810937\pi\)
\(578\) −16.6892 −0.694181
\(579\) −23.2952 −0.968117
\(580\) −4.89154 −0.203110
\(581\) 25.5890 1.06161
\(582\) 7.63332 0.316411
\(583\) 0 0
\(584\) 6.94427 0.287356
\(585\) −6.75608 −0.279330
\(586\) −13.9163 −0.574876
\(587\) −10.0592 −0.415187 −0.207593 0.978215i \(-0.566563\pi\)
−0.207593 + 0.978215i \(0.566563\pi\)
\(588\) 3.23122 0.133253
\(589\) −15.5122 −0.639167
\(590\) −10.3262 −0.425124
\(591\) 24.5896 1.01148
\(592\) −5.73294 −0.235622
\(593\) 23.6430 0.970900 0.485450 0.874264i \(-0.338656\pi\)
0.485450 + 0.874264i \(0.338656\pi\)
\(594\) 0 0
\(595\) 1.78309 0.0730994
\(596\) 6.08532 0.249264
\(597\) 10.5112 0.430194
\(598\) 17.5313 0.716909
\(599\) 31.4007 1.28300 0.641499 0.767124i \(-0.278313\pi\)
0.641499 + 0.767124i \(0.278313\pi\)
\(600\) 1.00000 0.0408248
\(601\) −17.1469 −0.699436 −0.349718 0.936855i \(-0.613723\pi\)
−0.349718 + 0.936855i \(0.613723\pi\)
\(602\) 4.22662 0.172264
\(603\) 15.1676 0.617674
\(604\) −18.2761 −0.743644
\(605\) 0 0
\(606\) −15.9739 −0.648894
\(607\) 1.40153 0.0568865 0.0284433 0.999595i \(-0.490945\pi\)
0.0284433 + 0.999595i \(0.490945\pi\)
\(608\) −6.41156 −0.260023
\(609\) 15.6462 0.634017
\(610\) 3.92512 0.158923
\(611\) −40.4834 −1.63778
\(612\) −0.557453 −0.0225337
\(613\) −12.0383 −0.486223 −0.243111 0.969998i \(-0.578168\pi\)
−0.243111 + 0.969998i \(0.578168\pi\)
\(614\) 24.0813 0.971843
\(615\) 1.31979 0.0532190
\(616\) 0 0
\(617\) −5.25177 −0.211428 −0.105714 0.994397i \(-0.533713\pi\)
−0.105714 + 0.994397i \(0.533713\pi\)
\(618\) 18.2904 0.735748
\(619\) 42.0324 1.68942 0.844712 0.535221i \(-0.179772\pi\)
0.844712 + 0.535221i \(0.179772\pi\)
\(620\) −2.41941 −0.0971657
\(621\) 2.59489 0.104130
\(622\) −30.3816 −1.21819
\(623\) −41.8299 −1.67588
\(624\) 6.75608 0.270460
\(625\) 1.00000 0.0400000
\(626\) 16.2012 0.647531
\(627\) 0 0
\(628\) 18.4874 0.737729
\(629\) 3.19585 0.127427
\(630\) −3.19863 −0.127436
\(631\) −16.9950 −0.676561 −0.338281 0.941045i \(-0.609845\pi\)
−0.338281 + 0.941045i \(0.609845\pi\)
\(632\) −11.6724 −0.464302
\(633\) 7.06863 0.280953
\(634\) −33.0640 −1.31314
\(635\) −16.2155 −0.643493
\(636\) −2.43470 −0.0965419
\(637\) 21.8304 0.864950
\(638\) 0 0
\(639\) −6.35097 −0.251241
\(640\) −1.00000 −0.0395285
\(641\) −49.9080 −1.97125 −0.985624 0.168954i \(-0.945961\pi\)
−0.985624 + 0.168954i \(0.945961\pi\)
\(642\) −17.1149 −0.675471
\(643\) −26.7579 −1.05523 −0.527614 0.849484i \(-0.676913\pi\)
−0.527614 + 0.849484i \(0.676913\pi\)
\(644\) 8.30010 0.327070
\(645\) −1.32139 −0.0520295
\(646\) 3.57414 0.140623
\(647\) −7.85550 −0.308832 −0.154416 0.988006i \(-0.549350\pi\)
−0.154416 + 0.988006i \(0.549350\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 6.75608 0.264995
\(651\) 7.73878 0.303307
\(652\) 11.5480 0.452255
\(653\) 0.290394 0.0113640 0.00568200 0.999984i \(-0.498191\pi\)
0.00568200 + 0.999984i \(0.498191\pi\)
\(654\) −8.94427 −0.349749
\(655\) 17.2888 0.675529
\(656\) −1.31979 −0.0515291
\(657\) 6.94427 0.270922
\(658\) −19.1666 −0.747194
\(659\) 17.7202 0.690282 0.345141 0.938551i \(-0.387831\pi\)
0.345141 + 0.938551i \(0.387831\pi\)
\(660\) 0 0
\(661\) 22.9494 0.892630 0.446315 0.894876i \(-0.352736\pi\)
0.446315 + 0.894876i \(0.352736\pi\)
\(662\) −0.733928 −0.0285249
\(663\) −3.76620 −0.146267
\(664\) 8.00000 0.310460
\(665\) 20.5082 0.795273
\(666\) −5.73294 −0.222147
\(667\) 12.6930 0.491476
\(668\) −2.95212 −0.114221
\(669\) 0.801373 0.0309829
\(670\) −15.1676 −0.585977
\(671\) 0 0
\(672\) 3.19863 0.123390
\(673\) 35.3081 1.36103 0.680515 0.732735i \(-0.261756\pi\)
0.680515 + 0.732735i \(0.261756\pi\)
\(674\) −16.3816 −0.630994
\(675\) 1.00000 0.0384900
\(676\) 32.6446 1.25556
\(677\) 42.2238 1.62279 0.811397 0.584496i \(-0.198708\pi\)
0.811397 + 0.584496i \(0.198708\pi\)
\(678\) 4.81666 0.184983
\(679\) 24.4162 0.937006
\(680\) 0.557453 0.0213774
\(681\) 4.86939 0.186595
\(682\) 0 0
\(683\) 34.7325 1.32900 0.664502 0.747287i \(-0.268644\pi\)
0.664502 + 0.747287i \(0.268644\pi\)
\(684\) −6.41156 −0.245152
\(685\) −7.93157 −0.303050
\(686\) −12.0549 −0.460260
\(687\) −1.16119 −0.0443020
\(688\) 1.32139 0.0503773
\(689\) −16.4490 −0.626657
\(690\) −2.59489 −0.0987859
\(691\) 16.7185 0.636002 0.318001 0.948090i \(-0.396988\pi\)
0.318001 + 0.948090i \(0.396988\pi\)
\(692\) −1.84140 −0.0699994
\(693\) 0 0
\(694\) −16.2012 −0.614990
\(695\) −8.21778 −0.311718
\(696\) 4.89154 0.185413
\(697\) 0.735720 0.0278674
\(698\) 4.19604 0.158823
\(699\) −12.0527 −0.455876
\(700\) 3.19863 0.120897
\(701\) 0.639577 0.0241565 0.0120782 0.999927i \(-0.496155\pi\)
0.0120782 + 0.999927i \(0.496155\pi\)
\(702\) 6.75608 0.254992
\(703\) 36.7571 1.38632
\(704\) 0 0
\(705\) 5.99215 0.225677
\(706\) 3.32139 0.125002
\(707\) −51.0944 −1.92160
\(708\) 10.3262 0.388084
\(709\) 12.6396 0.474689 0.237345 0.971426i \(-0.423723\pi\)
0.237345 + 0.971426i \(0.423723\pi\)
\(710\) 6.35097 0.238348
\(711\) −11.6724 −0.437748
\(712\) −13.0775 −0.490099
\(713\) 6.27811 0.235117
\(714\) −1.78309 −0.0667303
\(715\) 0 0
\(716\) −18.5623 −0.693706
\(717\) 28.2604 1.05540
\(718\) −22.1803 −0.827763
\(719\) 41.0960 1.53262 0.766311 0.642469i \(-0.222090\pi\)
0.766311 + 0.642469i \(0.222090\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 58.5042 2.17881
\(722\) 22.1080 0.822776
\(723\) −24.7857 −0.921789
\(724\) 9.52786 0.354100
\(725\) 4.89154 0.181667
\(726\) 0 0
\(727\) −15.8668 −0.588467 −0.294234 0.955734i \(-0.595064\pi\)
−0.294234 + 0.955734i \(0.595064\pi\)
\(728\) 21.6102 0.800927
\(729\) 1.00000 0.0370370
\(730\) −6.94427 −0.257019
\(731\) −0.736611 −0.0272445
\(732\) −3.92512 −0.145077
\(733\) −20.7873 −0.767795 −0.383898 0.923376i \(-0.625418\pi\)
−0.383898 + 0.923376i \(0.625418\pi\)
\(734\) −17.0497 −0.629317
\(735\) −3.23122 −0.119185
\(736\) 2.59489 0.0956491
\(737\) 0 0
\(738\) −1.31979 −0.0485821
\(739\) 34.6150 1.27333 0.636667 0.771139i \(-0.280312\pi\)
0.636667 + 0.771139i \(0.280312\pi\)
\(740\) 5.73294 0.210747
\(741\) −43.3170 −1.59129
\(742\) −7.78768 −0.285895
\(743\) 41.2454 1.51315 0.756573 0.653910i \(-0.226872\pi\)
0.756573 + 0.653910i \(0.226872\pi\)
\(744\) 2.41941 0.0886998
\(745\) −6.08532 −0.222949
\(746\) −0.231215 −0.00846540
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) −54.7442 −2.00031
\(750\) −1.00000 −0.0365148
\(751\) 3.35796 0.122534 0.0612668 0.998121i \(-0.480486\pi\)
0.0612668 + 0.998121i \(0.480486\pi\)
\(752\) −5.99215 −0.218511
\(753\) −17.8615 −0.650911
\(754\) 33.0477 1.20352
\(755\) 18.2761 0.665135
\(756\) 3.19863 0.116333
\(757\) 14.7247 0.535177 0.267589 0.963533i \(-0.413773\pi\)
0.267589 + 0.963533i \(0.413773\pi\)
\(758\) −11.9723 −0.434852
\(759\) 0 0
\(760\) 6.41156 0.232572
\(761\) 0.229814 0.00833074 0.00416537 0.999991i \(-0.498674\pi\)
0.00416537 + 0.999991i \(0.498674\pi\)
\(762\) 16.2155 0.587426
\(763\) −28.6094 −1.03573
\(764\) 0.913695 0.0330563
\(765\) 0.557453 0.0201548
\(766\) 3.07102 0.110960
\(767\) 69.7649 2.51906
\(768\) 1.00000 0.0360844
\(769\) −26.3568 −0.950451 −0.475226 0.879864i \(-0.657634\pi\)
−0.475226 + 0.879864i \(0.657634\pi\)
\(770\) 0 0
\(771\) 4.32237 0.155666
\(772\) −23.2952 −0.838414
\(773\) −54.7575 −1.96949 −0.984744 0.174007i \(-0.944329\pi\)
−0.984744 + 0.174007i \(0.944329\pi\)
\(774\) 1.32139 0.0474962
\(775\) 2.41941 0.0869077
\(776\) 7.63332 0.274020
\(777\) −18.3375 −0.657856
\(778\) 19.7569 0.708321
\(779\) 8.46190 0.303179
\(780\) −6.75608 −0.241906
\(781\) 0 0
\(782\) −1.44653 −0.0517279
\(783\) 4.89154 0.174809
\(784\) 3.23122 0.115401
\(785\) −18.4874 −0.659844
\(786\) −17.2888 −0.616671
\(787\) 7.45825 0.265858 0.132929 0.991126i \(-0.457562\pi\)
0.132929 + 0.991126i \(0.457562\pi\)
\(788\) 24.5896 0.875969
\(789\) −23.0282 −0.819825
\(790\) 11.6724 0.415284
\(791\) 15.4067 0.547799
\(792\) 0 0
\(793\) −26.5184 −0.941697
\(794\) −11.4546 −0.406508
\(795\) 2.43470 0.0863497
\(796\) 10.5112 0.372559
\(797\) 35.0946 1.24311 0.621557 0.783369i \(-0.286500\pi\)
0.621557 + 0.783369i \(0.286500\pi\)
\(798\) −20.5082 −0.725982
\(799\) 3.34034 0.118173
\(800\) 1.00000 0.0353553
\(801\) −13.0775 −0.462070
\(802\) 22.3001 0.787444
\(803\) 0 0
\(804\) 15.1676 0.534921
\(805\) −8.30010 −0.292540
\(806\) 16.3457 0.575753
\(807\) −11.8567 −0.417375
\(808\) −15.9739 −0.561958
\(809\) −22.7625 −0.800288 −0.400144 0.916452i \(-0.631040\pi\)
−0.400144 + 0.916452i \(0.631040\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 21.7665 0.764326 0.382163 0.924095i \(-0.375179\pi\)
0.382163 + 0.924095i \(0.375179\pi\)
\(812\) 15.6462 0.549075
\(813\) −11.4268 −0.400757
\(814\) 0 0
\(815\) −11.5480 −0.404509
\(816\) −0.557453 −0.0195148
\(817\) −8.47214 −0.296403
\(818\) 36.2811 1.26854
\(819\) 21.6102 0.755121
\(820\) 1.31979 0.0460890
\(821\) 35.5251 1.23983 0.619917 0.784668i \(-0.287166\pi\)
0.619917 + 0.784668i \(0.287166\pi\)
\(822\) 7.93157 0.276645
\(823\) 39.6319 1.38148 0.690741 0.723102i \(-0.257284\pi\)
0.690741 + 0.723102i \(0.257284\pi\)
\(824\) 18.2904 0.637176
\(825\) 0 0
\(826\) 33.0298 1.14925
\(827\) 17.6236 0.612833 0.306417 0.951898i \(-0.400870\pi\)
0.306417 + 0.951898i \(0.400870\pi\)
\(828\) 2.59489 0.0901788
\(829\) 7.04973 0.244847 0.122424 0.992478i \(-0.460933\pi\)
0.122424 + 0.992478i \(0.460933\pi\)
\(830\) −8.00000 −0.277684
\(831\) −15.0384 −0.521677
\(832\) 6.75608 0.234225
\(833\) −1.80125 −0.0624097
\(834\) 8.21778 0.284558
\(835\) 2.95212 0.102162
\(836\) 0 0
\(837\) 2.41941 0.0836269
\(838\) 9.06157 0.313027
\(839\) −22.5599 −0.778855 −0.389427 0.921057i \(-0.627327\pi\)
−0.389427 + 0.921057i \(0.627327\pi\)
\(840\) −3.19863 −0.110363
\(841\) −5.07281 −0.174924
\(842\) −2.42586 −0.0836005
\(843\) 13.2666 0.456928
\(844\) 7.06863 0.243312
\(845\) −32.6446 −1.12301
\(846\) −5.99215 −0.206014
\(847\) 0 0
\(848\) −2.43470 −0.0836078
\(849\) 15.1214 0.518964
\(850\) −0.557453 −0.0191205
\(851\) −14.8764 −0.509956
\(852\) −6.35097 −0.217581
\(853\) −18.3884 −0.629607 −0.314804 0.949157i \(-0.601939\pi\)
−0.314804 + 0.949157i \(0.601939\pi\)
\(854\) −12.5550 −0.429623
\(855\) 6.41156 0.219271
\(856\) −17.1149 −0.584975
\(857\) −21.9179 −0.748700 −0.374350 0.927287i \(-0.622134\pi\)
−0.374350 + 0.927287i \(0.622134\pi\)
\(858\) 0 0
\(859\) −32.0841 −1.09470 −0.547348 0.836905i \(-0.684363\pi\)
−0.547348 + 0.836905i \(0.684363\pi\)
\(860\) −1.32139 −0.0450589
\(861\) −4.22151 −0.143869
\(862\) −2.65847 −0.0905480
\(863\) 0.406288 0.0138302 0.00691511 0.999976i \(-0.497799\pi\)
0.00691511 + 0.999976i \(0.497799\pi\)
\(864\) 1.00000 0.0340207
\(865\) 1.84140 0.0626094
\(866\) −19.1918 −0.652163
\(867\) −16.6892 −0.566796
\(868\) 7.73878 0.262671
\(869\) 0 0
\(870\) −4.89154 −0.165839
\(871\) 102.474 3.47219
\(872\) −8.94427 −0.302891
\(873\) 7.63332 0.258349
\(874\) −16.6373 −0.562765
\(875\) −3.19863 −0.108133
\(876\) 6.94427 0.234625
\(877\) 6.38542 0.215620 0.107810 0.994172i \(-0.465616\pi\)
0.107810 + 0.994172i \(0.465616\pi\)
\(878\) −12.6955 −0.428452
\(879\) −13.9163 −0.469384
\(880\) 0 0
\(881\) −39.6551 −1.33601 −0.668006 0.744155i \(-0.732852\pi\)
−0.668006 + 0.744155i \(0.732852\pi\)
\(882\) 3.23122 0.108801
\(883\) −4.90523 −0.165074 −0.0825371 0.996588i \(-0.526302\pi\)
−0.0825371 + 0.996588i \(0.526302\pi\)
\(884\) −3.76620 −0.126671
\(885\) −10.3262 −0.347113
\(886\) 16.9189 0.568400
\(887\) 7.61724 0.255762 0.127881 0.991790i \(-0.459182\pi\)
0.127881 + 0.991790i \(0.459182\pi\)
\(888\) −5.73294 −0.192385
\(889\) 51.8674 1.73958
\(890\) 13.0775 0.438358
\(891\) 0 0
\(892\) 0.801373 0.0268319
\(893\) 38.4190 1.28564
\(894\) 6.08532 0.203523
\(895\) 18.5623 0.620469
\(896\) 3.19863 0.106859
\(897\) 17.5313 0.585354
\(898\) −19.2809 −0.643413
\(899\) 11.8346 0.394707
\(900\) 1.00000 0.0333333
\(901\) 1.35723 0.0452159
\(902\) 0 0
\(903\) 4.22662 0.140653
\(904\) 4.81666 0.160200
\(905\) −9.52786 −0.316717
\(906\) −18.2761 −0.607183
\(907\) −51.7186 −1.71729 −0.858645 0.512571i \(-0.828693\pi\)
−0.858645 + 0.512571i \(0.828693\pi\)
\(908\) 4.86939 0.161596
\(909\) −15.9739 −0.529820
\(910\) −21.6102 −0.716371
\(911\) −32.2012 −1.06687 −0.533437 0.845840i \(-0.679100\pi\)
−0.533437 + 0.845840i \(0.679100\pi\)
\(912\) −6.41156 −0.212308
\(913\) 0 0
\(914\) −39.5219 −1.30727
\(915\) 3.92512 0.129760
\(916\) −1.16119 −0.0383667
\(917\) −55.3004 −1.82618
\(918\) −0.557453 −0.0183987
\(919\) 31.5512 1.04078 0.520389 0.853929i \(-0.325787\pi\)
0.520389 + 0.853929i \(0.325787\pi\)
\(920\) −2.59489 −0.0855511
\(921\) 24.0813 0.793507
\(922\) −13.8398 −0.455790
\(923\) −42.9077 −1.41232
\(924\) 0 0
\(925\) −5.73294 −0.188498
\(926\) 12.5919 0.413795
\(927\) 18.2904 0.600735
\(928\) 4.89154 0.160573
\(929\) 26.7857 0.878809 0.439405 0.898289i \(-0.355189\pi\)
0.439405 + 0.898289i \(0.355189\pi\)
\(930\) −2.41941 −0.0793355
\(931\) −20.7171 −0.678976
\(932\) −12.0527 −0.394800
\(933\) −30.3816 −0.994647
\(934\) −11.7368 −0.384040
\(935\) 0 0
\(936\) 6.75608 0.220829
\(937\) −21.3572 −0.697710 −0.348855 0.937177i \(-0.613429\pi\)
−0.348855 + 0.937177i \(0.613429\pi\)
\(938\) 48.5156 1.58409
\(939\) 16.2012 0.528707
\(940\) 5.99215 0.195442
\(941\) −41.5778 −1.35540 −0.677699 0.735340i \(-0.737023\pi\)
−0.677699 + 0.735340i \(0.737023\pi\)
\(942\) 18.4874 0.602353
\(943\) −3.42471 −0.111524
\(944\) 10.3262 0.336090
\(945\) −3.19863 −0.104051
\(946\) 0 0
\(947\) 8.59847 0.279413 0.139706 0.990193i \(-0.455384\pi\)
0.139706 + 0.990193i \(0.455384\pi\)
\(948\) −11.6724 −0.379101
\(949\) 46.9161 1.52296
\(950\) −6.41156 −0.208018
\(951\) −33.0640 −1.07217
\(952\) −1.78309 −0.0577901
\(953\) 5.86067 0.189846 0.0949229 0.995485i \(-0.469740\pi\)
0.0949229 + 0.995485i \(0.469740\pi\)
\(954\) −2.43470 −0.0788262
\(955\) −0.913695 −0.0295665
\(956\) 28.2604 0.914006
\(957\) 0 0
\(958\) −21.4007 −0.691425
\(959\) 25.3701 0.819244
\(960\) −1.00000 −0.0322749
\(961\) −25.1465 −0.811176
\(962\) −38.7322 −1.24878
\(963\) −17.1149 −0.551520
\(964\) −24.7857 −0.798293
\(965\) 23.2952 0.749901
\(966\) 8.30010 0.267051
\(967\) 24.7496 0.795894 0.397947 0.917408i \(-0.369723\pi\)
0.397947 + 0.917408i \(0.369723\pi\)
\(968\) 0 0
\(969\) 3.57414 0.114818
\(970\) −7.63332 −0.245091
\(971\) 43.1418 1.38449 0.692243 0.721664i \(-0.256622\pi\)
0.692243 + 0.721664i \(0.256622\pi\)
\(972\) 1.00000 0.0320750
\(973\) 26.2856 0.842678
\(974\) −15.8502 −0.507874
\(975\) 6.75608 0.216368
\(976\) −3.92512 −0.125640
\(977\) −42.5522 −1.36137 −0.680683 0.732579i \(-0.738317\pi\)
−0.680683 + 0.732579i \(0.738317\pi\)
\(978\) 11.5480 0.369264
\(979\) 0 0
\(980\) −3.23122 −0.103217
\(981\) −8.94427 −0.285569
\(982\) −27.6698 −0.882978
\(983\) 18.3473 0.585188 0.292594 0.956237i \(-0.405482\pi\)
0.292594 + 0.956237i \(0.405482\pi\)
\(984\) −1.31979 −0.0420733
\(985\) −24.5896 −0.783491
\(986\) −2.72681 −0.0868392
\(987\) −19.1666 −0.610081
\(988\) −43.3170 −1.37810
\(989\) 3.42886 0.109031
\(990\) 0 0
\(991\) 1.48883 0.0472941 0.0236471 0.999720i \(-0.492472\pi\)
0.0236471 + 0.999720i \(0.492472\pi\)
\(992\) 2.41941 0.0768162
\(993\) −0.733928 −0.0232905
\(994\) −20.3144 −0.644334
\(995\) −10.5112 −0.333227
\(996\) 8.00000 0.253490
\(997\) 55.9405 1.77165 0.885826 0.464017i \(-0.153592\pi\)
0.885826 + 0.464017i \(0.153592\pi\)
\(998\) −7.06176 −0.223536
\(999\) −5.73294 −0.181382
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 3630.2.a.bs.1.3 4
11.7 odd 10 330.2.m.f.181.2 yes 8
11.8 odd 10 330.2.m.f.31.2 8
11.10 odd 2 3630.2.a.bq.1.2 4
33.8 even 10 990.2.n.i.361.2 8
33.29 even 10 990.2.n.i.181.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.m.f.31.2 8 11.8 odd 10
330.2.m.f.181.2 yes 8 11.7 odd 10
990.2.n.i.181.2 8 33.29 even 10
990.2.n.i.361.2 8 33.8 even 10
3630.2.a.bq.1.2 4 11.10 odd 2
3630.2.a.bs.1.3 4 1.1 even 1 trivial