Properties

Label 370.2.a.e.1.1
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73205 q^{6} -1.26795 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73205 q^{6} -1.26795 q^{7} -1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +1.46410 q^{11} -2.73205 q^{12} +1.46410 q^{13} +1.26795 q^{14} -2.73205 q^{15} +1.00000 q^{16} -1.46410 q^{17} -4.46410 q^{18} -4.19615 q^{19} +1.00000 q^{20} +3.46410 q^{21} -1.46410 q^{22} -8.00000 q^{23} +2.73205 q^{24} +1.00000 q^{25} -1.46410 q^{26} -4.00000 q^{27} -1.26795 q^{28} -8.92820 q^{29} +2.73205 q^{30} -2.73205 q^{31} -1.00000 q^{32} -4.00000 q^{33} +1.46410 q^{34} -1.26795 q^{35} +4.46410 q^{36} +1.00000 q^{37} +4.19615 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} -3.46410 q^{42} -6.92820 q^{43} +1.46410 q^{44} +4.46410 q^{45} +8.00000 q^{46} -1.26795 q^{47} -2.73205 q^{48} -5.39230 q^{49} -1.00000 q^{50} +4.00000 q^{51} +1.46410 q^{52} -6.00000 q^{53} +4.00000 q^{54} +1.46410 q^{55} +1.26795 q^{56} +11.4641 q^{57} +8.92820 q^{58} +0.196152 q^{59} -2.73205 q^{60} +8.92820 q^{61} +2.73205 q^{62} -5.66025 q^{63} +1.00000 q^{64} +1.46410 q^{65} +4.00000 q^{66} +13.6603 q^{67} -1.46410 q^{68} +21.8564 q^{69} +1.26795 q^{70} -10.9282 q^{71} -4.46410 q^{72} +12.9282 q^{73} -1.00000 q^{74} -2.73205 q^{75} -4.19615 q^{76} -1.85641 q^{77} +4.00000 q^{78} +5.26795 q^{79} +1.00000 q^{80} -2.46410 q^{81} +2.00000 q^{82} -5.26795 q^{83} +3.46410 q^{84} -1.46410 q^{85} +6.92820 q^{86} +24.3923 q^{87} -1.46410 q^{88} -2.00000 q^{89} -4.46410 q^{90} -1.85641 q^{91} -8.00000 q^{92} +7.46410 q^{93} +1.26795 q^{94} -4.19615 q^{95} +2.73205 q^{96} -2.00000 q^{97} +5.39230 q^{98} +6.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - 4 q^{13} + 6 q^{14} - 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{20} + 4 q^{22} - 16 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} - 8 q^{27} - 6 q^{28} - 4 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 8 q^{33} - 4 q^{34} - 6 q^{35} + 2 q^{36} + 2 q^{37} - 2 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 4 q^{44} + 2 q^{45} + 16 q^{46} - 6 q^{47} - 2 q^{48} + 10 q^{49} - 2 q^{50} + 8 q^{51} - 4 q^{52} - 12 q^{53} + 8 q^{54} - 4 q^{55} + 6 q^{56} + 16 q^{57} + 4 q^{58} - 10 q^{59} - 2 q^{60} + 4 q^{61} + 2 q^{62} + 6 q^{63} + 2 q^{64} - 4 q^{65} + 8 q^{66} + 10 q^{67} + 4 q^{68} + 16 q^{69} + 6 q^{70} - 8 q^{71} - 2 q^{72} + 12 q^{73} - 2 q^{74} - 2 q^{75} + 2 q^{76} + 24 q^{77} + 8 q^{78} + 14 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 14 q^{83} + 4 q^{85} + 28 q^{87} + 4 q^{88} - 4 q^{89} - 2 q^{90} + 24 q^{91} - 16 q^{92} + 8 q^{93} + 6 q^{94} + 2 q^{95} + 2 q^{96} - 4 q^{97} - 10 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.73205 1.11536
\(7\) −1.26795 −0.479240 −0.239620 0.970867i \(-0.577023\pi\)
−0.239620 + 0.970867i \(0.577023\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.46410 1.48803
\(10\) −1.00000 −0.316228
\(11\) 1.46410 0.441443 0.220722 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(12\) −2.73205 −0.788675
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 1.26795 0.338874
\(15\) −2.73205 −0.705412
\(16\) 1.00000 0.250000
\(17\) −1.46410 −0.355097 −0.177548 0.984112i \(-0.556817\pi\)
−0.177548 + 0.984112i \(0.556817\pi\)
\(18\) −4.46410 −1.05220
\(19\) −4.19615 −0.962663 −0.481332 0.876539i \(-0.659847\pi\)
−0.481332 + 0.876539i \(0.659847\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.46410 0.755929
\(22\) −1.46410 −0.312148
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 2.73205 0.557678
\(25\) 1.00000 0.200000
\(26\) −1.46410 −0.287134
\(27\) −4.00000 −0.769800
\(28\) −1.26795 −0.239620
\(29\) −8.92820 −1.65793 −0.828963 0.559304i \(-0.811069\pi\)
−0.828963 + 0.559304i \(0.811069\pi\)
\(30\) 2.73205 0.498802
\(31\) −2.73205 −0.490691 −0.245345 0.969436i \(-0.578901\pi\)
−0.245345 + 0.969436i \(0.578901\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 1.46410 0.251091
\(35\) −1.26795 −0.214323
\(36\) 4.46410 0.744017
\(37\) 1.00000 0.164399
\(38\) 4.19615 0.680706
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −3.46410 −0.534522
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 1.46410 0.220722
\(45\) 4.46410 0.665469
\(46\) 8.00000 1.17954
\(47\) −1.26795 −0.184949 −0.0924747 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) −2.73205 −0.394338
\(49\) −5.39230 −0.770329
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) 1.46410 0.203034
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 4.00000 0.544331
\(55\) 1.46410 0.197419
\(56\) 1.26795 0.169437
\(57\) 11.4641 1.51846
\(58\) 8.92820 1.17233
\(59\) 0.196152 0.0255369 0.0127684 0.999918i \(-0.495936\pi\)
0.0127684 + 0.999918i \(0.495936\pi\)
\(60\) −2.73205 −0.352706
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 2.73205 0.346971
\(63\) −5.66025 −0.713125
\(64\) 1.00000 0.125000
\(65\) 1.46410 0.181599
\(66\) 4.00000 0.492366
\(67\) 13.6603 1.66887 0.834433 0.551110i \(-0.185795\pi\)
0.834433 + 0.551110i \(0.185795\pi\)
\(68\) −1.46410 −0.177548
\(69\) 21.8564 2.63120
\(70\) 1.26795 0.151549
\(71\) −10.9282 −1.29694 −0.648470 0.761241i \(-0.724591\pi\)
−0.648470 + 0.761241i \(0.724591\pi\)
\(72\) −4.46410 −0.526099
\(73\) 12.9282 1.51313 0.756566 0.653917i \(-0.226876\pi\)
0.756566 + 0.653917i \(0.226876\pi\)
\(74\) −1.00000 −0.116248
\(75\) −2.73205 −0.315470
\(76\) −4.19615 −0.481332
\(77\) −1.85641 −0.211557
\(78\) 4.00000 0.452911
\(79\) 5.26795 0.592691 0.296345 0.955081i \(-0.404232\pi\)
0.296345 + 0.955081i \(0.404232\pi\)
\(80\) 1.00000 0.111803
\(81\) −2.46410 −0.273789
\(82\) 2.00000 0.220863
\(83\) −5.26795 −0.578233 −0.289116 0.957294i \(-0.593361\pi\)
−0.289116 + 0.957294i \(0.593361\pi\)
\(84\) 3.46410 0.377964
\(85\) −1.46410 −0.158804
\(86\) 6.92820 0.747087
\(87\) 24.3923 2.61513
\(88\) −1.46410 −0.156074
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −4.46410 −0.470558
\(91\) −1.85641 −0.194604
\(92\) −8.00000 −0.834058
\(93\) 7.46410 0.773991
\(94\) 1.26795 0.130779
\(95\) −4.19615 −0.430516
\(96\) 2.73205 0.278839
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 5.39230 0.544705
\(99\) 6.53590 0.656883
\(100\) 1.00000 0.100000
\(101\) −2.53590 −0.252331 −0.126166 0.992009i \(-0.540267\pi\)
−0.126166 + 0.992009i \(0.540267\pi\)
\(102\) −4.00000 −0.396059
\(103\) −13.4641 −1.32666 −0.663329 0.748328i \(-0.730857\pi\)
−0.663329 + 0.748328i \(0.730857\pi\)
\(104\) −1.46410 −0.143567
\(105\) 3.46410 0.338062
\(106\) 6.00000 0.582772
\(107\) 6.73205 0.650812 0.325406 0.945574i \(-0.394499\pi\)
0.325406 + 0.945574i \(0.394499\pi\)
\(108\) −4.00000 −0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −1.46410 −0.139597
\(111\) −2.73205 −0.259315
\(112\) −1.26795 −0.119810
\(113\) −17.4641 −1.64288 −0.821442 0.570292i \(-0.806830\pi\)
−0.821442 + 0.570292i \(0.806830\pi\)
\(114\) −11.4641 −1.07371
\(115\) −8.00000 −0.746004
\(116\) −8.92820 −0.828963
\(117\) 6.53590 0.604244
\(118\) −0.196152 −0.0180573
\(119\) 1.85641 0.170177
\(120\) 2.73205 0.249401
\(121\) −8.85641 −0.805128
\(122\) −8.92820 −0.808322
\(123\) 5.46410 0.492681
\(124\) −2.73205 −0.245345
\(125\) 1.00000 0.0894427
\(126\) 5.66025 0.504256
\(127\) −13.6603 −1.21215 −0.606076 0.795407i \(-0.707257\pi\)
−0.606076 + 0.795407i \(0.707257\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 18.9282 1.66654
\(130\) −1.46410 −0.128410
\(131\) 12.5885 1.09986 0.549929 0.835211i \(-0.314655\pi\)
0.549929 + 0.835211i \(0.314655\pi\)
\(132\) −4.00000 −0.348155
\(133\) 5.32051 0.461347
\(134\) −13.6603 −1.18007
\(135\) −4.00000 −0.344265
\(136\) 1.46410 0.125546
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −21.8564 −1.86054
\(139\) −6.92820 −0.587643 −0.293821 0.955860i \(-0.594927\pi\)
−0.293821 + 0.955860i \(0.594927\pi\)
\(140\) −1.26795 −0.107161
\(141\) 3.46410 0.291730
\(142\) 10.9282 0.917074
\(143\) 2.14359 0.179256
\(144\) 4.46410 0.372008
\(145\) −8.92820 −0.741447
\(146\) −12.9282 −1.06995
\(147\) 14.7321 1.21508
\(148\) 1.00000 0.0821995
\(149\) −16.3923 −1.34291 −0.671455 0.741045i \(-0.734330\pi\)
−0.671455 + 0.741045i \(0.734330\pi\)
\(150\) 2.73205 0.223071
\(151\) 8.39230 0.682956 0.341478 0.939890i \(-0.389073\pi\)
0.341478 + 0.939890i \(0.389073\pi\)
\(152\) 4.19615 0.340353
\(153\) −6.53590 −0.528396
\(154\) 1.85641 0.149593
\(155\) −2.73205 −0.219444
\(156\) −4.00000 −0.320256
\(157\) 16.9282 1.35102 0.675509 0.737352i \(-0.263924\pi\)
0.675509 + 0.737352i \(0.263924\pi\)
\(158\) −5.26795 −0.419096
\(159\) 16.3923 1.29999
\(160\) −1.00000 −0.0790569
\(161\) 10.1436 0.799427
\(162\) 2.46410 0.193598
\(163\) 23.3205 1.82660 0.913302 0.407284i \(-0.133524\pi\)
0.913302 + 0.407284i \(0.133524\pi\)
\(164\) −2.00000 −0.156174
\(165\) −4.00000 −0.311400
\(166\) 5.26795 0.408872
\(167\) 5.46410 0.422825 0.211412 0.977397i \(-0.432194\pi\)
0.211412 + 0.977397i \(0.432194\pi\)
\(168\) −3.46410 −0.267261
\(169\) −10.8564 −0.835108
\(170\) 1.46410 0.112291
\(171\) −18.7321 −1.43248
\(172\) −6.92820 −0.528271
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) −24.3923 −1.84918
\(175\) −1.26795 −0.0958479
\(176\) 1.46410 0.110361
\(177\) −0.535898 −0.0402806
\(178\) 2.00000 0.149906
\(179\) −17.6603 −1.31999 −0.659995 0.751270i \(-0.729441\pi\)
−0.659995 + 0.751270i \(0.729441\pi\)
\(180\) 4.46410 0.332734
\(181\) −1.46410 −0.108826 −0.0544129 0.998519i \(-0.517329\pi\)
−0.0544129 + 0.998519i \(0.517329\pi\)
\(182\) 1.85641 0.137606
\(183\) −24.3923 −1.80313
\(184\) 8.00000 0.589768
\(185\) 1.00000 0.0735215
\(186\) −7.46410 −0.547294
\(187\) −2.14359 −0.156755
\(188\) −1.26795 −0.0924747
\(189\) 5.07180 0.368919
\(190\) 4.19615 0.304421
\(191\) −5.26795 −0.381175 −0.190588 0.981670i \(-0.561039\pi\)
−0.190588 + 0.981670i \(0.561039\pi\)
\(192\) −2.73205 −0.197169
\(193\) −11.8564 −0.853443 −0.426721 0.904383i \(-0.640332\pi\)
−0.426721 + 0.904383i \(0.640332\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) −5.39230 −0.385165
\(197\) 18.7846 1.33835 0.669174 0.743106i \(-0.266648\pi\)
0.669174 + 0.743106i \(0.266648\pi\)
\(198\) −6.53590 −0.464486
\(199\) 26.0526 1.84682 0.923408 0.383819i \(-0.125391\pi\)
0.923408 + 0.383819i \(0.125391\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −37.3205 −2.63239
\(202\) 2.53590 0.178425
\(203\) 11.3205 0.794544
\(204\) 4.00000 0.280056
\(205\) −2.00000 −0.139686
\(206\) 13.4641 0.938088
\(207\) −35.7128 −2.48221
\(208\) 1.46410 0.101517
\(209\) −6.14359 −0.424961
\(210\) −3.46410 −0.239046
\(211\) 9.85641 0.678543 0.339272 0.940688i \(-0.389819\pi\)
0.339272 + 0.940688i \(0.389819\pi\)
\(212\) −6.00000 −0.412082
\(213\) 29.8564 2.04573
\(214\) −6.73205 −0.460194
\(215\) −6.92820 −0.472500
\(216\) 4.00000 0.272166
\(217\) 3.46410 0.235159
\(218\) −2.00000 −0.135457
\(219\) −35.3205 −2.38674
\(220\) 1.46410 0.0987097
\(221\) −2.14359 −0.144194
\(222\) 2.73205 0.183363
\(223\) −22.0526 −1.47675 −0.738374 0.674391i \(-0.764406\pi\)
−0.738374 + 0.674391i \(0.764406\pi\)
\(224\) 1.26795 0.0847184
\(225\) 4.46410 0.297607
\(226\) 17.4641 1.16169
\(227\) 3.60770 0.239451 0.119726 0.992807i \(-0.461799\pi\)
0.119726 + 0.992807i \(0.461799\pi\)
\(228\) 11.4641 0.759229
\(229\) 15.8564 1.04782 0.523910 0.851773i \(-0.324473\pi\)
0.523910 + 0.851773i \(0.324473\pi\)
\(230\) 8.00000 0.527504
\(231\) 5.07180 0.333700
\(232\) 8.92820 0.586165
\(233\) 15.0718 0.987386 0.493693 0.869636i \(-0.335647\pi\)
0.493693 + 0.869636i \(0.335647\pi\)
\(234\) −6.53590 −0.427265
\(235\) −1.26795 −0.0827119
\(236\) 0.196152 0.0127684
\(237\) −14.3923 −0.934881
\(238\) −1.85641 −0.120333
\(239\) −17.2679 −1.11697 −0.558485 0.829514i \(-0.688617\pi\)
−0.558485 + 0.829514i \(0.688617\pi\)
\(240\) −2.73205 −0.176353
\(241\) −8.92820 −0.575116 −0.287558 0.957763i \(-0.592843\pi\)
−0.287558 + 0.957763i \(0.592843\pi\)
\(242\) 8.85641 0.569311
\(243\) 18.7321 1.20166
\(244\) 8.92820 0.571570
\(245\) −5.39230 −0.344502
\(246\) −5.46410 −0.348378
\(247\) −6.14359 −0.390907
\(248\) 2.73205 0.173485
\(249\) 14.3923 0.912075
\(250\) −1.00000 −0.0632456
\(251\) 22.7321 1.43483 0.717417 0.696644i \(-0.245324\pi\)
0.717417 + 0.696644i \(0.245324\pi\)
\(252\) −5.66025 −0.356562
\(253\) −11.7128 −0.736378
\(254\) 13.6603 0.857121
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −25.4641 −1.58841 −0.794204 0.607652i \(-0.792112\pi\)
−0.794204 + 0.607652i \(0.792112\pi\)
\(258\) −18.9282 −1.17842
\(259\) −1.26795 −0.0787865
\(260\) 1.46410 0.0907997
\(261\) −39.8564 −2.46705
\(262\) −12.5885 −0.777717
\(263\) 30.0526 1.85312 0.926560 0.376147i \(-0.122751\pi\)
0.926560 + 0.376147i \(0.122751\pi\)
\(264\) 4.00000 0.246183
\(265\) −6.00000 −0.368577
\(266\) −5.32051 −0.326221
\(267\) 5.46410 0.334398
\(268\) 13.6603 0.834433
\(269\) −0.392305 −0.0239192 −0.0119596 0.999928i \(-0.503807\pi\)
−0.0119596 + 0.999928i \(0.503807\pi\)
\(270\) 4.00000 0.243432
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) −1.46410 −0.0887742
\(273\) 5.07180 0.306959
\(274\) −2.00000 −0.120824
\(275\) 1.46410 0.0882886
\(276\) 21.8564 1.31560
\(277\) −26.2487 −1.57713 −0.788566 0.614950i \(-0.789176\pi\)
−0.788566 + 0.614950i \(0.789176\pi\)
\(278\) 6.92820 0.415526
\(279\) −12.1962 −0.730165
\(280\) 1.26795 0.0757745
\(281\) 4.92820 0.293992 0.146996 0.989137i \(-0.453040\pi\)
0.146996 + 0.989137i \(0.453040\pi\)
\(282\) −3.46410 −0.206284
\(283\) −4.39230 −0.261095 −0.130548 0.991442i \(-0.541674\pi\)
−0.130548 + 0.991442i \(0.541674\pi\)
\(284\) −10.9282 −0.648470
\(285\) 11.4641 0.679075
\(286\) −2.14359 −0.126753
\(287\) 2.53590 0.149689
\(288\) −4.46410 −0.263050
\(289\) −14.8564 −0.873906
\(290\) 8.92820 0.524282
\(291\) 5.46410 0.320311
\(292\) 12.9282 0.756566
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −14.7321 −0.859191
\(295\) 0.196152 0.0114204
\(296\) −1.00000 −0.0581238
\(297\) −5.85641 −0.339823
\(298\) 16.3923 0.949581
\(299\) −11.7128 −0.677369
\(300\) −2.73205 −0.157735
\(301\) 8.78461 0.506336
\(302\) −8.39230 −0.482923
\(303\) 6.92820 0.398015
\(304\) −4.19615 −0.240666
\(305\) 8.92820 0.511227
\(306\) 6.53590 0.373632
\(307\) −12.5885 −0.718461 −0.359231 0.933249i \(-0.616961\pi\)
−0.359231 + 0.933249i \(0.616961\pi\)
\(308\) −1.85641 −0.105779
\(309\) 36.7846 2.09260
\(310\) 2.73205 0.155170
\(311\) 27.1244 1.53808 0.769041 0.639200i \(-0.220734\pi\)
0.769041 + 0.639200i \(0.220734\pi\)
\(312\) 4.00000 0.226455
\(313\) 3.85641 0.217977 0.108988 0.994043i \(-0.465239\pi\)
0.108988 + 0.994043i \(0.465239\pi\)
\(314\) −16.9282 −0.955314
\(315\) −5.66025 −0.318919
\(316\) 5.26795 0.296345
\(317\) −31.8564 −1.78923 −0.894617 0.446834i \(-0.852552\pi\)
−0.894617 + 0.446834i \(0.852552\pi\)
\(318\) −16.3923 −0.919235
\(319\) −13.0718 −0.731880
\(320\) 1.00000 0.0559017
\(321\) −18.3923 −1.02656
\(322\) −10.1436 −0.565280
\(323\) 6.14359 0.341839
\(324\) −2.46410 −0.136895
\(325\) 1.46410 0.0812137
\(326\) −23.3205 −1.29160
\(327\) −5.46410 −0.302166
\(328\) 2.00000 0.110432
\(329\) 1.60770 0.0886351
\(330\) 4.00000 0.220193
\(331\) −8.87564 −0.487850 −0.243925 0.969794i \(-0.578435\pi\)
−0.243925 + 0.969794i \(0.578435\pi\)
\(332\) −5.26795 −0.289116
\(333\) 4.46410 0.244631
\(334\) −5.46410 −0.298982
\(335\) 13.6603 0.746339
\(336\) 3.46410 0.188982
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 10.8564 0.590511
\(339\) 47.7128 2.59140
\(340\) −1.46410 −0.0794021
\(341\) −4.00000 −0.216612
\(342\) 18.7321 1.01291
\(343\) 15.7128 0.848412
\(344\) 6.92820 0.373544
\(345\) 21.8564 1.17671
\(346\) −10.0000 −0.537603
\(347\) 17.0718 0.916462 0.458231 0.888833i \(-0.348483\pi\)
0.458231 + 0.888833i \(0.348483\pi\)
\(348\) 24.3923 1.30756
\(349\) 19.3205 1.03420 0.517102 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(350\) 1.26795 0.0677747
\(351\) −5.85641 −0.312592
\(352\) −1.46410 −0.0780369
\(353\) 15.8564 0.843951 0.421976 0.906607i \(-0.361337\pi\)
0.421976 + 0.906607i \(0.361337\pi\)
\(354\) 0.535898 0.0284827
\(355\) −10.9282 −0.580009
\(356\) −2.00000 −0.106000
\(357\) −5.07180 −0.268428
\(358\) 17.6603 0.933373
\(359\) 8.39230 0.442929 0.221464 0.975168i \(-0.428916\pi\)
0.221464 + 0.975168i \(0.428916\pi\)
\(360\) −4.46410 −0.235279
\(361\) −1.39230 −0.0732792
\(362\) 1.46410 0.0769515
\(363\) 24.1962 1.26997
\(364\) −1.85641 −0.0973021
\(365\) 12.9282 0.676693
\(366\) 24.3923 1.27501
\(367\) −21.6603 −1.13066 −0.565328 0.824866i \(-0.691250\pi\)
−0.565328 + 0.824866i \(0.691250\pi\)
\(368\) −8.00000 −0.417029
\(369\) −8.92820 −0.464784
\(370\) −1.00000 −0.0519875
\(371\) 7.60770 0.394972
\(372\) 7.46410 0.386996
\(373\) 30.7846 1.59397 0.796983 0.604001i \(-0.206428\pi\)
0.796983 + 0.604001i \(0.206428\pi\)
\(374\) 2.14359 0.110843
\(375\) −2.73205 −0.141082
\(376\) 1.26795 0.0653895
\(377\) −13.0718 −0.673232
\(378\) −5.07180 −0.260865
\(379\) 11.6077 0.596247 0.298124 0.954527i \(-0.403639\pi\)
0.298124 + 0.954527i \(0.403639\pi\)
\(380\) −4.19615 −0.215258
\(381\) 37.3205 1.91199
\(382\) 5.26795 0.269532
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 2.73205 0.139419
\(385\) −1.85641 −0.0946112
\(386\) 11.8564 0.603475
\(387\) −30.9282 −1.57217
\(388\) −2.00000 −0.101535
\(389\) 15.8564 0.803952 0.401976 0.915650i \(-0.368324\pi\)
0.401976 + 0.915650i \(0.368324\pi\)
\(390\) 4.00000 0.202548
\(391\) 11.7128 0.592342
\(392\) 5.39230 0.272353
\(393\) −34.3923 −1.73486
\(394\) −18.7846 −0.946355
\(395\) 5.26795 0.265059
\(396\) 6.53590 0.328441
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −26.0526 −1.30590
\(399\) −14.5359 −0.727705
\(400\) 1.00000 0.0500000
\(401\) −19.0718 −0.952400 −0.476200 0.879337i \(-0.657986\pi\)
−0.476200 + 0.879337i \(0.657986\pi\)
\(402\) 37.3205 1.86138
\(403\) −4.00000 −0.199254
\(404\) −2.53590 −0.126166
\(405\) −2.46410 −0.122442
\(406\) −11.3205 −0.561827
\(407\) 1.46410 0.0725728
\(408\) −4.00000 −0.198030
\(409\) 16.9282 0.837046 0.418523 0.908206i \(-0.362548\pi\)
0.418523 + 0.908206i \(0.362548\pi\)
\(410\) 2.00000 0.0987730
\(411\) −5.46410 −0.269524
\(412\) −13.4641 −0.663329
\(413\) −0.248711 −0.0122383
\(414\) 35.7128 1.75519
\(415\) −5.26795 −0.258593
\(416\) −1.46410 −0.0717835
\(417\) 18.9282 0.926918
\(418\) 6.14359 0.300493
\(419\) −34.2487 −1.67316 −0.836580 0.547846i \(-0.815448\pi\)
−0.836580 + 0.547846i \(0.815448\pi\)
\(420\) 3.46410 0.169031
\(421\) −27.8564 −1.35764 −0.678819 0.734306i \(-0.737508\pi\)
−0.678819 + 0.734306i \(0.737508\pi\)
\(422\) −9.85641 −0.479802
\(423\) −5.66025 −0.275211
\(424\) 6.00000 0.291386
\(425\) −1.46410 −0.0710194
\(426\) −29.8564 −1.44655
\(427\) −11.3205 −0.547838
\(428\) 6.73205 0.325406
\(429\) −5.85641 −0.282750
\(430\) 6.92820 0.334108
\(431\) 8.19615 0.394795 0.197397 0.980324i \(-0.436751\pi\)
0.197397 + 0.980324i \(0.436751\pi\)
\(432\) −4.00000 −0.192450
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −3.46410 −0.166282
\(435\) 24.3923 1.16952
\(436\) 2.00000 0.0957826
\(437\) 33.5692 1.60583
\(438\) 35.3205 1.68768
\(439\) −31.5167 −1.50421 −0.752104 0.659044i \(-0.770961\pi\)
−0.752104 + 0.659044i \(0.770961\pi\)
\(440\) −1.46410 −0.0697983
\(441\) −24.0718 −1.14628
\(442\) 2.14359 0.101960
\(443\) −39.1244 −1.85885 −0.929427 0.369006i \(-0.879698\pi\)
−0.929427 + 0.369006i \(0.879698\pi\)
\(444\) −2.73205 −0.129657
\(445\) −2.00000 −0.0948091
\(446\) 22.0526 1.04422
\(447\) 44.7846 2.11824
\(448\) −1.26795 −0.0599050
\(449\) 33.7128 1.59101 0.795503 0.605950i \(-0.207207\pi\)
0.795503 + 0.605950i \(0.207207\pi\)
\(450\) −4.46410 −0.210440
\(451\) −2.92820 −0.137884
\(452\) −17.4641 −0.821442
\(453\) −22.9282 −1.07726
\(454\) −3.60770 −0.169318
\(455\) −1.85641 −0.0870297
\(456\) −11.4641 −0.536856
\(457\) −4.14359 −0.193829 −0.0969146 0.995293i \(-0.530897\pi\)
−0.0969146 + 0.995293i \(0.530897\pi\)
\(458\) −15.8564 −0.740921
\(459\) 5.85641 0.273354
\(460\) −8.00000 −0.373002
\(461\) −26.7846 −1.24748 −0.623742 0.781630i \(-0.714388\pi\)
−0.623742 + 0.781630i \(0.714388\pi\)
\(462\) −5.07180 −0.235961
\(463\) 5.07180 0.235706 0.117853 0.993031i \(-0.462399\pi\)
0.117853 + 0.993031i \(0.462399\pi\)
\(464\) −8.92820 −0.414481
\(465\) 7.46410 0.346139
\(466\) −15.0718 −0.698188
\(467\) 37.1769 1.72034 0.860171 0.510005i \(-0.170357\pi\)
0.860171 + 0.510005i \(0.170357\pi\)
\(468\) 6.53590 0.302122
\(469\) −17.3205 −0.799787
\(470\) 1.26795 0.0584861
\(471\) −46.2487 −2.13103
\(472\) −0.196152 −0.00902865
\(473\) −10.1436 −0.466403
\(474\) 14.3923 0.661060
\(475\) −4.19615 −0.192533
\(476\) 1.85641 0.0850883
\(477\) −26.7846 −1.22638
\(478\) 17.2679 0.789818
\(479\) −34.0526 −1.55590 −0.777951 0.628325i \(-0.783741\pi\)
−0.777951 + 0.628325i \(0.783741\pi\)
\(480\) 2.73205 0.124700
\(481\) 1.46410 0.0667573
\(482\) 8.92820 0.406669
\(483\) −27.7128 −1.26098
\(484\) −8.85641 −0.402564
\(485\) −2.00000 −0.0908153
\(486\) −18.7321 −0.849703
\(487\) −16.3923 −0.742806 −0.371403 0.928472i \(-0.621123\pi\)
−0.371403 + 0.928472i \(0.621123\pi\)
\(488\) −8.92820 −0.404161
\(489\) −63.7128 −2.88119
\(490\) 5.39230 0.243600
\(491\) −1.07180 −0.0483695 −0.0241848 0.999708i \(-0.507699\pi\)
−0.0241848 + 0.999708i \(0.507699\pi\)
\(492\) 5.46410 0.246341
\(493\) 13.0718 0.588724
\(494\) 6.14359 0.276413
\(495\) 6.53590 0.293767
\(496\) −2.73205 −0.122673
\(497\) 13.8564 0.621545
\(498\) −14.3923 −0.644935
\(499\) 40.5885 1.81699 0.908494 0.417897i \(-0.137233\pi\)
0.908494 + 0.417897i \(0.137233\pi\)
\(500\) 1.00000 0.0447214
\(501\) −14.9282 −0.666943
\(502\) −22.7321 −1.01458
\(503\) −5.07180 −0.226140 −0.113070 0.993587i \(-0.536068\pi\)
−0.113070 + 0.993587i \(0.536068\pi\)
\(504\) 5.66025 0.252128
\(505\) −2.53590 −0.112846
\(506\) 11.7128 0.520698
\(507\) 29.6603 1.31726
\(508\) −13.6603 −0.606076
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) −4.00000 −0.177123
\(511\) −16.3923 −0.725153
\(512\) −1.00000 −0.0441942
\(513\) 16.7846 0.741059
\(514\) 25.4641 1.12317
\(515\) −13.4641 −0.593299
\(516\) 18.9282 0.833268
\(517\) −1.85641 −0.0816447
\(518\) 1.26795 0.0557105
\(519\) −27.3205 −1.19924
\(520\) −1.46410 −0.0642051
\(521\) 33.4641 1.46609 0.733044 0.680181i \(-0.238099\pi\)
0.733044 + 0.680181i \(0.238099\pi\)
\(522\) 39.8564 1.74447
\(523\) −4.78461 −0.209216 −0.104608 0.994514i \(-0.533359\pi\)
−0.104608 + 0.994514i \(0.533359\pi\)
\(524\) 12.5885 0.549929
\(525\) 3.46410 0.151186
\(526\) −30.0526 −1.31035
\(527\) 4.00000 0.174243
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) 0.875644 0.0379997
\(532\) 5.32051 0.230673
\(533\) −2.92820 −0.126835
\(534\) −5.46410 −0.236455
\(535\) 6.73205 0.291052
\(536\) −13.6603 −0.590033
\(537\) 48.2487 2.08209
\(538\) 0.392305 0.0169135
\(539\) −7.89488 −0.340057
\(540\) −4.00000 −0.172133
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −16.7846 −0.720961
\(543\) 4.00000 0.171656
\(544\) 1.46410 0.0627728
\(545\) 2.00000 0.0856706
\(546\) −5.07180 −0.217053
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 2.00000 0.0854358
\(549\) 39.8564 1.70103
\(550\) −1.46410 −0.0624295
\(551\) 37.4641 1.59602
\(552\) −21.8564 −0.930270
\(553\) −6.67949 −0.284041
\(554\) 26.2487 1.11520
\(555\) −2.73205 −0.115969
\(556\) −6.92820 −0.293821
\(557\) −32.1051 −1.36034 −0.680169 0.733056i \(-0.738094\pi\)
−0.680169 + 0.733056i \(0.738094\pi\)
\(558\) 12.1962 0.516304
\(559\) −10.1436 −0.429028
\(560\) −1.26795 −0.0535806
\(561\) 5.85641 0.247258
\(562\) −4.92820 −0.207884
\(563\) 17.0718 0.719490 0.359745 0.933051i \(-0.382864\pi\)
0.359745 + 0.933051i \(0.382864\pi\)
\(564\) 3.46410 0.145865
\(565\) −17.4641 −0.734720
\(566\) 4.39230 0.184622
\(567\) 3.12436 0.131211
\(568\) 10.9282 0.458537
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −11.4641 −0.480178
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 2.14359 0.0896281
\(573\) 14.3923 0.601247
\(574\) −2.53590 −0.105846
\(575\) −8.00000 −0.333623
\(576\) 4.46410 0.186004
\(577\) −3.60770 −0.150190 −0.0750952 0.997176i \(-0.523926\pi\)
−0.0750952 + 0.997176i \(0.523926\pi\)
\(578\) 14.8564 0.617945
\(579\) 32.3923 1.34618
\(580\) −8.92820 −0.370723
\(581\) 6.67949 0.277112
\(582\) −5.46410 −0.226494
\(583\) −8.78461 −0.363821
\(584\) −12.9282 −0.534973
\(585\) 6.53590 0.270226
\(586\) −6.00000 −0.247858
\(587\) 3.21539 0.132713 0.0663567 0.997796i \(-0.478862\pi\)
0.0663567 + 0.997796i \(0.478862\pi\)
\(588\) 14.7321 0.607540
\(589\) 11.4641 0.472370
\(590\) −0.196152 −0.00807547
\(591\) −51.3205 −2.11104
\(592\) 1.00000 0.0410997
\(593\) 6.78461 0.278611 0.139305 0.990249i \(-0.455513\pi\)
0.139305 + 0.990249i \(0.455513\pi\)
\(594\) 5.85641 0.240291
\(595\) 1.85641 0.0761052
\(596\) −16.3923 −0.671455
\(597\) −71.1769 −2.91308
\(598\) 11.7128 0.478973
\(599\) 2.53590 0.103614 0.0518070 0.998657i \(-0.483502\pi\)
0.0518070 + 0.998657i \(0.483502\pi\)
\(600\) 2.73205 0.111536
\(601\) 48.3923 1.97396 0.986982 0.160833i \(-0.0514180\pi\)
0.986982 + 0.160833i \(0.0514180\pi\)
\(602\) −8.78461 −0.358034
\(603\) 60.9808 2.48333
\(604\) 8.39230 0.341478
\(605\) −8.85641 −0.360064
\(606\) −6.92820 −0.281439
\(607\) −40.7846 −1.65540 −0.827698 0.561174i \(-0.810350\pi\)
−0.827698 + 0.561174i \(0.810350\pi\)
\(608\) 4.19615 0.170176
\(609\) −30.9282 −1.25327
\(610\) −8.92820 −0.361492
\(611\) −1.85641 −0.0751022
\(612\) −6.53590 −0.264198
\(613\) 3.07180 0.124069 0.0620344 0.998074i \(-0.480241\pi\)
0.0620344 + 0.998074i \(0.480241\pi\)
\(614\) 12.5885 0.508029
\(615\) 5.46410 0.220334
\(616\) 1.85641 0.0747967
\(617\) 12.9282 0.520470 0.260235 0.965545i \(-0.416200\pi\)
0.260235 + 0.965545i \(0.416200\pi\)
\(618\) −36.7846 −1.47969
\(619\) 9.85641 0.396162 0.198081 0.980186i \(-0.436529\pi\)
0.198081 + 0.980186i \(0.436529\pi\)
\(620\) −2.73205 −0.109722
\(621\) 32.0000 1.28412
\(622\) −27.1244 −1.08759
\(623\) 2.53590 0.101599
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −3.85641 −0.154133
\(627\) 16.7846 0.670313
\(628\) 16.9282 0.675509
\(629\) −1.46410 −0.0583776
\(630\) 5.66025 0.225510
\(631\) −20.9808 −0.835231 −0.417615 0.908624i \(-0.637134\pi\)
−0.417615 + 0.908624i \(0.637134\pi\)
\(632\) −5.26795 −0.209548
\(633\) −26.9282 −1.07030
\(634\) 31.8564 1.26518
\(635\) −13.6603 −0.542091
\(636\) 16.3923 0.649997
\(637\) −7.89488 −0.312807
\(638\) 13.0718 0.517517
\(639\) −48.7846 −1.92989
\(640\) −1.00000 −0.0395285
\(641\) −13.4641 −0.531800 −0.265900 0.964001i \(-0.585669\pi\)
−0.265900 + 0.964001i \(0.585669\pi\)
\(642\) 18.3923 0.725886
\(643\) 41.4641 1.63518 0.817592 0.575798i \(-0.195308\pi\)
0.817592 + 0.575798i \(0.195308\pi\)
\(644\) 10.1436 0.399714
\(645\) 18.9282 0.745297
\(646\) −6.14359 −0.241716
\(647\) 19.7128 0.774991 0.387495 0.921872i \(-0.373340\pi\)
0.387495 + 0.921872i \(0.373340\pi\)
\(648\) 2.46410 0.0967991
\(649\) 0.287187 0.0112731
\(650\) −1.46410 −0.0574268
\(651\) −9.46410 −0.370927
\(652\) 23.3205 0.913302
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 5.46410 0.213663
\(655\) 12.5885 0.491872
\(656\) −2.00000 −0.0780869
\(657\) 57.7128 2.25159
\(658\) −1.60770 −0.0626745
\(659\) 6.92820 0.269884 0.134942 0.990853i \(-0.456915\pi\)
0.134942 + 0.990853i \(0.456915\pi\)
\(660\) −4.00000 −0.155700
\(661\) −44.9282 −1.74750 −0.873752 0.486371i \(-0.838320\pi\)
−0.873752 + 0.486371i \(0.838320\pi\)
\(662\) 8.87564 0.344962
\(663\) 5.85641 0.227444
\(664\) 5.26795 0.204436
\(665\) 5.32051 0.206320
\(666\) −4.46410 −0.172980
\(667\) 71.4256 2.76561
\(668\) 5.46410 0.211412
\(669\) 60.2487 2.32935
\(670\) −13.6603 −0.527742
\(671\) 13.0718 0.504631
\(672\) −3.46410 −0.133631
\(673\) −19.0718 −0.735164 −0.367582 0.929991i \(-0.619814\pi\)
−0.367582 + 0.929991i \(0.619814\pi\)
\(674\) −26.0000 −1.00148
\(675\) −4.00000 −0.153960
\(676\) −10.8564 −0.417554
\(677\) −31.8564 −1.22434 −0.612171 0.790726i \(-0.709703\pi\)
−0.612171 + 0.790726i \(0.709703\pi\)
\(678\) −47.7128 −1.83240
\(679\) 2.53590 0.0973188
\(680\) 1.46410 0.0561457
\(681\) −9.85641 −0.377698
\(682\) 4.00000 0.153168
\(683\) 36.7846 1.40752 0.703762 0.710436i \(-0.251502\pi\)
0.703762 + 0.710436i \(0.251502\pi\)
\(684\) −18.7321 −0.716238
\(685\) 2.00000 0.0764161
\(686\) −15.7128 −0.599918
\(687\) −43.3205 −1.65278
\(688\) −6.92820 −0.264135
\(689\) −8.78461 −0.334667
\(690\) −21.8564 −0.832059
\(691\) 20.3923 0.775760 0.387880 0.921710i \(-0.373208\pi\)
0.387880 + 0.921710i \(0.373208\pi\)
\(692\) 10.0000 0.380143
\(693\) −8.28719 −0.314804
\(694\) −17.0718 −0.648037
\(695\) −6.92820 −0.262802
\(696\) −24.3923 −0.924588
\(697\) 2.92820 0.110914
\(698\) −19.3205 −0.731292
\(699\) −41.1769 −1.55745
\(700\) −1.26795 −0.0479240
\(701\) 15.8564 0.598888 0.299444 0.954114i \(-0.403199\pi\)
0.299444 + 0.954114i \(0.403199\pi\)
\(702\) 5.85641 0.221036
\(703\) −4.19615 −0.158261
\(704\) 1.46410 0.0551804
\(705\) 3.46410 0.130466
\(706\) −15.8564 −0.596764
\(707\) 3.21539 0.120927
\(708\) −0.535898 −0.0201403
\(709\) 16.1436 0.606285 0.303143 0.952945i \(-0.401964\pi\)
0.303143 + 0.952945i \(0.401964\pi\)
\(710\) 10.9282 0.410128
\(711\) 23.5167 0.881944
\(712\) 2.00000 0.0749532
\(713\) 21.8564 0.818529
\(714\) 5.07180 0.189807
\(715\) 2.14359 0.0801659
\(716\) −17.6603 −0.659995
\(717\) 47.1769 1.76185
\(718\) −8.39230 −0.313198
\(719\) 8.39230 0.312980 0.156490 0.987680i \(-0.449982\pi\)
0.156490 + 0.987680i \(0.449982\pi\)
\(720\) 4.46410 0.166367
\(721\) 17.0718 0.635787
\(722\) 1.39230 0.0518162
\(723\) 24.3923 0.907160
\(724\) −1.46410 −0.0544129
\(725\) −8.92820 −0.331585
\(726\) −24.1962 −0.898003
\(727\) −32.7846 −1.21591 −0.607957 0.793970i \(-0.708011\pi\)
−0.607957 + 0.793970i \(0.708011\pi\)
\(728\) 1.85641 0.0688030
\(729\) −43.7846 −1.62165
\(730\) −12.9282 −0.478494
\(731\) 10.1436 0.375174
\(732\) −24.3923 −0.901566
\(733\) 0.143594 0.00530375 0.00265187 0.999996i \(-0.499156\pi\)
0.00265187 + 0.999996i \(0.499156\pi\)
\(734\) 21.6603 0.799495
\(735\) 14.7321 0.543400
\(736\) 8.00000 0.294884
\(737\) 20.0000 0.736709
\(738\) 8.92820 0.328652
\(739\) −6.92820 −0.254858 −0.127429 0.991848i \(-0.540673\pi\)
−0.127429 + 0.991848i \(0.540673\pi\)
\(740\) 1.00000 0.0367607
\(741\) 16.7846 0.616598
\(742\) −7.60770 −0.279287
\(743\) −7.12436 −0.261367 −0.130684 0.991424i \(-0.541717\pi\)
−0.130684 + 0.991424i \(0.541717\pi\)
\(744\) −7.46410 −0.273647
\(745\) −16.3923 −0.600568
\(746\) −30.7846 −1.12710
\(747\) −23.5167 −0.860430
\(748\) −2.14359 −0.0783775
\(749\) −8.53590 −0.311895
\(750\) 2.73205 0.0997604
\(751\) 0.392305 0.0143154 0.00715770 0.999974i \(-0.497722\pi\)
0.00715770 + 0.999974i \(0.497722\pi\)
\(752\) −1.26795 −0.0462373
\(753\) −62.1051 −2.26324
\(754\) 13.0718 0.476047
\(755\) 8.39230 0.305427
\(756\) 5.07180 0.184459
\(757\) −53.7128 −1.95223 −0.976113 0.217265i \(-0.930286\pi\)
−0.976113 + 0.217265i \(0.930286\pi\)
\(758\) −11.6077 −0.421610
\(759\) 32.0000 1.16153
\(760\) 4.19615 0.152210
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −37.3205 −1.35198
\(763\) −2.53590 −0.0918057
\(764\) −5.26795 −0.190588
\(765\) −6.53590 −0.236306
\(766\) 8.00000 0.289052
\(767\) 0.287187 0.0103697
\(768\) −2.73205 −0.0985844
\(769\) 20.9282 0.754690 0.377345 0.926073i \(-0.376837\pi\)
0.377345 + 0.926073i \(0.376837\pi\)
\(770\) 1.85641 0.0669002
\(771\) 69.5692 2.50547
\(772\) −11.8564 −0.426721
\(773\) −38.7846 −1.39499 −0.697493 0.716592i \(-0.745701\pi\)
−0.697493 + 0.716592i \(0.745701\pi\)
\(774\) 30.9282 1.11169
\(775\) −2.73205 −0.0981382
\(776\) 2.00000 0.0717958
\(777\) 3.46410 0.124274
\(778\) −15.8564 −0.568480
\(779\) 8.39230 0.300686
\(780\) −4.00000 −0.143223
\(781\) −16.0000 −0.572525
\(782\) −11.7128 −0.418849
\(783\) 35.7128 1.27627
\(784\) −5.39230 −0.192582
\(785\) 16.9282 0.604193
\(786\) 34.3923 1.22673
\(787\) 11.8038 0.420762 0.210381 0.977620i \(-0.432530\pi\)
0.210381 + 0.977620i \(0.432530\pi\)
\(788\) 18.7846 0.669174
\(789\) −82.1051 −2.92302
\(790\) −5.26795 −0.187425
\(791\) 22.1436 0.787336
\(792\) −6.53590 −0.232243
\(793\) 13.0718 0.464193
\(794\) 22.0000 0.780751
\(795\) 16.3923 0.581375
\(796\) 26.0526 0.923408
\(797\) 17.4641 0.618610 0.309305 0.950963i \(-0.399904\pi\)
0.309305 + 0.950963i \(0.399904\pi\)
\(798\) 14.5359 0.514565
\(799\) 1.85641 0.0656749
\(800\) −1.00000 −0.0353553
\(801\) −8.92820 −0.315463
\(802\) 19.0718 0.673449
\(803\) 18.9282 0.667962
\(804\) −37.3205 −1.31619
\(805\) 10.1436 0.357515
\(806\) 4.00000 0.140894
\(807\) 1.07180 0.0377290
\(808\) 2.53590 0.0892126
\(809\) −39.5692 −1.39118 −0.695590 0.718439i \(-0.744857\pi\)
−0.695590 + 0.718439i \(0.744857\pi\)
\(810\) 2.46410 0.0865797
\(811\) −14.1436 −0.496649 −0.248324 0.968677i \(-0.579880\pi\)
−0.248324 + 0.968677i \(0.579880\pi\)
\(812\) 11.3205 0.397272
\(813\) −45.8564 −1.60825
\(814\) −1.46410 −0.0513167
\(815\) 23.3205 0.816882
\(816\) 4.00000 0.140028
\(817\) 29.0718 1.01709
\(818\) −16.9282 −0.591881
\(819\) −8.28719 −0.289578
\(820\) −2.00000 −0.0698430
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 5.46410 0.190582
\(823\) −10.4449 −0.364085 −0.182043 0.983291i \(-0.558271\pi\)
−0.182043 + 0.983291i \(0.558271\pi\)
\(824\) 13.4641 0.469044
\(825\) −4.00000 −0.139262
\(826\) 0.248711 0.00865377
\(827\) 4.39230 0.152735 0.0763677 0.997080i \(-0.475668\pi\)
0.0763677 + 0.997080i \(0.475668\pi\)
\(828\) −35.7128 −1.24111
\(829\) 31.5692 1.09644 0.548222 0.836333i \(-0.315305\pi\)
0.548222 + 0.836333i \(0.315305\pi\)
\(830\) 5.26795 0.182853
\(831\) 71.7128 2.48769
\(832\) 1.46410 0.0507586
\(833\) 7.89488 0.273541
\(834\) −18.9282 −0.655430
\(835\) 5.46410 0.189093
\(836\) −6.14359 −0.212481
\(837\) 10.9282 0.377734
\(838\) 34.2487 1.18310
\(839\) 32.7846 1.13185 0.565925 0.824457i \(-0.308519\pi\)
0.565925 + 0.824457i \(0.308519\pi\)
\(840\) −3.46410 −0.119523
\(841\) 50.7128 1.74872
\(842\) 27.8564 0.959995
\(843\) −13.4641 −0.463728
\(844\) 9.85641 0.339272
\(845\) −10.8564 −0.373472
\(846\) 5.66025 0.194604
\(847\) 11.2295 0.385849
\(848\) −6.00000 −0.206041
\(849\) 12.0000 0.411839
\(850\) 1.46410 0.0502183
\(851\) −8.00000 −0.274236
\(852\) 29.8564 1.02286
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 11.3205 0.387380
\(855\) −18.7321 −0.640623
\(856\) −6.73205 −0.230097
\(857\) −4.14359 −0.141542 −0.0707712 0.997493i \(-0.522546\pi\)
−0.0707712 + 0.997493i \(0.522546\pi\)
\(858\) 5.85641 0.199934
\(859\) −14.4449 −0.492852 −0.246426 0.969162i \(-0.579256\pi\)
−0.246426 + 0.969162i \(0.579256\pi\)
\(860\) −6.92820 −0.236250
\(861\) −6.92820 −0.236113
\(862\) −8.19615 −0.279162
\(863\) 38.4449 1.30868 0.654339 0.756201i \(-0.272947\pi\)
0.654339 + 0.756201i \(0.272947\pi\)
\(864\) 4.00000 0.136083
\(865\) 10.0000 0.340010
\(866\) 34.0000 1.15537
\(867\) 40.5885 1.37846
\(868\) 3.46410 0.117579
\(869\) 7.71281 0.261639
\(870\) −24.3923 −0.826977
\(871\) 20.0000 0.677674
\(872\) −2.00000 −0.0677285
\(873\) −8.92820 −0.302174
\(874\) −33.5692 −1.13550
\(875\) −1.26795 −0.0428645
\(876\) −35.3205 −1.19337
\(877\) −50.7846 −1.71487 −0.857437 0.514589i \(-0.827945\pi\)
−0.857437 + 0.514589i \(0.827945\pi\)
\(878\) 31.5167 1.06364
\(879\) −16.3923 −0.552899
\(880\) 1.46410 0.0493549
\(881\) 47.3205 1.59427 0.797134 0.603802i \(-0.206348\pi\)
0.797134 + 0.603802i \(0.206348\pi\)
\(882\) 24.0718 0.810540
\(883\) −34.2487 −1.15256 −0.576280 0.817252i \(-0.695496\pi\)
−0.576280 + 0.817252i \(0.695496\pi\)
\(884\) −2.14359 −0.0720969
\(885\) −0.535898 −0.0180140
\(886\) 39.1244 1.31441
\(887\) 20.1962 0.678120 0.339060 0.940765i \(-0.389891\pi\)
0.339060 + 0.940765i \(0.389891\pi\)
\(888\) 2.73205 0.0916816
\(889\) 17.3205 0.580911
\(890\) 2.00000 0.0670402
\(891\) −3.60770 −0.120862
\(892\) −22.0526 −0.738374
\(893\) 5.32051 0.178044
\(894\) −44.7846 −1.49782
\(895\) −17.6603 −0.590317
\(896\) 1.26795 0.0423592
\(897\) 32.0000 1.06845
\(898\) −33.7128 −1.12501
\(899\) 24.3923 0.813529
\(900\) 4.46410 0.148803
\(901\) 8.78461 0.292658
\(902\) 2.92820 0.0974985
\(903\) −24.0000 −0.798670
\(904\) 17.4641 0.580847
\(905\) −1.46410 −0.0486684
\(906\) 22.9282 0.761739
\(907\) 5.75129 0.190968 0.0954842 0.995431i \(-0.469560\pi\)
0.0954842 + 0.995431i \(0.469560\pi\)
\(908\) 3.60770 0.119726
\(909\) −11.3205 −0.375478
\(910\) 1.85641 0.0615393
\(911\) −27.9090 −0.924665 −0.462333 0.886707i \(-0.652987\pi\)
−0.462333 + 0.886707i \(0.652987\pi\)
\(912\) 11.4641 0.379614
\(913\) −7.71281 −0.255257
\(914\) 4.14359 0.137058
\(915\) −24.3923 −0.806385
\(916\) 15.8564 0.523910
\(917\) −15.9615 −0.527096
\(918\) −5.85641 −0.193290
\(919\) −13.2679 −0.437669 −0.218835 0.975762i \(-0.570226\pi\)
−0.218835 + 0.975762i \(0.570226\pi\)
\(920\) 8.00000 0.263752
\(921\) 34.3923 1.13326
\(922\) 26.7846 0.882104
\(923\) −16.0000 −0.526646
\(924\) 5.07180 0.166850
\(925\) 1.00000 0.0328798
\(926\) −5.07180 −0.166670
\(927\) −60.1051 −1.97411
\(928\) 8.92820 0.293083
\(929\) −15.8564 −0.520232 −0.260116 0.965577i \(-0.583761\pi\)
−0.260116 + 0.965577i \(0.583761\pi\)
\(930\) −7.46410 −0.244758
\(931\) 22.6269 0.741568
\(932\) 15.0718 0.493693
\(933\) −74.1051 −2.42609
\(934\) −37.1769 −1.21647
\(935\) −2.14359 −0.0701030
\(936\) −6.53590 −0.213633
\(937\) 3.85641 0.125983 0.0629917 0.998014i \(-0.479936\pi\)
0.0629917 + 0.998014i \(0.479936\pi\)
\(938\) 17.3205 0.565535
\(939\) −10.5359 −0.343826
\(940\) −1.26795 −0.0413559
\(941\) 28.3923 0.925563 0.462781 0.886472i \(-0.346852\pi\)
0.462781 + 0.886472i \(0.346852\pi\)
\(942\) 46.2487 1.50686
\(943\) 16.0000 0.521032
\(944\) 0.196152 0.00638422
\(945\) 5.07180 0.164986
\(946\) 10.1436 0.329797
\(947\) −45.1769 −1.46805 −0.734026 0.679121i \(-0.762361\pi\)
−0.734026 + 0.679121i \(0.762361\pi\)
\(948\) −14.3923 −0.467440
\(949\) 18.9282 0.614435
\(950\) 4.19615 0.136141
\(951\) 87.0333 2.82225
\(952\) −1.85641 −0.0601665
\(953\) −11.8564 −0.384067 −0.192033 0.981388i \(-0.561508\pi\)
−0.192033 + 0.981388i \(0.561508\pi\)
\(954\) 26.7846 0.867184
\(955\) −5.26795 −0.170467
\(956\) −17.2679 −0.558485
\(957\) 35.7128 1.15443
\(958\) 34.0526 1.10019
\(959\) −2.53590 −0.0818884
\(960\) −2.73205 −0.0881766
\(961\) −23.5359 −0.759223
\(962\) −1.46410 −0.0472045
\(963\) 30.0526 0.968430
\(964\) −8.92820 −0.287558
\(965\) −11.8564 −0.381671
\(966\) 27.7128 0.891645
\(967\) −23.6077 −0.759172 −0.379586 0.925156i \(-0.623934\pi\)
−0.379586 + 0.925156i \(0.623934\pi\)
\(968\) 8.85641 0.284656
\(969\) −16.7846 −0.539199
\(970\) 2.00000 0.0642161
\(971\) 14.9282 0.479069 0.239534 0.970888i \(-0.423005\pi\)
0.239534 + 0.970888i \(0.423005\pi\)
\(972\) 18.7321 0.600831
\(973\) 8.78461 0.281622
\(974\) 16.3923 0.525243
\(975\) −4.00000 −0.128103
\(976\) 8.92820 0.285785
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 63.7128 2.03731
\(979\) −2.92820 −0.0935858
\(980\) −5.39230 −0.172251
\(981\) 8.92820 0.285056
\(982\) 1.07180 0.0342024
\(983\) −38.4449 −1.22620 −0.613100 0.790005i \(-0.710078\pi\)
−0.613100 + 0.790005i \(0.710078\pi\)
\(984\) −5.46410 −0.174189
\(985\) 18.7846 0.598527
\(986\) −13.0718 −0.416291
\(987\) −4.39230 −0.139809
\(988\) −6.14359 −0.195454
\(989\) 55.4256 1.76243
\(990\) −6.53590 −0.207724
\(991\) 5.94744 0.188927 0.0944633 0.995528i \(-0.469886\pi\)
0.0944633 + 0.995528i \(0.469886\pi\)
\(992\) 2.73205 0.0867427
\(993\) 24.2487 0.769510
\(994\) −13.8564 −0.439499
\(995\) 26.0526 0.825922
\(996\) 14.3923 0.456038
\(997\) −4.14359 −0.131229 −0.0656145 0.997845i \(-0.520901\pi\)
−0.0656145 + 0.997845i \(0.520901\pi\)
\(998\) −40.5885 −1.28481
\(999\) −4.00000 −0.126554
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 370.2.a.e.1.1 2
3.2 odd 2 3330.2.a.bd.1.2 2
4.3 odd 2 2960.2.a.q.1.2 2
5.2 odd 4 1850.2.b.l.149.2 4
5.3 odd 4 1850.2.b.l.149.3 4
5.4 even 2 1850.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.1 2 1.1 even 1 trivial
1850.2.a.x.1.2 2 5.4 even 2
1850.2.b.l.149.2 4 5.2 odd 4
1850.2.b.l.149.3 4 5.3 odd 4
2960.2.a.q.1.2 2 4.3 odd 2
3330.2.a.bd.1.2 2 3.2 odd 2