Properties

Label 231.2.a.e.1.3
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.47283 q^{2} +1.00000 q^{3} +4.11491 q^{4} -2.58774 q^{5} +2.47283 q^{6} -1.00000 q^{7} +5.22982 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.47283 q^{2} +1.00000 q^{3} +4.11491 q^{4} -2.58774 q^{5} +2.47283 q^{6} -1.00000 q^{7} +5.22982 q^{8} +1.00000 q^{9} -6.39905 q^{10} -1.00000 q^{11} +4.11491 q^{12} -5.87189 q^{13} -2.47283 q^{14} -2.58774 q^{15} +4.70265 q^{16} +7.51396 q^{17} +2.47283 q^{18} -2.35793 q^{19} -10.6483 q^{20} -1.00000 q^{21} -2.47283 q^{22} +6.94567 q^{23} +5.22982 q^{24} +1.69641 q^{25} -14.5202 q^{26} +1.00000 q^{27} -4.11491 q^{28} -5.87189 q^{29} -6.39905 q^{30} -3.66152 q^{31} +1.16924 q^{32} -1.00000 q^{33} +18.5808 q^{34} +2.58774 q^{35} +4.11491 q^{36} +3.30359 q^{37} -5.83076 q^{38} -5.87189 q^{39} -13.5334 q^{40} +5.28415 q^{41} -2.47283 q^{42} +7.40530 q^{43} -4.11491 q^{44} -2.58774 q^{45} +17.1755 q^{46} +7.53341 q^{47} +4.70265 q^{48} +1.00000 q^{49} +4.19493 q^{50} +7.51396 q^{51} -24.1623 q^{52} -4.22982 q^{53} +2.47283 q^{54} +2.58774 q^{55} -5.22982 q^{56} -2.35793 q^{57} -14.5202 q^{58} -0.926221 q^{59} -10.6483 q^{60} -2.00000 q^{61} -9.05433 q^{62} -1.00000 q^{63} -6.51396 q^{64} +15.1949 q^{65} -2.47283 q^{66} -10.1017 q^{67} +30.9193 q^{68} +6.94567 q^{69} +6.39905 q^{70} +4.45963 q^{71} +5.22982 q^{72} -2.12811 q^{73} +8.16924 q^{74} +1.69641 q^{75} -9.70265 q^{76} +1.00000 q^{77} -14.5202 q^{78} -4.45963 q^{79} -12.1692 q^{80} +1.00000 q^{81} +13.0668 q^{82} -10.6894 q^{83} -4.11491 q^{84} -19.4442 q^{85} +18.3121 q^{86} -5.87189 q^{87} -5.22982 q^{88} +15.8913 q^{89} -6.39905 q^{90} +5.87189 q^{91} +28.5808 q^{92} -3.66152 q^{93} +18.6289 q^{94} +6.10170 q^{95} +1.16924 q^{96} +10.1212 q^{97} +2.47283 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 11 q^{10} - 3 q^{11} + 6 q^{12} - 4 q^{13} - 2 q^{14} + 4 q^{15} - 4 q^{16} + 8 q^{17} + 2 q^{18} - 8 q^{19} - 3 q^{20} - 3 q^{21} - 2 q^{22} + 10 q^{23} + 3 q^{24} + 15 q^{25} - q^{26} + 3 q^{27} - 6 q^{28} - 4 q^{29} - 11 q^{30} - 2 q^{31} + 8 q^{32} - 3 q^{33} - 4 q^{34} - 4 q^{35} + 6 q^{36} - 13 q^{38} - 4 q^{39} - 18 q^{40} + 14 q^{41} - 2 q^{42} - 14 q^{43} - 6 q^{44} + 4 q^{45} + 28 q^{46} - 4 q^{48} + 3 q^{49} - 19 q^{50} + 8 q^{51} - 29 q^{52} + 2 q^{54} - 4 q^{55} - 3 q^{56} - 8 q^{57} - q^{58} - 3 q^{60} - 6 q^{61} - 38 q^{62} - 3 q^{63} - 5 q^{64} + 14 q^{65} - 2 q^{66} - 4 q^{67} + 42 q^{68} + 10 q^{69} + 11 q^{70} - 12 q^{71} + 3 q^{72} - 20 q^{73} + 29 q^{74} + 15 q^{75} - 11 q^{76} + 3 q^{77} - q^{78} + 12 q^{79} - 41 q^{80} + 3 q^{81} - 6 q^{82} + 6 q^{83} - 6 q^{84} - 6 q^{85} + 24 q^{86} - 4 q^{87} - 3 q^{88} + 26 q^{89} - 11 q^{90} + 4 q^{91} + 26 q^{92} - 2 q^{93} + 35 q^{94} - 8 q^{95} + 8 q^{96} - 4 q^{97} + 2 q^{98} - 3 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.47283 1.74856 0.874279 0.485424i \(-0.161335\pi\)
0.874279 + 0.485424i \(0.161335\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.11491 2.05745
\(5\) −2.58774 −1.15727 −0.578637 0.815586i \(-0.696415\pi\)
−0.578637 + 0.815586i \(0.696415\pi\)
\(6\) 2.47283 1.00953
\(7\) −1.00000 −0.377964
\(8\) 5.22982 1.84902
\(9\) 1.00000 0.333333
\(10\) −6.39905 −2.02356
\(11\) −1.00000 −0.301511
\(12\) 4.11491 1.18787
\(13\) −5.87189 −1.62857 −0.814284 0.580466i \(-0.802870\pi\)
−0.814284 + 0.580466i \(0.802870\pi\)
\(14\) −2.47283 −0.660893
\(15\) −2.58774 −0.668152
\(16\) 4.70265 1.17566
\(17\) 7.51396 1.82240 0.911202 0.411960i \(-0.135156\pi\)
0.911202 + 0.411960i \(0.135156\pi\)
\(18\) 2.47283 0.582853
\(19\) −2.35793 −0.540945 −0.270473 0.962728i \(-0.587180\pi\)
−0.270473 + 0.962728i \(0.587180\pi\)
\(20\) −10.6483 −2.38104
\(21\) −1.00000 −0.218218
\(22\) −2.47283 −0.527210
\(23\) 6.94567 1.44827 0.724136 0.689657i \(-0.242239\pi\)
0.724136 + 0.689657i \(0.242239\pi\)
\(24\) 5.22982 1.06753
\(25\) 1.69641 0.339281
\(26\) −14.5202 −2.84765
\(27\) 1.00000 0.192450
\(28\) −4.11491 −0.777644
\(29\) −5.87189 −1.09038 −0.545191 0.838312i \(-0.683543\pi\)
−0.545191 + 0.838312i \(0.683543\pi\)
\(30\) −6.39905 −1.16830
\(31\) −3.66152 −0.657629 −0.328814 0.944395i \(-0.606649\pi\)
−0.328814 + 0.944395i \(0.606649\pi\)
\(32\) 1.16924 0.206694
\(33\) −1.00000 −0.174078
\(34\) 18.5808 3.18658
\(35\) 2.58774 0.437408
\(36\) 4.11491 0.685818
\(37\) 3.30359 0.543108 0.271554 0.962423i \(-0.412463\pi\)
0.271554 + 0.962423i \(0.412463\pi\)
\(38\) −5.83076 −0.945874
\(39\) −5.87189 −0.940255
\(40\) −13.5334 −2.13982
\(41\) 5.28415 0.825245 0.412623 0.910902i \(-0.364613\pi\)
0.412623 + 0.910902i \(0.364613\pi\)
\(42\) −2.47283 −0.381567
\(43\) 7.40530 1.12930 0.564649 0.825331i \(-0.309012\pi\)
0.564649 + 0.825331i \(0.309012\pi\)
\(44\) −4.11491 −0.620346
\(45\) −2.58774 −0.385758
\(46\) 17.1755 2.53239
\(47\) 7.53341 1.09886 0.549430 0.835540i \(-0.314845\pi\)
0.549430 + 0.835540i \(0.314845\pi\)
\(48\) 4.70265 0.678769
\(49\) 1.00000 0.142857
\(50\) 4.19493 0.593253
\(51\) 7.51396 1.05217
\(52\) −24.1623 −3.35071
\(53\) −4.22982 −0.581010 −0.290505 0.956874i \(-0.593823\pi\)
−0.290505 + 0.956874i \(0.593823\pi\)
\(54\) 2.47283 0.336510
\(55\) 2.58774 0.348931
\(56\) −5.22982 −0.698863
\(57\) −2.35793 −0.312315
\(58\) −14.5202 −1.90660
\(59\) −0.926221 −0.120584 −0.0602918 0.998181i \(-0.519203\pi\)
−0.0602918 + 0.998181i \(0.519203\pi\)
\(60\) −10.6483 −1.37469
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −9.05433 −1.14990
\(63\) −1.00000 −0.125988
\(64\) −6.51396 −0.814245
\(65\) 15.1949 1.88470
\(66\) −2.47283 −0.304385
\(67\) −10.1017 −1.23412 −0.617060 0.786916i \(-0.711676\pi\)
−0.617060 + 0.786916i \(0.711676\pi\)
\(68\) 30.9193 3.74951
\(69\) 6.94567 0.836160
\(70\) 6.39905 0.764833
\(71\) 4.45963 0.529261 0.264630 0.964350i \(-0.414750\pi\)
0.264630 + 0.964350i \(0.414750\pi\)
\(72\) 5.22982 0.616340
\(73\) −2.12811 −0.249077 −0.124538 0.992215i \(-0.539745\pi\)
−0.124538 + 0.992215i \(0.539745\pi\)
\(74\) 8.16924 0.949655
\(75\) 1.69641 0.195884
\(76\) −9.70265 −1.11297
\(77\) 1.00000 0.113961
\(78\) −14.5202 −1.64409
\(79\) −4.45963 −0.501748 −0.250874 0.968020i \(-0.580718\pi\)
−0.250874 + 0.968020i \(0.580718\pi\)
\(80\) −12.1692 −1.36056
\(81\) 1.00000 0.111111
\(82\) 13.0668 1.44299
\(83\) −10.6894 −1.17332 −0.586660 0.809834i \(-0.699557\pi\)
−0.586660 + 0.809834i \(0.699557\pi\)
\(84\) −4.11491 −0.448973
\(85\) −19.4442 −2.10902
\(86\) 18.3121 1.97464
\(87\) −5.87189 −0.629533
\(88\) −5.22982 −0.557500
\(89\) 15.8913 1.68448 0.842239 0.539104i \(-0.181237\pi\)
0.842239 + 0.539104i \(0.181237\pi\)
\(90\) −6.39905 −0.674520
\(91\) 5.87189 0.615541
\(92\) 28.5808 2.97975
\(93\) −3.66152 −0.379682
\(94\) 18.6289 1.92142
\(95\) 6.10170 0.626022
\(96\) 1.16924 0.119335
\(97\) 10.1212 1.02765 0.513824 0.857896i \(-0.328229\pi\)
0.513824 + 0.857896i \(0.328229\pi\)
\(98\) 2.47283 0.249794
\(99\) −1.00000 −0.100504
\(100\) 6.98055 0.698055
\(101\) −13.4053 −1.33388 −0.666939 0.745113i \(-0.732396\pi\)
−0.666939 + 0.745113i \(0.732396\pi\)
\(102\) 18.5808 1.83977
\(103\) −4.71585 −0.464667 −0.232333 0.972636i \(-0.574636\pi\)
−0.232333 + 0.972636i \(0.574636\pi\)
\(104\) −30.7089 −3.01125
\(105\) 2.58774 0.252538
\(106\) −10.4596 −1.01593
\(107\) 7.07378 0.683848 0.341924 0.939728i \(-0.388921\pi\)
0.341924 + 0.939728i \(0.388921\pi\)
\(108\) 4.11491 0.395957
\(109\) 18.8370 1.80426 0.902129 0.431467i \(-0.142004\pi\)
0.902129 + 0.431467i \(0.142004\pi\)
\(110\) 6.39905 0.610126
\(111\) 3.30359 0.313563
\(112\) −4.70265 −0.444359
\(113\) −9.02792 −0.849276 −0.424638 0.905363i \(-0.639599\pi\)
−0.424638 + 0.905363i \(0.639599\pi\)
\(114\) −5.83076 −0.546101
\(115\) −17.9736 −1.67605
\(116\) −24.1623 −2.24341
\(117\) −5.87189 −0.542856
\(118\) −2.29039 −0.210848
\(119\) −7.51396 −0.688804
\(120\) −13.5334 −1.23543
\(121\) 1.00000 0.0909091
\(122\) −4.94567 −0.447760
\(123\) 5.28415 0.476456
\(124\) −15.0668 −1.35304
\(125\) 8.54885 0.764632
\(126\) −2.47283 −0.220298
\(127\) −10.2298 −0.907749 −0.453875 0.891066i \(-0.649959\pi\)
−0.453875 + 0.891066i \(0.649959\pi\)
\(128\) −18.4464 −1.63045
\(129\) 7.40530 0.652000
\(130\) 37.5745 3.29550
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −4.11491 −0.358157
\(133\) 2.35793 0.204458
\(134\) −24.9798 −2.15793
\(135\) −2.58774 −0.222717
\(136\) 39.2966 3.36966
\(137\) −10.8370 −0.925868 −0.462934 0.886393i \(-0.653203\pi\)
−0.462934 + 0.886393i \(0.653203\pi\)
\(138\) 17.1755 1.46207
\(139\) 0.459630 0.0389853 0.0194927 0.999810i \(-0.493795\pi\)
0.0194927 + 0.999810i \(0.493795\pi\)
\(140\) 10.6483 0.899947
\(141\) 7.53341 0.634428
\(142\) 11.0279 0.925443
\(143\) 5.87189 0.491032
\(144\) 4.70265 0.391887
\(145\) 15.1949 1.26187
\(146\) −5.26247 −0.435525
\(147\) 1.00000 0.0824786
\(148\) 13.5940 1.11742
\(149\) 11.0474 0.905036 0.452518 0.891755i \(-0.350526\pi\)
0.452518 + 0.891755i \(0.350526\pi\)
\(150\) 4.19493 0.342515
\(151\) −16.1212 −1.31192 −0.655960 0.754795i \(-0.727736\pi\)
−0.655960 + 0.754795i \(0.727736\pi\)
\(152\) −12.3315 −1.00022
\(153\) 7.51396 0.607468
\(154\) 2.47283 0.199267
\(155\) 9.47507 0.761056
\(156\) −24.1623 −1.93453
\(157\) 13.0668 1.04285 0.521423 0.853298i \(-0.325401\pi\)
0.521423 + 0.853298i \(0.325401\pi\)
\(158\) −11.0279 −0.877335
\(159\) −4.22982 −0.335446
\(160\) −3.02569 −0.239202
\(161\) −6.94567 −0.547395
\(162\) 2.47283 0.194284
\(163\) −10.1017 −0.791227 −0.395613 0.918417i \(-0.629468\pi\)
−0.395613 + 0.918417i \(0.629468\pi\)
\(164\) 21.7438 1.69790
\(165\) 2.58774 0.201455
\(166\) −26.4332 −2.05162
\(167\) −10.4860 −0.811434 −0.405717 0.913999i \(-0.632978\pi\)
−0.405717 + 0.913999i \(0.632978\pi\)
\(168\) −5.22982 −0.403489
\(169\) 21.4791 1.65224
\(170\) −48.0823 −3.68774
\(171\) −2.35793 −0.180315
\(172\) 30.4721 2.32348
\(173\) 14.4985 1.10230 0.551151 0.834405i \(-0.314189\pi\)
0.551151 + 0.834405i \(0.314189\pi\)
\(174\) −14.5202 −1.10077
\(175\) −1.69641 −0.128236
\(176\) −4.70265 −0.354476
\(177\) −0.926221 −0.0696190
\(178\) 39.2966 2.94541
\(179\) 14.3510 1.07264 0.536321 0.844014i \(-0.319814\pi\)
0.536321 + 0.844014i \(0.319814\pi\)
\(180\) −10.6483 −0.793679
\(181\) 9.66152 0.718135 0.359068 0.933312i \(-0.383095\pi\)
0.359068 + 0.933312i \(0.383095\pi\)
\(182\) 14.5202 1.07631
\(183\) −2.00000 −0.147844
\(184\) 36.3246 2.67788
\(185\) −8.54885 −0.628524
\(186\) −9.05433 −0.663896
\(187\) −7.51396 −0.549475
\(188\) 30.9993 2.26086
\(189\) −1.00000 −0.0727393
\(190\) 15.0885 1.09463
\(191\) −9.55286 −0.691220 −0.345610 0.938378i \(-0.612328\pi\)
−0.345610 + 0.938378i \(0.612328\pi\)
\(192\) −6.51396 −0.470105
\(193\) −22.1212 −1.59232 −0.796158 0.605089i \(-0.793137\pi\)
−0.796158 + 0.605089i \(0.793137\pi\)
\(194\) 25.0279 1.79690
\(195\) 15.1949 1.08813
\(196\) 4.11491 0.293922
\(197\) −26.2423 −1.86969 −0.934843 0.355061i \(-0.884460\pi\)
−0.934843 + 0.355061i \(0.884460\pi\)
\(198\) −2.47283 −0.175737
\(199\) 1.05433 0.0747396 0.0373698 0.999302i \(-0.488102\pi\)
0.0373698 + 0.999302i \(0.488102\pi\)
\(200\) 8.87189 0.627337
\(201\) −10.1017 −0.712519
\(202\) −33.1491 −2.33236
\(203\) 5.87189 0.412126
\(204\) 30.9193 2.16478
\(205\) −13.6740 −0.955034
\(206\) −11.6615 −0.812497
\(207\) 6.94567 0.482757
\(208\) −27.6134 −1.91465
\(209\) 2.35793 0.163101
\(210\) 6.39905 0.441577
\(211\) −8.71585 −0.600024 −0.300012 0.953935i \(-0.596991\pi\)
−0.300012 + 0.953935i \(0.596991\pi\)
\(212\) −17.4053 −1.19540
\(213\) 4.45963 0.305569
\(214\) 17.4923 1.19575
\(215\) −19.1630 −1.30691
\(216\) 5.22982 0.355844
\(217\) 3.66152 0.248560
\(218\) 46.5808 3.15485
\(219\) −2.12811 −0.143804
\(220\) 10.6483 0.717909
\(221\) −44.1212 −2.96791
\(222\) 8.16924 0.548283
\(223\) −23.7438 −1.59000 −0.795000 0.606609i \(-0.792529\pi\)
−0.795000 + 0.606609i \(0.792529\pi\)
\(224\) −1.16924 −0.0781231
\(225\) 1.69641 0.113094
\(226\) −22.3246 −1.48501
\(227\) −13.1755 −0.874488 −0.437244 0.899343i \(-0.644045\pi\)
−0.437244 + 0.899343i \(0.644045\pi\)
\(228\) −9.70265 −0.642574
\(229\) 13.3664 0.883277 0.441638 0.897193i \(-0.354397\pi\)
0.441638 + 0.897193i \(0.354397\pi\)
\(230\) −44.4457 −2.93066
\(231\) 1.00000 0.0657952
\(232\) −30.7089 −2.01614
\(233\) 7.13659 0.467533 0.233767 0.972293i \(-0.424895\pi\)
0.233767 + 0.972293i \(0.424895\pi\)
\(234\) −14.5202 −0.949216
\(235\) −19.4945 −1.27168
\(236\) −3.81131 −0.248095
\(237\) −4.45963 −0.289684
\(238\) −18.5808 −1.20441
\(239\) −19.9930 −1.29324 −0.646621 0.762811i \(-0.723818\pi\)
−0.646621 + 0.762811i \(0.723818\pi\)
\(240\) −12.1692 −0.785521
\(241\) −9.19493 −0.592297 −0.296149 0.955142i \(-0.595702\pi\)
−0.296149 + 0.955142i \(0.595702\pi\)
\(242\) 2.47283 0.158960
\(243\) 1.00000 0.0641500
\(244\) −8.22982 −0.526860
\(245\) −2.58774 −0.165325
\(246\) 13.0668 0.833110
\(247\) 13.8455 0.880967
\(248\) −19.1491 −1.21597
\(249\) −10.6894 −0.677416
\(250\) 21.1399 1.33700
\(251\) −9.42474 −0.594885 −0.297442 0.954740i \(-0.596134\pi\)
−0.297442 + 0.954740i \(0.596134\pi\)
\(252\) −4.11491 −0.259215
\(253\) −6.94567 −0.436670
\(254\) −25.2966 −1.58725
\(255\) −19.4442 −1.21764
\(256\) −32.5870 −2.03669
\(257\) −0.0194469 −0.00121307 −0.000606533 1.00000i \(-0.500193\pi\)
−0.000606533 1.00000i \(0.500193\pi\)
\(258\) 18.3121 1.14006
\(259\) −3.30359 −0.205275
\(260\) 62.5257 3.87768
\(261\) −5.87189 −0.363461
\(262\) 9.89134 0.611089
\(263\) 27.9930 1.72612 0.863062 0.505097i \(-0.168543\pi\)
0.863062 + 0.505097i \(0.168543\pi\)
\(264\) −5.22982 −0.321873
\(265\) 10.9457 0.672387
\(266\) 5.83076 0.357507
\(267\) 15.8913 0.972534
\(268\) −41.5676 −2.53914
\(269\) 30.9193 1.88518 0.942590 0.333951i \(-0.108382\pi\)
0.942590 + 0.333951i \(0.108382\pi\)
\(270\) −6.39905 −0.389434
\(271\) −22.3579 −1.35815 −0.679074 0.734070i \(-0.737618\pi\)
−0.679074 + 0.734070i \(0.737618\pi\)
\(272\) 35.3355 2.14253
\(273\) 5.87189 0.355383
\(274\) −26.7981 −1.61893
\(275\) −1.69641 −0.102297
\(276\) 28.5808 1.72036
\(277\) 24.7717 1.48839 0.744194 0.667964i \(-0.232834\pi\)
0.744194 + 0.667964i \(0.232834\pi\)
\(278\) 1.13659 0.0681681
\(279\) −3.66152 −0.219210
\(280\) 13.5334 0.808776
\(281\) −1.19493 −0.0712835 −0.0356418 0.999365i \(-0.511348\pi\)
−0.0356418 + 0.999365i \(0.511348\pi\)
\(282\) 18.6289 1.10933
\(283\) −15.5723 −0.925677 −0.462839 0.886443i \(-0.653169\pi\)
−0.462839 + 0.886443i \(0.653169\pi\)
\(284\) 18.3510 1.08893
\(285\) 6.10170 0.361434
\(286\) 14.5202 0.858598
\(287\) −5.28415 −0.311913
\(288\) 1.16924 0.0688981
\(289\) 39.4596 2.32115
\(290\) 37.5745 2.20645
\(291\) 10.1212 0.593312
\(292\) −8.75698 −0.512464
\(293\) −9.74378 −0.569238 −0.284619 0.958641i \(-0.591867\pi\)
−0.284619 + 0.958641i \(0.591867\pi\)
\(294\) 2.47283 0.144219
\(295\) 2.39682 0.139548
\(296\) 17.2772 1.00422
\(297\) −1.00000 −0.0580259
\(298\) 27.3183 1.58251
\(299\) −40.7842 −2.35861
\(300\) 6.98055 0.403022
\(301\) −7.40530 −0.426834
\(302\) −39.8649 −2.29397
\(303\) −13.4053 −0.770114
\(304\) −11.0885 −0.635969
\(305\) 5.17548 0.296347
\(306\) 18.5808 1.06219
\(307\) 6.35097 0.362469 0.181234 0.983440i \(-0.441991\pi\)
0.181234 + 0.983440i \(0.441991\pi\)
\(308\) 4.11491 0.234469
\(309\) −4.71585 −0.268275
\(310\) 23.4303 1.33075
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) −30.7089 −1.73855
\(313\) 16.9457 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(314\) 32.3121 1.82348
\(315\) 2.58774 0.145803
\(316\) −18.3510 −1.03232
\(317\) 20.0125 1.12401 0.562007 0.827133i \(-0.310030\pi\)
0.562007 + 0.827133i \(0.310030\pi\)
\(318\) −10.4596 −0.586547
\(319\) 5.87189 0.328763
\(320\) 16.8565 0.942304
\(321\) 7.07378 0.394820
\(322\) −17.1755 −0.957152
\(323\) −17.7174 −0.985821
\(324\) 4.11491 0.228606
\(325\) −9.96111 −0.552543
\(326\) −24.9798 −1.38351
\(327\) 18.8370 1.04169
\(328\) 27.6351 1.52589
\(329\) −7.53341 −0.415330
\(330\) 6.39905 0.352256
\(331\) 7.54037 0.414456 0.207228 0.978293i \(-0.433556\pi\)
0.207228 + 0.978293i \(0.433556\pi\)
\(332\) −43.9861 −2.41405
\(333\) 3.30359 0.181036
\(334\) −25.9302 −1.41884
\(335\) 26.1406 1.42821
\(336\) −4.70265 −0.256551
\(337\) −12.6894 −0.691238 −0.345619 0.938375i \(-0.612331\pi\)
−0.345619 + 0.938375i \(0.612331\pi\)
\(338\) 53.1142 2.88903
\(339\) −9.02792 −0.490330
\(340\) −80.0111 −4.33921
\(341\) 3.66152 0.198282
\(342\) −5.83076 −0.315291
\(343\) −1.00000 −0.0539949
\(344\) 38.7283 2.08809
\(345\) −17.9736 −0.967666
\(346\) 35.8524 1.92744
\(347\) 1.13659 0.0610153 0.0305076 0.999535i \(-0.490288\pi\)
0.0305076 + 0.999535i \(0.490288\pi\)
\(348\) −24.1623 −1.29523
\(349\) −14.7911 −0.791752 −0.395876 0.918304i \(-0.629559\pi\)
−0.395876 + 0.918304i \(0.629559\pi\)
\(350\) −4.19493 −0.224228
\(351\) −5.87189 −0.313418
\(352\) −1.16924 −0.0623207
\(353\) −7.52092 −0.400298 −0.200149 0.979765i \(-0.564143\pi\)
−0.200149 + 0.979765i \(0.564143\pi\)
\(354\) −2.29039 −0.121733
\(355\) −11.5404 −0.612499
\(356\) 65.3914 3.46574
\(357\) −7.51396 −0.397681
\(358\) 35.4876 1.87558
\(359\) −17.5962 −0.928693 −0.464346 0.885654i \(-0.653711\pi\)
−0.464346 + 0.885654i \(0.653711\pi\)
\(360\) −13.5334 −0.713273
\(361\) −13.4402 −0.707378
\(362\) 23.8913 1.25570
\(363\) 1.00000 0.0524864
\(364\) 24.1623 1.26645
\(365\) 5.50700 0.288250
\(366\) −4.94567 −0.258514
\(367\) 14.3121 0.747084 0.373542 0.927613i \(-0.378143\pi\)
0.373542 + 0.927613i \(0.378143\pi\)
\(368\) 32.6630 1.70268
\(369\) 5.28415 0.275082
\(370\) −21.1399 −1.09901
\(371\) 4.22982 0.219601
\(372\) −15.0668 −0.781178
\(373\) −16.4332 −0.850880 −0.425440 0.904987i \(-0.639881\pi\)
−0.425440 + 0.904987i \(0.639881\pi\)
\(374\) −18.5808 −0.960789
\(375\) 8.54885 0.441461
\(376\) 39.3983 2.03181
\(377\) 34.4791 1.77576
\(378\) −2.47283 −0.127189
\(379\) 19.5334 1.00336 0.501682 0.865052i \(-0.332715\pi\)
0.501682 + 0.865052i \(0.332715\pi\)
\(380\) 25.1079 1.28801
\(381\) −10.2298 −0.524089
\(382\) −23.6226 −1.20864
\(383\) 2.35097 0.120129 0.0600644 0.998195i \(-0.480869\pi\)
0.0600644 + 0.998195i \(0.480869\pi\)
\(384\) −18.4464 −0.941340
\(385\) −2.58774 −0.131884
\(386\) −54.7019 −2.78426
\(387\) 7.40530 0.376432
\(388\) 41.6476 2.11434
\(389\) −1.74378 −0.0884130 −0.0442065 0.999022i \(-0.514076\pi\)
−0.0442065 + 0.999022i \(0.514076\pi\)
\(390\) 37.5745 1.90266
\(391\) 52.1895 2.63934
\(392\) 5.22982 0.264146
\(393\) 4.00000 0.201773
\(394\) −64.8929 −3.26925
\(395\) 11.5404 0.580659
\(396\) −4.11491 −0.206782
\(397\) 6.54189 0.328328 0.164164 0.986433i \(-0.447507\pi\)
0.164164 + 0.986433i \(0.447507\pi\)
\(398\) 2.60719 0.130687
\(399\) 2.35793 0.118044
\(400\) 7.97760 0.398880
\(401\) 5.66152 0.282723 0.141361 0.989958i \(-0.454852\pi\)
0.141361 + 0.989958i \(0.454852\pi\)
\(402\) −24.9798 −1.24588
\(403\) 21.5000 1.07099
\(404\) −55.1616 −2.74439
\(405\) −2.58774 −0.128586
\(406\) 14.5202 0.720626
\(407\) −3.30359 −0.163753
\(408\) 39.2966 1.94547
\(409\) −5.48755 −0.271342 −0.135671 0.990754i \(-0.543319\pi\)
−0.135671 + 0.990754i \(0.543319\pi\)
\(410\) −33.8135 −1.66993
\(411\) −10.8370 −0.534550
\(412\) −19.4053 −0.956030
\(413\) 0.926221 0.0455764
\(414\) 17.1755 0.844129
\(415\) 27.6615 1.35785
\(416\) −6.86565 −0.336616
\(417\) 0.459630 0.0225082
\(418\) 5.83076 0.285192
\(419\) 8.24926 0.403003 0.201501 0.979488i \(-0.435418\pi\)
0.201501 + 0.979488i \(0.435418\pi\)
\(420\) 10.6483 0.519585
\(421\) 32.2617 1.57234 0.786171 0.618009i \(-0.212061\pi\)
0.786171 + 0.618009i \(0.212061\pi\)
\(422\) −21.5529 −1.04918
\(423\) 7.53341 0.366287
\(424\) −22.1212 −1.07430
\(425\) 12.7467 0.618307
\(426\) 11.0279 0.534305
\(427\) 2.00000 0.0967868
\(428\) 29.1079 1.40699
\(429\) 5.87189 0.283497
\(430\) −47.3869 −2.28520
\(431\) −23.2772 −1.12122 −0.560611 0.828079i \(-0.689434\pi\)
−0.560611 + 0.828079i \(0.689434\pi\)
\(432\) 4.70265 0.226256
\(433\) −15.9302 −0.765558 −0.382779 0.923840i \(-0.625033\pi\)
−0.382779 + 0.923840i \(0.625033\pi\)
\(434\) 9.05433 0.434622
\(435\) 15.1949 0.728541
\(436\) 77.5125 3.71218
\(437\) −16.3774 −0.783436
\(438\) −5.26247 −0.251450
\(439\) 22.8565 1.09088 0.545439 0.838150i \(-0.316363\pi\)
0.545439 + 0.838150i \(0.316363\pi\)
\(440\) 13.5334 0.645180
\(441\) 1.00000 0.0476190
\(442\) −109.104 −5.18956
\(443\) 21.5962 1.02607 0.513034 0.858368i \(-0.328522\pi\)
0.513034 + 0.858368i \(0.328522\pi\)
\(444\) 13.5940 0.645142
\(445\) −41.1227 −1.94940
\(446\) −58.7144 −2.78021
\(447\) 11.0474 0.522523
\(448\) 6.51396 0.307756
\(449\) 18.6630 0.880763 0.440382 0.897811i \(-0.354843\pi\)
0.440382 + 0.897811i \(0.354843\pi\)
\(450\) 4.19493 0.197751
\(451\) −5.28415 −0.248821
\(452\) −37.1491 −1.74735
\(453\) −16.1212 −0.757438
\(454\) −32.5808 −1.52909
\(455\) −15.1949 −0.712349
\(456\) −12.3315 −0.577476
\(457\) 36.4721 1.70609 0.853047 0.521834i \(-0.174752\pi\)
0.853047 + 0.521834i \(0.174752\pi\)
\(458\) 33.0529 1.54446
\(459\) 7.51396 0.350722
\(460\) −73.9597 −3.44839
\(461\) 7.17548 0.334196 0.167098 0.985940i \(-0.446560\pi\)
0.167098 + 0.985940i \(0.446560\pi\)
\(462\) 2.47283 0.115047
\(463\) −0.452670 −0.0210373 −0.0105187 0.999945i \(-0.503348\pi\)
−0.0105187 + 0.999945i \(0.503348\pi\)
\(464\) −27.6134 −1.28192
\(465\) 9.47507 0.439396
\(466\) 17.6476 0.817509
\(467\) −9.68097 −0.447982 −0.223991 0.974591i \(-0.571909\pi\)
−0.223991 + 0.974591i \(0.571909\pi\)
\(468\) −24.1623 −1.11690
\(469\) 10.1017 0.466453
\(470\) −48.2067 −2.22361
\(471\) 13.0668 0.602087
\(472\) −4.84396 −0.222962
\(473\) −7.40530 −0.340496
\(474\) −11.0279 −0.506529
\(475\) −4.00000 −0.183533
\(476\) −30.9193 −1.41718
\(477\) −4.22982 −0.193670
\(478\) −49.4395 −2.26131
\(479\) 24.3635 1.11319 0.556597 0.830782i \(-0.312107\pi\)
0.556597 + 0.830782i \(0.312107\pi\)
\(480\) −3.02569 −0.138103
\(481\) −19.3983 −0.884488
\(482\) −22.7375 −1.03567
\(483\) −6.94567 −0.316039
\(484\) 4.11491 0.187041
\(485\) −26.1909 −1.18927
\(486\) 2.47283 0.112170
\(487\) −19.2702 −0.873217 −0.436609 0.899652i \(-0.643821\pi\)
−0.436609 + 0.899652i \(0.643821\pi\)
\(488\) −10.4596 −0.473485
\(489\) −10.1017 −0.456815
\(490\) −6.39905 −0.289080
\(491\) 29.6421 1.33773 0.668864 0.743385i \(-0.266781\pi\)
0.668864 + 0.743385i \(0.266781\pi\)
\(492\) 21.7438 0.980285
\(493\) −44.1212 −1.98712
\(494\) 34.2376 1.54042
\(495\) 2.58774 0.116310
\(496\) −17.2188 −0.773149
\(497\) −4.45963 −0.200042
\(498\) −26.4332 −1.18450
\(499\) 29.3719 1.31487 0.657434 0.753512i \(-0.271642\pi\)
0.657434 + 0.753512i \(0.271642\pi\)
\(500\) 35.1777 1.57320
\(501\) −10.4860 −0.468482
\(502\) −23.3058 −1.04019
\(503\) 12.5947 0.561570 0.280785 0.959771i \(-0.409405\pi\)
0.280785 + 0.959771i \(0.409405\pi\)
\(504\) −5.22982 −0.232954
\(505\) 34.6894 1.54366
\(506\) −17.1755 −0.763543
\(507\) 21.4791 0.953919
\(508\) −42.0947 −1.86765
\(509\) −15.2144 −0.674365 −0.337183 0.941439i \(-0.609474\pi\)
−0.337183 + 0.941439i \(0.609474\pi\)
\(510\) −48.0823 −2.12912
\(511\) 2.12811 0.0941421
\(512\) −43.6894 −1.93082
\(513\) −2.35793 −0.104105
\(514\) −0.0480890 −0.00212112
\(515\) 12.2034 0.537746
\(516\) 30.4721 1.34146
\(517\) −7.53341 −0.331319
\(518\) −8.16924 −0.358936
\(519\) 14.4985 0.636415
\(520\) 79.4667 3.48484
\(521\) 20.5180 0.898909 0.449454 0.893303i \(-0.351618\pi\)
0.449454 + 0.893303i \(0.351618\pi\)
\(522\) −14.5202 −0.635532
\(523\) −29.8844 −1.30675 −0.653376 0.757033i \(-0.726648\pi\)
−0.653376 + 0.757033i \(0.726648\pi\)
\(524\) 16.4596 0.719042
\(525\) −1.69641 −0.0740372
\(526\) 69.2221 3.01823
\(527\) −27.5125 −1.19846
\(528\) −4.70265 −0.204657
\(529\) 25.2423 1.09749
\(530\) 27.0668 1.17571
\(531\) −0.926221 −0.0401946
\(532\) 9.70265 0.420663
\(533\) −31.0279 −1.34397
\(534\) 39.2966 1.70053
\(535\) −18.3051 −0.791399
\(536\) −52.8300 −2.28191
\(537\) 14.3510 0.619290
\(538\) 76.4582 3.29635
\(539\) −1.00000 −0.0430730
\(540\) −10.6483 −0.458231
\(541\) 26.9582 1.15902 0.579511 0.814965i \(-0.303244\pi\)
0.579511 + 0.814965i \(0.303244\pi\)
\(542\) −55.2874 −2.37480
\(543\) 9.66152 0.414616
\(544\) 8.78562 0.376680
\(545\) −48.7453 −2.08802
\(546\) 14.5202 0.621407
\(547\) 26.0558 1.11407 0.557034 0.830490i \(-0.311939\pi\)
0.557034 + 0.830490i \(0.311939\pi\)
\(548\) −44.5933 −1.90493
\(549\) −2.00000 −0.0853579
\(550\) −4.19493 −0.178872
\(551\) 13.8455 0.589837
\(552\) 36.3246 1.54608
\(553\) 4.45963 0.189643
\(554\) 61.2563 2.60253
\(555\) −8.54885 −0.362878
\(556\) 1.89134 0.0802105
\(557\) 8.01945 0.339795 0.169897 0.985462i \(-0.445656\pi\)
0.169897 + 0.985462i \(0.445656\pi\)
\(558\) −9.05433 −0.383300
\(559\) −43.4831 −1.83914
\(560\) 12.1692 0.514244
\(561\) −7.51396 −0.317240
\(562\) −2.95486 −0.124643
\(563\) 21.9736 0.926077 0.463038 0.886338i \(-0.346759\pi\)
0.463038 + 0.886338i \(0.346759\pi\)
\(564\) 30.9993 1.30531
\(565\) 23.3619 0.982844
\(566\) −38.5077 −1.61860
\(567\) −1.00000 −0.0419961
\(568\) 23.3230 0.978613
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 15.0885 0.631988
\(571\) 37.6740 1.57661 0.788304 0.615286i \(-0.210959\pi\)
0.788304 + 0.615286i \(0.210959\pi\)
\(572\) 24.1623 1.01028
\(573\) −9.55286 −0.399076
\(574\) −13.0668 −0.545398
\(575\) 11.7827 0.491371
\(576\) −6.51396 −0.271415
\(577\) −7.80908 −0.325096 −0.162548 0.986701i \(-0.551971\pi\)
−0.162548 + 0.986701i \(0.551971\pi\)
\(578\) 97.5771 4.05867
\(579\) −22.1212 −0.919324
\(580\) 62.5257 2.59624
\(581\) 10.6894 0.443473
\(582\) 25.0279 1.03744
\(583\) 4.22982 0.175181
\(584\) −11.1296 −0.460547
\(585\) 15.1949 0.628233
\(586\) −24.0947 −0.995345
\(587\) −43.2633 −1.78567 −0.892833 0.450388i \(-0.851286\pi\)
−0.892833 + 0.450388i \(0.851286\pi\)
\(588\) 4.11491 0.169696
\(589\) 8.63360 0.355741
\(590\) 5.92694 0.244008
\(591\) −26.2423 −1.07946
\(592\) 15.5356 0.638511
\(593\) −29.7438 −1.22143 −0.610715 0.791850i \(-0.709118\pi\)
−0.610715 + 0.791850i \(0.709118\pi\)
\(594\) −2.47283 −0.101462
\(595\) 19.4442 0.797134
\(596\) 45.4589 1.86207
\(597\) 1.05433 0.0431509
\(598\) −100.853 −4.12417
\(599\) 11.5404 0.471527 0.235763 0.971810i \(-0.424241\pi\)
0.235763 + 0.971810i \(0.424241\pi\)
\(600\) 8.87189 0.362193
\(601\) 7.06129 0.288036 0.144018 0.989575i \(-0.453998\pi\)
0.144018 + 0.989575i \(0.453998\pi\)
\(602\) −18.3121 −0.746344
\(603\) −10.1017 −0.411373
\(604\) −66.3370 −2.69922
\(605\) −2.58774 −0.105207
\(606\) −33.1491 −1.34659
\(607\) −17.3859 −0.705670 −0.352835 0.935686i \(-0.614782\pi\)
−0.352835 + 0.935686i \(0.614782\pi\)
\(608\) −2.75698 −0.111810
\(609\) 5.87189 0.237941
\(610\) 12.7981 0.518180
\(611\) −44.2353 −1.78957
\(612\) 30.9193 1.24984
\(613\) 39.4611 1.59382 0.796910 0.604098i \(-0.206466\pi\)
0.796910 + 0.604098i \(0.206466\pi\)
\(614\) 15.7049 0.633798
\(615\) −13.6740 −0.551389
\(616\) 5.22982 0.210715
\(617\) −4.40378 −0.177290 −0.0886448 0.996063i \(-0.528254\pi\)
−0.0886448 + 0.996063i \(0.528254\pi\)
\(618\) −11.6615 −0.469095
\(619\) 16.4163 0.659825 0.329913 0.944011i \(-0.392981\pi\)
0.329913 + 0.944011i \(0.392981\pi\)
\(620\) 38.9890 1.56584
\(621\) 6.94567 0.278720
\(622\) 19.7827 0.793213
\(623\) −15.8913 −0.636673
\(624\) −27.6134 −1.10542
\(625\) −30.6042 −1.22417
\(626\) 41.9038 1.67481
\(627\) 2.35793 0.0941665
\(628\) 53.7688 2.14561
\(629\) 24.8231 0.989761
\(630\) 6.39905 0.254944
\(631\) 44.4596 1.76991 0.884955 0.465677i \(-0.154189\pi\)
0.884955 + 0.465677i \(0.154189\pi\)
\(632\) −23.3230 −0.927741
\(633\) −8.71585 −0.346424
\(634\) 49.4876 1.96540
\(635\) 26.4721 1.05051
\(636\) −17.4053 −0.690165
\(637\) −5.87189 −0.232653
\(638\) 14.5202 0.574860
\(639\) 4.45963 0.176420
\(640\) 47.7346 1.88688
\(641\) 10.5419 0.416380 0.208190 0.978088i \(-0.433243\pi\)
0.208190 + 0.978088i \(0.433243\pi\)
\(642\) 17.4923 0.690365
\(643\) −23.4053 −0.923015 −0.461507 0.887136i \(-0.652691\pi\)
−0.461507 + 0.887136i \(0.652691\pi\)
\(644\) −28.5808 −1.12624
\(645\) −19.1630 −0.754542
\(646\) −43.8121 −1.72376
\(647\) 11.4945 0.451896 0.225948 0.974139i \(-0.427452\pi\)
0.225948 + 0.974139i \(0.427452\pi\)
\(648\) 5.22982 0.205447
\(649\) 0.926221 0.0363574
\(650\) −24.6322 −0.966153
\(651\) 3.66152 0.143506
\(652\) −41.5676 −1.62791
\(653\) 19.8774 0.777863 0.388932 0.921267i \(-0.372844\pi\)
0.388932 + 0.921267i \(0.372844\pi\)
\(654\) 46.5808 1.82145
\(655\) −10.3510 −0.404446
\(656\) 24.8495 0.970210
\(657\) −2.12811 −0.0830255
\(658\) −18.6289 −0.726229
\(659\) 15.0738 0.587191 0.293596 0.955930i \(-0.405148\pi\)
0.293596 + 0.955930i \(0.405148\pi\)
\(660\) 10.6483 0.414485
\(661\) −4.74226 −0.184453 −0.0922263 0.995738i \(-0.529398\pi\)
−0.0922263 + 0.995738i \(0.529398\pi\)
\(662\) 18.6461 0.724701
\(663\) −44.1212 −1.71352
\(664\) −55.9038 −2.16949
\(665\) −6.10170 −0.236614
\(666\) 8.16924 0.316552
\(667\) −40.7842 −1.57917
\(668\) −43.1491 −1.66949
\(669\) −23.7438 −0.917987
\(670\) 64.6414 2.49731
\(671\) 2.00000 0.0772091
\(672\) −1.16924 −0.0451044
\(673\) 9.87885 0.380802 0.190401 0.981706i \(-0.439021\pi\)
0.190401 + 0.981706i \(0.439021\pi\)
\(674\) −31.3789 −1.20867
\(675\) 1.69641 0.0652947
\(676\) 88.3844 3.39940
\(677\) −29.0279 −1.11563 −0.557817 0.829964i \(-0.688361\pi\)
−0.557817 + 0.829964i \(0.688361\pi\)
\(678\) −22.3246 −0.857369
\(679\) −10.1212 −0.388414
\(680\) −101.690 −3.89962
\(681\) −13.1755 −0.504886
\(682\) 9.05433 0.346708
\(683\) −37.9597 −1.45249 −0.726243 0.687438i \(-0.758735\pi\)
−0.726243 + 0.687438i \(0.758735\pi\)
\(684\) −9.70265 −0.370990
\(685\) 28.0434 1.07148
\(686\) −2.47283 −0.0944132
\(687\) 13.3664 0.509960
\(688\) 34.8245 1.32767
\(689\) 24.8370 0.946214
\(690\) −44.4457 −1.69202
\(691\) −15.5264 −0.590654 −0.295327 0.955396i \(-0.595429\pi\)
−0.295327 + 0.955396i \(0.595429\pi\)
\(692\) 59.6601 2.26794
\(693\) 1.00000 0.0379869
\(694\) 2.81060 0.106689
\(695\) −1.18940 −0.0451167
\(696\) −30.7089 −1.16402
\(697\) 39.7049 1.50393
\(698\) −36.5761 −1.38442
\(699\) 7.13659 0.269931
\(700\) −6.98055 −0.263840
\(701\) −27.4317 −1.03608 −0.518041 0.855356i \(-0.673338\pi\)
−0.518041 + 0.855356i \(0.673338\pi\)
\(702\) −14.5202 −0.548030
\(703\) −7.78963 −0.293792
\(704\) 6.51396 0.245504
\(705\) −19.4945 −0.734206
\(706\) −18.5980 −0.699945
\(707\) 13.4053 0.504158
\(708\) −3.81131 −0.143238
\(709\) −44.5180 −1.67191 −0.835954 0.548800i \(-0.815085\pi\)
−0.835954 + 0.548800i \(0.815085\pi\)
\(710\) −28.5374 −1.07099
\(711\) −4.45963 −0.167249
\(712\) 83.1087 3.11463
\(713\) −25.4317 −0.952425
\(714\) −18.5808 −0.695368
\(715\) −15.1949 −0.568258
\(716\) 59.0529 2.20691
\(717\) −19.9930 −0.746654
\(718\) −43.5125 −1.62387
\(719\) 17.3859 0.648383 0.324191 0.945992i \(-0.394908\pi\)
0.324191 + 0.945992i \(0.394908\pi\)
\(720\) −12.1692 −0.453521
\(721\) 4.71585 0.175628
\(722\) −33.2353 −1.23689
\(723\) −9.19493 −0.341963
\(724\) 39.7563 1.47753
\(725\) −9.96111 −0.369946
\(726\) 2.47283 0.0917755
\(727\) −18.6506 −0.691711 −0.345855 0.938288i \(-0.612411\pi\)
−0.345855 + 0.938288i \(0.612411\pi\)
\(728\) 30.7089 1.13815
\(729\) 1.00000 0.0370370
\(730\) 13.6179 0.504021
\(731\) 55.6431 2.05804
\(732\) −8.22982 −0.304183
\(733\) 16.3510 0.603937 0.301968 0.953318i \(-0.402356\pi\)
0.301968 + 0.953318i \(0.402356\pi\)
\(734\) 35.3914 1.30632
\(735\) −2.58774 −0.0954503
\(736\) 8.12115 0.299350
\(737\) 10.1017 0.372101
\(738\) 13.0668 0.480996
\(739\) −32.8370 −1.20793 −0.603964 0.797011i \(-0.706413\pi\)
−0.603964 + 0.797011i \(0.706413\pi\)
\(740\) −35.1777 −1.29316
\(741\) 13.8455 0.508626
\(742\) 10.4596 0.383985
\(743\) 20.9123 0.767198 0.383599 0.923500i \(-0.374685\pi\)
0.383599 + 0.923500i \(0.374685\pi\)
\(744\) −19.1491 −0.702039
\(745\) −28.5877 −1.04737
\(746\) −40.6366 −1.48781
\(747\) −10.6894 −0.391106
\(748\) −30.9193 −1.13052
\(749\) −7.07378 −0.258470
\(750\) 21.1399 0.771919
\(751\) −39.2633 −1.43274 −0.716368 0.697722i \(-0.754197\pi\)
−0.716368 + 0.697722i \(0.754197\pi\)
\(752\) 35.4270 1.29189
\(753\) −9.42474 −0.343457
\(754\) 85.2610 3.10502
\(755\) 41.7174 1.51825
\(756\) −4.11491 −0.149658
\(757\) −10.3454 −0.376011 −0.188006 0.982168i \(-0.560202\pi\)
−0.188006 + 0.982168i \(0.560202\pi\)
\(758\) 48.3029 1.75444
\(759\) −6.94567 −0.252112
\(760\) 31.9108 1.15753
\(761\) 30.9582 1.12223 0.561116 0.827737i \(-0.310372\pi\)
0.561116 + 0.827737i \(0.310372\pi\)
\(762\) −25.2966 −0.916400
\(763\) −18.8370 −0.681945
\(764\) −39.3091 −1.42215
\(765\) −19.4442 −0.703006
\(766\) 5.81355 0.210052
\(767\) 5.43867 0.196379
\(768\) −32.5870 −1.17588
\(769\) −28.8998 −1.04215 −0.521077 0.853510i \(-0.674470\pi\)
−0.521077 + 0.853510i \(0.674470\pi\)
\(770\) −6.39905 −0.230606
\(771\) −0.0194469 −0.000700364 0
\(772\) −91.0265 −3.27612
\(773\) −26.7523 −0.962212 −0.481106 0.876663i \(-0.659765\pi\)
−0.481106 + 0.876663i \(0.659765\pi\)
\(774\) 18.3121 0.658214
\(775\) −6.21142 −0.223121
\(776\) 52.9317 1.90014
\(777\) −3.30359 −0.118516
\(778\) −4.31207 −0.154595
\(779\) −12.4596 −0.446413
\(780\) 62.5257 2.23878
\(781\) −4.45963 −0.159578
\(782\) 129.056 4.61503
\(783\) −5.87189 −0.209844
\(784\) 4.70265 0.167952
\(785\) −33.8135 −1.20686
\(786\) 9.89134 0.352812
\(787\) 22.1406 0.789227 0.394614 0.918847i \(-0.370878\pi\)
0.394614 + 0.918847i \(0.370878\pi\)
\(788\) −107.985 −3.84679
\(789\) 27.9930 0.996579
\(790\) 28.5374 1.01532
\(791\) 9.02792 0.320996
\(792\) −5.22982 −0.185833
\(793\) 11.7438 0.417034
\(794\) 16.1770 0.574100
\(795\) 10.9457 0.388203
\(796\) 4.33848 0.153773
\(797\) 8.18396 0.289891 0.144945 0.989440i \(-0.453699\pi\)
0.144945 + 0.989440i \(0.453699\pi\)
\(798\) 5.83076 0.206407
\(799\) 56.6058 2.00257
\(800\) 1.98351 0.0701275
\(801\) 15.8913 0.561493
\(802\) 14.0000 0.494357
\(803\) 2.12811 0.0750994
\(804\) −41.5676 −1.46598
\(805\) 17.9736 0.633486
\(806\) 53.1660 1.87269
\(807\) 30.9193 1.08841
\(808\) −70.1072 −2.46636
\(809\) 17.6685 0.621191 0.310595 0.950542i \(-0.399472\pi\)
0.310595 + 0.950542i \(0.399472\pi\)
\(810\) −6.39905 −0.224840
\(811\) −43.0210 −1.51067 −0.755335 0.655339i \(-0.772526\pi\)
−0.755335 + 0.655339i \(0.772526\pi\)
\(812\) 24.1623 0.847930
\(813\) −22.3579 −0.784127
\(814\) −8.16924 −0.286332
\(815\) 26.1406 0.915665
\(816\) 35.3355 1.23699
\(817\) −17.4611 −0.610888
\(818\) −13.5698 −0.474457
\(819\) 5.87189 0.205180
\(820\) −56.2673 −1.96494
\(821\) 18.0364 0.629475 0.314737 0.949179i \(-0.398084\pi\)
0.314737 + 0.949179i \(0.398084\pi\)
\(822\) −26.7981 −0.934691
\(823\) −8.21037 −0.286195 −0.143098 0.989709i \(-0.545706\pi\)
−0.143098 + 0.989709i \(0.545706\pi\)
\(824\) −24.6630 −0.859178
\(825\) −1.69641 −0.0590613
\(826\) 2.29039 0.0796929
\(827\) 6.83148 0.237554 0.118777 0.992921i \(-0.462103\pi\)
0.118777 + 0.992921i \(0.462103\pi\)
\(828\) 28.5808 0.993251
\(829\) −25.7438 −0.894118 −0.447059 0.894504i \(-0.647529\pi\)
−0.447059 + 0.894504i \(0.647529\pi\)
\(830\) 68.4023 2.37428
\(831\) 24.7717 0.859321
\(832\) 38.2493 1.32605
\(833\) 7.51396 0.260343
\(834\) 1.13659 0.0393569
\(835\) 27.1352 0.939051
\(836\) 9.70265 0.335573
\(837\) −3.66152 −0.126561
\(838\) 20.3991 0.704674
\(839\) −30.6002 −1.05644 −0.528219 0.849108i \(-0.677140\pi\)
−0.528219 + 0.849108i \(0.677140\pi\)
\(840\) 13.5334 0.466947
\(841\) 5.47908 0.188934
\(842\) 79.7779 2.74933
\(843\) −1.19493 −0.0411556
\(844\) −35.8649 −1.23452
\(845\) −55.5823 −1.91209
\(846\) 18.6289 0.640474
\(847\) −1.00000 −0.0343604
\(848\) −19.8913 −0.683071
\(849\) −15.5723 −0.534440
\(850\) 31.5205 1.08115
\(851\) 22.9457 0.786567
\(852\) 18.3510 0.628694
\(853\) −30.4596 −1.04292 −0.521459 0.853276i \(-0.674612\pi\)
−0.521459 + 0.853276i \(0.674612\pi\)
\(854\) 4.94567 0.169237
\(855\) 6.10170 0.208674
\(856\) 36.9946 1.26445
\(857\) −41.2702 −1.40976 −0.704882 0.709325i \(-0.749000\pi\)
−0.704882 + 0.709325i \(0.749000\pi\)
\(858\) 14.5202 0.495712
\(859\) −28.9193 −0.986712 −0.493356 0.869827i \(-0.664230\pi\)
−0.493356 + 0.869827i \(0.664230\pi\)
\(860\) −78.8540 −2.68890
\(861\) −5.28415 −0.180083
\(862\) −57.5606 −1.96052
\(863\) 37.0015 1.25955 0.629773 0.776779i \(-0.283148\pi\)
0.629773 + 0.776779i \(0.283148\pi\)
\(864\) 1.16924 0.0397783
\(865\) −37.5184 −1.27566
\(866\) −39.3928 −1.33862
\(867\) 39.4596 1.34012
\(868\) 15.0668 0.511401
\(869\) 4.45963 0.151283
\(870\) 37.5745 1.27390
\(871\) 59.3161 2.00985
\(872\) 98.5140 3.33611
\(873\) 10.1212 0.342549
\(874\) −40.4985 −1.36988
\(875\) −8.54885 −0.289004
\(876\) −8.75698 −0.295871
\(877\) −6.12562 −0.206847 −0.103424 0.994637i \(-0.532980\pi\)
−0.103424 + 0.994637i \(0.532980\pi\)
\(878\) 56.5202 1.90746
\(879\) −9.74378 −0.328649
\(880\) 12.1692 0.410225
\(881\) 25.4123 0.856161 0.428080 0.903741i \(-0.359190\pi\)
0.428080 + 0.903741i \(0.359190\pi\)
\(882\) 2.47283 0.0832646
\(883\) −17.0040 −0.572230 −0.286115 0.958195i \(-0.592364\pi\)
−0.286115 + 0.958195i \(0.592364\pi\)
\(884\) −181.554 −6.10634
\(885\) 2.39682 0.0805682
\(886\) 53.4039 1.79414
\(887\) 35.2269 1.18280 0.591401 0.806378i \(-0.298575\pi\)
0.591401 + 0.806378i \(0.298575\pi\)
\(888\) 17.2772 0.579784
\(889\) 10.2298 0.343097
\(890\) −101.690 −3.40864
\(891\) −1.00000 −0.0335013
\(892\) −97.7034 −3.27135
\(893\) −17.7632 −0.594424
\(894\) 27.3183 0.913661
\(895\) −37.1366 −1.24134
\(896\) 18.4464 0.616252
\(897\) −40.7842 −1.36174
\(898\) 46.1506 1.54007
\(899\) 21.5000 0.717067
\(900\) 6.98055 0.232685
\(901\) −31.7827 −1.05883
\(902\) −13.0668 −0.435077
\(903\) −7.40530 −0.246433
\(904\) −47.2144 −1.57033
\(905\) −25.0015 −0.831079
\(906\) −39.8649 −1.32442
\(907\) −9.72682 −0.322974 −0.161487 0.986875i \(-0.551629\pi\)
−0.161487 + 0.986875i \(0.551629\pi\)
\(908\) −54.2159 −1.79922
\(909\) −13.4053 −0.444626
\(910\) −37.5745 −1.24558
\(911\) −42.7717 −1.41709 −0.708545 0.705666i \(-0.750648\pi\)
−0.708545 + 0.705666i \(0.750648\pi\)
\(912\) −11.0885 −0.367177
\(913\) 10.6894 0.353769
\(914\) 90.1895 2.98320
\(915\) 5.17548 0.171096
\(916\) 55.0015 1.81730
\(917\) −4.00000 −0.132092
\(918\) 18.5808 0.613257
\(919\) 6.26871 0.206786 0.103393 0.994641i \(-0.467030\pi\)
0.103393 + 0.994641i \(0.467030\pi\)
\(920\) −93.9986 −3.09904
\(921\) 6.35097 0.209271
\(922\) 17.7438 0.584360
\(923\) −26.1865 −0.861938
\(924\) 4.11491 0.135371
\(925\) 5.60424 0.184266
\(926\) −1.11938 −0.0367850
\(927\) −4.71585 −0.154889
\(928\) −6.86565 −0.225376
\(929\) 20.0194 0.656817 0.328408 0.944536i \(-0.393488\pi\)
0.328408 + 0.944536i \(0.393488\pi\)
\(930\) 23.4303 0.768309
\(931\) −2.35793 −0.0772779
\(932\) 29.3664 0.961929
\(933\) 8.00000 0.261908
\(934\) −23.9394 −0.783322
\(935\) 19.4442 0.635893
\(936\) −30.7089 −1.00375
\(937\) −11.2144 −0.366358 −0.183179 0.983080i \(-0.558639\pi\)
−0.183179 + 0.983080i \(0.558639\pi\)
\(938\) 24.9798 0.815621
\(939\) 16.9457 0.553001
\(940\) −80.2181 −2.61643
\(941\) 31.3400 1.02165 0.510827 0.859683i \(-0.329339\pi\)
0.510827 + 0.859683i \(0.329339\pi\)
\(942\) 32.3121 1.05278
\(943\) 36.7019 1.19518
\(944\) −4.35569 −0.141766
\(945\) 2.58774 0.0841792
\(946\) −18.3121 −0.595377
\(947\) −23.2841 −0.756633 −0.378317 0.925676i \(-0.623497\pi\)
−0.378317 + 0.925676i \(0.623497\pi\)
\(948\) −18.3510 −0.596012
\(949\) 12.4960 0.405638
\(950\) −9.89134 −0.320917
\(951\) 20.0125 0.648949
\(952\) −39.2966 −1.27361
\(953\) 26.8300 0.869110 0.434555 0.900645i \(-0.356906\pi\)
0.434555 + 0.900645i \(0.356906\pi\)
\(954\) −10.4596 −0.338643
\(955\) 24.7203 0.799931
\(956\) −82.2695 −2.66079
\(957\) 5.87189 0.189811
\(958\) 60.2468 1.94648
\(959\) 10.8370 0.349945
\(960\) 16.8565 0.544040
\(961\) −17.5933 −0.567525
\(962\) −47.9689 −1.54658
\(963\) 7.07378 0.227949
\(964\) −37.8363 −1.21862
\(965\) 57.2438 1.84274
\(966\) −17.1755 −0.552612
\(967\) 36.2034 1.16422 0.582112 0.813109i \(-0.302227\pi\)
0.582112 + 0.813109i \(0.302227\pi\)
\(968\) 5.22982 0.168093
\(969\) −17.7174 −0.569164
\(970\) −64.7658 −2.07950
\(971\) 6.06281 0.194565 0.0972824 0.995257i \(-0.468985\pi\)
0.0972824 + 0.995257i \(0.468985\pi\)
\(972\) 4.11491 0.131986
\(973\) −0.459630 −0.0147351
\(974\) −47.6521 −1.52687
\(975\) −9.96111 −0.319011
\(976\) −9.40530 −0.301056
\(977\) −4.28263 −0.137013 −0.0685067 0.997651i \(-0.521823\pi\)
−0.0685067 + 0.997651i \(0.521823\pi\)
\(978\) −24.9798 −0.798767
\(979\) −15.8913 −0.507889
\(980\) −10.6483 −0.340148
\(981\) 18.8370 0.601419
\(982\) 73.2999 2.33909
\(983\) −17.1894 −0.548257 −0.274128 0.961693i \(-0.588389\pi\)
−0.274128 + 0.961693i \(0.588389\pi\)
\(984\) 27.6351 0.880975
\(985\) 67.9083 2.16374
\(986\) −109.104 −3.47459
\(987\) −7.53341 −0.239791
\(988\) 56.9729 1.81255
\(989\) 51.4347 1.63553
\(990\) 6.39905 0.203375
\(991\) 35.4945 1.12752 0.563760 0.825939i \(-0.309354\pi\)
0.563760 + 0.825939i \(0.309354\pi\)
\(992\) −4.28120 −0.135928
\(993\) 7.54037 0.239286
\(994\) −11.0279 −0.349785
\(995\) −2.72834 −0.0864942
\(996\) −43.9861 −1.39375
\(997\) 30.4068 0.962993 0.481497 0.876448i \(-0.340093\pi\)
0.481497 + 0.876448i \(0.340093\pi\)
\(998\) 72.6319 2.29912
\(999\) 3.30359 0.104521
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 231.2.a.e.1.3 3
3.2 odd 2 693.2.a.l.1.1 3
4.3 odd 2 3696.2.a.bo.1.1 3
5.4 even 2 5775.2.a.bp.1.1 3
7.6 odd 2 1617.2.a.t.1.3 3
11.10 odd 2 2541.2.a.bg.1.1 3
21.20 even 2 4851.2.a.bi.1.1 3
33.32 even 2 7623.2.a.cd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.3 3 1.1 even 1 trivial
693.2.a.l.1.1 3 3.2 odd 2
1617.2.a.t.1.3 3 7.6 odd 2
2541.2.a.bg.1.1 3 11.10 odd 2
3696.2.a.bo.1.1 3 4.3 odd 2
4851.2.a.bi.1.1 3 21.20 even 2
5775.2.a.bp.1.1 3 5.4 even 2
7623.2.a.cd.1.3 3 33.32 even 2