Properties

Label 8470.2.a.bq.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.618034 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} -1.00000 q^{10} -0.618034 q^{12} -3.38197 q^{13} +1.00000 q^{14} -0.618034 q^{15} +1.00000 q^{16} -2.38197 q^{17} +2.61803 q^{18} +0.763932 q^{19} +1.00000 q^{20} +0.618034 q^{21} -0.472136 q^{23} +0.618034 q^{24} +1.00000 q^{25} +3.38197 q^{26} +3.47214 q^{27} -1.00000 q^{28} -0.854102 q^{29} +0.618034 q^{30} +0.763932 q^{31} -1.00000 q^{32} +2.38197 q^{34} -1.00000 q^{35} -2.61803 q^{36} +9.70820 q^{37} -0.763932 q^{38} +2.09017 q^{39} -1.00000 q^{40} +8.94427 q^{41} -0.618034 q^{42} +1.23607 q^{43} -2.61803 q^{45} +0.472136 q^{46} -1.90983 q^{47} -0.618034 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.47214 q^{51} -3.38197 q^{52} -2.47214 q^{53} -3.47214 q^{54} +1.00000 q^{56} -0.472136 q^{57} +0.854102 q^{58} +9.23607 q^{59} -0.618034 q^{60} -1.70820 q^{61} -0.763932 q^{62} +2.61803 q^{63} +1.00000 q^{64} -3.38197 q^{65} +3.23607 q^{67} -2.38197 q^{68} +0.291796 q^{69} +1.00000 q^{70} +8.32624 q^{71} +2.61803 q^{72} +11.3262 q^{73} -9.70820 q^{74} -0.618034 q^{75} +0.763932 q^{76} -2.09017 q^{78} -5.85410 q^{79} +1.00000 q^{80} +5.70820 q^{81} -8.94427 q^{82} -11.3820 q^{83} +0.618034 q^{84} -2.38197 q^{85} -1.23607 q^{86} +0.527864 q^{87} +13.7082 q^{89} +2.61803 q^{90} +3.38197 q^{91} -0.472136 q^{92} -0.472136 q^{93} +1.90983 q^{94} +0.763932 q^{95} +0.618034 q^{96} -5.61803 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + q^{3} + 2q^{4} + 2q^{5} - q^{6} - 2q^{7} - 2q^{8} - 3q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + q^{3} + 2q^{4} + 2q^{5} - q^{6} - 2q^{7} - 2q^{8} - 3q^{9} - 2q^{10} + q^{12} - 9q^{13} + 2q^{14} + q^{15} + 2q^{16} - 7q^{17} + 3q^{18} + 6q^{19} + 2q^{20} - q^{21} + 8q^{23} - q^{24} + 2q^{25} + 9q^{26} - 2q^{27} - 2q^{28} + 5q^{29} - q^{30} + 6q^{31} - 2q^{32} + 7q^{34} - 2q^{35} - 3q^{36} + 6q^{37} - 6q^{38} - 7q^{39} - 2q^{40} + q^{42} - 2q^{43} - 3q^{45} - 8q^{46} - 15q^{47} + q^{48} + 2q^{49} - 2q^{50} - 6q^{51} - 9q^{52} + 4q^{53} + 2q^{54} + 2q^{56} + 8q^{57} - 5q^{58} + 14q^{59} + q^{60} + 10q^{61} - 6q^{62} + 3q^{63} + 2q^{64} - 9q^{65} + 2q^{67} - 7q^{68} + 14q^{69} + 2q^{70} + q^{71} + 3q^{72} + 7q^{73} - 6q^{74} + q^{75} + 6q^{76} + 7q^{78} - 5q^{79} + 2q^{80} - 2q^{81} - 25q^{83} - q^{84} - 7q^{85} + 2q^{86} + 10q^{87} + 14q^{89} + 3q^{90} + 9q^{91} + 8q^{92} + 8q^{93} + 15q^{94} + 6q^{95} - q^{96} - 9q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) −0.618034 −0.356822 −0.178411 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.618034 0.252311
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.61803 −0.872678
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −0.618034 −0.178411
\(13\) −3.38197 −0.937989 −0.468994 0.883201i \(-0.655384\pi\)
−0.468994 + 0.883201i \(0.655384\pi\)
\(14\) 1.00000 0.267261
\(15\) −0.618034 −0.159576
\(16\) 1.00000 0.250000
\(17\) −2.38197 −0.577712 −0.288856 0.957373i \(-0.593275\pi\)
−0.288856 + 0.957373i \(0.593275\pi\)
\(18\) 2.61803 0.617077
\(19\) 0.763932 0.175258 0.0876290 0.996153i \(-0.472071\pi\)
0.0876290 + 0.996153i \(0.472071\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.618034 0.134866
\(22\) 0 0
\(23\) −0.472136 −0.0984472 −0.0492236 0.998788i \(-0.515675\pi\)
−0.0492236 + 0.998788i \(0.515675\pi\)
\(24\) 0.618034 0.126156
\(25\) 1.00000 0.200000
\(26\) 3.38197 0.663258
\(27\) 3.47214 0.668213
\(28\) −1.00000 −0.188982
\(29\) −0.854102 −0.158603 −0.0793014 0.996851i \(-0.525269\pi\)
−0.0793014 + 0.996851i \(0.525269\pi\)
\(30\) 0.618034 0.112837
\(31\) 0.763932 0.137206 0.0686031 0.997644i \(-0.478146\pi\)
0.0686031 + 0.997644i \(0.478146\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.38197 0.408504
\(35\) −1.00000 −0.169031
\(36\) −2.61803 −0.436339
\(37\) 9.70820 1.59602 0.798009 0.602645i \(-0.205886\pi\)
0.798009 + 0.602645i \(0.205886\pi\)
\(38\) −0.763932 −0.123926
\(39\) 2.09017 0.334695
\(40\) −1.00000 −0.158114
\(41\) 8.94427 1.39686 0.698430 0.715678i \(-0.253882\pi\)
0.698430 + 0.715678i \(0.253882\pi\)
\(42\) −0.618034 −0.0953647
\(43\) 1.23607 0.188499 0.0942493 0.995549i \(-0.469955\pi\)
0.0942493 + 0.995549i \(0.469955\pi\)
\(44\) 0 0
\(45\) −2.61803 −0.390273
\(46\) 0.472136 0.0696126
\(47\) −1.90983 −0.278577 −0.139289 0.990252i \(-0.544482\pi\)
−0.139289 + 0.990252i \(0.544482\pi\)
\(48\) −0.618034 −0.0892055
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 1.47214 0.206140
\(52\) −3.38197 −0.468994
\(53\) −2.47214 −0.339574 −0.169787 0.985481i \(-0.554308\pi\)
−0.169787 + 0.985481i \(0.554308\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −0.472136 −0.0625359
\(58\) 0.854102 0.112149
\(59\) 9.23607 1.20243 0.601217 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) −0.618034 −0.0797878
\(61\) −1.70820 −0.218713 −0.109357 0.994003i \(-0.534879\pi\)
−0.109357 + 0.994003i \(0.534879\pi\)
\(62\) −0.763932 −0.0970195
\(63\) 2.61803 0.329841
\(64\) 1.00000 0.125000
\(65\) −3.38197 −0.419481
\(66\) 0 0
\(67\) 3.23607 0.395349 0.197674 0.980268i \(-0.436661\pi\)
0.197674 + 0.980268i \(0.436661\pi\)
\(68\) −2.38197 −0.288856
\(69\) 0.291796 0.0351281
\(70\) 1.00000 0.119523
\(71\) 8.32624 0.988143 0.494071 0.869421i \(-0.335508\pi\)
0.494071 + 0.869421i \(0.335508\pi\)
\(72\) 2.61803 0.308538
\(73\) 11.3262 1.32564 0.662818 0.748781i \(-0.269360\pi\)
0.662818 + 0.748781i \(0.269360\pi\)
\(74\) −9.70820 −1.12856
\(75\) −0.618034 −0.0713644
\(76\) 0.763932 0.0876290
\(77\) 0 0
\(78\) −2.09017 −0.236665
\(79\) −5.85410 −0.658638 −0.329319 0.944219i \(-0.606819\pi\)
−0.329319 + 0.944219i \(0.606819\pi\)
\(80\) 1.00000 0.111803
\(81\) 5.70820 0.634245
\(82\) −8.94427 −0.987730
\(83\) −11.3820 −1.24933 −0.624667 0.780892i \(-0.714765\pi\)
−0.624667 + 0.780892i \(0.714765\pi\)
\(84\) 0.618034 0.0674330
\(85\) −2.38197 −0.258360
\(86\) −1.23607 −0.133289
\(87\) 0.527864 0.0565930
\(88\) 0 0
\(89\) 13.7082 1.45307 0.726533 0.687131i \(-0.241130\pi\)
0.726533 + 0.687131i \(0.241130\pi\)
\(90\) 2.61803 0.275965
\(91\) 3.38197 0.354526
\(92\) −0.472136 −0.0492236
\(93\) −0.472136 −0.0489582
\(94\) 1.90983 0.196984
\(95\) 0.763932 0.0783778
\(96\) 0.618034 0.0630778
\(97\) −5.61803 −0.570425 −0.285212 0.958464i \(-0.592064\pi\)
−0.285212 + 0.958464i \(0.592064\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.763932 0.0760141 0.0380070 0.999277i \(-0.487899\pi\)
0.0380070 + 0.999277i \(0.487899\pi\)
\(102\) −1.47214 −0.145763
\(103\) −16.5623 −1.63193 −0.815966 0.578100i \(-0.803795\pi\)
−0.815966 + 0.578100i \(0.803795\pi\)
\(104\) 3.38197 0.331629
\(105\) 0.618034 0.0603139
\(106\) 2.47214 0.240115
\(107\) 15.2361 1.47293 0.736463 0.676478i \(-0.236494\pi\)
0.736463 + 0.676478i \(0.236494\pi\)
\(108\) 3.47214 0.334106
\(109\) −12.4721 −1.19461 −0.597307 0.802013i \(-0.703763\pi\)
−0.597307 + 0.802013i \(0.703763\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) −10.4721 −0.985136 −0.492568 0.870274i \(-0.663942\pi\)
−0.492568 + 0.870274i \(0.663942\pi\)
\(114\) 0.472136 0.0442196
\(115\) −0.472136 −0.0440269
\(116\) −0.854102 −0.0793014
\(117\) 8.85410 0.818562
\(118\) −9.23607 −0.850249
\(119\) 2.38197 0.218354
\(120\) 0.618034 0.0564185
\(121\) 0 0
\(122\) 1.70820 0.154654
\(123\) −5.52786 −0.498431
\(124\) 0.763932 0.0686031
\(125\) 1.00000 0.0894427
\(126\) −2.61803 −0.233233
\(127\) 2.18034 0.193474 0.0967369 0.995310i \(-0.469159\pi\)
0.0967369 + 0.995310i \(0.469159\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.763932 −0.0672605
\(130\) 3.38197 0.296618
\(131\) −15.4164 −1.34694 −0.673469 0.739216i \(-0.735196\pi\)
−0.673469 + 0.739216i \(0.735196\pi\)
\(132\) 0 0
\(133\) −0.763932 −0.0662413
\(134\) −3.23607 −0.279554
\(135\) 3.47214 0.298834
\(136\) 2.38197 0.204252
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −0.291796 −0.0248393
\(139\) 8.94427 0.758643 0.379322 0.925265i \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 1.18034 0.0994026
\(142\) −8.32624 −0.698722
\(143\) 0 0
\(144\) −2.61803 −0.218169
\(145\) −0.854102 −0.0709293
\(146\) −11.3262 −0.937366
\(147\) −0.618034 −0.0509746
\(148\) 9.70820 0.798009
\(149\) −14.1459 −1.15888 −0.579439 0.815016i \(-0.696728\pi\)
−0.579439 + 0.815016i \(0.696728\pi\)
\(150\) 0.618034 0.0504623
\(151\) −23.6180 −1.92201 −0.961004 0.276534i \(-0.910814\pi\)
−0.961004 + 0.276534i \(0.910814\pi\)
\(152\) −0.763932 −0.0619631
\(153\) 6.23607 0.504156
\(154\) 0 0
\(155\) 0.763932 0.0613605
\(156\) 2.09017 0.167348
\(157\) 0.0901699 0.00719634 0.00359817 0.999994i \(-0.498855\pi\)
0.00359817 + 0.999994i \(0.498855\pi\)
\(158\) 5.85410 0.465727
\(159\) 1.52786 0.121168
\(160\) −1.00000 −0.0790569
\(161\) 0.472136 0.0372095
\(162\) −5.70820 −0.448479
\(163\) 16.6525 1.30432 0.652161 0.758080i \(-0.273862\pi\)
0.652161 + 0.758080i \(0.273862\pi\)
\(164\) 8.94427 0.698430
\(165\) 0 0
\(166\) 11.3820 0.883412
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) −0.618034 −0.0476824
\(169\) −1.56231 −0.120177
\(170\) 2.38197 0.182688
\(171\) −2.00000 −0.152944
\(172\) 1.23607 0.0942493
\(173\) −22.5066 −1.71114 −0.855572 0.517684i \(-0.826794\pi\)
−0.855572 + 0.517684i \(0.826794\pi\)
\(174\) −0.527864 −0.0400173
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −5.70820 −0.429055
\(178\) −13.7082 −1.02747
\(179\) 0.562306 0.0420287 0.0210144 0.999779i \(-0.493310\pi\)
0.0210144 + 0.999779i \(0.493310\pi\)
\(180\) −2.61803 −0.195137
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) −3.38197 −0.250688
\(183\) 1.05573 0.0780417
\(184\) 0.472136 0.0348063
\(185\) 9.70820 0.713761
\(186\) 0.472136 0.0346187
\(187\) 0 0
\(188\) −1.90983 −0.139289
\(189\) −3.47214 −0.252561
\(190\) −0.763932 −0.0554215
\(191\) −5.85410 −0.423588 −0.211794 0.977314i \(-0.567931\pi\)
−0.211794 + 0.977314i \(0.567931\pi\)
\(192\) −0.618034 −0.0446028
\(193\) 4.94427 0.355896 0.177948 0.984040i \(-0.443054\pi\)
0.177948 + 0.984040i \(0.443054\pi\)
\(194\) 5.61803 0.403351
\(195\) 2.09017 0.149680
\(196\) 1.00000 0.0714286
\(197\) −11.4164 −0.813385 −0.406693 0.913565i \(-0.633318\pi\)
−0.406693 + 0.913565i \(0.633318\pi\)
\(198\) 0 0
\(199\) 6.94427 0.492266 0.246133 0.969236i \(-0.420840\pi\)
0.246133 + 0.969236i \(0.420840\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.00000 −0.141069
\(202\) −0.763932 −0.0537501
\(203\) 0.854102 0.0599462
\(204\) 1.47214 0.103070
\(205\) 8.94427 0.624695
\(206\) 16.5623 1.15395
\(207\) 1.23607 0.0859127
\(208\) −3.38197 −0.234497
\(209\) 0 0
\(210\) −0.618034 −0.0426484
\(211\) 0.326238 0.0224591 0.0112296 0.999937i \(-0.496425\pi\)
0.0112296 + 0.999937i \(0.496425\pi\)
\(212\) −2.47214 −0.169787
\(213\) −5.14590 −0.352591
\(214\) −15.2361 −1.04152
\(215\) 1.23607 0.0842991
\(216\) −3.47214 −0.236249
\(217\) −0.763932 −0.0518591
\(218\) 12.4721 0.844720
\(219\) −7.00000 −0.473016
\(220\) 0 0
\(221\) 8.05573 0.541887
\(222\) 6.00000 0.402694
\(223\) −9.52786 −0.638033 −0.319016 0.947749i \(-0.603353\pi\)
−0.319016 + 0.947749i \(0.603353\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.61803 −0.174536
\(226\) 10.4721 0.696596
\(227\) −13.8541 −0.919529 −0.459765 0.888041i \(-0.652066\pi\)
−0.459765 + 0.888041i \(0.652066\pi\)
\(228\) −0.472136 −0.0312680
\(229\) −5.70820 −0.377209 −0.188604 0.982053i \(-0.560396\pi\)
−0.188604 + 0.982053i \(0.560396\pi\)
\(230\) 0.472136 0.0311317
\(231\) 0 0
\(232\) 0.854102 0.0560745
\(233\) −14.6525 −0.959916 −0.479958 0.877292i \(-0.659348\pi\)
−0.479958 + 0.877292i \(0.659348\pi\)
\(234\) −8.85410 −0.578811
\(235\) −1.90983 −0.124584
\(236\) 9.23607 0.601217
\(237\) 3.61803 0.235017
\(238\) −2.38197 −0.154400
\(239\) −2.14590 −0.138807 −0.0694033 0.997589i \(-0.522110\pi\)
−0.0694033 + 0.997589i \(0.522110\pi\)
\(240\) −0.618034 −0.0398939
\(241\) −9.70820 −0.625360 −0.312680 0.949858i \(-0.601227\pi\)
−0.312680 + 0.949858i \(0.601227\pi\)
\(242\) 0 0
\(243\) −13.9443 −0.894525
\(244\) −1.70820 −0.109357
\(245\) 1.00000 0.0638877
\(246\) 5.52786 0.352444
\(247\) −2.58359 −0.164390
\(248\) −0.763932 −0.0485097
\(249\) 7.03444 0.445790
\(250\) −1.00000 −0.0632456
\(251\) 5.23607 0.330498 0.165249 0.986252i \(-0.447157\pi\)
0.165249 + 0.986252i \(0.447157\pi\)
\(252\) 2.61803 0.164921
\(253\) 0 0
\(254\) −2.18034 −0.136807
\(255\) 1.47214 0.0921887
\(256\) 1.00000 0.0625000
\(257\) 20.2705 1.26444 0.632220 0.774789i \(-0.282144\pi\)
0.632220 + 0.774789i \(0.282144\pi\)
\(258\) 0.763932 0.0475603
\(259\) −9.70820 −0.603238
\(260\) −3.38197 −0.209741
\(261\) 2.23607 0.138409
\(262\) 15.4164 0.952429
\(263\) −27.7082 −1.70856 −0.854281 0.519812i \(-0.826002\pi\)
−0.854281 + 0.519812i \(0.826002\pi\)
\(264\) 0 0
\(265\) −2.47214 −0.151862
\(266\) 0.763932 0.0468397
\(267\) −8.47214 −0.518486
\(268\) 3.23607 0.197674
\(269\) 6.29180 0.383618 0.191809 0.981432i \(-0.438565\pi\)
0.191809 + 0.981432i \(0.438565\pi\)
\(270\) −3.47214 −0.211307
\(271\) 25.1246 1.52621 0.763106 0.646274i \(-0.223674\pi\)
0.763106 + 0.646274i \(0.223674\pi\)
\(272\) −2.38197 −0.144428
\(273\) −2.09017 −0.126503
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) 0.291796 0.0175641
\(277\) 13.4164 0.806114 0.403057 0.915175i \(-0.367948\pi\)
0.403057 + 0.915175i \(0.367948\pi\)
\(278\) −8.94427 −0.536442
\(279\) −2.00000 −0.119737
\(280\) 1.00000 0.0597614
\(281\) 26.3607 1.57255 0.786273 0.617879i \(-0.212008\pi\)
0.786273 + 0.617879i \(0.212008\pi\)
\(282\) −1.18034 −0.0702882
\(283\) −1.32624 −0.0788367 −0.0394183 0.999223i \(-0.512550\pi\)
−0.0394183 + 0.999223i \(0.512550\pi\)
\(284\) 8.32624 0.494071
\(285\) −0.472136 −0.0279669
\(286\) 0 0
\(287\) −8.94427 −0.527964
\(288\) 2.61803 0.154269
\(289\) −11.3262 −0.666249
\(290\) 0.854102 0.0501546
\(291\) 3.47214 0.203540
\(292\) 11.3262 0.662818
\(293\) −1.41641 −0.0827474 −0.0413737 0.999144i \(-0.513173\pi\)
−0.0413737 + 0.999144i \(0.513173\pi\)
\(294\) 0.618034 0.0360445
\(295\) 9.23607 0.537745
\(296\) −9.70820 −0.564278
\(297\) 0 0
\(298\) 14.1459 0.819450
\(299\) 1.59675 0.0923423
\(300\) −0.618034 −0.0356822
\(301\) −1.23607 −0.0712458
\(302\) 23.6180 1.35907
\(303\) −0.472136 −0.0271235
\(304\) 0.763932 0.0438145
\(305\) −1.70820 −0.0978115
\(306\) −6.23607 −0.356492
\(307\) −22.4508 −1.28134 −0.640669 0.767817i \(-0.721343\pi\)
−0.640669 + 0.767817i \(0.721343\pi\)
\(308\) 0 0
\(309\) 10.2361 0.582310
\(310\) −0.763932 −0.0433884
\(311\) −12.1803 −0.690684 −0.345342 0.938477i \(-0.612237\pi\)
−0.345342 + 0.938477i \(0.612237\pi\)
\(312\) −2.09017 −0.118333
\(313\) −23.8885 −1.35026 −0.675130 0.737699i \(-0.735913\pi\)
−0.675130 + 0.737699i \(0.735913\pi\)
\(314\) −0.0901699 −0.00508858
\(315\) 2.61803 0.147510
\(316\) −5.85410 −0.329319
\(317\) −17.8885 −1.00472 −0.502360 0.864658i \(-0.667535\pi\)
−0.502360 + 0.864658i \(0.667535\pi\)
\(318\) −1.52786 −0.0856784
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −9.41641 −0.525573
\(322\) −0.472136 −0.0263111
\(323\) −1.81966 −0.101249
\(324\) 5.70820 0.317122
\(325\) −3.38197 −0.187598
\(326\) −16.6525 −0.922295
\(327\) 7.70820 0.426265
\(328\) −8.94427 −0.493865
\(329\) 1.90983 0.105292
\(330\) 0 0
\(331\) −8.61803 −0.473690 −0.236845 0.971547i \(-0.576113\pi\)
−0.236845 + 0.971547i \(0.576113\pi\)
\(332\) −11.3820 −0.624667
\(333\) −25.4164 −1.39281
\(334\) 20.0000 1.09435
\(335\) 3.23607 0.176805
\(336\) 0.618034 0.0337165
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 1.56231 0.0849782
\(339\) 6.47214 0.351518
\(340\) −2.38197 −0.129180
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −1.00000 −0.0539949
\(344\) −1.23607 −0.0666443
\(345\) 0.291796 0.0157098
\(346\) 22.5066 1.20996
\(347\) 9.88854 0.530845 0.265422 0.964132i \(-0.414489\pi\)
0.265422 + 0.964132i \(0.414489\pi\)
\(348\) 0.527864 0.0282965
\(349\) 9.05573 0.484742 0.242371 0.970184i \(-0.422075\pi\)
0.242371 + 0.970184i \(0.422075\pi\)
\(350\) 1.00000 0.0534522
\(351\) −11.7426 −0.626776
\(352\) 0 0
\(353\) 31.4508 1.67396 0.836980 0.547234i \(-0.184319\pi\)
0.836980 + 0.547234i \(0.184319\pi\)
\(354\) 5.70820 0.303388
\(355\) 8.32624 0.441911
\(356\) 13.7082 0.726533
\(357\) −1.47214 −0.0779137
\(358\) −0.562306 −0.0297188
\(359\) 28.9787 1.52944 0.764719 0.644364i \(-0.222878\pi\)
0.764719 + 0.644364i \(0.222878\pi\)
\(360\) 2.61803 0.137983
\(361\) −18.4164 −0.969285
\(362\) 0 0
\(363\) 0 0
\(364\) 3.38197 0.177263
\(365\) 11.3262 0.592842
\(366\) −1.05573 −0.0551838
\(367\) 19.6738 1.02696 0.513481 0.858101i \(-0.328356\pi\)
0.513481 + 0.858101i \(0.328356\pi\)
\(368\) −0.472136 −0.0246118
\(369\) −23.4164 −1.21901
\(370\) −9.70820 −0.504705
\(371\) 2.47214 0.128347
\(372\) −0.472136 −0.0244791
\(373\) −28.6525 −1.48357 −0.741784 0.670638i \(-0.766020\pi\)
−0.741784 + 0.670638i \(0.766020\pi\)
\(374\) 0 0
\(375\) −0.618034 −0.0319151
\(376\) 1.90983 0.0984920
\(377\) 2.88854 0.148768
\(378\) 3.47214 0.178587
\(379\) 22.8541 1.17394 0.586968 0.809610i \(-0.300321\pi\)
0.586968 + 0.809610i \(0.300321\pi\)
\(380\) 0.763932 0.0391889
\(381\) −1.34752 −0.0690358
\(382\) 5.85410 0.299522
\(383\) 2.14590 0.109650 0.0548251 0.998496i \(-0.482540\pi\)
0.0548251 + 0.998496i \(0.482540\pi\)
\(384\) 0.618034 0.0315389
\(385\) 0 0
\(386\) −4.94427 −0.251657
\(387\) −3.23607 −0.164499
\(388\) −5.61803 −0.285212
\(389\) 13.2705 0.672842 0.336421 0.941712i \(-0.390784\pi\)
0.336421 + 0.941712i \(0.390784\pi\)
\(390\) −2.09017 −0.105840
\(391\) 1.12461 0.0568741
\(392\) −1.00000 −0.0505076
\(393\) 9.52786 0.480617
\(394\) 11.4164 0.575150
\(395\) −5.85410 −0.294552
\(396\) 0 0
\(397\) −10.4377 −0.523853 −0.261926 0.965088i \(-0.584358\pi\)
−0.261926 + 0.965088i \(0.584358\pi\)
\(398\) −6.94427 −0.348085
\(399\) 0.472136 0.0236364
\(400\) 1.00000 0.0500000
\(401\) −34.3820 −1.71695 −0.858477 0.512853i \(-0.828589\pi\)
−0.858477 + 0.512853i \(0.828589\pi\)
\(402\) 2.00000 0.0997509
\(403\) −2.58359 −0.128698
\(404\) 0.763932 0.0380070
\(405\) 5.70820 0.283643
\(406\) −0.854102 −0.0423884
\(407\) 0 0
\(408\) −1.47214 −0.0728816
\(409\) −0.763932 −0.0377740 −0.0188870 0.999822i \(-0.506012\pi\)
−0.0188870 + 0.999822i \(0.506012\pi\)
\(410\) −8.94427 −0.441726
\(411\) 9.88854 0.487766
\(412\) −16.5623 −0.815966
\(413\) −9.23607 −0.454477
\(414\) −1.23607 −0.0607494
\(415\) −11.3820 −0.558719
\(416\) 3.38197 0.165815
\(417\) −5.52786 −0.270701
\(418\) 0 0
\(419\) −10.9443 −0.534663 −0.267331 0.963605i \(-0.586142\pi\)
−0.267331 + 0.963605i \(0.586142\pi\)
\(420\) 0.618034 0.0301570
\(421\) 4.32624 0.210848 0.105424 0.994427i \(-0.466380\pi\)
0.105424 + 0.994427i \(0.466380\pi\)
\(422\) −0.326238 −0.0158810
\(423\) 5.00000 0.243108
\(424\) 2.47214 0.120058
\(425\) −2.38197 −0.115542
\(426\) 5.14590 0.249320
\(427\) 1.70820 0.0826658
\(428\) 15.2361 0.736463
\(429\) 0 0
\(430\) −1.23607 −0.0596085
\(431\) −4.56231 −0.219759 −0.109879 0.993945i \(-0.535046\pi\)
−0.109879 + 0.993945i \(0.535046\pi\)
\(432\) 3.47214 0.167053
\(433\) −7.09017 −0.340732 −0.170366 0.985381i \(-0.554495\pi\)
−0.170366 + 0.985381i \(0.554495\pi\)
\(434\) 0.763932 0.0366699
\(435\) 0.527864 0.0253091
\(436\) −12.4721 −0.597307
\(437\) −0.360680 −0.0172537
\(438\) 7.00000 0.334473
\(439\) −20.6525 −0.985689 −0.492844 0.870117i \(-0.664043\pi\)
−0.492844 + 0.870117i \(0.664043\pi\)
\(440\) 0 0
\(441\) −2.61803 −0.124668
\(442\) −8.05573 −0.383172
\(443\) −10.7639 −0.511410 −0.255705 0.966755i \(-0.582308\pi\)
−0.255705 + 0.966755i \(0.582308\pi\)
\(444\) −6.00000 −0.284747
\(445\) 13.7082 0.649831
\(446\) 9.52786 0.451157
\(447\) 8.74265 0.413513
\(448\) −1.00000 −0.0472456
\(449\) 18.5066 0.873379 0.436690 0.899612i \(-0.356151\pi\)
0.436690 + 0.899612i \(0.356151\pi\)
\(450\) 2.61803 0.123415
\(451\) 0 0
\(452\) −10.4721 −0.492568
\(453\) 14.5967 0.685815
\(454\) 13.8541 0.650205
\(455\) 3.38197 0.158549
\(456\) 0.472136 0.0221098
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 5.70820 0.266727
\(459\) −8.27051 −0.386034
\(460\) −0.472136 −0.0220135
\(461\) 15.1246 0.704423 0.352212 0.935920i \(-0.385430\pi\)
0.352212 + 0.935920i \(0.385430\pi\)
\(462\) 0 0
\(463\) 8.18034 0.380173 0.190086 0.981767i \(-0.439123\pi\)
0.190086 + 0.981767i \(0.439123\pi\)
\(464\) −0.854102 −0.0396507
\(465\) −0.472136 −0.0218948
\(466\) 14.6525 0.678763
\(467\) 19.3820 0.896890 0.448445 0.893810i \(-0.351978\pi\)
0.448445 + 0.893810i \(0.351978\pi\)
\(468\) 8.85410 0.409281
\(469\) −3.23607 −0.149428
\(470\) 1.90983 0.0880939
\(471\) −0.0557281 −0.00256781
\(472\) −9.23607 −0.425124
\(473\) 0 0
\(474\) −3.61803 −0.166182
\(475\) 0.763932 0.0350516
\(476\) 2.38197 0.109177
\(477\) 6.47214 0.296339
\(478\) 2.14590 0.0981511
\(479\) −5.12461 −0.234149 −0.117075 0.993123i \(-0.537352\pi\)
−0.117075 + 0.993123i \(0.537352\pi\)
\(480\) 0.618034 0.0282093
\(481\) −32.8328 −1.49705
\(482\) 9.70820 0.442197
\(483\) −0.291796 −0.0132772
\(484\) 0 0
\(485\) −5.61803 −0.255102
\(486\) 13.9443 0.632525
\(487\) 34.1803 1.54886 0.774430 0.632660i \(-0.218037\pi\)
0.774430 + 0.632660i \(0.218037\pi\)
\(488\) 1.70820 0.0773268
\(489\) −10.2918 −0.465411
\(490\) −1.00000 −0.0451754
\(491\) 16.9443 0.764684 0.382342 0.924021i \(-0.375118\pi\)
0.382342 + 0.924021i \(0.375118\pi\)
\(492\) −5.52786 −0.249215
\(493\) 2.03444 0.0916267
\(494\) 2.58359 0.116241
\(495\) 0 0
\(496\) 0.763932 0.0343016
\(497\) −8.32624 −0.373483
\(498\) −7.03444 −0.315221
\(499\) −0.201626 −0.00902602 −0.00451301 0.999990i \(-0.501437\pi\)
−0.00451301 + 0.999990i \(0.501437\pi\)
\(500\) 1.00000 0.0447214
\(501\) 12.3607 0.552234
\(502\) −5.23607 −0.233697
\(503\) 20.9787 0.935395 0.467697 0.883889i \(-0.345084\pi\)
0.467697 + 0.883889i \(0.345084\pi\)
\(504\) −2.61803 −0.116617
\(505\) 0.763932 0.0339945
\(506\) 0 0
\(507\) 0.965558 0.0428819
\(508\) 2.18034 0.0967369
\(509\) 15.8885 0.704247 0.352124 0.935953i \(-0.385460\pi\)
0.352124 + 0.935953i \(0.385460\pi\)
\(510\) −1.47214 −0.0651873
\(511\) −11.3262 −0.501043
\(512\) −1.00000 −0.0441942
\(513\) 2.65248 0.117110
\(514\) −20.2705 −0.894094
\(515\) −16.5623 −0.729822
\(516\) −0.763932 −0.0336302
\(517\) 0 0
\(518\) 9.70820 0.426554
\(519\) 13.9098 0.610574
\(520\) 3.38197 0.148309
\(521\) 34.5410 1.51327 0.756635 0.653838i \(-0.226842\pi\)
0.756635 + 0.653838i \(0.226842\pi\)
\(522\) −2.23607 −0.0978700
\(523\) −32.7426 −1.43174 −0.715868 0.698236i \(-0.753969\pi\)
−0.715868 + 0.698236i \(0.753969\pi\)
\(524\) −15.4164 −0.673469
\(525\) 0.618034 0.0269732
\(526\) 27.7082 1.20814
\(527\) −1.81966 −0.0792656
\(528\) 0 0
\(529\) −22.7771 −0.990308
\(530\) 2.47214 0.107383
\(531\) −24.1803 −1.04934
\(532\) −0.763932 −0.0331207
\(533\) −30.2492 −1.31024
\(534\) 8.47214 0.366625
\(535\) 15.2361 0.658713
\(536\) −3.23607 −0.139777
\(537\) −0.347524 −0.0149968
\(538\) −6.29180 −0.271259
\(539\) 0 0
\(540\) 3.47214 0.149417
\(541\) −3.09017 −0.132857 −0.0664284 0.997791i \(-0.521160\pi\)
−0.0664284 + 0.997791i \(0.521160\pi\)
\(542\) −25.1246 −1.07919
\(543\) 0 0
\(544\) 2.38197 0.102126
\(545\) −12.4721 −0.534248
\(546\) 2.09017 0.0894510
\(547\) −0.180340 −0.00771078 −0.00385539 0.999993i \(-0.501227\pi\)
−0.00385539 + 0.999993i \(0.501227\pi\)
\(548\) −16.0000 −0.683486
\(549\) 4.47214 0.190866
\(550\) 0 0
\(551\) −0.652476 −0.0277964
\(552\) −0.291796 −0.0124197
\(553\) 5.85410 0.248942
\(554\) −13.4164 −0.570009
\(555\) −6.00000 −0.254686
\(556\) 8.94427 0.379322
\(557\) −3.59675 −0.152399 −0.0761995 0.997093i \(-0.524279\pi\)
−0.0761995 + 0.997093i \(0.524279\pi\)
\(558\) 2.00000 0.0846668
\(559\) −4.18034 −0.176810
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −26.3607 −1.11196
\(563\) 3.43769 0.144882 0.0724408 0.997373i \(-0.476921\pi\)
0.0724408 + 0.997373i \(0.476921\pi\)
\(564\) 1.18034 0.0497013
\(565\) −10.4721 −0.440566
\(566\) 1.32624 0.0557459
\(567\) −5.70820 −0.239722
\(568\) −8.32624 −0.349361
\(569\) 5.79837 0.243080 0.121540 0.992587i \(-0.461217\pi\)
0.121540 + 0.992587i \(0.461217\pi\)
\(570\) 0.472136 0.0197756
\(571\) 37.1459 1.55451 0.777254 0.629187i \(-0.216612\pi\)
0.777254 + 0.629187i \(0.216612\pi\)
\(572\) 0 0
\(573\) 3.61803 0.151146
\(574\) 8.94427 0.373327
\(575\) −0.472136 −0.0196894
\(576\) −2.61803 −0.109085
\(577\) −9.21478 −0.383616 −0.191808 0.981432i \(-0.561435\pi\)
−0.191808 + 0.981432i \(0.561435\pi\)
\(578\) 11.3262 0.471109
\(579\) −3.05573 −0.126992
\(580\) −0.854102 −0.0354647
\(581\) 11.3820 0.472204
\(582\) −3.47214 −0.143925
\(583\) 0 0
\(584\) −11.3262 −0.468683
\(585\) 8.85410 0.366072
\(586\) 1.41641 0.0585113
\(587\) −18.5066 −0.763848 −0.381924 0.924194i \(-0.624738\pi\)
−0.381924 + 0.924194i \(0.624738\pi\)
\(588\) −0.618034 −0.0254873
\(589\) 0.583592 0.0240465
\(590\) −9.23607 −0.380243
\(591\) 7.05573 0.290234
\(592\) 9.70820 0.399005
\(593\) 9.79837 0.402371 0.201185 0.979553i \(-0.435521\pi\)
0.201185 + 0.979553i \(0.435521\pi\)
\(594\) 0 0
\(595\) 2.38197 0.0976511
\(596\) −14.1459 −0.579439
\(597\) −4.29180 −0.175652
\(598\) −1.59675 −0.0652959
\(599\) −35.3951 −1.44621 −0.723103 0.690741i \(-0.757285\pi\)
−0.723103 + 0.690741i \(0.757285\pi\)
\(600\) 0.618034 0.0252311
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 1.23607 0.0503784
\(603\) −8.47214 −0.345012
\(604\) −23.6180 −0.961004
\(605\) 0 0
\(606\) 0.472136 0.0191792
\(607\) −21.6869 −0.880245 −0.440122 0.897938i \(-0.645065\pi\)
−0.440122 + 0.897938i \(0.645065\pi\)
\(608\) −0.763932 −0.0309815
\(609\) −0.527864 −0.0213901
\(610\) 1.70820 0.0691632
\(611\) 6.45898 0.261302
\(612\) 6.23607 0.252078
\(613\) 5.05573 0.204199 0.102099 0.994774i \(-0.467444\pi\)
0.102099 + 0.994774i \(0.467444\pi\)
\(614\) 22.4508 0.906043
\(615\) −5.52786 −0.222905
\(616\) 0 0
\(617\) −26.1803 −1.05398 −0.526990 0.849871i \(-0.676680\pi\)
−0.526990 + 0.849871i \(0.676680\pi\)
\(618\) −10.2361 −0.411755
\(619\) 27.4164 1.10196 0.550979 0.834519i \(-0.314254\pi\)
0.550979 + 0.834519i \(0.314254\pi\)
\(620\) 0.763932 0.0306802
\(621\) −1.63932 −0.0657837
\(622\) 12.1803 0.488387
\(623\) −13.7082 −0.549208
\(624\) 2.09017 0.0836738
\(625\) 1.00000 0.0400000
\(626\) 23.8885 0.954778
\(627\) 0 0
\(628\) 0.0901699 0.00359817
\(629\) −23.1246 −0.922039
\(630\) −2.61803 −0.104305
\(631\) −46.4508 −1.84918 −0.924590 0.380965i \(-0.875592\pi\)
−0.924590 + 0.380965i \(0.875592\pi\)
\(632\) 5.85410 0.232864
\(633\) −0.201626 −0.00801392
\(634\) 17.8885 0.710445
\(635\) 2.18034 0.0865241
\(636\) 1.52786 0.0605838
\(637\) −3.38197 −0.133998
\(638\) 0 0
\(639\) −21.7984 −0.862330
\(640\) −1.00000 −0.0395285
\(641\) −14.7984 −0.584501 −0.292250 0.956342i \(-0.594404\pi\)
−0.292250 + 0.956342i \(0.594404\pi\)
\(642\) 9.41641 0.371636
\(643\) −21.8541 −0.861842 −0.430921 0.902390i \(-0.641811\pi\)
−0.430921 + 0.902390i \(0.641811\pi\)
\(644\) 0.472136 0.0186048
\(645\) −0.763932 −0.0300798
\(646\) 1.81966 0.0715936
\(647\) 0.742646 0.0291964 0.0145982 0.999893i \(-0.495353\pi\)
0.0145982 + 0.999893i \(0.495353\pi\)
\(648\) −5.70820 −0.224239
\(649\) 0 0
\(650\) 3.38197 0.132652
\(651\) 0.472136 0.0185045
\(652\) 16.6525 0.652161
\(653\) −27.7082 −1.08431 −0.542153 0.840280i \(-0.682391\pi\)
−0.542153 + 0.840280i \(0.682391\pi\)
\(654\) −7.70820 −0.301415
\(655\) −15.4164 −0.602369
\(656\) 8.94427 0.349215
\(657\) −29.6525 −1.15685
\(658\) −1.90983 −0.0744529
\(659\) −47.7426 −1.85979 −0.929895 0.367826i \(-0.880102\pi\)
−0.929895 + 0.367826i \(0.880102\pi\)
\(660\) 0 0
\(661\) −1.05573 −0.0410631 −0.0205315 0.999789i \(-0.506536\pi\)
−0.0205315 + 0.999789i \(0.506536\pi\)
\(662\) 8.61803 0.334949
\(663\) −4.97871 −0.193357
\(664\) 11.3820 0.441706
\(665\) −0.763932 −0.0296240
\(666\) 25.4164 0.984866
\(667\) 0.403252 0.0156140
\(668\) −20.0000 −0.773823
\(669\) 5.88854 0.227664
\(670\) −3.23607 −0.125020
\(671\) 0 0
\(672\) −0.618034 −0.0238412
\(673\) 31.4164 1.21101 0.605507 0.795840i \(-0.292970\pi\)
0.605507 + 0.795840i \(0.292970\pi\)
\(674\) 8.00000 0.308148
\(675\) 3.47214 0.133643
\(676\) −1.56231 −0.0600887
\(677\) −50.9230 −1.95713 −0.978565 0.205940i \(-0.933975\pi\)
−0.978565 + 0.205940i \(0.933975\pi\)
\(678\) −6.47214 −0.248561
\(679\) 5.61803 0.215600
\(680\) 2.38197 0.0913442
\(681\) 8.56231 0.328108
\(682\) 0 0
\(683\) 5.59675 0.214154 0.107077 0.994251i \(-0.465851\pi\)
0.107077 + 0.994251i \(0.465851\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −16.0000 −0.611329
\(686\) 1.00000 0.0381802
\(687\) 3.52786 0.134596
\(688\) 1.23607 0.0471246
\(689\) 8.36068 0.318517
\(690\) −0.291796 −0.0111085
\(691\) −6.47214 −0.246212 −0.123106 0.992394i \(-0.539285\pi\)
−0.123106 + 0.992394i \(0.539285\pi\)
\(692\) −22.5066 −0.855572
\(693\) 0 0
\(694\) −9.88854 −0.375364
\(695\) 8.94427 0.339276
\(696\) −0.527864 −0.0200086
\(697\) −21.3050 −0.806983
\(698\) −9.05573 −0.342764
\(699\) 9.05573 0.342519
\(700\) −1.00000 −0.0377964
\(701\) −48.4721 −1.83077 −0.915384 0.402583i \(-0.868113\pi\)
−0.915384 + 0.402583i \(0.868113\pi\)
\(702\) 11.7426 0.443198
\(703\) 7.41641 0.279715
\(704\) 0 0
\(705\) 1.18034 0.0444542
\(706\) −31.4508 −1.18367
\(707\) −0.763932 −0.0287306
\(708\) −5.70820 −0.214527
\(709\) 36.0344 1.35330 0.676651 0.736304i \(-0.263431\pi\)
0.676651 + 0.736304i \(0.263431\pi\)
\(710\) −8.32624 −0.312478
\(711\) 15.3262 0.574779
\(712\) −13.7082 −0.513737
\(713\) −0.360680 −0.0135076
\(714\) 1.47214 0.0550933
\(715\) 0 0
\(716\) 0.562306 0.0210144
\(717\) 1.32624 0.0495293
\(718\) −28.9787 −1.08148
\(719\) 12.3607 0.460976 0.230488 0.973075i \(-0.425968\pi\)
0.230488 + 0.973075i \(0.425968\pi\)
\(720\) −2.61803 −0.0975684
\(721\) 16.5623 0.616813
\(722\) 18.4164 0.685388
\(723\) 6.00000 0.223142
\(724\) 0 0
\(725\) −0.854102 −0.0317206
\(726\) 0 0
\(727\) 23.6738 0.878011 0.439006 0.898484i \(-0.355331\pi\)
0.439006 + 0.898484i \(0.355331\pi\)
\(728\) −3.38197 −0.125344
\(729\) −8.50658 −0.315058
\(730\) −11.3262 −0.419203
\(731\) −2.94427 −0.108898
\(732\) 1.05573 0.0390208
\(733\) −47.7426 −1.76341 −0.881707 0.471797i \(-0.843606\pi\)
−0.881707 + 0.471797i \(0.843606\pi\)
\(734\) −19.6738 −0.726172
\(735\) −0.618034 −0.0227965
\(736\) 0.472136 0.0174032
\(737\) 0 0
\(738\) 23.4164 0.861970
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 9.70820 0.356881
\(741\) 1.59675 0.0586580
\(742\) −2.47214 −0.0907550
\(743\) −21.1246 −0.774987 −0.387493 0.921872i \(-0.626659\pi\)
−0.387493 + 0.921872i \(0.626659\pi\)
\(744\) 0.472136 0.0173093
\(745\) −14.1459 −0.518266
\(746\) 28.6525 1.04904
\(747\) 29.7984 1.09027
\(748\) 0 0
\(749\) −15.2361 −0.556714
\(750\) 0.618034 0.0225674
\(751\) 18.2705 0.666700 0.333350 0.942803i \(-0.391821\pi\)
0.333350 + 0.942803i \(0.391821\pi\)
\(752\) −1.90983 −0.0696443
\(753\) −3.23607 −0.117929
\(754\) −2.88854 −0.105195
\(755\) −23.6180 −0.859548
\(756\) −3.47214 −0.126280
\(757\) 18.6525 0.677936 0.338968 0.940798i \(-0.389922\pi\)
0.338968 + 0.940798i \(0.389922\pi\)
\(758\) −22.8541 −0.830098
\(759\) 0 0
\(760\) −0.763932 −0.0277107
\(761\) −45.4164 −1.64634 −0.823172 0.567792i \(-0.807798\pi\)
−0.823172 + 0.567792i \(0.807798\pi\)
\(762\) 1.34752 0.0488156
\(763\) 12.4721 0.451522
\(764\) −5.85410 −0.211794
\(765\) 6.23607 0.225466
\(766\) −2.14590 −0.0775344
\(767\) −31.2361 −1.12787
\(768\) −0.618034 −0.0223014
\(769\) 8.29180 0.299010 0.149505 0.988761i \(-0.452232\pi\)
0.149505 + 0.988761i \(0.452232\pi\)
\(770\) 0 0
\(771\) −12.5279 −0.451180
\(772\) 4.94427 0.177948
\(773\) 1.49342 0.0537147 0.0268573 0.999639i \(-0.491450\pi\)
0.0268573 + 0.999639i \(0.491450\pi\)
\(774\) 3.23607 0.116318
\(775\) 0.763932 0.0274412
\(776\) 5.61803 0.201676
\(777\) 6.00000 0.215249
\(778\) −13.2705 −0.475771
\(779\) 6.83282 0.244811
\(780\) 2.09017 0.0748401
\(781\) 0 0
\(782\) −1.12461 −0.0402160
\(783\) −2.96556 −0.105980
\(784\) 1.00000 0.0357143
\(785\) 0.0901699 0.00321830
\(786\) −9.52786 −0.339848
\(787\) 27.0902 0.965660 0.482830 0.875714i \(-0.339609\pi\)
0.482830 + 0.875714i \(0.339609\pi\)
\(788\) −11.4164 −0.406693
\(789\) 17.1246 0.609652
\(790\) 5.85410 0.208280
\(791\) 10.4721 0.372346
\(792\) 0 0
\(793\) 5.77709 0.205150
\(794\) 10.4377 0.370420
\(795\) 1.52786 0.0541878
\(796\) 6.94427 0.246133
\(797\) −48.3262 −1.71180 −0.855902 0.517139i \(-0.826997\pi\)
−0.855902 + 0.517139i \(0.826997\pi\)
\(798\) −0.472136 −0.0167134
\(799\) 4.54915 0.160937
\(800\) −1.00000 −0.0353553
\(801\) −35.8885 −1.26806
\(802\) 34.3820 1.21407
\(803\) 0 0
\(804\) −2.00000 −0.0705346
\(805\) 0.472136 0.0166406
\(806\) 2.58359 0.0910032
\(807\) −3.88854 −0.136883
\(808\) −0.763932 −0.0268750
\(809\) −32.6869 −1.14921 −0.574605 0.818431i \(-0.694844\pi\)
−0.574605 + 0.818431i \(0.694844\pi\)
\(810\) −5.70820 −0.200566
\(811\) 53.8885 1.89228 0.946141 0.323754i \(-0.104945\pi\)
0.946141 + 0.323754i \(0.104945\pi\)
\(812\) 0.854102 0.0299731
\(813\) −15.5279 −0.544586
\(814\) 0 0
\(815\) 16.6525 0.583311
\(816\) 1.47214 0.0515351
\(817\) 0.944272 0.0330359
\(818\) 0.763932 0.0267103
\(819\) −8.85410 −0.309387
\(820\) 8.94427 0.312348
\(821\) 37.2148 1.29880 0.649402 0.760445i \(-0.275019\pi\)
0.649402 + 0.760445i \(0.275019\pi\)
\(822\) −9.88854 −0.344903
\(823\) 4.36068 0.152004 0.0760019 0.997108i \(-0.475785\pi\)
0.0760019 + 0.997108i \(0.475785\pi\)
\(824\) 16.5623 0.576975
\(825\) 0 0
\(826\) 9.23607 0.321364
\(827\) 15.0557 0.523539 0.261769 0.965130i \(-0.415694\pi\)
0.261769 + 0.965130i \(0.415694\pi\)
\(828\) 1.23607 0.0429563
\(829\) −5.88854 −0.204518 −0.102259 0.994758i \(-0.532607\pi\)
−0.102259 + 0.994758i \(0.532607\pi\)
\(830\) 11.3820 0.395074
\(831\) −8.29180 −0.287639
\(832\) −3.38197 −0.117249
\(833\) −2.38197 −0.0825302
\(834\) 5.52786 0.191414
\(835\) −20.0000 −0.692129
\(836\) 0 0
\(837\) 2.65248 0.0916830
\(838\) 10.9443 0.378064
\(839\) 44.3607 1.53150 0.765750 0.643138i \(-0.222368\pi\)
0.765750 + 0.643138i \(0.222368\pi\)
\(840\) −0.618034 −0.0213242
\(841\) −28.2705 −0.974845
\(842\) −4.32624 −0.149092
\(843\) −16.2918 −0.561119
\(844\) 0.326238 0.0112296
\(845\) −1.56231 −0.0537450
\(846\) −5.00000 −0.171904
\(847\) 0 0
\(848\) −2.47214 −0.0848935
\(849\) 0.819660 0.0281307
\(850\) 2.38197 0.0817008
\(851\) −4.58359 −0.157124
\(852\) −5.14590 −0.176296
\(853\) 25.2705 0.865246 0.432623 0.901575i \(-0.357588\pi\)
0.432623 + 0.901575i \(0.357588\pi\)
\(854\) −1.70820 −0.0584535
\(855\) −2.00000 −0.0683986
\(856\) −15.2361 −0.520758
\(857\) −6.58359 −0.224891 −0.112446 0.993658i \(-0.535868\pi\)
−0.112446 + 0.993658i \(0.535868\pi\)
\(858\) 0 0
\(859\) 28.5836 0.975260 0.487630 0.873051i \(-0.337862\pi\)
0.487630 + 0.873051i \(0.337862\pi\)
\(860\) 1.23607 0.0421496
\(861\) 5.52786 0.188389
\(862\) 4.56231 0.155393
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −3.47214 −0.118124
\(865\) −22.5066 −0.765247
\(866\) 7.09017 0.240934
\(867\) 7.00000 0.237732
\(868\) −0.763932 −0.0259295
\(869\) 0 0
\(870\) −0.527864 −0.0178963
\(871\) −10.9443 −0.370833
\(872\) 12.4721 0.422360
\(873\) 14.7082 0.497797
\(874\) 0.360680 0.0122002
\(875\) −1.00000 −0.0338062
\(876\) −7.00000 −0.236508
\(877\) 40.5410 1.36897 0.684486 0.729026i \(-0.260026\pi\)
0.684486 + 0.729026i \(0.260026\pi\)
\(878\) 20.6525 0.696987
\(879\) 0.875388 0.0295261
\(880\) 0 0
\(881\) −47.4853 −1.59982 −0.799910 0.600120i \(-0.795120\pi\)
−0.799910 + 0.600120i \(0.795120\pi\)
\(882\) 2.61803 0.0881538
\(883\) −38.7639 −1.30451 −0.652255 0.758000i \(-0.726177\pi\)
−0.652255 + 0.758000i \(0.726177\pi\)
\(884\) 8.05573 0.270943
\(885\) −5.70820 −0.191879
\(886\) 10.7639 0.361621
\(887\) −29.1459 −0.978623 −0.489312 0.872109i \(-0.662752\pi\)
−0.489312 + 0.872109i \(0.662752\pi\)
\(888\) 6.00000 0.201347
\(889\) −2.18034 −0.0731263
\(890\) −13.7082 −0.459500
\(891\) 0 0
\(892\) −9.52786 −0.319016
\(893\) −1.45898 −0.0488229
\(894\) −8.74265 −0.292398
\(895\) 0.562306 0.0187958
\(896\) 1.00000 0.0334077
\(897\) −0.986844 −0.0329498
\(898\) −18.5066 −0.617573
\(899\) −0.652476 −0.0217613
\(900\) −2.61803 −0.0872678
\(901\) 5.88854 0.196176
\(902\) 0 0
\(903\) 0.763932 0.0254221
\(904\) 10.4721 0.348298
\(905\) 0 0
\(906\) −14.5967 −0.484944
\(907\) 14.3607 0.476839 0.238419 0.971162i \(-0.423371\pi\)
0.238419 + 0.971162i \(0.423371\pi\)
\(908\) −13.8541 −0.459765
\(909\) −2.00000 −0.0663358
\(910\) −3.38197 −0.112111
\(911\) −38.4508 −1.27393 −0.636967 0.770891i \(-0.719811\pi\)
−0.636967 + 0.770891i \(0.719811\pi\)
\(912\) −0.472136 −0.0156340
\(913\) 0 0
\(914\) −14.0000 −0.463079
\(915\) 1.05573 0.0349013
\(916\) −5.70820 −0.188604
\(917\) 15.4164 0.509095
\(918\) 8.27051 0.272967
\(919\) −14.4934 −0.478094 −0.239047 0.971008i \(-0.576835\pi\)
−0.239047 + 0.971008i \(0.576835\pi\)
\(920\) 0.472136 0.0155659
\(921\) 13.8754 0.457210
\(922\) −15.1246 −0.498103
\(923\) −28.1591 −0.926867
\(924\) 0 0
\(925\) 9.70820 0.319204
\(926\) −8.18034 −0.268823
\(927\) 43.3607 1.42415
\(928\) 0.854102 0.0280373
\(929\) −43.7771 −1.43628 −0.718140 0.695899i \(-0.755006\pi\)
−0.718140 + 0.695899i \(0.755006\pi\)
\(930\) 0.472136 0.0154819
\(931\) 0.763932 0.0250369
\(932\) −14.6525 −0.479958
\(933\) 7.52786 0.246451
\(934\) −19.3820 −0.634197
\(935\) 0 0
\(936\) −8.85410 −0.289405
\(937\) −36.3951 −1.18898 −0.594488 0.804104i \(-0.702645\pi\)
−0.594488 + 0.804104i \(0.702645\pi\)
\(938\) 3.23607 0.105661
\(939\) 14.7639 0.481803
\(940\) −1.90983 −0.0622918
\(941\) −11.3475 −0.369919 −0.184959 0.982746i \(-0.559215\pi\)
−0.184959 + 0.982746i \(0.559215\pi\)
\(942\) 0.0557281 0.00181572
\(943\) −4.22291 −0.137517
\(944\) 9.23607 0.300608
\(945\) −3.47214 −0.112949
\(946\) 0 0
\(947\) 1.34752 0.0437887 0.0218943 0.999760i \(-0.493030\pi\)
0.0218943 + 0.999760i \(0.493030\pi\)
\(948\) 3.61803 0.117508
\(949\) −38.3050 −1.24343
\(950\) −0.763932 −0.0247852
\(951\) 11.0557 0.358507
\(952\) −2.38197 −0.0772000
\(953\) −29.4164 −0.952891 −0.476445 0.879204i \(-0.658075\pi\)
−0.476445 + 0.879204i \(0.658075\pi\)
\(954\) −6.47214 −0.209543
\(955\) −5.85410 −0.189434
\(956\) −2.14590 −0.0694033
\(957\) 0 0
\(958\) 5.12461 0.165569
\(959\) 16.0000 0.516667
\(960\) −0.618034 −0.0199470
\(961\) −30.4164 −0.981174
\(962\) 32.8328 1.05857
\(963\) −39.8885 −1.28539
\(964\) −9.70820 −0.312680
\(965\) 4.94427 0.159162
\(966\) 0.291796 0.00938838
\(967\) 17.7082 0.569457 0.284729 0.958608i \(-0.408096\pi\)
0.284729 + 0.958608i \(0.408096\pi\)
\(968\) 0 0
\(969\) 1.12461 0.0361277
\(970\) 5.61803 0.180384
\(971\) −2.36068 −0.0757578 −0.0378789 0.999282i \(-0.512060\pi\)
−0.0378789 + 0.999282i \(0.512060\pi\)
\(972\) −13.9443 −0.447263
\(973\) −8.94427 −0.286740
\(974\) −34.1803 −1.09521
\(975\) 2.09017 0.0669390
\(976\) −1.70820 −0.0546783
\(977\) −58.9017 −1.88443 −0.942216 0.335006i \(-0.891262\pi\)
−0.942216 + 0.335006i \(0.891262\pi\)
\(978\) 10.2918 0.329095
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 32.6525 1.04251
\(982\) −16.9443 −0.540713
\(983\) −0.965558 −0.0307965 −0.0153983 0.999881i \(-0.504902\pi\)
−0.0153983 + 0.999881i \(0.504902\pi\)
\(984\) 5.52786 0.176222
\(985\) −11.4164 −0.363757
\(986\) −2.03444 −0.0647898
\(987\) −1.18034 −0.0375706
\(988\) −2.58359 −0.0821950
\(989\) −0.583592 −0.0185572
\(990\) 0 0
\(991\) −53.9230 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(992\) −0.763932 −0.0242549
\(993\) 5.32624 0.169023
\(994\) 8.32624 0.264092
\(995\) 6.94427 0.220148
\(996\) 7.03444 0.222895
\(997\) 2.56231 0.0811490 0.0405745 0.999177i \(-0.487081\pi\)
0.0405745 + 0.999177i \(0.487081\pi\)
\(998\) 0.201626 0.00638236
\(999\) 33.7082 1.06648
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 8470.2.a.bq.1.1 2
11.2 odd 10 770.2.n.a.631.1 yes 4
11.6 odd 10 770.2.n.a.421.1 4
11.10 odd 2 8470.2.a.cc.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.a.421.1 4 11.6 odd 10
770.2.n.a.631.1 yes 4 11.2 odd 10
8470.2.a.bq.1.1 2 1.1 even 1 trivial
8470.2.a.cc.1.1 2 11.10 odd 2