Properties

Label 4235.2.a.k.1.1
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
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 385)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{2} -1.61803 q^{3} +3.00000 q^{4} -1.00000 q^{5} +3.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q-2.23607 q^{2} -1.61803 q^{3} +3.00000 q^{4} -1.00000 q^{5} +3.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} -0.381966 q^{9} +2.23607 q^{10} -4.85410 q^{12} -2.38197 q^{13} -2.23607 q^{14} +1.61803 q^{15} -1.00000 q^{16} -3.38197 q^{17} +0.854102 q^{18} -1.23607 q^{19} -3.00000 q^{20} -1.61803 q^{21} +4.47214 q^{23} +3.61803 q^{24} +1.00000 q^{25} +5.32624 q^{26} +5.47214 q^{27} +3.00000 q^{28} +5.85410 q^{29} -3.61803 q^{30} -1.23607 q^{31} +6.70820 q^{32} +7.56231 q^{34} -1.00000 q^{35} -1.14590 q^{36} -2.76393 q^{37} +2.76393 q^{38} +3.85410 q^{39} +2.23607 q^{40} +6.47214 q^{41} +3.61803 q^{42} +0.763932 q^{43} +0.381966 q^{45} -10.0000 q^{46} -7.85410 q^{47} +1.61803 q^{48} +1.00000 q^{49} -2.23607 q^{50} +5.47214 q^{51} -7.14590 q^{52} -10.4721 q^{53} -12.2361 q^{54} -2.23607 q^{56} +2.00000 q^{57} -13.0902 q^{58} -4.76393 q^{59} +4.85410 q^{60} +9.23607 q^{61} +2.76393 q^{62} -0.381966 q^{63} -13.0000 q^{64} +2.38197 q^{65} -14.1803 q^{67} -10.1459 q^{68} -7.23607 q^{69} +2.23607 q^{70} +0.673762 q^{71} +0.854102 q^{72} -8.61803 q^{73} +6.18034 q^{74} -1.61803 q^{75} -3.70820 q^{76} -8.61803 q^{78} +13.7984 q^{79} +1.00000 q^{80} -7.70820 q^{81} -14.4721 q^{82} +10.5623 q^{83} -4.85410 q^{84} +3.38197 q^{85} -1.70820 q^{86} -9.47214 q^{87} +17.2361 q^{89} -0.854102 q^{90} -2.38197 q^{91} +13.4164 q^{92} +2.00000 q^{93} +17.5623 q^{94} +1.23607 q^{95} -10.8541 q^{96} +0.326238 q^{97} -2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 6 q^{4} - 2 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{9} + O(q^{10}) \) \( 2 q - q^{3} + 6 q^{4} - 2 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{9} - 3 q^{12} - 7 q^{13} + q^{15} - 2 q^{16} - 9 q^{17} - 5 q^{18} + 2 q^{19} - 6 q^{20} - q^{21} + 5 q^{24} + 2 q^{25} - 5 q^{26} + 2 q^{27} + 6 q^{28} + 5 q^{29} - 5 q^{30} + 2 q^{31} - 5 q^{34} - 2 q^{35} - 9 q^{36} - 10 q^{37} + 10 q^{38} + q^{39} + 4 q^{41} + 5 q^{42} + 6 q^{43} + 3 q^{45} - 20 q^{46} - 9 q^{47} + q^{48} + 2 q^{49} + 2 q^{51} - 21 q^{52} - 12 q^{53} - 20 q^{54} + 4 q^{57} - 15 q^{58} - 14 q^{59} + 3 q^{60} + 14 q^{61} + 10 q^{62} - 3 q^{63} - 26 q^{64} + 7 q^{65} - 6 q^{67} - 27 q^{68} - 10 q^{69} + 17 q^{71} - 5 q^{72} - 15 q^{73} - 10 q^{74} - q^{75} + 6 q^{76} - 15 q^{78} + 3 q^{79} + 2 q^{80} - 2 q^{81} - 20 q^{82} + q^{83} - 3 q^{84} + 9 q^{85} + 10 q^{86} - 10 q^{87} + 30 q^{89} + 5 q^{90} - 7 q^{91} + 4 q^{93} + 15 q^{94} - 2 q^{95} - 15 q^{96} - 15 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23607 −1.58114 −0.790569 0.612372i \(-0.790215\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 3.00000 1.50000
\(5\) −1.00000 −0.447214
\(6\) 3.61803 1.47706
\(7\) 1.00000 0.377964
\(8\) −2.23607 −0.790569
\(9\) −0.381966 −0.127322
\(10\) 2.23607 0.707107
\(11\) 0 0
\(12\) −4.85410 −1.40126
\(13\) −2.38197 −0.660639 −0.330319 0.943869i \(-0.607156\pi\)
−0.330319 + 0.943869i \(0.607156\pi\)
\(14\) −2.23607 −0.597614
\(15\) 1.61803 0.417775
\(16\) −1.00000 −0.250000
\(17\) −3.38197 −0.820247 −0.410124 0.912030i \(-0.634514\pi\)
−0.410124 + 0.912030i \(0.634514\pi\)
\(18\) 0.854102 0.201314
\(19\) −1.23607 −0.283573 −0.141787 0.989897i \(-0.545285\pi\)
−0.141787 + 0.989897i \(0.545285\pi\)
\(20\) −3.00000 −0.670820
\(21\) −1.61803 −0.353084
\(22\) 0 0
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 3.61803 0.738528
\(25\) 1.00000 0.200000
\(26\) 5.32624 1.04456
\(27\) 5.47214 1.05311
\(28\) 3.00000 0.566947
\(29\) 5.85410 1.08708 0.543540 0.839383i \(-0.317084\pi\)
0.543540 + 0.839383i \(0.317084\pi\)
\(30\) −3.61803 −0.660560
\(31\) −1.23607 −0.222004 −0.111002 0.993820i \(-0.535406\pi\)
−0.111002 + 0.993820i \(0.535406\pi\)
\(32\) 6.70820 1.18585
\(33\) 0 0
\(34\) 7.56231 1.29692
\(35\) −1.00000 −0.169031
\(36\) −1.14590 −0.190983
\(37\) −2.76393 −0.454388 −0.227194 0.973850i \(-0.572955\pi\)
−0.227194 + 0.973850i \(0.572955\pi\)
\(38\) 2.76393 0.448369
\(39\) 3.85410 0.617150
\(40\) 2.23607 0.353553
\(41\) 6.47214 1.01078 0.505389 0.862892i \(-0.331349\pi\)
0.505389 + 0.862892i \(0.331349\pi\)
\(42\) 3.61803 0.558275
\(43\) 0.763932 0.116499 0.0582493 0.998302i \(-0.481448\pi\)
0.0582493 + 0.998302i \(0.481448\pi\)
\(44\) 0 0
\(45\) 0.381966 0.0569401
\(46\) −10.0000 −1.47442
\(47\) −7.85410 −1.14564 −0.572819 0.819682i \(-0.694150\pi\)
−0.572819 + 0.819682i \(0.694150\pi\)
\(48\) 1.61803 0.233543
\(49\) 1.00000 0.142857
\(50\) −2.23607 −0.316228
\(51\) 5.47214 0.766252
\(52\) −7.14590 −0.990958
\(53\) −10.4721 −1.43846 −0.719229 0.694773i \(-0.755505\pi\)
−0.719229 + 0.694773i \(0.755505\pi\)
\(54\) −12.2361 −1.66512
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 2.00000 0.264906
\(58\) −13.0902 −1.71882
\(59\) −4.76393 −0.620211 −0.310106 0.950702i \(-0.600364\pi\)
−0.310106 + 0.950702i \(0.600364\pi\)
\(60\) 4.85410 0.626662
\(61\) 9.23607 1.18256 0.591279 0.806467i \(-0.298623\pi\)
0.591279 + 0.806467i \(0.298623\pi\)
\(62\) 2.76393 0.351020
\(63\) −0.381966 −0.0481232
\(64\) −13.0000 −1.62500
\(65\) 2.38197 0.295447
\(66\) 0 0
\(67\) −14.1803 −1.73240 −0.866202 0.499694i \(-0.833446\pi\)
−0.866202 + 0.499694i \(0.833446\pi\)
\(68\) −10.1459 −1.23037
\(69\) −7.23607 −0.871120
\(70\) 2.23607 0.267261
\(71\) 0.673762 0.0799608 0.0399804 0.999200i \(-0.487270\pi\)
0.0399804 + 0.999200i \(0.487270\pi\)
\(72\) 0.854102 0.100657
\(73\) −8.61803 −1.00866 −0.504332 0.863510i \(-0.668261\pi\)
−0.504332 + 0.863510i \(0.668261\pi\)
\(74\) 6.18034 0.718450
\(75\) −1.61803 −0.186834
\(76\) −3.70820 −0.425360
\(77\) 0 0
\(78\) −8.61803 −0.975800
\(79\) 13.7984 1.55244 0.776219 0.630463i \(-0.217135\pi\)
0.776219 + 0.630463i \(0.217135\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.70820 −0.856467
\(82\) −14.4721 −1.59818
\(83\) 10.5623 1.15936 0.579682 0.814843i \(-0.303177\pi\)
0.579682 + 0.814843i \(0.303177\pi\)
\(84\) −4.85410 −0.529626
\(85\) 3.38197 0.366826
\(86\) −1.70820 −0.184200
\(87\) −9.47214 −1.01552
\(88\) 0 0
\(89\) 17.2361 1.82702 0.913510 0.406817i \(-0.133361\pi\)
0.913510 + 0.406817i \(0.133361\pi\)
\(90\) −0.854102 −0.0900303
\(91\) −2.38197 −0.249698
\(92\) 13.4164 1.39876
\(93\) 2.00000 0.207390
\(94\) 17.5623 1.81141
\(95\) 1.23607 0.126818
\(96\) −10.8541 −1.10779
\(97\) 0.326238 0.0331244 0.0165622 0.999863i \(-0.494728\pi\)
0.0165622 + 0.999863i \(0.494728\pi\)
\(98\) −2.23607 −0.225877
\(99\) 0 0
\(100\) 3.00000 0.300000
\(101\) 0.291796 0.0290348 0.0145174 0.999895i \(-0.495379\pi\)
0.0145174 + 0.999895i \(0.495379\pi\)
\(102\) −12.2361 −1.21155
\(103\) −1.67376 −0.164921 −0.0824603 0.996594i \(-0.526278\pi\)
−0.0824603 + 0.996594i \(0.526278\pi\)
\(104\) 5.32624 0.522281
\(105\) 1.61803 0.157904
\(106\) 23.4164 2.27440
\(107\) 2.76393 0.267199 0.133600 0.991035i \(-0.457346\pi\)
0.133600 + 0.991035i \(0.457346\pi\)
\(108\) 16.4164 1.57967
\(109\) 20.4721 1.96087 0.980437 0.196831i \(-0.0630649\pi\)
0.980437 + 0.196831i \(0.0630649\pi\)
\(110\) 0 0
\(111\) 4.47214 0.424476
\(112\) −1.00000 −0.0944911
\(113\) 1.52786 0.143729 0.0718647 0.997414i \(-0.477105\pi\)
0.0718647 + 0.997414i \(0.477105\pi\)
\(114\) −4.47214 −0.418854
\(115\) −4.47214 −0.417029
\(116\) 17.5623 1.63062
\(117\) 0.909830 0.0841138
\(118\) 10.6525 0.980640
\(119\) −3.38197 −0.310024
\(120\) −3.61803 −0.330280
\(121\) 0 0
\(122\) −20.6525 −1.86979
\(123\) −10.4721 −0.944241
\(124\) −3.70820 −0.333007
\(125\) −1.00000 −0.0894427
\(126\) 0.854102 0.0760895
\(127\) −4.76393 −0.422731 −0.211365 0.977407i \(-0.567791\pi\)
−0.211365 + 0.977407i \(0.567791\pi\)
\(128\) 15.6525 1.38350
\(129\) −1.23607 −0.108830
\(130\) −5.32624 −0.467142
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −1.23607 −0.107181
\(134\) 31.7082 2.73917
\(135\) −5.47214 −0.470966
\(136\) 7.56231 0.648462
\(137\) 15.4164 1.31711 0.658556 0.752531i \(-0.271167\pi\)
0.658556 + 0.752531i \(0.271167\pi\)
\(138\) 16.1803 1.37736
\(139\) −14.4721 −1.22751 −0.613755 0.789496i \(-0.710342\pi\)
−0.613755 + 0.789496i \(0.710342\pi\)
\(140\) −3.00000 −0.253546
\(141\) 12.7082 1.07022
\(142\) −1.50658 −0.126429
\(143\) 0 0
\(144\) 0.381966 0.0318305
\(145\) −5.85410 −0.486157
\(146\) 19.2705 1.59484
\(147\) −1.61803 −0.133453
\(148\) −8.29180 −0.681581
\(149\) 11.1459 0.913108 0.456554 0.889696i \(-0.349084\pi\)
0.456554 + 0.889696i \(0.349084\pi\)
\(150\) 3.61803 0.295411
\(151\) 2.61803 0.213053 0.106526 0.994310i \(-0.466027\pi\)
0.106526 + 0.994310i \(0.466027\pi\)
\(152\) 2.76393 0.224184
\(153\) 1.29180 0.104436
\(154\) 0 0
\(155\) 1.23607 0.0992834
\(156\) 11.5623 0.925725
\(157\) −4.90983 −0.391847 −0.195924 0.980619i \(-0.562770\pi\)
−0.195924 + 0.980619i \(0.562770\pi\)
\(158\) −30.8541 −2.45462
\(159\) 16.9443 1.34377
\(160\) −6.70820 −0.530330
\(161\) 4.47214 0.352454
\(162\) 17.2361 1.35419
\(163\) 0.180340 0.0141253 0.00706266 0.999975i \(-0.497752\pi\)
0.00706266 + 0.999975i \(0.497752\pi\)
\(164\) 19.4164 1.51617
\(165\) 0 0
\(166\) −23.6180 −1.83311
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 3.61803 0.279137
\(169\) −7.32624 −0.563557
\(170\) −7.56231 −0.580002
\(171\) 0.472136 0.0361051
\(172\) 2.29180 0.174748
\(173\) −18.5623 −1.41127 −0.705633 0.708578i \(-0.749337\pi\)
−0.705633 + 0.708578i \(0.749337\pi\)
\(174\) 21.1803 1.60568
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 7.70820 0.579384
\(178\) −38.5410 −2.88877
\(179\) 17.3820 1.29919 0.649595 0.760281i \(-0.274939\pi\)
0.649595 + 0.760281i \(0.274939\pi\)
\(180\) 1.14590 0.0854102
\(181\) 14.4721 1.07571 0.537853 0.843039i \(-0.319236\pi\)
0.537853 + 0.843039i \(0.319236\pi\)
\(182\) 5.32624 0.394807
\(183\) −14.9443 −1.10471
\(184\) −10.0000 −0.737210
\(185\) 2.76393 0.203208
\(186\) −4.47214 −0.327913
\(187\) 0 0
\(188\) −23.5623 −1.71846
\(189\) 5.47214 0.398039
\(190\) −2.76393 −0.200517
\(191\) 11.9098 0.861765 0.430883 0.902408i \(-0.358202\pi\)
0.430883 + 0.902408i \(0.358202\pi\)
\(192\) 21.0344 1.51803
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −0.729490 −0.0523743
\(195\) −3.85410 −0.275998
\(196\) 3.00000 0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) −2.23607 −0.158114
\(201\) 22.9443 1.61836
\(202\) −0.652476 −0.0459080
\(203\) 5.85410 0.410877
\(204\) 16.4164 1.14938
\(205\) −6.47214 −0.452034
\(206\) 3.74265 0.260762
\(207\) −1.70820 −0.118728
\(208\) 2.38197 0.165160
\(209\) 0 0
\(210\) −3.61803 −0.249668
\(211\) −23.3262 −1.60584 −0.802922 0.596084i \(-0.796723\pi\)
−0.802922 + 0.596084i \(0.796723\pi\)
\(212\) −31.4164 −2.15769
\(213\) −1.09017 −0.0746972
\(214\) −6.18034 −0.422479
\(215\) −0.763932 −0.0520997
\(216\) −12.2361 −0.832559
\(217\) −1.23607 −0.0839098
\(218\) −45.7771 −3.10042
\(219\) 13.9443 0.942267
\(220\) 0 0
\(221\) 8.05573 0.541887
\(222\) −10.0000 −0.671156
\(223\) 10.4721 0.701266 0.350633 0.936513i \(-0.385966\pi\)
0.350633 + 0.936513i \(0.385966\pi\)
\(224\) 6.70820 0.448211
\(225\) −0.381966 −0.0254644
\(226\) −3.41641 −0.227256
\(227\) −13.7984 −0.915830 −0.457915 0.888996i \(-0.651404\pi\)
−0.457915 + 0.888996i \(0.651404\pi\)
\(228\) 6.00000 0.397360
\(229\) −29.2361 −1.93197 −0.965987 0.258591i \(-0.916742\pi\)
−0.965987 + 0.258591i \(0.916742\pi\)
\(230\) 10.0000 0.659380
\(231\) 0 0
\(232\) −13.0902 −0.859412
\(233\) 24.6525 1.61504 0.807519 0.589842i \(-0.200810\pi\)
0.807519 + 0.589842i \(0.200810\pi\)
\(234\) −2.03444 −0.132996
\(235\) 7.85410 0.512345
\(236\) −14.2918 −0.930317
\(237\) −22.3262 −1.45024
\(238\) 7.56231 0.490191
\(239\) 21.9787 1.42168 0.710842 0.703351i \(-0.248314\pi\)
0.710842 + 0.703351i \(0.248314\pi\)
\(240\) −1.61803 −0.104444
\(241\) −23.1246 −1.48959 −0.744794 0.667295i \(-0.767452\pi\)
−0.744794 + 0.667295i \(0.767452\pi\)
\(242\) 0 0
\(243\) −3.94427 −0.253025
\(244\) 27.7082 1.77384
\(245\) −1.00000 −0.0638877
\(246\) 23.4164 1.49298
\(247\) 2.94427 0.187340
\(248\) 2.76393 0.175510
\(249\) −17.0902 −1.08305
\(250\) 2.23607 0.141421
\(251\) 5.70820 0.360299 0.180149 0.983639i \(-0.442342\pi\)
0.180149 + 0.983639i \(0.442342\pi\)
\(252\) −1.14590 −0.0721848
\(253\) 0 0
\(254\) 10.6525 0.668396
\(255\) −5.47214 −0.342678
\(256\) −9.00000 −0.562500
\(257\) −4.61803 −0.288065 −0.144033 0.989573i \(-0.546007\pi\)
−0.144033 + 0.989573i \(0.546007\pi\)
\(258\) 2.76393 0.172075
\(259\) −2.76393 −0.171742
\(260\) 7.14590 0.443170
\(261\) −2.23607 −0.138409
\(262\) 17.8885 1.10516
\(263\) 8.18034 0.504421 0.252211 0.967672i \(-0.418842\pi\)
0.252211 + 0.967672i \(0.418842\pi\)
\(264\) 0 0
\(265\) 10.4721 0.643298
\(266\) 2.76393 0.169468
\(267\) −27.8885 −1.70675
\(268\) −42.5410 −2.59861
\(269\) 12.6525 0.771435 0.385718 0.922617i \(-0.373954\pi\)
0.385718 + 0.922617i \(0.373954\pi\)
\(270\) 12.2361 0.744663
\(271\) −18.7639 −1.13983 −0.569914 0.821704i \(-0.693023\pi\)
−0.569914 + 0.821704i \(0.693023\pi\)
\(272\) 3.38197 0.205062
\(273\) 3.85410 0.233261
\(274\) −34.4721 −2.08254
\(275\) 0 0
\(276\) −21.7082 −1.30668
\(277\) −17.4164 −1.04645 −0.523225 0.852194i \(-0.675271\pi\)
−0.523225 + 0.852194i \(0.675271\pi\)
\(278\) 32.3607 1.94086
\(279\) 0.472136 0.0282660
\(280\) 2.23607 0.133631
\(281\) −0.472136 −0.0281653 −0.0140826 0.999901i \(-0.504483\pi\)
−0.0140826 + 0.999901i \(0.504483\pi\)
\(282\) −28.4164 −1.69217
\(283\) 27.5623 1.63841 0.819205 0.573501i \(-0.194415\pi\)
0.819205 + 0.573501i \(0.194415\pi\)
\(284\) 2.02129 0.119941
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 6.47214 0.382038
\(288\) −2.56231 −0.150985
\(289\) −5.56231 −0.327194
\(290\) 13.0902 0.768681
\(291\) −0.527864 −0.0309439
\(292\) −25.8541 −1.51300
\(293\) −17.4164 −1.01748 −0.508739 0.860921i \(-0.669888\pi\)
−0.508739 + 0.860921i \(0.669888\pi\)
\(294\) 3.61803 0.211008
\(295\) 4.76393 0.277367
\(296\) 6.18034 0.359225
\(297\) 0 0
\(298\) −24.9230 −1.44375
\(299\) −10.6525 −0.616049
\(300\) −4.85410 −0.280252
\(301\) 0.763932 0.0440323
\(302\) −5.85410 −0.336866
\(303\) −0.472136 −0.0271235
\(304\) 1.23607 0.0708934
\(305\) −9.23607 −0.528856
\(306\) −2.88854 −0.165127
\(307\) 0.437694 0.0249805 0.0124903 0.999922i \(-0.496024\pi\)
0.0124903 + 0.999922i \(0.496024\pi\)
\(308\) 0 0
\(309\) 2.70820 0.154064
\(310\) −2.76393 −0.156981
\(311\) −27.1246 −1.53810 −0.769048 0.639191i \(-0.779269\pi\)
−0.769048 + 0.639191i \(0.779269\pi\)
\(312\) −8.61803 −0.487900
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 10.9787 0.619565
\(315\) 0.381966 0.0215213
\(316\) 41.3951 2.32866
\(317\) −30.4721 −1.71149 −0.855743 0.517401i \(-0.826899\pi\)
−0.855743 + 0.517401i \(0.826899\pi\)
\(318\) −37.8885 −2.12468
\(319\) 0 0
\(320\) 13.0000 0.726722
\(321\) −4.47214 −0.249610
\(322\) −10.0000 −0.557278
\(323\) 4.18034 0.232600
\(324\) −23.1246 −1.28470
\(325\) −2.38197 −0.132128
\(326\) −0.403252 −0.0223341
\(327\) −33.1246 −1.83180
\(328\) −14.4721 −0.799090
\(329\) −7.85410 −0.433011
\(330\) 0 0
\(331\) 19.5066 1.07218 0.536089 0.844161i \(-0.319901\pi\)
0.536089 + 0.844161i \(0.319901\pi\)
\(332\) 31.6869 1.73905
\(333\) 1.05573 0.0578535
\(334\) −26.8328 −1.46823
\(335\) 14.1803 0.774755
\(336\) 1.61803 0.0882710
\(337\) 4.58359 0.249684 0.124842 0.992177i \(-0.460158\pi\)
0.124842 + 0.992177i \(0.460158\pi\)
\(338\) 16.3820 0.891061
\(339\) −2.47214 −0.134268
\(340\) 10.1459 0.550239
\(341\) 0 0
\(342\) −1.05573 −0.0570872
\(343\) 1.00000 0.0539949
\(344\) −1.70820 −0.0921002
\(345\) 7.23607 0.389577
\(346\) 41.5066 2.23141
\(347\) 10.4721 0.562174 0.281087 0.959682i \(-0.409305\pi\)
0.281087 + 0.959682i \(0.409305\pi\)
\(348\) −28.4164 −1.52328
\(349\) 31.8885 1.70695 0.853477 0.521130i \(-0.174489\pi\)
0.853477 + 0.521130i \(0.174489\pi\)
\(350\) −2.23607 −0.119523
\(351\) −13.0344 −0.695727
\(352\) 0 0
\(353\) 6.56231 0.349276 0.174638 0.984633i \(-0.444124\pi\)
0.174638 + 0.984633i \(0.444124\pi\)
\(354\) −17.2361 −0.916087
\(355\) −0.673762 −0.0357596
\(356\) 51.7082 2.74053
\(357\) 5.47214 0.289616
\(358\) −38.8673 −2.05420
\(359\) −33.9787 −1.79333 −0.896664 0.442712i \(-0.854016\pi\)
−0.896664 + 0.442712i \(0.854016\pi\)
\(360\) −0.854102 −0.0450151
\(361\) −17.4721 −0.919586
\(362\) −32.3607 −1.70084
\(363\) 0 0
\(364\) −7.14590 −0.374547
\(365\) 8.61803 0.451089
\(366\) 33.4164 1.74670
\(367\) 24.4508 1.27632 0.638162 0.769902i \(-0.279695\pi\)
0.638162 + 0.769902i \(0.279695\pi\)
\(368\) −4.47214 −0.233126
\(369\) −2.47214 −0.128694
\(370\) −6.18034 −0.321301
\(371\) −10.4721 −0.543686
\(372\) 6.00000 0.311086
\(373\) 29.1246 1.50802 0.754008 0.656866i \(-0.228118\pi\)
0.754008 + 0.656866i \(0.228118\pi\)
\(374\) 0 0
\(375\) 1.61803 0.0835549
\(376\) 17.5623 0.905707
\(377\) −13.9443 −0.718167
\(378\) −12.2361 −0.629355
\(379\) −17.7426 −0.911378 −0.455689 0.890139i \(-0.650607\pi\)
−0.455689 + 0.890139i \(0.650607\pi\)
\(380\) 3.70820 0.190227
\(381\) 7.70820 0.394903
\(382\) −26.6312 −1.36257
\(383\) 2.20163 0.112498 0.0562489 0.998417i \(-0.482086\pi\)
0.0562489 + 0.998417i \(0.482086\pi\)
\(384\) −25.3262 −1.29242
\(385\) 0 0
\(386\) 8.94427 0.455251
\(387\) −0.291796 −0.0148328
\(388\) 0.978714 0.0496867
\(389\) −13.4377 −0.681318 −0.340659 0.940187i \(-0.610650\pi\)
−0.340659 + 0.940187i \(0.610650\pi\)
\(390\) 8.61803 0.436391
\(391\) −15.1246 −0.764884
\(392\) −2.23607 −0.112938
\(393\) 12.9443 0.652952
\(394\) −26.8328 −1.35182
\(395\) −13.7984 −0.694272
\(396\) 0 0
\(397\) −34.1591 −1.71439 −0.857197 0.514989i \(-0.827796\pi\)
−0.857197 + 0.514989i \(0.827796\pi\)
\(398\) 40.2492 2.01751
\(399\) 2.00000 0.100125
\(400\) −1.00000 −0.0500000
\(401\) −22.5066 −1.12392 −0.561962 0.827163i \(-0.689953\pi\)
−0.561962 + 0.827163i \(0.689953\pi\)
\(402\) −51.3050 −2.55886
\(403\) 2.94427 0.146665
\(404\) 0.875388 0.0435522
\(405\) 7.70820 0.383024
\(406\) −13.0902 −0.649654
\(407\) 0 0
\(408\) −12.2361 −0.605776
\(409\) −1.23607 −0.0611196 −0.0305598 0.999533i \(-0.509729\pi\)
−0.0305598 + 0.999533i \(0.509729\pi\)
\(410\) 14.4721 0.714728
\(411\) −24.9443 −1.23041
\(412\) −5.02129 −0.247381
\(413\) −4.76393 −0.234418
\(414\) 3.81966 0.187726
\(415\) −10.5623 −0.518483
\(416\) −15.9787 −0.783421
\(417\) 23.4164 1.14671
\(418\) 0 0
\(419\) 27.3050 1.33393 0.666967 0.745087i \(-0.267592\pi\)
0.666967 + 0.745087i \(0.267592\pi\)
\(420\) 4.85410 0.236856
\(421\) 3.50658 0.170900 0.0854501 0.996342i \(-0.472767\pi\)
0.0854501 + 0.996342i \(0.472767\pi\)
\(422\) 52.1591 2.53906
\(423\) 3.00000 0.145865
\(424\) 23.4164 1.13720
\(425\) −3.38197 −0.164049
\(426\) 2.43769 0.118107
\(427\) 9.23607 0.446965
\(428\) 8.29180 0.400799
\(429\) 0 0
\(430\) 1.70820 0.0823769
\(431\) −10.3262 −0.497397 −0.248699 0.968581i \(-0.580003\pi\)
−0.248699 + 0.968581i \(0.580003\pi\)
\(432\) −5.47214 −0.263278
\(433\) −17.0344 −0.818623 −0.409312 0.912395i \(-0.634231\pi\)
−0.409312 + 0.912395i \(0.634231\pi\)
\(434\) 2.76393 0.132673
\(435\) 9.47214 0.454154
\(436\) 61.4164 2.94131
\(437\) −5.52786 −0.264434
\(438\) −31.1803 −1.48985
\(439\) −17.7082 −0.845166 −0.422583 0.906324i \(-0.638877\pi\)
−0.422583 + 0.906324i \(0.638877\pi\)
\(440\) 0 0
\(441\) −0.381966 −0.0181889
\(442\) −18.0132 −0.856798
\(443\) −34.6525 −1.64639 −0.823194 0.567760i \(-0.807810\pi\)
−0.823194 + 0.567760i \(0.807810\pi\)
\(444\) 13.4164 0.636715
\(445\) −17.2361 −0.817068
\(446\) −23.4164 −1.10880
\(447\) −18.0344 −0.853000
\(448\) −13.0000 −0.614192
\(449\) 24.2705 1.14540 0.572698 0.819766i \(-0.305897\pi\)
0.572698 + 0.819766i \(0.305897\pi\)
\(450\) 0.854102 0.0402628
\(451\) 0 0
\(452\) 4.58359 0.215594
\(453\) −4.23607 −0.199028
\(454\) 30.8541 1.44805
\(455\) 2.38197 0.111668
\(456\) −4.47214 −0.209427
\(457\) −33.7771 −1.58003 −0.790013 0.613090i \(-0.789926\pi\)
−0.790013 + 0.613090i \(0.789926\pi\)
\(458\) 65.3738 3.05472
\(459\) −18.5066 −0.863813
\(460\) −13.4164 −0.625543
\(461\) −29.7082 −1.38365 −0.691825 0.722066i \(-0.743193\pi\)
−0.691825 + 0.722066i \(0.743193\pi\)
\(462\) 0 0
\(463\) 23.1246 1.07469 0.537346 0.843362i \(-0.319427\pi\)
0.537346 + 0.843362i \(0.319427\pi\)
\(464\) −5.85410 −0.271770
\(465\) −2.00000 −0.0927478
\(466\) −55.1246 −2.55360
\(467\) −16.6738 −0.771570 −0.385785 0.922589i \(-0.626069\pi\)
−0.385785 + 0.922589i \(0.626069\pi\)
\(468\) 2.72949 0.126171
\(469\) −14.1803 −0.654787
\(470\) −17.5623 −0.810089
\(471\) 7.94427 0.366053
\(472\) 10.6525 0.490320
\(473\) 0 0
\(474\) 49.9230 2.29304
\(475\) −1.23607 −0.0567147
\(476\) −10.1459 −0.465036
\(477\) 4.00000 0.183147
\(478\) −49.1459 −2.24788
\(479\) 8.65248 0.395342 0.197671 0.980268i \(-0.436662\pi\)
0.197671 + 0.980268i \(0.436662\pi\)
\(480\) 10.8541 0.495420
\(481\) 6.58359 0.300186
\(482\) 51.7082 2.35524
\(483\) −7.23607 −0.329252
\(484\) 0 0
\(485\) −0.326238 −0.0148137
\(486\) 8.81966 0.400068
\(487\) −24.1803 −1.09572 −0.547858 0.836571i \(-0.684557\pi\)
−0.547858 + 0.836571i \(0.684557\pi\)
\(488\) −20.6525 −0.934894
\(489\) −0.291796 −0.0131955
\(490\) 2.23607 0.101015
\(491\) −37.8885 −1.70989 −0.854943 0.518722i \(-0.826408\pi\)
−0.854943 + 0.518722i \(0.826408\pi\)
\(492\) −31.4164 −1.41636
\(493\) −19.7984 −0.891674
\(494\) −6.58359 −0.296210
\(495\) 0 0
\(496\) 1.23607 0.0555011
\(497\) 0.673762 0.0302224
\(498\) 38.2148 1.71245
\(499\) 6.03444 0.270139 0.135069 0.990836i \(-0.456874\pi\)
0.135069 + 0.990836i \(0.456874\pi\)
\(500\) −3.00000 −0.134164
\(501\) −19.4164 −0.867461
\(502\) −12.7639 −0.569682
\(503\) 8.09017 0.360723 0.180361 0.983600i \(-0.442273\pi\)
0.180361 + 0.983600i \(0.442273\pi\)
\(504\) 0.854102 0.0380447
\(505\) −0.291796 −0.0129848
\(506\) 0 0
\(507\) 11.8541 0.526459
\(508\) −14.2918 −0.634096
\(509\) −18.3607 −0.813823 −0.406911 0.913468i \(-0.633394\pi\)
−0.406911 + 0.913468i \(0.633394\pi\)
\(510\) 12.2361 0.541822
\(511\) −8.61803 −0.381239
\(512\) −11.1803 −0.494106
\(513\) −6.76393 −0.298635
\(514\) 10.3262 0.455471
\(515\) 1.67376 0.0737548
\(516\) −3.70820 −0.163245
\(517\) 0 0
\(518\) 6.18034 0.271549
\(519\) 30.0344 1.31837
\(520\) −5.32624 −0.233571
\(521\) 8.18034 0.358387 0.179194 0.983814i \(-0.442651\pi\)
0.179194 + 0.983814i \(0.442651\pi\)
\(522\) 5.00000 0.218844
\(523\) −7.85410 −0.343436 −0.171718 0.985146i \(-0.554932\pi\)
−0.171718 + 0.985146i \(0.554932\pi\)
\(524\) −24.0000 −1.04844
\(525\) −1.61803 −0.0706168
\(526\) −18.2918 −0.797560
\(527\) 4.18034 0.182098
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) −23.4164 −1.01714
\(531\) 1.81966 0.0789665
\(532\) −3.70820 −0.160771
\(533\) −15.4164 −0.667759
\(534\) 62.3607 2.69861
\(535\) −2.76393 −0.119495
\(536\) 31.7082 1.36959
\(537\) −28.1246 −1.21367
\(538\) −28.2918 −1.21975
\(539\) 0 0
\(540\) −16.4164 −0.706450
\(541\) −12.8541 −0.552641 −0.276321 0.961066i \(-0.589115\pi\)
−0.276321 + 0.961066i \(0.589115\pi\)
\(542\) 41.9574 1.80223
\(543\) −23.4164 −1.00489
\(544\) −22.6869 −0.972694
\(545\) −20.4721 −0.876930
\(546\) −8.61803 −0.368818
\(547\) −40.0689 −1.71322 −0.856611 0.515963i \(-0.827434\pi\)
−0.856611 + 0.515963i \(0.827434\pi\)
\(548\) 46.2492 1.97567
\(549\) −3.52786 −0.150566
\(550\) 0 0
\(551\) −7.23607 −0.308267
\(552\) 16.1803 0.688681
\(553\) 13.7984 0.586767
\(554\) 38.9443 1.65458
\(555\) −4.47214 −0.189832
\(556\) −43.4164 −1.84127
\(557\) 4.29180 0.181849 0.0909246 0.995858i \(-0.471018\pi\)
0.0909246 + 0.995858i \(0.471018\pi\)
\(558\) −1.05573 −0.0446925
\(559\) −1.81966 −0.0769634
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 1.05573 0.0445332
\(563\) 21.3820 0.901142 0.450571 0.892740i \(-0.351220\pi\)
0.450571 + 0.892740i \(0.351220\pi\)
\(564\) 38.1246 1.60534
\(565\) −1.52786 −0.0642777
\(566\) −61.6312 −2.59055
\(567\) −7.70820 −0.323714
\(568\) −1.50658 −0.0632146
\(569\) 13.2016 0.553441 0.276720 0.960950i \(-0.410752\pi\)
0.276720 + 0.960950i \(0.410752\pi\)
\(570\) 4.47214 0.187317
\(571\) −10.9787 −0.459445 −0.229722 0.973256i \(-0.573782\pi\)
−0.229722 + 0.973256i \(0.573782\pi\)
\(572\) 0 0
\(573\) −19.2705 −0.805037
\(574\) −14.4721 −0.604055
\(575\) 4.47214 0.186501
\(576\) 4.96556 0.206898
\(577\) −18.2148 −0.758291 −0.379146 0.925337i \(-0.623782\pi\)
−0.379146 + 0.925337i \(0.623782\pi\)
\(578\) 12.4377 0.517340
\(579\) 6.47214 0.268973
\(580\) −17.5623 −0.729235
\(581\) 10.5623 0.438198
\(582\) 1.18034 0.0489267
\(583\) 0 0
\(584\) 19.2705 0.797419
\(585\) −0.909830 −0.0376168
\(586\) 38.9443 1.60877
\(587\) 40.2705 1.66214 0.831071 0.556166i \(-0.187728\pi\)
0.831071 + 0.556166i \(0.187728\pi\)
\(588\) −4.85410 −0.200180
\(589\) 1.52786 0.0629545
\(590\) −10.6525 −0.438555
\(591\) −19.4164 −0.798684
\(592\) 2.76393 0.113597
\(593\) 14.7984 0.607696 0.303848 0.952720i \(-0.401728\pi\)
0.303848 + 0.952720i \(0.401728\pi\)
\(594\) 0 0
\(595\) 3.38197 0.138647
\(596\) 33.4377 1.36966
\(597\) 29.1246 1.19199
\(598\) 23.8197 0.974058
\(599\) −7.27051 −0.297065 −0.148532 0.988908i \(-0.547455\pi\)
−0.148532 + 0.988908i \(0.547455\pi\)
\(600\) 3.61803 0.147706
\(601\) −11.4164 −0.465685 −0.232842 0.972514i \(-0.574803\pi\)
−0.232842 + 0.972514i \(0.574803\pi\)
\(602\) −1.70820 −0.0696212
\(603\) 5.41641 0.220573
\(604\) 7.85410 0.319579
\(605\) 0 0
\(606\) 1.05573 0.0428860
\(607\) 13.0902 0.531314 0.265657 0.964068i \(-0.414411\pi\)
0.265657 + 0.964068i \(0.414411\pi\)
\(608\) −8.29180 −0.336277
\(609\) −9.47214 −0.383830
\(610\) 20.6525 0.836194
\(611\) 18.7082 0.756853
\(612\) 3.87539 0.156653
\(613\) −12.8328 −0.518313 −0.259156 0.965835i \(-0.583444\pi\)
−0.259156 + 0.965835i \(0.583444\pi\)
\(614\) −0.978714 −0.0394977
\(615\) 10.4721 0.422277
\(616\) 0 0
\(617\) −7.23607 −0.291313 −0.145657 0.989335i \(-0.546529\pi\)
−0.145657 + 0.989335i \(0.546529\pi\)
\(618\) −6.05573 −0.243597
\(619\) −13.5279 −0.543731 −0.271865 0.962335i \(-0.587641\pi\)
−0.271865 + 0.962335i \(0.587641\pi\)
\(620\) 3.70820 0.148925
\(621\) 24.4721 0.982033
\(622\) 60.6525 2.43194
\(623\) 17.2361 0.690548
\(624\) −3.85410 −0.154288
\(625\) 1.00000 0.0400000
\(626\) 40.2492 1.60868
\(627\) 0 0
\(628\) −14.7295 −0.587771
\(629\) 9.34752 0.372710
\(630\) −0.854102 −0.0340282
\(631\) 28.5066 1.13483 0.567414 0.823432i \(-0.307944\pi\)
0.567414 + 0.823432i \(0.307944\pi\)
\(632\) −30.8541 −1.22731
\(633\) 37.7426 1.50014
\(634\) 68.1378 2.70610
\(635\) 4.76393 0.189051
\(636\) 50.8328 2.01565
\(637\) −2.38197 −0.0943769
\(638\) 0 0
\(639\) −0.257354 −0.0101808
\(640\) −15.6525 −0.618718
\(641\) −25.9787 −1.02610 −0.513049 0.858359i \(-0.671484\pi\)
−0.513049 + 0.858359i \(0.671484\pi\)
\(642\) 10.0000 0.394669
\(643\) −13.7984 −0.544155 −0.272077 0.962275i \(-0.587711\pi\)
−0.272077 + 0.962275i \(0.587711\pi\)
\(644\) 13.4164 0.528681
\(645\) 1.23607 0.0486701
\(646\) −9.34752 −0.367773
\(647\) −40.1459 −1.57830 −0.789149 0.614202i \(-0.789478\pi\)
−0.789149 + 0.614202i \(0.789478\pi\)
\(648\) 17.2361 0.677097
\(649\) 0 0
\(650\) 5.32624 0.208912
\(651\) 2.00000 0.0783862
\(652\) 0.541020 0.0211880
\(653\) −15.8197 −0.619071 −0.309536 0.950888i \(-0.600174\pi\)
−0.309536 + 0.950888i \(0.600174\pi\)
\(654\) 74.0689 2.89632
\(655\) 8.00000 0.312586
\(656\) −6.47214 −0.252694
\(657\) 3.29180 0.128425
\(658\) 17.5623 0.684650
\(659\) −25.0344 −0.975203 −0.487602 0.873066i \(-0.662128\pi\)
−0.487602 + 0.873066i \(0.662128\pi\)
\(660\) 0 0
\(661\) −33.4164 −1.29975 −0.649874 0.760042i \(-0.725178\pi\)
−0.649874 + 0.760042i \(0.725178\pi\)
\(662\) −43.6180 −1.69526
\(663\) −13.0344 −0.506216
\(664\) −23.6180 −0.916557
\(665\) 1.23607 0.0479327
\(666\) −2.36068 −0.0914745
\(667\) 26.1803 1.01371
\(668\) 36.0000 1.39288
\(669\) −16.9443 −0.655103
\(670\) −31.7082 −1.22499
\(671\) 0 0
\(672\) −10.8541 −0.418706
\(673\) 13.8885 0.535364 0.267682 0.963507i \(-0.413742\pi\)
0.267682 + 0.963507i \(0.413742\pi\)
\(674\) −10.2492 −0.394785
\(675\) 5.47214 0.210623
\(676\) −21.9787 −0.845335
\(677\) −42.1459 −1.61980 −0.809899 0.586569i \(-0.800478\pi\)
−0.809899 + 0.586569i \(0.800478\pi\)
\(678\) 5.52786 0.212296
\(679\) 0.326238 0.0125199
\(680\) −7.56231 −0.290001
\(681\) 22.3262 0.855543
\(682\) 0 0
\(683\) 39.2361 1.50133 0.750663 0.660685i \(-0.229734\pi\)
0.750663 + 0.660685i \(0.229734\pi\)
\(684\) 1.41641 0.0541577
\(685\) −15.4164 −0.589031
\(686\) −2.23607 −0.0853735
\(687\) 47.3050 1.80480
\(688\) −0.763932 −0.0291246
\(689\) 24.9443 0.950301
\(690\) −16.1803 −0.615975
\(691\) 26.8328 1.02077 0.510384 0.859946i \(-0.329503\pi\)
0.510384 + 0.859946i \(0.329503\pi\)
\(692\) −55.6869 −2.11690
\(693\) 0 0
\(694\) −23.4164 −0.888875
\(695\) 14.4721 0.548959
\(696\) 21.1803 0.802839
\(697\) −21.8885 −0.829088
\(698\) −71.3050 −2.69893
\(699\) −39.8885 −1.50872
\(700\) 3.00000 0.113389
\(701\) −11.3050 −0.426982 −0.213491 0.976945i \(-0.568483\pi\)
−0.213491 + 0.976945i \(0.568483\pi\)
\(702\) 29.1459 1.10004
\(703\) 3.41641 0.128852
\(704\) 0 0
\(705\) −12.7082 −0.478619
\(706\) −14.6738 −0.552254
\(707\) 0.291796 0.0109741
\(708\) 23.1246 0.869076
\(709\) 42.7426 1.60523 0.802617 0.596495i \(-0.203440\pi\)
0.802617 + 0.596495i \(0.203440\pi\)
\(710\) 1.50658 0.0565409
\(711\) −5.27051 −0.197660
\(712\) −38.5410 −1.44439
\(713\) −5.52786 −0.207020
\(714\) −12.2361 −0.457923
\(715\) 0 0
\(716\) 52.1459 1.94878
\(717\) −35.5623 −1.32810
\(718\) 75.9787 2.83550
\(719\) −13.8885 −0.517955 −0.258978 0.965883i \(-0.583386\pi\)
−0.258978 + 0.965883i \(0.583386\pi\)
\(720\) −0.381966 −0.0142350
\(721\) −1.67376 −0.0623342
\(722\) 39.0689 1.45399
\(723\) 37.4164 1.39153
\(724\) 43.4164 1.61356
\(725\) 5.85410 0.217416
\(726\) 0 0
\(727\) −21.4377 −0.795080 −0.397540 0.917585i \(-0.630136\pi\)
−0.397540 + 0.917585i \(0.630136\pi\)
\(728\) 5.32624 0.197404
\(729\) 29.5066 1.09284
\(730\) −19.2705 −0.713234
\(731\) −2.58359 −0.0955576
\(732\) −44.8328 −1.65707
\(733\) 47.8673 1.76802 0.884009 0.467470i \(-0.154835\pi\)
0.884009 + 0.467470i \(0.154835\pi\)
\(734\) −54.6738 −2.01805
\(735\) 1.61803 0.0596821
\(736\) 30.0000 1.10581
\(737\) 0 0
\(738\) 5.52786 0.203483
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 8.29180 0.304812
\(741\) −4.76393 −0.175007
\(742\) 23.4164 0.859643
\(743\) −36.6525 −1.34465 −0.672324 0.740257i \(-0.734704\pi\)
−0.672324 + 0.740257i \(0.734704\pi\)
\(744\) −4.47214 −0.163956
\(745\) −11.1459 −0.408354
\(746\) −65.1246 −2.38438
\(747\) −4.03444 −0.147613
\(748\) 0 0
\(749\) 2.76393 0.100992
\(750\) −3.61803 −0.132112
\(751\) 21.6738 0.790887 0.395443 0.918490i \(-0.370591\pi\)
0.395443 + 0.918490i \(0.370591\pi\)
\(752\) 7.85410 0.286410
\(753\) −9.23607 −0.336581
\(754\) 31.1803 1.13552
\(755\) −2.61803 −0.0952800
\(756\) 16.4164 0.597059
\(757\) −18.1803 −0.660776 −0.330388 0.943845i \(-0.607180\pi\)
−0.330388 + 0.943845i \(0.607180\pi\)
\(758\) 39.6738 1.44102
\(759\) 0 0
\(760\) −2.76393 −0.100258
\(761\) 4.83282 0.175189 0.0875947 0.996156i \(-0.472082\pi\)
0.0875947 + 0.996156i \(0.472082\pi\)
\(762\) −17.2361 −0.624397
\(763\) 20.4721 0.741141
\(764\) 35.7295 1.29265
\(765\) −1.29180 −0.0467050
\(766\) −4.92299 −0.177875
\(767\) 11.3475 0.409735
\(768\) 14.5623 0.525472
\(769\) 1.70820 0.0615994 0.0307997 0.999526i \(-0.490195\pi\)
0.0307997 + 0.999526i \(0.490195\pi\)
\(770\) 0 0
\(771\) 7.47214 0.269102
\(772\) −12.0000 −0.431889
\(773\) −29.6180 −1.06529 −0.532643 0.846340i \(-0.678801\pi\)
−0.532643 + 0.846340i \(0.678801\pi\)
\(774\) 0.652476 0.0234528
\(775\) −1.23607 −0.0444009
\(776\) −0.729490 −0.0261872
\(777\) 4.47214 0.160437
\(778\) 30.0476 1.07726
\(779\) −8.00000 −0.286630
\(780\) −11.5623 −0.413997
\(781\) 0 0
\(782\) 33.8197 1.20939
\(783\) 32.0344 1.14482
\(784\) −1.00000 −0.0357143
\(785\) 4.90983 0.175239
\(786\) −28.9443 −1.03241
\(787\) −10.9656 −0.390880 −0.195440 0.980716i \(-0.562613\pi\)
−0.195440 + 0.980716i \(0.562613\pi\)
\(788\) 36.0000 1.28245
\(789\) −13.2361 −0.471216
\(790\) 30.8541 1.09774
\(791\) 1.52786 0.0543246
\(792\) 0 0
\(793\) −22.0000 −0.781243
\(794\) 76.3820 2.71069
\(795\) −16.9443 −0.600951
\(796\) −54.0000 −1.91398
\(797\) −43.1033 −1.52680 −0.763399 0.645927i \(-0.776471\pi\)
−0.763399 + 0.645927i \(0.776471\pi\)
\(798\) −4.47214 −0.158312
\(799\) 26.5623 0.939707
\(800\) 6.70820 0.237171
\(801\) −6.58359 −0.232620
\(802\) 50.3262 1.77708
\(803\) 0 0
\(804\) 68.8328 2.42755
\(805\) −4.47214 −0.157622
\(806\) −6.58359 −0.231897
\(807\) −20.4721 −0.720653
\(808\) −0.652476 −0.0229540
\(809\) −8.09017 −0.284435 −0.142218 0.989835i \(-0.545423\pi\)
−0.142218 + 0.989835i \(0.545423\pi\)
\(810\) −17.2361 −0.605614
\(811\) −4.58359 −0.160952 −0.0804758 0.996757i \(-0.525644\pi\)
−0.0804758 + 0.996757i \(0.525644\pi\)
\(812\) 17.5623 0.616316
\(813\) 30.3607 1.06480
\(814\) 0 0
\(815\) −0.180340 −0.00631703
\(816\) −5.47214 −0.191563
\(817\) −0.944272 −0.0330359
\(818\) 2.76393 0.0966386
\(819\) 0.909830 0.0317920
\(820\) −19.4164 −0.678050
\(821\) −28.3262 −0.988593 −0.494296 0.869294i \(-0.664574\pi\)
−0.494296 + 0.869294i \(0.664574\pi\)
\(822\) 55.7771 1.94545
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 3.74265 0.130381
\(825\) 0 0
\(826\) 10.6525 0.370647
\(827\) 2.11146 0.0734225 0.0367113 0.999326i \(-0.488312\pi\)
0.0367113 + 0.999326i \(0.488312\pi\)
\(828\) −5.12461 −0.178093
\(829\) −1.88854 −0.0655918 −0.0327959 0.999462i \(-0.510441\pi\)
−0.0327959 + 0.999462i \(0.510441\pi\)
\(830\) 23.6180 0.819794
\(831\) 28.1803 0.977565
\(832\) 30.9656 1.07354
\(833\) −3.38197 −0.117178
\(834\) −52.3607 −1.81310
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) −6.76393 −0.233796
\(838\) −61.0557 −2.10914
\(839\) 52.3607 1.80769 0.903846 0.427859i \(-0.140732\pi\)
0.903846 + 0.427859i \(0.140732\pi\)
\(840\) −3.61803 −0.124834
\(841\) 5.27051 0.181742
\(842\) −7.84095 −0.270217
\(843\) 0.763932 0.0263112
\(844\) −69.9787 −2.40877
\(845\) 7.32624 0.252030
\(846\) −6.70820 −0.230633
\(847\) 0 0
\(848\) 10.4721 0.359615
\(849\) −44.5967 −1.53056
\(850\) 7.56231 0.259385
\(851\) −12.3607 −0.423719
\(852\) −3.27051 −0.112046
\(853\) −38.5623 −1.32035 −0.660174 0.751113i \(-0.729518\pi\)
−0.660174 + 0.751113i \(0.729518\pi\)
\(854\) −20.6525 −0.706713
\(855\) −0.472136 −0.0161467
\(856\) −6.18034 −0.211240
\(857\) −41.1935 −1.40714 −0.703572 0.710624i \(-0.748413\pi\)
−0.703572 + 0.710624i \(0.748413\pi\)
\(858\) 0 0
\(859\) −28.9443 −0.987566 −0.493783 0.869585i \(-0.664386\pi\)
−0.493783 + 0.869585i \(0.664386\pi\)
\(860\) −2.29180 −0.0781496
\(861\) −10.4721 −0.356889
\(862\) 23.0902 0.786454
\(863\) 20.9443 0.712951 0.356476 0.934305i \(-0.383978\pi\)
0.356476 + 0.934305i \(0.383978\pi\)
\(864\) 36.7082 1.24884
\(865\) 18.5623 0.631137
\(866\) 38.0902 1.29436
\(867\) 9.00000 0.305656
\(868\) −3.70820 −0.125865
\(869\) 0 0
\(870\) −21.1803 −0.718081
\(871\) 33.7771 1.14449
\(872\) −45.7771 −1.55021
\(873\) −0.124612 −0.00421747
\(874\) 12.3607 0.418106
\(875\) −1.00000 −0.0338062
\(876\) 41.8328 1.41340
\(877\) 21.0132 0.709564 0.354782 0.934949i \(-0.384555\pi\)
0.354782 + 0.934949i \(0.384555\pi\)
\(878\) 39.5967 1.33633
\(879\) 28.1803 0.950499
\(880\) 0 0
\(881\) −0.763932 −0.0257375 −0.0128688 0.999917i \(-0.504096\pi\)
−0.0128688 + 0.999917i \(0.504096\pi\)
\(882\) 0.854102 0.0287591
\(883\) 12.1803 0.409901 0.204951 0.978772i \(-0.434297\pi\)
0.204951 + 0.978772i \(0.434297\pi\)
\(884\) 24.1672 0.812830
\(885\) −7.70820 −0.259108
\(886\) 77.4853 2.60317
\(887\) 27.8541 0.935249 0.467625 0.883927i \(-0.345110\pi\)
0.467625 + 0.883927i \(0.345110\pi\)
\(888\) −10.0000 −0.335578
\(889\) −4.76393 −0.159777
\(890\) 38.5410 1.29190
\(891\) 0 0
\(892\) 31.4164 1.05190
\(893\) 9.70820 0.324873
\(894\) 40.3262 1.34871
\(895\) −17.3820 −0.581015
\(896\) 15.6525 0.522913
\(897\) 17.2361 0.575496
\(898\) −54.2705 −1.81103
\(899\) −7.23607 −0.241336
\(900\) −1.14590 −0.0381966
\(901\) 35.4164 1.17989
\(902\) 0 0
\(903\) −1.23607 −0.0411338
\(904\) −3.41641 −0.113628
\(905\) −14.4721 −0.481070
\(906\) 9.47214 0.314691
\(907\) −7.88854 −0.261935 −0.130967 0.991387i \(-0.541808\pi\)
−0.130967 + 0.991387i \(0.541808\pi\)
\(908\) −41.3951 −1.37375
\(909\) −0.111456 −0.00369677
\(910\) −5.32624 −0.176563
\(911\) 36.5066 1.20952 0.604758 0.796409i \(-0.293270\pi\)
0.604758 + 0.796409i \(0.293270\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) 75.5279 2.49824
\(915\) 14.9443 0.494042
\(916\) −87.7082 −2.89796
\(917\) −8.00000 −0.264183
\(918\) 41.3820 1.36581
\(919\) 31.2705 1.03152 0.515759 0.856733i \(-0.327510\pi\)
0.515759 + 0.856733i \(0.327510\pi\)
\(920\) 10.0000 0.329690
\(921\) −0.708204 −0.0233361
\(922\) 66.4296 2.18774
\(923\) −1.60488 −0.0528252
\(924\) 0 0
\(925\) −2.76393 −0.0908775
\(926\) −51.7082 −1.69924
\(927\) 0.639320 0.0209980
\(928\) 39.2705 1.28912
\(929\) 17.8885 0.586904 0.293452 0.955974i \(-0.405196\pi\)
0.293452 + 0.955974i \(0.405196\pi\)
\(930\) 4.47214 0.146647
\(931\) −1.23607 −0.0405105
\(932\) 73.9574 2.42256
\(933\) 43.8885 1.43685
\(934\) 37.2837 1.21996
\(935\) 0 0
\(936\) −2.03444 −0.0664978
\(937\) −56.4508 −1.84417 −0.922084 0.386989i \(-0.873515\pi\)
−0.922084 + 0.386989i \(0.873515\pi\)
\(938\) 31.7082 1.03531
\(939\) 29.1246 0.950446
\(940\) 23.5623 0.768518
\(941\) −36.5410 −1.19120 −0.595602 0.803280i \(-0.703086\pi\)
−0.595602 + 0.803280i \(0.703086\pi\)
\(942\) −17.7639 −0.578780
\(943\) 28.9443 0.942555
\(944\) 4.76393 0.155053
\(945\) −5.47214 −0.178009
\(946\) 0 0
\(947\) −24.0689 −0.782134 −0.391067 0.920362i \(-0.627894\pi\)
−0.391067 + 0.920362i \(0.627894\pi\)
\(948\) −66.9787 −2.17537
\(949\) 20.5279 0.666363
\(950\) 2.76393 0.0896738
\(951\) 49.3050 1.59882
\(952\) 7.56231 0.245096
\(953\) −25.0557 −0.811635 −0.405817 0.913954i \(-0.633013\pi\)
−0.405817 + 0.913954i \(0.633013\pi\)
\(954\) −8.94427 −0.289581
\(955\) −11.9098 −0.385393
\(956\) 65.9361 2.13253
\(957\) 0 0
\(958\) −19.3475 −0.625090
\(959\) 15.4164 0.497822
\(960\) −21.0344 −0.678884
\(961\) −29.4721 −0.950714
\(962\) −14.7214 −0.474636
\(963\) −1.05573 −0.0340204
\(964\) −69.3738 −2.23438
\(965\) 4.00000 0.128765
\(966\) 16.1803 0.520594
\(967\) 21.2361 0.682906 0.341453 0.939899i \(-0.389081\pi\)
0.341453 + 0.939899i \(0.389081\pi\)
\(968\) 0 0
\(969\) −6.76393 −0.217289
\(970\) 0.729490 0.0234225
\(971\) −29.4164 −0.944017 −0.472009 0.881594i \(-0.656471\pi\)
−0.472009 + 0.881594i \(0.656471\pi\)
\(972\) −11.8328 −0.379538
\(973\) −14.4721 −0.463955
\(974\) 54.0689 1.73248
\(975\) 3.85410 0.123430
\(976\) −9.23607 −0.295639
\(977\) 36.7639 1.17618 0.588091 0.808795i \(-0.299880\pi\)
0.588091 + 0.808795i \(0.299880\pi\)
\(978\) 0.652476 0.0208639
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) −7.81966 −0.249663
\(982\) 84.7214 2.70357
\(983\) 18.9787 0.605327 0.302663 0.953097i \(-0.402124\pi\)
0.302663 + 0.953097i \(0.402124\pi\)
\(984\) 23.4164 0.746488
\(985\) −12.0000 −0.382352
\(986\) 44.2705 1.40986
\(987\) 12.7082 0.404507
\(988\) 8.83282 0.281009
\(989\) 3.41641 0.108635
\(990\) 0 0
\(991\) 7.14590 0.226997 0.113498 0.993538i \(-0.463794\pi\)
0.113498 + 0.993538i \(0.463794\pi\)
\(992\) −8.29180 −0.263265
\(993\) −31.5623 −1.00160
\(994\) −1.50658 −0.0477857
\(995\) 18.0000 0.570638
\(996\) −51.2705 −1.62457
\(997\) −26.2148 −0.830230 −0.415115 0.909769i \(-0.636259\pi\)
−0.415115 + 0.909769i \(0.636259\pi\)
\(998\) −13.4934 −0.427127
\(999\) −15.1246 −0.478522
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 4235.2.a.k.1.1 2
11.7 odd 10 385.2.n.b.71.1 4
11.8 odd 10 385.2.n.b.141.1 yes 4
11.10 odd 2 4235.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.n.b.71.1 4 11.7 odd 10
385.2.n.b.141.1 yes 4 11.8 odd 10
4235.2.a.j.1.2 2 11.10 odd 2
4235.2.a.k.1.1 2 1.1 even 1 trivial