Properties

Label 690.2.a.l.1.2
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.12311 q^{11} -1.00000 q^{12} +2.00000 q^{13} +3.12311 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.12311 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -3.12311 q^{21} -3.12311 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +3.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} +1.00000 q^{32} +3.12311 q^{33} -1.12311 q^{34} +3.12311 q^{35} +1.00000 q^{36} +1.12311 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -3.12311 q^{42} -3.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +2.75379 q^{49} +1.00000 q^{50} +1.12311 q^{51} +2.00000 q^{52} +12.2462 q^{53} -1.00000 q^{54} -3.12311 q^{55} +3.12311 q^{56} -4.00000 q^{57} +2.00000 q^{58} +2.24621 q^{59} -1.00000 q^{60} +9.12311 q^{61} +3.12311 q^{63} +1.00000 q^{64} +2.00000 q^{65} +3.12311 q^{66} -8.00000 q^{67} -1.12311 q^{68} -1.00000 q^{69} +3.12311 q^{70} -10.2462 q^{71} +1.00000 q^{72} -4.24621 q^{73} +1.12311 q^{74} -1.00000 q^{75} +4.00000 q^{76} -9.75379 q^{77} -2.00000 q^{78} +3.12311 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -13.3693 q^{83} -3.12311 q^{84} -1.12311 q^{85} -2.00000 q^{87} -3.12311 q^{88} -5.12311 q^{89} +1.00000 q^{90} +6.24621 q^{91} +1.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -16.2462 q^{97} +2.75379 q^{98} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{11} - 2q^{12} + 4q^{13} - 2q^{14} - 2q^{15} + 2q^{16} + 6q^{17} + 2q^{18} + 8q^{19} + 2q^{20} + 2q^{21} + 2q^{22} + 2q^{23} - 2q^{24} + 2q^{25} + 4q^{26} - 2q^{27} - 2q^{28} + 4q^{29} - 2q^{30} + 2q^{32} - 2q^{33} + 6q^{34} - 2q^{35} + 2q^{36} - 6q^{37} + 8q^{38} - 4q^{39} + 2q^{40} + 4q^{41} + 2q^{42} + 2q^{44} + 2q^{45} + 2q^{46} - 16q^{47} - 2q^{48} + 22q^{49} + 2q^{50} - 6q^{51} + 4q^{52} + 8q^{53} - 2q^{54} + 2q^{55} - 2q^{56} - 8q^{57} + 4q^{58} - 12q^{59} - 2q^{60} + 10q^{61} - 2q^{63} + 2q^{64} + 4q^{65} - 2q^{66} - 16q^{67} + 6q^{68} - 2q^{69} - 2q^{70} - 4q^{71} + 2q^{72} + 8q^{73} - 6q^{74} - 2q^{75} + 8q^{76} - 36q^{77} - 4q^{78} - 2q^{79} + 2q^{80} + 2q^{81} + 4q^{82} - 2q^{83} + 2q^{84} + 6q^{85} - 4q^{87} + 2q^{88} - 2q^{89} + 2q^{90} - 4q^{91} + 2q^{92} - 16q^{94} + 8q^{95} - 2q^{96} - 16q^{97} + 22q^{98} + 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 3.12311 1.18042 0.590211 0.807249i \(-0.299044\pi\)
0.590211 + 0.807249i \(0.299044\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −3.12311 −0.941652 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 3.12311 0.834685
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.12311 −0.272393 −0.136197 0.990682i \(-0.543488\pi\)
−0.136197 + 0.990682i \(0.543488\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.12311 −0.681518
\(22\) −3.12311 −0.665848
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 3.12311 0.590211
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.12311 0.543663
\(34\) −1.12311 −0.192611
\(35\) 3.12311 0.527901
\(36\) 1.00000 0.166667
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −3.12311 −0.481906
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −3.12311 −0.470826
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.75379 0.393398
\(50\) 1.00000 0.141421
\(51\) 1.12311 0.157266
\(52\) 2.00000 0.277350
\(53\) 12.2462 1.68215 0.841073 0.540921i \(-0.181924\pi\)
0.841073 + 0.540921i \(0.181924\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.12311 −0.421119
\(56\) 3.12311 0.417343
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) 2.24621 0.292432 0.146216 0.989253i \(-0.453291\pi\)
0.146216 + 0.989253i \(0.453291\pi\)
\(60\) −1.00000 −0.129099
\(61\) 9.12311 1.16809 0.584047 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) 0 0
\(63\) 3.12311 0.393474
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 3.12311 0.384428
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −1.12311 −0.136197
\(69\) −1.00000 −0.120386
\(70\) 3.12311 0.373283
\(71\) −10.2462 −1.21600 −0.608001 0.793936i \(-0.708028\pi\)
−0.608001 + 0.793936i \(0.708028\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.24621 −0.496981 −0.248491 0.968634i \(-0.579935\pi\)
−0.248491 + 0.968634i \(0.579935\pi\)
\(74\) 1.12311 0.130558
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) −9.75379 −1.11155
\(78\) −2.00000 −0.226455
\(79\) 3.12311 0.351377 0.175688 0.984446i \(-0.443785\pi\)
0.175688 + 0.984446i \(0.443785\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −13.3693 −1.46747 −0.733737 0.679434i \(-0.762225\pi\)
−0.733737 + 0.679434i \(0.762225\pi\)
\(84\) −3.12311 −0.340759
\(85\) −1.12311 −0.121818
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) −3.12311 −0.332924
\(89\) −5.12311 −0.543048 −0.271524 0.962432i \(-0.587528\pi\)
−0.271524 + 0.962432i \(0.587528\pi\)
\(90\) 1.00000 0.105409
\(91\) 6.24621 0.654781
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) −16.2462 −1.64955 −0.824776 0.565459i \(-0.808699\pi\)
−0.824776 + 0.565459i \(0.808699\pi\)
\(98\) 2.75379 0.278175
\(99\) −3.12311 −0.313884
\(100\) 1.00000 0.100000
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 1.12311 0.111204
\(103\) −9.36932 −0.923186 −0.461593 0.887092i \(-0.652722\pi\)
−0.461593 + 0.887092i \(0.652722\pi\)
\(104\) 2.00000 0.196116
\(105\) −3.12311 −0.304784
\(106\) 12.2462 1.18946
\(107\) −13.3693 −1.29246 −0.646230 0.763142i \(-0.723655\pi\)
−0.646230 + 0.763142i \(0.723655\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.87689 0.275557 0.137778 0.990463i \(-0.456004\pi\)
0.137778 + 0.990463i \(0.456004\pi\)
\(110\) −3.12311 −0.297776
\(111\) −1.12311 −0.106600
\(112\) 3.12311 0.295106
\(113\) −9.12311 −0.858230 −0.429115 0.903250i \(-0.641174\pi\)
−0.429115 + 0.903250i \(0.641174\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.00000 0.0932505
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 2.24621 0.206781
\(119\) −3.50758 −0.321539
\(120\) −1.00000 −0.0912871
\(121\) −1.24621 −0.113292
\(122\) 9.12311 0.825967
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 3.12311 0.278228
\(127\) −5.75379 −0.510566 −0.255283 0.966866i \(-0.582169\pi\)
−0.255283 + 0.966866i \(0.582169\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 3.12311 0.271831
\(133\) 12.4924 1.08323
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) −1.12311 −0.0963055
\(137\) −10.8769 −0.929276 −0.464638 0.885501i \(-0.653816\pi\)
−0.464638 + 0.885501i \(0.653816\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 16.4924 1.39887 0.699435 0.714697i \(-0.253435\pi\)
0.699435 + 0.714697i \(0.253435\pi\)
\(140\) 3.12311 0.263951
\(141\) 8.00000 0.673722
\(142\) −10.2462 −0.859843
\(143\) −6.24621 −0.522334
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) −4.24621 −0.351419
\(147\) −2.75379 −0.227129
\(148\) 1.12311 0.0923187
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 6.24621 0.508309 0.254155 0.967164i \(-0.418203\pi\)
0.254155 + 0.967164i \(0.418203\pi\)
\(152\) 4.00000 0.324443
\(153\) −1.12311 −0.0907977
\(154\) −9.75379 −0.785983
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 9.12311 0.728103 0.364052 0.931379i \(-0.381393\pi\)
0.364052 + 0.931379i \(0.381393\pi\)
\(158\) 3.12311 0.248461
\(159\) −12.2462 −0.971188
\(160\) 1.00000 0.0790569
\(161\) 3.12311 0.246135
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 2.00000 0.156174
\(165\) 3.12311 0.243133
\(166\) −13.3693 −1.03766
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −3.12311 −0.240953
\(169\) −9.00000 −0.692308
\(170\) −1.12311 −0.0861383
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 20.2462 1.53929 0.769645 0.638471i \(-0.220433\pi\)
0.769645 + 0.638471i \(0.220433\pi\)
\(174\) −2.00000 −0.151620
\(175\) 3.12311 0.236085
\(176\) −3.12311 −0.235413
\(177\) −2.24621 −0.168836
\(178\) −5.12311 −0.383993
\(179\) −8.49242 −0.634753 −0.317377 0.948300i \(-0.602802\pi\)
−0.317377 + 0.948300i \(0.602802\pi\)
\(180\) 1.00000 0.0745356
\(181\) 10.8769 0.808473 0.404237 0.914654i \(-0.367537\pi\)
0.404237 + 0.914654i \(0.367537\pi\)
\(182\) 6.24621 0.463000
\(183\) −9.12311 −0.674399
\(184\) 1.00000 0.0737210
\(185\) 1.12311 0.0825724
\(186\) 0 0
\(187\) 3.50758 0.256499
\(188\) −8.00000 −0.583460
\(189\) −3.12311 −0.227173
\(190\) 4.00000 0.290191
\(191\) 12.4924 0.903920 0.451960 0.892038i \(-0.350725\pi\)
0.451960 + 0.892038i \(0.350725\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.24621 0.593575 0.296788 0.954944i \(-0.404085\pi\)
0.296788 + 0.954944i \(0.404085\pi\)
\(194\) −16.2462 −1.16641
\(195\) −2.00000 −0.143223
\(196\) 2.75379 0.196699
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −3.12311 −0.221949
\(199\) 19.1231 1.35560 0.677801 0.735246i \(-0.262933\pi\)
0.677801 + 0.735246i \(0.262933\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) 0.246211 0.0173234
\(203\) 6.24621 0.438398
\(204\) 1.12311 0.0786331
\(205\) 2.00000 0.139686
\(206\) −9.36932 −0.652791
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) −12.4924 −0.864119
\(210\) −3.12311 −0.215515
\(211\) −16.4924 −1.13539 −0.567693 0.823241i \(-0.692164\pi\)
−0.567693 + 0.823241i \(0.692164\pi\)
\(212\) 12.2462 0.841073
\(213\) 10.2462 0.702059
\(214\) −13.3693 −0.913908
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.87689 0.194848
\(219\) 4.24621 0.286932
\(220\) −3.12311 −0.210560
\(221\) −2.24621 −0.151097
\(222\) −1.12311 −0.0753779
\(223\) −10.2462 −0.686137 −0.343069 0.939310i \(-0.611466\pi\)
−0.343069 + 0.939310i \(0.611466\pi\)
\(224\) 3.12311 0.208671
\(225\) 1.00000 0.0666667
\(226\) −9.12311 −0.606860
\(227\) −5.36932 −0.356374 −0.178187 0.983997i \(-0.557023\pi\)
−0.178187 + 0.983997i \(0.557023\pi\)
\(228\) −4.00000 −0.264906
\(229\) 23.3693 1.54429 0.772144 0.635448i \(-0.219184\pi\)
0.772144 + 0.635448i \(0.219184\pi\)
\(230\) 1.00000 0.0659380
\(231\) 9.75379 0.641752
\(232\) 2.00000 0.131306
\(233\) 28.7386 1.88273 0.941365 0.337389i \(-0.109544\pi\)
0.941365 + 0.337389i \(0.109544\pi\)
\(234\) 2.00000 0.130744
\(235\) −8.00000 −0.521862
\(236\) 2.24621 0.146216
\(237\) −3.12311 −0.202868
\(238\) −3.50758 −0.227362
\(239\) −18.2462 −1.18025 −0.590125 0.807312i \(-0.700921\pi\)
−0.590125 + 0.807312i \(0.700921\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 30.4924 1.96419 0.982095 0.188387i \(-0.0603260\pi\)
0.982095 + 0.188387i \(0.0603260\pi\)
\(242\) −1.24621 −0.0801095
\(243\) −1.00000 −0.0641500
\(244\) 9.12311 0.584047
\(245\) 2.75379 0.175933
\(246\) −2.00000 −0.127515
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 13.3693 0.847246
\(250\) 1.00000 0.0632456
\(251\) 17.3693 1.09634 0.548171 0.836366i \(-0.315324\pi\)
0.548171 + 0.836366i \(0.315324\pi\)
\(252\) 3.12311 0.196737
\(253\) −3.12311 −0.196348
\(254\) −5.75379 −0.361025
\(255\) 1.12311 0.0703316
\(256\) 1.00000 0.0625000
\(257\) 8.24621 0.514385 0.257192 0.966360i \(-0.417203\pi\)
0.257192 + 0.966360i \(0.417203\pi\)
\(258\) 0 0
\(259\) 3.50758 0.217950
\(260\) 2.00000 0.124035
\(261\) 2.00000 0.123797
\(262\) −4.00000 −0.247121
\(263\) −6.24621 −0.385158 −0.192579 0.981281i \(-0.561685\pi\)
−0.192579 + 0.981281i \(0.561685\pi\)
\(264\) 3.12311 0.192214
\(265\) 12.2462 0.752279
\(266\) 12.4924 0.765960
\(267\) 5.12311 0.313529
\(268\) −8.00000 −0.488678
\(269\) 0.246211 0.0150118 0.00750588 0.999972i \(-0.497611\pi\)
0.00750588 + 0.999972i \(0.497611\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −6.24621 −0.379430 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(272\) −1.12311 −0.0680983
\(273\) −6.24621 −0.378038
\(274\) −10.8769 −0.657097
\(275\) −3.12311 −0.188330
\(276\) −1.00000 −0.0601929
\(277\) 3.75379 0.225543 0.112772 0.993621i \(-0.464027\pi\)
0.112772 + 0.993621i \(0.464027\pi\)
\(278\) 16.4924 0.989150
\(279\) 0 0
\(280\) 3.12311 0.186641
\(281\) −21.1231 −1.26010 −0.630049 0.776555i \(-0.716965\pi\)
−0.630049 + 0.776555i \(0.716965\pi\)
\(282\) 8.00000 0.476393
\(283\) 14.2462 0.846849 0.423425 0.905931i \(-0.360828\pi\)
0.423425 + 0.905931i \(0.360828\pi\)
\(284\) −10.2462 −0.608001
\(285\) −4.00000 −0.236940
\(286\) −6.24621 −0.369346
\(287\) 6.24621 0.368702
\(288\) 1.00000 0.0589256
\(289\) −15.7386 −0.925802
\(290\) 2.00000 0.117444
\(291\) 16.2462 0.952370
\(292\) −4.24621 −0.248491
\(293\) 28.2462 1.65016 0.825081 0.565015i \(-0.191130\pi\)
0.825081 + 0.565015i \(0.191130\pi\)
\(294\) −2.75379 −0.160604
\(295\) 2.24621 0.130779
\(296\) 1.12311 0.0652792
\(297\) 3.12311 0.181221
\(298\) −10.0000 −0.579284
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 6.24621 0.359429
\(303\) −0.246211 −0.0141445
\(304\) 4.00000 0.229416
\(305\) 9.12311 0.522388
\(306\) −1.12311 −0.0642037
\(307\) −32.4924 −1.85444 −0.927220 0.374516i \(-0.877809\pi\)
−0.927220 + 0.374516i \(0.877809\pi\)
\(308\) −9.75379 −0.555774
\(309\) 9.36932 0.533002
\(310\) 0 0
\(311\) −22.7386 −1.28939 −0.644695 0.764440i \(-0.723016\pi\)
−0.644695 + 0.764440i \(0.723016\pi\)
\(312\) −2.00000 −0.113228
\(313\) −22.4924 −1.27135 −0.635673 0.771958i \(-0.719278\pi\)
−0.635673 + 0.771958i \(0.719278\pi\)
\(314\) 9.12311 0.514847
\(315\) 3.12311 0.175967
\(316\) 3.12311 0.175688
\(317\) −28.7386 −1.61412 −0.807061 0.590468i \(-0.798943\pi\)
−0.807061 + 0.590468i \(0.798943\pi\)
\(318\) −12.2462 −0.686733
\(319\) −6.24621 −0.349721
\(320\) 1.00000 0.0559017
\(321\) 13.3693 0.746203
\(322\) 3.12311 0.174044
\(323\) −4.49242 −0.249965
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −12.0000 −0.664619
\(327\) −2.87689 −0.159093
\(328\) 2.00000 0.110432
\(329\) −24.9848 −1.37746
\(330\) 3.12311 0.171921
\(331\) 0.492423 0.0270660 0.0135330 0.999908i \(-0.495692\pi\)
0.0135330 + 0.999908i \(0.495692\pi\)
\(332\) −13.3693 −0.733737
\(333\) 1.12311 0.0615458
\(334\) −8.00000 −0.437741
\(335\) −8.00000 −0.437087
\(336\) −3.12311 −0.170379
\(337\) 26.4924 1.44313 0.721567 0.692345i \(-0.243422\pi\)
0.721567 + 0.692345i \(0.243422\pi\)
\(338\) −9.00000 −0.489535
\(339\) 9.12311 0.495499
\(340\) −1.12311 −0.0609090
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −13.2614 −0.716046
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 20.2462 1.08844
\(347\) 14.7386 0.791211 0.395606 0.918420i \(-0.370535\pi\)
0.395606 + 0.918420i \(0.370535\pi\)
\(348\) −2.00000 −0.107211
\(349\) 24.7386 1.32423 0.662114 0.749403i \(-0.269659\pi\)
0.662114 + 0.749403i \(0.269659\pi\)
\(350\) 3.12311 0.166937
\(351\) −2.00000 −0.106752
\(352\) −3.12311 −0.166462
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) −2.24621 −0.119385
\(355\) −10.2462 −0.543812
\(356\) −5.12311 −0.271524
\(357\) 3.50758 0.185641
\(358\) −8.49242 −0.448838
\(359\) 20.4924 1.08155 0.540774 0.841168i \(-0.318131\pi\)
0.540774 + 0.841168i \(0.318131\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 10.8769 0.571677
\(363\) 1.24621 0.0654091
\(364\) 6.24621 0.327390
\(365\) −4.24621 −0.222257
\(366\) −9.12311 −0.476872
\(367\) 4.87689 0.254572 0.127286 0.991866i \(-0.459373\pi\)
0.127286 + 0.991866i \(0.459373\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.00000 0.104116
\(370\) 1.12311 0.0583875
\(371\) 38.2462 1.98564
\(372\) 0 0
\(373\) −11.3693 −0.588681 −0.294340 0.955701i \(-0.595100\pi\)
−0.294340 + 0.955701i \(0.595100\pi\)
\(374\) 3.50758 0.181373
\(375\) −1.00000 −0.0516398
\(376\) −8.00000 −0.412568
\(377\) 4.00000 0.206010
\(378\) −3.12311 −0.160635
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 4.00000 0.205196
\(381\) 5.75379 0.294776
\(382\) 12.4924 0.639168
\(383\) 30.2462 1.54551 0.772755 0.634705i \(-0.218878\pi\)
0.772755 + 0.634705i \(0.218878\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −9.75379 −0.497099
\(386\) 8.24621 0.419721
\(387\) 0 0
\(388\) −16.2462 −0.824776
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −2.00000 −0.101274
\(391\) −1.12311 −0.0567979
\(392\) 2.75379 0.139087
\(393\) 4.00000 0.201773
\(394\) −10.0000 −0.503793
\(395\) 3.12311 0.157140
\(396\) −3.12311 −0.156942
\(397\) −1.50758 −0.0756631 −0.0378316 0.999284i \(-0.512045\pi\)
−0.0378316 + 0.999284i \(0.512045\pi\)
\(398\) 19.1231 0.958555
\(399\) −12.4924 −0.625403
\(400\) 1.00000 0.0500000
\(401\) 7.36932 0.368006 0.184003 0.982926i \(-0.441094\pi\)
0.184003 + 0.982926i \(0.441094\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) 0.246211 0.0122495
\(405\) 1.00000 0.0496904
\(406\) 6.24621 0.309994
\(407\) −3.50758 −0.173864
\(408\) 1.12311 0.0556020
\(409\) 0.246211 0.0121744 0.00608718 0.999981i \(-0.498062\pi\)
0.00608718 + 0.999981i \(0.498062\pi\)
\(410\) 2.00000 0.0987730
\(411\) 10.8769 0.536518
\(412\) −9.36932 −0.461593
\(413\) 7.01515 0.345193
\(414\) 1.00000 0.0491473
\(415\) −13.3693 −0.656274
\(416\) 2.00000 0.0980581
\(417\) −16.4924 −0.807637
\(418\) −12.4924 −0.611024
\(419\) 39.6155 1.93535 0.967673 0.252210i \(-0.0811573\pi\)
0.967673 + 0.252210i \(0.0811573\pi\)
\(420\) −3.12311 −0.152392
\(421\) −11.3693 −0.554107 −0.277053 0.960855i \(-0.589358\pi\)
−0.277053 + 0.960855i \(0.589358\pi\)
\(422\) −16.4924 −0.802839
\(423\) −8.00000 −0.388973
\(424\) 12.2462 0.594729
\(425\) −1.12311 −0.0544786
\(426\) 10.2462 0.496431
\(427\) 28.4924 1.37884
\(428\) −13.3693 −0.646230
\(429\) 6.24621 0.301570
\(430\) 0 0
\(431\) 9.75379 0.469823 0.234912 0.972017i \(-0.424520\pi\)
0.234912 + 0.972017i \(0.424520\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −22.4924 −1.08092 −0.540458 0.841371i \(-0.681749\pi\)
−0.540458 + 0.841371i \(0.681749\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 2.87689 0.137778
\(437\) 4.00000 0.191346
\(438\) 4.24621 0.202892
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −3.12311 −0.148888
\(441\) 2.75379 0.131133
\(442\) −2.24621 −0.106841
\(443\) 8.49242 0.403487 0.201744 0.979438i \(-0.435339\pi\)
0.201744 + 0.979438i \(0.435339\pi\)
\(444\) −1.12311 −0.0533002
\(445\) −5.12311 −0.242858
\(446\) −10.2462 −0.485172
\(447\) 10.0000 0.472984
\(448\) 3.12311 0.147553
\(449\) 22.4924 1.06148 0.530742 0.847534i \(-0.321913\pi\)
0.530742 + 0.847534i \(0.321913\pi\)
\(450\) 1.00000 0.0471405
\(451\) −6.24621 −0.294123
\(452\) −9.12311 −0.429115
\(453\) −6.24621 −0.293473
\(454\) −5.36932 −0.251995
\(455\) 6.24621 0.292827
\(456\) −4.00000 −0.187317
\(457\) 16.7386 0.783000 0.391500 0.920178i \(-0.371956\pi\)
0.391500 + 0.920178i \(0.371956\pi\)
\(458\) 23.3693 1.09198
\(459\) 1.12311 0.0524221
\(460\) 1.00000 0.0466252
\(461\) −16.7386 −0.779596 −0.389798 0.920900i \(-0.627455\pi\)
−0.389798 + 0.920900i \(0.627455\pi\)
\(462\) 9.75379 0.453787
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 28.7386 1.33129
\(467\) 21.3693 0.988854 0.494427 0.869219i \(-0.335378\pi\)
0.494427 + 0.869219i \(0.335378\pi\)
\(468\) 2.00000 0.0924500
\(469\) −24.9848 −1.15369
\(470\) −8.00000 −0.369012
\(471\) −9.12311 −0.420371
\(472\) 2.24621 0.103390
\(473\) 0 0
\(474\) −3.12311 −0.143449
\(475\) 4.00000 0.183533
\(476\) −3.50758 −0.160770
\(477\) 12.2462 0.560715
\(478\) −18.2462 −0.834562
\(479\) 12.4924 0.570793 0.285397 0.958409i \(-0.407875\pi\)
0.285397 + 0.958409i \(0.407875\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 2.24621 0.102418
\(482\) 30.4924 1.38889
\(483\) −3.12311 −0.142106
\(484\) −1.24621 −0.0566460
\(485\) −16.2462 −0.737702
\(486\) −1.00000 −0.0453609
\(487\) 7.50758 0.340201 0.170100 0.985427i \(-0.445591\pi\)
0.170100 + 0.985427i \(0.445591\pi\)
\(488\) 9.12311 0.412984
\(489\) 12.0000 0.542659
\(490\) 2.75379 0.124403
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −2.24621 −0.101164
\(494\) 8.00000 0.359937
\(495\) −3.12311 −0.140373
\(496\) 0 0
\(497\) −32.0000 −1.43540
\(498\) 13.3693 0.599093
\(499\) 36.9848 1.65567 0.827835 0.560972i \(-0.189573\pi\)
0.827835 + 0.560972i \(0.189573\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) 17.3693 0.775231
\(503\) 42.7386 1.90562 0.952811 0.303565i \(-0.0981769\pi\)
0.952811 + 0.303565i \(0.0981769\pi\)
\(504\) 3.12311 0.139114
\(505\) 0.246211 0.0109563
\(506\) −3.12311 −0.138839
\(507\) 9.00000 0.399704
\(508\) −5.75379 −0.255283
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 1.12311 0.0497320
\(511\) −13.2614 −0.586648
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 8.24621 0.363725
\(515\) −9.36932 −0.412861
\(516\) 0 0
\(517\) 24.9848 1.09883
\(518\) 3.50758 0.154114
\(519\) −20.2462 −0.888710
\(520\) 2.00000 0.0877058
\(521\) 17.1231 0.750177 0.375088 0.926989i \(-0.377612\pi\)
0.375088 + 0.926989i \(0.377612\pi\)
\(522\) 2.00000 0.0875376
\(523\) 4.49242 0.196440 0.0982200 0.995165i \(-0.468685\pi\)
0.0982200 + 0.995165i \(0.468685\pi\)
\(524\) −4.00000 −0.174741
\(525\) −3.12311 −0.136304
\(526\) −6.24621 −0.272348
\(527\) 0 0
\(528\) 3.12311 0.135916
\(529\) 1.00000 0.0434783
\(530\) 12.2462 0.531941
\(531\) 2.24621 0.0974773
\(532\) 12.4924 0.541615
\(533\) 4.00000 0.173259
\(534\) 5.12311 0.221698
\(535\) −13.3693 −0.578006
\(536\) −8.00000 −0.345547
\(537\) 8.49242 0.366475
\(538\) 0.246211 0.0106149
\(539\) −8.60037 −0.370444
\(540\) −1.00000 −0.0430331
\(541\) −32.2462 −1.38637 −0.693186 0.720758i \(-0.743794\pi\)
−0.693186 + 0.720758i \(0.743794\pi\)
\(542\) −6.24621 −0.268298
\(543\) −10.8769 −0.466772
\(544\) −1.12311 −0.0481528
\(545\) 2.87689 0.123233
\(546\) −6.24621 −0.267313
\(547\) −32.4924 −1.38928 −0.694638 0.719360i \(-0.744435\pi\)
−0.694638 + 0.719360i \(0.744435\pi\)
\(548\) −10.8769 −0.464638
\(549\) 9.12311 0.389365
\(550\) −3.12311 −0.133170
\(551\) 8.00000 0.340811
\(552\) −1.00000 −0.0425628
\(553\) 9.75379 0.414773
\(554\) 3.75379 0.159483
\(555\) −1.12311 −0.0476732
\(556\) 16.4924 0.699435
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3.12311 0.131975
\(561\) −3.50758 −0.148090
\(562\) −21.1231 −0.891024
\(563\) −13.3693 −0.563450 −0.281725 0.959495i \(-0.590907\pi\)
−0.281725 + 0.959495i \(0.590907\pi\)
\(564\) 8.00000 0.336861
\(565\) −9.12311 −0.383812
\(566\) 14.2462 0.598813
\(567\) 3.12311 0.131158
\(568\) −10.2462 −0.429921
\(569\) 18.8769 0.791361 0.395680 0.918388i \(-0.370509\pi\)
0.395680 + 0.918388i \(0.370509\pi\)
\(570\) −4.00000 −0.167542
\(571\) −14.7386 −0.616793 −0.308396 0.951258i \(-0.599792\pi\)
−0.308396 + 0.951258i \(0.599792\pi\)
\(572\) −6.24621 −0.261167
\(573\) −12.4924 −0.521878
\(574\) 6.24621 0.260712
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −18.4924 −0.769850 −0.384925 0.922948i \(-0.625773\pi\)
−0.384925 + 0.922948i \(0.625773\pi\)
\(578\) −15.7386 −0.654641
\(579\) −8.24621 −0.342701
\(580\) 2.00000 0.0830455
\(581\) −41.7538 −1.73224
\(582\) 16.2462 0.673427
\(583\) −38.2462 −1.58400
\(584\) −4.24621 −0.175709
\(585\) 2.00000 0.0826898
\(586\) 28.2462 1.16684
\(587\) −26.2462 −1.08330 −0.541649 0.840605i \(-0.682200\pi\)
−0.541649 + 0.840605i \(0.682200\pi\)
\(588\) −2.75379 −0.113564
\(589\) 0 0
\(590\) 2.24621 0.0924751
\(591\) 10.0000 0.411345
\(592\) 1.12311 0.0461594
\(593\) 24.2462 0.995673 0.497836 0.867271i \(-0.334128\pi\)
0.497836 + 0.867271i \(0.334128\pi\)
\(594\) 3.12311 0.128143
\(595\) −3.50758 −0.143797
\(596\) −10.0000 −0.409616
\(597\) −19.1231 −0.782657
\(598\) 2.00000 0.0817861
\(599\) 10.2462 0.418649 0.209324 0.977846i \(-0.432874\pi\)
0.209324 + 0.977846i \(0.432874\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 6.24621 0.254155
\(605\) −1.24621 −0.0506657
\(606\) −0.246211 −0.0100016
\(607\) −22.7386 −0.922933 −0.461466 0.887158i \(-0.652676\pi\)
−0.461466 + 0.887158i \(0.652676\pi\)
\(608\) 4.00000 0.162221
\(609\) −6.24621 −0.253109
\(610\) 9.12311 0.369384
\(611\) −16.0000 −0.647291
\(612\) −1.12311 −0.0453989
\(613\) 34.8769 1.40866 0.704332 0.709870i \(-0.251247\pi\)
0.704332 + 0.709870i \(0.251247\pi\)
\(614\) −32.4924 −1.31129
\(615\) −2.00000 −0.0806478
\(616\) −9.75379 −0.392991
\(617\) −34.1080 −1.37313 −0.686567 0.727066i \(-0.740883\pi\)
−0.686567 + 0.727066i \(0.740883\pi\)
\(618\) 9.36932 0.376889
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −22.7386 −0.911736
\(623\) −16.0000 −0.641026
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −22.4924 −0.898978
\(627\) 12.4924 0.498899
\(628\) 9.12311 0.364052
\(629\) −1.26137 −0.0502940
\(630\) 3.12311 0.124428
\(631\) −14.6307 −0.582438 −0.291219 0.956656i \(-0.594061\pi\)
−0.291219 + 0.956656i \(0.594061\pi\)
\(632\) 3.12311 0.124230
\(633\) 16.4924 0.655515
\(634\) −28.7386 −1.14136
\(635\) −5.75379 −0.228332
\(636\) −12.2462 −0.485594
\(637\) 5.50758 0.218218
\(638\) −6.24621 −0.247290
\(639\) −10.2462 −0.405334
\(640\) 1.00000 0.0395285
\(641\) 15.3693 0.607052 0.303526 0.952823i \(-0.401836\pi\)
0.303526 + 0.952823i \(0.401836\pi\)
\(642\) 13.3693 0.527645
\(643\) 3.50758 0.138325 0.0691627 0.997605i \(-0.477967\pi\)
0.0691627 + 0.997605i \(0.477967\pi\)
\(644\) 3.12311 0.123068
\(645\) 0 0
\(646\) −4.49242 −0.176752
\(647\) 44.4924 1.74918 0.874589 0.484865i \(-0.161131\pi\)
0.874589 + 0.484865i \(0.161131\pi\)
\(648\) 1.00000 0.0392837
\(649\) −7.01515 −0.275369
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −22.4924 −0.880197 −0.440098 0.897950i \(-0.645056\pi\)
−0.440098 + 0.897950i \(0.645056\pi\)
\(654\) −2.87689 −0.112495
\(655\) −4.00000 −0.156293
\(656\) 2.00000 0.0780869
\(657\) −4.24621 −0.165660
\(658\) −24.9848 −0.974011
\(659\) 17.3693 0.676613 0.338306 0.941036i \(-0.390146\pi\)
0.338306 + 0.941036i \(0.390146\pi\)
\(660\) 3.12311 0.121567
\(661\) −23.8617 −0.928114 −0.464057 0.885805i \(-0.653607\pi\)
−0.464057 + 0.885805i \(0.653607\pi\)
\(662\) 0.492423 0.0191385
\(663\) 2.24621 0.0872356
\(664\) −13.3693 −0.518830
\(665\) 12.4924 0.484435
\(666\) 1.12311 0.0435195
\(667\) 2.00000 0.0774403
\(668\) −8.00000 −0.309529
\(669\) 10.2462 0.396141
\(670\) −8.00000 −0.309067
\(671\) −28.4924 −1.09994
\(672\) −3.12311 −0.120476
\(673\) 18.9848 0.731812 0.365906 0.930652i \(-0.380759\pi\)
0.365906 + 0.930652i \(0.380759\pi\)
\(674\) 26.4924 1.02045
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −13.5076 −0.519138 −0.259569 0.965725i \(-0.583581\pi\)
−0.259569 + 0.965725i \(0.583581\pi\)
\(678\) 9.12311 0.350371
\(679\) −50.7386 −1.94717
\(680\) −1.12311 −0.0430691
\(681\) 5.36932 0.205753
\(682\) 0 0
\(683\) 34.2462 1.31039 0.655197 0.755458i \(-0.272585\pi\)
0.655197 + 0.755458i \(0.272585\pi\)
\(684\) 4.00000 0.152944
\(685\) −10.8769 −0.415585
\(686\) −13.2614 −0.506321
\(687\) −23.3693 −0.891595
\(688\) 0 0
\(689\) 24.4924 0.933087
\(690\) −1.00000 −0.0380693
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 20.2462 0.769645
\(693\) −9.75379 −0.370516
\(694\) 14.7386 0.559471
\(695\) 16.4924 0.625593
\(696\) −2.00000 −0.0758098
\(697\) −2.24621 −0.0850813
\(698\) 24.7386 0.936371
\(699\) −28.7386 −1.08699
\(700\) 3.12311 0.118042
\(701\) −19.7538 −0.746090 −0.373045 0.927813i \(-0.621686\pi\)
−0.373045 + 0.927813i \(0.621686\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 4.49242 0.169435
\(704\) −3.12311 −0.117706
\(705\) 8.00000 0.301297
\(706\) −14.0000 −0.526897
\(707\) 0.768944 0.0289191
\(708\) −2.24621 −0.0844178
\(709\) −11.3693 −0.426984 −0.213492 0.976945i \(-0.568484\pi\)
−0.213492 + 0.976945i \(0.568484\pi\)
\(710\) −10.2462 −0.384533
\(711\) 3.12311 0.117126
\(712\) −5.12311 −0.191997
\(713\) 0 0
\(714\) 3.50758 0.131268
\(715\) −6.24621 −0.233595
\(716\) −8.49242 −0.317377
\(717\) 18.2462 0.681417
\(718\) 20.4924 0.764770
\(719\) −26.2462 −0.978819 −0.489409 0.872054i \(-0.662788\pi\)
−0.489409 + 0.872054i \(0.662788\pi\)
\(720\) 1.00000 0.0372678
\(721\) −29.2614 −1.08975
\(722\) −3.00000 −0.111648
\(723\) −30.4924 −1.13403
\(724\) 10.8769 0.404237
\(725\) 2.00000 0.0742781
\(726\) 1.24621 0.0462512
\(727\) −29.8617 −1.10751 −0.553755 0.832679i \(-0.686806\pi\)
−0.553755 + 0.832679i \(0.686806\pi\)
\(728\) 6.24621 0.231500
\(729\) 1.00000 0.0370370
\(730\) −4.24621 −0.157159
\(731\) 0 0
\(732\) −9.12311 −0.337200
\(733\) 43.8617 1.62007 0.810035 0.586381i \(-0.199448\pi\)
0.810035 + 0.586381i \(0.199448\pi\)
\(734\) 4.87689 0.180009
\(735\) −2.75379 −0.101575
\(736\) 1.00000 0.0368605
\(737\) 24.9848 0.920329
\(738\) 2.00000 0.0736210
\(739\) 7.50758 0.276171 0.138085 0.990420i \(-0.455905\pi\)
0.138085 + 0.990420i \(0.455905\pi\)
\(740\) 1.12311 0.0412862
\(741\) −8.00000 −0.293887
\(742\) 38.2462 1.40406
\(743\) 34.7386 1.27444 0.637218 0.770683i \(-0.280085\pi\)
0.637218 + 0.770683i \(0.280085\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −11.3693 −0.416260
\(747\) −13.3693 −0.489158
\(748\) 3.50758 0.128250
\(749\) −41.7538 −1.52565
\(750\) −1.00000 −0.0365148
\(751\) −13.8617 −0.505822 −0.252911 0.967490i \(-0.581388\pi\)
−0.252911 + 0.967490i \(0.581388\pi\)
\(752\) −8.00000 −0.291730
\(753\) −17.3693 −0.632973
\(754\) 4.00000 0.145671
\(755\) 6.24621 0.227323
\(756\) −3.12311 −0.113586
\(757\) −47.8617 −1.73956 −0.869782 0.493436i \(-0.835741\pi\)
−0.869782 + 0.493436i \(0.835741\pi\)
\(758\) 4.00000 0.145287
\(759\) 3.12311 0.113362
\(760\) 4.00000 0.145095
\(761\) −31.7538 −1.15107 −0.575537 0.817776i \(-0.695207\pi\)
−0.575537 + 0.817776i \(0.695207\pi\)
\(762\) 5.75379 0.208438
\(763\) 8.98485 0.325273
\(764\) 12.4924 0.451960
\(765\) −1.12311 −0.0406060
\(766\) 30.2462 1.09284
\(767\) 4.49242 0.162212
\(768\) −1.00000 −0.0360844
\(769\) 28.7386 1.03634 0.518171 0.855277i \(-0.326613\pi\)
0.518171 + 0.855277i \(0.326613\pi\)
\(770\) −9.75379 −0.351502
\(771\) −8.24621 −0.296980
\(772\) 8.24621 0.296788
\(773\) −19.7538 −0.710494 −0.355247 0.934772i \(-0.615603\pi\)
−0.355247 + 0.934772i \(0.615603\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −16.2462 −0.583205
\(777\) −3.50758 −0.125834
\(778\) −18.0000 −0.645331
\(779\) 8.00000 0.286630
\(780\) −2.00000 −0.0716115
\(781\) 32.0000 1.14505
\(782\) −1.12311 −0.0401622
\(783\) −2.00000 −0.0714742
\(784\) 2.75379 0.0983496
\(785\) 9.12311 0.325618
\(786\) 4.00000 0.142675
\(787\) 48.0000 1.71102 0.855508 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) −10.0000 −0.356235
\(789\) 6.24621 0.222371
\(790\) 3.12311 0.111115
\(791\) −28.4924 −1.01307
\(792\) −3.12311 −0.110975
\(793\) 18.2462 0.647942
\(794\) −1.50758 −0.0535019
\(795\) −12.2462 −0.434328
\(796\) 19.1231 0.677801
\(797\) 10.4924 0.371661 0.185830 0.982582i \(-0.440503\pi\)
0.185830 + 0.982582i \(0.440503\pi\)
\(798\) −12.4924 −0.442227
\(799\) 8.98485 0.317861
\(800\) 1.00000 0.0353553
\(801\) −5.12311 −0.181016
\(802\) 7.36932 0.260220
\(803\) 13.2614 0.467983
\(804\) 8.00000 0.282138
\(805\) 3.12311 0.110075
\(806\) 0 0
\(807\) −0.246211 −0.00866705
\(808\) 0.246211 0.00866168
\(809\) −12.2462 −0.430554 −0.215277 0.976553i \(-0.569065\pi\)
−0.215277 + 0.976553i \(0.569065\pi\)
\(810\) 1.00000 0.0351364
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 6.24621 0.219199
\(813\) 6.24621 0.219064
\(814\) −3.50758 −0.122941
\(815\) −12.0000 −0.420342
\(816\) 1.12311 0.0393166
\(817\) 0 0
\(818\) 0.246211 0.00860857
\(819\) 6.24621 0.218260
\(820\) 2.00000 0.0698430
\(821\) −36.2462 −1.26500 −0.632501 0.774560i \(-0.717971\pi\)
−0.632501 + 0.774560i \(0.717971\pi\)
\(822\) 10.8769 0.379375
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −9.36932 −0.326396
\(825\) 3.12311 0.108733
\(826\) 7.01515 0.244088
\(827\) 8.87689 0.308680 0.154340 0.988018i \(-0.450675\pi\)
0.154340 + 0.988018i \(0.450675\pi\)
\(828\) 1.00000 0.0347524
\(829\) 15.7538 0.547152 0.273576 0.961850i \(-0.411794\pi\)
0.273576 + 0.961850i \(0.411794\pi\)
\(830\) −13.3693 −0.464056
\(831\) −3.75379 −0.130217
\(832\) 2.00000 0.0693375
\(833\) −3.09280 −0.107159
\(834\) −16.4924 −0.571086
\(835\) −8.00000 −0.276851
\(836\) −12.4924 −0.432059
\(837\) 0 0
\(838\) 39.6155 1.36850
\(839\) −9.75379 −0.336738 −0.168369 0.985724i \(-0.553850\pi\)
−0.168369 + 0.985724i \(0.553850\pi\)
\(840\) −3.12311 −0.107757
\(841\) −25.0000 −0.862069
\(842\) −11.3693 −0.391813
\(843\) 21.1231 0.727518
\(844\) −16.4924 −0.567693
\(845\) −9.00000 −0.309609
\(846\) −8.00000 −0.275046
\(847\) −3.89205 −0.133732
\(848\) 12.2462 0.420537
\(849\) −14.2462 −0.488929
\(850\) −1.12311 −0.0385222
\(851\) 1.12311 0.0384996
\(852\) 10.2462 0.351029
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 28.4924 0.974991
\(855\) 4.00000 0.136797
\(856\) −13.3693 −0.456954
\(857\) −24.7386 −0.845056 −0.422528 0.906350i \(-0.638857\pi\)
−0.422528 + 0.906350i \(0.638857\pi\)
\(858\) 6.24621 0.213242
\(859\) 15.5076 0.529112 0.264556 0.964370i \(-0.414775\pi\)
0.264556 + 0.964370i \(0.414775\pi\)
\(860\) 0 0
\(861\) −6.24621 −0.212870
\(862\) 9.75379 0.332215
\(863\) −16.9848 −0.578171 −0.289085 0.957303i \(-0.593351\pi\)
−0.289085 + 0.957303i \(0.593351\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 20.2462 0.688392
\(866\) −22.4924 −0.764324
\(867\) 15.7386 0.534512
\(868\) 0 0
\(869\) −9.75379 −0.330875
\(870\) −2.00000 −0.0678064
\(871\) −16.0000 −0.542139
\(872\) 2.87689 0.0974239
\(873\) −16.2462 −0.549851
\(874\) 4.00000 0.135302
\(875\) 3.12311 0.105580
\(876\) 4.24621 0.143466
\(877\) 52.7386 1.78086 0.890429 0.455123i \(-0.150405\pi\)
0.890429 + 0.455123i \(0.150405\pi\)
\(878\) 0 0
\(879\) −28.2462 −0.952721
\(880\) −3.12311 −0.105280
\(881\) −9.61553 −0.323955 −0.161978 0.986794i \(-0.551787\pi\)
−0.161978 + 0.986794i \(0.551787\pi\)
\(882\) 2.75379 0.0927249
\(883\) 44.9848 1.51386 0.756930 0.653496i \(-0.226698\pi\)
0.756930 + 0.653496i \(0.226698\pi\)
\(884\) −2.24621 −0.0755483
\(885\) −2.24621 −0.0755056
\(886\) 8.49242 0.285309
\(887\) 19.5076 0.655000 0.327500 0.944851i \(-0.393794\pi\)
0.327500 + 0.944851i \(0.393794\pi\)
\(888\) −1.12311 −0.0376890
\(889\) −17.9697 −0.602684
\(890\) −5.12311 −0.171727
\(891\) −3.12311 −0.104628
\(892\) −10.2462 −0.343069
\(893\) −32.0000 −1.07084
\(894\) 10.0000 0.334450
\(895\) −8.49242 −0.283870
\(896\) 3.12311 0.104336
\(897\) −2.00000 −0.0667781
\(898\) 22.4924 0.750582
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −13.7538 −0.458205
\(902\) −6.24621 −0.207976
\(903\) 0 0
\(904\) −9.12311 −0.303430
\(905\) 10.8769 0.361560
\(906\) −6.24621 −0.207516
\(907\) 54.2462 1.80122 0.900608 0.434632i \(-0.143122\pi\)
0.900608 + 0.434632i \(0.143122\pi\)
\(908\) −5.36932 −0.178187
\(909\) 0.246211 0.00816631
\(910\) 6.24621 0.207060
\(911\) −30.2462 −1.00210 −0.501051 0.865418i \(-0.667053\pi\)
−0.501051 + 0.865418i \(0.667053\pi\)
\(912\) −4.00000 −0.132453
\(913\) 41.7538 1.38185
\(914\) 16.7386 0.553665
\(915\) −9.12311 −0.301601
\(916\) 23.3693 0.772144
\(917\) −12.4924 −0.412536
\(918\) 1.12311 0.0370680
\(919\) 19.1231 0.630813 0.315407 0.948957i \(-0.397859\pi\)
0.315407 + 0.948957i \(0.397859\pi\)
\(920\) 1.00000 0.0329690
\(921\) 32.4924 1.07066
\(922\) −16.7386 −0.551258
\(923\) −20.4924 −0.674516
\(924\) 9.75379 0.320876
\(925\) 1.12311 0.0369275
\(926\) −36.0000 −1.18303
\(927\) −9.36932 −0.307729
\(928\) 2.00000 0.0656532
\(929\) 34.9848 1.14782 0.573908 0.818920i \(-0.305427\pi\)
0.573908 + 0.818920i \(0.305427\pi\)
\(930\) 0 0
\(931\) 11.0152 0.361007
\(932\) 28.7386 0.941365
\(933\) 22.7386 0.744429
\(934\) 21.3693 0.699225
\(935\) 3.50758 0.114710
\(936\) 2.00000 0.0653720
\(937\) 32.7386 1.06952 0.534762 0.845003i \(-0.320401\pi\)
0.534762 + 0.845003i \(0.320401\pi\)
\(938\) −24.9848 −0.815784
\(939\) 22.4924 0.734012
\(940\) −8.00000 −0.260931
\(941\) 26.4924 0.863628 0.431814 0.901963i \(-0.357874\pi\)
0.431814 + 0.901963i \(0.357874\pi\)
\(942\) −9.12311 −0.297247
\(943\) 2.00000 0.0651290
\(944\) 2.24621 0.0731079
\(945\) −3.12311 −0.101595
\(946\) 0 0
\(947\) 5.75379 0.186973 0.0934865 0.995621i \(-0.470199\pi\)
0.0934865 + 0.995621i \(0.470199\pi\)
\(948\) −3.12311 −0.101434
\(949\) −8.49242 −0.275676
\(950\) 4.00000 0.129777
\(951\) 28.7386 0.931914
\(952\) −3.50758 −0.113681
\(953\) 23.8617 0.772958 0.386479 0.922298i \(-0.373691\pi\)
0.386479 + 0.922298i \(0.373691\pi\)
\(954\) 12.2462 0.396486
\(955\) 12.4924 0.404245
\(956\) −18.2462 −0.590125
\(957\) 6.24621 0.201911
\(958\) 12.4924 0.403612
\(959\) −33.9697 −1.09694
\(960\) −1.00000 −0.0322749
\(961\) −31.0000 −1.00000
\(962\) 2.24621 0.0724208
\(963\) −13.3693 −0.430820
\(964\) 30.4924 0.982095
\(965\) 8.24621 0.265455
\(966\) −3.12311 −0.100484
\(967\) 60.9848 1.96114 0.980570 0.196168i \(-0.0628500\pi\)
0.980570 + 0.196168i \(0.0628500\pi\)
\(968\) −1.24621 −0.0400547
\(969\) 4.49242 0.144317
\(970\) −16.2462 −0.521634
\(971\) 43.1231 1.38389 0.691943 0.721952i \(-0.256755\pi\)
0.691943 + 0.721952i \(0.256755\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 51.5076 1.65126
\(974\) 7.50758 0.240558
\(975\) −2.00000 −0.0640513
\(976\) 9.12311 0.292023
\(977\) −35.8617 −1.14732 −0.573659 0.819094i \(-0.694477\pi\)
−0.573659 + 0.819094i \(0.694477\pi\)
\(978\) 12.0000 0.383718
\(979\) 16.0000 0.511362
\(980\) 2.75379 0.0879666
\(981\) 2.87689 0.0918522
\(982\) −12.0000 −0.382935
\(983\) −9.75379 −0.311098 −0.155549 0.987828i \(-0.549715\pi\)
−0.155549 + 0.987828i \(0.549715\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −10.0000 −0.318626
\(986\) −2.24621 −0.0715339
\(987\) 24.9848 0.795276
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) −3.12311 −0.0992588
\(991\) 61.4773 1.95289 0.976445 0.215767i \(-0.0692252\pi\)
0.976445 + 0.215767i \(0.0692252\pi\)
\(992\) 0 0
\(993\) −0.492423 −0.0156266
\(994\) −32.0000 −1.01498
\(995\) 19.1231 0.606243
\(996\) 13.3693 0.423623
\(997\) 3.75379 0.118884 0.0594418 0.998232i \(-0.481068\pi\)
0.0594418 + 0.998232i \(0.481068\pi\)
\(998\) 36.9848 1.17073
\(999\) −1.12311 −0.0355335
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 690.2.a.l.1.2 2
3.2 odd 2 2070.2.a.t.1.2 2
4.3 odd 2 5520.2.a.bs.1.1 2
5.2 odd 4 3450.2.d.v.2899.4 4
5.3 odd 4 3450.2.d.v.2899.1 4
5.4 even 2 3450.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.l.1.2 2 1.1 even 1 trivial
2070.2.a.t.1.2 2 3.2 odd 2
3450.2.a.bi.1.1 2 5.4 even 2
3450.2.d.v.2899.1 4 5.3 odd 4
3450.2.d.v.2899.4 4 5.2 odd 4
5520.2.a.bs.1.1 2 4.3 odd 2