Properties

Label 8470.2.a.cy.1.6
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(6\)
Coefficient field: 6.6.13298000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.6
Root \(-0.770059\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.25498 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.25498 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.59492 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.25498 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.25498 q^{6} -1.00000 q^{7} -1.00000 q^{8} +7.59492 q^{9} -1.00000 q^{10} +3.25498 q^{12} -3.57888 q^{13} +1.00000 q^{14} +3.25498 q^{15} +1.00000 q^{16} -3.01604 q^{17} -7.59492 q^{18} -2.86689 q^{19} +1.00000 q^{20} -3.25498 q^{21} -9.00193 q^{23} -3.25498 q^{24} +1.00000 q^{25} +3.57888 q^{26} +14.9564 q^{27} -1.00000 q^{28} -1.56460 q^{29} -3.25498 q^{30} -0.473234 q^{31} -1.00000 q^{32} +3.01604 q^{34} -1.00000 q^{35} +7.59492 q^{36} -10.8935 q^{37} +2.86689 q^{38} -11.6492 q^{39} -1.00000 q^{40} -3.84241 q^{41} +3.25498 q^{42} -2.25195 q^{43} +7.59492 q^{45} +9.00193 q^{46} -9.61359 q^{47} +3.25498 q^{48} +1.00000 q^{49} -1.00000 q^{50} -9.81716 q^{51} -3.57888 q^{52} -5.62157 q^{53} -14.9564 q^{54} +1.00000 q^{56} -9.33169 q^{57} +1.56460 q^{58} -2.73647 q^{59} +3.25498 q^{60} +11.8983 q^{61} +0.473234 q^{62} -7.59492 q^{63} +1.00000 q^{64} -3.57888 q^{65} +0.752899 q^{67} -3.01604 q^{68} -29.3011 q^{69} +1.00000 q^{70} -1.90401 q^{71} -7.59492 q^{72} -1.66740 q^{73} +10.8935 q^{74} +3.25498 q^{75} -2.86689 q^{76} +11.6492 q^{78} +14.5396 q^{79} +1.00000 q^{80} +25.8981 q^{81} +3.84241 q^{82} +5.31576 q^{83} -3.25498 q^{84} -3.01604 q^{85} +2.25195 q^{86} -5.09274 q^{87} -11.1627 q^{89} -7.59492 q^{90} +3.57888 q^{91} -9.00193 q^{92} -1.54037 q^{93} +9.61359 q^{94} -2.86689 q^{95} -3.25498 q^{96} -1.05562 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} - 6q^{10} + q^{12} - 2q^{13} + 6q^{14} + q^{15} + 6q^{16} - 7q^{17} - 15q^{18} - 11q^{19} + 6q^{20} - q^{21} - 6q^{23} - q^{24} + 6q^{25} + 2q^{26} + 4q^{27} - 6q^{28} - 2q^{29} - q^{30} - 6q^{32} + 7q^{34} - 6q^{35} + 15q^{36} - 14q^{37} + 11q^{38} - 20q^{39} - 6q^{40} - 13q^{41} + q^{42} - 19q^{43} + 15q^{45} + 6q^{46} + 22q^{47} + q^{48} + 6q^{49} - 6q^{50} + 14q^{51} - 2q^{52} - 10q^{53} - 4q^{54} + 6q^{56} - 32q^{57} + 2q^{58} - 7q^{59} + q^{60} - 22q^{61} - 15q^{63} + 6q^{64} - 2q^{65} + 5q^{67} - 7q^{68} - 36q^{69} + 6q^{70} + 8q^{71} - 15q^{72} - 13q^{73} + 14q^{74} + q^{75} - 11q^{76} + 20q^{78} + 16q^{79} + 6q^{80} + 18q^{81} + 13q^{82} + 5q^{83} - q^{84} - 7q^{85} + 19q^{86} - 14q^{87} + q^{89} - 15q^{90} + 2q^{91} - 6q^{92} - 42q^{93} - 22q^{94} - 11q^{95} - q^{96} - 3q^{97} - 6q^{98} + O(q^{100}) \)

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
\(3\) 3.25498 1.87927 0.939633 0.342184i \(-0.111167\pi\)
0.939633 + 0.342184i \(0.111167\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −3.25498 −1.32884
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 7.59492 2.53164
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 3.25498 0.939633
\(13\) −3.57888 −0.992603 −0.496302 0.868150i \(-0.665309\pi\)
−0.496302 + 0.868150i \(0.665309\pi\)
\(14\) 1.00000 0.267261
\(15\) 3.25498 0.840433
\(16\) 1.00000 0.250000
\(17\) −3.01604 −0.731497 −0.365749 0.930714i \(-0.619187\pi\)
−0.365749 + 0.930714i \(0.619187\pi\)
\(18\) −7.59492 −1.79014
\(19\) −2.86689 −0.657710 −0.328855 0.944380i \(-0.606663\pi\)
−0.328855 + 0.944380i \(0.606663\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.25498 −0.710296
\(22\) 0 0
\(23\) −9.00193 −1.87703 −0.938516 0.345235i \(-0.887799\pi\)
−0.938516 + 0.345235i \(0.887799\pi\)
\(24\) −3.25498 −0.664421
\(25\) 1.00000 0.200000
\(26\) 3.57888 0.701876
\(27\) 14.9564 2.87836
\(28\) −1.00000 −0.188982
\(29\) −1.56460 −0.290539 −0.145269 0.989392i \(-0.546405\pi\)
−0.145269 + 0.989392i \(0.546405\pi\)
\(30\) −3.25498 −0.594276
\(31\) −0.473234 −0.0849954 −0.0424977 0.999097i \(-0.513532\pi\)
−0.0424977 + 0.999097i \(0.513532\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.01604 0.517247
\(35\) −1.00000 −0.169031
\(36\) 7.59492 1.26582
\(37\) −10.8935 −1.79088 −0.895442 0.445178i \(-0.853140\pi\)
−0.895442 + 0.445178i \(0.853140\pi\)
\(38\) 2.86689 0.465071
\(39\) −11.6492 −1.86537
\(40\) −1.00000 −0.158114
\(41\) −3.84241 −0.600084 −0.300042 0.953926i \(-0.597001\pi\)
−0.300042 + 0.953926i \(0.597001\pi\)
\(42\) 3.25498 0.502255
\(43\) −2.25195 −0.343420 −0.171710 0.985148i \(-0.554929\pi\)
−0.171710 + 0.985148i \(0.554929\pi\)
\(44\) 0 0
\(45\) 7.59492 1.13218
\(46\) 9.00193 1.32726
\(47\) −9.61359 −1.40229 −0.701143 0.713021i \(-0.747326\pi\)
−0.701143 + 0.713021i \(0.747326\pi\)
\(48\) 3.25498 0.469817
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −9.81716 −1.37468
\(52\) −3.57888 −0.496302
\(53\) −5.62157 −0.772182 −0.386091 0.922461i \(-0.626175\pi\)
−0.386091 + 0.922461i \(0.626175\pi\)
\(54\) −14.9564 −2.03531
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −9.33169 −1.23601
\(58\) 1.56460 0.205442
\(59\) −2.73647 −0.356258 −0.178129 0.984007i \(-0.557004\pi\)
−0.178129 + 0.984007i \(0.557004\pi\)
\(60\) 3.25498 0.420217
\(61\) 11.8983 1.52343 0.761713 0.647915i \(-0.224359\pi\)
0.761713 + 0.647915i \(0.224359\pi\)
\(62\) 0.473234 0.0601008
\(63\) −7.59492 −0.956870
\(64\) 1.00000 0.125000
\(65\) −3.57888 −0.443906
\(66\) 0 0
\(67\) 0.752899 0.0919813 0.0459907 0.998942i \(-0.485356\pi\)
0.0459907 + 0.998942i \(0.485356\pi\)
\(68\) −3.01604 −0.365749
\(69\) −29.3011 −3.52744
\(70\) 1.00000 0.119523
\(71\) −1.90401 −0.225965 −0.112982 0.993597i \(-0.536040\pi\)
−0.112982 + 0.993597i \(0.536040\pi\)
\(72\) −7.59492 −0.895070
\(73\) −1.66740 −0.195155 −0.0975774 0.995228i \(-0.531109\pi\)
−0.0975774 + 0.995228i \(0.531109\pi\)
\(74\) 10.8935 1.26635
\(75\) 3.25498 0.375853
\(76\) −2.86689 −0.328855
\(77\) 0 0
\(78\) 11.6492 1.31901
\(79\) 14.5396 1.63583 0.817915 0.575339i \(-0.195130\pi\)
0.817915 + 0.575339i \(0.195130\pi\)
\(80\) 1.00000 0.111803
\(81\) 25.8981 2.87756
\(82\) 3.84241 0.424323
\(83\) 5.31576 0.583481 0.291740 0.956498i \(-0.405766\pi\)
0.291740 + 0.956498i \(0.405766\pi\)
\(84\) −3.25498 −0.355148
\(85\) −3.01604 −0.327136
\(86\) 2.25195 0.242834
\(87\) −5.09274 −0.545999
\(88\) 0 0
\(89\) −11.1627 −1.18324 −0.591621 0.806216i \(-0.701512\pi\)
−0.591621 + 0.806216i \(0.701512\pi\)
\(90\) −7.59492 −0.800575
\(91\) 3.57888 0.375169
\(92\) −9.00193 −0.938516
\(93\) −1.54037 −0.159729
\(94\) 9.61359 0.991566
\(95\) −2.86689 −0.294137
\(96\) −3.25498 −0.332210
\(97\) −1.05562 −0.107182 −0.0535910 0.998563i \(-0.517067\pi\)
−0.0535910 + 0.998563i \(0.517067\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.63558 −0.560761 −0.280381 0.959889i \(-0.590461\pi\)
−0.280381 + 0.959889i \(0.590461\pi\)
\(102\) 9.81716 0.972044
\(103\) −7.68278 −0.757007 −0.378504 0.925600i \(-0.623561\pi\)
−0.378504 + 0.925600i \(0.623561\pi\)
\(104\) 3.57888 0.350938
\(105\) −3.25498 −0.317654
\(106\) 5.62157 0.546015
\(107\) −17.5406 −1.69571 −0.847856 0.530227i \(-0.822107\pi\)
−0.847856 + 0.530227i \(0.822107\pi\)
\(108\) 14.9564 1.43918
\(109\) −10.2643 −0.983138 −0.491569 0.870839i \(-0.663576\pi\)
−0.491569 + 0.870839i \(0.663576\pi\)
\(110\) 0 0
\(111\) −35.4582 −3.36555
\(112\) −1.00000 −0.0944911
\(113\) 10.7388 1.01023 0.505113 0.863053i \(-0.331451\pi\)
0.505113 + 0.863053i \(0.331451\pi\)
\(114\) 9.33169 0.873992
\(115\) −9.00193 −0.839434
\(116\) −1.56460 −0.145269
\(117\) −27.1813 −2.51291
\(118\) 2.73647 0.251913
\(119\) 3.01604 0.276480
\(120\) −3.25498 −0.297138
\(121\) 0 0
\(122\) −11.8983 −1.07722
\(123\) −12.5070 −1.12772
\(124\) −0.473234 −0.0424977
\(125\) 1.00000 0.0894427
\(126\) 7.59492 0.676609
\(127\) −0.276538 −0.0245388 −0.0122694 0.999925i \(-0.503906\pi\)
−0.0122694 + 0.999925i \(0.503906\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.33008 −0.645377
\(130\) 3.57888 0.313889
\(131\) 16.8708 1.47400 0.737002 0.675890i \(-0.236241\pi\)
0.737002 + 0.675890i \(0.236241\pi\)
\(132\) 0 0
\(133\) 2.86689 0.248591
\(134\) −0.752899 −0.0650406
\(135\) 14.9564 1.28724
\(136\) 3.01604 0.258623
\(137\) 13.4886 1.15241 0.576205 0.817305i \(-0.304533\pi\)
0.576205 + 0.817305i \(0.304533\pi\)
\(138\) 29.3011 2.49428
\(139\) 1.47583 0.125178 0.0625890 0.998039i \(-0.480064\pi\)
0.0625890 + 0.998039i \(0.480064\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −31.2921 −2.63527
\(142\) 1.90401 0.159781
\(143\) 0 0
\(144\) 7.59492 0.632910
\(145\) −1.56460 −0.129933
\(146\) 1.66740 0.137995
\(147\) 3.25498 0.268467
\(148\) −10.8935 −0.895442
\(149\) 16.4457 1.34729 0.673644 0.739056i \(-0.264728\pi\)
0.673644 + 0.739056i \(0.264728\pi\)
\(150\) −3.25498 −0.265768
\(151\) −7.87495 −0.640854 −0.320427 0.947273i \(-0.603826\pi\)
−0.320427 + 0.947273i \(0.603826\pi\)
\(152\) 2.86689 0.232536
\(153\) −22.9066 −1.85189
\(154\) 0 0
\(155\) −0.473234 −0.0380111
\(156\) −11.6492 −0.932683
\(157\) −11.7854 −0.940575 −0.470288 0.882513i \(-0.655850\pi\)
−0.470288 + 0.882513i \(0.655850\pi\)
\(158\) −14.5396 −1.15671
\(159\) −18.2981 −1.45113
\(160\) −1.00000 −0.0790569
\(161\) 9.00193 0.709452
\(162\) −25.8981 −2.03475
\(163\) −8.20930 −0.643002 −0.321501 0.946909i \(-0.604187\pi\)
−0.321501 + 0.946909i \(0.604187\pi\)
\(164\) −3.84241 −0.300042
\(165\) 0 0
\(166\) −5.31576 −0.412583
\(167\) −20.1999 −1.56312 −0.781558 0.623832i \(-0.785575\pi\)
−0.781558 + 0.623832i \(0.785575\pi\)
\(168\) 3.25498 0.251127
\(169\) −0.191604 −0.0147388
\(170\) 3.01604 0.231320
\(171\) −21.7738 −1.66509
\(172\) −2.25195 −0.171710
\(173\) 24.7276 1.88000 0.940001 0.341172i \(-0.110824\pi\)
0.940001 + 0.341172i \(0.110824\pi\)
\(174\) 5.09274 0.386080
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −8.90717 −0.669504
\(178\) 11.1627 0.836678
\(179\) 25.5970 1.91321 0.956606 0.291384i \(-0.0941157\pi\)
0.956606 + 0.291384i \(0.0941157\pi\)
\(180\) 7.59492 0.566092
\(181\) −2.24943 −0.167199 −0.0835995 0.996499i \(-0.526642\pi\)
−0.0835995 + 0.996499i \(0.526642\pi\)
\(182\) −3.57888 −0.265284
\(183\) 38.7289 2.86292
\(184\) 9.00193 0.663631
\(185\) −10.8935 −0.800908
\(186\) 1.54037 0.112945
\(187\) 0 0
\(188\) −9.61359 −0.701143
\(189\) −14.9564 −1.08792
\(190\) 2.86689 0.207986
\(191\) 26.0718 1.88649 0.943244 0.332099i \(-0.107757\pi\)
0.943244 + 0.332099i \(0.107757\pi\)
\(192\) 3.25498 0.234908
\(193\) 23.7468 1.70934 0.854668 0.519175i \(-0.173761\pi\)
0.854668 + 0.519175i \(0.173761\pi\)
\(194\) 1.05562 0.0757891
\(195\) −11.6492 −0.834217
\(196\) 1.00000 0.0714286
\(197\) −8.36713 −0.596133 −0.298067 0.954545i \(-0.596342\pi\)
−0.298067 + 0.954545i \(0.596342\pi\)
\(198\) 0 0
\(199\) 6.83066 0.484213 0.242106 0.970250i \(-0.422162\pi\)
0.242106 + 0.970250i \(0.422162\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.45068 0.172857
\(202\) 5.63558 0.396518
\(203\) 1.56460 0.109813
\(204\) −9.81716 −0.687339
\(205\) −3.84241 −0.268366
\(206\) 7.68278 0.535285
\(207\) −68.3690 −4.75197
\(208\) −3.57888 −0.248151
\(209\) 0 0
\(210\) 3.25498 0.224615
\(211\) −2.84244 −0.195682 −0.0978409 0.995202i \(-0.531194\pi\)
−0.0978409 + 0.995202i \(0.531194\pi\)
\(212\) −5.62157 −0.386091
\(213\) −6.19753 −0.424648
\(214\) 17.5406 1.19905
\(215\) −2.25195 −0.153582
\(216\) −14.9564 −1.01765
\(217\) 0.473234 0.0321252
\(218\) 10.2643 0.695183
\(219\) −5.42737 −0.366748
\(220\) 0 0
\(221\) 10.7941 0.726087
\(222\) 35.4582 2.37980
\(223\) 20.0124 1.34013 0.670064 0.742303i \(-0.266267\pi\)
0.670064 + 0.742303i \(0.266267\pi\)
\(224\) 1.00000 0.0668153
\(225\) 7.59492 0.506328
\(226\) −10.7388 −0.714337
\(227\) 12.3334 0.818599 0.409300 0.912400i \(-0.365773\pi\)
0.409300 + 0.912400i \(0.365773\pi\)
\(228\) −9.33169 −0.618006
\(229\) 5.54315 0.366302 0.183151 0.983085i \(-0.441370\pi\)
0.183151 + 0.983085i \(0.441370\pi\)
\(230\) 9.00193 0.593570
\(231\) 0 0
\(232\) 1.56460 0.102721
\(233\) −12.9621 −0.849173 −0.424586 0.905387i \(-0.639581\pi\)
−0.424586 + 0.905387i \(0.639581\pi\)
\(234\) 27.1813 1.77690
\(235\) −9.61359 −0.627121
\(236\) −2.73647 −0.178129
\(237\) 47.3261 3.07416
\(238\) −3.01604 −0.195501
\(239\) 21.3109 1.37849 0.689244 0.724529i \(-0.257943\pi\)
0.689244 + 0.724529i \(0.257943\pi\)
\(240\) 3.25498 0.210108
\(241\) −7.67156 −0.494169 −0.247084 0.968994i \(-0.579472\pi\)
−0.247084 + 0.968994i \(0.579472\pi\)
\(242\) 0 0
\(243\) 39.4286 2.52935
\(244\) 11.8983 0.761713
\(245\) 1.00000 0.0638877
\(246\) 12.5070 0.797416
\(247\) 10.2603 0.652845
\(248\) 0.473234 0.0300504
\(249\) 17.3027 1.09652
\(250\) −1.00000 −0.0632456
\(251\) 27.1182 1.71169 0.855843 0.517235i \(-0.173039\pi\)
0.855843 + 0.517235i \(0.173039\pi\)
\(252\) −7.59492 −0.478435
\(253\) 0 0
\(254\) 0.276538 0.0173515
\(255\) −9.81716 −0.614775
\(256\) 1.00000 0.0625000
\(257\) 6.73716 0.420252 0.210126 0.977674i \(-0.432613\pi\)
0.210126 + 0.977674i \(0.432613\pi\)
\(258\) 7.33008 0.456351
\(259\) 10.8935 0.676891
\(260\) −3.57888 −0.221953
\(261\) −11.8830 −0.735539
\(262\) −16.8708 −1.04228
\(263\) −32.1781 −1.98419 −0.992094 0.125496i \(-0.959948\pi\)
−0.992094 + 0.125496i \(0.959948\pi\)
\(264\) 0 0
\(265\) −5.62157 −0.345330
\(266\) −2.86689 −0.175780
\(267\) −36.3343 −2.22363
\(268\) 0.752899 0.0459907
\(269\) −30.0523 −1.83232 −0.916160 0.400812i \(-0.868728\pi\)
−0.916160 + 0.400812i \(0.868728\pi\)
\(270\) −14.9564 −0.910218
\(271\) 8.92828 0.542354 0.271177 0.962529i \(-0.412587\pi\)
0.271177 + 0.962529i \(0.412587\pi\)
\(272\) −3.01604 −0.182874
\(273\) 11.6492 0.705042
\(274\) −13.4886 −0.814877
\(275\) 0 0
\(276\) −29.3011 −1.76372
\(277\) −1.68071 −0.100984 −0.0504919 0.998724i \(-0.516079\pi\)
−0.0504919 + 0.998724i \(0.516079\pi\)
\(278\) −1.47583 −0.0885142
\(279\) −3.59418 −0.215178
\(280\) 1.00000 0.0597614
\(281\) −7.90100 −0.471334 −0.235667 0.971834i \(-0.575728\pi\)
−0.235667 + 0.971834i \(0.575728\pi\)
\(282\) 31.2921 1.86342
\(283\) −27.0439 −1.60759 −0.803795 0.594906i \(-0.797189\pi\)
−0.803795 + 0.594906i \(0.797189\pi\)
\(284\) −1.90401 −0.112982
\(285\) −9.33169 −0.552761
\(286\) 0 0
\(287\) 3.84241 0.226810
\(288\) −7.59492 −0.447535
\(289\) −7.90350 −0.464912
\(290\) 1.56460 0.0918764
\(291\) −3.43603 −0.201423
\(292\) −1.66740 −0.0975774
\(293\) −10.2007 −0.595934 −0.297967 0.954576i \(-0.596308\pi\)
−0.297967 + 0.954576i \(0.596308\pi\)
\(294\) −3.25498 −0.189835
\(295\) −2.73647 −0.159323
\(296\) 10.8935 0.633173
\(297\) 0 0
\(298\) −16.4457 −0.952676
\(299\) 32.2168 1.86315
\(300\) 3.25498 0.187927
\(301\) 2.25195 0.129800
\(302\) 7.87495 0.453152
\(303\) −18.3437 −1.05382
\(304\) −2.86689 −0.164427
\(305\) 11.8983 0.681296
\(306\) 22.9066 1.30948
\(307\) −15.3552 −0.876370 −0.438185 0.898885i \(-0.644379\pi\)
−0.438185 + 0.898885i \(0.644379\pi\)
\(308\) 0 0
\(309\) −25.0073 −1.42262
\(310\) 0.473234 0.0268779
\(311\) −26.3580 −1.49463 −0.747313 0.664473i \(-0.768656\pi\)
−0.747313 + 0.664473i \(0.768656\pi\)
\(312\) 11.6492 0.659506
\(313\) 8.60472 0.486368 0.243184 0.969980i \(-0.421808\pi\)
0.243184 + 0.969980i \(0.421808\pi\)
\(314\) 11.7854 0.665087
\(315\) −7.59492 −0.427925
\(316\) 14.5396 0.817915
\(317\) 2.87738 0.161610 0.0808049 0.996730i \(-0.474251\pi\)
0.0808049 + 0.996730i \(0.474251\pi\)
\(318\) 18.2981 1.02611
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −57.0943 −3.18669
\(322\) −9.00193 −0.501658
\(323\) 8.64666 0.481113
\(324\) 25.8981 1.43878
\(325\) −3.57888 −0.198521
\(326\) 8.20930 0.454671
\(327\) −33.4100 −1.84758
\(328\) 3.84241 0.212162
\(329\) 9.61359 0.530014
\(330\) 0 0
\(331\) −19.4901 −1.07128 −0.535638 0.844448i \(-0.679929\pi\)
−0.535638 + 0.844448i \(0.679929\pi\)
\(332\) 5.31576 0.291740
\(333\) −82.7355 −4.53388
\(334\) 20.1999 1.10529
\(335\) 0.752899 0.0411353
\(336\) −3.25498 −0.177574
\(337\) 28.1366 1.53270 0.766349 0.642424i \(-0.222071\pi\)
0.766349 + 0.642424i \(0.222071\pi\)
\(338\) 0.191604 0.0104219
\(339\) 34.9548 1.89848
\(340\) −3.01604 −0.163568
\(341\) 0 0
\(342\) 21.7738 1.17739
\(343\) −1.00000 −0.0539949
\(344\) 2.25195 0.121417
\(345\) −29.3011 −1.57752
\(346\) −24.7276 −1.32936
\(347\) 0.231696 0.0124381 0.00621904 0.999981i \(-0.498020\pi\)
0.00621904 + 0.999981i \(0.498020\pi\)
\(348\) −5.09274 −0.273000
\(349\) 23.7081 1.26906 0.634532 0.772897i \(-0.281193\pi\)
0.634532 + 0.772897i \(0.281193\pi\)
\(350\) 1.00000 0.0534522
\(351\) −53.5272 −2.85707
\(352\) 0 0
\(353\) 20.1884 1.07452 0.537259 0.843417i \(-0.319460\pi\)
0.537259 + 0.843417i \(0.319460\pi\)
\(354\) 8.90717 0.473411
\(355\) −1.90401 −0.101055
\(356\) −11.1627 −0.591621
\(357\) 9.81716 0.519579
\(358\) −25.5970 −1.35285
\(359\) −25.7430 −1.35866 −0.679332 0.733831i \(-0.737731\pi\)
−0.679332 + 0.733831i \(0.737731\pi\)
\(360\) −7.59492 −0.400288
\(361\) −10.7809 −0.567418
\(362\) 2.24943 0.118228
\(363\) 0 0
\(364\) 3.57888 0.187584
\(365\) −1.66740 −0.0872759
\(366\) −38.7289 −2.02439
\(367\) −18.8546 −0.984200 −0.492100 0.870539i \(-0.663771\pi\)
−0.492100 + 0.870539i \(0.663771\pi\)
\(368\) −9.00193 −0.469258
\(369\) −29.1828 −1.51920
\(370\) 10.8935 0.566327
\(371\) 5.62157 0.291857
\(372\) −1.54037 −0.0798645
\(373\) −8.46477 −0.438289 −0.219145 0.975692i \(-0.570327\pi\)
−0.219145 + 0.975692i \(0.570327\pi\)
\(374\) 0 0
\(375\) 3.25498 0.168087
\(376\) 9.61359 0.495783
\(377\) 5.59951 0.288390
\(378\) 14.9564 0.769274
\(379\) 16.2403 0.834206 0.417103 0.908859i \(-0.363045\pi\)
0.417103 + 0.908859i \(0.363045\pi\)
\(380\) −2.86689 −0.147068
\(381\) −0.900126 −0.0461149
\(382\) −26.0718 −1.33395
\(383\) 34.3382 1.75460 0.877301 0.479940i \(-0.159342\pi\)
0.877301 + 0.479940i \(0.159342\pi\)
\(384\) −3.25498 −0.166105
\(385\) 0 0
\(386\) −23.7468 −1.20868
\(387\) −17.1034 −0.869416
\(388\) −1.05562 −0.0535910
\(389\) −1.78492 −0.0904991 −0.0452496 0.998976i \(-0.514408\pi\)
−0.0452496 + 0.998976i \(0.514408\pi\)
\(390\) 11.6492 0.589880
\(391\) 27.1502 1.37304
\(392\) −1.00000 −0.0505076
\(393\) 54.9140 2.77005
\(394\) 8.36713 0.421530
\(395\) 14.5396 0.731565
\(396\) 0 0
\(397\) 1.25072 0.0627718 0.0313859 0.999507i \(-0.490008\pi\)
0.0313859 + 0.999507i \(0.490008\pi\)
\(398\) −6.83066 −0.342390
\(399\) 9.33169 0.467169
\(400\) 1.00000 0.0500000
\(401\) −19.4278 −0.970176 −0.485088 0.874465i \(-0.661212\pi\)
−0.485088 + 0.874465i \(0.661212\pi\)
\(402\) −2.45068 −0.122229
\(403\) 1.69365 0.0843667
\(404\) −5.63558 −0.280381
\(405\) 25.8981 1.28689
\(406\) −1.56460 −0.0776497
\(407\) 0 0
\(408\) 9.81716 0.486022
\(409\) 6.77531 0.335017 0.167509 0.985871i \(-0.446428\pi\)
0.167509 + 0.985871i \(0.446428\pi\)
\(410\) 3.84241 0.189763
\(411\) 43.9052 2.16569
\(412\) −7.68278 −0.378504
\(413\) 2.73647 0.134653
\(414\) 68.3690 3.36015
\(415\) 5.31576 0.260940
\(416\) 3.57888 0.175469
\(417\) 4.80379 0.235243
\(418\) 0 0
\(419\) 26.3741 1.28846 0.644228 0.764833i \(-0.277179\pi\)
0.644228 + 0.764833i \(0.277179\pi\)
\(420\) −3.25498 −0.158827
\(421\) 19.1809 0.934822 0.467411 0.884040i \(-0.345187\pi\)
0.467411 + 0.884040i \(0.345187\pi\)
\(422\) 2.84244 0.138368
\(423\) −73.0144 −3.55008
\(424\) 5.62157 0.273007
\(425\) −3.01604 −0.146299
\(426\) 6.19753 0.300272
\(427\) −11.8983 −0.575801
\(428\) −17.5406 −0.847856
\(429\) 0 0
\(430\) 2.25195 0.108599
\(431\) −27.4826 −1.32379 −0.661894 0.749597i \(-0.730247\pi\)
−0.661894 + 0.749597i \(0.730247\pi\)
\(432\) 14.9564 0.719590
\(433\) 6.92935 0.333003 0.166502 0.986041i \(-0.446753\pi\)
0.166502 + 0.986041i \(0.446753\pi\)
\(434\) −0.473234 −0.0227160
\(435\) −5.09274 −0.244178
\(436\) −10.2643 −0.491569
\(437\) 25.8076 1.23454
\(438\) 5.42737 0.259330
\(439\) 9.07635 0.433191 0.216595 0.976261i \(-0.430505\pi\)
0.216595 + 0.976261i \(0.430505\pi\)
\(440\) 0 0
\(441\) 7.59492 0.361663
\(442\) −10.7941 −0.513421
\(443\) −23.5991 −1.12122 −0.560612 0.828078i \(-0.689434\pi\)
−0.560612 + 0.828078i \(0.689434\pi\)
\(444\) −35.4582 −1.68277
\(445\) −11.1627 −0.529162
\(446\) −20.0124 −0.947614
\(447\) 53.5306 2.53191
\(448\) −1.00000 −0.0472456
\(449\) −30.0205 −1.41675 −0.708377 0.705834i \(-0.750572\pi\)
−0.708377 + 0.705834i \(0.750572\pi\)
\(450\) −7.59492 −0.358028
\(451\) 0 0
\(452\) 10.7388 0.505113
\(453\) −25.6328 −1.20434
\(454\) −12.3334 −0.578837
\(455\) 3.57888 0.167781
\(456\) 9.33169 0.436996
\(457\) 2.10098 0.0982798 0.0491399 0.998792i \(-0.484352\pi\)
0.0491399 + 0.998792i \(0.484352\pi\)
\(458\) −5.54315 −0.259014
\(459\) −45.1091 −2.10551
\(460\) −9.00193 −0.419717
\(461\) −33.6599 −1.56770 −0.783848 0.620953i \(-0.786746\pi\)
−0.783848 + 0.620953i \(0.786746\pi\)
\(462\) 0 0
\(463\) 31.2506 1.45234 0.726168 0.687517i \(-0.241299\pi\)
0.726168 + 0.687517i \(0.241299\pi\)
\(464\) −1.56460 −0.0726346
\(465\) −1.54037 −0.0714330
\(466\) 12.9621 0.600456
\(467\) 26.9260 1.24599 0.622994 0.782227i \(-0.285916\pi\)
0.622994 + 0.782227i \(0.285916\pi\)
\(468\) −27.1813 −1.25646
\(469\) −0.752899 −0.0347657
\(470\) 9.61359 0.443442
\(471\) −38.3612 −1.76759
\(472\) 2.73647 0.125956
\(473\) 0 0
\(474\) −47.3261 −2.17376
\(475\) −2.86689 −0.131542
\(476\) 3.01604 0.138240
\(477\) −42.6954 −1.95489
\(478\) −21.3109 −0.974738
\(479\) −35.8357 −1.63738 −0.818688 0.574238i \(-0.805298\pi\)
−0.818688 + 0.574238i \(0.805298\pi\)
\(480\) −3.25498 −0.148569
\(481\) 38.9866 1.77764
\(482\) 7.67156 0.349430
\(483\) 29.3011 1.33325
\(484\) 0 0
\(485\) −1.05562 −0.0479332
\(486\) −39.4286 −1.78852
\(487\) −10.3029 −0.466870 −0.233435 0.972372i \(-0.574997\pi\)
−0.233435 + 0.972372i \(0.574997\pi\)
\(488\) −11.8983 −0.538612
\(489\) −26.7211 −1.20837
\(490\) −1.00000 −0.0451754
\(491\) −20.0929 −0.906778 −0.453389 0.891313i \(-0.649785\pi\)
−0.453389 + 0.891313i \(0.649785\pi\)
\(492\) −12.5070 −0.563859
\(493\) 4.71889 0.212528
\(494\) −10.2603 −0.461631
\(495\) 0 0
\(496\) −0.473234 −0.0212489
\(497\) 1.90401 0.0854067
\(498\) −17.3027 −0.775353
\(499\) 26.1662 1.17136 0.585680 0.810542i \(-0.300827\pi\)
0.585680 + 0.810542i \(0.300827\pi\)
\(500\) 1.00000 0.0447214
\(501\) −65.7504 −2.93751
\(502\) −27.1182 −1.21034
\(503\) 1.78113 0.0794167 0.0397084 0.999211i \(-0.487357\pi\)
0.0397084 + 0.999211i \(0.487357\pi\)
\(504\) 7.59492 0.338305
\(505\) −5.63558 −0.250780
\(506\) 0 0
\(507\) −0.623670 −0.0276981
\(508\) −0.276538 −0.0122694
\(509\) −7.96822 −0.353185 −0.176592 0.984284i \(-0.556507\pi\)
−0.176592 + 0.984284i \(0.556507\pi\)
\(510\) 9.81716 0.434711
\(511\) 1.66740 0.0737616
\(512\) −1.00000 −0.0441942
\(513\) −42.8784 −1.89313
\(514\) −6.73716 −0.297163
\(515\) −7.68278 −0.338544
\(516\) −7.33008 −0.322689
\(517\) 0 0
\(518\) −10.8935 −0.478634
\(519\) 80.4878 3.53302
\(520\) 3.57888 0.156944
\(521\) −38.1254 −1.67030 −0.835152 0.550019i \(-0.814621\pi\)
−0.835152 + 0.550019i \(0.814621\pi\)
\(522\) 11.8830 0.520105
\(523\) 7.78040 0.340213 0.170107 0.985426i \(-0.445589\pi\)
0.170107 + 0.985426i \(0.445589\pi\)
\(524\) 16.8708 0.737002
\(525\) −3.25498 −0.142059
\(526\) 32.1781 1.40303
\(527\) 1.42729 0.0621739
\(528\) 0 0
\(529\) 58.0348 2.52325
\(530\) 5.62157 0.244185
\(531\) −20.7833 −0.901918
\(532\) 2.86689 0.124296
\(533\) 13.7515 0.595645
\(534\) 36.3343 1.57234
\(535\) −17.5406 −0.758345
\(536\) −0.752899 −0.0325203
\(537\) 83.3180 3.59544
\(538\) 30.0523 1.29565
\(539\) 0 0
\(540\) 14.9564 0.643621
\(541\) 9.67143 0.415807 0.207904 0.978149i \(-0.433336\pi\)
0.207904 + 0.978149i \(0.433336\pi\)
\(542\) −8.92828 −0.383502
\(543\) −7.32187 −0.314211
\(544\) 3.01604 0.129312
\(545\) −10.2643 −0.439672
\(546\) −11.6492 −0.498540
\(547\) 38.6572 1.65286 0.826432 0.563036i \(-0.190367\pi\)
0.826432 + 0.563036i \(0.190367\pi\)
\(548\) 13.4886 0.576205
\(549\) 90.3669 3.85677
\(550\) 0 0
\(551\) 4.48553 0.191090
\(552\) 29.3011 1.24714
\(553\) −14.5396 −0.618286
\(554\) 1.68071 0.0714064
\(555\) −35.4582 −1.50512
\(556\) 1.47583 0.0625890
\(557\) −1.28946 −0.0546360 −0.0273180 0.999627i \(-0.508697\pi\)
−0.0273180 + 0.999627i \(0.508697\pi\)
\(558\) 3.59418 0.152154
\(559\) 8.05948 0.340880
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 7.90100 0.333284
\(563\) 22.8803 0.964289 0.482144 0.876092i \(-0.339858\pi\)
0.482144 + 0.876092i \(0.339858\pi\)
\(564\) −31.2921 −1.31763
\(565\) 10.7388 0.451786
\(566\) 27.0439 1.13674
\(567\) −25.8981 −1.08762
\(568\) 1.90401 0.0798906
\(569\) 3.30473 0.138541 0.0692707 0.997598i \(-0.477933\pi\)
0.0692707 + 0.997598i \(0.477933\pi\)
\(570\) 9.33169 0.390861
\(571\) 1.86937 0.0782307 0.0391153 0.999235i \(-0.487546\pi\)
0.0391153 + 0.999235i \(0.487546\pi\)
\(572\) 0 0
\(573\) 84.8633 3.54521
\(574\) −3.84241 −0.160379
\(575\) −9.00193 −0.375406
\(576\) 7.59492 0.316455
\(577\) −15.3442 −0.638787 −0.319394 0.947622i \(-0.603479\pi\)
−0.319394 + 0.947622i \(0.603479\pi\)
\(578\) 7.90350 0.328742
\(579\) 77.2956 3.21230
\(580\) −1.56460 −0.0649664
\(581\) −5.31576 −0.220535
\(582\) 3.43603 0.142428
\(583\) 0 0
\(584\) 1.66740 0.0689976
\(585\) −27.1813 −1.12381
\(586\) 10.2007 0.421389
\(587\) 9.24503 0.381583 0.190792 0.981631i \(-0.438895\pi\)
0.190792 + 0.981631i \(0.438895\pi\)
\(588\) 3.25498 0.134233
\(589\) 1.35671 0.0559023
\(590\) 2.73647 0.112659
\(591\) −27.2349 −1.12029
\(592\) −10.8935 −0.447721
\(593\) −15.5998 −0.640606 −0.320303 0.947315i \(-0.603785\pi\)
−0.320303 + 0.947315i \(0.603785\pi\)
\(594\) 0 0
\(595\) 3.01604 0.123646
\(596\) 16.4457 0.673644
\(597\) 22.2337 0.909965
\(598\) −32.2168 −1.31744
\(599\) −23.2701 −0.950791 −0.475396 0.879772i \(-0.657695\pi\)
−0.475396 + 0.879772i \(0.657695\pi\)
\(600\) −3.25498 −0.132884
\(601\) −8.23116 −0.335756 −0.167878 0.985808i \(-0.553691\pi\)
−0.167878 + 0.985808i \(0.553691\pi\)
\(602\) −2.25195 −0.0917828
\(603\) 5.71821 0.232864
\(604\) −7.87495 −0.320427
\(605\) 0 0
\(606\) 18.3437 0.745163
\(607\) 37.1360 1.50730 0.753652 0.657274i \(-0.228291\pi\)
0.753652 + 0.657274i \(0.228291\pi\)
\(608\) 2.86689 0.116268
\(609\) 5.09274 0.206368
\(610\) −11.8983 −0.481749
\(611\) 34.4059 1.39191
\(612\) −22.9066 −0.925944
\(613\) 17.5909 0.710490 0.355245 0.934773i \(-0.384397\pi\)
0.355245 + 0.934773i \(0.384397\pi\)
\(614\) 15.3552 0.619687
\(615\) −12.5070 −0.504330
\(616\) 0 0
\(617\) 5.16685 0.208010 0.104005 0.994577i \(-0.466834\pi\)
0.104005 + 0.994577i \(0.466834\pi\)
\(618\) 25.0073 1.00594
\(619\) −17.4862 −0.702828 −0.351414 0.936220i \(-0.614299\pi\)
−0.351414 + 0.936220i \(0.614299\pi\)
\(620\) −0.473234 −0.0190056
\(621\) −134.636 −5.40278
\(622\) 26.3580 1.05686
\(623\) 11.1627 0.447223
\(624\) −11.6492 −0.466341
\(625\) 1.00000 0.0400000
\(626\) −8.60472 −0.343914
\(627\) 0 0
\(628\) −11.7854 −0.470288
\(629\) 32.8553 1.31003
\(630\) 7.59492 0.302589
\(631\) −18.6264 −0.741505 −0.370752 0.928732i \(-0.620900\pi\)
−0.370752 + 0.928732i \(0.620900\pi\)
\(632\) −14.5396 −0.578353
\(633\) −9.25210 −0.367738
\(634\) −2.87738 −0.114275
\(635\) −0.276538 −0.0109741
\(636\) −18.2981 −0.725567
\(637\) −3.57888 −0.141800
\(638\) 0 0
\(639\) −14.4608 −0.572062
\(640\) −1.00000 −0.0395285
\(641\) −14.4218 −0.569625 −0.284813 0.958583i \(-0.591931\pi\)
−0.284813 + 0.958583i \(0.591931\pi\)
\(642\) 57.0943 2.25333
\(643\) 23.0331 0.908338 0.454169 0.890915i \(-0.349936\pi\)
0.454169 + 0.890915i \(0.349936\pi\)
\(644\) 9.00193 0.354726
\(645\) −7.33008 −0.288621
\(646\) −8.64666 −0.340198
\(647\) 12.8918 0.506830 0.253415 0.967358i \(-0.418446\pi\)
0.253415 + 0.967358i \(0.418446\pi\)
\(648\) −25.8981 −1.01737
\(649\) 0 0
\(650\) 3.57888 0.140375
\(651\) 1.54037 0.0603719
\(652\) −8.20930 −0.321501
\(653\) −23.8203 −0.932161 −0.466081 0.884742i \(-0.654334\pi\)
−0.466081 + 0.884742i \(0.654334\pi\)
\(654\) 33.4100 1.30643
\(655\) 16.8708 0.659195
\(656\) −3.84241 −0.150021
\(657\) −12.6638 −0.494062
\(658\) −9.61359 −0.374777
\(659\) −19.4714 −0.758498 −0.379249 0.925295i \(-0.623818\pi\)
−0.379249 + 0.925295i \(0.623818\pi\)
\(660\) 0 0
\(661\) −28.6251 −1.11339 −0.556694 0.830718i \(-0.687930\pi\)
−0.556694 + 0.830718i \(0.687930\pi\)
\(662\) 19.4901 0.757506
\(663\) 35.1345 1.36451
\(664\) −5.31576 −0.206292
\(665\) 2.86689 0.111173
\(666\) 82.7355 3.20593
\(667\) 14.0844 0.545350
\(668\) −20.1999 −0.781558
\(669\) 65.1400 2.51846
\(670\) −0.752899 −0.0290870
\(671\) 0 0
\(672\) 3.25498 0.125564
\(673\) 5.76294 0.222145 0.111073 0.993812i \(-0.464571\pi\)
0.111073 + 0.993812i \(0.464571\pi\)
\(674\) −28.1366 −1.08378
\(675\) 14.9564 0.575672
\(676\) −0.191604 −0.00736940
\(677\) −20.1902 −0.775973 −0.387987 0.921665i \(-0.626829\pi\)
−0.387987 + 0.921665i \(0.626829\pi\)
\(678\) −34.9548 −1.34243
\(679\) 1.05562 0.0405110
\(680\) 3.01604 0.115660
\(681\) 40.1452 1.53837
\(682\) 0 0
\(683\) −19.0998 −0.730833 −0.365417 0.930844i \(-0.619073\pi\)
−0.365417 + 0.930844i \(0.619073\pi\)
\(684\) −21.7738 −0.832543
\(685\) 13.4886 0.515374
\(686\) 1.00000 0.0381802
\(687\) 18.0429 0.688378
\(688\) −2.25195 −0.0858549
\(689\) 20.1189 0.766470
\(690\) 29.3011 1.11548
\(691\) −1.77501 −0.0675246 −0.0337623 0.999430i \(-0.510749\pi\)
−0.0337623 + 0.999430i \(0.510749\pi\)
\(692\) 24.7276 0.940001
\(693\) 0 0
\(694\) −0.231696 −0.00879505
\(695\) 1.47583 0.0559813
\(696\) 5.09274 0.193040
\(697\) 11.5889 0.438960
\(698\) −23.7081 −0.897364
\(699\) −42.1913 −1.59582
\(700\) −1.00000 −0.0377964
\(701\) −8.50630 −0.321279 −0.160639 0.987013i \(-0.551356\pi\)
−0.160639 + 0.987013i \(0.551356\pi\)
\(702\) 53.5272 2.02025
\(703\) 31.2305 1.17788
\(704\) 0 0
\(705\) −31.2921 −1.17853
\(706\) −20.1884 −0.759799
\(707\) 5.63558 0.211948
\(708\) −8.90717 −0.334752
\(709\) 22.1030 0.830096 0.415048 0.909799i \(-0.363765\pi\)
0.415048 + 0.909799i \(0.363765\pi\)
\(710\) 1.90401 0.0714564
\(711\) 110.427 4.14133
\(712\) 11.1627 0.418339
\(713\) 4.26002 0.159539
\(714\) −9.81716 −0.367398
\(715\) 0 0
\(716\) 25.5970 0.956606
\(717\) 69.3667 2.59055
\(718\) 25.7430 0.960721
\(719\) 20.1432 0.751216 0.375608 0.926779i \(-0.377434\pi\)
0.375608 + 0.926779i \(0.377434\pi\)
\(720\) 7.59492 0.283046
\(721\) 7.68278 0.286122
\(722\) 10.7809 0.401225
\(723\) −24.9708 −0.928674
\(724\) −2.24943 −0.0835995
\(725\) −1.56460 −0.0581077
\(726\) 0 0
\(727\) −41.7541 −1.54857 −0.774287 0.632835i \(-0.781891\pi\)
−0.774287 + 0.632835i \(0.781891\pi\)
\(728\) −3.57888 −0.132642
\(729\) 50.6454 1.87575
\(730\) 1.66740 0.0617134
\(731\) 6.79199 0.251211
\(732\) 38.7289 1.43146
\(733\) −48.0596 −1.77512 −0.887560 0.460692i \(-0.847601\pi\)
−0.887560 + 0.460692i \(0.847601\pi\)
\(734\) 18.8546 0.695934
\(735\) 3.25498 0.120062
\(736\) 9.00193 0.331816
\(737\) 0 0
\(738\) 29.1828 1.07423
\(739\) −39.9689 −1.47028 −0.735140 0.677916i \(-0.762883\pi\)
−0.735140 + 0.677916i \(0.762883\pi\)
\(740\) −10.8935 −0.400454
\(741\) 33.3970 1.22687
\(742\) −5.62157 −0.206374
\(743\) −17.4381 −0.639742 −0.319871 0.947461i \(-0.603640\pi\)
−0.319871 + 0.947461i \(0.603640\pi\)
\(744\) 1.54037 0.0564727
\(745\) 16.4457 0.602525
\(746\) 8.46477 0.309917
\(747\) 40.3728 1.47716
\(748\) 0 0
\(749\) 17.5406 0.640919
\(750\) −3.25498 −0.118855
\(751\) 28.4323 1.03751 0.518754 0.854923i \(-0.326396\pi\)
0.518754 + 0.854923i \(0.326396\pi\)
\(752\) −9.61359 −0.350571
\(753\) 88.2693 3.21671
\(754\) −5.59951 −0.203922
\(755\) −7.87495 −0.286599
\(756\) −14.9564 −0.543959
\(757\) 19.9596 0.725445 0.362722 0.931897i \(-0.381847\pi\)
0.362722 + 0.931897i \(0.381847\pi\)
\(758\) −16.2403 −0.589873
\(759\) 0 0
\(760\) 2.86689 0.103993
\(761\) −21.9752 −0.796600 −0.398300 0.917255i \(-0.630400\pi\)
−0.398300 + 0.917255i \(0.630400\pi\)
\(762\) 0.900126 0.0326081
\(763\) 10.2643 0.371591
\(764\) 26.0718 0.943244
\(765\) −22.9066 −0.828190
\(766\) −34.3382 −1.24069
\(767\) 9.79351 0.353623
\(768\) 3.25498 0.117454
\(769\) −23.8878 −0.861417 −0.430708 0.902491i \(-0.641736\pi\)
−0.430708 + 0.902491i \(0.641736\pi\)
\(770\) 0 0
\(771\) 21.9293 0.789766
\(772\) 23.7468 0.854668
\(773\) 3.82132 0.137443 0.0687216 0.997636i \(-0.478108\pi\)
0.0687216 + 0.997636i \(0.478108\pi\)
\(774\) 17.1034 0.614770
\(775\) −0.473234 −0.0169991
\(776\) 1.05562 0.0378946
\(777\) 35.4582 1.27206
\(778\) 1.78492 0.0639925
\(779\) 11.0158 0.394681
\(780\) −11.6492 −0.417108
\(781\) 0 0
\(782\) −27.1502 −0.970889
\(783\) −23.4008 −0.836275
\(784\) 1.00000 0.0357143
\(785\) −11.7854 −0.420638
\(786\) −54.9140 −1.95872
\(787\) 31.8195 1.13424 0.567122 0.823634i \(-0.308057\pi\)
0.567122 + 0.823634i \(0.308057\pi\)
\(788\) −8.36713 −0.298067
\(789\) −104.739 −3.72882
\(790\) −14.5396 −0.517295
\(791\) −10.7388 −0.381829
\(792\) 0 0
\(793\) −42.5827 −1.51216
\(794\) −1.25072 −0.0443864
\(795\) −18.2981 −0.648967
\(796\) 6.83066 0.242106
\(797\) −37.3003 −1.32124 −0.660622 0.750719i \(-0.729707\pi\)
−0.660622 + 0.750719i \(0.729707\pi\)
\(798\) −9.33169 −0.330338
\(799\) 28.9950 1.02577
\(800\) −1.00000 −0.0353553
\(801\) −84.7797 −2.99554
\(802\) 19.4278 0.686018
\(803\) 0 0
\(804\) 2.45068 0.0864287
\(805\) 9.00193 0.317276
\(806\) −1.69365 −0.0596563
\(807\) −97.8197 −3.44342
\(808\) 5.63558 0.198259
\(809\) 34.4351 1.21067 0.605337 0.795969i \(-0.293038\pi\)
0.605337 + 0.795969i \(0.293038\pi\)
\(810\) −25.8981 −0.909966
\(811\) 29.0879 1.02142 0.510708 0.859754i \(-0.329383\pi\)
0.510708 + 0.859754i \(0.329383\pi\)
\(812\) 1.56460 0.0549066
\(813\) 29.0614 1.01923
\(814\) 0 0
\(815\) −8.20930 −0.287559
\(816\) −9.81716 −0.343669
\(817\) 6.45611 0.225871
\(818\) −6.77531 −0.236893
\(819\) 27.1813 0.949793
\(820\) −3.84241 −0.134183
\(821\) −7.44892 −0.259969 −0.129985 0.991516i \(-0.541493\pi\)
−0.129985 + 0.991516i \(0.541493\pi\)
\(822\) −43.9052 −1.53137
\(823\) 43.3944 1.51263 0.756317 0.654206i \(-0.226997\pi\)
0.756317 + 0.654206i \(0.226997\pi\)
\(824\) 7.68278 0.267642
\(825\) 0 0
\(826\) −2.73647 −0.0952140
\(827\) −7.19740 −0.250278 −0.125139 0.992139i \(-0.539938\pi\)
−0.125139 + 0.992139i \(0.539938\pi\)
\(828\) −68.3690 −2.37599
\(829\) −9.22493 −0.320395 −0.160198 0.987085i \(-0.551213\pi\)
−0.160198 + 0.987085i \(0.551213\pi\)
\(830\) −5.31576 −0.184513
\(831\) −5.47067 −0.189776
\(832\) −3.57888 −0.124075
\(833\) −3.01604 −0.104500
\(834\) −4.80379 −0.166342
\(835\) −20.1999 −0.699047
\(836\) 0 0
\(837\) −7.07788 −0.244647
\(838\) −26.3741 −0.911077
\(839\) −3.04238 −0.105034 −0.0525172 0.998620i \(-0.516724\pi\)
−0.0525172 + 0.998620i \(0.516724\pi\)
\(840\) 3.25498 0.112308
\(841\) −26.5520 −0.915587
\(842\) −19.1809 −0.661019
\(843\) −25.7176 −0.885762
\(844\) −2.84244 −0.0978409
\(845\) −0.191604 −0.00659139
\(846\) 73.0144 2.51029
\(847\) 0 0
\(848\) −5.62157 −0.193045
\(849\) −88.0273 −3.02109
\(850\) 3.01604 0.103449
\(851\) 98.0628 3.36155
\(852\) −6.19753 −0.212324
\(853\) −29.9859 −1.02670 −0.513349 0.858180i \(-0.671595\pi\)
−0.513349 + 0.858180i \(0.671595\pi\)
\(854\) 11.8983 0.407153
\(855\) −21.7738 −0.744649
\(856\) 17.5406 0.599525
\(857\) 7.77150 0.265469 0.132735 0.991152i \(-0.457624\pi\)
0.132735 + 0.991152i \(0.457624\pi\)
\(858\) 0 0
\(859\) −50.9695 −1.73906 −0.869529 0.493883i \(-0.835577\pi\)
−0.869529 + 0.493883i \(0.835577\pi\)
\(860\) −2.25195 −0.0767910
\(861\) 12.5070 0.426237
\(862\) 27.4826 0.936060
\(863\) −25.2461 −0.859386 −0.429693 0.902975i \(-0.641378\pi\)
−0.429693 + 0.902975i \(0.641378\pi\)
\(864\) −14.9564 −0.508827
\(865\) 24.7276 0.840762
\(866\) −6.92935 −0.235469
\(867\) −25.7258 −0.873693
\(868\) 0.473234 0.0160626
\(869\) 0 0
\(870\) 5.09274 0.172660
\(871\) −2.69454 −0.0913009
\(872\) 10.2643 0.347592
\(873\) −8.01735 −0.271346
\(874\) −25.8076 −0.872954
\(875\) −1.00000 −0.0338062
\(876\) −5.42737 −0.183374
\(877\) 1.25141 0.0422572 0.0211286 0.999777i \(-0.493274\pi\)
0.0211286 + 0.999777i \(0.493274\pi\)
\(878\) −9.07635 −0.306312
\(879\) −33.2033 −1.11992
\(880\) 0 0
\(881\) −16.6655 −0.561476 −0.280738 0.959784i \(-0.590579\pi\)
−0.280738 + 0.959784i \(0.590579\pi\)
\(882\) −7.59492 −0.255734
\(883\) −1.23966 −0.0417180 −0.0208590 0.999782i \(-0.506640\pi\)
−0.0208590 + 0.999782i \(0.506640\pi\)
\(884\) 10.7941 0.363043
\(885\) −8.90717 −0.299411
\(886\) 23.5991 0.792826
\(887\) 40.2814 1.35252 0.676259 0.736664i \(-0.263600\pi\)
0.676259 + 0.736664i \(0.263600\pi\)
\(888\) 35.4582 1.18990
\(889\) 0.276538 0.00927478
\(890\) 11.1627 0.374174
\(891\) 0 0
\(892\) 20.0124 0.670064
\(893\) 27.5611 0.922297
\(894\) −53.5306 −1.79033
\(895\) 25.5970 0.855615
\(896\) 1.00000 0.0334077
\(897\) 104.865 3.50135
\(898\) 30.0205 1.00180
\(899\) 0.740422 0.0246944
\(900\) 7.59492 0.253164
\(901\) 16.9549 0.564849
\(902\) 0 0
\(903\) 7.33008 0.243930
\(904\) −10.7388 −0.357169
\(905\) −2.24943 −0.0747737
\(906\) 25.6328 0.851594
\(907\) −8.61141 −0.285937 −0.142969 0.989727i \(-0.545665\pi\)
−0.142969 + 0.989727i \(0.545665\pi\)
\(908\) 12.3334 0.409300
\(909\) −42.8018 −1.41965
\(910\) −3.57888 −0.118639
\(911\) 5.14473 0.170453 0.0852263 0.996362i \(-0.472839\pi\)
0.0852263 + 0.996362i \(0.472839\pi\)
\(912\) −9.33169 −0.309003
\(913\) 0 0
\(914\) −2.10098 −0.0694943
\(915\) 38.7289 1.28034
\(916\) 5.54315 0.183151
\(917\) −16.8708 −0.557121
\(918\) 45.1091 1.48882
\(919\) −21.7516 −0.717521 −0.358760 0.933430i \(-0.616800\pi\)
−0.358760 + 0.933430i \(0.616800\pi\)
\(920\) 9.00193 0.296785
\(921\) −49.9811 −1.64693
\(922\) 33.6599 1.10853
\(923\) 6.81424 0.224293
\(924\) 0 0
\(925\) −10.8935 −0.358177
\(926\) −31.2506 −1.02696
\(927\) −58.3501 −1.91647
\(928\) 1.56460 0.0513604
\(929\) −46.1193 −1.51312 −0.756562 0.653922i \(-0.773122\pi\)
−0.756562 + 0.653922i \(0.773122\pi\)
\(930\) 1.54037 0.0505107
\(931\) −2.86689 −0.0939586
\(932\) −12.9621 −0.424586
\(933\) −85.7949 −2.80880
\(934\) −26.9260 −0.881047
\(935\) 0 0
\(936\) 27.1813 0.888450
\(937\) −11.2889 −0.368793 −0.184397 0.982852i \(-0.559033\pi\)
−0.184397 + 0.982852i \(0.559033\pi\)
\(938\) 0.752899 0.0245830
\(939\) 28.0082 0.914014
\(940\) −9.61359 −0.313561
\(941\) 12.7563 0.415845 0.207922 0.978145i \(-0.433330\pi\)
0.207922 + 0.978145i \(0.433330\pi\)
\(942\) 38.3612 1.24988
\(943\) 34.5891 1.12638
\(944\) −2.73647 −0.0890645
\(945\) −14.9564 −0.486532
\(946\) 0 0
\(947\) −47.3261 −1.53789 −0.768945 0.639315i \(-0.779218\pi\)
−0.768945 + 0.639315i \(0.779218\pi\)
\(948\) 47.3261 1.53708
\(949\) 5.96744 0.193711
\(950\) 2.86689 0.0930142
\(951\) 9.36583 0.303708
\(952\) −3.01604 −0.0977504
\(953\) 5.17931 0.167774 0.0838872 0.996475i \(-0.473266\pi\)
0.0838872 + 0.996475i \(0.473266\pi\)
\(954\) 42.6954 1.38231
\(955\) 26.0718 0.843663
\(956\) 21.3109 0.689244
\(957\) 0 0
\(958\) 35.8357 1.15780
\(959\) −13.4886 −0.435570
\(960\) 3.25498 0.105054
\(961\) −30.7760 −0.992776
\(962\) −38.9866 −1.25698
\(963\) −133.219 −4.29293
\(964\) −7.67156 −0.247084
\(965\) 23.7468 0.764438
\(966\) −29.3011 −0.942749
\(967\) −39.6832 −1.27612 −0.638062 0.769985i \(-0.720264\pi\)
−0.638062 + 0.769985i \(0.720264\pi\)
\(968\) 0 0
\(969\) 28.1447 0.904139
\(970\) 1.05562 0.0338939
\(971\) 1.43366 0.0460085 0.0230042 0.999735i \(-0.492677\pi\)
0.0230042 + 0.999735i \(0.492677\pi\)
\(972\) 39.4286 1.26467
\(973\) −1.47583 −0.0473128
\(974\) 10.3029 0.330127
\(975\) −11.6492 −0.373073
\(976\) 11.8983 0.380856
\(977\) −33.2754 −1.06458 −0.532288 0.846564i \(-0.678667\pi\)
−0.532288 + 0.846564i \(0.678667\pi\)
\(978\) 26.7211 0.854448
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −77.9562 −2.48895
\(982\) 20.0929 0.641189
\(983\) 41.4050 1.32062 0.660308 0.750995i \(-0.270426\pi\)
0.660308 + 0.750995i \(0.270426\pi\)
\(984\) 12.5070 0.398708
\(985\) −8.36713 −0.266599
\(986\) −4.71889 −0.150280
\(987\) 31.2921 0.996038
\(988\) 10.2603 0.326423
\(989\) 20.2719 0.644610
\(990\) 0 0
\(991\) −37.2508 −1.18331 −0.591655 0.806191i \(-0.701525\pi\)
−0.591655 + 0.806191i \(0.701525\pi\)
\(992\) 0.473234 0.0150252
\(993\) −63.4401 −2.01321
\(994\) −1.90401 −0.0603916
\(995\) 6.83066 0.216547
\(996\) 17.3027 0.548258
\(997\) −20.0494 −0.634972 −0.317486 0.948263i \(-0.602839\pi\)
−0.317486 + 0.948263i \(0.602839\pi\)
\(998\) −26.1662 −0.828277
\(999\) −162.928 −5.15481
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.cy.1.6 6
11.2 odd 10 770.2.n.i.631.3 yes 12
11.6 odd 10 770.2.n.i.421.3 12
11.10 odd 2 8470.2.a.de.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.421.3 12 11.6 odd 10
770.2.n.i.631.3 yes 12 11.2 odd 10
8470.2.a.cy.1.6 6 1.1 even 1 trivial
8470.2.a.de.1.6 6 11.10 odd 2