Properties

Label 8470.2.a.cq.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + x + 11\)
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 1.2
Root \(-1.48718\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.919131 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.15520 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.919131 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.919131 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.15520 q^{9} +1.00000 q^{10} -0.919131 q^{12} +3.72325 q^{13} +1.00000 q^{14} +0.919131 q^{15} +1.00000 q^{16} -1.31694 q^{17} +2.15520 q^{18} -2.70741 q^{19} -1.00000 q^{20} +0.919131 q^{21} -2.00000 q^{23} +0.919131 q^{24} +1.00000 q^{25} -3.72325 q^{26} +4.73830 q^{27} -1.00000 q^{28} -2.95002 q^{29} -0.919131 q^{30} +7.70820 q^{31} -1.00000 q^{32} +1.31694 q^{34} +1.00000 q^{35} -2.15520 q^{36} -8.04870 q^{37} +2.70741 q^{38} -3.42216 q^{39} +1.00000 q^{40} +3.24458 q^{41} -0.919131 q^{42} +6.48064 q^{43} +2.15520 q^{45} +2.00000 q^{46} -9.37090 q^{47} -0.919131 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.21044 q^{51} +3.72325 q^{52} +13.2848 q^{53} -4.73830 q^{54} +1.00000 q^{56} +2.48847 q^{57} +2.95002 q^{58} -2.83905 q^{59} +0.919131 q^{60} +1.44651 q^{61} -7.70820 q^{62} +2.15520 q^{63} +1.00000 q^{64} -3.72325 q^{65} -6.70741 q^{67} -1.31694 q^{68} +1.83826 q^{69} -1.00000 q^{70} +8.03365 q^{71} +2.15520 q^{72} -1.33071 q^{73} +8.04870 q^{74} -0.919131 q^{75} -2.70741 q^{76} +3.42216 q^{78} -7.72325 q^{79} -1.00000 q^{80} +2.11048 q^{81} -3.24458 q^{82} +0.285256 q^{83} +0.919131 q^{84} +1.31694 q^{85} -6.48064 q^{86} +2.71145 q^{87} -8.14015 q^{89} -2.15520 q^{90} -3.72325 q^{91} -2.00000 q^{92} -7.08485 q^{93} +9.37090 q^{94} +2.70741 q^{95} +0.919131 q^{96} +14.7984 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} - 2q^{6} - 4q^{7} - 4q^{8} + 6q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} - 2q^{6} - 4q^{7} - 4q^{8} + 6q^{9} + 4q^{10} + 2q^{12} - q^{13} + 4q^{14} - 2q^{15} + 4q^{16} - 2q^{17} - 6q^{18} + 3q^{19} - 4q^{20} - 2q^{21} - 8q^{23} - 2q^{24} + 4q^{25} + q^{26} + 14q^{27} - 4q^{28} - 15q^{29} + 2q^{30} + 4q^{31} - 4q^{32} + 2q^{34} + 4q^{35} + 6q^{36} + 2q^{37} - 3q^{38} + q^{39} + 4q^{40} - 11q^{41} + 2q^{42} - 7q^{43} - 6q^{45} + 8q^{46} + 5q^{47} + 2q^{48} + 4q^{49} - 4q^{50} - 18q^{51} - q^{52} + 10q^{53} - 14q^{54} + 4q^{56} + 34q^{57} + 15q^{58} + 13q^{59} - 2q^{60} - 26q^{61} - 4q^{62} - 6q^{63} + 4q^{64} + q^{65} - 13q^{67} - 2q^{68} - 4q^{69} - 4q^{70} - 13q^{71} - 6q^{72} + 18q^{73} - 2q^{74} + 2q^{75} + 3q^{76} - q^{78} - 15q^{79} - 4q^{80} - 8q^{81} + 11q^{82} + 2q^{83} - 2q^{84} + 2q^{85} + 7q^{86} - 26q^{87} - 7q^{89} + 6q^{90} + q^{91} - 8q^{92} + 2q^{93} - 5q^{94} - 3q^{95} - 2q^{96} + 10q^{97} - 4q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.919131 −0.530660 −0.265330 0.964158i \(-0.585481\pi\)
−0.265330 + 0.964158i \(0.585481\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.919131 0.375234
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.15520 −0.718400
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −0.919131 −0.265330
\(13\) 3.72325 1.03264 0.516322 0.856394i \(-0.327301\pi\)
0.516322 + 0.856394i \(0.327301\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.919131 0.237319
\(16\) 1.00000 0.250000
\(17\) −1.31694 −0.319404 −0.159702 0.987165i \(-0.551053\pi\)
−0.159702 + 0.987165i \(0.551053\pi\)
\(18\) 2.15520 0.507985
\(19\) −2.70741 −0.621123 −0.310561 0.950553i \(-0.600517\pi\)
−0.310561 + 0.950553i \(0.600517\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.919131 0.200571
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0.919131 0.187617
\(25\) 1.00000 0.200000
\(26\) −3.72325 −0.730190
\(27\) 4.73830 0.911887
\(28\) −1.00000 −0.188982
\(29\) −2.95002 −0.547805 −0.273902 0.961757i \(-0.588315\pi\)
−0.273902 + 0.961757i \(0.588315\pi\)
\(30\) −0.919131 −0.167810
\(31\) 7.70820 1.38443 0.692217 0.721689i \(-0.256634\pi\)
0.692217 + 0.721689i \(0.256634\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.31694 0.225853
\(35\) 1.00000 0.169031
\(36\) −2.15520 −0.359200
\(37\) −8.04870 −1.32320 −0.661599 0.749858i \(-0.730122\pi\)
−0.661599 + 0.749858i \(0.730122\pi\)
\(38\) 2.70741 0.439200
\(39\) −3.42216 −0.547984
\(40\) 1.00000 0.158114
\(41\) 3.24458 0.506718 0.253359 0.967372i \(-0.418465\pi\)
0.253359 + 0.967372i \(0.418465\pi\)
\(42\) −0.919131 −0.141825
\(43\) 6.48064 0.988289 0.494145 0.869380i \(-0.335481\pi\)
0.494145 + 0.869380i \(0.335481\pi\)
\(44\) 0 0
\(45\) 2.15520 0.321278
\(46\) 2.00000 0.294884
\(47\) −9.37090 −1.36689 −0.683443 0.730004i \(-0.739518\pi\)
−0.683443 + 0.730004i \(0.739518\pi\)
\(48\) −0.919131 −0.132665
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 1.21044 0.169495
\(52\) 3.72325 0.516322
\(53\) 13.2848 1.82480 0.912402 0.409296i \(-0.134226\pi\)
0.912402 + 0.409296i \(0.134226\pi\)
\(54\) −4.73830 −0.644801
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 2.48847 0.329605
\(58\) 2.95002 0.387357
\(59\) −2.83905 −0.369613 −0.184807 0.982775i \(-0.559166\pi\)
−0.184807 + 0.982775i \(0.559166\pi\)
\(60\) 0.919131 0.118659
\(61\) 1.44651 0.185206 0.0926030 0.995703i \(-0.470481\pi\)
0.0926030 + 0.995703i \(0.470481\pi\)
\(62\) −7.70820 −0.978943
\(63\) 2.15520 0.271530
\(64\) 1.00000 0.125000
\(65\) −3.72325 −0.461813
\(66\) 0 0
\(67\) −6.70741 −0.819441 −0.409720 0.912211i \(-0.634374\pi\)
−0.409720 + 0.912211i \(0.634374\pi\)
\(68\) −1.31694 −0.159702
\(69\) 1.83826 0.221301
\(70\) −1.00000 −0.119523
\(71\) 8.03365 0.953419 0.476709 0.879061i \(-0.341829\pi\)
0.476709 + 0.879061i \(0.341829\pi\)
\(72\) 2.15520 0.253993
\(73\) −1.33071 −0.155747 −0.0778736 0.996963i \(-0.524813\pi\)
−0.0778736 + 0.996963i \(0.524813\pi\)
\(74\) 8.04870 0.935642
\(75\) −0.919131 −0.106132
\(76\) −2.70741 −0.310561
\(77\) 0 0
\(78\) 3.42216 0.387483
\(79\) −7.72325 −0.868934 −0.434467 0.900688i \(-0.643063\pi\)
−0.434467 + 0.900688i \(0.643063\pi\)
\(80\) −1.00000 −0.111803
\(81\) 2.11048 0.234498
\(82\) −3.24458 −0.358304
\(83\) 0.285256 0.0313109 0.0156555 0.999877i \(-0.495017\pi\)
0.0156555 + 0.999877i \(0.495017\pi\)
\(84\) 0.919131 0.100285
\(85\) 1.31694 0.142842
\(86\) −6.48064 −0.698826
\(87\) 2.71145 0.290698
\(88\) 0 0
\(89\) −8.14015 −0.862854 −0.431427 0.902148i \(-0.641990\pi\)
−0.431427 + 0.902148i \(0.641990\pi\)
\(90\) −2.15520 −0.227178
\(91\) −3.72325 −0.390303
\(92\) −2.00000 −0.208514
\(93\) −7.08485 −0.734664
\(94\) 9.37090 0.966534
\(95\) 2.70741 0.277775
\(96\) 0.919131 0.0938084
\(97\) 14.7984 1.50255 0.751274 0.659991i \(-0.229440\pi\)
0.751274 + 0.659991i \(0.229440\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −14.1000 −1.40300 −0.701499 0.712670i \(-0.747486\pi\)
−0.701499 + 0.712670i \(0.747486\pi\)
\(102\) −1.21044 −0.119851
\(103\) 11.7627 1.15901 0.579504 0.814969i \(-0.303246\pi\)
0.579504 + 0.814969i \(0.303246\pi\)
\(104\) −3.72325 −0.365095
\(105\) −0.919131 −0.0896980
\(106\) −13.2848 −1.29033
\(107\) 19.8272 1.91677 0.958383 0.285484i \(-0.0921544\pi\)
0.958383 + 0.285484i \(0.0921544\pi\)
\(108\) 4.73830 0.455943
\(109\) −0.995533 −0.0953548 −0.0476774 0.998863i \(-0.515182\pi\)
−0.0476774 + 0.998863i \(0.515182\pi\)
\(110\) 0 0
\(111\) 7.39781 0.702169
\(112\) −1.00000 −0.0944911
\(113\) 11.0691 1.04129 0.520645 0.853773i \(-0.325691\pi\)
0.520645 + 0.853773i \(0.325691\pi\)
\(114\) −2.48847 −0.233066
\(115\) 2.00000 0.186501
\(116\) −2.95002 −0.273902
\(117\) −8.02435 −0.741851
\(118\) 2.83905 0.261356
\(119\) 1.31694 0.120723
\(120\) −0.919131 −0.0839048
\(121\) 0 0
\(122\) −1.44651 −0.130960
\(123\) −2.98219 −0.268895
\(124\) 7.70820 0.692217
\(125\) −1.00000 −0.0894427
\(126\) −2.15520 −0.192000
\(127\) 6.50223 0.576980 0.288490 0.957483i \(-0.406847\pi\)
0.288490 + 0.957483i \(0.406847\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.95656 −0.524446
\(130\) 3.72325 0.326551
\(131\) −21.7622 −1.90137 −0.950684 0.310160i \(-0.899617\pi\)
−0.950684 + 0.310160i \(0.899617\pi\)
\(132\) 0 0
\(133\) 2.70741 0.234762
\(134\) 6.70741 0.579432
\(135\) −4.73830 −0.407808
\(136\) 1.31694 0.112926
\(137\) −2.90408 −0.248112 −0.124056 0.992275i \(-0.539590\pi\)
−0.124056 + 0.992275i \(0.539590\pi\)
\(138\) −1.83826 −0.156483
\(139\) −1.26775 −0.107529 −0.0537645 0.998554i \(-0.517122\pi\)
−0.0537645 + 0.998554i \(0.517122\pi\)
\(140\) 1.00000 0.0845154
\(141\) 8.61308 0.725352
\(142\) −8.03365 −0.674169
\(143\) 0 0
\(144\) −2.15520 −0.179600
\(145\) 2.95002 0.244986
\(146\) 1.33071 0.110130
\(147\) −0.919131 −0.0758086
\(148\) −8.04870 −0.661599
\(149\) 5.85489 0.479652 0.239826 0.970816i \(-0.422910\pi\)
0.239826 + 0.970816i \(0.422910\pi\)
\(150\) 0.919131 0.0750467
\(151\) 10.2243 0.832039 0.416020 0.909356i \(-0.363425\pi\)
0.416020 + 0.909356i \(0.363425\pi\)
\(152\) 2.70741 0.219600
\(153\) 2.83826 0.229460
\(154\) 0 0
\(155\) −7.70820 −0.619138
\(156\) −3.42216 −0.273992
\(157\) 1.18004 0.0941772 0.0470886 0.998891i \(-0.485006\pi\)
0.0470886 + 0.998891i \(0.485006\pi\)
\(158\) 7.72325 0.614429
\(159\) −12.2104 −0.968351
\(160\) 1.00000 0.0790569
\(161\) 2.00000 0.157622
\(162\) −2.11048 −0.165815
\(163\) 13.5818 1.06381 0.531905 0.846804i \(-0.321476\pi\)
0.531905 + 0.846804i \(0.321476\pi\)
\(164\) 3.24458 0.253359
\(165\) 0 0
\(166\) −0.285256 −0.0221402
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −0.919131 −0.0709125
\(169\) 0.862611 0.0663547
\(170\) −1.31694 −0.101004
\(171\) 5.83501 0.446214
\(172\) 6.48064 0.494145
\(173\) −12.4741 −0.948389 −0.474194 0.880420i \(-0.657261\pi\)
−0.474194 + 0.880420i \(0.657261\pi\)
\(174\) −2.71145 −0.205555
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 2.60946 0.196139
\(178\) 8.14015 0.610130
\(179\) 12.2295 0.914078 0.457039 0.889447i \(-0.348910\pi\)
0.457039 + 0.889447i \(0.348910\pi\)
\(180\) 2.15520 0.160639
\(181\) 20.9718 1.55882 0.779411 0.626513i \(-0.215518\pi\)
0.779411 + 0.626513i \(0.215518\pi\)
\(182\) 3.72325 0.275986
\(183\) −1.32953 −0.0982815
\(184\) 2.00000 0.147442
\(185\) 8.04870 0.591752
\(186\) 7.08485 0.519486
\(187\) 0 0
\(188\) −9.37090 −0.683443
\(189\) −4.73830 −0.344661
\(190\) −2.70741 −0.196416
\(191\) 6.74632 0.488147 0.244073 0.969757i \(-0.421516\pi\)
0.244073 + 0.969757i \(0.421516\pi\)
\(192\) −0.919131 −0.0663325
\(193\) −19.9673 −1.43728 −0.718640 0.695382i \(-0.755235\pi\)
−0.718640 + 0.695382i \(0.755235\pi\)
\(194\) −14.7984 −1.06246
\(195\) 3.42216 0.245066
\(196\) 1.00000 0.0714286
\(197\) −21.1717 −1.50842 −0.754212 0.656631i \(-0.771981\pi\)
−0.754212 + 0.656631i \(0.771981\pi\)
\(198\) 0 0
\(199\) 8.27872 0.586863 0.293431 0.955980i \(-0.405203\pi\)
0.293431 + 0.955980i \(0.405203\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 6.16499 0.434845
\(202\) 14.1000 0.992070
\(203\) 2.95002 0.207051
\(204\) 1.21044 0.0847476
\(205\) −3.24458 −0.226611
\(206\) −11.7627 −0.819543
\(207\) 4.31040 0.299593
\(208\) 3.72325 0.258161
\(209\) 0 0
\(210\) 0.919131 0.0634260
\(211\) −15.5912 −1.07334 −0.536671 0.843792i \(-0.680318\pi\)
−0.536671 + 0.843792i \(0.680318\pi\)
\(212\) 13.2848 0.912402
\(213\) −7.38397 −0.505942
\(214\) −19.8272 −1.35536
\(215\) −6.48064 −0.441976
\(216\) −4.73830 −0.322401
\(217\) −7.70820 −0.523267
\(218\) 0.995533 0.0674260
\(219\) 1.22309 0.0826489
\(220\) 0 0
\(221\) −4.90329 −0.329831
\(222\) −7.39781 −0.496508
\(223\) −16.6313 −1.11372 −0.556858 0.830608i \(-0.687993\pi\)
−0.556858 + 0.830608i \(0.687993\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.15520 −0.143680
\(226\) −11.0691 −0.736304
\(227\) −1.19716 −0.0794582 −0.0397291 0.999210i \(-0.512649\pi\)
−0.0397291 + 0.999210i \(0.512649\pi\)
\(228\) 2.48847 0.164803
\(229\) 14.8613 0.982064 0.491032 0.871141i \(-0.336620\pi\)
0.491032 + 0.871141i \(0.336620\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 2.95002 0.193678
\(233\) 6.33199 0.414822 0.207411 0.978254i \(-0.433496\pi\)
0.207411 + 0.978254i \(0.433496\pi\)
\(234\) 8.02435 0.524568
\(235\) 9.37090 0.611290
\(236\) −2.83905 −0.184807
\(237\) 7.09868 0.461109
\(238\) −1.31694 −0.0853644
\(239\) 11.6906 0.756202 0.378101 0.925764i \(-0.376577\pi\)
0.378101 + 0.925764i \(0.376577\pi\)
\(240\) 0.919131 0.0593296
\(241\) −10.3945 −0.669566 −0.334783 0.942295i \(-0.608663\pi\)
−0.334783 + 0.942295i \(0.608663\pi\)
\(242\) 0 0
\(243\) −16.1547 −1.03633
\(244\) 1.44651 0.0926030
\(245\) −1.00000 −0.0638877
\(246\) 2.98219 0.190138
\(247\) −10.0804 −0.641399
\(248\) −7.70820 −0.489471
\(249\) −0.262188 −0.0166155
\(250\) 1.00000 0.0632456
\(251\) −22.3395 −1.41006 −0.705029 0.709179i \(-0.749066\pi\)
−0.705029 + 0.709179i \(0.749066\pi\)
\(252\) 2.15520 0.135765
\(253\) 0 0
\(254\) −6.50223 −0.407986
\(255\) −1.21044 −0.0758005
\(256\) 1.00000 0.0625000
\(257\) −5.28201 −0.329482 −0.164741 0.986337i \(-0.552679\pi\)
−0.164741 + 0.986337i \(0.552679\pi\)
\(258\) 5.95656 0.370839
\(259\) 8.04870 0.500122
\(260\) −3.72325 −0.230906
\(261\) 6.35788 0.393543
\(262\) 21.7622 1.34447
\(263\) −25.3847 −1.56529 −0.782645 0.622469i \(-0.786130\pi\)
−0.782645 + 0.622469i \(0.786130\pi\)
\(264\) 0 0
\(265\) −13.2848 −0.816077
\(266\) −2.70741 −0.166002
\(267\) 7.48186 0.457883
\(268\) −6.70741 −0.409720
\(269\) −15.3782 −0.937627 −0.468814 0.883297i \(-0.655318\pi\)
−0.468814 + 0.883297i \(0.655318\pi\)
\(270\) 4.73830 0.288364
\(271\) −10.3274 −0.627346 −0.313673 0.949531i \(-0.601560\pi\)
−0.313673 + 0.949531i \(0.601560\pi\)
\(272\) −1.31694 −0.0798510
\(273\) 3.42216 0.207118
\(274\) 2.90408 0.175442
\(275\) 0 0
\(276\) 1.83826 0.110650
\(277\) 22.7614 1.36760 0.683799 0.729670i \(-0.260326\pi\)
0.683799 + 0.729670i \(0.260326\pi\)
\(278\) 1.26775 0.0760345
\(279\) −16.6127 −0.994577
\(280\) −1.00000 −0.0597614
\(281\) 3.07890 0.183672 0.0918359 0.995774i \(-0.470726\pi\)
0.0918359 + 0.995774i \(0.470726\pi\)
\(282\) −8.61308 −0.512901
\(283\) 10.9791 0.652642 0.326321 0.945259i \(-0.394191\pi\)
0.326321 + 0.945259i \(0.394191\pi\)
\(284\) 8.03365 0.476709
\(285\) −2.48847 −0.147404
\(286\) 0 0
\(287\) −3.24458 −0.191521
\(288\) 2.15520 0.126996
\(289\) −15.2657 −0.897981
\(290\) −2.95002 −0.173231
\(291\) −13.6016 −0.797342
\(292\) −1.33071 −0.0778736
\(293\) −8.05126 −0.470360 −0.235180 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(294\) 0.919131 0.0536048
\(295\) 2.83905 0.165296
\(296\) 8.04870 0.467821
\(297\) 0 0
\(298\) −5.85489 −0.339165
\(299\) −7.44651 −0.430643
\(300\) −0.919131 −0.0530660
\(301\) −6.48064 −0.373538
\(302\) −10.2243 −0.588341
\(303\) 12.9597 0.744516
\(304\) −2.70741 −0.155281
\(305\) −1.44651 −0.0828267
\(306\) −2.83826 −0.162253
\(307\) 27.5963 1.57501 0.787503 0.616311i \(-0.211373\pi\)
0.787503 + 0.616311i \(0.211373\pi\)
\(308\) 0 0
\(309\) −10.8114 −0.615040
\(310\) 7.70820 0.437797
\(311\) 29.7870 1.68907 0.844533 0.535504i \(-0.179878\pi\)
0.844533 + 0.535504i \(0.179878\pi\)
\(312\) 3.42216 0.193741
\(313\) 30.9153 1.74744 0.873718 0.486433i \(-0.161702\pi\)
0.873718 + 0.486433i \(0.161702\pi\)
\(314\) −1.18004 −0.0665934
\(315\) −2.15520 −0.121432
\(316\) −7.72325 −0.434467
\(317\) −2.32189 −0.130411 −0.0652053 0.997872i \(-0.520770\pi\)
−0.0652053 + 0.997872i \(0.520770\pi\)
\(318\) 12.2104 0.684727
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −18.2238 −1.01715
\(322\) −2.00000 −0.111456
\(323\) 3.56549 0.198389
\(324\) 2.11048 0.117249
\(325\) 3.72325 0.206529
\(326\) −13.5818 −0.752228
\(327\) 0.915025 0.0506010
\(328\) −3.24458 −0.179152
\(329\) 9.37090 0.516634
\(330\) 0 0
\(331\) −9.95686 −0.547279 −0.273639 0.961832i \(-0.588227\pi\)
−0.273639 + 0.961832i \(0.588227\pi\)
\(332\) 0.285256 0.0156555
\(333\) 17.3465 0.950585
\(334\) 12.0000 0.656611
\(335\) 6.70741 0.366465
\(336\) 0.919131 0.0501427
\(337\) −20.7682 −1.13132 −0.565658 0.824640i \(-0.691378\pi\)
−0.565658 + 0.824640i \(0.691378\pi\)
\(338\) −0.862611 −0.0469198
\(339\) −10.1739 −0.552572
\(340\) 1.31694 0.0714210
\(341\) 0 0
\(342\) −5.83501 −0.315521
\(343\) −1.00000 −0.0539949
\(344\) −6.48064 −0.349413
\(345\) −1.83826 −0.0989687
\(346\) 12.4741 0.670612
\(347\) −15.7606 −0.846072 −0.423036 0.906113i \(-0.639036\pi\)
−0.423036 + 0.906113i \(0.639036\pi\)
\(348\) 2.71145 0.145349
\(349\) −23.2953 −1.24697 −0.623484 0.781836i \(-0.714283\pi\)
−0.623484 + 0.781836i \(0.714283\pi\)
\(350\) 1.00000 0.0534522
\(351\) 17.6419 0.941655
\(352\) 0 0
\(353\) 27.1303 1.44400 0.721999 0.691894i \(-0.243224\pi\)
0.721999 + 0.691894i \(0.243224\pi\)
\(354\) −2.60946 −0.138691
\(355\) −8.03365 −0.426382
\(356\) −8.14015 −0.431427
\(357\) −1.21044 −0.0640631
\(358\) −12.2295 −0.646351
\(359\) −32.0637 −1.69226 −0.846130 0.532977i \(-0.821073\pi\)
−0.846130 + 0.532977i \(0.821073\pi\)
\(360\) −2.15520 −0.113589
\(361\) −11.6699 −0.614206
\(362\) −20.9718 −1.10225
\(363\) 0 0
\(364\) −3.72325 −0.195151
\(365\) 1.33071 0.0696523
\(366\) 1.32953 0.0694955
\(367\) −11.6037 −0.605709 −0.302854 0.953037i \(-0.597940\pi\)
−0.302854 + 0.953037i \(0.597940\pi\)
\(368\) −2.00000 −0.104257
\(369\) −6.99271 −0.364026
\(370\) −8.04870 −0.418432
\(371\) −13.2848 −0.689711
\(372\) −7.08485 −0.367332
\(373\) 17.7252 0.917777 0.458889 0.888494i \(-0.348248\pi\)
0.458889 + 0.888494i \(0.348248\pi\)
\(374\) 0 0
\(375\) 0.919131 0.0474637
\(376\) 9.37090 0.483267
\(377\) −10.9837 −0.565688
\(378\) 4.73830 0.243712
\(379\) −12.0540 −0.619174 −0.309587 0.950871i \(-0.600191\pi\)
−0.309587 + 0.950871i \(0.600191\pi\)
\(380\) 2.70741 0.138887
\(381\) −5.97640 −0.306180
\(382\) −6.74632 −0.345172
\(383\) 6.75538 0.345184 0.172592 0.984993i \(-0.444786\pi\)
0.172592 + 0.984993i \(0.444786\pi\)
\(384\) 0.919131 0.0469042
\(385\) 0 0
\(386\) 19.9673 1.01631
\(387\) −13.9671 −0.709986
\(388\) 14.7984 0.751274
\(389\) −24.1441 −1.22416 −0.612078 0.790797i \(-0.709666\pi\)
−0.612078 + 0.790797i \(0.709666\pi\)
\(390\) −3.42216 −0.173288
\(391\) 2.63387 0.133201
\(392\) −1.00000 −0.0505076
\(393\) 20.0023 1.00898
\(394\) 21.1717 1.06662
\(395\) 7.72325 0.388599
\(396\) 0 0
\(397\) 10.5163 0.527798 0.263899 0.964550i \(-0.414991\pi\)
0.263899 + 0.964550i \(0.414991\pi\)
\(398\) −8.27872 −0.414975
\(399\) −2.48847 −0.124579
\(400\) 1.00000 0.0500000
\(401\) −37.1997 −1.85766 −0.928831 0.370503i \(-0.879185\pi\)
−0.928831 + 0.370503i \(0.879185\pi\)
\(402\) −6.16499 −0.307482
\(403\) 28.6996 1.42963
\(404\) −14.1000 −0.701499
\(405\) −2.11048 −0.104870
\(406\) −2.95002 −0.146407
\(407\) 0 0
\(408\) −1.21044 −0.0599256
\(409\) −26.3908 −1.30494 −0.652470 0.757815i \(-0.726267\pi\)
−0.652470 + 0.757815i \(0.726267\pi\)
\(410\) 3.24458 0.160238
\(411\) 2.66923 0.131663
\(412\) 11.7627 0.579504
\(413\) 2.83905 0.139701
\(414\) −4.31040 −0.211844
\(415\) −0.285256 −0.0140027
\(416\) −3.72325 −0.182547
\(417\) 1.16523 0.0570614
\(418\) 0 0
\(419\) 13.0447 0.637273 0.318637 0.947877i \(-0.396775\pi\)
0.318637 + 0.947877i \(0.396775\pi\)
\(420\) −0.919131 −0.0448490
\(421\) −13.4510 −0.655563 −0.327782 0.944753i \(-0.606301\pi\)
−0.327782 + 0.944753i \(0.606301\pi\)
\(422\) 15.5912 0.758967
\(423\) 20.1961 0.981970
\(424\) −13.2848 −0.645165
\(425\) −1.31694 −0.0638808
\(426\) 7.38397 0.357755
\(427\) −1.44651 −0.0700013
\(428\) 19.8272 0.958383
\(429\) 0 0
\(430\) 6.48064 0.312524
\(431\) −32.7451 −1.57728 −0.788638 0.614858i \(-0.789213\pi\)
−0.788638 + 0.614858i \(0.789213\pi\)
\(432\) 4.73830 0.227972
\(433\) 9.00434 0.432721 0.216361 0.976314i \(-0.430581\pi\)
0.216361 + 0.976314i \(0.430581\pi\)
\(434\) 7.70820 0.370006
\(435\) −2.71145 −0.130004
\(436\) −0.995533 −0.0476774
\(437\) 5.41482 0.259026
\(438\) −1.22309 −0.0584416
\(439\) −3.88952 −0.185637 −0.0928184 0.995683i \(-0.529588\pi\)
−0.0928184 + 0.995683i \(0.529588\pi\)
\(440\) 0 0
\(441\) −2.15520 −0.102629
\(442\) 4.90329 0.233226
\(443\) 5.63377 0.267669 0.133834 0.991004i \(-0.457271\pi\)
0.133834 + 0.991004i \(0.457271\pi\)
\(444\) 7.39781 0.351084
\(445\) 8.14015 0.385880
\(446\) 16.6313 0.787515
\(447\) −5.38141 −0.254532
\(448\) −1.00000 −0.0472456
\(449\) −5.62855 −0.265628 −0.132814 0.991141i \(-0.542401\pi\)
−0.132814 + 0.991141i \(0.542401\pi\)
\(450\) 2.15520 0.101597
\(451\) 0 0
\(452\) 11.0691 0.520645
\(453\) −9.39744 −0.441530
\(454\) 1.19716 0.0561855
\(455\) 3.72325 0.174549
\(456\) −2.48847 −0.116533
\(457\) −1.15770 −0.0541548 −0.0270774 0.999633i \(-0.508620\pi\)
−0.0270774 + 0.999633i \(0.508620\pi\)
\(458\) −14.8613 −0.694424
\(459\) −6.24005 −0.291260
\(460\) 2.00000 0.0932505
\(461\) −10.3088 −0.480129 −0.240065 0.970757i \(-0.577169\pi\)
−0.240065 + 0.970757i \(0.577169\pi\)
\(462\) 0 0
\(463\) −29.4740 −1.36977 −0.684887 0.728649i \(-0.740149\pi\)
−0.684887 + 0.728649i \(0.740149\pi\)
\(464\) −2.95002 −0.136951
\(465\) 7.08485 0.328552
\(466\) −6.33199 −0.293324
\(467\) 20.5977 0.953146 0.476573 0.879135i \(-0.341879\pi\)
0.476573 + 0.879135i \(0.341879\pi\)
\(468\) −8.02435 −0.370926
\(469\) 6.70741 0.309720
\(470\) −9.37090 −0.432247
\(471\) −1.08461 −0.0499761
\(472\) 2.83905 0.130678
\(473\) 0 0
\(474\) −7.09868 −0.326053
\(475\) −2.70741 −0.124225
\(476\) 1.31694 0.0603617
\(477\) −28.6313 −1.31094
\(478\) −11.6906 −0.534715
\(479\) 4.18526 0.191229 0.0956146 0.995418i \(-0.469518\pi\)
0.0956146 + 0.995418i \(0.469518\pi\)
\(480\) −0.919131 −0.0419524
\(481\) −29.9673 −1.36639
\(482\) 10.3945 0.473454
\(483\) −1.83826 −0.0836438
\(484\) 0 0
\(485\) −14.7984 −0.671960
\(486\) 16.1547 0.732792
\(487\) −8.06730 −0.365564 −0.182782 0.983153i \(-0.558510\pi\)
−0.182782 + 0.983153i \(0.558510\pi\)
\(488\) −1.44651 −0.0654802
\(489\) −12.4835 −0.564522
\(490\) 1.00000 0.0451754
\(491\) 23.5882 1.06452 0.532261 0.846580i \(-0.321342\pi\)
0.532261 + 0.846580i \(0.321342\pi\)
\(492\) −2.98219 −0.134448
\(493\) 3.88499 0.174971
\(494\) 10.0804 0.453538
\(495\) 0 0
\(496\) 7.70820 0.346109
\(497\) −8.03365 −0.360358
\(498\) 0.262188 0.0117489
\(499\) 7.22181 0.323293 0.161646 0.986849i \(-0.448320\pi\)
0.161646 + 0.986849i \(0.448320\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 11.0296 0.492765
\(502\) 22.3395 0.997061
\(503\) 11.3828 0.507532 0.253766 0.967266i \(-0.418331\pi\)
0.253766 + 0.967266i \(0.418331\pi\)
\(504\) −2.15520 −0.0960002
\(505\) 14.1000 0.627440
\(506\) 0 0
\(507\) −0.792852 −0.0352118
\(508\) 6.50223 0.288490
\(509\) 12.1422 0.538192 0.269096 0.963113i \(-0.413275\pi\)
0.269096 + 0.963113i \(0.413275\pi\)
\(510\) 1.21044 0.0535991
\(511\) 1.33071 0.0588669
\(512\) −1.00000 −0.0441942
\(513\) −12.8285 −0.566394
\(514\) 5.28201 0.232979
\(515\) −11.7627 −0.518324
\(516\) −5.95656 −0.262223
\(517\) 0 0
\(518\) −8.04870 −0.353640
\(519\) 11.4653 0.503272
\(520\) 3.72325 0.163275
\(521\) 12.1565 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(522\) −6.35788 −0.278277
\(523\) −29.3199 −1.28207 −0.641035 0.767512i \(-0.721495\pi\)
−0.641035 + 0.767512i \(0.721495\pi\)
\(524\) −21.7622 −0.950684
\(525\) 0.919131 0.0401142
\(526\) 25.3847 1.10683
\(527\) −10.1512 −0.442194
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 13.2848 0.577053
\(531\) 6.11872 0.265530
\(532\) 2.70741 0.117381
\(533\) 12.0804 0.523259
\(534\) −7.48186 −0.323772
\(535\) −19.8272 −0.857204
\(536\) 6.70741 0.289716
\(537\) −11.2405 −0.485065
\(538\) 15.3782 0.663002
\(539\) 0 0
\(540\) −4.73830 −0.203904
\(541\) 2.54640 0.109478 0.0547392 0.998501i \(-0.482567\pi\)
0.0547392 + 0.998501i \(0.482567\pi\)
\(542\) 10.3274 0.443600
\(543\) −19.2758 −0.827205
\(544\) 1.31694 0.0564632
\(545\) 0.995533 0.0426439
\(546\) −3.42216 −0.146455
\(547\) 32.1359 1.37403 0.687016 0.726642i \(-0.258920\pi\)
0.687016 + 0.726642i \(0.258920\pi\)
\(548\) −2.90408 −0.124056
\(549\) −3.11751 −0.133052
\(550\) 0 0
\(551\) 7.98692 0.340254
\(552\) −1.83826 −0.0782416
\(553\) 7.72325 0.328426
\(554\) −22.7614 −0.967038
\(555\) −7.39781 −0.314019
\(556\) −1.26775 −0.0537645
\(557\) 5.98042 0.253399 0.126699 0.991941i \(-0.459562\pi\)
0.126699 + 0.991941i \(0.459562\pi\)
\(558\) 16.6127 0.703272
\(559\) 24.1291 1.02055
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −3.07890 −0.129876
\(563\) −1.80284 −0.0759807 −0.0379903 0.999278i \(-0.512096\pi\)
−0.0379903 + 0.999278i \(0.512096\pi\)
\(564\) 8.61308 0.362676
\(565\) −11.0691 −0.465679
\(566\) −10.9791 −0.461488
\(567\) −2.11048 −0.0886317
\(568\) −8.03365 −0.337084
\(569\) −30.1377 −1.26344 −0.631718 0.775198i \(-0.717650\pi\)
−0.631718 + 0.775198i \(0.717650\pi\)
\(570\) 2.48847 0.104230
\(571\) −30.3322 −1.26936 −0.634681 0.772774i \(-0.718869\pi\)
−0.634681 + 0.772774i \(0.718869\pi\)
\(572\) 0 0
\(573\) −6.20075 −0.259040
\(574\) 3.24458 0.135426
\(575\) −2.00000 −0.0834058
\(576\) −2.15520 −0.0897999
\(577\) −15.7849 −0.657135 −0.328568 0.944480i \(-0.606566\pi\)
−0.328568 + 0.944480i \(0.606566\pi\)
\(578\) 15.2657 0.634968
\(579\) 18.3526 0.762708
\(580\) 2.95002 0.122493
\(581\) −0.285256 −0.0118344
\(582\) 13.6016 0.563806
\(583\) 0 0
\(584\) 1.33071 0.0550650
\(585\) 8.02435 0.331766
\(586\) 8.05126 0.332595
\(587\) 25.9219 1.06991 0.534956 0.844880i \(-0.320328\pi\)
0.534956 + 0.844880i \(0.320328\pi\)
\(588\) −0.919131 −0.0379043
\(589\) −20.8693 −0.859904
\(590\) −2.83905 −0.116882
\(591\) 19.4596 0.800460
\(592\) −8.04870 −0.330799
\(593\) 2.58469 0.106140 0.0530702 0.998591i \(-0.483099\pi\)
0.0530702 + 0.998591i \(0.483099\pi\)
\(594\) 0 0
\(595\) −1.31694 −0.0539892
\(596\) 5.85489 0.239826
\(597\) −7.60922 −0.311425
\(598\) 7.44651 0.304510
\(599\) 8.01529 0.327496 0.163748 0.986502i \(-0.447642\pi\)
0.163748 + 0.986502i \(0.447642\pi\)
\(600\) 0.919131 0.0375234
\(601\) −1.07488 −0.0438454 −0.0219227 0.999760i \(-0.506979\pi\)
−0.0219227 + 0.999760i \(0.506979\pi\)
\(602\) 6.48064 0.264131
\(603\) 14.4558 0.588686
\(604\) 10.2243 0.416020
\(605\) 0 0
\(606\) −12.9597 −0.526452
\(607\) −12.6752 −0.514472 −0.257236 0.966349i \(-0.582812\pi\)
−0.257236 + 0.966349i \(0.582812\pi\)
\(608\) 2.70741 0.109800
\(609\) −2.71145 −0.109874
\(610\) 1.44651 0.0585673
\(611\) −34.8902 −1.41151
\(612\) 2.83826 0.114730
\(613\) 21.0929 0.851935 0.425968 0.904738i \(-0.359934\pi\)
0.425968 + 0.904738i \(0.359934\pi\)
\(614\) −27.5963 −1.11370
\(615\) 2.98219 0.120254
\(616\) 0 0
\(617\) −15.4070 −0.620263 −0.310131 0.950694i \(-0.600373\pi\)
−0.310131 + 0.950694i \(0.600373\pi\)
\(618\) 10.8114 0.434899
\(619\) 18.6431 0.749328 0.374664 0.927161i \(-0.377758\pi\)
0.374664 + 0.927161i \(0.377758\pi\)
\(620\) −7.70820 −0.309569
\(621\) −9.47660 −0.380283
\(622\) −29.7870 −1.19435
\(623\) 8.14015 0.326128
\(624\) −3.42216 −0.136996
\(625\) 1.00000 0.0400000
\(626\) −30.9153 −1.23562
\(627\) 0 0
\(628\) 1.18004 0.0470886
\(629\) 10.5996 0.422635
\(630\) 2.15520 0.0858652
\(631\) 27.4709 1.09360 0.546799 0.837264i \(-0.315846\pi\)
0.546799 + 0.837264i \(0.315846\pi\)
\(632\) 7.72325 0.307214
\(633\) 14.3303 0.569580
\(634\) 2.32189 0.0922142
\(635\) −6.50223 −0.258033
\(636\) −12.2104 −0.484175
\(637\) 3.72325 0.147521
\(638\) 0 0
\(639\) −17.3141 −0.684936
\(640\) 1.00000 0.0395285
\(641\) −32.4224 −1.28061 −0.640304 0.768121i \(-0.721192\pi\)
−0.640304 + 0.768121i \(0.721192\pi\)
\(642\) 18.2238 0.719235
\(643\) 1.62945 0.0642591 0.0321296 0.999484i \(-0.489771\pi\)
0.0321296 + 0.999484i \(0.489771\pi\)
\(644\) 2.00000 0.0788110
\(645\) 5.95656 0.234539
\(646\) −3.56549 −0.140282
\(647\) 36.2914 1.42676 0.713382 0.700775i \(-0.247163\pi\)
0.713382 + 0.700775i \(0.247163\pi\)
\(648\) −2.11048 −0.0829074
\(649\) 0 0
\(650\) −3.72325 −0.146038
\(651\) 7.08485 0.277677
\(652\) 13.5818 0.531905
\(653\) −32.1275 −1.25725 −0.628623 0.777710i \(-0.716381\pi\)
−0.628623 + 0.777710i \(0.716381\pi\)
\(654\) −0.915025 −0.0357803
\(655\) 21.7622 0.850318
\(656\) 3.24458 0.126679
\(657\) 2.86793 0.111889
\(658\) −9.37090 −0.365316
\(659\) 24.7215 0.963012 0.481506 0.876443i \(-0.340090\pi\)
0.481506 + 0.876443i \(0.340090\pi\)
\(660\) 0 0
\(661\) −24.3370 −0.946598 −0.473299 0.880902i \(-0.656937\pi\)
−0.473299 + 0.880902i \(0.656937\pi\)
\(662\) 9.95686 0.386984
\(663\) 4.50676 0.175028
\(664\) −0.285256 −0.0110701
\(665\) −2.70741 −0.104989
\(666\) −17.3465 −0.672165
\(667\) 5.90004 0.228450
\(668\) −12.0000 −0.464294
\(669\) 15.2864 0.591004
\(670\) −6.70741 −0.259130
\(671\) 0 0
\(672\) −0.919131 −0.0354562
\(673\) −43.7873 −1.68787 −0.843937 0.536442i \(-0.819768\pi\)
−0.843937 + 0.536442i \(0.819768\pi\)
\(674\) 20.7682 0.799962
\(675\) 4.73830 0.182377
\(676\) 0.862611 0.0331773
\(677\) 20.9199 0.804018 0.402009 0.915636i \(-0.368312\pi\)
0.402009 + 0.915636i \(0.368312\pi\)
\(678\) 10.1739 0.390727
\(679\) −14.7984 −0.567909
\(680\) −1.31694 −0.0505022
\(681\) 1.10035 0.0421653
\(682\) 0 0
\(683\) −6.79955 −0.260178 −0.130089 0.991502i \(-0.541526\pi\)
−0.130089 + 0.991502i \(0.541526\pi\)
\(684\) 5.83501 0.223107
\(685\) 2.90408 0.110959
\(686\) 1.00000 0.0381802
\(687\) −13.6595 −0.521143
\(688\) 6.48064 0.247072
\(689\) 49.4625 1.88437
\(690\) 1.83826 0.0699814
\(691\) 33.0127 1.25586 0.627931 0.778269i \(-0.283902\pi\)
0.627931 + 0.778269i \(0.283902\pi\)
\(692\) −12.4741 −0.474194
\(693\) 0 0
\(694\) 15.7606 0.598263
\(695\) 1.26775 0.0480885
\(696\) −2.71145 −0.102777
\(697\) −4.27290 −0.161848
\(698\) 23.2953 0.881740
\(699\) −5.81992 −0.220130
\(700\) −1.00000 −0.0377964
\(701\) −34.2118 −1.29216 −0.646081 0.763269i \(-0.723593\pi\)
−0.646081 + 0.763269i \(0.723593\pi\)
\(702\) −17.6419 −0.665850
\(703\) 21.7911 0.821869
\(704\) 0 0
\(705\) −8.61308 −0.324387
\(706\) −27.1303 −1.02106
\(707\) 14.1000 0.530284
\(708\) 2.60946 0.0980695
\(709\) −10.7815 −0.404909 −0.202455 0.979292i \(-0.564892\pi\)
−0.202455 + 0.979292i \(0.564892\pi\)
\(710\) 8.03365 0.301498
\(711\) 16.6451 0.624242
\(712\) 8.14015 0.305065
\(713\) −15.4164 −0.577349
\(714\) 1.21044 0.0452995
\(715\) 0 0
\(716\) 12.2295 0.457039
\(717\) −10.7452 −0.401286
\(718\) 32.0637 1.19661
\(719\) −48.7392 −1.81767 −0.908833 0.417160i \(-0.863026\pi\)
−0.908833 + 0.417160i \(0.863026\pi\)
\(720\) 2.15520 0.0803195
\(721\) −11.7627 −0.438064
\(722\) 11.6699 0.434309
\(723\) 9.55386 0.355312
\(724\) 20.9718 0.779411
\(725\) −2.95002 −0.109561
\(726\) 0 0
\(727\) 11.9416 0.442891 0.221446 0.975173i \(-0.428922\pi\)
0.221446 + 0.975173i \(0.428922\pi\)
\(728\) 3.72325 0.137993
\(729\) 8.51686 0.315439
\(730\) −1.33071 −0.0492516
\(731\) −8.53460 −0.315664
\(732\) −1.32953 −0.0491408
\(733\) −38.9332 −1.43803 −0.719015 0.694995i \(-0.755407\pi\)
−0.719015 + 0.694995i \(0.755407\pi\)
\(734\) 11.6037 0.428301
\(735\) 0.919131 0.0339026
\(736\) 2.00000 0.0737210
\(737\) 0 0
\(738\) 6.99271 0.257405
\(739\) −29.3583 −1.07996 −0.539981 0.841677i \(-0.681569\pi\)
−0.539981 + 0.841677i \(0.681569\pi\)
\(740\) 8.04870 0.295876
\(741\) 9.26519 0.340365
\(742\) 13.2848 0.487699
\(743\) −27.1351 −0.995491 −0.497746 0.867323i \(-0.665839\pi\)
−0.497746 + 0.867323i \(0.665839\pi\)
\(744\) 7.08485 0.259743
\(745\) −5.85489 −0.214507
\(746\) −17.7252 −0.648966
\(747\) −0.614784 −0.0224938
\(748\) 0 0
\(749\) −19.8272 −0.724470
\(750\) −0.919131 −0.0335619
\(751\) −36.8507 −1.34470 −0.672350 0.740233i \(-0.734715\pi\)
−0.672350 + 0.740233i \(0.734715\pi\)
\(752\) −9.37090 −0.341721
\(753\) 20.5329 0.748262
\(754\) 10.9837 0.400002
\(755\) −10.2243 −0.372099
\(756\) −4.73830 −0.172330
\(757\) 1.04667 0.0380417 0.0190209 0.999819i \(-0.493945\pi\)
0.0190209 + 0.999819i \(0.493945\pi\)
\(758\) 12.0540 0.437822
\(759\) 0 0
\(760\) −2.70741 −0.0982082
\(761\) −2.62959 −0.0953227 −0.0476614 0.998864i \(-0.515177\pi\)
−0.0476614 + 0.998864i \(0.515177\pi\)
\(762\) 5.97640 0.216502
\(763\) 0.995533 0.0360407
\(764\) 6.74632 0.244073
\(765\) −2.83826 −0.102618
\(766\) −6.75538 −0.244082
\(767\) −10.5705 −0.381679
\(768\) −0.919131 −0.0331663
\(769\) −31.0270 −1.11886 −0.559431 0.828877i \(-0.688980\pi\)
−0.559431 + 0.828877i \(0.688980\pi\)
\(770\) 0 0
\(771\) 4.85485 0.174843
\(772\) −19.9673 −0.718640
\(773\) −8.84786 −0.318236 −0.159118 0.987260i \(-0.550865\pi\)
−0.159118 + 0.987260i \(0.550865\pi\)
\(774\) 13.9671 0.502036
\(775\) 7.70820 0.276887
\(776\) −14.7984 −0.531231
\(777\) −7.39781 −0.265395
\(778\) 24.1441 0.865609
\(779\) −8.78441 −0.314734
\(780\) 3.42216 0.122533
\(781\) 0 0
\(782\) −2.63387 −0.0941872
\(783\) −13.9781 −0.499536
\(784\) 1.00000 0.0357143
\(785\) −1.18004 −0.0421173
\(786\) −20.0023 −0.713457
\(787\) −6.15195 −0.219293 −0.109647 0.993971i \(-0.534972\pi\)
−0.109647 + 0.993971i \(0.534972\pi\)
\(788\) −21.1717 −0.754212
\(789\) 23.3319 0.830637
\(790\) −7.72325 −0.274781
\(791\) −11.0691 −0.393571
\(792\) 0 0
\(793\) 5.38571 0.191252
\(794\) −10.5163 −0.373210
\(795\) 12.2104 0.433060
\(796\) 8.27872 0.293431
\(797\) −47.3736 −1.67806 −0.839030 0.544085i \(-0.816877\pi\)
−0.839030 + 0.544085i \(0.816877\pi\)
\(798\) 2.48847 0.0880907
\(799\) 12.3409 0.436589
\(800\) −1.00000 −0.0353553
\(801\) 17.5436 0.619874
\(802\) 37.1997 1.31357
\(803\) 0 0
\(804\) 6.16499 0.217422
\(805\) −2.00000 −0.0704907
\(806\) −28.6996 −1.01090
\(807\) 14.1346 0.497562
\(808\) 14.1000 0.496035
\(809\) −9.79837 −0.344492 −0.172246 0.985054i \(-0.555102\pi\)
−0.172246 + 0.985054i \(0.555102\pi\)
\(810\) 2.11048 0.0741546
\(811\) −20.9999 −0.737406 −0.368703 0.929547i \(-0.620198\pi\)
−0.368703 + 0.929547i \(0.620198\pi\)
\(812\) 2.95002 0.103525
\(813\) 9.49224 0.332908
\(814\) 0 0
\(815\) −13.5818 −0.475750
\(816\) 1.21044 0.0423738
\(817\) −17.5458 −0.613849
\(818\) 26.3908 0.922732
\(819\) 8.02435 0.280393
\(820\) −3.24458 −0.113306
\(821\) −48.1483 −1.68039 −0.840194 0.542286i \(-0.817559\pi\)
−0.840194 + 0.542286i \(0.817559\pi\)
\(822\) −2.66923 −0.0931001
\(823\) −35.5348 −1.23867 −0.619333 0.785128i \(-0.712597\pi\)
−0.619333 + 0.785128i \(0.712597\pi\)
\(824\) −11.7627 −0.409771
\(825\) 0 0
\(826\) −2.83905 −0.0987833
\(827\) −44.0977 −1.53343 −0.766714 0.641988i \(-0.778110\pi\)
−0.766714 + 0.641988i \(0.778110\pi\)
\(828\) 4.31040 0.149797
\(829\) −6.77860 −0.235430 −0.117715 0.993047i \(-0.537557\pi\)
−0.117715 + 0.993047i \(0.537557\pi\)
\(830\) 0.285256 0.00990139
\(831\) −20.9207 −0.725730
\(832\) 3.72325 0.129081
\(833\) −1.31694 −0.0456292
\(834\) −1.16523 −0.0403485
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 36.5238 1.26245
\(838\) −13.0447 −0.450620
\(839\) −53.6528 −1.85230 −0.926150 0.377155i \(-0.876902\pi\)
−0.926150 + 0.377155i \(0.876902\pi\)
\(840\) 0.919131 0.0317130
\(841\) −20.2974 −0.699910
\(842\) 13.4510 0.463553
\(843\) −2.82991 −0.0974673
\(844\) −15.5912 −0.536671
\(845\) −0.862611 −0.0296747
\(846\) −20.1961 −0.694358
\(847\) 0 0
\(848\) 13.2848 0.456201
\(849\) −10.0913 −0.346331
\(850\) 1.31694 0.0451706
\(851\) 16.0974 0.551812
\(852\) −7.38397 −0.252971
\(853\) −8.65701 −0.296410 −0.148205 0.988957i \(-0.547350\pi\)
−0.148205 + 0.988957i \(0.547350\pi\)
\(854\) 1.44651 0.0494984
\(855\) −5.83501 −0.199553
\(856\) −19.8272 −0.677679
\(857\) −46.7535 −1.59707 −0.798534 0.601950i \(-0.794391\pi\)
−0.798534 + 0.601950i \(0.794391\pi\)
\(858\) 0 0
\(859\) −35.5377 −1.21253 −0.606265 0.795263i \(-0.707333\pi\)
−0.606265 + 0.795263i \(0.707333\pi\)
\(860\) −6.48064 −0.220988
\(861\) 2.98219 0.101633
\(862\) 32.7451 1.11530
\(863\) 40.0753 1.36418 0.682089 0.731269i \(-0.261072\pi\)
0.682089 + 0.731269i \(0.261072\pi\)
\(864\) −4.73830 −0.161200
\(865\) 12.4741 0.424132
\(866\) −9.00434 −0.305980
\(867\) 14.0312 0.476523
\(868\) −7.70820 −0.261633
\(869\) 0 0
\(870\) 2.71145 0.0919269
\(871\) −24.9734 −0.846191
\(872\) 0.995533 0.0337130
\(873\) −31.8934 −1.07943
\(874\) −5.41482 −0.183159
\(875\) 1.00000 0.0338062
\(876\) 1.22309 0.0413244
\(877\) 37.0839 1.25223 0.626117 0.779729i \(-0.284643\pi\)
0.626117 + 0.779729i \(0.284643\pi\)
\(878\) 3.88952 0.131265
\(879\) 7.40016 0.249601
\(880\) 0 0
\(881\) −24.7764 −0.834737 −0.417369 0.908737i \(-0.637048\pi\)
−0.417369 + 0.908737i \(0.637048\pi\)
\(882\) 2.15520 0.0725693
\(883\) −5.49721 −0.184996 −0.0924980 0.995713i \(-0.529485\pi\)
−0.0924980 + 0.995713i \(0.529485\pi\)
\(884\) −4.90329 −0.164915
\(885\) −2.60946 −0.0877161
\(886\) −5.63377 −0.189270
\(887\) 22.5044 0.755625 0.377813 0.925882i \(-0.376676\pi\)
0.377813 + 0.925882i \(0.376676\pi\)
\(888\) −7.39781 −0.248254
\(889\) −6.50223 −0.218078
\(890\) −8.14015 −0.272858
\(891\) 0 0
\(892\) −16.6313 −0.556858
\(893\) 25.3709 0.849004
\(894\) 5.38141 0.179981
\(895\) −12.2295 −0.408788
\(896\) 1.00000 0.0334077
\(897\) 6.84431 0.228525
\(898\) 5.62855 0.187827
\(899\) −22.7394 −0.758400
\(900\) −2.15520 −0.0718400
\(901\) −17.4952 −0.582850
\(902\) 0 0
\(903\) 5.95656 0.198222
\(904\) −11.0691 −0.368152
\(905\) −20.9718 −0.697126
\(906\) 9.39744 0.312209
\(907\) −10.0800 −0.334702 −0.167351 0.985897i \(-0.553521\pi\)
−0.167351 + 0.985897i \(0.553521\pi\)
\(908\) −1.19716 −0.0397291
\(909\) 30.3882 1.00791
\(910\) −3.72325 −0.123425
\(911\) 16.3242 0.540846 0.270423 0.962742i \(-0.412836\pi\)
0.270423 + 0.962742i \(0.412836\pi\)
\(912\) 2.48847 0.0824013
\(913\) 0 0
\(914\) 1.15770 0.0382932
\(915\) 1.32953 0.0439528
\(916\) 14.8613 0.491032
\(917\) 21.7622 0.718650
\(918\) 6.24005 0.205952
\(919\) 18.9580 0.625366 0.312683 0.949858i \(-0.398772\pi\)
0.312683 + 0.949858i \(0.398772\pi\)
\(920\) −2.00000 −0.0659380
\(921\) −25.3646 −0.835793
\(922\) 10.3088 0.339503
\(923\) 29.9113 0.984543
\(924\) 0 0
\(925\) −8.04870 −0.264640
\(926\) 29.4740 0.968577
\(927\) −25.3509 −0.832631
\(928\) 2.95002 0.0968392
\(929\) −15.5873 −0.511404 −0.255702 0.966756i \(-0.582307\pi\)
−0.255702 + 0.966756i \(0.582307\pi\)
\(930\) −7.08485 −0.232321
\(931\) −2.70741 −0.0887319
\(932\) 6.33199 0.207411
\(933\) −27.3781 −0.896320
\(934\) −20.5977 −0.673976
\(935\) 0 0
\(936\) 8.02435 0.262284
\(937\) 32.7215 1.06896 0.534482 0.845180i \(-0.320507\pi\)
0.534482 + 0.845180i \(0.320507\pi\)
\(938\) −6.70741 −0.219005
\(939\) −28.4152 −0.927295
\(940\) 9.37090 0.305645
\(941\) −14.5193 −0.473314 −0.236657 0.971593i \(-0.576052\pi\)
−0.236657 + 0.971593i \(0.576052\pi\)
\(942\) 1.08461 0.0353385
\(943\) −6.48915 −0.211316
\(944\) −2.83905 −0.0924033
\(945\) 4.73830 0.154137
\(946\) 0 0
\(947\) 16.5854 0.538954 0.269477 0.963007i \(-0.413149\pi\)
0.269477 + 0.963007i \(0.413149\pi\)
\(948\) 7.09868 0.230554
\(949\) −4.95455 −0.160832
\(950\) 2.70741 0.0878401
\(951\) 2.13412 0.0692037
\(952\) −1.31694 −0.0426822
\(953\) 29.5354 0.956745 0.478373 0.878157i \(-0.341227\pi\)
0.478373 + 0.878157i \(0.341227\pi\)
\(954\) 28.6313 0.926973
\(955\) −6.74632 −0.218306
\(956\) 11.6906 0.378101
\(957\) 0 0
\(958\) −4.18526 −0.135220
\(959\) 2.90408 0.0937777
\(960\) 0.919131 0.0296648
\(961\) 28.4164 0.916658
\(962\) 29.9673 0.966186
\(963\) −42.7315 −1.37700
\(964\) −10.3945 −0.334783
\(965\) 19.9673 0.642771
\(966\) 1.83826 0.0591451
\(967\) 36.7434 1.18159 0.590794 0.806823i \(-0.298815\pi\)
0.590794 + 0.806823i \(0.298815\pi\)
\(968\) 0 0
\(969\) −3.27715 −0.105277
\(970\) 14.7984 0.475147
\(971\) −22.2877 −0.715245 −0.357622 0.933866i \(-0.616412\pi\)
−0.357622 + 0.933866i \(0.616412\pi\)
\(972\) −16.1547 −0.518163
\(973\) 1.26775 0.0406422
\(974\) 8.06730 0.258493
\(975\) −3.42216 −0.109597
\(976\) 1.44651 0.0463015
\(977\) −40.5549 −1.29747 −0.648733 0.761016i \(-0.724701\pi\)
−0.648733 + 0.761016i \(0.724701\pi\)
\(978\) 12.4835 0.399177
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 2.14557 0.0685028
\(982\) −23.5882 −0.752731
\(983\) −39.8706 −1.27167 −0.635837 0.771824i \(-0.719345\pi\)
−0.635837 + 0.771824i \(0.719345\pi\)
\(984\) 2.98219 0.0950688
\(985\) 21.1717 0.674587
\(986\) −3.88499 −0.123723
\(987\) −8.61308 −0.274157
\(988\) −10.0804 −0.320700
\(989\) −12.9613 −0.412145
\(990\) 0 0
\(991\) 35.5410 1.12900 0.564498 0.825435i \(-0.309070\pi\)
0.564498 + 0.825435i \(0.309070\pi\)
\(992\) −7.70820 −0.244736
\(993\) 9.15166 0.290419
\(994\) 8.03365 0.254812
\(995\) −8.27872 −0.262453
\(996\) −0.262188 −0.00830774
\(997\) 47.4408 1.50246 0.751232 0.660038i \(-0.229460\pi\)
0.751232 + 0.660038i \(0.229460\pi\)
\(998\) −7.22181 −0.228602
\(999\) −38.1372 −1.20661
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.cq.1.2 4
11.3 even 5 770.2.n.e.141.2 yes 8
11.4 even 5 770.2.n.e.71.2 8
11.10 odd 2 8470.2.a.ct.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.e.71.2 8 11.4 even 5
770.2.n.e.141.2 yes 8 11.3 even 5
8470.2.a.cq.1.2 4 1.1 even 1 trivial
8470.2.a.ct.1.2 4 11.10 odd 2