Properties

Label 8470.2.a.cq.1.1
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 1.1
Root \(1.26498\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.04678 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.18929 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.04678 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.18929 q^{9} +1.00000 q^{10} -2.04678 q^{12} -3.50105 q^{13} +1.00000 q^{14} +2.04678 q^{15} +1.00000 q^{16} +4.28284 q^{17} -1.18929 q^{18} -4.44661 q^{19} -1.00000 q^{20} +2.04678 q^{21} -2.00000 q^{23} +2.04678 q^{24} +1.00000 q^{25} +3.50105 q^{26} +3.70611 q^{27} -1.00000 q^{28} -1.30628 q^{29} -2.04678 q^{30} -5.70820 q^{31} -1.00000 q^{32} -4.28284 q^{34} +1.00000 q^{35} +1.18929 q^{36} -0.327530 q^{37} +4.44661 q^{38} +7.16586 q^{39} +1.00000 q^{40} +3.87535 q^{41} -2.04678 q^{42} +2.63928 q^{43} -1.18929 q^{45} +2.00000 q^{46} +12.2258 q^{47} -2.04678 q^{48} +1.00000 q^{49} -1.00000 q^{50} -8.76602 q^{51} -3.50105 q^{52} +1.09146 q^{53} -3.70611 q^{54} +1.00000 q^{56} +9.10121 q^{57} +1.30628 q^{58} +10.0613 q^{59} +2.04678 q^{60} -13.0021 q^{61} +5.70820 q^{62} -1.18929 q^{63} +1.00000 q^{64} +3.50105 q^{65} -8.44661 q^{67} +4.28284 q^{68} +4.09355 q^{69} -1.00000 q^{70} -5.87963 q^{71} -1.18929 q^{72} +7.44187 q^{73} +0.327530 q^{74} -2.04678 q^{75} -4.44661 q^{76} -7.16586 q^{78} -0.498955 q^{79} -1.00000 q^{80} -11.1535 q^{81} -3.87535 q^{82} +12.6125 q^{83} +2.04678 q^{84} -4.28284 q^{85} -2.63928 q^{86} +2.67366 q^{87} +1.39645 q^{89} +1.18929 q^{90} +3.50105 q^{91} -2.00000 q^{92} +11.6834 q^{93} -12.2258 q^{94} +4.44661 q^{95} +2.04678 q^{96} -9.79837 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} - 2q^{6} - 4q^{7} - 4q^{8} + 6q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} - 2q^{6} - 4q^{7} - 4q^{8} + 6q^{9} + 4q^{10} + 2q^{12} - q^{13} + 4q^{14} - 2q^{15} + 4q^{16} - 2q^{17} - 6q^{18} + 3q^{19} - 4q^{20} - 2q^{21} - 8q^{23} - 2q^{24} + 4q^{25} + q^{26} + 14q^{27} - 4q^{28} - 15q^{29} + 2q^{30} + 4q^{31} - 4q^{32} + 2q^{34} + 4q^{35} + 6q^{36} + 2q^{37} - 3q^{38} + q^{39} + 4q^{40} - 11q^{41} + 2q^{42} - 7q^{43} - 6q^{45} + 8q^{46} + 5q^{47} + 2q^{48} + 4q^{49} - 4q^{50} - 18q^{51} - q^{52} + 10q^{53} - 14q^{54} + 4q^{56} + 34q^{57} + 15q^{58} + 13q^{59} - 2q^{60} - 26q^{61} - 4q^{62} - 6q^{63} + 4q^{64} + q^{65} - 13q^{67} - 2q^{68} - 4q^{69} - 4q^{70} - 13q^{71} - 6q^{72} + 18q^{73} - 2q^{74} + 2q^{75} + 3q^{76} - q^{78} - 15q^{79} - 4q^{80} - 8q^{81} + 11q^{82} + 2q^{83} - 2q^{84} + 2q^{85} + 7q^{86} - 26q^{87} - 7q^{89} + 6q^{90} + q^{91} - 8q^{92} + 2q^{93} - 5q^{94} - 3q^{95} - 2q^{96} + 10q^{97} - 4q^{98} + O(q^{100}) \)

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
\(3\) −2.04678 −1.18171 −0.590853 0.806779i \(-0.701209\pi\)
−0.590853 + 0.806779i \(0.701209\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.04678 0.835593
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.18929 0.396431
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.04678 −0.590853
\(13\) −3.50105 −0.971015 −0.485508 0.874232i \(-0.661365\pi\)
−0.485508 + 0.874232i \(0.661365\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.04678 0.528475
\(16\) 1.00000 0.250000
\(17\) 4.28284 1.03874 0.519371 0.854549i \(-0.326166\pi\)
0.519371 + 0.854549i \(0.326166\pi\)
\(18\) −1.18929 −0.280319
\(19\) −4.44661 −1.02012 −0.510061 0.860138i \(-0.670377\pi\)
−0.510061 + 0.860138i \(0.670377\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.04678 0.446643
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 2.04678 0.417796
\(25\) 1.00000 0.200000
\(26\) 3.50105 0.686611
\(27\) 3.70611 0.713242
\(28\) −1.00000 −0.188982
\(29\) −1.30628 −0.242570 −0.121285 0.992618i \(-0.538702\pi\)
−0.121285 + 0.992618i \(0.538702\pi\)
\(30\) −2.04678 −0.373688
\(31\) −5.70820 −1.02522 −0.512612 0.858620i \(-0.671322\pi\)
−0.512612 + 0.858620i \(0.671322\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.28284 −0.734502
\(35\) 1.00000 0.169031
\(36\) 1.18929 0.198215
\(37\) −0.327530 −0.0538456 −0.0269228 0.999638i \(-0.508571\pi\)
−0.0269228 + 0.999638i \(0.508571\pi\)
\(38\) 4.44661 0.721335
\(39\) 7.16586 1.14746
\(40\) 1.00000 0.158114
\(41\) 3.87535 0.605228 0.302614 0.953113i \(-0.402141\pi\)
0.302614 + 0.953113i \(0.402141\pi\)
\(42\) −2.04678 −0.315824
\(43\) 2.63928 0.402487 0.201243 0.979541i \(-0.435502\pi\)
0.201243 + 0.979541i \(0.435502\pi\)
\(44\) 0 0
\(45\) −1.18929 −0.177289
\(46\) 2.00000 0.294884
\(47\) 12.2258 1.78331 0.891655 0.452715i \(-0.149544\pi\)
0.891655 + 0.452715i \(0.149544\pi\)
\(48\) −2.04678 −0.295427
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −8.76602 −1.22749
\(52\) −3.50105 −0.485508
\(53\) 1.09146 0.149924 0.0749619 0.997186i \(-0.476116\pi\)
0.0749619 + 0.997186i \(0.476116\pi\)
\(54\) −3.70611 −0.504338
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 9.10121 1.20549
\(58\) 1.30628 0.171523
\(59\) 10.0613 1.30986 0.654932 0.755688i \(-0.272697\pi\)
0.654932 + 0.755688i \(0.272697\pi\)
\(60\) 2.04678 0.264238
\(61\) −13.0021 −1.66475 −0.832374 0.554215i \(-0.813019\pi\)
−0.832374 + 0.554215i \(0.813019\pi\)
\(62\) 5.70820 0.724943
\(63\) −1.18929 −0.149837
\(64\) 1.00000 0.125000
\(65\) 3.50105 0.434251
\(66\) 0 0
\(67\) −8.44661 −1.03192 −0.515959 0.856613i \(-0.672564\pi\)
−0.515959 + 0.856613i \(0.672564\pi\)
\(68\) 4.28284 0.519371
\(69\) 4.09355 0.492806
\(70\) −1.00000 −0.119523
\(71\) −5.87963 −0.697784 −0.348892 0.937163i \(-0.613442\pi\)
−0.348892 + 0.937163i \(0.613442\pi\)
\(72\) −1.18929 −0.140159
\(73\) 7.44187 0.871006 0.435503 0.900187i \(-0.356571\pi\)
0.435503 + 0.900187i \(0.356571\pi\)
\(74\) 0.327530 0.0380746
\(75\) −2.04678 −0.236341
\(76\) −4.44661 −0.510061
\(77\) 0 0
\(78\) −7.16586 −0.811373
\(79\) −0.498955 −0.0561368 −0.0280684 0.999606i \(-0.508936\pi\)
−0.0280684 + 0.999606i \(0.508936\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.1535 −1.23927
\(82\) −3.87535 −0.427961
\(83\) 12.6125 1.38440 0.692199 0.721707i \(-0.256642\pi\)
0.692199 + 0.721707i \(0.256642\pi\)
\(84\) 2.04678 0.223322
\(85\) −4.28284 −0.464540
\(86\) −2.63928 −0.284601
\(87\) 2.67366 0.286647
\(88\) 0 0
\(89\) 1.39645 0.148023 0.0740117 0.997257i \(-0.476420\pi\)
0.0740117 + 0.997257i \(0.476420\pi\)
\(90\) 1.18929 0.125362
\(91\) 3.50105 0.367009
\(92\) −2.00000 −0.208514
\(93\) 11.6834 1.21151
\(94\) −12.2258 −1.26099
\(95\) 4.44661 0.456212
\(96\) 2.04678 0.208898
\(97\) −9.79837 −0.994874 −0.497437 0.867500i \(-0.665725\pi\)
−0.497437 + 0.867500i \(0.665725\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −17.3874 −1.73011 −0.865057 0.501673i \(-0.832718\pi\)
−0.865057 + 0.501673i \(0.832718\pi\)
\(102\) 8.76602 0.867966
\(103\) 6.86988 0.676909 0.338455 0.940983i \(-0.390096\pi\)
0.338455 + 0.940983i \(0.390096\pi\)
\(104\) 3.50105 0.343306
\(105\) −2.04678 −0.199745
\(106\) −1.09146 −0.106012
\(107\) −1.75025 −0.169203 −0.0846013 0.996415i \(-0.526962\pi\)
−0.0846013 + 0.996415i \(0.526962\pi\)
\(108\) 3.70611 0.356621
\(109\) 5.88436 0.563620 0.281810 0.959470i \(-0.409065\pi\)
0.281810 + 0.959470i \(0.409065\pi\)
\(110\) 0 0
\(111\) 0.670380 0.0636297
\(112\) −1.00000 −0.0944911
\(113\) 17.1279 1.61126 0.805630 0.592419i \(-0.201827\pi\)
0.805630 + 0.592419i \(0.201827\pi\)
\(114\) −9.10121 −0.852407
\(115\) 2.00000 0.186501
\(116\) −1.30628 −0.121285
\(117\) −4.16376 −0.384940
\(118\) −10.0613 −0.926214
\(119\) −4.28284 −0.392608
\(120\) −2.04678 −0.186844
\(121\) 0 0
\(122\) 13.0021 1.17715
\(123\) −7.93198 −0.715202
\(124\) −5.70820 −0.512612
\(125\) −1.00000 −0.0894427
\(126\) 1.18929 0.105951
\(127\) 9.94218 0.882226 0.441113 0.897452i \(-0.354584\pi\)
0.441113 + 0.897452i \(0.354584\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.40202 −0.475621
\(130\) −3.50105 −0.307062
\(131\) 7.65112 0.668482 0.334241 0.942488i \(-0.391520\pi\)
0.334241 + 0.942488i \(0.391520\pi\)
\(132\) 0 0
\(133\) 4.44661 0.385570
\(134\) 8.44661 0.729676
\(135\) −3.70611 −0.318971
\(136\) −4.28284 −0.367251
\(137\) 2.16038 0.184574 0.0922870 0.995732i \(-0.470582\pi\)
0.0922870 + 0.995732i \(0.470582\pi\)
\(138\) −4.09355 −0.348466
\(139\) 21.1314 1.79234 0.896170 0.443711i \(-0.146338\pi\)
0.896170 + 0.443711i \(0.146338\pi\)
\(140\) 1.00000 0.0845154
\(141\) −25.0234 −2.10735
\(142\) 5.87963 0.493408
\(143\) 0 0
\(144\) 1.18929 0.0991077
\(145\) 1.30628 0.108481
\(146\) −7.44187 −0.615894
\(147\) −2.04678 −0.168815
\(148\) −0.327530 −0.0269228
\(149\) −16.0089 −1.31150 −0.655751 0.754977i \(-0.727648\pi\)
−0.655751 + 0.754977i \(0.727648\pi\)
\(150\) 2.04678 0.167119
\(151\) 12.9386 1.05293 0.526466 0.850196i \(-0.323517\pi\)
0.526466 + 0.850196i \(0.323517\pi\)
\(152\) 4.44661 0.360668
\(153\) 5.09355 0.411789
\(154\) 0 0
\(155\) 5.70820 0.458494
\(156\) 7.16586 0.573728
\(157\) 18.4955 1.47610 0.738050 0.674746i \(-0.235747\pi\)
0.738050 + 0.674746i \(0.235747\pi\)
\(158\) 0.498955 0.0396947
\(159\) −2.23398 −0.177166
\(160\) 1.00000 0.0790569
\(161\) 2.00000 0.157622
\(162\) 11.1535 0.876299
\(163\) 6.52922 0.511408 0.255704 0.966755i \(-0.417693\pi\)
0.255704 + 0.966755i \(0.417693\pi\)
\(164\) 3.87535 0.302614
\(165\) 0 0
\(166\) −12.6125 −0.978917
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −2.04678 −0.157912
\(169\) −0.742683 −0.0571295
\(170\) 4.28284 0.328479
\(171\) −5.28832 −0.404408
\(172\) 2.63928 0.201243
\(173\) −7.54354 −0.573525 −0.286762 0.958002i \(-0.592579\pi\)
−0.286762 + 0.958002i \(0.592579\pi\)
\(174\) −2.67366 −0.202690
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −20.5931 −1.54788
\(178\) −1.39645 −0.104668
\(179\) 6.66819 0.498404 0.249202 0.968452i \(-0.419832\pi\)
0.249202 + 0.968452i \(0.419832\pi\)
\(180\) −1.18929 −0.0886446
\(181\) −3.26233 −0.242487 −0.121244 0.992623i \(-0.538688\pi\)
−0.121244 + 0.992623i \(0.538688\pi\)
\(182\) −3.50105 −0.259515
\(183\) 26.6124 1.96724
\(184\) 2.00000 0.147442
\(185\) 0.327530 0.0240805
\(186\) −11.6834 −0.856670
\(187\) 0 0
\(188\) 12.2258 0.891655
\(189\) −3.70611 −0.269580
\(190\) −4.44661 −0.322591
\(191\) −13.7035 −0.991548 −0.495774 0.868452i \(-0.665116\pi\)
−0.495774 + 0.868452i \(0.665116\pi\)
\(192\) −2.04678 −0.147713
\(193\) 11.1467 0.802357 0.401178 0.916000i \(-0.368601\pi\)
0.401178 + 0.916000i \(0.368601\pi\)
\(194\) 9.79837 0.703482
\(195\) −7.16586 −0.513158
\(196\) 1.00000 0.0714286
\(197\) −3.51254 −0.250258 −0.125129 0.992140i \(-0.539935\pi\)
−0.125129 + 0.992140i \(0.539935\pi\)
\(198\) 0 0
\(199\) 19.5167 1.38350 0.691752 0.722135i \(-0.256839\pi\)
0.691752 + 0.722135i \(0.256839\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 17.2883 1.21942
\(202\) 17.3874 1.22338
\(203\) 1.30628 0.0916829
\(204\) −8.76602 −0.613744
\(205\) −3.87535 −0.270666
\(206\) −6.86988 −0.478647
\(207\) −2.37858 −0.165323
\(208\) −3.50105 −0.242754
\(209\) 0 0
\(210\) 2.04678 0.141241
\(211\) −14.3495 −0.987861 −0.493931 0.869501i \(-0.664440\pi\)
−0.493931 + 0.869501i \(0.664440\pi\)
\(212\) 1.09146 0.0749619
\(213\) 12.0343 0.824576
\(214\) 1.75025 0.119644
\(215\) −2.63928 −0.179998
\(216\) −3.70611 −0.252169
\(217\) 5.70820 0.387498
\(218\) −5.88436 −0.398539
\(219\) −15.2319 −1.02927
\(220\) 0 0
\(221\) −14.9944 −1.00863
\(222\) −0.670380 −0.0449930
\(223\) 13.2981 0.890504 0.445252 0.895405i \(-0.353114\pi\)
0.445252 + 0.895405i \(0.353114\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.18929 0.0792861
\(226\) −17.1279 −1.13933
\(227\) −18.9140 −1.25537 −0.627683 0.778469i \(-0.715997\pi\)
−0.627683 + 0.778469i \(0.715997\pi\)
\(228\) 9.10121 0.602743
\(229\) 3.89113 0.257133 0.128566 0.991701i \(-0.458962\pi\)
0.128566 + 0.991701i \(0.458962\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 1.30628 0.0857615
\(233\) 6.92431 0.453627 0.226813 0.973938i \(-0.427169\pi\)
0.226813 + 0.973938i \(0.427169\pi\)
\(234\) 4.16376 0.272194
\(235\) −12.2258 −0.797521
\(236\) 10.0613 0.654932
\(237\) 1.02125 0.0663372
\(238\) 4.28284 0.277616
\(239\) −26.6477 −1.72370 −0.861850 0.507164i \(-0.830694\pi\)
−0.861850 + 0.507164i \(0.830694\pi\)
\(240\) 2.04678 0.132119
\(241\) −0.0928140 −0.00597868 −0.00298934 0.999996i \(-0.500952\pi\)
−0.00298934 + 0.999996i \(0.500952\pi\)
\(242\) 0 0
\(243\) 11.7103 0.751216
\(244\) −13.0021 −0.832374
\(245\) −1.00000 −0.0638877
\(246\) 7.93198 0.505724
\(247\) 15.5678 0.990554
\(248\) 5.70820 0.362471
\(249\) −25.8149 −1.63595
\(250\) 1.00000 0.0632456
\(251\) 21.0063 1.32590 0.662952 0.748662i \(-0.269303\pi\)
0.662952 + 0.748662i \(0.269303\pi\)
\(252\) −1.18929 −0.0749184
\(253\) 0 0
\(254\) −9.94218 −0.623828
\(255\) 8.76602 0.548950
\(256\) 1.00000 0.0625000
\(257\) −4.23060 −0.263897 −0.131949 0.991257i \(-0.542123\pi\)
−0.131949 + 0.991257i \(0.542123\pi\)
\(258\) 5.40202 0.336315
\(259\) 0.327530 0.0203517
\(260\) 3.50105 0.217126
\(261\) −1.55355 −0.0961623
\(262\) −7.65112 −0.472688
\(263\) −16.4789 −1.01613 −0.508066 0.861318i \(-0.669639\pi\)
−0.508066 + 0.861318i \(0.669639\pi\)
\(264\) 0 0
\(265\) −1.09146 −0.0670480
\(266\) −4.44661 −0.272639
\(267\) −2.85822 −0.174920
\(268\) −8.44661 −0.515959
\(269\) 20.2848 1.23679 0.618394 0.785868i \(-0.287784\pi\)
0.618394 + 0.785868i \(0.287784\pi\)
\(270\) 3.70611 0.225547
\(271\) −13.8443 −0.840979 −0.420489 0.907298i \(-0.638142\pi\)
−0.420489 + 0.907298i \(0.638142\pi\)
\(272\) 4.28284 0.259686
\(273\) −7.16586 −0.433697
\(274\) −2.16038 −0.130513
\(275\) 0 0
\(276\) 4.09355 0.246403
\(277\) 8.50369 0.510937 0.255469 0.966817i \(-0.417770\pi\)
0.255469 + 0.966817i \(0.417770\pi\)
\(278\) −21.1314 −1.26738
\(279\) −6.78872 −0.406430
\(280\) −1.00000 −0.0597614
\(281\) −2.06245 −0.123036 −0.0615179 0.998106i \(-0.519594\pi\)
−0.0615179 + 0.998106i \(0.519594\pi\)
\(282\) 25.0234 1.49012
\(283\) −27.3214 −1.62409 −0.812044 0.583596i \(-0.801645\pi\)
−0.812044 + 0.583596i \(0.801645\pi\)
\(284\) −5.87963 −0.348892
\(285\) −9.10121 −0.539109
\(286\) 0 0
\(287\) −3.87535 −0.228755
\(288\) −1.18929 −0.0700797
\(289\) 1.34275 0.0789854
\(290\) −1.30628 −0.0767075
\(291\) 20.0551 1.17565
\(292\) 7.44187 0.435503
\(293\) −19.0599 −1.11349 −0.556746 0.830683i \(-0.687950\pi\)
−0.556746 + 0.830683i \(0.687950\pi\)
\(294\) 2.04678 0.119370
\(295\) −10.0613 −0.585789
\(296\) 0.327530 0.0190373
\(297\) 0 0
\(298\) 16.0089 0.927372
\(299\) 7.00209 0.404941
\(300\) −2.04678 −0.118171
\(301\) −2.63928 −0.152126
\(302\) −12.9386 −0.744535
\(303\) 35.5882 2.04449
\(304\) −4.44661 −0.255031
\(305\) 13.0021 0.744498
\(306\) −5.09355 −0.291179
\(307\) 27.8834 1.59139 0.795694 0.605699i \(-0.207106\pi\)
0.795694 + 0.605699i \(0.207106\pi\)
\(308\) 0 0
\(309\) −14.0611 −0.799908
\(310\) −5.70820 −0.324204
\(311\) 21.0336 1.19271 0.596354 0.802721i \(-0.296615\pi\)
0.596354 + 0.802721i \(0.296615\pi\)
\(312\) −7.16586 −0.405687
\(313\) 3.94821 0.223166 0.111583 0.993755i \(-0.464408\pi\)
0.111583 + 0.993755i \(0.464408\pi\)
\(314\) −18.4955 −1.04376
\(315\) 1.18929 0.0670090
\(316\) −0.498955 −0.0280684
\(317\) −28.1225 −1.57952 −0.789759 0.613417i \(-0.789794\pi\)
−0.789759 + 0.613417i \(0.789794\pi\)
\(318\) 2.23398 0.125275
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 3.58236 0.199948
\(322\) −2.00000 −0.111456
\(323\) −19.0441 −1.05964
\(324\) −11.1535 −0.619637
\(325\) −3.50105 −0.194203
\(326\) −6.52922 −0.361620
\(327\) −12.0440 −0.666033
\(328\) −3.87535 −0.213980
\(329\) −12.2258 −0.674028
\(330\) 0 0
\(331\) 30.2738 1.66400 0.831999 0.554777i \(-0.187196\pi\)
0.831999 + 0.554777i \(0.187196\pi\)
\(332\) 12.6125 0.692199
\(333\) −0.389529 −0.0213460
\(334\) 12.0000 0.656611
\(335\) 8.44661 0.461488
\(336\) 2.04678 0.111661
\(337\) 32.0764 1.74731 0.873656 0.486544i \(-0.161743\pi\)
0.873656 + 0.486544i \(0.161743\pi\)
\(338\) 0.742683 0.0403966
\(339\) −35.0570 −1.90404
\(340\) −4.28284 −0.232270
\(341\) 0 0
\(342\) 5.28832 0.285959
\(343\) −1.00000 −0.0539949
\(344\) −2.63928 −0.142301
\(345\) −4.09355 −0.220389
\(346\) 7.54354 0.405543
\(347\) −16.6585 −0.894275 −0.447138 0.894465i \(-0.647557\pi\)
−0.447138 + 0.894465i \(0.647557\pi\)
\(348\) 2.67366 0.143323
\(349\) 5.44944 0.291702 0.145851 0.989307i \(-0.453408\pi\)
0.145851 + 0.989307i \(0.453408\pi\)
\(350\) 1.00000 0.0534522
\(351\) −12.9753 −0.692569
\(352\) 0 0
\(353\) 12.9302 0.688207 0.344104 0.938932i \(-0.388183\pi\)
0.344104 + 0.938932i \(0.388183\pi\)
\(354\) 20.5931 1.09451
\(355\) 5.87963 0.312058
\(356\) 1.39645 0.0740117
\(357\) 8.76602 0.463947
\(358\) −6.66819 −0.352425
\(359\) −30.5347 −1.61156 −0.805780 0.592216i \(-0.798253\pi\)
−0.805780 + 0.592216i \(0.798253\pi\)
\(360\) 1.18929 0.0626812
\(361\) 0.772330 0.0406490
\(362\) 3.26233 0.171464
\(363\) 0 0
\(364\) 3.50105 0.183505
\(365\) −7.44187 −0.389526
\(366\) −26.6124 −1.39105
\(367\) 27.8437 1.45343 0.726715 0.686939i \(-0.241046\pi\)
0.726715 + 0.686939i \(0.241046\pi\)
\(368\) −2.00000 −0.104257
\(369\) 4.60892 0.239931
\(370\) −0.327530 −0.0170275
\(371\) −1.09146 −0.0566659
\(372\) 11.6834 0.605757
\(373\) 14.5146 0.751539 0.375770 0.926713i \(-0.377378\pi\)
0.375770 + 0.926713i \(0.377378\pi\)
\(374\) 0 0
\(375\) 2.04678 0.105695
\(376\) −12.2258 −0.630496
\(377\) 4.57335 0.235539
\(378\) 3.70611 0.190622
\(379\) −11.9208 −0.612329 −0.306165 0.951979i \(-0.599046\pi\)
−0.306165 + 0.951979i \(0.599046\pi\)
\(380\) 4.44661 0.228106
\(381\) −20.3494 −1.04253
\(382\) 13.7035 0.701131
\(383\) 31.7926 1.62453 0.812264 0.583290i \(-0.198235\pi\)
0.812264 + 0.583290i \(0.198235\pi\)
\(384\) 2.04678 0.104449
\(385\) 0 0
\(386\) −11.1467 −0.567352
\(387\) 3.13888 0.159558
\(388\) −9.79837 −0.497437
\(389\) 3.03309 0.153784 0.0768919 0.997039i \(-0.475500\pi\)
0.0768919 + 0.997039i \(0.475500\pi\)
\(390\) 7.16586 0.362857
\(391\) −8.56569 −0.433185
\(392\) −1.00000 −0.0505076
\(393\) −15.6601 −0.789949
\(394\) 3.51254 0.176959
\(395\) 0.498955 0.0251051
\(396\) 0 0
\(397\) −28.8927 −1.45008 −0.725041 0.688706i \(-0.758179\pi\)
−0.725041 + 0.688706i \(0.758179\pi\)
\(398\) −19.5167 −0.978285
\(399\) −9.10121 −0.455631
\(400\) 1.00000 0.0500000
\(401\) 36.2856 1.81202 0.906009 0.423258i \(-0.139114\pi\)
0.906009 + 0.423258i \(0.139114\pi\)
\(402\) −17.2883 −0.862263
\(403\) 19.9847 0.995508
\(404\) −17.3874 −0.865057
\(405\) 11.1535 0.554220
\(406\) −1.30628 −0.0648296
\(407\) 0 0
\(408\) 8.76602 0.433983
\(409\) 5.94636 0.294029 0.147014 0.989134i \(-0.453034\pi\)
0.147014 + 0.989134i \(0.453034\pi\)
\(410\) 3.87535 0.191390
\(411\) −4.42182 −0.218112
\(412\) 6.86988 0.338455
\(413\) −10.0613 −0.495082
\(414\) 2.37858 0.116901
\(415\) −12.6125 −0.619122
\(416\) 3.50105 0.171653
\(417\) −43.2512 −2.11802
\(418\) 0 0
\(419\) 7.10047 0.346881 0.173440 0.984844i \(-0.444512\pi\)
0.173440 + 0.984844i \(0.444512\pi\)
\(420\) −2.04678 −0.0998724
\(421\) −21.7460 −1.05983 −0.529917 0.848050i \(-0.677777\pi\)
−0.529917 + 0.848050i \(0.677777\pi\)
\(422\) 14.3495 0.698524
\(423\) 14.5400 0.706959
\(424\) −1.09146 −0.0530061
\(425\) 4.28284 0.207748
\(426\) −12.0343 −0.583063
\(427\) 13.0021 0.629215
\(428\) −1.75025 −0.0846013
\(429\) 0 0
\(430\) 2.63928 0.127278
\(431\) −18.7940 −0.905277 −0.452638 0.891694i \(-0.649517\pi\)
−0.452638 + 0.891694i \(0.649517\pi\)
\(432\) 3.70611 0.178310
\(433\) −32.4488 −1.55939 −0.779694 0.626161i \(-0.784625\pi\)
−0.779694 + 0.626161i \(0.784625\pi\)
\(434\) −5.70820 −0.274003
\(435\) −2.67366 −0.128192
\(436\) 5.88436 0.281810
\(437\) 8.89322 0.425420
\(438\) 15.2319 0.727806
\(439\) −17.1535 −0.818690 −0.409345 0.912380i \(-0.634243\pi\)
−0.409345 + 0.912380i \(0.634243\pi\)
\(440\) 0 0
\(441\) 1.18929 0.0566330
\(442\) 14.9944 0.713212
\(443\) 4.23861 0.201383 0.100691 0.994918i \(-0.467895\pi\)
0.100691 + 0.994918i \(0.467895\pi\)
\(444\) 0.670380 0.0318148
\(445\) −1.39645 −0.0661981
\(446\) −13.2981 −0.629682
\(447\) 32.7667 1.54981
\(448\) −1.00000 −0.0472456
\(449\) 13.1589 0.621008 0.310504 0.950572i \(-0.399502\pi\)
0.310504 + 0.950572i \(0.399502\pi\)
\(450\) −1.18929 −0.0560638
\(451\) 0 0
\(452\) 17.1279 0.805630
\(453\) −26.4825 −1.24426
\(454\) 18.9140 0.887678
\(455\) −3.50105 −0.164132
\(456\) −9.10121 −0.426203
\(457\) −0.679393 −0.0317806 −0.0158903 0.999874i \(-0.505058\pi\)
−0.0158903 + 0.999874i \(0.505058\pi\)
\(458\) −3.89113 −0.181820
\(459\) 15.8727 0.740875
\(460\) 2.00000 0.0932505
\(461\) −33.9310 −1.58033 −0.790163 0.612897i \(-0.790004\pi\)
−0.790163 + 0.612897i \(0.790004\pi\)
\(462\) 0 0
\(463\) −8.67985 −0.403387 −0.201693 0.979449i \(-0.564644\pi\)
−0.201693 + 0.979449i \(0.564644\pi\)
\(464\) −1.30628 −0.0606426
\(465\) −11.6834 −0.541805
\(466\) −6.92431 −0.320763
\(467\) 4.58156 0.212009 0.106005 0.994366i \(-0.466194\pi\)
0.106005 + 0.994366i \(0.466194\pi\)
\(468\) −4.16376 −0.192470
\(469\) 8.44661 0.390028
\(470\) 12.2258 0.563932
\(471\) −37.8561 −1.74432
\(472\) −10.0613 −0.463107
\(473\) 0 0
\(474\) −1.02125 −0.0469075
\(475\) −4.44661 −0.204024
\(476\) −4.28284 −0.196304
\(477\) 1.29807 0.0594344
\(478\) 26.6477 1.21884
\(479\) 38.8930 1.77707 0.888534 0.458811i \(-0.151725\pi\)
0.888534 + 0.458811i \(0.151725\pi\)
\(480\) −2.04678 −0.0934221
\(481\) 1.14670 0.0522849
\(482\) 0.0928140 0.00422756
\(483\) −4.09355 −0.186263
\(484\) 0 0
\(485\) 9.79837 0.444921
\(486\) −11.7103 −0.531190
\(487\) 19.7593 0.895377 0.447689 0.894189i \(-0.352247\pi\)
0.447689 + 0.894189i \(0.352247\pi\)
\(488\) 13.0021 0.588577
\(489\) −13.3638 −0.604334
\(490\) 1.00000 0.0451754
\(491\) −30.7082 −1.38584 −0.692920 0.721014i \(-0.743676\pi\)
−0.692920 + 0.721014i \(0.743676\pi\)
\(492\) −7.93198 −0.357601
\(493\) −5.59460 −0.251968
\(494\) −15.5678 −0.700427
\(495\) 0 0
\(496\) −5.70820 −0.256306
\(497\) 5.87963 0.263737
\(498\) 25.8149 1.15679
\(499\) −18.5980 −0.832563 −0.416281 0.909236i \(-0.636667\pi\)
−0.416281 + 0.909236i \(0.636667\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 24.5613 1.09732
\(502\) −21.0063 −0.937556
\(503\) −1.53678 −0.0685216 −0.0342608 0.999413i \(-0.510908\pi\)
−0.0342608 + 0.999413i \(0.510908\pi\)
\(504\) 1.18929 0.0529753
\(505\) 17.3874 0.773731
\(506\) 0 0
\(507\) 1.52011 0.0675103
\(508\) 9.94218 0.441113
\(509\) −19.0488 −0.844322 −0.422161 0.906521i \(-0.638728\pi\)
−0.422161 + 0.906521i \(0.638728\pi\)
\(510\) −8.76602 −0.388166
\(511\) −7.44187 −0.329209
\(512\) −1.00000 −0.0441942
\(513\) −16.4796 −0.727594
\(514\) 4.23060 0.186604
\(515\) −6.86988 −0.302723
\(516\) −5.40202 −0.237811
\(517\) 0 0
\(518\) −0.327530 −0.0143908
\(519\) 15.4399 0.677738
\(520\) −3.50105 −0.153531
\(521\) 18.1769 0.796344 0.398172 0.917311i \(-0.369645\pi\)
0.398172 + 0.917311i \(0.369645\pi\)
\(522\) 1.55355 0.0679970
\(523\) 17.2935 0.756192 0.378096 0.925766i \(-0.376579\pi\)
0.378096 + 0.925766i \(0.376579\pi\)
\(524\) 7.65112 0.334241
\(525\) 2.04678 0.0893286
\(526\) 16.4789 0.718514
\(527\) −24.4473 −1.06494
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 1.09146 0.0474101
\(531\) 11.9658 0.519270
\(532\) 4.44661 0.192785
\(533\) −13.5678 −0.587686
\(534\) 2.85822 0.123687
\(535\) 1.75025 0.0756697
\(536\) 8.44661 0.364838
\(537\) −13.6483 −0.588967
\(538\) −20.2848 −0.874541
\(539\) 0 0
\(540\) −3.70611 −0.159486
\(541\) −24.4783 −1.05241 −0.526203 0.850359i \(-0.676385\pi\)
−0.526203 + 0.850359i \(0.676385\pi\)
\(542\) 13.8443 0.594662
\(543\) 6.67726 0.286549
\(544\) −4.28284 −0.183625
\(545\) −5.88436 −0.252058
\(546\) 7.16586 0.306670
\(547\) −39.8203 −1.70259 −0.851297 0.524684i \(-0.824183\pi\)
−0.851297 + 0.524684i \(0.824183\pi\)
\(548\) 2.16038 0.0922870
\(549\) −15.4633 −0.659957
\(550\) 0 0
\(551\) 5.80852 0.247451
\(552\) −4.09355 −0.174233
\(553\) 0.498955 0.0212177
\(554\) −8.50369 −0.361287
\(555\) −0.670380 −0.0284560
\(556\) 21.1314 0.896170
\(557\) −22.9552 −0.972644 −0.486322 0.873780i \(-0.661662\pi\)
−0.486322 + 0.873780i \(0.661662\pi\)
\(558\) 6.78872 0.287389
\(559\) −9.24025 −0.390821
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 2.06245 0.0869994
\(563\) 15.9140 0.670696 0.335348 0.942094i \(-0.391146\pi\)
0.335348 + 0.942094i \(0.391146\pi\)
\(564\) −25.0234 −1.05368
\(565\) −17.1279 −0.720578
\(566\) 27.3214 1.14840
\(567\) 11.1535 0.468401
\(568\) 5.87963 0.246704
\(569\) −17.7349 −0.743484 −0.371742 0.928336i \(-0.621239\pi\)
−0.371742 + 0.928336i \(0.621239\pi\)
\(570\) 9.10121 0.381208
\(571\) −1.05280 −0.0440584 −0.0220292 0.999757i \(-0.507013\pi\)
−0.0220292 + 0.999757i \(0.507013\pi\)
\(572\) 0 0
\(573\) 28.0479 1.17172
\(574\) 3.87535 0.161754
\(575\) −2.00000 −0.0834058
\(576\) 1.18929 0.0495538
\(577\) −12.8223 −0.533798 −0.266899 0.963725i \(-0.585999\pi\)
−0.266899 + 0.963725i \(0.585999\pi\)
\(578\) −1.34275 −0.0558511
\(579\) −22.8148 −0.948150
\(580\) 1.30628 0.0542404
\(581\) −12.6125 −0.523253
\(582\) −20.0551 −0.831310
\(583\) 0 0
\(584\) −7.44187 −0.307947
\(585\) 4.16376 0.172150
\(586\) 19.0599 0.787358
\(587\) −9.76035 −0.402853 −0.201426 0.979504i \(-0.564558\pi\)
−0.201426 + 0.979504i \(0.564558\pi\)
\(588\) −2.04678 −0.0844076
\(589\) 25.3822 1.04585
\(590\) 10.0613 0.414216
\(591\) 7.18939 0.295732
\(592\) −0.327530 −0.0134614
\(593\) −25.4142 −1.04364 −0.521818 0.853057i \(-0.674746\pi\)
−0.521818 + 0.853057i \(0.674746\pi\)
\(594\) 0 0
\(595\) 4.28284 0.175579
\(596\) −16.0089 −0.655751
\(597\) −39.9464 −1.63490
\(598\) −7.00209 −0.286337
\(599\) −41.3324 −1.68879 −0.844397 0.535718i \(-0.820041\pi\)
−0.844397 + 0.535718i \(0.820041\pi\)
\(600\) 2.04678 0.0835593
\(601\) −39.2422 −1.60072 −0.800361 0.599518i \(-0.795359\pi\)
−0.800361 + 0.599518i \(0.795359\pi\)
\(602\) 2.63928 0.107569
\(603\) −10.0455 −0.409084
\(604\) 12.9386 0.526466
\(605\) 0 0
\(606\) −35.5882 −1.44567
\(607\) −7.82091 −0.317441 −0.158721 0.987324i \(-0.550737\pi\)
−0.158721 + 0.987324i \(0.550737\pi\)
\(608\) 4.44661 0.180334
\(609\) −2.67366 −0.108342
\(610\) −13.0021 −0.526439
\(611\) −42.8029 −1.73162
\(612\) 5.09355 0.205895
\(613\) −1.22930 −0.0496511 −0.0248256 0.999692i \(-0.507903\pi\)
−0.0248256 + 0.999692i \(0.507903\pi\)
\(614\) −27.8834 −1.12528
\(615\) 7.93198 0.319848
\(616\) 0 0
\(617\) −8.43129 −0.339431 −0.169715 0.985493i \(-0.554285\pi\)
−0.169715 + 0.985493i \(0.554285\pi\)
\(618\) 14.0611 0.565620
\(619\) 7.19522 0.289200 0.144600 0.989490i \(-0.453810\pi\)
0.144600 + 0.989490i \(0.453810\pi\)
\(620\) 5.70820 0.229247
\(621\) −7.41223 −0.297442
\(622\) −21.0336 −0.843372
\(623\) −1.39645 −0.0559476
\(624\) 7.16586 0.286864
\(625\) 1.00000 0.0400000
\(626\) −3.94821 −0.157802
\(627\) 0 0
\(628\) 18.4955 0.738050
\(629\) −1.40276 −0.0559317
\(630\) −1.18929 −0.0473825
\(631\) 9.16167 0.364721 0.182360 0.983232i \(-0.441626\pi\)
0.182360 + 0.983232i \(0.441626\pi\)
\(632\) 0.498955 0.0198474
\(633\) 29.3702 1.16736
\(634\) 28.1225 1.11689
\(635\) −9.94218 −0.394543
\(636\) −2.23398 −0.0885830
\(637\) −3.50105 −0.138716
\(638\) 0 0
\(639\) −6.99260 −0.276623
\(640\) 1.00000 0.0395285
\(641\) −7.82633 −0.309121 −0.154561 0.987983i \(-0.549396\pi\)
−0.154561 + 0.987983i \(0.549396\pi\)
\(642\) −3.58236 −0.141385
\(643\) −42.1180 −1.66097 −0.830487 0.557038i \(-0.811938\pi\)
−0.830487 + 0.557038i \(0.811938\pi\)
\(644\) 2.00000 0.0788110
\(645\) 5.40202 0.212704
\(646\) 19.0441 0.749281
\(647\) −20.5480 −0.807825 −0.403913 0.914798i \(-0.632350\pi\)
−0.403913 + 0.914798i \(0.632350\pi\)
\(648\) 11.1535 0.438149
\(649\) 0 0
\(650\) 3.50105 0.137322
\(651\) −11.6834 −0.457909
\(652\) 6.52922 0.255704
\(653\) −29.0694 −1.13757 −0.568786 0.822485i \(-0.692587\pi\)
−0.568786 + 0.822485i \(0.692587\pi\)
\(654\) 12.0440 0.470957
\(655\) −7.65112 −0.298954
\(656\) 3.87535 0.151307
\(657\) 8.85056 0.345293
\(658\) 12.2258 0.476610
\(659\) −16.3882 −0.638395 −0.319198 0.947688i \(-0.603413\pi\)
−0.319198 + 0.947688i \(0.603413\pi\)
\(660\) 0 0
\(661\) 37.7386 1.46786 0.733932 0.679223i \(-0.237683\pi\)
0.733932 + 0.679223i \(0.237683\pi\)
\(662\) −30.2738 −1.17662
\(663\) 30.6902 1.19191
\(664\) −12.6125 −0.489459
\(665\) −4.44661 −0.172432
\(666\) 0.389529 0.0150939
\(667\) 2.61256 0.101159
\(668\) −12.0000 −0.464294
\(669\) −27.2182 −1.05232
\(670\) −8.44661 −0.326321
\(671\) 0 0
\(672\) −2.04678 −0.0789561
\(673\) −21.0258 −0.810486 −0.405243 0.914209i \(-0.632813\pi\)
−0.405243 + 0.914209i \(0.632813\pi\)
\(674\) −32.0764 −1.23554
\(675\) 3.70611 0.142648
\(676\) −0.742683 −0.0285647
\(677\) 6.89196 0.264880 0.132440 0.991191i \(-0.457719\pi\)
0.132440 + 0.991191i \(0.457719\pi\)
\(678\) 35.0570 1.34636
\(679\) 9.79837 0.376027
\(680\) 4.28284 0.164240
\(681\) 38.7127 1.48348
\(682\) 0 0
\(683\) −1.37212 −0.0525026 −0.0262513 0.999655i \(-0.508357\pi\)
−0.0262513 + 0.999655i \(0.508357\pi\)
\(684\) −5.28832 −0.202204
\(685\) −2.16038 −0.0825440
\(686\) 1.00000 0.0381802
\(687\) −7.96427 −0.303856
\(688\) 2.63928 0.100622
\(689\) −3.82126 −0.145578
\(690\) 4.09355 0.155839
\(691\) −9.39673 −0.357469 −0.178734 0.983897i \(-0.557200\pi\)
−0.178734 + 0.983897i \(0.557200\pi\)
\(692\) −7.54354 −0.286762
\(693\) 0 0
\(694\) 16.6585 0.632348
\(695\) −21.1314 −0.801559
\(696\) −2.67366 −0.101345
\(697\) 16.5975 0.628676
\(698\) −5.44944 −0.206264
\(699\) −14.1725 −0.536054
\(700\) −1.00000 −0.0377964
\(701\) −13.5330 −0.511133 −0.255566 0.966792i \(-0.582262\pi\)
−0.255566 + 0.966792i \(0.582262\pi\)
\(702\) 12.9753 0.489720
\(703\) 1.45640 0.0549290
\(704\) 0 0
\(705\) 25.0234 0.942436
\(706\) −12.9302 −0.486636
\(707\) 17.3874 0.653922
\(708\) −20.5931 −0.773938
\(709\) −40.1756 −1.50883 −0.754413 0.656400i \(-0.772078\pi\)
−0.754413 + 0.656400i \(0.772078\pi\)
\(710\) −5.87963 −0.220659
\(711\) −0.593403 −0.0222544
\(712\) −1.39645 −0.0523342
\(713\) 11.4164 0.427548
\(714\) −8.76602 −0.328060
\(715\) 0 0
\(716\) 6.66819 0.249202
\(717\) 54.5420 2.03691
\(718\) 30.5347 1.13954
\(719\) 13.1839 0.491677 0.245838 0.969311i \(-0.420937\pi\)
0.245838 + 0.969311i \(0.420937\pi\)
\(720\) −1.18929 −0.0443223
\(721\) −6.86988 −0.255848
\(722\) −0.772330 −0.0287432
\(723\) 0.189970 0.00706504
\(724\) −3.26233 −0.121244
\(725\) −1.30628 −0.0485141
\(726\) 0 0
\(727\) −40.5403 −1.50356 −0.751779 0.659415i \(-0.770804\pi\)
−0.751779 + 0.659415i \(0.770804\pi\)
\(728\) −3.50105 −0.129757
\(729\) 9.49203 0.351557
\(730\) 7.44187 0.275436
\(731\) 11.3036 0.418080
\(732\) 26.6124 0.983621
\(733\) −22.8799 −0.845089 −0.422545 0.906342i \(-0.638863\pi\)
−0.422545 + 0.906342i \(0.638863\pi\)
\(734\) −27.8437 −1.02773
\(735\) 2.04678 0.0754965
\(736\) 2.00000 0.0737210
\(737\) 0 0
\(738\) −4.60892 −0.169657
\(739\) −30.1038 −1.10738 −0.553692 0.832721i \(-0.686782\pi\)
−0.553692 + 0.832721i \(0.686782\pi\)
\(740\) 0.327530 0.0120402
\(741\) −31.8638 −1.17054
\(742\) 1.09146 0.0400688
\(743\) 29.6655 1.08832 0.544161 0.838981i \(-0.316848\pi\)
0.544161 + 0.838981i \(0.316848\pi\)
\(744\) −11.6834 −0.428335
\(745\) 16.0089 0.586521
\(746\) −14.5146 −0.531418
\(747\) 14.9999 0.548818
\(748\) 0 0
\(749\) 1.75025 0.0639526
\(750\) −2.04678 −0.0747377
\(751\) 47.4328 1.73085 0.865423 0.501041i \(-0.167049\pi\)
0.865423 + 0.501041i \(0.167049\pi\)
\(752\) 12.2258 0.445828
\(753\) −42.9951 −1.56683
\(754\) −4.57335 −0.166552
\(755\) −12.9386 −0.470885
\(756\) −3.70611 −0.134790
\(757\) −26.5518 −0.965043 −0.482522 0.875884i \(-0.660279\pi\)
−0.482522 + 0.875884i \(0.660279\pi\)
\(758\) 11.9208 0.432982
\(759\) 0 0
\(760\) −4.44661 −0.161295
\(761\) −48.7468 −1.76707 −0.883534 0.468366i \(-0.844843\pi\)
−0.883534 + 0.468366i \(0.844843\pi\)
\(762\) 20.3494 0.737182
\(763\) −5.88436 −0.213028
\(764\) −13.7035 −0.495774
\(765\) −5.09355 −0.184158
\(766\) −31.7926 −1.14871
\(767\) −35.2249 −1.27190
\(768\) −2.04678 −0.0738567
\(769\) −25.8289 −0.931415 −0.465708 0.884939i \(-0.654200\pi\)
−0.465708 + 0.884939i \(0.654200\pi\)
\(770\) 0 0
\(771\) 8.65908 0.311849
\(772\) 11.1467 0.401178
\(773\) 38.6257 1.38927 0.694634 0.719363i \(-0.255566\pi\)
0.694634 + 0.719363i \(0.255566\pi\)
\(774\) −3.13888 −0.112825
\(775\) −5.70820 −0.205045
\(776\) 9.79837 0.351741
\(777\) −0.670380 −0.0240497
\(778\) −3.03309 −0.108742
\(779\) −17.2322 −0.617407
\(780\) −7.16586 −0.256579
\(781\) 0 0
\(782\) 8.56569 0.306308
\(783\) −4.84123 −0.173011
\(784\) 1.00000 0.0357143
\(785\) −18.4955 −0.660132
\(786\) 15.6601 0.558579
\(787\) 10.5712 0.376821 0.188411 0.982090i \(-0.439666\pi\)
0.188411 + 0.982090i \(0.439666\pi\)
\(788\) −3.51254 −0.125129
\(789\) 33.7286 1.20077
\(790\) −0.498955 −0.0177520
\(791\) −17.1279 −0.608999
\(792\) 0 0
\(793\) 45.5209 1.61649
\(794\) 28.8927 1.02536
\(795\) 2.23398 0.0792310
\(796\) 19.5167 0.691752
\(797\) −40.3031 −1.42761 −0.713805 0.700345i \(-0.753029\pi\)
−0.713805 + 0.700345i \(0.753029\pi\)
\(798\) 9.10121 0.322179
\(799\) 52.3610 1.85240
\(800\) −1.00000 −0.0353553
\(801\) 1.66079 0.0586810
\(802\) −36.2856 −1.28129
\(803\) 0 0
\(804\) 17.2883 0.609712
\(805\) −2.00000 −0.0704907
\(806\) −19.9847 −0.703930
\(807\) −41.5185 −1.46152
\(808\) 17.3874 0.611688
\(809\) 14.7984 0.520283 0.260142 0.965570i \(-0.416231\pi\)
0.260142 + 0.965570i \(0.416231\pi\)
\(810\) −11.1535 −0.391893
\(811\) −30.8043 −1.08169 −0.540843 0.841124i \(-0.681895\pi\)
−0.540843 + 0.841124i \(0.681895\pi\)
\(812\) 1.30628 0.0458415
\(813\) 28.3361 0.993790
\(814\) 0 0
\(815\) −6.52922 −0.228708
\(816\) −8.76602 −0.306872
\(817\) −11.7359 −0.410586
\(818\) −5.94636 −0.207910
\(819\) 4.16376 0.145494
\(820\) −3.87535 −0.135333
\(821\) 26.5412 0.926296 0.463148 0.886281i \(-0.346720\pi\)
0.463148 + 0.886281i \(0.346720\pi\)
\(822\) 4.42182 0.154229
\(823\) 39.8431 1.38884 0.694422 0.719568i \(-0.255660\pi\)
0.694422 + 0.719568i \(0.255660\pi\)
\(824\) −6.86988 −0.239324
\(825\) 0 0
\(826\) 10.0613 0.350076
\(827\) 36.6888 1.27579 0.637896 0.770122i \(-0.279805\pi\)
0.637896 + 0.770122i \(0.279805\pi\)
\(828\) −2.37858 −0.0826615
\(829\) 16.8821 0.586339 0.293169 0.956061i \(-0.405290\pi\)
0.293169 + 0.956061i \(0.405290\pi\)
\(830\) 12.6125 0.437785
\(831\) −17.4051 −0.603778
\(832\) −3.50105 −0.121377
\(833\) 4.28284 0.148392
\(834\) 43.2512 1.49767
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) −21.1553 −0.731233
\(838\) −7.10047 −0.245282
\(839\) −40.8091 −1.40889 −0.704444 0.709760i \(-0.748804\pi\)
−0.704444 + 0.709760i \(0.748804\pi\)
\(840\) 2.04678 0.0706205
\(841\) −27.2936 −0.941160
\(842\) 21.7460 0.749416
\(843\) 4.22138 0.145392
\(844\) −14.3495 −0.493931
\(845\) 0.742683 0.0255491
\(846\) −14.5400 −0.499896
\(847\) 0 0
\(848\) 1.09146 0.0374810
\(849\) 55.9208 1.91920
\(850\) −4.28284 −0.146900
\(851\) 0.655059 0.0224552
\(852\) 12.0343 0.412288
\(853\) −0.0955818 −0.00327266 −0.00163633 0.999999i \(-0.500521\pi\)
−0.00163633 + 0.999999i \(0.500521\pi\)
\(854\) −13.0021 −0.444922
\(855\) 5.28832 0.180857
\(856\) 1.75025 0.0598222
\(857\) 51.9594 1.77490 0.887449 0.460906i \(-0.152475\pi\)
0.887449 + 0.460906i \(0.152475\pi\)
\(858\) 0 0
\(859\) 35.1186 1.19823 0.599115 0.800663i \(-0.295519\pi\)
0.599115 + 0.800663i \(0.295519\pi\)
\(860\) −2.63928 −0.0899988
\(861\) 7.93198 0.270321
\(862\) 18.7940 0.640127
\(863\) −23.0325 −0.784037 −0.392018 0.919957i \(-0.628223\pi\)
−0.392018 + 0.919957i \(0.628223\pi\)
\(864\) −3.70611 −0.126085
\(865\) 7.54354 0.256488
\(866\) 32.4488 1.10265
\(867\) −2.74831 −0.0933376
\(868\) 5.70820 0.193749
\(869\) 0 0
\(870\) 2.67366 0.0906457
\(871\) 29.5720 1.00201
\(872\) −5.88436 −0.199270
\(873\) −11.6531 −0.394399
\(874\) −8.89322 −0.300818
\(875\) 1.00000 0.0338062
\(876\) −15.2319 −0.514637
\(877\) −30.7254 −1.03752 −0.518762 0.854919i \(-0.673607\pi\)
−0.518762 + 0.854919i \(0.673607\pi\)
\(878\) 17.1535 0.578901
\(879\) 39.0114 1.31582
\(880\) 0 0
\(881\) −42.3788 −1.42778 −0.713890 0.700258i \(-0.753068\pi\)
−0.713890 + 0.700258i \(0.753068\pi\)
\(882\) −1.18929 −0.0400455
\(883\) 38.3269 1.28980 0.644901 0.764266i \(-0.276899\pi\)
0.644901 + 0.764266i \(0.276899\pi\)
\(884\) −14.9944 −0.504317
\(885\) 20.5931 0.692231
\(886\) −4.23861 −0.142399
\(887\) −25.5816 −0.858948 −0.429474 0.903079i \(-0.641301\pi\)
−0.429474 + 0.903079i \(0.641301\pi\)
\(888\) −0.670380 −0.0224965
\(889\) −9.94218 −0.333450
\(890\) 1.39645 0.0468091
\(891\) 0 0
\(892\) 13.2981 0.445252
\(893\) −54.3632 −1.81919
\(894\) −32.7667 −1.09588
\(895\) −6.66819 −0.222893
\(896\) 1.00000 0.0334077
\(897\) −14.3317 −0.478522
\(898\) −13.1589 −0.439119
\(899\) 7.45652 0.248689
\(900\) 1.18929 0.0396431
\(901\) 4.67456 0.155732
\(902\) 0 0
\(903\) 5.40202 0.179768
\(904\) −17.1279 −0.569667
\(905\) 3.26233 0.108444
\(906\) 26.4825 0.879822
\(907\) −49.7760 −1.65279 −0.826393 0.563094i \(-0.809611\pi\)
−0.826393 + 0.563094i \(0.809611\pi\)
\(908\) −18.9140 −0.627683
\(909\) −20.6787 −0.685871
\(910\) 3.50105 0.116059
\(911\) 22.3261 0.739696 0.369848 0.929092i \(-0.379410\pi\)
0.369848 + 0.929092i \(0.379410\pi\)
\(912\) 9.10121 0.301371
\(913\) 0 0
\(914\) 0.679393 0.0224723
\(915\) −26.6124 −0.879778
\(916\) 3.89113 0.128566
\(917\) −7.65112 −0.252662
\(918\) −15.8727 −0.523877
\(919\) −17.9670 −0.592677 −0.296338 0.955083i \(-0.595766\pi\)
−0.296338 + 0.955083i \(0.595766\pi\)
\(920\) −2.00000 −0.0659380
\(921\) −57.0710 −1.88055
\(922\) 33.9310 1.11746
\(923\) 20.5848 0.677558
\(924\) 0 0
\(925\) −0.327530 −0.0107691
\(926\) 8.67985 0.285238
\(927\) 8.17029 0.268348
\(928\) 1.30628 0.0428808
\(929\) 13.7491 0.451092 0.225546 0.974233i \(-0.427583\pi\)
0.225546 + 0.974233i \(0.427583\pi\)
\(930\) 11.6834 0.383114
\(931\) −4.44661 −0.145732
\(932\) 6.92431 0.226813
\(933\) −43.0512 −1.40943
\(934\) −4.58156 −0.149913
\(935\) 0 0
\(936\) 4.16376 0.136097
\(937\) −8.38824 −0.274032 −0.137016 0.990569i \(-0.543751\pi\)
−0.137016 + 0.990569i \(0.543751\pi\)
\(938\) −8.44661 −0.275792
\(939\) −8.08110 −0.263717
\(940\) −12.2258 −0.398760
\(941\) −28.1650 −0.918153 −0.459077 0.888397i \(-0.651820\pi\)
−0.459077 + 0.888397i \(0.651820\pi\)
\(942\) 37.8561 1.23342
\(943\) −7.75070 −0.252398
\(944\) 10.0613 0.327466
\(945\) 3.70611 0.120560
\(946\) 0 0
\(947\) −0.901036 −0.0292797 −0.0146399 0.999893i \(-0.504660\pi\)
−0.0146399 + 0.999893i \(0.504660\pi\)
\(948\) 1.02125 0.0331686
\(949\) −26.0543 −0.845760
\(950\) 4.44661 0.144267
\(951\) 57.5605 1.86653
\(952\) 4.28284 0.138808
\(953\) −5.45845 −0.176817 −0.0884083 0.996084i \(-0.528178\pi\)
−0.0884083 + 0.996084i \(0.528178\pi\)
\(954\) −1.29807 −0.0420265
\(955\) 13.7035 0.443434
\(956\) −26.6477 −0.861850
\(957\) 0 0
\(958\) −38.8930 −1.25658
\(959\) −2.16038 −0.0697624
\(960\) 2.04678 0.0660594
\(961\) 1.58359 0.0510836
\(962\) −1.14670 −0.0369710
\(963\) −2.08155 −0.0670771
\(964\) −0.0928140 −0.00298934
\(965\) −11.1467 −0.358825
\(966\) 4.09355 0.131708
\(967\) −36.7612 −1.18216 −0.591079 0.806613i \(-0.701298\pi\)
−0.591079 + 0.806613i \(0.701298\pi\)
\(968\) 0 0
\(969\) 38.9791 1.25219
\(970\) −9.79837 −0.314607
\(971\) −47.2855 −1.51746 −0.758731 0.651404i \(-0.774180\pi\)
−0.758731 + 0.651404i \(0.774180\pi\)
\(972\) 11.7103 0.375608
\(973\) −21.1314 −0.677441
\(974\) −19.7593 −0.633127
\(975\) 7.16586 0.229491
\(976\) −13.0021 −0.416187
\(977\) −44.3011 −1.41732 −0.708658 0.705552i \(-0.750699\pi\)
−0.708658 + 0.705552i \(0.750699\pi\)
\(978\) 13.3638 0.427329
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 6.99823 0.223436
\(982\) 30.7082 0.979937
\(983\) −2.98404 −0.0951761 −0.0475880 0.998867i \(-0.515153\pi\)
−0.0475880 + 0.998867i \(0.515153\pi\)
\(984\) 7.93198 0.252862
\(985\) 3.51254 0.111919
\(986\) 5.59460 0.178168
\(987\) 25.0234 0.796504
\(988\) 15.5678 0.495277
\(989\) −5.27857 −0.167849
\(990\) 0 0
\(991\) −47.4047 −1.50586 −0.752930 0.658100i \(-0.771360\pi\)
−0.752930 + 0.658100i \(0.771360\pi\)
\(992\) 5.70820 0.181236
\(993\) −61.9637 −1.96636
\(994\) −5.87963 −0.186491
\(995\) −19.5167 −0.618722
\(996\) −25.8149 −0.817976
\(997\) 16.7474 0.530394 0.265197 0.964194i \(-0.414563\pi\)
0.265197 + 0.964194i \(0.414563\pi\)
\(998\) 18.5980 0.588711
\(999\) −1.21386 −0.0384049
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.1.1 4
11.5 even 5 770.2.n.e.421.1 8
11.9 even 5 770.2.n.e.631.1 yes 8
11.10 odd 2 8470.2.a.ct.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.e.421.1 8 11.5 even 5
770.2.n.e.631.1 yes 8 11.9 even 5
8470.2.a.cq.1.1 4 1.1 even 1 trivial
8470.2.a.ct.1.1 4 11.10 odd 2