Properties

Label 8470.2.a.da.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.38498000.1
Defining polynomial: \(x^{6} - x^{5} - 12 x^{4} + 5 x^{3} + 38 x^{2} - x - 19\)
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 \(-2.29803\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.29803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.29803 q^{6} +1.00000 q^{7} +1.00000 q^{8} +7.87703 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.29803 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.29803 q^{6} +1.00000 q^{7} +1.00000 q^{8} +7.87703 q^{9} -1.00000 q^{10} -3.29803 q^{12} +1.08609 q^{13} +1.00000 q^{14} +3.29803 q^{15} +1.00000 q^{16} +0.971513 q^{17} +7.87703 q^{18} -1.04489 q^{19} -1.00000 q^{20} -3.29803 q^{21} +2.42388 q^{23} -3.29803 q^{24} +1.00000 q^{25} +1.08609 q^{26} -16.0846 q^{27} +1.00000 q^{28} -8.23533 q^{29} +3.29803 q^{30} -4.17984 q^{31} +1.00000 q^{32} +0.971513 q^{34} -1.00000 q^{35} +7.87703 q^{36} +6.08312 q^{37} -1.04489 q^{38} -3.58197 q^{39} -1.00000 q^{40} -8.53772 q^{41} -3.29803 q^{42} -3.06122 q^{43} -7.87703 q^{45} +2.42388 q^{46} -12.3941 q^{47} -3.29803 q^{48} +1.00000 q^{49} +1.00000 q^{50} -3.20408 q^{51} +1.08609 q^{52} -0.323543 q^{53} -16.0846 q^{54} +1.00000 q^{56} +3.44609 q^{57} -8.23533 q^{58} +11.8882 q^{59} +3.29803 q^{60} +13.1507 q^{61} -4.17984 q^{62} +7.87703 q^{63} +1.00000 q^{64} -1.08609 q^{65} +8.03278 q^{67} +0.971513 q^{68} -7.99404 q^{69} -1.00000 q^{70} +4.18166 q^{71} +7.87703 q^{72} -9.94125 q^{73} +6.08312 q^{74} -3.29803 q^{75} -1.04489 q^{76} -3.58197 q^{78} -11.4402 q^{79} -1.00000 q^{80} +29.4165 q^{81} -8.53772 q^{82} -0.382041 q^{83} -3.29803 q^{84} -0.971513 q^{85} -3.06122 q^{86} +27.1604 q^{87} -11.2206 q^{89} -7.87703 q^{90} +1.08609 q^{91} +2.42388 q^{92} +13.7852 q^{93} -12.3941 q^{94} +1.04489 q^{95} -3.29803 q^{96} +14.8572 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} + O(q^{10}) \) \( 6q + 6q^{2} - 5q^{3} + 6q^{4} - 6q^{5} - 5q^{6} + 6q^{7} + 6q^{8} + 11q^{9} - 6q^{10} - 5q^{12} + 6q^{14} + 5q^{15} + 6q^{16} - 15q^{17} + 11q^{18} - 13q^{19} - 6q^{20} - 5q^{21} - 2q^{23} - 5q^{24} + 6q^{25} - 26q^{27} + 6q^{28} - 4q^{29} + 5q^{30} - 2q^{31} + 6q^{32} - 15q^{34} - 6q^{35} + 11q^{36} - 13q^{38} - 30q^{39} - 6q^{40} - 17q^{41} - 5q^{42} + 15q^{43} - 11q^{45} - 2q^{46} - 18q^{47} - 5q^{48} + 6q^{49} + 6q^{50} + 6q^{51} + 22q^{53} - 26q^{54} + 6q^{56} - 2q^{57} - 4q^{58} - 7q^{59} + 5q^{60} - 20q^{61} - 2q^{62} + 11q^{63} + 6q^{64} - 29q^{67} - 15q^{68} + 12q^{69} - 6q^{70} - 2q^{71} + 11q^{72} + q^{73} - 5q^{75} - 13q^{76} - 30q^{78} - 12q^{79} - 6q^{80} + 22q^{81} - 17q^{82} - 35q^{83} - 5q^{84} + 15q^{85} + 15q^{86} - 29q^{89} - 11q^{90} - 2q^{92} + 8q^{93} - 18q^{94} + 13q^{95} - 5q^{96} - 19q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) −3.29803 −1.90412 −0.952060 0.305910i \(-0.901039\pi\)
−0.952060 + 0.305910i \(0.901039\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −3.29803 −1.34642
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 7.87703 2.62568
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −3.29803 −0.952060
\(13\) 1.08609 0.301228 0.150614 0.988593i \(-0.451875\pi\)
0.150614 + 0.988593i \(0.451875\pi\)
\(14\) 1.00000 0.267261
\(15\) 3.29803 0.851549
\(16\) 1.00000 0.250000
\(17\) 0.971513 0.235627 0.117813 0.993036i \(-0.462412\pi\)
0.117813 + 0.993036i \(0.462412\pi\)
\(18\) 7.87703 1.85663
\(19\) −1.04489 −0.239715 −0.119857 0.992791i \(-0.538244\pi\)
−0.119857 + 0.992791i \(0.538244\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.29803 −0.719690
\(22\) 0 0
\(23\) 2.42388 0.505414 0.252707 0.967543i \(-0.418679\pi\)
0.252707 + 0.967543i \(0.418679\pi\)
\(24\) −3.29803 −0.673208
\(25\) 1.00000 0.200000
\(26\) 1.08609 0.213000
\(27\) −16.0846 −3.09548
\(28\) 1.00000 0.188982
\(29\) −8.23533 −1.52926 −0.764631 0.644468i \(-0.777079\pi\)
−0.764631 + 0.644468i \(0.777079\pi\)
\(30\) 3.29803 0.602136
\(31\) −4.17984 −0.750721 −0.375360 0.926879i \(-0.622481\pi\)
−0.375360 + 0.926879i \(0.622481\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.971513 0.166613
\(35\) −1.00000 −0.169031
\(36\) 7.87703 1.31284
\(37\) 6.08312 1.00006 0.500029 0.866009i \(-0.333323\pi\)
0.500029 + 0.866009i \(0.333323\pi\)
\(38\) −1.04489 −0.169504
\(39\) −3.58197 −0.573575
\(40\) −1.00000 −0.158114
\(41\) −8.53772 −1.33337 −0.666684 0.745341i \(-0.732287\pi\)
−0.666684 + 0.745341i \(0.732287\pi\)
\(42\) −3.29803 −0.508898
\(43\) −3.06122 −0.466832 −0.233416 0.972377i \(-0.574990\pi\)
−0.233416 + 0.972377i \(0.574990\pi\)
\(44\) 0 0
\(45\) −7.87703 −1.17424
\(46\) 2.42388 0.357382
\(47\) −12.3941 −1.80786 −0.903930 0.427681i \(-0.859331\pi\)
−0.903930 + 0.427681i \(0.859331\pi\)
\(48\) −3.29803 −0.476030
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −3.20408 −0.448661
\(52\) 1.08609 0.150614
\(53\) −0.323543 −0.0444421 −0.0222211 0.999753i \(-0.507074\pi\)
−0.0222211 + 0.999753i \(0.507074\pi\)
\(54\) −16.0846 −2.18884
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 3.44609 0.456446
\(58\) −8.23533 −1.08135
\(59\) 11.8882 1.54771 0.773857 0.633360i \(-0.218325\pi\)
0.773857 + 0.633360i \(0.218325\pi\)
\(60\) 3.29803 0.425774
\(61\) 13.1507 1.68377 0.841884 0.539658i \(-0.181446\pi\)
0.841884 + 0.539658i \(0.181446\pi\)
\(62\) −4.17984 −0.530840
\(63\) 7.87703 0.992412
\(64\) 1.00000 0.125000
\(65\) −1.08609 −0.134713
\(66\) 0 0
\(67\) 8.03278 0.981361 0.490680 0.871340i \(-0.336748\pi\)
0.490680 + 0.871340i \(0.336748\pi\)
\(68\) 0.971513 0.117813
\(69\) −7.99404 −0.962370
\(70\) −1.00000 −0.119523
\(71\) 4.18166 0.496272 0.248136 0.968725i \(-0.420182\pi\)
0.248136 + 0.968725i \(0.420182\pi\)
\(72\) 7.87703 0.928317
\(73\) −9.94125 −1.16354 −0.581768 0.813355i \(-0.697639\pi\)
−0.581768 + 0.813355i \(0.697639\pi\)
\(74\) 6.08312 0.707148
\(75\) −3.29803 −0.380824
\(76\) −1.04489 −0.119857
\(77\) 0 0
\(78\) −3.58197 −0.405579
\(79\) −11.4402 −1.28712 −0.643562 0.765394i \(-0.722544\pi\)
−0.643562 + 0.765394i \(0.722544\pi\)
\(80\) −1.00000 −0.111803
\(81\) 29.4165 3.26850
\(82\) −8.53772 −0.942834
\(83\) −0.382041 −0.0419344 −0.0209672 0.999780i \(-0.506675\pi\)
−0.0209672 + 0.999780i \(0.506675\pi\)
\(84\) −3.29803 −0.359845
\(85\) −0.971513 −0.105375
\(86\) −3.06122 −0.330100
\(87\) 27.1604 2.91190
\(88\) 0 0
\(89\) −11.2206 −1.18938 −0.594692 0.803954i \(-0.702726\pi\)
−0.594692 + 0.803954i \(0.702726\pi\)
\(90\) −7.87703 −0.830312
\(91\) 1.08609 0.113854
\(92\) 2.42388 0.252707
\(93\) 13.7852 1.42946
\(94\) −12.3941 −1.27835
\(95\) 1.04489 0.107204
\(96\) −3.29803 −0.336604
\(97\) 14.8572 1.50852 0.754262 0.656573i \(-0.227995\pi\)
0.754262 + 0.656573i \(0.227995\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 5.85686 0.582779 0.291390 0.956604i \(-0.405882\pi\)
0.291390 + 0.956604i \(0.405882\pi\)
\(102\) −3.20408 −0.317252
\(103\) −11.1086 −1.09456 −0.547280 0.836949i \(-0.684337\pi\)
−0.547280 + 0.836949i \(0.684337\pi\)
\(104\) 1.08609 0.106500
\(105\) 3.29803 0.321855
\(106\) −0.323543 −0.0314253
\(107\) 7.35576 0.711108 0.355554 0.934656i \(-0.384292\pi\)
0.355554 + 0.934656i \(0.384292\pi\)
\(108\) −16.0846 −1.54774
\(109\) −1.17544 −0.112587 −0.0562933 0.998414i \(-0.517928\pi\)
−0.0562933 + 0.998414i \(0.517928\pi\)
\(110\) 0 0
\(111\) −20.0623 −1.90423
\(112\) 1.00000 0.0944911
\(113\) 11.3045 1.06344 0.531720 0.846920i \(-0.321546\pi\)
0.531720 + 0.846920i \(0.321546\pi\)
\(114\) 3.44609 0.322756
\(115\) −2.42388 −0.226028
\(116\) −8.23533 −0.764631
\(117\) 8.55519 0.790928
\(118\) 11.8882 1.09440
\(119\) 0.971513 0.0890585
\(120\) 3.29803 0.301068
\(121\) 0 0
\(122\) 13.1507 1.19060
\(123\) 28.1577 2.53889
\(124\) −4.17984 −0.375360
\(125\) −1.00000 −0.0894427
\(126\) 7.87703 0.701742
\(127\) 6.94515 0.616282 0.308141 0.951341i \(-0.400293\pi\)
0.308141 + 0.951341i \(0.400293\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.0960 0.888905
\(130\) −1.08609 −0.0952567
\(131\) −4.83817 −0.422713 −0.211356 0.977409i \(-0.567788\pi\)
−0.211356 + 0.977409i \(0.567788\pi\)
\(132\) 0 0
\(133\) −1.04489 −0.0906037
\(134\) 8.03278 0.693927
\(135\) 16.0846 1.38434
\(136\) 0.971513 0.0833066
\(137\) −16.9448 −1.44769 −0.723846 0.689962i \(-0.757627\pi\)
−0.723846 + 0.689962i \(0.757627\pi\)
\(138\) −7.99404 −0.680498
\(139\) −9.81201 −0.832244 −0.416122 0.909309i \(-0.636611\pi\)
−0.416122 + 0.909309i \(0.636611\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 40.8760 3.44238
\(142\) 4.18166 0.350917
\(143\) 0 0
\(144\) 7.87703 0.656419
\(145\) 8.23533 0.683907
\(146\) −9.94125 −0.822744
\(147\) −3.29803 −0.272017
\(148\) 6.08312 0.500029
\(149\) −17.2573 −1.41377 −0.706885 0.707328i \(-0.749900\pi\)
−0.706885 + 0.707328i \(0.749900\pi\)
\(150\) −3.29803 −0.269283
\(151\) 15.6657 1.27485 0.637426 0.770511i \(-0.279999\pi\)
0.637426 + 0.770511i \(0.279999\pi\)
\(152\) −1.04489 −0.0847520
\(153\) 7.65264 0.618679
\(154\) 0 0
\(155\) 4.17984 0.335732
\(156\) −3.58197 −0.286787
\(157\) 9.25792 0.738862 0.369431 0.929258i \(-0.379553\pi\)
0.369431 + 0.929258i \(0.379553\pi\)
\(158\) −11.4402 −0.910135
\(159\) 1.06706 0.0846232
\(160\) −1.00000 −0.0790569
\(161\) 2.42388 0.191029
\(162\) 29.4165 2.31118
\(163\) −7.31437 −0.572905 −0.286453 0.958094i \(-0.592476\pi\)
−0.286453 + 0.958094i \(0.592476\pi\)
\(164\) −8.53772 −0.666684
\(165\) 0 0
\(166\) −0.382041 −0.0296521
\(167\) −12.6498 −0.978870 −0.489435 0.872040i \(-0.662797\pi\)
−0.489435 + 0.872040i \(0.662797\pi\)
\(168\) −3.29803 −0.254449
\(169\) −11.8204 −0.909262
\(170\) −0.971513 −0.0745117
\(171\) −8.23066 −0.629414
\(172\) −3.06122 −0.233416
\(173\) 1.01451 0.0771316 0.0385658 0.999256i \(-0.487721\pi\)
0.0385658 + 0.999256i \(0.487721\pi\)
\(174\) 27.1604 2.05902
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −39.2078 −2.94704
\(178\) −11.2206 −0.841021
\(179\) −17.1959 −1.28528 −0.642641 0.766167i \(-0.722162\pi\)
−0.642641 + 0.766167i \(0.722162\pi\)
\(180\) −7.87703 −0.587119
\(181\) 20.3704 1.51412 0.757059 0.653346i \(-0.226635\pi\)
0.757059 + 0.653346i \(0.226635\pi\)
\(182\) 1.08609 0.0805066
\(183\) −43.3713 −3.20610
\(184\) 2.42388 0.178691
\(185\) −6.08312 −0.447240
\(186\) 13.7852 1.01078
\(187\) 0 0
\(188\) −12.3941 −0.903930
\(189\) −16.0846 −1.16998
\(190\) 1.04489 0.0758045
\(191\) −6.42615 −0.464980 −0.232490 0.972599i \(-0.574687\pi\)
−0.232490 + 0.972599i \(0.574687\pi\)
\(192\) −3.29803 −0.238015
\(193\) 14.0142 1.00876 0.504382 0.863480i \(-0.331720\pi\)
0.504382 + 0.863480i \(0.331720\pi\)
\(194\) 14.8572 1.06669
\(195\) 3.58197 0.256510
\(196\) 1.00000 0.0714286
\(197\) 21.3383 1.52029 0.760147 0.649752i \(-0.225127\pi\)
0.760147 + 0.649752i \(0.225127\pi\)
\(198\) 0 0
\(199\) −26.4689 −1.87633 −0.938166 0.346186i \(-0.887477\pi\)
−0.938166 + 0.346186i \(0.887477\pi\)
\(200\) 1.00000 0.0707107
\(201\) −26.4924 −1.86863
\(202\) 5.85686 0.412087
\(203\) −8.23533 −0.578007
\(204\) −3.20408 −0.224331
\(205\) 8.53772 0.596300
\(206\) −11.1086 −0.773971
\(207\) 19.0930 1.32705
\(208\) 1.08609 0.0753070
\(209\) 0 0
\(210\) 3.29803 0.227586
\(211\) 12.6514 0.870955 0.435477 0.900200i \(-0.356580\pi\)
0.435477 + 0.900200i \(0.356580\pi\)
\(212\) −0.323543 −0.0222211
\(213\) −13.7913 −0.944961
\(214\) 7.35576 0.502830
\(215\) 3.06122 0.208774
\(216\) −16.0846 −1.09442
\(217\) −4.17984 −0.283746
\(218\) −1.17544 −0.0796107
\(219\) 32.7866 2.21551
\(220\) 0 0
\(221\) 1.05515 0.0709773
\(222\) −20.0623 −1.34650
\(223\) −19.2003 −1.28575 −0.642873 0.765973i \(-0.722258\pi\)
−0.642873 + 0.765973i \(0.722258\pi\)
\(224\) 1.00000 0.0668153
\(225\) 7.87703 0.525135
\(226\) 11.3045 0.751966
\(227\) 2.24667 0.149116 0.0745582 0.997217i \(-0.476245\pi\)
0.0745582 + 0.997217i \(0.476245\pi\)
\(228\) 3.44609 0.228223
\(229\) −4.78822 −0.316414 −0.158207 0.987406i \(-0.550571\pi\)
−0.158207 + 0.987406i \(0.550571\pi\)
\(230\) −2.42388 −0.159826
\(231\) 0 0
\(232\) −8.23533 −0.540676
\(233\) 12.7428 0.834807 0.417404 0.908721i \(-0.362940\pi\)
0.417404 + 0.908721i \(0.362940\pi\)
\(234\) 8.55519 0.559270
\(235\) 12.3941 0.808499
\(236\) 11.8882 0.773857
\(237\) 37.7302 2.45084
\(238\) 0.971513 0.0629738
\(239\) −2.25974 −0.146170 −0.0730851 0.997326i \(-0.523284\pi\)
−0.0730851 + 0.997326i \(0.523284\pi\)
\(240\) 3.29803 0.212887
\(241\) 1.94631 0.125373 0.0626865 0.998033i \(-0.480033\pi\)
0.0626865 + 0.998033i \(0.480033\pi\)
\(242\) 0 0
\(243\) −48.7628 −3.12814
\(244\) 13.1507 0.841884
\(245\) −1.00000 −0.0638877
\(246\) 28.1577 1.79527
\(247\) −1.13485 −0.0722089
\(248\) −4.17984 −0.265420
\(249\) 1.25998 0.0798482
\(250\) −1.00000 −0.0632456
\(251\) −15.8786 −1.00225 −0.501125 0.865375i \(-0.667080\pi\)
−0.501125 + 0.865375i \(0.667080\pi\)
\(252\) 7.87703 0.496206
\(253\) 0 0
\(254\) 6.94515 0.435777
\(255\) 3.20408 0.200648
\(256\) 1.00000 0.0625000
\(257\) −7.07546 −0.441355 −0.220677 0.975347i \(-0.570827\pi\)
−0.220677 + 0.975347i \(0.570827\pi\)
\(258\) 10.0960 0.628551
\(259\) 6.08312 0.377986
\(260\) −1.08609 −0.0673567
\(261\) −64.8699 −4.01535
\(262\) −4.83817 −0.298903
\(263\) −26.1468 −1.61228 −0.806139 0.591726i \(-0.798447\pi\)
−0.806139 + 0.591726i \(0.798447\pi\)
\(264\) 0 0
\(265\) 0.323543 0.0198751
\(266\) −1.04489 −0.0640665
\(267\) 37.0060 2.26473
\(268\) 8.03278 0.490680
\(269\) 17.5426 1.06959 0.534796 0.844981i \(-0.320388\pi\)
0.534796 + 0.844981i \(0.320388\pi\)
\(270\) 16.0846 0.978878
\(271\) −27.3246 −1.65985 −0.829927 0.557873i \(-0.811618\pi\)
−0.829927 + 0.557873i \(0.811618\pi\)
\(272\) 0.971513 0.0589066
\(273\) −3.58197 −0.216791
\(274\) −16.9448 −1.02367
\(275\) 0 0
\(276\) −7.99404 −0.481185
\(277\) −4.29659 −0.258157 −0.129079 0.991634i \(-0.541202\pi\)
−0.129079 + 0.991634i \(0.541202\pi\)
\(278\) −9.81201 −0.588485
\(279\) −32.9247 −1.97115
\(280\) −1.00000 −0.0597614
\(281\) −25.8683 −1.54317 −0.771586 0.636125i \(-0.780536\pi\)
−0.771586 + 0.636125i \(0.780536\pi\)
\(282\) 40.8760 2.43413
\(283\) 27.1448 1.61359 0.806795 0.590831i \(-0.201200\pi\)
0.806795 + 0.590831i \(0.201200\pi\)
\(284\) 4.18166 0.248136
\(285\) −3.44609 −0.204129
\(286\) 0 0
\(287\) −8.53772 −0.503966
\(288\) 7.87703 0.464158
\(289\) −16.0562 −0.944480
\(290\) 8.23533 0.483595
\(291\) −48.9997 −2.87241
\(292\) −9.94125 −0.581768
\(293\) −28.7674 −1.68061 −0.840304 0.542115i \(-0.817624\pi\)
−0.840304 + 0.542115i \(0.817624\pi\)
\(294\) −3.29803 −0.192345
\(295\) −11.8882 −0.692159
\(296\) 6.08312 0.353574
\(297\) 0 0
\(298\) −17.2573 −0.999686
\(299\) 2.63256 0.152245
\(300\) −3.29803 −0.190412
\(301\) −3.06122 −0.176446
\(302\) 15.6657 0.901457
\(303\) −19.3161 −1.10968
\(304\) −1.04489 −0.0599287
\(305\) −13.1507 −0.753004
\(306\) 7.65264 0.437472
\(307\) −22.6960 −1.29533 −0.647666 0.761924i \(-0.724255\pi\)
−0.647666 + 0.761924i \(0.724255\pi\)
\(308\) 0 0
\(309\) 36.6365 2.08418
\(310\) 4.17984 0.237399
\(311\) 0.759043 0.0430414 0.0215207 0.999768i \(-0.493149\pi\)
0.0215207 + 0.999768i \(0.493149\pi\)
\(312\) −3.58197 −0.202789
\(313\) 21.6267 1.22241 0.611207 0.791471i \(-0.290684\pi\)
0.611207 + 0.791471i \(0.290684\pi\)
\(314\) 9.25792 0.522454
\(315\) −7.87703 −0.443820
\(316\) −11.4402 −0.643562
\(317\) 0.0857876 0.00481831 0.00240916 0.999997i \(-0.499233\pi\)
0.00240916 + 0.999997i \(0.499233\pi\)
\(318\) 1.06706 0.0598376
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −24.2596 −1.35404
\(322\) 2.42388 0.135078
\(323\) −1.01513 −0.0564832
\(324\) 29.4165 1.63425
\(325\) 1.08609 0.0602456
\(326\) −7.31437 −0.405105
\(327\) 3.87663 0.214378
\(328\) −8.53772 −0.471417
\(329\) −12.3941 −0.683307
\(330\) 0 0
\(331\) −22.6026 −1.24235 −0.621177 0.783670i \(-0.713345\pi\)
−0.621177 + 0.783670i \(0.713345\pi\)
\(332\) −0.382041 −0.0209672
\(333\) 47.9169 2.62583
\(334\) −12.6498 −0.692165
\(335\) −8.03278 −0.438878
\(336\) −3.29803 −0.179923
\(337\) 15.2690 0.831753 0.415876 0.909421i \(-0.363475\pi\)
0.415876 + 0.909421i \(0.363475\pi\)
\(338\) −11.8204 −0.642945
\(339\) −37.2827 −2.02492
\(340\) −0.971513 −0.0526877
\(341\) 0 0
\(342\) −8.23066 −0.445063
\(343\) 1.00000 0.0539949
\(344\) −3.06122 −0.165050
\(345\) 7.99404 0.430385
\(346\) 1.01451 0.0545402
\(347\) 5.89915 0.316683 0.158342 0.987384i \(-0.449385\pi\)
0.158342 + 0.987384i \(0.449385\pi\)
\(348\) 27.1604 1.45595
\(349\) 16.9015 0.904718 0.452359 0.891836i \(-0.350583\pi\)
0.452359 + 0.891836i \(0.350583\pi\)
\(350\) 1.00000 0.0534522
\(351\) −17.4694 −0.932447
\(352\) 0 0
\(353\) −17.3914 −0.925648 −0.462824 0.886450i \(-0.653164\pi\)
−0.462824 + 0.886450i \(0.653164\pi\)
\(354\) −39.2078 −2.08387
\(355\) −4.18166 −0.221939
\(356\) −11.2206 −0.594692
\(357\) −3.20408 −0.169578
\(358\) −17.1959 −0.908832
\(359\) 22.4481 1.18476 0.592382 0.805657i \(-0.298188\pi\)
0.592382 + 0.805657i \(0.298188\pi\)
\(360\) −7.87703 −0.415156
\(361\) −17.9082 −0.942537
\(362\) 20.3704 1.07064
\(363\) 0 0
\(364\) 1.08609 0.0569268
\(365\) 9.94125 0.520349
\(366\) −43.3713 −2.26705
\(367\) −4.81275 −0.251224 −0.125612 0.992079i \(-0.540089\pi\)
−0.125612 + 0.992079i \(0.540089\pi\)
\(368\) 2.42388 0.126354
\(369\) −67.2519 −3.50099
\(370\) −6.08312 −0.316246
\(371\) −0.323543 −0.0167975
\(372\) 13.7852 0.714731
\(373\) −6.77161 −0.350621 −0.175310 0.984513i \(-0.556093\pi\)
−0.175310 + 0.984513i \(0.556093\pi\)
\(374\) 0 0
\(375\) 3.29803 0.170310
\(376\) −12.3941 −0.639175
\(377\) −8.94433 −0.460657
\(378\) −16.0846 −0.827303
\(379\) −35.5405 −1.82559 −0.912797 0.408413i \(-0.866082\pi\)
−0.912797 + 0.408413i \(0.866082\pi\)
\(380\) 1.04489 0.0536019
\(381\) −22.9053 −1.17348
\(382\) −6.42615 −0.328790
\(383\) 21.0463 1.07542 0.537709 0.843131i \(-0.319290\pi\)
0.537709 + 0.843131i \(0.319290\pi\)
\(384\) −3.29803 −0.168302
\(385\) 0 0
\(386\) 14.0142 0.713304
\(387\) −24.1134 −1.22575
\(388\) 14.8572 0.754262
\(389\) 35.1161 1.78046 0.890230 0.455512i \(-0.150544\pi\)
0.890230 + 0.455512i \(0.150544\pi\)
\(390\) 3.58197 0.181380
\(391\) 2.35483 0.119089
\(392\) 1.00000 0.0505076
\(393\) 15.9564 0.804896
\(394\) 21.3383 1.07501
\(395\) 11.4402 0.575620
\(396\) 0 0
\(397\) 10.4987 0.526914 0.263457 0.964671i \(-0.415137\pi\)
0.263457 + 0.964671i \(0.415137\pi\)
\(398\) −26.4689 −1.32677
\(399\) 3.44609 0.172520
\(400\) 1.00000 0.0500000
\(401\) −21.9027 −1.09377 −0.546884 0.837208i \(-0.684186\pi\)
−0.546884 + 0.837208i \(0.684186\pi\)
\(402\) −26.4924 −1.32132
\(403\) −4.53969 −0.226138
\(404\) 5.85686 0.291390
\(405\) −29.4165 −1.46172
\(406\) −8.23533 −0.408712
\(407\) 0 0
\(408\) −3.20408 −0.158626
\(409\) 9.24769 0.457269 0.228634 0.973512i \(-0.426574\pi\)
0.228634 + 0.973512i \(0.426574\pi\)
\(410\) 8.53772 0.421648
\(411\) 55.8845 2.75658
\(412\) −11.1086 −0.547280
\(413\) 11.8882 0.584981
\(414\) 19.0930 0.938369
\(415\) 0.382041 0.0187536
\(416\) 1.08609 0.0532501
\(417\) 32.3603 1.58469
\(418\) 0 0
\(419\) 2.02744 0.0990469 0.0495235 0.998773i \(-0.484230\pi\)
0.0495235 + 0.998773i \(0.484230\pi\)
\(420\) 3.29803 0.160928
\(421\) 9.29834 0.453173 0.226587 0.973991i \(-0.427243\pi\)
0.226587 + 0.973991i \(0.427243\pi\)
\(422\) 12.6514 0.615858
\(423\) −97.6284 −4.74685
\(424\) −0.323543 −0.0157127
\(425\) 0.971513 0.0471253
\(426\) −13.7913 −0.668189
\(427\) 13.1507 0.636405
\(428\) 7.35576 0.355554
\(429\) 0 0
\(430\) 3.06122 0.147625
\(431\) −18.1249 −0.873048 −0.436524 0.899693i \(-0.643791\pi\)
−0.436524 + 0.899693i \(0.643791\pi\)
\(432\) −16.0846 −0.773871
\(433\) −33.4904 −1.60945 −0.804723 0.593650i \(-0.797686\pi\)
−0.804723 + 0.593650i \(0.797686\pi\)
\(434\) −4.17984 −0.200639
\(435\) −27.1604 −1.30224
\(436\) −1.17544 −0.0562933
\(437\) −2.53270 −0.121155
\(438\) 32.7866 1.56660
\(439\) −39.7487 −1.89710 −0.948551 0.316623i \(-0.897451\pi\)
−0.948551 + 0.316623i \(0.897451\pi\)
\(440\) 0 0
\(441\) 7.87703 0.375097
\(442\) 1.05515 0.0501886
\(443\) 36.0159 1.71117 0.855584 0.517665i \(-0.173198\pi\)
0.855584 + 0.517665i \(0.173198\pi\)
\(444\) −20.0623 −0.952116
\(445\) 11.2206 0.531909
\(446\) −19.2003 −0.909160
\(447\) 56.9150 2.69199
\(448\) 1.00000 0.0472456
\(449\) 5.64047 0.266190 0.133095 0.991103i \(-0.457508\pi\)
0.133095 + 0.991103i \(0.457508\pi\)
\(450\) 7.87703 0.371327
\(451\) 0 0
\(452\) 11.3045 0.531720
\(453\) −51.6659 −2.42747
\(454\) 2.24667 0.105441
\(455\) −1.08609 −0.0509168
\(456\) 3.44609 0.161378
\(457\) −21.2353 −0.993345 −0.496673 0.867938i \(-0.665445\pi\)
−0.496673 + 0.867938i \(0.665445\pi\)
\(458\) −4.78822 −0.223739
\(459\) −15.6264 −0.729378
\(460\) −2.42388 −0.113014
\(461\) −36.7740 −1.71273 −0.856367 0.516367i \(-0.827284\pi\)
−0.856367 + 0.516367i \(0.827284\pi\)
\(462\) 0 0
\(463\) 21.8388 1.01494 0.507469 0.861670i \(-0.330581\pi\)
0.507469 + 0.861670i \(0.330581\pi\)
\(464\) −8.23533 −0.382315
\(465\) −13.7852 −0.639275
\(466\) 12.7428 0.590298
\(467\) 34.4719 1.59517 0.797584 0.603208i \(-0.206111\pi\)
0.797584 + 0.603208i \(0.206111\pi\)
\(468\) 8.55519 0.395464
\(469\) 8.03278 0.370920
\(470\) 12.3941 0.571695
\(471\) −30.5329 −1.40688
\(472\) 11.8882 0.547200
\(473\) 0 0
\(474\) 37.7302 1.73301
\(475\) −1.04489 −0.0479430
\(476\) 0.971513 0.0445292
\(477\) −2.54856 −0.116691
\(478\) −2.25974 −0.103358
\(479\) −5.73461 −0.262021 −0.131010 0.991381i \(-0.541822\pi\)
−0.131010 + 0.991381i \(0.541822\pi\)
\(480\) 3.29803 0.150534
\(481\) 6.60683 0.301246
\(482\) 1.94631 0.0886521
\(483\) −7.99404 −0.363742
\(484\) 0 0
\(485\) −14.8572 −0.674633
\(486\) −48.7628 −2.21193
\(487\) 6.98339 0.316448 0.158224 0.987403i \(-0.449423\pi\)
0.158224 + 0.987403i \(0.449423\pi\)
\(488\) 13.1507 0.595302
\(489\) 24.1230 1.09088
\(490\) −1.00000 −0.0451754
\(491\) −42.0589 −1.89809 −0.949045 0.315139i \(-0.897949\pi\)
−0.949045 + 0.315139i \(0.897949\pi\)
\(492\) 28.1577 1.26945
\(493\) −8.00073 −0.360335
\(494\) −1.13485 −0.0510594
\(495\) 0 0
\(496\) −4.17984 −0.187680
\(497\) 4.18166 0.187573
\(498\) 1.25998 0.0564612
\(499\) −19.9215 −0.891808 −0.445904 0.895081i \(-0.647118\pi\)
−0.445904 + 0.895081i \(0.647118\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 41.7194 1.86389
\(502\) −15.8786 −0.708698
\(503\) 8.42225 0.375530 0.187765 0.982214i \(-0.439876\pi\)
0.187765 + 0.982214i \(0.439876\pi\)
\(504\) 7.87703 0.350871
\(505\) −5.85686 −0.260627
\(506\) 0 0
\(507\) 38.9841 1.73134
\(508\) 6.94515 0.308141
\(509\) −7.87220 −0.348929 −0.174465 0.984663i \(-0.555819\pi\)
−0.174465 + 0.984663i \(0.555819\pi\)
\(510\) 3.20408 0.141879
\(511\) −9.94125 −0.439775
\(512\) 1.00000 0.0441942
\(513\) 16.8067 0.742034
\(514\) −7.07546 −0.312085
\(515\) 11.1086 0.489502
\(516\) 10.0960 0.444453
\(517\) 0 0
\(518\) 6.08312 0.267277
\(519\) −3.34588 −0.146868
\(520\) −1.08609 −0.0476283
\(521\) 23.4996 1.02953 0.514767 0.857330i \(-0.327878\pi\)
0.514767 + 0.857330i \(0.327878\pi\)
\(522\) −64.8699 −2.83928
\(523\) −8.57857 −0.375115 −0.187557 0.982254i \(-0.560057\pi\)
−0.187557 + 0.982254i \(0.560057\pi\)
\(524\) −4.83817 −0.211356
\(525\) −3.29803 −0.143938
\(526\) −26.1468 −1.14005
\(527\) −4.06077 −0.176890
\(528\) 0 0
\(529\) −17.1248 −0.744556
\(530\) 0.323543 0.0140538
\(531\) 93.6439 4.06380
\(532\) −1.04489 −0.0453019
\(533\) −9.27276 −0.401648
\(534\) 37.0060 1.60141
\(535\) −7.35576 −0.318017
\(536\) 8.03278 0.346963
\(537\) 56.7127 2.44733
\(538\) 17.5426 0.756316
\(539\) 0 0
\(540\) 16.0846 0.692171
\(541\) −18.8588 −0.810804 −0.405402 0.914139i \(-0.632868\pi\)
−0.405402 + 0.914139i \(0.632868\pi\)
\(542\) −27.3246 −1.17369
\(543\) −67.1822 −2.88306
\(544\) 0.971513 0.0416533
\(545\) 1.17544 0.0503502
\(546\) −3.58197 −0.153294
\(547\) 32.1205 1.37338 0.686688 0.726953i \(-0.259064\pi\)
0.686688 + 0.726953i \(0.259064\pi\)
\(548\) −16.9448 −0.723846
\(549\) 103.588 4.42103
\(550\) 0 0
\(551\) 8.60504 0.366587
\(552\) −7.99404 −0.340249
\(553\) −11.4402 −0.486487
\(554\) −4.29659 −0.182545
\(555\) 20.0623 0.851598
\(556\) −9.81201 −0.416122
\(557\) −24.7492 −1.04866 −0.524329 0.851516i \(-0.675684\pi\)
−0.524329 + 0.851516i \(0.675684\pi\)
\(558\) −32.9247 −1.39381
\(559\) −3.32478 −0.140623
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −25.8683 −1.09119
\(563\) 4.50739 0.189964 0.0949820 0.995479i \(-0.469721\pi\)
0.0949820 + 0.995479i \(0.469721\pi\)
\(564\) 40.8760 1.72119
\(565\) −11.3045 −0.475585
\(566\) 27.1448 1.14098
\(567\) 29.4165 1.23538
\(568\) 4.18166 0.175459
\(569\) −35.3172 −1.48058 −0.740288 0.672290i \(-0.765311\pi\)
−0.740288 + 0.672290i \(0.765311\pi\)
\(570\) −3.44609 −0.144341
\(571\) −21.6537 −0.906179 −0.453089 0.891465i \(-0.649678\pi\)
−0.453089 + 0.891465i \(0.649678\pi\)
\(572\) 0 0
\(573\) 21.1937 0.885378
\(574\) −8.53772 −0.356358
\(575\) 2.42388 0.101083
\(576\) 7.87703 0.328210
\(577\) −6.00586 −0.250027 −0.125014 0.992155i \(-0.539897\pi\)
−0.125014 + 0.992155i \(0.539897\pi\)
\(578\) −16.0562 −0.667848
\(579\) −46.2193 −1.92081
\(580\) 8.23533 0.341953
\(581\) −0.382041 −0.0158497
\(582\) −48.9997 −2.03110
\(583\) 0 0
\(584\) −9.94125 −0.411372
\(585\) −8.55519 −0.353714
\(586\) −28.7674 −1.18837
\(587\) 5.77309 0.238281 0.119140 0.992877i \(-0.461986\pi\)
0.119140 + 0.992877i \(0.461986\pi\)
\(588\) −3.29803 −0.136009
\(589\) 4.36748 0.179959
\(590\) −11.8882 −0.489430
\(591\) −70.3745 −2.89482
\(592\) 6.08312 0.250015
\(593\) 10.5859 0.434710 0.217355 0.976093i \(-0.430257\pi\)
0.217355 + 0.976093i \(0.430257\pi\)
\(594\) 0 0
\(595\) −0.971513 −0.0398282
\(596\) −17.2573 −0.706885
\(597\) 87.2954 3.57276
\(598\) 2.63256 0.107653
\(599\) −21.4656 −0.877062 −0.438531 0.898716i \(-0.644501\pi\)
−0.438531 + 0.898716i \(0.644501\pi\)
\(600\) −3.29803 −0.134642
\(601\) 21.6912 0.884801 0.442401 0.896818i \(-0.354127\pi\)
0.442401 + 0.896818i \(0.354127\pi\)
\(602\) −3.06122 −0.124766
\(603\) 63.2745 2.57674
\(604\) 15.6657 0.637426
\(605\) 0 0
\(606\) −19.3161 −0.784664
\(607\) 3.63740 0.147637 0.0738187 0.997272i \(-0.476481\pi\)
0.0738187 + 0.997272i \(0.476481\pi\)
\(608\) −1.04489 −0.0423760
\(609\) 27.1604 1.10059
\(610\) −13.1507 −0.532454
\(611\) −13.4611 −0.544578
\(612\) 7.65264 0.309340
\(613\) −33.4826 −1.35235 −0.676174 0.736742i \(-0.736363\pi\)
−0.676174 + 0.736742i \(0.736363\pi\)
\(614\) −22.6960 −0.915938
\(615\) −28.1577 −1.13543
\(616\) 0 0
\(617\) 19.6244 0.790048 0.395024 0.918671i \(-0.370736\pi\)
0.395024 + 0.918671i \(0.370736\pi\)
\(618\) 36.6365 1.47374
\(619\) 4.83357 0.194278 0.0971389 0.995271i \(-0.469031\pi\)
0.0971389 + 0.995271i \(0.469031\pi\)
\(620\) 4.17984 0.167866
\(621\) −38.9872 −1.56450
\(622\) 0.759043 0.0304348
\(623\) −11.2206 −0.449545
\(624\) −3.58197 −0.143394
\(625\) 1.00000 0.0400000
\(626\) 21.6267 0.864377
\(627\) 0 0
\(628\) 9.25792 0.369431
\(629\) 5.90983 0.235640
\(630\) −7.87703 −0.313828
\(631\) 10.2621 0.408528 0.204264 0.978916i \(-0.434520\pi\)
0.204264 + 0.978916i \(0.434520\pi\)
\(632\) −11.4402 −0.455067
\(633\) −41.7246 −1.65840
\(634\) 0.0857876 0.00340706
\(635\) −6.94515 −0.275610
\(636\) 1.06706 0.0423116
\(637\) 1.08609 0.0430326
\(638\) 0 0
\(639\) 32.9391 1.30305
\(640\) −1.00000 −0.0395285
\(641\) −10.9243 −0.431485 −0.215742 0.976450i \(-0.569217\pi\)
−0.215742 + 0.976450i \(0.569217\pi\)
\(642\) −24.2596 −0.957448
\(643\) −3.61607 −0.142604 −0.0713020 0.997455i \(-0.522715\pi\)
−0.0713020 + 0.997455i \(0.522715\pi\)
\(644\) 2.42388 0.0955143
\(645\) −10.0960 −0.397531
\(646\) −1.01513 −0.0399397
\(647\) −15.1540 −0.595766 −0.297883 0.954602i \(-0.596281\pi\)
−0.297883 + 0.954602i \(0.596281\pi\)
\(648\) 29.4165 1.15559
\(649\) 0 0
\(650\) 1.08609 0.0426001
\(651\) 13.7852 0.540286
\(652\) −7.31437 −0.286453
\(653\) −10.3572 −0.405307 −0.202654 0.979250i \(-0.564957\pi\)
−0.202654 + 0.979250i \(0.564957\pi\)
\(654\) 3.87663 0.151588
\(655\) 4.83817 0.189043
\(656\) −8.53772 −0.333342
\(657\) −78.3075 −3.05507
\(658\) −12.3941 −0.483171
\(659\) −36.6439 −1.42744 −0.713722 0.700429i \(-0.752992\pi\)
−0.713722 + 0.700429i \(0.752992\pi\)
\(660\) 0 0
\(661\) −23.5064 −0.914292 −0.457146 0.889392i \(-0.651128\pi\)
−0.457146 + 0.889392i \(0.651128\pi\)
\(662\) −22.6026 −0.878477
\(663\) −3.47993 −0.135149
\(664\) −0.382041 −0.0148261
\(665\) 1.04489 0.0405192
\(666\) 47.9169 1.85674
\(667\) −19.9615 −0.772911
\(668\) −12.6498 −0.489435
\(669\) 63.3232 2.44822
\(670\) −8.03278 −0.310334
\(671\) 0 0
\(672\) −3.29803 −0.127224
\(673\) 2.53720 0.0978020 0.0489010 0.998804i \(-0.484428\pi\)
0.0489010 + 0.998804i \(0.484428\pi\)
\(674\) 15.2690 0.588138
\(675\) −16.0846 −0.619097
\(676\) −11.8204 −0.454631
\(677\) −28.9289 −1.11183 −0.555913 0.831240i \(-0.687631\pi\)
−0.555913 + 0.831240i \(0.687631\pi\)
\(678\) −37.2827 −1.43183
\(679\) 14.8572 0.570169
\(680\) −0.971513 −0.0372558
\(681\) −7.40958 −0.283936
\(682\) 0 0
\(683\) −5.90465 −0.225935 −0.112968 0.993599i \(-0.536036\pi\)
−0.112968 + 0.993599i \(0.536036\pi\)
\(684\) −8.23066 −0.314707
\(685\) 16.9448 0.647427
\(686\) 1.00000 0.0381802
\(687\) 15.7917 0.602491
\(688\) −3.06122 −0.116708
\(689\) −0.351398 −0.0133872
\(690\) 7.99404 0.304328
\(691\) −47.9809 −1.82528 −0.912639 0.408766i \(-0.865959\pi\)
−0.912639 + 0.408766i \(0.865959\pi\)
\(692\) 1.01451 0.0385658
\(693\) 0 0
\(694\) 5.89915 0.223929
\(695\) 9.81201 0.372191
\(696\) 27.1604 1.02951
\(697\) −8.29451 −0.314177
\(698\) 16.9015 0.639732
\(699\) −42.0261 −1.58957
\(700\) 1.00000 0.0377964
\(701\) −31.0997 −1.17462 −0.587310 0.809362i \(-0.699813\pi\)
−0.587310 + 0.809362i \(0.699813\pi\)
\(702\) −17.4694 −0.659340
\(703\) −6.35621 −0.239729
\(704\) 0 0
\(705\) −40.8760 −1.53948
\(706\) −17.3914 −0.654532
\(707\) 5.85686 0.220270
\(708\) −39.2078 −1.47352
\(709\) 1.54403 0.0579872 0.0289936 0.999580i \(-0.490770\pi\)
0.0289936 + 0.999580i \(0.490770\pi\)
\(710\) −4.18166 −0.156935
\(711\) −90.1149 −3.37957
\(712\) −11.2206 −0.420511
\(713\) −10.1314 −0.379425
\(714\) −3.20408 −0.119910
\(715\) 0 0
\(716\) −17.1959 −0.642641
\(717\) 7.45269 0.278326
\(718\) 22.4481 0.837754
\(719\) −27.9067 −1.04074 −0.520371 0.853940i \(-0.674207\pi\)
−0.520371 + 0.853940i \(0.674207\pi\)
\(720\) −7.87703 −0.293560
\(721\) −11.1086 −0.413705
\(722\) −17.9082 −0.666474
\(723\) −6.41901 −0.238725
\(724\) 20.3704 0.757059
\(725\) −8.23533 −0.305852
\(726\) 0 0
\(727\) 19.3666 0.718267 0.359133 0.933286i \(-0.383072\pi\)
0.359133 + 0.933286i \(0.383072\pi\)
\(728\) 1.08609 0.0402533
\(729\) 72.5719 2.68785
\(730\) 9.94125 0.367942
\(731\) −2.97402 −0.109998
\(732\) −43.3713 −1.60305
\(733\) −5.23132 −0.193223 −0.0966116 0.995322i \(-0.530800\pi\)
−0.0966116 + 0.995322i \(0.530800\pi\)
\(734\) −4.81275 −0.177642
\(735\) 3.29803 0.121650
\(736\) 2.42388 0.0893455
\(737\) 0 0
\(738\) −67.2519 −2.47558
\(739\) 10.9383 0.402373 0.201187 0.979553i \(-0.435520\pi\)
0.201187 + 0.979553i \(0.435520\pi\)
\(740\) −6.08312 −0.223620
\(741\) 3.74278 0.137494
\(742\) −0.323543 −0.0118777
\(743\) 32.0747 1.17671 0.588353 0.808604i \(-0.299776\pi\)
0.588353 + 0.808604i \(0.299776\pi\)
\(744\) 13.7852 0.505391
\(745\) 17.2573 0.632257
\(746\) −6.77161 −0.247926
\(747\) −3.00935 −0.110106
\(748\) 0 0
\(749\) 7.35576 0.268774
\(750\) 3.29803 0.120427
\(751\) 2.29206 0.0836383 0.0418192 0.999125i \(-0.486685\pi\)
0.0418192 + 0.999125i \(0.486685\pi\)
\(752\) −12.3941 −0.451965
\(753\) 52.3683 1.90841
\(754\) −8.94433 −0.325733
\(755\) −15.6657 −0.570132
\(756\) −16.0846 −0.584992
\(757\) 49.1920 1.78791 0.893956 0.448155i \(-0.147919\pi\)
0.893956 + 0.448155i \(0.147919\pi\)
\(758\) −35.5405 −1.29089
\(759\) 0 0
\(760\) 1.04489 0.0379023
\(761\) −32.2555 −1.16926 −0.584630 0.811300i \(-0.698760\pi\)
−0.584630 + 0.811300i \(0.698760\pi\)
\(762\) −22.9053 −0.829773
\(763\) −1.17544 −0.0425537
\(764\) −6.42615 −0.232490
\(765\) −7.65264 −0.276682
\(766\) 21.0463 0.760435
\(767\) 12.9117 0.466215
\(768\) −3.29803 −0.119008
\(769\) 36.9722 1.33325 0.666625 0.745393i \(-0.267738\pi\)
0.666625 + 0.745393i \(0.267738\pi\)
\(770\) 0 0
\(771\) 23.3351 0.840393
\(772\) 14.0142 0.504382
\(773\) −44.1296 −1.58723 −0.793616 0.608419i \(-0.791804\pi\)
−0.793616 + 0.608419i \(0.791804\pi\)
\(774\) −24.1134 −0.866737
\(775\) −4.17984 −0.150144
\(776\) 14.8572 0.533344
\(777\) −20.0623 −0.719732
\(778\) 35.1161 1.25897
\(779\) 8.92101 0.319628
\(780\) 3.58197 0.128255
\(781\) 0 0
\(782\) 2.35483 0.0842087
\(783\) 132.462 4.73381
\(784\) 1.00000 0.0357143
\(785\) −9.25792 −0.330429
\(786\) 15.9564 0.569148
\(787\) −27.4346 −0.977938 −0.488969 0.872301i \(-0.662627\pi\)
−0.488969 + 0.872301i \(0.662627\pi\)
\(788\) 21.3383 0.760147
\(789\) 86.2329 3.06997
\(790\) 11.4402 0.407025
\(791\) 11.3045 0.401943
\(792\) 0 0
\(793\) 14.2828 0.507198
\(794\) 10.4987 0.372584
\(795\) −1.06706 −0.0378446
\(796\) −26.4689 −0.938166
\(797\) −3.49222 −0.123701 −0.0618505 0.998085i \(-0.519700\pi\)
−0.0618505 + 0.998085i \(0.519700\pi\)
\(798\) 3.44609 0.121990
\(799\) −12.0410 −0.425980
\(800\) 1.00000 0.0353553
\(801\) −88.3852 −3.12294
\(802\) −21.9027 −0.773411
\(803\) 0 0
\(804\) −26.4924 −0.934315
\(805\) −2.42388 −0.0854306
\(806\) −4.53969 −0.159904
\(807\) −57.8562 −2.03663
\(808\) 5.85686 0.206044
\(809\) −22.9517 −0.806939 −0.403469 0.914993i \(-0.632196\pi\)
−0.403469 + 0.914993i \(0.632196\pi\)
\(810\) −29.4165 −1.03359
\(811\) −25.3841 −0.891356 −0.445678 0.895193i \(-0.647037\pi\)
−0.445678 + 0.895193i \(0.647037\pi\)
\(812\) −8.23533 −0.289003
\(813\) 90.1176 3.16056
\(814\) 0 0
\(815\) 7.31437 0.256211
\(816\) −3.20408 −0.112165
\(817\) 3.19865 0.111907
\(818\) 9.24769 0.323338
\(819\) 8.55519 0.298943
\(820\) 8.53772 0.298150
\(821\) −0.156645 −0.00546695 −0.00273348 0.999996i \(-0.500870\pi\)
−0.00273348 + 0.999996i \(0.500870\pi\)
\(822\) 55.8845 1.94920
\(823\) 20.2569 0.706112 0.353056 0.935602i \(-0.385143\pi\)
0.353056 + 0.935602i \(0.385143\pi\)
\(824\) −11.1086 −0.386986
\(825\) 0 0
\(826\) 11.8882 0.413644
\(827\) −5.20676 −0.181057 −0.0905284 0.995894i \(-0.528856\pi\)
−0.0905284 + 0.995894i \(0.528856\pi\)
\(828\) 19.0930 0.663527
\(829\) 38.9470 1.35268 0.676342 0.736587i \(-0.263564\pi\)
0.676342 + 0.736587i \(0.263564\pi\)
\(830\) 0.382041 0.0132608
\(831\) 14.1703 0.491563
\(832\) 1.08609 0.0376535
\(833\) 0.971513 0.0336609
\(834\) 32.3603 1.12055
\(835\) 12.6498 0.437764
\(836\) 0 0
\(837\) 67.2310 2.32384
\(838\) 2.02744 0.0700367
\(839\) 9.50578 0.328176 0.164088 0.986446i \(-0.447532\pi\)
0.164088 + 0.986446i \(0.447532\pi\)
\(840\) 3.29803 0.113793
\(841\) 38.8206 1.33864
\(842\) 9.29834 0.320442
\(843\) 85.3144 2.93839
\(844\) 12.6514 0.435477
\(845\) 11.8204 0.406634
\(846\) −97.6284 −3.35653
\(847\) 0 0
\(848\) −0.323543 −0.0111105
\(849\) −89.5245 −3.07247
\(850\) 0.971513 0.0333226
\(851\) 14.7448 0.505444
\(852\) −13.7913 −0.472481
\(853\) −1.34852 −0.0461725 −0.0230863 0.999733i \(-0.507349\pi\)
−0.0230863 + 0.999733i \(0.507349\pi\)
\(854\) 13.1507 0.450006
\(855\) 8.23066 0.281482
\(856\) 7.35576 0.251415
\(857\) 24.4121 0.833900 0.416950 0.908929i \(-0.363099\pi\)
0.416950 + 0.908929i \(0.363099\pi\)
\(858\) 0 0
\(859\) 4.39769 0.150047 0.0750236 0.997182i \(-0.476097\pi\)
0.0750236 + 0.997182i \(0.476097\pi\)
\(860\) 3.06122 0.104387
\(861\) 28.1577 0.959612
\(862\) −18.1249 −0.617338
\(863\) −20.6638 −0.703405 −0.351703 0.936112i \(-0.614397\pi\)
−0.351703 + 0.936112i \(0.614397\pi\)
\(864\) −16.0846 −0.547210
\(865\) −1.01451 −0.0344943
\(866\) −33.4904 −1.13805
\(867\) 52.9538 1.79840
\(868\) −4.17984 −0.141873
\(869\) 0 0
\(870\) −27.1604 −0.920823
\(871\) 8.72435 0.295613
\(872\) −1.17544 −0.0398053
\(873\) 117.031 3.96090
\(874\) −2.53270 −0.0856698
\(875\) −1.00000 −0.0338062
\(876\) 32.7866 1.10776
\(877\) 17.7328 0.598794 0.299397 0.954129i \(-0.403215\pi\)
0.299397 + 0.954129i \(0.403215\pi\)
\(878\) −39.7487 −1.34145
\(879\) 94.8758 3.20008
\(880\) 0 0
\(881\) 52.1966 1.75855 0.879274 0.476317i \(-0.158029\pi\)
0.879274 + 0.476317i \(0.158029\pi\)
\(882\) 7.87703 0.265233
\(883\) 3.26731 0.109954 0.0549768 0.998488i \(-0.482492\pi\)
0.0549768 + 0.998488i \(0.482492\pi\)
\(884\) 1.05515 0.0354887
\(885\) 39.2078 1.31795
\(886\) 36.0159 1.20998
\(887\) −43.5164 −1.46114 −0.730569 0.682839i \(-0.760745\pi\)
−0.730569 + 0.682839i \(0.760745\pi\)
\(888\) −20.0623 −0.673248
\(889\) 6.94515 0.232933
\(890\) 11.2206 0.376116
\(891\) 0 0
\(892\) −19.2003 −0.642873
\(893\) 12.9505 0.433371
\(894\) 56.9150 1.90352
\(895\) 17.1959 0.574796
\(896\) 1.00000 0.0334077
\(897\) −8.68228 −0.289893
\(898\) 5.64047 0.188225
\(899\) 34.4223 1.14805
\(900\) 7.87703 0.262568
\(901\) −0.314327 −0.0104717
\(902\) 0 0
\(903\) 10.0960 0.335975
\(904\) 11.3045 0.375983
\(905\) −20.3704 −0.677134
\(906\) −51.6659 −1.71648
\(907\) 19.0450 0.632377 0.316189 0.948696i \(-0.397597\pi\)
0.316189 + 0.948696i \(0.397597\pi\)
\(908\) 2.24667 0.0745582
\(909\) 46.1346 1.53019
\(910\) −1.08609 −0.0360036
\(911\) 13.1363 0.435226 0.217613 0.976035i \(-0.430173\pi\)
0.217613 + 0.976035i \(0.430173\pi\)
\(912\) 3.44609 0.114112
\(913\) 0 0
\(914\) −21.2353 −0.702401
\(915\) 43.3713 1.43381
\(916\) −4.78822 −0.158207
\(917\) −4.83817 −0.159770
\(918\) −15.6264 −0.515748
\(919\) −20.2209 −0.667024 −0.333512 0.942746i \(-0.608234\pi\)
−0.333512 + 0.942746i \(0.608234\pi\)
\(920\) −2.42388 −0.0799130
\(921\) 74.8523 2.46647
\(922\) −36.7740 −1.21109
\(923\) 4.54167 0.149491
\(924\) 0 0
\(925\) 6.08312 0.200012
\(926\) 21.8388 0.717669
\(927\) −87.5026 −2.87396
\(928\) −8.23533 −0.270338
\(929\) −26.7617 −0.878022 −0.439011 0.898482i \(-0.644671\pi\)
−0.439011 + 0.898482i \(0.644671\pi\)
\(930\) −13.7852 −0.452036
\(931\) −1.04489 −0.0342450
\(932\) 12.7428 0.417404
\(933\) −2.50335 −0.0819560
\(934\) 34.4719 1.12795
\(935\) 0 0
\(936\) 8.55519 0.279635
\(937\) 38.9586 1.27272 0.636361 0.771391i \(-0.280439\pi\)
0.636361 + 0.771391i \(0.280439\pi\)
\(938\) 8.03278 0.262280
\(939\) −71.3256 −2.32762
\(940\) 12.3941 0.404250
\(941\) 17.2136 0.561149 0.280574 0.959832i \(-0.409475\pi\)
0.280574 + 0.959832i \(0.409475\pi\)
\(942\) −30.5329 −0.994817
\(943\) −20.6944 −0.673903
\(944\) 11.8882 0.386929
\(945\) 16.0846 0.523232
\(946\) 0 0
\(947\) −9.19053 −0.298652 −0.149326 0.988788i \(-0.547710\pi\)
−0.149326 + 0.988788i \(0.547710\pi\)
\(948\) 37.7302 1.22542
\(949\) −10.7971 −0.350489
\(950\) −1.04489 −0.0339008
\(951\) −0.282931 −0.00917465
\(952\) 0.971513 0.0314869
\(953\) −15.1329 −0.490204 −0.245102 0.969497i \(-0.578822\pi\)
−0.245102 + 0.969497i \(0.578822\pi\)
\(954\) −2.54856 −0.0825127
\(955\) 6.42615 0.207945
\(956\) −2.25974 −0.0730851
\(957\) 0 0
\(958\) −5.73461 −0.185277
\(959\) −16.9448 −0.547176
\(960\) 3.29803 0.106444
\(961\) −13.5290 −0.436419
\(962\) 6.60683 0.213013
\(963\) 57.9416 1.86714
\(964\) 1.94631 0.0626865
\(965\) −14.0142 −0.451133
\(966\) −7.99404 −0.257204
\(967\) −34.7020 −1.11594 −0.557970 0.829861i \(-0.688420\pi\)
−0.557970 + 0.829861i \(0.688420\pi\)
\(968\) 0 0
\(969\) 3.34793 0.107551
\(970\) −14.8572 −0.477037
\(971\) −8.12170 −0.260638 −0.130319 0.991472i \(-0.541600\pi\)
−0.130319 + 0.991472i \(0.541600\pi\)
\(972\) −48.7628 −1.56407
\(973\) −9.81201 −0.314559
\(974\) 6.98339 0.223762
\(975\) −3.58197 −0.114715
\(976\) 13.1507 0.420942
\(977\) 23.7897 0.761101 0.380550 0.924760i \(-0.375735\pi\)
0.380550 + 0.924760i \(0.375735\pi\)
\(978\) 24.1230 0.771369
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −9.25896 −0.295616
\(982\) −42.0589 −1.34215
\(983\) −23.2293 −0.740901 −0.370451 0.928852i \(-0.620797\pi\)
−0.370451 + 0.928852i \(0.620797\pi\)
\(984\) 28.1577 0.897635
\(985\) −21.3383 −0.679896
\(986\) −8.00073 −0.254795
\(987\) 40.8760 1.30110
\(988\) −1.13485 −0.0361044
\(989\) −7.42005 −0.235944
\(990\) 0 0
\(991\) 19.2128 0.610315 0.305157 0.952302i \(-0.401291\pi\)
0.305157 + 0.952302i \(0.401291\pi\)
\(992\) −4.17984 −0.132710
\(993\) 74.5443 2.36559
\(994\) 4.18166 0.132634
\(995\) 26.4689 0.839121
\(996\) 1.25998 0.0399241
\(997\) 12.0455 0.381486 0.190743 0.981640i \(-0.438910\pi\)
0.190743 + 0.981640i \(0.438910\pi\)
\(998\) −19.9215 −0.630603
\(999\) −97.8445 −3.09566
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.da.1.1 6
11.3 even 5 770.2.n.h.141.3 yes 12
11.4 even 5 770.2.n.h.71.3 12
11.10 odd 2 8470.2.a.cu.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.h.71.3 12 11.4 even 5
770.2.n.h.141.3 yes 12 11.3 even 5
8470.2.a.cu.1.1 6 11.10 odd 2
8470.2.a.da.1.1 6 1.1 even 1 trivial