Properties

Label 8470.2.a.db.1.3
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: 6.6.4642000.1
Defining polynomial: \(x^{6} - x^{5} - 8 x^{4} + 5 x^{3} + 14 x^{2} - 9 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.3
Root \(0.803425\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.803425 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.803425 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.35451 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.803425 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.803425 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.35451 q^{9} +1.00000 q^{10} -0.803425 q^{12} -2.64314 q^{13} -1.00000 q^{14} -0.803425 q^{15} +1.00000 q^{16} +1.60450 q^{17} -2.35451 q^{18} +1.90940 q^{19} +1.00000 q^{20} +0.803425 q^{21} -4.22916 q^{23} -0.803425 q^{24} +1.00000 q^{25} -2.64314 q^{26} +4.30195 q^{27} -1.00000 q^{28} +8.97936 q^{29} -0.803425 q^{30} +2.47600 q^{31} +1.00000 q^{32} +1.60450 q^{34} -1.00000 q^{35} -2.35451 q^{36} -5.60877 q^{37} +1.90940 q^{38} +2.12356 q^{39} +1.00000 q^{40} -9.69579 q^{41} +0.803425 q^{42} -4.00169 q^{43} -2.35451 q^{45} -4.22916 q^{46} -6.54502 q^{47} -0.803425 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.28909 q^{51} -2.64314 q^{52} +10.0478 q^{53} +4.30195 q^{54} -1.00000 q^{56} -1.53406 q^{57} +8.97936 q^{58} +4.16663 q^{59} -0.803425 q^{60} +13.2629 q^{61} +2.47600 q^{62} +2.35451 q^{63} +1.00000 q^{64} -2.64314 q^{65} -10.5029 q^{67} +1.60450 q^{68} +3.39781 q^{69} -1.00000 q^{70} +4.58489 q^{71} -2.35451 q^{72} +3.69293 q^{73} -5.60877 q^{74} -0.803425 q^{75} +1.90940 q^{76} +2.12356 q^{78} -13.9477 q^{79} +1.00000 q^{80} +3.60723 q^{81} -9.69579 q^{82} -13.5603 q^{83} +0.803425 q^{84} +1.60450 q^{85} -4.00169 q^{86} -7.21424 q^{87} -3.41009 q^{89} -2.35451 q^{90} +2.64314 q^{91} -4.22916 q^{92} -1.98928 q^{93} -6.54502 q^{94} +1.90940 q^{95} -0.803425 q^{96} -17.4713 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + O(q^{10}) \) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98} + O(q^{100}) \)

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
\(3\) −0.803425 −0.463858 −0.231929 0.972733i \(-0.574504\pi\)
−0.231929 + 0.972733i \(0.574504\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.803425 −0.327997
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.35451 −0.784836
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −0.803425 −0.231929
\(13\) −2.64314 −0.733075 −0.366537 0.930403i \(-0.619457\pi\)
−0.366537 + 0.930403i \(0.619457\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.803425 −0.207443
\(16\) 1.00000 0.250000
\(17\) 1.60450 0.389147 0.194574 0.980888i \(-0.437668\pi\)
0.194574 + 0.980888i \(0.437668\pi\)
\(18\) −2.35451 −0.554963
\(19\) 1.90940 0.438046 0.219023 0.975720i \(-0.429713\pi\)
0.219023 + 0.975720i \(0.429713\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.803425 0.175322
\(22\) 0 0
\(23\) −4.22916 −0.881840 −0.440920 0.897546i \(-0.645348\pi\)
−0.440920 + 0.897546i \(0.645348\pi\)
\(24\) −0.803425 −0.163998
\(25\) 1.00000 0.200000
\(26\) −2.64314 −0.518362
\(27\) 4.30195 0.827910
\(28\) −1.00000 −0.188982
\(29\) 8.97936 1.66742 0.833712 0.552199i \(-0.186211\pi\)
0.833712 + 0.552199i \(0.186211\pi\)
\(30\) −0.803425 −0.146685
\(31\) 2.47600 0.444702 0.222351 0.974967i \(-0.428627\pi\)
0.222351 + 0.974967i \(0.428627\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.60450 0.275169
\(35\) −1.00000 −0.169031
\(36\) −2.35451 −0.392418
\(37\) −5.60877 −0.922076 −0.461038 0.887380i \(-0.652523\pi\)
−0.461038 + 0.887380i \(0.652523\pi\)
\(38\) 1.90940 0.309746
\(39\) 2.12356 0.340042
\(40\) 1.00000 0.158114
\(41\) −9.69579 −1.51423 −0.757114 0.653283i \(-0.773391\pi\)
−0.757114 + 0.653283i \(0.773391\pi\)
\(42\) 0.803425 0.123971
\(43\) −4.00169 −0.610252 −0.305126 0.952312i \(-0.598699\pi\)
−0.305126 + 0.952312i \(0.598699\pi\)
\(44\) 0 0
\(45\) −2.35451 −0.350989
\(46\) −4.22916 −0.623555
\(47\) −6.54502 −0.954690 −0.477345 0.878716i \(-0.658401\pi\)
−0.477345 + 0.878716i \(0.658401\pi\)
\(48\) −0.803425 −0.115964
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −1.28909 −0.180509
\(52\) −2.64314 −0.366537
\(53\) 10.0478 1.38017 0.690087 0.723727i \(-0.257572\pi\)
0.690087 + 0.723727i \(0.257572\pi\)
\(54\) 4.30195 0.585421
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −1.53406 −0.203191
\(58\) 8.97936 1.17905
\(59\) 4.16663 0.542449 0.271225 0.962516i \(-0.412571\pi\)
0.271225 + 0.962516i \(0.412571\pi\)
\(60\) −0.803425 −0.103722
\(61\) 13.2629 1.69814 0.849068 0.528284i \(-0.177164\pi\)
0.849068 + 0.528284i \(0.177164\pi\)
\(62\) 2.47600 0.314452
\(63\) 2.35451 0.296640
\(64\) 1.00000 0.125000
\(65\) −2.64314 −0.327841
\(66\) 0 0
\(67\) −10.5029 −1.28314 −0.641570 0.767065i \(-0.721716\pi\)
−0.641570 + 0.767065i \(0.721716\pi\)
\(68\) 1.60450 0.194574
\(69\) 3.39781 0.409048
\(70\) −1.00000 −0.119523
\(71\) 4.58489 0.544126 0.272063 0.962279i \(-0.412294\pi\)
0.272063 + 0.962279i \(0.412294\pi\)
\(72\) −2.35451 −0.277481
\(73\) 3.69293 0.432224 0.216112 0.976369i \(-0.430662\pi\)
0.216112 + 0.976369i \(0.430662\pi\)
\(74\) −5.60877 −0.652006
\(75\) −0.803425 −0.0927715
\(76\) 1.90940 0.219023
\(77\) 0 0
\(78\) 2.12356 0.240446
\(79\) −13.9477 −1.56924 −0.784620 0.619977i \(-0.787142\pi\)
−0.784620 + 0.619977i \(0.787142\pi\)
\(80\) 1.00000 0.111803
\(81\) 3.60723 0.400804
\(82\) −9.69579 −1.07072
\(83\) −13.5603 −1.48843 −0.744215 0.667940i \(-0.767176\pi\)
−0.744215 + 0.667940i \(0.767176\pi\)
\(84\) 0.803425 0.0876609
\(85\) 1.60450 0.174032
\(86\) −4.00169 −0.431513
\(87\) −7.21424 −0.773448
\(88\) 0 0
\(89\) −3.41009 −0.361469 −0.180735 0.983532i \(-0.557847\pi\)
−0.180735 + 0.983532i \(0.557847\pi\)
\(90\) −2.35451 −0.248187
\(91\) 2.64314 0.277076
\(92\) −4.22916 −0.440920
\(93\) −1.98928 −0.206279
\(94\) −6.54502 −0.675068
\(95\) 1.90940 0.195900
\(96\) −0.803425 −0.0819992
\(97\) −17.4713 −1.77394 −0.886968 0.461830i \(-0.847193\pi\)
−0.886968 + 0.461830i \(0.847193\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.6494 −1.05965 −0.529827 0.848106i \(-0.677743\pi\)
−0.529827 + 0.848106i \(0.677743\pi\)
\(102\) −1.28909 −0.127639
\(103\) −6.86666 −0.676592 −0.338296 0.941040i \(-0.609851\pi\)
−0.338296 + 0.941040i \(0.609851\pi\)
\(104\) −2.64314 −0.259181
\(105\) 0.803425 0.0784063
\(106\) 10.0478 0.975930
\(107\) −2.88270 −0.278681 −0.139341 0.990245i \(-0.544498\pi\)
−0.139341 + 0.990245i \(0.544498\pi\)
\(108\) 4.30195 0.413955
\(109\) −18.4938 −1.77139 −0.885693 0.464271i \(-0.846316\pi\)
−0.885693 + 0.464271i \(0.846316\pi\)
\(110\) 0 0
\(111\) 4.50623 0.427712
\(112\) −1.00000 −0.0944911
\(113\) 11.9566 1.12478 0.562391 0.826871i \(-0.309882\pi\)
0.562391 + 0.826871i \(0.309882\pi\)
\(114\) −1.53406 −0.143678
\(115\) −4.22916 −0.394371
\(116\) 8.97936 0.833712
\(117\) 6.22329 0.575343
\(118\) 4.16663 0.383570
\(119\) −1.60450 −0.147084
\(120\) −0.803425 −0.0733423
\(121\) 0 0
\(122\) 13.2629 1.20076
\(123\) 7.78984 0.702386
\(124\) 2.47600 0.222351
\(125\) 1.00000 0.0894427
\(126\) 2.35451 0.209756
\(127\) −13.4085 −1.18982 −0.594908 0.803794i \(-0.702811\pi\)
−0.594908 + 0.803794i \(0.702811\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.21506 0.283070
\(130\) −2.64314 −0.231819
\(131\) −8.49928 −0.742585 −0.371293 0.928516i \(-0.621085\pi\)
−0.371293 + 0.928516i \(0.621085\pi\)
\(132\) 0 0
\(133\) −1.90940 −0.165566
\(134\) −10.5029 −0.907317
\(135\) 4.30195 0.370253
\(136\) 1.60450 0.137584
\(137\) 10.0747 0.860737 0.430369 0.902653i \(-0.358384\pi\)
0.430369 + 0.902653i \(0.358384\pi\)
\(138\) 3.39781 0.289241
\(139\) 13.8200 1.17220 0.586100 0.810238i \(-0.300662\pi\)
0.586100 + 0.810238i \(0.300662\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 5.25844 0.442840
\(142\) 4.58489 0.384755
\(143\) 0 0
\(144\) −2.35451 −0.196209
\(145\) 8.97936 0.745695
\(146\) 3.69293 0.305629
\(147\) −0.803425 −0.0662654
\(148\) −5.60877 −0.461038
\(149\) −18.3638 −1.50442 −0.752211 0.658922i \(-0.771013\pi\)
−0.752211 + 0.658922i \(0.771013\pi\)
\(150\) −0.803425 −0.0655994
\(151\) 20.5718 1.67411 0.837055 0.547118i \(-0.184275\pi\)
0.837055 + 0.547118i \(0.184275\pi\)
\(152\) 1.90940 0.154873
\(153\) −3.77780 −0.305417
\(154\) 0 0
\(155\) 2.47600 0.198877
\(156\) 2.12356 0.170021
\(157\) −4.64886 −0.371019 −0.185510 0.982642i \(-0.559394\pi\)
−0.185510 + 0.982642i \(0.559394\pi\)
\(158\) −13.9477 −1.10962
\(159\) −8.07267 −0.640204
\(160\) 1.00000 0.0790569
\(161\) 4.22916 0.333304
\(162\) 3.60723 0.283411
\(163\) 4.99400 0.391160 0.195580 0.980688i \(-0.437341\pi\)
0.195580 + 0.980688i \(0.437341\pi\)
\(164\) −9.69579 −0.757114
\(165\) 0 0
\(166\) −13.5603 −1.05248
\(167\) −1.86847 −0.144586 −0.0722932 0.997383i \(-0.523032\pi\)
−0.0722932 + 0.997383i \(0.523032\pi\)
\(168\) 0.803425 0.0619856
\(169\) −6.01382 −0.462602
\(170\) 1.60450 0.123059
\(171\) −4.49570 −0.343795
\(172\) −4.00169 −0.305126
\(173\) −11.7799 −0.895613 −0.447806 0.894131i \(-0.647795\pi\)
−0.447806 + 0.894131i \(0.647795\pi\)
\(174\) −7.21424 −0.546910
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −3.34758 −0.251619
\(178\) −3.41009 −0.255597
\(179\) −10.3705 −0.775128 −0.387564 0.921843i \(-0.626683\pi\)
−0.387564 + 0.921843i \(0.626683\pi\)
\(180\) −2.35451 −0.175495
\(181\) −22.6738 −1.68533 −0.842664 0.538440i \(-0.819014\pi\)
−0.842664 + 0.538440i \(0.819014\pi\)
\(182\) 2.64314 0.195922
\(183\) −10.6557 −0.787693
\(184\) −4.22916 −0.311778
\(185\) −5.60877 −0.412365
\(186\) −1.98928 −0.145861
\(187\) 0 0
\(188\) −6.54502 −0.477345
\(189\) −4.30195 −0.312921
\(190\) 1.90940 0.138522
\(191\) −8.48643 −0.614056 −0.307028 0.951700i \(-0.599335\pi\)
−0.307028 + 0.951700i \(0.599335\pi\)
\(192\) −0.803425 −0.0579822
\(193\) −25.1578 −1.81090 −0.905451 0.424452i \(-0.860467\pi\)
−0.905451 + 0.424452i \(0.860467\pi\)
\(194\) −17.4713 −1.25436
\(195\) 2.12356 0.152072
\(196\) 1.00000 0.0714286
\(197\) −13.2603 −0.944757 −0.472379 0.881396i \(-0.656604\pi\)
−0.472379 + 0.881396i \(0.656604\pi\)
\(198\) 0 0
\(199\) 3.59813 0.255065 0.127532 0.991834i \(-0.459294\pi\)
0.127532 + 0.991834i \(0.459294\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.43833 0.595194
\(202\) −10.6494 −0.749288
\(203\) −8.97936 −0.630227
\(204\) −1.28909 −0.0902545
\(205\) −9.69579 −0.677183
\(206\) −6.86666 −0.478423
\(207\) 9.95758 0.692100
\(208\) −2.64314 −0.183269
\(209\) 0 0
\(210\) 0.803425 0.0554416
\(211\) −10.0058 −0.688828 −0.344414 0.938818i \(-0.611922\pi\)
−0.344414 + 0.938818i \(0.611922\pi\)
\(212\) 10.0478 0.690087
\(213\) −3.68361 −0.252397
\(214\) −2.88270 −0.197057
\(215\) −4.00169 −0.272913
\(216\) 4.30195 0.292710
\(217\) −2.47600 −0.168082
\(218\) −18.4938 −1.25256
\(219\) −2.96699 −0.200491
\(220\) 0 0
\(221\) −4.24090 −0.285274
\(222\) 4.50623 0.302438
\(223\) 25.0397 1.67678 0.838391 0.545069i \(-0.183497\pi\)
0.838391 + 0.545069i \(0.183497\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.35451 −0.156967
\(226\) 11.9566 0.795341
\(227\) −0.182924 −0.0121411 −0.00607054 0.999982i \(-0.501932\pi\)
−0.00607054 + 0.999982i \(0.501932\pi\)
\(228\) −1.53406 −0.101596
\(229\) −0.134371 −0.00887950 −0.00443975 0.999990i \(-0.501413\pi\)
−0.00443975 + 0.999990i \(0.501413\pi\)
\(230\) −4.22916 −0.278862
\(231\) 0 0
\(232\) 8.97936 0.589524
\(233\) −2.93010 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(234\) 6.22329 0.406829
\(235\) −6.54502 −0.426950
\(236\) 4.16663 0.271225
\(237\) 11.2059 0.727904
\(238\) −1.60450 −0.104004
\(239\) 8.27410 0.535207 0.267604 0.963529i \(-0.413768\pi\)
0.267604 + 0.963529i \(0.413768\pi\)
\(240\) −0.803425 −0.0518609
\(241\) 16.5842 1.06828 0.534142 0.845395i \(-0.320635\pi\)
0.534142 + 0.845395i \(0.320635\pi\)
\(242\) 0 0
\(243\) −15.8040 −1.01383
\(244\) 13.2629 0.849068
\(245\) 1.00000 0.0638877
\(246\) 7.78984 0.496662
\(247\) −5.04681 −0.321121
\(248\) 2.47600 0.157226
\(249\) 10.8946 0.690420
\(250\) 1.00000 0.0632456
\(251\) −13.5486 −0.855180 −0.427590 0.903973i \(-0.640637\pi\)
−0.427590 + 0.903973i \(0.640637\pi\)
\(252\) 2.35451 0.148320
\(253\) 0 0
\(254\) −13.4085 −0.841327
\(255\) −1.28909 −0.0807261
\(256\) 1.00000 0.0625000
\(257\) 18.4226 1.14917 0.574584 0.818446i \(-0.305164\pi\)
0.574584 + 0.818446i \(0.305164\pi\)
\(258\) 3.21506 0.200161
\(259\) 5.60877 0.348512
\(260\) −2.64314 −0.163920
\(261\) −21.1420 −1.30865
\(262\) −8.49928 −0.525087
\(263\) −19.1658 −1.18181 −0.590907 0.806740i \(-0.701230\pi\)
−0.590907 + 0.806740i \(0.701230\pi\)
\(264\) 0 0
\(265\) 10.0478 0.617232
\(266\) −1.90940 −0.117073
\(267\) 2.73975 0.167670
\(268\) −10.5029 −0.641570
\(269\) 9.08666 0.554023 0.277012 0.960867i \(-0.410656\pi\)
0.277012 + 0.960867i \(0.410656\pi\)
\(270\) 4.30195 0.261808
\(271\) 19.1739 1.16473 0.582366 0.812927i \(-0.302127\pi\)
0.582366 + 0.812927i \(0.302127\pi\)
\(272\) 1.60450 0.0972869
\(273\) −2.12356 −0.128524
\(274\) 10.0747 0.608633
\(275\) 0 0
\(276\) 3.39781 0.204524
\(277\) −20.6691 −1.24189 −0.620944 0.783855i \(-0.713251\pi\)
−0.620944 + 0.783855i \(0.713251\pi\)
\(278\) 13.8200 0.828871
\(279\) −5.82976 −0.349018
\(280\) −1.00000 −0.0597614
\(281\) −1.37153 −0.0818184 −0.0409092 0.999163i \(-0.513025\pi\)
−0.0409092 + 0.999163i \(0.513025\pi\)
\(282\) 5.25844 0.313135
\(283\) −6.43513 −0.382528 −0.191264 0.981539i \(-0.561259\pi\)
−0.191264 + 0.981539i \(0.561259\pi\)
\(284\) 4.58489 0.272063
\(285\) −1.53406 −0.0908699
\(286\) 0 0
\(287\) 9.69579 0.572324
\(288\) −2.35451 −0.138741
\(289\) −14.4256 −0.848564
\(290\) 8.97936 0.527286
\(291\) 14.0368 0.822854
\(292\) 3.69293 0.216112
\(293\) 4.59731 0.268578 0.134289 0.990942i \(-0.457125\pi\)
0.134289 + 0.990942i \(0.457125\pi\)
\(294\) −0.803425 −0.0468567
\(295\) 4.16663 0.242591
\(296\) −5.60877 −0.326003
\(297\) 0 0
\(298\) −18.3638 −1.06379
\(299\) 11.1782 0.646454
\(300\) −0.803425 −0.0463858
\(301\) 4.00169 0.230654
\(302\) 20.5718 1.18377
\(303\) 8.55598 0.491528
\(304\) 1.90940 0.109512
\(305\) 13.2629 0.759429
\(306\) −3.77780 −0.215962
\(307\) −24.8042 −1.41565 −0.707825 0.706388i \(-0.750323\pi\)
−0.707825 + 0.706388i \(0.750323\pi\)
\(308\) 0 0
\(309\) 5.51685 0.313843
\(310\) 2.47600 0.140627
\(311\) 25.8023 1.46311 0.731556 0.681782i \(-0.238795\pi\)
0.731556 + 0.681782i \(0.238795\pi\)
\(312\) 2.12356 0.120223
\(313\) 16.5081 0.933093 0.466547 0.884497i \(-0.345498\pi\)
0.466547 + 0.884497i \(0.345498\pi\)
\(314\) −4.64886 −0.262350
\(315\) 2.35451 0.132662
\(316\) −13.9477 −0.784620
\(317\) 2.33084 0.130913 0.0654566 0.997855i \(-0.479150\pi\)
0.0654566 + 0.997855i \(0.479150\pi\)
\(318\) −8.07267 −0.452693
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 2.31603 0.129268
\(322\) 4.22916 0.235682
\(323\) 3.06363 0.170465
\(324\) 3.60723 0.200402
\(325\) −2.64314 −0.146615
\(326\) 4.99400 0.276592
\(327\) 14.8584 0.821671
\(328\) −9.69579 −0.535361
\(329\) 6.54502 0.360839
\(330\) 0 0
\(331\) 6.54411 0.359697 0.179848 0.983694i \(-0.442439\pi\)
0.179848 + 0.983694i \(0.442439\pi\)
\(332\) −13.5603 −0.744215
\(333\) 13.2059 0.723678
\(334\) −1.86847 −0.102238
\(335\) −10.5029 −0.573837
\(336\) 0.803425 0.0438304
\(337\) 16.6761 0.908407 0.454204 0.890898i \(-0.349924\pi\)
0.454204 + 0.890898i \(0.349924\pi\)
\(338\) −6.01382 −0.327109
\(339\) −9.60623 −0.521739
\(340\) 1.60450 0.0870160
\(341\) 0 0
\(342\) −4.49570 −0.243099
\(343\) −1.00000 −0.0539949
\(344\) −4.00169 −0.215757
\(345\) 3.39781 0.182932
\(346\) −11.7799 −0.633294
\(347\) 29.2816 1.57192 0.785959 0.618279i \(-0.212170\pi\)
0.785959 + 0.618279i \(0.212170\pi\)
\(348\) −7.21424 −0.386724
\(349\) −16.7836 −0.898406 −0.449203 0.893430i \(-0.648292\pi\)
−0.449203 + 0.893430i \(0.648292\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −11.3706 −0.606920
\(352\) 0 0
\(353\) 22.9110 1.21943 0.609715 0.792621i \(-0.291284\pi\)
0.609715 + 0.792621i \(0.291284\pi\)
\(354\) −3.34758 −0.177922
\(355\) 4.58489 0.243340
\(356\) −3.41009 −0.180735
\(357\) 1.28909 0.0682260
\(358\) −10.3705 −0.548098
\(359\) 8.36518 0.441497 0.220749 0.975331i \(-0.429150\pi\)
0.220749 + 0.975331i \(0.429150\pi\)
\(360\) −2.35451 −0.124093
\(361\) −15.3542 −0.808115
\(362\) −22.6738 −1.19171
\(363\) 0 0
\(364\) 2.64314 0.138538
\(365\) 3.69293 0.193297
\(366\) −10.6557 −0.556983
\(367\) −1.80322 −0.0941273 −0.0470637 0.998892i \(-0.514986\pi\)
−0.0470637 + 0.998892i \(0.514986\pi\)
\(368\) −4.22916 −0.220460
\(369\) 22.8288 1.18842
\(370\) −5.60877 −0.291586
\(371\) −10.0478 −0.521657
\(372\) −1.98928 −0.103139
\(373\) 9.60451 0.497303 0.248651 0.968593i \(-0.420013\pi\)
0.248651 + 0.968593i \(0.420013\pi\)
\(374\) 0 0
\(375\) −0.803425 −0.0414887
\(376\) −6.54502 −0.337534
\(377\) −23.7337 −1.22235
\(378\) −4.30195 −0.221268
\(379\) −5.94046 −0.305141 −0.152570 0.988293i \(-0.548755\pi\)
−0.152570 + 0.988293i \(0.548755\pi\)
\(380\) 1.90940 0.0979502
\(381\) 10.7728 0.551905
\(382\) −8.48643 −0.434203
\(383\) −29.4258 −1.50359 −0.751794 0.659399i \(-0.770811\pi\)
−0.751794 + 0.659399i \(0.770811\pi\)
\(384\) −0.803425 −0.0409996
\(385\) 0 0
\(386\) −25.1578 −1.28050
\(387\) 9.42201 0.478948
\(388\) −17.4713 −0.886968
\(389\) −18.8617 −0.956324 −0.478162 0.878272i \(-0.658697\pi\)
−0.478162 + 0.878272i \(0.658697\pi\)
\(390\) 2.12356 0.107531
\(391\) −6.78566 −0.343166
\(392\) 1.00000 0.0505076
\(393\) 6.82853 0.344454
\(394\) −13.2603 −0.668044
\(395\) −13.9477 −0.701785
\(396\) 0 0
\(397\) 25.4542 1.27751 0.638756 0.769409i \(-0.279449\pi\)
0.638756 + 0.769409i \(0.279449\pi\)
\(398\) 3.59813 0.180358
\(399\) 1.53406 0.0767991
\(400\) 1.00000 0.0500000
\(401\) 14.9867 0.748399 0.374199 0.927348i \(-0.377918\pi\)
0.374199 + 0.927348i \(0.377918\pi\)
\(402\) 8.43833 0.420866
\(403\) −6.54440 −0.326000
\(404\) −10.6494 −0.529827
\(405\) 3.60723 0.179245
\(406\) −8.97936 −0.445638
\(407\) 0 0
\(408\) −1.28909 −0.0638196
\(409\) 4.85775 0.240200 0.120100 0.992762i \(-0.461678\pi\)
0.120100 + 0.992762i \(0.461678\pi\)
\(410\) −9.69579 −0.478841
\(411\) −8.09424 −0.399260
\(412\) −6.86666 −0.338296
\(413\) −4.16663 −0.205027
\(414\) 9.95758 0.489388
\(415\) −13.5603 −0.665647
\(416\) −2.64314 −0.129590
\(417\) −11.1034 −0.543734
\(418\) 0 0
\(419\) −1.77493 −0.0867109 −0.0433554 0.999060i \(-0.513805\pi\)
−0.0433554 + 0.999060i \(0.513805\pi\)
\(420\) 0.803425 0.0392031
\(421\) −16.9245 −0.824851 −0.412426 0.910991i \(-0.635318\pi\)
−0.412426 + 0.910991i \(0.635318\pi\)
\(422\) −10.0058 −0.487075
\(423\) 15.4103 0.749275
\(424\) 10.0478 0.487965
\(425\) 1.60450 0.0778295
\(426\) −3.68361 −0.178472
\(427\) −13.2629 −0.641835
\(428\) −2.88270 −0.139341
\(429\) 0 0
\(430\) −4.00169 −0.192979
\(431\) 4.15621 0.200198 0.100099 0.994977i \(-0.468084\pi\)
0.100099 + 0.994977i \(0.468084\pi\)
\(432\) 4.30195 0.206977
\(433\) 22.2410 1.06883 0.534417 0.845221i \(-0.320531\pi\)
0.534417 + 0.845221i \(0.320531\pi\)
\(434\) −2.47600 −0.118852
\(435\) −7.21424 −0.345896
\(436\) −18.4938 −0.885693
\(437\) −8.07515 −0.386287
\(438\) −2.96699 −0.141768
\(439\) −21.6172 −1.03173 −0.515866 0.856669i \(-0.672530\pi\)
−0.515866 + 0.856669i \(0.672530\pi\)
\(440\) 0 0
\(441\) −2.35451 −0.112119
\(442\) −4.24090 −0.201719
\(443\) −12.9462 −0.615092 −0.307546 0.951533i \(-0.599508\pi\)
−0.307546 + 0.951533i \(0.599508\pi\)
\(444\) 4.50623 0.213856
\(445\) −3.41009 −0.161654
\(446\) 25.0397 1.18566
\(447\) 14.7540 0.697838
\(448\) −1.00000 −0.0472456
\(449\) 13.8561 0.653910 0.326955 0.945040i \(-0.393977\pi\)
0.326955 + 0.945040i \(0.393977\pi\)
\(450\) −2.35451 −0.110993
\(451\) 0 0
\(452\) 11.9566 0.562391
\(453\) −16.5279 −0.776549
\(454\) −0.182924 −0.00858503
\(455\) 2.64314 0.123912
\(456\) −1.53406 −0.0718389
\(457\) −9.98796 −0.467217 −0.233608 0.972331i \(-0.575053\pi\)
−0.233608 + 0.972331i \(0.575053\pi\)
\(458\) −0.134371 −0.00627876
\(459\) 6.90246 0.322179
\(460\) −4.22916 −0.197185
\(461\) 27.8198 1.29570 0.647850 0.761768i \(-0.275668\pi\)
0.647850 + 0.761768i \(0.275668\pi\)
\(462\) 0 0
\(463\) −10.2727 −0.477413 −0.238707 0.971092i \(-0.576723\pi\)
−0.238707 + 0.971092i \(0.576723\pi\)
\(464\) 8.97936 0.416856
\(465\) −1.98928 −0.0922506
\(466\) −2.93010 −0.135734
\(467\) 9.03790 0.418224 0.209112 0.977892i \(-0.432943\pi\)
0.209112 + 0.977892i \(0.432943\pi\)
\(468\) 6.22329 0.287672
\(469\) 10.5029 0.484981
\(470\) −6.54502 −0.301899
\(471\) 3.73501 0.172100
\(472\) 4.16663 0.191785
\(473\) 0 0
\(474\) 11.2059 0.514706
\(475\) 1.90940 0.0876093
\(476\) −1.60450 −0.0735420
\(477\) −23.6577 −1.08321
\(478\) 8.27410 0.378449
\(479\) −39.4300 −1.80160 −0.900801 0.434233i \(-0.857020\pi\)
−0.900801 + 0.434233i \(0.857020\pi\)
\(480\) −0.803425 −0.0366712
\(481\) 14.8247 0.675950
\(482\) 16.5842 0.755391
\(483\) −3.39781 −0.154606
\(484\) 0 0
\(485\) −17.4713 −0.793329
\(486\) −15.8040 −0.716883
\(487\) −18.9186 −0.857282 −0.428641 0.903475i \(-0.641007\pi\)
−0.428641 + 0.903475i \(0.641007\pi\)
\(488\) 13.2629 0.600381
\(489\) −4.01230 −0.181443
\(490\) 1.00000 0.0451754
\(491\) −1.51922 −0.0685615 −0.0342808 0.999412i \(-0.510914\pi\)
−0.0342808 + 0.999412i \(0.510914\pi\)
\(492\) 7.78984 0.351193
\(493\) 14.4073 0.648874
\(494\) −5.04681 −0.227067
\(495\) 0 0
\(496\) 2.47600 0.111176
\(497\) −4.58489 −0.205660
\(498\) 10.8946 0.488201
\(499\) 11.8172 0.529012 0.264506 0.964384i \(-0.414791\pi\)
0.264506 + 0.964384i \(0.414791\pi\)
\(500\) 1.00000 0.0447214
\(501\) 1.50117 0.0670675
\(502\) −13.5486 −0.604703
\(503\) −16.0491 −0.715592 −0.357796 0.933800i \(-0.616472\pi\)
−0.357796 + 0.933800i \(0.616472\pi\)
\(504\) 2.35451 0.104878
\(505\) −10.6494 −0.473891
\(506\) 0 0
\(507\) 4.83166 0.214581
\(508\) −13.4085 −0.594908
\(509\) 40.2765 1.78522 0.892612 0.450826i \(-0.148870\pi\)
0.892612 + 0.450826i \(0.148870\pi\)
\(510\) −1.28909 −0.0570820
\(511\) −3.69293 −0.163365
\(512\) 1.00000 0.0441942
\(513\) 8.21414 0.362663
\(514\) 18.4226 0.812584
\(515\) −6.86666 −0.302581
\(516\) 3.21506 0.141535
\(517\) 0 0
\(518\) 5.60877 0.246435
\(519\) 9.46431 0.415437
\(520\) −2.64314 −0.115909
\(521\) −27.4901 −1.20436 −0.602182 0.798359i \(-0.705702\pi\)
−0.602182 + 0.798359i \(0.705702\pi\)
\(522\) −21.1420 −0.925359
\(523\) 20.2364 0.884875 0.442438 0.896799i \(-0.354114\pi\)
0.442438 + 0.896799i \(0.354114\pi\)
\(524\) −8.49928 −0.371293
\(525\) 0.803425 0.0350643
\(526\) −19.1658 −0.835668
\(527\) 3.97273 0.173055
\(528\) 0 0
\(529\) −5.11424 −0.222358
\(530\) 10.0478 0.436449
\(531\) −9.81037 −0.425734
\(532\) −1.90940 −0.0827830
\(533\) 25.6273 1.11004
\(534\) 2.73975 0.118561
\(535\) −2.88270 −0.124630
\(536\) −10.5029 −0.453658
\(537\) 8.33193 0.359549
\(538\) 9.08666 0.391754
\(539\) 0 0
\(540\) 4.30195 0.185126
\(541\) 26.5254 1.14042 0.570209 0.821500i \(-0.306862\pi\)
0.570209 + 0.821500i \(0.306862\pi\)
\(542\) 19.1739 0.823590
\(543\) 18.2167 0.781752
\(544\) 1.60450 0.0687922
\(545\) −18.4938 −0.792188
\(546\) −2.12356 −0.0908801
\(547\) 3.17578 0.135787 0.0678933 0.997693i \(-0.478372\pi\)
0.0678933 + 0.997693i \(0.478372\pi\)
\(548\) 10.0747 0.430369
\(549\) −31.2275 −1.33276
\(550\) 0 0
\(551\) 17.1452 0.730409
\(552\) 3.39781 0.144620
\(553\) 13.9477 0.593117
\(554\) −20.6691 −0.878148
\(555\) 4.50623 0.191279
\(556\) 13.8200 0.586100
\(557\) −29.2227 −1.23821 −0.619103 0.785310i \(-0.712503\pi\)
−0.619103 + 0.785310i \(0.712503\pi\)
\(558\) −5.82976 −0.246793
\(559\) 10.5770 0.447360
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −1.37153 −0.0578543
\(563\) −6.15480 −0.259394 −0.129697 0.991554i \(-0.541400\pi\)
−0.129697 + 0.991554i \(0.541400\pi\)
\(564\) 5.25844 0.221420
\(565\) 11.9566 0.503018
\(566\) −6.43513 −0.270488
\(567\) −3.60723 −0.151490
\(568\) 4.58489 0.192378
\(569\) −3.59047 −0.150520 −0.0752601 0.997164i \(-0.523979\pi\)
−0.0752601 + 0.997164i \(0.523979\pi\)
\(570\) −1.53406 −0.0642547
\(571\) −28.1128 −1.17649 −0.588243 0.808684i \(-0.700180\pi\)
−0.588243 + 0.808684i \(0.700180\pi\)
\(572\) 0 0
\(573\) 6.81821 0.284835
\(574\) 9.69579 0.404695
\(575\) −4.22916 −0.176368
\(576\) −2.35451 −0.0981045
\(577\) 28.5378 1.18804 0.594022 0.804448i \(-0.297539\pi\)
0.594022 + 0.804448i \(0.297539\pi\)
\(578\) −14.4256 −0.600026
\(579\) 20.2124 0.840000
\(580\) 8.97936 0.372847
\(581\) 13.5603 0.562574
\(582\) 14.0368 0.581846
\(583\) 0 0
\(584\) 3.69293 0.152814
\(585\) 6.22329 0.257301
\(586\) 4.59731 0.189913
\(587\) −19.5107 −0.805291 −0.402646 0.915356i \(-0.631909\pi\)
−0.402646 + 0.915356i \(0.631909\pi\)
\(588\) −0.803425 −0.0331327
\(589\) 4.72767 0.194800
\(590\) 4.16663 0.171538
\(591\) 10.6537 0.438233
\(592\) −5.60877 −0.230519
\(593\) 4.84298 0.198877 0.0994386 0.995044i \(-0.468295\pi\)
0.0994386 + 0.995044i \(0.468295\pi\)
\(594\) 0 0
\(595\) −1.60450 −0.0657779
\(596\) −18.3638 −0.752211
\(597\) −2.89083 −0.118314
\(598\) 11.1782 0.457112
\(599\) −3.67149 −0.150013 −0.0750064 0.997183i \(-0.523898\pi\)
−0.0750064 + 0.997183i \(0.523898\pi\)
\(600\) −0.803425 −0.0327997
\(601\) −1.73373 −0.0707203 −0.0353601 0.999375i \(-0.511258\pi\)
−0.0353601 + 0.999375i \(0.511258\pi\)
\(602\) 4.00169 0.163097
\(603\) 24.7293 1.00705
\(604\) 20.5718 0.837055
\(605\) 0 0
\(606\) 8.55598 0.347563
\(607\) −28.4589 −1.15511 −0.577555 0.816351i \(-0.695993\pi\)
−0.577555 + 0.816351i \(0.695993\pi\)
\(608\) 1.90940 0.0774364
\(609\) 7.21424 0.292336
\(610\) 13.2629 0.536997
\(611\) 17.2994 0.699859
\(612\) −3.77780 −0.152708
\(613\) −8.07904 −0.326309 −0.163155 0.986601i \(-0.552167\pi\)
−0.163155 + 0.986601i \(0.552167\pi\)
\(614\) −24.8042 −1.00102
\(615\) 7.78984 0.314117
\(616\) 0 0
\(617\) 2.53152 0.101915 0.0509577 0.998701i \(-0.483773\pi\)
0.0509577 + 0.998701i \(0.483773\pi\)
\(618\) 5.51685 0.221920
\(619\) −27.6305 −1.11056 −0.555281 0.831663i \(-0.687389\pi\)
−0.555281 + 0.831663i \(0.687389\pi\)
\(620\) 2.47600 0.0994385
\(621\) −18.1936 −0.730084
\(622\) 25.8023 1.03458
\(623\) 3.41009 0.136622
\(624\) 2.12356 0.0850106
\(625\) 1.00000 0.0400000
\(626\) 16.5081 0.659797
\(627\) 0 0
\(628\) −4.64886 −0.185510
\(629\) −8.99925 −0.358823
\(630\) 2.35451 0.0938058
\(631\) −32.2983 −1.28577 −0.642887 0.765961i \(-0.722263\pi\)
−0.642887 + 0.765961i \(0.722263\pi\)
\(632\) −13.9477 −0.554810
\(633\) 8.03891 0.319518
\(634\) 2.33084 0.0925696
\(635\) −13.4085 −0.532102
\(636\) −8.07267 −0.320102
\(637\) −2.64314 −0.104725
\(638\) 0 0
\(639\) −10.7952 −0.427050
\(640\) 1.00000 0.0395285
\(641\) 31.3973 1.24012 0.620060 0.784554i \(-0.287108\pi\)
0.620060 + 0.784554i \(0.287108\pi\)
\(642\) 2.31603 0.0914066
\(643\) 4.12543 0.162691 0.0813456 0.996686i \(-0.474078\pi\)
0.0813456 + 0.996686i \(0.474078\pi\)
\(644\) 4.22916 0.166652
\(645\) 3.21506 0.126593
\(646\) 3.06363 0.120537
\(647\) 20.9288 0.822794 0.411397 0.911456i \(-0.365041\pi\)
0.411397 + 0.911456i \(0.365041\pi\)
\(648\) 3.60723 0.141705
\(649\) 0 0
\(650\) −2.64314 −0.103672
\(651\) 1.98928 0.0779660
\(652\) 4.99400 0.195580
\(653\) −2.00850 −0.0785988 −0.0392994 0.999227i \(-0.512513\pi\)
−0.0392994 + 0.999227i \(0.512513\pi\)
\(654\) 14.8584 0.581009
\(655\) −8.49928 −0.332094
\(656\) −9.69579 −0.378557
\(657\) −8.69502 −0.339225
\(658\) 6.54502 0.255152
\(659\) 11.7132 0.456282 0.228141 0.973628i \(-0.426735\pi\)
0.228141 + 0.973628i \(0.426735\pi\)
\(660\) 0 0
\(661\) 35.8654 1.39500 0.697502 0.716583i \(-0.254295\pi\)
0.697502 + 0.716583i \(0.254295\pi\)
\(662\) 6.54411 0.254344
\(663\) 3.40725 0.132327
\(664\) −13.5603 −0.526240
\(665\) −1.90940 −0.0740434
\(666\) 13.2059 0.511718
\(667\) −37.9751 −1.47040
\(668\) −1.86847 −0.0722932
\(669\) −20.1175 −0.777788
\(670\) −10.5029 −0.405764
\(671\) 0 0
\(672\) 0.803425 0.0309928
\(673\) −33.3617 −1.28600 −0.642999 0.765867i \(-0.722310\pi\)
−0.642999 + 0.765867i \(0.722310\pi\)
\(674\) 16.6761 0.642341
\(675\) 4.30195 0.165582
\(676\) −6.01382 −0.231301
\(677\) −13.2733 −0.510134 −0.255067 0.966923i \(-0.582097\pi\)
−0.255067 + 0.966923i \(0.582097\pi\)
\(678\) −9.60623 −0.368925
\(679\) 17.4713 0.670485
\(680\) 1.60450 0.0615296
\(681\) 0.146965 0.00563173
\(682\) 0 0
\(683\) −15.3702 −0.588123 −0.294061 0.955787i \(-0.595007\pi\)
−0.294061 + 0.955787i \(0.595007\pi\)
\(684\) −4.49570 −0.171897
\(685\) 10.0747 0.384933
\(686\) −1.00000 −0.0381802
\(687\) 0.107957 0.00411883
\(688\) −4.00169 −0.152563
\(689\) −26.5578 −1.01177
\(690\) 3.39781 0.129352
\(691\) 20.7798 0.790501 0.395251 0.918573i \(-0.370658\pi\)
0.395251 + 0.918573i \(0.370658\pi\)
\(692\) −11.7799 −0.447806
\(693\) 0 0
\(694\) 29.2816 1.11151
\(695\) 13.8200 0.524224
\(696\) −7.21424 −0.273455
\(697\) −15.5569 −0.589258
\(698\) −16.7836 −0.635269
\(699\) 2.35411 0.0890407
\(700\) −1.00000 −0.0377964
\(701\) −4.18595 −0.158101 −0.0790506 0.996871i \(-0.525189\pi\)
−0.0790506 + 0.996871i \(0.525189\pi\)
\(702\) −11.3706 −0.429157
\(703\) −10.7094 −0.403912
\(704\) 0 0
\(705\) 5.25844 0.198044
\(706\) 22.9110 0.862267
\(707\) 10.6494 0.400511
\(708\) −3.34758 −0.125810
\(709\) 5.48528 0.206004 0.103002 0.994681i \(-0.467155\pi\)
0.103002 + 0.994681i \(0.467155\pi\)
\(710\) 4.58489 0.172068
\(711\) 32.8400 1.23160
\(712\) −3.41009 −0.127799
\(713\) −10.4714 −0.392156
\(714\) 1.28909 0.0482431
\(715\) 0 0
\(716\) −10.3705 −0.387564
\(717\) −6.64762 −0.248260
\(718\) 8.36518 0.312186
\(719\) −16.6013 −0.619122 −0.309561 0.950880i \(-0.600182\pi\)
−0.309561 + 0.950880i \(0.600182\pi\)
\(720\) −2.35451 −0.0877473
\(721\) 6.86666 0.255728
\(722\) −15.3542 −0.571424
\(723\) −13.3242 −0.495532
\(724\) −22.6738 −0.842664
\(725\) 8.97936 0.333485
\(726\) 0 0
\(727\) −22.9499 −0.851164 −0.425582 0.904920i \(-0.639931\pi\)
−0.425582 + 0.904920i \(0.639931\pi\)
\(728\) 2.64314 0.0979612
\(729\) 1.87562 0.0694673
\(730\) 3.69293 0.136681
\(731\) −6.42070 −0.237478
\(732\) −10.6557 −0.393847
\(733\) −15.9482 −0.589060 −0.294530 0.955642i \(-0.595163\pi\)
−0.294530 + 0.955642i \(0.595163\pi\)
\(734\) −1.80322 −0.0665581
\(735\) −0.803425 −0.0296348
\(736\) −4.22916 −0.155889
\(737\) 0 0
\(738\) 22.8288 0.840340
\(739\) −41.6561 −1.53234 −0.766171 0.642636i \(-0.777841\pi\)
−0.766171 + 0.642636i \(0.777841\pi\)
\(740\) −5.60877 −0.206182
\(741\) 4.05473 0.148954
\(742\) −10.0478 −0.368867
\(743\) −41.9839 −1.54024 −0.770120 0.637899i \(-0.779804\pi\)
−0.770120 + 0.637899i \(0.779804\pi\)
\(744\) −1.98928 −0.0729305
\(745\) −18.3638 −0.672798
\(746\) 9.60451 0.351646
\(747\) 31.9277 1.16817
\(748\) 0 0
\(749\) 2.88270 0.105332
\(750\) −0.803425 −0.0293369
\(751\) 30.9563 1.12961 0.564806 0.825223i \(-0.308951\pi\)
0.564806 + 0.825223i \(0.308951\pi\)
\(752\) −6.54502 −0.238672
\(753\) 10.8853 0.396682
\(754\) −23.7337 −0.864330
\(755\) 20.5718 0.748685
\(756\) −4.30195 −0.156460
\(757\) 19.9821 0.726261 0.363130 0.931738i \(-0.381708\pi\)
0.363130 + 0.931738i \(0.381708\pi\)
\(758\) −5.94046 −0.215767
\(759\) 0 0
\(760\) 1.90940 0.0692612
\(761\) 8.32474 0.301771 0.150886 0.988551i \(-0.451787\pi\)
0.150886 + 0.988551i \(0.451787\pi\)
\(762\) 10.7728 0.390256
\(763\) 18.4938 0.669521
\(764\) −8.48643 −0.307028
\(765\) −3.77780 −0.136587
\(766\) −29.4258 −1.06320
\(767\) −11.0130 −0.397656
\(768\) −0.803425 −0.0289911
\(769\) −38.3934 −1.38450 −0.692251 0.721657i \(-0.743381\pi\)
−0.692251 + 0.721657i \(0.743381\pi\)
\(770\) 0 0
\(771\) −14.8011 −0.533050
\(772\) −25.1578 −0.905451
\(773\) 28.1213 1.01145 0.505726 0.862694i \(-0.331225\pi\)
0.505726 + 0.862694i \(0.331225\pi\)
\(774\) 9.42201 0.338667
\(775\) 2.47600 0.0889405
\(776\) −17.4713 −0.627181
\(777\) −4.50623 −0.161660
\(778\) −18.8617 −0.676223
\(779\) −18.5131 −0.663302
\(780\) 2.12356 0.0760358
\(781\) 0 0
\(782\) −6.78566 −0.242655
\(783\) 38.6287 1.38048
\(784\) 1.00000 0.0357143
\(785\) −4.64886 −0.165925
\(786\) 6.82853 0.243566
\(787\) −30.1966 −1.07639 −0.538197 0.842819i \(-0.680894\pi\)
−0.538197 + 0.842819i \(0.680894\pi\)
\(788\) −13.2603 −0.472379
\(789\) 15.3983 0.548193
\(790\) −13.9477 −0.496237
\(791\) −11.9566 −0.425128
\(792\) 0 0
\(793\) −35.0556 −1.24486
\(794\) 25.4542 0.903337
\(795\) −8.07267 −0.286308
\(796\) 3.59813 0.127532
\(797\) 46.4224 1.64436 0.822182 0.569224i \(-0.192756\pi\)
0.822182 + 0.569224i \(0.192756\pi\)
\(798\) 1.53406 0.0543051
\(799\) −10.5015 −0.371515
\(800\) 1.00000 0.0353553
\(801\) 8.02909 0.283694
\(802\) 14.9867 0.529198
\(803\) 0 0
\(804\) 8.43833 0.297597
\(805\) 4.22916 0.149058
\(806\) −6.54440 −0.230517
\(807\) −7.30045 −0.256988
\(808\) −10.6494 −0.374644
\(809\) −12.0678 −0.424282 −0.212141 0.977239i \(-0.568044\pi\)
−0.212141 + 0.977239i \(0.568044\pi\)
\(810\) 3.60723 0.126745
\(811\) 14.9538 0.525098 0.262549 0.964919i \(-0.415437\pi\)
0.262549 + 0.964919i \(0.415437\pi\)
\(812\) −8.97936 −0.315114
\(813\) −15.4048 −0.540270
\(814\) 0 0
\(815\) 4.99400 0.174932
\(816\) −1.28909 −0.0451273
\(817\) −7.64083 −0.267319
\(818\) 4.85775 0.169847
\(819\) −6.22329 −0.217459
\(820\) −9.69579 −0.338592
\(821\) 23.8592 0.832693 0.416347 0.909206i \(-0.363310\pi\)
0.416347 + 0.909206i \(0.363310\pi\)
\(822\) −8.09424 −0.282319
\(823\) −42.3899 −1.47762 −0.738809 0.673914i \(-0.764612\pi\)
−0.738809 + 0.673914i \(0.764612\pi\)
\(824\) −6.86666 −0.239212
\(825\) 0 0
\(826\) −4.16663 −0.144976
\(827\) −52.9595 −1.84158 −0.920790 0.390058i \(-0.872455\pi\)
−0.920790 + 0.390058i \(0.872455\pi\)
\(828\) 9.95758 0.346050
\(829\) 44.8068 1.55620 0.778102 0.628137i \(-0.216182\pi\)
0.778102 + 0.628137i \(0.216182\pi\)
\(830\) −13.5603 −0.470683
\(831\) 16.6061 0.576060
\(832\) −2.64314 −0.0916343
\(833\) 1.60450 0.0555925
\(834\) −11.1034 −0.384478
\(835\) −1.86847 −0.0646610
\(836\) 0 0
\(837\) 10.6516 0.368174
\(838\) −1.77493 −0.0613138
\(839\) −40.6109 −1.40204 −0.701022 0.713140i \(-0.747272\pi\)
−0.701022 + 0.713140i \(0.747272\pi\)
\(840\) 0.803425 0.0277208
\(841\) 51.6288 1.78030
\(842\) −16.9245 −0.583258
\(843\) 1.10192 0.0379521
\(844\) −10.0058 −0.344414
\(845\) −6.01382 −0.206882
\(846\) 15.4103 0.529817
\(847\) 0 0
\(848\) 10.0478 0.345043
\(849\) 5.17014 0.177439
\(850\) 1.60450 0.0550338
\(851\) 23.7204 0.813123
\(852\) −3.68361 −0.126198
\(853\) −0.116843 −0.00400061 −0.00200031 0.999998i \(-0.500637\pi\)
−0.00200031 + 0.999998i \(0.500637\pi\)
\(854\) −13.2629 −0.453846
\(855\) −4.49570 −0.153750
\(856\) −2.88270 −0.0985287
\(857\) 22.2623 0.760465 0.380233 0.924891i \(-0.375844\pi\)
0.380233 + 0.924891i \(0.375844\pi\)
\(858\) 0 0
\(859\) 37.0142 1.26291 0.631454 0.775413i \(-0.282458\pi\)
0.631454 + 0.775413i \(0.282458\pi\)
\(860\) −4.00169 −0.136457
\(861\) −7.78984 −0.265477
\(862\) 4.15621 0.141561
\(863\) 44.0257 1.49865 0.749326 0.662201i \(-0.230377\pi\)
0.749326 + 0.662201i \(0.230377\pi\)
\(864\) 4.30195 0.146355
\(865\) −11.7799 −0.400530
\(866\) 22.2410 0.755780
\(867\) 11.5899 0.393613
\(868\) −2.47600 −0.0840408
\(869\) 0 0
\(870\) −7.21424 −0.244586
\(871\) 27.7607 0.940637
\(872\) −18.4938 −0.626280
\(873\) 41.1362 1.39225
\(874\) −8.07515 −0.273146
\(875\) −1.00000 −0.0338062
\(876\) −2.96699 −0.100245
\(877\) −29.4099 −0.993100 −0.496550 0.868008i \(-0.665400\pi\)
−0.496550 + 0.868008i \(0.665400\pi\)
\(878\) −21.6172 −0.729545
\(879\) −3.69359 −0.124582
\(880\) 0 0
\(881\) 34.0756 1.14804 0.574019 0.818842i \(-0.305384\pi\)
0.574019 + 0.818842i \(0.305384\pi\)
\(882\) −2.35451 −0.0792804
\(883\) −18.4353 −0.620397 −0.310199 0.950672i \(-0.600396\pi\)
−0.310199 + 0.950672i \(0.600396\pi\)
\(884\) −4.24090 −0.142637
\(885\) −3.34758 −0.112528
\(886\) −12.9462 −0.434936
\(887\) 2.19623 0.0737421 0.0368711 0.999320i \(-0.488261\pi\)
0.0368711 + 0.999320i \(0.488261\pi\)
\(888\) 4.50623 0.151219
\(889\) 13.4085 0.449708
\(890\) −3.41009 −0.114307
\(891\) 0 0
\(892\) 25.0397 0.838391
\(893\) −12.4971 −0.418198
\(894\) 14.7540 0.493446
\(895\) −10.3705 −0.346648
\(896\) −1.00000 −0.0334077
\(897\) −8.98088 −0.299863
\(898\) 13.8561 0.462384
\(899\) 22.2329 0.741508
\(900\) −2.35451 −0.0784836
\(901\) 16.1217 0.537091
\(902\) 0 0
\(903\) −3.21506 −0.106990
\(904\) 11.9566 0.397671
\(905\) −22.6738 −0.753702
\(906\) −16.5279 −0.549103
\(907\) 4.84606 0.160911 0.0804554 0.996758i \(-0.474363\pi\)
0.0804554 + 0.996758i \(0.474363\pi\)
\(908\) −0.182924 −0.00607054
\(909\) 25.0741 0.831654
\(910\) 2.64314 0.0876192
\(911\) 59.6573 1.97653 0.988267 0.152736i \(-0.0488085\pi\)
0.988267 + 0.152736i \(0.0488085\pi\)
\(912\) −1.53406 −0.0507978
\(913\) 0 0
\(914\) −9.98796 −0.330372
\(915\) −10.6557 −0.352267
\(916\) −0.134371 −0.00443975
\(917\) 8.49928 0.280671
\(918\) 6.90246 0.227815
\(919\) −16.9163 −0.558018 −0.279009 0.960288i \(-0.590006\pi\)
−0.279009 + 0.960288i \(0.590006\pi\)
\(920\) −4.22916 −0.139431
\(921\) 19.9283 0.656660
\(922\) 27.8198 0.916198
\(923\) −12.1185 −0.398885
\(924\) 0 0
\(925\) −5.60877 −0.184415
\(926\) −10.2727 −0.337582
\(927\) 16.1676 0.531014
\(928\) 8.97936 0.294762
\(929\) 23.9924 0.787164 0.393582 0.919290i \(-0.371236\pi\)
0.393582 + 0.919290i \(0.371236\pi\)
\(930\) −1.98928 −0.0652310
\(931\) 1.90940 0.0625781
\(932\) −2.93010 −0.0959785
\(933\) −20.7302 −0.678676
\(934\) 9.03790 0.295729
\(935\) 0 0
\(936\) 6.22329 0.203415
\(937\) 30.5167 0.996937 0.498468 0.866908i \(-0.333896\pi\)
0.498468 + 0.866908i \(0.333896\pi\)
\(938\) 10.5029 0.342933
\(939\) −13.2630 −0.432822
\(940\) −6.54502 −0.213475
\(941\) 17.6656 0.575882 0.287941 0.957648i \(-0.407029\pi\)
0.287941 + 0.957648i \(0.407029\pi\)
\(942\) 3.73501 0.121693
\(943\) 41.0050 1.33531
\(944\) 4.16663 0.135612
\(945\) −4.30195 −0.139942
\(946\) 0 0
\(947\) 4.02455 0.130780 0.0653902 0.997860i \(-0.479171\pi\)
0.0653902 + 0.997860i \(0.479171\pi\)
\(948\) 11.2059 0.363952
\(949\) −9.76091 −0.316853
\(950\) 1.90940 0.0619491
\(951\) −1.87266 −0.0607251
\(952\) −1.60450 −0.0520020
\(953\) −51.5532 −1.66997 −0.834985 0.550272i \(-0.814524\pi\)
−0.834985 + 0.550272i \(0.814524\pi\)
\(954\) −23.6577 −0.765945
\(955\) −8.48643 −0.274614
\(956\) 8.27410 0.267604
\(957\) 0 0
\(958\) −39.4300 −1.27392
\(959\) −10.0747 −0.325328
\(960\) −0.803425 −0.0259304
\(961\) −24.8694 −0.802240
\(962\) 14.8247 0.477969
\(963\) 6.78734 0.218719
\(964\) 16.5842 0.534142
\(965\) −25.1578 −0.809860
\(966\) −3.39781 −0.109323
\(967\) 25.3238 0.814359 0.407179 0.913348i \(-0.366512\pi\)
0.407179 + 0.913348i \(0.366512\pi\)
\(968\) 0 0
\(969\) −2.46139 −0.0790713
\(970\) −17.4713 −0.560968
\(971\) −41.2627 −1.32418 −0.662091 0.749423i \(-0.730331\pi\)
−0.662091 + 0.749423i \(0.730331\pi\)
\(972\) −15.8040 −0.506913
\(973\) −13.8200 −0.443050
\(974\) −18.9186 −0.606190
\(975\) 2.12356 0.0680085
\(976\) 13.2629 0.424534
\(977\) 14.0714 0.450184 0.225092 0.974337i \(-0.427732\pi\)
0.225092 + 0.974337i \(0.427732\pi\)
\(978\) −4.01230 −0.128299
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 43.5438 1.39025
\(982\) −1.51922 −0.0484803
\(983\) 51.4025 1.63949 0.819743 0.572732i \(-0.194117\pi\)
0.819743 + 0.572732i \(0.194117\pi\)
\(984\) 7.78984 0.248331
\(985\) −13.2603 −0.422508
\(986\) 14.4073 0.458823
\(987\) −5.25844 −0.167378
\(988\) −5.04681 −0.160560
\(989\) 16.9238 0.538145
\(990\) 0 0
\(991\) 45.9460 1.45952 0.729761 0.683702i \(-0.239631\pi\)
0.729761 + 0.683702i \(0.239631\pi\)
\(992\) 2.47600 0.0786130
\(993\) −5.25770 −0.166848
\(994\) −4.58489 −0.145424
\(995\) 3.59813 0.114068
\(996\) 10.8946 0.345210
\(997\) 12.2089 0.386659 0.193329 0.981134i \(-0.438071\pi\)
0.193329 + 0.981134i \(0.438071\pi\)
\(998\) 11.8172 0.374068
\(999\) −24.1286 −0.763396
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.db.1.3 6
11.5 even 5 770.2.n.g.421.2 12
11.9 even 5 770.2.n.g.631.2 yes 12
11.10 odd 2 8470.2.a.cv.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.g.421.2 12 11.5 even 5
770.2.n.g.631.2 yes 12 11.9 even 5
8470.2.a.cv.1.3 6 11.10 odd 2
8470.2.a.db.1.3 6 1.1 even 1 trivial