Properties

Label 8470.2.a.dh.1.3
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5\)
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 1.3
Root \(-1.12455\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.12455 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.73538 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.12455 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.12455 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.73538 q^{9} -1.00000 q^{10} -1.12455 q^{12} +7.06945 q^{13} -1.00000 q^{14} +1.12455 q^{15} +1.00000 q^{16} +3.44477 q^{17} -1.73538 q^{18} +3.43616 q^{19} -1.00000 q^{20} +1.12455 q^{21} +3.63913 q^{23} -1.12455 q^{24} +1.00000 q^{25} +7.06945 q^{26} +5.32519 q^{27} -1.00000 q^{28} -8.69298 q^{29} +1.12455 q^{30} -4.52145 q^{31} +1.00000 q^{32} +3.44477 q^{34} +1.00000 q^{35} -1.73538 q^{36} -0.780576 q^{37} +3.43616 q^{38} -7.94999 q^{39} -1.00000 q^{40} -5.01886 q^{41} +1.12455 q^{42} -5.39500 q^{43} +1.73538 q^{45} +3.63913 q^{46} +9.82775 q^{47} -1.12455 q^{48} +1.00000 q^{49} +1.00000 q^{50} -3.87383 q^{51} +7.06945 q^{52} +11.3718 q^{53} +5.32519 q^{54} -1.00000 q^{56} -3.86415 q^{57} -8.69298 q^{58} -10.5247 q^{59} +1.12455 q^{60} +0.456235 q^{61} -4.52145 q^{62} +1.73538 q^{63} +1.00000 q^{64} -7.06945 q^{65} +10.4937 q^{67} +3.44477 q^{68} -4.09241 q^{69} +1.00000 q^{70} -6.81376 q^{71} -1.73538 q^{72} -15.1151 q^{73} -0.780576 q^{74} -1.12455 q^{75} +3.43616 q^{76} -7.94999 q^{78} -16.2368 q^{79} -1.00000 q^{80} -0.782335 q^{81} -5.01886 q^{82} +7.40344 q^{83} +1.12455 q^{84} -3.44477 q^{85} -5.39500 q^{86} +9.77573 q^{87} -4.97143 q^{89} +1.73538 q^{90} -7.06945 q^{91} +3.63913 q^{92} +5.08461 q^{93} +9.82775 q^{94} -3.43616 q^{95} -1.12455 q^{96} +2.88993 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - q^{13} - 8q^{14} + 8q^{16} + 6q^{17} + 8q^{18} + 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} - 8q^{28} + 3q^{29} - 8q^{31} + 8q^{32} + 6q^{34} + 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} + 35q^{39} - 8q^{40} + 11q^{41} - 5q^{43} - 8q^{45} + 10q^{46} - 15q^{47} + 8q^{49} + 8q^{50} - 6q^{51} - q^{52} - 16q^{53} - 8q^{56} + 38q^{57} + 3q^{58} - 9q^{59} + 32q^{61} - 8q^{62} - 8q^{63} + 8q^{64} + q^{65} + 33q^{67} + 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} + 8q^{72} - 34q^{73} - 6q^{74} + 5q^{76} + 35q^{78} + 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} + 50q^{83} - 6q^{85} - 5q^{86} - 12q^{87} + q^{89} - 8q^{90} + q^{91} + 10q^{92} + 26q^{93} - 15q^{94} - 5q^{95} - 4q^{97} + 8q^{98} + O(q^{100}) \)

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
\(3\) −1.12455 −0.649262 −0.324631 0.945841i \(-0.605240\pi\)
−0.324631 + 0.945841i \(0.605240\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.12455 −0.459097
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −1.73538 −0.578459
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.12455 −0.324631
\(13\) 7.06945 1.96071 0.980357 0.197232i \(-0.0631953\pi\)
0.980357 + 0.197232i \(0.0631953\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.12455 0.290359
\(16\) 1.00000 0.250000
\(17\) 3.44477 0.835479 0.417740 0.908567i \(-0.362822\pi\)
0.417740 + 0.908567i \(0.362822\pi\)
\(18\) −1.73538 −0.409032
\(19\) 3.43616 0.788308 0.394154 0.919044i \(-0.371038\pi\)
0.394154 + 0.919044i \(0.371038\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.12455 0.245398
\(22\) 0 0
\(23\) 3.63913 0.758812 0.379406 0.925230i \(-0.376128\pi\)
0.379406 + 0.925230i \(0.376128\pi\)
\(24\) −1.12455 −0.229549
\(25\) 1.00000 0.200000
\(26\) 7.06945 1.38643
\(27\) 5.32519 1.02483
\(28\) −1.00000 −0.188982
\(29\) −8.69298 −1.61425 −0.807123 0.590383i \(-0.798977\pi\)
−0.807123 + 0.590383i \(0.798977\pi\)
\(30\) 1.12455 0.205315
\(31\) −4.52145 −0.812076 −0.406038 0.913856i \(-0.633090\pi\)
−0.406038 + 0.913856i \(0.633090\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.44477 0.590773
\(35\) 1.00000 0.169031
\(36\) −1.73538 −0.289230
\(37\) −0.780576 −0.128326 −0.0641630 0.997939i \(-0.520438\pi\)
−0.0641630 + 0.997939i \(0.520438\pi\)
\(38\) 3.43616 0.557418
\(39\) −7.94999 −1.27302
\(40\) −1.00000 −0.158114
\(41\) −5.01886 −0.783814 −0.391907 0.920005i \(-0.628185\pi\)
−0.391907 + 0.920005i \(0.628185\pi\)
\(42\) 1.12455 0.173523
\(43\) −5.39500 −0.822729 −0.411365 0.911471i \(-0.634948\pi\)
−0.411365 + 0.911471i \(0.634948\pi\)
\(44\) 0 0
\(45\) 1.73538 0.258695
\(46\) 3.63913 0.536561
\(47\) 9.82775 1.43352 0.716762 0.697318i \(-0.245623\pi\)
0.716762 + 0.697318i \(0.245623\pi\)
\(48\) −1.12455 −0.162315
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −3.87383 −0.542445
\(52\) 7.06945 0.980357
\(53\) 11.3718 1.56204 0.781020 0.624507i \(-0.214700\pi\)
0.781020 + 0.624507i \(0.214700\pi\)
\(54\) 5.32519 0.724667
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −3.86415 −0.511819
\(58\) −8.69298 −1.14144
\(59\) −10.5247 −1.37020 −0.685099 0.728450i \(-0.740241\pi\)
−0.685099 + 0.728450i \(0.740241\pi\)
\(60\) 1.12455 0.145179
\(61\) 0.456235 0.0584149 0.0292074 0.999573i \(-0.490702\pi\)
0.0292074 + 0.999573i \(0.490702\pi\)
\(62\) −4.52145 −0.574224
\(63\) 1.73538 0.218637
\(64\) 1.00000 0.125000
\(65\) −7.06945 −0.876858
\(66\) 0 0
\(67\) 10.4937 1.28201 0.641006 0.767536i \(-0.278517\pi\)
0.641006 + 0.767536i \(0.278517\pi\)
\(68\) 3.44477 0.417740
\(69\) −4.09241 −0.492668
\(70\) 1.00000 0.119523
\(71\) −6.81376 −0.808645 −0.404322 0.914617i \(-0.632493\pi\)
−0.404322 + 0.914617i \(0.632493\pi\)
\(72\) −1.73538 −0.204516
\(73\) −15.1151 −1.76909 −0.884547 0.466452i \(-0.845532\pi\)
−0.884547 + 0.466452i \(0.845532\pi\)
\(74\) −0.780576 −0.0907401
\(75\) −1.12455 −0.129852
\(76\) 3.43616 0.394154
\(77\) 0 0
\(78\) −7.94999 −0.900159
\(79\) −16.2368 −1.82678 −0.913390 0.407087i \(-0.866545\pi\)
−0.913390 + 0.407087i \(0.866545\pi\)
\(80\) −1.00000 −0.111803
\(81\) −0.782335 −0.0869261
\(82\) −5.01886 −0.554240
\(83\) 7.40344 0.812633 0.406317 0.913732i \(-0.366813\pi\)
0.406317 + 0.913732i \(0.366813\pi\)
\(84\) 1.12455 0.122699
\(85\) −3.44477 −0.373638
\(86\) −5.39500 −0.581757
\(87\) 9.77573 1.04807
\(88\) 0 0
\(89\) −4.97143 −0.526970 −0.263485 0.964663i \(-0.584872\pi\)
−0.263485 + 0.964663i \(0.584872\pi\)
\(90\) 1.73538 0.182925
\(91\) −7.06945 −0.741080
\(92\) 3.63913 0.379406
\(93\) 5.08461 0.527250
\(94\) 9.82775 1.01365
\(95\) −3.43616 −0.352542
\(96\) −1.12455 −0.114774
\(97\) 2.88993 0.293427 0.146714 0.989179i \(-0.453130\pi\)
0.146714 + 0.989179i \(0.453130\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.2266 1.21659 0.608295 0.793711i \(-0.291854\pi\)
0.608295 + 0.793711i \(0.291854\pi\)
\(102\) −3.87383 −0.383566
\(103\) 3.70633 0.365196 0.182598 0.983188i \(-0.441549\pi\)
0.182598 + 0.983188i \(0.441549\pi\)
\(104\) 7.06945 0.693217
\(105\) −1.12455 −0.109745
\(106\) 11.3718 1.10453
\(107\) −2.85107 −0.275623 −0.137812 0.990458i \(-0.544007\pi\)
−0.137812 + 0.990458i \(0.544007\pi\)
\(108\) 5.32519 0.512417
\(109\) 9.80121 0.938786 0.469393 0.882989i \(-0.344473\pi\)
0.469393 + 0.882989i \(0.344473\pi\)
\(110\) 0 0
\(111\) 0.877800 0.0833171
\(112\) −1.00000 −0.0944911
\(113\) 10.8032 1.01628 0.508140 0.861275i \(-0.330333\pi\)
0.508140 + 0.861275i \(0.330333\pi\)
\(114\) −3.86415 −0.361910
\(115\) −3.63913 −0.339351
\(116\) −8.69298 −0.807123
\(117\) −12.2682 −1.13419
\(118\) −10.5247 −0.968877
\(119\) −3.44477 −0.315781
\(120\) 1.12455 0.102657
\(121\) 0 0
\(122\) 0.456235 0.0413055
\(123\) 5.64398 0.508901
\(124\) −4.52145 −0.406038
\(125\) −1.00000 −0.0894427
\(126\) 1.73538 0.154600
\(127\) 21.2610 1.88661 0.943303 0.331934i \(-0.107701\pi\)
0.943303 + 0.331934i \(0.107701\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.06697 0.534167
\(130\) −7.06945 −0.620032
\(131\) 10.2637 0.896747 0.448373 0.893846i \(-0.352003\pi\)
0.448373 + 0.893846i \(0.352003\pi\)
\(132\) 0 0
\(133\) −3.43616 −0.297953
\(134\) 10.4937 0.906519
\(135\) −5.32519 −0.458319
\(136\) 3.44477 0.295386
\(137\) −0.760911 −0.0650090 −0.0325045 0.999472i \(-0.510348\pi\)
−0.0325045 + 0.999472i \(0.510348\pi\)
\(138\) −4.09241 −0.348369
\(139\) 17.6377 1.49601 0.748007 0.663691i \(-0.231011\pi\)
0.748007 + 0.663691i \(0.231011\pi\)
\(140\) 1.00000 0.0845154
\(141\) −11.0518 −0.930733
\(142\) −6.81376 −0.571798
\(143\) 0 0
\(144\) −1.73538 −0.144615
\(145\) 8.69298 0.721913
\(146\) −15.1151 −1.25094
\(147\) −1.12455 −0.0927517
\(148\) −0.780576 −0.0641630
\(149\) 0.778180 0.0637510 0.0318755 0.999492i \(-0.489852\pi\)
0.0318755 + 0.999492i \(0.489852\pi\)
\(150\) −1.12455 −0.0918195
\(151\) 13.4154 1.09173 0.545864 0.837874i \(-0.316202\pi\)
0.545864 + 0.837874i \(0.316202\pi\)
\(152\) 3.43616 0.278709
\(153\) −5.97797 −0.483290
\(154\) 0 0
\(155\) 4.52145 0.363171
\(156\) −7.94999 −0.636508
\(157\) −24.0361 −1.91829 −0.959146 0.282912i \(-0.908700\pi\)
−0.959146 + 0.282912i \(0.908700\pi\)
\(158\) −16.2368 −1.29173
\(159\) −12.7882 −1.01417
\(160\) −1.00000 −0.0790569
\(161\) −3.63913 −0.286804
\(162\) −0.782335 −0.0614660
\(163\) 0.905391 0.0709157 0.0354579 0.999371i \(-0.488711\pi\)
0.0354579 + 0.999371i \(0.488711\pi\)
\(164\) −5.01886 −0.391907
\(165\) 0 0
\(166\) 7.40344 0.574618
\(167\) −3.64082 −0.281735 −0.140867 0.990028i \(-0.544989\pi\)
−0.140867 + 0.990028i \(0.544989\pi\)
\(168\) 1.12455 0.0867613
\(169\) 36.9772 2.84440
\(170\) −3.44477 −0.264202
\(171\) −5.96303 −0.456004
\(172\) −5.39500 −0.411365
\(173\) −3.17934 −0.241721 −0.120860 0.992670i \(-0.538565\pi\)
−0.120860 + 0.992670i \(0.538565\pi\)
\(174\) 9.77573 0.741096
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 11.8356 0.889618
\(178\) −4.97143 −0.372624
\(179\) 4.48137 0.334953 0.167477 0.985876i \(-0.446438\pi\)
0.167477 + 0.985876i \(0.446438\pi\)
\(180\) 1.73538 0.129347
\(181\) −9.02937 −0.671148 −0.335574 0.942014i \(-0.608930\pi\)
−0.335574 + 0.942014i \(0.608930\pi\)
\(182\) −7.06945 −0.524023
\(183\) −0.513061 −0.0379265
\(184\) 3.63913 0.268281
\(185\) 0.780576 0.0573891
\(186\) 5.08461 0.372822
\(187\) 0 0
\(188\) 9.82775 0.716762
\(189\) −5.32519 −0.387351
\(190\) −3.43616 −0.249285
\(191\) 2.90870 0.210466 0.105233 0.994448i \(-0.466441\pi\)
0.105233 + 0.994448i \(0.466441\pi\)
\(192\) −1.12455 −0.0811577
\(193\) −7.77320 −0.559527 −0.279764 0.960069i \(-0.590256\pi\)
−0.279764 + 0.960069i \(0.590256\pi\)
\(194\) 2.88993 0.207485
\(195\) 7.94999 0.569310
\(196\) 1.00000 0.0714286
\(197\) 26.2257 1.86851 0.934254 0.356609i \(-0.116067\pi\)
0.934254 + 0.356609i \(0.116067\pi\)
\(198\) 0 0
\(199\) 5.64963 0.400491 0.200246 0.979746i \(-0.435826\pi\)
0.200246 + 0.979746i \(0.435826\pi\)
\(200\) 1.00000 0.0707107
\(201\) −11.8008 −0.832361
\(202\) 12.2266 0.860260
\(203\) 8.69298 0.610128
\(204\) −3.87383 −0.271222
\(205\) 5.01886 0.350532
\(206\) 3.70633 0.258233
\(207\) −6.31527 −0.438942
\(208\) 7.06945 0.490178
\(209\) 0 0
\(210\) −1.12455 −0.0776016
\(211\) 7.42973 0.511484 0.255742 0.966745i \(-0.417680\pi\)
0.255742 + 0.966745i \(0.417680\pi\)
\(212\) 11.3718 0.781020
\(213\) 7.66245 0.525022
\(214\) −2.85107 −0.194895
\(215\) 5.39500 0.367936
\(216\) 5.32519 0.362333
\(217\) 4.52145 0.306936
\(218\) 9.80121 0.663822
\(219\) 16.9978 1.14860
\(220\) 0 0
\(221\) 24.3526 1.63814
\(222\) 0.877800 0.0589141
\(223\) 1.51167 0.101229 0.0506145 0.998718i \(-0.483882\pi\)
0.0506145 + 0.998718i \(0.483882\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.73538 −0.115692
\(226\) 10.8032 0.718618
\(227\) 13.8558 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(228\) −3.86415 −0.255909
\(229\) 20.5463 1.35774 0.678870 0.734258i \(-0.262470\pi\)
0.678870 + 0.734258i \(0.262470\pi\)
\(230\) −3.63913 −0.239957
\(231\) 0 0
\(232\) −8.69298 −0.570722
\(233\) 16.2345 1.06356 0.531780 0.846883i \(-0.321523\pi\)
0.531780 + 0.846883i \(0.321523\pi\)
\(234\) −12.2682 −0.801995
\(235\) −9.82775 −0.641092
\(236\) −10.5247 −0.685099
\(237\) 18.2591 1.18606
\(238\) −3.44477 −0.223291
\(239\) −0.392921 −0.0254160 −0.0127080 0.999919i \(-0.504045\pi\)
−0.0127080 + 0.999919i \(0.504045\pi\)
\(240\) 1.12455 0.0725897
\(241\) −3.32681 −0.214298 −0.107149 0.994243i \(-0.534172\pi\)
−0.107149 + 0.994243i \(0.534172\pi\)
\(242\) 0 0
\(243\) −15.0958 −0.968395
\(244\) 0.456235 0.0292074
\(245\) −1.00000 −0.0638877
\(246\) 5.64398 0.359847
\(247\) 24.2918 1.54565
\(248\) −4.52145 −0.287112
\(249\) −8.32557 −0.527612
\(250\) −1.00000 −0.0632456
\(251\) 17.2755 1.09042 0.545211 0.838299i \(-0.316450\pi\)
0.545211 + 0.838299i \(0.316450\pi\)
\(252\) 1.73538 0.109318
\(253\) 0 0
\(254\) 21.2610 1.33403
\(255\) 3.87383 0.242589
\(256\) 1.00000 0.0625000
\(257\) −0.448545 −0.0279795 −0.0139897 0.999902i \(-0.504453\pi\)
−0.0139897 + 0.999902i \(0.504453\pi\)
\(258\) 6.06697 0.377713
\(259\) 0.780576 0.0485026
\(260\) −7.06945 −0.438429
\(261\) 15.0856 0.933775
\(262\) 10.2637 0.634096
\(263\) 26.1440 1.61211 0.806053 0.591843i \(-0.201600\pi\)
0.806053 + 0.591843i \(0.201600\pi\)
\(264\) 0 0
\(265\) −11.3718 −0.698565
\(266\) −3.43616 −0.210684
\(267\) 5.59064 0.342142
\(268\) 10.4937 0.641006
\(269\) 14.1617 0.863453 0.431727 0.902004i \(-0.357904\pi\)
0.431727 + 0.902004i \(0.357904\pi\)
\(270\) −5.32519 −0.324081
\(271\) −0.934211 −0.0567493 −0.0283746 0.999597i \(-0.509033\pi\)
−0.0283746 + 0.999597i \(0.509033\pi\)
\(272\) 3.44477 0.208870
\(273\) 7.94999 0.481155
\(274\) −0.760911 −0.0459683
\(275\) 0 0
\(276\) −4.09241 −0.246334
\(277\) −25.3249 −1.52162 −0.760812 0.648973i \(-0.775199\pi\)
−0.760812 + 0.648973i \(0.775199\pi\)
\(278\) 17.6377 1.05784
\(279\) 7.84641 0.469752
\(280\) 1.00000 0.0597614
\(281\) 27.7779 1.65709 0.828545 0.559922i \(-0.189169\pi\)
0.828545 + 0.559922i \(0.189169\pi\)
\(282\) −11.0518 −0.658127
\(283\) −19.5153 −1.16006 −0.580032 0.814594i \(-0.696960\pi\)
−0.580032 + 0.814594i \(0.696960\pi\)
\(284\) −6.81376 −0.404322
\(285\) 3.86415 0.228892
\(286\) 0 0
\(287\) 5.01886 0.296254
\(288\) −1.73538 −0.102258
\(289\) −5.13357 −0.301975
\(290\) 8.69298 0.510469
\(291\) −3.24988 −0.190511
\(292\) −15.1151 −0.884547
\(293\) −13.3505 −0.779945 −0.389972 0.920827i \(-0.627515\pi\)
−0.389972 + 0.920827i \(0.627515\pi\)
\(294\) −1.12455 −0.0655854
\(295\) 10.5247 0.612771
\(296\) −0.780576 −0.0453701
\(297\) 0 0
\(298\) 0.778180 0.0450788
\(299\) 25.7267 1.48781
\(300\) −1.12455 −0.0649262
\(301\) 5.39500 0.310962
\(302\) 13.4154 0.771968
\(303\) −13.7495 −0.789886
\(304\) 3.43616 0.197077
\(305\) −0.456235 −0.0261239
\(306\) −5.97797 −0.341738
\(307\) 25.7848 1.47162 0.735809 0.677189i \(-0.236802\pi\)
0.735809 + 0.677189i \(0.236802\pi\)
\(308\) 0 0
\(309\) −4.16798 −0.237108
\(310\) 4.52145 0.256801
\(311\) 5.11540 0.290068 0.145034 0.989427i \(-0.453671\pi\)
0.145034 + 0.989427i \(0.453671\pi\)
\(312\) −7.94999 −0.450079
\(313\) 16.5571 0.935861 0.467930 0.883765i \(-0.345000\pi\)
0.467930 + 0.883765i \(0.345000\pi\)
\(314\) −24.0361 −1.35644
\(315\) −1.73538 −0.0977774
\(316\) −16.2368 −0.913390
\(317\) 15.3500 0.862143 0.431072 0.902318i \(-0.358136\pi\)
0.431072 + 0.902318i \(0.358136\pi\)
\(318\) −12.7882 −0.717128
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 3.20618 0.178952
\(322\) −3.63913 −0.202801
\(323\) 11.8368 0.658615
\(324\) −0.782335 −0.0434630
\(325\) 7.06945 0.392143
\(326\) 0.905391 0.0501450
\(327\) −11.0220 −0.609518
\(328\) −5.01886 −0.277120
\(329\) −9.82775 −0.541821
\(330\) 0 0
\(331\) −6.59151 −0.362302 −0.181151 0.983455i \(-0.557982\pi\)
−0.181151 + 0.983455i \(0.557982\pi\)
\(332\) 7.40344 0.406317
\(333\) 1.35459 0.0742313
\(334\) −3.64082 −0.199217
\(335\) −10.4937 −0.573333
\(336\) 1.12455 0.0613495
\(337\) −19.1547 −1.04342 −0.521711 0.853123i \(-0.674706\pi\)
−0.521711 + 0.853123i \(0.674706\pi\)
\(338\) 36.9772 2.01129
\(339\) −12.1488 −0.659832
\(340\) −3.44477 −0.186819
\(341\) 0 0
\(342\) −5.96303 −0.322444
\(343\) −1.00000 −0.0539949
\(344\) −5.39500 −0.290879
\(345\) 4.09241 0.220328
\(346\) −3.17934 −0.170922
\(347\) −9.00265 −0.483288 −0.241644 0.970365i \(-0.577687\pi\)
−0.241644 + 0.970365i \(0.577687\pi\)
\(348\) 9.77573 0.524034
\(349\) −6.72881 −0.360185 −0.180092 0.983650i \(-0.557640\pi\)
−0.180092 + 0.983650i \(0.557640\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 37.6462 2.00940
\(352\) 0 0
\(353\) −29.9182 −1.59238 −0.796192 0.605044i \(-0.793156\pi\)
−0.796192 + 0.605044i \(0.793156\pi\)
\(354\) 11.8356 0.629055
\(355\) 6.81376 0.361637
\(356\) −4.97143 −0.263485
\(357\) 3.87383 0.205025
\(358\) 4.48137 0.236848
\(359\) −23.5622 −1.24356 −0.621782 0.783190i \(-0.713591\pi\)
−0.621782 + 0.783190i \(0.713591\pi\)
\(360\) 1.73538 0.0914624
\(361\) −7.19282 −0.378570
\(362\) −9.02937 −0.474573
\(363\) 0 0
\(364\) −7.06945 −0.370540
\(365\) 15.1151 0.791163
\(366\) −0.513061 −0.0268181
\(367\) −10.1699 −0.530862 −0.265431 0.964130i \(-0.585514\pi\)
−0.265431 + 0.964130i \(0.585514\pi\)
\(368\) 3.63913 0.189703
\(369\) 8.70961 0.453404
\(370\) 0.780576 0.0405802
\(371\) −11.3718 −0.590395
\(372\) 5.08461 0.263625
\(373\) 10.7767 0.557996 0.278998 0.960292i \(-0.409998\pi\)
0.278998 + 0.960292i \(0.409998\pi\)
\(374\) 0 0
\(375\) 1.12455 0.0580717
\(376\) 9.82775 0.506827
\(377\) −61.4546 −3.16507
\(378\) −5.32519 −0.273898
\(379\) −5.39945 −0.277351 −0.138676 0.990338i \(-0.544285\pi\)
−0.138676 + 0.990338i \(0.544285\pi\)
\(380\) −3.43616 −0.176271
\(381\) −23.9091 −1.22490
\(382\) 2.90870 0.148822
\(383\) 7.17505 0.366628 0.183314 0.983054i \(-0.441318\pi\)
0.183314 + 0.983054i \(0.441318\pi\)
\(384\) −1.12455 −0.0573872
\(385\) 0 0
\(386\) −7.77320 −0.395646
\(387\) 9.36235 0.475915
\(388\) 2.88993 0.146714
\(389\) 16.5388 0.838549 0.419274 0.907860i \(-0.362285\pi\)
0.419274 + 0.907860i \(0.362285\pi\)
\(390\) 7.94999 0.402563
\(391\) 12.5360 0.633972
\(392\) 1.00000 0.0505076
\(393\) −11.5421 −0.582224
\(394\) 26.2257 1.32123
\(395\) 16.2368 0.816960
\(396\) 0 0
\(397\) 16.7413 0.840221 0.420111 0.907473i \(-0.361991\pi\)
0.420111 + 0.907473i \(0.361991\pi\)
\(398\) 5.64963 0.283190
\(399\) 3.86415 0.193449
\(400\) 1.00000 0.0500000
\(401\) −20.3412 −1.01579 −0.507896 0.861419i \(-0.669576\pi\)
−0.507896 + 0.861419i \(0.669576\pi\)
\(402\) −11.8008 −0.588568
\(403\) −31.9641 −1.59225
\(404\) 12.2266 0.608295
\(405\) 0.782335 0.0388745
\(406\) 8.69298 0.431425
\(407\) 0 0
\(408\) −3.87383 −0.191783
\(409\) −0.443589 −0.0219341 −0.0109670 0.999940i \(-0.503491\pi\)
−0.0109670 + 0.999940i \(0.503491\pi\)
\(410\) 5.01886 0.247864
\(411\) 0.855686 0.0422079
\(412\) 3.70633 0.182598
\(413\) 10.5247 0.517886
\(414\) −6.31527 −0.310379
\(415\) −7.40344 −0.363421
\(416\) 7.06945 0.346608
\(417\) −19.8346 −0.971305
\(418\) 0 0
\(419\) −18.6545 −0.911330 −0.455665 0.890151i \(-0.650599\pi\)
−0.455665 + 0.890151i \(0.650599\pi\)
\(420\) −1.12455 −0.0548726
\(421\) 12.2716 0.598081 0.299040 0.954240i \(-0.403333\pi\)
0.299040 + 0.954240i \(0.403333\pi\)
\(422\) 7.42973 0.361674
\(423\) −17.0549 −0.829235
\(424\) 11.3718 0.552264
\(425\) 3.44477 0.167096
\(426\) 7.66245 0.371247
\(427\) −0.456235 −0.0220787
\(428\) −2.85107 −0.137812
\(429\) 0 0
\(430\) 5.39500 0.260170
\(431\) 21.9795 1.05872 0.529358 0.848398i \(-0.322433\pi\)
0.529358 + 0.848398i \(0.322433\pi\)
\(432\) 5.32519 0.256208
\(433\) 22.2345 1.06852 0.534262 0.845319i \(-0.320590\pi\)
0.534262 + 0.845319i \(0.320590\pi\)
\(434\) 4.52145 0.217036
\(435\) −9.77573 −0.468710
\(436\) 9.80121 0.469393
\(437\) 12.5046 0.598178
\(438\) 16.9978 0.812186
\(439\) −11.0302 −0.526444 −0.263222 0.964735i \(-0.584785\pi\)
−0.263222 + 0.964735i \(0.584785\pi\)
\(440\) 0 0
\(441\) −1.73538 −0.0826370
\(442\) 24.3526 1.15834
\(443\) 30.8702 1.46669 0.733345 0.679857i \(-0.237958\pi\)
0.733345 + 0.679857i \(0.237958\pi\)
\(444\) 0.877800 0.0416586
\(445\) 4.97143 0.235668
\(446\) 1.51167 0.0715797
\(447\) −0.875106 −0.0413911
\(448\) −1.00000 −0.0472456
\(449\) −13.9952 −0.660475 −0.330238 0.943898i \(-0.607129\pi\)
−0.330238 + 0.943898i \(0.607129\pi\)
\(450\) −1.73538 −0.0818065
\(451\) 0 0
\(452\) 10.8032 0.508140
\(453\) −15.0863 −0.708817
\(454\) 13.8558 0.650284
\(455\) 7.06945 0.331421
\(456\) −3.86415 −0.180955
\(457\) −7.06533 −0.330502 −0.165251 0.986252i \(-0.552843\pi\)
−0.165251 + 0.986252i \(0.552843\pi\)
\(458\) 20.5463 0.960067
\(459\) 18.3440 0.856227
\(460\) −3.63913 −0.169676
\(461\) 10.6011 0.493744 0.246872 0.969048i \(-0.420597\pi\)
0.246872 + 0.969048i \(0.420597\pi\)
\(462\) 0 0
\(463\) 4.59137 0.213379 0.106690 0.994292i \(-0.465975\pi\)
0.106690 + 0.994292i \(0.465975\pi\)
\(464\) −8.69298 −0.403561
\(465\) −5.08461 −0.235793
\(466\) 16.2345 0.752050
\(467\) −30.9287 −1.43121 −0.715606 0.698504i \(-0.753849\pi\)
−0.715606 + 0.698504i \(0.753849\pi\)
\(468\) −12.2682 −0.567096
\(469\) −10.4937 −0.484555
\(470\) −9.82775 −0.453320
\(471\) 27.0299 1.24547
\(472\) −10.5247 −0.484438
\(473\) 0 0
\(474\) 18.2591 0.838670
\(475\) 3.43616 0.157662
\(476\) −3.44477 −0.157891
\(477\) −19.7344 −0.903576
\(478\) −0.392921 −0.0179718
\(479\) 3.01362 0.137696 0.0688480 0.997627i \(-0.478068\pi\)
0.0688480 + 0.997627i \(0.478068\pi\)
\(480\) 1.12455 0.0513287
\(481\) −5.51825 −0.251610
\(482\) −3.32681 −0.151532
\(483\) 4.09241 0.186211
\(484\) 0 0
\(485\) −2.88993 −0.131225
\(486\) −15.0958 −0.684759
\(487\) 3.00551 0.136193 0.0680963 0.997679i \(-0.478307\pi\)
0.0680963 + 0.997679i \(0.478307\pi\)
\(488\) 0.456235 0.0206528
\(489\) −1.01816 −0.0460429
\(490\) −1.00000 −0.0451754
\(491\) 34.6723 1.56474 0.782369 0.622815i \(-0.214011\pi\)
0.782369 + 0.622815i \(0.214011\pi\)
\(492\) 5.64398 0.254450
\(493\) −29.9453 −1.34867
\(494\) 24.2918 1.09294
\(495\) 0 0
\(496\) −4.52145 −0.203019
\(497\) 6.81376 0.305639
\(498\) −8.32557 −0.373078
\(499\) −5.25216 −0.235119 −0.117559 0.993066i \(-0.537507\pi\)
−0.117559 + 0.993066i \(0.537507\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 4.09430 0.182920
\(502\) 17.2755 0.771044
\(503\) 33.6525 1.50049 0.750245 0.661160i \(-0.229936\pi\)
0.750245 + 0.661160i \(0.229936\pi\)
\(504\) 1.73538 0.0772998
\(505\) −12.2266 −0.544076
\(506\) 0 0
\(507\) −41.5828 −1.84676
\(508\) 21.2610 0.943303
\(509\) −20.1633 −0.893724 −0.446862 0.894603i \(-0.647458\pi\)
−0.446862 + 0.894603i \(0.647458\pi\)
\(510\) 3.87383 0.171536
\(511\) 15.1151 0.668654
\(512\) 1.00000 0.0441942
\(513\) 18.2982 0.807885
\(514\) −0.448545 −0.0197845
\(515\) −3.70633 −0.163321
\(516\) 6.06697 0.267083
\(517\) 0 0
\(518\) 0.780576 0.0342965
\(519\) 3.57534 0.156940
\(520\) −7.06945 −0.310016
\(521\) −34.1002 −1.49396 −0.746979 0.664848i \(-0.768496\pi\)
−0.746979 + 0.664848i \(0.768496\pi\)
\(522\) 15.0856 0.660279
\(523\) −37.5694 −1.64280 −0.821398 0.570355i \(-0.806806\pi\)
−0.821398 + 0.570355i \(0.806806\pi\)
\(524\) 10.2637 0.448373
\(525\) 1.12455 0.0490796
\(526\) 26.1440 1.13993
\(527\) −15.5753 −0.678472
\(528\) 0 0
\(529\) −9.75670 −0.424204
\(530\) −11.3718 −0.493960
\(531\) 18.2643 0.792604
\(532\) −3.43616 −0.148976
\(533\) −35.4806 −1.53683
\(534\) 5.59064 0.241931
\(535\) 2.85107 0.123263
\(536\) 10.4937 0.453260
\(537\) −5.03954 −0.217472
\(538\) 14.1617 0.610554
\(539\) 0 0
\(540\) −5.32519 −0.229160
\(541\) −22.6441 −0.973547 −0.486773 0.873528i \(-0.661826\pi\)
−0.486773 + 0.873528i \(0.661826\pi\)
\(542\) −0.934211 −0.0401278
\(543\) 10.1540 0.435751
\(544\) 3.44477 0.147693
\(545\) −9.80121 −0.419838
\(546\) 7.94999 0.340228
\(547\) 24.3655 1.04179 0.520896 0.853620i \(-0.325598\pi\)
0.520896 + 0.853620i \(0.325598\pi\)
\(548\) −0.760911 −0.0325045
\(549\) −0.791739 −0.0337906
\(550\) 0 0
\(551\) −29.8704 −1.27252
\(552\) −4.09241 −0.174184
\(553\) 16.2368 0.690458
\(554\) −25.3249 −1.07595
\(555\) −0.877800 −0.0372605
\(556\) 17.6377 0.748007
\(557\) 25.1506 1.06567 0.532833 0.846220i \(-0.321127\pi\)
0.532833 + 0.846220i \(0.321127\pi\)
\(558\) 7.84641 0.332165
\(559\) −38.1397 −1.61314
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 27.7779 1.17174
\(563\) −20.1972 −0.851212 −0.425606 0.904909i \(-0.639939\pi\)
−0.425606 + 0.904909i \(0.639939\pi\)
\(564\) −11.0518 −0.465366
\(565\) −10.8032 −0.454494
\(566\) −19.5153 −0.820289
\(567\) 0.782335 0.0328550
\(568\) −6.81376 −0.285899
\(569\) 25.8942 1.08554 0.542771 0.839881i \(-0.317375\pi\)
0.542771 + 0.839881i \(0.317375\pi\)
\(570\) 3.86415 0.161851
\(571\) 22.4564 0.939771 0.469885 0.882727i \(-0.344295\pi\)
0.469885 + 0.882727i \(0.344295\pi\)
\(572\) 0 0
\(573\) −3.27099 −0.136648
\(574\) 5.01886 0.209483
\(575\) 3.63913 0.151762
\(576\) −1.73538 −0.0723074
\(577\) −33.8577 −1.40952 −0.704758 0.709448i \(-0.748945\pi\)
−0.704758 + 0.709448i \(0.748945\pi\)
\(578\) −5.13357 −0.213528
\(579\) 8.74139 0.363280
\(580\) 8.69298 0.360956
\(581\) −7.40344 −0.307146
\(582\) −3.24988 −0.134712
\(583\) 0 0
\(584\) −15.1151 −0.625469
\(585\) 12.2682 0.507226
\(586\) −13.3505 −0.551504
\(587\) 28.5545 1.17857 0.589285 0.807925i \(-0.299409\pi\)
0.589285 + 0.807925i \(0.299409\pi\)
\(588\) −1.12455 −0.0463758
\(589\) −15.5364 −0.640166
\(590\) 10.5247 0.433295
\(591\) −29.4923 −1.21315
\(592\) −0.780576 −0.0320815
\(593\) 34.9417 1.43488 0.717442 0.696618i \(-0.245313\pi\)
0.717442 + 0.696618i \(0.245313\pi\)
\(594\) 0 0
\(595\) 3.44477 0.141222
\(596\) 0.778180 0.0318755
\(597\) −6.35331 −0.260024
\(598\) 25.7267 1.05204
\(599\) −31.0639 −1.26924 −0.634618 0.772826i \(-0.718843\pi\)
−0.634618 + 0.772826i \(0.718843\pi\)
\(600\) −1.12455 −0.0459097
\(601\) −43.2602 −1.76462 −0.882310 0.470668i \(-0.844013\pi\)
−0.882310 + 0.470668i \(0.844013\pi\)
\(602\) 5.39500 0.219884
\(603\) −18.2106 −0.741591
\(604\) 13.4154 0.545864
\(605\) 0 0
\(606\) −13.7495 −0.558534
\(607\) −33.1013 −1.34354 −0.671771 0.740759i \(-0.734466\pi\)
−0.671771 + 0.740759i \(0.734466\pi\)
\(608\) 3.43616 0.139355
\(609\) −9.77573 −0.396133
\(610\) −0.456235 −0.0184724
\(611\) 69.4768 2.81073
\(612\) −5.97797 −0.241645
\(613\) −17.0269 −0.687708 −0.343854 0.939023i \(-0.611733\pi\)
−0.343854 + 0.939023i \(0.611733\pi\)
\(614\) 25.7848 1.04059
\(615\) −5.64398 −0.227587
\(616\) 0 0
\(617\) 3.82168 0.153855 0.0769276 0.997037i \(-0.475489\pi\)
0.0769276 + 0.997037i \(0.475489\pi\)
\(618\) −4.16798 −0.167661
\(619\) −28.4586 −1.14385 −0.571924 0.820307i \(-0.693803\pi\)
−0.571924 + 0.820307i \(0.693803\pi\)
\(620\) 4.52145 0.181586
\(621\) 19.3791 0.777656
\(622\) 5.11540 0.205109
\(623\) 4.97143 0.199176
\(624\) −7.94999 −0.318254
\(625\) 1.00000 0.0400000
\(626\) 16.5571 0.661753
\(627\) 0 0
\(628\) −24.0361 −0.959146
\(629\) −2.68890 −0.107214
\(630\) −1.73538 −0.0691391
\(631\) 8.90784 0.354615 0.177308 0.984155i \(-0.443261\pi\)
0.177308 + 0.984155i \(0.443261\pi\)
\(632\) −16.2368 −0.645864
\(633\) −8.35514 −0.332087
\(634\) 15.3500 0.609627
\(635\) −21.2610 −0.843716
\(636\) −12.7882 −0.507086
\(637\) 7.06945 0.280102
\(638\) 0 0
\(639\) 11.8244 0.467768
\(640\) −1.00000 −0.0395285
\(641\) 28.2250 1.11482 0.557410 0.830237i \(-0.311795\pi\)
0.557410 + 0.830237i \(0.311795\pi\)
\(642\) 3.20618 0.126538
\(643\) 27.7470 1.09423 0.547117 0.837056i \(-0.315725\pi\)
0.547117 + 0.837056i \(0.315725\pi\)
\(644\) −3.63913 −0.143402
\(645\) −6.06697 −0.238887
\(646\) 11.8368 0.465711
\(647\) 26.0679 1.02484 0.512418 0.858736i \(-0.328750\pi\)
0.512418 + 0.858736i \(0.328750\pi\)
\(648\) −0.782335 −0.0307330
\(649\) 0 0
\(650\) 7.06945 0.277287
\(651\) −5.08461 −0.199282
\(652\) 0.905391 0.0354579
\(653\) 10.8574 0.424884 0.212442 0.977174i \(-0.431858\pi\)
0.212442 + 0.977174i \(0.431858\pi\)
\(654\) −11.0220 −0.430994
\(655\) −10.2637 −0.401037
\(656\) −5.01886 −0.195954
\(657\) 26.2305 1.02335
\(658\) −9.82775 −0.383126
\(659\) −8.76914 −0.341597 −0.170799 0.985306i \(-0.554635\pi\)
−0.170799 + 0.985306i \(0.554635\pi\)
\(660\) 0 0
\(661\) −18.7665 −0.729934 −0.364967 0.931020i \(-0.618920\pi\)
−0.364967 + 0.931020i \(0.618920\pi\)
\(662\) −6.59151 −0.256186
\(663\) −27.3859 −1.06358
\(664\) 7.40344 0.287309
\(665\) 3.43616 0.133248
\(666\) 1.35459 0.0524894
\(667\) −31.6349 −1.22491
\(668\) −3.64082 −0.140867
\(669\) −1.69996 −0.0657241
\(670\) −10.4937 −0.405408
\(671\) 0 0
\(672\) 1.12455 0.0433806
\(673\) 13.0246 0.502061 0.251031 0.967979i \(-0.419230\pi\)
0.251031 + 0.967979i \(0.419230\pi\)
\(674\) −19.1547 −0.737810
\(675\) 5.32519 0.204967
\(676\) 36.9772 1.42220
\(677\) −10.0804 −0.387422 −0.193711 0.981059i \(-0.562052\pi\)
−0.193711 + 0.981059i \(0.562052\pi\)
\(678\) −12.1488 −0.466571
\(679\) −2.88993 −0.110905
\(680\) −3.44477 −0.132101
\(681\) −15.5816 −0.597087
\(682\) 0 0
\(683\) −8.98900 −0.343955 −0.171977 0.985101i \(-0.555016\pi\)
−0.171977 + 0.985101i \(0.555016\pi\)
\(684\) −5.96303 −0.228002
\(685\) 0.760911 0.0290729
\(686\) −1.00000 −0.0381802
\(687\) −23.1055 −0.881529
\(688\) −5.39500 −0.205682
\(689\) 80.3925 3.06271
\(690\) 4.09241 0.155795
\(691\) 45.5777 1.73386 0.866929 0.498432i \(-0.166090\pi\)
0.866929 + 0.498432i \(0.166090\pi\)
\(692\) −3.17934 −0.120860
\(693\) 0 0
\(694\) −9.00265 −0.341736
\(695\) −17.6377 −0.669038
\(696\) 9.77573 0.370548
\(697\) −17.2888 −0.654860
\(698\) −6.72881 −0.254689
\(699\) −18.2566 −0.690529
\(700\) −1.00000 −0.0377964
\(701\) 11.8996 0.449442 0.224721 0.974423i \(-0.427853\pi\)
0.224721 + 0.974423i \(0.427853\pi\)
\(702\) 37.6462 1.42086
\(703\) −2.68218 −0.101160
\(704\) 0 0
\(705\) 11.0518 0.416236
\(706\) −29.9182 −1.12599
\(707\) −12.2266 −0.459828
\(708\) 11.8356 0.444809
\(709\) −51.2048 −1.92304 −0.961518 0.274740i \(-0.911408\pi\)
−0.961518 + 0.274740i \(0.911408\pi\)
\(710\) 6.81376 0.255716
\(711\) 28.1769 1.05672
\(712\) −4.97143 −0.186312
\(713\) −16.4541 −0.616213
\(714\) 3.87383 0.144974
\(715\) 0 0
\(716\) 4.48137 0.167477
\(717\) 0.441861 0.0165016
\(718\) −23.5622 −0.879333
\(719\) −27.8658 −1.03922 −0.519610 0.854403i \(-0.673923\pi\)
−0.519610 + 0.854403i \(0.673923\pi\)
\(720\) 1.73538 0.0646737
\(721\) −3.70633 −0.138031
\(722\) −7.19282 −0.267689
\(723\) 3.74118 0.139136
\(724\) −9.02937 −0.335574
\(725\) −8.69298 −0.322849
\(726\) 0 0
\(727\) 41.3933 1.53519 0.767597 0.640933i \(-0.221453\pi\)
0.767597 + 0.640933i \(0.221453\pi\)
\(728\) −7.06945 −0.262011
\(729\) 19.3230 0.715668
\(730\) 15.1151 0.559436
\(731\) −18.5845 −0.687373
\(732\) −0.513061 −0.0189633
\(733\) −5.84935 −0.216051 −0.108025 0.994148i \(-0.534453\pi\)
−0.108025 + 0.994148i \(0.534453\pi\)
\(734\) −10.1699 −0.375376
\(735\) 1.12455 0.0414798
\(736\) 3.63913 0.134140
\(737\) 0 0
\(738\) 8.70961 0.320605
\(739\) 48.5521 1.78602 0.893009 0.450039i \(-0.148590\pi\)
0.893009 + 0.450039i \(0.148590\pi\)
\(740\) 0.780576 0.0286945
\(741\) −27.3174 −1.00353
\(742\) −11.3718 −0.417473
\(743\) 33.8585 1.24215 0.621073 0.783752i \(-0.286697\pi\)
0.621073 + 0.783752i \(0.286697\pi\)
\(744\) 5.08461 0.186411
\(745\) −0.778180 −0.0285103
\(746\) 10.7767 0.394562
\(747\) −12.8478 −0.470075
\(748\) 0 0
\(749\) 2.85107 0.104176
\(750\) 1.12455 0.0410629
\(751\) 34.9061 1.27374 0.636870 0.770971i \(-0.280229\pi\)
0.636870 + 0.770971i \(0.280229\pi\)
\(752\) 9.82775 0.358381
\(753\) −19.4273 −0.707969
\(754\) −61.4546 −2.23804
\(755\) −13.4154 −0.488235
\(756\) −5.32519 −0.193675
\(757\) −18.7354 −0.680948 −0.340474 0.940254i \(-0.610588\pi\)
−0.340474 + 0.940254i \(0.610588\pi\)
\(758\) −5.39945 −0.196117
\(759\) 0 0
\(760\) −3.43616 −0.124643
\(761\) −26.6630 −0.966531 −0.483266 0.875474i \(-0.660549\pi\)
−0.483266 + 0.875474i \(0.660549\pi\)
\(762\) −23.9091 −0.866136
\(763\) −9.80121 −0.354828
\(764\) 2.90870 0.105233
\(765\) 5.97797 0.216134
\(766\) 7.17505 0.259245
\(767\) −74.4038 −2.68657
\(768\) −1.12455 −0.0405789
\(769\) −38.2777 −1.38033 −0.690165 0.723652i \(-0.742462\pi\)
−0.690165 + 0.723652i \(0.742462\pi\)
\(770\) 0 0
\(771\) 0.504413 0.0181660
\(772\) −7.77320 −0.279764
\(773\) 7.52232 0.270559 0.135280 0.990807i \(-0.456807\pi\)
0.135280 + 0.990807i \(0.456807\pi\)
\(774\) 9.36235 0.336523
\(775\) −4.52145 −0.162415
\(776\) 2.88993 0.103742
\(777\) −0.877800 −0.0314909
\(778\) 16.5388 0.592943
\(779\) −17.2456 −0.617887
\(780\) 7.94999 0.284655
\(781\) 0 0
\(782\) 12.5360 0.448286
\(783\) −46.2918 −1.65433
\(784\) 1.00000 0.0357143
\(785\) 24.0361 0.857886
\(786\) −11.5421 −0.411694
\(787\) 0.215236 0.00767235 0.00383618 0.999993i \(-0.498779\pi\)
0.00383618 + 0.999993i \(0.498779\pi\)
\(788\) 26.2257 0.934254
\(789\) −29.4003 −1.04668
\(790\) 16.2368 0.577678
\(791\) −10.8032 −0.384118
\(792\) 0 0
\(793\) 3.22533 0.114535
\(794\) 16.7413 0.594126
\(795\) 12.7882 0.453552
\(796\) 5.64963 0.200246
\(797\) 28.2639 1.00116 0.500580 0.865690i \(-0.333120\pi\)
0.500580 + 0.865690i \(0.333120\pi\)
\(798\) 3.86415 0.136789
\(799\) 33.8543 1.19768
\(800\) 1.00000 0.0353553
\(801\) 8.62730 0.304831
\(802\) −20.3412 −0.718273
\(803\) 0 0
\(804\) −11.8008 −0.416181
\(805\) 3.63913 0.128263
\(806\) −31.9641 −1.12589
\(807\) −15.9256 −0.560607
\(808\) 12.2266 0.430130
\(809\) −4.57788 −0.160950 −0.0804748 0.996757i \(-0.525644\pi\)
−0.0804748 + 0.996757i \(0.525644\pi\)
\(810\) 0.782335 0.0274884
\(811\) 44.4278 1.56007 0.780035 0.625736i \(-0.215201\pi\)
0.780035 + 0.625736i \(0.215201\pi\)
\(812\) 8.69298 0.305064
\(813\) 1.05057 0.0368451
\(814\) 0 0
\(815\) −0.905391 −0.0317145
\(816\) −3.87383 −0.135611
\(817\) −18.5381 −0.648564
\(818\) −0.443589 −0.0155097
\(819\) 12.2682 0.428684
\(820\) 5.01886 0.175266
\(821\) 51.5288 1.79837 0.899184 0.437571i \(-0.144161\pi\)
0.899184 + 0.437571i \(0.144161\pi\)
\(822\) 0.855686 0.0298455
\(823\) 3.01093 0.104954 0.0524772 0.998622i \(-0.483288\pi\)
0.0524772 + 0.998622i \(0.483288\pi\)
\(824\) 3.70633 0.129116
\(825\) 0 0
\(826\) 10.5247 0.366201
\(827\) −17.5581 −0.610555 −0.305277 0.952263i \(-0.598749\pi\)
−0.305277 + 0.952263i \(0.598749\pi\)
\(828\) −6.31527 −0.219471
\(829\) −54.9773 −1.90944 −0.954720 0.297507i \(-0.903845\pi\)
−0.954720 + 0.297507i \(0.903845\pi\)
\(830\) −7.40344 −0.256977
\(831\) 28.4792 0.987932
\(832\) 7.06945 0.245089
\(833\) 3.44477 0.119354
\(834\) −19.8346 −0.686816
\(835\) 3.64082 0.125996
\(836\) 0 0
\(837\) −24.0776 −0.832242
\(838\) −18.6545 −0.644408
\(839\) 8.14300 0.281127 0.140564 0.990072i \(-0.455109\pi\)
0.140564 + 0.990072i \(0.455109\pi\)
\(840\) −1.12455 −0.0388008
\(841\) 46.5679 1.60579
\(842\) 12.2716 0.422907
\(843\) −31.2378 −1.07589
\(844\) 7.42973 0.255742
\(845\) −36.9772 −1.27205
\(846\) −17.0549 −0.586358
\(847\) 0 0
\(848\) 11.3718 0.390510
\(849\) 21.9460 0.753185
\(850\) 3.44477 0.118155
\(851\) −2.84062 −0.0973753
\(852\) 7.66245 0.262511
\(853\) −18.4464 −0.631593 −0.315796 0.948827i \(-0.602272\pi\)
−0.315796 + 0.948827i \(0.602272\pi\)
\(854\) −0.456235 −0.0156120
\(855\) 5.96303 0.203931
\(856\) −2.85107 −0.0974476
\(857\) 36.0108 1.23011 0.615054 0.788485i \(-0.289134\pi\)
0.615054 + 0.788485i \(0.289134\pi\)
\(858\) 0 0
\(859\) 25.6553 0.875349 0.437674 0.899134i \(-0.355802\pi\)
0.437674 + 0.899134i \(0.355802\pi\)
\(860\) 5.39500 0.183968
\(861\) −5.64398 −0.192346
\(862\) 21.9795 0.748626
\(863\) 15.4519 0.525990 0.262995 0.964797i \(-0.415290\pi\)
0.262995 + 0.964797i \(0.415290\pi\)
\(864\) 5.32519 0.181167
\(865\) 3.17934 0.108101
\(866\) 22.2345 0.755561
\(867\) 5.77298 0.196061
\(868\) 4.52145 0.153468
\(869\) 0 0
\(870\) −9.77573 −0.331428
\(871\) 74.1848 2.51366
\(872\) 9.80121 0.331911
\(873\) −5.01511 −0.169736
\(874\) 12.5046 0.422976
\(875\) 1.00000 0.0338062
\(876\) 16.9978 0.574302
\(877\) 27.9195 0.942774 0.471387 0.881926i \(-0.343753\pi\)
0.471387 + 0.881926i \(0.343753\pi\)
\(878\) −11.0302 −0.372252
\(879\) 15.0134 0.506388
\(880\) 0 0
\(881\) 23.1798 0.780946 0.390473 0.920614i \(-0.372311\pi\)
0.390473 + 0.920614i \(0.372311\pi\)
\(882\) −1.73538 −0.0584332
\(883\) 24.4908 0.824181 0.412090 0.911143i \(-0.364799\pi\)
0.412090 + 0.911143i \(0.364799\pi\)
\(884\) 24.3526 0.819068
\(885\) −11.8356 −0.397849
\(886\) 30.8702 1.03711
\(887\) −22.6184 −0.759453 −0.379726 0.925099i \(-0.623982\pi\)
−0.379726 + 0.925099i \(0.623982\pi\)
\(888\) 0.877800 0.0294570
\(889\) −21.2610 −0.713070
\(890\) 4.97143 0.166643
\(891\) 0 0
\(892\) 1.51167 0.0506145
\(893\) 33.7697 1.13006
\(894\) −0.875106 −0.0292679
\(895\) −4.48137 −0.149796
\(896\) −1.00000 −0.0334077
\(897\) −28.9311 −0.965980
\(898\) −13.9952 −0.467027
\(899\) 39.3048 1.31089
\(900\) −1.73538 −0.0578459
\(901\) 39.1733 1.30505
\(902\) 0 0
\(903\) −6.06697 −0.201896
\(904\) 10.8032 0.359309
\(905\) 9.02937 0.300146
\(906\) −15.0863 −0.501209
\(907\) 29.1264 0.967127 0.483563 0.875309i \(-0.339342\pi\)
0.483563 + 0.875309i \(0.339342\pi\)
\(908\) 13.8558 0.459820
\(909\) −21.2177 −0.703748
\(910\) 7.06945 0.234350
\(911\) −0.781467 −0.0258912 −0.0129456 0.999916i \(-0.504121\pi\)
−0.0129456 + 0.999916i \(0.504121\pi\)
\(912\) −3.86415 −0.127955
\(913\) 0 0
\(914\) −7.06533 −0.233700
\(915\) 0.513061 0.0169613
\(916\) 20.5463 0.678870
\(917\) −10.2637 −0.338938
\(918\) 18.3440 0.605444
\(919\) 0.902346 0.0297656 0.0148828 0.999889i \(-0.495262\pi\)
0.0148828 + 0.999889i \(0.495262\pi\)
\(920\) −3.63913 −0.119979
\(921\) −28.9965 −0.955466
\(922\) 10.6011 0.349130
\(923\) −48.1696 −1.58552
\(924\) 0 0
\(925\) −0.780576 −0.0256652
\(926\) 4.59137 0.150882
\(927\) −6.43189 −0.211251
\(928\) −8.69298 −0.285361
\(929\) 56.9745 1.86927 0.934637 0.355603i \(-0.115724\pi\)
0.934637 + 0.355603i \(0.115724\pi\)
\(930\) −5.08461 −0.166731
\(931\) 3.43616 0.112615
\(932\) 16.2345 0.531780
\(933\) −5.75254 −0.188330
\(934\) −30.9287 −1.01202
\(935\) 0 0
\(936\) −12.2682 −0.400998
\(937\) 35.7088 1.16656 0.583278 0.812273i \(-0.301770\pi\)
0.583278 + 0.812273i \(0.301770\pi\)
\(938\) −10.4937 −0.342632
\(939\) −18.6193 −0.607619
\(940\) −9.82775 −0.320546
\(941\) 9.23591 0.301082 0.150541 0.988604i \(-0.451898\pi\)
0.150541 + 0.988604i \(0.451898\pi\)
\(942\) 27.0299 0.880683
\(943\) −18.2643 −0.594768
\(944\) −10.5247 −0.342550
\(945\) 5.32519 0.173228
\(946\) 0 0
\(947\) −26.7339 −0.868736 −0.434368 0.900735i \(-0.643028\pi\)
−0.434368 + 0.900735i \(0.643028\pi\)
\(948\) 18.2591 0.593029
\(949\) −106.856 −3.46869
\(950\) 3.43616 0.111484
\(951\) −17.2619 −0.559757
\(952\) −3.44477 −0.111646
\(953\) −41.1619 −1.33337 −0.666683 0.745341i \(-0.732287\pi\)
−0.666683 + 0.745341i \(0.732287\pi\)
\(954\) −19.7344 −0.638924
\(955\) −2.90870 −0.0941234
\(956\) −0.392921 −0.0127080
\(957\) 0 0
\(958\) 3.01362 0.0973658
\(959\) 0.760911 0.0245711
\(960\) 1.12455 0.0362948
\(961\) −10.5565 −0.340533
\(962\) −5.51825 −0.177915
\(963\) 4.94768 0.159437
\(964\) −3.32681 −0.107149
\(965\) 7.77320 0.250228
\(966\) 4.09241 0.131671
\(967\) 31.2558 1.00512 0.502559 0.864543i \(-0.332392\pi\)
0.502559 + 0.864543i \(0.332392\pi\)
\(968\) 0 0
\(969\) −13.3111 −0.427614
\(970\) −2.88993 −0.0927899
\(971\) −10.2903 −0.330232 −0.165116 0.986274i \(-0.552800\pi\)
−0.165116 + 0.986274i \(0.552800\pi\)
\(972\) −15.0958 −0.484198
\(973\) −17.6377 −0.565440
\(974\) 3.00551 0.0963028
\(975\) −7.94999 −0.254603
\(976\) 0.456235 0.0146037
\(977\) −44.9080 −1.43673 −0.718367 0.695665i \(-0.755110\pi\)
−0.718367 + 0.695665i \(0.755110\pi\)
\(978\) −1.01816 −0.0325572
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −17.0088 −0.543049
\(982\) 34.6723 1.10644
\(983\) 11.9639 0.381588 0.190794 0.981630i \(-0.438894\pi\)
0.190794 + 0.981630i \(0.438894\pi\)
\(984\) 5.64398 0.179924
\(985\) −26.2257 −0.835622
\(986\) −29.9453 −0.953653
\(987\) 11.0518 0.351784
\(988\) 24.2918 0.772824
\(989\) −19.6331 −0.624297
\(990\) 0 0
\(991\) 25.7714 0.818656 0.409328 0.912387i \(-0.365763\pi\)
0.409328 + 0.912387i \(0.365763\pi\)
\(992\) −4.52145 −0.143556
\(993\) 7.41252 0.235229
\(994\) 6.81376 0.216119
\(995\) −5.64963 −0.179105
\(996\) −8.32557 −0.263806
\(997\) 9.16694 0.290320 0.145160 0.989408i \(-0.453630\pi\)
0.145160 + 0.989408i \(0.453630\pi\)
\(998\) −5.25216 −0.166254
\(999\) −4.15672 −0.131513
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.dh.1.3 8
11.5 even 5 770.2.n.k.421.2 16
11.9 even 5 770.2.n.k.631.2 yes 16
11.10 odd 2 8470.2.a.dg.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.421.2 16 11.5 even 5
770.2.n.k.631.2 yes 16 11.9 even 5
8470.2.a.dg.1.3 8 11.10 odd 2
8470.2.a.dh.1.3 8 1.1 even 1 trivial