Properties

Label 8470.2.a.dh.1.2
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 1.2
Root \(-2.53932\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.53932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.53932 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.44815 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.53932 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.53932 q^{6} -1.00000 q^{7} +1.00000 q^{8} +3.44815 q^{9} -1.00000 q^{10} -2.53932 q^{12} -4.11459 q^{13} -1.00000 q^{14} +2.53932 q^{15} +1.00000 q^{16} -6.84829 q^{17} +3.44815 q^{18} -6.62155 q^{19} -1.00000 q^{20} +2.53932 q^{21} -3.13877 q^{23} -2.53932 q^{24} +1.00000 q^{25} -4.11459 q^{26} -1.13800 q^{27} -1.00000 q^{28} +5.49561 q^{29} +2.53932 q^{30} -5.57409 q^{31} +1.00000 q^{32} -6.84829 q^{34} +1.00000 q^{35} +3.44815 q^{36} -7.78523 q^{37} -6.62155 q^{38} +10.4483 q^{39} -1.00000 q^{40} +2.90755 q^{41} +2.53932 q^{42} +3.23479 q^{43} -3.44815 q^{45} -3.13877 q^{46} -1.49988 q^{47} -2.53932 q^{48} +1.00000 q^{49} +1.00000 q^{50} +17.3900 q^{51} -4.11459 q^{52} -11.1313 q^{53} -1.13800 q^{54} -1.00000 q^{56} +16.8142 q^{57} +5.49561 q^{58} -7.42428 q^{59} +2.53932 q^{60} -2.28166 q^{61} -5.57409 q^{62} -3.44815 q^{63} +1.00000 q^{64} +4.11459 q^{65} +14.1969 q^{67} -6.84829 q^{68} +7.97035 q^{69} +1.00000 q^{70} +0.0608812 q^{71} +3.44815 q^{72} -3.91284 q^{73} -7.78523 q^{74} -2.53932 q^{75} -6.62155 q^{76} +10.4483 q^{78} -4.23983 q^{79} -1.00000 q^{80} -7.45471 q^{81} +2.90755 q^{82} +10.2468 q^{83} +2.53932 q^{84} +6.84829 q^{85} +3.23479 q^{86} -13.9551 q^{87} -6.54394 q^{89} -3.44815 q^{90} +4.11459 q^{91} -3.13877 q^{92} +14.1544 q^{93} -1.49988 q^{94} +6.62155 q^{95} -2.53932 q^{96} -0.691019 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - q^{13} - 8q^{14} + 8q^{16} + 6q^{17} + 8q^{18} + 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} - 8q^{28} + 3q^{29} - 8q^{31} + 8q^{32} + 6q^{34} + 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} + 35q^{39} - 8q^{40} + 11q^{41} - 5q^{43} - 8q^{45} + 10q^{46} - 15q^{47} + 8q^{49} + 8q^{50} - 6q^{51} - q^{52} - 16q^{53} - 8q^{56} + 38q^{57} + 3q^{58} - 9q^{59} + 32q^{61} - 8q^{62} - 8q^{63} + 8q^{64} + q^{65} + 33q^{67} + 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} + 8q^{72} - 34q^{73} - 6q^{74} + 5q^{76} + 35q^{78} + 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} + 50q^{83} - 6q^{85} - 5q^{86} - 12q^{87} + q^{89} - 8q^{90} + q^{91} + 10q^{92} + 26q^{93} - 15q^{94} - 5q^{95} - 4q^{97} + 8q^{98} + O(q^{100}) \)

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
\(3\) −2.53932 −1.46608 −0.733039 0.680187i \(-0.761899\pi\)
−0.733039 + 0.680187i \(0.761899\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.53932 −1.03667
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 3.44815 1.14938
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.53932 −0.733039
\(13\) −4.11459 −1.14118 −0.570591 0.821234i \(-0.693286\pi\)
−0.570591 + 0.821234i \(0.693286\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.53932 0.655650
\(16\) 1.00000 0.250000
\(17\) −6.84829 −1.66095 −0.830477 0.557052i \(-0.811932\pi\)
−0.830477 + 0.557052i \(0.811932\pi\)
\(18\) 3.44815 0.812737
\(19\) −6.62155 −1.51909 −0.759544 0.650456i \(-0.774578\pi\)
−0.759544 + 0.650456i \(0.774578\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.53932 0.554125
\(22\) 0 0
\(23\) −3.13877 −0.654479 −0.327240 0.944941i \(-0.606118\pi\)
−0.327240 + 0.944941i \(0.606118\pi\)
\(24\) −2.53932 −0.518337
\(25\) 1.00000 0.200000
\(26\) −4.11459 −0.806938
\(27\) −1.13800 −0.219007
\(28\) −1.00000 −0.188982
\(29\) 5.49561 1.02051 0.510255 0.860023i \(-0.329551\pi\)
0.510255 + 0.860023i \(0.329551\pi\)
\(30\) 2.53932 0.463614
\(31\) −5.57409 −1.00114 −0.500568 0.865697i \(-0.666875\pi\)
−0.500568 + 0.865697i \(0.666875\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.84829 −1.17447
\(35\) 1.00000 0.169031
\(36\) 3.44815 0.574692
\(37\) −7.78523 −1.27988 −0.639942 0.768424i \(-0.721041\pi\)
−0.639942 + 0.768424i \(0.721041\pi\)
\(38\) −6.62155 −1.07416
\(39\) 10.4483 1.67306
\(40\) −1.00000 −0.158114
\(41\) 2.90755 0.454084 0.227042 0.973885i \(-0.427095\pi\)
0.227042 + 0.973885i \(0.427095\pi\)
\(42\) 2.53932 0.391826
\(43\) 3.23479 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(44\) 0 0
\(45\) −3.44815 −0.514020
\(46\) −3.13877 −0.462787
\(47\) −1.49988 −0.218780 −0.109390 0.993999i \(-0.534890\pi\)
−0.109390 + 0.993999i \(0.534890\pi\)
\(48\) −2.53932 −0.366519
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 17.3900 2.43509
\(52\) −4.11459 −0.570591
\(53\) −11.1313 −1.52900 −0.764499 0.644625i \(-0.777013\pi\)
−0.764499 + 0.644625i \(0.777013\pi\)
\(54\) −1.13800 −0.154862
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 16.8142 2.22710
\(58\) 5.49561 0.721610
\(59\) −7.42428 −0.966560 −0.483280 0.875466i \(-0.660555\pi\)
−0.483280 + 0.875466i \(0.660555\pi\)
\(60\) 2.53932 0.327825
\(61\) −2.28166 −0.292137 −0.146069 0.989274i \(-0.546662\pi\)
−0.146069 + 0.989274i \(0.546662\pi\)
\(62\) −5.57409 −0.707910
\(63\) −3.44815 −0.434426
\(64\) 1.00000 0.125000
\(65\) 4.11459 0.510352
\(66\) 0 0
\(67\) 14.1969 1.73442 0.867212 0.497939i \(-0.165910\pi\)
0.867212 + 0.497939i \(0.165910\pi\)
\(68\) −6.84829 −0.830477
\(69\) 7.97035 0.959518
\(70\) 1.00000 0.119523
\(71\) 0.0608812 0.00722526 0.00361263 0.999993i \(-0.498850\pi\)
0.00361263 + 0.999993i \(0.498850\pi\)
\(72\) 3.44815 0.406368
\(73\) −3.91284 −0.457963 −0.228982 0.973431i \(-0.573540\pi\)
−0.228982 + 0.973431i \(0.573540\pi\)
\(74\) −7.78523 −0.905014
\(75\) −2.53932 −0.293215
\(76\) −6.62155 −0.759544
\(77\) 0 0
\(78\) 10.4483 1.18303
\(79\) −4.23983 −0.477018 −0.238509 0.971140i \(-0.576659\pi\)
−0.238509 + 0.971140i \(0.576659\pi\)
\(80\) −1.00000 −0.111803
\(81\) −7.45471 −0.828301
\(82\) 2.90755 0.321086
\(83\) 10.2468 1.12473 0.562364 0.826890i \(-0.309892\pi\)
0.562364 + 0.826890i \(0.309892\pi\)
\(84\) 2.53932 0.277063
\(85\) 6.84829 0.742802
\(86\) 3.23479 0.348817
\(87\) −13.9551 −1.49615
\(88\) 0 0
\(89\) −6.54394 −0.693656 −0.346828 0.937929i \(-0.612741\pi\)
−0.346828 + 0.937929i \(0.612741\pi\)
\(90\) −3.44815 −0.363467
\(91\) 4.11459 0.431326
\(92\) −3.13877 −0.327240
\(93\) 14.1544 1.46774
\(94\) −1.49988 −0.154701
\(95\) 6.62155 0.679357
\(96\) −2.53932 −0.259168
\(97\) −0.691019 −0.0701624 −0.0350812 0.999384i \(-0.511169\pi\)
−0.0350812 + 0.999384i \(0.511169\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 14.6190 1.45465 0.727324 0.686295i \(-0.240764\pi\)
0.727324 + 0.686295i \(0.240764\pi\)
\(102\) 17.3900 1.72187
\(103\) 1.07358 0.105783 0.0528914 0.998600i \(-0.483156\pi\)
0.0528914 + 0.998600i \(0.483156\pi\)
\(104\) −4.11459 −0.403469
\(105\) −2.53932 −0.247812
\(106\) −11.1313 −1.08116
\(107\) −18.8781 −1.82501 −0.912507 0.409062i \(-0.865856\pi\)
−0.912507 + 0.409062i \(0.865856\pi\)
\(108\) −1.13800 −0.109504
\(109\) 4.22315 0.404504 0.202252 0.979333i \(-0.435174\pi\)
0.202252 + 0.979333i \(0.435174\pi\)
\(110\) 0 0
\(111\) 19.7692 1.87641
\(112\) −1.00000 −0.0944911
\(113\) 7.36545 0.692883 0.346442 0.938072i \(-0.387390\pi\)
0.346442 + 0.938072i \(0.387390\pi\)
\(114\) 16.8142 1.57480
\(115\) 3.13877 0.292692
\(116\) 5.49561 0.510255
\(117\) −14.1877 −1.31166
\(118\) −7.42428 −0.683461
\(119\) 6.84829 0.627782
\(120\) 2.53932 0.231807
\(121\) 0 0
\(122\) −2.28166 −0.206572
\(123\) −7.38321 −0.665722
\(124\) −5.57409 −0.500568
\(125\) −1.00000 −0.0894427
\(126\) −3.44815 −0.307186
\(127\) −15.0031 −1.33131 −0.665656 0.746259i \(-0.731848\pi\)
−0.665656 + 0.746259i \(0.731848\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.21418 −0.723218
\(130\) 4.11459 0.360874
\(131\) −6.77296 −0.591757 −0.295878 0.955226i \(-0.595612\pi\)
−0.295878 + 0.955226i \(0.595612\pi\)
\(132\) 0 0
\(133\) 6.62155 0.574161
\(134\) 14.1969 1.22642
\(135\) 1.13800 0.0979431
\(136\) −6.84829 −0.587236
\(137\) −9.15100 −0.781822 −0.390911 0.920428i \(-0.627840\pi\)
−0.390911 + 0.920428i \(0.627840\pi\)
\(138\) 7.97035 0.678481
\(139\) 15.9131 1.34973 0.674865 0.737941i \(-0.264202\pi\)
0.674865 + 0.737941i \(0.264202\pi\)
\(140\) 1.00000 0.0845154
\(141\) 3.80868 0.320748
\(142\) 0.0608812 0.00510903
\(143\) 0 0
\(144\) 3.44815 0.287346
\(145\) −5.49561 −0.456386
\(146\) −3.91284 −0.323829
\(147\) −2.53932 −0.209440
\(148\) −7.78523 −0.639942
\(149\) 9.63068 0.788976 0.394488 0.918901i \(-0.370922\pi\)
0.394488 + 0.918901i \(0.370922\pi\)
\(150\) −2.53932 −0.207335
\(151\) 20.3543 1.65641 0.828205 0.560425i \(-0.189362\pi\)
0.828205 + 0.560425i \(0.189362\pi\)
\(152\) −6.62155 −0.537079
\(153\) −23.6139 −1.90907
\(154\) 0 0
\(155\) 5.57409 0.447721
\(156\) 10.4483 0.836531
\(157\) −6.01660 −0.480177 −0.240089 0.970751i \(-0.577177\pi\)
−0.240089 + 0.970751i \(0.577177\pi\)
\(158\) −4.23983 −0.337303
\(159\) 28.2659 2.24163
\(160\) −1.00000 −0.0790569
\(161\) 3.13877 0.247370
\(162\) −7.45471 −0.585698
\(163\) −8.61451 −0.674740 −0.337370 0.941372i \(-0.609537\pi\)
−0.337370 + 0.941372i \(0.609537\pi\)
\(164\) 2.90755 0.227042
\(165\) 0 0
\(166\) 10.2468 0.795303
\(167\) −9.01590 −0.697671 −0.348836 0.937184i \(-0.613423\pi\)
−0.348836 + 0.937184i \(0.613423\pi\)
\(168\) 2.53932 0.195913
\(169\) 3.92987 0.302298
\(170\) 6.84829 0.525240
\(171\) −22.8321 −1.74601
\(172\) 3.23479 0.246651
\(173\) −12.9388 −0.983715 −0.491858 0.870676i \(-0.663682\pi\)
−0.491858 + 0.870676i \(0.663682\pi\)
\(174\) −13.9551 −1.05794
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 18.8526 1.41705
\(178\) −6.54394 −0.490489
\(179\) 15.7602 1.17797 0.588987 0.808142i \(-0.299527\pi\)
0.588987 + 0.808142i \(0.299527\pi\)
\(180\) −3.44815 −0.257010
\(181\) −14.6094 −1.08591 −0.542953 0.839763i \(-0.682694\pi\)
−0.542953 + 0.839763i \(0.682694\pi\)
\(182\) 4.11459 0.304994
\(183\) 5.79388 0.428296
\(184\) −3.13877 −0.231393
\(185\) 7.78523 0.572381
\(186\) 14.1544 1.03785
\(187\) 0 0
\(188\) −1.49988 −0.109390
\(189\) 1.13800 0.0827770
\(190\) 6.62155 0.480378
\(191\) −17.9665 −1.30001 −0.650004 0.759931i \(-0.725233\pi\)
−0.650004 + 0.759931i \(0.725233\pi\)
\(192\) −2.53932 −0.183260
\(193\) 23.8180 1.71446 0.857229 0.514935i \(-0.172184\pi\)
0.857229 + 0.514935i \(0.172184\pi\)
\(194\) −0.691019 −0.0496123
\(195\) −10.4483 −0.748216
\(196\) 1.00000 0.0714286
\(197\) −1.85551 −0.132200 −0.0660998 0.997813i \(-0.521056\pi\)
−0.0660998 + 0.997813i \(0.521056\pi\)
\(198\) 0 0
\(199\) 21.8206 1.54682 0.773411 0.633905i \(-0.218549\pi\)
0.773411 + 0.633905i \(0.218549\pi\)
\(200\) 1.00000 0.0707107
\(201\) −36.0504 −2.54280
\(202\) 14.6190 1.02859
\(203\) −5.49561 −0.385717
\(204\) 17.3900 1.21754
\(205\) −2.90755 −0.203072
\(206\) 1.07358 0.0747997
\(207\) −10.8230 −0.752248
\(208\) −4.11459 −0.285296
\(209\) 0 0
\(210\) −2.53932 −0.175230
\(211\) −26.5722 −1.82930 −0.914651 0.404244i \(-0.867535\pi\)
−0.914651 + 0.404244i \(0.867535\pi\)
\(212\) −11.1313 −0.764499
\(213\) −0.154597 −0.0105928
\(214\) −18.8781 −1.29048
\(215\) −3.23479 −0.220611
\(216\) −1.13800 −0.0774308
\(217\) 5.57409 0.378394
\(218\) 4.22315 0.286028
\(219\) 9.93595 0.671409
\(220\) 0 0
\(221\) 28.1779 1.89545
\(222\) 19.7692 1.32682
\(223\) −9.89233 −0.662439 −0.331220 0.943554i \(-0.607460\pi\)
−0.331220 + 0.943554i \(0.607460\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 3.44815 0.229877
\(226\) 7.36545 0.489942
\(227\) 13.9299 0.924563 0.462282 0.886733i \(-0.347031\pi\)
0.462282 + 0.886733i \(0.347031\pi\)
\(228\) 16.8142 1.11355
\(229\) −10.1930 −0.673570 −0.336785 0.941581i \(-0.609340\pi\)
−0.336785 + 0.941581i \(0.609340\pi\)
\(230\) 3.13877 0.206965
\(231\) 0 0
\(232\) 5.49561 0.360805
\(233\) −13.2121 −0.865554 −0.432777 0.901501i \(-0.642466\pi\)
−0.432777 + 0.901501i \(0.642466\pi\)
\(234\) −14.1877 −0.927481
\(235\) 1.49988 0.0978414
\(236\) −7.42428 −0.483280
\(237\) 10.7663 0.699345
\(238\) 6.84829 0.443909
\(239\) 5.10128 0.329974 0.164987 0.986296i \(-0.447242\pi\)
0.164987 + 0.986296i \(0.447242\pi\)
\(240\) 2.53932 0.163912
\(241\) 24.0075 1.54646 0.773229 0.634127i \(-0.218640\pi\)
0.773229 + 0.634127i \(0.218640\pi\)
\(242\) 0 0
\(243\) 22.3439 1.43336
\(244\) −2.28166 −0.146069
\(245\) −1.00000 −0.0638877
\(246\) −7.38321 −0.470736
\(247\) 27.2450 1.73356
\(248\) −5.57409 −0.353955
\(249\) −26.0198 −1.64894
\(250\) −1.00000 −0.0632456
\(251\) 15.4967 0.978140 0.489070 0.872245i \(-0.337336\pi\)
0.489070 + 0.872245i \(0.337336\pi\)
\(252\) −3.44815 −0.217213
\(253\) 0 0
\(254\) −15.0031 −0.941380
\(255\) −17.3900 −1.08900
\(256\) 1.00000 0.0625000
\(257\) −4.12482 −0.257299 −0.128650 0.991690i \(-0.541064\pi\)
−0.128650 + 0.991690i \(0.541064\pi\)
\(258\) −8.21418 −0.511392
\(259\) 7.78523 0.483750
\(260\) 4.11459 0.255176
\(261\) 18.9497 1.17296
\(262\) −6.77296 −0.418435
\(263\) 0.308382 0.0190156 0.00950781 0.999955i \(-0.496974\pi\)
0.00950781 + 0.999955i \(0.496974\pi\)
\(264\) 0 0
\(265\) 11.1313 0.683788
\(266\) 6.62155 0.405993
\(267\) 16.6172 1.01695
\(268\) 14.1969 0.867212
\(269\) −17.1417 −1.04515 −0.522573 0.852594i \(-0.675028\pi\)
−0.522573 + 0.852594i \(0.675028\pi\)
\(270\) 1.13800 0.0692562
\(271\) 26.6258 1.61740 0.808701 0.588221i \(-0.200171\pi\)
0.808701 + 0.588221i \(0.200171\pi\)
\(272\) −6.84829 −0.415239
\(273\) −10.4483 −0.632358
\(274\) −9.15100 −0.552832
\(275\) 0 0
\(276\) 7.97035 0.479759
\(277\) −15.7129 −0.944098 −0.472049 0.881572i \(-0.656485\pi\)
−0.472049 + 0.881572i \(0.656485\pi\)
\(278\) 15.9131 0.954404
\(279\) −19.2203 −1.15069
\(280\) 1.00000 0.0597614
\(281\) −17.4342 −1.04004 −0.520020 0.854154i \(-0.674075\pi\)
−0.520020 + 0.854154i \(0.674075\pi\)
\(282\) 3.80868 0.226803
\(283\) 15.0195 0.892814 0.446407 0.894830i \(-0.352703\pi\)
0.446407 + 0.894830i \(0.352703\pi\)
\(284\) 0.0608812 0.00361263
\(285\) −16.8142 −0.995990
\(286\) 0 0
\(287\) −2.90755 −0.171627
\(288\) 3.44815 0.203184
\(289\) 29.8991 1.75877
\(290\) −5.49561 −0.322714
\(291\) 1.75472 0.102863
\(292\) −3.91284 −0.228982
\(293\) 7.50667 0.438544 0.219272 0.975664i \(-0.429632\pi\)
0.219272 + 0.975664i \(0.429632\pi\)
\(294\) −2.53932 −0.148096
\(295\) 7.42428 0.432259
\(296\) −7.78523 −0.452507
\(297\) 0 0
\(298\) 9.63068 0.557891
\(299\) 12.9148 0.746881
\(300\) −2.53932 −0.146608
\(301\) −3.23479 −0.186450
\(302\) 20.3543 1.17126
\(303\) −37.1224 −2.13263
\(304\) −6.62155 −0.379772
\(305\) 2.28166 0.130648
\(306\) −23.6139 −1.34992
\(307\) −5.08116 −0.289997 −0.144998 0.989432i \(-0.546318\pi\)
−0.144998 + 0.989432i \(0.546318\pi\)
\(308\) 0 0
\(309\) −2.72616 −0.155086
\(310\) 5.57409 0.316587
\(311\) 10.8436 0.614884 0.307442 0.951567i \(-0.400527\pi\)
0.307442 + 0.951567i \(0.400527\pi\)
\(312\) 10.4483 0.591517
\(313\) 33.4044 1.88813 0.944064 0.329764i \(-0.106969\pi\)
0.944064 + 0.329764i \(0.106969\pi\)
\(314\) −6.01660 −0.339537
\(315\) 3.44815 0.194281
\(316\) −4.23983 −0.238509
\(317\) −27.7007 −1.55583 −0.777914 0.628371i \(-0.783722\pi\)
−0.777914 + 0.628371i \(0.783722\pi\)
\(318\) 28.2659 1.58507
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 47.9375 2.67561
\(322\) 3.13877 0.174917
\(323\) 45.3463 2.52314
\(324\) −7.45471 −0.414151
\(325\) −4.11459 −0.228237
\(326\) −8.61451 −0.477113
\(327\) −10.7239 −0.593035
\(328\) 2.90755 0.160543
\(329\) 1.49988 0.0826911
\(330\) 0 0
\(331\) −28.8317 −1.58473 −0.792367 0.610045i \(-0.791152\pi\)
−0.792367 + 0.610045i \(0.791152\pi\)
\(332\) 10.2468 0.562364
\(333\) −26.8446 −1.47108
\(334\) −9.01590 −0.493328
\(335\) −14.1969 −0.775658
\(336\) 2.53932 0.138531
\(337\) −16.9730 −0.924577 −0.462288 0.886730i \(-0.652972\pi\)
−0.462288 + 0.886730i \(0.652972\pi\)
\(338\) 3.92987 0.213757
\(339\) −18.7032 −1.01582
\(340\) 6.84829 0.371401
\(341\) 0 0
\(342\) −22.8321 −1.23462
\(343\) −1.00000 −0.0539949
\(344\) 3.23479 0.174408
\(345\) −7.97035 −0.429109
\(346\) −12.9388 −0.695592
\(347\) 22.8463 1.22645 0.613227 0.789907i \(-0.289871\pi\)
0.613227 + 0.789907i \(0.289871\pi\)
\(348\) −13.9551 −0.748073
\(349\) −15.1421 −0.810540 −0.405270 0.914197i \(-0.632822\pi\)
−0.405270 + 0.914197i \(0.632822\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 4.68239 0.249927
\(352\) 0 0
\(353\) −9.01014 −0.479561 −0.239781 0.970827i \(-0.577076\pi\)
−0.239781 + 0.970827i \(0.577076\pi\)
\(354\) 18.8526 1.00201
\(355\) −0.0608812 −0.00323124
\(356\) −6.54394 −0.346828
\(357\) −17.3900 −0.920377
\(358\) 15.7602 0.832954
\(359\) 36.8929 1.94713 0.973566 0.228408i \(-0.0733519\pi\)
0.973566 + 0.228408i \(0.0733519\pi\)
\(360\) −3.44815 −0.181733
\(361\) 24.8450 1.30763
\(362\) −14.6094 −0.767852
\(363\) 0 0
\(364\) 4.11459 0.215663
\(365\) 3.91284 0.204807
\(366\) 5.79388 0.302851
\(367\) 34.5081 1.80131 0.900654 0.434537i \(-0.143088\pi\)
0.900654 + 0.434537i \(0.143088\pi\)
\(368\) −3.13877 −0.163620
\(369\) 10.0257 0.521916
\(370\) 7.78523 0.404735
\(371\) 11.1313 0.577907
\(372\) 14.1544 0.733871
\(373\) 2.31601 0.119918 0.0599592 0.998201i \(-0.480903\pi\)
0.0599592 + 0.998201i \(0.480903\pi\)
\(374\) 0 0
\(375\) 2.53932 0.131130
\(376\) −1.49988 −0.0773504
\(377\) −22.6122 −1.16459
\(378\) 1.13800 0.0585322
\(379\) −7.92393 −0.407025 −0.203513 0.979072i \(-0.565236\pi\)
−0.203513 + 0.979072i \(0.565236\pi\)
\(380\) 6.62155 0.339678
\(381\) 38.0977 1.95181
\(382\) −17.9665 −0.919244
\(383\) 19.2045 0.981304 0.490652 0.871356i \(-0.336759\pi\)
0.490652 + 0.871356i \(0.336759\pi\)
\(384\) −2.53932 −0.129584
\(385\) 0 0
\(386\) 23.8180 1.21231
\(387\) 11.1540 0.566992
\(388\) −0.691019 −0.0350812
\(389\) −7.65932 −0.388343 −0.194172 0.980968i \(-0.562202\pi\)
−0.194172 + 0.980968i \(0.562202\pi\)
\(390\) −10.4483 −0.529069
\(391\) 21.4952 1.08706
\(392\) 1.00000 0.0505076
\(393\) 17.1987 0.867561
\(394\) −1.85551 −0.0934793
\(395\) 4.23983 0.213329
\(396\) 0 0
\(397\) −20.0239 −1.00497 −0.502486 0.864585i \(-0.667581\pi\)
−0.502486 + 0.864585i \(0.667581\pi\)
\(398\) 21.8206 1.09377
\(399\) −16.8142 −0.841765
\(400\) 1.00000 0.0500000
\(401\) 20.7562 1.03651 0.518257 0.855225i \(-0.326581\pi\)
0.518257 + 0.855225i \(0.326581\pi\)
\(402\) −36.0504 −1.79803
\(403\) 22.9351 1.14248
\(404\) 14.6190 0.727324
\(405\) 7.45471 0.370428
\(406\) −5.49561 −0.272743
\(407\) 0 0
\(408\) 17.3900 0.860934
\(409\) 32.8553 1.62459 0.812295 0.583247i \(-0.198218\pi\)
0.812295 + 0.583247i \(0.198218\pi\)
\(410\) −2.90755 −0.143594
\(411\) 23.2373 1.14621
\(412\) 1.07358 0.0528914
\(413\) 7.42428 0.365325
\(414\) −10.8230 −0.531919
\(415\) −10.2468 −0.502994
\(416\) −4.11459 −0.201734
\(417\) −40.4084 −1.97881
\(418\) 0 0
\(419\) −16.3625 −0.799361 −0.399680 0.916655i \(-0.630879\pi\)
−0.399680 + 0.916655i \(0.630879\pi\)
\(420\) −2.53932 −0.123906
\(421\) 26.0472 1.26946 0.634731 0.772733i \(-0.281111\pi\)
0.634731 + 0.772733i \(0.281111\pi\)
\(422\) −26.5722 −1.29351
\(423\) −5.17181 −0.251462
\(424\) −11.1313 −0.540582
\(425\) −6.84829 −0.332191
\(426\) −0.154597 −0.00749024
\(427\) 2.28166 0.110417
\(428\) −18.8781 −0.912507
\(429\) 0 0
\(430\) −3.23479 −0.155996
\(431\) 18.2575 0.879433 0.439716 0.898137i \(-0.355079\pi\)
0.439716 + 0.898137i \(0.355079\pi\)
\(432\) −1.13800 −0.0547518
\(433\) 32.6096 1.56712 0.783560 0.621316i \(-0.213402\pi\)
0.783560 + 0.621316i \(0.213402\pi\)
\(434\) 5.57409 0.267565
\(435\) 13.9551 0.669097
\(436\) 4.22315 0.202252
\(437\) 20.7836 0.994212
\(438\) 9.93595 0.474758
\(439\) −29.2540 −1.39622 −0.698108 0.715993i \(-0.745974\pi\)
−0.698108 + 0.715993i \(0.745974\pi\)
\(440\) 0 0
\(441\) 3.44815 0.164198
\(442\) 28.1779 1.34029
\(443\) 11.4091 0.542063 0.271032 0.962570i \(-0.412635\pi\)
0.271032 + 0.962570i \(0.412635\pi\)
\(444\) 19.7692 0.938204
\(445\) 6.54394 0.310212
\(446\) −9.89233 −0.468415
\(447\) −24.4554 −1.15670
\(448\) −1.00000 −0.0472456
\(449\) 13.6315 0.643312 0.321656 0.946857i \(-0.395760\pi\)
0.321656 + 0.946857i \(0.395760\pi\)
\(450\) 3.44815 0.162547
\(451\) 0 0
\(452\) 7.36545 0.346442
\(453\) −51.6861 −2.42843
\(454\) 13.9299 0.653765
\(455\) −4.11459 −0.192895
\(456\) 16.8142 0.787399
\(457\) −6.66357 −0.311708 −0.155854 0.987780i \(-0.549813\pi\)
−0.155854 + 0.987780i \(0.549813\pi\)
\(458\) −10.1930 −0.476286
\(459\) 7.79333 0.363761
\(460\) 3.13877 0.146346
\(461\) 18.1977 0.847553 0.423776 0.905767i \(-0.360704\pi\)
0.423776 + 0.905767i \(0.360704\pi\)
\(462\) 0 0
\(463\) 11.3971 0.529666 0.264833 0.964294i \(-0.414683\pi\)
0.264833 + 0.964294i \(0.414683\pi\)
\(464\) 5.49561 0.255128
\(465\) −14.1544 −0.656394
\(466\) −13.2121 −0.612039
\(467\) 22.9793 1.06336 0.531679 0.846946i \(-0.321561\pi\)
0.531679 + 0.846946i \(0.321561\pi\)
\(468\) −14.1877 −0.655828
\(469\) −14.1969 −0.655551
\(470\) 1.49988 0.0691843
\(471\) 15.2781 0.703977
\(472\) −7.42428 −0.341730
\(473\) 0 0
\(474\) 10.7663 0.494512
\(475\) −6.62155 −0.303818
\(476\) 6.84829 0.313891
\(477\) −38.3823 −1.75740
\(478\) 5.10128 0.233327
\(479\) 1.36694 0.0624573 0.0312286 0.999512i \(-0.490058\pi\)
0.0312286 + 0.999512i \(0.490058\pi\)
\(480\) 2.53932 0.115904
\(481\) 32.0330 1.46058
\(482\) 24.0075 1.09351
\(483\) −7.97035 −0.362664
\(484\) 0 0
\(485\) 0.691019 0.0313776
\(486\) 22.3439 1.01354
\(487\) −12.2098 −0.553277 −0.276638 0.960974i \(-0.589220\pi\)
−0.276638 + 0.960974i \(0.589220\pi\)
\(488\) −2.28166 −0.103286
\(489\) 21.8750 0.989221
\(490\) −1.00000 −0.0451754
\(491\) 39.8546 1.79861 0.899307 0.437318i \(-0.144072\pi\)
0.899307 + 0.437318i \(0.144072\pi\)
\(492\) −7.38321 −0.332861
\(493\) −37.6356 −1.69502
\(494\) 27.2450 1.22581
\(495\) 0 0
\(496\) −5.57409 −0.250284
\(497\) −0.0608812 −0.00273089
\(498\) −26.0198 −1.16598
\(499\) 36.7505 1.64518 0.822589 0.568636i \(-0.192529\pi\)
0.822589 + 0.568636i \(0.192529\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 22.8943 1.02284
\(502\) 15.4967 0.691650
\(503\) 11.9782 0.534081 0.267041 0.963685i \(-0.413954\pi\)
0.267041 + 0.963685i \(0.413954\pi\)
\(504\) −3.44815 −0.153593
\(505\) −14.6190 −0.650538
\(506\) 0 0
\(507\) −9.97920 −0.443192
\(508\) −15.0031 −0.665656
\(509\) −23.6608 −1.04875 −0.524374 0.851488i \(-0.675701\pi\)
−0.524374 + 0.851488i \(0.675701\pi\)
\(510\) −17.3900 −0.770043
\(511\) 3.91284 0.173094
\(512\) 1.00000 0.0441942
\(513\) 7.53530 0.332691
\(514\) −4.12482 −0.181938
\(515\) −1.07358 −0.0473075
\(516\) −8.21418 −0.361609
\(517\) 0 0
\(518\) 7.78523 0.342063
\(519\) 32.8556 1.44220
\(520\) 4.11459 0.180437
\(521\) −9.40537 −0.412057 −0.206028 0.978546i \(-0.566054\pi\)
−0.206028 + 0.978546i \(0.566054\pi\)
\(522\) 18.9497 0.829406
\(523\) −28.4936 −1.24594 −0.622968 0.782247i \(-0.714074\pi\)
−0.622968 + 0.782247i \(0.714074\pi\)
\(524\) −6.77296 −0.295878
\(525\) 2.53932 0.110825
\(526\) 0.308382 0.0134461
\(527\) 38.1730 1.66284
\(528\) 0 0
\(529\) −13.1481 −0.571657
\(530\) 11.1313 0.483511
\(531\) −25.6000 −1.11095
\(532\) 6.62155 0.287081
\(533\) −11.9634 −0.518192
\(534\) 16.6172 0.719095
\(535\) 18.8781 0.816171
\(536\) 14.1969 0.613211
\(537\) −40.0203 −1.72700
\(538\) −17.1417 −0.739030
\(539\) 0 0
\(540\) 1.13800 0.0489715
\(541\) −2.39709 −0.103059 −0.0515294 0.998671i \(-0.516410\pi\)
−0.0515294 + 0.998671i \(0.516410\pi\)
\(542\) 26.6258 1.14368
\(543\) 37.0979 1.59202
\(544\) −6.84829 −0.293618
\(545\) −4.22315 −0.180900
\(546\) −10.4483 −0.447145
\(547\) 13.0332 0.557260 0.278630 0.960399i \(-0.410120\pi\)
0.278630 + 0.960399i \(0.410120\pi\)
\(548\) −9.15100 −0.390911
\(549\) −7.86752 −0.335777
\(550\) 0 0
\(551\) −36.3895 −1.55024
\(552\) 7.97035 0.339241
\(553\) 4.23983 0.180296
\(554\) −15.7129 −0.667578
\(555\) −19.7692 −0.839155
\(556\) 15.9131 0.674865
\(557\) 20.1348 0.853138 0.426569 0.904455i \(-0.359722\pi\)
0.426569 + 0.904455i \(0.359722\pi\)
\(558\) −19.2203 −0.813660
\(559\) −13.3099 −0.562947
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −17.4342 −0.735419
\(563\) 10.8907 0.458988 0.229494 0.973310i \(-0.426293\pi\)
0.229494 + 0.973310i \(0.426293\pi\)
\(564\) 3.80868 0.160374
\(565\) −7.36545 −0.309867
\(566\) 15.0195 0.631315
\(567\) 7.45471 0.313069
\(568\) 0.0608812 0.00255452
\(569\) −22.5186 −0.944027 −0.472014 0.881591i \(-0.656473\pi\)
−0.472014 + 0.881591i \(0.656473\pi\)
\(570\) −16.8142 −0.704271
\(571\) 14.1198 0.590893 0.295447 0.955359i \(-0.404532\pi\)
0.295447 + 0.955359i \(0.404532\pi\)
\(572\) 0 0
\(573\) 45.6226 1.90591
\(574\) −2.90755 −0.121359
\(575\) −3.13877 −0.130896
\(576\) 3.44815 0.143673
\(577\) −12.5952 −0.524344 −0.262172 0.965021i \(-0.584439\pi\)
−0.262172 + 0.965021i \(0.584439\pi\)
\(578\) 29.8991 1.24364
\(579\) −60.4816 −2.51353
\(580\) −5.49561 −0.228193
\(581\) −10.2468 −0.425107
\(582\) 1.75472 0.0727355
\(583\) 0 0
\(584\) −3.91284 −0.161914
\(585\) 14.1877 0.586590
\(586\) 7.50667 0.310098
\(587\) −15.4018 −0.635701 −0.317851 0.948141i \(-0.602961\pi\)
−0.317851 + 0.948141i \(0.602961\pi\)
\(588\) −2.53932 −0.104720
\(589\) 36.9091 1.52081
\(590\) 7.42428 0.305653
\(591\) 4.71174 0.193815
\(592\) −7.78523 −0.319971
\(593\) 27.0005 1.10878 0.554389 0.832258i \(-0.312952\pi\)
0.554389 + 0.832258i \(0.312952\pi\)
\(594\) 0 0
\(595\) −6.84829 −0.280753
\(596\) 9.63068 0.394488
\(597\) −55.4095 −2.26776
\(598\) 12.9148 0.528124
\(599\) 38.5299 1.57429 0.787145 0.616768i \(-0.211558\pi\)
0.787145 + 0.616768i \(0.211558\pi\)
\(600\) −2.53932 −0.103667
\(601\) 15.9195 0.649371 0.324686 0.945822i \(-0.394742\pi\)
0.324686 + 0.945822i \(0.394742\pi\)
\(602\) −3.23479 −0.131840
\(603\) 48.9529 1.99352
\(604\) 20.3543 0.828205
\(605\) 0 0
\(606\) −37.1224 −1.50799
\(607\) 20.2644 0.822509 0.411254 0.911521i \(-0.365091\pi\)
0.411254 + 0.911521i \(0.365091\pi\)
\(608\) −6.62155 −0.268539
\(609\) 13.9551 0.565490
\(610\) 2.28166 0.0923819
\(611\) 6.17139 0.249668
\(612\) −23.6139 −0.954537
\(613\) −28.0631 −1.13346 −0.566728 0.823905i \(-0.691791\pi\)
−0.566728 + 0.823905i \(0.691791\pi\)
\(614\) −5.08116 −0.205059
\(615\) 7.38321 0.297720
\(616\) 0 0
\(617\) 38.0285 1.53097 0.765484 0.643455i \(-0.222500\pi\)
0.765484 + 0.643455i \(0.222500\pi\)
\(618\) −2.72616 −0.109662
\(619\) −34.7958 −1.39856 −0.699281 0.714847i \(-0.746496\pi\)
−0.699281 + 0.714847i \(0.746496\pi\)
\(620\) 5.57409 0.223861
\(621\) 3.57191 0.143336
\(622\) 10.8436 0.434788
\(623\) 6.54394 0.262177
\(624\) 10.4483 0.418266
\(625\) 1.00000 0.0400000
\(626\) 33.4044 1.33511
\(627\) 0 0
\(628\) −6.01660 −0.240089
\(629\) 53.3155 2.12583
\(630\) 3.44815 0.137378
\(631\) −29.7828 −1.18564 −0.592818 0.805337i \(-0.701985\pi\)
−0.592818 + 0.805337i \(0.701985\pi\)
\(632\) −4.23983 −0.168651
\(633\) 67.4752 2.68190
\(634\) −27.7007 −1.10014
\(635\) 15.0031 0.595381
\(636\) 28.2659 1.12081
\(637\) −4.11459 −0.163026
\(638\) 0 0
\(639\) 0.209927 0.00830460
\(640\) −1.00000 −0.0395285
\(641\) 0.579841 0.0229023 0.0114512 0.999934i \(-0.496355\pi\)
0.0114512 + 0.999934i \(0.496355\pi\)
\(642\) 47.9375 1.89194
\(643\) 47.4766 1.87229 0.936147 0.351609i \(-0.114365\pi\)
0.936147 + 0.351609i \(0.114365\pi\)
\(644\) 3.13877 0.123685
\(645\) 8.21418 0.323433
\(646\) 45.3463 1.78413
\(647\) −41.5395 −1.63309 −0.816543 0.577284i \(-0.804112\pi\)
−0.816543 + 0.577284i \(0.804112\pi\)
\(648\) −7.45471 −0.292849
\(649\) 0 0
\(650\) −4.11459 −0.161388
\(651\) −14.1544 −0.554755
\(652\) −8.61451 −0.337370
\(653\) −26.4931 −1.03676 −0.518378 0.855152i \(-0.673464\pi\)
−0.518378 + 0.855152i \(0.673464\pi\)
\(654\) −10.7239 −0.419339
\(655\) 6.77296 0.264642
\(656\) 2.90755 0.113521
\(657\) −13.4921 −0.526375
\(658\) 1.49988 0.0584714
\(659\) −46.8092 −1.82343 −0.911713 0.410827i \(-0.865240\pi\)
−0.911713 + 0.410827i \(0.865240\pi\)
\(660\) 0 0
\(661\) −35.7060 −1.38880 −0.694401 0.719588i \(-0.744331\pi\)
−0.694401 + 0.719588i \(0.744331\pi\)
\(662\) −28.8317 −1.12058
\(663\) −71.5528 −2.77888
\(664\) 10.2468 0.397652
\(665\) −6.62155 −0.256773
\(666\) −26.8446 −1.04021
\(667\) −17.2495 −0.667903
\(668\) −9.01590 −0.348836
\(669\) 25.1198 0.971188
\(670\) −14.1969 −0.548473
\(671\) 0 0
\(672\) 2.53932 0.0979564
\(673\) 15.6243 0.602272 0.301136 0.953581i \(-0.402634\pi\)
0.301136 + 0.953581i \(0.402634\pi\)
\(674\) −16.9730 −0.653775
\(675\) −1.13800 −0.0438015
\(676\) 3.92987 0.151149
\(677\) 11.3333 0.435574 0.217787 0.975996i \(-0.430116\pi\)
0.217787 + 0.975996i \(0.430116\pi\)
\(678\) −18.7032 −0.718294
\(679\) 0.691019 0.0265189
\(680\) 6.84829 0.262620
\(681\) −35.3726 −1.35548
\(682\) 0 0
\(683\) −12.5315 −0.479504 −0.239752 0.970834i \(-0.577066\pi\)
−0.239752 + 0.970834i \(0.577066\pi\)
\(684\) −22.8321 −0.873007
\(685\) 9.15100 0.349642
\(686\) −1.00000 −0.0381802
\(687\) 25.8832 0.987506
\(688\) 3.23479 0.123325
\(689\) 45.8006 1.74487
\(690\) −7.97035 −0.303426
\(691\) 3.41506 0.129915 0.0649574 0.997888i \(-0.479309\pi\)
0.0649574 + 0.997888i \(0.479309\pi\)
\(692\) −12.9388 −0.491858
\(693\) 0 0
\(694\) 22.8463 0.867234
\(695\) −15.9131 −0.603618
\(696\) −13.9551 −0.528968
\(697\) −19.9118 −0.754212
\(698\) −15.1421 −0.573138
\(699\) 33.5498 1.26897
\(700\) −1.00000 −0.0377964
\(701\) 35.1062 1.32594 0.662971 0.748645i \(-0.269295\pi\)
0.662971 + 0.748645i \(0.269295\pi\)
\(702\) 4.68239 0.176725
\(703\) 51.5503 1.94426
\(704\) 0 0
\(705\) −3.80868 −0.143443
\(706\) −9.01014 −0.339101
\(707\) −14.6190 −0.549805
\(708\) 18.8526 0.708526
\(709\) −28.9326 −1.08659 −0.543293 0.839543i \(-0.682823\pi\)
−0.543293 + 0.839543i \(0.682823\pi\)
\(710\) −0.0608812 −0.00228483
\(711\) −14.6196 −0.548276
\(712\) −6.54394 −0.245244
\(713\) 17.4958 0.655223
\(714\) −17.3900 −0.650805
\(715\) 0 0
\(716\) 15.7602 0.588987
\(717\) −12.9538 −0.483768
\(718\) 36.8929 1.37683
\(719\) −7.06973 −0.263656 −0.131828 0.991273i \(-0.542085\pi\)
−0.131828 + 0.991273i \(0.542085\pi\)
\(720\) −3.44815 −0.128505
\(721\) −1.07358 −0.0399821
\(722\) 24.8450 0.924634
\(723\) −60.9627 −2.26723
\(724\) −14.6094 −0.542953
\(725\) 5.49561 0.204102
\(726\) 0 0
\(727\) −28.5131 −1.05749 −0.528747 0.848779i \(-0.677338\pi\)
−0.528747 + 0.848779i \(0.677338\pi\)
\(728\) 4.11459 0.152497
\(729\) −34.3742 −1.27312
\(730\) 3.91284 0.144821
\(731\) −22.1528 −0.819351
\(732\) 5.79388 0.214148
\(733\) 40.1218 1.48193 0.740967 0.671542i \(-0.234368\pi\)
0.740967 + 0.671542i \(0.234368\pi\)
\(734\) 34.5081 1.27372
\(735\) 2.53932 0.0936643
\(736\) −3.13877 −0.115697
\(737\) 0 0
\(738\) 10.0257 0.369050
\(739\) −36.5692 −1.34522 −0.672610 0.739998i \(-0.734827\pi\)
−0.672610 + 0.739998i \(0.734827\pi\)
\(740\) 7.78523 0.286191
\(741\) −69.1838 −2.54153
\(742\) 11.1313 0.408642
\(743\) −0.0491486 −0.00180309 −0.000901543 1.00000i \(-0.500287\pi\)
−0.000901543 1.00000i \(0.500287\pi\)
\(744\) 14.1544 0.518925
\(745\) −9.63068 −0.352841
\(746\) 2.31601 0.0847951
\(747\) 35.3324 1.29274
\(748\) 0 0
\(749\) 18.8781 0.689790
\(750\) 2.53932 0.0927229
\(751\) 6.39915 0.233508 0.116754 0.993161i \(-0.462751\pi\)
0.116754 + 0.993161i \(0.462751\pi\)
\(752\) −1.49988 −0.0546950
\(753\) −39.3510 −1.43403
\(754\) −22.6122 −0.823488
\(755\) −20.3543 −0.740769
\(756\) 1.13800 0.0413885
\(757\) −19.2059 −0.698050 −0.349025 0.937114i \(-0.613487\pi\)
−0.349025 + 0.937114i \(0.613487\pi\)
\(758\) −7.92393 −0.287810
\(759\) 0 0
\(760\) 6.62155 0.240189
\(761\) −2.87463 −0.104205 −0.0521026 0.998642i \(-0.516592\pi\)
−0.0521026 + 0.998642i \(0.516592\pi\)
\(762\) 38.0977 1.38014
\(763\) −4.22315 −0.152888
\(764\) −17.9665 −0.650004
\(765\) 23.6139 0.853764
\(766\) 19.2045 0.693887
\(767\) 30.5479 1.10302
\(768\) −2.53932 −0.0916298
\(769\) −35.4758 −1.27929 −0.639646 0.768670i \(-0.720919\pi\)
−0.639646 + 0.768670i \(0.720919\pi\)
\(770\) 0 0
\(771\) 10.4742 0.377220
\(772\) 23.8180 0.857229
\(773\) −31.2613 −1.12439 −0.562195 0.827005i \(-0.690043\pi\)
−0.562195 + 0.827005i \(0.690043\pi\)
\(774\) 11.1540 0.400924
\(775\) −5.57409 −0.200227
\(776\) −0.691019 −0.0248062
\(777\) −19.7692 −0.709216
\(778\) −7.65932 −0.274600
\(779\) −19.2525 −0.689793
\(780\) −10.4483 −0.374108
\(781\) 0 0
\(782\) 21.4952 0.768668
\(783\) −6.25398 −0.223499
\(784\) 1.00000 0.0357143
\(785\) 6.01660 0.214742
\(786\) 17.1987 0.613458
\(787\) −12.9176 −0.460462 −0.230231 0.973136i \(-0.573948\pi\)
−0.230231 + 0.973136i \(0.573948\pi\)
\(788\) −1.85551 −0.0660998
\(789\) −0.783080 −0.0278784
\(790\) 4.23983 0.150846
\(791\) −7.36545 −0.261885
\(792\) 0 0
\(793\) 9.38812 0.333382
\(794\) −20.0239 −0.710622
\(795\) −28.2659 −1.00249
\(796\) 21.8206 0.773411
\(797\) 10.1101 0.358119 0.179060 0.983838i \(-0.442694\pi\)
0.179060 + 0.983838i \(0.442694\pi\)
\(798\) −16.8142 −0.595218
\(799\) 10.2716 0.363384
\(800\) 1.00000 0.0353553
\(801\) −22.5645 −0.797276
\(802\) 20.7562 0.732926
\(803\) 0 0
\(804\) −36.0504 −1.27140
\(805\) −3.13877 −0.110627
\(806\) 22.9351 0.807854
\(807\) 43.5282 1.53227
\(808\) 14.6190 0.514296
\(809\) 5.06439 0.178054 0.0890272 0.996029i \(-0.471624\pi\)
0.0890272 + 0.996029i \(0.471624\pi\)
\(810\) 7.45471 0.261932
\(811\) −9.52150 −0.334345 −0.167173 0.985928i \(-0.553464\pi\)
−0.167173 + 0.985928i \(0.553464\pi\)
\(812\) −5.49561 −0.192858
\(813\) −67.6114 −2.37124
\(814\) 0 0
\(815\) 8.61451 0.301753
\(816\) 17.3900 0.608772
\(817\) −21.4193 −0.749368
\(818\) 32.8553 1.14876
\(819\) 14.1877 0.495759
\(820\) −2.90755 −0.101536
\(821\) 17.4006 0.607286 0.303643 0.952786i \(-0.401797\pi\)
0.303643 + 0.952786i \(0.401797\pi\)
\(822\) 23.2373 0.810494
\(823\) 18.4859 0.644379 0.322189 0.946675i \(-0.395581\pi\)
0.322189 + 0.946675i \(0.395581\pi\)
\(824\) 1.07358 0.0373998
\(825\) 0 0
\(826\) 7.42428 0.258324
\(827\) −55.2703 −1.92194 −0.960968 0.276660i \(-0.910772\pi\)
−0.960968 + 0.276660i \(0.910772\pi\)
\(828\) −10.8230 −0.376124
\(829\) −20.0696 −0.697046 −0.348523 0.937300i \(-0.613317\pi\)
−0.348523 + 0.937300i \(0.613317\pi\)
\(830\) −10.2468 −0.355670
\(831\) 39.9002 1.38412
\(832\) −4.11459 −0.142648
\(833\) −6.84829 −0.237279
\(834\) −40.4084 −1.39923
\(835\) 9.01590 0.312008
\(836\) 0 0
\(837\) 6.34329 0.219256
\(838\) −16.3625 −0.565233
\(839\) −10.7623 −0.371556 −0.185778 0.982592i \(-0.559481\pi\)
−0.185778 + 0.982592i \(0.559481\pi\)
\(840\) −2.53932 −0.0876149
\(841\) 1.20178 0.0414407
\(842\) 26.0472 0.897645
\(843\) 44.2711 1.52478
\(844\) −26.5722 −0.914651
\(845\) −3.92987 −0.135192
\(846\) −5.17181 −0.177811
\(847\) 0 0
\(848\) −11.1313 −0.382249
\(849\) −38.1392 −1.30893
\(850\) −6.84829 −0.234894
\(851\) 24.4361 0.837657
\(852\) −0.154597 −0.00529640
\(853\) −14.6593 −0.501923 −0.250962 0.967997i \(-0.580747\pi\)
−0.250962 + 0.967997i \(0.580747\pi\)
\(854\) 2.28166 0.0780769
\(855\) 22.8321 0.780841
\(856\) −18.8781 −0.645240
\(857\) 34.6532 1.18373 0.591865 0.806037i \(-0.298392\pi\)
0.591865 + 0.806037i \(0.298392\pi\)
\(858\) 0 0
\(859\) 9.13441 0.311662 0.155831 0.987784i \(-0.450194\pi\)
0.155831 + 0.987784i \(0.450194\pi\)
\(860\) −3.23479 −0.110306
\(861\) 7.38321 0.251619
\(862\) 18.2575 0.621853
\(863\) 23.4381 0.797841 0.398921 0.916986i \(-0.369385\pi\)
0.398921 + 0.916986i \(0.369385\pi\)
\(864\) −1.13800 −0.0387154
\(865\) 12.9388 0.439931
\(866\) 32.6096 1.10812
\(867\) −75.9234 −2.57849
\(868\) 5.57409 0.189197
\(869\) 0 0
\(870\) 13.9551 0.473123
\(871\) −58.4143 −1.97929
\(872\) 4.22315 0.143014
\(873\) −2.38274 −0.0806435
\(874\) 20.7836 0.703014
\(875\) 1.00000 0.0338062
\(876\) 9.93595 0.335705
\(877\) −50.9962 −1.72202 −0.861010 0.508589i \(-0.830167\pi\)
−0.861010 + 0.508589i \(0.830167\pi\)
\(878\) −29.2540 −0.987274
\(879\) −19.0618 −0.642940
\(880\) 0 0
\(881\) −1.75183 −0.0590207 −0.0295103 0.999564i \(-0.509395\pi\)
−0.0295103 + 0.999564i \(0.509395\pi\)
\(882\) 3.44815 0.116105
\(883\) −40.8658 −1.37524 −0.687621 0.726070i \(-0.741345\pi\)
−0.687621 + 0.726070i \(0.741345\pi\)
\(884\) 28.1779 0.947726
\(885\) −18.8526 −0.633725
\(886\) 11.4091 0.383297
\(887\) −23.5552 −0.790906 −0.395453 0.918486i \(-0.629412\pi\)
−0.395453 + 0.918486i \(0.629412\pi\)
\(888\) 19.7692 0.663410
\(889\) 15.0031 0.503189
\(890\) 6.54394 0.219353
\(891\) 0 0
\(892\) −9.89233 −0.331220
\(893\) 9.93153 0.332346
\(894\) −24.4554 −0.817911
\(895\) −15.7602 −0.526806
\(896\) −1.00000 −0.0334077
\(897\) −32.7947 −1.09498
\(898\) 13.6315 0.454891
\(899\) −30.6330 −1.02167
\(900\) 3.44815 0.114938
\(901\) 76.2302 2.53960
\(902\) 0 0
\(903\) 8.21418 0.273351
\(904\) 7.36545 0.244971
\(905\) 14.6094 0.485632
\(906\) −51.6861 −1.71716
\(907\) −41.4267 −1.37555 −0.687776 0.725923i \(-0.741413\pi\)
−0.687776 + 0.725923i \(0.741413\pi\)
\(908\) 13.9299 0.462282
\(909\) 50.4086 1.67195
\(910\) −4.11459 −0.136397
\(911\) 45.1138 1.49469 0.747344 0.664437i \(-0.231329\pi\)
0.747344 + 0.664437i \(0.231329\pi\)
\(912\) 16.8142 0.556775
\(913\) 0 0
\(914\) −6.66357 −0.220411
\(915\) −5.79388 −0.191540
\(916\) −10.1930 −0.336785
\(917\) 6.77296 0.223663
\(918\) 7.79333 0.257218
\(919\) 5.91882 0.195244 0.0976219 0.995224i \(-0.468876\pi\)
0.0976219 + 0.995224i \(0.468876\pi\)
\(920\) 3.13877 0.103482
\(921\) 12.9027 0.425158
\(922\) 18.1977 0.599310
\(923\) −0.250501 −0.00824535
\(924\) 0 0
\(925\) −7.78523 −0.255977
\(926\) 11.3971 0.374531
\(927\) 3.70186 0.121585
\(928\) 5.49561 0.180402
\(929\) −55.2709 −1.81338 −0.906690 0.421798i \(-0.861399\pi\)
−0.906690 + 0.421798i \(0.861399\pi\)
\(930\) −14.1544 −0.464141
\(931\) −6.62155 −0.217013
\(932\) −13.2121 −0.432777
\(933\) −27.5354 −0.901467
\(934\) 22.9793 0.751907
\(935\) 0 0
\(936\) −14.1877 −0.463740
\(937\) −16.0968 −0.525859 −0.262930 0.964815i \(-0.584689\pi\)
−0.262930 + 0.964815i \(0.584689\pi\)
\(938\) −14.1969 −0.463544
\(939\) −84.8244 −2.76814
\(940\) 1.49988 0.0489207
\(941\) −49.8464 −1.62495 −0.812474 0.582998i \(-0.801880\pi\)
−0.812474 + 0.582998i \(0.801880\pi\)
\(942\) 15.2781 0.497787
\(943\) −9.12615 −0.297188
\(944\) −7.42428 −0.241640
\(945\) −1.13800 −0.0370190
\(946\) 0 0
\(947\) −31.1447 −1.01207 −0.506033 0.862514i \(-0.668889\pi\)
−0.506033 + 0.862514i \(0.668889\pi\)
\(948\) 10.7663 0.349673
\(949\) 16.0997 0.522619
\(950\) −6.62155 −0.214832
\(951\) 70.3410 2.28096
\(952\) 6.84829 0.221954
\(953\) −38.8286 −1.25778 −0.628891 0.777494i \(-0.716491\pi\)
−0.628891 + 0.777494i \(0.716491\pi\)
\(954\) −38.3823 −1.24267
\(955\) 17.9665 0.581381
\(956\) 5.10128 0.164987
\(957\) 0 0
\(958\) 1.36694 0.0441640
\(959\) 9.15100 0.295501
\(960\) 2.53932 0.0819562
\(961\) 0.0704520 0.00227265
\(962\) 32.0330 1.03279
\(963\) −65.0944 −2.09764
\(964\) 24.0075 0.773229
\(965\) −23.8180 −0.766729
\(966\) −7.97035 −0.256442
\(967\) 18.4412 0.593030 0.296515 0.955028i \(-0.404176\pi\)
0.296515 + 0.955028i \(0.404176\pi\)
\(968\) 0 0
\(969\) −115.149 −3.69911
\(970\) 0.691019 0.0221873
\(971\) 6.48256 0.208035 0.104018 0.994575i \(-0.466830\pi\)
0.104018 + 0.994575i \(0.466830\pi\)
\(972\) 22.3439 0.716681
\(973\) −15.9131 −0.510150
\(974\) −12.2098 −0.391226
\(975\) 10.4483 0.334612
\(976\) −2.28166 −0.0730343
\(977\) −27.1448 −0.868438 −0.434219 0.900807i \(-0.642976\pi\)
−0.434219 + 0.900807i \(0.642976\pi\)
\(978\) 21.8750 0.699485
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 14.5621 0.464930
\(982\) 39.8546 1.27181
\(983\) −9.91081 −0.316106 −0.158053 0.987431i \(-0.550522\pi\)
−0.158053 + 0.987431i \(0.550522\pi\)
\(984\) −7.38321 −0.235368
\(985\) 1.85551 0.0591215
\(986\) −37.6356 −1.19856
\(987\) −3.80868 −0.121231
\(988\) 27.2450 0.866779
\(989\) −10.1533 −0.322856
\(990\) 0 0
\(991\) 23.8739 0.758380 0.379190 0.925319i \(-0.376203\pi\)
0.379190 + 0.925319i \(0.376203\pi\)
\(992\) −5.57409 −0.176977
\(993\) 73.2130 2.32334
\(994\) −0.0608812 −0.00193103
\(995\) −21.8206 −0.691760
\(996\) −26.0198 −0.824470
\(997\) −17.9566 −0.568691 −0.284345 0.958722i \(-0.591776\pi\)
−0.284345 + 0.958722i \(0.591776\pi\)
\(998\) 36.7505 1.16332
\(999\) 8.85955 0.280304
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.1.2 8
11.3 even 5 770.2.n.k.141.4 yes 16
11.4 even 5 770.2.n.k.71.4 16
11.10 odd 2 8470.2.a.dg.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.71.4 16 11.4 even 5
770.2.n.k.141.4 yes 16 11.3 even 5
8470.2.a.dg.1.2 8 11.10 odd 2
8470.2.a.dh.1.2 8 1.1 even 1 trivial