Properties

Label 5586.2.a.bw.1.3
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.29268.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 5 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.43292\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.11806 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.11806 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.11806 q^{10} +3.35203 q^{11} -1.00000 q^{12} -4.23612 q^{13} -2.11806 q^{15} +1.00000 q^{16} +6.95628 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.11806 q^{20} -3.35203 q^{22} -7.18070 q^{23} +1.00000 q^{24} -0.513812 q^{25} +4.23612 q^{26} -1.00000 q^{27} +7.29877 q^{29} +2.11806 q^{30} +4.54883 q^{31} -1.00000 q^{32} -3.35203 q^{33} -6.95628 q^{34} +1.00000 q^{36} -0.252218 q^{37} -1.00000 q^{38} +4.23612 q^{39} -2.11806 q^{40} -10.3359 q^{41} +3.17133 q^{43} +3.35203 q^{44} +2.11806 q^{45} +7.18070 q^{46} +7.15524 q^{47} -1.00000 q^{48} +0.513812 q^{50} -6.95628 q^{51} -4.23612 q^{52} +1.75733 q^{53} +1.00000 q^{54} +7.09982 q^{55} -1.00000 q^{57} -7.29877 q^{58} +13.0465 q^{59} -2.11806 q^{60} +6.91911 q^{61} -4.54883 q^{62} +1.00000 q^{64} -8.97238 q^{65} +3.35203 q^{66} -4.30616 q^{67} +6.95628 q^{68} +7.18070 q^{69} -10.4518 q^{71} -1.00000 q^{72} -12.7667 q^{73} +0.252218 q^{74} +0.513812 q^{75} +1.00000 q^{76} -4.23612 q^{78} +2.27114 q^{79} +2.11806 q^{80} +1.00000 q^{81} +10.3359 q^{82} +16.5699 q^{83} +14.7338 q^{85} -3.17133 q^{86} -7.29877 q^{87} -3.35203 q^{88} +3.79365 q^{89} -2.11806 q^{90} -7.18070 q^{92} -4.54883 q^{93} -7.15524 q^{94} +2.11806 q^{95} +1.00000 q^{96} -6.18070 q^{97} +3.35203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} - 2q^{11} - 4q^{12} + 4q^{16} + 10q^{17} - 4q^{18} + 4q^{19} + 2q^{22} - 5q^{23} + 4q^{24} + 4q^{25} - 4q^{27} - 3q^{29} + 9q^{31} - 4q^{32} + 2q^{33} - 10q^{34} + 4q^{36} - 14q^{37} - 4q^{38} + 4q^{41} + 21q^{43} - 2q^{44} + 5q^{46} + 7q^{47} - 4q^{48} - 4q^{50} - 10q^{51} - 7q^{53} + 4q^{54} - 4q^{57} + 3q^{58} + 7q^{59} + 23q^{61} - 9q^{62} + 4q^{64} - 48q^{65} - 2q^{66} + 6q^{67} + 10q^{68} + 5q^{69} + 2q^{71} - 4q^{72} - 5q^{73} + 14q^{74} - 4q^{75} + 4q^{76} - 11q^{79} + 4q^{81} - 4q^{82} + 14q^{83} + 6q^{85} - 21q^{86} + 3q^{87} + 2q^{88} + 10q^{89} - 5q^{92} - 9q^{93} - 7q^{94} + 4q^{96} - q^{97} - 2q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.11806 0.947226 0.473613 0.880733i \(-0.342949\pi\)
0.473613 + 0.880733i \(0.342949\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.11806 −0.669790
\(11\) 3.35203 1.01068 0.505338 0.862922i \(-0.331368\pi\)
0.505338 + 0.862922i \(0.331368\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.23612 −1.17489 −0.587445 0.809264i \(-0.699866\pi\)
−0.587445 + 0.809264i \(0.699866\pi\)
\(14\) 0 0
\(15\) −2.11806 −0.546881
\(16\) 1.00000 0.250000
\(17\) 6.95628 1.68715 0.843573 0.537014i \(-0.180448\pi\)
0.843573 + 0.537014i \(0.180448\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 2.11806 0.473613
\(21\) 0 0
\(22\) −3.35203 −0.714656
\(23\) −7.18070 −1.49728 −0.748640 0.662977i \(-0.769293\pi\)
−0.748640 + 0.662977i \(0.769293\pi\)
\(24\) 1.00000 0.204124
\(25\) −0.513812 −0.102762
\(26\) 4.23612 0.830772
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.29877 1.35535 0.677673 0.735363i \(-0.262988\pi\)
0.677673 + 0.735363i \(0.262988\pi\)
\(30\) 2.11806 0.386704
\(31\) 4.54883 0.816994 0.408497 0.912760i \(-0.366053\pi\)
0.408497 + 0.912760i \(0.366053\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.35203 −0.583514
\(34\) −6.95628 −1.19299
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.252218 −0.0414644 −0.0207322 0.999785i \(-0.506600\pi\)
−0.0207322 + 0.999785i \(0.506600\pi\)
\(38\) −1.00000 −0.162221
\(39\) 4.23612 0.678323
\(40\) −2.11806 −0.334895
\(41\) −10.3359 −1.61420 −0.807101 0.590413i \(-0.798965\pi\)
−0.807101 + 0.590413i \(0.798965\pi\)
\(42\) 0 0
\(43\) 3.17133 0.483623 0.241812 0.970323i \(-0.422258\pi\)
0.241812 + 0.970323i \(0.422258\pi\)
\(44\) 3.35203 0.505338
\(45\) 2.11806 0.315742
\(46\) 7.18070 1.05874
\(47\) 7.15524 1.04370 0.521849 0.853038i \(-0.325242\pi\)
0.521849 + 0.853038i \(0.325242\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0.513812 0.0726639
\(51\) −6.95628 −0.974075
\(52\) −4.23612 −0.587445
\(53\) 1.75733 0.241388 0.120694 0.992690i \(-0.461488\pi\)
0.120694 + 0.992690i \(0.461488\pi\)
\(54\) 1.00000 0.136083
\(55\) 7.09982 0.957339
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) −7.29877 −0.958375
\(59\) 13.0465 1.69852 0.849258 0.527978i \(-0.177050\pi\)
0.849258 + 0.527978i \(0.177050\pi\)
\(60\) −2.11806 −0.273441
\(61\) 6.91911 0.885901 0.442951 0.896546i \(-0.353932\pi\)
0.442951 + 0.896546i \(0.353932\pi\)
\(62\) −4.54883 −0.577702
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.97238 −1.11289
\(66\) 3.35203 0.412607
\(67\) −4.30616 −0.526081 −0.263041 0.964785i \(-0.584725\pi\)
−0.263041 + 0.964785i \(0.584725\pi\)
\(68\) 6.95628 0.843573
\(69\) 7.18070 0.864455
\(70\) 0 0
\(71\) −10.4518 −1.24041 −0.620203 0.784441i \(-0.712950\pi\)
−0.620203 + 0.784441i \(0.712950\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.7667 −1.49423 −0.747115 0.664695i \(-0.768562\pi\)
−0.747115 + 0.664695i \(0.768562\pi\)
\(74\) 0.252218 0.0293197
\(75\) 0.513812 0.0593298
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −4.23612 −0.479647
\(79\) 2.27114 0.255524 0.127762 0.991805i \(-0.459221\pi\)
0.127762 + 0.991805i \(0.459221\pi\)
\(80\) 2.11806 0.236807
\(81\) 1.00000 0.111111
\(82\) 10.3359 1.14141
\(83\) 16.5699 1.81878 0.909392 0.415940i \(-0.136547\pi\)
0.909392 + 0.415940i \(0.136547\pi\)
\(84\) 0 0
\(85\) 14.7338 1.59811
\(86\) −3.17133 −0.341973
\(87\) −7.29877 −0.782510
\(88\) −3.35203 −0.357328
\(89\) 3.79365 0.402126 0.201063 0.979578i \(-0.435560\pi\)
0.201063 + 0.979578i \(0.435560\pi\)
\(90\) −2.11806 −0.223263
\(91\) 0 0
\(92\) −7.18070 −0.748640
\(93\) −4.54883 −0.471692
\(94\) −7.15524 −0.738006
\(95\) 2.11806 0.217309
\(96\) 1.00000 0.102062
\(97\) −6.18070 −0.627555 −0.313778 0.949496i \(-0.601595\pi\)
−0.313778 + 0.949496i \(0.601595\pi\)
\(98\) 0 0
\(99\) 3.35203 0.336892
\(100\) −0.513812 −0.0513812
\(101\) 13.2296 1.31639 0.658196 0.752846i \(-0.271320\pi\)
0.658196 + 0.752846i \(0.271320\pi\)
\(102\) 6.95628 0.688775
\(103\) −7.45400 −0.734465 −0.367232 0.930129i \(-0.619695\pi\)
−0.367232 + 0.930129i \(0.619695\pi\)
\(104\) 4.23612 0.415386
\(105\) 0 0
\(106\) −1.75733 −0.170687
\(107\) 8.60708 0.832078 0.416039 0.909347i \(-0.363418\pi\)
0.416039 + 0.909347i \(0.363418\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.30746 0.604145 0.302073 0.953285i \(-0.402321\pi\)
0.302073 + 0.953285i \(0.402321\pi\)
\(110\) −7.09982 −0.676941
\(111\) 0.252218 0.0239395
\(112\) 0 0
\(113\) 1.97453 0.185748 0.0928741 0.995678i \(-0.470395\pi\)
0.0928741 + 0.995678i \(0.470395\pi\)
\(114\) 1.00000 0.0936586
\(115\) −15.2092 −1.41826
\(116\) 7.29877 0.677673
\(117\) −4.23612 −0.391630
\(118\) −13.0465 −1.20103
\(119\) 0 0
\(120\) 2.11806 0.193352
\(121\) 0.236125 0.0214659
\(122\) −6.91911 −0.626427
\(123\) 10.3359 0.931960
\(124\) 4.54883 0.408497
\(125\) −11.6786 −1.04457
\(126\) 0 0
\(127\) 12.2296 1.08520 0.542600 0.839991i \(-0.317440\pi\)
0.542600 + 0.839991i \(0.317440\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.17133 −0.279220
\(130\) 8.97238 0.786930
\(131\) −1.02108 −0.0892122 −0.0446061 0.999005i \(-0.514203\pi\)
−0.0446061 + 0.999005i \(0.514203\pi\)
\(132\) −3.35203 −0.291757
\(133\) 0 0
\(134\) 4.30616 0.371996
\(135\) −2.11806 −0.182294
\(136\) −6.95628 −0.596496
\(137\) −6.12761 −0.523517 −0.261759 0.965133i \(-0.584302\pi\)
−0.261759 + 0.965133i \(0.584302\pi\)
\(138\) −7.18070 −0.611262
\(139\) 18.3476 1.55623 0.778113 0.628124i \(-0.216177\pi\)
0.778113 + 0.628124i \(0.216177\pi\)
\(140\) 0 0
\(141\) −7.15524 −0.602580
\(142\) 10.4518 0.877100
\(143\) −14.1996 −1.18743
\(144\) 1.00000 0.0833333
\(145\) 15.4592 1.28382
\(146\) 12.7667 1.05658
\(147\) 0 0
\(148\) −0.252218 −0.0207322
\(149\) −21.9102 −1.79496 −0.897478 0.441058i \(-0.854603\pi\)
−0.897478 + 0.441058i \(0.854603\pi\)
\(150\) −0.513812 −0.0419525
\(151\) 0.349200 0.0284175 0.0142088 0.999899i \(-0.495477\pi\)
0.0142088 + 0.999899i \(0.495477\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 6.95628 0.562382
\(154\) 0 0
\(155\) 9.63471 0.773878
\(156\) 4.23612 0.339161
\(157\) 13.0487 1.04140 0.520700 0.853740i \(-0.325671\pi\)
0.520700 + 0.853740i \(0.325671\pi\)
\(158\) −2.27114 −0.180682
\(159\) −1.75733 −0.139365
\(160\) −2.11806 −0.167448
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 15.4614 1.21103 0.605515 0.795834i \(-0.292967\pi\)
0.605515 + 0.795834i \(0.292967\pi\)
\(164\) −10.3359 −0.807101
\(165\) −7.09982 −0.552720
\(166\) −16.5699 −1.28607
\(167\) −4.68797 −0.362766 −0.181383 0.983413i \(-0.558057\pi\)
−0.181383 + 0.983413i \(0.558057\pi\)
\(168\) 0 0
\(169\) 4.94475 0.380366
\(170\) −14.7338 −1.13003
\(171\) 1.00000 0.0764719
\(172\) 3.17133 0.241812
\(173\) −21.5283 −1.63677 −0.818385 0.574670i \(-0.805130\pi\)
−0.818385 + 0.574670i \(0.805130\pi\)
\(174\) 7.29877 0.553318
\(175\) 0 0
\(176\) 3.35203 0.252669
\(177\) −13.0465 −0.980639
\(178\) −3.79365 −0.284346
\(179\) 1.83822 0.137395 0.0686976 0.997638i \(-0.478116\pi\)
0.0686976 + 0.997638i \(0.478116\pi\)
\(180\) 2.11806 0.157871
\(181\) 20.3848 1.51519 0.757596 0.652724i \(-0.226374\pi\)
0.757596 + 0.652724i \(0.226374\pi\)
\(182\) 0 0
\(183\) −6.91911 −0.511475
\(184\) 7.18070 0.529369
\(185\) −0.534213 −0.0392761
\(186\) 4.54883 0.333536
\(187\) 23.3177 1.70516
\(188\) 7.15524 0.521849
\(189\) 0 0
\(190\) −2.11806 −0.153660
\(191\) −5.49789 −0.397814 −0.198907 0.980018i \(-0.563739\pi\)
−0.198907 + 0.980018i \(0.563739\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 7.89871 0.568562 0.284281 0.958741i \(-0.408245\pi\)
0.284281 + 0.958741i \(0.408245\pi\)
\(194\) 6.18070 0.443749
\(195\) 8.97238 0.642525
\(196\) 0 0
\(197\) −1.93736 −0.138031 −0.0690155 0.997616i \(-0.521986\pi\)
−0.0690155 + 0.997616i \(0.521986\pi\)
\(198\) −3.35203 −0.238219
\(199\) 8.73823 0.619437 0.309718 0.950828i \(-0.399765\pi\)
0.309718 + 0.950828i \(0.399765\pi\)
\(200\) 0.513812 0.0363320
\(201\) 4.30616 0.303733
\(202\) −13.2296 −0.930830
\(203\) 0 0
\(204\) −6.95628 −0.487037
\(205\) −21.8922 −1.52902
\(206\) 7.45400 0.519345
\(207\) −7.18070 −0.499093
\(208\) −4.23612 −0.293722
\(209\) 3.35203 0.231865
\(210\) 0 0
\(211\) −23.3870 −1.61003 −0.805013 0.593257i \(-0.797842\pi\)
−0.805013 + 0.593257i \(0.797842\pi\)
\(212\) 1.75733 0.120694
\(213\) 10.4518 0.716149
\(214\) −8.60708 −0.588368
\(215\) 6.71707 0.458101
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.30746 −0.427195
\(219\) 12.7667 0.862694
\(220\) 7.09982 0.478669
\(221\) −29.4677 −1.98221
\(222\) −0.252218 −0.0169278
\(223\) 9.80320 0.656471 0.328236 0.944596i \(-0.393546\pi\)
0.328236 + 0.944596i \(0.393546\pi\)
\(224\) 0 0
\(225\) −0.513812 −0.0342541
\(226\) −1.97453 −0.131344
\(227\) −18.2835 −1.21352 −0.606760 0.794885i \(-0.707531\pi\)
−0.606760 + 0.794885i \(0.707531\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −1.44463 −0.0954636 −0.0477318 0.998860i \(-0.515199\pi\)
−0.0477318 + 0.998860i \(0.515199\pi\)
\(230\) 15.2092 1.00286
\(231\) 0 0
\(232\) −7.29877 −0.479188
\(233\) −15.6808 −1.02728 −0.513640 0.858006i \(-0.671703\pi\)
−0.513640 + 0.858006i \(0.671703\pi\)
\(234\) 4.23612 0.276924
\(235\) 15.1552 0.988619
\(236\) 13.0465 0.849258
\(237\) −2.27114 −0.147527
\(238\) 0 0
\(239\) 3.76818 0.243744 0.121872 0.992546i \(-0.461110\pi\)
0.121872 + 0.992546i \(0.461110\pi\)
\(240\) −2.11806 −0.136720
\(241\) 2.99561 0.192964 0.0964821 0.995335i \(-0.469241\pi\)
0.0964821 + 0.995335i \(0.469241\pi\)
\(242\) −0.236125 −0.0151787
\(243\) −1.00000 −0.0641500
\(244\) 6.91911 0.442951
\(245\) 0 0
\(246\) −10.3359 −0.658995
\(247\) −4.23612 −0.269538
\(248\) −4.54883 −0.288851
\(249\) −16.5699 −1.05008
\(250\) 11.6786 0.738619
\(251\) 26.2922 1.65955 0.829775 0.558098i \(-0.188469\pi\)
0.829775 + 0.558098i \(0.188469\pi\)
\(252\) 0 0
\(253\) −24.0700 −1.51327
\(254\) −12.2296 −0.767352
\(255\) −14.7338 −0.922669
\(256\) 1.00000 0.0625000
\(257\) −19.7317 −1.23083 −0.615414 0.788204i \(-0.711011\pi\)
−0.615414 + 0.788204i \(0.711011\pi\)
\(258\) 3.17133 0.197438
\(259\) 0 0
\(260\) −8.97238 −0.556443
\(261\) 7.29877 0.451782
\(262\) 1.02108 0.0630825
\(263\) −4.44247 −0.273935 −0.136967 0.990576i \(-0.543736\pi\)
−0.136967 + 0.990576i \(0.543736\pi\)
\(264\) 3.35203 0.206303
\(265\) 3.72214 0.228649
\(266\) 0 0
\(267\) −3.79365 −0.232168
\(268\) −4.30616 −0.263041
\(269\) 19.4796 1.18769 0.593847 0.804578i \(-0.297608\pi\)
0.593847 + 0.804578i \(0.297608\pi\)
\(270\) 2.11806 0.128901
\(271\) 1.57578 0.0957215 0.0478608 0.998854i \(-0.484760\pi\)
0.0478608 + 0.998854i \(0.484760\pi\)
\(272\) 6.95628 0.421787
\(273\) 0 0
\(274\) 6.12761 0.370183
\(275\) −1.72231 −0.103859
\(276\) 7.18070 0.432228
\(277\) 9.80612 0.589192 0.294596 0.955622i \(-0.404815\pi\)
0.294596 + 0.955622i \(0.404815\pi\)
\(278\) −18.3476 −1.10042
\(279\) 4.54883 0.272331
\(280\) 0 0
\(281\) −3.60641 −0.215140 −0.107570 0.994198i \(-0.534307\pi\)
−0.107570 + 0.994198i \(0.534307\pi\)
\(282\) 7.15524 0.426088
\(283\) −21.5327 −1.27999 −0.639994 0.768380i \(-0.721063\pi\)
−0.639994 + 0.768380i \(0.721063\pi\)
\(284\) −10.4518 −0.620203
\(285\) −2.11806 −0.125463
\(286\) 14.1996 0.839642
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 31.3899 1.84646
\(290\) −15.4592 −0.907798
\(291\) 6.18070 0.362319
\(292\) −12.7667 −0.747115
\(293\) −24.2946 −1.41930 −0.709651 0.704553i \(-0.751148\pi\)
−0.709651 + 0.704553i \(0.751148\pi\)
\(294\) 0 0
\(295\) 27.6334 1.60888
\(296\) 0.252218 0.0146599
\(297\) −3.35203 −0.194505
\(298\) 21.9102 1.26923
\(299\) 30.4184 1.75914
\(300\) 0.513812 0.0296649
\(301\) 0 0
\(302\) −0.349200 −0.0200942
\(303\) −13.2296 −0.760020
\(304\) 1.00000 0.0573539
\(305\) 14.6551 0.839149
\(306\) −6.95628 −0.397664
\(307\) 5.79365 0.330661 0.165331 0.986238i \(-0.447131\pi\)
0.165331 + 0.986238i \(0.447131\pi\)
\(308\) 0 0
\(309\) 7.45400 0.424043
\(310\) −9.63471 −0.547215
\(311\) 18.6786 1.05917 0.529583 0.848258i \(-0.322348\pi\)
0.529583 + 0.848258i \(0.322348\pi\)
\(312\) −4.23612 −0.239823
\(313\) 25.7369 1.45473 0.727366 0.686249i \(-0.240744\pi\)
0.727366 + 0.686249i \(0.240744\pi\)
\(314\) −13.0487 −0.736381
\(315\) 0 0
\(316\) 2.27114 0.127762
\(317\) −20.7528 −1.16559 −0.582796 0.812619i \(-0.698041\pi\)
−0.582796 + 0.812619i \(0.698041\pi\)
\(318\) 1.75733 0.0985463
\(319\) 24.4657 1.36982
\(320\) 2.11806 0.118403
\(321\) −8.60708 −0.480401
\(322\) 0 0
\(323\) 6.95628 0.387058
\(324\) 1.00000 0.0555556
\(325\) 2.17657 0.120734
\(326\) −15.4614 −0.856328
\(327\) −6.30746 −0.348804
\(328\) 10.3359 0.570707
\(329\) 0 0
\(330\) 7.09982 0.390832
\(331\) −35.0035 −1.92397 −0.961983 0.273108i \(-0.911948\pi\)
−0.961983 + 0.273108i \(0.911948\pi\)
\(332\) 16.5699 0.909392
\(333\) −0.252218 −0.0138215
\(334\) 4.68797 0.256514
\(335\) −9.12072 −0.498318
\(336\) 0 0
\(337\) −9.06565 −0.493837 −0.246919 0.969036i \(-0.579418\pi\)
−0.246919 + 0.969036i \(0.579418\pi\)
\(338\) −4.94475 −0.268959
\(339\) −1.97453 −0.107242
\(340\) 14.7338 0.799055
\(341\) 15.2478 0.825716
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −3.17133 −0.170987
\(345\) 15.2092 0.818835
\(346\) 21.5283 1.15737
\(347\) −17.6560 −0.947822 −0.473911 0.880573i \(-0.657158\pi\)
−0.473911 + 0.880573i \(0.657158\pi\)
\(348\) −7.29877 −0.391255
\(349\) 19.9745 1.06921 0.534606 0.845101i \(-0.320460\pi\)
0.534606 + 0.845101i \(0.320460\pi\)
\(350\) 0 0
\(351\) 4.23612 0.226108
\(352\) −3.35203 −0.178664
\(353\) 22.5028 1.19770 0.598851 0.800860i \(-0.295624\pi\)
0.598851 + 0.800860i \(0.295624\pi\)
\(354\) 13.0465 0.693416
\(355\) −22.1377 −1.17495
\(356\) 3.79365 0.201063
\(357\) 0 0
\(358\) −1.83822 −0.0971530
\(359\) −9.83589 −0.519119 −0.259559 0.965727i \(-0.583577\pi\)
−0.259559 + 0.965727i \(0.583577\pi\)
\(360\) −2.11806 −0.111632
\(361\) 1.00000 0.0526316
\(362\) −20.3848 −1.07140
\(363\) −0.236125 −0.0123933
\(364\) 0 0
\(365\) −27.0407 −1.41537
\(366\) 6.91911 0.361668
\(367\) 35.3346 1.84445 0.922224 0.386655i \(-0.126370\pi\)
0.922224 + 0.386655i \(0.126370\pi\)
\(368\) −7.18070 −0.374320
\(369\) −10.3359 −0.538068
\(370\) 0.534213 0.0277724
\(371\) 0 0
\(372\) −4.54883 −0.235846
\(373\) 33.8294 1.75162 0.875811 0.482654i \(-0.160327\pi\)
0.875811 + 0.482654i \(0.160327\pi\)
\(374\) −23.3177 −1.20573
\(375\) 11.6786 0.603080
\(376\) −7.15524 −0.369003
\(377\) −30.9185 −1.59238
\(378\) 0 0
\(379\) 25.4999 1.30984 0.654920 0.755698i \(-0.272702\pi\)
0.654920 + 0.755698i \(0.272702\pi\)
\(380\) 2.11806 0.108654
\(381\) −12.2296 −0.626540
\(382\) 5.49789 0.281297
\(383\) 35.7186 1.82514 0.912568 0.408926i \(-0.134096\pi\)
0.912568 + 0.408926i \(0.134096\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −7.89871 −0.402034
\(387\) 3.17133 0.161208
\(388\) −6.18070 −0.313778
\(389\) 22.0248 1.11670 0.558350 0.829606i \(-0.311435\pi\)
0.558350 + 0.829606i \(0.311435\pi\)
\(390\) −8.97238 −0.454334
\(391\) −49.9510 −2.52613
\(392\) 0 0
\(393\) 1.02108 0.0515067
\(394\) 1.93736 0.0976027
\(395\) 4.81042 0.242039
\(396\) 3.35203 0.168446
\(397\) 19.7317 0.990305 0.495153 0.868806i \(-0.335112\pi\)
0.495153 + 0.868806i \(0.335112\pi\)
\(398\) −8.73823 −0.438008
\(399\) 0 0
\(400\) −0.513812 −0.0256906
\(401\) 14.8528 0.741711 0.370856 0.928691i \(-0.379064\pi\)
0.370856 + 0.928691i \(0.379064\pi\)
\(402\) −4.30616 −0.214772
\(403\) −19.2694 −0.959878
\(404\) 13.2296 0.658196
\(405\) 2.11806 0.105247
\(406\) 0 0
\(407\) −0.845443 −0.0419070
\(408\) 6.95628 0.344387
\(409\) 6.03632 0.298477 0.149238 0.988801i \(-0.452318\pi\)
0.149238 + 0.988801i \(0.452318\pi\)
\(410\) 21.8922 1.08118
\(411\) 6.12761 0.302253
\(412\) −7.45400 −0.367232
\(413\) 0 0
\(414\) 7.18070 0.352912
\(415\) 35.0961 1.72280
\(416\) 4.23612 0.207693
\(417\) −18.3476 −0.898488
\(418\) −3.35203 −0.163953
\(419\) 14.1370 0.690637 0.345319 0.938486i \(-0.387771\pi\)
0.345319 + 0.938486i \(0.387771\pi\)
\(420\) 0 0
\(421\) −35.7565 −1.74266 −0.871331 0.490695i \(-0.836743\pi\)
−0.871331 + 0.490695i \(0.836743\pi\)
\(422\) 23.3870 1.13846
\(423\) 7.15524 0.347899
\(424\) −1.75733 −0.0853436
\(425\) −3.57422 −0.173375
\(426\) −10.4518 −0.506394
\(427\) 0 0
\(428\) 8.60708 0.416039
\(429\) 14.1996 0.685565
\(430\) −6.71707 −0.323926
\(431\) 17.4242 0.839295 0.419648 0.907687i \(-0.362154\pi\)
0.419648 + 0.907687i \(0.362154\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −0.745628 −0.0358326 −0.0179163 0.999839i \(-0.505703\pi\)
−0.0179163 + 0.999839i \(0.505703\pi\)
\(434\) 0 0
\(435\) −15.4592 −0.741214
\(436\) 6.30746 0.302073
\(437\) −7.18070 −0.343500
\(438\) −12.7667 −0.610017
\(439\) 11.7289 0.559788 0.279894 0.960031i \(-0.409701\pi\)
0.279894 + 0.960031i \(0.409701\pi\)
\(440\) −7.09982 −0.338470
\(441\) 0 0
\(442\) 29.4677 1.40163
\(443\) 6.95611 0.330495 0.165247 0.986252i \(-0.447158\pi\)
0.165247 + 0.986252i \(0.447158\pi\)
\(444\) 0.252218 0.0119697
\(445\) 8.03519 0.380905
\(446\) −9.80320 −0.464195
\(447\) 21.9102 1.03632
\(448\) 0 0
\(449\) −27.5823 −1.30169 −0.650844 0.759211i \(-0.725585\pi\)
−0.650844 + 0.759211i \(0.725585\pi\)
\(450\) 0.513812 0.0242213
\(451\) −34.6464 −1.63144
\(452\) 1.97453 0.0928741
\(453\) −0.349200 −0.0164069
\(454\) 18.2835 0.858088
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 20.7733 0.971732 0.485866 0.874033i \(-0.338504\pi\)
0.485866 + 0.874033i \(0.338504\pi\)
\(458\) 1.44463 0.0675030
\(459\) −6.95628 −0.324692
\(460\) −15.2092 −0.709132
\(461\) 33.6610 1.56775 0.783875 0.620918i \(-0.213240\pi\)
0.783875 + 0.620918i \(0.213240\pi\)
\(462\) 0 0
\(463\) −18.4933 −0.859458 −0.429729 0.902958i \(-0.641391\pi\)
−0.429729 + 0.902958i \(0.641391\pi\)
\(464\) 7.29877 0.338837
\(465\) −9.63471 −0.446799
\(466\) 15.6808 0.726397
\(467\) −13.4329 −0.621602 −0.310801 0.950475i \(-0.600597\pi\)
−0.310801 + 0.950475i \(0.600597\pi\)
\(468\) −4.23612 −0.195815
\(469\) 0 0
\(470\) −15.1552 −0.699059
\(471\) −13.0487 −0.601253
\(472\) −13.0465 −0.600516
\(473\) 10.6304 0.488786
\(474\) 2.27114 0.104317
\(475\) −0.513812 −0.0235753
\(476\) 0 0
\(477\) 1.75733 0.0804627
\(478\) −3.76818 −0.172353
\(479\) −4.05893 −0.185457 −0.0927287 0.995691i \(-0.529559\pi\)
−0.0927287 + 0.995691i \(0.529559\pi\)
\(480\) 2.11806 0.0966759
\(481\) 1.06843 0.0487161
\(482\) −2.99561 −0.136446
\(483\) 0 0
\(484\) 0.236125 0.0107329
\(485\) −13.0911 −0.594437
\(486\) 1.00000 0.0453609
\(487\) 37.5078 1.69964 0.849820 0.527073i \(-0.176711\pi\)
0.849820 + 0.527073i \(0.176711\pi\)
\(488\) −6.91911 −0.313213
\(489\) −15.4614 −0.699189
\(490\) 0 0
\(491\) −27.2230 −1.22856 −0.614279 0.789089i \(-0.710553\pi\)
−0.614279 + 0.789089i \(0.710553\pi\)
\(492\) 10.3359 0.465980
\(493\) 50.7723 2.28667
\(494\) 4.23612 0.190592
\(495\) 7.09982 0.319113
\(496\) 4.54883 0.204249
\(497\) 0 0
\(498\) 16.5699 0.742515
\(499\) 6.93651 0.310521 0.155260 0.987874i \(-0.450378\pi\)
0.155260 + 0.987874i \(0.450378\pi\)
\(500\) −11.6786 −0.522283
\(501\) 4.68797 0.209443
\(502\) −26.2922 −1.17348
\(503\) 24.0509 1.07237 0.536187 0.844099i \(-0.319864\pi\)
0.536187 + 0.844099i \(0.319864\pi\)
\(504\) 0 0
\(505\) 28.0211 1.24692
\(506\) 24.0700 1.07004
\(507\) −4.94475 −0.219604
\(508\) 12.2296 0.542600
\(509\) 32.3685 1.43471 0.717355 0.696707i \(-0.245352\pi\)
0.717355 + 0.696707i \(0.245352\pi\)
\(510\) 14.7338 0.652426
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 19.7317 0.870327
\(515\) −15.7880 −0.695704
\(516\) −3.17133 −0.139610
\(517\) 23.9846 1.05484
\(518\) 0 0
\(519\) 21.5283 0.944990
\(520\) 8.97238 0.393465
\(521\) 12.5146 0.548273 0.274137 0.961691i \(-0.411608\pi\)
0.274137 + 0.961691i \(0.411608\pi\)
\(522\) −7.29877 −0.319458
\(523\) −15.9658 −0.698134 −0.349067 0.937098i \(-0.613501\pi\)
−0.349067 + 0.937098i \(0.613501\pi\)
\(524\) −1.02108 −0.0446061
\(525\) 0 0
\(526\) 4.44247 0.193701
\(527\) 31.6430 1.37839
\(528\) −3.35203 −0.145879
\(529\) 28.5625 1.24185
\(530\) −3.72214 −0.161679
\(531\) 13.0465 0.566172
\(532\) 0 0
\(533\) 43.7843 1.89651
\(534\) 3.79365 0.164167
\(535\) 18.2303 0.788166
\(536\) 4.30616 0.185998
\(537\) −1.83822 −0.0793251
\(538\) −19.4796 −0.839827
\(539\) 0 0
\(540\) −2.11806 −0.0911469
\(541\) −40.2696 −1.73132 −0.865662 0.500628i \(-0.833102\pi\)
−0.865662 + 0.500628i \(0.833102\pi\)
\(542\) −1.57578 −0.0676853
\(543\) −20.3848 −0.874796
\(544\) −6.95628 −0.298248
\(545\) 13.3596 0.572263
\(546\) 0 0
\(547\) 7.22088 0.308743 0.154371 0.988013i \(-0.450665\pi\)
0.154371 + 0.988013i \(0.450665\pi\)
\(548\) −6.12761 −0.261759
\(549\) 6.91911 0.295300
\(550\) 1.72231 0.0734397
\(551\) 7.29877 0.310938
\(552\) −7.18070 −0.305631
\(553\) 0 0
\(554\) −9.80612 −0.416622
\(555\) 0.534213 0.0226761
\(556\) 18.3476 0.778113
\(557\) 2.72532 0.115476 0.0577378 0.998332i \(-0.481611\pi\)
0.0577378 + 0.998332i \(0.481611\pi\)
\(558\) −4.54883 −0.192567
\(559\) −13.4341 −0.568204
\(560\) 0 0
\(561\) −23.3177 −0.984474
\(562\) 3.60641 0.152127
\(563\) 18.3468 0.773225 0.386613 0.922242i \(-0.373645\pi\)
0.386613 + 0.922242i \(0.373645\pi\)
\(564\) −7.15524 −0.301290
\(565\) 4.18218 0.175946
\(566\) 21.5327 0.905088
\(567\) 0 0
\(568\) 10.4518 0.438550
\(569\) 36.2931 1.52148 0.760742 0.649054i \(-0.224835\pi\)
0.760742 + 0.649054i \(0.224835\pi\)
\(570\) 2.11806 0.0887159
\(571\) 45.8110 1.91713 0.958566 0.284871i \(-0.0919508\pi\)
0.958566 + 0.284871i \(0.0919508\pi\)
\(572\) −14.1996 −0.593716
\(573\) 5.49789 0.229678
\(574\) 0 0
\(575\) 3.68953 0.153864
\(576\) 1.00000 0.0416667
\(577\) −15.8893 −0.661478 −0.330739 0.943722i \(-0.607298\pi\)
−0.330739 + 0.943722i \(0.607298\pi\)
\(578\) −31.3899 −1.30565
\(579\) −7.89871 −0.328259
\(580\) 15.4592 0.641910
\(581\) 0 0
\(582\) −6.18070 −0.256198
\(583\) 5.89063 0.243965
\(584\) 12.7667 0.528290
\(585\) −8.97238 −0.370962
\(586\) 24.2946 1.00360
\(587\) −4.19680 −0.173220 −0.0866102 0.996242i \(-0.527603\pi\)
−0.0866102 + 0.996242i \(0.527603\pi\)
\(588\) 0 0
\(589\) 4.54883 0.187431
\(590\) −27.6334 −1.13765
\(591\) 1.93736 0.0796923
\(592\) −0.252218 −0.0103661
\(593\) −24.7295 −1.01552 −0.507758 0.861499i \(-0.669526\pi\)
−0.507758 + 0.861499i \(0.669526\pi\)
\(594\) 3.35203 0.137536
\(595\) 0 0
\(596\) −21.9102 −0.897478
\(597\) −8.73823 −0.357632
\(598\) −30.4184 −1.24390
\(599\) −4.53928 −0.185470 −0.0927350 0.995691i \(-0.529561\pi\)
−0.0927350 + 0.995691i \(0.529561\pi\)
\(600\) −0.513812 −0.0209763
\(601\) 9.37441 0.382390 0.191195 0.981552i \(-0.438764\pi\)
0.191195 + 0.981552i \(0.438764\pi\)
\(602\) 0 0
\(603\) −4.30616 −0.175360
\(604\) 0.349200 0.0142088
\(605\) 0.500127 0.0203331
\(606\) 13.2296 0.537415
\(607\) 20.7608 0.842654 0.421327 0.906909i \(-0.361564\pi\)
0.421327 + 0.906909i \(0.361564\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −14.6551 −0.593368
\(611\) −30.3105 −1.22623
\(612\) 6.95628 0.281191
\(613\) 3.25738 0.131564 0.0657822 0.997834i \(-0.479046\pi\)
0.0657822 + 0.997834i \(0.479046\pi\)
\(614\) −5.79365 −0.233813
\(615\) 21.8922 0.882777
\(616\) 0 0
\(617\) −9.82446 −0.395518 −0.197759 0.980251i \(-0.563366\pi\)
−0.197759 + 0.980251i \(0.563366\pi\)
\(618\) −7.45400 −0.299844
\(619\) −13.7702 −0.553470 −0.276735 0.960946i \(-0.589252\pi\)
−0.276735 + 0.960946i \(0.589252\pi\)
\(620\) 9.63471 0.386939
\(621\) 7.18070 0.288152
\(622\) −18.6786 −0.748944
\(623\) 0 0
\(624\) 4.23612 0.169581
\(625\) −22.1669 −0.886678
\(626\) −25.7369 −1.02865
\(627\) −3.35203 −0.133867
\(628\) 13.0487 0.520700
\(629\) −1.75450 −0.0699565
\(630\) 0 0
\(631\) 45.3832 1.80668 0.903338 0.428930i \(-0.141109\pi\)
0.903338 + 0.428930i \(0.141109\pi\)
\(632\) −2.27114 −0.0903412
\(633\) 23.3870 0.929549
\(634\) 20.7528 0.824198
\(635\) 25.9030 1.02793
\(636\) −1.75733 −0.0696827
\(637\) 0 0
\(638\) −24.4657 −0.968607
\(639\) −10.4518 −0.413469
\(640\) −2.11806 −0.0837238
\(641\) −21.0779 −0.832525 −0.416263 0.909244i \(-0.636660\pi\)
−0.416263 + 0.909244i \(0.636660\pi\)
\(642\) 8.60708 0.339694
\(643\) 28.6923 1.13151 0.565757 0.824572i \(-0.308584\pi\)
0.565757 + 0.824572i \(0.308584\pi\)
\(644\) 0 0
\(645\) −6.71707 −0.264484
\(646\) −6.95628 −0.273691
\(647\) 0.851482 0.0334752 0.0167376 0.999860i \(-0.494672\pi\)
0.0167376 + 0.999860i \(0.494672\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 43.7325 1.71665
\(650\) −2.17657 −0.0853721
\(651\) 0 0
\(652\) 15.4614 0.605515
\(653\) −1.01471 −0.0397087 −0.0198544 0.999803i \(-0.506320\pi\)
−0.0198544 + 0.999803i \(0.506320\pi\)
\(654\) 6.30746 0.246641
\(655\) −2.16271 −0.0845041
\(656\) −10.3359 −0.403551
\(657\) −12.7667 −0.498077
\(658\) 0 0
\(659\) 10.9687 0.427279 0.213639 0.976913i \(-0.431468\pi\)
0.213639 + 0.976913i \(0.431468\pi\)
\(660\) −7.09982 −0.276360
\(661\) −20.2667 −0.788282 −0.394141 0.919050i \(-0.628958\pi\)
−0.394141 + 0.919050i \(0.628958\pi\)
\(662\) 35.0035 1.36045
\(663\) 29.4677 1.14443
\(664\) −16.5699 −0.643037
\(665\) 0 0
\(666\) 0.252218 0.00977325
\(667\) −52.4103 −2.02933
\(668\) −4.68797 −0.181383
\(669\) −9.80320 −0.379014
\(670\) 9.12072 0.352364
\(671\) 23.1931 0.895359
\(672\) 0 0
\(673\) 40.5792 1.56421 0.782107 0.623145i \(-0.214145\pi\)
0.782107 + 0.623145i \(0.214145\pi\)
\(674\) 9.06565 0.349196
\(675\) 0.513812 0.0197766
\(676\) 4.94475 0.190183
\(677\) −9.11282 −0.350234 −0.175117 0.984548i \(-0.556030\pi\)
−0.175117 + 0.984548i \(0.556030\pi\)
\(678\) 1.97453 0.0758314
\(679\) 0 0
\(680\) −14.7338 −0.565017
\(681\) 18.2835 0.700626
\(682\) −15.2478 −0.583870
\(683\) −4.07959 −0.156101 −0.0780505 0.996949i \(-0.524870\pi\)
−0.0780505 + 0.996949i \(0.524870\pi\)
\(684\) 1.00000 0.0382360
\(685\) −12.9787 −0.495889
\(686\) 0 0
\(687\) 1.44463 0.0551159
\(688\) 3.17133 0.120906
\(689\) −7.44428 −0.283604
\(690\) −15.2092 −0.579004
\(691\) −1.46638 −0.0557839 −0.0278920 0.999611i \(-0.508879\pi\)
−0.0278920 + 0.999611i \(0.508879\pi\)
\(692\) −21.5283 −0.818385
\(693\) 0 0
\(694\) 17.6560 0.670211
\(695\) 38.8615 1.47410
\(696\) 7.29877 0.276659
\(697\) −71.8997 −2.72340
\(698\) −19.9745 −0.756047
\(699\) 15.6808 0.593100
\(700\) 0 0
\(701\) 18.9890 0.717204 0.358602 0.933491i \(-0.383254\pi\)
0.358602 + 0.933491i \(0.383254\pi\)
\(702\) −4.23612 −0.159882
\(703\) −0.252218 −0.00951258
\(704\) 3.35203 0.126334
\(705\) −15.1552 −0.570779
\(706\) −22.5028 −0.846904
\(707\) 0 0
\(708\) −13.0465 −0.490319
\(709\) −20.6079 −0.773947 −0.386974 0.922091i \(-0.626480\pi\)
−0.386974 + 0.922091i \(0.626480\pi\)
\(710\) 22.1377 0.830812
\(711\) 2.27114 0.0851745
\(712\) −3.79365 −0.142173
\(713\) −32.6638 −1.22327
\(714\) 0 0
\(715\) −30.0757 −1.12477
\(716\) 1.83822 0.0686976
\(717\) −3.76818 −0.140725
\(718\) 9.83589 0.367072
\(719\) 5.02797 0.187512 0.0937559 0.995595i \(-0.470113\pi\)
0.0937559 + 0.995595i \(0.470113\pi\)
\(720\) 2.11806 0.0789355
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) −2.99561 −0.111408
\(724\) 20.3848 0.757596
\(725\) −3.75019 −0.139279
\(726\) 0.236125 0.00876342
\(727\) −12.6690 −0.469866 −0.234933 0.972012i \(-0.575487\pi\)
−0.234933 + 0.972012i \(0.575487\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 27.0407 1.00082
\(731\) 22.0607 0.815943
\(732\) −6.91911 −0.255738
\(733\) 3.30832 0.122195 0.0610977 0.998132i \(-0.480540\pi\)
0.0610977 + 0.998132i \(0.480540\pi\)
\(734\) −35.3346 −1.30422
\(735\) 0 0
\(736\) 7.18070 0.264684
\(737\) −14.4344 −0.531698
\(738\) 10.3359 0.380471
\(739\) −20.4897 −0.753726 −0.376863 0.926269i \(-0.622997\pi\)
−0.376863 + 0.926269i \(0.622997\pi\)
\(740\) −0.534213 −0.0196381
\(741\) 4.23612 0.155618
\(742\) 0 0
\(743\) 13.7973 0.506172 0.253086 0.967444i \(-0.418554\pi\)
0.253086 + 0.967444i \(0.418554\pi\)
\(744\) 4.54883 0.166768
\(745\) −46.4073 −1.70023
\(746\) −33.8294 −1.23858
\(747\) 16.5699 0.606261
\(748\) 23.3177 0.852579
\(749\) 0 0
\(750\) −11.6786 −0.426442
\(751\) −8.81343 −0.321607 −0.160803 0.986986i \(-0.551408\pi\)
−0.160803 + 0.986986i \(0.551408\pi\)
\(752\) 7.15524 0.260925
\(753\) −26.2922 −0.958142
\(754\) 30.9185 1.12598
\(755\) 0.739628 0.0269178
\(756\) 0 0
\(757\) −16.4445 −0.597684 −0.298842 0.954303i \(-0.596600\pi\)
−0.298842 + 0.954303i \(0.596600\pi\)
\(758\) −25.4999 −0.926197
\(759\) 24.0700 0.873684
\(760\) −2.11806 −0.0768302
\(761\) −27.6653 −1.00287 −0.501434 0.865196i \(-0.667194\pi\)
−0.501434 + 0.865196i \(0.667194\pi\)
\(762\) 12.2296 0.443031
\(763\) 0 0
\(764\) −5.49789 −0.198907
\(765\) 14.7338 0.532703
\(766\) −35.7186 −1.29057
\(767\) −55.2668 −1.99557
\(768\) −1.00000 −0.0360844
\(769\) −6.46278 −0.233054 −0.116527 0.993188i \(-0.537176\pi\)
−0.116527 + 0.993188i \(0.537176\pi\)
\(770\) 0 0
\(771\) 19.7317 0.710619
\(772\) 7.89871 0.284281
\(773\) −24.5329 −0.882387 −0.441194 0.897412i \(-0.645445\pi\)
−0.441194 + 0.897412i \(0.645445\pi\)
\(774\) −3.17133 −0.113991
\(775\) −2.33724 −0.0839562
\(776\) 6.18070 0.221874
\(777\) 0 0
\(778\) −22.0248 −0.789626
\(779\) −10.3359 −0.370323
\(780\) 8.97238 0.321263
\(781\) −35.0349 −1.25365
\(782\) 49.9510 1.78624
\(783\) −7.29877 −0.260837
\(784\) 0 0
\(785\) 27.6380 0.986441
\(786\) −1.02108 −0.0364207
\(787\) 1.21281 0.0432320 0.0216160 0.999766i \(-0.493119\pi\)
0.0216160 + 0.999766i \(0.493119\pi\)
\(788\) −1.93736 −0.0690155
\(789\) 4.44247 0.158156
\(790\) −4.81042 −0.171147
\(791\) 0 0
\(792\) −3.35203 −0.119109
\(793\) −29.3102 −1.04084
\(794\) −19.7317 −0.700251
\(795\) −3.72214 −0.132011
\(796\) 8.73823 0.309718
\(797\) 3.35867 0.118970 0.0594851 0.998229i \(-0.481054\pi\)
0.0594851 + 0.998229i \(0.481054\pi\)
\(798\) 0 0
\(799\) 49.7738 1.76087
\(800\) 0.513812 0.0181660
\(801\) 3.79365 0.134042
\(802\) −14.8528 −0.524469
\(803\) −42.7944 −1.51018
\(804\) 4.30616 0.151867
\(805\) 0 0
\(806\) 19.2694 0.678736
\(807\) −19.4796 −0.685716
\(808\) −13.2296 −0.465415
\(809\) −11.8885 −0.417977 −0.208988 0.977918i \(-0.567017\pi\)
−0.208988 + 0.977918i \(0.567017\pi\)
\(810\) −2.11806 −0.0744211
\(811\) −27.5357 −0.966908 −0.483454 0.875370i \(-0.660618\pi\)
−0.483454 + 0.875370i \(0.660618\pi\)
\(812\) 0 0
\(813\) −1.57578 −0.0552648
\(814\) 0.845443 0.0296328
\(815\) 32.7482 1.14712
\(816\) −6.95628 −0.243519
\(817\) 3.17133 0.110951
\(818\) −6.03632 −0.211055
\(819\) 0 0
\(820\) −21.8922 −0.764508
\(821\) 38.6740 1.34973 0.674867 0.737940i \(-0.264201\pi\)
0.674867 + 0.737940i \(0.264201\pi\)
\(822\) −6.12761 −0.213725
\(823\) 35.1566 1.22548 0.612741 0.790284i \(-0.290067\pi\)
0.612741 + 0.790284i \(0.290067\pi\)
\(824\) 7.45400 0.259672
\(825\) 1.72231 0.0599633
\(826\) 0 0
\(827\) 0.861710 0.0299646 0.0149823 0.999888i \(-0.495231\pi\)
0.0149823 + 0.999888i \(0.495231\pi\)
\(828\) −7.18070 −0.249547
\(829\) 3.85131 0.133761 0.0668807 0.997761i \(-0.478695\pi\)
0.0668807 + 0.997761i \(0.478695\pi\)
\(830\) −35.0961 −1.21820
\(831\) −9.80612 −0.340170
\(832\) −4.23612 −0.146861
\(833\) 0 0
\(834\) 18.3476 0.635327
\(835\) −9.92942 −0.343622
\(836\) 3.35203 0.115932
\(837\) −4.54883 −0.157231
\(838\) −14.1370 −0.488354
\(839\) −14.4709 −0.499593 −0.249796 0.968298i \(-0.580364\pi\)
−0.249796 + 0.968298i \(0.580364\pi\)
\(840\) 0 0
\(841\) 24.2720 0.836965
\(842\) 35.7565 1.23225
\(843\) 3.60641 0.124211
\(844\) −23.3870 −0.805013
\(845\) 10.4733 0.360292
\(846\) −7.15524 −0.246002
\(847\) 0 0
\(848\) 1.75733 0.0603470
\(849\) 21.5327 0.739002
\(850\) 3.57422 0.122595
\(851\) 1.81110 0.0620838
\(852\) 10.4518 0.358074
\(853\) −22.3323 −0.764642 −0.382321 0.924030i \(-0.624875\pi\)
−0.382321 + 0.924030i \(0.624875\pi\)
\(854\) 0 0
\(855\) 2.11806 0.0724362
\(856\) −8.60708 −0.294184
\(857\) −31.6095 −1.07976 −0.539879 0.841742i \(-0.681530\pi\)
−0.539879 + 0.841742i \(0.681530\pi\)
\(858\) −14.1996 −0.484767
\(859\) 23.7120 0.809042 0.404521 0.914529i \(-0.367438\pi\)
0.404521 + 0.914529i \(0.367438\pi\)
\(860\) 6.71707 0.229050
\(861\) 0 0
\(862\) −17.4242 −0.593471
\(863\) −27.9287 −0.950703 −0.475351 0.879796i \(-0.657679\pi\)
−0.475351 + 0.879796i \(0.657679\pi\)
\(864\) 1.00000 0.0340207
\(865\) −45.5984 −1.55039
\(866\) 0.745628 0.0253375
\(867\) −31.3899 −1.06606
\(868\) 0 0
\(869\) 7.61295 0.258252
\(870\) 15.4592 0.524117
\(871\) 18.2414 0.618088
\(872\) −6.30746 −0.213598
\(873\) −6.18070 −0.209185
\(874\) 7.18070 0.242891
\(875\) 0 0
\(876\) 12.7667 0.431347
\(877\) −12.2884 −0.414949 −0.207474 0.978240i \(-0.566524\pi\)
−0.207474 + 0.978240i \(0.566524\pi\)
\(878\) −11.7289 −0.395830
\(879\) 24.2946 0.819435
\(880\) 7.09982 0.239335
\(881\) −20.8380 −0.702049 −0.351024 0.936366i \(-0.614167\pi\)
−0.351024 + 0.936366i \(0.614167\pi\)
\(882\) 0 0
\(883\) −43.9662 −1.47958 −0.739790 0.672838i \(-0.765075\pi\)
−0.739790 + 0.672838i \(0.765075\pi\)
\(884\) −29.4677 −0.991106
\(885\) −27.6334 −0.928887
\(886\) −6.95611 −0.233695
\(887\) 0.211414 0.00709860 0.00354930 0.999994i \(-0.498870\pi\)
0.00354930 + 0.999994i \(0.498870\pi\)
\(888\) −0.252218 −0.00846388
\(889\) 0 0
\(890\) −8.03519 −0.269340
\(891\) 3.35203 0.112297
\(892\) 9.80320 0.328236
\(893\) 7.15524 0.239441
\(894\) −21.9102 −0.732788
\(895\) 3.89347 0.130144
\(896\) 0 0
\(897\) −30.4184 −1.01564
\(898\) 27.5823 0.920433
\(899\) 33.2009 1.10731
\(900\) −0.513812 −0.0171271
\(901\) 12.2245 0.407257
\(902\) 34.6464 1.15360
\(903\) 0 0
\(904\) −1.97453 −0.0656719
\(905\) 43.1763 1.43523
\(906\) 0.349200 0.0116014
\(907\) 29.9876 0.995722 0.497861 0.867257i \(-0.334119\pi\)
0.497861 + 0.867257i \(0.334119\pi\)
\(908\) −18.2835 −0.606760
\(909\) 13.2296 0.438798
\(910\) 0 0
\(911\) 36.8337 1.22035 0.610177 0.792265i \(-0.291098\pi\)
0.610177 + 0.792265i \(0.291098\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 55.5429 1.83820
\(914\) −20.7733 −0.687118
\(915\) −14.6551 −0.484483
\(916\) −1.44463 −0.0477318
\(917\) 0 0
\(918\) 6.95628 0.229592
\(919\) 20.6736 0.681959 0.340980 0.940071i \(-0.389241\pi\)
0.340980 + 0.940071i \(0.389241\pi\)
\(920\) 15.2092 0.501432
\(921\) −5.79365 −0.190907
\(922\) −33.6610 −1.10857
\(923\) 44.2753 1.45734
\(924\) 0 0
\(925\) 0.129592 0.00426097
\(926\) 18.4933 0.607729
\(927\) −7.45400 −0.244822
\(928\) −7.29877 −0.239594
\(929\) −52.4833 −1.72192 −0.860960 0.508673i \(-0.830136\pi\)
−0.860960 + 0.508673i \(0.830136\pi\)
\(930\) 9.63471 0.315934
\(931\) 0 0
\(932\) −15.6808 −0.513640
\(933\) −18.6786 −0.611510
\(934\) 13.4329 0.439539
\(935\) 49.3883 1.61517
\(936\) 4.23612 0.138462
\(937\) 8.21848 0.268486 0.134243 0.990948i \(-0.457140\pi\)
0.134243 + 0.990948i \(0.457140\pi\)
\(938\) 0 0
\(939\) −25.7369 −0.839890
\(940\) 15.1552 0.494309
\(941\) 21.0823 0.687265 0.343632 0.939104i \(-0.388343\pi\)
0.343632 + 0.939104i \(0.388343\pi\)
\(942\) 13.0487 0.425150
\(943\) 74.2193 2.41691
\(944\) 13.0465 0.424629
\(945\) 0 0
\(946\) −10.6304 −0.345624
\(947\) 54.1725 1.76037 0.880185 0.474631i \(-0.157418\pi\)
0.880185 + 0.474631i \(0.157418\pi\)
\(948\) −2.27114 −0.0737633
\(949\) 54.0814 1.75556
\(950\) 0.513812 0.0166702
\(951\) 20.7528 0.672955
\(952\) 0 0
\(953\) 21.4081 0.693477 0.346739 0.937962i \(-0.387289\pi\)
0.346739 + 0.937962i \(0.387289\pi\)
\(954\) −1.75733 −0.0568957
\(955\) −11.6449 −0.376819
\(956\) 3.76818 0.121872
\(957\) −24.4657 −0.790864
\(958\) 4.05893 0.131138
\(959\) 0 0
\(960\) −2.11806 −0.0683602
\(961\) −10.3081 −0.332521
\(962\) −1.06843 −0.0344475
\(963\) 8.60708 0.277359
\(964\) 2.99561 0.0964821
\(965\) 16.7300 0.538556
\(966\) 0 0
\(967\) 41.4314 1.33234 0.666171 0.745799i \(-0.267932\pi\)
0.666171 + 0.745799i \(0.267932\pi\)
\(968\) −0.236125 −0.00758934
\(969\) −6.95628 −0.223468
\(970\) 13.0911 0.420330
\(971\) 17.7463 0.569506 0.284753 0.958601i \(-0.408088\pi\)
0.284753 + 0.958601i \(0.408088\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −37.5078 −1.20183
\(975\) −2.17657 −0.0697060
\(976\) 6.91911 0.221475
\(977\) −32.7975 −1.04929 −0.524643 0.851323i \(-0.675801\pi\)
−0.524643 + 0.851323i \(0.675801\pi\)
\(978\) 15.4614 0.494401
\(979\) 12.7164 0.406419
\(980\) 0 0
\(981\) 6.30746 0.201382
\(982\) 27.2230 0.868722
\(983\) −20.0570 −0.639717 −0.319859 0.947465i \(-0.603635\pi\)
−0.319859 + 0.947465i \(0.603635\pi\)
\(984\) −10.3359 −0.329498
\(985\) −4.10345 −0.130747
\(986\) −50.7723 −1.61692
\(987\) 0 0
\(988\) −4.23612 −0.134769
\(989\) −22.7724 −0.724119
\(990\) −7.09982 −0.225647
\(991\) 20.3566 0.646648 0.323324 0.946288i \(-0.395200\pi\)
0.323324 + 0.946288i \(0.395200\pi\)
\(992\) −4.54883 −0.144426
\(993\) 35.0035 1.11080
\(994\) 0 0
\(995\) 18.5081 0.586747
\(996\) −16.5699 −0.525038
\(997\) 0.263842 0.00835596 0.00417798 0.999991i \(-0.498670\pi\)
0.00417798 + 0.999991i \(0.498670\pi\)
\(998\) −6.93651 −0.219571
\(999\) 0.252218 0.00797982
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 5586.2.a.bw.1.3 4
7.2 even 3 798.2.j.l.571.2 yes 8
7.4 even 3 798.2.j.l.457.2 8
7.6 odd 2 5586.2.a.bz.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.l.457.2 8 7.4 even 3
798.2.j.l.571.2 yes 8 7.2 even 3
5586.2.a.bw.1.3 4 1.1 even 1 trivial
5586.2.a.bz.1.2 4 7.6 odd 2