Properties

Label 5586.2.a.bw.1.2
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
Root \(-1.25548\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.14924 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.14924 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.14924 q^{10} -2.16830 q^{11} -1.00000 q^{12} +2.29847 q^{13} +1.14924 q^{15} +1.00000 q^{16} -4.99679 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.14924 q^{20} +2.16830 q^{22} -6.91567 q^{23} +1.00000 q^{24} -3.67925 q^{25} -2.29847 q^{26} -1.00000 q^{27} +3.76643 q^{29} -1.14924 q^{30} +4.38565 q^{31} -1.00000 q^{32} +2.16830 q^{33} +4.99679 q^{34} +1.00000 q^{36} +0.660191 q^{37} -1.00000 q^{38} -2.29847 q^{39} +1.14924 q^{40} +0.806583 q^{41} -2.08397 q^{43} -2.16830 q^{44} -1.14924 q^{45} +6.91567 q^{46} -3.72225 q^{47} -1.00000 q^{48} +3.67925 q^{50} +4.99679 q^{51} +2.29847 q^{52} -11.2713 q^{53} +1.00000 q^{54} +2.49189 q^{55} -1.00000 q^{57} -3.76643 q^{58} +10.4266 q^{59} +1.14924 q^{60} +2.57622 q^{61} -4.38565 q^{62} +1.00000 q^{64} -2.64149 q^{65} -2.16830 q^{66} +8.88568 q^{67} -4.99679 q^{68} +6.91567 q^{69} -0.323591 q^{71} -1.00000 q^{72} -4.72830 q^{73} -0.660191 q^{74} +3.67925 q^{75} +1.00000 q^{76} +2.29847 q^{78} -7.59208 q^{79} -1.14924 q^{80} +1.00000 q^{81} -0.806583 q^{82} +3.17435 q^{83} +5.74250 q^{85} +2.08397 q^{86} -3.76643 q^{87} +2.16830 q^{88} +12.3776 q^{89} +1.14924 q^{90} -6.91567 q^{92} -4.38565 q^{93} +3.72225 q^{94} -1.14924 q^{95} +1.00000 q^{96} -5.91567 q^{97} -2.16830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} - 2q^{11} - 4q^{12} + 4q^{16} + 10q^{17} - 4q^{18} + 4q^{19} + 2q^{22} - 5q^{23} + 4q^{24} + 4q^{25} - 4q^{27} - 3q^{29} + 9q^{31} - 4q^{32} + 2q^{33} - 10q^{34} + 4q^{36} - 14q^{37} - 4q^{38} + 4q^{41} + 21q^{43} - 2q^{44} + 5q^{46} + 7q^{47} - 4q^{48} - 4q^{50} - 10q^{51} - 7q^{53} + 4q^{54} - 4q^{57} + 3q^{58} + 7q^{59} + 23q^{61} - 9q^{62} + 4q^{64} - 48q^{65} - 2q^{66} + 6q^{67} + 10q^{68} + 5q^{69} + 2q^{71} - 4q^{72} - 5q^{73} + 14q^{74} - 4q^{75} + 4q^{76} - 11q^{79} + 4q^{81} - 4q^{82} + 14q^{83} + 6q^{85} - 21q^{86} + 3q^{87} + 2q^{88} + 10q^{89} - 5q^{92} - 9q^{93} - 7q^{94} + 4q^{96} - q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) −1.14924 −0.513954 −0.256977 0.966417i \(-0.582727\pi\)
−0.256977 + 0.966417i \(0.582727\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.14924 0.363421
\(11\) −2.16830 −0.653767 −0.326884 0.945065i \(-0.605998\pi\)
−0.326884 + 0.945065i \(0.605998\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.29847 0.637482 0.318741 0.947842i \(-0.396740\pi\)
0.318741 + 0.947842i \(0.396740\pi\)
\(14\) 0 0
\(15\) 1.14924 0.296732
\(16\) 1.00000 0.250000
\(17\) −4.99679 −1.21190 −0.605950 0.795503i \(-0.707207\pi\)
−0.605950 + 0.795503i \(0.707207\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) −1.14924 −0.256977
\(21\) 0 0
\(22\) 2.16830 0.462283
\(23\) −6.91567 −1.44202 −0.721008 0.692926i \(-0.756321\pi\)
−0.721008 + 0.692926i \(0.756321\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.67925 −0.735851
\(26\) −2.29847 −0.450768
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.76643 0.699409 0.349704 0.936860i \(-0.386282\pi\)
0.349704 + 0.936860i \(0.386282\pi\)
\(30\) −1.14924 −0.209821
\(31\) 4.38565 0.787686 0.393843 0.919178i \(-0.371145\pi\)
0.393843 + 0.919178i \(0.371145\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.16830 0.377453
\(34\) 4.99679 0.856943
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.660191 0.108535 0.0542674 0.998526i \(-0.482718\pi\)
0.0542674 + 0.998526i \(0.482718\pi\)
\(38\) −1.00000 −0.162221
\(39\) −2.29847 −0.368050
\(40\) 1.14924 0.181710
\(41\) 0.806583 0.125967 0.0629836 0.998015i \(-0.479938\pi\)
0.0629836 + 0.998015i \(0.479938\pi\)
\(42\) 0 0
\(43\) −2.08397 −0.317802 −0.158901 0.987295i \(-0.550795\pi\)
−0.158901 + 0.987295i \(0.550795\pi\)
\(44\) −2.16830 −0.326884
\(45\) −1.14924 −0.171318
\(46\) 6.91567 1.01966
\(47\) −3.72225 −0.542946 −0.271473 0.962446i \(-0.587511\pi\)
−0.271473 + 0.962446i \(0.587511\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 3.67925 0.520325
\(51\) 4.99679 0.699691
\(52\) 2.29847 0.318741
\(53\) −11.2713 −1.54824 −0.774118 0.633041i \(-0.781806\pi\)
−0.774118 + 0.633041i \(0.781806\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.49189 0.336006
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) −3.76643 −0.494557
\(59\) 10.4266 1.35743 0.678715 0.734401i \(-0.262537\pi\)
0.678715 + 0.734401i \(0.262537\pi\)
\(60\) 1.14924 0.148366
\(61\) 2.57622 0.329852 0.164926 0.986306i \(-0.447262\pi\)
0.164926 + 0.986306i \(0.447262\pi\)
\(62\) −4.38565 −0.556978
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.64149 −0.327637
\(66\) −2.16830 −0.266899
\(67\) 8.88568 1.08556 0.542779 0.839875i \(-0.317372\pi\)
0.542779 + 0.839875i \(0.317372\pi\)
\(68\) −4.99679 −0.605950
\(69\) 6.91567 0.832549
\(70\) 0 0
\(71\) −0.323591 −0.0384031 −0.0192016 0.999816i \(-0.506112\pi\)
−0.0192016 + 0.999816i \(0.506112\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.72830 −0.553406 −0.276703 0.960955i \(-0.589242\pi\)
−0.276703 + 0.960955i \(0.589242\pi\)
\(74\) −0.660191 −0.0767457
\(75\) 3.67925 0.424844
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 2.29847 0.260251
\(79\) −7.59208 −0.854175 −0.427088 0.904210i \(-0.640460\pi\)
−0.427088 + 0.904210i \(0.640460\pi\)
\(80\) −1.14924 −0.128489
\(81\) 1.00000 0.111111
\(82\) −0.806583 −0.0890722
\(83\) 3.17435 0.348431 0.174215 0.984708i \(-0.444261\pi\)
0.174215 + 0.984708i \(0.444261\pi\)
\(84\) 0 0
\(85\) 5.74250 0.622861
\(86\) 2.08397 0.224720
\(87\) −3.76643 −0.403804
\(88\) 2.16830 0.231142
\(89\) 12.3776 1.31202 0.656010 0.754752i \(-0.272243\pi\)
0.656010 + 0.754752i \(0.272243\pi\)
\(90\) 1.14924 0.121140
\(91\) 0 0
\(92\) −6.91567 −0.721008
\(93\) −4.38565 −0.454771
\(94\) 3.72225 0.383921
\(95\) −1.14924 −0.117909
\(96\) 1.00000 0.102062
\(97\) −5.91567 −0.600645 −0.300323 0.953838i \(-0.597094\pi\)
−0.300323 + 0.953838i \(0.597094\pi\)
\(98\) 0 0
\(99\) −2.16830 −0.217922
\(100\) −3.67925 −0.367925
\(101\) −12.8683 −1.28044 −0.640221 0.768191i \(-0.721157\pi\)
−0.640221 + 0.768191i \(0.721157\pi\)
\(102\) −4.99679 −0.494756
\(103\) 6.95582 0.685377 0.342689 0.939449i \(-0.388662\pi\)
0.342689 + 0.939449i \(0.388662\pi\)
\(104\) −2.29847 −0.225384
\(105\) 0 0
\(106\) 11.2713 1.09477
\(107\) −12.3987 −1.19862 −0.599312 0.800516i \(-0.704559\pi\)
−0.599312 + 0.800516i \(0.704559\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.0568 1.72953 0.864765 0.502176i \(-0.167467\pi\)
0.864765 + 0.502176i \(0.167467\pi\)
\(110\) −2.49189 −0.237592
\(111\) −0.660191 −0.0626626
\(112\) 0 0
\(113\) −8.63792 −0.812587 −0.406294 0.913743i \(-0.633179\pi\)
−0.406294 + 0.913743i \(0.633179\pi\)
\(114\) 1.00000 0.0936586
\(115\) 7.94774 0.741131
\(116\) 3.76643 0.349704
\(117\) 2.29847 0.212494
\(118\) −10.4266 −0.959848
\(119\) 0 0
\(120\) −1.14924 −0.104911
\(121\) −6.29847 −0.572589
\(122\) −2.57622 −0.233240
\(123\) −0.806583 −0.0727271
\(124\) 4.38565 0.393843
\(125\) 9.97452 0.892148
\(126\) 0 0
\(127\) −13.8683 −1.23061 −0.615305 0.788289i \(-0.710967\pi\)
−0.615305 + 0.788289i \(0.710967\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.08397 0.183483
\(130\) 2.64149 0.231674
\(131\) 12.2113 1.06691 0.533453 0.845830i \(-0.320894\pi\)
0.533453 + 0.845830i \(0.320894\pi\)
\(132\) 2.16830 0.188726
\(133\) 0 0
\(134\) −8.88568 −0.767606
\(135\) 1.14924 0.0989106
\(136\) 4.99679 0.428471
\(137\) 11.0808 0.946693 0.473347 0.880876i \(-0.343046\pi\)
0.473347 + 0.880876i \(0.343046\pi\)
\(138\) −6.91567 −0.588701
\(139\) −11.0175 −0.934494 −0.467247 0.884127i \(-0.654754\pi\)
−0.467247 + 0.884127i \(0.654754\pi\)
\(140\) 0 0
\(141\) 3.72225 0.313470
\(142\) 0.323591 0.0271551
\(143\) −4.98378 −0.416765
\(144\) 1.00000 0.0833333
\(145\) −4.32852 −0.359464
\(146\) 4.72830 0.391317
\(147\) 0 0
\(148\) 0.660191 0.0542674
\(149\) −21.2170 −1.73816 −0.869082 0.494669i \(-0.835289\pi\)
−0.869082 + 0.494669i \(0.835289\pi\)
\(150\) −3.67925 −0.300410
\(151\) 9.40187 0.765113 0.382557 0.923932i \(-0.375044\pi\)
0.382557 + 0.923932i \(0.375044\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −4.99679 −0.403967
\(154\) 0 0
\(155\) −5.04015 −0.404835
\(156\) −2.29847 −0.184025
\(157\) 6.14721 0.490601 0.245300 0.969447i \(-0.421113\pi\)
0.245300 + 0.969447i \(0.421113\pi\)
\(158\) 7.59208 0.603993
\(159\) 11.2713 0.893875
\(160\) 1.14924 0.0908552
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −8.60793 −0.674225 −0.337113 0.941464i \(-0.609450\pi\)
−0.337113 + 0.941464i \(0.609450\pi\)
\(164\) 0.806583 0.0629836
\(165\) −2.49189 −0.193993
\(166\) −3.17435 −0.246378
\(167\) 11.9749 0.926644 0.463322 0.886190i \(-0.346657\pi\)
0.463322 + 0.886190i \(0.346657\pi\)
\(168\) 0 0
\(169\) −7.71702 −0.593617
\(170\) −5.74250 −0.440429
\(171\) 1.00000 0.0764719
\(172\) −2.08397 −0.158901
\(173\) 8.10185 0.615972 0.307986 0.951391i \(-0.400345\pi\)
0.307986 + 0.951391i \(0.400345\pi\)
\(174\) 3.76643 0.285532
\(175\) 0 0
\(176\) −2.16830 −0.163442
\(177\) −10.4266 −0.783713
\(178\) −12.3776 −0.927738
\(179\) −6.84755 −0.511810 −0.255905 0.966702i \(-0.582373\pi\)
−0.255905 + 0.966702i \(0.582373\pi\)
\(180\) −1.14924 −0.0856591
\(181\) −16.5905 −1.23316 −0.616582 0.787291i \(-0.711483\pi\)
−0.616582 + 0.787291i \(0.711483\pi\)
\(182\) 0 0
\(183\) −2.57622 −0.190440
\(184\) 6.91567 0.509830
\(185\) −0.758716 −0.0557819
\(186\) 4.38565 0.321572
\(187\) 10.8345 0.792300
\(188\) −3.72225 −0.271473
\(189\) 0 0
\(190\) 1.14924 0.0833744
\(191\) 15.8902 1.14977 0.574887 0.818233i \(-0.305046\pi\)
0.574887 + 0.818233i \(0.305046\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.49676 0.467647 0.233824 0.972279i \(-0.424876\pi\)
0.233824 + 0.972279i \(0.424876\pi\)
\(194\) 5.91567 0.424720
\(195\) 2.64149 0.189161
\(196\) 0 0
\(197\) 1.06491 0.0758713 0.0379357 0.999280i \(-0.487922\pi\)
0.0379357 + 0.999280i \(0.487922\pi\)
\(198\) 2.16830 0.154094
\(199\) 23.5917 1.67237 0.836186 0.548446i \(-0.184780\pi\)
0.836186 + 0.548446i \(0.184780\pi\)
\(200\) 3.67925 0.260163
\(201\) −8.88568 −0.626748
\(202\) 12.8683 0.905409
\(203\) 0 0
\(204\) 4.99679 0.349845
\(205\) −0.926955 −0.0647414
\(206\) −6.95582 −0.484635
\(207\) −6.91567 −0.480672
\(208\) 2.29847 0.159370
\(209\) −2.16830 −0.149984
\(210\) 0 0
\(211\) 17.8699 1.23022 0.615109 0.788442i \(-0.289112\pi\)
0.615109 + 0.788442i \(0.289112\pi\)
\(212\) −11.2713 −0.774118
\(213\) 0.323591 0.0221721
\(214\) 12.3987 0.847555
\(215\) 2.39497 0.163336
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −18.0568 −1.22296
\(219\) 4.72830 0.319509
\(220\) 2.49189 0.168003
\(221\) −11.4850 −0.772564
\(222\) 0.660191 0.0443091
\(223\) 4.44605 0.297729 0.148865 0.988858i \(-0.452438\pi\)
0.148865 + 0.988858i \(0.452438\pi\)
\(224\) 0 0
\(225\) −3.67925 −0.245284
\(226\) 8.63792 0.574586
\(227\) 20.0938 1.33367 0.666835 0.745205i \(-0.267649\pi\)
0.666835 + 0.745205i \(0.267649\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 17.9555 1.18653 0.593265 0.805007i \(-0.297839\pi\)
0.593265 + 0.805007i \(0.297839\pi\)
\(230\) −7.94774 −0.524059
\(231\) 0 0
\(232\) −3.76643 −0.247278
\(233\) 10.2539 0.671757 0.335879 0.941905i \(-0.390967\pi\)
0.335879 + 0.941905i \(0.390967\pi\)
\(234\) −2.29847 −0.150256
\(235\) 4.27775 0.279050
\(236\) 10.4266 0.678715
\(237\) 7.59208 0.493158
\(238\) 0 0
\(239\) 1.73965 0.112529 0.0562644 0.998416i \(-0.482081\pi\)
0.0562644 + 0.998416i \(0.482081\pi\)
\(240\) 1.14924 0.0741829
\(241\) −20.8492 −1.34302 −0.671508 0.740997i \(-0.734353\pi\)
−0.671508 + 0.740997i \(0.734353\pi\)
\(242\) 6.29847 0.404881
\(243\) −1.00000 −0.0641500
\(244\) 2.57622 0.164926
\(245\) 0 0
\(246\) 0.806583 0.0514259
\(247\) 2.29847 0.146248
\(248\) −4.38565 −0.278489
\(249\) −3.17435 −0.201166
\(250\) −9.97452 −0.630844
\(251\) 3.19663 0.201769 0.100885 0.994898i \(-0.467833\pi\)
0.100885 + 0.994898i \(0.467833\pi\)
\(252\) 0 0
\(253\) 14.9952 0.942743
\(254\) 13.8683 0.870173
\(255\) −5.74250 −0.359609
\(256\) 1.00000 0.0625000
\(257\) −15.0219 −0.937041 −0.468520 0.883453i \(-0.655213\pi\)
−0.468520 + 0.883453i \(0.655213\pi\)
\(258\) −2.08397 −0.129742
\(259\) 0 0
\(260\) −2.64149 −0.163818
\(261\) 3.76643 0.233136
\(262\) −12.2113 −0.754416
\(263\) 10.6760 0.658313 0.329157 0.944275i \(-0.393236\pi\)
0.329157 + 0.944275i \(0.393236\pi\)
\(264\) −2.16830 −0.133450
\(265\) 12.9534 0.795723
\(266\) 0 0
\(267\) −12.3776 −0.757495
\(268\) 8.88568 0.542779
\(269\) −3.24906 −0.198099 −0.0990494 0.995083i \(-0.531580\pi\)
−0.0990494 + 0.995083i \(0.531580\pi\)
\(270\) −1.14924 −0.0699403
\(271\) 18.0349 1.09554 0.547772 0.836628i \(-0.315476\pi\)
0.547772 + 0.836628i \(0.315476\pi\)
\(272\) −4.99679 −0.302975
\(273\) 0 0
\(274\) −11.0808 −0.669413
\(275\) 7.97773 0.481075
\(276\) 6.91567 0.416274
\(277\) 22.2839 1.33891 0.669455 0.742852i \(-0.266528\pi\)
0.669455 + 0.742852i \(0.266528\pi\)
\(278\) 11.0175 0.660787
\(279\) 4.38565 0.262562
\(280\) 0 0
\(281\) 7.10790 0.424022 0.212011 0.977267i \(-0.431999\pi\)
0.212011 + 0.977267i \(0.431999\pi\)
\(282\) −3.72225 −0.221657
\(283\) −15.7474 −0.936084 −0.468042 0.883706i \(-0.655040\pi\)
−0.468042 + 0.883706i \(0.655040\pi\)
\(284\) −0.323591 −0.0192016
\(285\) 1.14924 0.0680749
\(286\) 4.98378 0.294697
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 7.96793 0.468702
\(290\) 4.32852 0.254180
\(291\) 5.91567 0.346783
\(292\) −4.72830 −0.276703
\(293\) 22.0139 1.28607 0.643034 0.765837i \(-0.277675\pi\)
0.643034 + 0.765837i \(0.277675\pi\)
\(294\) 0 0
\(295\) −11.9827 −0.697657
\(296\) −0.660191 −0.0383728
\(297\) 2.16830 0.125818
\(298\) 21.2170 1.22907
\(299\) −15.8955 −0.919260
\(300\) 3.67925 0.212422
\(301\) 0 0
\(302\) −9.40187 −0.541017
\(303\) 12.8683 0.739263
\(304\) 1.00000 0.0573539
\(305\) −2.96069 −0.169529
\(306\) 4.99679 0.285648
\(307\) 14.3776 0.820571 0.410286 0.911957i \(-0.365429\pi\)
0.410286 + 0.911957i \(0.365429\pi\)
\(308\) 0 0
\(309\) −6.95582 −0.395703
\(310\) 5.04015 0.286261
\(311\) −2.97452 −0.168670 −0.0843348 0.996437i \(-0.526877\pi\)
−0.0843348 + 0.996437i \(0.526877\pi\)
\(312\) 2.29847 0.130125
\(313\) −16.7588 −0.947265 −0.473633 0.880723i \(-0.657058\pi\)
−0.473633 + 0.880723i \(0.657058\pi\)
\(314\) −6.14721 −0.346907
\(315\) 0 0
\(316\) −7.59208 −0.427088
\(317\) −2.81061 −0.157860 −0.0789298 0.996880i \(-0.525150\pi\)
−0.0789298 + 0.996880i \(0.525150\pi\)
\(318\) −11.2713 −0.632065
\(319\) −8.16675 −0.457250
\(320\) −1.14924 −0.0642443
\(321\) 12.3987 0.692026
\(322\) 0 0
\(323\) −4.99679 −0.278029
\(324\) 1.00000 0.0555556
\(325\) −8.45667 −0.469092
\(326\) 8.60793 0.476749
\(327\) −18.0568 −0.998545
\(328\) −0.806583 −0.0445361
\(329\) 0 0
\(330\) 2.49189 0.137174
\(331\) −10.1391 −0.557294 −0.278647 0.960394i \(-0.589886\pi\)
−0.278647 + 0.960394i \(0.589886\pi\)
\(332\) 3.17435 0.174215
\(333\) 0.660191 0.0361782
\(334\) −11.9749 −0.655236
\(335\) −10.2118 −0.557928
\(336\) 0 0
\(337\) 21.4364 1.16772 0.583858 0.811856i \(-0.301542\pi\)
0.583858 + 0.811856i \(0.301542\pi\)
\(338\) 7.71702 0.419750
\(339\) 8.63792 0.469147
\(340\) 5.74250 0.311431
\(341\) −9.50941 −0.514963
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) 2.08397 0.112360
\(345\) −7.94774 −0.427892
\(346\) −8.10185 −0.435558
\(347\) 29.1826 1.56660 0.783302 0.621641i \(-0.213534\pi\)
0.783302 + 0.621641i \(0.213534\pi\)
\(348\) −3.76643 −0.201902
\(349\) 9.36208 0.501141 0.250570 0.968098i \(-0.419382\pi\)
0.250570 + 0.968098i \(0.419382\pi\)
\(350\) 0 0
\(351\) −2.29847 −0.122683
\(352\) 2.16830 0.115571
\(353\) 33.5994 1.78832 0.894159 0.447750i \(-0.147775\pi\)
0.894159 + 0.447750i \(0.147775\pi\)
\(354\) 10.4266 0.554169
\(355\) 0.371882 0.0197375
\(356\) 12.3776 0.656010
\(357\) 0 0
\(358\) 6.84755 0.361905
\(359\) −24.3630 −1.28583 −0.642915 0.765937i \(-0.722275\pi\)
−0.642915 + 0.765937i \(0.722275\pi\)
\(360\) 1.14924 0.0605701
\(361\) 1.00000 0.0526316
\(362\) 16.5905 0.871979
\(363\) 6.29847 0.330584
\(364\) 0 0
\(365\) 5.43394 0.284426
\(366\) 2.57622 0.134661
\(367\) −33.1571 −1.73079 −0.865394 0.501092i \(-0.832932\pi\)
−0.865394 + 0.501092i \(0.832932\pi\)
\(368\) −6.91567 −0.360504
\(369\) 0.806583 0.0419890
\(370\) 0.758716 0.0394438
\(371\) 0 0
\(372\) −4.38565 −0.227385
\(373\) −22.5460 −1.16739 −0.583693 0.811974i \(-0.698393\pi\)
−0.583693 + 0.811974i \(0.698393\pi\)
\(374\) −10.8345 −0.560241
\(375\) −9.97452 −0.515082
\(376\) 3.72225 0.191960
\(377\) 8.65704 0.445860
\(378\) 0 0
\(379\) 18.7616 0.963717 0.481858 0.876249i \(-0.339962\pi\)
0.481858 + 0.876249i \(0.339962\pi\)
\(380\) −1.14924 −0.0589546
\(381\) 13.8683 0.710494
\(382\) −15.8902 −0.813013
\(383\) −8.11770 −0.414795 −0.207398 0.978257i \(-0.566499\pi\)
−0.207398 + 0.978257i \(0.566499\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −6.49676 −0.330676
\(387\) −2.08397 −0.105934
\(388\) −5.91567 −0.300323
\(389\) −35.0034 −1.77474 −0.887371 0.461056i \(-0.847471\pi\)
−0.887371 + 0.461056i \(0.847471\pi\)
\(390\) −2.64149 −0.133757
\(391\) 34.5562 1.74758
\(392\) 0 0
\(393\) −12.2113 −0.615978
\(394\) −1.06491 −0.0536491
\(395\) 8.72510 0.439007
\(396\) −2.16830 −0.108961
\(397\) 15.0219 0.753928 0.376964 0.926228i \(-0.376968\pi\)
0.376964 + 0.926228i \(0.376968\pi\)
\(398\) −23.5917 −1.18255
\(399\) 0 0
\(400\) −3.67925 −0.183963
\(401\) −26.6287 −1.32977 −0.664886 0.746945i \(-0.731520\pi\)
−0.664886 + 0.746945i \(0.731520\pi\)
\(402\) 8.88568 0.443178
\(403\) 10.0803 0.502136
\(404\) −12.8683 −0.640221
\(405\) −1.14924 −0.0571060
\(406\) 0 0
\(407\) −1.43149 −0.0709565
\(408\) −4.99679 −0.247378
\(409\) 27.6489 1.36715 0.683575 0.729880i \(-0.260424\pi\)
0.683575 + 0.729880i \(0.260424\pi\)
\(410\) 0.926955 0.0457791
\(411\) −11.0808 −0.546574
\(412\) 6.95582 0.342689
\(413\) 0 0
\(414\) 6.91567 0.339887
\(415\) −3.64808 −0.179077
\(416\) −2.29847 −0.112692
\(417\) 11.0175 0.539530
\(418\) 2.16830 0.106055
\(419\) 1.91888 0.0937433 0.0468716 0.998901i \(-0.485075\pi\)
0.0468716 + 0.998901i \(0.485075\pi\)
\(420\) 0 0
\(421\) 25.9815 1.26626 0.633130 0.774046i \(-0.281770\pi\)
0.633130 + 0.774046i \(0.281770\pi\)
\(422\) −17.8699 −0.869895
\(423\) −3.72225 −0.180982
\(424\) 11.2713 0.547384
\(425\) 18.3845 0.891778
\(426\) −0.323591 −0.0156780
\(427\) 0 0
\(428\) −12.3987 −0.599312
\(429\) 4.98378 0.240619
\(430\) −2.39497 −0.115496
\(431\) 0.965082 0.0464864 0.0232432 0.999730i \(-0.492601\pi\)
0.0232432 + 0.999730i \(0.492601\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −5.93960 −0.285439 −0.142720 0.989763i \(-0.545585\pi\)
−0.142720 + 0.989763i \(0.545585\pi\)
\(434\) 0 0
\(435\) 4.32852 0.207537
\(436\) 18.0568 0.864765
\(437\) −6.91567 −0.330821
\(438\) −4.72830 −0.225927
\(439\) 21.5921 1.03053 0.515267 0.857030i \(-0.327693\pi\)
0.515267 + 0.857030i \(0.327693\pi\)
\(440\) −2.49189 −0.118796
\(441\) 0 0
\(442\) 11.4850 0.546286
\(443\) 13.9344 0.662042 0.331021 0.943623i \(-0.392607\pi\)
0.331021 + 0.943623i \(0.392607\pi\)
\(444\) −0.660191 −0.0313313
\(445\) −14.2248 −0.674318
\(446\) −4.44605 −0.210527
\(447\) 21.2170 1.00353
\(448\) 0 0
\(449\) 14.3273 0.676149 0.338074 0.941119i \(-0.390224\pi\)
0.338074 + 0.941119i \(0.390224\pi\)
\(450\) 3.67925 0.173442
\(451\) −1.74891 −0.0823532
\(452\) −8.63792 −0.406294
\(453\) −9.40187 −0.441738
\(454\) −20.0938 −0.943047
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 32.2981 1.51084 0.755421 0.655240i \(-0.227432\pi\)
0.755421 + 0.655240i \(0.227432\pi\)
\(458\) −17.9555 −0.839003
\(459\) 4.99679 0.233230
\(460\) 7.94774 0.370565
\(461\) 0.375849 0.0175050 0.00875252 0.999962i \(-0.497214\pi\)
0.00875252 + 0.999962i \(0.497214\pi\)
\(462\) 0 0
\(463\) 7.80824 0.362880 0.181440 0.983402i \(-0.441924\pi\)
0.181440 + 0.983402i \(0.441924\pi\)
\(464\) 3.76643 0.174852
\(465\) 5.04015 0.233731
\(466\) −10.2539 −0.475004
\(467\) −12.2555 −0.567116 −0.283558 0.958955i \(-0.591515\pi\)
−0.283558 + 0.958955i \(0.591515\pi\)
\(468\) 2.29847 0.106247
\(469\) 0 0
\(470\) −4.27775 −0.197318
\(471\) −6.14721 −0.283249
\(472\) −10.4266 −0.479924
\(473\) 4.51867 0.207769
\(474\) −7.59208 −0.348716
\(475\) −3.67925 −0.168816
\(476\) 0 0
\(477\) −11.2713 −0.516079
\(478\) −1.73965 −0.0795699
\(479\) 27.0751 1.23709 0.618546 0.785749i \(-0.287722\pi\)
0.618546 + 0.785749i \(0.287722\pi\)
\(480\) −1.14924 −0.0524553
\(481\) 1.51743 0.0691889
\(482\) 20.8492 0.949656
\(483\) 0 0
\(484\) −6.29847 −0.286294
\(485\) 6.79850 0.308704
\(486\) 1.00000 0.0453609
\(487\) 29.7498 1.34809 0.674047 0.738688i \(-0.264554\pi\)
0.674047 + 0.738688i \(0.264554\pi\)
\(488\) −2.57622 −0.116620
\(489\) 8.60793 0.389264
\(490\) 0 0
\(491\) 18.4381 0.832099 0.416050 0.909342i \(-0.363414\pi\)
0.416050 + 0.909342i \(0.363414\pi\)
\(492\) −0.806583 −0.0363636
\(493\) −18.8201 −0.847613
\(494\) −2.29847 −0.103413
\(495\) 2.49189 0.112002
\(496\) 4.38565 0.196922
\(497\) 0 0
\(498\) 3.17435 0.142246
\(499\) 33.1570 1.48431 0.742156 0.670227i \(-0.233803\pi\)
0.742156 + 0.670227i \(0.233803\pi\)
\(500\) 9.97452 0.446074
\(501\) −11.9749 −0.534998
\(502\) −3.19663 −0.142672
\(503\) 12.8678 0.573747 0.286873 0.957968i \(-0.407384\pi\)
0.286873 + 0.957968i \(0.407384\pi\)
\(504\) 0 0
\(505\) 14.7887 0.658089
\(506\) −14.9952 −0.666620
\(507\) 7.71702 0.342725
\(508\) −13.8683 −0.615305
\(509\) 8.70235 0.385725 0.192862 0.981226i \(-0.438223\pi\)
0.192862 + 0.981226i \(0.438223\pi\)
\(510\) 5.74250 0.254282
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 15.0219 0.662588
\(515\) −7.99389 −0.352253
\(516\) 2.08397 0.0917416
\(517\) 8.07096 0.354960
\(518\) 0 0
\(519\) −8.10185 −0.355632
\(520\) 2.64149 0.115837
\(521\) 37.7965 1.65590 0.827948 0.560805i \(-0.189508\pi\)
0.827948 + 0.560805i \(0.189508\pi\)
\(522\) −3.76643 −0.164852
\(523\) 42.3364 1.85124 0.925620 0.378455i \(-0.123545\pi\)
0.925620 + 0.378455i \(0.123545\pi\)
\(524\) 12.2113 0.533453
\(525\) 0 0
\(526\) −10.6760 −0.465498
\(527\) −21.9142 −0.954597
\(528\) 2.16830 0.0943632
\(529\) 24.8265 1.07941
\(530\) −12.9534 −0.562661
\(531\) 10.4266 0.452477
\(532\) 0 0
\(533\) 1.85391 0.0803018
\(534\) 12.3776 0.535630
\(535\) 14.2490 0.616038
\(536\) −8.88568 −0.383803
\(537\) 6.84755 0.295494
\(538\) 3.24906 0.140077
\(539\) 0 0
\(540\) 1.14924 0.0494553
\(541\) 8.01146 0.344440 0.172220 0.985059i \(-0.444906\pi\)
0.172220 + 0.985059i \(0.444906\pi\)
\(542\) −18.0349 −0.774666
\(543\) 16.5905 0.711968
\(544\) 4.99679 0.214236
\(545\) −20.7516 −0.888900
\(546\) 0 0
\(547\) −34.1587 −1.46052 −0.730260 0.683170i \(-0.760601\pi\)
−0.730260 + 0.683170i \(0.760601\pi\)
\(548\) 11.0808 0.473347
\(549\) 2.57622 0.109951
\(550\) −7.97773 −0.340171
\(551\) 3.76643 0.160455
\(552\) −6.91567 −0.294350
\(553\) 0 0
\(554\) −22.2839 −0.946753
\(555\) 0.758716 0.0322057
\(556\) −11.0175 −0.467247
\(557\) −40.4791 −1.71515 −0.857576 0.514357i \(-0.828031\pi\)
−0.857576 + 0.514357i \(0.828031\pi\)
\(558\) −4.38565 −0.185659
\(559\) −4.78995 −0.202593
\(560\) 0 0
\(561\) −10.8345 −0.457435
\(562\) −7.10790 −0.299829
\(563\) 18.2044 0.767224 0.383612 0.923494i \(-0.374680\pi\)
0.383612 + 0.923494i \(0.374680\pi\)
\(564\) 3.72225 0.156735
\(565\) 9.92702 0.417633
\(566\) 15.7474 0.661911
\(567\) 0 0
\(568\) 0.323591 0.0135776
\(569\) −16.0253 −0.671815 −0.335908 0.941895i \(-0.609043\pi\)
−0.335908 + 0.941895i \(0.609043\pi\)
\(570\) −1.14924 −0.0481362
\(571\) 7.02630 0.294041 0.147021 0.989133i \(-0.453032\pi\)
0.147021 + 0.989133i \(0.453032\pi\)
\(572\) −4.98378 −0.208382
\(573\) −15.8902 −0.663822
\(574\) 0 0
\(575\) 25.4445 1.06111
\(576\) 1.00000 0.0416667
\(577\) 22.9109 0.953794 0.476897 0.878959i \(-0.341762\pi\)
0.476897 + 0.878959i \(0.341762\pi\)
\(578\) −7.96793 −0.331422
\(579\) −6.49676 −0.269996
\(580\) −4.32852 −0.179732
\(581\) 0 0
\(582\) −5.91567 −0.245212
\(583\) 24.4396 1.01219
\(584\) 4.72830 0.195659
\(585\) −2.64149 −0.109212
\(586\) −22.0139 −0.909388
\(587\) −9.55395 −0.394334 −0.197167 0.980370i \(-0.563174\pi\)
−0.197167 + 0.980370i \(0.563174\pi\)
\(588\) 0 0
\(589\) 4.38565 0.180708
\(590\) 11.9827 0.493318
\(591\) −1.06491 −0.0438043
\(592\) 0.660191 0.0271337
\(593\) 8.10672 0.332903 0.166452 0.986050i \(-0.446769\pi\)
0.166452 + 0.986050i \(0.446769\pi\)
\(594\) −2.16830 −0.0889664
\(595\) 0 0
\(596\) −21.2170 −0.869082
\(597\) −23.5917 −0.965545
\(598\) 15.8955 0.650015
\(599\) −18.3172 −0.748419 −0.374210 0.927344i \(-0.622086\pi\)
−0.374210 + 0.927344i \(0.622086\pi\)
\(600\) −3.67925 −0.150205
\(601\) 15.5629 0.634824 0.317412 0.948288i \(-0.397186\pi\)
0.317412 + 0.948288i \(0.397186\pi\)
\(602\) 0 0
\(603\) 8.88568 0.361853
\(604\) 9.40187 0.382557
\(605\) 7.23844 0.294284
\(606\) −12.8683 −0.522738
\(607\) 28.3918 1.15239 0.576193 0.817314i \(-0.304538\pi\)
0.576193 + 0.817314i \(0.304538\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 2.96069 0.119875
\(611\) −8.55550 −0.346118
\(612\) −4.99679 −0.201983
\(613\) −35.4409 −1.43145 −0.715723 0.698384i \(-0.753903\pi\)
−0.715723 + 0.698384i \(0.753903\pi\)
\(614\) −14.3776 −0.580232
\(615\) 0.926955 0.0373784
\(616\) 0 0
\(617\) 27.6964 1.11502 0.557508 0.830172i \(-0.311758\pi\)
0.557508 + 0.830172i \(0.311758\pi\)
\(618\) 6.95582 0.279804
\(619\) −26.3914 −1.06076 −0.530380 0.847760i \(-0.677951\pi\)
−0.530380 + 0.847760i \(0.677951\pi\)
\(620\) −5.04015 −0.202417
\(621\) 6.91567 0.277516
\(622\) 2.97452 0.119267
\(623\) 0 0
\(624\) −2.29847 −0.0920126
\(625\) 6.93318 0.277327
\(626\) 16.7588 0.669818
\(627\) 2.16830 0.0865936
\(628\) 6.14721 0.245300
\(629\) −3.29884 −0.131533
\(630\) 0 0
\(631\) 21.3293 0.849105 0.424553 0.905403i \(-0.360431\pi\)
0.424553 + 0.905403i \(0.360431\pi\)
\(632\) 7.59208 0.301997
\(633\) −17.8699 −0.710266
\(634\) 2.81061 0.111624
\(635\) 15.9379 0.632478
\(636\) 11.2713 0.446937
\(637\) 0 0
\(638\) 8.16675 0.323325
\(639\) −0.323591 −0.0128010
\(640\) 1.14924 0.0454276
\(641\) 19.0070 0.750730 0.375365 0.926877i \(-0.377517\pi\)
0.375365 + 0.926877i \(0.377517\pi\)
\(642\) −12.3987 −0.489336
\(643\) 3.46630 0.136697 0.0683487 0.997661i \(-0.478227\pi\)
0.0683487 + 0.997661i \(0.478227\pi\)
\(644\) 0 0
\(645\) −2.39497 −0.0943020
\(646\) 4.99679 0.196596
\(647\) 12.3609 0.485957 0.242979 0.970032i \(-0.421876\pi\)
0.242979 + 0.970032i \(0.421876\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −22.6080 −0.887444
\(650\) 8.45667 0.331698
\(651\) 0 0
\(652\) −8.60793 −0.337113
\(653\) 50.7123 1.98452 0.992262 0.124161i \(-0.0396239\pi\)
0.992262 + 0.124161i \(0.0396239\pi\)
\(654\) 18.0568 0.706078
\(655\) −14.0337 −0.548341
\(656\) 0.806583 0.0314918
\(657\) −4.72830 −0.184469
\(658\) 0 0
\(659\) −29.4985 −1.14910 −0.574549 0.818470i \(-0.694822\pi\)
−0.574549 + 0.818470i \(0.694822\pi\)
\(660\) −2.49189 −0.0969967
\(661\) −37.8979 −1.47406 −0.737029 0.675861i \(-0.763772\pi\)
−0.737029 + 0.675861i \(0.763772\pi\)
\(662\) 10.1391 0.394066
\(663\) 11.4850 0.446040
\(664\) −3.17435 −0.123189
\(665\) 0 0
\(666\) −0.660191 −0.0255819
\(667\) −26.0474 −1.00856
\(668\) 11.9749 0.463322
\(669\) −4.44605 −0.171894
\(670\) 10.2118 0.394514
\(671\) −5.58602 −0.215646
\(672\) 0 0
\(673\) −0.234048 −0.00902187 −0.00451094 0.999990i \(-0.501436\pi\)
−0.00451094 + 0.999990i \(0.501436\pi\)
\(674\) −21.4364 −0.825700
\(675\) 3.67925 0.141615
\(676\) −7.71702 −0.296808
\(677\) −11.2235 −0.431353 −0.215676 0.976465i \(-0.569196\pi\)
−0.215676 + 0.976465i \(0.569196\pi\)
\(678\) −8.63792 −0.331737
\(679\) 0 0
\(680\) −5.74250 −0.220215
\(681\) −20.0938 −0.769995
\(682\) 9.50941 0.364134
\(683\) 16.5187 0.632072 0.316036 0.948747i \(-0.397648\pi\)
0.316036 + 0.948747i \(0.397648\pi\)
\(684\) 1.00000 0.0382360
\(685\) −12.7344 −0.486557
\(686\) 0 0
\(687\) −17.9555 −0.685043
\(688\) −2.08397 −0.0794506
\(689\) −25.9069 −0.986973
\(690\) 7.94774 0.302565
\(691\) 41.4575 1.57712 0.788559 0.614959i \(-0.210827\pi\)
0.788559 + 0.614959i \(0.210827\pi\)
\(692\) 8.10185 0.307986
\(693\) 0 0
\(694\) −29.1826 −1.10776
\(695\) 12.6617 0.480287
\(696\) 3.76643 0.142766
\(697\) −4.03033 −0.152660
\(698\) −9.36208 −0.354360
\(699\) −10.2539 −0.387839
\(700\) 0 0
\(701\) 26.9202 1.01676 0.508380 0.861133i \(-0.330244\pi\)
0.508380 + 0.861133i \(0.330244\pi\)
\(702\) 2.29847 0.0867503
\(703\) 0.660191 0.0248996
\(704\) −2.16830 −0.0817209
\(705\) −4.27775 −0.161109
\(706\) −33.5994 −1.26453
\(707\) 0 0
\(708\) −10.4266 −0.391856
\(709\) 29.6206 1.11242 0.556212 0.831040i \(-0.312254\pi\)
0.556212 + 0.831040i \(0.312254\pi\)
\(710\) −0.371882 −0.0139565
\(711\) −7.59208 −0.284725
\(712\) −12.3776 −0.463869
\(713\) −30.3297 −1.13586
\(714\) 0 0
\(715\) 5.72755 0.214198
\(716\) −6.84755 −0.255905
\(717\) −1.73965 −0.0649685
\(718\) 24.3630 0.909220
\(719\) −26.5038 −0.988425 −0.494213 0.869341i \(-0.664544\pi\)
−0.494213 + 0.869341i \(0.664544\pi\)
\(720\) −1.14924 −0.0428295
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 20.8492 0.775391
\(724\) −16.5905 −0.616582
\(725\) −13.8577 −0.514661
\(726\) −6.29847 −0.233758
\(727\) 27.4510 1.01810 0.509051 0.860736i \(-0.329996\pi\)
0.509051 + 0.860736i \(0.329996\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −5.43394 −0.201119
\(731\) 10.4132 0.385144
\(732\) −2.57622 −0.0952199
\(733\) −14.1651 −0.523200 −0.261600 0.965176i \(-0.584250\pi\)
−0.261600 + 0.965176i \(0.584250\pi\)
\(734\) 33.1571 1.22385
\(735\) 0 0
\(736\) 6.91567 0.254915
\(737\) −19.2668 −0.709703
\(738\) −0.806583 −0.0296907
\(739\) 7.54018 0.277370 0.138685 0.990337i \(-0.455712\pi\)
0.138685 + 0.990337i \(0.455712\pi\)
\(740\) −0.758716 −0.0278910
\(741\) −2.29847 −0.0844365
\(742\) 0 0
\(743\) −53.8226 −1.97456 −0.987279 0.158997i \(-0.949174\pi\)
−0.987279 + 0.158997i \(0.949174\pi\)
\(744\) 4.38565 0.160786
\(745\) 24.3833 0.893337
\(746\) 22.5460 0.825467
\(747\) 3.17435 0.116144
\(748\) 10.8345 0.396150
\(749\) 0 0
\(750\) 9.97452 0.364218
\(751\) 20.7762 0.758136 0.379068 0.925369i \(-0.376245\pi\)
0.379068 + 0.925369i \(0.376245\pi\)
\(752\) −3.72225 −0.135737
\(753\) −3.19663 −0.116492
\(754\) −8.65704 −0.315271
\(755\) −10.8050 −0.393233
\(756\) 0 0
\(757\) −15.9757 −0.580647 −0.290323 0.956929i \(-0.593763\pi\)
−0.290323 + 0.956929i \(0.593763\pi\)
\(758\) −18.7616 −0.681451
\(759\) −14.9952 −0.544293
\(760\) 1.14924 0.0416872
\(761\) 14.1830 0.514132 0.257066 0.966394i \(-0.417244\pi\)
0.257066 + 0.966394i \(0.417244\pi\)
\(762\) −13.8683 −0.502395
\(763\) 0 0
\(764\) 15.8902 0.574887
\(765\) 5.74250 0.207620
\(766\) 8.11770 0.293305
\(767\) 23.9653 0.865338
\(768\) −1.00000 −0.0360844
\(769\) −39.7426 −1.43316 −0.716578 0.697507i \(-0.754292\pi\)
−0.716578 + 0.697507i \(0.754292\pi\)
\(770\) 0 0
\(771\) 15.0219 0.541001
\(772\) 6.49676 0.233824
\(773\) 0.183792 0.00661053 0.00330527 0.999995i \(-0.498948\pi\)
0.00330527 + 0.999995i \(0.498948\pi\)
\(774\) 2.08397 0.0749067
\(775\) −16.1359 −0.579620
\(776\) 5.91567 0.212360
\(777\) 0 0
\(778\) 35.0034 1.25493
\(779\) 0.806583 0.0288988
\(780\) 2.64149 0.0945806
\(781\) 0.701641 0.0251067
\(782\) −34.5562 −1.23573
\(783\) −3.76643 −0.134601
\(784\) 0 0
\(785\) −7.06460 −0.252146
\(786\) 12.2113 0.435563
\(787\) −20.2158 −0.720616 −0.360308 0.932833i \(-0.617328\pi\)
−0.360308 + 0.932833i \(0.617328\pi\)
\(788\) 1.06491 0.0379357
\(789\) −10.6760 −0.380077
\(790\) −8.72510 −0.310425
\(791\) 0 0
\(792\) 2.16830 0.0770472
\(793\) 5.92138 0.210274
\(794\) −15.0219 −0.533108
\(795\) −12.9534 −0.459411
\(796\) 23.5917 0.836186
\(797\) −33.9377 −1.20213 −0.601067 0.799198i \(-0.705258\pi\)
−0.601067 + 0.799198i \(0.705258\pi\)
\(798\) 0 0
\(799\) 18.5993 0.657996
\(800\) 3.67925 0.130081
\(801\) 12.3776 0.437340
\(802\) 26.6287 0.940291
\(803\) 10.2524 0.361799
\(804\) −8.88568 −0.313374
\(805\) 0 0
\(806\) −10.0803 −0.355064
\(807\) 3.24906 0.114372
\(808\) 12.8683 0.452705
\(809\) −34.7190 −1.22066 −0.610328 0.792149i \(-0.708962\pi\)
−0.610328 + 0.792149i \(0.708962\pi\)
\(810\) 1.14924 0.0403801
\(811\) 44.1620 1.55074 0.775369 0.631509i \(-0.217564\pi\)
0.775369 + 0.631509i \(0.217564\pi\)
\(812\) 0 0
\(813\) −18.0349 −0.632512
\(814\) 1.43149 0.0501738
\(815\) 9.89255 0.346521
\(816\) 4.99679 0.174923
\(817\) −2.08397 −0.0729088
\(818\) −27.6489 −0.966721
\(819\) 0 0
\(820\) −0.926955 −0.0323707
\(821\) 12.1074 0.422552 0.211276 0.977426i \(-0.432238\pi\)
0.211276 + 0.977426i \(0.432238\pi\)
\(822\) 11.0808 0.386486
\(823\) 3.69623 0.128843 0.0644213 0.997923i \(-0.479480\pi\)
0.0644213 + 0.997923i \(0.479480\pi\)
\(824\) −6.95582 −0.242317
\(825\) −7.97773 −0.277749
\(826\) 0 0
\(827\) −11.8614 −0.412461 −0.206230 0.978503i \(-0.566120\pi\)
−0.206230 + 0.978503i \(0.566120\pi\)
\(828\) −6.91567 −0.240336
\(829\) 34.2921 1.19101 0.595506 0.803351i \(-0.296952\pi\)
0.595506 + 0.803351i \(0.296952\pi\)
\(830\) 3.64808 0.126627
\(831\) −22.2839 −0.773021
\(832\) 2.29847 0.0796852
\(833\) 0 0
\(834\) −11.0175 −0.381505
\(835\) −13.7620 −0.476253
\(836\) −2.16830 −0.0749922
\(837\) −4.38565 −0.151590
\(838\) −1.91888 −0.0662865
\(839\) 23.5395 0.812672 0.406336 0.913724i \(-0.366806\pi\)
0.406336 + 0.913724i \(0.366806\pi\)
\(840\) 0 0
\(841\) −14.8140 −0.510827
\(842\) −25.9815 −0.895381
\(843\) −7.10790 −0.244809
\(844\) 17.8699 0.615109
\(845\) 8.86868 0.305092
\(846\) 3.72225 0.127974
\(847\) 0 0
\(848\) −11.2713 −0.387059
\(849\) 15.7474 0.540448
\(850\) −18.3845 −0.630582
\(851\) −4.56566 −0.156509
\(852\) 0.323591 0.0110860
\(853\) −54.9855 −1.88267 −0.941334 0.337477i \(-0.890427\pi\)
−0.941334 + 0.337477i \(0.890427\pi\)
\(854\) 0 0
\(855\) −1.14924 −0.0393031
\(856\) 12.3987 0.423777
\(857\) 58.1333 1.98579 0.992897 0.118976i \(-0.0379612\pi\)
0.992897 + 0.118976i \(0.0379612\pi\)
\(858\) −4.98378 −0.170143
\(859\) 11.6517 0.397550 0.198775 0.980045i \(-0.436304\pi\)
0.198775 + 0.980045i \(0.436304\pi\)
\(860\) 2.39497 0.0816679
\(861\) 0 0
\(862\) −0.965082 −0.0328708
\(863\) −9.64470 −0.328309 −0.164155 0.986435i \(-0.552490\pi\)
−0.164155 + 0.986435i \(0.552490\pi\)
\(864\) 1.00000 0.0340207
\(865\) −9.31094 −0.316582
\(866\) 5.93960 0.201836
\(867\) −7.96793 −0.270605
\(868\) 0 0
\(869\) 16.4619 0.558432
\(870\) −4.32852 −0.146751
\(871\) 20.4235 0.692024
\(872\) −18.0568 −0.611481
\(873\) −5.91567 −0.200215
\(874\) 6.91567 0.233926
\(875\) 0 0
\(876\) 4.72830 0.159755
\(877\) −51.9199 −1.75321 −0.876605 0.481211i \(-0.840197\pi\)
−0.876605 + 0.481211i \(0.840197\pi\)
\(878\) −21.5921 −0.728697
\(879\) −22.0139 −0.742512
\(880\) 2.49189 0.0840016
\(881\) 1.32443 0.0446213 0.0223106 0.999751i \(-0.492898\pi\)
0.0223106 + 0.999751i \(0.492898\pi\)
\(882\) 0 0
\(883\) −45.6432 −1.53602 −0.768009 0.640439i \(-0.778752\pi\)
−0.768009 + 0.640439i \(0.778752\pi\)
\(884\) −11.4850 −0.386282
\(885\) 11.9827 0.402793
\(886\) −13.9344 −0.468134
\(887\) 5.18089 0.173957 0.0869786 0.996210i \(-0.472279\pi\)
0.0869786 + 0.996210i \(0.472279\pi\)
\(888\) 0.660191 0.0221546
\(889\) 0 0
\(890\) 14.2248 0.476815
\(891\) −2.16830 −0.0726408
\(892\) 4.44605 0.148865
\(893\) −3.72225 −0.124560
\(894\) −21.2170 −0.709602
\(895\) 7.86946 0.263047
\(896\) 0 0
\(897\) 15.8955 0.530735
\(898\) −14.3273 −0.478109
\(899\) 16.5183 0.550915
\(900\) −3.67925 −0.122642
\(901\) 56.3205 1.87631
\(902\) 1.74891 0.0582325
\(903\) 0 0
\(904\) 8.63792 0.287293
\(905\) 19.0665 0.633790
\(906\) 9.40187 0.312356
\(907\) 58.5017 1.94252 0.971258 0.238028i \(-0.0765008\pi\)
0.971258 + 0.238028i \(0.0765008\pi\)
\(908\) 20.0938 0.666835
\(909\) −12.8683 −0.426814
\(910\) 0 0
\(911\) 23.2344 0.769790 0.384895 0.922960i \(-0.374238\pi\)
0.384895 + 0.922960i \(0.374238\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −6.88295 −0.227792
\(914\) −32.2981 −1.06833
\(915\) 2.96069 0.0978774
\(916\) 17.9555 0.593265
\(917\) 0 0
\(918\) −4.99679 −0.164919
\(919\) 17.8751 0.589644 0.294822 0.955552i \(-0.404740\pi\)
0.294822 + 0.955552i \(0.404740\pi\)
\(920\) −7.94774 −0.262029
\(921\) −14.3776 −0.473757
\(922\) −0.375849 −0.0123779
\(923\) −0.743764 −0.0244813
\(924\) 0 0
\(925\) −2.42901 −0.0798654
\(926\) −7.80824 −0.256595
\(927\) 6.95582 0.228459
\(928\) −3.76643 −0.123639
\(929\) 46.4492 1.52395 0.761974 0.647607i \(-0.224230\pi\)
0.761974 + 0.647607i \(0.224230\pi\)
\(930\) −5.04015 −0.165273
\(931\) 0 0
\(932\) 10.2539 0.335879
\(933\) 2.97452 0.0973814
\(934\) 12.2555 0.401011
\(935\) −12.4515 −0.407206
\(936\) −2.29847 −0.0751280
\(937\) −42.3561 −1.38371 −0.691857 0.722034i \(-0.743207\pi\)
−0.691857 + 0.722034i \(0.743207\pi\)
\(938\) 0 0
\(939\) 16.7588 0.546904
\(940\) 4.27775 0.139525
\(941\) −46.4969 −1.51576 −0.757878 0.652396i \(-0.773764\pi\)
−0.757878 + 0.652396i \(0.773764\pi\)
\(942\) 6.14721 0.200287
\(943\) −5.57806 −0.181647
\(944\) 10.4266 0.339358
\(945\) 0 0
\(946\) −4.51867 −0.146915
\(947\) 47.2657 1.53593 0.767964 0.640493i \(-0.221270\pi\)
0.767964 + 0.640493i \(0.221270\pi\)
\(948\) 7.59208 0.246579
\(949\) −10.8679 −0.352786
\(950\) 3.67925 0.119371
\(951\) 2.81061 0.0911403
\(952\) 0 0
\(953\) −0.673201 −0.0218071 −0.0109036 0.999941i \(-0.503471\pi\)
−0.0109036 + 0.999941i \(0.503471\pi\)
\(954\) 11.2713 0.364923
\(955\) −18.2616 −0.590931
\(956\) 1.73965 0.0562644
\(957\) 8.16675 0.263994
\(958\) −27.0751 −0.874756
\(959\) 0 0
\(960\) 1.14924 0.0370915
\(961\) −11.7661 −0.379551
\(962\) −1.51743 −0.0489240
\(963\) −12.3987 −0.399541
\(964\) −20.8492 −0.671508
\(965\) −7.46632 −0.240349
\(966\) 0 0
\(967\) 1.83609 0.0590447 0.0295224 0.999564i \(-0.490601\pi\)
0.0295224 + 0.999564i \(0.490601\pi\)
\(968\) 6.29847 0.202441
\(969\) 4.99679 0.160520
\(970\) −6.79850 −0.218287
\(971\) 12.6488 0.405921 0.202960 0.979187i \(-0.434944\pi\)
0.202960 + 0.979187i \(0.434944\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −29.7498 −0.953247
\(975\) 8.45667 0.270830
\(976\) 2.57622 0.0824629
\(977\) 21.3457 0.682909 0.341454 0.939898i \(-0.389081\pi\)
0.341454 + 0.939898i \(0.389081\pi\)
\(978\) −8.60793 −0.275251
\(979\) −26.8383 −0.857756
\(980\) 0 0
\(981\) 18.0568 0.576510
\(982\) −18.4381 −0.588383
\(983\) 25.7268 0.820558 0.410279 0.911960i \(-0.365431\pi\)
0.410279 + 0.911960i \(0.365431\pi\)
\(984\) 0.806583 0.0257129
\(985\) −1.22383 −0.0389944
\(986\) 18.8201 0.599353
\(987\) 0 0
\(988\) 2.29847 0.0731242
\(989\) 14.4120 0.458276
\(990\) −2.49189 −0.0791975
\(991\) −58.1823 −1.84822 −0.924111 0.382124i \(-0.875193\pi\)
−0.924111 + 0.382124i \(0.875193\pi\)
\(992\) −4.38565 −0.139245
\(993\) 10.1391 0.321754
\(994\) 0 0
\(995\) −27.1125 −0.859523
\(996\) −3.17435 −0.100583
\(997\) −51.2792 −1.62403 −0.812014 0.583638i \(-0.801629\pi\)
−0.812014 + 0.583638i \(0.801629\pi\)
\(998\) −33.1570 −1.04957
\(999\) −0.660191 −0.0208875
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.1.2 4
7.2 even 3 798.2.j.l.571.3 yes 8
7.4 even 3 798.2.j.l.457.3 8
7.6 odd 2 5586.2.a.bz.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.l.457.3 8 7.4 even 3
798.2.j.l.571.3 yes 8 7.2 even 3
5586.2.a.bw.1.2 4 1.1 even 1 trivial
5586.2.a.bz.1.3 4 7.6 odd 2