Properties

Label 735.2.a.g.1.1
Level $735$
Weight $2$
Character 735.1
Self dual yes
Analytic conductor $5.869$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 735.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} +1.00000 q^{5} +2.73205 q^{6} -9.46410 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} +1.00000 q^{5} +2.73205 q^{6} -9.46410 q^{8} +1.00000 q^{9} -2.73205 q^{10} +0.732051 q^{11} -5.46410 q^{12} +2.26795 q^{13} -1.00000 q^{15} +14.9282 q^{16} +3.26795 q^{17} -2.73205 q^{18} +4.46410 q^{19} +5.46410 q^{20} -2.00000 q^{22} -4.73205 q^{23} +9.46410 q^{24} +1.00000 q^{25} -6.19615 q^{26} -1.00000 q^{27} -4.19615 q^{29} +2.73205 q^{30} -0.464102 q^{31} -21.8564 q^{32} -0.732051 q^{33} -8.92820 q^{34} +5.46410 q^{36} -3.19615 q^{37} -12.1962 q^{38} -2.26795 q^{39} -9.46410 q^{40} -0.732051 q^{41} +3.19615 q^{43} +4.00000 q^{44} +1.00000 q^{45} +12.9282 q^{46} +2.00000 q^{47} -14.9282 q^{48} -2.73205 q^{50} -3.26795 q^{51} +12.3923 q^{52} +12.3923 q^{53} +2.73205 q^{54} +0.732051 q^{55} -4.46410 q^{57} +11.4641 q^{58} -0.196152 q^{59} -5.46410 q^{60} +4.00000 q^{61} +1.26795 q^{62} +29.8564 q^{64} +2.26795 q^{65} +2.00000 q^{66} -14.6603 q^{67} +17.8564 q^{68} +4.73205 q^{69} +6.19615 q^{71} -9.46410 q^{72} +12.6603 q^{73} +8.73205 q^{74} -1.00000 q^{75} +24.3923 q^{76} +6.19615 q^{78} -7.39230 q^{79} +14.9282 q^{80} +1.00000 q^{81} +2.00000 q^{82} +15.1244 q^{83} +3.26795 q^{85} -8.73205 q^{86} +4.19615 q^{87} -6.92820 q^{88} +15.1244 q^{89} -2.73205 q^{90} -25.8564 q^{92} +0.464102 q^{93} -5.46410 q^{94} +4.46410 q^{95} +21.8564 q^{96} +14.9282 q^{97} +0.732051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} - 12q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} - 12q^{8} + 2q^{9} - 2q^{10} - 2q^{11} - 4q^{12} + 8q^{13} - 2q^{15} + 16q^{16} + 10q^{17} - 2q^{18} + 2q^{19} + 4q^{20} - 4q^{22} - 6q^{23} + 12q^{24} + 2q^{25} - 2q^{26} - 2q^{27} + 2q^{29} + 2q^{30} + 6q^{31} - 16q^{32} + 2q^{33} - 4q^{34} + 4q^{36} + 4q^{37} - 14q^{38} - 8q^{39} - 12q^{40} + 2q^{41} - 4q^{43} + 8q^{44} + 2q^{45} + 12q^{46} + 4q^{47} - 16q^{48} - 2q^{50} - 10q^{51} + 4q^{52} + 4q^{53} + 2q^{54} - 2q^{55} - 2q^{57} + 16q^{58} + 10q^{59} - 4q^{60} + 8q^{61} + 6q^{62} + 32q^{64} + 8q^{65} + 4q^{66} - 12q^{67} + 8q^{68} + 6q^{69} + 2q^{71} - 12q^{72} + 8q^{73} + 14q^{74} - 2q^{75} + 28q^{76} + 2q^{78} + 6q^{79} + 16q^{80} + 2q^{81} + 4q^{82} + 6q^{83} + 10q^{85} - 14q^{86} - 2q^{87} + 6q^{89} - 2q^{90} - 24q^{92} - 6q^{93} - 4q^{94} + 2q^{95} + 16q^{96} + 16q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73205 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.46410 2.73205
\(5\) 1.00000 0.447214
\(6\) 2.73205 1.11536
\(7\) 0 0
\(8\) −9.46410 −3.34607
\(9\) 1.00000 0.333333
\(10\) −2.73205 −0.863950
\(11\) 0.732051 0.220722 0.110361 0.993892i \(-0.464799\pi\)
0.110361 + 0.993892i \(0.464799\pi\)
\(12\) −5.46410 −1.57735
\(13\) 2.26795 0.629016 0.314508 0.949255i \(-0.398160\pi\)
0.314508 + 0.949255i \(0.398160\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 14.9282 3.73205
\(17\) 3.26795 0.792594 0.396297 0.918122i \(-0.370295\pi\)
0.396297 + 0.918122i \(0.370295\pi\)
\(18\) −2.73205 −0.643951
\(19\) 4.46410 1.02414 0.512068 0.858945i \(-0.328880\pi\)
0.512068 + 0.858945i \(0.328880\pi\)
\(20\) 5.46410 1.22181
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −4.73205 −0.986701 −0.493350 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) 9.46410 1.93185
\(25\) 1.00000 0.200000
\(26\) −6.19615 −1.21517
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.19615 −0.779206 −0.389603 0.920983i \(-0.627388\pi\)
−0.389603 + 0.920983i \(0.627388\pi\)
\(30\) 2.73205 0.498802
\(31\) −0.464102 −0.0833551 −0.0416776 0.999131i \(-0.513270\pi\)
−0.0416776 + 0.999131i \(0.513270\pi\)
\(32\) −21.8564 −3.86370
\(33\) −0.732051 −0.127434
\(34\) −8.92820 −1.53117
\(35\) 0 0
\(36\) 5.46410 0.910684
\(37\) −3.19615 −0.525444 −0.262722 0.964872i \(-0.584620\pi\)
−0.262722 + 0.964872i \(0.584620\pi\)
\(38\) −12.1962 −1.97848
\(39\) −2.26795 −0.363163
\(40\) −9.46410 −1.49641
\(41\) −0.732051 −0.114327 −0.0571636 0.998365i \(-0.518206\pi\)
−0.0571636 + 0.998365i \(0.518206\pi\)
\(42\) 0 0
\(43\) 3.19615 0.487409 0.243704 0.969850i \(-0.421637\pi\)
0.243704 + 0.969850i \(0.421637\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) 12.9282 1.90616
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) −14.9282 −2.15470
\(49\) 0 0
\(50\) −2.73205 −0.386370
\(51\) −3.26795 −0.457604
\(52\) 12.3923 1.71850
\(53\) 12.3923 1.70221 0.851107 0.524992i \(-0.175932\pi\)
0.851107 + 0.524992i \(0.175932\pi\)
\(54\) 2.73205 0.371785
\(55\) 0.732051 0.0987097
\(56\) 0 0
\(57\) −4.46410 −0.591285
\(58\) 11.4641 1.50531
\(59\) −0.196152 −0.0255369 −0.0127684 0.999918i \(-0.504064\pi\)
−0.0127684 + 0.999918i \(0.504064\pi\)
\(60\) −5.46410 −0.705412
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 1.26795 0.161030
\(63\) 0 0
\(64\) 29.8564 3.73205
\(65\) 2.26795 0.281304
\(66\) 2.00000 0.246183
\(67\) −14.6603 −1.79104 −0.895518 0.445026i \(-0.853194\pi\)
−0.895518 + 0.445026i \(0.853194\pi\)
\(68\) 17.8564 2.16541
\(69\) 4.73205 0.569672
\(70\) 0 0
\(71\) 6.19615 0.735348 0.367674 0.929955i \(-0.380154\pi\)
0.367674 + 0.929955i \(0.380154\pi\)
\(72\) −9.46410 −1.11536
\(73\) 12.6603 1.48177 0.740885 0.671632i \(-0.234406\pi\)
0.740885 + 0.671632i \(0.234406\pi\)
\(74\) 8.73205 1.01508
\(75\) −1.00000 −0.115470
\(76\) 24.3923 2.79799
\(77\) 0 0
\(78\) 6.19615 0.701576
\(79\) −7.39230 −0.831699 −0.415850 0.909433i \(-0.636516\pi\)
−0.415850 + 0.909433i \(0.636516\pi\)
\(80\) 14.9282 1.66902
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 15.1244 1.66011 0.830057 0.557679i \(-0.188308\pi\)
0.830057 + 0.557679i \(0.188308\pi\)
\(84\) 0 0
\(85\) 3.26795 0.354459
\(86\) −8.73205 −0.941601
\(87\) 4.19615 0.449875
\(88\) −6.92820 −0.738549
\(89\) 15.1244 1.60318 0.801589 0.597875i \(-0.203988\pi\)
0.801589 + 0.597875i \(0.203988\pi\)
\(90\) −2.73205 −0.287983
\(91\) 0 0
\(92\) −25.8564 −2.69572
\(93\) 0.464102 0.0481251
\(94\) −5.46410 −0.563579
\(95\) 4.46410 0.458007
\(96\) 21.8564 2.23071
\(97\) 14.9282 1.51573 0.757865 0.652412i \(-0.226243\pi\)
0.757865 + 0.652412i \(0.226243\pi\)
\(98\) 0 0
\(99\) 0.732051 0.0735739
\(100\) 5.46410 0.546410
\(101\) −7.26795 −0.723188 −0.361594 0.932336i \(-0.617767\pi\)
−0.361594 + 0.932336i \(0.617767\pi\)
\(102\) 8.92820 0.884024
\(103\) −9.19615 −0.906124 −0.453062 0.891479i \(-0.649668\pi\)
−0.453062 + 0.891479i \(0.649668\pi\)
\(104\) −21.4641 −2.10473
\(105\) 0 0
\(106\) −33.8564 −3.28842
\(107\) 2.19615 0.212310 0.106155 0.994350i \(-0.466146\pi\)
0.106155 + 0.994350i \(0.466146\pi\)
\(108\) −5.46410 −0.525783
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) −2.00000 −0.190693
\(111\) 3.19615 0.303365
\(112\) 0 0
\(113\) 8.92820 0.839895 0.419947 0.907548i \(-0.362049\pi\)
0.419947 + 0.907548i \(0.362049\pi\)
\(114\) 12.1962 1.14227
\(115\) −4.73205 −0.441266
\(116\) −22.9282 −2.12883
\(117\) 2.26795 0.209672
\(118\) 0.535898 0.0493334
\(119\) 0 0
\(120\) 9.46410 0.863950
\(121\) −10.4641 −0.951282
\(122\) −10.9282 −0.989393
\(123\) 0.732051 0.0660068
\(124\) −2.53590 −0.227730
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.80385 0.426273 0.213136 0.977022i \(-0.431632\pi\)
0.213136 + 0.977022i \(0.431632\pi\)
\(128\) −37.8564 −3.34607
\(129\) −3.19615 −0.281406
\(130\) −6.19615 −0.543439
\(131\) 15.4641 1.35110 0.675552 0.737312i \(-0.263905\pi\)
0.675552 + 0.737312i \(0.263905\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 40.0526 3.46001
\(135\) −1.00000 −0.0860663
\(136\) −30.9282 −2.65207
\(137\) 2.19615 0.187630 0.0938150 0.995590i \(-0.470094\pi\)
0.0938150 + 0.995590i \(0.470094\pi\)
\(138\) −12.9282 −1.10052
\(139\) 5.92820 0.502824 0.251412 0.967880i \(-0.419105\pi\)
0.251412 + 0.967880i \(0.419105\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) −16.9282 −1.42058
\(143\) 1.66025 0.138837
\(144\) 14.9282 1.24402
\(145\) −4.19615 −0.348471
\(146\) −34.5885 −2.86256
\(147\) 0 0
\(148\) −17.4641 −1.43554
\(149\) 5.85641 0.479776 0.239888 0.970801i \(-0.422889\pi\)
0.239888 + 0.970801i \(0.422889\pi\)
\(150\) 2.73205 0.223071
\(151\) −8.92820 −0.726567 −0.363283 0.931679i \(-0.618344\pi\)
−0.363283 + 0.931679i \(0.618344\pi\)
\(152\) −42.2487 −3.42682
\(153\) 3.26795 0.264198
\(154\) 0 0
\(155\) −0.464102 −0.0372775
\(156\) −12.3923 −0.992178
\(157\) 6.39230 0.510161 0.255081 0.966920i \(-0.417898\pi\)
0.255081 + 0.966920i \(0.417898\pi\)
\(158\) 20.1962 1.60672
\(159\) −12.3923 −0.982774
\(160\) −21.8564 −1.72790
\(161\) 0 0
\(162\) −2.73205 −0.214650
\(163\) 21.8564 1.71193 0.855963 0.517037i \(-0.172965\pi\)
0.855963 + 0.517037i \(0.172965\pi\)
\(164\) −4.00000 −0.312348
\(165\) −0.732051 −0.0569901
\(166\) −41.3205 −3.20709
\(167\) −17.6603 −1.36659 −0.683296 0.730142i \(-0.739454\pi\)
−0.683296 + 0.730142i \(0.739454\pi\)
\(168\) 0 0
\(169\) −7.85641 −0.604339
\(170\) −8.92820 −0.684762
\(171\) 4.46410 0.341378
\(172\) 17.4641 1.33163
\(173\) 14.5359 1.10514 0.552572 0.833465i \(-0.313646\pi\)
0.552572 + 0.833465i \(0.313646\pi\)
\(174\) −11.4641 −0.869091
\(175\) 0 0
\(176\) 10.9282 0.823744
\(177\) 0.196152 0.0147437
\(178\) −41.3205 −3.09710
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 5.46410 0.407270
\(181\) −24.3205 −1.80773 −0.903865 0.427819i \(-0.859282\pi\)
−0.903865 + 0.427819i \(0.859282\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 44.7846 3.30157
\(185\) −3.19615 −0.234986
\(186\) −1.26795 −0.0929705
\(187\) 2.39230 0.174943
\(188\) 10.9282 0.797021
\(189\) 0 0
\(190\) −12.1962 −0.884802
\(191\) −8.92820 −0.646022 −0.323011 0.946395i \(-0.604695\pi\)
−0.323011 + 0.946395i \(0.604695\pi\)
\(192\) −29.8564 −2.15470
\(193\) 1.19615 0.0861009 0.0430505 0.999073i \(-0.486292\pi\)
0.0430505 + 0.999073i \(0.486292\pi\)
\(194\) −40.7846 −2.92816
\(195\) −2.26795 −0.162411
\(196\) 0 0
\(197\) −0.339746 −0.0242059 −0.0121029 0.999927i \(-0.503853\pi\)
−0.0121029 + 0.999927i \(0.503853\pi\)
\(198\) −2.00000 −0.142134
\(199\) 22.0000 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) −9.46410 −0.669213
\(201\) 14.6603 1.03405
\(202\) 19.8564 1.39709
\(203\) 0 0
\(204\) −17.8564 −1.25020
\(205\) −0.732051 −0.0511286
\(206\) 25.1244 1.75050
\(207\) −4.73205 −0.328900
\(208\) 33.8564 2.34752
\(209\) 3.26795 0.226049
\(210\) 0 0
\(211\) 7.07180 0.486843 0.243421 0.969921i \(-0.421730\pi\)
0.243421 + 0.969921i \(0.421730\pi\)
\(212\) 67.7128 4.65054
\(213\) −6.19615 −0.424553
\(214\) −6.00000 −0.410152
\(215\) 3.19615 0.217976
\(216\) 9.46410 0.643951
\(217\) 0 0
\(218\) −30.0526 −2.03542
\(219\) −12.6603 −0.855501
\(220\) 4.00000 0.269680
\(221\) 7.41154 0.498554
\(222\) −8.73205 −0.586057
\(223\) −20.3923 −1.36557 −0.682785 0.730619i \(-0.739231\pi\)
−0.682785 + 0.730619i \(0.739231\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −24.3923 −1.62255
\(227\) −1.66025 −0.110195 −0.0550975 0.998481i \(-0.517547\pi\)
−0.0550975 + 0.998481i \(0.517547\pi\)
\(228\) −24.3923 −1.61542
\(229\) −3.00000 −0.198246 −0.0991228 0.995075i \(-0.531604\pi\)
−0.0991228 + 0.995075i \(0.531604\pi\)
\(230\) 12.9282 0.852460
\(231\) 0 0
\(232\) 39.7128 2.60727
\(233\) 17.3205 1.13470 0.567352 0.823475i \(-0.307968\pi\)
0.567352 + 0.823475i \(0.307968\pi\)
\(234\) −6.19615 −0.405055
\(235\) 2.00000 0.130466
\(236\) −1.07180 −0.0697680
\(237\) 7.39230 0.480182
\(238\) 0 0
\(239\) 7.07180 0.457437 0.228718 0.973493i \(-0.426547\pi\)
0.228718 + 0.973493i \(0.426547\pi\)
\(240\) −14.9282 −0.963611
\(241\) 13.4641 0.867299 0.433650 0.901082i \(-0.357226\pi\)
0.433650 + 0.901082i \(0.357226\pi\)
\(242\) 28.5885 1.83774
\(243\) −1.00000 −0.0641500
\(244\) 21.8564 1.39921
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 10.1244 0.644197
\(248\) 4.39230 0.278912
\(249\) −15.1244 −0.958467
\(250\) −2.73205 −0.172790
\(251\) −24.5885 −1.55201 −0.776005 0.630727i \(-0.782757\pi\)
−0.776005 + 0.630727i \(0.782757\pi\)
\(252\) 0 0
\(253\) −3.46410 −0.217786
\(254\) −13.1244 −0.823495
\(255\) −3.26795 −0.204647
\(256\) 43.7128 2.73205
\(257\) −5.66025 −0.353077 −0.176538 0.984294i \(-0.556490\pi\)
−0.176538 + 0.984294i \(0.556490\pi\)
\(258\) 8.73205 0.543634
\(259\) 0 0
\(260\) 12.3923 0.768538
\(261\) −4.19615 −0.259735
\(262\) −42.2487 −2.61013
\(263\) 8.39230 0.517492 0.258746 0.965945i \(-0.416691\pi\)
0.258746 + 0.965945i \(0.416691\pi\)
\(264\) 6.92820 0.426401
\(265\) 12.3923 0.761253
\(266\) 0 0
\(267\) −15.1244 −0.925596
\(268\) −80.1051 −4.89320
\(269\) −12.5359 −0.764327 −0.382164 0.924095i \(-0.624821\pi\)
−0.382164 + 0.924095i \(0.624821\pi\)
\(270\) 2.73205 0.166267
\(271\) 3.07180 0.186598 0.0932992 0.995638i \(-0.470259\pi\)
0.0932992 + 0.995638i \(0.470259\pi\)
\(272\) 48.7846 2.95800
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0.732051 0.0441443
\(276\) 25.8564 1.55637
\(277\) −14.6603 −0.880849 −0.440425 0.897790i \(-0.645172\pi\)
−0.440425 + 0.897790i \(0.645172\pi\)
\(278\) −16.1962 −0.971381
\(279\) −0.464102 −0.0277850
\(280\) 0 0
\(281\) 13.8564 0.826604 0.413302 0.910594i \(-0.364375\pi\)
0.413302 + 0.910594i \(0.364375\pi\)
\(282\) 5.46410 0.325383
\(283\) −24.1244 −1.43404 −0.717022 0.697050i \(-0.754495\pi\)
−0.717022 + 0.697050i \(0.754495\pi\)
\(284\) 33.8564 2.00901
\(285\) −4.46410 −0.264431
\(286\) −4.53590 −0.268213
\(287\) 0 0
\(288\) −21.8564 −1.28790
\(289\) −6.32051 −0.371795
\(290\) 11.4641 0.673195
\(291\) −14.9282 −0.875107
\(292\) 69.1769 4.04827
\(293\) 18.9282 1.10580 0.552899 0.833248i \(-0.313522\pi\)
0.552899 + 0.833248i \(0.313522\pi\)
\(294\) 0 0
\(295\) −0.196152 −0.0114204
\(296\) 30.2487 1.75817
\(297\) −0.732051 −0.0424779
\(298\) −16.0000 −0.926855
\(299\) −10.7321 −0.620651
\(300\) −5.46410 −0.315470
\(301\) 0 0
\(302\) 24.3923 1.40362
\(303\) 7.26795 0.417533
\(304\) 66.6410 3.82212
\(305\) 4.00000 0.229039
\(306\) −8.92820 −0.510391
\(307\) −32.1244 −1.83343 −0.916717 0.399537i \(-0.869171\pi\)
−0.916717 + 0.399537i \(0.869171\pi\)
\(308\) 0 0
\(309\) 9.19615 0.523151
\(310\) 1.26795 0.0720147
\(311\) 9.12436 0.517395 0.258697 0.965958i \(-0.416707\pi\)
0.258697 + 0.965958i \(0.416707\pi\)
\(312\) 21.4641 1.21517
\(313\) −12.6603 −0.715600 −0.357800 0.933798i \(-0.616473\pi\)
−0.357800 + 0.933798i \(0.616473\pi\)
\(314\) −17.4641 −0.985556
\(315\) 0 0
\(316\) −40.3923 −2.27224
\(317\) −28.4449 −1.59762 −0.798811 0.601582i \(-0.794537\pi\)
−0.798811 + 0.601582i \(0.794537\pi\)
\(318\) 33.8564 1.89857
\(319\) −3.07180 −0.171988
\(320\) 29.8564 1.66902
\(321\) −2.19615 −0.122577
\(322\) 0 0
\(323\) 14.5885 0.811723
\(324\) 5.46410 0.303561
\(325\) 2.26795 0.125803
\(326\) −59.7128 −3.30719
\(327\) −11.0000 −0.608301
\(328\) 6.92820 0.382546
\(329\) 0 0
\(330\) 2.00000 0.110096
\(331\) 8.07180 0.443666 0.221833 0.975085i \(-0.428796\pi\)
0.221833 + 0.975085i \(0.428796\pi\)
\(332\) 82.6410 4.53551
\(333\) −3.19615 −0.175148
\(334\) 48.2487 2.64005
\(335\) −14.6603 −0.800975
\(336\) 0 0
\(337\) 17.9808 0.979475 0.489737 0.871870i \(-0.337093\pi\)
0.489737 + 0.871870i \(0.337093\pi\)
\(338\) 21.4641 1.16749
\(339\) −8.92820 −0.484913
\(340\) 17.8564 0.968400
\(341\) −0.339746 −0.0183983
\(342\) −12.1962 −0.659492
\(343\) 0 0
\(344\) −30.2487 −1.63090
\(345\) 4.73205 0.254765
\(346\) −39.7128 −2.13497
\(347\) −21.0718 −1.13119 −0.565597 0.824682i \(-0.691354\pi\)
−0.565597 + 0.824682i \(0.691354\pi\)
\(348\) 22.9282 1.22908
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) −2.26795 −0.121054
\(352\) −16.0000 −0.852803
\(353\) −3.12436 −0.166293 −0.0831463 0.996537i \(-0.526497\pi\)
−0.0831463 + 0.996537i \(0.526497\pi\)
\(354\) −0.535898 −0.0284827
\(355\) 6.19615 0.328858
\(356\) 82.6410 4.37997
\(357\) 0 0
\(358\) 27.3205 1.44393
\(359\) −1.26795 −0.0669198 −0.0334599 0.999440i \(-0.510653\pi\)
−0.0334599 + 0.999440i \(0.510653\pi\)
\(360\) −9.46410 −0.498802
\(361\) 0.928203 0.0488528
\(362\) 66.4449 3.49226
\(363\) 10.4641 0.549223
\(364\) 0 0
\(365\) 12.6603 0.662668
\(366\) 10.9282 0.571226
\(367\) 11.1962 0.584434 0.292217 0.956352i \(-0.405607\pi\)
0.292217 + 0.956352i \(0.405607\pi\)
\(368\) −70.6410 −3.68242
\(369\) −0.732051 −0.0381090
\(370\) 8.73205 0.453958
\(371\) 0 0
\(372\) 2.53590 0.131480
\(373\) 26.5167 1.37298 0.686490 0.727139i \(-0.259150\pi\)
0.686490 + 0.727139i \(0.259150\pi\)
\(374\) −6.53590 −0.337963
\(375\) −1.00000 −0.0516398
\(376\) −18.9282 −0.976148
\(377\) −9.51666 −0.490133
\(378\) 0 0
\(379\) 6.32051 0.324663 0.162331 0.986736i \(-0.448099\pi\)
0.162331 + 0.986736i \(0.448099\pi\)
\(380\) 24.3923 1.25130
\(381\) −4.80385 −0.246109
\(382\) 24.3923 1.24802
\(383\) −23.3205 −1.19162 −0.595811 0.803125i \(-0.703169\pi\)
−0.595811 + 0.803125i \(0.703169\pi\)
\(384\) 37.8564 1.93185
\(385\) 0 0
\(386\) −3.26795 −0.166334
\(387\) 3.19615 0.162470
\(388\) 81.5692 4.14105
\(389\) 5.41154 0.274376 0.137188 0.990545i \(-0.456194\pi\)
0.137188 + 0.990545i \(0.456194\pi\)
\(390\) 6.19615 0.313754
\(391\) −15.4641 −0.782053
\(392\) 0 0
\(393\) −15.4641 −0.780061
\(394\) 0.928203 0.0467622
\(395\) −7.39230 −0.371947
\(396\) 4.00000 0.201008
\(397\) −31.1962 −1.56569 −0.782845 0.622217i \(-0.786232\pi\)
−0.782845 + 0.622217i \(0.786232\pi\)
\(398\) −60.1051 −3.01280
\(399\) 0 0
\(400\) 14.9282 0.746410
\(401\) −16.3923 −0.818593 −0.409296 0.912402i \(-0.634226\pi\)
−0.409296 + 0.912402i \(0.634226\pi\)
\(402\) −40.0526 −1.99764
\(403\) −1.05256 −0.0524317
\(404\) −39.7128 −1.97579
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −2.33975 −0.115977
\(408\) 30.9282 1.53117
\(409\) 3.14359 0.155441 0.0777203 0.996975i \(-0.475236\pi\)
0.0777203 + 0.996975i \(0.475236\pi\)
\(410\) 2.00000 0.0987730
\(411\) −2.19615 −0.108328
\(412\) −50.2487 −2.47558
\(413\) 0 0
\(414\) 12.9282 0.635387
\(415\) 15.1244 0.742425
\(416\) −49.5692 −2.43033
\(417\) −5.92820 −0.290305
\(418\) −8.92820 −0.436693
\(419\) −35.4641 −1.73253 −0.866267 0.499581i \(-0.833487\pi\)
−0.866267 + 0.499581i \(0.833487\pi\)
\(420\) 0 0
\(421\) 0.0717968 0.00349916 0.00174958 0.999998i \(-0.499443\pi\)
0.00174958 + 0.999998i \(0.499443\pi\)
\(422\) −19.3205 −0.940508
\(423\) 2.00000 0.0972433
\(424\) −117.282 −5.69572
\(425\) 3.26795 0.158519
\(426\) 16.9282 0.820174
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −1.66025 −0.0801578
\(430\) −8.73205 −0.421097
\(431\) 17.3205 0.834300 0.417150 0.908838i \(-0.363029\pi\)
0.417150 + 0.908838i \(0.363029\pi\)
\(432\) −14.9282 −0.718234
\(433\) 15.1962 0.730280 0.365140 0.930953i \(-0.381021\pi\)
0.365140 + 0.930953i \(0.381021\pi\)
\(434\) 0 0
\(435\) 4.19615 0.201190
\(436\) 60.1051 2.87851
\(437\) −21.1244 −1.01051
\(438\) 34.5885 1.65270
\(439\) 0.535898 0.0255770 0.0127885 0.999918i \(-0.495929\pi\)
0.0127885 + 0.999918i \(0.495929\pi\)
\(440\) −6.92820 −0.330289
\(441\) 0 0
\(442\) −20.2487 −0.963133
\(443\) 9.46410 0.449653 0.224827 0.974399i \(-0.427818\pi\)
0.224827 + 0.974399i \(0.427818\pi\)
\(444\) 17.4641 0.828810
\(445\) 15.1244 0.716963
\(446\) 55.7128 2.63808
\(447\) −5.85641 −0.276999
\(448\) 0 0
\(449\) −35.8564 −1.69217 −0.846084 0.533049i \(-0.821046\pi\)
−0.846084 + 0.533049i \(0.821046\pi\)
\(450\) −2.73205 −0.128790
\(451\) −0.535898 −0.0252345
\(452\) 48.7846 2.29464
\(453\) 8.92820 0.419484
\(454\) 4.53590 0.212880
\(455\) 0 0
\(456\) 42.2487 1.97848
\(457\) −16.6603 −0.779334 −0.389667 0.920956i \(-0.627410\pi\)
−0.389667 + 0.920956i \(0.627410\pi\)
\(458\) 8.19615 0.382981
\(459\) −3.26795 −0.152535
\(460\) −25.8564 −1.20556
\(461\) 16.9808 0.790873 0.395436 0.918493i \(-0.370593\pi\)
0.395436 + 0.918493i \(0.370593\pi\)
\(462\) 0 0
\(463\) 25.7321 1.19587 0.597935 0.801545i \(-0.295988\pi\)
0.597935 + 0.801545i \(0.295988\pi\)
\(464\) −62.6410 −2.90804
\(465\) 0.464102 0.0215222
\(466\) −47.3205 −2.19208
\(467\) −0.143594 −0.00664472 −0.00332236 0.999994i \(-0.501058\pi\)
−0.00332236 + 0.999994i \(0.501058\pi\)
\(468\) 12.3923 0.572834
\(469\) 0 0
\(470\) −5.46410 −0.252040
\(471\) −6.39230 −0.294542
\(472\) 1.85641 0.0854480
\(473\) 2.33975 0.107582
\(474\) −20.1962 −0.927640
\(475\) 4.46410 0.204827
\(476\) 0 0
\(477\) 12.3923 0.567405
\(478\) −19.3205 −0.883699
\(479\) 8.78461 0.401379 0.200690 0.979655i \(-0.435682\pi\)
0.200690 + 0.979655i \(0.435682\pi\)
\(480\) 21.8564 0.997604
\(481\) −7.24871 −0.330513
\(482\) −36.7846 −1.67549
\(483\) 0 0
\(484\) −57.1769 −2.59895
\(485\) 14.9282 0.677855
\(486\) 2.73205 0.123928
\(487\) −0.411543 −0.0186488 −0.00932439 0.999957i \(-0.502968\pi\)
−0.00932439 + 0.999957i \(0.502968\pi\)
\(488\) −37.8564 −1.71368
\(489\) −21.8564 −0.988381
\(490\) 0 0
\(491\) −38.2487 −1.72614 −0.863070 0.505084i \(-0.831461\pi\)
−0.863070 + 0.505084i \(0.831461\pi\)
\(492\) 4.00000 0.180334
\(493\) −13.7128 −0.617594
\(494\) −27.6603 −1.24449
\(495\) 0.732051 0.0329032
\(496\) −6.92820 −0.311086
\(497\) 0 0
\(498\) 41.3205 1.85162
\(499\) −13.5359 −0.605950 −0.302975 0.952998i \(-0.597980\pi\)
−0.302975 + 0.952998i \(0.597980\pi\)
\(500\) 5.46410 0.244362
\(501\) 17.6603 0.789002
\(502\) 67.1769 2.99825
\(503\) 14.3923 0.641721 0.320861 0.947126i \(-0.396028\pi\)
0.320861 + 0.947126i \(0.396028\pi\)
\(504\) 0 0
\(505\) −7.26795 −0.323419
\(506\) 9.46410 0.420731
\(507\) 7.85641 0.348915
\(508\) 26.2487 1.16460
\(509\) 4.53590 0.201050 0.100525 0.994935i \(-0.467948\pi\)
0.100525 + 0.994935i \(0.467948\pi\)
\(510\) 8.92820 0.395347
\(511\) 0 0
\(512\) −43.7128 −1.93185
\(513\) −4.46410 −0.197095
\(514\) 15.4641 0.682092
\(515\) −9.19615 −0.405231
\(516\) −17.4641 −0.768814
\(517\) 1.46410 0.0643911
\(518\) 0 0
\(519\) −14.5359 −0.638055
\(520\) −21.4641 −0.941263
\(521\) −5.46410 −0.239387 −0.119693 0.992811i \(-0.538191\pi\)
−0.119693 + 0.992811i \(0.538191\pi\)
\(522\) 11.4641 0.501770
\(523\) −27.7321 −1.21264 −0.606319 0.795222i \(-0.707355\pi\)
−0.606319 + 0.795222i \(0.707355\pi\)
\(524\) 84.4974 3.69129
\(525\) 0 0
\(526\) −22.9282 −0.999717
\(527\) −1.51666 −0.0660668
\(528\) −10.9282 −0.475589
\(529\) −0.607695 −0.0264215
\(530\) −33.8564 −1.47063
\(531\) −0.196152 −0.00851229
\(532\) 0 0
\(533\) −1.66025 −0.0719136
\(534\) 41.3205 1.78811
\(535\) 2.19615 0.0949479
\(536\) 138.746 5.99292
\(537\) 10.0000 0.431532
\(538\) 34.2487 1.47657
\(539\) 0 0
\(540\) −5.46410 −0.235137
\(541\) 5.78461 0.248700 0.124350 0.992238i \(-0.460315\pi\)
0.124350 + 0.992238i \(0.460315\pi\)
\(542\) −8.39230 −0.360480
\(543\) 24.3205 1.04369
\(544\) −71.4256 −3.06235
\(545\) 11.0000 0.471188
\(546\) 0 0
\(547\) −26.2487 −1.12231 −0.561157 0.827709i \(-0.689644\pi\)
−0.561157 + 0.827709i \(0.689644\pi\)
\(548\) 12.0000 0.512615
\(549\) 4.00000 0.170716
\(550\) −2.00000 −0.0852803
\(551\) −18.7321 −0.798012
\(552\) −44.7846 −1.90616
\(553\) 0 0
\(554\) 40.0526 1.70167
\(555\) 3.19615 0.135669
\(556\) 32.3923 1.37374
\(557\) 14.7846 0.626444 0.313222 0.949680i \(-0.398592\pi\)
0.313222 + 0.949680i \(0.398592\pi\)
\(558\) 1.26795 0.0536766
\(559\) 7.24871 0.306588
\(560\) 0 0
\(561\) −2.39230 −0.101003
\(562\) −37.8564 −1.59688
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) −10.9282 −0.460160
\(565\) 8.92820 0.375612
\(566\) 65.9090 2.77036
\(567\) 0 0
\(568\) −58.6410 −2.46052
\(569\) 32.4449 1.36016 0.680080 0.733138i \(-0.261945\pi\)
0.680080 + 0.733138i \(0.261945\pi\)
\(570\) 12.1962 0.510841
\(571\) 18.6077 0.778708 0.389354 0.921088i \(-0.372698\pi\)
0.389354 + 0.921088i \(0.372698\pi\)
\(572\) 9.07180 0.379311
\(573\) 8.92820 0.372981
\(574\) 0 0
\(575\) −4.73205 −0.197340
\(576\) 29.8564 1.24402
\(577\) 28.6603 1.19314 0.596571 0.802560i \(-0.296529\pi\)
0.596571 + 0.802560i \(0.296529\pi\)
\(578\) 17.2679 0.718252
\(579\) −1.19615 −0.0497104
\(580\) −22.9282 −0.952042
\(581\) 0 0
\(582\) 40.7846 1.69058
\(583\) 9.07180 0.375715
\(584\) −119.818 −4.95810
\(585\) 2.26795 0.0937682
\(586\) −51.7128 −2.13624
\(587\) 40.7321 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(588\) 0 0
\(589\) −2.07180 −0.0853669
\(590\) 0.535898 0.0220626
\(591\) 0.339746 0.0139753
\(592\) −47.7128 −1.96098
\(593\) 27.9090 1.14608 0.573042 0.819526i \(-0.305763\pi\)
0.573042 + 0.819526i \(0.305763\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 32.0000 1.31077
\(597\) −22.0000 −0.900400
\(598\) 29.3205 1.19900
\(599\) −38.2487 −1.56280 −0.781400 0.624030i \(-0.785494\pi\)
−0.781400 + 0.624030i \(0.785494\pi\)
\(600\) 9.46410 0.386370
\(601\) −0.0717968 −0.00292865 −0.00146433 0.999999i \(-0.500466\pi\)
−0.00146433 + 0.999999i \(0.500466\pi\)
\(602\) 0 0
\(603\) −14.6603 −0.597012
\(604\) −48.7846 −1.98502
\(605\) −10.4641 −0.425426
\(606\) −19.8564 −0.806611
\(607\) 3.19615 0.129728 0.0648639 0.997894i \(-0.479339\pi\)
0.0648639 + 0.997894i \(0.479339\pi\)
\(608\) −97.5692 −3.95695
\(609\) 0 0
\(610\) −10.9282 −0.442470
\(611\) 4.53590 0.183503
\(612\) 17.8564 0.721802
\(613\) −26.9282 −1.08762 −0.543810 0.839208i \(-0.683019\pi\)
−0.543810 + 0.839208i \(0.683019\pi\)
\(614\) 87.7654 3.54192
\(615\) 0.732051 0.0295191
\(616\) 0 0
\(617\) −36.2487 −1.45932 −0.729659 0.683811i \(-0.760321\pi\)
−0.729659 + 0.683811i \(0.760321\pi\)
\(618\) −25.1244 −1.01065
\(619\) −30.0718 −1.20869 −0.604344 0.796724i \(-0.706565\pi\)
−0.604344 + 0.796724i \(0.706565\pi\)
\(620\) −2.53590 −0.101844
\(621\) 4.73205 0.189891
\(622\) −24.9282 −0.999530
\(623\) 0 0
\(624\) −33.8564 −1.35534
\(625\) 1.00000 0.0400000
\(626\) 34.5885 1.38243
\(627\) −3.26795 −0.130509
\(628\) 34.9282 1.39379
\(629\) −10.4449 −0.416464
\(630\) 0 0
\(631\) 48.7846 1.94208 0.971042 0.238908i \(-0.0767893\pi\)
0.971042 + 0.238908i \(0.0767893\pi\)
\(632\) 69.9615 2.78292
\(633\) −7.07180 −0.281079
\(634\) 77.7128 3.08637
\(635\) 4.80385 0.190635
\(636\) −67.7128 −2.68499
\(637\) 0 0
\(638\) 8.39230 0.332255
\(639\) 6.19615 0.245116
\(640\) −37.8564 −1.49641
\(641\) 3.80385 0.150243 0.0751215 0.997174i \(-0.476066\pi\)
0.0751215 + 0.997174i \(0.476066\pi\)
\(642\) 6.00000 0.236801
\(643\) 4.51666 0.178120 0.0890599 0.996026i \(-0.471614\pi\)
0.0890599 + 0.996026i \(0.471614\pi\)
\(644\) 0 0
\(645\) −3.19615 −0.125848
\(646\) −39.8564 −1.56813
\(647\) −27.9090 −1.09721 −0.548607 0.836080i \(-0.684842\pi\)
−0.548607 + 0.836080i \(0.684842\pi\)
\(648\) −9.46410 −0.371785
\(649\) −0.143594 −0.00563654
\(650\) −6.19615 −0.243033
\(651\) 0 0
\(652\) 119.426 4.67707
\(653\) −44.5885 −1.74488 −0.872441 0.488720i \(-0.837464\pi\)
−0.872441 + 0.488720i \(0.837464\pi\)
\(654\) 30.0526 1.17515
\(655\) 15.4641 0.604232
\(656\) −10.9282 −0.426675
\(657\) 12.6603 0.493924
\(658\) 0 0
\(659\) 2.92820 0.114067 0.0570333 0.998372i \(-0.481836\pi\)
0.0570333 + 0.998372i \(0.481836\pi\)
\(660\) −4.00000 −0.155700
\(661\) 10.4641 0.407006 0.203503 0.979074i \(-0.434767\pi\)
0.203503 + 0.979074i \(0.434767\pi\)
\(662\) −22.0526 −0.857097
\(663\) −7.41154 −0.287840
\(664\) −143.138 −5.55485
\(665\) 0 0
\(666\) 8.73205 0.338360
\(667\) 19.8564 0.768843
\(668\) −96.4974 −3.73360
\(669\) 20.3923 0.788412
\(670\) 40.0526 1.54737
\(671\) 2.92820 0.113042
\(672\) 0 0
\(673\) −27.3397 −1.05387 −0.526935 0.849906i \(-0.676659\pi\)
−0.526935 + 0.849906i \(0.676659\pi\)
\(674\) −49.1244 −1.89220
\(675\) −1.00000 −0.0384900
\(676\) −42.9282 −1.65108
\(677\) −33.1244 −1.27307 −0.636536 0.771247i \(-0.719633\pi\)
−0.636536 + 0.771247i \(0.719633\pi\)
\(678\) 24.3923 0.936781
\(679\) 0 0
\(680\) −30.9282 −1.18604
\(681\) 1.66025 0.0636211
\(682\) 0.928203 0.0355427
\(683\) −28.0526 −1.07340 −0.536701 0.843773i \(-0.680330\pi\)
−0.536701 + 0.843773i \(0.680330\pi\)
\(684\) 24.3923 0.932663
\(685\) 2.19615 0.0839107
\(686\) 0 0
\(687\) 3.00000 0.114457
\(688\) 47.7128 1.81903
\(689\) 28.1051 1.07072
\(690\) −12.9282 −0.492168
\(691\) 8.85641 0.336914 0.168457 0.985709i \(-0.446122\pi\)
0.168457 + 0.985709i \(0.446122\pi\)
\(692\) 79.4256 3.01931
\(693\) 0 0
\(694\) 57.5692 2.18530
\(695\) 5.92820 0.224870
\(696\) −39.7128 −1.50531
\(697\) −2.39230 −0.0906150
\(698\) −60.1051 −2.27501
\(699\) −17.3205 −0.655122
\(700\) 0 0
\(701\) −8.58846 −0.324382 −0.162191 0.986759i \(-0.551856\pi\)
−0.162191 + 0.986759i \(0.551856\pi\)
\(702\) 6.19615 0.233859
\(703\) −14.2679 −0.538126
\(704\) 21.8564 0.823744
\(705\) −2.00000 −0.0753244
\(706\) 8.53590 0.321253
\(707\) 0 0
\(708\) 1.07180 0.0402806
\(709\) −1.07180 −0.0402522 −0.0201261 0.999797i \(-0.506407\pi\)
−0.0201261 + 0.999797i \(0.506407\pi\)
\(710\) −16.9282 −0.635304
\(711\) −7.39230 −0.277233
\(712\) −143.138 −5.36434
\(713\) 2.19615 0.0822466
\(714\) 0 0
\(715\) 1.66025 0.0620900
\(716\) −54.6410 −2.04203
\(717\) −7.07180 −0.264101
\(718\) 3.46410 0.129279
\(719\) −20.5359 −0.765860 −0.382930 0.923777i \(-0.625085\pi\)
−0.382930 + 0.923777i \(0.625085\pi\)
\(720\) 14.9282 0.556341
\(721\) 0 0
\(722\) −2.53590 −0.0943764
\(723\) −13.4641 −0.500735
\(724\) −132.890 −4.93881
\(725\) −4.19615 −0.155841
\(726\) −28.5885 −1.06102
\(727\) −13.3397 −0.494744 −0.247372 0.968921i \(-0.579567\pi\)
−0.247372 + 0.968921i \(0.579567\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −34.5885 −1.28018
\(731\) 10.4449 0.386317
\(732\) −21.8564 −0.807836
\(733\) −1.33975 −0.0494846 −0.0247423 0.999694i \(-0.507877\pi\)
−0.0247423 + 0.999694i \(0.507877\pi\)
\(734\) −30.5885 −1.12904
\(735\) 0 0
\(736\) 103.426 3.81232
\(737\) −10.7321 −0.395320
\(738\) 2.00000 0.0736210
\(739\) 27.7846 1.02207 0.511037 0.859559i \(-0.329262\pi\)
0.511037 + 0.859559i \(0.329262\pi\)
\(740\) −17.4641 −0.641993
\(741\) −10.1244 −0.371927
\(742\) 0 0
\(743\) −15.9090 −0.583643 −0.291822 0.956473i \(-0.594261\pi\)
−0.291822 + 0.956473i \(0.594261\pi\)
\(744\) −4.39230 −0.161030
\(745\) 5.85641 0.214562
\(746\) −72.4449 −2.65239
\(747\) 15.1244 0.553371
\(748\) 13.0718 0.477952
\(749\) 0 0
\(750\) 2.73205 0.0997604
\(751\) 18.0718 0.659449 0.329725 0.944077i \(-0.393044\pi\)
0.329725 + 0.944077i \(0.393044\pi\)
\(752\) 29.8564 1.08875
\(753\) 24.5885 0.896053
\(754\) 26.0000 0.946864
\(755\) −8.92820 −0.324931
\(756\) 0 0
\(757\) −27.8564 −1.01246 −0.506229 0.862399i \(-0.668961\pi\)
−0.506229 + 0.862399i \(0.668961\pi\)
\(758\) −17.2679 −0.627200
\(759\) 3.46410 0.125739
\(760\) −42.2487 −1.53252
\(761\) 46.7321 1.69404 0.847018 0.531565i \(-0.178396\pi\)
0.847018 + 0.531565i \(0.178396\pi\)
\(762\) 13.1244 0.475445
\(763\) 0 0
\(764\) −48.7846 −1.76497
\(765\) 3.26795 0.118153
\(766\) 63.7128 2.30204
\(767\) −0.444864 −0.0160631
\(768\) −43.7128 −1.57735
\(769\) 52.3205 1.88673 0.943363 0.331763i \(-0.107643\pi\)
0.943363 + 0.331763i \(0.107643\pi\)
\(770\) 0 0
\(771\) 5.66025 0.203849
\(772\) 6.53590 0.235232
\(773\) 43.5167 1.56519 0.782593 0.622534i \(-0.213897\pi\)
0.782593 + 0.622534i \(0.213897\pi\)
\(774\) −8.73205 −0.313867
\(775\) −0.464102 −0.0166710
\(776\) −141.282 −5.07173
\(777\) 0 0
\(778\) −14.7846 −0.530054
\(779\) −3.26795 −0.117086
\(780\) −12.3923 −0.443716
\(781\) 4.53590 0.162307
\(782\) 42.2487 1.51081
\(783\) 4.19615 0.149958
\(784\) 0 0
\(785\) 6.39230 0.228151
\(786\) 42.2487 1.50696
\(787\) 13.4641 0.479943 0.239972 0.970780i \(-0.422862\pi\)
0.239972 + 0.970780i \(0.422862\pi\)
\(788\) −1.85641 −0.0661317
\(789\) −8.39230 −0.298774
\(790\) 20.1962 0.718547
\(791\) 0 0
\(792\) −6.92820 −0.246183
\(793\) 9.07180 0.322149
\(794\) 85.2295 3.02468
\(795\) −12.3923 −0.439510
\(796\) 120.210 4.26074
\(797\) −3.94744 −0.139826 −0.0699128 0.997553i \(-0.522272\pi\)
−0.0699128 + 0.997553i \(0.522272\pi\)
\(798\) 0 0
\(799\) 6.53590 0.231223
\(800\) −21.8564 −0.772741
\(801\) 15.1244 0.534393
\(802\) 44.7846 1.58140
\(803\) 9.26795 0.327059
\(804\) 80.1051 2.82509
\(805\) 0 0
\(806\) 2.87564 0.101290
\(807\) 12.5359 0.441285
\(808\) 68.7846 2.41983
\(809\) 25.7128 0.904014 0.452007 0.892014i \(-0.350708\pi\)
0.452007 + 0.892014i \(0.350708\pi\)
\(810\) −2.73205 −0.0959945
\(811\) −3.46410 −0.121641 −0.0608205 0.998149i \(-0.519372\pi\)
−0.0608205 + 0.998149i \(0.519372\pi\)
\(812\) 0 0
\(813\) −3.07180 −0.107733
\(814\) 6.39230 0.224050
\(815\) 21.8564 0.765597
\(816\) −48.7846 −1.70780
\(817\) 14.2679 0.499172
\(818\) −8.58846 −0.300288
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −25.5167 −0.890538 −0.445269 0.895397i \(-0.646892\pi\)
−0.445269 + 0.895397i \(0.646892\pi\)
\(822\) 6.00000 0.209274
\(823\) 39.1769 1.36562 0.682811 0.730595i \(-0.260757\pi\)
0.682811 + 0.730595i \(0.260757\pi\)
\(824\) 87.0333 3.03195
\(825\) −0.732051 −0.0254867
\(826\) 0 0
\(827\) −3.75129 −0.130445 −0.0652225 0.997871i \(-0.520776\pi\)
−0.0652225 + 0.997871i \(0.520776\pi\)
\(828\) −25.8564 −0.898572
\(829\) −4.60770 −0.160032 −0.0800159 0.996794i \(-0.525497\pi\)
−0.0800159 + 0.996794i \(0.525497\pi\)
\(830\) −41.3205 −1.43426
\(831\) 14.6603 0.508559
\(832\) 67.7128 2.34752
\(833\) 0 0
\(834\) 16.1962 0.560827
\(835\) −17.6603 −0.611158
\(836\) 17.8564 0.617577
\(837\) 0.464102 0.0160417
\(838\) 96.8897 3.34700
\(839\) 18.4449 0.636787 0.318394 0.947959i \(-0.396857\pi\)
0.318394 + 0.947959i \(0.396857\pi\)
\(840\) 0 0
\(841\) −11.3923 −0.392838
\(842\) −0.196152 −0.00675986
\(843\) −13.8564 −0.477240
\(844\) 38.6410 1.33008
\(845\) −7.85641 −0.270269
\(846\) −5.46410 −0.187860
\(847\) 0 0
\(848\) 184.995 6.35275
\(849\) 24.1244 0.827946
\(850\) −8.92820 −0.306235
\(851\) 15.1244 0.518456
\(852\) −33.8564 −1.15990
\(853\) −31.9808 −1.09500 −0.547500 0.836806i \(-0.684420\pi\)
−0.547500 + 0.836806i \(0.684420\pi\)
\(854\) 0 0
\(855\) 4.46410 0.152669
\(856\) −20.7846 −0.710403
\(857\) 29.1244 0.994869 0.497435 0.867502i \(-0.334275\pi\)
0.497435 + 0.867502i \(0.334275\pi\)
\(858\) 4.53590 0.154853
\(859\) 7.46410 0.254672 0.127336 0.991860i \(-0.459357\pi\)
0.127336 + 0.991860i \(0.459357\pi\)
\(860\) 17.4641 0.595521
\(861\) 0 0
\(862\) −47.3205 −1.61174
\(863\) 14.3923 0.489920 0.244960 0.969533i \(-0.421225\pi\)
0.244960 + 0.969533i \(0.421225\pi\)
\(864\) 21.8564 0.743570
\(865\) 14.5359 0.494235
\(866\) −41.5167 −1.41079
\(867\) 6.32051 0.214656
\(868\) 0 0
\(869\) −5.41154 −0.183574
\(870\) −11.4641 −0.388669
\(871\) −33.2487 −1.12659
\(872\) −104.105 −3.52544
\(873\) 14.9282 0.505243
\(874\) 57.7128 1.95217
\(875\) 0 0
\(876\) −69.1769 −2.33727
\(877\) −4.14359 −0.139919 −0.0699596 0.997550i \(-0.522287\pi\)
−0.0699596 + 0.997550i \(0.522287\pi\)
\(878\) −1.46410 −0.0494110
\(879\) −18.9282 −0.638432
\(880\) 10.9282 0.368390
\(881\) 9.85641 0.332071 0.166035 0.986120i \(-0.446903\pi\)
0.166035 + 0.986120i \(0.446903\pi\)
\(882\) 0 0
\(883\) 53.5885 1.80340 0.901698 0.432367i \(-0.142322\pi\)
0.901698 + 0.432367i \(0.142322\pi\)
\(884\) 40.4974 1.36208
\(885\) 0.196152 0.00659359
\(886\) −25.8564 −0.868663
\(887\) −25.2679 −0.848415 −0.424207 0.905565i \(-0.639447\pi\)
−0.424207 + 0.905565i \(0.639447\pi\)
\(888\) −30.2487 −1.01508
\(889\) 0 0
\(890\) −41.3205 −1.38507
\(891\) 0.732051 0.0245246
\(892\) −111.426 −3.73081
\(893\) 8.92820 0.298771
\(894\) 16.0000 0.535120
\(895\) −10.0000 −0.334263
\(896\) 0 0
\(897\) 10.7321 0.358333
\(898\) 97.9615 3.26902
\(899\) 1.94744 0.0649508
\(900\) 5.46410 0.182137
\(901\) 40.4974 1.34916
\(902\) 1.46410 0.0487493
\(903\) 0 0
\(904\) −84.4974 −2.81034
\(905\) −24.3205 −0.808441
\(906\) −24.3923 −0.810380
\(907\) 33.5885 1.11529 0.557643 0.830081i \(-0.311706\pi\)
0.557643 + 0.830081i \(0.311706\pi\)
\(908\) −9.07180 −0.301058
\(909\) −7.26795 −0.241063
\(910\) 0 0
\(911\) −14.7321 −0.488095 −0.244047 0.969763i \(-0.578475\pi\)
−0.244047 + 0.969763i \(0.578475\pi\)
\(912\) −66.6410 −2.20670
\(913\) 11.0718 0.366423
\(914\) 45.5167 1.50556
\(915\) −4.00000 −0.132236
\(916\) −16.3923 −0.541617
\(917\) 0 0
\(918\) 8.92820 0.294675
\(919\) −30.8564 −1.01786 −0.508929 0.860808i \(-0.669959\pi\)
−0.508929 + 0.860808i \(0.669959\pi\)
\(920\) 44.7846 1.47650
\(921\) 32.1244 1.05853
\(922\) −46.3923 −1.52785
\(923\) 14.0526 0.462546
\(924\) 0 0
\(925\) −3.19615 −0.105089
\(926\) −70.3013 −2.31024
\(927\) −9.19615 −0.302041
\(928\) 91.7128 3.01062
\(929\) −52.4449 −1.72066 −0.860330 0.509737i \(-0.829743\pi\)
−0.860330 + 0.509737i \(0.829743\pi\)
\(930\) −1.26795 −0.0415777
\(931\) 0 0
\(932\) 94.6410 3.10007
\(933\) −9.12436 −0.298718
\(934\) 0.392305 0.0128366
\(935\) 2.39230 0.0782367
\(936\) −21.4641 −0.701576
\(937\) 31.7321 1.03664 0.518320 0.855186i \(-0.326557\pi\)
0.518320 + 0.855186i \(0.326557\pi\)
\(938\) 0 0
\(939\) 12.6603 0.413152
\(940\) 10.9282 0.356439
\(941\) −30.0526 −0.979685 −0.489843 0.871811i \(-0.662946\pi\)
−0.489843 + 0.871811i \(0.662946\pi\)
\(942\) 17.4641 0.569011
\(943\) 3.46410 0.112807
\(944\) −2.92820 −0.0953049
\(945\) 0 0
\(946\) −6.39230 −0.207832
\(947\) −5.66025 −0.183934 −0.0919668 0.995762i \(-0.529315\pi\)
−0.0919668 + 0.995762i \(0.529315\pi\)
\(948\) 40.3923 1.31188
\(949\) 28.7128 0.932057
\(950\) −12.1962 −0.395695
\(951\) 28.4449 0.922388
\(952\) 0 0
\(953\) −36.1051 −1.16956 −0.584780 0.811192i \(-0.698819\pi\)
−0.584780 + 0.811192i \(0.698819\pi\)
\(954\) −33.8564 −1.09614
\(955\) −8.92820 −0.288910
\(956\) 38.6410 1.24974
\(957\) 3.07180 0.0992971
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) −29.8564 −0.963611
\(961\) −30.7846 −0.993052
\(962\) 19.8038 0.638502
\(963\) 2.19615 0.0707700
\(964\) 73.5692 2.36951
\(965\) 1.19615 0.0385055
\(966\) 0 0
\(967\) −10.1244 −0.325577 −0.162789 0.986661i \(-0.552049\pi\)
−0.162789 + 0.986661i \(0.552049\pi\)
\(968\) 99.0333 3.18305
\(969\) −14.5885 −0.468649
\(970\) −40.7846 −1.30951
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −5.46410 −0.175261
\(973\) 0 0
\(974\) 1.12436 0.0360267
\(975\) −2.26795 −0.0726325
\(976\) 59.7128 1.91136
\(977\) −16.5885 −0.530712 −0.265356 0.964151i \(-0.585489\pi\)
−0.265356 + 0.964151i \(0.585489\pi\)
\(978\) 59.7128 1.90941
\(979\) 11.0718 0.353856
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) 104.497 3.33465
\(983\) 9.80385 0.312694 0.156347 0.987702i \(-0.450028\pi\)
0.156347 + 0.987702i \(0.450028\pi\)
\(984\) −6.92820 −0.220863
\(985\) −0.339746 −0.0108252
\(986\) 37.4641 1.19310
\(987\) 0 0
\(988\) 55.3205 1.75998
\(989\) −15.1244 −0.480927
\(990\) −2.00000 −0.0635642
\(991\) −21.1051 −0.670426 −0.335213 0.942142i \(-0.608808\pi\)
−0.335213 + 0.942142i \(0.608808\pi\)
\(992\) 10.1436 0.322059
\(993\) −8.07180 −0.256151
\(994\) 0 0
\(995\) 22.0000 0.697447
\(996\) −82.6410 −2.61858
\(997\) −55.9808 −1.77293 −0.886464 0.462797i \(-0.846846\pi\)
−0.886464 + 0.462797i \(0.846846\pi\)
\(998\) 36.9808 1.17061
\(999\) 3.19615 0.101122
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 735.2.a.g.1.1 2
3.2 odd 2 2205.2.a.z.1.2 2
5.4 even 2 3675.2.a.bg.1.2 2
7.2 even 3 105.2.i.d.46.2 yes 4
7.3 odd 6 735.2.i.l.226.2 4
7.4 even 3 105.2.i.d.16.2 4
7.5 odd 6 735.2.i.l.361.2 4
7.6 odd 2 735.2.a.h.1.1 2
21.2 odd 6 315.2.j.c.46.1 4
21.11 odd 6 315.2.j.c.226.1 4
21.20 even 2 2205.2.a.ba.1.2 2
28.11 odd 6 1680.2.bg.o.961.1 4
28.23 odd 6 1680.2.bg.o.1201.1 4
35.2 odd 12 525.2.r.a.424.1 4
35.4 even 6 525.2.i.f.226.1 4
35.9 even 6 525.2.i.f.151.1 4
35.18 odd 12 525.2.r.a.499.1 4
35.23 odd 12 525.2.r.f.424.2 4
35.32 odd 12 525.2.r.f.499.2 4
35.34 odd 2 3675.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.i.d.16.2 4 7.4 even 3
105.2.i.d.46.2 yes 4 7.2 even 3
315.2.j.c.46.1 4 21.2 odd 6
315.2.j.c.226.1 4 21.11 odd 6
525.2.i.f.151.1 4 35.9 even 6
525.2.i.f.226.1 4 35.4 even 6
525.2.r.a.424.1 4 35.2 odd 12
525.2.r.a.499.1 4 35.18 odd 12
525.2.r.f.424.2 4 35.23 odd 12
525.2.r.f.499.2 4 35.32 odd 12
735.2.a.g.1.1 2 1.1 even 1 trivial
735.2.a.h.1.1 2 7.6 odd 2
735.2.i.l.226.2 4 7.3 odd 6
735.2.i.l.361.2 4 7.5 odd 6
1680.2.bg.o.961.1 4 28.11 odd 6
1680.2.bg.o.1201.1 4 28.23 odd 6
2205.2.a.z.1.2 2 3.2 odd 2
2205.2.a.ba.1.2 2 21.20 even 2
3675.2.a.be.1.2 2 35.34 odd 2
3675.2.a.bg.1.2 2 5.4 even 2