Properties

Label 3330.2.a.x.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} -2.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -9.00000 q^{29} +7.00000 q^{31} +1.00000 q^{32} +7.00000 q^{34} +1.00000 q^{35} -1.00000 q^{37} -2.00000 q^{38} +1.00000 q^{40} +11.0000 q^{41} -11.0000 q^{43} +1.00000 q^{44} -8.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +11.0000 q^{53} +1.00000 q^{55} +1.00000 q^{56} -9.00000 q^{58} +10.0000 q^{59} -1.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} +7.00000 q^{68} +1.00000 q^{70} +4.00000 q^{73} -1.00000 q^{74} -2.00000 q^{76} +1.00000 q^{77} +12.0000 q^{79} +1.00000 q^{80} +11.0000 q^{82} +6.00000 q^{83} +7.00000 q^{85} -11.0000 q^{86} +1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} -8.00000 q^{94} -2.00000 q^{95} -19.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 7.00000 0.848875
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 11.0000 1.21475
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 7.00000 0.759257
\(86\) −11.0000 −1.18616
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −19.0000 −1.92916 −0.964579 0.263795i \(-0.915026\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 20.0000 1.99007 0.995037 0.0995037i \(-0.0317255\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 13.0000 1.22294 0.611469 0.791269i \(-0.290579\pi\)
0.611469 + 0.791269i \(0.290579\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 7.00000 0.641689
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) 0 0
\(124\) 7.00000 0.628619
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 21.0000 1.78120 0.890598 0.454791i \(-0.150286\pi\)
0.890598 + 0.454791i \(0.150286\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 7.00000 0.562254
\(156\) 0 0
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) 11.0000 0.858956
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 7.00000 0.536875
\(171\) 0 0
\(172\) −11.0000 −0.838742
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 7.00000 0.511891
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) −19.0000 −1.37479 −0.687396 0.726283i \(-0.741246\pi\)
−0.687396 + 0.726283i \(0.741246\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −19.0000 −1.36412
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 20.0000 1.40720
\(203\) −9.00000 −0.631676
\(204\) 0 0
\(205\) 11.0000 0.768273
\(206\) −12.0000 −0.836080
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 11.0000 0.755483
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) −11.0000 −0.750194
\(216\) 0 0
\(217\) 7.00000 0.475191
\(218\) 7.00000 0.474100
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 14.0000 0.941742
\(222\) 0 0
\(223\) 3.00000 0.200895 0.100447 0.994942i \(-0.467973\pi\)
0.100447 + 0.994942i \(0.467973\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 13.0000 0.864747
\(227\) 23.0000 1.52656 0.763282 0.646066i \(-0.223587\pi\)
0.763282 + 0.646066i \(0.223587\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 7.00000 0.453743
\(239\) −27.0000 −1.74648 −0.873242 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 7.00000 0.444500
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 0 0
\(265\) 11.0000 0.675725
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 21.0000 1.25950
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 11.0000 0.649309
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) −9.00000 −0.528498
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −6.00000 −0.345261
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 7.00000 0.397573
\(311\) 9.00000 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 23.0000 1.29797
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) −5.00000 −0.280828 −0.140414 0.990093i \(-0.544843\pi\)
−0.140414 + 0.990093i \(0.544843\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −14.0000 −0.778981
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 13.0000 0.720003
\(327\) 0 0
\(328\) 11.0000 0.607373
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 7.00000 0.379628
\(341\) 7.00000 0.379071
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 31.0000 1.64996 0.824982 0.565159i \(-0.191185\pi\)
0.824982 + 0.565159i \(0.191185\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 7.00000 0.361961
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −18.0000 −0.927047
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) −19.0000 −0.972125
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) −19.0000 −0.964579
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 14.0000 0.697390
\(404\) 20.0000 0.995037
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) −1.00000 −0.0495682
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 11.0000 0.543251
\(411\) 0 0
\(412\) −12.0000 −0.591198
\(413\) 10.0000 0.492068
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −2.00000 −0.0978232
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −5.00000 −0.243396
\(423\) 0 0
\(424\) 11.0000 0.534207
\(425\) 7.00000 0.339550
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −11.0000 −0.530467
\(431\) 1.00000 0.0481683 0.0240842 0.999710i \(-0.492333\pi\)
0.0240842 + 0.999710i \(0.492333\pi\)
\(432\) 0 0
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) 7.00000 0.336011
\(435\) 0 0
\(436\) 7.00000 0.335239
\(437\) 0 0
\(438\) 0 0
\(439\) 1.00000 0.0477274 0.0238637 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 14.0000 0.665912
\(443\) 30.0000 1.42534 0.712672 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 3.00000 0.142054
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) 11.0000 0.517970
\(452\) 13.0000 0.611469
\(453\) 0 0
\(454\) 23.0000 1.07944
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) 0 0
\(461\) 13.0000 0.605470 0.302735 0.953075i \(-0.402100\pi\)
0.302735 + 0.953075i \(0.402100\pi\)
\(462\) 0 0
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) −11.0000 −0.505781
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 7.00000 0.320844
\(477\) 0 0
\(478\) −27.0000 −1.23495
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −19.0000 −0.862746
\(486\) 0 0
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −63.0000 −2.83738
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) 0 0
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 16.0000 0.714115
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) −1.00000 −0.0439375
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −21.0000 −0.915644
\(527\) 49.0000 2.13447
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 11.0000 0.477809
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 22.0000 0.952926
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 7.00000 0.300123
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) −11.0000 −0.470326 −0.235163 0.971956i \(-0.575562\pi\)
−0.235163 + 0.971956i \(0.575562\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 21.0000 0.890598
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −22.0000 −0.930501
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −13.0000 −0.547885 −0.273942 0.961746i \(-0.588328\pi\)
−0.273942 + 0.961746i \(0.588328\pi\)
\(564\) 0 0
\(565\) 13.0000 0.546914
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 11.0000 0.460336 0.230168 0.973151i \(-0.426072\pi\)
0.230168 + 0.973151i \(0.426072\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 11.0000 0.459131
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 32.0000 1.33102
\(579\) 0 0
\(580\) −9.00000 −0.373705
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 11.0000 0.455573
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) −43.0000 −1.77480 −0.887400 0.461000i \(-0.847491\pi\)
−0.887400 + 0.461000i \(0.847491\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) 10.0000 0.411693
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 7.00000 0.286972
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −23.0000 −0.938190 −0.469095 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) −11.0000 −0.448327
\(603\) 0 0
\(604\) −6.00000 −0.244137
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −1.00000 −0.0404888
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 33.0000 1.33286 0.666429 0.745569i \(-0.267822\pi\)
0.666429 + 0.745569i \(0.267822\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) −19.0000 −0.763674 −0.381837 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) 7.00000 0.281127
\(621\) 0 0
\(622\) 9.00000 0.360867
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −18.0000 −0.719425
\(627\) 0 0
\(628\) 23.0000 0.917800
\(629\) −7.00000 −0.279108
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) −5.00000 −0.198575
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) −9.00000 −0.356313
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 27.0000 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(642\) 0 0
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −14.0000 −0.550823
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 11.0000 0.429478
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 41.0000 1.59472 0.797358 0.603507i \(-0.206231\pi\)
0.797358 + 0.603507i \(0.206231\pi\)
\(662\) −8.00000 −0.310929
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −19.0000 −0.729153
\(680\) 7.00000 0.268438
\(681\) 0 0
\(682\) 7.00000 0.268044
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) −11.0000 −0.419371
\(689\) 22.0000 0.838133
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) 3.00000 0.114043
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 21.0000 0.796575
\(696\) 0 0
\(697\) 77.0000 2.91658
\(698\) −32.0000 −1.21122
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 31.0000 1.16670
\(707\) 20.0000 0.752177
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −14.0000 −0.522475
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) −26.0000 −0.966282
\(725\) −9.00000 −0.334252
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) −77.0000 −2.84795
\(732\) 0 0
\(733\) −15.0000 −0.554038 −0.277019 0.960864i \(-0.589346\pi\)
−0.277019 + 0.960864i \(0.589346\pi\)
\(734\) 17.0000 0.627481
\(735\) 0 0
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −31.0000 −1.14035 −0.570177 0.821522i \(-0.693125\pi\)
−0.570177 + 0.821522i \(0.693125\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 11.0000 0.403823
\(743\) −21.0000 −0.770415 −0.385208 0.922830i \(-0.625870\pi\)
−0.385208 + 0.922830i \(0.625870\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −6.00000 −0.219676
\(747\) 0 0
\(748\) 7.00000 0.255945
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −18.0000 −0.655521
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) 43.0000 1.55875 0.779374 0.626559i \(-0.215537\pi\)
0.779374 + 0.626559i \(0.215537\pi\)
\(762\) 0 0
\(763\) 7.00000 0.253417
\(764\) −19.0000 −0.687396
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) 7.00000 0.251447
\(776\) −19.0000 −0.682060
\(777\) 0 0
\(778\) 21.0000 0.752886
\(779\) −22.0000 −0.788232
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 23.0000 0.820905
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) 13.0000 0.462227
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 48.0000 1.70025 0.850124 0.526583i \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(798\) 0 0
\(799\) −56.0000 −1.98114
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) 4.00000 0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) 14.0000 0.493129
\(807\) 0 0
\(808\) 20.0000 0.703598
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) −9.00000 −0.315838
\(813\) 0 0
\(814\) −1.00000 −0.0350500
\(815\) 13.0000 0.455370
\(816\) 0 0
\(817\) 22.0000 0.769683
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) 11.0000 0.384137
\(821\) −46.0000 −1.60541 −0.802706 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 6.00000 0.208263
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −42.0000 −1.45521
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) 26.0000 0.897620 0.448810 0.893627i \(-0.351848\pi\)
0.448810 + 0.893627i \(0.351848\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −2.00000 −0.0689246
\(843\) 0 0
\(844\) −5.00000 −0.172107
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 11.0000 0.377742
\(849\) 0 0
\(850\) 7.00000 0.240098
\(851\) 0 0
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) −1.00000 −0.0342193
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −31.0000 −1.05894 −0.529470 0.848329i \(-0.677609\pi\)
−0.529470 + 0.848329i \(0.677609\pi\)
\(858\) 0 0
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) −11.0000 −0.375097
\(861\) 0 0
\(862\) 1.00000 0.0340601
\(863\) −27.0000 −0.919091 −0.459545 0.888154i \(-0.651988\pi\)
−0.459545 + 0.888154i \(0.651988\pi\)
\(864\) 0 0
\(865\) 3.00000 0.102003
\(866\) 20.0000 0.679628
\(867\) 0 0
\(868\) 7.00000 0.237595
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 7.00000 0.237050
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 1.00000 0.0337484
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) 0 0
\(883\) 19.0000 0.639401 0.319700 0.947519i \(-0.396418\pi\)
0.319700 + 0.947519i \(0.396418\pi\)
\(884\) 14.0000 0.470871
\(885\) 0 0
\(886\) 30.0000 1.00787
\(887\) 47.0000 1.57811 0.789053 0.614325i \(-0.210572\pi\)
0.789053 + 0.614325i \(0.210572\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 3.00000 0.100447
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) −63.0000 −2.10117
\(900\) 0 0
\(901\) 77.0000 2.56524
\(902\) 11.0000 0.366260
\(903\) 0 0
\(904\) 13.0000 0.432374
\(905\) −26.0000 −0.864269
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 23.0000 0.763282
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) −11.0000 −0.363848
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 13.0000 0.428132
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 2.00000 0.0657241
\(927\) 0 0
\(928\) −9.00000 −0.295439
\(929\) 21.0000 0.688988 0.344494 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) −3.00000 −0.0981630
\(935\) 7.00000 0.228924
\(936\) 0 0
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −11.0000 −0.357641
\(947\) 47.0000 1.52729 0.763647 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) 7.00000 0.226871
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −19.0000 −0.614826
\(956\) −27.0000 −0.873242
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) 6.00000 0.193247
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −10.0000 −0.321412
\(969\) 0 0
\(970\) −19.0000 −0.610053
\(971\) −49.0000 −1.57248 −0.786242 0.617918i \(-0.787976\pi\)
−0.786242 + 0.617918i \(0.787976\pi\)
\(972\) 0 0
\(973\) 21.0000 0.673229
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −5.00000 −0.159964 −0.0799821 0.996796i \(-0.525486\pi\)
−0.0799821 + 0.996796i \(0.525486\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) −51.0000 −1.62665 −0.813324 0.581811i \(-0.802344\pi\)
−0.813324 + 0.581811i \(0.802344\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) −63.0000 −2.00633
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 37.0000 1.17534 0.587672 0.809099i \(-0.300045\pi\)
0.587672 + 0.809099i \(0.300045\pi\)
\(992\) 7.00000 0.222250
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −28.0000 −0.886325
\(999\) 0 0
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 3330.2.a.x.1.1 1
3.2 odd 2 1110.2.a.b.1.1 1
12.11 even 2 8880.2.a.s.1.1 1
15.14 odd 2 5550.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.b.1.1 1 3.2 odd 2
3330.2.a.x.1.1 1 1.1 even 1 trivial
5550.2.a.bk.1.1 1 15.14 odd 2
8880.2.a.s.1.1 1 12.11 even 2