Properties

Label 3330.2.a.b.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +5.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -5.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -3.00000 q^{28} +1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +3.00000 q^{35} -1.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} +7.00000 q^{41} +3.00000 q^{43} +5.00000 q^{44} -4.00000 q^{46} +2.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -5.00000 q^{53} -5.00000 q^{55} +3.00000 q^{56} -1.00000 q^{58} -6.00000 q^{59} +5.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} -3.00000 q^{70} +12.0000 q^{71} +1.00000 q^{74} -6.00000 q^{76} -15.0000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -7.00000 q^{82} -6.00000 q^{83} +3.00000 q^{85} -3.00000 q^{86} -5.00000 q^{88} +18.0000 q^{89} +6.00000 q^{91} +4.00000 q^{92} +6.00000 q^{95} -13.0000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −15.0000 −1.70941
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −7.00000 −0.773021
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −3.00000 −0.323498
\(87\) 0 0
\(88\) −5.00000 −0.533002
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 5.00000 0.485643
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 5.00000 0.476731
\(111\) 0 0
\(112\) −3.00000 −0.283473
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 9.00000 0.825029
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −5.00000 −0.452679
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 0 0
\(133\) 18.0000 1.56080
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −10.0000 −0.836242
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 15.0000 1.20873
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −3.00000 −0.230089
\(171\) 0 0
\(172\) 3.00000 0.228748
\(173\) −13.0000 −0.988372 −0.494186 0.869356i \(-0.664534\pi\)
−0.494186 + 0.869356i \(0.664534\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 13.0000 0.933346
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 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
\(202\) −12.0000 −0.844317
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −7.00000 −0.488901
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) 19.0000 1.30801 0.654007 0.756489i \(-0.273087\pi\)
0.654007 + 0.756489i \(0.273087\pi\)
\(212\) −5.00000 −0.343401
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) −5.00000 −0.338643
\(219\) 0 0
\(220\) −5.00000 −0.337100
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 9.00000 0.598671
\(227\) 21.0000 1.39382 0.696909 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) 0 0
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) −9.00000 −0.583383
\(239\) −29.0000 −1.87585 −0.937927 0.346833i \(-0.887257\pi\)
−0.937927 + 0.346833i \(0.887257\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 20.0000 1.25739
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 10.0000 0.617802
\(263\) −5.00000 −0.308313 −0.154157 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 0 0
\(265\) 5.00000 0.307148
\(266\) −18.0000 −1.10365
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −5.00000 −0.299880
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) −21.0000 −1.23959
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 1.00000 0.0587220
\(291\) 0 0
\(292\) 0 0
\(293\) −7.00000 −0.408944 −0.204472 0.978872i \(-0.565548\pi\)
−0.204472 + 0.978872i \(0.565548\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) −5.00000 −0.286299
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) −15.0000 −0.854704
\(309\) 0 0
\(310\) −3.00000 −0.170389
\(311\) 31.0000 1.75785 0.878924 0.476961i \(-0.158262\pi\)
0.878924 + 0.476961i \(0.158262\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −11.0000 −0.620766
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 27.0000 1.51647 0.758236 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) −11.0000 −0.609234
\(327\) 0 0
\(328\) −7.00000 −0.386510
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 3.00000 0.162698
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) −3.00000 −0.161749
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) 37.0000 1.96931 0.984656 0.174509i \(-0.0558337\pi\)
0.984656 + 0.174509i \(0.0558337\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) −3.00000 −0.156599 −0.0782994 0.996930i \(-0.524949\pi\)
−0.0782994 + 0.996930i \(0.524949\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) 15.0000 0.778761
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 15.0000 0.775632
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 32.0000 1.64373 0.821865 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −3.00000 −0.153493
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 15.0000 0.764471
\(386\) −18.0000 −0.916176
\(387\) 0 0
\(388\) −13.0000 −0.659975
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) −2.00000 −0.101015
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 7.00000 0.345705
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 18.0000 0.885722
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 30.0000 1.46735
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −19.0000 −0.924906
\(423\) 0 0
\(424\) 5.00000 0.242821
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) −15.0000 −0.725901
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 3.00000 0.144673
\(431\) 7.00000 0.337178 0.168589 0.985686i \(-0.446079\pi\)
0.168589 + 0.985686i \(0.446079\pi\)
\(432\) 0 0
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) −9.00000 −0.432014
\(435\) 0 0
\(436\) 5.00000 0.239457
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 5.00000 0.238366
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) −14.0000 −0.665160 −0.332580 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) −7.00000 −0.331460
\(447\) 0 0
\(448\) −3.00000 −0.141737
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) 35.0000 1.64809
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) −21.0000 −0.985579
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −5.00000 −0.233890 −0.116945 0.993138i \(-0.537310\pi\)
−0.116945 + 0.993138i \(0.537310\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 15.0000 0.689701
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 9.00000 0.412514
\(477\) 0 0
\(478\) 29.0000 1.32643
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 13.0000 0.590300
\(486\) 0 0
\(487\) −6.00000 −0.271886 −0.135943 0.990717i \(-0.543406\pi\)
−0.135943 + 0.990717i \(0.543406\pi\)
\(488\) −5.00000 −0.226339
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −36.0000 −1.61482
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) −20.0000 −0.889108
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 26.0000 1.14681
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) −3.00000 −0.131812
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 5.00000 0.218010
\(527\) 9.00000 0.392046
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −5.00000 −0.217186
\(531\) 0 0
\(532\) 18.0000 0.780399
\(533\) −14.0000 −0.606407
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −16.0000 −0.689809
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) 19.0000 0.812381 0.406191 0.913788i \(-0.366857\pi\)
0.406191 + 0.913788i \(0.366857\pi\)
\(548\) 10.0000 0.427179
\(549\) 0 0
\(550\) −5.00000 −0.213201
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 16.0000 0.679775
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 17.0000 0.716465 0.358232 0.933632i \(-0.383380\pi\)
0.358232 + 0.933632i \(0.383380\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −46.0000 −1.92842 −0.964210 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) −21.0000 −0.878823 −0.439411 0.898286i \(-0.644813\pi\)
−0.439411 + 0.898286i \(0.644813\pi\)
\(572\) −10.0000 −0.418121
\(573\) 0 0
\(574\) 21.0000 0.876523
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) −1.00000 −0.0415227
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) −25.0000 −1.03539
\(584\) 0 0
\(585\) 0 0
\(586\) 7.00000 0.289167
\(587\) −1.00000 −0.0412744 −0.0206372 0.999787i \(-0.506569\pi\)
−0.0206372 + 0.999787i \(0.506569\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −9.00000 −0.368964
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 9.00000 0.366813
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 5.00000 0.202444
\(611\) 0 0
\(612\) 0 0
\(613\) −43.0000 −1.73675 −0.868377 0.495905i \(-0.834836\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 15.0000 0.604367
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) −27.0000 −1.08522 −0.542611 0.839984i \(-0.682564\pi\)
−0.542611 + 0.839984i \(0.682564\pi\)
\(620\) 3.00000 0.120483
\(621\) 0 0
\(622\) −31.0000 −1.24299
\(623\) −54.0000 −2.16346
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 30.0000 1.19904
\(627\) 0 0
\(628\) 11.0000 0.438948
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −27.0000 −1.07231
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) −5.00000 −0.197952
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 23.0000 0.908445 0.454223 0.890888i \(-0.349917\pi\)
0.454223 + 0.890888i \(0.349917\pi\)
\(642\) 0 0
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −18.0000 −0.708201
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) −30.0000 −1.17760
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) −16.0000 −0.626128 −0.313064 0.949732i \(-0.601356\pi\)
−0.313064 + 0.949732i \(0.601356\pi\)
\(654\) 0 0
\(655\) 10.0000 0.390732
\(656\) 7.00000 0.273304
\(657\) 0 0
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −29.0000 −1.12797 −0.563985 0.825785i \(-0.690732\pi\)
−0.563985 + 0.825785i \(0.690732\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −18.0000 −0.698010
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 25.0000 0.965114
\(672\) 0 0
\(673\) 48.0000 1.85026 0.925132 0.379646i \(-0.123954\pi\)
0.925132 + 0.379646i \(0.123954\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 0 0
\(679\) 39.0000 1.49668
\(680\) −3.00000 −0.115045
\(681\) 0 0
\(682\) 15.0000 0.574380
\(683\) 29.0000 1.10965 0.554827 0.831966i \(-0.312784\pi\)
0.554827 + 0.831966i \(0.312784\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 3.00000 0.114374
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) −13.0000 −0.494186
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) −21.0000 −0.795432
\(698\) 20.0000 0.757011
\(699\) 0 0
\(700\) −3.00000 −0.113389
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −37.0000 −1.39251
\(707\) −36.0000 −1.35392
\(708\) 0 0
\(709\) 49.0000 1.84023 0.920117 0.391644i \(-0.128094\pi\)
0.920117 + 0.391644i \(0.128094\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) −18.0000 −0.674579
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 10.0000 0.373979
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −30.0000 −1.11959
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −22.0000 −0.815935 −0.407967 0.912996i \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) −6.00000 −0.222375
\(729\) 0 0
\(730\) 0 0
\(731\) −9.00000 −0.332877
\(732\) 0 0
\(733\) −51.0000 −1.88373 −0.941864 0.335994i \(-0.890928\pi\)
−0.941864 + 0.335994i \(0.890928\pi\)
\(734\) 3.00000 0.110732
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) 1.00000 0.0367856 0.0183928 0.999831i \(-0.494145\pi\)
0.0183928 + 0.999831i \(0.494145\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) −15.0000 −0.550667
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) −15.0000 −0.548454
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) −32.0000 −1.16229
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) −25.0000 −0.906249 −0.453125 0.891447i \(-0.649691\pi\)
−0.453125 + 0.891447i \(0.649691\pi\)
\(762\) 0 0
\(763\) −15.0000 −0.543036
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) −15.0000 −0.540562
\(771\) 0 0
\(772\) 18.0000 0.647834
\(773\) 17.0000 0.611448 0.305724 0.952120i \(-0.401102\pi\)
0.305724 + 0.952120i \(0.401102\pi\)
\(774\) 0 0
\(775\) −3.00000 −0.107763
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) −3.00000 −0.107555
\(779\) −42.0000 −1.50481
\(780\) 0 0
\(781\) 60.0000 2.14697
\(782\) 12.0000 0.429119
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) −11.0000 −0.392607
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 27.0000 0.960009
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −40.0000 −1.41687 −0.708436 0.705775i \(-0.750599\pi\)
−0.708436 + 0.705775i \(0.750599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −2.00000 −0.0706225
\(803\) 0 0
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) −6.00000 −0.211341
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −3.00000 −0.105279
\(813\) 0 0
\(814\) 5.00000 0.175250
\(815\) −11.0000 −0.385313
\(816\) 0 0
\(817\) −18.0000 −0.629740
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) −7.00000 −0.244451
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 56.0000 1.95204 0.976019 0.217687i \(-0.0698512\pi\)
0.976019 + 0.217687i \(0.0698512\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −18.0000 −0.626300
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) −6.00000 −0.208263
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) −30.0000 −1.03757
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 26.0000 0.897620 0.448810 0.893627i \(-0.351848\pi\)
0.448810 + 0.893627i \(0.351848\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −2.00000 −0.0689246
\(843\) 0 0
\(844\) 19.0000 0.654007
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −42.0000 −1.44314
\(848\) −5.00000 −0.171701
\(849\) 0 0
\(850\) 3.00000 0.102899
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 15.0000 0.513289
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) −3.00000 −0.102299
\(861\) 0 0
\(862\) −7.00000 −0.238421
\(863\) −27.0000 −0.919091 −0.459545 0.888154i \(-0.651988\pi\)
−0.459545 + 0.888154i \(0.651988\pi\)
\(864\) 0 0
\(865\) 13.0000 0.442013
\(866\) 8.00000 0.271851
\(867\) 0 0
\(868\) 9.00000 0.305480
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −5.00000 −0.169321
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 27.0000 0.911725 0.455863 0.890050i \(-0.349331\pi\)
0.455863 + 0.890050i \(0.349331\pi\)
\(878\) 29.0000 0.978703
\(879\) 0 0
\(880\) −5.00000 −0.168550
\(881\) 39.0000 1.31394 0.656972 0.753915i \(-0.271837\pi\)
0.656972 + 0.753915i \(0.271837\pi\)
\(882\) 0 0
\(883\) −3.00000 −0.100958 −0.0504790 0.998725i \(-0.516075\pi\)
−0.0504790 + 0.998725i \(0.516075\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 14.0000 0.470339
\(887\) 7.00000 0.235037 0.117518 0.993071i \(-0.462506\pi\)
0.117518 + 0.993071i \(0.462506\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) 7.00000 0.234377
\(893\) 0 0
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) −35.0000 −1.16537
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 21.0000 0.696909
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −30.0000 −0.992855
\(914\) 5.00000 0.165385
\(915\) 0 0
\(916\) 12.0000 0.396491
\(917\) 30.0000 0.990687
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) 21.0000 0.691598
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) −34.0000 −1.11731
\(927\) 0 0
\(928\) −1.00000 −0.0328266
\(929\) 1.00000 0.0328089 0.0164045 0.999865i \(-0.494778\pi\)
0.0164045 + 0.999865i \(0.494778\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) 0 0
\(934\) −15.0000 −0.490815
\(935\) 15.0000 0.490552
\(936\) 0 0
\(937\) −40.0000 −1.30674 −0.653372 0.757037i \(-0.726646\pi\)
−0.653372 + 0.757037i \(0.726646\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 28.0000 0.911805
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −15.0000 −0.487692
\(947\) 29.0000 0.942373 0.471187 0.882034i \(-0.343826\pi\)
0.471187 + 0.882034i \(0.343826\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) 0 0
\(952\) −9.00000 −0.291692
\(953\) 40.0000 1.29573 0.647864 0.761756i \(-0.275663\pi\)
0.647864 + 0.761756i \(0.275663\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) −29.0000 −0.937927
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) −13.0000 −0.417405
\(971\) 35.0000 1.12320 0.561602 0.827408i \(-0.310185\pi\)
0.561602 + 0.827408i \(0.310185\pi\)
\(972\) 0 0
\(973\) −15.0000 −0.480878
\(974\) 6.00000 0.192252
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 25.0000 0.799821 0.399910 0.916554i \(-0.369041\pi\)
0.399910 + 0.916554i \(0.369041\pi\)
\(978\) 0 0
\(979\) 90.0000 2.87641
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 4.00000 0.127645
\(983\) 29.0000 0.924956 0.462478 0.886631i \(-0.346960\pi\)
0.462478 + 0.886631i \(0.346960\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 3.00000 0.0955395
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) 7.00000 0.222362 0.111181 0.993800i \(-0.464537\pi\)
0.111181 + 0.993800i \(0.464537\pi\)
\(992\) 3.00000 0.0952501
\(993\) 0 0
\(994\) 36.0000 1.14185
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 0 0
\(999\) 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.b.1.1 1
3.2 odd 2 1110.2.a.j.1.1 1
12.11 even 2 8880.2.a.bc.1.1 1
15.14 odd 2 5550.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.j.1.1 1 3.2 odd 2
3330.2.a.b.1.1 1 1.1 even 1 trivial
5550.2.a.t.1.1 1 15.14 odd 2
8880.2.a.bc.1.1 1 12.11 even 2