Properties

Label 3380.2.f.e.3041.1
Level $3380$
Weight $2$
Character 3380.3041
Analytic conductor $26.989$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3041.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3380.3041
Dual form 3380.2.f.e.3041.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000i q^{5} -1.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} -1.00000i q^{15} +3.00000 q^{17} +7.00000i q^{19} -1.00000i q^{21} +3.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +4.00000i q^{31} +3.00000i q^{33} -1.00000 q^{35} -7.00000i q^{37} +9.00000i q^{41} -11.0000 q^{43} +2.00000i q^{45} +6.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +7.00000i q^{57} -3.00000i q^{59} +11.0000 q^{61} +2.00000i q^{63} +7.00000i q^{67} +3.00000 q^{69} +3.00000i q^{71} +2.00000i q^{73} -1.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +15.0000i q^{89} +4.00000i q^{93} +7.00000 q^{95} +7.00000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 4 q^{9} + 6 q^{17} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{35} - 22 q^{43} + 12 q^{49} + 6 q^{51} - 12 q^{53} + 6 q^{55} + 22 q^{61} + 6 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{87} + 14 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 1.00000i − 0.258199i
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 7.00000i 1.60591i 0.596040 + 0.802955i \(0.296740\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(20\) 0 0
\(21\) − 1.00000i − 0.218218i
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 3.00000i 0.522233i
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000i 1.40556i 0.711405 + 0.702782i \(0.248059\pi\)
−0.711405 + 0.702782i \(0.751941\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 7.00000i 0.927173i
\(58\) 0 0
\(59\) − 3.00000i − 0.390567i −0.980747 0.195283i \(-0.937437\pi\)
0.980747 0.195283i \(-0.0625627\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 3.00000i 0.356034i 0.984027 + 0.178017i \(0.0569683\pi\)
−0.984027 + 0.178017i \(0.943032\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) − 3.00000i − 0.325396i
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 15.0000i 1.59000i 0.606612 + 0.794998i \(0.292528\pi\)
−0.606612 + 0.794998i \(0.707472\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 7.00000 0.718185
\(96\) 0 0
\(97\) 7.00000i 0.710742i 0.934725 + 0.355371i \(0.115646\pi\)
−0.934725 + 0.355371i \(0.884354\pi\)
\(98\) 0 0
\(99\) − 6.00000i − 0.603023i
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) − 7.00000i − 0.664411i
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) − 3.00000i − 0.279751i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 3.00000i − 0.275010i
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 9.00000i 0.811503i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 19.0000 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 7.00000 0.606977
\(134\) 0 0
\(135\) 5.00000i 0.430331i
\(136\) 0 0
\(137\) − 15.0000i − 1.28154i −0.767734 0.640768i \(-0.778616\pi\)
0.767734 0.640768i \(-0.221384\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 3.00000i − 0.249136i
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) 21.0000i 1.72039i 0.509968 + 0.860194i \(0.329657\pi\)
−0.509968 + 0.860194i \(0.670343\pi\)
\(150\) 0 0
\(151\) 8.00000i 0.651031i 0.945537 + 0.325515i \(0.105538\pi\)
−0.945537 + 0.325515i \(0.894462\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) − 3.00000i − 0.236433i
\(162\) 0 0
\(163\) − 1.00000i − 0.0783260i −0.999233 0.0391630i \(-0.987531\pi\)
0.999233 0.0391630i \(-0.0124692\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) − 3.00000i − 0.232147i −0.993241 0.116073i \(-0.962969\pi\)
0.993241 0.116073i \(-0.0370308\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 14.0000i − 1.07061i
\(172\) 0 0
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 0 0
\(175\) 1.00000i 0.0755929i
\(176\) 0 0
\(177\) − 3.00000i − 0.225494i
\(178\) 0 0
\(179\) 21.0000 1.56961 0.784807 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 11.0000 0.813143
\(184\) 0 0
\(185\) −7.00000 −0.514650
\(186\) 0 0
\(187\) 9.00000i 0.658145i
\(188\) 0 0
\(189\) 5.00000i 0.363696i
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) 5.00000i 0.359908i 0.983675 + 0.179954i \(0.0575949\pi\)
−0.983675 + 0.179954i \(0.942405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 21.0000i − 1.49619i −0.663593 0.748094i \(-0.730969\pi\)
0.663593 0.748094i \(-0.269031\pi\)
\(198\) 0 0
\(199\) −17.0000 −1.20510 −0.602549 0.798082i \(-0.705848\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) 7.00000i 0.493742i
\(202\) 0 0
\(203\) − 3.00000i − 0.210559i
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) 0 0
\(213\) 3.00000i 0.205557i
\(214\) 0 0
\(215\) 11.0000i 0.750194i
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.0000i 1.27233i 0.771551 + 0.636167i \(0.219481\pi\)
−0.771551 + 0.636167i \(0.780519\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) − 27.0000i − 1.79205i −0.444001 0.896026i \(-0.646441\pi\)
0.444001 0.896026i \(-0.353559\pi\)
\(228\) 0 0
\(229\) − 22.0000i − 1.45380i −0.686743 0.726900i \(-0.740960\pi\)
0.686743 0.726900i \(-0.259040\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) − 1.00000i − 0.0644157i −0.999481 0.0322078i \(-0.989746\pi\)
0.999481 0.0322078i \(-0.0102538\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) − 6.00000i − 0.383326i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 9.00000i 0.565825i
\(254\) 0 0
\(255\) − 3.00000i − 0.187867i
\(256\) 0 0
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −3.00000 −0.184988 −0.0924940 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 0 0
\(267\) 15.0000i 0.917985i
\(268\) 0 0
\(269\) 27.0000 1.64622 0.823110 0.567883i \(-0.192237\pi\)
0.823110 + 0.567883i \(0.192237\pi\)
\(270\) 0 0
\(271\) 23.0000i 1.39715i 0.715537 + 0.698575i \(0.246182\pi\)
−0.715537 + 0.698575i \(0.753818\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 3.00000i − 0.180907i
\(276\) 0 0
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 0 0
\(279\) − 8.00000i − 0.478947i
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 0 0
\(285\) 7.00000 0.414644
\(286\) 0 0
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 7.00000i 0.410347i
\(292\) 0 0
\(293\) − 27.0000i − 1.57736i −0.614806 0.788678i \(-0.710766\pi\)
0.614806 0.788678i \(-0.289234\pi\)
\(294\) 0 0
\(295\) −3.00000 −0.174667
\(296\) 0 0
\(297\) − 15.0000i − 0.870388i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 11.0000i 0.634029i
\(302\) 0 0
\(303\) 9.00000 0.517036
\(304\) 0 0
\(305\) − 11.0000i − 0.629858i
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 9.00000i 0.503903i
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 21.0000i 1.16847i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.00000i − 0.110600i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 19.0000i 1.04433i 0.852843 + 0.522167i \(0.174876\pi\)
−0.852843 + 0.522167i \(0.825124\pi\)
\(332\) 0 0
\(333\) 14.0000i 0.767195i
\(334\) 0 0
\(335\) 7.00000 0.382451
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 0 0
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) − 13.0000i − 0.701934i
\(344\) 0 0
\(345\) − 3.00000i − 0.161515i
\(346\) 0 0
\(347\) −33.0000 −1.77153 −0.885766 0.464131i \(-0.846367\pi\)
−0.885766 + 0.464131i \(0.846367\pi\)
\(348\) 0 0
\(349\) − 1.00000i − 0.0535288i −0.999642 0.0267644i \(-0.991480\pi\)
0.999642 0.0267644i \(-0.00852039\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 9.00000i − 0.479022i −0.970894 0.239511i \(-0.923013\pi\)
0.970894 0.239511i \(-0.0769871\pi\)
\(354\) 0 0
\(355\) 3.00000 0.159223
\(356\) 0 0
\(357\) − 3.00000i − 0.158777i
\(358\) 0 0
\(359\) − 24.0000i − 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) 0 0
\(361\) −30.0000 −1.57895
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 5.00000 0.260998 0.130499 0.991448i \(-0.458342\pi\)
0.130499 + 0.991448i \(0.458342\pi\)
\(368\) 0 0
\(369\) − 18.0000i − 0.937043i
\(370\) 0 0
\(371\) 6.00000i 0.311504i
\(372\) 0 0
\(373\) −31.0000 −1.60512 −0.802560 0.596572i \(-0.796529\pi\)
−0.802560 + 0.596572i \(0.796529\pi\)
\(374\) 0 0
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.00000i 0.0513665i 0.999670 + 0.0256833i \(0.00817614\pi\)
−0.999670 + 0.0256833i \(0.991824\pi\)
\(380\) 0 0
\(381\) 19.0000 0.973399
\(382\) 0 0
\(383\) 9.00000i 0.459879i 0.973205 + 0.229939i \(0.0738528\pi\)
−0.973205 + 0.229939i \(0.926147\pi\)
\(384\) 0 0
\(385\) − 3.00000i − 0.152894i
\(386\) 0 0
\(387\) 22.0000 1.11832
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) − 8.00000i − 0.402524i
\(396\) 0 0
\(397\) 5.00000i 0.250943i 0.992097 + 0.125471i \(0.0400443\pi\)
−0.992097 + 0.125471i \(0.959956\pi\)
\(398\) 0 0
\(399\) 7.00000 0.350438
\(400\) 0 0
\(401\) − 33.0000i − 1.64794i −0.566632 0.823971i \(-0.691754\pi\)
0.566632 0.823971i \(-0.308246\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 21.0000 1.04093
\(408\) 0 0
\(409\) 25.0000i 1.23617i 0.786111 + 0.618085i \(0.212091\pi\)
−0.786111 + 0.618085i \(0.787909\pi\)
\(410\) 0 0
\(411\) − 15.0000i − 0.739895i
\(412\) 0 0
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) − 2.00000i − 0.0974740i −0.998812 0.0487370i \(-0.984480\pi\)
0.998812 0.0487370i \(-0.0155196\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) − 11.0000i − 0.532327i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.00000i 0.433515i 0.976226 + 0.216757i \(0.0695480\pi\)
−0.976226 + 0.216757i \(0.930452\pi\)
\(432\) 0 0
\(433\) −29.0000 −1.39365 −0.696826 0.717241i \(-0.745405\pi\)
−0.696826 + 0.717241i \(0.745405\pi\)
\(434\) 0 0
\(435\) − 3.00000i − 0.143839i
\(436\) 0 0
\(437\) 21.0000i 1.00457i
\(438\) 0 0
\(439\) −11.0000 −0.525001 −0.262501 0.964932i \(-0.584547\pi\)
−0.262501 + 0.964932i \(0.584547\pi\)
\(440\) 0 0
\(441\) −12.0000 −0.571429
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 15.0000 0.711068
\(446\) 0 0
\(447\) 21.0000i 0.993266i
\(448\) 0 0
\(449\) 3.00000i 0.141579i 0.997491 + 0.0707894i \(0.0225518\pi\)
−0.997491 + 0.0707894i \(0.977448\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 5.00000i − 0.233890i −0.993138 0.116945i \(-0.962690\pi\)
0.993138 0.116945i \(-0.0373101\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) − 27.0000i − 1.25752i −0.777601 0.628758i \(-0.783564\pi\)
0.777601 0.628758i \(-0.216436\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) − 33.0000i − 1.51734i
\(474\) 0 0
\(475\) − 7.00000i − 0.321182i
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 39.0000i 1.78196i 0.454047 + 0.890978i \(0.349980\pi\)
−0.454047 + 0.890978i \(0.650020\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 3.00000i − 0.136505i
\(484\) 0 0
\(485\) 7.00000 0.317854
\(486\) 0 0
\(487\) − 11.0000i − 0.498458i −0.968445 0.249229i \(-0.919823\pi\)
0.968445 0.249229i \(-0.0801771\pi\)
\(488\) 0 0
\(489\) − 1.00000i − 0.0452216i
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) 3.00000 0.134568
\(498\) 0 0
\(499\) − 32.0000i − 1.43252i −0.697835 0.716258i \(-0.745853\pi\)
0.697835 0.716258i \(-0.254147\pi\)
\(500\) 0 0
\(501\) − 3.00000i − 0.134030i
\(502\) 0 0
\(503\) 15.0000 0.668817 0.334408 0.942428i \(-0.391463\pi\)
0.334408 + 0.942428i \(0.391463\pi\)
\(504\) 0 0
\(505\) − 9.00000i − 0.400495i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.00000i 0.398918i 0.979906 + 0.199459i \(0.0639185\pi\)
−0.979906 + 0.199459i \(0.936082\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) − 35.0000i − 1.54529i
\(514\) 0 0
\(515\) 8.00000i 0.352522i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 0 0
\(525\) 1.00000i 0.0436436i
\(526\) 0 0
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 9.00000i − 0.389104i
\(536\) 0 0
\(537\) 21.0000 0.906217
\(538\) 0 0
\(539\) 18.0000i 0.775315i
\(540\) 0 0
\(541\) − 22.0000i − 0.945854i −0.881102 0.472927i \(-0.843197\pi\)
0.881102 0.472927i \(-0.156803\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) −22.0000 −0.938937
\(550\) 0 0
\(551\) 21.0000i 0.894630i
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 0 0
\(555\) −7.00000 −0.297133
\(556\) 0 0
\(557\) − 39.0000i − 1.65248i −0.563316 0.826242i \(-0.690475\pi\)
0.563316 0.826242i \(-0.309525\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 9.00000i 0.379980i
\(562\) 0 0
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) − 9.00000i − 0.378633i
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) 5.00000i 0.207793i
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) − 18.0000i − 0.745484i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 33.0000i − 1.36206i −0.732257 0.681028i \(-0.761533\pi\)
0.732257 0.681028i \(-0.238467\pi\)
\(588\) 0 0
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) − 21.0000i − 0.863825i
\(592\) 0 0
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) −17.0000 −0.695764
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) − 14.0000i − 0.570124i
\(604\) 0 0
\(605\) − 2.00000i − 0.0813116i
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 0 0
\(609\) − 3.00000i − 0.121566i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 43.0000i 1.73675i 0.495905 + 0.868377i \(0.334836\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) − 33.0000i − 1.32853i −0.747497 0.664265i \(-0.768745\pi\)
0.747497 0.664265i \(-0.231255\pi\)
\(618\) 0 0
\(619\) 44.0000i 1.76851i 0.467005 + 0.884255i \(0.345333\pi\)
−0.467005 + 0.884255i \(0.654667\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) 0 0
\(623\) 15.0000 0.600962
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −21.0000 −0.838659
\(628\) 0 0
\(629\) − 21.0000i − 0.837325i
\(630\) 0 0
\(631\) 17.0000i 0.676759i 0.941010 + 0.338380i \(0.109879\pi\)
−0.941010 + 0.338380i \(0.890121\pi\)
\(632\) 0 0
\(633\) 11.0000 0.437211
\(634\) 0 0
\(635\) − 19.0000i − 0.753992i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 6.00000i − 0.237356i
\(640\) 0 0
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 0 0
\(643\) − 11.0000i − 0.433798i −0.976194 0.216899i \(-0.930406\pi\)
0.976194 0.216899i \(-0.0695942\pi\)
\(644\) 0 0
\(645\) 11.0000i 0.433125i
\(646\) 0 0
\(647\) −45.0000 −1.76913 −0.884566 0.466415i \(-0.845546\pi\)
−0.884566 + 0.466415i \(0.845546\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) −39.0000 −1.52619 −0.763094 0.646288i \(-0.776321\pi\)
−0.763094 + 0.646288i \(0.776321\pi\)
\(654\) 0 0
\(655\) 12.0000i 0.468879i
\(656\) 0 0
\(657\) − 4.00000i − 0.156055i
\(658\) 0 0
\(659\) −39.0000 −1.51922 −0.759612 0.650376i \(-0.774611\pi\)
−0.759612 + 0.650376i \(0.774611\pi\)
\(660\) 0 0
\(661\) − 1.00000i − 0.0388955i −0.999811 0.0194477i \(-0.993809\pi\)
0.999811 0.0194477i \(-0.00619080\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 7.00000i − 0.271448i
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) 0 0
\(669\) 19.0000i 0.734582i
\(670\) 0 0
\(671\) 33.0000i 1.27395i
\(672\) 0 0
\(673\) −17.0000 −0.655302 −0.327651 0.944799i \(-0.606257\pi\)
−0.327651 + 0.944799i \(0.606257\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) 0 0
\(681\) − 27.0000i − 1.03464i
\(682\) 0 0
\(683\) 51.0000i 1.95146i 0.218975 + 0.975730i \(0.429729\pi\)
−0.218975 + 0.975730i \(0.570271\pi\)
\(684\) 0 0
\(685\) −15.0000 −0.573121
\(686\) 0 0
\(687\) − 22.0000i − 0.839352i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 23.0000i − 0.874961i −0.899228 0.437481i \(-0.855871\pi\)
0.899228 0.437481i \(-0.144129\pi\)
\(692\) 0 0
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) − 5.00000i − 0.189661i
\(696\) 0 0
\(697\) 27.0000i 1.02270i
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 49.0000 1.84807
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.00000i − 0.338480i
\(708\) 0 0
\(709\) − 37.0000i − 1.38956i −0.719220 0.694782i \(-0.755501\pi\)
0.719220 0.694782i \(-0.244499\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −9.00000 −0.335643 −0.167822 0.985817i \(-0.553673\pi\)
−0.167822 + 0.985817i \(0.553673\pi\)
\(720\) 0 0
\(721\) 8.00000i 0.297936i
\(722\) 0 0
\(723\) − 1.00000i − 0.0371904i
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 52.0000 1.92857 0.964287 0.264861i \(-0.0853260\pi\)
0.964287 + 0.264861i \(0.0853260\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −33.0000 −1.22055
\(732\) 0 0
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 0 0
\(735\) − 6.00000i − 0.221313i
\(736\) 0 0
\(737\) −21.0000 −0.773545
\(738\) 0 0
\(739\) 47.0000i 1.72892i 0.502699 + 0.864461i \(0.332340\pi\)
−0.502699 + 0.864461i \(0.667660\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 21.0000i − 0.770415i −0.922830 0.385208i \(-0.874130\pi\)
0.922830 0.385208i \(-0.125870\pi\)
\(744\) 0 0
\(745\) 21.0000 0.769380
\(746\) 0 0
\(747\) − 24.0000i − 0.878114i
\(748\) 0 0
\(749\) − 9.00000i − 0.328853i
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 0 0
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 29.0000 1.05402 0.527011 0.849858i \(-0.323312\pi\)
0.527011 + 0.849858i \(0.323312\pi\)
\(758\) 0 0
\(759\) 9.00000i 0.326679i
\(760\) 0 0
\(761\) 3.00000i 0.108750i 0.998521 + 0.0543750i \(0.0173166\pi\)
−0.998521 + 0.0543750i \(0.982683\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) 6.00000i 0.216930i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 13.0000i 0.468792i 0.972141 + 0.234396i \(0.0753112\pi\)
−0.972141 + 0.234396i \(0.924689\pi\)
\(770\) 0 0
\(771\) −9.00000 −0.324127
\(772\) 0 0
\(773\) 27.0000i 0.971123i 0.874203 + 0.485561i \(0.161385\pi\)
−0.874203 + 0.485561i \(0.838615\pi\)
\(774\) 0 0
\(775\) − 4.00000i − 0.143684i
\(776\) 0 0
\(777\) −7.00000 −0.251124
\(778\) 0 0
\(779\) −63.0000 −2.25721
\(780\) 0 0
\(781\) −9.00000 −0.322045
\(782\) 0 0
\(783\) −15.0000 −0.536056
\(784\) 0 0
\(785\) 10.0000i 0.356915i
\(786\) 0 0
\(787\) − 37.0000i − 1.31891i −0.751745 0.659454i \(-0.770788\pi\)
0.751745 0.659454i \(-0.229212\pi\)
\(788\) 0 0
\(789\) −3.00000 −0.106803
\(790\) 0 0
\(791\) − 9.00000i − 0.320003i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 6.00000i 0.212798i
\(796\) 0 0
\(797\) 51.0000 1.80651 0.903256 0.429101i \(-0.141170\pi\)
0.903256 + 0.429101i \(0.141170\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) − 30.0000i − 1.06000i
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) 27.0000 0.950445
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) − 32.0000i − 1.12367i −0.827249 0.561836i \(-0.810095\pi\)
0.827249 0.561836i \(-0.189905\pi\)
\(812\) 0 0
\(813\) 23.0000i 0.806645i
\(814\) 0 0
\(815\) −1.00000 −0.0350285
\(816\) 0 0
\(817\) − 77.0000i − 2.69389i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 15.0000i − 0.523504i −0.965135 0.261752i \(-0.915700\pi\)
0.965135 0.261752i \(-0.0843002\pi\)
\(822\) 0 0
\(823\) 13.0000 0.453152 0.226576 0.973994i \(-0.427247\pi\)
0.226576 + 0.973994i \(0.427247\pi\)
\(824\) 0 0
\(825\) − 3.00000i − 0.104447i
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 0 0
\(831\) 19.0000 0.659103
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) 0 0
\(837\) − 20.0000i − 0.691301i
\(838\) 0 0
\(839\) 21.0000i 0.725001i 0.931984 + 0.362500i \(0.118077\pi\)
−0.931984 + 0.362500i \(0.881923\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 6.00000i 0.206651i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) 0 0
\(849\) −5.00000 −0.171600
\(850\) 0 0
\(851\) − 21.0000i − 0.719871i
\(852\) 0 0
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) −14.0000 −0.478790
\(856\) 0 0
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 9.00000 0.306719
\(862\) 0 0
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 0 0
\(865\) − 3.00000i − 0.102003i
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 24.0000i 0.814144i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 14.0000i − 0.473828i
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) − 41.0000i − 1.38447i −0.721671 0.692236i \(-0.756626\pi\)
0.721671 0.692236i \(-0.243374\pi\)
\(878\) 0 0
\(879\) − 27.0000i − 0.910687i
\(880\) 0 0
\(881\) −27.0000 −0.909653 −0.454827 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) −3.00000 −0.100844
\(886\) 0 0
\(887\) 9.00000 0.302190 0.151095 0.988519i \(-0.451720\pi\)
0.151095 + 0.988519i \(0.451720\pi\)
\(888\) 0 0
\(889\) − 19.0000i − 0.637240i
\(890\) 0 0
\(891\) 3.00000i 0.100504i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) − 21.0000i − 0.701953i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000i 0.400222i
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) 11.0000i 0.366057i
\(904\) 0 0
\(905\) 2.00000i 0.0664822i
\(906\) 0 0
\(907\) 19.0000 0.630885 0.315442 0.948945i \(-0.397847\pi\)
0.315442 + 0.948945i \(0.397847\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) − 11.0000i − 0.363649i
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 20.0000i 0.659022i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 7.00000i 0.230159i
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) − 51.0000i − 1.67326i −0.547772 0.836628i \(-0.684524\pi\)
0.547772 0.836628i \(-0.315476\pi\)
\(930\) 0 0
\(931\) 42.0000i 1.37649i
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) − 42.0000i − 1.36916i −0.728937 0.684580i \(-0.759985\pi\)
0.728937 0.684580i \(-0.240015\pi\)
\(942\) 0 0
\(943\) 27.0000i 0.879241i
\(944\) 0 0
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) 21.0000i 0.682408i 0.939989 + 0.341204i \(0.110835\pi\)
−0.939989 + 0.341204i \(0.889165\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 18.0000i 0.583690i
\(952\) 0 0
\(953\) 51.0000 1.65205 0.826026 0.563632i \(-0.190596\pi\)
0.826026 + 0.563632i \(0.190596\pi\)
\(954\) 0 0
\(955\) 3.00000i 0.0970777i
\(956\) 0 0
\(957\) 9.00000i 0.290929i
\(958\) 0 0
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) 5.00000 0.160956
\(966\) 0 0
\(967\) 40.0000i 1.28631i 0.765735 + 0.643157i \(0.222376\pi\)
−0.765735 + 0.643157i \(0.777624\pi\)
\(968\) 0 0
\(969\) 21.0000i 0.674617i
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) − 5.00000i − 0.160293i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 9.00000i − 0.287936i −0.989582 0.143968i \(-0.954014\pi\)
0.989582 0.143968i \(-0.0459862\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) 4.00000i 0.127710i
\(982\) 0 0
\(983\) − 36.0000i − 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) 0 0
\(985\) −21.0000 −0.669116
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −33.0000 −1.04934
\(990\) 0 0
\(991\) −61.0000 −1.93773 −0.968864 0.247592i \(-0.920361\pi\)
−0.968864 + 0.247592i \(0.920361\pi\)
\(992\) 0 0
\(993\) 19.0000i 0.602947i
\(994\) 0 0
\(995\) 17.0000i 0.538936i
\(996\) 0 0
\(997\) −31.0000 −0.981780 −0.490890 0.871222i \(-0.663328\pi\)
−0.490890 + 0.871222i \(0.663328\pi\)
\(998\) 0 0
\(999\) 35.0000i 1.10735i
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 3380.2.f.e.3041.1 2
13.5 odd 4 3380.2.a.g.1.1 1
13.7 odd 12 260.2.i.b.81.1 yes 2
13.8 odd 4 3380.2.a.h.1.1 1
13.11 odd 12 260.2.i.b.61.1 2
13.12 even 2 inner 3380.2.f.e.3041.2 2
39.11 even 12 2340.2.q.b.1621.1 2
39.20 even 12 2340.2.q.b.2161.1 2
52.7 even 12 1040.2.q.j.81.1 2
52.11 even 12 1040.2.q.j.321.1 2
65.7 even 12 1300.2.bb.a.549.1 4
65.24 odd 12 1300.2.i.e.1101.1 2
65.33 even 12 1300.2.bb.a.549.2 4
65.37 even 12 1300.2.bb.a.1049.2 4
65.59 odd 12 1300.2.i.e.601.1 2
65.63 even 12 1300.2.bb.a.1049.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.b.61.1 2 13.11 odd 12
260.2.i.b.81.1 yes 2 13.7 odd 12
1040.2.q.j.81.1 2 52.7 even 12
1040.2.q.j.321.1 2 52.11 even 12
1300.2.i.e.601.1 2 65.59 odd 12
1300.2.i.e.1101.1 2 65.24 odd 12
1300.2.bb.a.549.1 4 65.7 even 12
1300.2.bb.a.549.2 4 65.33 even 12
1300.2.bb.a.1049.1 4 65.63 even 12
1300.2.bb.a.1049.2 4 65.37 even 12
2340.2.q.b.1621.1 2 39.11 even 12
2340.2.q.b.2161.1 2 39.20 even 12
3380.2.a.g.1.1 1 13.5 odd 4
3380.2.a.h.1.1 1 13.8 odd 4
3380.2.f.e.3041.1 2 1.1 even 1 trivial
3380.2.f.e.3041.2 2 13.12 even 2 inner