Properties

Label 690.2.a.f.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} -4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -12.0000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -6.00000 q^{58} +1.00000 q^{60} +6.00000 q^{61} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +2.00000 q^{85} +12.0000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} +1.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 2.00000 0.242536
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 12.0000 1.29399
\(87\) 6.00000 0.643268
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 7.00000 0.707107
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 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
\(129\) −12.0000 −1.05654
\(130\) 2.00000 0.175412
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −12.0000 −1.00702
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 6.00000 0.496564
\(147\) −7.00000 −0.577350
\(148\) 2.00000 0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −4.00000 −0.318223
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 2.00000 0.156174
\(165\) 4.00000 0.311400
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) −1.00000 −0.0737210
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 10.0000 0.717958
\(195\) −2.00000 −0.143223
\(196\) −7.00000 −0.500000
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −4.00000 −0.284268
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000 0.846415
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −2.00000 −0.137361
\(213\) 12.0000 0.822226
\(214\) 4.00000 0.273434
\(215\) −12.0000 −0.818393
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) −6.00000 −0.405442
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) −2.00000 −0.134231
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 2.00000 0.130744
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 1.00000 0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) −7.00000 −0.447214
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 16.0000 1.00393
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −4.00000 −0.246183
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 12.0000 0.733017
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) 4.00000 0.241209
\(276\) 1.00000 0.0601929
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −8.00000 −0.476393
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) −6.00000 −0.351123
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 4.00000 0.232104
\(298\) 10.0000 0.579284
\(299\) −2.00000 −0.115663
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 2.00000 0.113228
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 2.00000 0.112154
\(319\) 24.0000 1.34374
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 20.0000 1.10770
\(327\) −2.00000 −0.110600
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −4.00000 −0.219529
\(333\) 2.00000 0.109599
\(334\) −8.00000 −0.437741
\(335\) 12.0000 0.655630
\(336\) 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
\(339\) −6.00000 −0.325875
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 1.00000 0.0538382
\(346\) 14.0000 0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −4.00000 −0.213201
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 8.00000 0.422813
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) −6.00000 −0.315353
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) −6.00000 −0.313625
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.00000 0.104116
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −8.00000 −0.413670
\(375\) 1.00000 0.0516398
\(376\) −8.00000 −0.412568
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −16.0000 −0.818631
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) −12.0000 −0.609994
\(388\) −10.0000 −0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 2.00000 0.101274
\(391\) 2.00000 0.101144
\(392\) 7.00000 0.353553
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) 4.00000 0.201262
\(396\) 4.00000 0.201008
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) −2.00000 −0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −14.0000 −0.690569
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −4.00000 −0.196352
\(416\) 2.00000 0.0980581
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −4.00000 −0.194717
\(423\) 8.00000 0.388973
\(424\) 2.00000 0.0971286
\(425\) 2.00000 0.0970143
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −8.00000 −0.386244
\(430\) 12.0000 0.578691
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) 4.00000 0.190261
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 2.00000 0.0949158
\(445\) −6.00000 −0.284427
\(446\) −16.0000 −0.757622
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 8.00000 0.376705
\(452\) −6.00000 −0.282216
\(453\) −16.0000 −0.751746
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −14.0000 −0.654177
\(459\) 2.00000 0.0933520
\(460\) 1.00000 0.0466252
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) −48.0000 −2.20704
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 12.0000 0.548867
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −4.00000 −0.182384
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −10.0000 −0.454077
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −6.00000 −0.271607
\(489\) −20.0000 −0.904431
\(490\) 7.00000 0.316228
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) −12.0000 −0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −4.00000 −0.177822
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −22.0000 −0.970378
\(515\) 8.00000 0.352522
\(516\) −12.0000 −0.528271
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 2.00000 0.0877058
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 6.00000 0.259645
\(535\) −4.00000 −0.172935
\(536\) −12.0000 −0.518321
\(537\) −8.00000 −0.345225
\(538\) 18.0000 0.776035
\(539\) −28.0000 −1.20605
\(540\) 1.00000 0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 8.00000 0.343629
\(543\) 6.00000 0.257485
\(544\) −2.00000 −0.0857493
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) −14.0000 −0.598050
\(549\) 6.00000 0.256074
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −14.0000 −0.594803
\(555\) 2.00000 0.0848953
\(556\) 4.00000 0.169638
\(557\) −26.0000 −1.10166 −0.550828 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −10.0000 −0.421825
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 8.00000 0.336861
\(565\) −6.00000 −0.252422
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) −8.00000 −0.334497
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 13.0000 0.540729
\(579\) −6.00000 −0.249351
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −8.00000 −0.331326
\(584\) 6.00000 0.248282
\(585\) −2.00000 −0.0826898
\(586\) 18.0000 0.743573
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −7.00000 −0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) −22.0000 −0.904959
\(592\) 2.00000 0.0821995
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −4.00000 −0.163709
\(598\) 2.00000 0.0817861
\(599\) 44.0000 1.79779 0.898896 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) −16.0000 −0.651031
\(605\) 5.00000 0.203279
\(606\) −6.00000 −0.243733
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −16.0000 −0.647291
\(612\) 2.00000 0.0808452
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −12.0000 −0.484281
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −8.00000 −0.321807
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −4.00000 −0.159111
\(633\) 4.00000 0.158986
\(634\) 6.00000 0.238290
\(635\) −16.0000 −0.634941
\(636\) −2.00000 −0.0793052
\(637\) 14.0000 0.554700
\(638\) −24.0000 −0.950169
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 4.00000 0.155700
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 12.0000 0.466393
\(663\) −4.00000 −0.155347
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 6.00000 0.232321
\(668\) 8.00000 0.309529
\(669\) 16.0000 0.618596
\(670\) −12.0000 −0.463600
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −12.0000 −0.457496
\(689\) 4.00000 0.152388
\(690\) −1.00000 −0.0380693
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 4.00000 0.151729
\(696\) −6.00000 −0.227429
\(697\) 4.00000 0.151511
\(698\) 2.00000 0.0757011
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 8.00000 0.301297
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 0 0
\(709\) −50.0000 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) −12.0000 −0.450352
\(711\) 4.00000 0.150012
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −8.00000 −0.298974
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) −22.0000 −0.818189
\(724\) 6.00000 0.222988
\(725\) 6.00000 0.222834
\(726\) −5.00000 −0.185567
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −24.0000 −0.887672
\(732\) 6.00000 0.221766
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −16.0000 −0.590571
\(735\) −7.00000 −0.258199
\(736\) −1.00000 −0.0368605
\(737\) 48.0000 1.76810
\(738\) −2.00000 −0.0736210
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −10.0000 −0.366126
\(747\) −4.00000 −0.146352
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) 12.0000 0.437014
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) −8.00000 −0.290573
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 16.0000 0.578860
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) −6.00000 −0.215945
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) −2.00000 −0.0716115
\(781\) 48.0000 1.71758
\(782\) −2.00000 −0.0715199
\(783\) 6.00000 0.214423
\(784\) −7.00000 −0.250000
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −22.0000 −0.783718
\(789\) 8.00000 0.284808
\(790\) −4.00000 −0.142314
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) −12.0000 −0.426132
\(794\) 10.0000 0.354887
\(795\) −2.00000 −0.0709327
\(796\) −4.00000 −0.141776
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −18.0000 −0.635602
\(803\) −24.0000 −0.846942
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) −6.00000 −0.211079
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −8.00000 −0.280400
\(815\) −20.0000 −0.700569
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 14.0000 0.488306
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −8.00000 −0.278693
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 1.00000 0.0347524
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 4.00000 0.138842
\(831\) 14.0000 0.485655
\(832\) −2.00000 −0.0693375
\(833\) −14.0000 −0.485071
\(834\) −4.00000 −0.138509
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 56.0000 1.93333 0.966667 0.256036i \(-0.0824164\pi\)
0.966667 + 0.256036i \(0.0824164\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 2.00000 0.0689246
\(843\) 10.0000 0.344418
\(844\) 4.00000 0.137686
\(845\) −9.00000 −0.309609
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) 28.0000 0.960958
\(850\) −2.00000 −0.0685994
\(851\) 2.00000 0.0685591
\(852\) 12.0000 0.411113
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 8.00000 0.273115
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −14.0000 −0.476014
\(866\) 26.0000 0.883516
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) −6.00000 −0.203419
\(871\) −24.0000 −0.813209
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 4.00000 0.134840
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 7.00000 0.235702
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 4.00000 0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 14.0000 0.467186
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −4.00000 −0.133259
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 6.00000 0.199447
\(906\) 16.0000 0.531564
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −12.0000 −0.398234
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 2.00000 0.0661541
\(915\) 6.00000 0.198354
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 12.0000 0.395413
\(922\) −30.0000 −0.987997
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −8.00000 −0.262896
\(927\) 8.00000 0.262754
\(928\) −6.00000 −0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −26.0000 −0.851658
\(933\) −12.0000 −0.392862
\(934\) −36.0000 −1.17796
\(935\) 8.00000 0.261628
\(936\) 2.00000 0.0653720
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 8.00000 0.260931
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 14.0000 0.456145
\(943\) 2.00000 0.0651290
\(944\) 0 0
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 4.00000 0.129914
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 2.00000 0.0647524
\(955\) 16.0000 0.517748
\(956\) −12.0000 −0.388108
\(957\) 24.0000 0.775810
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) −4.00000 −0.128898
\(964\) −22.0000 −0.708572
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) −2.00000 −0.0640513
\(976\) 6.00000 0.192055
\(977\) 26.0000 0.831814 0.415907 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) 20.0000 0.639529
\(979\) −24.0000 −0.767043
\(980\) −7.00000 −0.223607
\(981\) −2.00000 −0.0638551
\(982\) 32.0000 1.02116
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −22.0000 −0.700978
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) −12.0000 −0.381578
\(990\) −4.00000 −0.127128
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) −4.00000 −0.126745
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −12.0000 −0.379853
\(999\) 2.00000 0.0632772
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 690.2.a.f.1.1 1
3.2 odd 2 2070.2.a.m.1.1 1
4.3 odd 2 5520.2.a.j.1.1 1
5.2 odd 4 3450.2.d.i.2899.1 2
5.3 odd 4 3450.2.d.i.2899.2 2
5.4 even 2 3450.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.f.1.1 1 1.1 even 1 trivial
2070.2.a.m.1.1 1 3.2 odd 2
3450.2.a.q.1.1 1 5.4 even 2
3450.2.d.i.2899.1 2 5.2 odd 4
3450.2.d.i.2899.2 2 5.3 odd 4
5520.2.a.j.1.1 1 4.3 odd 2