Properties

Label 7098.2.a.v.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +3.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{27} -1.00000 q^{28} -9.00000 q^{29} -3.00000 q^{30} +1.00000 q^{32} -4.00000 q^{33} -5.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +7.00000 q^{37} +4.00000 q^{38} +3.00000 q^{40} +7.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} -4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} +5.00000 q^{51} +7.00000 q^{53} -1.00000 q^{54} +12.0000 q^{55} -1.00000 q^{56} -4.00000 q^{57} -9.00000 q^{58} -8.00000 q^{59} -3.00000 q^{60} +7.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +12.0000 q^{67} -5.00000 q^{68} +4.00000 q^{69} -3.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} -1.00000 q^{73} +7.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} -4.00000 q^{77} -16.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +7.00000 q^{82} -8.00000 q^{83} +1.00000 q^{84} -15.0000 q^{85} +8.00000 q^{86} +9.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -4.00000 q^{92} +12.0000 q^{94} +12.0000 q^{95} -1.00000 q^{96} +18.0000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 3.00000 0.670820
\(21\) 1.00000 0.218218
\(22\) 4.00000 0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) −3.00000 −0.547723
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) −5.00000 −0.857493
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 4.00000 0.603023
\(45\) 3.00000 0.447214
\(46\) −4.00000 −0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 5.00000 0.700140
\(52\) 0 0
\(53\) 7.00000 0.961524 0.480762 0.876851i \(-0.340360\pi\)
0.480762 + 0.876851i \(0.340360\pi\)
\(54\) −1.00000 −0.136083
\(55\) 12.0000 1.61808
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −9.00000 −1.18176
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −3.00000 −0.387298
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −5.00000 −0.606339
\(69\) 4.00000 0.481543
\(70\) −3.00000 −0.358569
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 7.00000 0.813733
\(75\) −4.00000 −0.461880
\(76\) 4.00000 0.458831
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 7.00000 0.773021
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 1.00000 0.109109
\(85\) −15.0000 −1.62698
\(86\) 8.00000 0.862662
\(87\) 9.00000 0.964901
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 12.0000 1.23117
\(96\) −1.00000 −0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.00000 0.402015
\(100\) 4.00000 0.400000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 5.00000 0.495074
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 7.00000 0.679900
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 12.0000 1.14416
\(111\) −7.00000 −0.664411
\(112\) −1.00000 −0.0944911
\(113\) −5.00000 −0.470360 −0.235180 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) −4.00000 −0.374634
\(115\) −12.0000 −1.11901
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) 5.00000 0.458349
\(120\) −3.00000 −0.273861
\(121\) 5.00000 0.454545
\(122\) 7.00000 0.633750
\(123\) −7.00000 −0.631169
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) −4.00000 −0.346844
\(134\) 12.0000 1.03664
\(135\) −3.00000 −0.258199
\(136\) −5.00000 −0.428746
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 4.00000 0.340503
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −3.00000 −0.253546
\(141\) −12.0000 −1.01058
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −27.0000 −2.24223
\(146\) −1.00000 −0.0827606
\(147\) −1.00000 −0.0824786
\(148\) 7.00000 0.575396
\(149\) 7.00000 0.573462 0.286731 0.958011i \(-0.407431\pi\)
0.286731 + 0.958011i \(0.407431\pi\)
\(150\) −4.00000 −0.326599
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000 0.324443
\(153\) −5.00000 −0.404226
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) −16.0000 −1.27289
\(159\) −7.00000 −0.555136
\(160\) 3.00000 0.237171
\(161\) 4.00000 0.315244
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 7.00000 0.546608
\(165\) −12.0000 −0.934199
\(166\) −8.00000 −0.620920
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) −15.0000 −1.15045
\(171\) 4.00000 0.305888
\(172\) 8.00000 0.609994
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 9.00000 0.682288
\(175\) −4.00000 −0.302372
\(176\) 4.00000 0.301511
\(177\) 8.00000 0.601317
\(178\) −6.00000 −0.449719
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 3.00000 0.223607
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) −4.00000 −0.294884
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) −20.0000 −1.46254
\(188\) 12.0000 0.875190
\(189\) 1.00000 0.0727393
\(190\) 12.0000 0.870572
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 4.00000 0.282843
\(201\) −12.0000 −0.846415
\(202\) 3.00000 0.211079
\(203\) 9.00000 0.631676
\(204\) 5.00000 0.350070
\(205\) 21.0000 1.46670
\(206\) −4.00000 −0.278693
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 3.00000 0.207020
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 7.00000 0.480762
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 24.0000 1.63679
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 1.00000 0.0675737
\(220\) 12.0000 0.809040
\(221\) 0 0
\(222\) −7.00000 −0.469809
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.00000 0.266667
\(226\) −5.00000 −0.332595
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −12.0000 −0.791257
\(231\) 4.00000 0.263181
\(232\) −9.00000 −0.590879
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 36.0000 2.34838
\(236\) −8.00000 −0.520756
\(237\) 16.0000 1.03931
\(238\) 5.00000 0.324102
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) −3.00000 −0.193649
\(241\) −9.00000 −0.579741 −0.289870 0.957066i \(-0.593612\pi\)
−0.289870 + 0.957066i \(0.593612\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 7.00000 0.448129
\(245\) 3.00000 0.191663
\(246\) −7.00000 −0.446304
\(247\) 0 0
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) −3.00000 −0.189737
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −16.0000 −1.00591
\(254\) −8.00000 −0.501965
\(255\) 15.0000 0.939336
\(256\) 1.00000 0.0625000
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) −8.00000 −0.498058
\(259\) −7.00000 −0.434959
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) −12.0000 −0.741362
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −4.00000 −0.246183
\(265\) 21.0000 1.29002
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) 12.0000 0.733017
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −3.00000 −0.182574
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) 16.0000 0.964836
\(276\) 4.00000 0.240772
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 15.0000 0.894825 0.447412 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) −12.0000 −0.714590
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) −7.00000 −0.413197
\(288\) 1.00000 0.0589256
\(289\) 8.00000 0.470588
\(290\) −27.0000 −1.58549
\(291\) −18.0000 −1.05518
\(292\) −1.00000 −0.0585206
\(293\) −33.0000 −1.92788 −0.963940 0.266119i \(-0.914259\pi\)
−0.963940 + 0.266119i \(0.914259\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −24.0000 −1.39733
\(296\) 7.00000 0.406867
\(297\) −4.00000 −0.232104
\(298\) 7.00000 0.405499
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) −8.00000 −0.461112
\(302\) −8.00000 −0.460348
\(303\) −3.00000 −0.172345
\(304\) 4.00000 0.229416
\(305\) 21.0000 1.20246
\(306\) −5.00000 −0.285831
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −4.00000 −0.227921
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −13.0000 −0.733632
\(315\) −3.00000 −0.169031
\(316\) −16.0000 −0.900070
\(317\) 35.0000 1.96580 0.982898 0.184151i \(-0.0589536\pi\)
0.982898 + 0.184151i \(0.0589536\pi\)
\(318\) −7.00000 −0.392541
\(319\) −36.0000 −2.01561
\(320\) 3.00000 0.167705
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −20.0000 −1.11283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) −14.0000 −0.774202
\(328\) 7.00000 0.386510
\(329\) −12.0000 −0.661581
\(330\) −12.0000 −0.660578
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −8.00000 −0.439057
\(333\) 7.00000 0.383598
\(334\) −8.00000 −0.437741
\(335\) 36.0000 1.96689
\(336\) 1.00000 0.0545545
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) 0 0
\(339\) 5.00000 0.271563
\(340\) −15.0000 −0.813489
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) 12.0000 0.646058
\(346\) 14.0000 0.752645
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 9.00000 0.482451
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −25.0000 −1.33062 −0.665308 0.746569i \(-0.731700\pi\)
−0.665308 + 0.746569i \(0.731700\pi\)
\(354\) 8.00000 0.425195
\(355\) 36.0000 1.91068
\(356\) −6.00000 −0.317999
\(357\) −5.00000 −0.264628
\(358\) 20.0000 1.05703
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) −5.00000 −0.262794
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −3.00000 −0.157027
\(366\) −7.00000 −0.365896
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −4.00000 −0.208514
\(369\) 7.00000 0.364405
\(370\) 21.0000 1.09174
\(371\) −7.00000 −0.363422
\(372\) 0 0
\(373\) 15.0000 0.776671 0.388335 0.921518i \(-0.373050\pi\)
0.388335 + 0.921518i \(0.373050\pi\)
\(374\) −20.0000 −1.03418
\(375\) 3.00000 0.154919
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 12.0000 0.615587
\(381\) 8.00000 0.409852
\(382\) −24.0000 −1.22795
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −12.0000 −0.611577
\(386\) 19.0000 0.967075
\(387\) 8.00000 0.406663
\(388\) 18.0000 0.913812
\(389\) 7.00000 0.354914 0.177457 0.984129i \(-0.443213\pi\)
0.177457 + 0.984129i \(0.443213\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 22.0000 1.10834
\(395\) −48.0000 −2.41514
\(396\) 4.00000 0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 4.00000 0.200000
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 3.00000 0.149256
\(405\) 3.00000 0.149071
\(406\) 9.00000 0.446663
\(407\) 28.0000 1.38791
\(408\) 5.00000 0.247537
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 21.0000 1.03712
\(411\) −3.00000 −0.147979
\(412\) −4.00000 −0.197066
\(413\) 8.00000 0.393654
\(414\) −4.00000 −0.196589
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 16.0000 0.782586
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 3.00000 0.146385
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) −8.00000 −0.389434
\(423\) 12.0000 0.583460
\(424\) 7.00000 0.339950
\(425\) −20.0000 −0.970143
\(426\) −12.0000 −0.581402
\(427\) −7.00000 −0.338754
\(428\) 0 0
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 31.0000 1.48976 0.744882 0.667196i \(-0.232506\pi\)
0.744882 + 0.667196i \(0.232506\pi\)
\(434\) 0 0
\(435\) 27.0000 1.29455
\(436\) 14.0000 0.670478
\(437\) −16.0000 −0.765384
\(438\) 1.00000 0.0477818
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 12.0000 0.572078
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) −7.00000 −0.332205
\(445\) −18.0000 −0.853282
\(446\) −28.0000 −1.32584
\(447\) −7.00000 −0.331089
\(448\) −1.00000 −0.0472456
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 4.00000 0.188562
\(451\) 28.0000 1.31847
\(452\) −5.00000 −0.235180
\(453\) 8.00000 0.375873
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −1.00000 −0.0467780 −0.0233890 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 6.00000 0.280362
\(459\) 5.00000 0.233380
\(460\) −12.0000 −0.559503
\(461\) 19.0000 0.884918 0.442459 0.896789i \(-0.354106\pi\)
0.442459 + 0.896789i \(0.354106\pi\)
\(462\) 4.00000 0.186097
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 36.0000 1.66056
\(471\) 13.0000 0.599008
\(472\) −8.00000 −0.368230
\(473\) 32.0000 1.47136
\(474\) 16.0000 0.734904
\(475\) 16.0000 0.734130
\(476\) 5.00000 0.229175
\(477\) 7.00000 0.320508
\(478\) −4.00000 −0.182956
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) −3.00000 −0.136931
\(481\) 0 0
\(482\) −9.00000 −0.409939
\(483\) −4.00000 −0.182006
\(484\) 5.00000 0.227273
\(485\) 54.0000 2.45201
\(486\) −1.00000 −0.0453609
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 7.00000 0.316875
\(489\) −4.00000 −0.180886
\(490\) 3.00000 0.135526
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −7.00000 −0.315584
\(493\) 45.0000 2.02670
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 8.00000 0.358489
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −3.00000 −0.134164
\(501\) 8.00000 0.357414
\(502\) −12.0000 −0.535586
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 9.00000 0.400495
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 15.0000 0.664211
\(511\) 1.00000 0.0442374
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 7.00000 0.308757
\(515\) −12.0000 −0.528783
\(516\) −8.00000 −0.352180
\(517\) 48.0000 2.11104
\(518\) −7.00000 −0.307562
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) −9.00000 −0.393919
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −12.0000 −0.524222
\(525\) 4.00000 0.174574
\(526\) 8.00000 0.348817
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 21.0000 0.912182
\(531\) −8.00000 −0.347170
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) −20.0000 −0.863064
\(538\) 14.0000 0.603583
\(539\) 4.00000 0.172292
\(540\) −3.00000 −0.129099
\(541\) 43.0000 1.84871 0.924357 0.381528i \(-0.124602\pi\)
0.924357 + 0.381528i \(0.124602\pi\)
\(542\) 24.0000 1.03089
\(543\) 5.00000 0.214571
\(544\) −5.00000 −0.214373
\(545\) 42.0000 1.79908
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 3.00000 0.128154
\(549\) 7.00000 0.298753
\(550\) 16.0000 0.682242
\(551\) −36.0000 −1.53365
\(552\) 4.00000 0.170251
\(553\) 16.0000 0.680389
\(554\) 19.0000 0.807233
\(555\) −21.0000 −0.891400
\(556\) 4.00000 0.169638
\(557\) −33.0000 −1.39825 −0.699127 0.714997i \(-0.746428\pi\)
−0.699127 + 0.714997i \(0.746428\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −3.00000 −0.126773
\(561\) 20.0000 0.844401
\(562\) 15.0000 0.632737
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −12.0000 −0.505291
\(565\) −15.0000 −0.631055
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) 12.0000 0.503509
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −12.0000 −0.502625
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) −7.00000 −0.292174
\(575\) −16.0000 −0.667246
\(576\) 1.00000 0.0416667
\(577\) −5.00000 −0.208153 −0.104076 0.994569i \(-0.533189\pi\)
−0.104076 + 0.994569i \(0.533189\pi\)
\(578\) 8.00000 0.332756
\(579\) −19.0000 −0.789613
\(580\) −27.0000 −1.12111
\(581\) 8.00000 0.331896
\(582\) −18.0000 −0.746124
\(583\) 28.0000 1.15964
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) −33.0000 −1.36322
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) −22.0000 −0.904959
\(592\) 7.00000 0.287698
\(593\) −33.0000 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) −4.00000 −0.164122
\(595\) 15.0000 0.614940
\(596\) 7.00000 0.286731
\(597\) 0 0
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) −4.00000 −0.163299
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) −8.00000 −0.326056
\(603\) 12.0000 0.488678
\(604\) −8.00000 −0.325515
\(605\) 15.0000 0.609837
\(606\) −3.00000 −0.121867
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 4.00000 0.162221
\(609\) −9.00000 −0.364698
\(610\) 21.0000 0.850265
\(611\) 0 0
\(612\) −5.00000 −0.202113
\(613\) −1.00000 −0.0403896 −0.0201948 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) 4.00000 0.161427
\(615\) −21.0000 −0.846802
\(616\) −4.00000 −0.161165
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) 4.00000 0.160904
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 8.00000 0.320771
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −6.00000 −0.239808
\(627\) −16.0000 −0.638978
\(628\) −13.0000 −0.518756
\(629\) −35.0000 −1.39554
\(630\) −3.00000 −0.119523
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −16.0000 −0.636446
\(633\) 8.00000 0.317971
\(634\) 35.0000 1.39003
\(635\) −24.0000 −0.952411
\(636\) −7.00000 −0.277568
\(637\) 0 0
\(638\) −36.0000 −1.42525
\(639\) 12.0000 0.474713
\(640\) 3.00000 0.118585
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 4.00000 0.157622
\(645\) −24.0000 −0.944999
\(646\) −20.0000 −0.786889
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.00000 0.0392837
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −14.0000 −0.547443
\(655\) −36.0000 −1.40664
\(656\) 7.00000 0.273304
\(657\) −1.00000 −0.0390137
\(658\) −12.0000 −0.467809
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −12.0000 −0.467099
\(661\) −1.00000 −0.0388955 −0.0194477 0.999811i \(-0.506191\pi\)
−0.0194477 + 0.999811i \(0.506191\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) −12.0000 −0.465340
\(666\) 7.00000 0.271244
\(667\) 36.0000 1.39393
\(668\) −8.00000 −0.309529
\(669\) 28.0000 1.08254
\(670\) 36.0000 1.39080
\(671\) 28.0000 1.08093
\(672\) 1.00000 0.0385758
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −9.00000 −0.346667
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 5.00000 0.192024
\(679\) −18.0000 −0.690777
\(680\) −15.0000 −0.575224
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 4.00000 0.152944
\(685\) 9.00000 0.343872
\(686\) −1.00000 −0.0381802
\(687\) −6.00000 −0.228914
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) 12.0000 0.456832
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 14.0000 0.532200
\(693\) −4.00000 −0.151947
\(694\) 20.0000 0.759190
\(695\) 12.0000 0.455186
\(696\) 9.00000 0.341144
\(697\) −35.0000 −1.32572
\(698\) 14.0000 0.529908
\(699\) 6.00000 0.226941
\(700\) −4.00000 −0.151186
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 28.0000 1.05604
\(704\) 4.00000 0.150756
\(705\) −36.0000 −1.35584
\(706\) −25.0000 −0.940887
\(707\) −3.00000 −0.112827
\(708\) 8.00000 0.300658
\(709\) −1.00000 −0.0375558 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 36.0000 1.35106
\(711\) −16.0000 −0.600047
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −5.00000 −0.187120
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 4.00000 0.149383
\(718\) −12.0000 −0.447836
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 3.00000 0.111803
\(721\) 4.00000 0.148968
\(722\) −3.00000 −0.111648
\(723\) 9.00000 0.334714
\(724\) −5.00000 −0.185824
\(725\) −36.0000 −1.33701
\(726\) −5.00000 −0.185567
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −3.00000 −0.111035
\(731\) −40.0000 −1.47945
\(732\) −7.00000 −0.258727
\(733\) 23.0000 0.849524 0.424762 0.905305i \(-0.360358\pi\)
0.424762 + 0.905305i \(0.360358\pi\)
\(734\) −4.00000 −0.147643
\(735\) −3.00000 −0.110657
\(736\) −4.00000 −0.147442
\(737\) 48.0000 1.76810
\(738\) 7.00000 0.257674
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 21.0000 0.771975
\(741\) 0 0
\(742\) −7.00000 −0.256978
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 21.0000 0.769380
\(746\) 15.0000 0.549189
\(747\) −8.00000 −0.292705
\(748\) −20.0000 −0.731272
\(749\) 0 0
\(750\) 3.00000 0.109545
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 12.0000 0.437595
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 1.00000 0.0363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −28.0000 −1.01701
\(759\) 16.0000 0.580763
\(760\) 12.0000 0.435286
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 8.00000 0.289809
\(763\) −14.0000 −0.506834
\(764\) −24.0000 −0.868290
\(765\) −15.0000 −0.542326
\(766\) −28.0000 −1.01168
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) −12.0000 −0.432450
\(771\) −7.00000 −0.252099
\(772\) 19.0000 0.683825
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 7.00000 0.251124
\(778\) 7.00000 0.250962
\(779\) 28.0000 1.00320
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 20.0000 0.715199
\(783\) 9.00000 0.321634
\(784\) 1.00000 0.0357143
\(785\) −39.0000 −1.39197
\(786\) 12.0000 0.428026
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 22.0000 0.783718
\(789\) −8.00000 −0.284808
\(790\) −48.0000 −1.70776
\(791\) 5.00000 0.177780
\(792\) 4.00000 0.142134
\(793\) 0 0
\(794\) −2.00000 −0.0709773
\(795\) −21.0000 −0.744793
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 4.00000 0.141598
\(799\) −60.0000 −2.12265
\(800\) 4.00000 0.141421
\(801\) −6.00000 −0.212000
\(802\) −17.0000 −0.600291
\(803\) −4.00000 −0.141157
\(804\) −12.0000 −0.423207
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 3.00000 0.105540
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 3.00000 0.105409
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 9.00000 0.315838
\(813\) −24.0000 −0.841717
\(814\) 28.0000 0.981399
\(815\) 12.0000 0.420342
\(816\) 5.00000 0.175035
\(817\) 32.0000 1.11954
\(818\) −13.0000 −0.454534
\(819\) 0 0
\(820\) 21.0000 0.733352
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) −3.00000 −0.104637
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −4.00000 −0.139347
\(825\) −16.0000 −0.557048
\(826\) 8.00000 0.278356
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) −4.00000 −0.139010
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) −24.0000 −0.833052
\(831\) −19.0000 −0.659103
\(832\) 0 0
\(833\) −5.00000 −0.173240
\(834\) −4.00000 −0.138509
\(835\) −24.0000 −0.830554
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −16.0000 −0.552711
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 3.00000 0.103510
\(841\) 52.0000 1.79310
\(842\) −1.00000 −0.0344623
\(843\) −15.0000 −0.516627
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −5.00000 −0.171802
\(848\) 7.00000 0.240381
\(849\) 4.00000 0.137280
\(850\) −20.0000 −0.685994
\(851\) −28.0000 −0.959828
\(852\) −12.0000 −0.411113
\(853\) −49.0000 −1.67773 −0.838864 0.544341i \(-0.816780\pi\)
−0.838864 + 0.544341i \(0.816780\pi\)
\(854\) −7.00000 −0.239535
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) 31.0000 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 24.0000 0.818393
\(861\) 7.00000 0.238559
\(862\) −4.00000 −0.136241
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 42.0000 1.42804
\(866\) 31.0000 1.05342
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 27.0000 0.915386
\(871\) 0 0
\(872\) 14.0000 0.474100
\(873\) 18.0000 0.609208
\(874\) −16.0000 −0.541208
\(875\) 3.00000 0.101419
\(876\) 1.00000 0.0337869
\(877\) 3.00000 0.101303 0.0506514 0.998716i \(-0.483870\pi\)
0.0506514 + 0.998716i \(0.483870\pi\)
\(878\) −16.0000 −0.539974
\(879\) 33.0000 1.11306
\(880\) 12.0000 0.404520
\(881\) −41.0000 −1.38133 −0.690663 0.723177i \(-0.742681\pi\)
−0.690663 + 0.723177i \(0.742681\pi\)
\(882\) 1.00000 0.0336718
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) −16.0000 −0.537531
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −7.00000 −0.234905
\(889\) 8.00000 0.268311
\(890\) −18.0000 −0.603361
\(891\) 4.00000 0.134005
\(892\) −28.0000 −0.937509
\(893\) 48.0000 1.60626
\(894\) −7.00000 −0.234115
\(895\) 60.0000 2.00558
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) 4.00000 0.133333
\(901\) −35.0000 −1.16602
\(902\) 28.0000 0.932298
\(903\) 8.00000 0.266223
\(904\) −5.00000 −0.166298
\(905\) −15.0000 −0.498617
\(906\) 8.00000 0.265782
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 24.0000 0.796468
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −4.00000 −0.132453
\(913\) −32.0000 −1.05905
\(914\) −1.00000 −0.0330771
\(915\) −21.0000 −0.694239
\(916\) 6.00000 0.198246
\(917\) 12.0000 0.396275
\(918\) 5.00000 0.165025
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −12.0000 −0.395628
\(921\) −4.00000 −0.131804
\(922\) 19.0000 0.625732
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) 28.0000 0.920634
\(926\) 32.0000 1.05159
\(927\) −4.00000 −0.131377
\(928\) −9.00000 −0.295439
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −6.00000 −0.196537
\(933\) −8.00000 −0.261908
\(934\) −12.0000 −0.392652
\(935\) −60.0000 −1.96221
\(936\) 0 0
\(937\) −49.0000 −1.60076 −0.800380 0.599493i \(-0.795369\pi\)
−0.800380 + 0.599493i \(0.795369\pi\)
\(938\) −12.0000 −0.391814
\(939\) 6.00000 0.195803
\(940\) 36.0000 1.17419
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 13.0000 0.423563
\(943\) −28.0000 −0.911805
\(944\) −8.00000 −0.260378
\(945\) 3.00000 0.0975900
\(946\) 32.0000 1.04041
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) 16.0000 0.519109
\(951\) −35.0000 −1.13495
\(952\) 5.00000 0.162051
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 7.00000 0.226633
\(955\) −72.0000 −2.32987
\(956\) −4.00000 −0.129369
\(957\) 36.0000 1.16371
\(958\) 28.0000 0.904639
\(959\) −3.00000 −0.0968751
\(960\) −3.00000 −0.0968246
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −9.00000 −0.289870
\(965\) 57.0000 1.83489
\(966\) −4.00000 −0.128698
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 5.00000 0.160706
\(969\) 20.0000 0.642493
\(970\) 54.0000 1.73384
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) 7.00000 0.224065
\(977\) 47.0000 1.50366 0.751832 0.659355i \(-0.229171\pi\)
0.751832 + 0.659355i \(0.229171\pi\)
\(978\) −4.00000 −0.127906
\(979\) −24.0000 −0.767043
\(980\) 3.00000 0.0958315
\(981\) 14.0000 0.446986
\(982\) 20.0000 0.638226
\(983\) −28.0000 −0.893061 −0.446531 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) −7.00000 −0.223152
\(985\) 66.0000 2.10293
\(986\) 45.0000 1.43309
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 12.0000 0.381385
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 39.0000 1.23514 0.617571 0.786515i \(-0.288117\pi\)
0.617571 + 0.786515i \(0.288117\pi\)
\(998\) 4.00000 0.126618
\(999\) −7.00000 −0.221470
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 7098.2.a.v.1.1 1
13.3 even 3 546.2.l.b.295.1 yes 2
13.9 even 3 546.2.l.b.211.1 2
13.12 even 2 7098.2.a.a.1.1 1
39.29 odd 6 1638.2.r.o.1387.1 2
39.35 odd 6 1638.2.r.o.757.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.b.211.1 2 13.9 even 3
546.2.l.b.295.1 yes 2 13.3 even 3
1638.2.r.o.757.1 2 39.35 odd 6
1638.2.r.o.1387.1 2 39.29 odd 6
7098.2.a.a.1.1 1 13.12 even 2
7098.2.a.v.1.1 1 1.1 even 1 trivial