Properties

Label 429.2.a.b.1.1
Level $429$
Weight $2$
Character 429.1
Self dual yes
Analytic conductor $3.426$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.42558224671\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 429.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -10.0000 q^{29} +2.00000 q^{30} -5.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} -6.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -1.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{57} +10.0000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +14.0000 q^{61} +7.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} -12.0000 q^{67} +6.00000 q^{68} -8.00000 q^{69} +3.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} +12.0000 q^{85} -4.00000 q^{86} -10.0000 q^{87} -3.00000 q^{88} +2.00000 q^{89} +2.00000 q^{90} +8.00000 q^{92} -8.00000 q^{94} +8.00000 q^{95} -5.00000 q^{96} -14.0000 q^{97} +7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 0.612372
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) −1.00000 −0.174078
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 1.00000 0.160128
\(40\) −6.00000 −0.948683
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.00000 −0.298142
\(46\) 8.00000 1.17954
\(47\) 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
\(51\) −6.00000 −0.840168
\(52\) −1.00000 −0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 10.0000 1.31306
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 6.00000 0.727607
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) −4.00000 −0.431331
\(87\) −10.0000 −1.07211
\(88\) −3.00000 −0.319801
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 8.00000 0.820783
\(96\) −5.00000 −0.510310
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 7.00000 0.707107
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 6.00000 0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −2.00000 −0.190693
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 4.00000 0.374634
\(115\) 16.0000 1.49201
\(116\) 10.0000 0.928477
\(117\) 1.00000 0.0924500
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −6.00000 −0.547723
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −2.00000 −0.172133
\(136\) −18.0000 −1.54349
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 8.00000 0.681005
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) −1.00000 −0.0833333
\(145\) 20.0000 1.66091
\(146\) 6.00000 0.496564
\(147\) −7.00000 −0.577350
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −12.0000 −0.973329
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) −10.0000 −0.793052
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −10.0000 −0.780869
\(165\) 2.00000 0.155700
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −12.0000 −0.920358
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −12.0000 −0.901975
\(178\) −2.00000 −0.149906
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) −24.0000 −1.76930
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 7.00000 0.505181
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 14.0000 1.00514
\(195\) −2.00000 −0.143223
\(196\) 7.00000 0.500000
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −3.00000 −0.212132
\(201\) −12.0000 −0.846415
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −20.0000 −1.39686
\(206\) −8.00000 −0.557386
\(207\) −8.00000 −0.556038
\(208\) −1.00000 −0.0693375
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −8.00000 −0.545595
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) −6.00000 −0.405442
\(220\) −2.00000 −0.134840
\(221\) −6.00000 −0.403604
\(222\) −6.00000 −0.402694
\(223\) 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
\(226\) 14.0000 0.931266
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) −30.0000 −1.96960
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −16.0000 −1.04372
\(236\) 12.0000 0.781133
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 2.00000 0.129099
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 14.0000 0.894427
\(246\) −10.0000 −0.637577
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) −12.0000 −0.758947
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 8.00000 0.501965
\(255\) 12.0000 0.751469
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −10.0000 −0.618984
\(262\) 12.0000 0.741362
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −3.00000 −0.184637
\(265\) 20.0000 1.22859
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) 12.0000 0.733017
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 2.00000 0.121716
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 1.00000 0.0603023
\(276\) 8.00000 0.481543
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) −8.00000 −0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 1.00000 0.0591312
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) 19.0000 1.11765
\(290\) −20.0000 −1.17444
\(291\) −14.0000 −0.820695
\(292\) 6.00000 0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 7.00000 0.408248
\(295\) 24.0000 1.39733
\(296\) 18.0000 1.04623
\(297\) −1.00000 −0.0580259
\(298\) 10.0000 0.579284
\(299\) −8.00000 −0.462652
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) −28.0000 −1.60328
\(306\) 6.00000 0.342997
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 3.00000 0.169842
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 10.0000 0.560772
\(319\) 10.0000 0.559893
\(320\) −14.0000 −0.782624
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) −1.00000 −0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 12.0000 0.664619
\(327\) 14.0000 0.774202
\(328\) 30.0000 1.65647
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −14.0000 −0.760376
\(340\) −12.0000 −0.650791
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 16.0000 0.861411
\(346\) −6.00000 −0.322562
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 10.0000 0.536056
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 5.00000 0.266501
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) −14.0000 −0.731792
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 8.00000 0.417029
\(369\) 10.0000 0.520579
\(370\) 12.0000 0.623850
\(371\) 0 0
\(372\) 0 0
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) −6.00000 −0.310253
\(375\) 12.0000 0.619677
\(376\) 24.0000 1.23771
\(377\) −10.0000 −0.515026
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −8.00000 −0.410391
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 4.00000 0.203331
\(388\) 14.0000 0.710742
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 2.00000 0.101274
\(391\) 48.0000 2.42746
\(392\) −21.0000 −1.06066
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) −16.0000 −0.805047
\(396\) 1.00000 0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) −18.0000 −0.891133
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 20.0000 0.987730
\(411\) 18.0000 0.887875
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) −24.0000 −1.17811
\(416\) −5.00000 −0.245145
\(417\) −12.0000 −0.587643
\(418\) −4.00000 −0.195646
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 20.0000 0.973585
\(423\) 8.00000 0.388973
\(424\) −30.0000 −1.45693
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −1.00000 −0.0482805
\(430\) 8.00000 0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 20.0000 0.958927
\(436\) −14.0000 −0.670478
\(437\) 32.0000 1.53077
\(438\) 6.00000 0.286691
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 6.00000 0.286039
\(441\) −7.00000 −0.333333
\(442\) 6.00000 0.285391
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −6.00000 −0.284747
\(445\) −4.00000 −0.189618
\(446\) −16.0000 −0.757622
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 1.00000 0.0471405
\(451\) −10.0000 −0.470882
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) −16.0000 −0.746004
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 16.0000 0.738025
\(471\) −2.00000 −0.0921551
\(472\) −36.0000 −1.65703
\(473\) −4.00000 −0.183920
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −8.00000 −0.365911
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 10.0000 0.456435
\(481\) 6.00000 0.273576
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 28.0000 1.27141
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 42.0000 1.90125
\(489\) −12.0000 −0.542659
\(490\) −14.0000 −0.632456
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −10.0000 −0.450835
\(493\) 60.0000 2.70226
\(494\) 4.00000 0.179969
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) −8.00000 −0.355643
\(507\) 1.00000 0.0444116
\(508\) 8.00000 0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) −12.0000 −0.531369
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) −6.00000 −0.263117
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 10.0000 0.437688
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) 41.0000 1.78261
\(530\) −20.0000 −0.868744
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 10.0000 0.433148
\(534\) −2.00000 −0.0865485
\(535\) −24.0000 −1.03761
\(536\) −36.0000 −1.55496
\(537\) 20.0000 0.863064
\(538\) 2.00000 0.0862261
\(539\) 7.00000 0.301511
\(540\) 2.00000 0.0860663
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 8.00000 0.343629
\(543\) −10.0000 −0.429141
\(544\) 30.0000 1.28624
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −18.0000 −0.768922
\(549\) 14.0000 0.597505
\(550\) −1.00000 −0.0426401
\(551\) 40.0000 1.70406
\(552\) −24.0000 −1.02151
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −12.0000 −0.509372
\(556\) 12.0000 0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 6.00000 0.253095
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) −8.00000 −0.336861
\(565\) 28.0000 1.17797
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) −8.00000 −0.335083
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 7.00000 0.291667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −19.0000 −0.790296
\(579\) 18.0000 0.748054
\(580\) −20.0000 −0.830455
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) 10.0000 0.414158
\(584\) −18.0000 −0.744845
\(585\) −2.00000 −0.0826898
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 7.00000 0.288675
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) 6.00000 0.246807
\(592\) −6.00000 −0.246598
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −24.0000 −0.982255
\(598\) 8.00000 0.327144
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) −3.00000 −0.122474
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 2.00000 0.0812444
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) 28.0000 1.13369
\(611\) 8.00000 0.323645
\(612\) 6.00000 0.242536
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 20.0000 0.807134
\(615\) −20.0000 −0.806478
\(616\) 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
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) 4.00000 0.159745
\(628\) 2.00000 0.0798087
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 24.0000 0.954669
\(633\) −20.0000 −0.794929
\(634\) −22.0000 −0.873732
\(635\) 16.0000 0.634941
\(636\) 10.0000 0.396526
\(637\) −7.00000 −0.277350
\(638\) −10.0000 −0.395904
\(639\) 0 0
\(640\) −6.00000 −0.237171
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −12.0000 −0.473602
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −24.0000 −0.944267
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 3.00000 0.117851
\(649\) 12.0000 0.471041
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −14.0000 −0.547443
\(655\) 24.0000 0.937758
\(656\) −10.0000 −0.390434
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 20.0000 0.777322
\(663\) −6.00000 −0.233021
\(664\) 36.0000 1.39707
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 80.0000 3.09761
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) −24.0000 −0.927201
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −34.0000 −1.30963
\(675\) −1.00000 −0.0384900
\(676\) −1.00000 −0.0384615
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) 36.0000 1.38054
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 4.00000 0.152944
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) −10.0000 −0.380970
\(690\) −16.0000 −0.609110
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 24.0000 0.910372
\(696\) −30.0000 −1.13715
\(697\) −60.0000 −2.27266
\(698\) 2.00000 0.0757011
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −24.0000 −0.905177
\(704\) −7.00000 −0.263822
\(705\) −16.0000 −0.602595
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −20.0000 −0.747435
\(717\) 8.00000 0.298765
\(718\) 16.0000 0.597115
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) 10.0000 0.371647
\(725\) 10.0000 0.371391
\(726\) −1.00000 −0.0371135
\(727\) −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
\(730\) −12.0000 −0.444140
\(731\) −24.0000 −0.887672
\(732\) −14.0000 −0.517455
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 16.0000 0.590571
\(735\) 14.0000 0.516398
\(736\) 40.0000 1.47442
\(737\) 12.0000 0.442026
\(738\) −10.0000 −0.368105
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 12.0000 0.441129
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) −38.0000 −1.39128
\(747\) 12.0000 0.439057
\(748\) −6.00000 −0.219382
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −8.00000 −0.291730
\(753\) −20.0000 −0.728841
\(754\) 10.0000 0.364179
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 20.0000 0.726433
\(759\) 8.00000 0.290382
\(760\) 24.0000 0.870572
\(761\) −54.0000 −1.95750 −0.978749 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) 8.00000 0.289052
\(767\) −12.0000 −0.433295
\(768\) −17.0000 −0.613435
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −18.0000 −0.647834
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −42.0000 −1.50771
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) −40.0000 −1.43315
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) −48.0000 −1.71648
\(783\) −10.0000 −0.357371
\(784\) 7.00000 0.250000
\(785\) 4.00000 0.142766
\(786\) 12.0000 0.428026
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −6.00000 −0.213741
\(789\) 8.00000 0.284808
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) −3.00000 −0.106600
\(793\) 14.0000 0.497155
\(794\) −14.0000 −0.496841
\(795\) 20.0000 0.709327
\(796\) 24.0000 0.850657
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 5.00000 0.176777
\(801\) 2.00000 0.0706665
\(802\) 22.0000 0.776847
\(803\) 6.00000 0.211735
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) −2.00000 −0.0704033
\(808\) −6.00000 −0.211079
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 2.00000 0.0702728
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 6.00000 0.210300
\(815\) 24.0000 0.840683
\(816\) 6.00000 0.210042
\(817\) −16.0000 −0.559769
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 20.0000 0.698430
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −18.0000 −0.627822
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 24.0000 0.836080
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 8.00000 0.278019
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 24.0000 0.833052
\(831\) −10.0000 −0.346896
\(832\) 7.00000 0.242681
\(833\) 42.0000 1.45521
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −22.0000 −0.758170
\(843\) −6.00000 −0.206651
\(844\) 20.0000 0.688428
\(845\) −2.00000 −0.0688021
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) −6.00000 −0.205798
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 36.0000 1.23045
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 1.00000 0.0341394
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −5.00000 −0.170103
\(865\) −12.0000 −0.408012
\(866\) 14.0000 0.475739
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) −20.0000 −0.678064
\(871\) −12.0000 −0.406604
\(872\) 42.0000 1.42230
\(873\) −14.0000 −0.473828
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 16.0000 0.539974
\(879\) 6.00000 0.202375
\(880\) −2.00000 −0.0674200
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 7.00000 0.235702
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 6.00000 0.201802
\(885\) 24.0000 0.806751
\(886\) 4.00000 0.134383
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 18.0000 0.604040
\(889\) 0 0
\(890\) 4.00000 0.134080
\(891\) −1.00000 −0.0335013
\(892\) −16.0000 −0.535720
\(893\) −32.0000 −1.07084
\(894\) 10.0000 0.334450
\(895\) −40.0000 −1.33705
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 22.0000 0.734150
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 60.0000 1.99889
\(902\) 10.0000 0.332964
\(903\) 0 0
\(904\) −42.0000 −1.39690
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 20.0000 0.663723
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 4.00000 0.132453
\(913\) −12.0000 −0.397142
\(914\) 6.00000 0.198462
\(915\) −28.0000 −0.925651
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 48.0000 1.58251
\(921\) −20.0000 −0.659022
\(922\) 2.00000 0.0658665
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −16.0000 −0.525793
\(927\) 8.00000 0.262754
\(928\) 50.0000 1.64133
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −18.0000 −0.589610
\(933\) −8.00000 −0.261908
\(934\) 12.0000 0.392652
\(935\) −12.0000 −0.392442
\(936\) 3.00000 0.0980581
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 16.0000 0.521862
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 2.00000 0.0651635
\(943\) −80.0000 −2.60516
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −8.00000 −0.259828
\(949\) −6.00000 −0.194768
\(950\) −4.00000 −0.129777
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 10.0000 0.323254
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −14.0000 −0.451848
\(961\) −31.0000 −1.00000
\(962\) −6.00000 −0.193448
\(963\) 12.0000 0.386695
\(964\) −2.00000 −0.0644157
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 3.00000 0.0964237
\(969\) 24.0000 0.770991
\(970\) −28.0000 −0.899026
\(971\) −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
\(974\) −8.00000 −0.256337
\(975\) −1.00000 −0.0320256
\(976\) −14.0000 −0.448129
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 12.0000 0.383718
\(979\) −2.00000 −0.0639203
\(980\) −14.0000 −0.447214
\(981\) 14.0000 0.446986
\(982\) −12.0000 −0.382935
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 30.0000 0.956365
\(985\) −12.0000 −0.382352
\(986\) −60.0000 −1.91079
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −32.0000 −1.01754
\(990\) −2.00000 −0.0635642
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 48.0000 1.52170
\(996\) −12.0000 −0.380235
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 28.0000 0.886325
\(999\) 6.00000 0.189832
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 429.2.a.b.1.1 1
3.2 odd 2 1287.2.a.e.1.1 1
4.3 odd 2 6864.2.a.e.1.1 1
11.10 odd 2 4719.2.a.k.1.1 1
13.12 even 2 5577.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.a.b.1.1 1 1.1 even 1 trivial
1287.2.a.e.1.1 1 3.2 odd 2
4719.2.a.k.1.1 1 11.10 odd 2
5577.2.a.g.1.1 1 13.12 even 2
6864.2.a.e.1.1 1 4.3 odd 2