Properties

Label 930.2.a.d.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
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 \(\chi\) \(=\) 930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} -2.00000 q^{21} +4.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} -10.0000 q^{67} -6.00000 q^{68} -2.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -1.00000 q^{75} -8.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} -4.00000 q^{87} +4.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -1.00000 q^{93} +12.0000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) 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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 1.00000 0.182574
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000 0.308607
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −4.00000 −0.554700
\(53\) −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
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −1.00000 −0.127000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −4.00000 −0.492366
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −4.00000 −0.464991
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) −6.00000 −0.650791
\(86\) 8.00000 0.862662
\(87\) −4.00000 −0.428845
\(88\) 4.00000 0.426401
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 0.303046
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −6.00000 −0.594089
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 4.00000 0.392232
\(105\) −2.00000 −0.195180
\(106\) 2.00000 0.194257
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 4.00000 0.381385
\(111\) −4.00000 −0.379663
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) −4.00000 −0.369800
\(118\) −6.00000 −0.552345
\(119\) −12.0000 −1.10004
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 6.00000 0.541002
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 4.00000 0.350823
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 2.00000 0.169031
\(141\) 12.0000 1.01058
\(142\) 14.0000 1.17485
\(143\) 16.0000 1.33799
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −4.00000 −0.331042
\(147\) 3.00000 0.247436
\(148\) 4.00000 0.328798
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 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
\(153\) −6.00000 −0.485071
\(154\) 8.00000 0.644658
\(155\) 1.00000 0.0803219
\(156\) 4.00000 0.320256
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 8.00000 0.636446
\(159\) 2.00000 0.158610
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) −6.00000 −0.468521
\(165\) 4.00000 0.311400
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 4.00000 0.303239
\(175\) 2.00000 0.151186
\(176\) −4.00000 −0.301511
\(177\) −6.00000 −0.450988
\(178\) 8.00000 0.599625
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 1.00000 0.0745356
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 8.00000 0.592999
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 1.00000 0.0733236
\(187\) 24.0000 1.75505
\(188\) −12.0000 −0.875190
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) 4.00000 0.286446
\(196\) −3.00000 −0.214286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 10.0000 0.705346
\(202\) −14.0000 −0.985037
\(203\) 8.00000 0.561490
\(204\) 6.00000 0.420084
\(205\) −6.00000 −0.419058
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −2.00000 −0.137361
\(213\) 14.0000 0.959264
\(214\) 16.0000 1.09374
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 2.00000 0.135769
\(218\) −18.0000 −1.21911
\(219\) −4.00000 −0.270295
\(220\) −4.00000 −0.269680
\(221\) 24.0000 1.61441
\(222\) 4.00000 0.268462
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) −4.00000 −0.262613
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) −12.0000 −0.782794
\(236\) 6.00000 0.390567
\(237\) 8.00000 0.519656
\(238\) 12.0000 0.777844
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −3.00000 −0.191663
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −1.00000 −0.0635001
\(249\) 4.00000 0.253490
\(250\) −1.00000 −0.0632456
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 12.0000 0.752947
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −8.00000 −0.498058
\(259\) 8.00000 0.497096
\(260\) −4.00000 −0.248069
\(261\) 4.00000 0.247594
\(262\) −14.0000 −0.864923
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −4.00000 −0.246183
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −10.0000 −0.610847
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 1.00000 0.0608581
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −6.00000 −0.363803
\(273\) 8.00000 0.484182
\(274\) 18.0000 1.08742
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 12.0000 0.719712
\(279\) 1.00000 0.0598684
\(280\) −2.00000 −0.119523
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −12.0000 −0.714590
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −4.00000 −0.234888
\(291\) 2.00000 0.117242
\(292\) 4.00000 0.234082
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) −3.00000 −0.174964
\(295\) 6.00000 0.349334
\(296\) −4.00000 −0.232495
\(297\) 4.00000 0.232104
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −16.0000 −0.922225
\(302\) 16.0000 0.920697
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) 10.0000 0.572598
\(306\) 6.00000 0.342997
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) −8.00000 −0.455842
\(309\) −14.0000 −0.796432
\(310\) −1.00000 −0.0567962
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) −4.00000 −0.226455
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 6.00000 0.338600
\(315\) 2.00000 0.112687
\(316\) −8.00000 −0.450035
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) −2.00000 −0.112154
\(319\) −16.0000 −0.895828
\(320\) 1.00000 0.0559017
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −6.00000 −0.332309
\(327\) −18.0000 −0.995402
\(328\) 6.00000 0.331295
\(329\) −24.0000 −1.32316
\(330\) −4.00000 −0.220193
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −4.00000 −0.219529
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) −10.0000 −0.546358
\(336\) −2.00000 −0.109109
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) −3.00000 −0.163178
\(339\) 6.00000 0.325875
\(340\) −6.00000 −0.325396
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −4.00000 −0.214423
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −2.00000 −0.106904
\(351\) 4.00000 0.213504
\(352\) 4.00000 0.213201
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 6.00000 0.318896
\(355\) −14.0000 −0.743043
\(356\) −8.00000 −0.423999
\(357\) 12.0000 0.635107
\(358\) 4.00000 0.211407
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) −18.0000 −0.946059
\(363\) −5.00000 −0.262432
\(364\) −8.00000 −0.419314
\(365\) 4.00000 0.209370
\(366\) 10.0000 0.522708
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) −4.00000 −0.207950
\(371\) −4.00000 −0.207670
\(372\) −1.00000 −0.0518476
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −24.0000 −1.24101
\(375\) −1.00000 −0.0516398
\(376\) 12.0000 0.618853
\(377\) −16.0000 −0.824042
\(378\) 2.00000 0.102869
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 10.0000 0.511645
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.00000 −0.407718
\(386\) 2.00000 0.101797
\(387\) −8.00000 −0.406663
\(388\) −2.00000 −0.101535
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) −14.0000 −0.706207
\(394\) −2.00000 −0.100759
\(395\) −8.00000 −0.402524
\(396\) −4.00000 −0.201008
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −4.00000 −0.199750 −0.0998752 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) −10.0000 −0.498755
\(403\) −4.00000 −0.199254
\(404\) 14.0000 0.696526
\(405\) 1.00000 0.0496904
\(406\) −8.00000 −0.397033
\(407\) −16.0000 −0.793091
\(408\) −6.00000 −0.297044
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 6.00000 0.296319
\(411\) 18.0000 0.887875
\(412\) 14.0000 0.689730
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 4.00000 0.196116
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −16.0000 −0.778868
\(423\) −12.0000 −0.583460
\(424\) 2.00000 0.0971286
\(425\) −6.00000 −0.291043
\(426\) −14.0000 −0.678302
\(427\) 20.0000 0.967868
\(428\) −16.0000 −0.773389
\(429\) −16.0000 −0.772487
\(430\) 8.00000 0.385794
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) −2.00000 −0.0960031
\(435\) −4.00000 −0.191785
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 4.00000 0.190693
\(441\) −3.00000 −0.142857
\(442\) −24.0000 −1.14156
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.00000 −0.189832
\(445\) −8.00000 −0.379236
\(446\) −16.0000 −0.757622
\(447\) −14.0000 −0.662177
\(448\) 2.00000 0.0944911
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) −20.0000 −0.938647
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) 26.0000 1.21490
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) −8.00000 −0.372194
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) 4.00000 0.185695
\(465\) −1.00000 −0.0463739
\(466\) −6.00000 −0.277945
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) −4.00000 −0.184900
\(469\) −20.0000 −0.923514
\(470\) 12.0000 0.553519
\(471\) 6.00000 0.276465
\(472\) −6.00000 −0.276172
\(473\) 32.0000 1.47136
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) −2.00000 −0.0915737
\(478\) −20.0000 −0.914779
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 1.00000 0.0456435
\(481\) −16.0000 −0.729537
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) −10.0000 −0.452679
\(489\) −6.00000 −0.271329
\(490\) 3.00000 0.135526
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −24.0000 −1.08091
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 1.00000 0.0449013
\(497\) −28.0000 −1.25597
\(498\) −4.00000 −0.179244
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 28.0000 1.24970
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) −12.0000 −0.532414
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) −6.00000 −0.265684
\(511\) 8.00000 0.353899
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 14.0000 0.616914
\(516\) 8.00000 0.352180
\(517\) 48.0000 2.11104
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) 4.00000 0.175412
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −4.00000 −0.175075
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 14.0000 0.611593
\(525\) −2.00000 −0.0872872
\(526\) −24.0000 −1.04645
\(527\) −6.00000 −0.261364
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) 2.00000 0.0868744
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) −8.00000 −0.346194
\(535\) −16.0000 −0.691740
\(536\) 10.0000 0.431934
\(537\) 4.00000 0.172613
\(538\) 12.0000 0.517357
\(539\) 12.0000 0.516877
\(540\) −1.00000 −0.0430331
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 24.0000 1.03089
\(543\) −18.0000 −0.772454
\(544\) 6.00000 0.257248
\(545\) 18.0000 0.771035
\(546\) −8.00000 −0.342368
\(547\) 46.0000 1.96682 0.983409 0.181402i \(-0.0580636\pi\)
0.983409 + 0.181402i \(0.0580636\pi\)
\(548\) −18.0000 −0.768922
\(549\) 10.0000 0.426790
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 8.00000 0.339887
\(555\) −4.00000 −0.169791
\(556\) −12.0000 −0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 32.0000 1.35346
\(560\) 2.00000 0.0845154
\(561\) −24.0000 −1.01328
\(562\) −6.00000 −0.253095
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 12.0000 0.505291
\(565\) −6.00000 −0.252422
\(566\) −14.0000 −0.588464
\(567\) 2.00000 0.0839921
\(568\) 14.0000 0.587427
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 16.0000 0.668994
\(573\) 10.0000 0.417756
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) −19.0000 −0.790296
\(579\) 2.00000 0.0831172
\(580\) 4.00000 0.166091
\(581\) −8.00000 −0.331896
\(582\) −2.00000 −0.0829027
\(583\) 8.00000 0.331326
\(584\) −4.00000 −0.165521
\(585\) −4.00000 −0.165380
\(586\) −26.0000 −1.07405
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) −6.00000 −0.247016
\(591\) −2.00000 −0.0822690
\(592\) 4.00000 0.164399
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) −4.00000 −0.164122
\(595\) −12.0000 −0.491952
\(596\) 14.0000 0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 1.00000 0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 16.0000 0.652111
\(603\) −10.0000 −0.407231
\(604\) −16.0000 −0.651031
\(605\) 5.00000 0.203279
\(606\) 14.0000 0.568711
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) −10.0000 −0.404888
\(611\) 48.0000 1.94187
\(612\) −6.00000 −0.242536
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 6.00000 0.241943
\(616\) 8.00000 0.322329
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 14.0000 0.563163
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) −16.0000 −0.641026
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) −24.0000 −0.956943
\(630\) −2.00000 −0.0796819
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000 0.318223
\(633\) −16.0000 −0.635943
\(634\) −26.0000 −1.03259
\(635\) −12.0000 −0.476205
\(636\) 2.00000 0.0793052
\(637\) 12.0000 0.475457
\(638\) 16.0000 0.633446
\(639\) −14.0000 −0.553831
\(640\) −1.00000 −0.0395285
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) −16.0000 −0.631470
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) 4.00000 0.156893
\(651\) −2.00000 −0.0783862
\(652\) 6.00000 0.234978
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 18.0000 0.703856
\(655\) 14.0000 0.547025
\(656\) −6.00000 −0.234261
\(657\) 4.00000 0.156055
\(658\) 24.0000 0.935617
\(659\) −22.0000 −0.856998 −0.428499 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 4.00000 0.155700
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 28.0000 1.08825
\(663\) −24.0000 −0.932083
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 10.0000 0.386334
\(671\) −40.0000 −1.54418
\(672\) 2.00000 0.0771517
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) −8.00000 −0.308148
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) −6.00000 −0.230429
\(679\) −4.00000 −0.153506
\(680\) 6.00000 0.230089
\(681\) −20.0000 −0.766402
\(682\) 4.00000 0.153168
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 20.0000 0.763604
\(687\) 26.0000 0.991962
\(688\) −8.00000 −0.304997
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 6.00000 0.228086
\(693\) −8.00000 −0.303895
\(694\) −4.00000 −0.151838
\(695\) −12.0000 −0.455186
\(696\) 4.00000 0.151620
\(697\) 36.0000 1.36360
\(698\) −2.00000 −0.0757011
\(699\) −6.00000 −0.226941
\(700\) 2.00000 0.0755929
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) −4.00000 −0.150970
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 12.0000 0.451946
\(706\) −10.0000 −0.376355
\(707\) 28.0000 1.05305
\(708\) −6.00000 −0.225494
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 14.0000 0.525411
\(711\) −8.00000 −0.300023
\(712\) 8.00000 0.299813
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) 16.0000 0.598366
\(716\) −4.00000 −0.149487
\(717\) −20.0000 −0.746914
\(718\) −30.0000 −1.11959
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 1.00000 0.0372678
\(721\) 28.0000 1.04277
\(722\) 19.0000 0.707107
\(723\) −10.0000 −0.371904
\(724\) 18.0000 0.668965
\(725\) 4.00000 0.148556
\(726\) 5.00000 0.185567
\(727\) −30.0000 −1.11264 −0.556319 0.830969i \(-0.687787\pi\)
−0.556319 + 0.830969i \(0.687787\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 48.0000 1.77534
\(732\) −10.0000 −0.369611
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 40.0000 1.47342
\(738\) 6.00000 0.220863
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 1.00000 0.0366618
\(745\) 14.0000 0.512920
\(746\) −18.0000 −0.659027
\(747\) −4.00000 −0.146352
\(748\) 24.0000 0.877527
\(749\) −32.0000 −1.16925
\(750\) 1.00000 0.0365148
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −12.0000 −0.437595
\(753\) 28.0000 1.02038
\(754\) 16.0000 0.582686
\(755\) −16.0000 −0.582300
\(756\) −2.00000 −0.0727393
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) −12.0000 −0.434714
\(763\) 36.0000 1.30329
\(764\) −10.0000 −0.361787
\(765\) −6.00000 −0.216930
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 8.00000 0.288300
\(771\) 2.00000 0.0720282
\(772\) −2.00000 −0.0719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 8.00000 0.287554
\(775\) 1.00000 0.0359211
\(776\) 2.00000 0.0717958
\(777\) −8.00000 −0.286998
\(778\) −36.0000 −1.29066
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 56.0000 2.00384
\(782\) 0 0
\(783\) −4.00000 −0.142948
\(784\) −3.00000 −0.107143
\(785\) −6.00000 −0.214149
\(786\) 14.0000 0.499363
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 2.00000 0.0712470
\(789\) −24.0000 −0.854423
\(790\) 8.00000 0.284627
\(791\) −12.0000 −0.426671
\(792\) 4.00000 0.142134
\(793\) −40.0000 −1.42044
\(794\) 18.0000 0.638796
\(795\) 2.00000 0.0709327
\(796\) 0 0
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) −1.00000 −0.0353553
\(801\) −8.00000 −0.282666
\(802\) 4.00000 0.141245
\(803\) −16.0000 −0.564628
\(804\) 10.0000 0.352673
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 12.0000 0.422420
\(808\) −14.0000 −0.492518
\(809\) 20.0000 0.703163 0.351581 0.936157i \(-0.385644\pi\)
0.351581 + 0.936157i \(0.385644\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 8.00000 0.280745
\(813\) 24.0000 0.841717
\(814\) 16.0000 0.560800
\(815\) 6.00000 0.210171
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) −6.00000 −0.209785
\(819\) −8.00000 −0.279543
\(820\) −6.00000 −0.209529
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) −18.0000 −0.627822
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −14.0000 −0.487713
\(825\) 4.00000 0.139262
\(826\) −12.0000 −0.417533
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 4.00000 0.138842
\(831\) 8.00000 0.277517
\(832\) −4.00000 −0.138675
\(833\) 18.0000 0.623663
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 30.0000 1.03633
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 2.00000 0.0690066
\(841\) −13.0000 −0.448276
\(842\) 30.0000 1.03387
\(843\) −6.00000 −0.206651
\(844\) 16.0000 0.550743
\(845\) 3.00000 0.103203
\(846\) 12.0000 0.412568
\(847\) 10.0000 0.343604
\(848\) −2.00000 −0.0686803
\(849\) −14.0000 −0.480479
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 14.0000 0.479632
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 16.0000 0.546231
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −8.00000 −0.272798
\(861\) 12.0000 0.408959
\(862\) 10.0000 0.340601
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.00000 0.204006
\(866\) 4.00000 0.135926
\(867\) −19.0000 −0.645274
\(868\) 2.00000 0.0678844
\(869\) 32.0000 1.08553
\(870\) 4.00000 0.135613
\(871\) 40.0000 1.35535
\(872\) −18.0000 −0.609557
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) −4.00000 −0.135147
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −24.0000 −0.809961
\(879\) −26.0000 −0.876958
\(880\) −4.00000 −0.134840
\(881\) −56.0000 −1.88669 −0.943344 0.331816i \(-0.892339\pi\)
−0.943344 + 0.331816i \(0.892339\pi\)
\(882\) 3.00000 0.101015
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 24.0000 0.807207
\(885\) −6.00000 −0.201688
\(886\) 12.0000 0.403148
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 4.00000 0.134231
\(889\) −24.0000 −0.804934
\(890\) 8.00000 0.268161
\(891\) −4.00000 −0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) −4.00000 −0.133705
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) 4.00000 0.133407
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) −24.0000 −0.799113
\(903\) 16.0000 0.532447
\(904\) 6.00000 0.199557
\(905\) 18.0000 0.598340
\(906\) −16.0000 −0.531564
\(907\) 58.0000 1.92586 0.962929 0.269754i \(-0.0869425\pi\)
0.962929 + 0.269754i \(0.0869425\pi\)
\(908\) 20.0000 0.663723
\(909\) 14.0000 0.464351
\(910\) 8.00000 0.265197
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 4.00000 0.132308
\(915\) −10.0000 −0.330590
\(916\) −26.0000 −0.859064
\(917\) 28.0000 0.924641
\(918\) −6.00000 −0.198030
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) −24.0000 −0.790398
\(923\) 56.0000 1.84326
\(924\) 8.00000 0.263181
\(925\) 4.00000 0.131519
\(926\) −12.0000 −0.394344
\(927\) 14.0000 0.459820
\(928\) −4.00000 −0.131306
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 1.00000 0.0327913
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) −6.00000 −0.196431
\(934\) 24.0000 0.785304
\(935\) 24.0000 0.784884
\(936\) 4.00000 0.130744
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 20.0000 0.653023
\(939\) 4.00000 0.130535
\(940\) −12.0000 −0.391397
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) −6.00000 −0.195491
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) −2.00000 −0.0650600
\(946\) −32.0000 −1.04041
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 8.00000 0.259828
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) −26.0000 −0.843108
\(952\) 12.0000 0.388922
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) 2.00000 0.0647524
\(955\) −10.0000 −0.323592
\(956\) 20.0000 0.646846
\(957\) 16.0000 0.517207
\(958\) 14.0000 0.452319
\(959\) −36.0000 −1.16250
\(960\) −1.00000 −0.0322749
\(961\) 1.00000 0.0322581
\(962\) 16.0000 0.515861
\(963\) −16.0000 −0.515593
\(964\) 10.0000 0.322078
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −24.0000 −0.769405
\(974\) 28.0000 0.897178
\(975\) 4.00000 0.128103
\(976\) 10.0000 0.320092
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 6.00000 0.191859
\(979\) 32.0000 1.02272
\(980\) −3.00000 −0.0958315
\(981\) 18.0000 0.574696
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 2.00000 0.0637253
\(986\) 24.0000 0.764316
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 28.0000 0.888553
\(994\) 28.0000 0.888106
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 4.00000 0.126618
\(999\) −4.00000 −0.126554
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 930.2.a.d.1.1 1
3.2 odd 2 2790.2.a.t.1.1 1
4.3 odd 2 7440.2.a.y.1.1 1
5.2 odd 4 4650.2.d.p.3349.1 2
5.3 odd 4 4650.2.d.p.3349.2 2
5.4 even 2 4650.2.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.d.1.1 1 1.1 even 1 trivial
2790.2.a.t.1.1 1 3.2 odd 2
4650.2.a.bl.1.1 1 5.4 even 2
4650.2.d.p.3349.1 2 5.2 odd 4
4650.2.d.p.3349.2 2 5.3 odd 4
7440.2.a.y.1.1 1 4.3 odd 2