Properties

Label 930.2.a.b.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -1.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} -4.00000 q^{66} +16.0000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +6.00000 q^{78} -16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} -2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -18.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} +1.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 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
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) −6.00000 −0.960769
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 6.00000 0.832050
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) −4.00000 −0.492366
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 7.00000 0.707107
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −4.00000 −0.381385
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −4.00000 −0.374634
\(115\) 4.00000 0.373002
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) 6.00000 0.541002
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 6.00000 0.526235
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −4.00000 −0.340503
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −24.0000 −2.00698
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 6.00000 0.496564
\(147\) 7.00000 0.577350
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) −6.00000 −0.480384
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 16.0000 1.27289
\(159\) −2.00000 −0.158610
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) −6.00000 −0.468521
\(165\) −4.00000 −0.311400
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 4.00000 0.300658
\(178\) 18.0000 1.34916
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 4.00000 0.294884
\(185\) 2.00000 0.147043
\(186\) −1.00000 −0.0733236
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 14.0000 1.00514
\(195\) 6.00000 0.429669
\(196\) −7.00000 −0.500000
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 4.00000 0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −16.0000 −1.12855
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 6.00000 0.419058
\(206\) −8.00000 −0.557386
\(207\) −4.00000 −0.278019
\(208\) 6.00000 0.416025
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 2.00000 0.137361
\(213\) 12.0000 0.822226
\(214\) −4.00000 −0.273434
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 6.00000 0.405442
\(220\) 4.00000 0.269680
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 10.0000 0.665190
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 4.00000 0.264906
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 7.00000 0.447214
\(246\) −6.00000 −0.382546
\(247\) −24.0000 −1.52708
\(248\) 1.00000 0.0635001
\(249\) 12.0000 0.760469
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 8.00000 0.501965
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) −6.00000 −0.372104
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −4.00000 −0.246183
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 16.0000 0.977356
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −4.00000 −0.241209
\(276\) 4.00000 0.240772
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) −8.00000 −0.479808
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −12.0000 −0.712069
\(285\) −4.00000 −0.236940
\(286\) 24.0000 1.41915
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 14.0000 0.820695
\(292\) −6.00000 −0.351123
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −7.00000 −0.408248
\(295\) 4.00000 0.232889
\(296\) 2.00000 0.116248
\(297\) 4.00000 0.232104
\(298\) −10.0000 −0.579284
\(299\) −24.0000 −1.38796
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −2.00000 −0.114897
\(304\) −4.00000 −0.229416
\(305\) 6.00000 0.343559
\(306\) −2.00000 −0.114332
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) −1.00000 −0.0567962
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 6.00000 0.339683
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 2.00000 0.112154
\(319\) −8.00000 −0.447914
\(320\) −1.00000 −0.0559017
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) 24.0000 1.32924
\(327\) −6.00000 −0.331801
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) −12.0000 −0.658586
\(333\) −2.00000 −0.109599
\(334\) −12.0000 −0.656611
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −23.0000 −1.25104
\(339\) 10.0000 0.543125
\(340\) −2.00000 −0.108465
\(341\) 4.00000 0.216612
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −4.00000 −0.215353
\(346\) 14.0000 0.752645
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −2.00000 −0.107211
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 4.00000 0.213201
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −4.00000 −0.212598
\(355\) 12.0000 0.636894
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 6.00000 0.315353
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) −6.00000 −0.313625
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −4.00000 −0.208514
\(369\) −6.00000 −0.312348
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) 1.00000 0.0518476
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 8.00000 0.413670
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 4.00000 0.205196
\(381\) 8.00000 0.409852
\(382\) −4.00000 −0.204658
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −4.00000 −0.203331
\(388\) −14.0000 −0.710742
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −6.00000 −0.303822
\(391\) −8.00000 −0.404577
\(392\) 7.00000 0.353553
\(393\) 4.00000 0.201773
\(394\) −2.00000 −0.100759
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 16.0000 0.798007
\(403\) −6.00000 −0.298881
\(404\) 2.00000 0.0995037
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −6.00000 −0.296319
\(411\) −18.0000 −0.887875
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 12.0000 0.589057
\(416\) −6.00000 −0.294174
\(417\) −8.00000 −0.391762
\(418\) −16.0000 −0.782586
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 24.0000 1.15873
\(430\) −4.00000 −0.192897
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 6.00000 0.287348
\(437\) 16.0000 0.765384
\(438\) −6.00000 −0.286691
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) −12.0000 −0.570782
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 2.00000 0.0949158
\(445\) 18.0000 0.853282
\(446\) 8.00000 0.378811
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) −10.0000 −0.470360
\(453\) −16.0000 −0.751746
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 4.00000 0.186501
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 2.00000 0.0928477
\(465\) −1.00000 −0.0463739
\(466\) −22.0000 −1.01913
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 4.00000 0.184115
\(473\) 16.0000 0.735681
\(474\) −16.0000 −0.734904
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −12.0000 −0.547153
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 14.0000 0.635707
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 6.00000 0.271607
\(489\) 24.0000 1.08532
\(490\) −7.00000 −0.316228
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 6.00000 0.270501
\(493\) 4.00000 0.180151
\(494\) 24.0000 1.07981
\(495\) 4.00000 0.179787
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) −20.0000 −0.892644
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) −16.0000 −0.711287
\(507\) −23.0000 −1.02147
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) −22.0000 −0.970378
\(515\) −8.00000 −0.352522
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 6.00000 0.263117
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −2.00000 −0.0871214
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) 2.00000 0.0868744
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) −18.0000 −0.778936
\(535\) −4.00000 −0.172935
\(536\) −16.0000 −0.691095
\(537\) 4.00000 0.172613
\(538\) −18.0000 −0.776035
\(539\) 28.0000 1.20605
\(540\) 1.00000 0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) 6.00000 0.257485
\(544\) −2.00000 −0.0857493
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 18.0000 0.768922
\(549\) −6.00000 −0.256074
\(550\) 4.00000 0.170561
\(551\) −8.00000 −0.340811
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) −2.00000 −0.0848953
\(556\) 8.00000 0.339276
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 1.00000 0.0423334
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −10.0000 −0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 4.00000 0.167542
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) −24.0000 −1.00349
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 13.0000 0.540729
\(579\) 22.0000 0.914289
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) −8.00000 −0.331326
\(584\) 6.00000 0.248282
\(585\) −6.00000 −0.248069
\(586\) 6.00000 0.247858
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 7.00000 0.288675
\(589\) 4.00000 0.164817
\(590\) −4.00000 −0.164677
\(591\) −2.00000 −0.0822690
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −16.0000 −0.654836
\(598\) 24.0000 0.981433
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 16.0000 0.651570
\(604\) 16.0000 0.651031
\(605\) −5.00000 −0.203279
\(606\) 2.00000 0.0812444
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 16.0000 0.645707
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 46.0000 1.85189 0.925945 0.377658i \(-0.123271\pi\)
0.925945 + 0.377658i \(0.123271\pi\)
\(618\) 8.00000 0.321807
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 1.00000 0.0401610
\(621\) 4.00000 0.160514
\(622\) 28.0000 1.12270
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −26.0000 −1.03917
\(627\) −16.0000 −0.638978
\(628\) 18.0000 0.718278
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 16.0000 0.636446
\(633\) 4.00000 0.158986
\(634\) −2.00000 −0.0794301
\(635\) 8.00000 0.317470
\(636\) −2.00000 −0.0793052
\(637\) −42.0000 −1.66410
\(638\) 8.00000 0.316723
\(639\) −12.0000 −0.474713
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 4.00000 0.157867
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 8.00000 0.314756
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 16.0000 0.628055
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 6.00000 0.234619
\(655\) 4.00000 0.156293
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) −4.00000 −0.155700
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) −24.0000 −0.932786
\(663\) −12.0000 −0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) −8.00000 −0.309761
\(668\) 12.0000 0.464294
\(669\) 8.00000 0.309298
\(670\) 16.0000 0.618134
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −2.00000 −0.0770371
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) −10.0000 −0.384048
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) −20.0000 −0.766402
\(682\) −4.00000 −0.153168
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −4.00000 −0.152944
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 4.00000 0.152277
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −8.00000 −0.303457
\(696\) 2.00000 0.0758098
\(697\) −12.0000 −0.454532
\(698\) 2.00000 0.0757011
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 6.00000 0.226455
\(703\) 8.00000 0.301726
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) −12.0000 −0.450352
\(711\) −16.0000 −0.600047
\(712\) 18.0000 0.674579
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 20.0000 0.746393
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 14.0000 0.520666
\(724\) −6.00000 −0.222988
\(725\) 2.00000 0.0742781
\(726\) 5.00000 0.185567
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) −8.00000 −0.295891
\(732\) 6.00000 0.221766
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −16.0000 −0.590571
\(735\) −7.00000 −0.258199
\(736\) 4.00000 0.147442
\(737\) −64.0000 −2.35747
\(738\) 6.00000 0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 2.00000 0.0735215
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) −1.00000 −0.0366618
\(745\) −10.0000 −0.366372
\(746\) 38.0000 1.39128
\(747\) −12.0000 −0.439057
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) −20.0000 −0.728841
\(754\) −12.0000 −0.437014
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 20.0000 0.726433
\(759\) −16.0000 −0.580763
\(760\) −4.00000 −0.145095
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 4.00000 0.144715
\(765\) −2.00000 −0.0723102
\(766\) −4.00000 −0.144526
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) −22.0000 −0.791797
\(773\) 50.0000 1.79838 0.899188 0.437564i \(-0.144158\pi\)
0.899188 + 0.437564i \(0.144158\pi\)
\(774\) 4.00000 0.143777
\(775\) −1.00000 −0.0359211
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 24.0000 0.859889
\(780\) 6.00000 0.214834
\(781\) 48.0000 1.71758
\(782\) 8.00000 0.286079
\(783\) −2.00000 −0.0714742
\(784\) −7.00000 −0.250000
\(785\) −18.0000 −0.642448
\(786\) −4.00000 −0.142675
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 2.00000 0.0712470
\(789\) 12.0000 0.427211
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −36.0000 −1.27840
\(794\) 14.0000 0.496841
\(795\) 2.00000 0.0709327
\(796\) 16.0000 0.567105
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) −14.0000 −0.494357
\(803\) 24.0000 0.846942
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) −18.0000 −0.633630
\(808\) −2.00000 −0.0703598
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 1.00000 0.0351364
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) −8.00000 −0.280400
\(815\) 24.0000 0.840683
\(816\) −2.00000 −0.0700140
\(817\) 16.0000 0.559769
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) 18.0000 0.627822
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −8.00000 −0.278693
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 52.0000 1.80822 0.904109 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) −4.00000 −0.139010
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) −12.0000 −0.416526
\(831\) −6.00000 −0.208138
\(832\) 6.00000 0.208013
\(833\) −14.0000 −0.485071
\(834\) 8.00000 0.277017
\(835\) −12.0000 −0.415277
\(836\) 16.0000 0.553372
\(837\) 1.00000 0.0345651
\(838\) 12.0000 0.414533
\(839\) 28.0000 0.966667 0.483334 0.875436i \(-0.339426\pi\)
0.483334 + 0.875436i \(0.339426\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −14.0000 −0.482472
\(843\) −10.0000 −0.344418
\(844\) −4.00000 −0.137686
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 8.00000 0.274236
\(852\) 12.0000 0.411113
\(853\) 58.0000 1.98588 0.992941 0.118609i \(-0.0378434\pi\)
0.992941 + 0.118609i \(0.0378434\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) −4.00000 −0.136717
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) −24.0000 −0.819346
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) −2.00000 −0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) −2.00000 −0.0678064
\(871\) 96.0000 3.25284
\(872\) −6.00000 −0.203186
\(873\) −14.0000 −0.473828
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 32.0000 1.07995
\(879\) 6.00000 0.202375
\(880\) 4.00000 0.134840
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 7.00000 0.235702
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 12.0000 0.403604
\(885\) −4.00000 −0.134459
\(886\) 4.00000 0.134383
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) −18.0000 −0.603361
\(891\) −4.00000 −0.134005
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 24.0000 0.801337
\(898\) 18.0000 0.600668
\(899\) −2.00000 −0.0667037
\(900\) 1.00000 0.0333333
\(901\) 4.00000 0.133259
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 6.00000 0.199447
\(906\) 16.0000 0.531564
\(907\) −56.0000 −1.85945 −0.929725 0.368255i \(-0.879955\pi\)
−0.929725 + 0.368255i \(0.879955\pi\)
\(908\) 20.0000 0.663723
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 4.00000 0.132453
\(913\) 48.0000 1.58857
\(914\) 22.0000 0.727695
\(915\) −6.00000 −0.198354
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −4.00000 −0.131876
\(921\) 16.0000 0.527218
\(922\) 14.0000 0.461065
\(923\) −72.0000 −2.36991
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 24.0000 0.788689
\(927\) 8.00000 0.262754
\(928\) −2.00000 −0.0656532
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 1.00000 0.0327913
\(931\) 28.0000 0.917663
\(932\) 22.0000 0.720634
\(933\) 28.0000 0.916679
\(934\) 4.00000 0.130884
\(935\) 8.00000 0.261628
\(936\) −6.00000 −0.196116
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 18.0000 0.586472
\(943\) 24.0000 0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −44.0000 −1.42981 −0.714904 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) 16.0000 0.519656
\(949\) −36.0000 −1.16861
\(950\) 4.00000 0.129777
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −4.00000 −0.129437
\(956\) 0 0
\(957\) 8.00000 0.258603
\(958\) −36.0000 −1.16311
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) 1.00000 0.0322581
\(962\) 12.0000 0.386896
\(963\) 4.00000 0.128898
\(964\) −14.0000 −0.450910
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −5.00000 −0.160706
\(969\) 8.00000 0.256997
\(970\) −14.0000 −0.449513
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) −6.00000 −0.192154
\(976\) −6.00000 −0.192055
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) −24.0000 −0.767435
\(979\) 72.0000 2.30113
\(980\) 7.00000 0.223607
\(981\) 6.00000 0.191565
\(982\) −20.0000 −0.638226
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) −6.00000 −0.191273
\(985\) −2.00000 −0.0637253
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 16.0000 0.508770
\(990\) −4.00000 −0.127128
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 1.00000 0.0317500
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 12.0000 0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −16.0000 −0.506471
\(999\) 2.00000 0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 930.2.a.b.1.1 1
3.2 odd 2 2790.2.a.ba.1.1 1
4.3 odd 2 7440.2.a.q.1.1 1
5.2 odd 4 4650.2.d.o.3349.1 2
5.3 odd 4 4650.2.d.o.3349.2 2
5.4 even 2 4650.2.a.bp.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.b.1.1 1 1.1 even 1 trivial
2790.2.a.ba.1.1 1 3.2 odd 2
4650.2.a.bp.1.1 1 5.4 even 2
4650.2.d.o.3349.1 2 5.2 odd 4
4650.2.d.o.3349.2 2 5.3 odd 4
7440.2.a.q.1.1 1 4.3 odd 2