Properties

Label 663.2.a.a.1.1
Level $663$
Weight $2$
Character 663.1
Self dual yes
Analytic conductor $5.294$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29408165401\)
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 \(\chi\) \(=\) 663.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -1.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} -6.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -8.00000 q^{55} +4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} +14.0000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} +3.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +12.0000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +8.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} +5.00000 q^{96} -6.00000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 −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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 1.00000 0.242536
\(18\) −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
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.00000 −0.883883
\(33\) −4.00000 −0.696311
\(34\) −1.00000 −0.171499
\(35\) 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
\(39\) −1.00000 −0.160128
\(40\) −6.00000 −0.948683
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 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
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 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
\(51\) −1.00000 −0.140028
\(52\) −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
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 2.00000 0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −2.00000 −0.258199
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 2.00000 0.232495
\(75\) 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 \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) 12.0000 1.27920
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) −8.00000 −0.825137
\(95\) 8.00000 0.820783
\(96\) 5.00000 0.510310
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 7.00000 0.707107
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 1.00000 0.0990148
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 8.00000 0.762770
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 1.00000 0.0924500
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 6.00000 0.547723
\(121\) 5.00000 0.454545
\(122\) −14.0000 −1.26750
\(123\) −2.00000 −0.180334
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 3.00000 0.265165
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) −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
\(134\) 4.00000 0.345547
\(135\) 2.00000 0.172133
\(136\) 3.00000 0.257248
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) −1.00000 −0.0833333
\(145\) 4.00000 0.332182
\(146\) 14.0000 1.15865
\(147\) 7.00000 0.577350
\(148\) 2.00000 0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −12.0000 −0.973329
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 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
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −2.00000 −0.156174
\(165\) 8.00000 0.622799
\(166\) 4.00000 0.310460
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −4.00000 −0.300658
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 2.00000 0.149071
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) −8.00000 −0.586588
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −7.00000 −0.505181
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 6.00000 0.430775
\(195\) 2.00000 0.143223
\(196\) 7.00000 0.500000
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −4.00000 −0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −3.00000 −0.212132
\(201\) 4.00000 0.282138
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −4.00000 −0.279372
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 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
\(218\) 10.0000 0.677285
\(219\) 14.0000 0.946032
\(220\) 8.00000 0.539360
\(221\) 1.00000 0.0672673
\(222\) −2.00000 −0.134231
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −2.00000 −0.133038
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −16.0000 −1.04372
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −2.00000 −0.129099
\(241\) 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
\(244\) −14.0000 −0.896258
\(245\) 14.0000 0.894427
\(246\) 2.00000 0.127515
\(247\) −4.00000 −0.254514
\(248\) −24.0000 −1.52400
\(249\) 4.00000 0.253490
\(250\) −12.0000 −0.758947
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) −17.0000 −1.06250
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −2.00000 −0.123797
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −12.0000 −0.738549
\(265\) 20.0000 1.22859
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −2.00000 −0.121716
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −20.0000 −1.19952
\(279\) −8.00000 −0.478947
\(280\) 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
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) 1.00000 0.0588235
\(290\) −4.00000 −0.234888
\(291\) 6.00000 0.351726
\(292\) 14.0000 0.819288
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −7.00000 −0.408248
\(295\) −8.00000 −0.465778
\(296\) −6.00000 −0.348743
\(297\) −4.00000 −0.232104
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) −6.00000 −0.344691
\(304\) 4.00000 0.229416
\(305\) −28.0000 −1.60328
\(306\) −1.00000 −0.0571662
\(307\) −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
\(310\) −16.0000 −0.908739
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −3.00000 −0.169842
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) −10.0000 −0.560772
\(319\) −8.00000 −0.447914
\(320\) −14.0000 −0.782624
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) −1.00000 −0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 4.00000 0.221540
\(327\) 10.0000 0.553001
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) −2.00000 −0.109599
\(334\) −16.0000 −0.875481
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −2.00000 −0.108625
\(340\) 2.00000 0.108465
\(341\) −32.0000 −1.73290
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −2.00000 −0.107211
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −20.0000 −1.06600
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 28.0000 1.46559
\(366\) 14.0000 0.731792
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −4.00000 −0.206835
\(375\) −12.0000 −0.619677
\(376\) 24.0000 1.23771
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −4.00000 −0.203331
\(388\) 6.00000 0.304604
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −21.0000 −1.06066
\(393\) 4.00000 0.201773
\(394\) 2.00000 0.100759
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −4.00000 −0.199502
\(403\) −8.00000 −0.398508
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) −3.00000 −0.148522
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 4.00000 0.197546
\(411\) 22.0000 1.08518
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) −5.00000 −0.245145
\(417\) −20.0000 −0.979404
\(418\) 16.0000 0.782586
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −12.0000 −0.584151
\(423\) 8.00000 0.388973
\(424\) −30.0000 −1.45693
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −4.00000 −0.193122
\(430\) −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
\(435\) −4.00000 −0.191785
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) −14.0000 −0.668946
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −24.0000 −1.14416
\(441\) −7.00000 −0.333333
\(442\) −1.00000 −0.0475651
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 12.0000 0.568855
\(446\) −8.00000 −0.378811
\(447\) 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
\(451\) 8.00000 0.376705
\(452\) −2.00000 −0.0940721
\(453\) 16.0000 0.751746
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 10.0000 0.467269
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 2.00000 0.0928477
\(465\) −16.0000 −0.741982
\(466\) 6.00000 0.277945
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 16.0000 0.738025
\(471\) 2.00000 0.0921551
\(472\) 12.0000 0.552345
\(473\) −16.0000 −0.735681
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 24.0000 1.09773
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −10.0000 −0.456435
\(481\) −2.00000 −0.0911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 12.0000 0.544892
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 42.0000 1.90125
\(489\) 4.00000 0.180886
\(490\) −14.0000 −0.632456
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 2.00000 0.0901670
\(493\) −2.00000 −0.0900755
\(494\) 4.00000 0.179969
\(495\) −8.00000 −0.359573
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −12.0000 −0.536656
\(501\) −16.0000 −0.714827
\(502\) −12.0000 −0.535586
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) 30.0000 1.32324
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) −6.00000 −0.263117
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 2.00000 0.0875376
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −8.00000 −0.348485
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) −20.0000 −0.868744
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) −6.00000 −0.259645
\(535\) 24.0000 1.03761
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) 18.0000 0.776035
\(539\) −28.0000 −1.20605
\(540\) −2.00000 −0.0860663
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −8.00000 −0.343629
\(543\) −6.00000 −0.257485
\(544\) −5.00000 −0.214373
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 22.0000 0.939793
\(549\) 14.0000 0.597505
\(550\) 4.00000 0.170561
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −4.00000 −0.169791
\(556\) −20.0000 −0.848189
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 8.00000 0.338667
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 6.00000 0.253095
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 8.00000 0.336861
\(565\) −4.00000 −0.168281
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 8.00000 0.335083
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −4.00000 −0.167248
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 22.0000 0.914289
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) −40.0000 −1.65663
\(584\) −42.0000 −1.73797
\(585\) −2.00000 −0.0826898
\(586\) 26.0000 1.07405
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −7.00000 −0.288675
\(589\) 32.0000 1.31854
\(590\) 8.00000 0.329355
\(591\) 2.00000 0.0822690
\(592\) 2.00000 0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 3.00000 0.122474
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 16.0000 0.651031
\(605\) −10.0000 −0.406558
\(606\) 6.00000 0.243733
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) 28.0000 1.13369
\(611\) 8.00000 0.323645
\(612\) −1.00000 −0.0404226
\(613\) −42.0000 −1.69636 −0.848182 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) 20.0000 0.807134
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) −8.00000 −0.321807
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −19.0000 −0.760000
\(626\) 22.0000 0.879297
\(627\) 16.0000 0.638978
\(628\) 2.00000 0.0798087
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) −24.0000 −0.954669
\(633\) −12.0000 −0.476957
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) −7.00000 −0.277350
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) −6.00000 −0.237171
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) −12.0000 −0.473602
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 4.00000 0.157378
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 3.00000 0.117851
\(649\) 16.0000 0.628055
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −10.0000 −0.391031
\(655\) 8.00000 0.312586
\(656\) −2.00000 −0.0780869
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) −8.00000 −0.311400
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −4.00000 −0.155464
\(663\) −1.00000 −0.0388368
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) −8.00000 −0.309298
\(670\) −8.00000 −0.309067
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −18.0000 −0.693334
\(675\) 1.00000 0.0384900
\(676\) −1.00000 −0.0384615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 2.00000 0.0768095
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) −12.0000 −0.459841
\(682\) 32.0000 1.22534
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) 44.0000 1.68115
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −40.0000 −1.51729
\(696\) 6.00000 0.227429
\(697\) 2.00000 0.0757554
\(698\) −30.0000 −1.13552
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 1.00000 0.0377426
\(703\) 8.00000 0.301726
\(704\) 28.0000 1.05529
\(705\) 16.0000 0.602595
\(706\) 14.0000 0.526897
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −18.0000 −0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 12.0000 0.448461
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −10.0000 −0.371904
\(724\) −6.00000 −0.222988
\(725\) 2.00000 0.0742781
\(726\) 5.00000 0.185567
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −28.0000 −1.03633
\(731\) −4.00000 −0.147945
\(732\) 14.0000 0.517455
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −8.00000 −0.295285
\(735\) −14.0000 −0.516398
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) −2.00000 −0.0736210
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −4.00000 −0.147043
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 24.0000 0.879883
\(745\) 20.0000 0.732743
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −8.00000 −0.291730
\(753\) −12.0000 −0.437304
\(754\) 2.00000 0.0728357
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) −2.00000 −0.0723102
\(766\) 24.0000 0.867155
\(767\) 4.00000 0.144432
\(768\) 17.0000 0.613435
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 22.0000 0.791797
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) −38.0000 −1.36237
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 7.00000 0.250000
\(785\) 4.00000 0.142766
\(786\) −4.00000 −0.142675
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 2.00000 0.0712470
\(789\) −24.0000 −0.854423
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 12.0000 0.426401
\(793\) 14.0000 0.497155
\(794\) 26.0000 0.922705
\(795\) −20.0000 −0.709327
\(796\) −16.0000 −0.567105
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) 6.00000 0.211867
\(803\) −56.0000 −1.97620
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 18.0000 0.633630
\(808\) 18.0000 0.633238
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 2.00000 0.0702728
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 8.00000 0.280400
\(815\) 8.00000 0.280228
\(816\) 1.00000 0.0350070
\(817\) 16.0000 0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −22.0000 −0.767338
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −24.0000 −0.836080
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −8.00000 −0.277684
\(831\) 10.0000 0.346896
\(832\) 7.00000 0.242681
\(833\) −7.00000 −0.242536
\(834\) 20.0000 0.692543
\(835\) −32.0000 −1.10741
\(836\) 16.0000 0.553372
\(837\) 8.00000 0.276520
\(838\) −28.0000 −0.967244
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 6.00000 0.206651
\(844\) −12.0000 −0.413057
\(845\) −2.00000 −0.0688021
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) −20.0000 −0.686398
\(850\) 1.00000 0.0342997
\(851\) 0 0
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) −36.0000 −1.23045
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 4.00000 0.136558
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 5.00000 0.170103
\(865\) −28.0000 −0.952029
\(866\) 14.0000 0.475739
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 4.00000 0.135613
\(871\) −4.00000 −0.135535
\(872\) −30.0000 −1.01593
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) −16.0000 −0.539974
\(879\) 26.0000 0.876958
\(880\) 8.00000 0.269680
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 7.00000 0.235702
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −1.00000 −0.0336336
\(885\) 8.00000 0.268917
\(886\) 20.0000 0.671913
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) 4.00000 0.134005
\(892\) −8.00000 −0.267860
\(893\) −32.0000 −1.07084
\(894\) −10.0000 −0.334450
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) 16.0000 0.533630
\(900\) 1.00000 0.0333333
\(901\) −10.0000 −0.333148
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −12.0000 −0.398893
\(906\) −16.0000 −0.531564
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −12.0000 −0.398234
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −4.00000 −0.132453
\(913\) −16.0000 −0.529523
\(914\) 22.0000 0.727695
\(915\) 28.0000 0.925651
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −24.0000 −0.788689
\(927\) −8.00000 −0.262754
\(928\) 10.0000 0.328266
\(929\) −22.0000 −0.721797 −0.360898 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(930\) 16.0000 0.524661
\(931\) 28.0000 0.917663
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −4.00000 −0.130884
\(935\) −8.00000 −0.261628
\(936\) 3.00000 0.0980581
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 16.0000 0.521862
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −8.00000 −0.259828
\(949\) −14.0000 −0.454459
\(950\) −4.00000 −0.129777
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 10.0000 0.323762
\(955\) −32.0000 −1.03550
\(956\) 24.0000 0.776215
\(957\) 8.00000 0.258603
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 14.0000 0.451848
\(961\) 33.0000 1.06452
\(962\) 2.00000 0.0644826
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) 44.0000 1.41641
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 15.0000 0.482118
\(969\) 4.00000 0.128499
\(970\) −12.0000 −0.385297
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 1.00000 0.0320256
\(976\) −14.0000 −0.448129
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −4.00000 −0.127906
\(979\) −24.0000 −0.767043
\(980\) −14.0000 −0.447214
\(981\) −10.0000 −0.319275
\(982\) −28.0000 −0.893516
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) −6.00000 −0.191273
\(985\) 4.00000 0.127451
\(986\) 2.00000 0.0636930
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 8.00000 0.254257
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 40.0000 1.27000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) −4.00000 −0.126745
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 663.2.a.a.1.1 1
3.2 odd 2 1989.2.a.e.1.1 1
13.12 even 2 8619.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
663.2.a.a.1.1 1 1.1 even 1 trivial
1989.2.a.e.1.1 1 3.2 odd 2
8619.2.a.i.1.1 1 13.12 even 2