Properties

Label 663.2.a.c.1.1
Level $663$
Weight $2$
Character 663.1
Self dual yes
Analytic conductor $5.294$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +4.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} +3.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} +4.00000 q^{30} +10.0000 q^{31} +5.00000 q^{32} -6.00000 q^{33} +1.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} +12.0000 q^{40} -2.00000 q^{42} +12.0000 q^{43} -6.00000 q^{44} -4.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +11.0000 q^{50} -1.00000 q^{51} +1.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -24.0000 q^{55} -6.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} -4.00000 q^{60} -10.0000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} -6.00000 q^{66} +12.0000 q^{67} -1.00000 q^{68} -8.00000 q^{70} +2.00000 q^{71} -3.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -11.0000 q^{75} -4.00000 q^{76} +12.0000 q^{77} +1.00000 q^{78} -4.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{85} +12.0000 q^{86} +6.00000 q^{87} -18.0000 q^{88} -6.00000 q^{89} -4.00000 q^{90} -2.00000 q^{91} -10.0000 q^{93} +8.00000 q^{94} -16.0000 q^{95} -5.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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 \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −4.00000 −1.26491
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 4.00000 1.03280
\(16\) −1.00000 −0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 4.00000 0.894427
\(21\) −2.00000 −0.436436
\(22\) 6.00000 1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 4.00000 0.730297
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 5.00000 0.883883
\(33\) −6.00000 −1.04447
\(34\) 1.00000 0.171499
\(35\) −8.00000 −1.35225
\(36\) −1.00000 −0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 4.00000 0.648886
\(39\) 1.00000 0.160128
\(40\) 12.0000 1.89737
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −6.00000 −0.904534
\(45\) −4.00000 −0.596285
\(46\) 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
\(49\) −3.00000 −0.428571
\(50\) 11.0000 1.55563
\(51\) −1.00000 −0.140028
\(52\) 1.00000 0.138675
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) −24.0000 −3.23616
\(56\) −6.00000 −0.801784
\(57\) −4.00000 −0.529813
\(58\) −6.00000 −0.787839
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −4.00000 −0.516398
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 10.0000 1.27000
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) −6.00000 −0.738549
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −3.00000 −0.353553
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −4.00000 −0.464991
\(75\) −11.0000 −1.27017
\(76\) −4.00000 −0.458831
\(77\) 12.0000 1.36753
\(78\) 1.00000 0.113228
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) −4.00000 −0.433861
\(86\) 12.0000 1.29399
\(87\) 6.00000 0.643268
\(88\) −18.0000 −1.91881
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −4.00000 −0.421637
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 8.00000 0.825137
\(95\) −16.0000 −1.64157
\(96\) −5.00000 −0.510310
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −3.00000 −0.303046
\(99\) 6.00000 0.603023
\(100\) −11.0000 −1.10000
\(101\) 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 \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 3.00000 0.294174
\(105\) 8.00000 0.780720
\(106\) 2.00000 0.194257
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −24.0000 −2.28831
\(111\) 4.00000 0.379663
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −1.00000 −0.0924500
\(118\) −8.00000 −0.736460
\(119\) 2.00000 0.183340
\(120\) −12.0000 −1.09545
\(121\) 25.0000 2.27273
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) −24.0000 −2.14663
\(126\) 2.00000 0.178174
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −3.00000 −0.265165
\(129\) −12.0000 −1.05654
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 6.00000 0.522233
\(133\) 8.00000 0.693688
\(134\) 12.0000 1.03664
\(135\) 4.00000 0.344265
\(136\) −3.00000 −0.257248
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 8.00000 0.676123
\(141\) −8.00000 −0.673722
\(142\) 2.00000 0.167836
\(143\) −6.00000 −0.501745
\(144\) −1.00000 −0.0833333
\(145\) 24.0000 1.99309
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) 4.00000 0.328798
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −11.0000 −0.898146
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −12.0000 −0.973329
\(153\) 1.00000 0.0808452
\(154\) 12.0000 0.966988
\(155\) −40.0000 −3.21288
\(156\) −1.00000 −0.0800641
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −4.00000 −0.318223
\(159\) −2.00000 −0.158610
\(160\) −20.0000 −1.58114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 0 0
\(165\) 24.0000 1.86840
\(166\) −12.0000 −0.931381
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 6.00000 0.462910
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) 4.00000 0.305888
\(172\) −12.0000 −0.914991
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 6.00000 0.454859
\(175\) 22.0000 1.66304
\(176\) −6.00000 −0.452267
\(177\) 8.00000 0.601317
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 4.00000 0.298142
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 16.0000 1.17634
\(186\) −10.0000 −0.733236
\(187\) 6.00000 0.438763
\(188\) −8.00000 −0.583460
\(189\) −2.00000 −0.145479
\(190\) −16.0000 −1.16076
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −7.00000 −0.505181
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 8.00000 0.574367
\(195\) −4.00000 −0.286446
\(196\) 3.00000 0.214286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 6.00000 0.426401
\(199\) −28.0000 −1.98487 −0.992434 0.122782i \(-0.960818\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) −33.0000 −2.33345
\(201\) −12.0000 −0.846415
\(202\) 6.00000 0.422159
\(203\) −12.0000 −0.842235
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 24.0000 1.66011
\(210\) 8.00000 0.552052
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −2.00000 −0.137361
\(213\) −2.00000 −0.137038
\(214\) 16.0000 1.09374
\(215\) −48.0000 −3.27357
\(216\) 3.00000 0.204124
\(217\) 20.0000 1.35769
\(218\) −16.0000 −1.08366
\(219\) −4.00000 −0.270295
\(220\) 24.0000 1.61808
\(221\) −1.00000 −0.0672673
\(222\) 4.00000 0.268462
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 10.0000 0.668153
\(225\) 11.0000 0.733333
\(226\) 6.00000 0.399114
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 4.00000 0.264906
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 18.0000 1.18176
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −32.0000 −2.08745
\(236\) 8.00000 0.520756
\(237\) 4.00000 0.259828
\(238\) 2.00000 0.129641
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) −4.00000 −0.258199
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 25.0000 1.60706
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 12.0000 0.766652
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) −30.0000 −1.90500
\(249\) 12.0000 0.760469
\(250\) −24.0000 −1.51789
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 4.00000 0.250490
\(256\) −17.0000 −1.06250
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −12.0000 −0.747087
\(259\) −8.00000 −0.497096
\(260\) −4.00000 −0.248069
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 18.0000 1.10782
\(265\) −8.00000 −0.491436
\(266\) 8.00000 0.490511
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 4.00000 0.243432
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 2.00000 0.121046
\(274\) −6.00000 −0.362473
\(275\) 66.0000 3.97995
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 16.0000 0.959616
\(279\) 10.0000 0.598684
\(280\) 24.0000 1.43427
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −8.00000 −0.476393
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −2.00000 −0.118678
\(285\) 16.0000 0.947758
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) 1.00000 0.0588235
\(290\) 24.0000 1.40933
\(291\) −8.00000 −0.468968
\(292\) −4.00000 −0.234082
\(293\) 2.00000 0.116841 0.0584206 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 3.00000 0.174964
\(295\) 32.0000 1.86311
\(296\) 12.0000 0.697486
\(297\) −6.00000 −0.348155
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 11.0000 0.635085
\(301\) 24.0000 1.38334
\(302\) 20.0000 1.15087
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) 40.0000 2.29039
\(306\) 1.00000 0.0571662
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −12.0000 −0.683763
\(309\) −8.00000 −0.455104
\(310\) −40.0000 −2.27185
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −3.00000 −0.169842
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 2.00000 0.112867
\(315\) −8.00000 −0.450749
\(316\) 4.00000 0.225018
\(317\) −4.00000 −0.224662 −0.112331 0.993671i \(-0.535832\pi\)
−0.112331 + 0.993671i \(0.535832\pi\)
\(318\) −2.00000 −0.112154
\(319\) −36.0000 −2.01561
\(320\) −28.0000 −1.56525
\(321\) −16.0000 −0.893033
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) −1.00000 −0.0555556
\(325\) −11.0000 −0.610170
\(326\) 6.00000 0.332309
\(327\) 16.0000 0.884802
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 24.0000 1.32116
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 12.0000 0.658586
\(333\) −4.00000 −0.219199
\(334\) −14.0000 −0.766046
\(335\) −48.0000 −2.62252
\(336\) 2.00000 0.109109
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 1.00000 0.0543928
\(339\) −6.00000 −0.325875
\(340\) 4.00000 0.216930
\(341\) 60.0000 3.24918
\(342\) 4.00000 0.216295
\(343\) −20.0000 −1.07990
\(344\) −36.0000 −1.94099
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) −6.00000 −0.321634
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 22.0000 1.17595
\(351\) 1.00000 0.0533761
\(352\) 30.0000 1.59901
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 8.00000 0.425195
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) −2.00000 −0.105851
\(358\) 12.0000 0.634220
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 12.0000 0.632456
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) −25.0000 −1.31216
\(364\) 2.00000 0.104828
\(365\) −16.0000 −0.837478
\(366\) 10.0000 0.522708
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 16.0000 0.831800
\(371\) 4.00000 0.207670
\(372\) 10.0000 0.518476
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 6.00000 0.310253
\(375\) 24.0000 1.23935
\(376\) −24.0000 −1.23771
\(377\) 6.00000 0.309016
\(378\) −2.00000 −0.102869
\(379\) 34.0000 1.74646 0.873231 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) 16.0000 0.820783
\(381\) −16.0000 −0.819705
\(382\) 8.00000 0.409316
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 3.00000 0.153093
\(385\) −48.0000 −2.44631
\(386\) −4.00000 −0.203595
\(387\) 12.0000 0.609994
\(388\) −8.00000 −0.406138
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 12.0000 0.605320
\(394\) −12.0000 −0.604551
\(395\) 16.0000 0.805047
\(396\) −6.00000 −0.301511
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) −28.0000 −1.40351
\(399\) −8.00000 −0.400501
\(400\) −11.0000 −0.550000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −12.0000 −0.598506
\(403\) −10.0000 −0.498135
\(404\) −6.00000 −0.298511
\(405\) −4.00000 −0.198762
\(406\) −12.0000 −0.595550
\(407\) −24.0000 −1.18964
\(408\) 3.00000 0.148522
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −8.00000 −0.394132
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 48.0000 2.35623
\(416\) −5.00000 −0.245145
\(417\) −16.0000 −0.783523
\(418\) 24.0000 1.17388
\(419\) 8.00000 0.390826 0.195413 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) −8.00000 −0.390360
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −8.00000 −0.389434
\(423\) 8.00000 0.388973
\(424\) −6.00000 −0.291386
\(425\) 11.0000 0.533578
\(426\) −2.00000 −0.0969003
\(427\) −20.0000 −0.967868
\(428\) −16.0000 −0.773389
\(429\) 6.00000 0.289683
\(430\) −48.0000 −2.31477
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 1.00000 0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 20.0000 0.960031
\(435\) −24.0000 −1.15071
\(436\) 16.0000 0.766261
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 72.0000 3.43247
\(441\) −3.00000 −0.142857
\(442\) −1.00000 −0.0475651
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −4.00000 −0.189832
\(445\) 24.0000 1.13771
\(446\) −12.0000 −0.568216
\(447\) −18.0000 −0.851371
\(448\) 14.0000 0.661438
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 11.0000 0.518545
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −20.0000 −0.939682
\(454\) 6.00000 0.281594
\(455\) 8.00000 0.375046
\(456\) 12.0000 0.561951
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 6.00000 0.280362
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) −12.0000 −0.558291
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 6.00000 0.278543
\(465\) 40.0000 1.85496
\(466\) 14.0000 0.648537
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 1.00000 0.0462250
\(469\) 24.0000 1.10822
\(470\) −32.0000 −1.47605
\(471\) −2.00000 −0.0921551
\(472\) 24.0000 1.10469
\(473\) 72.0000 3.31056
\(474\) 4.00000 0.183726
\(475\) 44.0000 2.01886
\(476\) −2.00000 −0.0916698
\(477\) 2.00000 0.0915737
\(478\) −12.0000 −0.548867
\(479\) 34.0000 1.55350 0.776750 0.629809i \(-0.216867\pi\)
0.776750 + 0.629809i \(0.216867\pi\)
\(480\) 20.0000 0.912871
\(481\) 4.00000 0.182384
\(482\) −8.00000 −0.364390
\(483\) 0 0
\(484\) −25.0000 −1.13636
\(485\) −32.0000 −1.45305
\(486\) −1.00000 −0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 30.0000 1.35804
\(489\) −6.00000 −0.271329
\(490\) 12.0000 0.542105
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) −4.00000 −0.179969
\(495\) −24.0000 −1.07872
\(496\) −10.0000 −0.449013
\(497\) 4.00000 0.179425
\(498\) 12.0000 0.537733
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 24.0000 1.07331
\(501\) 14.0000 0.625474
\(502\) −20.0000 −0.892644
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) −6.00000 −0.267261
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) −16.0000 −0.709885
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 4.00000 0.177123
\(511\) 8.00000 0.353899
\(512\) −11.0000 −0.486136
\(513\) −4.00000 −0.176604
\(514\) 14.0000 0.617514
\(515\) −32.0000 −1.41009
\(516\) 12.0000 0.528271
\(517\) 48.0000 2.11104
\(518\) −8.00000 −0.351500
\(519\) −2.00000 −0.0877903
\(520\) −12.0000 −0.526235
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −6.00000 −0.262613
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 12.0000 0.524222
\(525\) −22.0000 −0.960159
\(526\) 16.0000 0.697633
\(527\) 10.0000 0.435607
\(528\) 6.00000 0.261116
\(529\) −23.0000 −1.00000
\(530\) −8.00000 −0.347498
\(531\) −8.00000 −0.347170
\(532\) −8.00000 −0.346844
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) −64.0000 −2.76696
\(536\) −36.0000 −1.55496
\(537\) −12.0000 −0.517838
\(538\) −30.0000 −1.29339
\(539\) −18.0000 −0.775315
\(540\) −4.00000 −0.172133
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) −8.00000 −0.343629
\(543\) 10.0000 0.429141
\(544\) 5.00000 0.214373
\(545\) 64.0000 2.74146
\(546\) 2.00000 0.0855921
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) 66.0000 2.81425
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 2.00000 0.0849719
\(555\) −16.0000 −0.679162
\(556\) −16.0000 −0.678551
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 10.0000 0.423334
\(559\) −12.0000 −0.507546
\(560\) 8.00000 0.338062
\(561\) −6.00000 −0.253320
\(562\) 2.00000 0.0843649
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 8.00000 0.336861
\(565\) −24.0000 −1.00969
\(566\) 12.0000 0.504398
\(567\) 2.00000 0.0839921
\(568\) −6.00000 −0.251754
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 16.0000 0.670166
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 6.00000 0.250873
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 1.00000 0.0415945
\(579\) 4.00000 0.166234
\(580\) −24.0000 −0.996546
\(581\) −24.0000 −0.995688
\(582\) −8.00000 −0.331611
\(583\) 12.0000 0.496989
\(584\) −12.0000 −0.496564
\(585\) 4.00000 0.165380
\(586\) 2.00000 0.0826192
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −3.00000 −0.123718
\(589\) 40.0000 1.64817
\(590\) 32.0000 1.31742
\(591\) 12.0000 0.493614
\(592\) 4.00000 0.164399
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −6.00000 −0.246183
\(595\) −8.00000 −0.327968
\(596\) −18.0000 −0.737309
\(597\) 28.0000 1.14596
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 33.0000 1.34722
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 24.0000 0.978167
\(603\) 12.0000 0.488678
\(604\) −20.0000 −0.813788
\(605\) −100.000 −4.06558
\(606\) −6.00000 −0.243733
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 20.0000 0.811107
\(609\) 12.0000 0.486265
\(610\) 40.0000 1.61955
\(611\) −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 \(0.822303\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −36.0000 −1.45048
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) −8.00000 −0.321807
\(619\) 18.0000 0.723481 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(620\) 40.0000 1.60644
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −12.0000 −0.480770
\(624\) −1.00000 −0.0400320
\(625\) 41.0000 1.64000
\(626\) −10.0000 −0.399680
\(627\) −24.0000 −0.958468
\(628\) −2.00000 −0.0798087
\(629\) −4.00000 −0.159490
\(630\) −8.00000 −0.318728
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 12.0000 0.477334
\(633\) 8.00000 0.317971
\(634\) −4.00000 −0.158860
\(635\) −64.0000 −2.53976
\(636\) 2.00000 0.0793052
\(637\) 3.00000 0.118864
\(638\) −36.0000 −1.42525
\(639\) 2.00000 0.0791188
\(640\) 12.0000 0.474342
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −16.0000 −0.631470
\(643\) −46.0000 −1.81406 −0.907031 0.421063i \(-0.861657\pi\)
−0.907031 + 0.421063i \(0.861657\pi\)
\(644\) 0 0
\(645\) 48.0000 1.89000
\(646\) 4.00000 0.157378
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) −3.00000 −0.117851
\(649\) −48.0000 −1.88416
\(650\) −11.0000 −0.431455
\(651\) −20.0000 −0.783862
\(652\) −6.00000 −0.234978
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 16.0000 0.625650
\(655\) 48.0000 1.87552
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 16.0000 0.623745
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −24.0000 −0.934199
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −8.00000 −0.310929
\(663\) 1.00000 0.0388368
\(664\) 36.0000 1.39707
\(665\) −32.0000 −1.24091
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 14.0000 0.541676
\(669\) 12.0000 0.463947
\(670\) −48.0000 −1.85440
\(671\) −60.0000 −2.31627
\(672\) −10.0000 −0.385758
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 2.00000 0.0770371
\(675\) −11.0000 −0.423390
\(676\) −1.00000 −0.0384615
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) −6.00000 −0.230429
\(679\) 16.0000 0.614024
\(680\) 12.0000 0.460179
\(681\) −6.00000 −0.229920
\(682\) 60.0000 2.29752
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) −4.00000 −0.152944
\(685\) 24.0000 0.916993
\(686\) −20.0000 −0.763604
\(687\) −6.00000 −0.228914
\(688\) −12.0000 −0.457496
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −6.00000 −0.228251 −0.114125 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 12.0000 0.455842
\(694\) −32.0000 −1.21470
\(695\) −64.0000 −2.42766
\(696\) −18.0000 −0.682288
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) −14.0000 −0.529529
\(700\) −22.0000 −0.831522
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 1.00000 0.0377426
\(703\) −16.0000 −0.603451
\(704\) 42.0000 1.58293
\(705\) 32.0000 1.20519
\(706\) −26.0000 −0.978523
\(707\) 12.0000 0.451306
\(708\) −8.00000 −0.300658
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) −8.00000 −0.300235
\(711\) −4.00000 −0.150012
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 24.0000 0.897549
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) −4.00000 −0.149279
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 4.00000 0.149071
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) 8.00000 0.297523
\(724\) 10.0000 0.371647
\(725\) −66.0000 −2.45118
\(726\) −25.0000 −0.927837
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) −16.0000 −0.592187
\(731\) 12.0000 0.443836
\(732\) −10.0000 −0.369611
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 0 0
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) 72.0000 2.65215
\(738\) 0 0
\(739\) −32.0000 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) −16.0000 −0.588172
\(741\) 4.00000 0.146944
\(742\) 4.00000 0.146845
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 30.0000 1.09985
\(745\) −72.0000 −2.63788
\(746\) 14.0000 0.512576
\(747\) −12.0000 −0.439057
\(748\) −6.00000 −0.219382
\(749\) 32.0000 1.16925
\(750\) 24.0000 0.876356
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −8.00000 −0.291730
\(753\) 20.0000 0.728841
\(754\) 6.00000 0.218507
\(755\) −80.0000 −2.91150
\(756\) 2.00000 0.0727393
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 34.0000 1.23494
\(759\) 0 0
\(760\) 48.0000 1.74114
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) −16.0000 −0.579619
\(763\) −32.0000 −1.15848
\(764\) −8.00000 −0.289430
\(765\) −4.00000 −0.144620
\(766\) 12.0000 0.433578
\(767\) 8.00000 0.288863
\(768\) 17.0000 0.613435
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) −48.0000 −1.72980
\(771\) −14.0000 −0.504198
\(772\) 4.00000 0.143963
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 12.0000 0.431331
\(775\) 110.000 3.95132
\(776\) −24.0000 −0.861550
\(777\) 8.00000 0.286998
\(778\) 10.0000 0.358517
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 3.00000 0.107143
\(785\) −8.00000 −0.285532
\(786\) 12.0000 0.428026
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) 12.0000 0.427482
\(789\) −16.0000 −0.569615
\(790\) 16.0000 0.569254
\(791\) 12.0000 0.426671
\(792\) −18.0000 −0.639602
\(793\) 10.0000 0.355110
\(794\) −20.0000 −0.709773
\(795\) 8.00000 0.283731
\(796\) 28.0000 0.992434
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −8.00000 −0.283197
\(799\) 8.00000 0.283020
\(800\) 55.0000 1.94454
\(801\) −6.00000 −0.212000
\(802\) 12.0000 0.423735
\(803\) 24.0000 0.846942
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −10.0000 −0.352235
\(807\) 30.0000 1.05605
\(808\) −18.0000 −0.633238
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) −4.00000 −0.140546
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 12.0000 0.421117
\(813\) 8.00000 0.280572
\(814\) −24.0000 −0.841200
\(815\) −24.0000 −0.840683
\(816\) 1.00000 0.0350070
\(817\) 48.0000 1.67931
\(818\) −14.0000 −0.489499
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 6.00000 0.209274
\(823\) 36.0000 1.25488 0.627441 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) −24.0000 −0.836080
\(825\) −66.0000 −2.29783
\(826\) −16.0000 −0.556711
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 48.0000 1.66610
\(831\) −2.00000 −0.0693792
\(832\) −7.00000 −0.242681
\(833\) −3.00000 −0.103944
\(834\) −16.0000 −0.554035
\(835\) 56.0000 1.93796
\(836\) −24.0000 −0.830057
\(837\) −10.0000 −0.345651
\(838\) 8.00000 0.276355
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) −24.0000 −0.828079
\(841\) 7.00000 0.241379
\(842\) 2.00000 0.0689246
\(843\) −2.00000 −0.0688837
\(844\) 8.00000 0.275371
\(845\) −4.00000 −0.137604
\(846\) 8.00000 0.275046
\(847\) 50.0000 1.71802
\(848\) −2.00000 −0.0686803
\(849\) −12.0000 −0.411839
\(850\) 11.0000 0.377297
\(851\) 0 0
\(852\) 2.00000 0.0685189
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) −20.0000 −0.684386
\(855\) −16.0000 −0.547188
\(856\) −48.0000 −1.64061
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) 6.00000 0.204837
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 48.0000 1.63679
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −5.00000 −0.170103
\(865\) −8.00000 −0.272008
\(866\) 18.0000 0.611665
\(867\) −1.00000 −0.0339618
\(868\) −20.0000 −0.678844
\(869\) −24.0000 −0.814144
\(870\) −24.0000 −0.813676
\(871\) −12.0000 −0.406604
\(872\) 48.0000 1.62549
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 4.00000 0.135147
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 32.0000 1.07995
\(879\) −2.00000 −0.0674583
\(880\) 24.0000 0.809040
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) −3.00000 −0.101015
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 1.00000 0.0336336
\(885\) −32.0000 −1.07567
\(886\) −4.00000 −0.134383
\(887\) 40.0000 1.34307 0.671534 0.740973i \(-0.265636\pi\)
0.671534 + 0.740973i \(0.265636\pi\)
\(888\) −12.0000 −0.402694
\(889\) 32.0000 1.07325
\(890\) 24.0000 0.804482
\(891\) 6.00000 0.201008
\(892\) 12.0000 0.401790
\(893\) 32.0000 1.07084
\(894\) −18.0000 −0.602010
\(895\) −48.0000 −1.60446
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) −60.0000 −2.00111
\(900\) −11.0000 −0.366667
\(901\) 2.00000 0.0666297
\(902\) 0 0
\(903\) −24.0000 −0.798670
\(904\) −18.0000 −0.598671
\(905\) 40.0000 1.32964
\(906\) −20.0000 −0.664455
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −6.00000 −0.199117
\(909\) 6.00000 0.199007
\(910\) 8.00000 0.265197
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 4.00000 0.132453
\(913\) −72.0000 −2.38285
\(914\) 10.0000 0.330771
\(915\) −40.0000 −1.32236
\(916\) −6.00000 −0.198246
\(917\) −24.0000 −0.792550
\(918\) −1.00000 −0.0330049
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 14.0000 0.461065
\(923\) −2.00000 −0.0658308
\(924\) 12.0000 0.394771
\(925\) −44.0000 −1.44671
\(926\) −28.0000 −0.920137
\(927\) 8.00000 0.262754
\(928\) −30.0000 −0.984798
\(929\) 20.0000 0.656179 0.328089 0.944647i \(-0.393595\pi\)
0.328089 + 0.944647i \(0.393595\pi\)
\(930\) 40.0000 1.31165
\(931\) −12.0000 −0.393284
\(932\) −14.0000 −0.458585
\(933\) 12.0000 0.392862
\(934\) 28.0000 0.916188
\(935\) −24.0000 −0.784884
\(936\) 3.00000 0.0980581
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 24.0000 0.783628
\(939\) 10.0000 0.326338
\(940\) 32.0000 1.04372
\(941\) −44.0000 −1.43436 −0.717180 0.696888i \(-0.754567\pi\)
−0.717180 + 0.696888i \(0.754567\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) 8.00000 0.260378
\(945\) 8.00000 0.260240
\(946\) 72.0000 2.34092
\(947\) 2.00000 0.0649913 0.0324956 0.999472i \(-0.489654\pi\)
0.0324956 + 0.999472i \(0.489654\pi\)
\(948\) −4.00000 −0.129914
\(949\) −4.00000 −0.129845
\(950\) 44.0000 1.42755
\(951\) 4.00000 0.129709
\(952\) −6.00000 −0.194461
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 2.00000 0.0647524
\(955\) −32.0000 −1.03550
\(956\) 12.0000 0.388108
\(957\) 36.0000 1.16371
\(958\) 34.0000 1.09849
\(959\) −12.0000 −0.387500
\(960\) 28.0000 0.903696
\(961\) 69.0000 2.22581
\(962\) 4.00000 0.128965
\(963\) 16.0000 0.515593
\(964\) 8.00000 0.257663
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −75.0000 −2.41059
\(969\) −4.00000 −0.128499
\(970\) −32.0000 −1.02746
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 32.0000 1.02587
\(974\) −2.00000 −0.0640841
\(975\) 11.0000 0.352282
\(976\) 10.0000 0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −6.00000 −0.191859
\(979\) −36.0000 −1.15056
\(980\) −12.0000 −0.383326
\(981\) −16.0000 −0.510841
\(982\) −36.0000 −1.14881
\(983\) 46.0000 1.46717 0.733586 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) −6.00000 −0.191079
\(987\) −16.0000 −0.509286
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) −24.0000 −0.762770
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 50.0000 1.58750
\(993\) 8.00000 0.253872
\(994\) 4.00000 0.126872
\(995\) 112.000 3.55064
\(996\) −12.0000 −0.380235
\(997\) −34.0000 −1.07679 −0.538395 0.842692i \(-0.680969\pi\)
−0.538395 + 0.842692i \(0.680969\pi\)
\(998\) −10.0000 −0.316544
\(999\) 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 663.2.a.c.1.1 1
3.2 odd 2 1989.2.a.b.1.1 1
13.12 even 2 8619.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
663.2.a.c.1.1 1 1.1 even 1 trivial
1989.2.a.b.1.1 1 3.2 odd 2
8619.2.a.e.1.1 1 13.12 even 2