Properties

Label 8470.2.a.l.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
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 \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +7.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +9.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -7.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +1.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} +7.00000 q^{39} +1.00000 q^{40} -12.0000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +2.00000 q^{45} -9.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +7.00000 q^{52} -6.00000 q^{53} +5.00000 q^{54} +1.00000 q^{56} -5.00000 q^{57} +9.00000 q^{59} -1.00000 q^{60} -14.0000 q^{61} -2.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -7.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} +9.00000 q^{69} -1.00000 q^{70} +2.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -5.00000 q^{76} -7.00000 q^{78} -11.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -9.00000 q^{83} -1.00000 q^{84} -6.00000 q^{85} -4.00000 q^{86} +12.0000 q^{89} -2.00000 q^{90} -7.00000 q^{91} +9.00000 q^{92} +2.00000 q^{93} -6.00000 q^{94} +5.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 7.00000 1.94145 0.970725 0.240192i \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 2.00000 0.471405
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −7.00000 −1.37281
\(27\) −5.00000 −0.962250
\(28\) −1.00000 −0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 1.00000 0.182574
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 5.00000 0.811107
\(39\) 7.00000 1.12090
\(40\) 1.00000 0.158114
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) −9.00000 −1.32698
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) 7.00000 0.970725
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) −1.00000 −0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −2.00000 −0.254000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −7.00000 −0.868243
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.00000 0.727607
\(69\) 9.00000 1.08347
\(70\) −1.00000 −0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.00000 0.235702
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −5.00000 −0.573539
\(77\) 0 0
\(78\) −7.00000 −0.792594
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) −1.00000 −0.109109
\(85\) −6.00000 −0.650791
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) −2.00000 −0.210819
\(91\) −7.00000 −0.733799
\(92\) 9.00000 0.938315
\(93\) 2.00000 0.207390
\(94\) −6.00000 −0.618853
\(95\) 5.00000 0.512989
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −7.00000 −0.686406
\(105\) 1.00000 0.0975900
\(106\) 6.00000 0.582772
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −5.00000 −0.481125
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 5.00000 0.468293
\(115\) −9.00000 −0.839254
\(116\) 0 0
\(117\) −14.0000 −1.29430
\(118\) −9.00000 −0.828517
\(119\) −6.00000 −0.550019
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) 14.0000 1.26750
\(123\) −12.0000 −1.08200
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 7.00000 0.613941
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 5.00000 0.433555
\(134\) −2.00000 −0.172774
\(135\) 5.00000 0.430331
\(136\) −6.00000 −0.514496
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) −9.00000 −0.766131
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 1.00000 0.0845154
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 5.00000 0.405554
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 7.00000 0.560449
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 11.0000 0.875113
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) −9.00000 −0.709299
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) 36.0000 2.76923
\(170\) 6.00000 0.460179
\(171\) 10.0000 0.764719
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) −12.0000 −0.899438
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 2.00000 0.149071
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 7.00000 0.518875
\(183\) −14.0000 −1.03491
\(184\) −9.00000 −0.663489
\(185\) −2.00000 −0.147043
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 5.00000 0.363696
\(190\) −5.00000 −0.362738
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) 13.0000 0.935760 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) −8.00000 −0.574367
\(195\) −7.00000 −0.501280
\(196\) 1.00000 0.0714286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.00000 0.141069
\(202\) 9.00000 0.633238
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 12.0000 0.838116
\(206\) −8.00000 −0.557386
\(207\) −18.0000 −1.25109
\(208\) 7.00000 0.485363
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) −4.00000 −0.272798
\(216\) 5.00000 0.340207
\(217\) −2.00000 −0.135769
\(218\) 2.00000 0.135457
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 42.0000 2.82523
\(222\) −2.00000 −0.134231
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.00000 −0.133333
\(226\) 15.0000 0.997785
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −5.00000 −0.331133
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 9.00000 0.593442
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 14.0000 0.915209
\(235\) −6.00000 −0.391397
\(236\) 9.00000 0.585850
\(237\) −11.0000 −0.714527
\(238\) 6.00000 0.388922
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) −14.0000 −0.896258
\(245\) −1.00000 −0.0638877
\(246\) 12.0000 0.765092
\(247\) −35.0000 −2.22700
\(248\) −2.00000 −0.127000
\(249\) −9.00000 −0.570352
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 5.00000 0.313728
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −4.00000 −0.249029
\(259\) −2.00000 −0.124274
\(260\) −7.00000 −0.434122
\(261\) 0 0
\(262\) −3.00000 −0.185341
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −5.00000 −0.306570
\(267\) 12.0000 0.734388
\(268\) 2.00000 0.122169
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) −5.00000 −0.304290
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 6.00000 0.363803
\(273\) −7.00000 −0.423659
\(274\) 3.00000 0.181237
\(275\) 0 0
\(276\) 9.00000 0.541736
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 5.00000 0.299880
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) 15.0000 0.894825 0.447412 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) −6.00000 −0.357295
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 0 0
\(285\) 5.00000 0.296174
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 2.00000 0.117851
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 10.0000 0.585206
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −9.00000 −0.524000
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 63.0000 3.64338
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 5.00000 0.287718
\(303\) −9.00000 −0.517036
\(304\) −5.00000 −0.286770
\(305\) 14.0000 0.801638
\(306\) 12.0000 0.685994
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 2.00000 0.113592
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −7.00000 −0.396297
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −17.0000 −0.959366
\(315\) −2.00000 −0.112687
\(316\) −11.0000 −0.618798
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 18.0000 1.00466
\(322\) 9.00000 0.501550
\(323\) −30.0000 −1.66924
\(324\) 1.00000 0.0555556
\(325\) 7.00000 0.388290
\(326\) −2.00000 −0.110770
\(327\) −2.00000 −0.110600
\(328\) 12.0000 0.662589
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −9.00000 −0.493939
\(333\) −4.00000 −0.219199
\(334\) 12.0000 0.656611
\(335\) −2.00000 −0.109272
\(336\) −1.00000 −0.0545545
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) −36.0000 −1.95814
\(339\) −15.0000 −0.814688
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) −10.0000 −0.540738
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) −9.00000 −0.484544
\(346\) 6.00000 0.322562
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 1.00000 0.0534522
\(351\) −35.0000 −1.86816
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −9.00000 −0.478345
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) −6.00000 −0.317554
\(358\) 12.0000 0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −2.00000 −0.105409
\(361\) 6.00000 0.315789
\(362\) 13.0000 0.683265
\(363\) 0 0
\(364\) −7.00000 −0.366900
\(365\) −10.0000 −0.523424
\(366\) 14.0000 0.731792
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 9.00000 0.469157
\(369\) 24.0000 1.24939
\(370\) 2.00000 0.103975
\(371\) 6.00000 0.311504
\(372\) 2.00000 0.103695
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) −5.00000 −0.257172
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 5.00000 0.256495
\(381\) −5.00000 −0.256158
\(382\) −3.00000 −0.153493
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −13.0000 −0.661683
\(387\) −8.00000 −0.406663
\(388\) 8.00000 0.406138
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 7.00000 0.354459
\(391\) 54.0000 2.73090
\(392\) −1.00000 −0.0505076
\(393\) 3.00000 0.151330
\(394\) −12.0000 −0.604551
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 16.0000 0.802008
\(399\) 5.00000 0.250313
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 14.0000 0.697390
\(404\) −9.00000 −0.447767
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) −12.0000 −0.592638
\(411\) −3.00000 −0.147979
\(412\) 8.00000 0.394132
\(413\) −9.00000 −0.442861
\(414\) 18.0000 0.884652
\(415\) 9.00000 0.441793
\(416\) −7.00000 −0.343203
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 1.00000 0.0487950
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −4.00000 −0.194717
\(423\) −12.0000 −0.583460
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 14.0000 0.677507
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) −5.00000 −0.240563
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −45.0000 −2.15264
\(438\) −10.0000 −0.477818
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) −42.0000 −1.99774
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 2.00000 0.0949158
\(445\) −12.0000 −0.568855
\(446\) −2.00000 −0.0947027
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 2.00000 0.0942809
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) −5.00000 −0.234920
\(454\) −24.0000 −1.12638
\(455\) 7.00000 0.328165
\(456\) 5.00000 0.234146
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) −26.0000 −1.21490
\(459\) −30.0000 −1.40028
\(460\) −9.00000 −0.419627
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 3.00000 0.138972
\(467\) 21.0000 0.971764 0.485882 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) −14.0000 −0.647150
\(469\) −2.00000 −0.0923514
\(470\) 6.00000 0.276759
\(471\) 17.0000 0.783319
\(472\) −9.00000 −0.414259
\(473\) 0 0
\(474\) 11.0000 0.505247
\(475\) −5.00000 −0.229416
\(476\) −6.00000 −0.275010
\(477\) 12.0000 0.549442
\(478\) −15.0000 −0.686084
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) 14.0000 0.638345
\(482\) 20.0000 0.910975
\(483\) −9.00000 −0.409514
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) −16.0000 −0.725775
\(487\) −25.0000 −1.13286 −0.566429 0.824110i \(-0.691675\pi\)
−0.566429 + 0.824110i \(0.691675\pi\)
\(488\) 14.0000 0.633750
\(489\) 2.00000 0.0904431
\(490\) 1.00000 0.0451754
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) −12.0000 −0.541002
\(493\) 0 0
\(494\) 35.0000 1.57472
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 9.00000 0.403300
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) −24.0000 −1.07117
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 9.00000 0.400495
\(506\) 0 0
\(507\) 36.0000 1.59882
\(508\) −5.00000 −0.221839
\(509\) −9.00000 −0.398918 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) 6.00000 0.265684
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 25.0000 1.10378
\(514\) −18.0000 −0.793946
\(515\) −8.00000 −0.352522
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) −6.00000 −0.263371
\(520\) 7.00000 0.306970
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) 7.00000 0.306089 0.153044 0.988219i \(-0.451092\pi\)
0.153044 + 0.988219i \(0.451092\pi\)
\(524\) 3.00000 0.131056
\(525\) −1.00000 −0.0436436
\(526\) −21.0000 −0.915644
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) −6.00000 −0.260623
\(531\) −18.0000 −0.781133
\(532\) 5.00000 0.216777
\(533\) −84.0000 −3.63844
\(534\) −12.0000 −0.519291
\(535\) −18.0000 −0.778208
\(536\) −2.00000 −0.0863868
\(537\) −12.0000 −0.517838
\(538\) −15.0000 −0.646696
\(539\) 0 0
\(540\) 5.00000 0.215166
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 20.0000 0.859074
\(543\) −13.0000 −0.557883
\(544\) −6.00000 −0.257248
\(545\) 2.00000 0.0856706
\(546\) 7.00000 0.299572
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −3.00000 −0.128154
\(549\) 28.0000 1.19501
\(550\) 0 0
\(551\) 0 0
\(552\) −9.00000 −0.383065
\(553\) 11.0000 0.467768
\(554\) −28.0000 −1.18961
\(555\) −2.00000 −0.0848953
\(556\) −5.00000 −0.212047
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 4.00000 0.169334
\(559\) 28.0000 1.18427
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −15.0000 −0.632737
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 6.00000 0.252646
\(565\) 15.0000 0.631055
\(566\) 5.00000 0.210166
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) −5.00000 −0.209427
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) −12.0000 −0.500870
\(575\) 9.00000 0.375326
\(576\) −2.00000 −0.0833333
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −19.0000 −0.790296
\(579\) 13.0000 0.540262
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) −8.00000 −0.331611
\(583\) 0 0
\(584\) −10.0000 −0.413803
\(585\) 14.0000 0.578829
\(586\) −9.00000 −0.371787
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 1.00000 0.0412393
\(589\) −10.0000 −0.412043
\(590\) 9.00000 0.370524
\(591\) 12.0000 0.493614
\(592\) 2.00000 0.0821995
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 6.00000 0.245770
\(597\) −16.0000 −0.654836
\(598\) −63.0000 −2.57626
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) −5.00000 −0.203447
\(605\) 0 0
\(606\) 9.00000 0.365600
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) 42.0000 1.69914
\(612\) −12.0000 −0.485071
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) −28.0000 −1.12999
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) −8.00000 −0.321807
\(619\) −25.0000 −1.00483 −0.502417 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −45.0000 −1.80579
\(622\) 12.0000 0.481156
\(623\) −12.0000 −0.480770
\(624\) 7.00000 0.280224
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 17.0000 0.678374
\(629\) 12.0000 0.478471
\(630\) 2.00000 0.0796819
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 11.0000 0.437557
\(633\) 4.00000 0.158986
\(634\) −18.0000 −0.714871
\(635\) 5.00000 0.198419
\(636\) −6.00000 −0.237915
\(637\) 7.00000 0.277350
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 27.0000 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(642\) −18.0000 −0.710403
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) −9.00000 −0.354650
\(645\) −4.00000 −0.157500
\(646\) 30.0000 1.18033
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −7.00000 −0.274563
\(651\) −2.00000 −0.0783862
\(652\) 2.00000 0.0783260
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 2.00000 0.0782062
\(655\) −3.00000 −0.117220
\(656\) −12.0000 −0.468521
\(657\) −20.0000 −0.780274
\(658\) 6.00000 0.233904
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) 4.00000 0.155464
\(663\) 42.0000 1.63114
\(664\) 9.00000 0.349268
\(665\) −5.00000 −0.193892
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) 2.00000 0.0773245
\(670\) 2.00000 0.0772667
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −41.0000 −1.58043 −0.790217 0.612827i \(-0.790032\pi\)
−0.790217 + 0.612827i \(0.790032\pi\)
\(674\) 23.0000 0.885927
\(675\) −5.00000 −0.192450
\(676\) 36.0000 1.38462
\(677\) −33.0000 −1.26829 −0.634147 0.773213i \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) 15.0000 0.576072
\(679\) −8.00000 −0.307012
\(680\) 6.00000 0.230089
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 10.0000 0.382360
\(685\) 3.00000 0.114624
\(686\) 1.00000 0.0381802
\(687\) 26.0000 0.991962
\(688\) 4.00000 0.152499
\(689\) −42.0000 −1.60007
\(690\) 9.00000 0.342624
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) 5.00000 0.189661
\(696\) 0 0
\(697\) −72.0000 −2.72719
\(698\) −25.0000 −0.946264
\(699\) −3.00000 −0.113470
\(700\) −1.00000 −0.0377964
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 35.0000 1.32099
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) −18.0000 −0.677439
\(707\) 9.00000 0.338480
\(708\) 9.00000 0.338241
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 22.0000 0.825064
\(712\) −12.0000 −0.449719
\(713\) 18.0000 0.674105
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 15.0000 0.560185
\(718\) −24.0000 −0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 2.00000 0.0745356
\(721\) −8.00000 −0.297936
\(722\) −6.00000 −0.223297
\(723\) −20.0000 −0.743808
\(724\) −13.0000 −0.483141
\(725\) 0 0
\(726\) 0 0
\(727\) 20.0000 0.741759 0.370879 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 7.00000 0.259437
\(729\) 13.0000 0.481481
\(730\) 10.0000 0.370117
\(731\) 24.0000 0.887672
\(732\) −14.0000 −0.517455
\(733\) 49.0000 1.80986 0.904928 0.425564i \(-0.139924\pi\)
0.904928 + 0.425564i \(0.139924\pi\)
\(734\) 28.0000 1.03350
\(735\) −1.00000 −0.0368856
\(736\) −9.00000 −0.331744
\(737\) 0 0
\(738\) −24.0000 −0.883452
\(739\) −38.0000 −1.39785 −0.698926 0.715194i \(-0.746338\pi\)
−0.698926 + 0.715194i \(0.746338\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −35.0000 −1.28576
\(742\) −6.00000 −0.220267
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) −2.00000 −0.0733236
\(745\) −6.00000 −0.219823
\(746\) −22.0000 −0.805477
\(747\) 18.0000 0.658586
\(748\) 0 0
\(749\) −18.0000 −0.657706
\(750\) 1.00000 0.0365148
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 6.00000 0.218797
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 5.00000 0.181848
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) −5.00000 −0.181369
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 5.00000 0.181131
\(763\) 2.00000 0.0724049
\(764\) 3.00000 0.108536
\(765\) 12.0000 0.433861
\(766\) −24.0000 −0.867155
\(767\) 63.0000 2.27480
\(768\) 1.00000 0.0360844
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 13.0000 0.467880
\(773\) 27.0000 0.971123 0.485561 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(774\) 8.00000 0.287554
\(775\) 2.00000 0.0718421
\(776\) −8.00000 −0.287183
\(777\) −2.00000 −0.0717496
\(778\) −18.0000 −0.645331
\(779\) 60.0000 2.14972
\(780\) −7.00000 −0.250640
\(781\) 0 0
\(782\) −54.0000 −1.93104
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −17.0000 −0.606756
\(786\) −3.00000 −0.107006
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 12.0000 0.427482
\(789\) 21.0000 0.747620
\(790\) −11.0000 −0.391362
\(791\) 15.0000 0.533339
\(792\) 0 0
\(793\) −98.0000 −3.48008
\(794\) 34.0000 1.20661
\(795\) 6.00000 0.212798
\(796\) −16.0000 −0.567105
\(797\) −27.0000 −0.956389 −0.478195 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) −5.00000 −0.176998
\(799\) 36.0000 1.27359
\(800\) −1.00000 −0.0353553
\(801\) −24.0000 −0.847998
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 9.00000 0.317208
\(806\) −14.0000 −0.493129
\(807\) 15.0000 0.528025
\(808\) 9.00000 0.316619
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 1.00000 0.0351364
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 6.00000 0.210042
\(817\) −20.0000 −0.699711
\(818\) −4.00000 −0.139857
\(819\) 14.0000 0.489200
\(820\) 12.0000 0.419058
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 3.00000 0.104637
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 9.00000 0.313150
\(827\) 18.0000 0.625921 0.312961 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) −18.0000 −0.625543
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) −9.00000 −0.312395
\(831\) 28.0000 0.971309
\(832\) 7.00000 0.242681
\(833\) 6.00000 0.207888
\(834\) 5.00000 0.173136
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 3.00000 0.103633
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −29.0000 −1.00000
\(842\) −8.00000 −0.275698
\(843\) 15.0000 0.516627
\(844\) 4.00000 0.137686
\(845\) −36.0000 −1.23844
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −5.00000 −0.171600
\(850\) −6.00000 −0.205798
\(851\) 18.0000 0.617032
\(852\) 0 0
\(853\) 7.00000 0.239675 0.119838 0.992793i \(-0.461763\pi\)
0.119838 + 0.992793i \(0.461763\pi\)
\(854\) −14.0000 −0.479070
\(855\) −10.0000 −0.341993
\(856\) −18.0000 −0.615227
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −4.00000 −0.136399
\(861\) 12.0000 0.408959
\(862\) 33.0000 1.12398
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 5.00000 0.170103
\(865\) 6.00000 0.204006
\(866\) −8.00000 −0.271851
\(867\) 19.0000 0.645274
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 2.00000 0.0677285
\(873\) −16.0000 −0.541518
\(874\) 45.0000 1.52215
\(875\) 1.00000 0.0338062
\(876\) 10.0000 0.337869
\(877\) 52.0000 1.75592 0.877958 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) −28.0000 −0.944954
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 2.00000 0.0673435
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 42.0000 1.41261
\(885\) −9.00000 −0.302532
\(886\) 24.0000 0.806296
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 5.00000 0.167695
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) −30.0000 −1.00391
\(894\) −6.00000 −0.200670
\(895\) 12.0000 0.401116
\(896\) 1.00000 0.0334077
\(897\) 63.0000 2.10351
\(898\) 27.0000 0.901002
\(899\) 0 0
\(900\) −2.00000 −0.0666667
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 15.0000 0.498893
\(905\) 13.0000 0.432135
\(906\) 5.00000 0.166114
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 24.0000 0.796468
\(909\) 18.0000 0.597022
\(910\) −7.00000 −0.232048
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) −5.00000 −0.165567
\(913\) 0 0
\(914\) −1.00000 −0.0330771
\(915\) 14.0000 0.462826
\(916\) 26.0000 0.859064
\(917\) −3.00000 −0.0990687
\(918\) 30.0000 0.990148
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 9.00000 0.296721
\(921\) 28.0000 0.922631
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −5.00000 −0.164310
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 2.00000 0.0655826
\(931\) −5.00000 −0.163868
\(932\) −3.00000 −0.0982683
\(933\) −12.0000 −0.392862
\(934\) −21.0000 −0.687141
\(935\) 0 0
\(936\) 14.0000 0.457604
\(937\) −20.0000 −0.653372 −0.326686 0.945133i \(-0.605932\pi\)
−0.326686 + 0.945133i \(0.605932\pi\)
\(938\) 2.00000 0.0653023
\(939\) −10.0000 −0.326338
\(940\) −6.00000 −0.195698
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) −17.0000 −0.553890
\(943\) −108.000 −3.51696
\(944\) 9.00000 0.292925
\(945\) −5.00000 −0.162650
\(946\) 0 0
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) −11.0000 −0.357263
\(949\) 70.0000 2.27230
\(950\) 5.00000 0.162221
\(951\) 18.0000 0.583690
\(952\) 6.00000 0.194461
\(953\) 33.0000 1.06897 0.534487 0.845176i \(-0.320505\pi\)
0.534487 + 0.845176i \(0.320505\pi\)
\(954\) −12.0000 −0.388514
\(955\) −3.00000 −0.0970777
\(956\) 15.0000 0.485135
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 3.00000 0.0968751
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) −14.0000 −0.451378
\(963\) −36.0000 −1.16008
\(964\) −20.0000 −0.644157
\(965\) −13.0000 −0.418485
\(966\) 9.00000 0.289570
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) −30.0000 −0.963739
\(970\) 8.00000 0.256865
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 16.0000 0.513200
\(973\) 5.00000 0.160293
\(974\) 25.0000 0.801052
\(975\) 7.00000 0.224179
\(976\) −14.0000 −0.448129
\(977\) 45.0000 1.43968 0.719839 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 4.00000 0.127710
\(982\) 6.00000 0.191468
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 12.0000 0.382546
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) −35.0000 −1.11350
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) 23.0000 0.730619 0.365310 0.930886i \(-0.380963\pi\)
0.365310 + 0.930886i \(0.380963\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) −9.00000 −0.285176
\(997\) −35.0000 −1.10846 −0.554231 0.832363i \(-0.686987\pi\)
−0.554231 + 0.832363i \(0.686987\pi\)
\(998\) −14.0000 −0.443162
\(999\) −10.0000 −0.316386
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 8470.2.a.l.1.1 1
11.10 odd 2 8470.2.a.bc.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.l.1.1 1 1.1 even 1 trivial
8470.2.a.bc.1.1 yes 1 11.10 odd 2