Properties

Label 8470.2.a.bc.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 $1$

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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 \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −2.00000 −0.471405
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) −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 \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 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 \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 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 \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 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 \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 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 \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) −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 \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −5.00000 −0.481125
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 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 \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 7.00000 0.613941
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 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 \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −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 \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 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 \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 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 \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 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 \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 8.00000 0.574367
\(195\) 7.00000 0.501280
\(196\) 1.00000 0.0714286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 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 \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −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 \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 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 \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 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 \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 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 \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 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 \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −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 \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 5.00000 0.299880
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 6.00000 0.357295
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) 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 \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 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 \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 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 \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 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 \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 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 \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 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 \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 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 \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) −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 \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) −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 \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 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 \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 26.0000 1.21490
\(459\) 30.0000 1.40028
\(460\) −9.00000 −0.419627
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 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 \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −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 \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 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 \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) −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 \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) −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 \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 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 \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −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 \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −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 \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 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 \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) −5.00000 −0.209427
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 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 \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 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 \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −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 \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 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 \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(614\) −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 \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 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 \(-0.209968\pi\)
0.790217 + 0.612827i \(0.209968\pi\)
\(674\) 23.0000 0.885927
\(675\) −5.00000 −0.192450
\(676\) 36.0000 1.38462
\(677\) 33.0000 1.26829 0.634147 0.773213i \(-0.281352\pi\)
0.634147 + 0.773213i \(0.281352\pi\)
\(678\) −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 \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 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 \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) −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 \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −35.0000 −1.28576
\(742\) −6.00000 −0.220267
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 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 \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 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 \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 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 \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) −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 \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 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 \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −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 \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) −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 \(-0.538237\pi\)
−0.119838 + 0.992793i \(0.538237\pi\)
\(854\) 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 \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) −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 \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 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 \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −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 \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) 2.00000 0.0653023
\(939\) −10.0000 −0.326338
\(940\) −6.00000 −0.195698
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 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 \(-0.679495\pi\)
−0.534487 + 0.845176i \(0.679495\pi\)
\(954\) 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 \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 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 \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) 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.bc.1.1 yes 1
11.10 odd 2 8470.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.l.1.1 1 11.10 odd 2
8470.2.a.bc.1.1 yes 1 1.1 even 1 trivial