Properties

Label 8470.2.a.k.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -5.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} -5.00000 q^{39} +1.00000 q^{40} +1.00000 q^{42} -8.00000 q^{43} +2.00000 q^{45} +3.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -5.00000 q^{52} +6.00000 q^{53} +5.00000 q^{54} +1.00000 q^{56} -5.00000 q^{57} +9.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} +14.0000 q^{67} -6.00000 q^{68} -3.00000 q^{69} -1.00000 q^{70} +2.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -5.00000 q^{76} +5.00000 q^{78} +1.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{83} -1.00000 q^{84} +6.00000 q^{85} +8.00000 q^{86} -2.00000 q^{90} +5.00000 q^{91} -3.00000 q^{92} -10.0000 q^{93} +6.00000 q^{94} +5.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 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 \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 5.00000 0.980581
\(27\) −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
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −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
\(39\) −5.00000 −0.800641
\(40\) 1.00000 0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 0.154303
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 3.00000 0.442326
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) −5.00000 −0.693375
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 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
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 10.0000 1.27000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −6.00000 −0.727607
\(69\) −3.00000 −0.361158
\(70\) −1.00000 −0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.00000 0.235702
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −5.00000 −0.573539
\(77\) 0 0
\(78\) 5.00000 0.566139
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −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
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −2.00000 −0.210819
\(91\) 5.00000 0.524142
\(92\) −3.00000 −0.312772
\(93\) −10.0000 −1.03695
\(94\) 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
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 6.00000 0.594089
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 5.00000 0.490290
\(105\) 1.00000 0.0975900
\(106\) −6.00000 −0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −5.00000 −0.481125
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 5.00000 0.468293
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 10.0000 0.924500
\(118\) −9.00000 −0.828517
\(119\) 6.00000 0.550019
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) −5.00000 −0.438529
\(131\) −21.0000 −1.83478 −0.917389 0.397991i \(-0.869707\pi\)
−0.917389 + 0.397991i \(0.869707\pi\)
\(132\) 0 0
\(133\) 5.00000 0.433555
\(134\) −14.0000 −1.20942
\(135\) 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 \(0.540905\pi\)
\(138\) 3.00000 0.255377
\(139\) 19.0000 1.61156 0.805779 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 1.00000 0.0845154
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 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
\(151\) 7.00000 0.569652 0.284826 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 5.00000 0.405554
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 10.0000 0.803219
\(156\) −5.00000 −0.400320
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) 3.00000 0.236433
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) 12.0000 0.923077
\(170\) −6.00000 −0.460179
\(171\) 10.0000 0.764719
\(172\) −8.00000 −0.609994
\(173\) −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
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 2.00000 0.149071
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) −5.00000 −0.370625
\(183\) 10.0000 0.739221
\(184\) 3.00000 0.221163
\(185\) −2.00000 −0.147043
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 5.00000 0.363696
\(190\) −5.00000 −0.362738
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 1.00000 0.0721688
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) −8.00000 −0.574367
\(195\) 5.00000 0.358057
\(196\) 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
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 14.0000 0.987484
\(202\) −3.00000 −0.211079
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 6.00000 0.417029
\(208\) −5.00000 −0.346688
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 8.00000 0.545595
\(216\) 5.00000 0.340207
\(217\) 10.0000 0.678844
\(218\) −10.0000 −0.677285
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) −2.00000 −0.134231
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.00000 −0.133333
\(226\) −9.00000 −0.598671
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) −5.00000 −0.331133
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) −10.0000 −0.653720
\(235\) 6.00000 0.391397
\(236\) 9.00000 0.585850
\(237\) 1.00000 0.0649570
\(238\) −6.00000 −0.388922
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 25.0000 1.59071
\(248\) 10.0000 0.635001
\(249\) −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
\(254\) −7.00000 −0.439219
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 8.00000 0.498058
\(259\) −2.00000 −0.124274
\(260\) 5.00000 0.310087
\(261\) 0 0
\(262\) 21.0000 1.29738
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −5.00000 −0.306570
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) −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
\(273\) 5.00000 0.302614
\(274\) 3.00000 0.181237
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) −32.0000 −1.92269 −0.961347 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) −19.0000 −1.13954
\(279\) 20.0000 1.19737
\(280\) −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
\(283\) −29.0000 −1.72387 −0.861936 0.507018i \(-0.830748\pi\)
−0.861936 + 0.507018i \(0.830748\pi\)
\(284\) 0 0
\(285\) 5.00000 0.296174
\(286\) 0 0
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −2.00000 −0.117041
\(293\) −27.0000 −1.57736 −0.788678 0.614806i \(-0.789234\pi\)
−0.788678 + 0.614806i \(0.789234\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −9.00000 −0.524000
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 15.0000 0.867472
\(300\) 1.00000 0.0577350
\(301\) 8.00000 0.461112
\(302\) −7.00000 −0.402805
\(303\) 3.00000 0.172345
\(304\) −5.00000 −0.286770
\(305\) −10.0000 −0.572598
\(306\) −12.0000 −0.685994
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) −10.0000 −0.567962
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 5.00000 0.283069
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −5.00000 −0.282166
\(315\) −2.00000 −0.112687
\(316\) 1.00000 0.0562544
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −6.00000 −0.334887
\(322\) −3.00000 −0.167183
\(323\) 30.0000 1.66924
\(324\) 1.00000 0.0555556
\(325\) −5.00000 −0.277350
\(326\) −2.00000 −0.110770
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −9.00000 −0.493939
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −14.0000 −0.764902
\(336\) −1.00000 −0.0545545
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) −12.0000 −0.652714
\(339\) 9.00000 0.488813
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) −10.0000 −0.540738
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) 3.00000 0.161515
\(346\) 6.00000 0.322562
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) −11.0000 −0.588817 −0.294408 0.955680i \(-0.595123\pi\)
−0.294408 + 0.955680i \(0.595123\pi\)
\(350\) 1.00000 0.0534522
\(351\) 25.0000 1.33440
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −9.00000 −0.478345
\(355\) 0 0
\(356\) 0 0
\(357\) 6.00000 0.317554
\(358\) 0 0
\(359\) 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
\(362\) 25.0000 1.31397
\(363\) 0 0
\(364\) 5.00000 0.262071
\(365\) 2.00000 0.104685
\(366\) −10.0000 −0.522708
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 2.00000 0.103975
\(371\) −6.00000 −0.311504
\(372\) −10.0000 −0.518476
\(373\) 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
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 5.00000 0.256495
\(381\) 7.00000 0.358621
\(382\) −15.0000 −0.767467
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 11.0000 0.559885
\(387\) 16.0000 0.813326
\(388\) 8.00000 0.406138
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −5.00000 −0.253185
\(391\) 18.0000 0.910299
\(392\) −1.00000 −0.0505076
\(393\) −21.0000 −1.05931
\(394\) 12.0000 0.604551
\(395\) −1.00000 −0.0503155
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −8.00000 −0.401004
\(399\) 5.00000 0.250313
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −14.0000 −0.698257
\(403\) 50.0000 2.49068
\(404\) 3.00000 0.149256
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) −4.00000 −0.197066
\(413\) −9.00000 −0.442861
\(414\) −6.00000 −0.294884
\(415\) 9.00000 0.441793
\(416\) 5.00000 0.245145
\(417\) 19.0000 0.930434
\(418\) 0 0
\(419\) 21.0000 1.02592 0.512959 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 1.00000 0.0487950
\(421\) −40.0000 −1.94948 −0.974740 0.223341i \(-0.928304\pi\)
−0.974740 + 0.223341i \(0.928304\pi\)
\(422\) 20.0000 0.973585
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) −5.00000 −0.240563
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 15.0000 0.717547
\(438\) 2.00000 0.0955637
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) −30.0000 −1.42695
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 10.0000 0.473514
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) −3.00000 −0.141579 −0.0707894 0.997491i \(-0.522552\pi\)
−0.0707894 + 0.997491i \(0.522552\pi\)
\(450\) 2.00000 0.0942809
\(451\) 0 0
\(452\) 9.00000 0.423324
\(453\) 7.00000 0.328889
\(454\) 24.0000 1.12638
\(455\) −5.00000 −0.234404
\(456\) 5.00000 0.234146
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 22.0000 1.02799
\(459\) 30.0000 1.40028
\(460\) 3.00000 0.139876
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −7.00000 −0.325318 −0.162659 0.986682i \(-0.552007\pi\)
−0.162659 + 0.986682i \(0.552007\pi\)
\(464\) 0 0
\(465\) 10.0000 0.463739
\(466\) −21.0000 −0.972806
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 10.0000 0.462250
\(469\) −14.0000 −0.646460
\(470\) −6.00000 −0.276759
\(471\) 5.00000 0.230388
\(472\) −9.00000 −0.414259
\(473\) 0 0
\(474\) −1.00000 −0.0459315
\(475\) −5.00000 −0.229416
\(476\) 6.00000 0.275010
\(477\) −12.0000 −0.549442
\(478\) −3.00000 −0.137217
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 1.00000 0.0456435
\(481\) −10.0000 −0.455961
\(482\) −4.00000 −0.182195
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) −16.0000 −0.725775
\(487\) −13.0000 −0.589086 −0.294543 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) −10.0000 −0.452679
\(489\) 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 \(0.456772\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −25.0000 −1.12480
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) 9.00000 0.403300
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 7.00000 0.310575
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) −6.00000 −0.265684
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 25.0000 1.10378
\(514\) −6.00000 −0.264649
\(515\) 4.00000 0.176261
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) −6.00000 −0.263371
\(520\) −5.00000 −0.219265
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 0 0
\(523\) −17.0000 −0.743358 −0.371679 0.928361i \(-0.621218\pi\)
−0.371679 + 0.928361i \(0.621218\pi\)
\(524\) −21.0000 −0.917389
\(525\) −1.00000 −0.0436436
\(526\) −9.00000 −0.392419
\(527\) 60.0000 2.61364
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 6.00000 0.260623
\(531\) −18.0000 −0.781133
\(532\) 5.00000 0.216777
\(533\) 0 0
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 21.0000 0.905374
\(539\) 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
\(543\) −25.0000 −1.07285
\(544\) 6.00000 0.257248
\(545\) −10.0000 −0.428353
\(546\) −5.00000 −0.213980
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −3.00000 −0.128154
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 0 0
\(552\) 3.00000 0.127688
\(553\) −1.00000 −0.0425243
\(554\) 32.0000 1.35955
\(555\) −2.00000 −0.0848953
\(556\) 19.0000 0.805779
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −20.0000 −0.846668
\(559\) 40.0000 1.69182
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −15.0000 −0.632737
\(563\) −9.00000 −0.379305 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) −6.00000 −0.252646
\(565\) −9.00000 −0.378633
\(566\) 29.0000 1.21896
\(567\) −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
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) −2.00000 −0.0833333
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) −19.0000 −0.790296
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) −8.00000 −0.331611
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) −10.0000 −0.413449
\(586\) 27.0000 1.11536
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 1.00000 0.0412393
\(589\) 50.0000 2.06021
\(590\) 9.00000 0.370524
\(591\) −12.0000 −0.493614
\(592\) 2.00000 0.0821995
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 6.00000 0.245770
\(597\) 8.00000 0.327418
\(598\) −15.0000 −0.613396
\(599\) 3.00000 0.122577 0.0612883 0.998120i \(-0.480479\pi\)
0.0612883 + 0.998120i \(0.480479\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) −8.00000 −0.326056
\(603\) −28.0000 −1.14025
\(604\) 7.00000 0.284826
\(605\) 0 0
\(606\) −3.00000 −0.121867
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 30.0000 1.21367
\(612\) 12.0000 0.485071
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 4.00000 0.160904
\(619\) 23.0000 0.924448 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 10.0000 0.401610
\(621\) 15.0000 0.601929
\(622\) 0 0
\(623\) 0 0
\(624\) −5.00000 −0.200160
\(625\) 1.00000 0.0400000
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) 5.00000 0.199522
\(629\) −12.0000 −0.478471
\(630\) 2.00000 0.0796819
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −20.0000 −0.794929
\(634\) −30.0000 −1.19145
\(635\) −7.00000 −0.277787
\(636\) 6.00000 0.237915
\(637\) −5.00000 −0.198107
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −21.0000 −0.829450 −0.414725 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 6.00000 0.236801
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 3.00000 0.118217
\(645\) 8.00000 0.315000
\(646\) −30.0000 −1.18033
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 5.00000 0.196116
\(651\) 10.0000 0.391931
\(652\) 2.00000 0.0783260
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) −10.0000 −0.391031
\(655\) 21.0000 0.820538
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) −6.00000 −0.233904
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) −8.00000 −0.310929
\(663\) 30.0000 1.16510
\(664\) 9.00000 0.349268
\(665\) −5.00000 −0.193892
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) −10.0000 −0.386622
\(670\) 14.0000 0.540867
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −17.0000 −0.655302 −0.327651 0.944799i \(-0.606257\pi\)
−0.327651 + 0.944799i \(0.606257\pi\)
\(674\) 23.0000 0.885927
\(675\) −5.00000 −0.192450
\(676\) 12.0000 0.461538
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) −9.00000 −0.345643
\(679\) −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
\(687\) −22.0000 −0.839352
\(688\) −8.00000 −0.304997
\(689\) −30.0000 −1.14291
\(690\) −3.00000 −0.114208
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) −19.0000 −0.720711
\(696\) 0 0
\(697\) 0 0
\(698\) 11.0000 0.416356
\(699\) 21.0000 0.794293
\(700\) −1.00000 −0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −25.0000 −0.943564
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) 6.00000 0.225973
\(706\) −30.0000 −1.12906
\(707\) −3.00000 −0.112827
\(708\) 9.00000 0.338241
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) 0 0
\(713\) 30.0000 1.12351
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) 3.00000 0.112037
\(718\) −24.0000 −0.895672
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 2.00000 0.0745356
\(721\) 4.00000 0.148968
\(722\) −6.00000 −0.223297
\(723\) 4.00000 0.148762
\(724\) −25.0000 −0.929118
\(725\) 0 0
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −5.00000 −0.185312
\(729\) 13.0000 0.481481
\(730\) −2.00000 −0.0740233
\(731\) 48.0000 1.77534
\(732\) 10.0000 0.369611
\(733\) 13.0000 0.480166 0.240083 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(734\) 16.0000 0.590571
\(735\) −1.00000 −0.0368856
\(736\) 3.00000 0.110581
\(737\) 0 0
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 25.0000 0.918398
\(742\) 6.00000 0.220267
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 10.0000 0.366618
\(745\) −6.00000 −0.219823
\(746\) −22.0000 −0.805477
\(747\) 18.0000 0.658586
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 1.00000 0.0365148
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) −6.00000 −0.218797
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) −7.00000 −0.254756
\(756\) 5.00000 0.181848
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 34.0000 1.23494
\(759\) 0 0
\(760\) −5.00000 −0.181369
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −7.00000 −0.253583
\(763\) −10.0000 −0.362024
\(764\) 15.0000 0.542681
\(765\) −12.0000 −0.433861
\(766\) −12.0000 −0.433578
\(767\) −45.0000 −1.62486
\(768\) 1.00000 0.0360844
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −11.0000 −0.395899
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) −16.0000 −0.575108
\(775\) −10.0000 −0.359211
\(776\) −8.00000 −0.287183
\(777\) −2.00000 −0.0717496
\(778\) −30.0000 −1.07555
\(779\) 0 0
\(780\) 5.00000 0.179029
\(781\) 0 0
\(782\) −18.0000 −0.643679
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −5.00000 −0.178458
\(786\) 21.0000 0.749045
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −12.0000 −0.427482
\(789\) 9.00000 0.320408
\(790\) 1.00000 0.0355784
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) −50.0000 −1.77555
\(794\) −38.0000 −1.34857
\(795\) −6.00000 −0.212798
\(796\) 8.00000 0.283552
\(797\) 33.0000 1.16892 0.584460 0.811423i \(-0.301306\pi\)
0.584460 + 0.811423i \(0.301306\pi\)
\(798\) −5.00000 −0.176998
\(799\) 36.0000 1.27359
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) 14.0000 0.493742
\(805\) −3.00000 −0.105736
\(806\) −50.0000 −1.76117
\(807\) −21.0000 −0.739235
\(808\) −3.00000 −0.105540
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 1.00000 0.0351364
\(811\) −32.0000 −1.12367 −0.561836 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) −6.00000 −0.210042
\(817\) 40.0000 1.39942
\(818\) −28.0000 −0.978997
\(819\) −10.0000 −0.349428
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 3.00000 0.104637
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 9.00000 0.313150
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 6.00000 0.208514
\(829\) 17.0000 0.590434 0.295217 0.955430i \(-0.404608\pi\)
0.295217 + 0.955430i \(0.404608\pi\)
\(830\) −9.00000 −0.312395
\(831\) −32.0000 −1.11007
\(832\) −5.00000 −0.173344
\(833\) −6.00000 −0.207888
\(834\) −19.0000 −0.657916
\(835\) 0 0
\(836\) 0 0
\(837\) 50.0000 1.72825
\(838\) −21.0000 −0.725433
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −29.0000 −1.00000
\(842\) 40.0000 1.37849
\(843\) 15.0000 0.516627
\(844\) −20.0000 −0.688428
\(845\) −12.0000 −0.412813
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −29.0000 −0.995277
\(850\) 6.00000 0.205798
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 10.0000 0.342193
\(855\) −10.0000 −0.341993
\(856\) 6.00000 0.205076
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −3.00000 −0.102180
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 5.00000 0.170103
\(865\) 6.00000 0.204006
\(866\) 4.00000 0.135926
\(867\) 19.0000 0.645274
\(868\) 10.0000 0.339422
\(869\) 0 0
\(870\) 0 0
\(871\) −70.0000 −2.37186
\(872\) −10.0000 −0.338643
\(873\) −16.0000 −0.541518
\(874\) −15.0000 −0.507383
\(875\) 1.00000 0.0338062
\(876\) −2.00000 −0.0675737
\(877\) 40.0000 1.35070 0.675352 0.737496i \(-0.263992\pi\)
0.675352 + 0.737496i \(0.263992\pi\)
\(878\) −16.0000 −0.539974
\(879\) −27.0000 −0.910687
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 2.00000 0.0673435
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 30.0000 1.00901
\(885\) −9.00000 −0.302532
\(886\) 24.0000 0.806296
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −7.00000 −0.234772
\(890\) 0 0
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) 30.0000 1.00391
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 15.0000 0.500835
\(898\) 3.00000 0.100111
\(899\) 0 0
\(900\) −2.00000 −0.0666667
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) −9.00000 −0.299336
\(905\) 25.0000 0.831028
\(906\) −7.00000 −0.232559
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −24.0000 −0.796468
\(909\) −6.00000 −0.199007
\(910\) 5.00000 0.165748
\(911\) 45.0000 1.49092 0.745458 0.666552i \(-0.232231\pi\)
0.745458 + 0.666552i \(0.232231\pi\)
\(912\) −5.00000 −0.165567
\(913\) 0 0
\(914\) −1.00000 −0.0330771
\(915\) −10.0000 −0.330590
\(916\) −22.0000 −0.726900
\(917\) 21.0000 0.693481
\(918\) −30.0000 −0.990148
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 4.00000 0.131804
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 7.00000 0.230034
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) −10.0000 −0.327913
\(931\) −5.00000 −0.163868
\(932\) 21.0000 0.687878
\(933\) 0 0
\(934\) 3.00000 0.0981630
\(935\) 0 0
\(936\) −10.0000 −0.326860
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) 14.0000 0.457116
\(939\) −34.0000 −1.10955
\(940\) 6.00000 0.195698
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) −5.00000 −0.162909
\(943\) 0 0
\(944\) 9.00000 0.292925
\(945\) −5.00000 −0.162650
\(946\) 0 0
\(947\) 6.00000 0.194974 0.0974869 0.995237i \(-0.468920\pi\)
0.0974869 + 0.995237i \(0.468920\pi\)
\(948\) 1.00000 0.0324785
\(949\) 10.0000 0.324614
\(950\) 5.00000 0.162221
\(951\) 30.0000 0.972817
\(952\) −6.00000 −0.194461
\(953\) 57.0000 1.84641 0.923206 0.384307i \(-0.125559\pi\)
0.923206 + 0.384307i \(0.125559\pi\)
\(954\) 12.0000 0.388514
\(955\) −15.0000 −0.485389
\(956\) 3.00000 0.0970269
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) 3.00000 0.0968751
\(960\) −1.00000 −0.0322749
\(961\) 69.0000 2.22581
\(962\) 10.0000 0.322413
\(963\) 12.0000 0.386695
\(964\) 4.00000 0.128831
\(965\) 11.0000 0.354103
\(966\) −3.00000 −0.0965234
\(967\) −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
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 16.0000 0.513200
\(973\) −19.0000 −0.609112
\(974\) 13.0000 0.416547
\(975\) −5.00000 −0.160128
\(976\) 10.0000 0.320092
\(977\) −27.0000 −0.863807 −0.431903 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −20.0000 −0.638551
\(982\) −6.00000 −0.191468
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 25.0000 0.795356
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 35.0000 1.11181 0.555906 0.831245i \(-0.312372\pi\)
0.555906 + 0.831245i \(0.312372\pi\)
\(992\) 10.0000 0.317500
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) −9.00000 −0.285176
\(997\) 25.0000 0.791758 0.395879 0.918303i \(-0.370440\pi\)
0.395879 + 0.918303i \(0.370440\pi\)
\(998\) 34.0000 1.07625
\(999\) −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.k.1.1 1
11.10 odd 2 8470.2.a.bd.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.k.1.1 1 1.1 even 1 trivial
8470.2.a.bd.1.1 yes 1 11.10 odd 2