Properties

Label 8470.2.a.ck.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.77846 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.77846 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.71982 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.77846 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.77846 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.71982 q^{9} -1.00000 q^{10} -2.77846 q^{12} -2.77846 q^{13} +1.00000 q^{14} +2.77846 q^{15} +1.00000 q^{16} +6.71982 q^{17} +4.71982 q^{18} +7.49828 q^{19} -1.00000 q^{20} -2.77846 q^{21} -3.83709 q^{23} -2.77846 q^{24} +1.00000 q^{25} -2.77846 q^{26} -4.77846 q^{27} +1.00000 q^{28} -5.43965 q^{29} +2.77846 q^{30} +1.00000 q^{32} +6.71982 q^{34} -1.00000 q^{35} +4.71982 q^{36} -0.117266 q^{37} +7.49828 q^{38} +7.71982 q^{39} -1.00000 q^{40} +5.55691 q^{41} -2.77846 q^{42} +10.3940 q^{43} -4.71982 q^{45} -3.83709 q^{46} +2.11727 q^{47} -2.77846 q^{48} +1.00000 q^{49} +1.00000 q^{50} -18.6707 q^{51} -2.77846 q^{52} -2.71982 q^{53} -4.77846 q^{54} +1.00000 q^{56} -20.8337 q^{57} -5.43965 q^{58} -3.49828 q^{59} +2.77846 q^{60} -4.27674 q^{61} +4.71982 q^{63} +1.00000 q^{64} +2.77846 q^{65} -12.1595 q^{67} +6.71982 q^{68} +10.6612 q^{69} -1.00000 q^{70} +3.16291 q^{71} +4.71982 q^{72} +8.83709 q^{73} -0.117266 q^{74} -2.77846 q^{75} +7.49828 q^{76} +7.71982 q^{78} -8.55691 q^{79} -1.00000 q^{80} -0.882734 q^{81} +5.55691 q^{82} -0.778457 q^{83} -2.77846 q^{84} -6.71982 q^{85} +10.3940 q^{86} +15.1138 q^{87} +9.55691 q^{89} -4.71982 q^{90} -2.77846 q^{91} -3.83709 q^{92} +2.11727 q^{94} -7.49828 q^{95} -2.77846 q^{96} -4.83709 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} + 3q^{8} + 5q^{9} - 3q^{10} + 3q^{14} + 3q^{16} + 11q^{17} + 5q^{18} + 5q^{19} - 3q^{20} - 4q^{23} + 3q^{25} - 6q^{27} + 3q^{28} + 2q^{29} + 3q^{32} + 11q^{34} - 3q^{35} + 5q^{36} - 2q^{37} + 5q^{38} + 14q^{39} - 3q^{40} + 7q^{43} - 5q^{45} - 4q^{46} + 8q^{47} + 3q^{49} + 3q^{50} - 6q^{51} + q^{53} - 6q^{54} + 3q^{56} - 20q^{57} + 2q^{58} + 7q^{59} + 13q^{61} + 5q^{63} + 3q^{64} - 9q^{67} + 11q^{68} + 22q^{69} - 3q^{70} + 17q^{71} + 5q^{72} + 19q^{73} - 2q^{74} + 5q^{76} + 14q^{78} - 9q^{79} - 3q^{80} - q^{81} + 6q^{83} - 11q^{85} + 7q^{86} + 12q^{87} + 12q^{89} - 5q^{90} - 4q^{92} + 8q^{94} - 5q^{95} - 7q^{97} + 3q^{98} + O(q^{100}) \)

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
\(3\) −2.77846 −1.60414 −0.802071 0.597228i \(-0.796269\pi\)
−0.802071 + 0.597228i \(0.796269\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.77846 −1.13430
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 4.71982 1.57327
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.77846 −0.802071
\(13\) −2.77846 −0.770605 −0.385303 0.922790i \(-0.625903\pi\)
−0.385303 + 0.922790i \(0.625903\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.77846 0.717395
\(16\) 1.00000 0.250000
\(17\) 6.71982 1.62980 0.814898 0.579604i \(-0.196793\pi\)
0.814898 + 0.579604i \(0.196793\pi\)
\(18\) 4.71982 1.11247
\(19\) 7.49828 1.72022 0.860112 0.510106i \(-0.170394\pi\)
0.860112 + 0.510106i \(0.170394\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.77846 −0.606309
\(22\) 0 0
\(23\) −3.83709 −0.800089 −0.400044 0.916496i \(-0.631005\pi\)
−0.400044 + 0.916496i \(0.631005\pi\)
\(24\) −2.77846 −0.567150
\(25\) 1.00000 0.200000
\(26\) −2.77846 −0.544900
\(27\) −4.77846 −0.919615
\(28\) 1.00000 0.188982
\(29\) −5.43965 −1.01012 −0.505059 0.863085i \(-0.668529\pi\)
−0.505059 + 0.863085i \(0.668529\pi\)
\(30\) 2.77846 0.507275
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.71982 1.15244
\(35\) −1.00000 −0.169031
\(36\) 4.71982 0.786637
\(37\) −0.117266 −0.0192785 −0.00963923 0.999954i \(-0.503068\pi\)
−0.00963923 + 0.999954i \(0.503068\pi\)
\(38\) 7.49828 1.21638
\(39\) 7.71982 1.23616
\(40\) −1.00000 −0.158114
\(41\) 5.55691 0.867844 0.433922 0.900950i \(-0.357129\pi\)
0.433922 + 0.900950i \(0.357129\pi\)
\(42\) −2.77846 −0.428725
\(43\) 10.3940 1.58507 0.792535 0.609826i \(-0.208761\pi\)
0.792535 + 0.609826i \(0.208761\pi\)
\(44\) 0 0
\(45\) −4.71982 −0.703590
\(46\) −3.83709 −0.565748
\(47\) 2.11727 0.308835 0.154418 0.988006i \(-0.450650\pi\)
0.154418 + 0.988006i \(0.450650\pi\)
\(48\) −2.77846 −0.401036
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −18.6707 −2.61443
\(52\) −2.77846 −0.385303
\(53\) −2.71982 −0.373597 −0.186798 0.982398i \(-0.559811\pi\)
−0.186798 + 0.982398i \(0.559811\pi\)
\(54\) −4.77846 −0.650266
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −20.8337 −2.75948
\(58\) −5.43965 −0.714261
\(59\) −3.49828 −0.455437 −0.227719 0.973727i \(-0.573127\pi\)
−0.227719 + 0.973727i \(0.573127\pi\)
\(60\) 2.77846 0.358697
\(61\) −4.27674 −0.547580 −0.273790 0.961789i \(-0.588277\pi\)
−0.273790 + 0.961789i \(0.588277\pi\)
\(62\) 0 0
\(63\) 4.71982 0.594642
\(64\) 1.00000 0.125000
\(65\) 2.77846 0.344625
\(66\) 0 0
\(67\) −12.1595 −1.48552 −0.742758 0.669560i \(-0.766483\pi\)
−0.742758 + 0.669560i \(0.766483\pi\)
\(68\) 6.71982 0.814898
\(69\) 10.6612 1.28346
\(70\) −1.00000 −0.119523
\(71\) 3.16291 0.375368 0.187684 0.982229i \(-0.439902\pi\)
0.187684 + 0.982229i \(0.439902\pi\)
\(72\) 4.71982 0.556237
\(73\) 8.83709 1.03430 0.517152 0.855894i \(-0.326992\pi\)
0.517152 + 0.855894i \(0.326992\pi\)
\(74\) −0.117266 −0.0136319
\(75\) −2.77846 −0.320829
\(76\) 7.49828 0.860112
\(77\) 0 0
\(78\) 7.71982 0.874098
\(79\) −8.55691 −0.962728 −0.481364 0.876521i \(-0.659858\pi\)
−0.481364 + 0.876521i \(0.659858\pi\)
\(80\) −1.00000 −0.111803
\(81\) −0.882734 −0.0980815
\(82\) 5.55691 0.613659
\(83\) −0.778457 −0.0854468 −0.0427234 0.999087i \(-0.513603\pi\)
−0.0427234 + 0.999087i \(0.513603\pi\)
\(84\) −2.77846 −0.303155
\(85\) −6.71982 −0.728867
\(86\) 10.3940 1.12081
\(87\) 15.1138 1.62037
\(88\) 0 0
\(89\) 9.55691 1.01303 0.506515 0.862231i \(-0.330933\pi\)
0.506515 + 0.862231i \(0.330933\pi\)
\(90\) −4.71982 −0.497513
\(91\) −2.77846 −0.291261
\(92\) −3.83709 −0.400044
\(93\) 0 0
\(94\) 2.11727 0.218379
\(95\) −7.49828 −0.769307
\(96\) −2.77846 −0.283575
\(97\) −4.83709 −0.491132 −0.245566 0.969380i \(-0.578974\pi\)
−0.245566 + 0.969380i \(0.578974\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −1.22154 −0.121548 −0.0607740 0.998152i \(-0.519357\pi\)
−0.0607740 + 0.998152i \(0.519357\pi\)
\(102\) −18.6707 −1.84868
\(103\) −8.27674 −0.815531 −0.407766 0.913087i \(-0.633692\pi\)
−0.407766 + 0.913087i \(0.633692\pi\)
\(104\) −2.77846 −0.272450
\(105\) 2.77846 0.271150
\(106\) −2.71982 −0.264173
\(107\) −16.2767 −1.57353 −0.786766 0.617252i \(-0.788246\pi\)
−0.786766 + 0.617252i \(0.788246\pi\)
\(108\) −4.77846 −0.459807
\(109\) −1.88273 −0.180333 −0.0901666 0.995927i \(-0.528740\pi\)
−0.0901666 + 0.995927i \(0.528740\pi\)
\(110\) 0 0
\(111\) 0.325819 0.0309254
\(112\) 1.00000 0.0944911
\(113\) 15.1595 1.42608 0.713042 0.701122i \(-0.247317\pi\)
0.713042 + 0.701122i \(0.247317\pi\)
\(114\) −20.8337 −1.95125
\(115\) 3.83709 0.357811
\(116\) −5.43965 −0.505059
\(117\) −13.1138 −1.21237
\(118\) −3.49828 −0.322043
\(119\) 6.71982 0.616005
\(120\) 2.77846 0.253637
\(121\) 0 0
\(122\) −4.27674 −0.387198
\(123\) −15.4396 −1.39215
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 4.71982 0.420475
\(127\) 11.8371 1.05037 0.525186 0.850987i \(-0.323996\pi\)
0.525186 + 0.850987i \(0.323996\pi\)
\(128\) 1.00000 0.0883883
\(129\) −28.8793 −2.54268
\(130\) 2.77846 0.243687
\(131\) 18.0096 1.57350 0.786751 0.617271i \(-0.211762\pi\)
0.786751 + 0.617271i \(0.211762\pi\)
\(132\) 0 0
\(133\) 7.49828 0.650183
\(134\) −12.1595 −1.05042
\(135\) 4.77846 0.411264
\(136\) 6.71982 0.576220
\(137\) −12.9509 −1.10647 −0.553236 0.833025i \(-0.686607\pi\)
−0.553236 + 0.833025i \(0.686607\pi\)
\(138\) 10.6612 0.907541
\(139\) 18.8207 1.59635 0.798174 0.602427i \(-0.205800\pi\)
0.798174 + 0.602427i \(0.205800\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −5.88273 −0.495416
\(142\) 3.16291 0.265425
\(143\) 0 0
\(144\) 4.71982 0.393319
\(145\) 5.43965 0.451738
\(146\) 8.83709 0.731363
\(147\) −2.77846 −0.229163
\(148\) −0.117266 −0.00963923
\(149\) 16.6707 1.36572 0.682860 0.730549i \(-0.260736\pi\)
0.682860 + 0.730549i \(0.260736\pi\)
\(150\) −2.77846 −0.226860
\(151\) 16.9509 1.37945 0.689723 0.724073i \(-0.257732\pi\)
0.689723 + 0.724073i \(0.257732\pi\)
\(152\) 7.49828 0.608191
\(153\) 31.7164 2.56412
\(154\) 0 0
\(155\) 0 0
\(156\) 7.71982 0.618081
\(157\) −6.66119 −0.531621 −0.265810 0.964025i \(-0.585640\pi\)
−0.265810 + 0.964025i \(0.585640\pi\)
\(158\) −8.55691 −0.680752
\(159\) 7.55691 0.599302
\(160\) −1.00000 −0.0790569
\(161\) −3.83709 −0.302405
\(162\) −0.882734 −0.0693541
\(163\) −16.5113 −1.29326 −0.646631 0.762803i \(-0.723823\pi\)
−0.646631 + 0.762803i \(0.723823\pi\)
\(164\) 5.55691 0.433922
\(165\) 0 0
\(166\) −0.778457 −0.0604200
\(167\) −4.51127 −0.349093 −0.174546 0.984649i \(-0.555846\pi\)
−0.174546 + 0.984649i \(0.555846\pi\)
\(168\) −2.77846 −0.214363
\(169\) −5.28018 −0.406167
\(170\) −6.71982 −0.515387
\(171\) 35.3906 2.70638
\(172\) 10.3940 0.792535
\(173\) 10.5535 0.802366 0.401183 0.915998i \(-0.368599\pi\)
0.401183 + 0.915998i \(0.368599\pi\)
\(174\) 15.1138 1.14578
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 9.71982 0.730587
\(178\) 9.55691 0.716321
\(179\) 8.32582 0.622301 0.311150 0.950361i \(-0.399286\pi\)
0.311150 + 0.950361i \(0.399286\pi\)
\(180\) −4.71982 −0.351795
\(181\) −21.9345 −1.63038 −0.815189 0.579195i \(-0.803367\pi\)
−0.815189 + 0.579195i \(0.803367\pi\)
\(182\) −2.77846 −0.205953
\(183\) 11.8827 0.878397
\(184\) −3.83709 −0.282874
\(185\) 0.117266 0.00862159
\(186\) 0 0
\(187\) 0 0
\(188\) 2.11727 0.154418
\(189\) −4.77846 −0.347582
\(190\) −7.49828 −0.543982
\(191\) 11.7198 0.848017 0.424008 0.905658i \(-0.360623\pi\)
0.424008 + 0.905658i \(0.360623\pi\)
\(192\) −2.77846 −0.200518
\(193\) −18.5991 −1.33879 −0.669397 0.742905i \(-0.733447\pi\)
−0.669397 + 0.742905i \(0.733447\pi\)
\(194\) −4.83709 −0.347283
\(195\) −7.71982 −0.552828
\(196\) 1.00000 0.0714286
\(197\) −10.0682 −0.717328 −0.358664 0.933467i \(-0.616768\pi\)
−0.358664 + 0.933467i \(0.616768\pi\)
\(198\) 0 0
\(199\) −16.5535 −1.17344 −0.586722 0.809788i \(-0.699582\pi\)
−0.586722 + 0.809788i \(0.699582\pi\)
\(200\) 1.00000 0.0707107
\(201\) 33.7846 2.38298
\(202\) −1.22154 −0.0859475
\(203\) −5.43965 −0.381788
\(204\) −18.6707 −1.30721
\(205\) −5.55691 −0.388112
\(206\) −8.27674 −0.576668
\(207\) −18.1104 −1.25876
\(208\) −2.77846 −0.192651
\(209\) 0 0
\(210\) 2.77846 0.191732
\(211\) −26.8793 −1.85045 −0.925224 0.379423i \(-0.876123\pi\)
−0.925224 + 0.379423i \(0.876123\pi\)
\(212\) −2.71982 −0.186798
\(213\) −8.78801 −0.602144
\(214\) −16.2767 −1.11265
\(215\) −10.3940 −0.708865
\(216\) −4.77846 −0.325133
\(217\) 0 0
\(218\) −1.88273 −0.127515
\(219\) −24.5535 −1.65917
\(220\) 0 0
\(221\) −18.6707 −1.25593
\(222\) 0.325819 0.0218676
\(223\) −5.83365 −0.390650 −0.195325 0.980739i \(-0.562576\pi\)
−0.195325 + 0.980739i \(0.562576\pi\)
\(224\) 1.00000 0.0668153
\(225\) 4.71982 0.314655
\(226\) 15.1595 1.00839
\(227\) −5.88273 −0.390451 −0.195225 0.980758i \(-0.562544\pi\)
−0.195225 + 0.980758i \(0.562544\pi\)
\(228\) −20.8337 −1.37974
\(229\) 25.9509 1.71488 0.857442 0.514580i \(-0.172052\pi\)
0.857442 + 0.514580i \(0.172052\pi\)
\(230\) 3.83709 0.253010
\(231\) 0 0
\(232\) −5.43965 −0.357130
\(233\) 17.2767 1.13184 0.565918 0.824461i \(-0.308522\pi\)
0.565918 + 0.824461i \(0.308522\pi\)
\(234\) −13.1138 −0.857278
\(235\) −2.11727 −0.138115
\(236\) −3.49828 −0.227719
\(237\) 23.7750 1.54435
\(238\) 6.71982 0.435581
\(239\) 26.9440 1.74287 0.871433 0.490515i \(-0.163191\pi\)
0.871433 + 0.490515i \(0.163191\pi\)
\(240\) 2.77846 0.179349
\(241\) 24.9966 1.61017 0.805085 0.593159i \(-0.202120\pi\)
0.805085 + 0.593159i \(0.202120\pi\)
\(242\) 0 0
\(243\) 16.7880 1.07695
\(244\) −4.27674 −0.273790
\(245\) −1.00000 −0.0638877
\(246\) −15.4396 −0.984396
\(247\) −20.8337 −1.32561
\(248\) 0 0
\(249\) 2.16291 0.137069
\(250\) −1.00000 −0.0632456
\(251\) 9.83365 0.620695 0.310347 0.950623i \(-0.399555\pi\)
0.310347 + 0.950623i \(0.399555\pi\)
\(252\) 4.71982 0.297321
\(253\) 0 0
\(254\) 11.8371 0.742725
\(255\) 18.6707 1.16921
\(256\) 1.00000 0.0625000
\(257\) −8.15947 −0.508974 −0.254487 0.967076i \(-0.581907\pi\)
−0.254487 + 0.967076i \(0.581907\pi\)
\(258\) −28.8793 −1.79795
\(259\) −0.117266 −0.00728657
\(260\) 2.77846 0.172313
\(261\) −25.6742 −1.58919
\(262\) 18.0096 1.11263
\(263\) 28.2733 1.74341 0.871703 0.490034i \(-0.163016\pi\)
0.871703 + 0.490034i \(0.163016\pi\)
\(264\) 0 0
\(265\) 2.71982 0.167077
\(266\) 7.49828 0.459749
\(267\) −26.5535 −1.62505
\(268\) −12.1595 −0.742758
\(269\) 3.06207 0.186698 0.0933489 0.995633i \(-0.470243\pi\)
0.0933489 + 0.995633i \(0.470243\pi\)
\(270\) 4.77846 0.290808
\(271\) 22.6707 1.37715 0.688575 0.725165i \(-0.258237\pi\)
0.688575 + 0.725165i \(0.258237\pi\)
\(272\) 6.71982 0.407449
\(273\) 7.71982 0.467225
\(274\) −12.9509 −0.782394
\(275\) 0 0
\(276\) 10.6612 0.641728
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 18.8207 1.12879
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −24.6673 −1.47153 −0.735764 0.677238i \(-0.763177\pi\)
−0.735764 + 0.677238i \(0.763177\pi\)
\(282\) −5.88273 −0.350312
\(283\) 21.2147 1.26108 0.630541 0.776156i \(-0.282833\pi\)
0.630541 + 0.776156i \(0.282833\pi\)
\(284\) 3.16291 0.187684
\(285\) 20.8337 1.23408
\(286\) 0 0
\(287\) 5.55691 0.328014
\(288\) 4.71982 0.278118
\(289\) 28.1560 1.65624
\(290\) 5.43965 0.319427
\(291\) 13.4396 0.787846
\(292\) 8.83709 0.517152
\(293\) 17.4492 1.01939 0.509697 0.860354i \(-0.329758\pi\)
0.509697 + 0.860354i \(0.329758\pi\)
\(294\) −2.77846 −0.162043
\(295\) 3.49828 0.203678
\(296\) −0.117266 −0.00681597
\(297\) 0 0
\(298\) 16.6707 0.965710
\(299\) 10.6612 0.616553
\(300\) −2.77846 −0.160414
\(301\) 10.3940 0.599100
\(302\) 16.9509 0.975416
\(303\) 3.39400 0.194980
\(304\) 7.49828 0.430056
\(305\) 4.27674 0.244885
\(306\) 31.7164 1.81311
\(307\) −2.43621 −0.139042 −0.0695209 0.997580i \(-0.522147\pi\)
−0.0695209 + 0.997580i \(0.522147\pi\)
\(308\) 0 0
\(309\) 22.9966 1.30823
\(310\) 0 0
\(311\) 23.7846 1.34870 0.674350 0.738412i \(-0.264424\pi\)
0.674350 + 0.738412i \(0.264424\pi\)
\(312\) 7.71982 0.437049
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −6.66119 −0.375913
\(315\) −4.71982 −0.265932
\(316\) −8.55691 −0.481364
\(317\) −20.7129 −1.16336 −0.581678 0.813419i \(-0.697603\pi\)
−0.581678 + 0.813419i \(0.697603\pi\)
\(318\) 7.55691 0.423771
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 45.2242 2.52417
\(322\) −3.83709 −0.213833
\(323\) 50.3871 2.80361
\(324\) −0.882734 −0.0490408
\(325\) −2.77846 −0.154121
\(326\) −16.5113 −0.914475
\(327\) 5.23109 0.289280
\(328\) 5.55691 0.306829
\(329\) 2.11727 0.116729
\(330\) 0 0
\(331\) 16.1173 0.885885 0.442943 0.896550i \(-0.353935\pi\)
0.442943 + 0.896550i \(0.353935\pi\)
\(332\) −0.778457 −0.0427234
\(333\) −0.553476 −0.0303303
\(334\) −4.51127 −0.246846
\(335\) 12.1595 0.664343
\(336\) −2.77846 −0.151577
\(337\) −6.28018 −0.342103 −0.171051 0.985262i \(-0.554716\pi\)
−0.171051 + 0.985262i \(0.554716\pi\)
\(338\) −5.28018 −0.287204
\(339\) −42.1199 −2.28764
\(340\) −6.71982 −0.364434
\(341\) 0 0
\(342\) 35.3906 1.91370
\(343\) 1.00000 0.0539949
\(344\) 10.3940 0.560407
\(345\) −10.6612 −0.573979
\(346\) 10.5535 0.567358
\(347\) 28.7198 1.54176 0.770880 0.636980i \(-0.219817\pi\)
0.770880 + 0.636980i \(0.219817\pi\)
\(348\) 15.1138 0.810186
\(349\) 1.33193 0.0712968 0.0356484 0.999364i \(-0.488650\pi\)
0.0356484 + 0.999364i \(0.488650\pi\)
\(350\) 1.00000 0.0534522
\(351\) 13.2767 0.708660
\(352\) 0 0
\(353\) −14.5113 −0.772357 −0.386179 0.922424i \(-0.626205\pi\)
−0.386179 + 0.922424i \(0.626205\pi\)
\(354\) 9.71982 0.516603
\(355\) −3.16291 −0.167870
\(356\) 9.55691 0.506515
\(357\) −18.6707 −0.988160
\(358\) 8.32582 0.440033
\(359\) 14.8302 0.782709 0.391354 0.920240i \(-0.372007\pi\)
0.391354 + 0.920240i \(0.372007\pi\)
\(360\) −4.71982 −0.248757
\(361\) 37.2242 1.95917
\(362\) −21.9345 −1.15285
\(363\) 0 0
\(364\) −2.77846 −0.145631
\(365\) −8.83709 −0.462554
\(366\) 11.8827 0.621120
\(367\) −18.9475 −0.989050 −0.494525 0.869163i \(-0.664658\pi\)
−0.494525 + 0.869163i \(0.664658\pi\)
\(368\) −3.83709 −0.200022
\(369\) 26.2277 1.36536
\(370\) 0.117266 0.00609639
\(371\) −2.71982 −0.141206
\(372\) 0 0
\(373\) 11.0716 0.573267 0.286633 0.958040i \(-0.407464\pi\)
0.286633 + 0.958040i \(0.407464\pi\)
\(374\) 0 0
\(375\) 2.77846 0.143479
\(376\) 2.11727 0.109190
\(377\) 15.1138 0.778402
\(378\) −4.77846 −0.245777
\(379\) 30.0844 1.54533 0.772666 0.634813i \(-0.218923\pi\)
0.772666 + 0.634813i \(0.218923\pi\)
\(380\) −7.49828 −0.384654
\(381\) −32.8888 −1.68495
\(382\) 11.7198 0.599638
\(383\) 37.2733 1.90458 0.952288 0.305200i \(-0.0987232\pi\)
0.952288 + 0.305200i \(0.0987232\pi\)
\(384\) −2.77846 −0.141788
\(385\) 0 0
\(386\) −18.5991 −0.946670
\(387\) 49.0579 2.49375
\(388\) −4.83709 −0.245566
\(389\) 24.3449 1.23434 0.617168 0.786831i \(-0.288280\pi\)
0.617168 + 0.786831i \(0.288280\pi\)
\(390\) −7.71982 −0.390908
\(391\) −25.7846 −1.30398
\(392\) 1.00000 0.0505076
\(393\) −50.0388 −2.52412
\(394\) −10.0682 −0.507228
\(395\) 8.55691 0.430545
\(396\) 0 0
\(397\) 14.3189 0.718647 0.359324 0.933213i \(-0.383007\pi\)
0.359324 + 0.933213i \(0.383007\pi\)
\(398\) −16.5535 −0.829751
\(399\) −20.8337 −1.04299
\(400\) 1.00000 0.0500000
\(401\) 17.7655 0.887165 0.443583 0.896234i \(-0.353707\pi\)
0.443583 + 0.896234i \(0.353707\pi\)
\(402\) 33.7846 1.68502
\(403\) 0 0
\(404\) −1.22154 −0.0607740
\(405\) 0.882734 0.0438634
\(406\) −5.43965 −0.269965
\(407\) 0 0
\(408\) −18.6707 −0.924340
\(409\) −9.88273 −0.488670 −0.244335 0.969691i \(-0.578570\pi\)
−0.244335 + 0.969691i \(0.578570\pi\)
\(410\) −5.55691 −0.274436
\(411\) 35.9836 1.77494
\(412\) −8.27674 −0.407766
\(413\) −3.49828 −0.172139
\(414\) −18.1104 −0.890077
\(415\) 0.778457 0.0382130
\(416\) −2.77846 −0.136225
\(417\) −52.2924 −2.56077
\(418\) 0 0
\(419\) 8.05863 0.393690 0.196845 0.980435i \(-0.436930\pi\)
0.196845 + 0.980435i \(0.436930\pi\)
\(420\) 2.77846 0.135575
\(421\) −4.65164 −0.226707 −0.113354 0.993555i \(-0.536159\pi\)
−0.113354 + 0.993555i \(0.536159\pi\)
\(422\) −26.8793 −1.30846
\(423\) 9.99312 0.485882
\(424\) −2.71982 −0.132086
\(425\) 6.71982 0.325959
\(426\) −8.78801 −0.425780
\(427\) −4.27674 −0.206966
\(428\) −16.2767 −0.786766
\(429\) 0 0
\(430\) −10.3940 −0.501243
\(431\) −37.8793 −1.82458 −0.912291 0.409543i \(-0.865688\pi\)
−0.912291 + 0.409543i \(0.865688\pi\)
\(432\) −4.77846 −0.229904
\(433\) 15.4656 0.743231 0.371615 0.928387i \(-0.378804\pi\)
0.371615 + 0.928387i \(0.378804\pi\)
\(434\) 0 0
\(435\) −15.1138 −0.724653
\(436\) −1.88273 −0.0901666
\(437\) −28.7716 −1.37633
\(438\) −24.5535 −1.17321
\(439\) 33.6413 1.60561 0.802806 0.596240i \(-0.203339\pi\)
0.802806 + 0.596240i \(0.203339\pi\)
\(440\) 0 0
\(441\) 4.71982 0.224754
\(442\) −18.6707 −0.888077
\(443\) −11.8759 −0.564239 −0.282120 0.959379i \(-0.591037\pi\)
−0.282120 + 0.959379i \(0.591037\pi\)
\(444\) 0.325819 0.0154627
\(445\) −9.55691 −0.453041
\(446\) −5.83365 −0.276231
\(447\) −46.3189 −2.19081
\(448\) 1.00000 0.0472456
\(449\) −17.8793 −0.843776 −0.421888 0.906648i \(-0.638632\pi\)
−0.421888 + 0.906648i \(0.638632\pi\)
\(450\) 4.71982 0.222495
\(451\) 0 0
\(452\) 15.1595 0.713042
\(453\) −47.0974 −2.21283
\(454\) −5.88273 −0.276090
\(455\) 2.77846 0.130256
\(456\) −20.8337 −0.975625
\(457\) 10.5078 0.491536 0.245768 0.969329i \(-0.420960\pi\)
0.245768 + 0.969329i \(0.420960\pi\)
\(458\) 25.9509 1.21261
\(459\) −32.1104 −1.49878
\(460\) 3.83709 0.178905
\(461\) −41.3837 −1.92743 −0.963715 0.266932i \(-0.913990\pi\)
−0.963715 + 0.266932i \(0.913990\pi\)
\(462\) 0 0
\(463\) 28.7424 1.33577 0.667886 0.744264i \(-0.267200\pi\)
0.667886 + 0.744264i \(0.267200\pi\)
\(464\) −5.43965 −0.252529
\(465\) 0 0
\(466\) 17.2767 0.800329
\(467\) 16.3614 0.757113 0.378557 0.925578i \(-0.376421\pi\)
0.378557 + 0.925578i \(0.376421\pi\)
\(468\) −13.1138 −0.606187
\(469\) −12.1595 −0.561472
\(470\) −2.11727 −0.0976622
\(471\) 18.5078 0.852796
\(472\) −3.49828 −0.161021
\(473\) 0 0
\(474\) 23.7750 1.09202
\(475\) 7.49828 0.344045
\(476\) 6.71982 0.308003
\(477\) −12.8371 −0.587770
\(478\) 26.9440 1.23239
\(479\) 41.9018 1.91454 0.957272 0.289189i \(-0.0933857\pi\)
0.957272 + 0.289189i \(0.0933857\pi\)
\(480\) 2.77846 0.126819
\(481\) 0.325819 0.0148561
\(482\) 24.9966 1.13856
\(483\) 10.6612 0.485101
\(484\) 0 0
\(485\) 4.83709 0.219641
\(486\) 16.7880 0.761520
\(487\) −16.9509 −0.768119 −0.384060 0.923308i \(-0.625474\pi\)
−0.384060 + 0.923308i \(0.625474\pi\)
\(488\) −4.27674 −0.193599
\(489\) 45.8759 2.07458
\(490\) −1.00000 −0.0451754
\(491\) −32.9966 −1.48911 −0.744557 0.667559i \(-0.767339\pi\)
−0.744557 + 0.667559i \(0.767339\pi\)
\(492\) −15.4396 −0.696073
\(493\) −36.5535 −1.64629
\(494\) −20.8337 −0.937350
\(495\) 0 0
\(496\) 0 0
\(497\) 3.16291 0.141876
\(498\) 2.16291 0.0969223
\(499\) 7.67418 0.343544 0.171772 0.985137i \(-0.445051\pi\)
0.171772 + 0.985137i \(0.445051\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.5344 0.559994
\(502\) 9.83365 0.438897
\(503\) 29.0647 1.29593 0.647967 0.761669i \(-0.275620\pi\)
0.647967 + 0.761669i \(0.275620\pi\)
\(504\) 4.71982 0.210238
\(505\) 1.22154 0.0543579
\(506\) 0 0
\(507\) 14.6707 0.651551
\(508\) 11.8371 0.525186
\(509\) −36.2603 −1.60721 −0.803605 0.595163i \(-0.797087\pi\)
−0.803605 + 0.595163i \(0.797087\pi\)
\(510\) 18.6707 0.826754
\(511\) 8.83709 0.390930
\(512\) 1.00000 0.0441942
\(513\) −35.8302 −1.58194
\(514\) −8.15947 −0.359899
\(515\) 8.27674 0.364717
\(516\) −28.8793 −1.27134
\(517\) 0 0
\(518\) −0.117266 −0.00515239
\(519\) −29.3224 −1.28711
\(520\) 2.77846 0.121843
\(521\) 17.6742 0.774320 0.387160 0.922013i \(-0.373456\pi\)
0.387160 + 0.922013i \(0.373456\pi\)
\(522\) −25.6742 −1.12373
\(523\) −16.1268 −0.705177 −0.352588 0.935779i \(-0.614698\pi\)
−0.352588 + 0.935779i \(0.614698\pi\)
\(524\) 18.0096 0.786751
\(525\) −2.77846 −0.121262
\(526\) 28.2733 1.23277
\(527\) 0 0
\(528\) 0 0
\(529\) −8.27674 −0.359858
\(530\) 2.71982 0.118142
\(531\) −16.5113 −0.716528
\(532\) 7.49828 0.325092
\(533\) −15.4396 −0.668765
\(534\) −26.5535 −1.14908
\(535\) 16.2767 0.703705
\(536\) −12.1595 −0.525209
\(537\) −23.1329 −0.998260
\(538\) 3.06207 0.132015
\(539\) 0 0
\(540\) 4.77846 0.205632
\(541\) 40.9897 1.76228 0.881142 0.472852i \(-0.156775\pi\)
0.881142 + 0.472852i \(0.156775\pi\)
\(542\) 22.6707 0.973792
\(543\) 60.9440 2.61536
\(544\) 6.71982 0.288110
\(545\) 1.88273 0.0806475
\(546\) 7.71982 0.330378
\(547\) −19.6251 −0.839109 −0.419554 0.907730i \(-0.637814\pi\)
−0.419554 + 0.907730i \(0.637814\pi\)
\(548\) −12.9509 −0.553236
\(549\) −20.1855 −0.861494
\(550\) 0 0
\(551\) −40.7880 −1.73763
\(552\) 10.6612 0.453770
\(553\) −8.55691 −0.363877
\(554\) −10.0000 −0.424859
\(555\) −0.325819 −0.0138303
\(556\) 18.8207 0.798174
\(557\) −21.4396 −0.908427 −0.454214 0.890893i \(-0.650080\pi\)
−0.454214 + 0.890893i \(0.650080\pi\)
\(558\) 0 0
\(559\) −28.8793 −1.22146
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −24.6673 −1.04053
\(563\) 5.89229 0.248330 0.124165 0.992262i \(-0.460375\pi\)
0.124165 + 0.992262i \(0.460375\pi\)
\(564\) −5.88273 −0.247708
\(565\) −15.1595 −0.637764
\(566\) 21.2147 0.891719
\(567\) −0.882734 −0.0370713
\(568\) 3.16291 0.132713
\(569\) −36.2311 −1.51889 −0.759443 0.650574i \(-0.774529\pi\)
−0.759443 + 0.650574i \(0.774529\pi\)
\(570\) 20.8337 0.872626
\(571\) 37.5500 1.57142 0.785710 0.618595i \(-0.212298\pi\)
0.785710 + 0.618595i \(0.212298\pi\)
\(572\) 0 0
\(573\) −32.5630 −1.36034
\(574\) 5.55691 0.231941
\(575\) −3.83709 −0.160018
\(576\) 4.71982 0.196659
\(577\) 42.7129 1.77816 0.889082 0.457749i \(-0.151344\pi\)
0.889082 + 0.457749i \(0.151344\pi\)
\(578\) 28.1560 1.17114
\(579\) 51.6769 2.14762
\(580\) 5.43965 0.225869
\(581\) −0.778457 −0.0322958
\(582\) 13.4396 0.557091
\(583\) 0 0
\(584\) 8.83709 0.365681
\(585\) 13.1138 0.542190
\(586\) 17.4492 0.720820
\(587\) 5.54736 0.228964 0.114482 0.993425i \(-0.463479\pi\)
0.114482 + 0.993425i \(0.463479\pi\)
\(588\) −2.77846 −0.114582
\(589\) 0 0
\(590\) 3.49828 0.144022
\(591\) 27.9740 1.15070
\(592\) −0.117266 −0.00481962
\(593\) −16.2699 −0.668123 −0.334062 0.942551i \(-0.608419\pi\)
−0.334062 + 0.942551i \(0.608419\pi\)
\(594\) 0 0
\(595\) −6.71982 −0.275486
\(596\) 16.6707 0.682860
\(597\) 45.9931 1.88237
\(598\) 10.6612 0.435969
\(599\) −17.4362 −0.712424 −0.356212 0.934405i \(-0.615932\pi\)
−0.356212 + 0.934405i \(0.615932\pi\)
\(600\) −2.77846 −0.113430
\(601\) −1.12070 −0.0457145 −0.0228572 0.999739i \(-0.507276\pi\)
−0.0228572 + 0.999739i \(0.507276\pi\)
\(602\) 10.3940 0.423628
\(603\) −57.3906 −2.33712
\(604\) 16.9509 0.689723
\(605\) 0 0
\(606\) 3.39400 0.137872
\(607\) −24.5113 −0.994882 −0.497441 0.867498i \(-0.665727\pi\)
−0.497441 + 0.867498i \(0.665727\pi\)
\(608\) 7.49828 0.304095
\(609\) 15.1138 0.612443
\(610\) 4.27674 0.173160
\(611\) −5.88273 −0.237990
\(612\) 31.7164 1.28206
\(613\) 4.71982 0.190632 0.0953159 0.995447i \(-0.469614\pi\)
0.0953159 + 0.995447i \(0.469614\pi\)
\(614\) −2.43621 −0.0983174
\(615\) 15.4396 0.622587
\(616\) 0 0
\(617\) 20.4622 0.823777 0.411888 0.911234i \(-0.364869\pi\)
0.411888 + 0.911234i \(0.364869\pi\)
\(618\) 22.9966 0.925057
\(619\) 17.8854 0.718875 0.359438 0.933169i \(-0.382969\pi\)
0.359438 + 0.933169i \(0.382969\pi\)
\(620\) 0 0
\(621\) 18.3354 0.735773
\(622\) 23.7846 0.953674
\(623\) 9.55691 0.382890
\(624\) 7.71982 0.309040
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −6.66119 −0.265810
\(629\) −0.788009 −0.0314200
\(630\) −4.71982 −0.188042
\(631\) 39.6182 1.57718 0.788588 0.614922i \(-0.210812\pi\)
0.788588 + 0.614922i \(0.210812\pi\)
\(632\) −8.55691 −0.340376
\(633\) 74.6830 2.96838
\(634\) −20.7129 −0.822616
\(635\) −11.8371 −0.469741
\(636\) 7.55691 0.299651
\(637\) −2.77846 −0.110086
\(638\) 0 0
\(639\) 14.9284 0.590557
\(640\) −1.00000 −0.0395285
\(641\) −4.55691 −0.179987 −0.0899936 0.995942i \(-0.528685\pi\)
−0.0899936 + 0.995942i \(0.528685\pi\)
\(642\) 45.2242 1.78486
\(643\) −33.8827 −1.33620 −0.668102 0.744069i \(-0.732893\pi\)
−0.668102 + 0.744069i \(0.732893\pi\)
\(644\) −3.83709 −0.151203
\(645\) 28.8793 1.13712
\(646\) 50.3871 1.98246
\(647\) 10.8111 0.425029 0.212514 0.977158i \(-0.431835\pi\)
0.212514 + 0.977158i \(0.431835\pi\)
\(648\) −0.882734 −0.0346771
\(649\) 0 0
\(650\) −2.77846 −0.108980
\(651\) 0 0
\(652\) −16.5113 −0.646631
\(653\) 11.3646 0.444731 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(654\) 5.23109 0.204552
\(655\) −18.0096 −0.703691
\(656\) 5.55691 0.216961
\(657\) 41.7095 1.62724
\(658\) 2.11727 0.0825396
\(659\) 18.2277 0.710049 0.355024 0.934857i \(-0.384473\pi\)
0.355024 + 0.934857i \(0.384473\pi\)
\(660\) 0 0
\(661\) −11.8854 −0.462289 −0.231144 0.972919i \(-0.574247\pi\)
−0.231144 + 0.972919i \(0.574247\pi\)
\(662\) 16.1173 0.626415
\(663\) 51.8759 2.01469
\(664\) −0.778457 −0.0302100
\(665\) −7.49828 −0.290771
\(666\) −0.553476 −0.0214468
\(667\) 20.8724 0.808183
\(668\) −4.51127 −0.174546
\(669\) 16.2086 0.626659
\(670\) 12.1595 0.469761
\(671\) 0 0
\(672\) −2.77846 −0.107181
\(673\) −1.51127 −0.0582552 −0.0291276 0.999576i \(-0.509273\pi\)
−0.0291276 + 0.999576i \(0.509273\pi\)
\(674\) −6.28018 −0.241903
\(675\) −4.77846 −0.183923
\(676\) −5.28018 −0.203084
\(677\) −9.10428 −0.349906 −0.174953 0.984577i \(-0.555977\pi\)
−0.174953 + 0.984577i \(0.555977\pi\)
\(678\) −42.1199 −1.61761
\(679\) −4.83709 −0.185630
\(680\) −6.71982 −0.257693
\(681\) 16.3449 0.626339
\(682\) 0 0
\(683\) 16.7129 0.639503 0.319752 0.947501i \(-0.396401\pi\)
0.319752 + 0.947501i \(0.396401\pi\)
\(684\) 35.3906 1.35319
\(685\) 12.9509 0.494829
\(686\) 1.00000 0.0381802
\(687\) −72.1035 −2.75092
\(688\) 10.3940 0.396268
\(689\) 7.55691 0.287896
\(690\) −10.6612 −0.405865
\(691\) 29.1820 1.11014 0.555068 0.831805i \(-0.312692\pi\)
0.555068 + 0.831805i \(0.312692\pi\)
\(692\) 10.5535 0.401183
\(693\) 0 0
\(694\) 28.7198 1.09019
\(695\) −18.8207 −0.713908
\(696\) 15.1138 0.572888
\(697\) 37.3415 1.41441
\(698\) 1.33193 0.0504144
\(699\) −48.0027 −1.81563
\(700\) 1.00000 0.0377964
\(701\) −4.99656 −0.188718 −0.0943588 0.995538i \(-0.530080\pi\)
−0.0943588 + 0.995538i \(0.530080\pi\)
\(702\) 13.2767 0.501098
\(703\) −0.879296 −0.0331633
\(704\) 0 0
\(705\) 5.88273 0.221557
\(706\) −14.5113 −0.546139
\(707\) −1.22154 −0.0459408
\(708\) 9.71982 0.365293
\(709\) 17.3484 0.651531 0.325766 0.945451i \(-0.394378\pi\)
0.325766 + 0.945451i \(0.394378\pi\)
\(710\) −3.16291 −0.118702
\(711\) −40.3871 −1.51464
\(712\) 9.55691 0.358161
\(713\) 0 0
\(714\) −18.6707 −0.698735
\(715\) 0 0
\(716\) 8.32582 0.311150
\(717\) −74.8629 −2.79581
\(718\) 14.8302 0.553459
\(719\) −5.20512 −0.194118 −0.0970590 0.995279i \(-0.530944\pi\)
−0.0970590 + 0.995279i \(0.530944\pi\)
\(720\) −4.71982 −0.175897
\(721\) −8.27674 −0.308242
\(722\) 37.2242 1.38534
\(723\) −69.4519 −2.58294
\(724\) −21.9345 −0.815189
\(725\) −5.43965 −0.202023
\(726\) 0 0
\(727\) 20.9966 0.778719 0.389360 0.921086i \(-0.372696\pi\)
0.389360 + 0.921086i \(0.372696\pi\)
\(728\) −2.77846 −0.102976
\(729\) −43.9966 −1.62950
\(730\) −8.83709 −0.327075
\(731\) 69.8459 2.58334
\(732\) 11.8827 0.439198
\(733\) 6.80444 0.251328 0.125664 0.992073i \(-0.459894\pi\)
0.125664 + 0.992073i \(0.459894\pi\)
\(734\) −18.9475 −0.699364
\(735\) 2.77846 0.102485
\(736\) −3.83709 −0.141437
\(737\) 0 0
\(738\) 26.2277 0.965453
\(739\) −3.23797 −0.119111 −0.0595553 0.998225i \(-0.518968\pi\)
−0.0595553 + 0.998225i \(0.518968\pi\)
\(740\) 0.117266 0.00431080
\(741\) 57.8854 2.12647
\(742\) −2.71982 −0.0998479
\(743\) −16.5535 −0.607288 −0.303644 0.952786i \(-0.598203\pi\)
−0.303644 + 0.952786i \(0.598203\pi\)
\(744\) 0 0
\(745\) −16.6707 −0.610769
\(746\) 11.0716 0.405361
\(747\) −3.67418 −0.134431
\(748\) 0 0
\(749\) −16.2767 −0.594739
\(750\) 2.77846 0.101455
\(751\) −9.04221 −0.329955 −0.164977 0.986297i \(-0.552755\pi\)
−0.164977 + 0.986297i \(0.552755\pi\)
\(752\) 2.11727 0.0772088
\(753\) −27.3224 −0.995683
\(754\) 15.1138 0.550413
\(755\) −16.9509 −0.616907
\(756\) −4.77846 −0.173791
\(757\) −36.7198 −1.33460 −0.667302 0.744787i \(-0.732551\pi\)
−0.667302 + 0.744787i \(0.732551\pi\)
\(758\) 30.0844 1.09272
\(759\) 0 0
\(760\) −7.49828 −0.271991
\(761\) −1.23109 −0.0446272 −0.0223136 0.999751i \(-0.507103\pi\)
−0.0223136 + 0.999751i \(0.507103\pi\)
\(762\) −32.8888 −1.19144
\(763\) −1.88273 −0.0681595
\(764\) 11.7198 0.424008
\(765\) −31.7164 −1.14671
\(766\) 37.2733 1.34674
\(767\) 9.71982 0.350963
\(768\) −2.77846 −0.100259
\(769\) −6.34492 −0.228804 −0.114402 0.993435i \(-0.536495\pi\)
−0.114402 + 0.993435i \(0.536495\pi\)
\(770\) 0 0
\(771\) 22.6707 0.816467
\(772\) −18.5991 −0.669397
\(773\) 36.2372 1.30336 0.651681 0.758493i \(-0.274064\pi\)
0.651681 + 0.758493i \(0.274064\pi\)
\(774\) 49.0579 1.76335
\(775\) 0 0
\(776\) −4.83709 −0.173641
\(777\) 0.325819 0.0116887
\(778\) 24.3449 0.872808
\(779\) 41.6673 1.49289
\(780\) −7.71982 −0.276414
\(781\) 0 0
\(782\) −25.7846 −0.922054
\(783\) 25.9931 0.928918
\(784\) 1.00000 0.0357143
\(785\) 6.66119 0.237748
\(786\) −50.0388 −1.78482
\(787\) 7.55691 0.269375 0.134687 0.990888i \(-0.456997\pi\)
0.134687 + 0.990888i \(0.456997\pi\)
\(788\) −10.0682 −0.358664
\(789\) −78.5562 −2.79667
\(790\) 8.55691 0.304441
\(791\) 15.1595 0.539009
\(792\) 0 0
\(793\) 11.8827 0.421968
\(794\) 14.3189 0.508160
\(795\) −7.55691 −0.268016
\(796\) −16.5535 −0.586722
\(797\) 16.4267 0.581862 0.290931 0.956744i \(-0.406035\pi\)
0.290931 + 0.956744i \(0.406035\pi\)
\(798\) −20.8337 −0.737503
\(799\) 14.2277 0.503338
\(800\) 1.00000 0.0353553
\(801\) 45.1070 1.59378
\(802\) 17.7655 0.627320
\(803\) 0 0
\(804\) 33.7846 1.19149
\(805\) 3.83709 0.135240
\(806\) 0 0
\(807\) −8.50783 −0.299490
\(808\) −1.22154 −0.0429737
\(809\) 29.4819 1.03653 0.518263 0.855221i \(-0.326579\pi\)
0.518263 + 0.855221i \(0.326579\pi\)
\(810\) 0.882734 0.0310161
\(811\) −12.5275 −0.439900 −0.219950 0.975511i \(-0.570589\pi\)
−0.219950 + 0.975511i \(0.570589\pi\)
\(812\) −5.43965 −0.190894
\(813\) −62.9897 −2.20914
\(814\) 0 0
\(815\) 16.5113 0.578365
\(816\) −18.6707 −0.653607
\(817\) 77.9372 2.72668
\(818\) −9.88273 −0.345542
\(819\) −13.1138 −0.458234
\(820\) −5.55691 −0.194056
\(821\) −4.44996 −0.155305 −0.0776524 0.996980i \(-0.524742\pi\)
−0.0776524 + 0.996980i \(0.524742\pi\)
\(822\) 35.9836 1.25507
\(823\) −45.4396 −1.58393 −0.791963 0.610569i \(-0.790941\pi\)
−0.791963 + 0.610569i \(0.790941\pi\)
\(824\) −8.27674 −0.288334
\(825\) 0 0
\(826\) −3.49828 −0.121721
\(827\) −35.8528 −1.24672 −0.623361 0.781934i \(-0.714233\pi\)
−0.623361 + 0.781934i \(0.714233\pi\)
\(828\) −18.1104 −0.629380
\(829\) −23.8854 −0.829575 −0.414787 0.909918i \(-0.636144\pi\)
−0.414787 + 0.909918i \(0.636144\pi\)
\(830\) 0.778457 0.0270206
\(831\) 27.7846 0.963836
\(832\) −2.77846 −0.0963257
\(833\) 6.71982 0.232828
\(834\) −52.2924 −1.81074
\(835\) 4.51127 0.156119
\(836\) 0 0
\(837\) 0 0
\(838\) 8.05863 0.278381
\(839\) −9.22422 −0.318455 −0.159228 0.987242i \(-0.550900\pi\)
−0.159228 + 0.987242i \(0.550900\pi\)
\(840\) 2.77846 0.0958659
\(841\) 0.589769 0.0203369
\(842\) −4.65164 −0.160306
\(843\) 68.5370 2.36054
\(844\) −26.8793 −0.925224
\(845\) 5.28018 0.181644
\(846\) 9.99312 0.343571
\(847\) 0 0
\(848\) −2.71982 −0.0933991
\(849\) −58.9440 −2.02295
\(850\) 6.71982 0.230488
\(851\) 0.449961 0.0154245
\(852\) −8.78801 −0.301072
\(853\) −49.8923 −1.70828 −0.854140 0.520044i \(-0.825916\pi\)
−0.854140 + 0.520044i \(0.825916\pi\)
\(854\) −4.27674 −0.146347
\(855\) −35.3906 −1.21033
\(856\) −16.2767 −0.556327
\(857\) 8.44309 0.288410 0.144205 0.989548i \(-0.453937\pi\)
0.144205 + 0.989548i \(0.453937\pi\)
\(858\) 0 0
\(859\) 30.3871 1.03680 0.518398 0.855140i \(-0.326529\pi\)
0.518398 + 0.855140i \(0.326529\pi\)
\(860\) −10.3940 −0.354433
\(861\) −15.4396 −0.526182
\(862\) −37.8793 −1.29017
\(863\) 0.788009 0.0268241 0.0134121 0.999910i \(-0.495731\pi\)
0.0134121 + 0.999910i \(0.495731\pi\)
\(864\) −4.77846 −0.162566
\(865\) −10.5535 −0.358829
\(866\) 15.4656 0.525543
\(867\) −78.2303 −2.65684
\(868\) 0 0
\(869\) 0 0
\(870\) −15.1138 −0.512407
\(871\) 33.7846 1.14475
\(872\) −1.88273 −0.0637574
\(873\) −22.8302 −0.772686
\(874\) −28.7716 −0.973213
\(875\) −1.00000 −0.0338062
\(876\) −24.5535 −0.829585
\(877\) 27.6251 0.932833 0.466417 0.884565i \(-0.345545\pi\)
0.466417 + 0.884565i \(0.345545\pi\)
\(878\) 33.6413 1.13534
\(879\) −48.4819 −1.63525
\(880\) 0 0
\(881\) 19.9087 0.670742 0.335371 0.942086i \(-0.391138\pi\)
0.335371 + 0.942086i \(0.391138\pi\)
\(882\) 4.71982 0.158925
\(883\) 1.60600 0.0540461 0.0270230 0.999635i \(-0.491397\pi\)
0.0270230 + 0.999635i \(0.491397\pi\)
\(884\) −18.6707 −0.627965
\(885\) −9.71982 −0.326728
\(886\) −11.8759 −0.398977
\(887\) 36.5466 1.22711 0.613557 0.789650i \(-0.289738\pi\)
0.613557 + 0.789650i \(0.289738\pi\)
\(888\) 0.325819 0.0109338
\(889\) 11.8371 0.397003
\(890\) −9.55691 −0.320348
\(891\) 0 0
\(892\) −5.83365 −0.195325
\(893\) 15.8759 0.531265
\(894\) −46.3189 −1.54914
\(895\) −8.32582 −0.278301
\(896\) 1.00000 0.0334077
\(897\) −29.6217 −0.989038
\(898\) −17.8793 −0.596640
\(899\) 0 0
\(900\) 4.71982 0.157327
\(901\) −18.2767 −0.608886
\(902\) 0 0
\(903\) −28.8793 −0.961043
\(904\) 15.1595 0.504197
\(905\) 21.9345 0.729127
\(906\) −47.0974 −1.56471
\(907\) 29.9440 0.994276 0.497138 0.867672i \(-0.334384\pi\)
0.497138 + 0.867672i \(0.334384\pi\)
\(908\) −5.88273 −0.195225
\(909\) −5.76547 −0.191228
\(910\) 2.77846 0.0921050
\(911\) 0.993124 0.0329037 0.0164518 0.999865i \(-0.494763\pi\)
0.0164518 + 0.999865i \(0.494763\pi\)
\(912\) −20.8337 −0.689871
\(913\) 0 0
\(914\) 10.5078 0.347568
\(915\) −11.8827 −0.392831
\(916\) 25.9509 0.857442
\(917\) 18.0096 0.594728
\(918\) −32.1104 −1.05980
\(919\) 9.10695 0.300411 0.150205 0.988655i \(-0.452007\pi\)
0.150205 + 0.988655i \(0.452007\pi\)
\(920\) 3.83709 0.126505
\(921\) 6.76891 0.223043
\(922\) −41.3837 −1.36290
\(923\) −8.78801 −0.289261
\(924\) 0 0
\(925\) −0.117266 −0.00385569
\(926\) 28.7424 0.944533
\(927\) −39.0647 −1.28305
\(928\) −5.43965 −0.178565
\(929\) 15.2964 0.501859 0.250929 0.968005i \(-0.419264\pi\)
0.250929 + 0.968005i \(0.419264\pi\)
\(930\) 0 0
\(931\) 7.49828 0.245746
\(932\) 17.2767 0.565918
\(933\) −66.0844 −2.16351
\(934\) 16.3614 0.535360
\(935\) 0 0
\(936\) −13.1138 −0.428639
\(937\) −40.4293 −1.32077 −0.660384 0.750928i \(-0.729607\pi\)
−0.660384 + 0.750928i \(0.729607\pi\)
\(938\) −12.1595 −0.397021
\(939\) 27.7846 0.906715
\(940\) −2.11727 −0.0690576
\(941\) −54.5888 −1.77954 −0.889772 0.456405i \(-0.849137\pi\)
−0.889772 + 0.456405i \(0.849137\pi\)
\(942\) 18.5078 0.603018
\(943\) −21.3224 −0.694352
\(944\) −3.49828 −0.113859
\(945\) 4.77846 0.155443
\(946\) 0 0
\(947\) 47.4588 1.54220 0.771101 0.636713i \(-0.219706\pi\)
0.771101 + 0.636713i \(0.219706\pi\)
\(948\) 23.7750 0.772177
\(949\) −24.5535 −0.797040
\(950\) 7.49828 0.243276
\(951\) 57.5500 1.86619
\(952\) 6.71982 0.217791
\(953\) 35.1855 1.13977 0.569884 0.821725i \(-0.306988\pi\)
0.569884 + 0.821725i \(0.306988\pi\)
\(954\) −12.8371 −0.415616
\(955\) −11.7198 −0.379245
\(956\) 26.9440 0.871433
\(957\) 0 0
\(958\) 41.9018 1.35379
\(959\) −12.9509 −0.418207
\(960\) 2.77846 0.0896743
\(961\) −31.0000 −1.00000
\(962\) 0.325819 0.0105048
\(963\) −76.8233 −2.47560
\(964\) 24.9966 0.805085
\(965\) 18.5991 0.598727
\(966\) 10.6612 0.343018
\(967\) −25.2051 −0.810542 −0.405271 0.914197i \(-0.632823\pi\)
−0.405271 + 0.914197i \(0.632823\pi\)
\(968\) 0 0
\(969\) −139.998 −4.49740
\(970\) 4.83709 0.155310
\(971\) 13.1465 0.421891 0.210945 0.977498i \(-0.432346\pi\)
0.210945 + 0.977498i \(0.432346\pi\)
\(972\) 16.7880 0.538476
\(973\) 18.8207 0.603363
\(974\) −16.9509 −0.543142
\(975\) 7.71982 0.247232
\(976\) −4.27674 −0.136895
\(977\) 58.7164 1.87850 0.939252 0.343229i \(-0.111521\pi\)
0.939252 + 0.343229i \(0.111521\pi\)
\(978\) 45.8759 1.46695
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −8.88617 −0.283714
\(982\) −32.9966 −1.05296
\(983\) −4.43621 −0.141493 −0.0707466 0.997494i \(-0.522538\pi\)
−0.0707466 + 0.997494i \(0.522538\pi\)
\(984\) −15.4396 −0.492198
\(985\) 10.0682 0.320799
\(986\) −36.5535 −1.16410
\(987\) −5.88273 −0.187249
\(988\) −20.8337 −0.662807
\(989\) −39.8827 −1.26820
\(990\) 0 0
\(991\) 14.8827 0.472766 0.236383 0.971660i \(-0.424038\pi\)
0.236383 + 0.971660i \(0.424038\pi\)
\(992\) 0 0
\(993\) −44.7811 −1.42109
\(994\) 3.16291 0.100321
\(995\) 16.5535 0.524780
\(996\) 2.16291 0.0685344
\(997\) −15.0061 −0.475248 −0.237624 0.971357i \(-0.576369\pi\)
−0.237624 + 0.971357i \(0.576369\pi\)
\(998\) 7.67418 0.242922
\(999\) 0.560352 0.0177288
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.ck.1.1 yes 3
11.10 odd 2 8470.2.a.cg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cg.1.1 3 11.10 odd 2
8470.2.a.ck.1.1 yes 3 1.1 even 1 trivial