Properties

Label 3850.2.c.ba.1849.6
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.6
Root \(0.264658 + 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.ba.1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +2.24914i q^{3} -1.00000 q^{4} -2.24914 q^{6} -1.00000i q^{7} -1.00000i q^{8} -2.05863 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +2.24914i q^{3} -1.00000 q^{4} -2.24914 q^{6} -1.00000i q^{7} -1.00000i q^{8} -2.05863 q^{9} +1.00000 q^{11} -2.24914i q^{12} -0.941367i q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.49828i q^{17} -2.05863i q^{18} +4.36641 q^{19} +2.24914 q^{21} +1.00000i q^{22} -6.24914i q^{23} +2.24914 q^{24} +0.941367 q^{26} +2.11727i q^{27} +1.00000i q^{28} -8.74742 q^{29} -9.55691 q^{31} +1.00000i q^{32} +2.24914i q^{33} -6.49828 q^{34} +2.05863 q^{36} +4.24914i q^{37} +4.36641i q^{38} +2.11727 q^{39} -2.13187 q^{41} +2.24914i q^{42} -7.67418i q^{43} -1.00000 q^{44} +6.24914 q^{46} +11.1138i q^{47} +2.24914i q^{48} -1.00000 q^{49} -14.6155 q^{51} +0.941367i q^{52} +4.74742i q^{53} -2.11727 q^{54} -1.00000 q^{56} +9.82066i q^{57} -8.74742i q^{58} +1.88273 q^{59} +9.11383 q^{61} -9.55691i q^{62} +2.05863i q^{63} -1.00000 q^{64} -2.24914 q^{66} +12.9966i q^{67} -6.49828i q^{68} +14.0552 q^{69} -14.6155 q^{71} +2.05863i q^{72} +10.4983i q^{73} -4.24914 q^{74} -4.36641 q^{76} -1.00000i q^{77} +2.11727i q^{78} -8.36641 q^{79} -10.9379 q^{81} -2.13187i q^{82} +8.49828i q^{83} -2.24914 q^{84} +7.67418 q^{86} -19.6742i q^{87} -1.00000i q^{88} -12.3810 q^{89} -0.941367 q^{91} +6.24914i q^{92} -21.4948i q^{93} -11.1138 q^{94} -2.24914 q^{96} -15.3630i q^{97} -1.00000i q^{98} -2.05863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 4q^{6} - 14q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 4q^{6} - 14q^{9} + 6q^{11} + 6q^{14} + 6q^{16} + 12q^{19} - 4q^{21} - 4q^{24} + 4q^{26} - 24q^{31} - 4q^{34} + 14q^{36} + 16q^{39} + 8q^{41} - 6q^{44} + 20q^{46} - 6q^{49} - 56q^{51} - 16q^{54} - 6q^{56} + 8q^{59} - 12q^{61} - 6q^{64} + 4q^{66} + 16q^{69} - 56q^{71} - 8q^{74} - 12q^{76} - 36q^{79} + 6q^{81} + 4q^{84} + 16q^{86} - 36q^{89} - 4q^{91} + 4q^{96} - 14q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 2.24914i 1.29854i 0.760557 + 0.649271i \(0.224926\pi\)
−0.760557 + 0.649271i \(0.775074\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −2.24914 −0.918208
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) −2.05863 −0.686211
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) − 2.24914i − 0.649271i
\(13\) − 0.941367i − 0.261088i −0.991443 0.130544i \(-0.958328\pi\)
0.991443 0.130544i \(-0.0416724\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.49828i 1.57606i 0.615634 + 0.788032i \(0.288900\pi\)
−0.615634 + 0.788032i \(0.711100\pi\)
\(18\) − 2.05863i − 0.485224i
\(19\) 4.36641 1.00172 0.500861 0.865528i \(-0.333017\pi\)
0.500861 + 0.865528i \(0.333017\pi\)
\(20\) 0 0
\(21\) 2.24914 0.490803
\(22\) 1.00000i 0.213201i
\(23\) − 6.24914i − 1.30304i −0.758633 0.651518i \(-0.774133\pi\)
0.758633 0.651518i \(-0.225867\pi\)
\(24\) 2.24914 0.459104
\(25\) 0 0
\(26\) 0.941367 0.184617
\(27\) 2.11727i 0.407468i
\(28\) 1.00000i 0.188982i
\(29\) −8.74742 −1.62436 −0.812178 0.583410i \(-0.801718\pi\)
−0.812178 + 0.583410i \(0.801718\pi\)
\(30\) 0 0
\(31\) −9.55691 −1.71647 −0.858236 0.513255i \(-0.828440\pi\)
−0.858236 + 0.513255i \(0.828440\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.24914i 0.391525i
\(34\) −6.49828 −1.11445
\(35\) 0 0
\(36\) 2.05863 0.343106
\(37\) 4.24914i 0.698554i 0.937019 + 0.349277i \(0.113573\pi\)
−0.937019 + 0.349277i \(0.886427\pi\)
\(38\) 4.36641i 0.708325i
\(39\) 2.11727 0.339034
\(40\) 0 0
\(41\) −2.13187 −0.332943 −0.166471 0.986046i \(-0.553237\pi\)
−0.166471 + 0.986046i \(0.553237\pi\)
\(42\) 2.24914i 0.347050i
\(43\) − 7.67418i − 1.17030i −0.810925 0.585151i \(-0.801035\pi\)
0.810925 0.585151i \(-0.198965\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.24914 0.921386
\(47\) 11.1138i 1.62112i 0.585657 + 0.810559i \(0.300837\pi\)
−0.585657 + 0.810559i \(0.699163\pi\)
\(48\) 2.24914i 0.324635i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −14.6155 −2.04659
\(52\) 0.941367i 0.130544i
\(53\) 4.74742i 0.652109i 0.945351 + 0.326054i \(0.105719\pi\)
−0.945351 + 0.326054i \(0.894281\pi\)
\(54\) −2.11727 −0.288123
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 9.82066i 1.30078i
\(58\) − 8.74742i − 1.14859i
\(59\) 1.88273 0.245111 0.122556 0.992462i \(-0.460891\pi\)
0.122556 + 0.992462i \(0.460891\pi\)
\(60\) 0 0
\(61\) 9.11383 1.16691 0.583453 0.812147i \(-0.301701\pi\)
0.583453 + 0.812147i \(0.301701\pi\)
\(62\) − 9.55691i − 1.21373i
\(63\) 2.05863i 0.259363i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.24914 −0.276850
\(67\) 12.9966i 1.58778i 0.608060 + 0.793891i \(0.291948\pi\)
−0.608060 + 0.793891i \(0.708052\pi\)
\(68\) − 6.49828i − 0.788032i
\(69\) 14.0552 1.69205
\(70\) 0 0
\(71\) −14.6155 −1.73455 −0.867273 0.497833i \(-0.834129\pi\)
−0.867273 + 0.497833i \(0.834129\pi\)
\(72\) 2.05863i 0.242612i
\(73\) 10.4983i 1.22873i 0.789022 + 0.614365i \(0.210588\pi\)
−0.789022 + 0.614365i \(0.789412\pi\)
\(74\) −4.24914 −0.493953
\(75\) 0 0
\(76\) −4.36641 −0.500861
\(77\) − 1.00000i − 0.113961i
\(78\) 2.11727i 0.239733i
\(79\) −8.36641 −0.941294 −0.470647 0.882322i \(-0.655980\pi\)
−0.470647 + 0.882322i \(0.655980\pi\)
\(80\) 0 0
\(81\) −10.9379 −1.21533
\(82\) − 2.13187i − 0.235426i
\(83\) 8.49828i 0.932808i 0.884572 + 0.466404i \(0.154451\pi\)
−0.884572 + 0.466404i \(0.845549\pi\)
\(84\) −2.24914 −0.245401
\(85\) 0 0
\(86\) 7.67418 0.827528
\(87\) − 19.6742i − 2.10929i
\(88\) − 1.00000i − 0.106600i
\(89\) −12.3810 −1.31238 −0.656192 0.754594i \(-0.727834\pi\)
−0.656192 + 0.754594i \(0.727834\pi\)
\(90\) 0 0
\(91\) −0.941367 −0.0986821
\(92\) 6.24914i 0.651518i
\(93\) − 21.4948i − 2.22891i
\(94\) −11.1138 −1.14630
\(95\) 0 0
\(96\) −2.24914 −0.229552
\(97\) − 15.3630i − 1.55987i −0.625859 0.779937i \(-0.715251\pi\)
0.625859 0.779937i \(-0.284749\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) −2.05863 −0.206900
\(100\) 0 0
\(101\) −16.8793 −1.67955 −0.839776 0.542932i \(-0.817314\pi\)
−0.839776 + 0.542932i \(0.817314\pi\)
\(102\) − 14.6155i − 1.44715i
\(103\) 6.61555i 0.651849i 0.945396 + 0.325925i \(0.105676\pi\)
−0.945396 + 0.325925i \(0.894324\pi\)
\(104\) −0.941367 −0.0923086
\(105\) 0 0
\(106\) −4.74742 −0.461110
\(107\) − 14.5535i − 1.40694i −0.710726 0.703469i \(-0.751633\pi\)
0.710726 0.703469i \(-0.248367\pi\)
\(108\) − 2.11727i − 0.203734i
\(109\) 0.249141 0.0238633 0.0119317 0.999929i \(-0.496202\pi\)
0.0119317 + 0.999929i \(0.496202\pi\)
\(110\) 0 0
\(111\) −9.55691 −0.907102
\(112\) − 1.00000i − 0.0944911i
\(113\) − 10.9966i − 1.03447i −0.855844 0.517235i \(-0.826961\pi\)
0.855844 0.517235i \(-0.173039\pi\)
\(114\) −9.82066 −0.919789
\(115\) 0 0
\(116\) 8.74742 0.812178
\(117\) 1.93793i 0.179162i
\(118\) 1.88273i 0.173320i
\(119\) 6.49828 0.595696
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 9.11383i 0.825127i
\(123\) − 4.79488i − 0.432340i
\(124\) 9.55691 0.858236
\(125\) 0 0
\(126\) −2.05863 −0.183398
\(127\) − 5.88273i − 0.522008i −0.965338 0.261004i \(-0.915946\pi\)
0.965338 0.261004i \(-0.0840536\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 17.2603 1.51969
\(130\) 0 0
\(131\) −1.25258 −0.109438 −0.0547191 0.998502i \(-0.517426\pi\)
−0.0547191 + 0.998502i \(0.517426\pi\)
\(132\) − 2.24914i − 0.195763i
\(133\) − 4.36641i − 0.378615i
\(134\) −12.9966 −1.12273
\(135\) 0 0
\(136\) 6.49828 0.557223
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 14.0552i 1.19646i
\(139\) −10.9820 −0.931477 −0.465739 0.884922i \(-0.654211\pi\)
−0.465739 + 0.884922i \(0.654211\pi\)
\(140\) 0 0
\(141\) −24.9966 −2.10509
\(142\) − 14.6155i − 1.22651i
\(143\) − 0.941367i − 0.0787210i
\(144\) −2.05863 −0.171553
\(145\) 0 0
\(146\) −10.4983 −0.868844
\(147\) − 2.24914i − 0.185506i
\(148\) − 4.24914i − 0.349277i
\(149\) −0.0146079 −0.00119673 −0.000598363 1.00000i \(-0.500190\pi\)
−0.000598363 1.00000i \(0.500190\pi\)
\(150\) 0 0
\(151\) −15.2457 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(152\) − 4.36641i − 0.354162i
\(153\) − 13.3776i − 1.08151i
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −2.11727 −0.169517
\(157\) − 5.50172i − 0.439085i −0.975603 0.219542i \(-0.929544\pi\)
0.975603 0.219542i \(-0.0704565\pi\)
\(158\) − 8.36641i − 0.665596i
\(159\) −10.6776 −0.846790
\(160\) 0 0
\(161\) −6.24914 −0.492501
\(162\) − 10.9379i − 0.859365i
\(163\) 18.2277i 1.42770i 0.700298 + 0.713850i \(0.253050\pi\)
−0.700298 + 0.713850i \(0.746950\pi\)
\(164\) 2.13187 0.166471
\(165\) 0 0
\(166\) −8.49828 −0.659595
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) − 2.24914i − 0.173525i
\(169\) 12.1138 0.931833
\(170\) 0 0
\(171\) −8.98883 −0.687393
\(172\) 7.67418i 0.585151i
\(173\) − 0.117266i − 0.00891559i −0.999990 0.00445780i \(-0.998581\pi\)
0.999990 0.00445780i \(-0.00141897\pi\)
\(174\) 19.6742 1.49150
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 4.23453i 0.318287i
\(178\) − 12.3810i − 0.927996i
\(179\) 22.5535 1.68573 0.842863 0.538128i \(-0.180868\pi\)
0.842863 + 0.538128i \(0.180868\pi\)
\(180\) 0 0
\(181\) 20.8793 1.55195 0.775973 0.630766i \(-0.217259\pi\)
0.775973 + 0.630766i \(0.217259\pi\)
\(182\) − 0.941367i − 0.0697788i
\(183\) 20.4983i 1.51528i
\(184\) −6.24914 −0.460693
\(185\) 0 0
\(186\) 21.4948 1.57608
\(187\) 6.49828i 0.475201i
\(188\) − 11.1138i − 0.810559i
\(189\) 2.11727 0.154008
\(190\) 0 0
\(191\) 3.11383 0.225309 0.112654 0.993634i \(-0.464065\pi\)
0.112654 + 0.993634i \(0.464065\pi\)
\(192\) − 2.24914i − 0.162318i
\(193\) 6.17246i 0.444304i 0.975012 + 0.222152i \(0.0713080\pi\)
−0.975012 + 0.222152i \(0.928692\pi\)
\(194\) 15.3630 1.10300
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.73281i 0.479693i 0.970811 + 0.239847i \(0.0770971\pi\)
−0.970811 + 0.239847i \(0.922903\pi\)
\(198\) − 2.05863i − 0.146301i
\(199\) 13.2311 0.937927 0.468964 0.883217i \(-0.344627\pi\)
0.468964 + 0.883217i \(0.344627\pi\)
\(200\) 0 0
\(201\) −29.2311 −2.06180
\(202\) − 16.8793i − 1.18762i
\(203\) 8.74742i 0.613949i
\(204\) 14.6155 1.02329
\(205\) 0 0
\(206\) −6.61555 −0.460927
\(207\) 12.8647i 0.894158i
\(208\) − 0.941367i − 0.0652720i
\(209\) 4.36641 0.302031
\(210\) 0 0
\(211\) 23.1138 1.59122 0.795611 0.605808i \(-0.207150\pi\)
0.795611 + 0.605808i \(0.207150\pi\)
\(212\) − 4.74742i − 0.326054i
\(213\) − 32.8724i − 2.25238i
\(214\) 14.5535 0.994855
\(215\) 0 0
\(216\) 2.11727 0.144062
\(217\) 9.55691i 0.648766i
\(218\) 0.249141i 0.0168739i
\(219\) −23.6121 −1.59556
\(220\) 0 0
\(221\) 6.11727 0.411492
\(222\) − 9.55691i − 0.641418i
\(223\) − 10.3810i − 0.695164i −0.937650 0.347582i \(-0.887003\pi\)
0.937650 0.347582i \(-0.112997\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 10.9966 0.731480
\(227\) 16.1104i 1.06928i 0.845079 + 0.534642i \(0.179554\pi\)
−0.845079 + 0.534642i \(0.820446\pi\)
\(228\) − 9.82066i − 0.650389i
\(229\) −24.2897 −1.60511 −0.802555 0.596578i \(-0.796527\pi\)
−0.802555 + 0.596578i \(0.796527\pi\)
\(230\) 0 0
\(231\) 2.24914 0.147983
\(232\) 8.74742i 0.574296i
\(233\) − 6.88617i − 0.451128i −0.974228 0.225564i \(-0.927578\pi\)
0.974228 0.225564i \(-0.0724225\pi\)
\(234\) −1.93793 −0.126686
\(235\) 0 0
\(236\) −1.88273 −0.122556
\(237\) − 18.8172i − 1.22231i
\(238\) 6.49828i 0.421221i
\(239\) 11.1353 0.720283 0.360142 0.932898i \(-0.382728\pi\)
0.360142 + 0.932898i \(0.382728\pi\)
\(240\) 0 0
\(241\) 2.36641 0.152434 0.0762168 0.997091i \(-0.475716\pi\)
0.0762168 + 0.997091i \(0.475716\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) − 18.2491i − 1.17068i
\(244\) −9.11383 −0.583453
\(245\) 0 0
\(246\) 4.79488 0.305711
\(247\) − 4.11039i − 0.261538i
\(248\) 9.55691i 0.606865i
\(249\) −19.1138 −1.21129
\(250\) 0 0
\(251\) 25.1070 1.58474 0.792368 0.610043i \(-0.208848\pi\)
0.792368 + 0.610043i \(0.208848\pi\)
\(252\) − 2.05863i − 0.129682i
\(253\) − 6.24914i − 0.392880i
\(254\) 5.88273 0.369116
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.86469i − 0.178694i −0.996001 0.0893472i \(-0.971522\pi\)
0.996001 0.0893472i \(-0.0284781\pi\)
\(258\) 17.2603i 1.07458i
\(259\) 4.24914 0.264029
\(260\) 0 0
\(261\) 18.0077 1.11465
\(262\) − 1.25258i − 0.0773846i
\(263\) − 3.76547i − 0.232189i −0.993238 0.116094i \(-0.962963\pi\)
0.993238 0.116094i \(-0.0370375\pi\)
\(264\) 2.24914 0.138425
\(265\) 0 0
\(266\) 4.36641 0.267722
\(267\) − 27.8466i − 1.70419i
\(268\) − 12.9966i − 0.793891i
\(269\) −8.94137 −0.545165 −0.272582 0.962132i \(-0.587878\pi\)
−0.272582 + 0.962132i \(0.587878\pi\)
\(270\) 0 0
\(271\) 21.4948 1.30572 0.652859 0.757479i \(-0.273569\pi\)
0.652859 + 0.757479i \(0.273569\pi\)
\(272\) 6.49828i 0.394016i
\(273\) − 2.11727i − 0.128143i
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −14.0552 −0.846023
\(277\) 7.64820i 0.459536i 0.973245 + 0.229768i \(0.0737967\pi\)
−0.973245 + 0.229768i \(0.926203\pi\)
\(278\) − 10.9820i − 0.658654i
\(279\) 19.6742 1.17786
\(280\) 0 0
\(281\) −28.6155 −1.70706 −0.853530 0.521043i \(-0.825543\pi\)
−0.853530 + 0.521043i \(0.825543\pi\)
\(282\) − 24.9966i − 1.48852i
\(283\) 2.87930i 0.171156i 0.996331 + 0.0855782i \(0.0272737\pi\)
−0.996331 + 0.0855782i \(0.972726\pi\)
\(284\) 14.6155 0.867273
\(285\) 0 0
\(286\) 0.941367 0.0556642
\(287\) 2.13187i 0.125841i
\(288\) − 2.05863i − 0.121306i
\(289\) −25.2277 −1.48398
\(290\) 0 0
\(291\) 34.5535 2.02556
\(292\) − 10.4983i − 0.614365i
\(293\) 12.9414i 0.756043i 0.925797 + 0.378021i \(0.123395\pi\)
−0.925797 + 0.378021i \(0.876605\pi\)
\(294\) 2.24914 0.131173
\(295\) 0 0
\(296\) 4.24914 0.246976
\(297\) 2.11727i 0.122856i
\(298\) − 0.0146079i 0 0.000846213i
\(299\) −5.88273 −0.340207
\(300\) 0 0
\(301\) −7.67418 −0.442332
\(302\) − 15.2457i − 0.877292i
\(303\) − 37.9639i − 2.18097i
\(304\) 4.36641 0.250431
\(305\) 0 0
\(306\) 13.3776 0.764745
\(307\) 0.498281i 0.0284384i 0.999899 + 0.0142192i \(0.00452626\pi\)
−0.999899 + 0.0142192i \(0.995474\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) −14.8793 −0.846454
\(310\) 0 0
\(311\) −23.4396 −1.32914 −0.664570 0.747226i \(-0.731385\pi\)
−0.664570 + 0.747226i \(0.731385\pi\)
\(312\) − 2.11727i − 0.119867i
\(313\) − 9.36984i − 0.529615i −0.964301 0.264807i \(-0.914692\pi\)
0.964301 0.264807i \(-0.0853084\pi\)
\(314\) 5.50172 0.310480
\(315\) 0 0
\(316\) 8.36641 0.470647
\(317\) − 9.97852i − 0.560449i −0.959934 0.280225i \(-0.909591\pi\)
0.959934 0.280225i \(-0.0904090\pi\)
\(318\) − 10.6776i − 0.598771i
\(319\) −8.74742 −0.489762
\(320\) 0 0
\(321\) 32.7328 1.82697
\(322\) − 6.24914i − 0.348251i
\(323\) 28.3741i 1.57878i
\(324\) 10.9379 0.607663
\(325\) 0 0
\(326\) −18.2277 −1.00954
\(327\) 0.560352i 0.0309875i
\(328\) 2.13187i 0.117713i
\(329\) 11.1138 0.612725
\(330\) 0 0
\(331\) −20.4362 −1.12328 −0.561638 0.827383i \(-0.689829\pi\)
−0.561638 + 0.827383i \(0.689829\pi\)
\(332\) − 8.49828i − 0.466404i
\(333\) − 8.74742i − 0.479356i
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 2.24914 0.122701
\(337\) 7.88273i 0.429400i 0.976680 + 0.214700i \(0.0688774\pi\)
−0.976680 + 0.214700i \(0.931123\pi\)
\(338\) 12.1138i 0.658905i
\(339\) 24.7328 1.34330
\(340\) 0 0
\(341\) −9.55691 −0.517536
\(342\) − 8.98883i − 0.486060i
\(343\) 1.00000i 0.0539949i
\(344\) −7.67418 −0.413764
\(345\) 0 0
\(346\) 0.117266 0.00630428
\(347\) 13.5569i 0.727773i 0.931443 + 0.363887i \(0.118550\pi\)
−0.931443 + 0.363887i \(0.881450\pi\)
\(348\) 19.6742i 1.05465i
\(349\) 10.7328 0.574514 0.287257 0.957853i \(-0.407257\pi\)
0.287257 + 0.957853i \(0.407257\pi\)
\(350\) 0 0
\(351\) 1.99312 0.106385
\(352\) 1.00000i 0.0533002i
\(353\) 20.8578i 1.11015i 0.831801 + 0.555075i \(0.187310\pi\)
−0.831801 + 0.555075i \(0.812690\pi\)
\(354\) −4.23453 −0.225063
\(355\) 0 0
\(356\) 12.3810 0.656192
\(357\) 14.6155i 0.773537i
\(358\) 22.5535i 1.19199i
\(359\) −0.366407 −0.0193382 −0.00966911 0.999953i \(-0.503078\pi\)
−0.00966911 + 0.999953i \(0.503078\pi\)
\(360\) 0 0
\(361\) 0.0655089 0.00344783
\(362\) 20.8793i 1.09739i
\(363\) 2.24914i 0.118049i
\(364\) 0.941367 0.0493410
\(365\) 0 0
\(366\) −20.4983 −1.07146
\(367\) 20.8432i 1.08801i 0.839083 + 0.544003i \(0.183092\pi\)
−0.839083 + 0.544003i \(0.816908\pi\)
\(368\) − 6.24914i − 0.325759i
\(369\) 4.38875 0.228469
\(370\) 0 0
\(371\) 4.74742 0.246474
\(372\) 21.4948i 1.11446i
\(373\) 5.37758i 0.278440i 0.990261 + 0.139220i \(0.0444596\pi\)
−0.990261 + 0.139220i \(0.955540\pi\)
\(374\) −6.49828 −0.336018
\(375\) 0 0
\(376\) 11.1138 0.573152
\(377\) 8.23453i 0.424100i
\(378\) 2.11727i 0.108900i
\(379\) −3.53093 −0.181372 −0.0906860 0.995880i \(-0.528906\pi\)
−0.0906860 + 0.995880i \(0.528906\pi\)
\(380\) 0 0
\(381\) 13.2311 0.677850
\(382\) 3.11383i 0.159317i
\(383\) − 1.38445i − 0.0707422i −0.999374 0.0353711i \(-0.988739\pi\)
0.999374 0.0353711i \(-0.0112613\pi\)
\(384\) 2.24914 0.114776
\(385\) 0 0
\(386\) −6.17246 −0.314170
\(387\) 15.7983i 0.803074i
\(388\) 15.3630i 0.779937i
\(389\) −15.7294 −0.797511 −0.398756 0.917057i \(-0.630558\pi\)
−0.398756 + 0.917057i \(0.630558\pi\)
\(390\) 0 0
\(391\) 40.6087 2.05367
\(392\) 1.00000i 0.0505076i
\(393\) − 2.81722i − 0.142110i
\(394\) −6.73281 −0.339194
\(395\) 0 0
\(396\) 2.05863 0.103450
\(397\) − 17.6121i − 0.883926i −0.897033 0.441963i \(-0.854282\pi\)
0.897033 0.441963i \(-0.145718\pi\)
\(398\) 13.2311i 0.663215i
\(399\) 9.82066 0.491648
\(400\) 0 0
\(401\) −32.8172 −1.63881 −0.819407 0.573212i \(-0.805697\pi\)
−0.819407 + 0.573212i \(0.805697\pi\)
\(402\) − 29.2311i − 1.45791i
\(403\) 8.99656i 0.448151i
\(404\) 16.8793 0.839776
\(405\) 0 0
\(406\) −8.74742 −0.434127
\(407\) 4.24914i 0.210622i
\(408\) 14.6155i 0.723577i
\(409\) 33.3561 1.64935 0.824676 0.565605i \(-0.191357\pi\)
0.824676 + 0.565605i \(0.191357\pi\)
\(410\) 0 0
\(411\) −22.4914 −1.10942
\(412\) − 6.61555i − 0.325925i
\(413\) − 1.88273i − 0.0926433i
\(414\) −12.8647 −0.632265
\(415\) 0 0
\(416\) 0.941367 0.0461543
\(417\) − 24.7000i − 1.20956i
\(418\) 4.36641i 0.213568i
\(419\) −12.3449 −0.603089 −0.301544 0.953452i \(-0.597502\pi\)
−0.301544 + 0.953452i \(0.597502\pi\)
\(420\) 0 0
\(421\) −8.87930 −0.432750 −0.216375 0.976310i \(-0.569423\pi\)
−0.216375 + 0.976310i \(0.569423\pi\)
\(422\) 23.1138i 1.12516i
\(423\) − 22.8793i − 1.11243i
\(424\) 4.74742 0.230555
\(425\) 0 0
\(426\) 32.8724 1.59267
\(427\) − 9.11383i − 0.441049i
\(428\) 14.5535i 0.703469i
\(429\) 2.11727 0.102223
\(430\) 0 0
\(431\) 18.2784 0.880437 0.440219 0.897891i \(-0.354901\pi\)
0.440219 + 0.897891i \(0.354901\pi\)
\(432\) 2.11727i 0.101867i
\(433\) − 6.86469i − 0.329896i −0.986302 0.164948i \(-0.947254\pi\)
0.986302 0.164948i \(-0.0527456\pi\)
\(434\) −9.55691 −0.458747
\(435\) 0 0
\(436\) −0.249141 −0.0119317
\(437\) − 27.2863i − 1.30528i
\(438\) − 23.6121i − 1.12823i
\(439\) 11.8466 0.565409 0.282705 0.959207i \(-0.408768\pi\)
0.282705 + 0.959207i \(0.408768\pi\)
\(440\) 0 0
\(441\) 2.05863 0.0980302
\(442\) 6.11727i 0.290969i
\(443\) − 17.8827i − 0.849634i −0.905279 0.424817i \(-0.860338\pi\)
0.905279 0.424817i \(-0.139662\pi\)
\(444\) 9.55691 0.453551
\(445\) 0 0
\(446\) 10.3810 0.491555
\(447\) − 0.0328552i − 0.00155400i
\(448\) 1.00000i 0.0472456i
\(449\) 16.8172 0.793654 0.396827 0.917893i \(-0.370111\pi\)
0.396827 + 0.917893i \(0.370111\pi\)
\(450\) 0 0
\(451\) −2.13187 −0.100386
\(452\) 10.9966i 0.517235i
\(453\) − 34.2897i − 1.61107i
\(454\) −16.1104 −0.756098
\(455\) 0 0
\(456\) 9.82066 0.459895
\(457\) − 12.2277i − 0.571986i −0.958232 0.285993i \(-0.907677\pi\)
0.958232 0.285993i \(-0.0923234\pi\)
\(458\) − 24.2897i − 1.13498i
\(459\) −13.7586 −0.642196
\(460\) 0 0
\(461\) −16.6155 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(462\) 2.24914i 0.104639i
\(463\) − 18.4837i − 0.859009i −0.903065 0.429505i \(-0.858688\pi\)
0.903065 0.429505i \(-0.141312\pi\)
\(464\) −8.74742 −0.406089
\(465\) 0 0
\(466\) 6.88617 0.318996
\(467\) 24.3956i 1.12889i 0.825469 + 0.564447i \(0.190911\pi\)
−0.825469 + 0.564447i \(0.809089\pi\)
\(468\) − 1.93793i − 0.0895808i
\(469\) 12.9966 0.600125
\(470\) 0 0
\(471\) 12.3741 0.570170
\(472\) − 1.88273i − 0.0866598i
\(473\) − 7.67418i − 0.352859i
\(474\) 18.8172 0.864304
\(475\) 0 0
\(476\) −6.49828 −0.297848
\(477\) − 9.77320i − 0.447484i
\(478\) 11.1353i 0.509317i
\(479\) 27.7655 1.26864 0.634318 0.773072i \(-0.281281\pi\)
0.634318 + 0.773072i \(0.281281\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 2.36641i 0.107787i
\(483\) − 14.0552i − 0.639534i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 18.2491 0.827798
\(487\) 10.0958i 0.457484i 0.973487 + 0.228742i \(0.0734612\pi\)
−0.973487 + 0.228742i \(0.926539\pi\)
\(488\) − 9.11383i − 0.412564i
\(489\) −40.9966 −1.85393
\(490\) 0 0
\(491\) −32.1104 −1.44912 −0.724561 0.689211i \(-0.757957\pi\)
−0.724561 + 0.689211i \(0.757957\pi\)
\(492\) 4.79488i 0.216170i
\(493\) − 56.8432i − 2.56009i
\(494\) 4.11039 0.184935
\(495\) 0 0
\(496\) −9.55691 −0.429118
\(497\) 14.6155i 0.655597i
\(498\) − 19.1138i − 0.856511i
\(499\) 8.79488 0.393713 0.196857 0.980432i \(-0.436927\pi\)
0.196857 + 0.980432i \(0.436927\pi\)
\(500\) 0 0
\(501\) 17.9931 0.803874
\(502\) 25.1070i 1.12058i
\(503\) 15.0034i 0.668970i 0.942401 + 0.334485i \(0.108562\pi\)
−0.942401 + 0.334485i \(0.891438\pi\)
\(504\) 2.05863 0.0916988
\(505\) 0 0
\(506\) 6.24914 0.277808
\(507\) 27.2457i 1.21002i
\(508\) 5.88273i 0.261004i
\(509\) −21.8759 −0.969630 −0.484815 0.874617i \(-0.661113\pi\)
−0.484815 + 0.874617i \(0.661113\pi\)
\(510\) 0 0
\(511\) 10.4983 0.464417
\(512\) 1.00000i 0.0441942i
\(513\) 9.24485i 0.408170i
\(514\) 2.86469 0.126356
\(515\) 0 0
\(516\) −17.2603 −0.759843
\(517\) 11.1138i 0.488786i
\(518\) 4.24914i 0.186697i
\(519\) 0.263748 0.0115773
\(520\) 0 0
\(521\) 14.0292 0.614631 0.307316 0.951608i \(-0.400569\pi\)
0.307316 + 0.951608i \(0.400569\pi\)
\(522\) 18.0077i 0.788177i
\(523\) − 1.14992i − 0.0502825i −0.999684 0.0251412i \(-0.991996\pi\)
0.999684 0.0251412i \(-0.00800355\pi\)
\(524\) 1.25258 0.0547191
\(525\) 0 0
\(526\) 3.76547 0.164182
\(527\) − 62.1035i − 2.70527i
\(528\) 2.24914i 0.0978813i
\(529\) −16.0518 −0.697902
\(530\) 0 0
\(531\) −3.87586 −0.168198
\(532\) 4.36641i 0.189308i
\(533\) 2.00688i 0.0869274i
\(534\) 27.8466 1.20504
\(535\) 0 0
\(536\) 12.9966 0.561366
\(537\) 50.7259i 2.18899i
\(538\) − 8.94137i − 0.385490i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −6.47680 −0.278459 −0.139230 0.990260i \(-0.544463\pi\)
−0.139230 + 0.990260i \(0.544463\pi\)
\(542\) 21.4948i 0.923283i
\(543\) 46.9605i 2.01527i
\(544\) −6.49828 −0.278612
\(545\) 0 0
\(546\) 2.11727 0.0906106
\(547\) 12.9966i 0.555693i 0.960626 + 0.277846i \(0.0896206\pi\)
−0.960626 + 0.277846i \(0.910379\pi\)
\(548\) − 10.0000i − 0.427179i
\(549\) −18.7620 −0.800744
\(550\) 0 0
\(551\) −38.1948 −1.62715
\(552\) − 14.0552i − 0.598229i
\(553\) 8.36641i 0.355776i
\(554\) −7.64820 −0.324941
\(555\) 0 0
\(556\) 10.9820 0.465739
\(557\) − 16.4914i − 0.698763i −0.936981 0.349382i \(-0.886392\pi\)
0.936981 0.349382i \(-0.113608\pi\)
\(558\) 19.6742i 0.832874i
\(559\) −7.22422 −0.305552
\(560\) 0 0
\(561\) −14.6155 −0.617069
\(562\) − 28.6155i − 1.20707i
\(563\) 16.7620i 0.706435i 0.935541 + 0.353218i \(0.114912\pi\)
−0.935541 + 0.353218i \(0.885088\pi\)
\(564\) 24.9966 1.05255
\(565\) 0 0
\(566\) −2.87930 −0.121026
\(567\) 10.9379i 0.459350i
\(568\) 14.6155i 0.613255i
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 12.7328 0.532852 0.266426 0.963855i \(-0.414157\pi\)
0.266426 + 0.963855i \(0.414157\pi\)
\(572\) 0.941367i 0.0393605i
\(573\) 7.00344i 0.292573i
\(574\) −2.13187 −0.0889827
\(575\) 0 0
\(576\) 2.05863 0.0857764
\(577\) − 11.8613i − 0.493790i −0.969042 0.246895i \(-0.920590\pi\)
0.969042 0.246895i \(-0.0794103\pi\)
\(578\) − 25.2277i − 1.04933i
\(579\) −13.8827 −0.576947
\(580\) 0 0
\(581\) 8.49828 0.352568
\(582\) 34.5535i 1.43229i
\(583\) 4.74742i 0.196618i
\(584\) 10.4983 0.434422
\(585\) 0 0
\(586\) −12.9414 −0.534603
\(587\) − 8.60094i − 0.354999i −0.984121 0.177499i \(-0.943199\pi\)
0.984121 0.177499i \(-0.0568008\pi\)
\(588\) 2.24914i 0.0927530i
\(589\) −41.7294 −1.71943
\(590\) 0 0
\(591\) −15.1430 −0.622902
\(592\) 4.24914i 0.174639i
\(593\) − 10.7328i − 0.440744i −0.975416 0.220372i \(-0.929273\pi\)
0.975416 0.220372i \(-0.0707271\pi\)
\(594\) −2.11727 −0.0868725
\(595\) 0 0
\(596\) 0.0146079 0.000598363 0
\(597\) 29.7586i 1.21794i
\(598\) − 5.88273i − 0.240563i
\(599\) −16.0812 −0.657059 −0.328529 0.944494i \(-0.606553\pi\)
−0.328529 + 0.944494i \(0.606553\pi\)
\(600\) 0 0
\(601\) −34.0889 −1.39052 −0.695258 0.718760i \(-0.744710\pi\)
−0.695258 + 0.718760i \(0.744710\pi\)
\(602\) − 7.67418i − 0.312776i
\(603\) − 26.7552i − 1.08955i
\(604\) 15.2457 0.620339
\(605\) 0 0
\(606\) 37.9639 1.54218
\(607\) 11.6742i 0.473840i 0.971529 + 0.236920i \(0.0761380\pi\)
−0.971529 + 0.236920i \(0.923862\pi\)
\(608\) 4.36641i 0.177081i
\(609\) −19.6742 −0.797238
\(610\) 0 0
\(611\) 10.4622 0.423255
\(612\) 13.3776i 0.540756i
\(613\) 34.2637i 1.38390i 0.721946 + 0.691950i \(0.243248\pi\)
−0.721946 + 0.691950i \(0.756752\pi\)
\(614\) −0.498281 −0.0201090
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 39.3707i 1.58500i 0.609869 + 0.792502i \(0.291222\pi\)
−0.609869 + 0.792502i \(0.708778\pi\)
\(618\) − 14.8793i − 0.598533i
\(619\) −16.1104 −0.647531 −0.323766 0.946137i \(-0.604949\pi\)
−0.323766 + 0.946137i \(0.604949\pi\)
\(620\) 0 0
\(621\) 13.2311 0.530946
\(622\) − 23.4396i − 0.939844i
\(623\) 12.3810i 0.496035i
\(624\) 2.11727 0.0847585
\(625\) 0 0
\(626\) 9.36984 0.374494
\(627\) 9.82066i 0.392199i
\(628\) 5.50172i 0.219542i
\(629\) −27.6121 −1.10097
\(630\) 0 0
\(631\) 13.8827 0.552663 0.276331 0.961062i \(-0.410881\pi\)
0.276331 + 0.961062i \(0.410881\pi\)
\(632\) 8.36641i 0.332798i
\(633\) 51.9862i 2.06627i
\(634\) 9.97852 0.396298
\(635\) 0 0
\(636\) 10.6776 0.423395
\(637\) 0.941367i 0.0372983i
\(638\) − 8.74742i − 0.346314i
\(639\) 30.0881 1.19026
\(640\) 0 0
\(641\) −39.3415 −1.55390 −0.776948 0.629565i \(-0.783233\pi\)
−0.776948 + 0.629565i \(0.783233\pi\)
\(642\) 32.7328i 1.29186i
\(643\) − 38.3595i − 1.51275i −0.654137 0.756376i \(-0.726968\pi\)
0.654137 0.756376i \(-0.273032\pi\)
\(644\) 6.24914 0.246251
\(645\) 0 0
\(646\) −28.3741 −1.11637
\(647\) 25.7294i 1.01153i 0.862672 + 0.505763i \(0.168789\pi\)
−0.862672 + 0.505763i \(0.831211\pi\)
\(648\) 10.9379i 0.429682i
\(649\) 1.88273 0.0739038
\(650\) 0 0
\(651\) −21.4948 −0.842449
\(652\) − 18.2277i − 0.713850i
\(653\) − 18.4768i − 0.723053i −0.932362 0.361526i \(-0.882256\pi\)
0.932362 0.361526i \(-0.117744\pi\)
\(654\) −0.560352 −0.0219115
\(655\) 0 0
\(656\) −2.13187 −0.0832357
\(657\) − 21.6121i − 0.843169i
\(658\) 11.1138i 0.433262i
\(659\) 39.3776 1.53393 0.766966 0.641687i \(-0.221765\pi\)
0.766966 + 0.641687i \(0.221765\pi\)
\(660\) 0 0
\(661\) −34.2208 −1.33103 −0.665517 0.746383i \(-0.731789\pi\)
−0.665517 + 0.746383i \(0.731789\pi\)
\(662\) − 20.4362i − 0.794276i
\(663\) 13.7586i 0.534339i
\(664\) 8.49828 0.329797
\(665\) 0 0
\(666\) 8.74742 0.338956
\(667\) 54.6639i 2.11659i
\(668\) 8.00000i 0.309529i
\(669\) 23.3484 0.902700
\(670\) 0 0
\(671\) 9.11383 0.351835
\(672\) 2.24914i 0.0867625i
\(673\) 24.4622i 0.942948i 0.881880 + 0.471474i \(0.156278\pi\)
−0.881880 + 0.471474i \(0.843722\pi\)
\(674\) −7.88273 −0.303632
\(675\) 0 0
\(676\) −12.1138 −0.465916
\(677\) − 39.1070i − 1.50300i −0.659732 0.751501i \(-0.729330\pi\)
0.659732 0.751501i \(-0.270670\pi\)
\(678\) 24.7328i 0.949858i
\(679\) −15.3630 −0.589577
\(680\) 0 0
\(681\) −36.2345 −1.38851
\(682\) − 9.55691i − 0.365953i
\(683\) − 23.3776i − 0.894518i −0.894404 0.447259i \(-0.852400\pi\)
0.894404 0.447259i \(-0.147600\pi\)
\(684\) 8.98883 0.343697
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) − 54.6310i − 2.08430i
\(688\) − 7.67418i − 0.292575i
\(689\) 4.46907 0.170258
\(690\) 0 0
\(691\) 8.49828 0.323290 0.161645 0.986849i \(-0.448320\pi\)
0.161645 + 0.986849i \(0.448320\pi\)
\(692\) 0.117266i 0.00445780i
\(693\) 2.05863i 0.0782010i
\(694\) −13.5569 −0.514613
\(695\) 0 0
\(696\) −19.6742 −0.745748
\(697\) − 13.8535i − 0.524739i
\(698\) 10.7328i 0.406243i
\(699\) 15.4880 0.585809
\(700\) 0 0
\(701\) 14.9751 0.565601 0.282800 0.959179i \(-0.408737\pi\)
0.282800 + 0.959179i \(0.408737\pi\)
\(702\) 1.99312i 0.0752256i
\(703\) 18.5535i 0.699758i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −20.8578 −0.784994
\(707\) 16.8793i 0.634811i
\(708\) − 4.23453i − 0.159143i
\(709\) −40.7259 −1.52949 −0.764747 0.644330i \(-0.777136\pi\)
−0.764747 + 0.644330i \(0.777136\pi\)
\(710\) 0 0
\(711\) 17.2234 0.645927
\(712\) 12.3810i 0.463998i
\(713\) 59.7225i 2.23663i
\(714\) −14.6155 −0.546973
\(715\) 0 0
\(716\) −22.5535 −0.842863
\(717\) 25.0449i 0.935318i
\(718\) − 0.366407i − 0.0136742i
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 6.61555 0.246376
\(722\) 0.0655089i 0.00243799i
\(723\) 5.32238i 0.197942i
\(724\) −20.8793 −0.775973
\(725\) 0 0
\(726\) −2.24914 −0.0834734
\(727\) − 51.6413i − 1.91527i −0.287984 0.957635i \(-0.592985\pi\)
0.287984 0.957635i \(-0.407015\pi\)
\(728\) 0.941367i 0.0348894i
\(729\) 8.23109 0.304855
\(730\) 0 0
\(731\) 49.8690 1.84447
\(732\) − 20.4983i − 0.757638i
\(733\) 42.0000i 1.55131i 0.631160 + 0.775653i \(0.282579\pi\)
−0.631160 + 0.775653i \(0.717421\pi\)
\(734\) −20.8432 −0.769337
\(735\) 0 0
\(736\) 6.24914 0.230346
\(737\) 12.9966i 0.478735i
\(738\) 4.38875i 0.161552i
\(739\) 49.1070 1.80643 0.903214 0.429190i \(-0.141201\pi\)
0.903214 + 0.429190i \(0.141201\pi\)
\(740\) 0 0
\(741\) 9.24485 0.339618
\(742\) 4.74742i 0.174283i
\(743\) 53.2311i 1.95286i 0.215836 + 0.976430i \(0.430752\pi\)
−0.215836 + 0.976430i \(0.569248\pi\)
\(744\) −21.4948 −0.788039
\(745\) 0 0
\(746\) −5.37758 −0.196887
\(747\) − 17.4948i − 0.640103i
\(748\) − 6.49828i − 0.237601i
\(749\) −14.5535 −0.531772
\(750\) 0 0
\(751\) −31.6121 −1.15354 −0.576771 0.816906i \(-0.695688\pi\)
−0.576771 + 0.816906i \(0.695688\pi\)
\(752\) 11.1138i 0.405280i
\(753\) 56.4691i 2.05785i
\(754\) −8.23453 −0.299884
\(755\) 0 0
\(756\) −2.11727 −0.0770042
\(757\) − 1.24570i − 0.0452758i −0.999744 0.0226379i \(-0.992794\pi\)
0.999744 0.0226379i \(-0.00720649\pi\)
\(758\) − 3.53093i − 0.128249i
\(759\) 14.0552 0.510171
\(760\) 0 0
\(761\) −10.6009 −0.384284 −0.192142 0.981367i \(-0.561543\pi\)
−0.192142 + 0.981367i \(0.561543\pi\)
\(762\) 13.2311i 0.479312i
\(763\) − 0.249141i − 0.00901949i
\(764\) −3.11383 −0.112654
\(765\) 0 0
\(766\) 1.38445 0.0500223
\(767\) − 1.77234i − 0.0639956i
\(768\) 2.24914i 0.0811589i
\(769\) 19.1284 0.689789 0.344895 0.938641i \(-0.387915\pi\)
0.344895 + 0.938641i \(0.387915\pi\)
\(770\) 0 0
\(771\) 6.44309 0.232042
\(772\) − 6.17246i − 0.222152i
\(773\) − 39.9931i − 1.43845i −0.694776 0.719226i \(-0.744496\pi\)
0.694776 0.719226i \(-0.255504\pi\)
\(774\) −15.7983 −0.567859
\(775\) 0 0
\(776\) −15.3630 −0.551498
\(777\) 9.55691i 0.342852i
\(778\) − 15.7294i − 0.563925i
\(779\) −9.30863 −0.333516
\(780\) 0 0
\(781\) −14.6155 −0.522985
\(782\) 40.6087i 1.45216i
\(783\) − 18.5206i − 0.661873i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 2.81722 0.100487
\(787\) 37.9931i 1.35431i 0.735841 + 0.677154i \(0.236787\pi\)
−0.735841 + 0.677154i \(0.763213\pi\)
\(788\) − 6.73281i − 0.239847i
\(789\) 8.46907 0.301507
\(790\) 0 0
\(791\) −10.9966 −0.390993
\(792\) 2.05863i 0.0731503i
\(793\) − 8.57946i − 0.304665i
\(794\) 17.6121 0.625030
\(795\) 0 0
\(796\) −13.2311 −0.468964
\(797\) − 20.3810i − 0.721933i −0.932579 0.360966i \(-0.882447\pi\)
0.932579 0.360966i \(-0.117553\pi\)
\(798\) 9.82066i 0.347648i
\(799\) −72.2208 −2.55499
\(800\) 0 0
\(801\) 25.4880 0.900573
\(802\) − 32.8172i − 1.15882i
\(803\) 10.4983i 0.370476i
\(804\) 29.2311 1.03090
\(805\) 0 0
\(806\) −8.99656 −0.316890
\(807\) − 20.1104i − 0.707919i
\(808\) 16.8793i 0.593812i
\(809\) 31.7294 1.11555 0.557773 0.829994i \(-0.311656\pi\)
0.557773 + 0.829994i \(0.311656\pi\)
\(810\) 0 0
\(811\) 43.5095 1.52782 0.763912 0.645321i \(-0.223276\pi\)
0.763912 + 0.645321i \(0.223276\pi\)
\(812\) − 8.74742i − 0.306974i
\(813\) 48.3449i 1.69553i
\(814\) −4.24914 −0.148932
\(815\) 0 0
\(816\) −14.6155 −0.511646
\(817\) − 33.5086i − 1.17232i
\(818\) 33.3561i 1.16627i
\(819\) 1.93793 0.0677167
\(820\) 0 0
\(821\) −24.7766 −0.864711 −0.432355 0.901703i \(-0.642317\pi\)
−0.432355 + 0.901703i \(0.642317\pi\)
\(822\) − 22.4914i − 0.784478i
\(823\) − 28.8647i − 1.00616i −0.864240 0.503080i \(-0.832200\pi\)
0.864240 0.503080i \(-0.167800\pi\)
\(824\) 6.61555 0.230464
\(825\) 0 0
\(826\) 1.88273 0.0655087
\(827\) 34.2277i 1.19021i 0.803647 + 0.595106i \(0.202890\pi\)
−0.803647 + 0.595106i \(0.797110\pi\)
\(828\) − 12.8647i − 0.447079i
\(829\) −23.6381 −0.820985 −0.410492 0.911864i \(-0.634643\pi\)
−0.410492 + 0.911864i \(0.634643\pi\)
\(830\) 0 0
\(831\) −17.2019 −0.596727
\(832\) 0.941367i 0.0326360i
\(833\) − 6.49828i − 0.225152i
\(834\) 24.7000 0.855290
\(835\) 0 0
\(836\) −4.36641 −0.151015
\(837\) − 20.2345i − 0.699408i
\(838\) − 12.3449i − 0.426448i
\(839\) 24.9053 0.859826 0.429913 0.902870i \(-0.358544\pi\)
0.429913 + 0.902870i \(0.358544\pi\)
\(840\) 0 0
\(841\) 47.5174 1.63853
\(842\) − 8.87930i − 0.306001i
\(843\) − 64.3604i − 2.21669i
\(844\) −23.1138 −0.795611
\(845\) 0 0
\(846\) 22.8793 0.786606
\(847\) − 1.00000i − 0.0343604i
\(848\) 4.74742i 0.163027i
\(849\) −6.47594 −0.222254
\(850\) 0 0
\(851\) 26.5535 0.910241
\(852\) 32.8724i 1.12619i
\(853\) − 22.9966i − 0.787387i −0.919242 0.393694i \(-0.871197\pi\)
0.919242 0.393694i \(-0.128803\pi\)
\(854\) 9.11383 0.311869
\(855\) 0 0
\(856\) −14.5535 −0.497428
\(857\) − 31.2603i − 1.06783i −0.845538 0.533916i \(-0.820720\pi\)
0.845538 0.533916i \(-0.179280\pi\)
\(858\) 2.11727i 0.0722823i
\(859\) −20.3449 −0.694160 −0.347080 0.937836i \(-0.612827\pi\)
−0.347080 + 0.937836i \(0.612827\pi\)
\(860\) 0 0
\(861\) −4.79488 −0.163409
\(862\) 18.2784i 0.622563i
\(863\) 58.3595i 1.98658i 0.115644 + 0.993291i \(0.463107\pi\)
−0.115644 + 0.993291i \(0.536893\pi\)
\(864\) −2.11727 −0.0720309
\(865\) 0 0
\(866\) 6.86469 0.233272
\(867\) − 56.7405i − 1.92701i
\(868\) − 9.55691i − 0.324383i
\(869\) −8.36641 −0.283811
\(870\) 0 0
\(871\) 12.2345 0.414551
\(872\) − 0.249141i − 0.00843696i
\(873\) 31.6267i 1.07040i
\(874\) 27.2863 0.922973
\(875\) 0 0
\(876\) 23.6121 0.797779
\(877\) 11.9639i 0.403992i 0.979386 + 0.201996i \(0.0647429\pi\)
−0.979386 + 0.201996i \(0.935257\pi\)
\(878\) 11.8466i 0.399805i
\(879\) −29.1070 −0.981753
\(880\) 0 0
\(881\) 5.88961 0.198426 0.0992130 0.995066i \(-0.468367\pi\)
0.0992130 + 0.995066i \(0.468367\pi\)
\(882\) 2.05863i 0.0693178i
\(883\) 19.4880i 0.655822i 0.944709 + 0.327911i \(0.106345\pi\)
−0.944709 + 0.327911i \(0.893655\pi\)
\(884\) −6.11727 −0.205746
\(885\) 0 0
\(886\) 17.8827 0.600782
\(887\) 50.4622i 1.69435i 0.531310 + 0.847177i \(0.321700\pi\)
−0.531310 + 0.847177i \(0.678300\pi\)
\(888\) 9.55691i 0.320709i
\(889\) −5.88273 −0.197301
\(890\) 0 0
\(891\) −10.9379 −0.366434
\(892\) 10.3810i 0.347582i
\(893\) 48.5275i 1.62391i
\(894\) 0.0328552 0.00109884
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) − 13.2311i − 0.441773i
\(898\) 16.8172i 0.561198i
\(899\) 83.5984 2.78816
\(900\) 0 0
\(901\) −30.8501 −1.02777
\(902\) − 2.13187i − 0.0709836i
\(903\) − 17.2603i − 0.574387i
\(904\) −10.9966 −0.365740
\(905\) 0 0
\(906\) 34.2897 1.13920
\(907\) − 16.8432i − 0.559269i −0.960106 0.279635i \(-0.909787\pi\)
0.960106 0.279635i \(-0.0902134\pi\)
\(908\) − 16.1104i − 0.534642i
\(909\) 34.7483 1.15253
\(910\) 0 0
\(911\) −38.6155 −1.27939 −0.639695 0.768629i \(-0.720939\pi\)
−0.639695 + 0.768629i \(0.720939\pi\)
\(912\) 9.82066i 0.325195i
\(913\) 8.49828i 0.281252i
\(914\) 12.2277 0.404455
\(915\) 0 0
\(916\) 24.2897 0.802555
\(917\) 1.25258i 0.0413638i
\(918\) − 13.7586i − 0.454101i
\(919\) −17.2818 −0.570074 −0.285037 0.958517i \(-0.592006\pi\)
−0.285037 + 0.958517i \(0.592006\pi\)
\(920\) 0 0
\(921\) −1.12070 −0.0369285
\(922\) − 16.6155i − 0.547204i
\(923\) 13.7586i 0.452870i
\(924\) −2.24914 −0.0739913
\(925\) 0 0
\(926\) 18.4837 0.607411
\(927\) − 13.6190i − 0.447306i
\(928\) − 8.74742i − 0.287148i
\(929\) −2.91539 −0.0956508 −0.0478254 0.998856i \(-0.515229\pi\)
−0.0478254 + 0.998856i \(0.515229\pi\)
\(930\) 0 0
\(931\) −4.36641 −0.143103
\(932\) 6.88617i 0.225564i
\(933\) − 52.7191i − 1.72594i
\(934\) −24.3956 −0.798249
\(935\) 0 0
\(936\) 1.93793 0.0633432
\(937\) 22.5795i 0.737639i 0.929501 + 0.368819i \(0.120238\pi\)
−0.929501 + 0.368819i \(0.879762\pi\)
\(938\) 12.9966i 0.424353i
\(939\) 21.0741 0.687727
\(940\) 0 0
\(941\) −21.7655 −0.709534 −0.354767 0.934955i \(-0.615440\pi\)
−0.354767 + 0.934955i \(0.615440\pi\)
\(942\) 12.3741i 0.403171i
\(943\) 13.3224i 0.433836i
\(944\) 1.88273 0.0612778
\(945\) 0 0
\(946\) 7.67418 0.249509
\(947\) − 1.49484i − 0.0485759i −0.999705 0.0242879i \(-0.992268\pi\)
0.999705 0.0242879i \(-0.00773185\pi\)
\(948\) 18.8172i 0.611155i
\(949\) 9.88273 0.320807
\(950\) 0 0
\(951\) 22.4431 0.727767
\(952\) − 6.49828i − 0.210610i
\(953\) 4.83098i 0.156491i 0.996934 + 0.0782453i \(0.0249317\pi\)
−0.996934 + 0.0782453i \(0.975068\pi\)
\(954\) 9.77320 0.316419
\(955\) 0 0
\(956\) −11.1353 −0.360142
\(957\) − 19.6742i − 0.635976i
\(958\) 27.7655i 0.897062i
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 60.3346 1.94628
\(962\) 4.00000i 0.128965i
\(963\) 29.9603i 0.965456i
\(964\) −2.36641 −0.0762168
\(965\) 0 0
\(966\) 14.0552 0.452218
\(967\) − 51.9278i − 1.66989i −0.550336 0.834943i \(-0.685501\pi\)
0.550336 0.834943i \(-0.314499\pi\)
\(968\) − 1.00000i − 0.0321412i
\(969\) −63.8174 −2.05011
\(970\) 0 0
\(971\) 59.6933 1.91565 0.957824 0.287354i \(-0.0927757\pi\)
0.957824 + 0.287354i \(0.0927757\pi\)
\(972\) 18.2491i 0.585341i
\(973\) 10.9820i 0.352065i
\(974\) −10.0958 −0.323490
\(975\) 0 0
\(976\) 9.11383 0.291727
\(977\) 31.2311i 0.999171i 0.866265 + 0.499586i \(0.166514\pi\)
−0.866265 + 0.499586i \(0.833486\pi\)
\(978\) − 40.9966i − 1.31093i
\(979\) −12.3810 −0.395699
\(980\) 0 0
\(981\) −0.512889 −0.0163753
\(982\) − 32.1104i − 1.02468i
\(983\) 22.6155i 0.721324i 0.932697 + 0.360662i \(0.117449\pi\)
−0.932697 + 0.360662i \(0.882551\pi\)
\(984\) −4.79488 −0.152855
\(985\) 0 0
\(986\) 56.8432 1.81026
\(987\) 24.9966i 0.795649i
\(988\) 4.11039i 0.130769i
\(989\) −47.9570 −1.52494
\(990\) 0 0
\(991\) 22.4102 0.711884 0.355942 0.934508i \(-0.384160\pi\)
0.355942 + 0.934508i \(0.384160\pi\)
\(992\) − 9.55691i − 0.303432i
\(993\) − 45.9639i − 1.45862i
\(994\) −14.6155 −0.463577
\(995\) 0 0
\(996\) 19.1138 0.605645
\(997\) − 27.3415i − 0.865914i −0.901415 0.432957i \(-0.857470\pi\)
0.901415 0.432957i \(-0.142530\pi\)
\(998\) 8.79488i 0.278397i
\(999\) −8.99656 −0.284639
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 3850.2.c.ba.1849.6 6
5.2 odd 4 3850.2.a.bt.1.3 3
5.3 odd 4 770.2.a.m.1.1 3
5.4 even 2 inner 3850.2.c.ba.1849.1 6
15.8 even 4 6930.2.a.ce.1.2 3
20.3 even 4 6160.2.a.bf.1.3 3
35.13 even 4 5390.2.a.ca.1.3 3
55.43 even 4 8470.2.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.1 3 5.3 odd 4
3850.2.a.bt.1.3 3 5.2 odd 4
3850.2.c.ba.1849.1 6 5.4 even 2 inner
3850.2.c.ba.1849.6 6 1.1 even 1 trivial
5390.2.a.ca.1.3 3 35.13 even 4
6160.2.a.bf.1.3 3 20.3 even 4
6930.2.a.ce.1.2 3 15.8 even 4
8470.2.a.ci.1.1 3 55.43 even 4