Properties

Label 3850.2.a.ba.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} +2.00000 q^{24} -2.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} +8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +12.0000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -12.0000 q^{47} +2.00000 q^{48} +1.00000 q^{49} +12.0000 q^{51} -2.00000 q^{52} -4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} +2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} +12.0000 q^{69} +12.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} +4.00000 q^{74} +2.00000 q^{76} +1.00000 q^{77} -4.00000 q^{78} +14.0000 q^{79} -11.0000 q^{81} +12.0000 q^{82} -12.0000 q^{83} -2.00000 q^{84} +4.00000 q^{86} -1.00000 q^{88} +6.00000 q^{89} +2.00000 q^{91} +6.00000 q^{92} +16.0000 q^{93} -12.0000 q^{94} +2.00000 q^{96} -8.00000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −4.00000 −0.769800
\(28\) −1.00000 −0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 2.00000 0.324443
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) −2.00000 −0.277350
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 1.00000 0.113961
\(78\) −4.00000 −0.452911
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 12.0000 1.32518
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 6.00000 0.625543
\(93\) 16.0000 1.65912
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 12.0000 1.18818
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −4.00000 −0.384900
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 24.0000 2.16401
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −2.00000 −0.174078
\(133\) −2.00000 −0.173422
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 12.0000 1.02151
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 12.0000 1.00702
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 2.00000 0.164957
\(148\) 4.00000 0.328798
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 2.00000 0.162221
\(153\) 6.00000 0.485071
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 14.0000 1.11378
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) −11.0000 −0.864242
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) 4.00000 0.295689
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 16.0000 1.17318
\(187\) −6.00000 −0.438763
\(188\) −12.0000 −0.875190
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 2.00000 0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 6.00000 0.417029
\(208\) −2.00000 −0.138675
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) −8.00000 −0.543075
\(218\) −16.0000 −1.08366
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 8.00000 0.536925
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 28.0000 1.81880
\(238\) −6.00000 −0.388922
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 24.0000 1.53018
\(247\) −4.00000 −0.254514
\(248\) 8.00000 0.508001
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −6.00000 −0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 8.00000 0.498058
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 12.0000 0.734388
\(268\) −8.00000 −0.488678
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000 0.363803
\(273\) 4.00000 0.242091
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 14.0000 0.839664
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −24.0000 −1.42918
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 14.0000 0.805609
\(303\) −12.0000 −0.689382
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 1.00000 0.0569803
\(309\) 32.0000 1.82042
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −4.00000 −0.226455
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) −6.00000 −0.334367
\(323\) 12.0000 0.667698
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) −32.0000 −1.76960
\(328\) 12.0000 0.662589
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −12.0000 −0.658586
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 2.00000 0.108148
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) −1.00000 −0.0533002
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −12.0000 −0.635107
\(358\) −12.0000 −0.634220
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) 2.00000 0.104973
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 6.00000 0.312772
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 16.0000 0.829561
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −24.0000 −1.22795
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 4.00000 0.203331
\(388\) −8.00000 −0.406138
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 1.00000 0.0505076
\(393\) −36.0000 −1.81596
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −16.0000 −0.802008
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −16.0000 −0.798007
\(403\) −16.0000 −0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 12.0000 0.594089
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 28.0000 1.37117
\(418\) −2.00000 −0.0978232
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) −2.00000 −0.0967868
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −4.00000 −0.192450
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 12.0000 0.574038
\(438\) −4.00000 −0.191127
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 6.00000 0.282216
\(453\) 28.0000 1.31555
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −10.0000 −0.467269
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 2.00000 0.0930484
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 28.0000 1.28608
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 20.0000 0.910975
\(483\) −12.0000 −0.546019
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 2.00000 0.0905357
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 24.0000 1.08200
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) −24.0000 −1.07547
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −18.0000 −0.799408
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −24.0000 −1.05859
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 12.0000 0.527759
\(518\) −4.00000 −0.175750
\(519\) 36.0000 1.58022
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 48.0000 2.09091
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) −24.0000 −1.03956
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) −24.0000 −1.03568
\(538\) −18.0000 −0.776035
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 20.0000 0.859074
\(543\) 4.00000 0.171656
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 12.0000 0.510754
\(553\) −14.0000 −0.595341
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 8.00000 0.338667
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 6.00000 0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −24.0000 −1.01058
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 11.0000 0.461957
\(568\) 12.0000 0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 2.00000 0.0836242
\(573\) −48.0000 −2.00523
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 19.0000 0.790296
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) −16.0000 −0.663221
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 30.0000 1.23823 0.619116 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 2.00000 0.0824786
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 4.00000 0.164399
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 0 0
\(597\) −32.0000 −1.30967
\(598\) −12.0000 −0.490716
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) −4.00000 −0.163028
\(603\) −8.00000 −0.325785
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 6.00000 0.242536
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 32.0000 1.28723
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) −4.00000 −0.159745
\(628\) −14.0000 −0.558661
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 14.0000 0.556890
\(633\) −8.00000 −0.317971
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −24.0000 −0.947204
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) −8.00000 −0.313304
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) −32.0000 −1.25130
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) −2.00000 −0.0780274
\(658\) 12.0000 0.467809
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −28.0000 −1.08825
\(663\) −24.0000 −0.932083
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) −2.00000 −0.0771517
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 12.0000 0.460857
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) −8.00000 −0.306336
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −20.0000 −0.763048
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 18.0000 0.684257
\(693\) 1.00000 0.0379869
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 72.0000 2.72719
\(698\) 14.0000 0.529908
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 8.00000 0.301941
\(703\) 8.00000 0.301726
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 6.00000 0.224860
\(713\) 48.0000 1.79761
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 36.0000 1.34444
\(718\) 30.0000 1.11959
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −15.0000 −0.558242
\(723\) 40.0000 1.48762
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −20.0000 −0.741759 −0.370879 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 4.00000 0.147844
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 8.00000 0.294684
\(738\) 12.0000 0.441726
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 16.0000 0.586588
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) −12.0000 −0.439057
\(748\) −6.00000 −0.219382
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) −28.0000 −1.01701
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) −16.0000 −0.579619
\(763\) 16.0000 0.579239
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −48.0000 −1.72868
\(772\) −2.00000 −0.0719816
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) −8.00000 −0.286998
\(778\) −6.00000 −0.215110
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 36.0000 1.28736
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −36.0000 −1.28408
\(787\) 16.0000 0.570338 0.285169 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) −6.00000 −0.213741
\(789\) −48.0000 −1.70885
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −1.00000 −0.0355335
\(793\) −4.00000 −0.142044
\(794\) −38.0000 −1.34857
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −4.00000 −0.141598
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 30.0000 1.05934
\(803\) 2.00000 0.0705785
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −36.0000 −1.26726
\(808\) −6.00000 −0.211079
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) 40.0000 1.40286
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 8.00000 0.279885
\(818\) −4.00000 −0.139857
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 12.0000 0.418548
\(823\) −50.0000 −1.74289 −0.871445 0.490493i \(-0.836817\pi\)
−0.871445 + 0.490493i \(0.836817\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 6.00000 0.208514
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) −2.00000 −0.0693375
\(833\) 6.00000 0.207888
\(834\) 28.0000 0.969561
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) −32.0000 −1.10608
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −10.0000 −0.344623
\(843\) 12.0000 0.413302
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 24.0000 0.822226
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 4.00000 0.136558
\(859\) −16.0000 −0.545913 −0.272956 0.962026i \(-0.588002\pi\)
−0.272956 + 0.962026i \(0.588002\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 30.0000 1.02180
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) 38.0000 1.29055
\(868\) −8.00000 −0.271538
\(869\) −14.0000 −0.474917
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −16.0000 −0.541828
\(873\) −8.00000 −0.270759
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) −28.0000 −0.944954
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 1.00000 0.0336718
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 8.00000 0.268462
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) −8.00000 −0.267860
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −24.0000 −0.801337
\(898\) −18.0000 −0.600668
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −12.0000 −0.399556
\(903\) −8.00000 −0.266223
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 28.0000 0.930238
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −24.0000 −0.796468
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 4.00000 0.132453
\(913\) 12.0000 0.397142
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 18.0000 0.594412
\(918\) −24.0000 −0.792118
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) −18.0000 −0.592798
\(923\) −24.0000 −0.789970
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) 22.0000 0.722965
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −6.00000 −0.196537
\(933\) 48.0000 1.57145
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 8.00000 0.261209
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −28.0000 −0.912289
\(943\) 72.0000 2.34464
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 28.0000 0.909398
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) −6.00000 −0.194461
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −8.00000 −0.257930
\(963\) −12.0000 −0.386695
\(964\) 20.0000 0.644157
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 1.00000 0.0321412
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) −10.0000 −0.320750
\(973\) −14.0000 −0.448819
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −16.0000 −0.511624
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −36.0000 −1.14881
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 24.0000 0.765092
\(985\) 0 0
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) −4.00000 −0.127257
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 8.00000 0.254000
\(993\) −56.0000 −1.77711
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −4.00000 −0.126618
\(999\) −16.0000 −0.506218
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.a.ba.1.1 1
5.2 odd 4 3850.2.c.o.1849.2 2
5.3 odd 4 3850.2.c.o.1849.1 2
5.4 even 2 770.2.a.a.1.1 1
15.14 odd 2 6930.2.a.bm.1.1 1
20.19 odd 2 6160.2.a.k.1.1 1
35.34 odd 2 5390.2.a.r.1.1 1
55.54 odd 2 8470.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.a.1.1 1 5.4 even 2
3850.2.a.ba.1.1 1 1.1 even 1 trivial
3850.2.c.o.1849.1 2 5.3 odd 4
3850.2.c.o.1849.2 2 5.2 odd 4
5390.2.a.r.1.1 1 35.34 odd 2
6160.2.a.k.1.1 1 20.19 odd 2
6930.2.a.bm.1.1 1 15.14 odd 2
8470.2.a.r.1.1 1 55.54 odd 2