Properties

Label 975.2.a.d.1.1
Level $975$
Weight $2$
Character 975.1
Self dual yes
Analytic conductor $7.785$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 975.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.00000 q^{14} -1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -3.00000 q^{21} +1.00000 q^{22} -3.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} +1.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} +1.00000 q^{33} -5.00000 q^{34} -1.00000 q^{36} +8.00000 q^{37} +8.00000 q^{38} +1.00000 q^{39} -2.00000 q^{41} +3.00000 q^{42} -8.00000 q^{43} +1.00000 q^{44} +11.0000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -5.00000 q^{51} +1.00000 q^{52} +11.0000 q^{53} +1.00000 q^{54} +9.00000 q^{56} +8.00000 q^{57} -1.00000 q^{58} +5.00000 q^{59} +1.00000 q^{61} -3.00000 q^{62} +3.00000 q^{63} +7.00000 q^{64} -1.00000 q^{66} -3.00000 q^{67} -5.00000 q^{68} +16.0000 q^{71} +3.00000 q^{72} +4.00000 q^{73} -8.00000 q^{74} +8.00000 q^{76} -3.00000 q^{77} -1.00000 q^{78} +12.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +3.00000 q^{83} +3.00000 q^{84} +8.00000 q^{86} -1.00000 q^{87} -3.00000 q^{88} -3.00000 q^{91} -3.00000 q^{93} -11.0000 q^{94} +5.00000 q^{96} +2.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −5.00000 −0.883883
\(33\) 1.00000 0.174078
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 8.00000 1.29777
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 3.00000 0.462910
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) 11.0000 1.60451 0.802257 0.596978i \(-0.203632\pi\)
0.802257 + 0.596978i \(0.203632\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 1.00000 0.138675
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 9.00000 1.20268
\(57\) 8.00000 1.05963
\(58\) −1.00000 −0.131306
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −3.00000 −0.381000
\(63\) 3.00000 0.377964
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) −5.00000 −0.606339
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 3.00000 0.353553
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) −3.00000 −0.341882
\(78\) −1.00000 −0.113228
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −1.00000 −0.107211
\(88\) −3.00000 −0.319801
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) −11.0000 −1.13456
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −2.00000 −0.202031
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 5.00000 0.495074
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −11.0000 −1.06841
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −3.00000 −0.283473
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) −1.00000 −0.0924500
\(118\) −5.00000 −0.460287
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) 2.00000 0.180334
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −24.0000 −2.08106
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) 15.0000 1.28624
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) −11.0000 −0.926367
\(142\) −16.0000 −1.34269
\(143\) 1.00000 0.0836242
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) −2.00000 −0.164957
\(148\) −8.00000 −0.657596
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) −24.0000 −1.94666
\(153\) 5.00000 0.404226
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) −12.0000 −0.954669
\(159\) −11.0000 −0.872357
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −9.00000 −0.694365
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 8.00000 0.609994
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −5.00000 −0.375823
\(178\) 0 0
\(179\) 26.0000 1.94333 0.971666 0.236360i \(-0.0759544\pi\)
0.971666 + 0.236360i \(0.0759544\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 3.00000 0.222375
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) −5.00000 −0.365636
\(188\) −11.0000 −0.802257
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) −7.00000 −0.505181
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) −9.00000 −0.633238
\(203\) 3.00000 0.210559
\(204\) 5.00000 0.350070
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −11.0000 −0.755483
\(213\) −16.0000 −1.09630
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 9.00000 0.610960
\(218\) 10.0000 0.677285
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 8.00000 0.536925
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −15.0000 −1.00223
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) −23.0000 −1.52656 −0.763282 0.646066i \(-0.776413\pi\)
−0.763282 + 0.646066i \(0.776413\pi\)
\(228\) −8.00000 −0.529813
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 3.00000 0.196960
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −5.00000 −0.325472
\(237\) −12.0000 −0.779484
\(238\) −15.0000 −0.972306
\(239\) −19.0000 −1.22901 −0.614504 0.788914i \(-0.710644\pi\)
−0.614504 + 0.788914i \(0.710644\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 10.0000 0.642824
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 8.00000 0.509028
\(248\) 9.00000 0.571501
\(249\) −3.00000 −0.190117
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) −8.00000 −0.498058
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −6.00000 −0.370681
\(263\) −14.0000 −0.863277 −0.431638 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 24.0000 1.47153
\(267\) 0 0
\(268\) 3.00000 0.183254
\(269\) 17.0000 1.03651 0.518254 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) 0 0
\(271\) −5.00000 −0.303728 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(272\) −5.00000 −0.303170
\(273\) 3.00000 0.181568
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 10.0000 0.599760
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 11.0000 0.655040
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) −6.00000 −0.354169
\(288\) −5.00000 −0.294628
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −4.00000 −0.234082
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 24.0000 1.39497
\(297\) 1.00000 0.0580259
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 17.0000 0.978240
\(303\) −9.00000 −0.517036
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −5.00000 −0.285831
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 3.00000 0.170941
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 3.00000 0.169842
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) −11.0000 −0.620766
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 16.0000 0.898650 0.449325 0.893368i \(-0.351665\pi\)
0.449325 + 0.893368i \(0.351665\pi\)
\(318\) 11.0000 0.616849
\(319\) −1.00000 −0.0559893
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) −40.0000 −2.22566
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 10.0000 0.553001
\(328\) −6.00000 −0.331295
\(329\) 33.0000 1.81935
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −3.00000 −0.164646
\(333\) 8.00000 0.438397
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) 8.00000 0.432590
\(343\) −15.0000 −0.809924
\(344\) −24.0000 −1.29399
\(345\) 0 0
\(346\) −21.0000 −1.12897
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 1.00000 0.0536056
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 5.00000 0.266501
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 5.00000 0.265747
\(355\) 0 0
\(356\) 0 0
\(357\) −15.0000 −0.793884
\(358\) −26.0000 −1.37414
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −5.00000 −0.262794
\(363\) 10.0000 0.524864
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) 1.00000 0.0522708
\(367\) 12.0000 0.626395 0.313197 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 33.0000 1.71327
\(372\) 3.00000 0.155543
\(373\) 31.0000 1.60512 0.802560 0.596572i \(-0.203471\pi\)
0.802560 + 0.596572i \(0.203471\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 33.0000 1.70185
\(377\) −1.00000 −0.0515026
\(378\) 3.00000 0.154303
\(379\) −35.0000 −1.79783 −0.898915 0.438124i \(-0.855643\pi\)
−0.898915 + 0.438124i \(0.855643\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −10.0000 −0.511645
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −8.00000 −0.406663
\(388\) −2.00000 −0.101535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) −6.00000 −0.302660
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −10.0000 −0.501255
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −3.00000 −0.149626
\(403\) −3.00000 −0.149441
\(404\) −9.00000 −0.447767
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) −8.00000 −0.396545
\(408\) −15.0000 −0.742611
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −14.0000 −0.689730
\(413\) 15.0000 0.738102
\(414\) 0 0
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) 10.0000 0.489702
\(418\) −8.00000 −0.391293
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 16.0000 0.778868
\(423\) 11.0000 0.534838
\(424\) 33.0000 1.60262
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 3.00000 0.145180
\(428\) 18.0000 0.870063
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −9.00000 −0.432014
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 5.00000 0.237826
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −10.0000 −0.472984
\(448\) 21.0000 0.992157
\(449\) −28.0000 −1.32140 −0.660701 0.750649i \(-0.729741\pi\)
−0.660701 + 0.750649i \(0.729741\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −2.00000 −0.0940721
\(453\) 17.0000 0.798730
\(454\) 23.0000 1.07944
\(455\) 0 0
\(456\) 24.0000 1.12390
\(457\) 34.0000 1.59045 0.795226 0.606313i \(-0.207352\pi\)
0.795226 + 0.606313i \(0.207352\pi\)
\(458\) 18.0000 0.841085
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) −3.00000 −0.139573
\(463\) 9.00000 0.418265 0.209133 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 1.00000 0.0462250
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) −11.0000 −0.506853
\(472\) 15.0000 0.690431
\(473\) 8.00000 0.367840
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) −15.0000 −0.687524
\(477\) 11.0000 0.503655
\(478\) 19.0000 0.869040
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −37.0000 −1.67663 −0.838315 0.545186i \(-0.816459\pi\)
−0.838315 + 0.545186i \(0.816459\pi\)
\(488\) 3.00000 0.135804
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 5.00000 0.225189
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 48.0000 2.15309
\(498\) 3.00000 0.134433
\(499\) −41.0000 −1.83541 −0.917706 0.397260i \(-0.869961\pi\)
−0.917706 + 0.397260i \(0.869961\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 30.0000 1.33897
\(503\) −26.0000 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 9.00000 0.400892
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 8.00000 0.354943
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 11.0000 0.486136
\(513\) 8.00000 0.353209
\(514\) −3.00000 −0.132324
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) −11.0000 −0.483779
\(518\) −24.0000 −1.05450
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) 15.0000 0.653410
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) 24.0000 1.04053
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) −26.0000 −1.12198
\(538\) −17.0000 −0.732922
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 12.0000 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) 5.00000 0.214768
\(543\) −5.00000 −0.214571
\(544\) −25.0000 −1.07187
\(545\) 0 0
\(546\) −3.00000 −0.128388
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −12.0000 −0.512615
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −3.00000 −0.127000
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 18.0000 0.759284
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 11.0000 0.463184
\(565\) 0 0
\(566\) −18.0000 −0.756596
\(567\) 3.00000 0.125988
\(568\) 48.0000 2.01404
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −1.00000 −0.0418121
\(573\) −10.0000 −0.417756
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −8.00000 −0.332756
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) 2.00000 0.0829027
\(583\) −11.0000 −0.455573
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) 45.0000 1.85735 0.928674 0.370896i \(-0.120949\pi\)
0.928674 + 0.370896i \(0.120949\pi\)
\(588\) 2.00000 0.0824786
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −8.00000 −0.328798
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) 46.0000 1.87951 0.939755 0.341850i \(-0.111053\pi\)
0.939755 + 0.341850i \(0.111053\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 24.0000 0.978167
\(603\) −3.00000 −0.122169
\(604\) 17.0000 0.691720
\(605\) 0 0
\(606\) 9.00000 0.365600
\(607\) −38.0000 −1.54237 −0.771186 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) 40.0000 1.62221
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) −11.0000 −0.445012
\(612\) −5.00000 −0.202113
\(613\) −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
\(616\) −9.00000 −0.362620
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 14.0000 0.563163
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −29.0000 −1.15907
\(627\) −8.00000 −0.319489
\(628\) −11.0000 −0.438948
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 36.0000 1.43200
\(633\) 16.0000 0.635943
\(634\) −16.0000 −0.635441
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) −2.00000 −0.0792429
\(638\) 1.00000 0.0395904
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) −18.0000 −0.710403
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 40.0000 1.57378
\(647\) −16.0000 −0.629025 −0.314512 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 3.00000 0.117851
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) −20.0000 −0.783260
\(653\) −35.0000 −1.36966 −0.684828 0.728705i \(-0.740123\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 4.00000 0.156055
\(658\) −33.0000 −1.28647
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 12.0000 0.466393
\(663\) 5.00000 0.194184
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −1.00000 −0.0386046
\(672\) 15.0000 0.578638
\(673\) −45.0000 −1.73462 −0.867311 0.497766i \(-0.834154\pi\)
−0.867311 + 0.497766i \(0.834154\pi\)
\(674\) 5.00000 0.192593
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 2.00000 0.0768095
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 23.0000 0.881362
\(682\) 3.00000 0.114876
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) 18.0000 0.686743
\(688\) 8.00000 0.304997
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) −29.0000 −1.10321 −0.551606 0.834105i \(-0.685985\pi\)
−0.551606 + 0.834105i \(0.685985\pi\)
\(692\) −21.0000 −0.798300
\(693\) −3.00000 −0.113961
\(694\) −2.00000 −0.0759190
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −10.0000 −0.378777
\(698\) −16.0000 −0.605609
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 31.0000 1.17085 0.585427 0.810725i \(-0.300927\pi\)
0.585427 + 0.810725i \(0.300927\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −64.0000 −2.41381
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 27.0000 1.01544
\(708\) 5.00000 0.187912
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 0 0
\(714\) 15.0000 0.561361
\(715\) 0 0
\(716\) −26.0000 −0.971666
\(717\) 19.0000 0.709568
\(718\) −15.0000 −0.559795
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) −45.0000 −1.67473
\(723\) −10.0000 −0.371904
\(724\) −5.00000 −0.185824
\(725\) 0 0
\(726\) −10.0000 −0.371135
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) −9.00000 −0.333562
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 1.00000 0.0369611
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 0 0
\(737\) 3.00000 0.110506
\(738\) 2.00000 0.0736210
\(739\) 7.00000 0.257499 0.128750 0.991677i \(-0.458904\pi\)
0.128750 + 0.991677i \(0.458904\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) −33.0000 −1.21147
\(743\) −13.0000 −0.476924 −0.238462 0.971152i \(-0.576643\pi\)
−0.238462 + 0.971152i \(0.576643\pi\)
\(744\) −9.00000 −0.329956
\(745\) 0 0
\(746\) −31.0000 −1.13499
\(747\) 3.00000 0.109764
\(748\) 5.00000 0.182818
\(749\) −54.0000 −1.97312
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −11.0000 −0.401129
\(753\) 30.0000 1.09326
\(754\) 1.00000 0.0364179
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) −23.0000 −0.835949 −0.417975 0.908459i \(-0.637260\pi\)
−0.417975 + 0.908459i \(0.637260\pi\)
\(758\) 35.0000 1.27126
\(759\) 0 0
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) −8.00000 −0.289809
\(763\) −30.0000 −1.08607
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −5.00000 −0.180540
\(768\) 17.0000 0.613435
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) 10.0000 0.359908
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) −24.0000 −0.860995
\(778\) 30.0000 1.07555
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) 6.00000 0.214013
\(787\) −17.0000 −0.605985 −0.302992 0.952993i \(-0.597986\pi\)
−0.302992 + 0.952993i \(0.597986\pi\)
\(788\) −6.00000 −0.213741
\(789\) 14.0000 0.498413
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) −3.00000 −0.106600
\(793\) −1.00000 −0.0355110
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 43.0000 1.52314 0.761569 0.648084i \(-0.224429\pi\)
0.761569 + 0.648084i \(0.224429\pi\)
\(798\) −24.0000 −0.849591
\(799\) 55.0000 1.94576
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.00000 −0.141157
\(804\) −3.00000 −0.105802
\(805\) 0 0
\(806\) 3.00000 0.105670
\(807\) −17.0000 −0.598428
\(808\) 27.0000 0.949857
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −33.0000 −1.15879 −0.579393 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) −3.00000 −0.105279
\(813\) 5.00000 0.175358
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 5.00000 0.175035
\(817\) 64.0000 2.23908
\(818\) −2.00000 −0.0699284
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 12.0000 0.418548
\(823\) −28.0000 −0.976019 −0.488009 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 42.0000 1.46314
\(825\) 0 0
\(826\) −15.0000 −0.521917
\(827\) −1.00000 −0.0347734 −0.0173867 0.999849i \(-0.505535\pi\)
−0.0173867 + 0.999849i \(0.505535\pi\)
\(828\) 0 0
\(829\) 51.0000 1.77130 0.885652 0.464350i \(-0.153712\pi\)
0.885652 + 0.464350i \(0.153712\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) −7.00000 −0.242681
\(833\) 10.0000 0.346479
\(834\) −10.0000 −0.346272
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) −3.00000 −0.103695
\(838\) 26.0000 0.898155
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −28.0000 −0.964944
\(843\) 18.0000 0.619953
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) −11.0000 −0.378188
\(847\) −30.0000 −1.03081
\(848\) −11.0000 −0.377742
\(849\) −18.0000 −0.617758
\(850\) 0 0
\(851\) 0 0
\(852\) 16.0000 0.548151
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) −3.00000 −0.102658
\(855\) 0 0
\(856\) −54.0000 −1.84568
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 1.00000 0.0341394
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 12.0000 0.408722
\(863\) 7.00000 0.238283 0.119141 0.992877i \(-0.461986\pi\)
0.119141 + 0.992877i \(0.461986\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) −8.00000 −0.271694
\(868\) −9.00000 −0.305480
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) −30.0000 −1.01593
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −24.0000 −0.810422 −0.405211 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) −4.00000 −0.134993
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) −25.0000 −0.842271 −0.421136 0.906998i \(-0.638368\pi\)
−0.421136 + 0.906998i \(0.638368\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 5.00000 0.168168
\(885\) 0 0
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) −24.0000 −0.805387
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −8.00000 −0.267860
\(893\) −88.0000 −2.94481
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 9.00000 0.300669
\(897\) 0 0
\(898\) 28.0000 0.934372
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) 55.0000 1.83232
\(902\) −2.00000 −0.0665927
\(903\) 24.0000 0.798670
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −17.0000 −0.564787
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 23.0000 0.763282
\(909\) 9.00000 0.298511
\(910\) 0 0
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) −8.00000 −0.264906
\(913\) −3.00000 −0.0992855
\(914\) −34.0000 −1.12462
\(915\) 0 0
\(916\) 18.0000 0.594737
\(917\) 18.0000 0.594412
\(918\) 5.00000 0.165025
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) −42.0000 −1.38320
\(923\) −16.0000 −0.526646
\(924\) −3.00000 −0.0986928
\(925\) 0 0
\(926\) −9.00000 −0.295758
\(927\) 14.0000 0.459820
\(928\) −5.00000 −0.164133
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) −16.0000 −0.524379
\(932\) −6.00000 −0.196537
\(933\) −6.00000 −0.196431
\(934\) 32.0000 1.04707
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) −45.0000 −1.47009 −0.735043 0.678021i \(-0.762838\pi\)
−0.735043 + 0.678021i \(0.762838\pi\)
\(938\) 9.00000 0.293860
\(939\) −29.0000 −0.946379
\(940\) 0 0
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) 11.0000 0.358399
\(943\) 0 0
\(944\) −5.00000 −0.162736
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −21.0000 −0.682408 −0.341204 0.939989i \(-0.610835\pi\)
−0.341204 + 0.939989i \(0.610835\pi\)
\(948\) 12.0000 0.389742
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −16.0000 −0.518836
\(952\) 45.0000 1.45846
\(953\) −37.0000 −1.19855 −0.599274 0.800544i \(-0.704544\pi\)
−0.599274 + 0.800544i \(0.704544\pi\)
\(954\) −11.0000 −0.356138
\(955\) 0 0
\(956\) 19.0000 0.614504
\(957\) 1.00000 0.0323254
\(958\) 15.0000 0.484628
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 8.00000 0.257930
\(963\) −18.0000 −0.580042
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −37.0000 −1.18984 −0.594920 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(968\) −30.0000 −0.964237
\(969\) 40.0000 1.28499
\(970\) 0 0
\(971\) 54.0000 1.73294 0.866471 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(972\) 1.00000 0.0320750
\(973\) −30.0000 −0.961756
\(974\) 37.0000 1.18556
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 26.0000 0.829693
\(983\) 21.0000 0.669796 0.334898 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −5.00000 −0.159232
\(987\) −33.0000 −1.05040
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −15.0000 −0.476250
\(993\) 12.0000 0.380808
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) 3.00000 0.0950586
\(997\) −3.00000 −0.0950110 −0.0475055 0.998871i \(-0.515127\pi\)
−0.0475055 + 0.998871i \(0.515127\pi\)
\(998\) 41.0000 1.29783
\(999\) −8.00000 −0.253109
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 975.2.a.d.1.1 1
3.2 odd 2 2925.2.a.o.1.1 1
5.2 odd 4 975.2.c.g.274.1 2
5.3 odd 4 975.2.c.g.274.2 2
5.4 even 2 975.2.a.k.1.1 yes 1
15.2 even 4 2925.2.c.i.2224.2 2
15.8 even 4 2925.2.c.i.2224.1 2
15.14 odd 2 2925.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
975.2.a.d.1.1 1 1.1 even 1 trivial
975.2.a.k.1.1 yes 1 5.4 even 2
975.2.c.g.274.1 2 5.2 odd 4
975.2.c.g.274.2 2 5.3 odd 4
2925.2.a.b.1.1 1 15.14 odd 2
2925.2.a.o.1.1 1 3.2 odd 2
2925.2.c.i.2224.1 2 15.8 even 4
2925.2.c.i.2224.2 2 15.2 even 4