Properties

Label 3525.2.a.c.1.1
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 141)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -6.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -2.00000 q^{18} +6.00000 q^{19} -3.00000 q^{21} -2.00000 q^{22} -3.00000 q^{23} -4.00000 q^{26} -1.00000 q^{27} +6.00000 q^{28} +3.00000 q^{29} +2.00000 q^{31} +8.00000 q^{32} -1.00000 q^{33} +4.00000 q^{34} +2.00000 q^{36} +7.00000 q^{37} -12.0000 q^{38} -2.00000 q^{39} +10.0000 q^{41} +6.00000 q^{42} +10.0000 q^{43} +2.00000 q^{44} +6.00000 q^{46} +1.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{51} +4.00000 q^{52} -4.00000 q^{53} +2.00000 q^{54} -6.00000 q^{57} -6.00000 q^{58} +8.00000 q^{59} -10.0000 q^{61} -4.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} +2.00000 q^{66} -10.0000 q^{67} -4.00000 q^{68} +3.00000 q^{69} -14.0000 q^{71} +10.0000 q^{73} -14.0000 q^{74} +12.0000 q^{76} +3.00000 q^{77} +4.00000 q^{78} +17.0000 q^{79} +1.00000 q^{81} -20.0000 q^{82} -8.00000 q^{83} -6.00000 q^{84} -20.0000 q^{86} -3.00000 q^{87} +6.00000 q^{89} +6.00000 q^{91} -6.00000 q^{92} -2.00000 q^{93} -2.00000 q^{94} -8.00000 q^{96} -1.00000 q^{97} -4.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −6.00000 −1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −2.00000 −0.471405
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) −2.00000 −0.426401
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 6.00000 1.13389
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 8.00000 1.41421
\(33\) −1.00000 −0.174078
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −12.0000 −1.94666
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 6.00000 0.925820
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 1.00000 0.145865
\(48\) 4.00000 0.577350
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 4.00000 0.554700
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) −6.00000 −0.787839
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 3.00000 0.377964
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −4.00000 −0.485071
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −14.0000 −1.62747
\(75\) 0 0
\(76\) 12.0000 1.37649
\(77\) 3.00000 0.341882
\(78\) 4.00000 0.452911
\(79\) 17.0000 1.91265 0.956325 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −20.0000 −2.20863
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −6.00000 −0.654654
\(85\) 0 0
\(86\) −20.0000 −2.15666
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −6.00000 −0.625543
\(93\) −2.00000 −0.207390
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) −4.00000 −0.404061
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −16.0000 −1.59206 −0.796030 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) −4.00000 −0.396059
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) −5.00000 −0.483368 −0.241684 0.970355i \(-0.577700\pi\)
−0.241684 + 0.970355i \(0.577700\pi\)
\(108\) −2.00000 −0.192450
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −12.0000 −1.13389
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −16.0000 −1.47292
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 20.0000 1.81071
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −2.00000 −0.174078
\(133\) 18.0000 1.56080
\(134\) 20.0000 1.72774
\(135\) 0 0
\(136\) 0 0
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) −6.00000 −0.510754
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 28.0000 2.34971
\(143\) 2.00000 0.167248
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) −20.0000 −1.65521
\(147\) −2.00000 −0.164957
\(148\) 14.0000 1.15079
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 3.00000 0.239426 0.119713 0.992809i \(-0.461803\pi\)
0.119713 + 0.992809i \(0.461803\pi\)
\(158\) −34.0000 −2.70489
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) −2.00000 −0.157135
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 20.0000 1.56174
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 20.0000 1.52499
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −8.00000 −0.601317
\(178\) −12.0000 −0.899438
\(179\) 5.00000 0.373718 0.186859 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) −12.0000 −0.889499
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −2.00000 −0.146254
\(188\) 2.00000 0.145865
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 8.00000 0.577350
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −2.00000 −0.142134
\(199\) 26.0000 1.84309 0.921546 0.388270i \(-0.126927\pi\)
0.921546 + 0.388270i \(0.126927\pi\)
\(200\) 0 0
\(201\) 10.0000 0.705346
\(202\) 32.0000 2.25151
\(203\) 9.00000 0.631676
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 22.0000 1.53281
\(207\) −3.00000 −0.208514
\(208\) −8.00000 −0.554700
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −8.00000 −0.549442
\(213\) 14.0000 0.959264
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) −12.0000 −0.812743
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 14.0000 0.939618
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 24.0000 1.60357
\(225\) 0 0
\(226\) −20.0000 −1.33038
\(227\) −7.00000 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) −12.0000 −0.794719
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −23.0000 −1.50678 −0.753390 0.657574i \(-0.771583\pi\)
−0.753390 + 0.657574i \(0.771583\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 16.0000 1.04151
\(237\) −17.0000 −1.10427
\(238\) 12.0000 0.777844
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 11.0000 0.708572 0.354286 0.935137i \(-0.384724\pi\)
0.354286 + 0.935137i \(0.384724\pi\)
\(242\) 20.0000 1.28565
\(243\) −1.00000 −0.0641500
\(244\) −20.0000 −1.28037
\(245\) 0 0
\(246\) 20.0000 1.27515
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 6.00000 0.377964
\(253\) −3.00000 −0.188608
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 31.0000 1.93373 0.966863 0.255294i \(-0.0821723\pi\)
0.966863 + 0.255294i \(0.0821723\pi\)
\(258\) 20.0000 1.24515
\(259\) 21.0000 1.30488
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 12.0000 0.741362
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −36.0000 −2.20730
\(267\) −6.00000 −0.367194
\(268\) −20.0000 −1.22169
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) 8.00000 0.485071
\(273\) −6.00000 −0.363137
\(274\) −44.0000 −2.65814
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 4.00000 0.239904
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 2.00000 0.119098
\(283\) 21.0000 1.24832 0.624160 0.781296i \(-0.285441\pi\)
0.624160 + 0.781296i \(0.285441\pi\)
\(284\) −28.0000 −1.66149
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 30.0000 1.77084
\(288\) 8.00000 0.471405
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 1.00000 0.0586210
\(292\) 20.0000 1.17041
\(293\) −27.0000 −1.57736 −0.788678 0.614806i \(-0.789234\pi\)
−0.788678 + 0.614806i \(0.789234\pi\)
\(294\) 4.00000 0.233285
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 24.0000 1.39028
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 30.0000 1.72917
\(302\) 28.0000 1.61122
\(303\) 16.0000 0.919176
\(304\) −24.0000 −1.37649
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 6.00000 0.341882
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) −23.0000 −1.30421 −0.652105 0.758129i \(-0.726114\pi\)
−0.652105 + 0.758129i \(0.726114\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 34.0000 1.91265
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −8.00000 −0.448618
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) 18.0000 1.00310
\(323\) −12.0000 −0.667698
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) −6.00000 −0.331801
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −16.0000 −0.878114
\(333\) 7.00000 0.383598
\(334\) 42.0000 2.29814
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 18.0000 0.979071
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) −12.0000 −0.648886
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 8.00000 0.426401
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 16.0000 0.850390
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 6.00000 0.317554
\(358\) −10.0000 −0.528516
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −24.0000 −1.26141
\(363\) 10.0000 0.524864
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) −20.0000 −1.04542
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 12.0000 0.625543
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) −4.00000 −0.207390
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 6.00000 0.308607
\(379\) 35.0000 1.79783 0.898915 0.438124i \(-0.144357\pi\)
0.898915 + 0.438124i \(0.144357\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 36.0000 1.84192
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) 10.0000 0.508329
\(388\) −2.00000 −0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 16.0000 0.806068
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −52.0000 −2.60652
\(399\) −18.0000 −0.901127
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) −20.0000 −0.997509
\(403\) 4.00000 0.199254
\(404\) −32.0000 −1.59206
\(405\) 0 0
\(406\) −18.0000 −0.893325
\(407\) 7.00000 0.346977
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) −22.0000 −1.08386
\(413\) 24.0000 1.18096
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 16.0000 0.784465
\(417\) 2.00000 0.0979404
\(418\) −12.0000 −0.586939
\(419\) 25.0000 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) −32.0000 −1.55774
\(423\) 1.00000 0.0486217
\(424\) 0 0
\(425\) 0 0
\(426\) −28.0000 −1.35660
\(427\) −30.0000 −1.45180
\(428\) −10.0000 −0.483368
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 4.00000 0.192450
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) −18.0000 −0.861057
\(438\) 20.0000 0.955637
\(439\) −15.0000 −0.715911 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 8.00000 0.380521
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −14.0000 −0.664411
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) 12.0000 0.567581
\(448\) −24.0000 −1.13389
\(449\) 7.00000 0.330350 0.165175 0.986264i \(-0.447181\pi\)
0.165175 + 0.986264i \(0.447181\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 20.0000 0.940721
\(453\) 14.0000 0.657777
\(454\) 14.0000 0.657053
\(455\) 0 0
\(456\) 0 0
\(457\) 13.0000 0.608114 0.304057 0.952654i \(-0.401659\pi\)
0.304057 + 0.952654i \(0.401659\pi\)
\(458\) −8.00000 −0.373815
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 31.0000 1.44381 0.721907 0.691990i \(-0.243266\pi\)
0.721907 + 0.691990i \(0.243266\pi\)
\(462\) 6.00000 0.279145
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) −12.0000 −0.557086
\(465\) 0 0
\(466\) 46.0000 2.13091
\(467\) 21.0000 0.971764 0.485882 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) 4.00000 0.184900
\(469\) −30.0000 −1.38527
\(470\) 0 0
\(471\) −3.00000 −0.138233
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 34.0000 1.56167
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) −4.00000 −0.183147
\(478\) −48.0000 −2.19547
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) −22.0000 −1.00207
\(483\) 9.00000 0.409514
\(484\) −20.0000 −0.909091
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) −20.0000 −0.901670
\(493\) −6.00000 −0.270226
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −42.0000 −1.88396
\(498\) −16.0000 −0.716977
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) 0 0
\(501\) 21.0000 0.938211
\(502\) −36.0000 −1.60676
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 30.0000 1.32712
\(512\) −32.0000 −1.41421
\(513\) −6.00000 −0.264906
\(514\) −62.0000 −2.73470
\(515\) 0 0
\(516\) −20.0000 −0.880451
\(517\) 1.00000 0.0439799
\(518\) −42.0000 −1.84537
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −6.00000 −0.262613
\(523\) 13.0000 0.568450 0.284225 0.958758i \(-0.408264\pi\)
0.284225 + 0.958758i \(0.408264\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −4.00000 −0.174243
\(528\) 4.00000 0.174078
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 36.0000 1.56080
\(533\) 20.0000 0.866296
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 0 0
\(537\) −5.00000 −0.215766
\(538\) 24.0000 1.03471
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) 30.0000 1.28861
\(543\) −12.0000 −0.514969
\(544\) −16.0000 −0.685994
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 44.0000 1.87959
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 51.0000 2.16874
\(554\) 52.0000 2.20927
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −3.00000 −0.127114 −0.0635570 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) −4.00000 −0.169334
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) −54.0000 −2.27785
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −2.00000 −0.0842152
\(565\) 0 0
\(566\) −42.0000 −1.76539
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 4.00000 0.167248
\(573\) 18.0000 0.751961
\(574\) −60.0000 −2.50435
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 26.0000 1.08146
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) −2.00000 −0.0829027
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 54.0000 2.23072
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −4.00000 −0.164957
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 8.00000 0.329076
\(592\) −28.0000 −1.15079
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −24.0000 −0.983078
\(597\) −26.0000 −1.06411
\(598\) 12.0000 0.490716
\(599\) −39.0000 −1.59350 −0.796748 0.604311i \(-0.793448\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) 0 0
\(601\) 7.00000 0.285536 0.142768 0.989756i \(-0.454400\pi\)
0.142768 + 0.989756i \(0.454400\pi\)
\(602\) −60.0000 −2.44542
\(603\) −10.0000 −0.407231
\(604\) −28.0000 −1.13930
\(605\) 0 0
\(606\) −32.0000 −1.29991
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 48.0000 1.94666
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 2.00000 0.0809113
\(612\) −4.00000 −0.161690
\(613\) −27.0000 −1.09052 −0.545260 0.838267i \(-0.683569\pi\)
−0.545260 + 0.838267i \(0.683569\pi\)
\(614\) 46.0000 1.85641
\(615\) 0 0
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) −22.0000 −0.884970
\(619\) −19.0000 −0.763674 −0.381837 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 46.0000 1.84443
\(623\) 18.0000 0.721155
\(624\) 8.00000 0.320256
\(625\) 0 0
\(626\) 12.0000 0.479616
\(627\) −6.00000 −0.239617
\(628\) 6.00000 0.239426
\(629\) −14.0000 −0.558217
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 0 0
\(633\) −16.0000 −0.635943
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 8.00000 0.317221
\(637\) 4.00000 0.158486
\(638\) −6.00000 −0.237542
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) 17.0000 0.671460 0.335730 0.941958i \(-0.391017\pi\)
0.335730 + 0.941958i \(0.391017\pi\)
\(642\) −10.0000 −0.394669
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −18.0000 −0.709299
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 12.0000 0.469956
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) −40.0000 −1.56174
\(657\) 10.0000 0.390137
\(658\) −6.00000 −0.233904
\(659\) −10.0000 −0.389545 −0.194772 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −8.00000 −0.310929
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) −14.0000 −0.542489
\(667\) −9.00000 −0.348481
\(668\) −42.0000 −1.62503
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) −24.0000 −0.925820
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 20.0000 0.768095
\(679\) −3.00000 −0.115129
\(680\) 0 0
\(681\) 7.00000 0.268241
\(682\) −4.00000 −0.153168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 12.0000 0.458831
\(685\) 0 0
\(686\) 30.0000 1.14541
\(687\) −4.00000 −0.152610
\(688\) −40.0000 −1.52499
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) 34.0000 1.29342 0.646710 0.762736i \(-0.276144\pi\)
0.646710 + 0.762736i \(0.276144\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) −32.0000 −1.21122
\(699\) 23.0000 0.869940
\(700\) 0 0
\(701\) −1.00000 −0.0377695 −0.0188847 0.999822i \(-0.506012\pi\)
−0.0188847 + 0.999822i \(0.506012\pi\)
\(702\) 4.00000 0.150970
\(703\) 42.0000 1.58406
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) −48.0000 −1.80523
\(708\) −16.0000 −0.601317
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) 17.0000 0.637550
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) −24.0000 −0.896296
\(718\) 30.0000 1.11959
\(719\) 2.00000 0.0745874 0.0372937 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 0 0
\(721\) −33.0000 −1.22898
\(722\) −34.0000 −1.26535
\(723\) −11.0000 −0.409094
\(724\) 24.0000 0.891953
\(725\) 0 0
\(726\) −20.0000 −0.742270
\(727\) −10.0000 −0.370879 −0.185440 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 20.0000 0.739221
\(733\) −41.0000 −1.51437 −0.757185 0.653201i \(-0.773426\pi\)
−0.757185 + 0.653201i \(0.773426\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −24.0000 −0.884652
\(737\) −10.0000 −0.368355
\(738\) −20.0000 −0.736210
\(739\) 17.0000 0.625355 0.312678 0.949859i \(-0.398774\pi\)
0.312678 + 0.949859i \(0.398774\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 24.0000 0.881068
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32.0000 −1.17160
\(747\) −8.00000 −0.292705
\(748\) −4.00000 −0.146254
\(749\) −15.0000 −0.548088
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) −4.00000 −0.145865
\(753\) −18.0000 −0.655956
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −6.00000 −0.218218
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) −70.0000 −2.54251
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 16.0000 0.579619
\(763\) 18.0000 0.651644
\(764\) −36.0000 −1.30243
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 16.0000 0.577727
\(768\) −16.0000 −0.577350
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −31.0000 −1.11644
\(772\) −8.00000 −0.287926
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) −20.0000 −0.718885
\(775\) 0 0
\(776\) 0 0
\(777\) −21.0000 −0.753371
\(778\) −12.0000 −0.430221
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) −14.0000 −0.500959
\(782\) −12.0000 −0.429119
\(783\) −3.00000 −0.107211
\(784\) −8.00000 −0.285714
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −16.0000 −0.569976
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 30.0000 1.06668
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −4.00000 −0.141955
\(795\) 0 0
\(796\) 52.0000 1.84309
\(797\) −29.0000 −1.02723 −0.513616 0.858020i \(-0.671695\pi\)
−0.513616 + 0.858020i \(0.671695\pi\)
\(798\) 36.0000 1.27439
\(799\) −2.00000 −0.0707549
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −8.00000 −0.282490
\(803\) 10.0000 0.352892
\(804\) 20.0000 0.705346
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) 53.0000 1.86338 0.931690 0.363253i \(-0.118334\pi\)
0.931690 + 0.363253i \(0.118334\pi\)
\(810\) 0 0
\(811\) −29.0000 −1.01833 −0.509164 0.860670i \(-0.670045\pi\)
−0.509164 + 0.860670i \(0.670045\pi\)
\(812\) 18.0000 0.631676
\(813\) 15.0000 0.526073
\(814\) −14.0000 −0.490700
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) 60.0000 2.09913
\(818\) 64.0000 2.23771
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 44.0000 1.53468
\(823\) 39.0000 1.35945 0.679727 0.733465i \(-0.262098\pi\)
0.679727 + 0.733465i \(0.262098\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) −6.00000 −0.208514
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) −16.0000 −0.554700
\(833\) −4.00000 −0.138592
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) −2.00000 −0.0691301
\(838\) −50.0000 −1.72722
\(839\) −31.0000 −1.07024 −0.535119 0.844776i \(-0.679733\pi\)
−0.535119 + 0.844776i \(0.679733\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 40.0000 1.37849
\(843\) −27.0000 −0.929929
\(844\) 32.0000 1.10149
\(845\) 0 0
\(846\) −2.00000 −0.0687614
\(847\) −30.0000 −1.03081
\(848\) 16.0000 0.549442
\(849\) −21.0000 −0.720718
\(850\) 0 0
\(851\) −21.0000 −0.719871
\(852\) 28.0000 0.959264
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 60.0000 2.05316
\(855\) 0 0
\(856\) 0 0
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) 4.00000 0.136558
\(859\) −18.0000 −0.614152 −0.307076 0.951685i \(-0.599351\pi\)
−0.307076 + 0.951685i \(0.599351\pi\)
\(860\) 0 0
\(861\) −30.0000 −1.02240
\(862\) −20.0000 −0.681203
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 12.0000 0.407307
\(869\) 17.0000 0.576686
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) −20.0000 −0.675737
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 30.0000 1.01245
\(879\) 27.0000 0.910687
\(880\) 0 0
\(881\) −53.0000 −1.78562 −0.892808 0.450438i \(-0.851268\pi\)
−0.892808 + 0.450438i \(0.851268\pi\)
\(882\) −4.00000 −0.134687
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 24.0000 0.803579
\(893\) 6.00000 0.200782
\(894\) −24.0000 −0.802680
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) −14.0000 −0.467186
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) −20.0000 −0.665927
\(903\) −30.0000 −0.998337
\(904\) 0 0
\(905\) 0 0
\(906\) −28.0000 −0.930238
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −14.0000 −0.464606
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) 50.0000 1.65657 0.828287 0.560304i \(-0.189316\pi\)
0.828287 + 0.560304i \(0.189316\pi\)
\(912\) 24.0000 0.794719
\(913\) −8.00000 −0.264761
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) −18.0000 −0.594412
\(918\) −4.00000 −0.132020
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) −62.0000 −2.04186
\(923\) −28.0000 −0.921631
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) −56.0000 −1.84027
\(927\) −11.0000 −0.361287
\(928\) 24.0000 0.787839
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −46.0000 −1.50678
\(933\) 23.0000 0.752986
\(934\) −42.0000 −1.37428
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 60.0000 1.95907
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 6.00000 0.195491
\(943\) −30.0000 −0.976934
\(944\) −32.0000 −1.04151
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) 38.0000 1.23483 0.617417 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) −34.0000 −1.10427
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) −31.0000 −1.00419 −0.502094 0.864813i \(-0.667437\pi\)
−0.502094 + 0.864813i \(0.667437\pi\)
\(954\) 8.00000 0.259010
\(955\) 0 0
\(956\) 48.0000 1.55243
\(957\) −3.00000 −0.0969762
\(958\) −24.0000 −0.775405
\(959\) 66.0000 2.13125
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −28.0000 −0.902756
\(963\) −5.00000 −0.161123
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) −18.0000 −0.579141
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) 49.0000 1.57248 0.786242 0.617918i \(-0.212024\pi\)
0.786242 + 0.617918i \(0.212024\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −6.00000 −0.192351
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 40.0000 1.28037
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 12.0000 0.383718
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −72.0000 −2.29761
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) −3.00000 −0.0954911
\(988\) 24.0000 0.763542
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) −57.0000 −1.81066 −0.905332 0.424704i \(-0.860378\pi\)
−0.905332 + 0.424704i \(0.860378\pi\)
\(992\) 16.0000 0.508001
\(993\) −4.00000 −0.126936
\(994\) 84.0000 2.66432
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) −40.0000 −1.26681 −0.633406 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) 44.0000 1.39280
\(999\) −7.00000 −0.221470
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 3525.2.a.c.1.1 1
5.4 even 2 141.2.a.e.1.1 1
15.14 odd 2 423.2.a.b.1.1 1
20.19 odd 2 2256.2.a.e.1.1 1
35.34 odd 2 6909.2.a.k.1.1 1
40.19 odd 2 9024.2.a.bq.1.1 1
40.29 even 2 9024.2.a.n.1.1 1
60.59 even 2 6768.2.a.n.1.1 1
235.234 odd 2 6627.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
141.2.a.e.1.1 1 5.4 even 2
423.2.a.b.1.1 1 15.14 odd 2
2256.2.a.e.1.1 1 20.19 odd 2
3525.2.a.c.1.1 1 1.1 even 1 trivial
6627.2.a.i.1.1 1 235.234 odd 2
6768.2.a.n.1.1 1 60.59 even 2
6909.2.a.k.1.1 1 35.34 odd 2
9024.2.a.n.1.1 1 40.29 even 2
9024.2.a.bq.1.1 1 40.19 odd 2