Properties

Label 546.2.a.f.1.1
Level $546$
Weight $2$
Character 546.1
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
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 \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +5.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} -3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -11.0000 q^{37} -1.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +1.00000 q^{42} -5.00000 q^{43} +5.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} -1.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -5.00000 q^{55} +1.00000 q^{56} -1.00000 q^{57} +9.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} -15.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} +5.00000 q^{66} -2.00000 q^{67} -3.00000 q^{68} +3.00000 q^{69} -1.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +11.0000 q^{73} -11.0000 q^{74} -4.00000 q^{75} -1.00000 q^{76} +5.00000 q^{77} -1.00000 q^{78} +10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -14.0000 q^{83} +1.00000 q^{84} +3.00000 q^{85} -5.00000 q^{86} +9.00000 q^{87} +5.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -1.00000 q^{91} +3.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 5.00000 1.06600
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.00000 0.870388
\(34\) −3.00000 −0.514496
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) −1.00000 −0.162221
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 0.154303
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 5.00000 0.753778
\(45\) −1.00000 −0.149071
\(46\) 3.00000 0.442326
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) −3.00000 −0.420084
\(52\) −1.00000 −0.138675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) −5.00000 −0.674200
\(56\) 1.00000 0.133631
\(57\) −1.00000 −0.132453
\(58\) 9.00000 1.18176
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 5.00000 0.615457
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −3.00000 −0.363803
\(69\) 3.00000 0.361158
\(70\) −1.00000 −0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −11.0000 −1.27872
\(75\) −4.00000 −0.461880
\(76\) −1.00000 −0.114708
\(77\) 5.00000 0.569803
\(78\) −1.00000 −0.113228
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 1.00000 0.109109
\(85\) 3.00000 0.325396
\(86\) −5.00000 −0.539164
\(87\) 9.00000 0.964901
\(88\) 5.00000 0.533002
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) −1.00000 −0.104828
\(92\) 3.00000 0.312772
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) 1.00000 0.102598
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 5.00000 0.502519
\(100\) −4.00000 −0.400000
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) −3.00000 −0.297044
\(103\) −15.0000 −1.47799 −0.738997 0.673709i \(-0.764700\pi\)
−0.738997 + 0.673709i \(0.764700\pi\)
\(104\) −1.00000 −0.0980581
\(105\) −1.00000 −0.0975900
\(106\) −2.00000 −0.194257
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 1.00000 0.0962250
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) −5.00000 −0.476731
\(111\) −11.0000 −1.04407
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −3.00000 −0.279751
\(116\) 9.00000 0.835629
\(117\) −1.00000 −0.0924500
\(118\) 4.00000 0.368230
\(119\) −3.00000 −0.275010
\(120\) −1.00000 −0.0912871
\(121\) 14.0000 1.27273
\(122\) −15.0000 −1.35804
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 9.00000 0.804984
\(126\) 1.00000 0.0890871
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.00000 −0.440225
\(130\) 1.00000 0.0877058
\(131\) 13.0000 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(132\) 5.00000 0.435194
\(133\) −1.00000 −0.0867110
\(134\) −2.00000 −0.172774
\(135\) −1.00000 −0.0860663
\(136\) −3.00000 −0.257248
\(137\) 19.0000 1.62328 0.811640 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) 3.00000 0.255377
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.00000 −0.673722
\(142\) −12.0000 −1.00702
\(143\) −5.00000 −0.418121
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) 11.0000 0.910366
\(147\) 1.00000 0.0824786
\(148\) −11.0000 −0.904194
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) −4.00000 −0.326599
\(151\) −23.0000 −1.87171 −0.935857 0.352381i \(-0.885372\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −3.00000 −0.242536
\(154\) 5.00000 0.402911
\(155\) −4.00000 −0.321288
\(156\) −1.00000 −0.0800641
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 10.0000 0.795557
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 3.00000 0.236433
\(162\) 1.00000 0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) −5.00000 −0.389249
\(166\) −14.0000 −1.08661
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 1.00000 0.0771517
\(169\) 1.00000 0.0769231
\(170\) 3.00000 0.230089
\(171\) −1.00000 −0.0764719
\(172\) −5.00000 −0.381246
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 9.00000 0.682288
\(175\) −4.00000 −0.302372
\(176\) 5.00000 0.376889
\(177\) 4.00000 0.300658
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −15.0000 −1.10883
\(184\) 3.00000 0.221163
\(185\) 11.0000 0.808736
\(186\) 4.00000 0.293294
\(187\) −15.0000 −1.09691
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) 1.00000 0.0725476
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) −14.0000 −1.00514
\(195\) 1.00000 0.0716115
\(196\) 1.00000 0.0714286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 5.00000 0.355335
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) −4.00000 −0.282843
\(201\) −2.00000 −0.141069
\(202\) 4.00000 0.281439
\(203\) 9.00000 0.631676
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −15.0000 −1.04510
\(207\) 3.00000 0.208514
\(208\) −1.00000 −0.0693375
\(209\) −5.00000 −0.345857
\(210\) −1.00000 −0.0690066
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) −2.00000 −0.137361
\(213\) −12.0000 −0.822226
\(214\) 10.0000 0.683586
\(215\) 5.00000 0.340997
\(216\) 1.00000 0.0680414
\(217\) 4.00000 0.271538
\(218\) −9.00000 −0.609557
\(219\) 11.0000 0.743311
\(220\) −5.00000 −0.337100
\(221\) 3.00000 0.201802
\(222\) −11.0000 −0.738272
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 1.00000 0.0668153
\(225\) −4.00000 −0.266667
\(226\) 2.00000 0.133038
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −3.00000 −0.197814
\(231\) 5.00000 0.328976
\(232\) 9.00000 0.590879
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) 10.0000 0.649570
\(238\) −3.00000 −0.194461
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 14.0000 0.899954
\(243\) 1.00000 0.0641500
\(244\) −15.0000 −0.960277
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 4.00000 0.254000
\(249\) −14.0000 −0.887214
\(250\) 9.00000 0.569210
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 1.00000 0.0629941
\(253\) 15.0000 0.943042
\(254\) 2.00000 0.125491
\(255\) 3.00000 0.187867
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −5.00000 −0.311286
\(259\) −11.0000 −0.683507
\(260\) 1.00000 0.0620174
\(261\) 9.00000 0.557086
\(262\) 13.0000 0.803143
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 5.00000 0.307729
\(265\) 2.00000 0.122859
\(266\) −1.00000 −0.0613139
\(267\) 6.00000 0.367194
\(268\) −2.00000 −0.122169
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) −3.00000 −0.181902
\(273\) −1.00000 −0.0605228
\(274\) 19.0000 1.14783
\(275\) −20.0000 −1.20605
\(276\) 3.00000 0.180579
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) −1.00000 −0.0597614
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −8.00000 −0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −12.0000 −0.712069
\(285\) 1.00000 0.0592349
\(286\) −5.00000 −0.295656
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) −9.00000 −0.528498
\(291\) −14.0000 −0.820695
\(292\) 11.0000 0.643726
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 1.00000 0.0583212
\(295\) −4.00000 −0.232889
\(296\) −11.0000 −0.639362
\(297\) 5.00000 0.290129
\(298\) −4.00000 −0.231714
\(299\) −3.00000 −0.173494
\(300\) −4.00000 −0.230940
\(301\) −5.00000 −0.288195
\(302\) −23.0000 −1.32350
\(303\) 4.00000 0.229794
\(304\) −1.00000 −0.0573539
\(305\) 15.0000 0.858898
\(306\) −3.00000 −0.171499
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 5.00000 0.284901
\(309\) −15.0000 −0.853320
\(310\) −4.00000 −0.227185
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −3.00000 −0.169300
\(315\) −1.00000 −0.0563436
\(316\) 10.0000 0.562544
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) −2.00000 −0.112154
\(319\) 45.0000 2.51952
\(320\) −1.00000 −0.0559017
\(321\) 10.0000 0.558146
\(322\) 3.00000 0.167183
\(323\) 3.00000 0.166924
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 10.0000 0.553849
\(327\) −9.00000 −0.497701
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) −5.00000 −0.275241
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −14.0000 −0.768350
\(333\) −11.0000 −0.602796
\(334\) −21.0000 −1.14907
\(335\) 2.00000 0.109272
\(336\) 1.00000 0.0545545
\(337\) −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
\(340\) 3.00000 0.162698
\(341\) 20.0000 1.08306
\(342\) −1.00000 −0.0540738
\(343\) 1.00000 0.0539949
\(344\) −5.00000 −0.269582
\(345\) −3.00000 −0.161515
\(346\) 6.00000 0.322562
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 9.00000 0.482451
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −4.00000 −0.213809
\(351\) −1.00000 −0.0533761
\(352\) 5.00000 0.266501
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) 12.0000 0.636894
\(356\) 6.00000 0.317999
\(357\) −3.00000 −0.158777
\(358\) 12.0000 0.634220
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.0000 −0.947368
\(362\) 14.0000 0.735824
\(363\) 14.0000 0.734809
\(364\) −1.00000 −0.0524142
\(365\) −11.0000 −0.575766
\(366\) −15.0000 −0.784063
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 11.0000 0.571863
\(371\) −2.00000 −0.103835
\(372\) 4.00000 0.207390
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −15.0000 −0.775632
\(375\) 9.00000 0.464758
\(376\) −8.00000 −0.412568
\(377\) −9.00000 −0.463524
\(378\) 1.00000 0.0514344
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 1.00000 0.0512989
\(381\) 2.00000 0.102463
\(382\) 3.00000 0.153493
\(383\) −29.0000 −1.48183 −0.740915 0.671598i \(-0.765608\pi\)
−0.740915 + 0.671598i \(0.765608\pi\)
\(384\) 1.00000 0.0510310
\(385\) −5.00000 −0.254824
\(386\) 12.0000 0.610784
\(387\) −5.00000 −0.254164
\(388\) −14.0000 −0.710742
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 1.00000 0.0506370
\(391\) −9.00000 −0.455150
\(392\) 1.00000 0.0505076
\(393\) 13.0000 0.655763
\(394\) −12.0000 −0.604551
\(395\) −10.0000 −0.503155
\(396\) 5.00000 0.251259
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 11.0000 0.551380
\(399\) −1.00000 −0.0500626
\(400\) −4.00000 −0.200000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −4.00000 −0.199254
\(404\) 4.00000 0.199007
\(405\) −1.00000 −0.0496904
\(406\) 9.00000 0.446663
\(407\) −55.0000 −2.72625
\(408\) −3.00000 −0.148522
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 19.0000 0.937201
\(412\) −15.0000 −0.738997
\(413\) 4.00000 0.196827
\(414\) 3.00000 0.147442
\(415\) 14.0000 0.687233
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) −5.00000 −0.244558
\(419\) 7.00000 0.341972 0.170986 0.985273i \(-0.445305\pi\)
0.170986 + 0.985273i \(0.445305\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 9.00000 0.438113
\(423\) −8.00000 −0.388973
\(424\) −2.00000 −0.0971286
\(425\) 12.0000 0.582086
\(426\) −12.0000 −0.581402
\(427\) −15.0000 −0.725901
\(428\) 10.0000 0.483368
\(429\) −5.00000 −0.241402
\(430\) 5.00000 0.241121
\(431\) −2.00000 −0.0963366 −0.0481683 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) 1.00000 0.0481125
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 4.00000 0.192006
\(435\) −9.00000 −0.431517
\(436\) −9.00000 −0.431022
\(437\) −3.00000 −0.143509
\(438\) 11.0000 0.525600
\(439\) 13.0000 0.620456 0.310228 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) −5.00000 −0.238366
\(441\) 1.00000 0.0476190
\(442\) 3.00000 0.142695
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) −11.0000 −0.522037
\(445\) −6.00000 −0.284427
\(446\) −14.0000 −0.662919
\(447\) −4.00000 −0.189194
\(448\) 1.00000 0.0472456
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) −4.00000 −0.188562
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) −23.0000 −1.08063
\(454\) 18.0000 0.844782
\(455\) 1.00000 0.0468807
\(456\) −1.00000 −0.0468293
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) 6.00000 0.280362
\(459\) −3.00000 −0.140028
\(460\) −3.00000 −0.139876
\(461\) −35.0000 −1.63011 −0.815056 0.579382i \(-0.803294\pi\)
−0.815056 + 0.579382i \(0.803294\pi\)
\(462\) 5.00000 0.232621
\(463\) 9.00000 0.418265 0.209133 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) 9.00000 0.417815
\(465\) −4.00000 −0.185496
\(466\) 24.0000 1.11178
\(467\) −15.0000 −0.694117 −0.347059 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −2.00000 −0.0923514
\(470\) 8.00000 0.369012
\(471\) −3.00000 −0.138233
\(472\) 4.00000 0.184115
\(473\) −25.0000 −1.14950
\(474\) 10.0000 0.459315
\(475\) 4.00000 0.183533
\(476\) −3.00000 −0.137505
\(477\) −2.00000 −0.0915737
\(478\) −12.0000 −0.548867
\(479\) −31.0000 −1.41643 −0.708213 0.705999i \(-0.750498\pi\)
−0.708213 + 0.705999i \(0.750498\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 11.0000 0.501557
\(482\) 18.0000 0.819878
\(483\) 3.00000 0.136505
\(484\) 14.0000 0.636364
\(485\) 14.0000 0.635707
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −15.0000 −0.679018
\(489\) 10.0000 0.452216
\(490\) −1.00000 −0.0451754
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) −27.0000 −1.21602
\(494\) 1.00000 0.0449921
\(495\) −5.00000 −0.224733
\(496\) 4.00000 0.179605
\(497\) −12.0000 −0.538274
\(498\) −14.0000 −0.627355
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 9.00000 0.402492
\(501\) −21.0000 −0.938211
\(502\) −7.00000 −0.312425
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 1.00000 0.0445435
\(505\) −4.00000 −0.177998
\(506\) 15.0000 0.666831
\(507\) 1.00000 0.0444116
\(508\) 2.00000 0.0887357
\(509\) 13.0000 0.576215 0.288107 0.957598i \(-0.406974\pi\)
0.288107 + 0.957598i \(0.406974\pi\)
\(510\) 3.00000 0.132842
\(511\) 11.0000 0.486611
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 6.00000 0.264649
\(515\) 15.0000 0.660979
\(516\) −5.00000 −0.220113
\(517\) −40.0000 −1.75920
\(518\) −11.0000 −0.483312
\(519\) 6.00000 0.263371
\(520\) 1.00000 0.0438529
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) 9.00000 0.393919
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 13.0000 0.567908
\(525\) −4.00000 −0.174574
\(526\) 12.0000 0.523225
\(527\) −12.0000 −0.522728
\(528\) 5.00000 0.217597
\(529\) −14.0000 −0.608696
\(530\) 2.00000 0.0868744
\(531\) 4.00000 0.173585
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) −10.0000 −0.432338
\(536\) −2.00000 −0.0863868
\(537\) 12.0000 0.517838
\(538\) 18.0000 0.776035
\(539\) 5.00000 0.215365
\(540\) −1.00000 −0.0430331
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) −22.0000 −0.944981
\(543\) 14.0000 0.600798
\(544\) −3.00000 −0.128624
\(545\) 9.00000 0.385518
\(546\) −1.00000 −0.0427960
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 19.0000 0.811640
\(549\) −15.0000 −0.640184
\(550\) −20.0000 −0.852803
\(551\) −9.00000 −0.383413
\(552\) 3.00000 0.127688
\(553\) 10.0000 0.425243
\(554\) 26.0000 1.10463
\(555\) 11.0000 0.466924
\(556\) 0 0
\(557\) −44.0000 −1.86434 −0.932170 0.362021i \(-0.882087\pi\)
−0.932170 + 0.362021i \(0.882087\pi\)
\(558\) 4.00000 0.169334
\(559\) 5.00000 0.211477
\(560\) −1.00000 −0.0422577
\(561\) −15.0000 −0.633300
\(562\) −26.0000 −1.09674
\(563\) −3.00000 −0.126435 −0.0632175 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(564\) −8.00000 −0.336861
\(565\) −2.00000 −0.0841406
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −12.0000 −0.503509
\(569\) −46.0000 −1.92842 −0.964210 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 1.00000 0.0418854
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) −5.00000 −0.209061
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −8.00000 −0.332756
\(579\) 12.0000 0.498703
\(580\) −9.00000 −0.373705
\(581\) −14.0000 −0.580818
\(582\) −14.0000 −0.580319
\(583\) −10.0000 −0.414158
\(584\) 11.0000 0.455183
\(585\) 1.00000 0.0413449
\(586\) 14.0000 0.578335
\(587\) 14.0000 0.577842 0.288921 0.957353i \(-0.406704\pi\)
0.288921 + 0.957353i \(0.406704\pi\)
\(588\) 1.00000 0.0412393
\(589\) −4.00000 −0.164817
\(590\) −4.00000 −0.164677
\(591\) −12.0000 −0.493614
\(592\) −11.0000 −0.452097
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 5.00000 0.205152
\(595\) 3.00000 0.122988
\(596\) −4.00000 −0.163846
\(597\) 11.0000 0.450200
\(598\) −3.00000 −0.122679
\(599\) 5.00000 0.204294 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) −4.00000 −0.163299
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −5.00000 −0.203785
\(603\) −2.00000 −0.0814463
\(604\) −23.0000 −0.935857
\(605\) −14.0000 −0.569181
\(606\) 4.00000 0.162489
\(607\) −43.0000 −1.74532 −0.872658 0.488332i \(-0.837606\pi\)
−0.872658 + 0.488332i \(0.837606\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 9.00000 0.364698
\(610\) 15.0000 0.607332
\(611\) 8.00000 0.323645
\(612\) −3.00000 −0.121268
\(613\) −9.00000 −0.363507 −0.181753 0.983344i \(-0.558177\pi\)
−0.181753 + 0.983344i \(0.558177\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) 9.00000 0.362326 0.181163 0.983453i \(-0.442014\pi\)
0.181163 + 0.983453i \(0.442014\pi\)
\(618\) −15.0000 −0.603388
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) −4.00000 −0.160644
\(621\) 3.00000 0.120386
\(622\) −10.0000 −0.400963
\(623\) 6.00000 0.240385
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) −8.00000 −0.319744
\(627\) −5.00000 −0.199681
\(628\) −3.00000 −0.119713
\(629\) 33.0000 1.31580
\(630\) −1.00000 −0.0398410
\(631\) 37.0000 1.47295 0.736473 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) 10.0000 0.397779
\(633\) 9.00000 0.357718
\(634\) 24.0000 0.953162
\(635\) −2.00000 −0.0793676
\(636\) −2.00000 −0.0793052
\(637\) −1.00000 −0.0396214
\(638\) 45.0000 1.78157
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 10.0000 0.394669
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 3.00000 0.118217
\(645\) 5.00000 0.196875
\(646\) 3.00000 0.118033
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) 20.0000 0.785069
\(650\) 4.00000 0.156893
\(651\) 4.00000 0.156772
\(652\) 10.0000 0.391630
\(653\) 31.0000 1.21312 0.606562 0.795036i \(-0.292548\pi\)
0.606562 + 0.795036i \(0.292548\pi\)
\(654\) −9.00000 −0.351928
\(655\) −13.0000 −0.507952
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) −8.00000 −0.311872
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) −5.00000 −0.194625
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 10.0000 0.388661
\(663\) 3.00000 0.116510
\(664\) −14.0000 −0.543305
\(665\) 1.00000 0.0387783
\(666\) −11.0000 −0.426241
\(667\) 27.0000 1.04544
\(668\) −21.0000 −0.812514
\(669\) −14.0000 −0.541271
\(670\) 2.00000 0.0772667
\(671\) −75.0000 −2.89534
\(672\) 1.00000 0.0385758
\(673\) −33.0000 −1.27206 −0.636028 0.771666i \(-0.719424\pi\)
−0.636028 + 0.771666i \(0.719424\pi\)
\(674\) −5.00000 −0.192593
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 2.00000 0.0768095
\(679\) −14.0000 −0.537271
\(680\) 3.00000 0.115045
\(681\) 18.0000 0.689761
\(682\) 20.0000 0.765840
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −19.0000 −0.725953
\(686\) 1.00000 0.0381802
\(687\) 6.00000 0.228914
\(688\) −5.00000 −0.190623
\(689\) 2.00000 0.0761939
\(690\) −3.00000 −0.114208
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 6.00000 0.228086
\(693\) 5.00000 0.189934
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) 0 0
\(698\) 14.0000 0.529908
\(699\) 24.0000 0.907763
\(700\) −4.00000 −0.151186
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 11.0000 0.414873
\(704\) 5.00000 0.188445
\(705\) 8.00000 0.301297
\(706\) 18.0000 0.677439
\(707\) 4.00000 0.150435
\(708\) 4.00000 0.150329
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 12.0000 0.450352
\(711\) 10.0000 0.375029
\(712\) 6.00000 0.224860
\(713\) 12.0000 0.449404
\(714\) −3.00000 −0.112272
\(715\) 5.00000 0.186989
\(716\) 12.0000 0.448461
\(717\) −12.0000 −0.448148
\(718\) −4.00000 −0.149279
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −15.0000 −0.558629
\(722\) −18.0000 −0.669891
\(723\) 18.0000 0.669427
\(724\) 14.0000 0.520306
\(725\) −36.0000 −1.33701
\(726\) 14.0000 0.519589
\(727\) −35.0000 −1.29808 −0.649039 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) −11.0000 −0.407128
\(731\) 15.0000 0.554795
\(732\) −15.0000 −0.554416
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 32.0000 1.18114
\(735\) −1.00000 −0.0368856
\(736\) 3.00000 0.110581
\(737\) −10.0000 −0.368355
\(738\) 0 0
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 11.0000 0.404368
\(741\) 1.00000 0.0367359
\(742\) −2.00000 −0.0734223
\(743\) −26.0000 −0.953847 −0.476924 0.878945i \(-0.658248\pi\)
−0.476924 + 0.878945i \(0.658248\pi\)
\(744\) 4.00000 0.146647
\(745\) 4.00000 0.146549
\(746\) −4.00000 −0.146450
\(747\) −14.0000 −0.512233
\(748\) −15.0000 −0.548454
\(749\) 10.0000 0.365392
\(750\) 9.00000 0.328634
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −8.00000 −0.291730
\(753\) −7.00000 −0.255094
\(754\) −9.00000 −0.327761
\(755\) 23.0000 0.837056
\(756\) 1.00000 0.0363696
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 16.0000 0.581146
\(759\) 15.0000 0.544466
\(760\) 1.00000 0.0362738
\(761\) 20.0000 0.724999 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 2.00000 0.0724524
\(763\) −9.00000 −0.325822
\(764\) 3.00000 0.108536
\(765\) 3.00000 0.108465
\(766\) −29.0000 −1.04781
\(767\) −4.00000 −0.144432
\(768\) 1.00000 0.0360844
\(769\) 49.0000 1.76699 0.883493 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) −5.00000 −0.180187
\(771\) 6.00000 0.216085
\(772\) 12.0000 0.431889
\(773\) −3.00000 −0.107903 −0.0539513 0.998544i \(-0.517182\pi\)
−0.0539513 + 0.998544i \(0.517182\pi\)
\(774\) −5.00000 −0.179721
\(775\) −16.0000 −0.574737
\(776\) −14.0000 −0.502571
\(777\) −11.0000 −0.394623
\(778\) −2.00000 −0.0717035
\(779\) 0 0
\(780\) 1.00000 0.0358057
\(781\) −60.0000 −2.14697
\(782\) −9.00000 −0.321839
\(783\) 9.00000 0.321634
\(784\) 1.00000 0.0357143
\(785\) 3.00000 0.107075
\(786\) 13.0000 0.463695
\(787\) 39.0000 1.39020 0.695100 0.718913i \(-0.255360\pi\)
0.695100 + 0.718913i \(0.255360\pi\)
\(788\) −12.0000 −0.427482
\(789\) 12.0000 0.427211
\(790\) −10.0000 −0.355784
\(791\) 2.00000 0.0711118
\(792\) 5.00000 0.177667
\(793\) 15.0000 0.532666
\(794\) 6.00000 0.212932
\(795\) 2.00000 0.0709327
\(796\) 11.0000 0.389885
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) −1.00000 −0.0353996
\(799\) 24.0000 0.849059
\(800\) −4.00000 −0.141421
\(801\) 6.00000 0.212000
\(802\) −18.0000 −0.635602
\(803\) 55.0000 1.94091
\(804\) −2.00000 −0.0705346
\(805\) −3.00000 −0.105736
\(806\) −4.00000 −0.140894
\(807\) 18.0000 0.633630
\(808\) 4.00000 0.140720
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 35.0000 1.22902 0.614508 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) 9.00000 0.315838
\(813\) −22.0000 −0.771574
\(814\) −55.0000 −1.92775
\(815\) −10.0000 −0.350285
\(816\) −3.00000 −0.105021
\(817\) 5.00000 0.174928
\(818\) 25.0000 0.874105
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −32.0000 −1.11681 −0.558404 0.829569i \(-0.688586\pi\)
−0.558404 + 0.829569i \(0.688586\pi\)
\(822\) 19.0000 0.662701
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −15.0000 −0.522550
\(825\) −20.0000 −0.696311
\(826\) 4.00000 0.139178
\(827\) −19.0000 −0.660695 −0.330347 0.943859i \(-0.607166\pi\)
−0.330347 + 0.943859i \(0.607166\pi\)
\(828\) 3.00000 0.104257
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 14.0000 0.485947
\(831\) 26.0000 0.901930
\(832\) −1.00000 −0.0346688
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 21.0000 0.726735
\(836\) −5.00000 −0.172929
\(837\) 4.00000 0.138260
\(838\) 7.00000 0.241811
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 52.0000 1.79310
\(842\) 30.0000 1.03387
\(843\) −26.0000 −0.895488
\(844\) 9.00000 0.309793
\(845\) −1.00000 −0.0344010
\(846\) −8.00000 −0.275046
\(847\) 14.0000 0.481046
\(848\) −2.00000 −0.0686803
\(849\) 4.00000 0.137280
\(850\) 12.0000 0.411597
\(851\) −33.0000 −1.13123
\(852\) −12.0000 −0.411113
\(853\) −28.0000 −0.958702 −0.479351 0.877623i \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) −15.0000 −0.513289
\(855\) 1.00000 0.0341993
\(856\) 10.0000 0.341793
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −5.00000 −0.170697
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 5.00000 0.170499
\(861\) 0 0
\(862\) −2.00000 −0.0681203
\(863\) 38.0000 1.29354 0.646768 0.762687i \(-0.276120\pi\)
0.646768 + 0.762687i \(0.276120\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) 28.0000 0.951479
\(867\) −8.00000 −0.271694
\(868\) 4.00000 0.135769
\(869\) 50.0000 1.69613
\(870\) −9.00000 −0.305129
\(871\) 2.00000 0.0677674
\(872\) −9.00000 −0.304778
\(873\) −14.0000 −0.473828
\(874\) −3.00000 −0.101477
\(875\) 9.00000 0.304256
\(876\) 11.0000 0.371656
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 13.0000 0.438729
\(879\) 14.0000 0.472208
\(880\) −5.00000 −0.168550
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 1.00000 0.0336718
\(883\) 51.0000 1.71629 0.858143 0.513410i \(-0.171618\pi\)
0.858143 + 0.513410i \(0.171618\pi\)
\(884\) 3.00000 0.100901
\(885\) −4.00000 −0.134459
\(886\) −18.0000 −0.604722
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −11.0000 −0.369136
\(889\) 2.00000 0.0670778
\(890\) −6.00000 −0.201120
\(891\) 5.00000 0.167506
\(892\) −14.0000 −0.468755
\(893\) 8.00000 0.267710
\(894\) −4.00000 −0.133780
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) −3.00000 −0.100167
\(898\) 23.0000 0.767520
\(899\) 36.0000 1.20067
\(900\) −4.00000 −0.133333
\(901\) 6.00000 0.199889
\(902\) 0 0
\(903\) −5.00000 −0.166390
\(904\) 2.00000 0.0665190
\(905\) −14.0000 −0.465376
\(906\) −23.0000 −0.764124
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 18.0000 0.597351
\(909\) 4.00000 0.132672
\(910\) 1.00000 0.0331497
\(911\) −19.0000 −0.629498 −0.314749 0.949175i \(-0.601920\pi\)
−0.314749 + 0.949175i \(0.601920\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −70.0000 −2.31666
\(914\) −32.0000 −1.05847
\(915\) 15.0000 0.495885
\(916\) 6.00000 0.198246
\(917\) 13.0000 0.429298
\(918\) −3.00000 −0.0990148
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) −35.0000 −1.15266
\(923\) 12.0000 0.394985
\(924\) 5.00000 0.164488
\(925\) 44.0000 1.44671
\(926\) 9.00000 0.295758
\(927\) −15.0000 −0.492665
\(928\) 9.00000 0.295439
\(929\) −8.00000 −0.262471 −0.131236 0.991351i \(-0.541894\pi\)
−0.131236 + 0.991351i \(0.541894\pi\)
\(930\) −4.00000 −0.131165
\(931\) −1.00000 −0.0327737
\(932\) 24.0000 0.786146
\(933\) −10.0000 −0.327385
\(934\) −15.0000 −0.490815
\(935\) 15.0000 0.490552
\(936\) −1.00000 −0.0326860
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) −2.00000 −0.0653023
\(939\) −8.00000 −0.261070
\(940\) 8.00000 0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −3.00000 −0.0977453
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) −1.00000 −0.0325300
\(946\) −25.0000 −0.812820
\(947\) −39.0000 −1.26733 −0.633665 0.773608i \(-0.718450\pi\)
−0.633665 + 0.773608i \(0.718450\pi\)
\(948\) 10.0000 0.324785
\(949\) −11.0000 −0.357075
\(950\) 4.00000 0.129777
\(951\) 24.0000 0.778253
\(952\) −3.00000 −0.0972306
\(953\) 16.0000 0.518291 0.259145 0.965838i \(-0.416559\pi\)
0.259145 + 0.965838i \(0.416559\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −3.00000 −0.0970777
\(956\) −12.0000 −0.388108
\(957\) 45.0000 1.45464
\(958\) −31.0000 −1.00156
\(959\) 19.0000 0.613542
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 11.0000 0.354654
\(963\) 10.0000 0.322245
\(964\) 18.0000 0.579741
\(965\) −12.0000 −0.386294
\(966\) 3.00000 0.0965234
\(967\) −5.00000 −0.160789 −0.0803946 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(968\) 14.0000 0.449977
\(969\) 3.00000 0.0963739
\(970\) 14.0000 0.449513
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 4.00000 0.128103
\(976\) −15.0000 −0.480138
\(977\) −51.0000 −1.63163 −0.815817 0.578310i \(-0.803713\pi\)
−0.815817 + 0.578310i \(0.803713\pi\)
\(978\) 10.0000 0.319765
\(979\) 30.0000 0.958804
\(980\) −1.00000 −0.0319438
\(981\) −9.00000 −0.287348
\(982\) −26.0000 −0.829693
\(983\) 25.0000 0.797376 0.398688 0.917087i \(-0.369466\pi\)
0.398688 + 0.917087i \(0.369466\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) −27.0000 −0.859855
\(987\) −8.00000 −0.254643
\(988\) 1.00000 0.0318142
\(989\) −15.0000 −0.476972
\(990\) −5.00000 −0.158910
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 4.00000 0.127000
\(993\) 10.0000 0.317340
\(994\) −12.0000 −0.380617
\(995\) −11.0000 −0.348723
\(996\) −14.0000 −0.443607
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −4.00000 −0.126618
\(999\) −11.0000 −0.348025
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 546.2.a.f.1.1 1
3.2 odd 2 1638.2.a.h.1.1 1
4.3 odd 2 4368.2.a.e.1.1 1
7.6 odd 2 3822.2.a.w.1.1 1
13.12 even 2 7098.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.f.1.1 1 1.1 even 1 trivial
1638.2.a.h.1.1 1 3.2 odd 2
3822.2.a.w.1.1 1 7.6 odd 2
4368.2.a.e.1.1 1 4.3 odd 2
7098.2.a.n.1.1 1 13.12 even 2