Properties

Label 546.2.a.d.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -7.00000 q^{19} +3.00000 q^{20} +1.00000 q^{21} -3.00000 q^{22} +9.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -9.00000 q^{29} -3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +3.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +7.00000 q^{38} +1.00000 q^{39} -3.00000 q^{40} +12.0000 q^{41} -1.00000 q^{42} -1.00000 q^{43} +3.00000 q^{44} +3.00000 q^{45} -9.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +9.00000 q^{55} -1.00000 q^{56} -7.00000 q^{57} +9.00000 q^{58} +12.0000 q^{59} +3.00000 q^{60} -1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{65} -3.00000 q^{66} +14.0000 q^{67} -3.00000 q^{68} +9.00000 q^{69} -3.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} -7.00000 q^{73} +7.00000 q^{74} +4.00000 q^{75} -7.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} -10.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} -6.00000 q^{83} +1.00000 q^{84} -9.00000 q^{85} +1.00000 q^{86} -9.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -3.00000 q^{90} +1.00000 q^{91} +9.00000 q^{92} -4.00000 q^{93} -21.0000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −1.00000 −0.267261
\(15\) 3.00000 0.774597
\(16\) 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
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 3.00000 0.670820
\(21\) 1.00000 0.218218
\(22\) −3.00000 −0.639602
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) −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 \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) −3.00000 −0.547723
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) 3.00000 0.514496
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 7.00000 1.13555
\(39\) 1.00000 0.160128
\(40\) −3.00000 −0.474342
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −1.00000 −0.154303
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 3.00000 0.452267
\(45\) 3.00000 0.447214
\(46\) −9.00000 −1.32698
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) −3.00000 −0.420084
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 9.00000 1.21356
\(56\) −1.00000 −0.133631
\(57\) −7.00000 −0.927173
\(58\) 9.00000 1.18176
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 3.00000 0.387298
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) −3.00000 −0.369274
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −3.00000 −0.363803
\(69\) 9.00000 1.08347
\(70\) −3.00000 −0.358569
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 7.00000 0.813733
\(75\) 4.00000 0.461880
\(76\) −7.00000 −0.802955
\(77\) 3.00000 0.341882
\(78\) −1.00000 −0.113228
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 1.00000 0.109109
\(85\) −9.00000 −0.976187
\(86\) 1.00000 0.107833
\(87\) −9.00000 −0.964901
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −3.00000 −0.316228
\(91\) 1.00000 0.104828
\(92\) 9.00000 0.938315
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −21.0000 −2.15455
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 3.00000 0.301511
\(100\) 4.00000 0.400000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 3.00000 0.297044
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 3.00000 0.292770
\(106\) 6.00000 0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) −9.00000 −0.858116
\(111\) −7.00000 −0.664411
\(112\) 1.00000 0.0944911
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 7.00000 0.655610
\(115\) 27.0000 2.51776
\(116\) −9.00000 −0.835629
\(117\) 1.00000 0.0924500
\(118\) −12.0000 −1.10469
\(119\) −3.00000 −0.275010
\(120\) −3.00000 −0.273861
\(121\) −2.00000 −0.181818
\(122\) 1.00000 0.0905357
\(123\) 12.0000 1.08200
\(124\) −4.00000 −0.359211
\(125\) −3.00000 −0.268328
\(126\) −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
\(129\) −1.00000 −0.0880451
\(130\) −3.00000 −0.263117
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 3.00000 0.261116
\(133\) −7.00000 −0.606977
\(134\) −14.0000 −1.20942
\(135\) 3.00000 0.258199
\(136\) 3.00000 0.257248
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) −9.00000 −0.766131
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) −27.0000 −2.24223
\(146\) 7.00000 0.579324
\(147\) 1.00000 0.0824786
\(148\) −7.00000 −0.575396
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) −4.00000 −0.326599
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 7.00000 0.567775
\(153\) −3.00000 −0.242536
\(154\) −3.00000 −0.241747
\(155\) −12.0000 −0.963863
\(156\) 1.00000 0.0800641
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 10.0000 0.795557
\(159\) −6.00000 −0.475831
\(160\) −3.00000 −0.237171
\(161\) 9.00000 0.709299
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 12.0000 0.937043
\(165\) 9.00000 0.700649
\(166\) 6.00000 0.465690
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 1.00000 0.0769231
\(170\) 9.00000 0.690268
\(171\) −7.00000 −0.535303
\(172\) −1.00000 −0.0762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 9.00000 0.682288
\(175\) 4.00000 0.302372
\(176\) 3.00000 0.226134
\(177\) 12.0000 0.901975
\(178\) 6.00000 0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 3.00000 0.223607
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −1.00000 −0.0741249
\(183\) −1.00000 −0.0739221
\(184\) −9.00000 −0.663489
\(185\) −21.0000 −1.54395
\(186\) 4.00000 0.293294
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 21.0000 1.52350
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 10.0000 0.717958
\(195\) 3.00000 0.214834
\(196\) 1.00000 0.0714286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) −3.00000 −0.213201
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −4.00000 −0.282843
\(201\) 14.0000 0.987484
\(202\) −12.0000 −0.844317
\(203\) −9.00000 −0.631676
\(204\) −3.00000 −0.210042
\(205\) 36.0000 2.51435
\(206\) 13.0000 0.905753
\(207\) 9.00000 0.625543
\(208\) 1.00000 0.0693375
\(209\) −21.0000 −1.45260
\(210\) −3.00000 −0.207020
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −6.00000 −0.412082
\(213\) 12.0000 0.822226
\(214\) 6.00000 0.410152
\(215\) −3.00000 −0.204598
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) −11.0000 −0.745014
\(219\) −7.00000 −0.473016
\(220\) 9.00000 0.606780
\(221\) −3.00000 −0.201802
\(222\) 7.00000 0.469809
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.00000 0.266667
\(226\) 18.0000 1.19734
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) −7.00000 −0.463586
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −27.0000 −1.78033
\(231\) 3.00000 0.197386
\(232\) 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
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −10.0000 −0.649570
\(238\) 3.00000 0.194461
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 3.00000 0.193649
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 3.00000 0.191663
\(246\) −12.0000 −0.765092
\(247\) −7.00000 −0.445399
\(248\) 4.00000 0.254000
\(249\) −6.00000 −0.380235
\(250\) 3.00000 0.189737
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 1.00000 0.0629941
\(253\) 27.0000 1.69748
\(254\) −2.00000 −0.125491
\(255\) −9.00000 −0.563602
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 1.00000 0.0622573
\(259\) −7.00000 −0.434959
\(260\) 3.00000 0.186052
\(261\) −9.00000 −0.557086
\(262\) 15.0000 0.926703
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −3.00000 −0.184637
\(265\) −18.0000 −1.10573
\(266\) 7.00000 0.429198
\(267\) −6.00000 −0.367194
\(268\) 14.0000 0.855186
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −3.00000 −0.182574
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −3.00000 −0.181902
\(273\) 1.00000 0.0605228
\(274\) −9.00000 −0.543710
\(275\) 12.0000 0.723627
\(276\) 9.00000 0.541736
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 4.00000 0.239904
\(279\) −4.00000 −0.239474
\(280\) −3.00000 −0.179284
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) −21.0000 −1.24393
\(286\) −3.00000 −0.177394
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 27.0000 1.58549
\(291\) −10.0000 −0.586210
\(292\) −7.00000 −0.409644
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 36.0000 2.09600
\(296\) 7.00000 0.406867
\(297\) 3.00000 0.174078
\(298\) 12.0000 0.695141
\(299\) 9.00000 0.520483
\(300\) 4.00000 0.230940
\(301\) −1.00000 −0.0576390
\(302\) −17.0000 −0.978240
\(303\) 12.0000 0.689382
\(304\) −7.00000 −0.401478
\(305\) −3.00000 −0.171780
\(306\) 3.00000 0.171499
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 3.00000 0.170941
\(309\) −13.0000 −0.739544
\(310\) 12.0000 0.681554
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 13.0000 0.733632
\(315\) 3.00000 0.169031
\(316\) −10.0000 −0.562544
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 6.00000 0.336463
\(319\) −27.0000 −1.51171
\(320\) 3.00000 0.167705
\(321\) −6.00000 −0.334887
\(322\) −9.00000 −0.501550
\(323\) 21.0000 1.16847
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −2.00000 −0.110770
\(327\) 11.0000 0.608301
\(328\) −12.0000 −0.662589
\(329\) 0 0
\(330\) −9.00000 −0.495434
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −6.00000 −0.329293
\(333\) −7.00000 −0.383598
\(334\) −3.00000 −0.164153
\(335\) 42.0000 2.29471
\(336\) 1.00000 0.0545545
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −18.0000 −0.977626
\(340\) −9.00000 −0.488094
\(341\) −12.0000 −0.649836
\(342\) 7.00000 0.378517
\(343\) 1.00000 0.0539949
\(344\) 1.00000 0.0539164
\(345\) 27.0000 1.45363
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −9.00000 −0.482451
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −4.00000 −0.213809
\(351\) 1.00000 0.0533761
\(352\) −3.00000 −0.159901
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −12.0000 −0.637793
\(355\) 36.0000 1.91068
\(356\) −6.00000 −0.317999
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −3.00000 −0.158114
\(361\) 30.0000 1.57895
\(362\) −2.00000 −0.105118
\(363\) −2.00000 −0.104973
\(364\) 1.00000 0.0524142
\(365\) −21.0000 −1.09919
\(366\) 1.00000 0.0522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 9.00000 0.469157
\(369\) 12.0000 0.624695
\(370\) 21.0000 1.09174
\(371\) −6.00000 −0.311504
\(372\) −4.00000 −0.207390
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 9.00000 0.465379
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) −1.00000 −0.0514344
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −21.0000 −1.07728
\(381\) 2.00000 0.102463
\(382\) 15.0000 0.767467
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 9.00000 0.458682
\(386\) 4.00000 0.203595
\(387\) −1.00000 −0.0508329
\(388\) −10.0000 −0.507673
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) −3.00000 −0.151911
\(391\) −27.0000 −1.36545
\(392\) −1.00000 −0.0505076
\(393\) −15.0000 −0.756650
\(394\) 12.0000 0.604551
\(395\) −30.0000 −1.50946
\(396\) 3.00000 0.150756
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 7.00000 0.350878
\(399\) −7.00000 −0.350438
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −14.0000 −0.698257
\(403\) −4.00000 −0.199254
\(404\) 12.0000 0.597022
\(405\) 3.00000 0.149071
\(406\) 9.00000 0.446663
\(407\) −21.0000 −1.04093
\(408\) 3.00000 0.148522
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) −36.0000 −1.77791
\(411\) 9.00000 0.443937
\(412\) −13.0000 −0.640464
\(413\) 12.0000 0.590481
\(414\) −9.00000 −0.442326
\(415\) −18.0000 −0.883585
\(416\) −1.00000 −0.0490290
\(417\) −4.00000 −0.195881
\(418\) 21.0000 1.02714
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 3.00000 0.146385
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −5.00000 −0.243396
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −12.0000 −0.582086
\(426\) −12.0000 −0.581402
\(427\) −1.00000 −0.0483934
\(428\) −6.00000 −0.290021
\(429\) 3.00000 0.144841
\(430\) 3.00000 0.144673
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 1.00000 0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 4.00000 0.192006
\(435\) −27.0000 −1.29455
\(436\) 11.0000 0.526804
\(437\) −63.0000 −3.01370
\(438\) 7.00000 0.334473
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) −9.00000 −0.429058
\(441\) 1.00000 0.0476190
\(442\) 3.00000 0.142695
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −7.00000 −0.332205
\(445\) −18.0000 −0.853282
\(446\) −26.0000 −1.23114
\(447\) −12.0000 −0.567581
\(448\) 1.00000 0.0472456
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) −4.00000 −0.188562
\(451\) 36.0000 1.69517
\(452\) −18.0000 −0.846649
\(453\) 17.0000 0.798730
\(454\) 18.0000 0.844782
\(455\) 3.00000 0.140642
\(456\) 7.00000 0.327805
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −14.0000 −0.654177
\(459\) −3.00000 −0.140028
\(460\) 27.0000 1.25888
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) −3.00000 −0.139573
\(463\) 41.0000 1.90543 0.952716 0.303863i \(-0.0982765\pi\)
0.952716 + 0.303863i \(0.0982765\pi\)
\(464\) −9.00000 −0.417815
\(465\) −12.0000 −0.556487
\(466\) −24.0000 −1.11178
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 1.00000 0.0462250
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) −12.0000 −0.552345
\(473\) −3.00000 −0.137940
\(474\) 10.0000 0.459315
\(475\) −28.0000 −1.28473
\(476\) −3.00000 −0.137505
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) −3.00000 −0.136931
\(481\) −7.00000 −0.319173
\(482\) 10.0000 0.455488
\(483\) 9.00000 0.409514
\(484\) −2.00000 −0.0909091
\(485\) −30.0000 −1.36223
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 1.00000 0.0452679
\(489\) 2.00000 0.0904431
\(490\) −3.00000 −0.135526
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 12.0000 0.541002
\(493\) 27.0000 1.21602
\(494\) 7.00000 0.314945
\(495\) 9.00000 0.404520
\(496\) −4.00000 −0.179605
\(497\) 12.0000 0.538274
\(498\) 6.00000 0.268866
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −3.00000 −0.134164
\(501\) 3.00000 0.134030
\(502\) −21.0000 −0.937276
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 36.0000 1.60198
\(506\) −27.0000 −1.20030
\(507\) 1.00000 0.0444116
\(508\) 2.00000 0.0887357
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 9.00000 0.398527
\(511\) −7.00000 −0.309662
\(512\) −1.00000 −0.0441942
\(513\) −7.00000 −0.309058
\(514\) 18.0000 0.793946
\(515\) −39.0000 −1.71855
\(516\) −1.00000 −0.0440225
\(517\) 0 0
\(518\) 7.00000 0.307562
\(519\) −6.00000 −0.263371
\(520\) −3.00000 −0.131559
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 9.00000 0.393919
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −15.0000 −0.655278
\(525\) 4.00000 0.174574
\(526\) −12.0000 −0.523225
\(527\) 12.0000 0.522728
\(528\) 3.00000 0.130558
\(529\) 58.0000 2.52174
\(530\) 18.0000 0.781870
\(531\) 12.0000 0.520756
\(532\) −7.00000 −0.303488
\(533\) 12.0000 0.519778
\(534\) 6.00000 0.259645
\(535\) −18.0000 −0.778208
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) 3.00000 0.129219
\(540\) 3.00000 0.129099
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 2.00000 0.0858282
\(544\) 3.00000 0.128624
\(545\) 33.0000 1.41356
\(546\) −1.00000 −0.0427960
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 9.00000 0.384461
\(549\) −1.00000 −0.0426790
\(550\) −12.0000 −0.511682
\(551\) 63.0000 2.68389
\(552\) −9.00000 −0.383065
\(553\) −10.0000 −0.425243
\(554\) −26.0000 −1.10463
\(555\) −21.0000 −0.891400
\(556\) −4.00000 −0.169638
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 4.00000 0.169334
\(559\) −1.00000 −0.0422955
\(560\) 3.00000 0.126773
\(561\) −9.00000 −0.379980
\(562\) 6.00000 0.253095
\(563\) −15.0000 −0.632175 −0.316087 0.948730i \(-0.602369\pi\)
−0.316087 + 0.948730i \(0.602369\pi\)
\(564\) 0 0
\(565\) −54.0000 −2.27180
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −12.0000 −0.503509
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 21.0000 0.879593
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 3.00000 0.125436
\(573\) −15.0000 −0.626634
\(574\) −12.0000 −0.500870
\(575\) 36.0000 1.50130
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 8.00000 0.332756
\(579\) −4.00000 −0.166234
\(580\) −27.0000 −1.12111
\(581\) −6.00000 −0.248922
\(582\) 10.0000 0.414513
\(583\) −18.0000 −0.745484
\(584\) 7.00000 0.289662
\(585\) 3.00000 0.124035
\(586\) 18.0000 0.743573
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 1.00000 0.0412393
\(589\) 28.0000 1.15372
\(590\) −36.0000 −1.48210
\(591\) −12.0000 −0.493614
\(592\) −7.00000 −0.287698
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −3.00000 −0.123091
\(595\) −9.00000 −0.368964
\(596\) −12.0000 −0.491539
\(597\) −7.00000 −0.286491
\(598\) −9.00000 −0.368037
\(599\) −9.00000 −0.367730 −0.183865 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) −4.00000 −0.163299
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 1.00000 0.0407570
\(603\) 14.0000 0.570124
\(604\) 17.0000 0.691720
\(605\) −6.00000 −0.243935
\(606\) −12.0000 −0.487467
\(607\) 23.0000 0.933541 0.466771 0.884378i \(-0.345417\pi\)
0.466771 + 0.884378i \(0.345417\pi\)
\(608\) 7.00000 0.283887
\(609\) −9.00000 −0.364698
\(610\) 3.00000 0.121466
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 16.0000 0.645707
\(615\) 36.0000 1.45166
\(616\) −3.00000 −0.120873
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 13.0000 0.522937
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) −12.0000 −0.481932
\(621\) 9.00000 0.361158
\(622\) −18.0000 −0.721734
\(623\) −6.00000 −0.240385
\(624\) 1.00000 0.0400320
\(625\) −29.0000 −1.16000
\(626\) −8.00000 −0.319744
\(627\) −21.0000 −0.838659
\(628\) −13.0000 −0.518756
\(629\) 21.0000 0.837325
\(630\) −3.00000 −0.119523
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 10.0000 0.397779
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) −6.00000 −0.237915
\(637\) 1.00000 0.0396214
\(638\) 27.0000 1.06894
\(639\) 12.0000 0.474713
\(640\) −3.00000 −0.118585
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 6.00000 0.236801
\(643\) −49.0000 −1.93237 −0.966186 0.257847i \(-0.916987\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) 9.00000 0.354650
\(645\) −3.00000 −0.118125
\(646\) −21.0000 −0.826234
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 36.0000 1.41312
\(650\) −4.00000 −0.156893
\(651\) −4.00000 −0.156772
\(652\) 2.00000 0.0783260
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) −11.0000 −0.430134
\(655\) −45.0000 −1.75830
\(656\) 12.0000 0.468521
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 9.00000 0.350325
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 10.0000 0.388661
\(663\) −3.00000 −0.116510
\(664\) 6.00000 0.232845
\(665\) −21.0000 −0.814345
\(666\) 7.00000 0.271244
\(667\) −81.0000 −3.13633
\(668\) 3.00000 0.116073
\(669\) 26.0000 1.00522
\(670\) −42.0000 −1.62260
\(671\) −3.00000 −0.115814
\(672\) −1.00000 −0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 13.0000 0.500741
\(675\) 4.00000 0.153960
\(676\) 1.00000 0.0384615
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 18.0000 0.691286
\(679\) −10.0000 −0.383765
\(680\) 9.00000 0.345134
\(681\) −18.0000 −0.689761
\(682\) 12.0000 0.459504
\(683\) 33.0000 1.26271 0.631355 0.775494i \(-0.282499\pi\)
0.631355 + 0.775494i \(0.282499\pi\)
\(684\) −7.00000 −0.267652
\(685\) 27.0000 1.03162
\(686\) −1.00000 −0.0381802
\(687\) 14.0000 0.534133
\(688\) −1.00000 −0.0381246
\(689\) −6.00000 −0.228582
\(690\) −27.0000 −1.02787
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −6.00000 −0.228086
\(693\) 3.00000 0.113961
\(694\) −12.0000 −0.455514
\(695\) −12.0000 −0.455186
\(696\) 9.00000 0.341144
\(697\) −36.0000 −1.36360
\(698\) −26.0000 −0.984115
\(699\) 24.0000 0.907763
\(700\) 4.00000 0.151186
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 49.0000 1.84807
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 12.0000 0.451306
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −36.0000 −1.35106
\(711\) −10.0000 −0.375029
\(712\) 6.00000 0.224860
\(713\) −36.0000 −1.34821
\(714\) 3.00000 0.112272
\(715\) 9.00000 0.336581
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) −24.0000 −0.895672
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 3.00000 0.111803
\(721\) −13.0000 −0.484145
\(722\) −30.0000 −1.11648
\(723\) −10.0000 −0.371904
\(724\) 2.00000 0.0743294
\(725\) −36.0000 −1.33701
\(726\) 2.00000 0.0742270
\(727\) −1.00000 −0.0370879 −0.0185440 0.999828i \(-0.505903\pi\)
−0.0185440 + 0.999828i \(0.505903\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) 21.0000 0.777245
\(731\) 3.00000 0.110959
\(732\) −1.00000 −0.0369611
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) −8.00000 −0.295285
\(735\) 3.00000 0.110657
\(736\) −9.00000 −0.331744
\(737\) 42.0000 1.54709
\(738\) −12.0000 −0.441726
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) −21.0000 −0.771975
\(741\) −7.00000 −0.257151
\(742\) 6.00000 0.220267
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 4.00000 0.146647
\(745\) −36.0000 −1.31894
\(746\) 4.00000 0.146450
\(747\) −6.00000 −0.219529
\(748\) −9.00000 −0.329073
\(749\) −6.00000 −0.219235
\(750\) 3.00000 0.109545
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 21.0000 0.765283
\(754\) 9.00000 0.327761
\(755\) 51.0000 1.85608
\(756\) 1.00000 0.0363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 16.0000 0.581146
\(759\) 27.0000 0.980038
\(760\) 21.0000 0.761750
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 11.0000 0.398227
\(764\) −15.0000 −0.542681
\(765\) −9.00000 −0.325396
\(766\) 21.0000 0.758761
\(767\) 12.0000 0.433295
\(768\) 1.00000 0.0360844
\(769\) −13.0000 −0.468792 −0.234396 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) −9.00000 −0.324337
\(771\) −18.0000 −0.648254
\(772\) −4.00000 −0.143963
\(773\) 33.0000 1.18693 0.593464 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) 1.00000 0.0359443
\(775\) −16.0000 −0.574737
\(776\) 10.0000 0.358979
\(777\) −7.00000 −0.251124
\(778\) 30.0000 1.07555
\(779\) −84.0000 −3.00961
\(780\) 3.00000 0.107417
\(781\) 36.0000 1.28818
\(782\) 27.0000 0.965518
\(783\) −9.00000 −0.321634
\(784\) 1.00000 0.0357143
\(785\) −39.0000 −1.39197
\(786\) 15.0000 0.535032
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) −12.0000 −0.427482
\(789\) 12.0000 0.427211
\(790\) 30.0000 1.06735
\(791\) −18.0000 −0.640006
\(792\) −3.00000 −0.106600
\(793\) −1.00000 −0.0355110
\(794\) 34.0000 1.20661
\(795\) −18.0000 −0.638394
\(796\) −7.00000 −0.248108
\(797\) 36.0000 1.27519 0.637593 0.770374i \(-0.279930\pi\)
0.637593 + 0.770374i \(0.279930\pi\)
\(798\) 7.00000 0.247797
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) −6.00000 −0.212000
\(802\) −18.0000 −0.635602
\(803\) −21.0000 −0.741074
\(804\) 14.0000 0.493742
\(805\) 27.0000 0.951625
\(806\) 4.00000 0.140894
\(807\) 6.00000 0.211210
\(808\) −12.0000 −0.422159
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) −3.00000 −0.105409
\(811\) −43.0000 −1.50993 −0.754967 0.655763i \(-0.772347\pi\)
−0.754967 + 0.655763i \(0.772347\pi\)
\(812\) −9.00000 −0.315838
\(813\) 2.00000 0.0701431
\(814\) 21.0000 0.736050
\(815\) 6.00000 0.210171
\(816\) −3.00000 −0.105021
\(817\) 7.00000 0.244899
\(818\) 13.0000 0.454534
\(819\) 1.00000 0.0349428
\(820\) 36.0000 1.25717
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −9.00000 −0.313911
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 13.0000 0.452876
\(825\) 12.0000 0.417786
\(826\) −12.0000 −0.417533
\(827\) 27.0000 0.938882 0.469441 0.882964i \(-0.344455\pi\)
0.469441 + 0.882964i \(0.344455\pi\)
\(828\) 9.00000 0.312772
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 18.0000 0.624789
\(831\) 26.0000 0.901930
\(832\) 1.00000 0.0346688
\(833\) −3.00000 −0.103944
\(834\) 4.00000 0.138509
\(835\) 9.00000 0.311458
\(836\) −21.0000 −0.726300
\(837\) −4.00000 −0.138260
\(838\) −3.00000 −0.103633
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −3.00000 −0.103510
\(841\) 52.0000 1.79310
\(842\) 10.0000 0.344623
\(843\) −6.00000 −0.206651
\(844\) 5.00000 0.172107
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) 12.0000 0.411597
\(851\) −63.0000 −2.15961
\(852\) 12.0000 0.411113
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 1.00000 0.0342193
\(855\) −21.0000 −0.718185
\(856\) 6.00000 0.205076
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −3.00000 −0.102418
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −3.00000 −0.102299
\(861\) 12.0000 0.408959
\(862\) −18.0000 −0.613082
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) 16.0000 0.543702
\(867\) −8.00000 −0.271694
\(868\) −4.00000 −0.135769
\(869\) −30.0000 −1.01768
\(870\) 27.0000 0.915386
\(871\) 14.0000 0.474372
\(872\) −11.0000 −0.372507
\(873\) −10.0000 −0.338449
\(874\) 63.0000 2.13101
\(875\) −3.00000 −0.101419
\(876\) −7.00000 −0.236508
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 1.00000 0.0337484
\(879\) −18.0000 −0.607125
\(880\) 9.00000 0.303390
\(881\) −3.00000 −0.101073 −0.0505363 0.998722i \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −25.0000 −0.841317 −0.420658 0.907219i \(-0.638201\pi\)
−0.420658 + 0.907219i \(0.638201\pi\)
\(884\) −3.00000 −0.100901
\(885\) 36.0000 1.21013
\(886\) −18.0000 −0.604722
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 7.00000 0.234905
\(889\) 2.00000 0.0670778
\(890\) 18.0000 0.603361
\(891\) 3.00000 0.100504
\(892\) 26.0000 0.870544
\(893\) 0 0
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 9.00000 0.300501
\(898\) −21.0000 −0.700779
\(899\) 36.0000 1.20067
\(900\) 4.00000 0.133333
\(901\) 18.0000 0.599667
\(902\) −36.0000 −1.19867
\(903\) −1.00000 −0.0332779
\(904\) 18.0000 0.598671
\(905\) 6.00000 0.199447
\(906\) −17.0000 −0.564787
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −18.0000 −0.597351
\(909\) 12.0000 0.398015
\(910\) −3.00000 −0.0994490
\(911\) −33.0000 −1.09334 −0.546669 0.837349i \(-0.684105\pi\)
−0.546669 + 0.837349i \(0.684105\pi\)
\(912\) −7.00000 −0.231793
\(913\) −18.0000 −0.595713
\(914\) 28.0000 0.926158
\(915\) −3.00000 −0.0991769
\(916\) 14.0000 0.462573
\(917\) −15.0000 −0.495344
\(918\) 3.00000 0.0990148
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) −27.0000 −0.890164
\(921\) −16.0000 −0.527218
\(922\) −33.0000 −1.08680
\(923\) 12.0000 0.394985
\(924\) 3.00000 0.0986928
\(925\) −28.0000 −0.920634
\(926\) −41.0000 −1.34734
\(927\) −13.0000 −0.426976
\(928\) 9.00000 0.295439
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 12.0000 0.393496
\(931\) −7.00000 −0.229416
\(932\) 24.0000 0.786146
\(933\) 18.0000 0.589294
\(934\) 3.00000 0.0981630
\(935\) −27.0000 −0.882994
\(936\) −1.00000 −0.0326860
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) −14.0000 −0.457116
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 13.0000 0.423563
\(943\) 108.000 3.51696
\(944\) 12.0000 0.390567
\(945\) 3.00000 0.0975900
\(946\) 3.00000 0.0975384
\(947\) −57.0000 −1.85225 −0.926126 0.377215i \(-0.876882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(948\) −10.0000 −0.324785
\(949\) −7.00000 −0.227230
\(950\) 28.0000 0.908440
\(951\) 0 0
\(952\) 3.00000 0.0972306
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 6.00000 0.194257
\(955\) −45.0000 −1.45617
\(956\) 24.0000 0.776215
\(957\) −27.0000 −0.872786
\(958\) 15.0000 0.484628
\(959\) 9.00000 0.290625
\(960\) 3.00000 0.0968246
\(961\) −15.0000 −0.483871
\(962\) 7.00000 0.225689
\(963\) −6.00000 −0.193347
\(964\) −10.0000 −0.322078
\(965\) −12.0000 −0.386294
\(966\) −9.00000 −0.289570
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) 2.00000 0.0642824
\(969\) 21.0000 0.674617
\(970\) 30.0000 0.963242
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) 16.0000 0.512673
\(975\) 4.00000 0.128103
\(976\) −1.00000 −0.0320092
\(977\) 15.0000 0.479893 0.239946 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(978\) −2.00000 −0.0639529
\(979\) −18.0000 −0.575282
\(980\) 3.00000 0.0958315
\(981\) 11.0000 0.351203
\(982\) −30.0000 −0.957338
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) −12.0000 −0.382546
\(985\) −36.0000 −1.14706
\(986\) −27.0000 −0.859855
\(987\) 0 0
\(988\) −7.00000 −0.222700
\(989\) −9.00000 −0.286183
\(990\) −9.00000 −0.286039
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 4.00000 0.127000
\(993\) −10.0000 −0.317340
\(994\) −12.0000 −0.380617
\(995\) −21.0000 −0.665745
\(996\) −6.00000 −0.190117
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −32.0000 −1.01294
\(999\) −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 546.2.a.d.1.1 1
3.2 odd 2 1638.2.a.l.1.1 1
4.3 odd 2 4368.2.a.l.1.1 1
7.6 odd 2 3822.2.a.a.1.1 1
13.12 even 2 7098.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.d.1.1 1 1.1 even 1 trivial
1638.2.a.l.1.1 1 3.2 odd 2
3822.2.a.a.1.1 1 7.6 odd 2
4368.2.a.l.1.1 1 4.3 odd 2
7098.2.a.w.1.1 1 13.12 even 2