Properties

Label 507.2.a.b.1.1
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} -3.00000 q^{24} -4.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +7.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} -1.00000 q^{37} -6.00000 q^{38} +3.00000 q^{40} -9.00000 q^{41} -2.00000 q^{42} +6.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} +7.00000 q^{51} -9.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -6.00000 q^{56} -6.00000 q^{57} +1.00000 q^{58} +1.00000 q^{60} +1.00000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} +2.00000 q^{67} +7.00000 q^{68} +6.00000 q^{69} +2.00000 q^{70} -6.00000 q^{71} +3.00000 q^{72} -11.0000 q^{73} +1.00000 q^{74} +4.00000 q^{75} -6.00000 q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +14.0000 q^{83} -2.00000 q^{84} -7.00000 q^{85} -6.00000 q^{86} +1.00000 q^{87} +6.00000 q^{88} +14.0000 q^{89} -1.00000 q^{90} +6.00000 q^{92} +4.00000 q^{93} +6.00000 q^{94} +6.00000 q^{95} +5.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) −1.00000 −0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −3.00000 −0.612372
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) −2.00000 −0.348155
\(34\) 7.00000 1.20049
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) −2.00000 −0.308607
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 4.00000 0.565685
\(51\) 7.00000 0.980196
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) −6.00000 −0.801784
\(57\) −6.00000 −0.794719
\(58\) 1.00000 0.131306
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 4.00000 0.508001
\(63\) −2.00000 −0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 7.00000 0.848875
\(69\) 6.00000 0.722315
\(70\) 2.00000 0.239046
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 3.00000 0.353553
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 1.00000 0.116248
\(75\) 4.00000 0.461880
\(76\) −6.00000 −0.688247
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 9.00000 0.993884
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) −2.00000 −0.218218
\(85\) −7.00000 −0.759257
\(86\) −6.00000 −0.646997
\(87\) 1.00000 0.107211
\(88\) 6.00000 0.639602
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 4.00000 0.414781
\(94\) 6.00000 0.618853
\(95\) 6.00000 0.615587
\(96\) 5.00000 0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 3.00000 0.303046
\(99\) 2.00000 0.201008
\(100\) 4.00000 0.400000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) −7.00000 −0.693103
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 9.00000 0.874157
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −2.00000 −0.190693
\(111\) 1.00000 0.0949158
\(112\) 2.00000 0.188982
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 6.00000 0.561951
\(115\) −6.00000 −0.559503
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) 0 0
\(119\) 14.0000 1.28338
\(120\) −3.00000 −0.273861
\(121\) −7.00000 −0.636364
\(122\) −1.00000 −0.0905357
\(123\) 9.00000 0.811503
\(124\) 4.00000 0.359211
\(125\) −9.00000 −0.804984
\(126\) 2.00000 0.178174
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 3.00000 0.265165
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 2.00000 0.174078
\(133\) −12.0000 −1.04053
\(134\) −2.00000 −0.172774
\(135\) −1.00000 −0.0860663
\(136\) −21.0000 −1.80074
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) −6.00000 −0.510754
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 2.00000 0.169031
\(141\) 6.00000 0.505291
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) −1.00000 −0.0830455
\(146\) 11.0000 0.910366
\(147\) 3.00000 0.247436
\(148\) 1.00000 0.0821995
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) −4.00000 −0.326599
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 18.0000 1.45999
\(153\) −7.00000 −0.565916
\(154\) 4.00000 0.322329
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 4.00000 0.318223
\(159\) 9.00000 0.713746
\(160\) −5.00000 −0.395285
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 9.00000 0.702782
\(165\) −2.00000 −0.155700
\(166\) −14.0000 −1.08661
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 6.00000 0.462910
\(169\) 0 0
\(170\) 7.00000 0.536875
\(171\) 6.00000 0.458831
\(172\) −6.00000 −0.457496
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 8.00000 0.604743
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) −18.0000 −1.32698
\(185\) −1.00000 −0.0735215
\(186\) −4.00000 −0.293294
\(187\) −14.0000 −1.02378
\(188\) 6.00000 0.437595
\(189\) 2.00000 0.145479
\(190\) −6.00000 −0.435286
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) −7.00000 −0.505181
\(193\) 9.00000 0.647834 0.323917 0.946085i \(-0.395000\pi\)
0.323917 + 0.946085i \(0.395000\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −2.00000 −0.142134
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) −12.0000 −0.848528
\(201\) −2.00000 −0.141069
\(202\) −3.00000 −0.211079
\(203\) 2.00000 0.140372
\(204\) −7.00000 −0.490098
\(205\) −9.00000 −0.628587
\(206\) −6.00000 −0.418040
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) −2.00000 −0.138013
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 9.00000 0.618123
\(213\) 6.00000 0.411113
\(214\) 6.00000 0.410152
\(215\) 6.00000 0.409197
\(216\) −3.00000 −0.204124
\(217\) 8.00000 0.543075
\(218\) −2.00000 −0.135457
\(219\) 11.0000 0.743311
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) −1.00000 −0.0671156
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 10.0000 0.668153
\(225\) −4.00000 −0.266667
\(226\) 15.0000 0.997785
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 6.00000 0.397360
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 6.00000 0.395628
\(231\) 4.00000 0.263181
\(232\) −3.00000 −0.196960
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) −14.0000 −0.907485
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 1.00000 0.0645497
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) −3.00000 −0.191663
\(246\) −9.00000 −0.573819
\(247\) 0 0
\(248\) −12.0000 −0.762001
\(249\) −14.0000 −0.887214
\(250\) 9.00000 0.569210
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 2.00000 0.125988
\(253\) −12.0000 −0.754434
\(254\) −20.0000 −1.25491
\(255\) 7.00000 0.438357
\(256\) −17.0000 −1.06250
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 6.00000 0.373544
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 8.00000 0.494242
\(263\) −30.0000 −1.84988 −0.924940 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) −6.00000 −0.369274
\(265\) −9.00000 −0.552866
\(266\) 12.0000 0.735767
\(267\) −14.0000 −0.856786
\(268\) −2.00000 −0.122169
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 1.00000 0.0608581
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) −8.00000 −0.482418
\(276\) −6.00000 −0.361158
\(277\) −31.0000 −1.86261 −0.931305 0.364241i \(-0.881328\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) −12.0000 −0.719712
\(279\) −4.00000 −0.239474
\(280\) −6.00000 −0.358569
\(281\) 19.0000 1.13344 0.566722 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) −6.00000 −0.357295
\(283\) −18.0000 −1.06999 −0.534994 0.844856i \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(284\) 6.00000 0.356034
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) 18.0000 1.06251
\(288\) −5.00000 −0.294628
\(289\) 32.0000 1.88235
\(290\) 1.00000 0.0587220
\(291\) −2.00000 −0.117242
\(292\) 11.0000 0.643726
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −3.00000 −0.174371
\(297\) −2.00000 −0.116052
\(298\) 3.00000 0.173785
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) −12.0000 −0.691669
\(302\) −2.00000 −0.115087
\(303\) −3.00000 −0.172345
\(304\) −6.00000 −0.344124
\(305\) 1.00000 0.0572598
\(306\) 7.00000 0.400163
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 4.00000 0.227921
\(309\) −6.00000 −0.341328
\(310\) 4.00000 0.227185
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 3.00000 0.169300
\(315\) −2.00000 −0.112687
\(316\) 4.00000 0.225018
\(317\) 25.0000 1.40414 0.702070 0.712108i \(-0.252259\pi\)
0.702070 + 0.712108i \(0.252259\pi\)
\(318\) −9.00000 −0.504695
\(319\) −2.00000 −0.111979
\(320\) 7.00000 0.391312
\(321\) 6.00000 0.334887
\(322\) −12.0000 −0.668734
\(323\) −42.0000 −2.33694
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −2.00000 −0.110600
\(328\) −27.0000 −1.49083
\(329\) 12.0000 0.661581
\(330\) 2.00000 0.110096
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −14.0000 −0.768350
\(333\) −1.00000 −0.0547997
\(334\) 16.0000 0.875481
\(335\) 2.00000 0.109272
\(336\) −2.00000 −0.109109
\(337\) −33.0000 −1.79762 −0.898812 0.438334i \(-0.855569\pi\)
−0.898812 + 0.438334i \(0.855569\pi\)
\(338\) 0 0
\(339\) 15.0000 0.814688
\(340\) 7.00000 0.379628
\(341\) −8.00000 −0.433224
\(342\) −6.00000 −0.324443
\(343\) 20.0000 1.07990
\(344\) 18.0000 0.970495
\(345\) 6.00000 0.323029
\(346\) 6.00000 0.322562
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) −10.0000 −0.533002
\(353\) 11.0000 0.585471 0.292735 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) −14.0000 −0.741999
\(357\) −14.0000 −0.740959
\(358\) 2.00000 0.105703
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 3.00000 0.158114
\(361\) 17.0000 0.894737
\(362\) 7.00000 0.367912
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) 1.00000 0.0522708
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) 6.00000 0.312772
\(369\) −9.00000 −0.468521
\(370\) 1.00000 0.0519875
\(371\) 18.0000 0.934513
\(372\) −4.00000 −0.207390
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) 14.0000 0.723923
\(375\) 9.00000 0.464758
\(376\) −18.0000 −0.928279
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) −6.00000 −0.307794
\(381\) −20.0000 −1.02463
\(382\) 4.00000 0.204658
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −3.00000 −0.153093
\(385\) −4.00000 −0.203859
\(386\) −9.00000 −0.458088
\(387\) 6.00000 0.304997
\(388\) −2.00000 −0.101535
\(389\) 19.0000 0.963338 0.481669 0.876353i \(-0.340031\pi\)
0.481669 + 0.876353i \(0.340031\pi\)
\(390\) 0 0
\(391\) 42.0000 2.12403
\(392\) −9.00000 −0.454569
\(393\) 8.00000 0.403547
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) −2.00000 −0.100504
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) −14.0000 −0.701757
\(399\) 12.0000 0.600751
\(400\) 4.00000 0.200000
\(401\) −1.00000 −0.0499376 −0.0249688 0.999688i \(-0.507949\pi\)
−0.0249688 + 0.999688i \(0.507949\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) −3.00000 −0.149256
\(405\) 1.00000 0.0496904
\(406\) −2.00000 −0.0992583
\(407\) −2.00000 −0.0991363
\(408\) 21.0000 1.03965
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 9.00000 0.444478
\(411\) −3.00000 −0.147979
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 14.0000 0.687233
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) −12.0000 −0.586939
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 8.00000 0.389434
\(423\) −6.00000 −0.291730
\(424\) −27.0000 −1.31124
\(425\) 28.0000 1.35820
\(426\) −6.00000 −0.290701
\(427\) −2.00000 −0.0967868
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 1.00000 0.0481125
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) −8.00000 −0.384012
\(435\) 1.00000 0.0479463
\(436\) −2.00000 −0.0957826
\(437\) −36.0000 −1.72211
\(438\) −11.0000 −0.525600
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 6.00000 0.286039
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 14.0000 0.663664
\(446\) 16.0000 0.757622
\(447\) 3.00000 0.141895
\(448\) −14.0000 −0.661438
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 4.00000 0.188562
\(451\) −18.0000 −0.847587
\(452\) 15.0000 0.705541
\(453\) −2.00000 −0.0939682
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) 13.0000 0.608114 0.304057 0.952654i \(-0.401659\pi\)
0.304057 + 0.952654i \(0.401659\pi\)
\(458\) −22.0000 −1.02799
\(459\) 7.00000 0.326732
\(460\) 6.00000 0.279751
\(461\) −19.0000 −0.884918 −0.442459 0.896789i \(-0.645894\pi\)
−0.442459 + 0.896789i \(0.645894\pi\)
\(462\) −4.00000 −0.186097
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 1.00000 0.0464238
\(465\) 4.00000 0.185496
\(466\) −10.0000 −0.463241
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 6.00000 0.276759
\(471\) 3.00000 0.138233
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) −4.00000 −0.183726
\(475\) −24.0000 −1.10120
\(476\) −14.0000 −0.641689
\(477\) −9.00000 −0.412082
\(478\) 30.0000 1.37217
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 5.00000 0.228218
\(481\) 0 0
\(482\) 7.00000 0.318841
\(483\) −12.0000 −0.546019
\(484\) 7.00000 0.318182
\(485\) 2.00000 0.0908153
\(486\) 1.00000 0.0453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 3.00000 0.135804
\(489\) −4.00000 −0.180886
\(490\) 3.00000 0.135526
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) −9.00000 −0.405751
\(493\) 7.00000 0.315264
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 4.00000 0.179605
\(497\) 12.0000 0.538274
\(498\) 14.0000 0.627355
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 9.00000 0.402492
\(501\) 16.0000 0.714827
\(502\) −12.0000 −0.535586
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) −6.00000 −0.267261
\(505\) 3.00000 0.133498
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) −20.0000 −0.887357
\(509\) −7.00000 −0.310270 −0.155135 0.987893i \(-0.549581\pi\)
−0.155135 + 0.987893i \(0.549581\pi\)
\(510\) −7.00000 −0.309965
\(511\) 22.0000 0.973223
\(512\) 11.0000 0.486136
\(513\) −6.00000 −0.264906
\(514\) 7.00000 0.308757
\(515\) 6.00000 0.264392
\(516\) 6.00000 0.264135
\(517\) −12.0000 −0.527759
\(518\) −2.00000 −0.0878750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 1.00000 0.0437688
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 8.00000 0.349482
\(525\) −8.00000 −0.349149
\(526\) 30.0000 1.30806
\(527\) 28.0000 1.21970
\(528\) 2.00000 0.0870388
\(529\) 13.0000 0.565217
\(530\) 9.00000 0.390935
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) 0 0
\(534\) 14.0000 0.605839
\(535\) −6.00000 −0.259403
\(536\) 6.00000 0.259161
\(537\) 2.00000 0.0863064
\(538\) 14.0000 0.603583
\(539\) −6.00000 −0.258438
\(540\) 1.00000 0.0430331
\(541\) −45.0000 −1.93470 −0.967351 0.253442i \(-0.918437\pi\)
−0.967351 + 0.253442i \(0.918437\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 35.0000 1.50061
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) −3.00000 −0.128154
\(549\) 1.00000 0.0426790
\(550\) 8.00000 0.341121
\(551\) −6.00000 −0.255609
\(552\) 18.0000 0.766131
\(553\) 8.00000 0.340195
\(554\) 31.0000 1.31706
\(555\) 1.00000 0.0424476
\(556\) −12.0000 −0.508913
\(557\) 9.00000 0.381342 0.190671 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 14.0000 0.591080
\(562\) −19.0000 −0.801467
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −6.00000 −0.252646
\(565\) −15.0000 −0.631055
\(566\) 18.0000 0.756596
\(567\) −2.00000 −0.0839921
\(568\) −18.0000 −0.755263
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 6.00000 0.251312
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) 4.00000 0.167102
\(574\) −18.0000 −0.751305
\(575\) 24.0000 1.00087
\(576\) 7.00000 0.291667
\(577\) −11.0000 −0.457936 −0.228968 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) −32.0000 −1.33102
\(579\) −9.00000 −0.374027
\(580\) 1.00000 0.0415227
\(581\) −28.0000 −1.16164
\(582\) 2.00000 0.0829027
\(583\) −18.0000 −0.745484
\(584\) −33.0000 −1.36555
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) −3.00000 −0.123718
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 1.00000 0.0410997
\(593\) −13.0000 −0.533846 −0.266923 0.963718i \(-0.586007\pi\)
−0.266923 + 0.963718i \(0.586007\pi\)
\(594\) 2.00000 0.0820610
\(595\) 14.0000 0.573944
\(596\) 3.00000 0.122885
\(597\) −14.0000 −0.572982
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 12.0000 0.489898
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 12.0000 0.489083
\(603\) 2.00000 0.0814463
\(604\) −2.00000 −0.0813788
\(605\) −7.00000 −0.284590
\(606\) 3.00000 0.121867
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −30.0000 −1.21666
\(609\) −2.00000 −0.0810441
\(610\) −1.00000 −0.0404888
\(611\) 0 0
\(612\) 7.00000 0.282958
\(613\) 23.0000 0.928961 0.464481 0.885583i \(-0.346241\pi\)
0.464481 + 0.885583i \(0.346241\pi\)
\(614\) 14.0000 0.564994
\(615\) 9.00000 0.362915
\(616\) −12.0000 −0.483494
\(617\) −13.0000 −0.523360 −0.261680 0.965155i \(-0.584277\pi\)
−0.261680 + 0.965155i \(0.584277\pi\)
\(618\) 6.00000 0.241355
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 4.00000 0.160644
\(621\) 6.00000 0.240772
\(622\) −18.0000 −0.721734
\(623\) −28.0000 −1.12180
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −6.00000 −0.239808
\(627\) −12.0000 −0.479234
\(628\) 3.00000 0.119713
\(629\) 7.00000 0.279108
\(630\) 2.00000 0.0796819
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −12.0000 −0.477334
\(633\) 8.00000 0.317971
\(634\) −25.0000 −0.992877
\(635\) 20.0000 0.793676
\(636\) −9.00000 −0.356873
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) −6.00000 −0.237356
\(640\) 3.00000 0.118585
\(641\) −31.0000 −1.22443 −0.612213 0.790693i \(-0.709721\pi\)
−0.612213 + 0.790693i \(0.709721\pi\)
\(642\) −6.00000 −0.236801
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) −12.0000 −0.472866
\(645\) −6.00000 −0.236250
\(646\) 42.0000 1.65247
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) −4.00000 −0.156652
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 2.00000 0.0782062
\(655\) −8.00000 −0.312586
\(656\) 9.00000 0.351391
\(657\) −11.0000 −0.429151
\(658\) −12.0000 −0.467809
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 2.00000 0.0778499
\(661\) −45.0000 −1.75030 −0.875149 0.483854i \(-0.839236\pi\)
−0.875149 + 0.483854i \(0.839236\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 42.0000 1.62992
\(665\) −12.0000 −0.465340
\(666\) 1.00000 0.0387492
\(667\) 6.00000 0.232321
\(668\) 16.0000 0.619059
\(669\) 16.0000 0.618596
\(670\) −2.00000 −0.0772667
\(671\) 2.00000 0.0772091
\(672\) −10.0000 −0.385758
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 33.0000 1.27111
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −15.0000 −0.576072
\(679\) −4.00000 −0.153506
\(680\) −21.0000 −0.805313
\(681\) −14.0000 −0.536481
\(682\) 8.00000 0.306336
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −6.00000 −0.229416
\(685\) 3.00000 0.114624
\(686\) −20.0000 −0.763604
\(687\) −22.0000 −0.839352
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) −6.00000 −0.228416
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 6.00000 0.228086
\(693\) −4.00000 −0.151947
\(694\) −18.0000 −0.683271
\(695\) 12.0000 0.455186
\(696\) 3.00000 0.113715
\(697\) 63.0000 2.38630
\(698\) −26.0000 −0.984115
\(699\) −10.0000 −0.378235
\(700\) −8.00000 −0.302372
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) 14.0000 0.527645
\(705\) 6.00000 0.225973
\(706\) −11.0000 −0.413990
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 11.0000 0.413114 0.206557 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(710\) 6.00000 0.225176
\(711\) −4.00000 −0.150012
\(712\) 42.0000 1.57402
\(713\) 24.0000 0.898807
\(714\) 14.0000 0.523937
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) 30.0000 1.12037
\(718\) −18.0000 −0.671754
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −12.0000 −0.446903
\(722\) −17.0000 −0.632674
\(723\) 7.00000 0.260333
\(724\) 7.00000 0.260153
\(725\) 4.00000 0.148556
\(726\) −7.00000 −0.259794
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.0000 0.407128
\(731\) −42.0000 −1.55343
\(732\) 1.00000 0.0369611
\(733\) 15.0000 0.554038 0.277019 0.960864i \(-0.410654\pi\)
0.277019 + 0.960864i \(0.410654\pi\)
\(734\) −10.0000 −0.369107
\(735\) 3.00000 0.110657
\(736\) 30.0000 1.10581
\(737\) 4.00000 0.147342
\(738\) 9.00000 0.331295
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) −18.0000 −0.660801
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 12.0000 0.439941
\(745\) −3.00000 −0.109911
\(746\) 11.0000 0.402739
\(747\) 14.0000 0.512233
\(748\) 14.0000 0.511891
\(749\) 12.0000 0.438470
\(750\) −9.00000 −0.328634
\(751\) −34.0000 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(752\) 6.00000 0.218797
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) −2.00000 −0.0727393
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) −36.0000 −1.30758
\(759\) 12.0000 0.435572
\(760\) 18.0000 0.652929
\(761\) −50.0000 −1.81250 −0.906249 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) 20.0000 0.724524
\(763\) −4.00000 −0.144810
\(764\) 4.00000 0.144715
\(765\) −7.00000 −0.253086
\(766\) 8.00000 0.289052
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 4.00000 0.144150
\(771\) 7.00000 0.252099
\(772\) −9.00000 −0.323917
\(773\) 14.0000 0.503545 0.251773 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) −6.00000 −0.215666
\(775\) 16.0000 0.574737
\(776\) 6.00000 0.215387
\(777\) −2.00000 −0.0717496
\(778\) −19.0000 −0.681183
\(779\) −54.0000 −1.93475
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) −42.0000 −1.50192
\(783\) 1.00000 0.0357371
\(784\) 3.00000 0.107143
\(785\) −3.00000 −0.107075
\(786\) −8.00000 −0.285351
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 6.00000 0.213741
\(789\) 30.0000 1.06803
\(790\) 4.00000 0.142314
\(791\) 30.0000 1.06668
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −34.0000 −1.20661
\(795\) 9.00000 0.319197
\(796\) −14.0000 −0.496217
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) −12.0000 −0.424795
\(799\) 42.0000 1.48585
\(800\) 20.0000 0.707107
\(801\) 14.0000 0.494666
\(802\) 1.00000 0.0353112
\(803\) −22.0000 −0.776363
\(804\) 2.00000 0.0705346
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 14.0000 0.492823
\(808\) 9.00000 0.316619
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) 4.00000 0.140114
\(816\) −7.00000 −0.245049
\(817\) 36.0000 1.25948
\(818\) 7.00000 0.244749
\(819\) 0 0
\(820\) 9.00000 0.314294
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 3.00000 0.104637
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 18.0000 0.627060
\(825\) 8.00000 0.278524
\(826\) 0 0
\(827\) −16.0000 −0.556375 −0.278187 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) 6.00000 0.208514
\(829\) 17.0000 0.590434 0.295217 0.955430i \(-0.404608\pi\)
0.295217 + 0.955430i \(0.404608\pi\)
\(830\) −14.0000 −0.485947
\(831\) 31.0000 1.07538
\(832\) 0 0
\(833\) 21.0000 0.727607
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) −12.0000 −0.415029
\(837\) 4.00000 0.138260
\(838\) −16.0000 −0.552711
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 6.00000 0.207020
\(841\) −28.0000 −0.965517
\(842\) −19.0000 −0.654783
\(843\) −19.0000 −0.654395
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 14.0000 0.481046
\(848\) 9.00000 0.309061
\(849\) 18.0000 0.617758
\(850\) −28.0000 −0.960392
\(851\) 6.00000 0.205677
\(852\) −6.00000 −0.205557
\(853\) −21.0000 −0.719026 −0.359513 0.933140i \(-0.617057\pi\)
−0.359513 + 0.933140i \(0.617057\pi\)
\(854\) 2.00000 0.0684386
\(855\) 6.00000 0.205196
\(856\) −18.0000 −0.615227
\(857\) −31.0000 −1.05894 −0.529470 0.848329i \(-0.677609\pi\)
−0.529470 + 0.848329i \(0.677609\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) −6.00000 −0.204598
\(861\) −18.0000 −0.613438
\(862\) −30.0000 −1.02180
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) 5.00000 0.170103
\(865\) −6.00000 −0.204006
\(866\) −19.0000 −0.645646
\(867\) −32.0000 −1.08678
\(868\) −8.00000 −0.271538
\(869\) −8.00000 −0.271381
\(870\) −1.00000 −0.0339032
\(871\) 0 0
\(872\) 6.00000 0.203186
\(873\) 2.00000 0.0676897
\(874\) 36.0000 1.21772
\(875\) 18.0000 0.608511
\(876\) −11.0000 −0.371656
\(877\) −17.0000 −0.574049 −0.287025 0.957923i \(-0.592666\pi\)
−0.287025 + 0.957923i \(0.592666\pi\)
\(878\) −14.0000 −0.472477
\(879\) −9.00000 −0.303562
\(880\) −2.00000 −0.0674200
\(881\) 37.0000 1.24656 0.623281 0.781998i \(-0.285799\pi\)
0.623281 + 0.781998i \(0.285799\pi\)
\(882\) 3.00000 0.101015
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 3.00000 0.100673
\(889\) −40.0000 −1.34156
\(890\) −14.0000 −0.469281
\(891\) 2.00000 0.0670025
\(892\) 16.0000 0.535720
\(893\) −36.0000 −1.20469
\(894\) −3.00000 −0.100335
\(895\) −2.00000 −0.0668526
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) 4.00000 0.133407
\(900\) 4.00000 0.133333
\(901\) 63.0000 2.09883
\(902\) 18.0000 0.599334
\(903\) 12.0000 0.399335
\(904\) −45.0000 −1.49668
\(905\) −7.00000 −0.232688
\(906\) 2.00000 0.0664455
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −14.0000 −0.464606
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 6.00000 0.198680
\(913\) 28.0000 0.926665
\(914\) −13.0000 −0.430002
\(915\) −1.00000 −0.0330590
\(916\) −22.0000 −0.726900
\(917\) 16.0000 0.528367
\(918\) −7.00000 −0.231034
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −18.0000 −0.593442
\(921\) 14.0000 0.461316
\(922\) 19.0000 0.625732
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 4.00000 0.131519
\(926\) −26.0000 −0.854413
\(927\) 6.00000 0.197066
\(928\) 5.00000 0.164133
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) −4.00000 −0.131165
\(931\) −18.0000 −0.589926
\(932\) −10.0000 −0.327561
\(933\) −18.0000 −0.589294
\(934\) −6.00000 −0.196326
\(935\) −14.0000 −0.457849
\(936\) 0 0
\(937\) −49.0000 −1.60076 −0.800380 0.599493i \(-0.795369\pi\)
−0.800380 + 0.599493i \(0.795369\pi\)
\(938\) 4.00000 0.130605
\(939\) −6.00000 −0.195803
\(940\) 6.00000 0.195698
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −3.00000 −0.0977453
\(943\) 54.0000 1.75848
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) −12.0000 −0.390154
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) 24.0000 0.778663
\(951\) −25.0000 −0.810681
\(952\) 42.0000 1.36123
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 9.00000 0.291386
\(955\) −4.00000 −0.129437
\(956\) 30.0000 0.970269
\(957\) 2.00000 0.0646508
\(958\) 24.0000 0.775405
\(959\) −6.00000 −0.193750
\(960\) −7.00000 −0.225924
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 7.00000 0.225455
\(965\) 9.00000 0.289720
\(966\) 12.0000 0.386094
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) −21.0000 −0.674966
\(969\) 42.0000 1.34923
\(970\) −2.00000 −0.0642161
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) −24.0000 −0.769405
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −33.0000 −1.05576 −0.527882 0.849318i \(-0.677014\pi\)
−0.527882 + 0.849318i \(0.677014\pi\)
\(978\) 4.00000 0.127906
\(979\) 28.0000 0.894884
\(980\) 3.00000 0.0958315
\(981\) 2.00000 0.0638551
\(982\) −6.00000 −0.191468
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 27.0000 0.860729
\(985\) −6.00000 −0.191176
\(986\) −7.00000 −0.222925
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) −2.00000 −0.0635642
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 20.0000 0.635001
\(993\) −4.00000 −0.126936
\(994\) −12.0000 −0.380617
\(995\) 14.0000 0.443830
\(996\) 14.0000 0.443607
\(997\) −35.0000 −1.10846 −0.554231 0.832363i \(-0.686987\pi\)
−0.554231 + 0.832363i \(0.686987\pi\)
\(998\) 24.0000 0.759707
\(999\) 1.00000 0.0316386
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 507.2.a.b.1.1 1
3.2 odd 2 1521.2.a.d.1.1 1
4.3 odd 2 8112.2.a.bc.1.1 1
13.2 odd 12 507.2.j.d.316.1 4
13.3 even 3 507.2.e.c.22.1 2
13.4 even 6 39.2.e.a.16.1 2
13.5 odd 4 507.2.b.b.337.2 2
13.6 odd 12 507.2.j.d.361.2 4
13.7 odd 12 507.2.j.d.361.1 4
13.8 odd 4 507.2.b.b.337.1 2
13.9 even 3 507.2.e.c.484.1 2
13.10 even 6 39.2.e.a.22.1 yes 2
13.11 odd 12 507.2.j.d.316.2 4
13.12 even 2 507.2.a.c.1.1 1
39.5 even 4 1521.2.b.c.1351.1 2
39.8 even 4 1521.2.b.c.1351.2 2
39.17 odd 6 117.2.g.b.55.1 2
39.23 odd 6 117.2.g.b.100.1 2
39.38 odd 2 1521.2.a.a.1.1 1
52.23 odd 6 624.2.q.c.529.1 2
52.43 odd 6 624.2.q.c.289.1 2
52.51 odd 2 8112.2.a.w.1.1 1
65.4 even 6 975.2.i.f.601.1 2
65.17 odd 12 975.2.bb.d.874.2 4
65.23 odd 12 975.2.bb.d.724.2 4
65.43 odd 12 975.2.bb.d.874.1 4
65.49 even 6 975.2.i.f.451.1 2
65.62 odd 12 975.2.bb.d.724.1 4
156.23 even 6 1872.2.t.j.1153.1 2
156.95 even 6 1872.2.t.j.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.a.16.1 2 13.4 even 6
39.2.e.a.22.1 yes 2 13.10 even 6
117.2.g.b.55.1 2 39.17 odd 6
117.2.g.b.100.1 2 39.23 odd 6
507.2.a.b.1.1 1 1.1 even 1 trivial
507.2.a.c.1.1 1 13.12 even 2
507.2.b.b.337.1 2 13.8 odd 4
507.2.b.b.337.2 2 13.5 odd 4
507.2.e.c.22.1 2 13.3 even 3
507.2.e.c.484.1 2 13.9 even 3
507.2.j.d.316.1 4 13.2 odd 12
507.2.j.d.316.2 4 13.11 odd 12
507.2.j.d.361.1 4 13.7 odd 12
507.2.j.d.361.2 4 13.6 odd 12
624.2.q.c.289.1 2 52.43 odd 6
624.2.q.c.529.1 2 52.23 odd 6
975.2.i.f.451.1 2 65.49 even 6
975.2.i.f.601.1 2 65.4 even 6
975.2.bb.d.724.1 4 65.62 odd 12
975.2.bb.d.724.2 4 65.23 odd 12
975.2.bb.d.874.1 4 65.43 odd 12
975.2.bb.d.874.2 4 65.17 odd 12
1521.2.a.a.1.1 1 39.38 odd 2
1521.2.a.d.1.1 1 3.2 odd 2
1521.2.b.c.1351.1 2 39.5 even 4
1521.2.b.c.1351.2 2 39.8 even 4
1872.2.t.j.289.1 2 156.95 even 6
1872.2.t.j.1153.1 2 156.23 even 6
8112.2.a.w.1.1 1 52.51 odd 2
8112.2.a.bc.1.1 1 4.3 odd 2