Properties

Label 462.2.a.c.1.1
Level $462$
Weight $2$
Character 462.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68908857338\)
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 \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +8.00000 q^{38} -2.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} +8.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -14.0000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +2.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +12.0000 q^{85} -8.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} -14.0000 q^{89} -2.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} -4.00000 q^{94} -16.0000 q^{95} +1.00000 q^{96} +18.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) −1.00000 −0.213201
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −6.00000 −1.02899
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 8.00000 1.29777
\(39\) −2.00000 −0.320256
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −1.00000 −0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.00000 0.298142
\(46\) −4.00000 −0.589768
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) 1.00000 0.133631
\(57\) 8.00000 1.05963
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −2.00000 −0.258199
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 1.00000 0.123091
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) −4.00000 −0.481543
\(70\) 2.00000 0.239046
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) −1.00000 −0.113961
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) 12.0000 1.30158
\(86\) −8.00000 −0.862662
\(87\) −2.00000 −0.214423
\(88\) −1.00000 −0.106600
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −2.00000 −0.210819
\(91\) −2.00000 −0.209657
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) −4.00000 −0.412568
\(95\) −16.0000 −1.64157
\(96\) 1.00000 0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000 0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 2.00000 0.195180
\(106\) −10.0000 −0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −2.00000 −0.190693
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −8.00000 −0.749269
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) −4.00000 −0.368230
\(119\) −6.00000 −0.550019
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) 14.0000 1.26750
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) −12.0000 −1.07331
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 8.00000 0.693688
\(134\) 4.00000 0.345547
\(135\) −2.00000 −0.172133
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 4.00000 0.340503
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −2.00000 −0.169031
\(141\) −4.00000 −0.336861
\(142\) 4.00000 0.335673
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 14.0000 1.15865
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 8.00000 0.648886
\(153\) 6.00000 0.485071
\(154\) 1.00000 0.0805823
\(155\) 16.0000 1.28515
\(156\) −2.00000 −0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 8.00000 0.636446
\(159\) −10.0000 −0.793052
\(160\) −2.00000 −0.158114
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 6.00000 0.468521
\(165\) −2.00000 −0.155700
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) −8.00000 −0.611775
\(172\) 8.00000 0.609994
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 2.00000 0.151620
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) −4.00000 −0.300658
\(178\) 14.0000 1.04934
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 2.00000 0.148250
\(183\) 14.0000 1.03491
\(184\) −4.00000 −0.294884
\(185\) 12.0000 0.882258
\(186\) 8.00000 0.586588
\(187\) 6.00000 0.438763
\(188\) 4.00000 0.291730
\(189\) 1.00000 0.0727393
\(190\) 16.0000 1.16076
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −18.0000 −1.29232
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) −2.00000 −0.140372
\(204\) −6.00000 −0.420084
\(205\) 12.0000 0.838116
\(206\) 16.0000 1.11477
\(207\) 4.00000 0.278019
\(208\) 2.00000 0.138675
\(209\) −8.00000 −0.553372
\(210\) −2.00000 −0.138013
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 10.0000 0.686803
\(213\) 4.00000 0.274075
\(214\) 12.0000 0.820303
\(215\) 16.0000 1.09119
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −10.0000 −0.677285
\(219\) 14.0000 0.946032
\(220\) 2.00000 0.134840
\(221\) 12.0000 0.807207
\(222\) 6.00000 0.402694
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −10.0000 −0.665190
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 8.00000 0.529813
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −8.00000 −0.527504
\(231\) 1.00000 0.0657952
\(232\) −2.00000 −0.131306
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −2.00000 −0.130744
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 6.00000 0.388922
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −2.00000 −0.129099
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) 2.00000 0.127775
\(246\) 6.00000 0.382546
\(247\) −16.0000 −1.01806
\(248\) −8.00000 −0.508001
\(249\) −4.00000 −0.253490
\(250\) 12.0000 0.758947
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) −8.00000 −0.501965
\(255\) −12.0000 −0.751469
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 8.00000 0.498058
\(259\) −6.00000 −0.372822
\(260\) 4.00000 0.248069
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 1.00000 0.0615457
\(265\) 20.0000 1.22859
\(266\) −8.00000 −0.490511
\(267\) 14.0000 0.856786
\(268\) −4.00000 −0.244339
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 2.00000 0.121716
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 2.00000 0.121046
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) −4.00000 −0.240772
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 16.0000 0.959616
\(279\) 8.00000 0.478947
\(280\) 2.00000 0.119523
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 4.00000 0.238197
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −4.00000 −0.237356
\(285\) 16.0000 0.947758
\(286\) −2.00000 −0.118262
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −4.00000 −0.234888
\(291\) −18.0000 −1.05518
\(292\) −14.0000 −0.819288
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 1.00000 0.0583212
\(295\) 8.00000 0.465778
\(296\) −6.00000 −0.348743
\(297\) −1.00000 −0.0580259
\(298\) −10.0000 −0.579284
\(299\) 8.00000 0.462652
\(300\) 1.00000 0.0577350
\(301\) −8.00000 −0.461112
\(302\) −16.0000 −0.920697
\(303\) 6.00000 0.344691
\(304\) −8.00000 −0.458831
\(305\) −28.0000 −1.60328
\(306\) −6.00000 −0.342997
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 16.0000 0.910208
\(310\) −16.0000 −0.908739
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 2.00000 0.113228
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 2.00000 0.112867
\(315\) −2.00000 −0.112687
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 10.0000 0.560772
\(319\) 2.00000 0.111979
\(320\) 2.00000 0.111803
\(321\) 12.0000 0.669775
\(322\) 4.00000 0.222911
\(323\) −48.0000 −2.67079
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −12.0000 −0.664619
\(327\) −10.0000 −0.553001
\(328\) −6.00000 −0.331295
\(329\) −4.00000 −0.220527
\(330\) 2.00000 0.110096
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) 8.00000 0.437741
\(335\) −8.00000 −0.437087
\(336\) 1.00000 0.0545545
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) −10.0000 −0.543125
\(340\) 12.0000 0.650791
\(341\) 8.00000 0.433224
\(342\) 8.00000 0.432590
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) −8.00000 −0.430706
\(346\) 14.0000 0.752645
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) −2.00000 −0.107211
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) −1.00000 −0.0533002
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 4.00000 0.212598
\(355\) −8.00000 −0.424596
\(356\) −14.0000 −0.741999
\(357\) 6.00000 0.317554
\(358\) 12.0000 0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −2.00000 −0.105409
\(361\) 45.0000 2.36842
\(362\) 10.0000 0.525588
\(363\) −1.00000 −0.0524864
\(364\) −2.00000 −0.104828
\(365\) −28.0000 −1.46559
\(366\) −14.0000 −0.731792
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 4.00000 0.208514
\(369\) 6.00000 0.312348
\(370\) −12.0000 −0.623850
\(371\) −10.0000 −0.519174
\(372\) −8.00000 −0.414781
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −6.00000 −0.310253
\(375\) 12.0000 0.619677
\(376\) −4.00000 −0.206284
\(377\) 4.00000 0.206010
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −16.0000 −0.820783
\(381\) −8.00000 −0.409852
\(382\) 4.00000 0.204658
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 1.00000 0.0510310
\(385\) −2.00000 −0.101929
\(386\) 6.00000 0.305392
\(387\) 8.00000 0.406663
\(388\) 18.0000 0.913812
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 4.00000 0.202548
\(391\) 24.0000 1.21373
\(392\) −1.00000 −0.0505076
\(393\) 4.00000 0.201773
\(394\) 6.00000 0.302276
\(395\) −16.0000 −0.805047
\(396\) 1.00000 0.0502519
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 8.00000 0.401004
\(399\) −8.00000 −0.400501
\(400\) −1.00000 −0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) −4.00000 −0.199502
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) 2.00000 0.0993808
\(406\) 2.00000 0.0992583
\(407\) 6.00000 0.297409
\(408\) 6.00000 0.297044
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −12.0000 −0.592638
\(411\) 6.00000 0.295958
\(412\) −16.0000 −0.788263
\(413\) −4.00000 −0.196827
\(414\) −4.00000 −0.196589
\(415\) 8.00000 0.392705
\(416\) −2.00000 −0.0980581
\(417\) 16.0000 0.783523
\(418\) 8.00000 0.391293
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −8.00000 −0.389434
\(423\) 4.00000 0.194487
\(424\) −10.0000 −0.485643
\(425\) −6.00000 −0.291043
\(426\) −4.00000 −0.193801
\(427\) 14.0000 0.677507
\(428\) −12.0000 −0.580042
\(429\) −2.00000 −0.0965609
\(430\) −16.0000 −0.771589
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 8.00000 0.384012
\(435\) −4.00000 −0.191785
\(436\) 10.0000 0.478913
\(437\) −32.0000 −1.53077
\(438\) −14.0000 −0.668946
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −6.00000 −0.284747
\(445\) −28.0000 −1.32733
\(446\) −16.0000 −0.757622
\(447\) −10.0000 −0.472984
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) 6.00000 0.282529
\(452\) 10.0000 0.470360
\(453\) −16.0000 −0.751746
\(454\) 4.00000 0.187729
\(455\) −4.00000 −0.187523
\(456\) −8.00000 −0.374634
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) 8.00000 0.373002
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 2.00000 0.0928477
\(465\) −16.0000 −0.741982
\(466\) 18.0000 0.833834
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 2.00000 0.0924500
\(469\) 4.00000 0.184703
\(470\) −8.00000 −0.369012
\(471\) 2.00000 0.0921551
\(472\) −4.00000 −0.184115
\(473\) 8.00000 0.367840
\(474\) −8.00000 −0.367452
\(475\) 8.00000 0.367065
\(476\) −6.00000 −0.275010
\(477\) 10.0000 0.457869
\(478\) 8.00000 0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 2.00000 0.0912871
\(481\) 12.0000 0.547153
\(482\) −2.00000 −0.0910975
\(483\) 4.00000 0.182006
\(484\) 1.00000 0.0454545
\(485\) 36.0000 1.63468
\(486\) 1.00000 0.0453609
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 14.0000 0.633750
\(489\) −12.0000 −0.542659
\(490\) −2.00000 −0.0903508
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −6.00000 −0.270501
\(493\) 12.0000 0.540453
\(494\) 16.0000 0.719874
\(495\) 2.00000 0.0898933
\(496\) 8.00000 0.359211
\(497\) 4.00000 0.179425
\(498\) 4.00000 0.179244
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −12.0000 −0.536656
\(501\) 8.00000 0.357414
\(502\) 20.0000 0.892644
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 1.00000 0.0445435
\(505\) −12.0000 −0.533993
\(506\) −4.00000 −0.177822
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 12.0000 0.531369
\(511\) 14.0000 0.619324
\(512\) −1.00000 −0.0441942
\(513\) 8.00000 0.353209
\(514\) −26.0000 −1.14681
\(515\) −32.0000 −1.41009
\(516\) −8.00000 −0.352180
\(517\) 4.00000 0.175920
\(518\) 6.00000 0.263625
\(519\) 14.0000 0.614532
\(520\) −4.00000 −0.175412
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −4.00000 −0.174741
\(525\) −1.00000 −0.0436436
\(526\) −24.0000 −1.04645
\(527\) 48.0000 2.09091
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) −20.0000 −0.868744
\(531\) 4.00000 0.173585
\(532\) 8.00000 0.346844
\(533\) 12.0000 0.519778
\(534\) −14.0000 −0.605839
\(535\) −24.0000 −1.03761
\(536\) 4.00000 0.172774
\(537\) 12.0000 0.517838
\(538\) −2.00000 −0.0862261
\(539\) 1.00000 0.0430730
\(540\) −2.00000 −0.0860663
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −8.00000 −0.343629
\(543\) 10.0000 0.429141
\(544\) −6.00000 −0.257248
\(545\) 20.0000 0.856706
\(546\) −2.00000 −0.0855921
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) −6.00000 −0.256307
\(549\) −14.0000 −0.597505
\(550\) 1.00000 0.0426401
\(551\) −16.0000 −0.681623
\(552\) 4.00000 0.170251
\(553\) 8.00000 0.340195
\(554\) 6.00000 0.254916
\(555\) −12.0000 −0.509372
\(556\) −16.0000 −0.678551
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −8.00000 −0.338667
\(559\) 16.0000 0.676728
\(560\) −2.00000 −0.0845154
\(561\) −6.00000 −0.253320
\(562\) −30.0000 −1.26547
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −4.00000 −0.168430
\(565\) 20.0000 0.841406
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 4.00000 0.167836
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) −16.0000 −0.670166
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 2.00000 0.0836242
\(573\) 4.00000 0.167102
\(574\) 6.00000 0.250435
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −19.0000 −0.790296
\(579\) 6.00000 0.249351
\(580\) 4.00000 0.166091
\(581\) −4.00000 −0.165948
\(582\) 18.0000 0.746124
\(583\) 10.0000 0.414158
\(584\) 14.0000 0.579324
\(585\) 4.00000 0.165380
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −64.0000 −2.63707
\(590\) −8.00000 −0.329355
\(591\) 6.00000 0.246807
\(592\) 6.00000 0.246598
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 1.00000 0.0410305
\(595\) −12.0000 −0.491952
\(596\) 10.0000 0.409616
\(597\) 8.00000 0.327418
\(598\) −8.00000 −0.327144
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) 16.0000 0.651031
\(605\) 2.00000 0.0813116
\(606\) −6.00000 −0.243733
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 8.00000 0.324443
\(609\) 2.00000 0.0810441
\(610\) 28.0000 1.13369
\(611\) 8.00000 0.323645
\(612\) 6.00000 0.242536
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 8.00000 0.322854
\(615\) −12.0000 −0.483887
\(616\) 1.00000 0.0402911
\(617\) −46.0000 −1.85189 −0.925945 0.377658i \(-0.876729\pi\)
−0.925945 + 0.377658i \(0.876729\pi\)
\(618\) −16.0000 −0.643614
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 16.0000 0.642575
\(621\) −4.00000 −0.160514
\(622\) −12.0000 −0.481156
\(623\) 14.0000 0.560898
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) 8.00000 0.319489
\(628\) −2.00000 −0.0798087
\(629\) 36.0000 1.43541
\(630\) 2.00000 0.0796819
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 8.00000 0.318223
\(633\) −8.00000 −0.317971
\(634\) −18.0000 −0.714871
\(635\) 16.0000 0.634941
\(636\) −10.0000 −0.396526
\(637\) 2.00000 0.0792429
\(638\) −2.00000 −0.0791808
\(639\) −4.00000 −0.158238
\(640\) −2.00000 −0.0790569
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −12.0000 −0.473602
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) −4.00000 −0.157622
\(645\) −16.0000 −0.629999
\(646\) 48.0000 1.88853
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.00000 0.157014
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) 12.0000 0.469956
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 10.0000 0.391031
\(655\) −8.00000 −0.312586
\(656\) 6.00000 0.234261
\(657\) −14.0000 −0.546192
\(658\) 4.00000 0.155936
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) −4.00000 −0.155464
\(663\) −12.0000 −0.466041
\(664\) −4.00000 −0.155230
\(665\) 16.0000 0.620453
\(666\) −6.00000 −0.232495
\(667\) 8.00000 0.309761
\(668\) −8.00000 −0.309529
\(669\) −16.0000 −0.618596
\(670\) 8.00000 0.309067
\(671\) −14.0000 −0.540464
\(672\) −1.00000 −0.0385758
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 22.0000 0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 10.0000 0.384048
\(679\) −18.0000 −0.690777
\(680\) −12.0000 −0.460179
\(681\) 4.00000 0.153280
\(682\) −8.00000 −0.306336
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −8.00000 −0.305888
\(685\) −12.0000 −0.458496
\(686\) 1.00000 0.0381802
\(687\) 10.0000 0.381524
\(688\) 8.00000 0.304997
\(689\) 20.0000 0.761939
\(690\) 8.00000 0.304555
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) −14.0000 −0.532200
\(693\) −1.00000 −0.0379869
\(694\) 36.0000 1.36654
\(695\) −32.0000 −1.21383
\(696\) 2.00000 0.0758098
\(697\) 36.0000 1.36360
\(698\) 22.0000 0.832712
\(699\) 18.0000 0.680823
\(700\) 1.00000 0.0377964
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 2.00000 0.0754851
\(703\) −48.0000 −1.81035
\(704\) 1.00000 0.0376889
\(705\) −8.00000 −0.301297
\(706\) −26.0000 −0.978523
\(707\) 6.00000 0.225653
\(708\) −4.00000 −0.150329
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 8.00000 0.300235
\(711\) −8.00000 −0.300023
\(712\) 14.0000 0.524672
\(713\) 32.0000 1.19841
\(714\) −6.00000 −0.224544
\(715\) 4.00000 0.149592
\(716\) −12.0000 −0.448461
\(717\) 8.00000 0.298765
\(718\) −24.0000 −0.895672
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 2.00000 0.0745356
\(721\) 16.0000 0.595871
\(722\) −45.0000 −1.67473
\(723\) −2.00000 −0.0743808
\(724\) −10.0000 −0.371647
\(725\) −2.00000 −0.0742781
\(726\) 1.00000 0.0371135
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) 48.0000 1.77534
\(732\) 14.0000 0.517455
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) −4.00000 −0.147442
\(737\) −4.00000 −0.147342
\(738\) −6.00000 −0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 12.0000 0.441129
\(741\) 16.0000 0.587775
\(742\) 10.0000 0.367112
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 8.00000 0.293294
\(745\) 20.0000 0.732743
\(746\) 22.0000 0.805477
\(747\) 4.00000 0.146352
\(748\) 6.00000 0.219382
\(749\) 12.0000 0.438470
\(750\) −12.0000 −0.438178
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 4.00000 0.145865
\(753\) 20.0000 0.728841
\(754\) −4.00000 −0.145671
\(755\) 32.0000 1.16460
\(756\) 1.00000 0.0363696
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) −20.0000 −0.726433
\(759\) −4.00000 −0.145191
\(760\) 16.0000 0.580381
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 8.00000 0.289809
\(763\) −10.0000 −0.362024
\(764\) −4.00000 −0.144715
\(765\) 12.0000 0.433861
\(766\) 28.0000 1.01168
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 2.00000 0.0720750
\(771\) −26.0000 −0.936367
\(772\) −6.00000 −0.215945
\(773\) −22.0000 −0.791285 −0.395643 0.918405i \(-0.629478\pi\)
−0.395643 + 0.918405i \(0.629478\pi\)
\(774\) −8.00000 −0.287554
\(775\) −8.00000 −0.287368
\(776\) −18.0000 −0.646162
\(777\) 6.00000 0.215249
\(778\) 30.0000 1.07555
\(779\) −48.0000 −1.71978
\(780\) −4.00000 −0.143223
\(781\) −4.00000 −0.143131
\(782\) −24.0000 −0.858238
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −4.00000 −0.142766
\(786\) −4.00000 −0.142675
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −6.00000 −0.213741
\(789\) −24.0000 −0.854423
\(790\) 16.0000 0.569254
\(791\) −10.0000 −0.355559
\(792\) −1.00000 −0.0355335
\(793\) −28.0000 −0.994309
\(794\) 18.0000 0.638796
\(795\) −20.0000 −0.709327
\(796\) −8.00000 −0.283552
\(797\) −22.0000 −0.779280 −0.389640 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 8.00000 0.283197
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) −34.0000 −1.20058
\(803\) −14.0000 −0.494049
\(804\) 4.00000 0.141069
\(805\) −8.00000 −0.281963
\(806\) −16.0000 −0.563576
\(807\) −2.00000 −0.0704033
\(808\) 6.00000 0.211079
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 56.0000 1.96643 0.983213 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) −2.00000 −0.0701862
\(813\) −8.00000 −0.280572
\(814\) −6.00000 −0.210300
\(815\) 24.0000 0.840683
\(816\) −6.00000 −0.210042
\(817\) −64.0000 −2.23908
\(818\) 22.0000 0.769212
\(819\) −2.00000 −0.0698857
\(820\) 12.0000 0.419058
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) −6.00000 −0.209274
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 16.0000 0.557386
\(825\) 1.00000 0.0348155
\(826\) 4.00000 0.139178
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 4.00000 0.139010
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) −8.00000 −0.277684
\(831\) 6.00000 0.208138
\(832\) 2.00000 0.0693375
\(833\) 6.00000 0.207888
\(834\) −16.0000 −0.554035
\(835\) −16.0000 −0.553703
\(836\) −8.00000 −0.276686
\(837\) −8.00000 −0.276520
\(838\) 12.0000 0.414533
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) −30.0000 −1.03325
\(844\) 8.00000 0.275371
\(845\) −18.0000 −0.619219
\(846\) −4.00000 −0.137523
\(847\) −1.00000 −0.0343604
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 24.0000 0.822709
\(852\) 4.00000 0.137038
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −14.0000 −0.479070
\(855\) −16.0000 −0.547188
\(856\) 12.0000 0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 2.00000 0.0682789
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 16.0000 0.545595
\(861\) 6.00000 0.204479
\(862\) −32.0000 −1.08992
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 1.00000 0.0340207
\(865\) −28.0000 −0.952029
\(866\) −2.00000 −0.0679628
\(867\) −19.0000 −0.645274
\(868\) −8.00000 −0.271538
\(869\) −8.00000 −0.271381
\(870\) 4.00000 0.135613
\(871\) −8.00000 −0.271070
\(872\) −10.0000 −0.338643
\(873\) 18.0000 0.609208
\(874\) 32.0000 1.08242
\(875\) 12.0000 0.405674
\(876\) 14.0000 0.473016
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 40.0000 1.34993
\(879\) 6.00000 0.202375
\(880\) 2.00000 0.0674200
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 12.0000 0.403604
\(885\) −8.00000 −0.268917
\(886\) −4.00000 −0.134383
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 6.00000 0.201347
\(889\) −8.00000 −0.268311
\(890\) 28.0000 0.938562
\(891\) 1.00000 0.0335013
\(892\) 16.0000 0.535720
\(893\) −32.0000 −1.07084
\(894\) 10.0000 0.334450
\(895\) −24.0000 −0.802232
\(896\) 1.00000 0.0334077
\(897\) −8.00000 −0.267112
\(898\) 6.00000 0.200223
\(899\) 16.0000 0.533630
\(900\) −1.00000 −0.0333333
\(901\) 60.0000 1.99889
\(902\) −6.00000 −0.199778
\(903\) 8.00000 0.266223
\(904\) −10.0000 −0.332595
\(905\) −20.0000 −0.664822
\(906\) 16.0000 0.531564
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −4.00000 −0.132745
\(909\) −6.00000 −0.199007
\(910\) 4.00000 0.132599
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 8.00000 0.264906
\(913\) 4.00000 0.132381
\(914\) −10.0000 −0.330771
\(915\) 28.0000 0.925651
\(916\) −10.0000 −0.330409
\(917\) 4.00000 0.132092
\(918\) 6.00000 0.198030
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −8.00000 −0.263752
\(921\) 8.00000 0.263609
\(922\) −2.00000 −0.0658665
\(923\) −8.00000 −0.263323
\(924\) 1.00000 0.0328976
\(925\) −6.00000 −0.197279
\(926\) 24.0000 0.788689
\(927\) −16.0000 −0.525509
\(928\) −2.00000 −0.0656532
\(929\) −22.0000 −0.721797 −0.360898 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(930\) 16.0000 0.524661
\(931\) −8.00000 −0.262189
\(932\) −18.0000 −0.589610
\(933\) −12.0000 −0.392862
\(934\) −20.0000 −0.654420
\(935\) 12.0000 0.392442
\(936\) −2.00000 −0.0653720
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) −4.00000 −0.130605
\(939\) 6.00000 0.195803
\(940\) 8.00000 0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 24.0000 0.781548
\(944\) 4.00000 0.130189
\(945\) 2.00000 0.0650600
\(946\) −8.00000 −0.260102
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 8.00000 0.259828
\(949\) −28.0000 −0.908918
\(950\) −8.00000 −0.259554
\(951\) −18.0000 −0.583690
\(952\) 6.00000 0.194461
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) −10.0000 −0.323762
\(955\) −8.00000 −0.258874
\(956\) −8.00000 −0.258738
\(957\) −2.00000 −0.0646508
\(958\) −24.0000 −0.775405
\(959\) 6.00000 0.193750
\(960\) −2.00000 −0.0645497
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) −12.0000 −0.386294
\(966\) −4.00000 −0.128698
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 48.0000 1.54198
\(970\) −36.0000 −1.15589
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.0000 0.512936
\(974\) 24.0000 0.769010
\(975\) 2.00000 0.0640513
\(976\) −14.0000 −0.448129
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 12.0000 0.383718
\(979\) −14.0000 −0.447442
\(980\) 2.00000 0.0638877
\(981\) 10.0000 0.319275
\(982\) −20.0000 −0.638226
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 6.00000 0.191273
\(985\) −12.0000 −0.382352
\(986\) −12.0000 −0.382158
\(987\) 4.00000 0.127321
\(988\) −16.0000 −0.509028
\(989\) 32.0000 1.01754
\(990\) −2.00000 −0.0635642
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −8.00000 −0.254000
\(993\) −4.00000 −0.126936
\(994\) −4.00000 −0.126872
\(995\) −16.0000 −0.507234
\(996\) −4.00000 −0.126745
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 28.0000 0.886325
\(999\) −6.00000 −0.189832
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 462.2.a.c.1.1 1
3.2 odd 2 1386.2.a.g.1.1 1
4.3 odd 2 3696.2.a.bb.1.1 1
7.6 odd 2 3234.2.a.i.1.1 1
11.10 odd 2 5082.2.a.v.1.1 1
21.20 even 2 9702.2.a.by.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.c.1.1 1 1.1 even 1 trivial
1386.2.a.g.1.1 1 3.2 odd 2
3234.2.a.i.1.1 1 7.6 odd 2
3696.2.a.bb.1.1 1 4.3 odd 2
5082.2.a.v.1.1 1 11.10 odd 2
9702.2.a.by.1.1 1 21.20 even 2