Properties

Label 231.2.a.a.1.1
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
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 \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} -5.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -6.00000 q^{40} +10.0000 q^{41} +1.00000 q^{42} -4.00000 q^{43} +1.00000 q^{44} -2.00000 q^{45} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +3.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -12.0000 q^{65} -1.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +2.00000 q^{70} +3.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +6.00000 q^{78} +16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -3.00000 q^{88} +18.0000 q^{89} +2.00000 q^{90} +6.00000 q^{91} -8.00000 q^{93} +8.00000 q^{94} -8.00000 q^{95} +5.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −5.00000 −0.883883
\(33\) 1.00000 0.174078
\(34\) −2.00000 −0.342997
\(35\) −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 \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −6.00000 −0.960769
\(40\) −6.00000 −0.948683
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −6.00000 −0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) 3.00000 0.400892
\(57\) −4.00000 −0.529813
\(58\) 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
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −8.00000 −1.01600
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −12.0000 −1.48842
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) 6.00000 0.679366
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 2.00000 0.214423
\(88\) −3.00000 −0.319801
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 2.00000 0.210819
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) −8.00000 −0.820783
\(96\) 5.00000 0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000 0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 18.0000 1.76505
\(105\) 2.00000 0.195180
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −2.00000 −0.190693
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) −4.00000 −0.368230
\(119\) 2.00000 0.183340
\(120\) 6.00000 0.547723
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −10.0000 −0.901670
\(124\) −8.00000 −0.718421
\(125\) 12.0000 1.07331
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) 4.00000 0.352180
\(130\) 12.0000 1.05247
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 4.00000 0.346844
\(134\) 12.0000 1.03664
\(135\) 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 \(0.582506\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 2.00000 0.169031
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) −1.00000 −0.0833333
\(145\) 4.00000 0.332182
\(146\) −2.00000 −0.165521
\(147\) −1.00000 −0.0824786
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 12.0000 0.973329
\(153\) 2.00000 0.161690
\(154\) 1.00000 0.0805823
\(155\) −16.0000 −1.28515
\(156\) 6.00000 0.480384
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −16.0000 −1.27289
\(159\) −6.00000 −0.475831
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −10.0000 −0.780869
\(165\) −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 \(0.600171\pi\)
\(168\) −3.00000 −0.231455
\(169\) 23.0000 1.76923
\(170\) 4.00000 0.306786
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −2.00000 −0.151620
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) −4.00000 −0.300658
\(178\) −18.0000 −1.34916
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −6.00000 −0.444750
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 8.00000 0.586588
\(187\) −2.00000 −0.146254
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) 8.00000 0.580381
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −7.00000 −0.505181
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 12.0000 0.859338
\(196\) −1.00000 −0.0714286
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 1.00000 0.0710669
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −3.00000 −0.212132
\(201\) 12.0000 0.846415
\(202\) −6.00000 −0.422159
\(203\) −2.00000 −0.140372
\(204\) 2.00000 0.140028
\(205\) −20.0000 −1.39686
\(206\) 0 0
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −4.00000 −0.276686
\(210\) −2.00000 −0.138013
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 8.00000 0.545595
\(216\) −3.00000 −0.204124
\(217\) 8.00000 0.543075
\(218\) 2.00000 0.135457
\(219\) −2.00000 −0.135147
\(220\) −2.00000 −0.134840
\(221\) 12.0000 0.807207
\(222\) 6.00000 0.402694
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −5.00000 −0.334077
\(225\) −1.00000 −0.0666667
\(226\) −18.0000 −1.19734
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 4.00000 0.264906
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) −6.00000 −0.393919
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −6.00000 −0.392232
\(235\) 16.0000 1.04372
\(236\) −4.00000 −0.260378
\(237\) −16.0000 −1.03931
\(238\) −2.00000 −0.129641
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −2.00000 −0.129099
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −2.00000 −0.127775
\(246\) 10.0000 0.637577
\(247\) 24.0000 1.52708
\(248\) 24.0000 1.52400
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 4.00000 0.250490
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −4.00000 −0.249029
\(259\) 6.00000 0.372822
\(260\) 12.0000 0.744208
\(261\) −2.00000 −0.123797
\(262\) −4.00000 −0.247121
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 3.00000 0.184637
\(265\) −12.0000 −0.737154
\(266\) −4.00000 −0.245256
\(267\) −18.0000 −1.10158
\(268\) 12.0000 0.733017
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) −2.00000 −0.121716
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −2.00000 −0.121268
\(273\) −6.00000 −0.363137
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 20.0000 1.19952
\(279\) 8.00000 0.478947
\(280\) −6.00000 −0.358569
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −8.00000 −0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 6.00000 0.354787
\(287\) 10.0000 0.590281
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) −2.00000 −0.117242
\(292\) −2.00000 −0.117041
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 1.00000 0.0583212
\(295\) −8.00000 −0.465778
\(296\) 18.0000 1.04623
\(297\) 1.00000 0.0580259
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −4.00000 −0.230556
\(302\) −8.00000 −0.460348
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) 20.0000 1.14520
\(306\) −2.00000 −0.114332
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) −18.0000 −1.01905
\(313\) −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
\(316\) −16.0000 −0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 6.00000 0.336463
\(319\) 2.00000 0.111979
\(320\) −14.0000 −0.782624
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) −1.00000 −0.0555556
\(325\) −6.00000 −0.332820
\(326\) 12.0000 0.664619
\(327\) 2.00000 0.110600
\(328\) 30.0000 1.65647
\(329\) −8.00000 −0.441054
\(330\) 2.00000 0.110096
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −4.00000 −0.219529
\(333\) 6.00000 0.328798
\(334\) 8.00000 0.437741
\(335\) 24.0000 1.31126
\(336\) 1.00000 0.0545545
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −23.0000 −1.25104
\(339\) −18.0000 −0.977626
\(340\) 4.00000 0.216930
\(341\) −8.00000 −0.433224
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −2.00000 −0.107211
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 1.00000 0.0534522
\(351\) −6.00000 −0.320256
\(352\) 5.00000 0.266501
\(353\) −22.0000 −1.17094 −0.585471 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) −2.00000 −0.105851
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) −1.00000 −0.0524864
\(364\) −6.00000 −0.314485
\(365\) −4.00000 −0.209370
\(366\) −10.0000 −0.522708
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 12.0000 0.623850
\(371\) 6.00000 0.311504
\(372\) 8.00000 0.414781
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 2.00000 0.103418
\(375\) −12.0000 −0.619677
\(376\) −24.0000 −1.23771
\(377\) −12.0000 −0.618031
\(378\) 1.00000 0.0514344
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 8.00000 0.410391
\(381\) 16.0000 0.819705
\(382\) 24.0000 1.22795
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −3.00000 −0.153093
\(385\) 2.00000 0.101929
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −12.0000 −0.607644
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) −4.00000 −0.201773
\(394\) 26.0000 1.30986
\(395\) −32.0000 −1.61009
\(396\) 1.00000 0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −16.0000 −0.802008
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) −12.0000 −0.598506
\(403\) 48.0000 2.39105
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) 2.00000 0.0992583
\(407\) −6.00000 −0.297409
\(408\) −6.00000 −0.297044
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 20.0000 0.987730
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) −30.0000 −1.47087
\(417\) 20.0000 0.979404
\(418\) 4.00000 0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 12.0000 0.584151
\(423\) −8.00000 −0.388973
\(424\) 18.0000 0.874157
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) 6.00000 0.289683
\(430\) −8.00000 −0.385794
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) −8.00000 −0.384012
\(435\) −4.00000 −0.191785
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 2.00000 0.0955637
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 6.00000 0.286039
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 6.00000 0.284747
\(445\) −36.0000 −1.70656
\(446\) 24.0000 1.13643
\(447\) 10.0000 0.472984
\(448\) 7.00000 0.330719
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 1.00000 0.0471405
\(451\) −10.0000 −0.470882
\(452\) −18.0000 −0.846649
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) −12.0000 −0.562569
\(456\) −12.0000 −0.561951
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 26.0000 1.21490
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 2.00000 0.0928477
\(465\) 16.0000 0.741982
\(466\) −26.0000 −1.20443
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −6.00000 −0.277350
\(469\) −12.0000 −0.554109
\(470\) −16.0000 −0.738025
\(471\) 2.00000 0.0921551
\(472\) 12.0000 0.552345
\(473\) 4.00000 0.183920
\(474\) 16.0000 0.734904
\(475\) −4.00000 −0.183533
\(476\) −2.00000 −0.0916698
\(477\) 6.00000 0.274721
\(478\) 24.0000 1.09773
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −10.0000 −0.456435
\(481\) 36.0000 1.64146
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) −4.00000 −0.181631
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −30.0000 −1.35804
\(489\) 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 \(0.350963\pi\)
\(492\) 10.0000 0.450835
\(493\) −4.00000 −0.180151
\(494\) −24.0000 −1.07981
\(495\) 2.00000 0.0898933
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −12.0000 −0.536656
\(501\) 8.00000 0.357414
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 3.00000 0.133631
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 16.0000 0.709885
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) −4.00000 −0.177123
\(511\) 2.00000 0.0884748
\(512\) 11.0000 0.486136
\(513\) −4.00000 −0.176604
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 8.00000 0.351840
\(518\) −6.00000 −0.263625
\(519\) −14.0000 −0.614532
\(520\) −36.0000 −1.57870
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 2.00000 0.0875376
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −4.00000 −0.174741
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) 12.0000 0.521247
\(531\) 4.00000 0.173585
\(532\) −4.00000 −0.173422
\(533\) 60.0000 2.59889
\(534\) 18.0000 0.778936
\(535\) 24.0000 1.03761
\(536\) −36.0000 −1.55496
\(537\) −12.0000 −0.517838
\(538\) 10.0000 0.431131
\(539\) −1.00000 −0.0430730
\(540\) −2.00000 −0.0860663
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −16.0000 −0.687259
\(543\) 10.0000 0.429141
\(544\) −10.0000 −0.428746
\(545\) 4.00000 0.171341
\(546\) 6.00000 0.256776
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) −1.00000 −0.0426401
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −22.0000 −0.934690
\(555\) 12.0000 0.509372
\(556\) 20.0000 0.848189
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) 2.00000 0.0845154
\(561\) 2.00000 0.0844401
\(562\) −10.0000 −0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −8.00000 −0.336861
\(565\) −36.0000 −1.51453
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −8.00000 −0.335083
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 6.00000 0.250873
\(573\) 24.0000 1.00261
\(574\) −10.0000 −0.417392
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 13.0000 0.540729
\(579\) −2.00000 −0.0831172
\(580\) −4.00000 −0.166091
\(581\) 4.00000 0.165948
\(582\) 2.00000 0.0829027
\(583\) −6.00000 −0.248495
\(584\) 6.00000 0.248282
\(585\) −12.0000 −0.496139
\(586\) 10.0000 0.413096
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 1.00000 0.0412393
\(589\) 32.0000 1.31854
\(590\) 8.00000 0.329355
\(591\) 26.0000 1.06950
\(592\) −6.00000 −0.246598
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −4.00000 −0.163984
\(596\) 10.0000 0.409616
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 3.00000 0.122474
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 4.00000 0.163028
\(603\) −12.0000 −0.488678
\(604\) −8.00000 −0.325515
\(605\) −2.00000 −0.0813116
\(606\) 6.00000 0.243733
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) −20.0000 −0.811107
\(609\) 2.00000 0.0810441
\(610\) −20.0000 −0.809776
\(611\) −48.0000 −1.94187
\(612\) −2.00000 −0.0808452
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −20.0000 −0.807134
\(615\) 20.0000 0.806478
\(616\) −3.00000 −0.120873
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) −32.0000 −1.28308
\(623\) 18.0000 0.721155
\(624\) 6.00000 0.240192
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) 4.00000 0.159745
\(628\) 2.00000 0.0798087
\(629\) 12.0000 0.478471
\(630\) 2.00000 0.0796819
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 48.0000 1.90934
\(633\) 12.0000 0.476957
\(634\) 18.0000 0.714871
\(635\) 32.0000 1.26988
\(636\) 6.00000 0.237915
\(637\) 6.00000 0.237729
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −6.00000 −0.237171
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −12.0000 −0.473602
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −8.00000 −0.314756
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 3.00000 0.117851
\(649\) −4.00000 −0.157014
\(650\) 6.00000 0.235339
\(651\) −8.00000 −0.313545
\(652\) 12.0000 0.469956
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −8.00000 −0.312586
\(656\) −10.0000 −0.390434
\(657\) 2.00000 0.0780274
\(658\) 8.00000 0.311872
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 2.00000 0.0778499
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −28.0000 −1.08825
\(663\) −12.0000 −0.466041
\(664\) 12.0000 0.465690
\(665\) −8.00000 −0.310227
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) 24.0000 0.927894
\(670\) −24.0000 −0.927201
\(671\) 10.0000 0.386046
\(672\) 5.00000 0.192879
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 1.00000 0.0384900
\(676\) −23.0000 −0.884615
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 18.0000 0.691286
\(679\) 2.00000 0.0767530
\(680\) −12.0000 −0.460179
\(681\) 12.0000 0.459841
\(682\) 8.00000 0.306336
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −4.00000 −0.152944
\(685\) 12.0000 0.458496
\(686\) −1.00000 −0.0381802
\(687\) 26.0000 0.991962
\(688\) 4.00000 0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −14.0000 −0.532200
\(693\) −1.00000 −0.0379869
\(694\) −4.00000 −0.151838
\(695\) 40.0000 1.51729
\(696\) 6.00000 0.227429
\(697\) 20.0000 0.757554
\(698\) 10.0000 0.378506
\(699\) −26.0000 −0.983410
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 6.00000 0.226455
\(703\) 24.0000 0.905177
\(704\) −7.00000 −0.263822
\(705\) −16.0000 −0.602595
\(706\) 22.0000 0.827981
\(707\) 6.00000 0.225653
\(708\) 4.00000 0.150329
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 54.0000 2.02374
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 12.0000 0.448775
\(716\) −12.0000 −0.448461
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −26.0000 −0.966950
\(724\) 10.0000 0.371647
\(725\) 2.00000 0.0742781
\(726\) 1.00000 0.0371135
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 18.0000 0.667124
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) −8.00000 −0.295891
\(732\) −10.0000 −0.369611
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 8.00000 0.295285
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) −10.0000 −0.368105
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 12.0000 0.441129
\(741\) −24.0000 −0.881662
\(742\) −6.00000 −0.220267
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −24.0000 −0.879883
\(745\) 20.0000 0.732743
\(746\) −22.0000 −0.805477
\(747\) 4.00000 0.146352
\(748\) 2.00000 0.0731272
\(749\) −12.0000 −0.438470
\(750\) 12.0000 0.438178
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) 12.0000 0.437014
\(755\) −16.0000 −0.582300
\(756\) 1.00000 0.0363696
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −16.0000 −0.579619
\(763\) −2.00000 −0.0724049
\(764\) 24.0000 0.868290
\(765\) −4.00000 −0.144620
\(766\) −8.00000 −0.289052
\(767\) 24.0000 0.866590
\(768\) 17.0000 0.613435
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 6.00000 0.216085
\(772\) −2.00000 −0.0719816
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) 6.00000 0.215387
\(777\) −6.00000 −0.215249
\(778\) −6.00000 −0.215110
\(779\) 40.0000 1.43315
\(780\) −12.0000 −0.429669
\(781\) 0 0
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) −1.00000 −0.0357143
\(785\) 4.00000 0.142766
\(786\) 4.00000 0.142675
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 26.0000 0.926212
\(789\) 0 0
\(790\) 32.0000 1.13851
\(791\) 18.0000 0.640006
\(792\) −3.00000 −0.106600
\(793\) −60.0000 −2.13066
\(794\) −14.0000 −0.496841
\(795\) 12.0000 0.425596
\(796\) −16.0000 −0.567105
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 4.00000 0.141598
\(799\) −16.0000 −0.566039
\(800\) 5.00000 0.176777
\(801\) 18.0000 0.635999
\(802\) −34.0000 −1.20058
\(803\) −2.00000 −0.0705785
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 10.0000 0.352017
\(808\) 18.0000 0.633238
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 2.00000 0.0702728
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 2.00000 0.0701862
\(813\) −16.0000 −0.561144
\(814\) 6.00000 0.210300
\(815\) 24.0000 0.840683
\(816\) 2.00000 0.0700140
\(817\) −16.0000 −0.559769
\(818\) 14.0000 0.489499
\(819\) 6.00000 0.209657
\(820\) 20.0000 0.698430
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −6.00000 −0.209274
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) −4.00000 −0.139178
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 8.00000 0.277684
\(831\) −22.0000 −0.763172
\(832\) 42.0000 1.45609
\(833\) 2.00000 0.0692959
\(834\) −20.0000 −0.692543
\(835\) 16.0000 0.553703
\(836\) 4.00000 0.138343
\(837\) −8.00000 −0.276520
\(838\) −12.0000 −0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 6.00000 0.207020
\(841\) −25.0000 −0.862069
\(842\) −6.00000 −0.206774
\(843\) −10.0000 −0.344418
\(844\) 12.0000 0.413057
\(845\) −46.0000 −1.58245
\(846\) 8.00000 0.275046
\(847\) 1.00000 0.0343604
\(848\) −6.00000 −0.206041
\(849\) 4.00000 0.137280
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 10.0000 0.342193
\(855\) −8.00000 −0.273594
\(856\) −36.0000 −1.23045
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) −6.00000 −0.204837
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) −8.00000 −0.272798
\(861\) −10.0000 −0.340799
\(862\) 24.0000 0.817443
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) 5.00000 0.170103
\(865\) −28.0000 −0.952029
\(866\) 30.0000 1.01944
\(867\) 13.0000 0.441503
\(868\) −8.00000 −0.271538
\(869\) −16.0000 −0.542763
\(870\) 4.00000 0.135613
\(871\) −72.0000 −2.43963
\(872\) −6.00000 −0.203186
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 2.00000 0.0675737
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) −8.00000 −0.269987
\(879\) 10.0000 0.337292
\(880\) −2.00000 −0.0674200
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) −12.0000 −0.403604
\(885\) 8.00000 0.268917
\(886\) 12.0000 0.403148
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −18.0000 −0.604040
\(889\) −16.0000 −0.536623
\(890\) 36.0000 1.20672
\(891\) −1.00000 −0.0335013
\(892\) 24.0000 0.803579
\(893\) −32.0000 −1.07084
\(894\) −10.0000 −0.334450
\(895\) −24.0000 −0.802232
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −34.0000 −1.13459
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 10.0000 0.332964
\(903\) 4.00000 0.133112
\(904\) 54.0000 1.79601
\(905\) 20.0000 0.664822
\(906\) 8.00000 0.265782
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) 12.0000 0.397796
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 4.00000 0.132453
\(913\) −4.00000 −0.132381
\(914\) 22.0000 0.727695
\(915\) −20.0000 −0.661180
\(916\) 26.0000 0.859064
\(917\) 4.00000 0.132092
\(918\) 2.00000 0.0660098
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 34.0000 1.11973
\(923\) 0 0
\(924\) −1.00000 −0.0328976
\(925\) −6.00000 −0.197279
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) −16.0000 −0.524661
\(931\) 4.00000 0.131095
\(932\) −26.0000 −0.851658
\(933\) −32.0000 −1.04763
\(934\) −12.0000 −0.392652
\(935\) 4.00000 0.130814
\(936\) 18.0000 0.588348
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) 12.0000 0.391814
\(939\) 6.00000 0.195803
\(940\) −16.0000 −0.521862
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 2.00000 0.0650600
\(946\) −4.00000 −0.130051
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 16.0000 0.519656
\(949\) 12.0000 0.389536
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) 6.00000 0.194461
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −6.00000 −0.194257
\(955\) 48.0000 1.55324
\(956\) 24.0000 0.776215
\(957\) −2.00000 −0.0646508
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 14.0000 0.451848
\(961\) 33.0000 1.06452
\(962\) −36.0000 −1.16069
\(963\) −12.0000 −0.386695
\(964\) −26.0000 −0.837404
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 3.00000 0.0964237
\(969\) −8.00000 −0.256997
\(970\) 4.00000 0.128432
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) −20.0000 −0.641171
\(974\) −8.00000 −0.256337
\(975\) 6.00000 0.192154
\(976\) 10.0000 0.320092
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −12.0000 −0.383718
\(979\) −18.0000 −0.575282
\(980\) 2.00000 0.0638877
\(981\) −2.00000 −0.0638551
\(982\) −20.0000 −0.638226
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) −30.0000 −0.956365
\(985\) 52.0000 1.65686
\(986\) 4.00000 0.127386
\(987\) 8.00000 0.254643
\(988\) −24.0000 −0.763542
\(989\) 0 0
\(990\) −2.00000 −0.0635642
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) −40.0000 −1.27000
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 4.00000 0.126745
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −20.0000 −0.633089
\(999\) −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 231.2.a.a.1.1 1
3.2 odd 2 693.2.a.d.1.1 1
4.3 odd 2 3696.2.a.t.1.1 1
5.4 even 2 5775.2.a.t.1.1 1
7.6 odd 2 1617.2.a.e.1.1 1
11.10 odd 2 2541.2.a.h.1.1 1
21.20 even 2 4851.2.a.p.1.1 1
33.32 even 2 7623.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.a.1.1 1 1.1 even 1 trivial
693.2.a.d.1.1 1 3.2 odd 2
1617.2.a.e.1.1 1 7.6 odd 2
2541.2.a.h.1.1 1 11.10 odd 2
3696.2.a.t.1.1 1 4.3 odd 2
4851.2.a.p.1.1 1 21.20 even 2
5775.2.a.t.1.1 1 5.4 even 2
7623.2.a.f.1.1 1 33.32 even 2