Properties

Label 65.2.a.a.1.1
Level $65$
Weight $2$
Character 65.1
Self dual yes
Analytic conductor $0.519$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.519027613138\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 65.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +8.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} -8.00000 q^{42} +10.0000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +4.00000 q^{47} +2.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} +1.00000 q^{52} +2.00000 q^{53} -4.00000 q^{54} -2.00000 q^{55} -12.0000 q^{56} +12.0000 q^{57} -2.00000 q^{58} +6.00000 q^{59} -2.00000 q^{60} +2.00000 q^{61} +10.0000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +1.00000 q^{65} +4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} +12.0000 q^{69} -4.00000 q^{70} +6.00000 q^{71} +3.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -2.00000 q^{75} +6.00000 q^{76} -8.00000 q^{77} -2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -16.0000 q^{83} -8.00000 q^{84} -2.00000 q^{85} -10.0000 q^{86} -4.00000 q^{87} +6.00000 q^{88} +2.00000 q^{89} +1.00000 q^{90} +4.00000 q^{91} +6.00000 q^{92} +20.0000 q^{93} -4.00000 q^{94} +6.00000 q^{95} +10.0000 q^{96} -2.00000 q^{97} -9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 \(0.615027\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.00000 0.816497
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 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 \(0.402509\pi\)
\(12\) 2.00000 0.577350
\(13\) −1.00000 −0.277350
\(14\) 4.00000 1.06904
\(15\) 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
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 8.00000 1.74574
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −6.00000 −1.22474
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 4.00000 0.769800
\(28\) 4.00000 0.755929
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 −0.365148
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −5.00000 −0.883883
\(33\) −4.00000 −0.696311
\(34\) −2.00000 −0.342997
\(35\) 4.00000 0.676123
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 6.00000 0.973329
\(39\) 2.00000 0.320256
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −8.00000 −1.23443
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 2.00000 0.288675
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −4.00000 −0.560112
\(52\) 1.00000 0.138675
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −4.00000 −0.544331
\(55\) −2.00000 −0.269680
\(56\) −12.0000 −1.60357
\(57\) 12.0000 1.58944
\(58\) −2.00000 −0.262613
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −2.00000 −0.258199
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 10.0000 1.27000
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 1.00000 0.124035
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) 12.0000 1.44463
\(70\) −4.00000 −0.478091
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 3.00000 0.353553
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000 0.232495
\(75\) −2.00000 −0.230940
\(76\) 6.00000 0.688247
\(77\) −8.00000 −0.911685
\(78\) −2.00000 −0.226455
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) −8.00000 −0.872872
\(85\) −2.00000 −0.216930
\(86\) −10.0000 −1.07833
\(87\) −4.00000 −0.428845
\(88\) 6.00000 0.639602
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.00000 0.419314
\(92\) 6.00000 0.625543
\(93\) 20.0000 2.07390
\(94\) −4.00000 −0.412568
\(95\) 6.00000 0.615587
\(96\) 10.0000 1.02062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −9.00000 −0.909137
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 4.00000 0.396059
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) −3.00000 −0.294174
\(105\) −8.00000 −0.780720
\(106\) −2.00000 −0.194257
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) −4.00000 −0.384900
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 2.00000 0.190693
\(111\) 4.00000 0.379663
\(112\) 4.00000 0.377964
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −12.0000 −1.12390
\(115\) 6.00000 0.559503
\(116\) −2.00000 −0.185695
\(117\) −1.00000 −0.0924500
\(118\) −6.00000 −0.552345
\(119\) −8.00000 −0.733359
\(120\) 6.00000 0.547723
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 12.0000 1.08200
\(124\) 10.0000 0.898027
\(125\) −1.00000 −0.0894427
\(126\) 4.00000 0.356348
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 3.00000 0.265165
\(129\) −20.0000 −1.76090
\(130\) −1.00000 −0.0877058
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 4.00000 0.348155
\(133\) 24.0000 2.08106
\(134\) 4.00000 0.345547
\(135\) −4.00000 −0.344265
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −12.0000 −1.02151
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −4.00000 −0.338062
\(141\) −8.00000 −0.673722
\(142\) −6.00000 −0.503509
\(143\) −2.00000 −0.167248
\(144\) −1.00000 −0.0833333
\(145\) −2.00000 −0.166091
\(146\) 6.00000 0.496564
\(147\) −18.0000 −1.48461
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 2.00000 0.163299
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −18.0000 −1.45999
\(153\) 2.00000 0.161690
\(154\) 8.00000 0.644658
\(155\) 10.0000 0.803219
\(156\) −2.00000 −0.160128
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 12.0000 0.954669
\(159\) −4.00000 −0.317221
\(160\) 5.00000 0.395285
\(161\) 24.0000 1.89146
\(162\) 11.0000 0.864242
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 6.00000 0.468521
\(165\) 4.00000 0.311400
\(166\) 16.0000 1.24184
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 24.0000 1.85164
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) −6.00000 −0.458831
\(172\) −10.0000 −0.762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 4.00000 0.303239
\(175\) −4.00000 −0.302372
\(176\) −2.00000 −0.150756
\(177\) −12.0000 −0.901975
\(178\) −2.00000 −0.149906
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −4.00000 −0.296500
\(183\) −4.00000 −0.295689
\(184\) −18.0000 −1.32698
\(185\) 2.00000 0.147043
\(186\) −20.0000 −1.46647
\(187\) 4.00000 0.292509
\(188\) −4.00000 −0.291730
\(189\) −16.0000 −1.16383
\(190\) −6.00000 −0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −14.0000 −1.01036
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) −9.00000 −0.642857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −2.00000 −0.142134
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 3.00000 0.212132
\(201\) 8.00000 0.564276
\(202\) 18.0000 1.26648
\(203\) −8.00000 −0.561490
\(204\) 4.00000 0.280056
\(205\) 6.00000 0.419058
\(206\) −2.00000 −0.139347
\(207\) −6.00000 −0.417029
\(208\) 1.00000 0.0693375
\(209\) −12.0000 −0.830057
\(210\) 8.00000 0.552052
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −2.00000 −0.137361
\(213\) −12.0000 −0.822226
\(214\) −10.0000 −0.683586
\(215\) −10.0000 −0.681994
\(216\) 12.0000 0.816497
\(217\) 40.0000 2.71538
\(218\) −10.0000 −0.677285
\(219\) 12.0000 0.810885
\(220\) 2.00000 0.134840
\(221\) −2.00000 −0.134535
\(222\) −4.00000 −0.268462
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 20.0000 1.33631
\(225\) 1.00000 0.0666667
\(226\) 14.0000 0.931266
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −12.0000 −0.794719
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −6.00000 −0.395628
\(231\) 16.0000 1.05272
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 1.00000 0.0653720
\(235\) −4.00000 −0.260931
\(236\) −6.00000 −0.390567
\(237\) 24.0000 1.55897
\(238\) 8.00000 0.518563
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −2.00000 −0.129099
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000 0.449977
\(243\) 10.0000 0.641500
\(244\) −2.00000 −0.128037
\(245\) −9.00000 −0.574989
\(246\) −12.0000 −0.765092
\(247\) 6.00000 0.381771
\(248\) −30.0000 −1.90500
\(249\) 32.0000 2.02792
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 4.00000 0.251976
\(253\) −12.0000 −0.754434
\(254\) 2.00000 0.125491
\(255\) 4.00000 0.250490
\(256\) −17.0000 −1.06250
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 20.0000 1.24515
\(259\) 8.00000 0.497096
\(260\) −1.00000 −0.0620174
\(261\) 2.00000 0.123797
\(262\) −20.0000 −1.23560
\(263\) 14.0000 0.863277 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) −12.0000 −0.738549
\(265\) −2.00000 −0.122859
\(266\) −24.0000 −1.47153
\(267\) −4.00000 −0.244796
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 4.00000 0.243432
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −2.00000 −0.121268
\(273\) −8.00000 −0.484182
\(274\) 2.00000 0.120824
\(275\) 2.00000 0.120605
\(276\) −12.0000 −0.722315
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 12.0000 0.717137
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 8.00000 0.476393
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) −6.00000 −0.356034
\(285\) −12.0000 −0.710819
\(286\) 2.00000 0.118262
\(287\) 24.0000 1.41668
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 4.00000 0.234484
\(292\) 6.00000 0.351123
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 18.0000 1.04978
\(295\) −6.00000 −0.349334
\(296\) −6.00000 −0.348743
\(297\) 8.00000 0.464207
\(298\) −18.0000 −1.04271
\(299\) 6.00000 0.346989
\(300\) 2.00000 0.115470
\(301\) −40.0000 −2.30556
\(302\) −10.0000 −0.575435
\(303\) 36.0000 2.06815
\(304\) 6.00000 0.344124
\(305\) −2.00000 −0.114520
\(306\) −2.00000 −0.114332
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 8.00000 0.455842
\(309\) −4.00000 −0.227552
\(310\) −10.0000 −0.567962
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 6.00000 0.339683
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 6.00000 0.338600
\(315\) 4.00000 0.225374
\(316\) 12.0000 0.675053
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 4.00000 0.224309
\(319\) 4.00000 0.223957
\(320\) −7.00000 −0.391312
\(321\) −20.0000 −1.11629
\(322\) −24.0000 −1.33747
\(323\) −12.0000 −0.667698
\(324\) 11.0000 0.611111
\(325\) −1.00000 −0.0554700
\(326\) 12.0000 0.664619
\(327\) −20.0000 −1.10600
\(328\) −18.0000 −0.993884
\(329\) −16.0000 −0.882109
\(330\) −4.00000 −0.220193
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 16.0000 0.878114
\(333\) −2.00000 −0.109599
\(334\) 12.0000 0.656611
\(335\) 4.00000 0.218543
\(336\) −8.00000 −0.436436
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 28.0000 1.52075
\(340\) 2.00000 0.108465
\(341\) −20.0000 −1.08306
\(342\) 6.00000 0.324443
\(343\) −8.00000 −0.431959
\(344\) 30.0000 1.61749
\(345\) −12.0000 −0.646058
\(346\) 6.00000 0.322562
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) 4.00000 0.214423
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 4.00000 0.213809
\(351\) −4.00000 −0.213504
\(352\) −10.0000 −0.533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 12.0000 0.637793
\(355\) −6.00000 −0.318447
\(356\) −2.00000 −0.106000
\(357\) 16.0000 0.846810
\(358\) −12.0000 −0.634220
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) −3.00000 −0.158114
\(361\) 17.0000 0.894737
\(362\) 22.0000 1.15629
\(363\) 14.0000 0.734809
\(364\) −4.00000 −0.209657
\(365\) 6.00000 0.314054
\(366\) 4.00000 0.209083
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 6.00000 0.312772
\(369\) −6.00000 −0.312348
\(370\) −2.00000 −0.103975
\(371\) −8.00000 −0.415339
\(372\) −20.0000 −1.03695
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) −4.00000 −0.206835
\(375\) 2.00000 0.103280
\(376\) 12.0000 0.618853
\(377\) −2.00000 −0.103005
\(378\) 16.0000 0.822951
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) −6.00000 −0.307794
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −6.00000 −0.306186
\(385\) 8.00000 0.407718
\(386\) 2.00000 0.101797
\(387\) 10.0000 0.508329
\(388\) 2.00000 0.101535
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 2.00000 0.101274
\(391\) −12.0000 −0.606866
\(392\) 27.0000 1.36371
\(393\) −40.0000 −2.01773
\(394\) 6.00000 0.302276
\(395\) 12.0000 0.603786
\(396\) −2.00000 −0.100504
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 16.0000 0.802008
\(399\) −48.0000 −2.40301
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −8.00000 −0.399004
\(403\) 10.0000 0.498135
\(404\) 18.0000 0.895533
\(405\) 11.0000 0.546594
\(406\) 8.00000 0.397033
\(407\) −4.00000 −0.198273
\(408\) −12.0000 −0.594089
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) −6.00000 −0.296319
\(411\) 4.00000 0.197305
\(412\) −2.00000 −0.0985329
\(413\) −24.0000 −1.18096
\(414\) 6.00000 0.294884
\(415\) 16.0000 0.785409
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 8.00000 0.390360
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −12.0000 −0.584151
\(423\) 4.00000 0.194487
\(424\) 6.00000 0.291386
\(425\) 2.00000 0.0970143
\(426\) 12.0000 0.581402
\(427\) −8.00000 −0.387147
\(428\) −10.0000 −0.483368
\(429\) 4.00000 0.193122
\(430\) 10.0000 0.482243
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) −4.00000 −0.192450
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) −40.0000 −1.92006
\(435\) 4.00000 0.191785
\(436\) −10.0000 −0.478913
\(437\) 36.0000 1.72211
\(438\) −12.0000 −0.573382
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −6.00000 −0.286039
\(441\) 9.00000 0.428571
\(442\) 2.00000 0.0951303
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) −4.00000 −0.189832
\(445\) −2.00000 −0.0948091
\(446\) −4.00000 −0.189405
\(447\) −36.0000 −1.70274
\(448\) −28.0000 −1.32288
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −12.0000 −0.565058
\(452\) 14.0000 0.658505
\(453\) −20.0000 −0.939682
\(454\) −4.00000 −0.187729
\(455\) −4.00000 −0.187523
\(456\) 36.0000 1.68585
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 22.0000 1.02799
\(459\) 8.00000 0.373408
\(460\) −6.00000 −0.279751
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) −16.0000 −0.744387
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −20.0000 −0.927478
\(466\) −10.0000 −0.463241
\(467\) −10.0000 −0.462745 −0.231372 0.972865i \(-0.574322\pi\)
−0.231372 + 0.972865i \(0.574322\pi\)
\(468\) 1.00000 0.0462250
\(469\) 16.0000 0.738811
\(470\) 4.00000 0.184506
\(471\) 12.0000 0.552931
\(472\) 18.0000 0.828517
\(473\) 20.0000 0.919601
\(474\) −24.0000 −1.10236
\(475\) −6.00000 −0.275299
\(476\) 8.00000 0.366679
\(477\) 2.00000 0.0915737
\(478\) 6.00000 0.274434
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) −10.0000 −0.456435
\(481\) 2.00000 0.0911922
\(482\) −10.0000 −0.455488
\(483\) −48.0000 −2.18408
\(484\) 7.00000 0.318182
\(485\) 2.00000 0.0908153
\(486\) −10.0000 −0.453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 6.00000 0.271607
\(489\) 24.0000 1.08532
\(490\) 9.00000 0.406579
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −12.0000 −0.541002
\(493\) 4.00000 0.180151
\(494\) −6.00000 −0.269953
\(495\) −2.00000 −0.0898933
\(496\) 10.0000 0.449013
\(497\) −24.0000 −1.07655
\(498\) −32.0000 −1.43395
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.0000 1.07224
\(502\) −24.0000 −1.07117
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) −12.0000 −0.534522
\(505\) 18.0000 0.800989
\(506\) 12.0000 0.533465
\(507\) −2.00000 −0.0888231
\(508\) 2.00000 0.0887357
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −4.00000 −0.177123
\(511\) 24.0000 1.06170
\(512\) 11.0000 0.486136
\(513\) −24.0000 −1.05963
\(514\) −2.00000 −0.0882162
\(515\) −2.00000 −0.0881305
\(516\) 20.0000 0.880451
\(517\) 8.00000 0.351840
\(518\) −8.00000 −0.351500
\(519\) 12.0000 0.526742
\(520\) 3.00000 0.131559
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) −20.0000 −0.873704
\(525\) 8.00000 0.349149
\(526\) −14.0000 −0.610429
\(527\) −20.0000 −0.871214
\(528\) 4.00000 0.174078
\(529\) 13.0000 0.565217
\(530\) 2.00000 0.0868744
\(531\) 6.00000 0.260378
\(532\) −24.0000 −1.04053
\(533\) 6.00000 0.259889
\(534\) 4.00000 0.173097
\(535\) −10.0000 −0.432338
\(536\) −12.0000 −0.518321
\(537\) −24.0000 −1.03568
\(538\) −6.00000 −0.258678
\(539\) 18.0000 0.775315
\(540\) 4.00000 0.172133
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 44.0000 1.88822
\(544\) −10.0000 −0.428746
\(545\) −10.0000 −0.428353
\(546\) 8.00000 0.342368
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) 2.00000 0.0854358
\(549\) 2.00000 0.0853579
\(550\) −2.00000 −0.0852803
\(551\) −12.0000 −0.511217
\(552\) 36.0000 1.53226
\(553\) 48.0000 2.04117
\(554\) 14.0000 0.594803
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) −26.0000 −1.10166 −0.550828 0.834619i \(-0.685688\pi\)
−0.550828 + 0.834619i \(0.685688\pi\)
\(558\) 10.0000 0.423334
\(559\) −10.0000 −0.422955
\(560\) −4.00000 −0.169031
\(561\) −8.00000 −0.337760
\(562\) 6.00000 0.253095
\(563\) −22.0000 −0.927189 −0.463595 0.886047i \(-0.653441\pi\)
−0.463595 + 0.886047i \(0.653441\pi\)
\(564\) 8.00000 0.336861
\(565\) 14.0000 0.588984
\(566\) −2.00000 −0.0840663
\(567\) 44.0000 1.84783
\(568\) 18.0000 0.755263
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 12.0000 0.502625
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) −6.00000 −0.250217
\(576\) 7.00000 0.291667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 13.0000 0.540729
\(579\) 4.00000 0.166234
\(580\) 2.00000 0.0830455
\(581\) 64.0000 2.65517
\(582\) −4.00000 −0.165805
\(583\) 4.00000 0.165663
\(584\) −18.0000 −0.744845
\(585\) 1.00000 0.0413449
\(586\) −22.0000 −0.908812
\(587\) −44.0000 −1.81607 −0.908037 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 18.0000 0.742307
\(589\) 60.0000 2.47226
\(590\) 6.00000 0.247016
\(591\) 12.0000 0.493614
\(592\) 2.00000 0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −8.00000 −0.328244
\(595\) 8.00000 0.327968
\(596\) −18.0000 −0.737309
\(597\) 32.0000 1.30967
\(598\) −6.00000 −0.245358
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) −6.00000 −0.244949
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 40.0000 1.63028
\(603\) −4.00000 −0.162893
\(604\) −10.0000 −0.406894
\(605\) 7.00000 0.284590
\(606\) −36.0000 −1.46240
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 30.0000 1.21666
\(609\) 16.0000 0.648353
\(610\) 2.00000 0.0809776
\(611\) −4.00000 −0.161823
\(612\) −2.00000 −0.0808452
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) −8.00000 −0.322854
\(615\) −12.0000 −0.483887
\(616\) −24.0000 −0.966988
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 4.00000 0.160904
\(619\) −2.00000 −0.0803868 −0.0401934 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) −10.0000 −0.401610
\(621\) −24.0000 −0.963087
\(622\) 4.00000 0.160385
\(623\) −8.00000 −0.320513
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 22.0000 0.879297
\(627\) 24.0000 0.958468
\(628\) 6.00000 0.239426
\(629\) −4.00000 −0.159490
\(630\) −4.00000 −0.159364
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) −36.0000 −1.43200
\(633\) −24.0000 −0.953914
\(634\) 18.0000 0.714871
\(635\) 2.00000 0.0793676
\(636\) 4.00000 0.158610
\(637\) −9.00000 −0.356593
\(638\) −4.00000 −0.158362
\(639\) 6.00000 0.237356
\(640\) −3.00000 −0.118585
\(641\) −46.0000 −1.81689 −0.908445 0.418004i \(-0.862730\pi\)
−0.908445 + 0.418004i \(0.862730\pi\)
\(642\) 20.0000 0.789337
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −24.0000 −0.945732
\(645\) 20.0000 0.787499
\(646\) 12.0000 0.472134
\(647\) 38.0000 1.49393 0.746967 0.664861i \(-0.231509\pi\)
0.746967 + 0.664861i \(0.231509\pi\)
\(648\) −33.0000 −1.29636
\(649\) 12.0000 0.471041
\(650\) 1.00000 0.0392232
\(651\) −80.0000 −3.13545
\(652\) 12.0000 0.469956
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 20.0000 0.782062
\(655\) −20.0000 −0.781465
\(656\) 6.00000 0.234261
\(657\) −6.00000 −0.234082
\(658\) 16.0000 0.623745
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −4.00000 −0.155700
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 18.0000 0.699590
\(663\) 4.00000 0.155347
\(664\) −48.0000 −1.86276
\(665\) −24.0000 −0.930680
\(666\) 2.00000 0.0774984
\(667\) −12.0000 −0.464642
\(668\) 12.0000 0.464294
\(669\) −8.00000 −0.309298
\(670\) −4.00000 −0.154533
\(671\) 4.00000 0.154418
\(672\) −40.0000 −1.54303
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −26.0000 −1.00148
\(675\) 4.00000 0.153960
\(676\) −1.00000 −0.0384615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −28.0000 −1.07533
\(679\) 8.00000 0.307012
\(680\) −6.00000 −0.230089
\(681\) −8.00000 −0.306561
\(682\) 20.0000 0.765840
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 6.00000 0.229416
\(685\) 2.00000 0.0764161
\(686\) 8.00000 0.305441
\(687\) 44.0000 1.67870
\(688\) −10.0000 −0.381246
\(689\) −2.00000 −0.0761939
\(690\) 12.0000 0.456832
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) 6.00000 0.228086
\(693\) −8.00000 −0.303895
\(694\) 22.0000 0.835109
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) −12.0000 −0.454532
\(698\) 30.0000 1.13552
\(699\) −20.0000 −0.756469
\(700\) 4.00000 0.151186
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 4.00000 0.150970
\(703\) 12.0000 0.452589
\(704\) 14.0000 0.527645
\(705\) 8.00000 0.301297
\(706\) −18.0000 −0.677439
\(707\) 72.0000 2.70784
\(708\) 12.0000 0.450988
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 6.00000 0.225176
\(711\) −12.0000 −0.450035
\(712\) 6.00000 0.224860
\(713\) 60.0000 2.24702
\(714\) −16.0000 −0.598785
\(715\) 2.00000 0.0747958
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) 10.0000 0.373197
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 1.00000 0.0372678
\(721\) −8.00000 −0.297936
\(722\) −17.0000 −0.632674
\(723\) −20.0000 −0.743808
\(724\) 22.0000 0.817624
\(725\) 2.00000 0.0742781
\(726\) −14.0000 −0.519589
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 12.0000 0.444750
\(729\) 13.0000 0.481481
\(730\) −6.00000 −0.222070
\(731\) 20.0000 0.739727
\(732\) 4.00000 0.147844
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 14.0000 0.516749
\(735\) 18.0000 0.663940
\(736\) 30.0000 1.10581
\(737\) −8.00000 −0.294684
\(738\) 6.00000 0.220863
\(739\) 6.00000 0.220714 0.110357 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −12.0000 −0.440831
\(742\) 8.00000 0.293689
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 60.0000 2.19971
\(745\) −18.0000 −0.659469
\(746\) −34.0000 −1.24483
\(747\) −16.0000 −0.585409
\(748\) −4.00000 −0.146254
\(749\) −40.0000 −1.46157
\(750\) −2.00000 −0.0730297
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) −4.00000 −0.145865
\(753\) −48.0000 −1.74922
\(754\) 2.00000 0.0728357
\(755\) −10.0000 −0.363937
\(756\) 16.0000 0.581914
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −10.0000 −0.363216
\(759\) 24.0000 0.871145
\(760\) 18.0000 0.652929
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) −4.00000 −0.144905
\(763\) −40.0000 −1.44810
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) −12.0000 −0.433578
\(767\) −6.00000 −0.216647
\(768\) 34.0000 1.22687
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −8.00000 −0.288300
\(771\) −4.00000 −0.144056
\(772\) 2.00000 0.0719816
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) −10.0000 −0.359443
\(775\) −10.0000 −0.359211
\(776\) −6.00000 −0.215387
\(777\) −16.0000 −0.573997
\(778\) 10.0000 0.358517
\(779\) 36.0000 1.28983
\(780\) 2.00000 0.0716115
\(781\) 12.0000 0.429394
\(782\) 12.0000 0.429119
\(783\) 8.00000 0.285897
\(784\) −9.00000 −0.321429
\(785\) 6.00000 0.214149
\(786\) 40.0000 1.42675
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 6.00000 0.213741
\(789\) −28.0000 −0.996826
\(790\) −12.0000 −0.426941
\(791\) 56.0000 1.99113
\(792\) 6.00000 0.213201
\(793\) −2.00000 −0.0710221
\(794\) −6.00000 −0.212932
\(795\) 4.00000 0.141865
\(796\) 16.0000 0.567105
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 48.0000 1.69918
\(799\) 8.00000 0.283020
\(800\) −5.00000 −0.176777
\(801\) 2.00000 0.0706665
\(802\) −10.0000 −0.353112
\(803\) −12.0000 −0.423471
\(804\) −8.00000 −0.282138
\(805\) −24.0000 −0.845889
\(806\) −10.0000 −0.352235
\(807\) −12.0000 −0.422420
\(808\) −54.0000 −1.89971
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −11.0000 −0.386501
\(811\) −42.0000 −1.47482 −0.737410 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) 8.00000 0.280745
\(813\) −4.00000 −0.140286
\(814\) 4.00000 0.140200
\(815\) 12.0000 0.420342
\(816\) 4.00000 0.140028
\(817\) −60.0000 −2.09913
\(818\) −18.0000 −0.629355
\(819\) 4.00000 0.139771
\(820\) −6.00000 −0.209529
\(821\) 50.0000 1.74501 0.872506 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) −4.00000 −0.139516
\(823\) −46.0000 −1.60346 −0.801730 0.597687i \(-0.796087\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) 6.00000 0.209020
\(825\) −4.00000 −0.139262
\(826\) 24.0000 0.835067
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 6.00000 0.208514
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) −16.0000 −0.555368
\(831\) 28.0000 0.971309
\(832\) −7.00000 −0.242681
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 12.0000 0.415029
\(837\) −40.0000 −1.38260
\(838\) 40.0000 1.38178
\(839\) 38.0000 1.31191 0.655953 0.754802i \(-0.272267\pi\)
0.655953 + 0.754802i \(0.272267\pi\)
\(840\) −24.0000 −0.828079
\(841\) −25.0000 −0.862069
\(842\) −10.0000 −0.344623
\(843\) 12.0000 0.413302
\(844\) −12.0000 −0.413057
\(845\) −1.00000 −0.0344010
\(846\) −4.00000 −0.137523
\(847\) 28.0000 0.962091
\(848\) −2.00000 −0.0686803
\(849\) −4.00000 −0.137280
\(850\) −2.00000 −0.0685994
\(851\) 12.0000 0.411355
\(852\) 12.0000 0.411113
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 8.00000 0.273754
\(855\) 6.00000 0.205196
\(856\) 30.0000 1.02538
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −4.00000 −0.136558
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 10.0000 0.340997
\(861\) −48.0000 −1.63584
\(862\) −14.0000 −0.476842
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −20.0000 −0.680414
\(865\) 6.00000 0.204006
\(866\) −10.0000 −0.339814
\(867\) 26.0000 0.883006
\(868\) −40.0000 −1.35769
\(869\) −24.0000 −0.814144
\(870\) −4.00000 −0.135613
\(871\) 4.00000 0.135535
\(872\) 30.0000 1.01593
\(873\) −2.00000 −0.0676897
\(874\) −36.0000 −1.21772
\(875\) 4.00000 0.135225
\(876\) −12.0000 −0.405442
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 0 0
\(879\) −44.0000 −1.48408
\(880\) 2.00000 0.0674200
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) −9.00000 −0.303046
\(883\) −22.0000 −0.740359 −0.370179 0.928960i \(-0.620704\pi\)
−0.370179 + 0.928960i \(0.620704\pi\)
\(884\) 2.00000 0.0672673
\(885\) 12.0000 0.403376
\(886\) −14.0000 −0.470339
\(887\) 58.0000 1.94745 0.973725 0.227728i \(-0.0731298\pi\)
0.973725 + 0.227728i \(0.0731298\pi\)
\(888\) 12.0000 0.402694
\(889\) 8.00000 0.268311
\(890\) 2.00000 0.0670402
\(891\) −22.0000 −0.737028
\(892\) −4.00000 −0.133930
\(893\) −24.0000 −0.803129
\(894\) 36.0000 1.20402
\(895\) −12.0000 −0.401116
\(896\) −12.0000 −0.400892
\(897\) −12.0000 −0.400668
\(898\) 6.00000 0.200223
\(899\) −20.0000 −0.667037
\(900\) −1.00000 −0.0333333
\(901\) 4.00000 0.133259
\(902\) 12.0000 0.399556
\(903\) 80.0000 2.66223
\(904\) −42.0000 −1.39690
\(905\) 22.0000 0.731305
\(906\) 20.0000 0.664455
\(907\) −34.0000 −1.12895 −0.564476 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) −4.00000 −0.132745
\(909\) −18.0000 −0.597022
\(910\) 4.00000 0.132599
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) −12.0000 −0.397360
\(913\) −32.0000 −1.05905
\(914\) −38.0000 −1.25693
\(915\) 4.00000 0.132236
\(916\) 22.0000 0.726900
\(917\) −80.0000 −2.64183
\(918\) −8.00000 −0.264039
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 18.0000 0.593442
\(921\) −16.0000 −0.527218
\(922\) −10.0000 −0.329332
\(923\) −6.00000 −0.197492
\(924\) −16.0000 −0.526361
\(925\) −2.00000 −0.0657596
\(926\) 24.0000 0.788689
\(927\) 2.00000 0.0656886
\(928\) −10.0000 −0.328266
\(929\) −38.0000 −1.24674 −0.623370 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 20.0000 0.655826
\(931\) −54.0000 −1.76978
\(932\) −10.0000 −0.327561
\(933\) 8.00000 0.261908
\(934\) 10.0000 0.327210
\(935\) −4.00000 −0.130814
\(936\) −3.00000 −0.0980581
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) −16.0000 −0.522419
\(939\) 44.0000 1.43589
\(940\) 4.00000 0.130466
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −12.0000 −0.390981
\(943\) 36.0000 1.17232
\(944\) −6.00000 −0.195283
\(945\) 16.0000 0.520480
\(946\) −20.0000 −0.650256
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −24.0000 −0.779484
\(949\) 6.00000 0.194768
\(950\) 6.00000 0.194666
\(951\) 36.0000 1.16738
\(952\) −24.0000 −0.777844
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) −8.00000 −0.258603
\(958\) −30.0000 −0.969256
\(959\) 8.00000 0.258333
\(960\) 14.0000 0.451848
\(961\) 69.0000 2.22581
\(962\) −2.00000 −0.0644826
\(963\) 10.0000 0.322245
\(964\) −10.0000 −0.322078
\(965\) 2.00000 0.0643823
\(966\) 48.0000 1.54437
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −21.0000 −0.674966
\(969\) 24.0000 0.770991
\(970\) −2.00000 −0.0642161
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −24.0000 −0.767435
\(979\) 4.00000 0.127841
\(980\) 9.00000 0.287494
\(981\) 10.0000 0.319275
\(982\) 24.0000 0.765871
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 36.0000 1.14764
\(985\) 6.00000 0.191176
\(986\) −4.00000 −0.127386
\(987\) 32.0000 1.01857
\(988\) −6.00000 −0.190885
\(989\) −60.0000 −1.90789
\(990\) 2.00000 0.0635642
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 50.0000 1.58750
\(993\) 36.0000 1.14243
\(994\) 24.0000 0.761234
\(995\) 16.0000 0.507234
\(996\) −32.0000 −1.01396
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −10.0000 −0.316544
\(999\) −8.00000 −0.253109
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 65.2.a.a.1.1 1
3.2 odd 2 585.2.a.h.1.1 1
4.3 odd 2 1040.2.a.f.1.1 1
5.2 odd 4 325.2.b.b.274.1 2
5.3 odd 4 325.2.b.b.274.2 2
5.4 even 2 325.2.a.d.1.1 1
7.6 odd 2 3185.2.a.e.1.1 1
8.3 odd 2 4160.2.a.f.1.1 1
8.5 even 2 4160.2.a.q.1.1 1
11.10 odd 2 7865.2.a.c.1.1 1
12.11 even 2 9360.2.a.ca.1.1 1
13.2 odd 12 845.2.m.b.316.1 4
13.3 even 3 845.2.e.b.191.1 2
13.4 even 6 845.2.e.a.146.1 2
13.5 odd 4 845.2.c.a.506.2 2
13.6 odd 12 845.2.m.b.361.2 4
13.7 odd 12 845.2.m.b.361.1 4
13.8 odd 4 845.2.c.a.506.1 2
13.9 even 3 845.2.e.b.146.1 2
13.10 even 6 845.2.e.a.191.1 2
13.11 odd 12 845.2.m.b.316.2 4
13.12 even 2 845.2.a.a.1.1 1
15.2 even 4 2925.2.c.h.2224.2 2
15.8 even 4 2925.2.c.h.2224.1 2
15.14 odd 2 2925.2.a.f.1.1 1
20.19 odd 2 5200.2.a.d.1.1 1
39.38 odd 2 7605.2.a.f.1.1 1
65.64 even 2 4225.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.a.1.1 1 1.1 even 1 trivial
325.2.a.d.1.1 1 5.4 even 2
325.2.b.b.274.1 2 5.2 odd 4
325.2.b.b.274.2 2 5.3 odd 4
585.2.a.h.1.1 1 3.2 odd 2
845.2.a.a.1.1 1 13.12 even 2
845.2.c.a.506.1 2 13.8 odd 4
845.2.c.a.506.2 2 13.5 odd 4
845.2.e.a.146.1 2 13.4 even 6
845.2.e.a.191.1 2 13.10 even 6
845.2.e.b.146.1 2 13.9 even 3
845.2.e.b.191.1 2 13.3 even 3
845.2.m.b.316.1 4 13.2 odd 12
845.2.m.b.316.2 4 13.11 odd 12
845.2.m.b.361.1 4 13.7 odd 12
845.2.m.b.361.2 4 13.6 odd 12
1040.2.a.f.1.1 1 4.3 odd 2
2925.2.a.f.1.1 1 15.14 odd 2
2925.2.c.h.2224.1 2 15.8 even 4
2925.2.c.h.2224.2 2 15.2 even 4
3185.2.a.e.1.1 1 7.6 odd 2
4160.2.a.f.1.1 1 8.3 odd 2
4160.2.a.q.1.1 1 8.5 even 2
4225.2.a.g.1.1 1 65.64 even 2
5200.2.a.d.1.1 1 20.19 odd 2
7605.2.a.f.1.1 1 39.38 odd 2
7865.2.a.c.1.1 1 11.10 odd 2
9360.2.a.ca.1.1 1 12.11 even 2