Properties

Label 1110.2.a.b.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} -1.00000 q^{30} +7.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +7.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -11.0000 q^{41} +1.00000 q^{42} -11.0000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +7.00000 q^{51} +2.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{57} -9.00000 q^{58} -10.0000 q^{59} +1.00000 q^{60} -1.00000 q^{61} -7.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -1.00000 q^{66} -8.00000 q^{67} -7.00000 q^{68} +1.00000 q^{70} -1.00000 q^{72} +4.00000 q^{73} +1.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +11.0000 q^{82} -6.00000 q^{83} -1.00000 q^{84} +7.00000 q^{85} +11.0000 q^{86} -9.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -7.00000 q^{93} -8.00000 q^{94} +2.00000 q^{95} +1.00000 q^{96} -19.0000 q^{97} +6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −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
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) −1.00000 −0.182574
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 7.00000 1.20049
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 2.00000 0.324443
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) −11.0000 −1.71791 −0.858956 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 1.00000 0.154303
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) 7.00000 0.980196
\(52\) 2.00000 0.277350
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) −9.00000 −1.18176
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 1.00000 0.129099
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −7.00000 −0.889001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −1.00000 −0.123091
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −7.00000 −0.848875
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 1.00000 0.116248
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) −1.00000 −0.113961
\(78\) 2.00000 0.226455
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 11.0000 1.21475
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −1.00000 −0.109109
\(85\) 7.00000 0.759257
\(86\) 11.0000 1.18616
\(87\) −9.00000 −0.964901
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −7.00000 −0.725866
\(94\) −8.00000 −0.825137
\(95\) 2.00000 0.205196
\(96\) 1.00000 0.102062
\(97\) −19.0000 −1.92916 −0.964579 0.263795i \(-0.915026\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 6.00000 0.606092
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −20.0000 −1.99007 −0.995037 0.0995037i \(-0.968274\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) −7.00000 −0.693103
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −2.00000 −0.196116
\(105\) 1.00000 0.0975900
\(106\) 11.0000 1.06841
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 1.00000 0.0949158
\(112\) 1.00000 0.0944911
\(113\) −13.0000 −1.22294 −0.611469 0.791269i \(-0.709421\pi\)
−0.611469 + 0.791269i \(0.709421\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) 2.00000 0.184900
\(118\) 10.0000 0.920575
\(119\) −7.00000 −0.641689
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) 1.00000 0.0905357
\(123\) 11.0000 0.991837
\(124\) 7.00000 0.628619
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.0000 0.968496
\(130\) 2.00000 0.175412
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 1.00000 0.0870388
\(133\) −2.00000 −0.173422
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) 7.00000 0.600245
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 21.0000 1.78120 0.890598 0.454791i \(-0.150286\pi\)
0.890598 + 0.454791i \(0.150286\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) −4.00000 −0.331042
\(147\) 6.00000 0.494872
\(148\) −1.00000 −0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 2.00000 0.162221
\(153\) −7.00000 −0.565916
\(154\) 1.00000 0.0805823
\(155\) −7.00000 −0.562254
\(156\) −2.00000 −0.160128
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) −12.0000 −0.954669
\(159\) 11.0000 0.872357
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) −11.0000 −0.858956
\(165\) −1.00000 −0.0778499
\(166\) 6.00000 0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) −7.00000 −0.536875
\(171\) −2.00000 −0.152944
\(172\) −11.0000 −0.838742
\(173\) −3.00000 −0.228086 −0.114043 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 9.00000 0.682288
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 10.0000 0.751646
\(178\) −6.00000 −0.449719
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) −2.00000 −0.148250
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 7.00000 0.513265
\(187\) 7.00000 0.511891
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) −2.00000 −0.145095
\(191\) 19.0000 1.37479 0.687396 0.726283i \(-0.258754\pi\)
0.687396 + 0.726283i \(0.258754\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 19.0000 1.36412
\(195\) 2.00000 0.143223
\(196\) −6.00000 −0.428571
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 1.00000 0.0710669
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 20.0000 1.40720
\(203\) 9.00000 0.631676
\(204\) 7.00000 0.490098
\(205\) 11.0000 0.768273
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 2.00000 0.138343
\(210\) −1.00000 −0.0690066
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −11.0000 −0.755483
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 11.0000 0.750194
\(216\) 1.00000 0.0680414
\(217\) 7.00000 0.475191
\(218\) −7.00000 −0.474100
\(219\) −4.00000 −0.270295
\(220\) 1.00000 0.0674200
\(221\) −14.0000 −0.941742
\(222\) −1.00000 −0.0671156
\(223\) 3.00000 0.200895 0.100447 0.994942i \(-0.467973\pi\)
0.100447 + 0.994942i \(0.467973\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 13.0000 0.864747
\(227\) −23.0000 −1.52656 −0.763282 0.646066i \(-0.776413\pi\)
−0.763282 + 0.646066i \(0.776413\pi\)
\(228\) 2.00000 0.132453
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) −9.00000 −0.590879
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −2.00000 −0.130744
\(235\) −8.00000 −0.521862
\(236\) −10.0000 −0.650945
\(237\) −12.0000 −0.779484
\(238\) 7.00000 0.453743
\(239\) 27.0000 1.74648 0.873242 0.487286i \(-0.162013\pi\)
0.873242 + 0.487286i \(0.162013\pi\)
\(240\) 1.00000 0.0645497
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 10.0000 0.642824
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 6.00000 0.383326
\(246\) −11.0000 −0.701334
\(247\) −4.00000 −0.254514
\(248\) −7.00000 −0.444500
\(249\) 6.00000 0.380235
\(250\) 1.00000 0.0632456
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) −7.00000 −0.438357
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −11.0000 −0.684830
\(259\) −1.00000 −0.0621370
\(260\) −2.00000 −0.124035
\(261\) 9.00000 0.557086
\(262\) 6.00000 0.370681
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 11.0000 0.675725
\(266\) 2.00000 0.122628
\(267\) −6.00000 −0.367194
\(268\) −8.00000 −0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −7.00000 −0.424437
\(273\) −2.00000 −0.121046
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −21.0000 −1.25950
\(279\) 7.00000 0.419079
\(280\) 1.00000 0.0597614
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 8.00000 0.476393
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 2.00000 0.118262
\(287\) −11.0000 −0.649309
\(288\) −1.00000 −0.0589256
\(289\) 32.0000 1.88235
\(290\) 9.00000 0.528498
\(291\) 19.0000 1.11380
\(292\) 4.00000 0.234082
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −6.00000 −0.349927
\(295\) 10.0000 0.582223
\(296\) 1.00000 0.0581238
\(297\) 1.00000 0.0580259
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −11.0000 −0.634029
\(302\) 6.00000 0.345261
\(303\) 20.0000 1.14897
\(304\) −2.00000 −0.114708
\(305\) 1.00000 0.0572598
\(306\) 7.00000 0.400163
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 12.0000 0.682656
\(310\) 7.00000 0.397573
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 2.00000 0.113228
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) −23.0000 −1.29797
\(315\) −1.00000 −0.0563436
\(316\) 12.0000 0.675053
\(317\) 5.00000 0.280828 0.140414 0.990093i \(-0.455157\pi\)
0.140414 + 0.990093i \(0.455157\pi\)
\(318\) −11.0000 −0.616849
\(319\) −9.00000 −0.503903
\(320\) −1.00000 −0.0559017
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 14.0000 0.778981
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −13.0000 −0.720003
\(327\) −7.00000 −0.387101
\(328\) 11.0000 0.607373
\(329\) 8.00000 0.441054
\(330\) 1.00000 0.0550482
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −6.00000 −0.329293
\(333\) −1.00000 −0.0547997
\(334\) −12.0000 −0.656611
\(335\) 8.00000 0.437087
\(336\) −1.00000 −0.0545545
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 9.00000 0.489535
\(339\) 13.0000 0.706063
\(340\) 7.00000 0.379628
\(341\) −7.00000 −0.379071
\(342\) 2.00000 0.108148
\(343\) −13.0000 −0.701934
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −9.00000 −0.482451
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) 1.00000 0.0533002
\(353\) −31.0000 −1.64996 −0.824982 0.565159i \(-0.808815\pi\)
−0.824982 + 0.565159i \(0.808815\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 7.00000 0.370479
\(358\) −24.0000 −1.26844
\(359\) 14.0000 0.738892 0.369446 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) 26.0000 1.36653
\(363\) 10.0000 0.524864
\(364\) 2.00000 0.104828
\(365\) −4.00000 −0.209370
\(366\) −1.00000 −0.0522708
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 0 0
\(369\) −11.0000 −0.572637
\(370\) −1.00000 −0.0519875
\(371\) −11.0000 −0.571092
\(372\) −7.00000 −0.362933
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −7.00000 −0.361961
\(375\) 1.00000 0.0516398
\(376\) −8.00000 −0.412568
\(377\) 18.0000 0.927047
\(378\) 1.00000 0.0514344
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 2.00000 0.102598
\(381\) 16.0000 0.819705
\(382\) −19.0000 −0.972125
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.00000 0.0509647
\(386\) −14.0000 −0.712581
\(387\) −11.0000 −0.559161
\(388\) −19.0000 −0.964579
\(389\) −21.0000 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 6.00000 0.302660
\(394\) 18.0000 0.906827
\(395\) −12.0000 −0.603786
\(396\) −1.00000 −0.0502519
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 4.00000 0.200502
\(399\) 2.00000 0.100125
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −8.00000 −0.399004
\(403\) 14.0000 0.697390
\(404\) −20.0000 −0.995037
\(405\) −1.00000 −0.0496904
\(406\) −9.00000 −0.446663
\(407\) 1.00000 0.0495682
\(408\) −7.00000 −0.346552
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) −11.0000 −0.543251
\(411\) 6.00000 0.295958
\(412\) −12.0000 −0.591198
\(413\) −10.0000 −0.492068
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) −2.00000 −0.0980581
\(417\) −21.0000 −1.02837
\(418\) −2.00000 −0.0978232
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 1.00000 0.0487950
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 5.00000 0.243396
\(423\) 8.00000 0.388973
\(424\) 11.0000 0.534207
\(425\) −7.00000 −0.339550
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) −6.00000 −0.290021
\(429\) 2.00000 0.0965609
\(430\) −11.0000 −0.530467
\(431\) −1.00000 −0.0481683 −0.0240842 0.999710i \(-0.507667\pi\)
−0.0240842 + 0.999710i \(0.507667\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) −7.00000 −0.336011
\(435\) 9.00000 0.431517
\(436\) 7.00000 0.335239
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 1.00000 0.0477274 0.0238637 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −6.00000 −0.285714
\(442\) 14.0000 0.665912
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 1.00000 0.0474579
\(445\) −6.00000 −0.284427
\(446\) −3.00000 −0.142054
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 11.0000 0.517970
\(452\) −13.0000 −0.611469
\(453\) 6.00000 0.281905
\(454\) 23.0000 1.07944
\(455\) −2.00000 −0.0937614
\(456\) −2.00000 −0.0936586
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 16.0000 0.747631
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) −13.0000 −0.605470 −0.302735 0.953075i \(-0.597900\pi\)
−0.302735 + 0.953075i \(0.597900\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) 9.00000 0.417815
\(465\) 7.00000 0.324617
\(466\) −24.0000 −1.11178
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 2.00000 0.0924500
\(469\) −8.00000 −0.369406
\(470\) 8.00000 0.369012
\(471\) −23.0000 −1.05978
\(472\) 10.0000 0.460287
\(473\) 11.0000 0.505781
\(474\) 12.0000 0.551178
\(475\) −2.00000 −0.0917663
\(476\) −7.00000 −0.320844
\(477\) −11.0000 −0.503655
\(478\) −27.0000 −1.23495
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −2.00000 −0.0911922
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 19.0000 0.862746
\(486\) 1.00000 0.0453609
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 1.00000 0.0452679
\(489\) −13.0000 −0.587880
\(490\) −6.00000 −0.271052
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 11.0000 0.495918
\(493\) −63.0000 −2.83738
\(494\) 4.00000 0.179969
\(495\) 1.00000 0.0449467
\(496\) 7.00000 0.314309
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) 16.0000 0.714115
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 7.00000 0.309965
\(511\) 4.00000 0.176950
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −14.0000 −0.617514
\(515\) 12.0000 0.528783
\(516\) 11.0000 0.484248
\(517\) −8.00000 −0.351840
\(518\) 1.00000 0.0439375
\(519\) 3.00000 0.131685
\(520\) 2.00000 0.0877058
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) −9.00000 −0.393919
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) −6.00000 −0.262111
\(525\) −1.00000 −0.0436436
\(526\) −21.0000 −0.915644
\(527\) −49.0000 −2.13447
\(528\) 1.00000 0.0435194
\(529\) −23.0000 −1.00000
\(530\) −11.0000 −0.477809
\(531\) −10.0000 −0.433963
\(532\) −2.00000 −0.0867110
\(533\) −22.0000 −0.952926
\(534\) 6.00000 0.259645
\(535\) 6.00000 0.259403
\(536\) 8.00000 0.345547
\(537\) −24.0000 −1.03568
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 1.00000 0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −8.00000 −0.343629
\(543\) 26.0000 1.11577
\(544\) 7.00000 0.300123
\(545\) −7.00000 −0.299847
\(546\) 2.00000 0.0855921
\(547\) −11.0000 −0.470326 −0.235163 0.971956i \(-0.575562\pi\)
−0.235163 + 0.971956i \(0.575562\pi\)
\(548\) −6.00000 −0.256307
\(549\) −1.00000 −0.0426790
\(550\) 1.00000 0.0426401
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) −8.00000 −0.339887
\(555\) −1.00000 −0.0424476
\(556\) 21.0000 0.890598
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −7.00000 −0.296334
\(559\) −22.0000 −0.930501
\(560\) −1.00000 −0.0422577
\(561\) −7.00000 −0.295540
\(562\) −6.00000 −0.253095
\(563\) 13.0000 0.547885 0.273942 0.961746i \(-0.411672\pi\)
0.273942 + 0.961746i \(0.411672\pi\)
\(564\) −8.00000 −0.336861
\(565\) 13.0000 0.546914
\(566\) −12.0000 −0.504398
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 2.00000 0.0837708
\(571\) 11.0000 0.460336 0.230168 0.973151i \(-0.426072\pi\)
0.230168 + 0.973151i \(0.426072\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −19.0000 −0.793736
\(574\) 11.0000 0.459131
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −32.0000 −1.33102
\(579\) −14.0000 −0.581820
\(580\) −9.00000 −0.373705
\(581\) −6.00000 −0.248922
\(582\) −19.0000 −0.787575
\(583\) 11.0000 0.455573
\(584\) −4.00000 −0.165521
\(585\) −2.00000 −0.0826898
\(586\) 9.00000 0.371787
\(587\) 43.0000 1.77480 0.887400 0.461000i \(-0.152509\pi\)
0.887400 + 0.461000i \(0.152509\pi\)
\(588\) 6.00000 0.247436
\(589\) −14.0000 −0.576860
\(590\) −10.0000 −0.411693
\(591\) 18.0000 0.740421
\(592\) −1.00000 −0.0410997
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 7.00000 0.286972
\(596\) 6.00000 0.245770
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 1.00000 0.0408248
\(601\) −23.0000 −0.938190 −0.469095 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 11.0000 0.448327
\(603\) −8.00000 −0.325785
\(604\) −6.00000 −0.244137
\(605\) 10.0000 0.406558
\(606\) −20.0000 −0.812444
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 2.00000 0.0811107
\(609\) −9.00000 −0.364698
\(610\) −1.00000 −0.0404888
\(611\) 16.0000 0.647291
\(612\) −7.00000 −0.282958
\(613\) 33.0000 1.33286 0.666429 0.745569i \(-0.267822\pi\)
0.666429 + 0.745569i \(0.267822\pi\)
\(614\) −10.0000 −0.403567
\(615\) −11.0000 −0.443563
\(616\) 1.00000 0.0402911
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) −12.0000 −0.482711
\(619\) −19.0000 −0.763674 −0.381837 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) −7.00000 −0.281127
\(621\) 0 0
\(622\) 9.00000 0.360867
\(623\) 6.00000 0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) −2.00000 −0.0798723
\(628\) 23.0000 0.917800
\(629\) 7.00000 0.279108
\(630\) 1.00000 0.0398410
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) −12.0000 −0.477334
\(633\) 5.00000 0.198732
\(634\) −5.00000 −0.198575
\(635\) 16.0000 0.634941
\(636\) 11.0000 0.436178
\(637\) −12.0000 −0.475457
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) −6.00000 −0.236801
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 0 0
\(645\) −11.0000 −0.433125
\(646\) −14.0000 −0.550823
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 10.0000 0.392534
\(650\) −2.00000 −0.0784465
\(651\) −7.00000 −0.274352
\(652\) 13.0000 0.509119
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 7.00000 0.273722
\(655\) 6.00000 0.234439
\(656\) −11.0000 −0.429478
\(657\) 4.00000 0.156055
\(658\) −8.00000 −0.311872
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 41.0000 1.59472 0.797358 0.603507i \(-0.206231\pi\)
0.797358 + 0.603507i \(0.206231\pi\)
\(662\) 8.00000 0.310929
\(663\) 14.0000 0.543715
\(664\) 6.00000 0.232845
\(665\) 2.00000 0.0775567
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) −3.00000 −0.115987
\(670\) −8.00000 −0.309067
\(671\) 1.00000 0.0386046
\(672\) 1.00000 0.0385758
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) −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 \(0.536783\pi\)
\(678\) −13.0000 −0.499262
\(679\) −19.0000 −0.729153
\(680\) −7.00000 −0.268438
\(681\) 23.0000 0.881362
\(682\) 7.00000 0.268044
\(683\) −15.0000 −0.573959 −0.286980 0.957937i \(-0.592651\pi\)
−0.286980 + 0.957937i \(0.592651\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 6.00000 0.229248
\(686\) 13.0000 0.496342
\(687\) 16.0000 0.610438
\(688\) −11.0000 −0.419371
\(689\) −22.0000 −0.838133
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) −3.00000 −0.114043
\(693\) −1.00000 −0.0379869
\(694\) −28.0000 −1.06287
\(695\) −21.0000 −0.796575
\(696\) 9.00000 0.341144
\(697\) 77.0000 2.91658
\(698\) 32.0000 1.21122
\(699\) −24.0000 −0.907763
\(700\) 1.00000 0.0377964
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000 0.0754851
\(703\) 2.00000 0.0754314
\(704\) −1.00000 −0.0376889
\(705\) 8.00000 0.301297
\(706\) 31.0000 1.16670
\(707\) −20.0000 −0.752177
\(708\) 10.0000 0.375823
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) −7.00000 −0.261968
\(715\) 2.00000 0.0747958
\(716\) 24.0000 0.896922
\(717\) −27.0000 −1.00833
\(718\) −14.0000 −0.522475
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −12.0000 −0.446903
\(722\) 15.0000 0.558242
\(723\) −6.00000 −0.223142
\(724\) −26.0000 −0.966282
\(725\) 9.00000 0.334252
\(726\) −10.0000 −0.371135
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) 77.0000 2.84795
\(732\) 1.00000 0.0369611
\(733\) −15.0000 −0.554038 −0.277019 0.960864i \(-0.589346\pi\)
−0.277019 + 0.960864i \(0.589346\pi\)
\(734\) −17.0000 −0.627481
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 11.0000 0.404916
\(739\) −31.0000 −1.14035 −0.570177 0.821522i \(-0.693125\pi\)
−0.570177 + 0.821522i \(0.693125\pi\)
\(740\) 1.00000 0.0367607
\(741\) 4.00000 0.146944
\(742\) 11.0000 0.403823
\(743\) 21.0000 0.770415 0.385208 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 7.00000 0.256632
\(745\) −6.00000 −0.219823
\(746\) 6.00000 0.219676
\(747\) −6.00000 −0.219529
\(748\) 7.00000 0.255945
\(749\) −6.00000 −0.219235
\(750\) −1.00000 −0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 8.00000 0.291730
\(753\) 16.0000 0.583072
\(754\) −18.0000 −0.655521
\(755\) 6.00000 0.218362
\(756\) −1.00000 −0.0363696
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) −43.0000 −1.55875 −0.779374 0.626559i \(-0.784463\pi\)
−0.779374 + 0.626559i \(0.784463\pi\)
\(762\) −16.0000 −0.579619
\(763\) 7.00000 0.253417
\(764\) 19.0000 0.687396
\(765\) 7.00000 0.253086
\(766\) −24.0000 −0.867155
\(767\) −20.0000 −0.722158
\(768\) −1.00000 −0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) −1.00000 −0.0360375
\(771\) −14.0000 −0.504198
\(772\) 14.0000 0.503871
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) 11.0000 0.395387
\(775\) 7.00000 0.251447
\(776\) 19.0000 0.682060
\(777\) 1.00000 0.0358748
\(778\) 21.0000 0.752886
\(779\) 22.0000 0.788232
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) −6.00000 −0.214286
\(785\) −23.0000 −0.820905
\(786\) −6.00000 −0.214013
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −18.0000 −0.641223
\(789\) −21.0000 −0.747620
\(790\) 12.0000 0.426941
\(791\) −13.0000 −0.462227
\(792\) 1.00000 0.0355335
\(793\) −2.00000 −0.0710221
\(794\) 14.0000 0.496841
\(795\) −11.0000 −0.390130
\(796\) −4.00000 −0.141776
\(797\) −48.0000 −1.70025 −0.850124 0.526583i \(-0.823473\pi\)
−0.850124 + 0.526583i \(0.823473\pi\)
\(798\) −2.00000 −0.0707992
\(799\) −56.0000 −1.98114
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) 6.00000 0.211867
\(803\) −4.00000 −0.141157
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −14.0000 −0.493129
\(807\) 0 0
\(808\) 20.0000 0.703598
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 1.00000 0.0351364
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 9.00000 0.315838
\(813\) −8.00000 −0.280572
\(814\) −1.00000 −0.0350500
\(815\) −13.0000 −0.455370
\(816\) 7.00000 0.245049
\(817\) 22.0000 0.769683
\(818\) 18.0000 0.629355
\(819\) 2.00000 0.0698857
\(820\) 11.0000 0.384137
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) −6.00000 −0.209274
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 12.0000 0.418040
\(825\) 1.00000 0.0348155
\(826\) 10.0000 0.347945
\(827\) 21.0000 0.730242 0.365121 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) −6.00000 −0.208263
\(831\) −8.00000 −0.277517
\(832\) 2.00000 0.0693375
\(833\) 42.0000 1.45521
\(834\) 21.0000 0.727171
\(835\) −12.0000 −0.415277
\(836\) 2.00000 0.0691714
\(837\) −7.00000 −0.241955
\(838\) −28.0000 −0.967244
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 52.0000 1.79310
\(842\) 2.00000 0.0689246
\(843\) −6.00000 −0.206651
\(844\) −5.00000 −0.172107
\(845\) 9.00000 0.309609
\(846\) −8.00000 −0.275046
\(847\) −10.0000 −0.343604
\(848\) −11.0000 −0.377742
\(849\) −12.0000 −0.411839
\(850\) 7.00000 0.240098
\(851\) 0 0
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 1.00000 0.0342193
\(855\) 2.00000 0.0683986
\(856\) 6.00000 0.205076
\(857\) 31.0000 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 11.0000 0.375097
\(861\) 11.0000 0.374879
\(862\) 1.00000 0.0340601
\(863\) 27.0000 0.919091 0.459545 0.888154i \(-0.348012\pi\)
0.459545 + 0.888154i \(0.348012\pi\)
\(864\) 1.00000 0.0340207
\(865\) 3.00000 0.102003
\(866\) −20.0000 −0.679628
\(867\) −32.0000 −1.08678
\(868\) 7.00000 0.237595
\(869\) −12.0000 −0.407072
\(870\) −9.00000 −0.305129
\(871\) −16.0000 −0.542139
\(872\) −7.00000 −0.237050
\(873\) −19.0000 −0.643053
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) −4.00000 −0.135147
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) −1.00000 −0.0337484
\(879\) 9.00000 0.303562
\(880\) 1.00000 0.0337100
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 6.00000 0.202031
\(883\) 19.0000 0.639401 0.319700 0.947519i \(-0.396418\pi\)
0.319700 + 0.947519i \(0.396418\pi\)
\(884\) −14.0000 −0.470871
\(885\) −10.0000 −0.336146
\(886\) 30.0000 1.00787
\(887\) −47.0000 −1.57811 −0.789053 0.614325i \(-0.789428\pi\)
−0.789053 + 0.614325i \(0.789428\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −16.0000 −0.536623
\(890\) 6.00000 0.201120
\(891\) −1.00000 −0.0335013
\(892\) 3.00000 0.100447
\(893\) −16.0000 −0.535420
\(894\) 6.00000 0.200670
\(895\) −24.0000 −0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) 63.0000 2.10117
\(900\) 1.00000 0.0333333
\(901\) 77.0000 2.56524
\(902\) −11.0000 −0.366260
\(903\) 11.0000 0.366057
\(904\) 13.0000 0.432374
\(905\) 26.0000 0.864269
\(906\) −6.00000 −0.199337
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −23.0000 −0.763282
\(909\) −20.0000 −0.663358
\(910\) 2.00000 0.0662994
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 2.00000 0.0662266
\(913\) 6.00000 0.198571
\(914\) 11.0000 0.363848
\(915\) −1.00000 −0.0330590
\(916\) −16.0000 −0.528655
\(917\) −6.00000 −0.198137
\(918\) −7.00000 −0.231034
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 13.0000 0.428132
\(923\) 0 0
\(924\) 1.00000 0.0328976
\(925\) −1.00000 −0.0328798
\(926\) −2.00000 −0.0657241
\(927\) −12.0000 −0.394132
\(928\) −9.00000 −0.295439
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) −7.00000 −0.229539
\(931\) 12.0000 0.393284
\(932\) 24.0000 0.786146
\(933\) 9.00000 0.294647
\(934\) −3.00000 −0.0981630
\(935\) −7.00000 −0.228924
\(936\) −2.00000 −0.0653720
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 8.00000 0.261209
\(939\) 18.0000 0.587408
\(940\) −8.00000 −0.260931
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 23.0000 0.749380
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) 1.00000 0.0325300
\(946\) −11.0000 −0.357641
\(947\) −47.0000 −1.52729 −0.763647 0.645634i \(-0.776593\pi\)
−0.763647 + 0.645634i \(0.776593\pi\)
\(948\) −12.0000 −0.389742
\(949\) 8.00000 0.259691
\(950\) 2.00000 0.0648886
\(951\) −5.00000 −0.162136
\(952\) 7.00000 0.226871
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 11.0000 0.356138
\(955\) −19.0000 −0.614826
\(956\) 27.0000 0.873242
\(957\) 9.00000 0.290929
\(958\) −24.0000 −0.775405
\(959\) −6.00000 −0.193750
\(960\) 1.00000 0.0322749
\(961\) 18.0000 0.580645
\(962\) 2.00000 0.0644826
\(963\) −6.00000 −0.193347
\(964\) 6.00000 0.193247
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) 10.0000 0.321412
\(969\) −14.0000 −0.449745
\(970\) −19.0000 −0.610053
\(971\) 49.0000 1.57248 0.786242 0.617918i \(-0.212024\pi\)
0.786242 + 0.617918i \(0.212024\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 21.0000 0.673229
\(974\) −18.0000 −0.576757
\(975\) −2.00000 −0.0640513
\(976\) −1.00000 −0.0320092
\(977\) 5.00000 0.159964 0.0799821 0.996796i \(-0.474514\pi\)
0.0799821 + 0.996796i \(0.474514\pi\)
\(978\) 13.0000 0.415694
\(979\) −6.00000 −0.191761
\(980\) 6.00000 0.191663
\(981\) 7.00000 0.223493
\(982\) 12.0000 0.382935
\(983\) 51.0000 1.62665 0.813324 0.581811i \(-0.197656\pi\)
0.813324 + 0.581811i \(0.197656\pi\)
\(984\) −11.0000 −0.350667
\(985\) 18.0000 0.573528
\(986\) 63.0000 2.00633
\(987\) −8.00000 −0.254643
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) −1.00000 −0.0317821
\(991\) 37.0000 1.17534 0.587672 0.809099i \(-0.300045\pi\)
0.587672 + 0.809099i \(0.300045\pi\)
\(992\) −7.00000 −0.222250
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 6.00000 0.190117
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 28.0000 0.886325
\(999\) 1.00000 0.0316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1110.2.a.b.1.1 1
3.2 odd 2 3330.2.a.x.1.1 1
4.3 odd 2 8880.2.a.s.1.1 1
5.4 even 2 5550.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.b.1.1 1 1.1 even 1 trivial
3330.2.a.x.1.1 1 3.2 odd 2
5550.2.a.bk.1.1 1 5.4 even 2
8880.2.a.s.1.1 1 4.3 odd 2