Properties

Label 3234.2.a.z.1.2
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +1.41421 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -2.58579 q^{19} -1.00000 q^{22} -3.65685 q^{23} -1.00000 q^{24} -5.00000 q^{25} -1.41421 q^{26} +1.00000 q^{27} +5.65685 q^{29} -11.0711 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -7.65685 q^{37} +2.58579 q^{38} +1.41421 q^{39} +1.65685 q^{41} -10.0000 q^{43} +1.00000 q^{44} +3.65685 q^{46} -1.41421 q^{47} +1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} +1.41421 q^{52} +7.65685 q^{53} -1.00000 q^{54} -2.58579 q^{57} -5.65685 q^{58} -1.17157 q^{59} -1.41421 q^{61} +11.0711 q^{62} +1.00000 q^{64} -1.00000 q^{66} +11.3137 q^{67} -4.00000 q^{68} -3.65685 q^{69} +2.34315 q^{71} -1.00000 q^{72} -4.00000 q^{73} +7.65685 q^{74} -5.00000 q^{75} -2.58579 q^{76} -1.41421 q^{78} -9.65685 q^{79} +1.00000 q^{81} -1.65685 q^{82} +0.928932 q^{83} +10.0000 q^{86} +5.65685 q^{87} -1.00000 q^{88} -5.41421 q^{89} -3.65685 q^{92} -11.0711 q^{93} +1.41421 q^{94} -1.00000 q^{96} +12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{22} + 4 q^{23} - 2 q^{24} - 10 q^{25} + 2 q^{27} - 8 q^{31} - 2 q^{32} + 2 q^{33} + 8 q^{34} + 2 q^{36} - 4 q^{37} + 8 q^{38} - 8 q^{41} - 20 q^{43} + 2 q^{44} - 4 q^{46} + 2 q^{48} + 10 q^{50} - 8 q^{51} + 4 q^{53} - 2 q^{54} - 8 q^{57} - 8 q^{59} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 8 q^{68} + 4 q^{69} + 16 q^{71} - 2 q^{72} - 8 q^{73} + 4 q^{74} - 10 q^{75} - 8 q^{76} - 8 q^{79} + 2 q^{81} + 8 q^{82} + 16 q^{83} + 20 q^{86} - 2 q^{88} - 8 q^{89} + 4 q^{92} - 8 q^{93} - 2 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.58579 −0.593220 −0.296610 0.954999i \(-0.595856\pi\)
−0.296610 + 0.954999i \(0.595856\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −3.65685 −0.762507 −0.381253 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −1.41421 −0.277350
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.65685 1.05045 0.525226 0.850963i \(-0.323981\pi\)
0.525226 + 0.850963i \(0.323981\pi\)
\(30\) 0 0
\(31\) −11.0711 −1.98842 −0.994211 0.107443i \(-0.965734\pi\)
−0.994211 + 0.107443i \(0.965734\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 2.58579 0.419470
\(39\) 1.41421 0.226455
\(40\) 0 0
\(41\) 1.65685 0.258757 0.129379 0.991595i \(-0.458702\pi\)
0.129379 + 0.991595i \(0.458702\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 3.65685 0.539174
\(47\) −1.41421 −0.206284 −0.103142 0.994667i \(-0.532890\pi\)
−0.103142 + 0.994667i \(0.532890\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −4.00000 −0.560112
\(52\) 1.41421 0.196116
\(53\) 7.65685 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −2.58579 −0.342496
\(58\) −5.65685 −0.742781
\(59\) −1.17157 −0.152526 −0.0762629 0.997088i \(-0.524299\pi\)
−0.0762629 + 0.997088i \(0.524299\pi\)
\(60\) 0 0
\(61\) −1.41421 −0.181071 −0.0905357 0.995893i \(-0.528858\pi\)
−0.0905357 + 0.995893i \(0.528858\pi\)
\(62\) 11.0711 1.40603
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 11.3137 1.38219 0.691095 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(68\) −4.00000 −0.485071
\(69\) −3.65685 −0.440234
\(70\) 0 0
\(71\) 2.34315 0.278080 0.139040 0.990287i \(-0.455598\pi\)
0.139040 + 0.990287i \(0.455598\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 7.65685 0.890091
\(75\) −5.00000 −0.577350
\(76\) −2.58579 −0.296610
\(77\) 0 0
\(78\) −1.41421 −0.160128
\(79\) −9.65685 −1.08648 −0.543240 0.839577i \(-0.682803\pi\)
−0.543240 + 0.839577i \(0.682803\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.65685 −0.182969
\(83\) 0.928932 0.101964 0.0509818 0.998700i \(-0.483765\pi\)
0.0509818 + 0.998700i \(0.483765\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 5.65685 0.606478
\(88\) −1.00000 −0.106600
\(89\) −5.41421 −0.573905 −0.286953 0.957945i \(-0.592642\pi\)
−0.286953 + 0.957945i \(0.592642\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.65685 −0.381253
\(93\) −11.0711 −1.14802
\(94\) 1.41421 0.145865
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 12.7279 1.29232 0.646162 0.763200i \(-0.276373\pi\)
0.646162 + 0.763200i \(0.276373\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −5.00000 −0.500000
\(101\) −8.24264 −0.820173 −0.410087 0.912047i \(-0.634502\pi\)
−0.410087 + 0.912047i \(0.634502\pi\)
\(102\) 4.00000 0.396059
\(103\) 5.89949 0.581295 0.290647 0.956830i \(-0.406129\pi\)
0.290647 + 0.956830i \(0.406129\pi\)
\(104\) −1.41421 −0.138675
\(105\) 0 0
\(106\) −7.65685 −0.743699
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −7.65685 −0.726756
\(112\) 0 0
\(113\) 5.65685 0.532152 0.266076 0.963952i \(-0.414273\pi\)
0.266076 + 0.963952i \(0.414273\pi\)
\(114\) 2.58579 0.242181
\(115\) 0 0
\(116\) 5.65685 0.525226
\(117\) 1.41421 0.130744
\(118\) 1.17157 0.107852
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.41421 0.128037
\(123\) 1.65685 0.149394
\(124\) −11.0711 −0.994211
\(125\) 0 0
\(126\) 0 0
\(127\) 7.31371 0.648987 0.324493 0.945888i \(-0.394806\pi\)
0.324493 + 0.945888i \(0.394806\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −12.2426 −1.06964 −0.534822 0.844965i \(-0.679621\pi\)
−0.534822 + 0.844965i \(0.679621\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −11.3137 −0.977356
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 13.3137 1.13747 0.568733 0.822522i \(-0.307434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(138\) 3.65685 0.311292
\(139\) −21.8995 −1.85749 −0.928745 0.370718i \(-0.879112\pi\)
−0.928745 + 0.370718i \(0.879112\pi\)
\(140\) 0 0
\(141\) −1.41421 −0.119098
\(142\) −2.34315 −0.196632
\(143\) 1.41421 0.118262
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −7.65685 −0.629390
\(149\) −20.9706 −1.71798 −0.858988 0.511996i \(-0.828906\pi\)
−0.858988 + 0.511996i \(0.828906\pi\)
\(150\) 5.00000 0.408248
\(151\) 9.65685 0.785864 0.392932 0.919568i \(-0.371461\pi\)
0.392932 + 0.919568i \(0.371461\pi\)
\(152\) 2.58579 0.209735
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 1.41421 0.113228
\(157\) −2.82843 −0.225733 −0.112867 0.993610i \(-0.536003\pi\)
−0.112867 + 0.993610i \(0.536003\pi\)
\(158\) 9.65685 0.768258
\(159\) 7.65685 0.607228
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −5.65685 −0.443079 −0.221540 0.975151i \(-0.571108\pi\)
−0.221540 + 0.975151i \(0.571108\pi\)
\(164\) 1.65685 0.129379
\(165\) 0 0
\(166\) −0.928932 −0.0720991
\(167\) −5.17157 −0.400188 −0.200094 0.979777i \(-0.564125\pi\)
−0.200094 + 0.979777i \(0.564125\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) −2.58579 −0.197740
\(172\) −10.0000 −0.762493
\(173\) −19.0711 −1.44995 −0.724973 0.688777i \(-0.758148\pi\)
−0.724973 + 0.688777i \(0.758148\pi\)
\(174\) −5.65685 −0.428845
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −1.17157 −0.0880608
\(178\) 5.41421 0.405812
\(179\) 13.6569 1.02076 0.510381 0.859949i \(-0.329505\pi\)
0.510381 + 0.859949i \(0.329505\pi\)
\(180\) 0 0
\(181\) 10.3431 0.768800 0.384400 0.923167i \(-0.374408\pi\)
0.384400 + 0.923167i \(0.374408\pi\)
\(182\) 0 0
\(183\) −1.41421 −0.104542
\(184\) 3.65685 0.269587
\(185\) 0 0
\(186\) 11.0711 0.811770
\(187\) −4.00000 −0.292509
\(188\) −1.41421 −0.103142
\(189\) 0 0
\(190\) 0 0
\(191\) 22.9706 1.66209 0.831046 0.556204i \(-0.187743\pi\)
0.831046 + 0.556204i \(0.187743\pi\)
\(192\) 1.00000 0.0721688
\(193\) −3.65685 −0.263226 −0.131613 0.991301i \(-0.542016\pi\)
−0.131613 + 0.991301i \(0.542016\pi\)
\(194\) −12.7279 −0.913812
\(195\) 0 0
\(196\) 0 0
\(197\) −1.31371 −0.0935979 −0.0467989 0.998904i \(-0.514902\pi\)
−0.0467989 + 0.998904i \(0.514902\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −13.4142 −0.950908 −0.475454 0.879740i \(-0.657716\pi\)
−0.475454 + 0.879740i \(0.657716\pi\)
\(200\) 5.00000 0.353553
\(201\) 11.3137 0.798007
\(202\) 8.24264 0.579950
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −5.89949 −0.411037
\(207\) −3.65685 −0.254169
\(208\) 1.41421 0.0980581
\(209\) −2.58579 −0.178863
\(210\) 0 0
\(211\) −5.31371 −0.365811 −0.182905 0.983131i \(-0.558550\pi\)
−0.182905 + 0.983131i \(0.558550\pi\)
\(212\) 7.65685 0.525875
\(213\) 2.34315 0.160550
\(214\) 5.31371 0.363238
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −5.65685 −0.380521
\(222\) 7.65685 0.513894
\(223\) 11.5563 0.773870 0.386935 0.922107i \(-0.373534\pi\)
0.386935 + 0.922107i \(0.373534\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) −5.65685 −0.376288
\(227\) 7.55635 0.501533 0.250766 0.968048i \(-0.419317\pi\)
0.250766 + 0.968048i \(0.419317\pi\)
\(228\) −2.58579 −0.171248
\(229\) 5.65685 0.373815 0.186908 0.982377i \(-0.440153\pi\)
0.186908 + 0.982377i \(0.440153\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.65685 −0.371391
\(233\) 23.6569 1.54981 0.774906 0.632076i \(-0.217797\pi\)
0.774906 + 0.632076i \(0.217797\pi\)
\(234\) −1.41421 −0.0924500
\(235\) 0 0
\(236\) −1.17157 −0.0762629
\(237\) −9.65685 −0.627280
\(238\) 0 0
\(239\) 25.6569 1.65960 0.829802 0.558058i \(-0.188453\pi\)
0.829802 + 0.558058i \(0.188453\pi\)
\(240\) 0 0
\(241\) −15.3137 −0.986443 −0.493221 0.869904i \(-0.664181\pi\)
−0.493221 + 0.869904i \(0.664181\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −1.41421 −0.0905357
\(245\) 0 0
\(246\) −1.65685 −0.105637
\(247\) −3.65685 −0.232680
\(248\) 11.0711 0.703014
\(249\) 0.928932 0.0588687
\(250\) 0 0
\(251\) −9.17157 −0.578905 −0.289452 0.957192i \(-0.593473\pi\)
−0.289452 + 0.957192i \(0.593473\pi\)
\(252\) 0 0
\(253\) −3.65685 −0.229904
\(254\) −7.31371 −0.458903
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −8.72792 −0.544433 −0.272216 0.962236i \(-0.587757\pi\)
−0.272216 + 0.962236i \(0.587757\pi\)
\(258\) 10.0000 0.622573
\(259\) 0 0
\(260\) 0 0
\(261\) 5.65685 0.350150
\(262\) 12.2426 0.756353
\(263\) 2.34315 0.144485 0.0722423 0.997387i \(-0.476985\pi\)
0.0722423 + 0.997387i \(0.476985\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) −5.41421 −0.331344
\(268\) 11.3137 0.691095
\(269\) 22.6274 1.37962 0.689809 0.723991i \(-0.257694\pi\)
0.689809 + 0.723991i \(0.257694\pi\)
\(270\) 0 0
\(271\) −26.8284 −1.62971 −0.814855 0.579664i \(-0.803184\pi\)
−0.814855 + 0.579664i \(0.803184\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) −13.3137 −0.804311
\(275\) −5.00000 −0.301511
\(276\) −3.65685 −0.220117
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 21.8995 1.31344
\(279\) −11.0711 −0.662807
\(280\) 0 0
\(281\) 14.9706 0.893069 0.446534 0.894766i \(-0.352658\pi\)
0.446534 + 0.894766i \(0.352658\pi\)
\(282\) 1.41421 0.0842152
\(283\) −24.2426 −1.44108 −0.720538 0.693416i \(-0.756105\pi\)
−0.720538 + 0.693416i \(0.756105\pi\)
\(284\) 2.34315 0.139040
\(285\) 0 0
\(286\) −1.41421 −0.0836242
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 12.7279 0.746124
\(292\) −4.00000 −0.234082
\(293\) −11.0711 −0.646779 −0.323389 0.946266i \(-0.604822\pi\)
−0.323389 + 0.946266i \(0.604822\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.65685 0.445046
\(297\) 1.00000 0.0580259
\(298\) 20.9706 1.21479
\(299\) −5.17157 −0.299080
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) −9.65685 −0.555690
\(303\) −8.24264 −0.473527
\(304\) −2.58579 −0.148305
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) −24.7279 −1.41130 −0.705649 0.708562i \(-0.749344\pi\)
−0.705649 + 0.708562i \(0.749344\pi\)
\(308\) 0 0
\(309\) 5.89949 0.335611
\(310\) 0 0
\(311\) −11.7574 −0.666699 −0.333349 0.942803i \(-0.608179\pi\)
−0.333349 + 0.942803i \(0.608179\pi\)
\(312\) −1.41421 −0.0800641
\(313\) 3.75736 0.212379 0.106189 0.994346i \(-0.466135\pi\)
0.106189 + 0.994346i \(0.466135\pi\)
\(314\) 2.82843 0.159617
\(315\) 0 0
\(316\) −9.65685 −0.543240
\(317\) 17.3137 0.972435 0.486217 0.873838i \(-0.338376\pi\)
0.486217 + 0.873838i \(0.338376\pi\)
\(318\) −7.65685 −0.429375
\(319\) 5.65685 0.316723
\(320\) 0 0
\(321\) −5.31371 −0.296582
\(322\) 0 0
\(323\) 10.3431 0.575508
\(324\) 1.00000 0.0555556
\(325\) −7.07107 −0.392232
\(326\) 5.65685 0.313304
\(327\) −10.0000 −0.553001
\(328\) −1.65685 −0.0914845
\(329\) 0 0
\(330\) 0 0
\(331\) −4.97056 −0.273207 −0.136603 0.990626i \(-0.543619\pi\)
−0.136603 + 0.990626i \(0.543619\pi\)
\(332\) 0.928932 0.0509818
\(333\) −7.65685 −0.419593
\(334\) 5.17157 0.282976
\(335\) 0 0
\(336\) 0 0
\(337\) 5.31371 0.289456 0.144728 0.989471i \(-0.453769\pi\)
0.144728 + 0.989471i \(0.453769\pi\)
\(338\) 11.0000 0.598321
\(339\) 5.65685 0.307238
\(340\) 0 0
\(341\) −11.0711 −0.599532
\(342\) 2.58579 0.139823
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 19.0711 1.02527
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 5.65685 0.303239
\(349\) −25.4142 −1.36039 −0.680196 0.733030i \(-0.738105\pi\)
−0.680196 + 0.733030i \(0.738105\pi\)
\(350\) 0 0
\(351\) 1.41421 0.0754851
\(352\) −1.00000 −0.0533002
\(353\) −5.89949 −0.313998 −0.156999 0.987599i \(-0.550182\pi\)
−0.156999 + 0.987599i \(0.550182\pi\)
\(354\) 1.17157 0.0622684
\(355\) 0 0
\(356\) −5.41421 −0.286953
\(357\) 0 0
\(358\) −13.6569 −0.721787
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −12.3137 −0.648090
\(362\) −10.3431 −0.543624
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) 1.41421 0.0739221
\(367\) −22.8701 −1.19381 −0.596904 0.802313i \(-0.703603\pi\)
−0.596904 + 0.802313i \(0.703603\pi\)
\(368\) −3.65685 −0.190627
\(369\) 1.65685 0.0862524
\(370\) 0 0
\(371\) 0 0
\(372\) −11.0711 −0.574008
\(373\) 8.62742 0.446711 0.223355 0.974737i \(-0.428299\pi\)
0.223355 + 0.974737i \(0.428299\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 1.41421 0.0729325
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −6.34315 −0.325826 −0.162913 0.986640i \(-0.552089\pi\)
−0.162913 + 0.986640i \(0.552089\pi\)
\(380\) 0 0
\(381\) 7.31371 0.374693
\(382\) −22.9706 −1.17528
\(383\) −23.5563 −1.20367 −0.601837 0.798619i \(-0.705564\pi\)
−0.601837 + 0.798619i \(0.705564\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 3.65685 0.186129
\(387\) −10.0000 −0.508329
\(388\) 12.7279 0.646162
\(389\) −32.6274 −1.65428 −0.827138 0.561999i \(-0.810032\pi\)
−0.827138 + 0.561999i \(0.810032\pi\)
\(390\) 0 0
\(391\) 14.6274 0.739740
\(392\) 0 0
\(393\) −12.2426 −0.617560
\(394\) 1.31371 0.0661837
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 27.7990 1.39519 0.697596 0.716492i \(-0.254253\pi\)
0.697596 + 0.716492i \(0.254253\pi\)
\(398\) 13.4142 0.672394
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −11.3137 −0.564276
\(403\) −15.6569 −0.779923
\(404\) −8.24264 −0.410087
\(405\) 0 0
\(406\) 0 0
\(407\) −7.65685 −0.379536
\(408\) 4.00000 0.198030
\(409\) −4.48528 −0.221783 −0.110891 0.993833i \(-0.535371\pi\)
−0.110891 + 0.993833i \(0.535371\pi\)
\(410\) 0 0
\(411\) 13.3137 0.656717
\(412\) 5.89949 0.290647
\(413\) 0 0
\(414\) 3.65685 0.179725
\(415\) 0 0
\(416\) −1.41421 −0.0693375
\(417\) −21.8995 −1.07242
\(418\) 2.58579 0.126475
\(419\) 40.2843 1.96802 0.984008 0.178126i \(-0.0570034\pi\)
0.984008 + 0.178126i \(0.0570034\pi\)
\(420\) 0 0
\(421\) 18.9706 0.924569 0.462284 0.886732i \(-0.347030\pi\)
0.462284 + 0.886732i \(0.347030\pi\)
\(422\) 5.31371 0.258667
\(423\) −1.41421 −0.0687614
\(424\) −7.65685 −0.371850
\(425\) 20.0000 0.970143
\(426\) −2.34315 −0.113526
\(427\) 0 0
\(428\) −5.31371 −0.256848
\(429\) 1.41421 0.0682789
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) 5.21320 0.250531 0.125265 0.992123i \(-0.460022\pi\)
0.125265 + 0.992123i \(0.460022\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 9.45584 0.452334
\(438\) 4.00000 0.191127
\(439\) 3.79899 0.181316 0.0906579 0.995882i \(-0.471103\pi\)
0.0906579 + 0.995882i \(0.471103\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5.65685 0.269069
\(443\) −28.2843 −1.34383 −0.671913 0.740630i \(-0.734527\pi\)
−0.671913 + 0.740630i \(0.734527\pi\)
\(444\) −7.65685 −0.363378
\(445\) 0 0
\(446\) −11.5563 −0.547209
\(447\) −20.9706 −0.991874
\(448\) 0 0
\(449\) 21.6569 1.02205 0.511025 0.859566i \(-0.329266\pi\)
0.511025 + 0.859566i \(0.329266\pi\)
\(450\) 5.00000 0.235702
\(451\) 1.65685 0.0780182
\(452\) 5.65685 0.266076
\(453\) 9.65685 0.453719
\(454\) −7.55635 −0.354637
\(455\) 0 0
\(456\) 2.58579 0.121091
\(457\) 8.34315 0.390276 0.195138 0.980776i \(-0.437485\pi\)
0.195138 + 0.980776i \(0.437485\pi\)
\(458\) −5.65685 −0.264327
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −18.1005 −0.843025 −0.421512 0.906823i \(-0.638501\pi\)
−0.421512 + 0.906823i \(0.638501\pi\)
\(462\) 0 0
\(463\) −4.68629 −0.217790 −0.108895 0.994053i \(-0.534731\pi\)
−0.108895 + 0.994053i \(0.534731\pi\)
\(464\) 5.65685 0.262613
\(465\) 0 0
\(466\) −23.6569 −1.09588
\(467\) −29.9411 −1.38551 −0.692755 0.721173i \(-0.743603\pi\)
−0.692755 + 0.721173i \(0.743603\pi\)
\(468\) 1.41421 0.0653720
\(469\) 0 0
\(470\) 0 0
\(471\) −2.82843 −0.130327
\(472\) 1.17157 0.0539260
\(473\) −10.0000 −0.459800
\(474\) 9.65685 0.443554
\(475\) 12.9289 0.593220
\(476\) 0 0
\(477\) 7.65685 0.350583
\(478\) −25.6569 −1.17352
\(479\) 0.970563 0.0443461 0.0221731 0.999754i \(-0.492942\pi\)
0.0221731 + 0.999754i \(0.492942\pi\)
\(480\) 0 0
\(481\) −10.8284 −0.493734
\(482\) 15.3137 0.697520
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −11.6569 −0.528222 −0.264111 0.964492i \(-0.585079\pi\)
−0.264111 + 0.964492i \(0.585079\pi\)
\(488\) 1.41421 0.0640184
\(489\) −5.65685 −0.255812
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 1.65685 0.0746968
\(493\) −22.6274 −1.01909
\(494\) 3.65685 0.164530
\(495\) 0 0
\(496\) −11.0711 −0.497106
\(497\) 0 0
\(498\) −0.928932 −0.0416264
\(499\) 34.6274 1.55014 0.775068 0.631878i \(-0.217716\pi\)
0.775068 + 0.631878i \(0.217716\pi\)
\(500\) 0 0
\(501\) −5.17157 −0.231049
\(502\) 9.17157 0.409347
\(503\) −16.4853 −0.735042 −0.367521 0.930015i \(-0.619793\pi\)
−0.367521 + 0.930015i \(0.619793\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.65685 0.162567
\(507\) −11.0000 −0.488527
\(508\) 7.31371 0.324493
\(509\) −25.9411 −1.14982 −0.574910 0.818217i \(-0.694963\pi\)
−0.574910 + 0.818217i \(0.694963\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −2.58579 −0.114165
\(514\) 8.72792 0.384972
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) −1.41421 −0.0621970
\(518\) 0 0
\(519\) −19.0711 −0.837127
\(520\) 0 0
\(521\) 32.7279 1.43384 0.716918 0.697157i \(-0.245552\pi\)
0.716918 + 0.697157i \(0.245552\pi\)
\(522\) −5.65685 −0.247594
\(523\) 40.2426 1.75969 0.879844 0.475263i \(-0.157647\pi\)
0.879844 + 0.475263i \(0.157647\pi\)
\(524\) −12.2426 −0.534822
\(525\) 0 0
\(526\) −2.34315 −0.102166
\(527\) 44.2843 1.92905
\(528\) 1.00000 0.0435194
\(529\) −9.62742 −0.418583
\(530\) 0 0
\(531\) −1.17157 −0.0508419
\(532\) 0 0
\(533\) 2.34315 0.101493
\(534\) 5.41421 0.234296
\(535\) 0 0
\(536\) −11.3137 −0.488678
\(537\) 13.6569 0.589337
\(538\) −22.6274 −0.975537
\(539\) 0 0
\(540\) 0 0
\(541\) 36.6274 1.57474 0.787368 0.616483i \(-0.211443\pi\)
0.787368 + 0.616483i \(0.211443\pi\)
\(542\) 26.8284 1.15238
\(543\) 10.3431 0.443867
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 9.65685 0.412897 0.206449 0.978457i \(-0.433809\pi\)
0.206449 + 0.978457i \(0.433809\pi\)
\(548\) 13.3137 0.568733
\(549\) −1.41421 −0.0603572
\(550\) 5.00000 0.213201
\(551\) −14.6274 −0.623149
\(552\) 3.65685 0.155646
\(553\) 0 0
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −21.8995 −0.928745
\(557\) −35.9411 −1.52287 −0.761437 0.648239i \(-0.775506\pi\)
−0.761437 + 0.648239i \(0.775506\pi\)
\(558\) 11.0711 0.468676
\(559\) −14.1421 −0.598149
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −14.9706 −0.631495
\(563\) 25.4142 1.07108 0.535541 0.844509i \(-0.320108\pi\)
0.535541 + 0.844509i \(0.320108\pi\)
\(564\) −1.41421 −0.0595491
\(565\) 0 0
\(566\) 24.2426 1.01899
\(567\) 0 0
\(568\) −2.34315 −0.0983162
\(569\) −29.3137 −1.22889 −0.614447 0.788958i \(-0.710621\pi\)
−0.614447 + 0.788958i \(0.710621\pi\)
\(570\) 0 0
\(571\) −16.2843 −0.681476 −0.340738 0.940158i \(-0.610677\pi\)
−0.340738 + 0.940158i \(0.610677\pi\)
\(572\) 1.41421 0.0591312
\(573\) 22.9706 0.959609
\(574\) 0 0
\(575\) 18.2843 0.762507
\(576\) 1.00000 0.0416667
\(577\) 36.2426 1.50880 0.754400 0.656414i \(-0.227928\pi\)
0.754400 + 0.656414i \(0.227928\pi\)
\(578\) 1.00000 0.0415945
\(579\) −3.65685 −0.151974
\(580\) 0 0
\(581\) 0 0
\(582\) −12.7279 −0.527589
\(583\) 7.65685 0.317115
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 11.0711 0.457342
\(587\) −20.4853 −0.845518 −0.422759 0.906242i \(-0.638938\pi\)
−0.422759 + 0.906242i \(0.638938\pi\)
\(588\) 0 0
\(589\) 28.6274 1.17957
\(590\) 0 0
\(591\) −1.31371 −0.0540387
\(592\) −7.65685 −0.314695
\(593\) 0.686292 0.0281826 0.0140913 0.999901i \(-0.495514\pi\)
0.0140913 + 0.999901i \(0.495514\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −20.9706 −0.858988
\(597\) −13.4142 −0.549007
\(598\) 5.17157 0.211481
\(599\) 0.970563 0.0396561 0.0198281 0.999803i \(-0.493688\pi\)
0.0198281 + 0.999803i \(0.493688\pi\)
\(600\) 5.00000 0.204124
\(601\) 37.4558 1.52786 0.763928 0.645302i \(-0.223268\pi\)
0.763928 + 0.645302i \(0.223268\pi\)
\(602\) 0 0
\(603\) 11.3137 0.460730
\(604\) 9.65685 0.392932
\(605\) 0 0
\(606\) 8.24264 0.334834
\(607\) 19.3137 0.783919 0.391960 0.919982i \(-0.371797\pi\)
0.391960 + 0.919982i \(0.371797\pi\)
\(608\) 2.58579 0.104867
\(609\) 0 0
\(610\) 0 0
\(611\) −2.00000 −0.0809113
\(612\) −4.00000 −0.161690
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) 24.7279 0.997938
\(615\) 0 0
\(616\) 0 0
\(617\) 25.6569 1.03291 0.516453 0.856316i \(-0.327252\pi\)
0.516453 + 0.856316i \(0.327252\pi\)
\(618\) −5.89949 −0.237312
\(619\) −15.3137 −0.615510 −0.307755 0.951466i \(-0.599578\pi\)
−0.307755 + 0.951466i \(0.599578\pi\)
\(620\) 0 0
\(621\) −3.65685 −0.146745
\(622\) 11.7574 0.471427
\(623\) 0 0
\(624\) 1.41421 0.0566139
\(625\) 25.0000 1.00000
\(626\) −3.75736 −0.150174
\(627\) −2.58579 −0.103266
\(628\) −2.82843 −0.112867
\(629\) 30.6274 1.22120
\(630\) 0 0
\(631\) −41.5980 −1.65599 −0.827995 0.560736i \(-0.810518\pi\)
−0.827995 + 0.560736i \(0.810518\pi\)
\(632\) 9.65685 0.384129
\(633\) −5.31371 −0.211201
\(634\) −17.3137 −0.687615
\(635\) 0 0
\(636\) 7.65685 0.303614
\(637\) 0 0
\(638\) −5.65685 −0.223957
\(639\) 2.34315 0.0926934
\(640\) 0 0
\(641\) −48.2843 −1.90711 −0.953557 0.301213i \(-0.902609\pi\)
−0.953557 + 0.301213i \(0.902609\pi\)
\(642\) 5.31371 0.209715
\(643\) 41.1716 1.62365 0.811824 0.583902i \(-0.198475\pi\)
0.811824 + 0.583902i \(0.198475\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −10.3431 −0.406946
\(647\) 1.89949 0.0746769 0.0373384 0.999303i \(-0.488112\pi\)
0.0373384 + 0.999303i \(0.488112\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −1.17157 −0.0459883
\(650\) 7.07107 0.277350
\(651\) 0 0
\(652\) −5.65685 −0.221540
\(653\) 12.6274 0.494149 0.247075 0.968996i \(-0.420531\pi\)
0.247075 + 0.968996i \(0.420531\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 1.65685 0.0646893
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −2.68629 −0.104643 −0.0523215 0.998630i \(-0.516662\pi\)
−0.0523215 + 0.998630i \(0.516662\pi\)
\(660\) 0 0
\(661\) −8.97056 −0.348914 −0.174457 0.984665i \(-0.555817\pi\)
−0.174457 + 0.984665i \(0.555817\pi\)
\(662\) 4.97056 0.193186
\(663\) −5.65685 −0.219694
\(664\) −0.928932 −0.0360496
\(665\) 0 0
\(666\) 7.65685 0.296697
\(667\) −20.6863 −0.800976
\(668\) −5.17157 −0.200094
\(669\) 11.5563 0.446794
\(670\) 0 0
\(671\) −1.41421 −0.0545951
\(672\) 0 0
\(673\) 8.34315 0.321605 0.160802 0.986987i \(-0.448592\pi\)
0.160802 + 0.986987i \(0.448592\pi\)
\(674\) −5.31371 −0.204676
\(675\) −5.00000 −0.192450
\(676\) −11.0000 −0.423077
\(677\) 21.4142 0.823015 0.411508 0.911406i \(-0.365002\pi\)
0.411508 + 0.911406i \(0.365002\pi\)
\(678\) −5.65685 −0.217250
\(679\) 0 0
\(680\) 0 0
\(681\) 7.55635 0.289560
\(682\) 11.0711 0.423933
\(683\) 8.28427 0.316989 0.158494 0.987360i \(-0.449336\pi\)
0.158494 + 0.987360i \(0.449336\pi\)
\(684\) −2.58579 −0.0988700
\(685\) 0 0
\(686\) 0 0
\(687\) 5.65685 0.215822
\(688\) −10.0000 −0.381246
\(689\) 10.8284 0.412530
\(690\) 0 0
\(691\) 12.4853 0.474962 0.237481 0.971392i \(-0.423678\pi\)
0.237481 + 0.971392i \(0.423678\pi\)
\(692\) −19.0711 −0.724973
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) −5.65685 −0.214423
\(697\) −6.62742 −0.251031
\(698\) 25.4142 0.961942
\(699\) 23.6569 0.894784
\(700\) 0 0
\(701\) −18.6863 −0.705771 −0.352886 0.935666i \(-0.614800\pi\)
−0.352886 + 0.935666i \(0.614800\pi\)
\(702\) −1.41421 −0.0533761
\(703\) 19.7990 0.746733
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 5.89949 0.222030
\(707\) 0 0
\(708\) −1.17157 −0.0440304
\(709\) −44.6274 −1.67602 −0.838009 0.545657i \(-0.816280\pi\)
−0.838009 + 0.545657i \(0.816280\pi\)
\(710\) 0 0
\(711\) −9.65685 −0.362160
\(712\) 5.41421 0.202906
\(713\) 40.4853 1.51619
\(714\) 0 0
\(715\) 0 0
\(716\) 13.6569 0.510381
\(717\) 25.6569 0.958173
\(718\) 4.00000 0.149279
\(719\) −17.4142 −0.649441 −0.324720 0.945810i \(-0.605270\pi\)
−0.324720 + 0.945810i \(0.605270\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12.3137 0.458269
\(723\) −15.3137 −0.569523
\(724\) 10.3431 0.384400
\(725\) −28.2843 −1.05045
\(726\) −1.00000 −0.0371135
\(727\) −31.3553 −1.16291 −0.581453 0.813580i \(-0.697515\pi\)
−0.581453 + 0.813580i \(0.697515\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 40.0000 1.47945
\(732\) −1.41421 −0.0522708
\(733\) 44.2426 1.63414 0.817070 0.576539i \(-0.195597\pi\)
0.817070 + 0.576539i \(0.195597\pi\)
\(734\) 22.8701 0.844149
\(735\) 0 0
\(736\) 3.65685 0.134793
\(737\) 11.3137 0.416746
\(738\) −1.65685 −0.0609896
\(739\) 17.6569 0.649518 0.324759 0.945797i \(-0.394717\pi\)
0.324759 + 0.945797i \(0.394717\pi\)
\(740\) 0 0
\(741\) −3.65685 −0.134338
\(742\) 0 0
\(743\) 28.9706 1.06283 0.531413 0.847113i \(-0.321661\pi\)
0.531413 + 0.847113i \(0.321661\pi\)
\(744\) 11.0711 0.405885
\(745\) 0 0
\(746\) −8.62742 −0.315872
\(747\) 0.928932 0.0339879
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) −6.97056 −0.254359 −0.127180 0.991880i \(-0.540592\pi\)
−0.127180 + 0.991880i \(0.540592\pi\)
\(752\) −1.41421 −0.0515711
\(753\) −9.17157 −0.334231
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) −29.3137 −1.06542 −0.532712 0.846296i \(-0.678827\pi\)
−0.532712 + 0.846296i \(0.678827\pi\)
\(758\) 6.34315 0.230393
\(759\) −3.65685 −0.132735
\(760\) 0 0
\(761\) 42.1421 1.52765 0.763826 0.645423i \(-0.223319\pi\)
0.763826 + 0.645423i \(0.223319\pi\)
\(762\) −7.31371 −0.264948
\(763\) 0 0
\(764\) 22.9706 0.831046
\(765\) 0 0
\(766\) 23.5563 0.851125
\(767\) −1.65685 −0.0598255
\(768\) 1.00000 0.0360844
\(769\) −0.201010 −0.00724861 −0.00362431 0.999993i \(-0.501154\pi\)
−0.00362431 + 0.999993i \(0.501154\pi\)
\(770\) 0 0
\(771\) −8.72792 −0.314328
\(772\) −3.65685 −0.131613
\(773\) −14.1421 −0.508657 −0.254329 0.967118i \(-0.581854\pi\)
−0.254329 + 0.967118i \(0.581854\pi\)
\(774\) 10.0000 0.359443
\(775\) 55.3553 1.98842
\(776\) −12.7279 −0.456906
\(777\) 0 0
\(778\) 32.6274 1.16975
\(779\) −4.28427 −0.153500
\(780\) 0 0
\(781\) 2.34315 0.0838443
\(782\) −14.6274 −0.523075
\(783\) 5.65685 0.202159
\(784\) 0 0
\(785\) 0 0
\(786\) 12.2426 0.436681
\(787\) 1.61522 0.0575765 0.0287883 0.999586i \(-0.490835\pi\)
0.0287883 + 0.999586i \(0.490835\pi\)
\(788\) −1.31371 −0.0467989
\(789\) 2.34315 0.0834182
\(790\) 0 0
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −2.00000 −0.0710221
\(794\) −27.7990 −0.986549
\(795\) 0 0
\(796\) −13.4142 −0.475454
\(797\) 45.1716 1.60006 0.800030 0.599961i \(-0.204817\pi\)
0.800030 + 0.599961i \(0.204817\pi\)
\(798\) 0 0
\(799\) 5.65685 0.200125
\(800\) 5.00000 0.176777
\(801\) −5.41421 −0.191302
\(802\) 6.00000 0.211867
\(803\) −4.00000 −0.141157
\(804\) 11.3137 0.399004
\(805\) 0 0
\(806\) 15.6569 0.551489
\(807\) 22.6274 0.796523
\(808\) 8.24264 0.289975
\(809\) −14.9706 −0.526337 −0.263168 0.964750i \(-0.584768\pi\)
−0.263168 + 0.964750i \(0.584768\pi\)
\(810\) 0 0
\(811\) −54.8701 −1.92675 −0.963374 0.268161i \(-0.913584\pi\)
−0.963374 + 0.268161i \(0.913584\pi\)
\(812\) 0 0
\(813\) −26.8284 −0.940914
\(814\) 7.65685 0.268373
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 25.8579 0.904652
\(818\) 4.48528 0.156824
\(819\) 0 0
\(820\) 0 0
\(821\) 36.9706 1.29028 0.645141 0.764064i \(-0.276799\pi\)
0.645141 + 0.764064i \(0.276799\pi\)
\(822\) −13.3137 −0.464369
\(823\) 25.5980 0.892289 0.446145 0.894961i \(-0.352797\pi\)
0.446145 + 0.894961i \(0.352797\pi\)
\(824\) −5.89949 −0.205519
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) −3.65685 −0.127084
\(829\) 3.31371 0.115090 0.0575449 0.998343i \(-0.481673\pi\)
0.0575449 + 0.998343i \(0.481673\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) 1.41421 0.0490290
\(833\) 0 0
\(834\) 21.8995 0.758317
\(835\) 0 0
\(836\) −2.58579 −0.0894313
\(837\) −11.0711 −0.382672
\(838\) −40.2843 −1.39160
\(839\) 12.7279 0.439417 0.219708 0.975566i \(-0.429489\pi\)
0.219708 + 0.975566i \(0.429489\pi\)
\(840\) 0 0
\(841\) 3.00000 0.103448
\(842\) −18.9706 −0.653769
\(843\) 14.9706 0.515614
\(844\) −5.31371 −0.182905
\(845\) 0 0
\(846\) 1.41421 0.0486217
\(847\) 0 0
\(848\) 7.65685 0.262937
\(849\) −24.2426 −0.832005
\(850\) −20.0000 −0.685994
\(851\) 28.0000 0.959828
\(852\) 2.34315 0.0802749
\(853\) 27.2721 0.933778 0.466889 0.884316i \(-0.345375\pi\)
0.466889 + 0.884316i \(0.345375\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 5.31371 0.181619
\(857\) 9.17157 0.313295 0.156647 0.987655i \(-0.449931\pi\)
0.156647 + 0.987655i \(0.449931\pi\)
\(858\) −1.41421 −0.0482805
\(859\) 38.4264 1.31109 0.655546 0.755155i \(-0.272439\pi\)
0.655546 + 0.755155i \(0.272439\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) −29.5980 −1.00753 −0.503763 0.863842i \(-0.668052\pi\)
−0.503763 + 0.863842i \(0.668052\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −5.21320 −0.177152
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −9.65685 −0.327586
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 10.0000 0.338643
\(873\) 12.7279 0.430775
\(874\) −9.45584 −0.319849
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 30.6863 1.03620 0.518101 0.855319i \(-0.326639\pi\)
0.518101 + 0.855319i \(0.326639\pi\)
\(878\) −3.79899 −0.128210
\(879\) −11.0711 −0.373418
\(880\) 0 0
\(881\) 2.10051 0.0707678 0.0353839 0.999374i \(-0.488735\pi\)
0.0353839 + 0.999374i \(0.488735\pi\)
\(882\) 0 0
\(883\) 18.6274 0.626862 0.313431 0.949611i \(-0.398521\pi\)
0.313431 + 0.949611i \(0.398521\pi\)
\(884\) −5.65685 −0.190261
\(885\) 0 0
\(886\) 28.2843 0.950229
\(887\) −52.2843 −1.75553 −0.877767 0.479088i \(-0.840968\pi\)
−0.877767 + 0.479088i \(0.840968\pi\)
\(888\) 7.65685 0.256947
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 11.5563 0.386935
\(893\) 3.65685 0.122372
\(894\) 20.9706 0.701361
\(895\) 0 0
\(896\) 0 0
\(897\) −5.17157 −0.172674
\(898\) −21.6569 −0.722699
\(899\) −62.6274 −2.08874
\(900\) −5.00000 −0.166667
\(901\) −30.6274 −1.02035
\(902\) −1.65685 −0.0551672
\(903\) 0 0
\(904\) −5.65685 −0.188144
\(905\) 0 0
\(906\) −9.65685 −0.320827
\(907\) 33.9411 1.12700 0.563498 0.826117i \(-0.309455\pi\)
0.563498 + 0.826117i \(0.309455\pi\)
\(908\) 7.55635 0.250766
\(909\) −8.24264 −0.273391
\(910\) 0 0
\(911\) −37.5980 −1.24568 −0.622838 0.782351i \(-0.714021\pi\)
−0.622838 + 0.782351i \(0.714021\pi\)
\(912\) −2.58579 −0.0856239
\(913\) 0.928932 0.0307432
\(914\) −8.34315 −0.275967
\(915\) 0 0
\(916\) 5.65685 0.186908
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) −30.6274 −1.01031 −0.505153 0.863030i \(-0.668564\pi\)
−0.505153 + 0.863030i \(0.668564\pi\)
\(920\) 0 0
\(921\) −24.7279 −0.814813
\(922\) 18.1005 0.596108
\(923\) 3.31371 0.109072
\(924\) 0 0
\(925\) 38.2843 1.25878
\(926\) 4.68629 0.154001
\(927\) 5.89949 0.193765
\(928\) −5.65685 −0.185695
\(929\) 20.9289 0.686656 0.343328 0.939216i \(-0.388446\pi\)
0.343328 + 0.939216i \(0.388446\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 23.6569 0.774906
\(933\) −11.7574 −0.384919
\(934\) 29.9411 0.979704
\(935\) 0 0
\(936\) −1.41421 −0.0462250
\(937\) 5.37258 0.175515 0.0877573 0.996142i \(-0.472030\pi\)
0.0877573 + 0.996142i \(0.472030\pi\)
\(938\) 0 0
\(939\) 3.75736 0.122617
\(940\) 0 0
\(941\) 57.6985 1.88092 0.940458 0.339909i \(-0.110396\pi\)
0.940458 + 0.339909i \(0.110396\pi\)
\(942\) 2.82843 0.0921551
\(943\) −6.05887 −0.197304
\(944\) −1.17157 −0.0381314
\(945\) 0 0
\(946\) 10.0000 0.325128
\(947\) −47.3137 −1.53749 −0.768744 0.639556i \(-0.779118\pi\)
−0.768744 + 0.639556i \(0.779118\pi\)
\(948\) −9.65685 −0.313640
\(949\) −5.65685 −0.183629
\(950\) −12.9289 −0.419470
\(951\) 17.3137 0.561435
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −7.65685 −0.247900
\(955\) 0 0
\(956\) 25.6569 0.829802
\(957\) 5.65685 0.182860
\(958\) −0.970563 −0.0313575
\(959\) 0 0
\(960\) 0 0
\(961\) 91.5685 2.95382
\(962\) 10.8284 0.349123
\(963\) −5.31371 −0.171232
\(964\) −15.3137 −0.493221
\(965\) 0 0
\(966\) 0 0
\(967\) −12.2843 −0.395036 −0.197518 0.980299i \(-0.563288\pi\)
−0.197518 + 0.980299i \(0.563288\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 10.3431 0.332270
\(970\) 0 0
\(971\) 54.9117 1.76220 0.881100 0.472930i \(-0.156804\pi\)
0.881100 + 0.472930i \(0.156804\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 11.6569 0.373510
\(975\) −7.07107 −0.226455
\(976\) −1.41421 −0.0452679
\(977\) 37.6569 1.20475 0.602375 0.798213i \(-0.294221\pi\)
0.602375 + 0.798213i \(0.294221\pi\)
\(978\) 5.65685 0.180886
\(979\) −5.41421 −0.173039
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −20.0000 −0.638226
\(983\) 41.0122 1.30809 0.654043 0.756457i \(-0.273072\pi\)
0.654043 + 0.756457i \(0.273072\pi\)
\(984\) −1.65685 −0.0528186
\(985\) 0 0
\(986\) 22.6274 0.720604
\(987\) 0 0
\(988\) −3.65685 −0.116340
\(989\) 36.5685 1.16281
\(990\) 0 0
\(991\) 20.6863 0.657122 0.328561 0.944483i \(-0.393436\pi\)
0.328561 + 0.944483i \(0.393436\pi\)
\(992\) 11.0711 0.351507
\(993\) −4.97056 −0.157736
\(994\) 0 0
\(995\) 0 0
\(996\) 0.928932 0.0294343
\(997\) 9.89949 0.313520 0.156760 0.987637i \(-0.449895\pi\)
0.156760 + 0.987637i \(0.449895\pi\)
\(998\) −34.6274 −1.09611
\(999\) −7.65685 −0.242252
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 3234.2.a.z.1.2 yes 2
3.2 odd 2 9702.2.a.dl.1.2 2
7.6 odd 2 3234.2.a.y.1.1 2
21.20 even 2 9702.2.a.de.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.y.1.1 2 7.6 odd 2
3234.2.a.z.1.2 yes 2 1.1 even 1 trivial
9702.2.a.de.1.1 2 21.20 even 2
9702.2.a.dl.1.2 2 3.2 odd 2