Properties

Label 966.2.a.d.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +3.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{29} +3.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +3.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} -3.00000 q^{39} -3.00000 q^{40} +5.00000 q^{41} +1.00000 q^{42} -7.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} +1.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} +3.00000 q^{52} +12.0000 q^{53} +1.00000 q^{54} +12.0000 q^{55} -1.00000 q^{56} -1.00000 q^{58} -2.00000 q^{59} -3.00000 q^{60} -6.00000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +9.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} +1.00000 q^{69} -3.00000 q^{70} +10.0000 q^{71} -1.00000 q^{72} +5.00000 q^{74} -4.00000 q^{75} +4.00000 q^{77} +3.00000 q^{78} +4.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} +7.00000 q^{86} -1.00000 q^{87} -4.00000 q^{88} +10.0000 q^{89} -3.00000 q^{90} +3.00000 q^{91} -1.00000 q^{92} +2.00000 q^{93} +3.00000 q^{94} +1.00000 q^{96} +19.0000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.00000 0.670820
\(21\) −1.00000 −0.218218
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 3.00000 0.547723
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) −3.00000 −0.474342
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 1.00000 0.154303
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 4.00000 0.603023
\(45\) 3.00000 0.447214
\(46\) 1.00000 0.147442
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 1.00000 0.136083
\(55\) 12.0000 1.61808
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) −3.00000 −0.387298
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 9.00000 1.11631
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) −3.00000 −0.358569
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 5.00000 0.581238
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 3.00000 0.339683
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −5.00000 −0.552158
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 7.00000 0.754829
\(87\) −1.00000 −0.107211
\(88\) −4.00000 −0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −3.00000 −0.316228
\(91\) 3.00000 0.314485
\(92\) −1.00000 −0.104257
\(93\) 2.00000 0.207390
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 19.0000 1.92916 0.964579 0.263795i \(-0.0849741\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.00000 0.402015
\(100\) 4.00000 0.400000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) −3.00000 −0.294174
\(105\) −3.00000 −0.292770
\(106\) −12.0000 −1.16554
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) −12.0000 −1.14416
\(111\) 5.00000 0.474579
\(112\) 1.00000 0.0944911
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 1.00000 0.0928477
\(117\) 3.00000 0.277350
\(118\) 2.00000 0.184115
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) −5.00000 −0.450835
\(124\) −2.00000 −0.179605
\(125\) −3.00000 −0.268328
\(126\) −1.00000 −0.0890871
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.00000 0.616316
\(130\) −9.00000 −0.789352
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 3.00000 0.253546
\(141\) 3.00000 0.252646
\(142\) −10.0000 −0.839181
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) −5.00000 −0.410997
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 4.00000 0.326599
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) −6.00000 −0.481932
\(156\) −3.00000 −0.240192
\(157\) −24.0000 −1.91541 −0.957704 0.287754i \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) −4.00000 −0.318223
\(159\) −12.0000 −0.951662
\(160\) −3.00000 −0.237171
\(161\) −1.00000 −0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 5.00000 0.390434
\(165\) −12.0000 −0.934199
\(166\) −4.00000 −0.310460
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) −7.00000 −0.533745
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 1.00000 0.0758098
\(175\) 4.00000 0.302372
\(176\) 4.00000 0.301511
\(177\) 2.00000 0.150329
\(178\) −10.0000 −0.749532
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 3.00000 0.223607
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) −3.00000 −0.222375
\(183\) 6.00000 0.443533
\(184\) 1.00000 0.0737210
\(185\) −15.0000 −1.10282
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) −3.00000 −0.218797
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) −19.0000 −1.36412
\(195\) −9.00000 −0.644503
\(196\) 1.00000 0.0714286
\(197\) −21.0000 −1.49619 −0.748094 0.663593i \(-0.769031\pi\)
−0.748094 + 0.663593i \(0.769031\pi\)
\(198\) −4.00000 −0.284268
\(199\) −23.0000 −1.63043 −0.815213 0.579161i \(-0.803380\pi\)
−0.815213 + 0.579161i \(0.803380\pi\)
\(200\) −4.00000 −0.282843
\(201\) 12.0000 0.846415
\(202\) −14.0000 −0.985037
\(203\) 1.00000 0.0701862
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) 1.00000 0.0696733
\(207\) −1.00000 −0.0695048
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) 3.00000 0.207020
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 12.0000 0.824163
\(213\) −10.0000 −0.685189
\(214\) −4.00000 −0.273434
\(215\) −21.0000 −1.43219
\(216\) 1.00000 0.0680414
\(217\) −2.00000 −0.135769
\(218\) 1.00000 0.0677285
\(219\) 0 0
\(220\) 12.0000 0.809040
\(221\) 0 0
\(222\) −5.00000 −0.335578
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.00000 0.266667
\(226\) −7.00000 −0.465633
\(227\) −5.00000 −0.331862 −0.165931 0.986137i \(-0.553063\pi\)
−0.165931 + 0.986137i \(0.553063\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 3.00000 0.197814
\(231\) −4.00000 −0.263181
\(232\) −1.00000 −0.0656532
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −3.00000 −0.196116
\(235\) −9.00000 −0.587095
\(236\) −2.00000 −0.130189
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −3.00000 −0.193649
\(241\) 9.00000 0.579741 0.289870 0.957066i \(-0.406388\pi\)
0.289870 + 0.957066i \(0.406388\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 3.00000 0.191663
\(246\) 5.00000 0.318788
\(247\) 0 0
\(248\) 2.00000 0.127000
\(249\) −4.00000 −0.253490
\(250\) 3.00000 0.189737
\(251\) 19.0000 1.19927 0.599635 0.800274i \(-0.295313\pi\)
0.599635 + 0.800274i \(0.295313\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −7.00000 −0.435801
\(259\) −5.00000 −0.310685
\(260\) 9.00000 0.558156
\(261\) 1.00000 0.0618984
\(262\) 18.0000 1.11204
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 4.00000 0.246183
\(265\) 36.0000 2.21146
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 3.00000 0.182574
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 17.0000 1.02701
\(275\) 16.0000 0.964836
\(276\) 1.00000 0.0601929
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) −7.00000 −0.419832
\(279\) −2.00000 −0.119737
\(280\) −3.00000 −0.179284
\(281\) 1.00000 0.0596550 0.0298275 0.999555i \(-0.490504\pi\)
0.0298275 + 0.999555i \(0.490504\pi\)
\(282\) −3.00000 −0.178647
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 5.00000 0.295141
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −3.00000 −0.176166
\(291\) −19.0000 −1.11380
\(292\) 0 0
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 1.00000 0.0583212
\(295\) −6.00000 −0.349334
\(296\) 5.00000 0.290619
\(297\) −4.00000 −0.232104
\(298\) −12.0000 −0.695141
\(299\) −3.00000 −0.173494
\(300\) −4.00000 −0.230940
\(301\) −7.00000 −0.403473
\(302\) 17.0000 0.978240
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) −18.0000 −1.03068
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 4.00000 0.227921
\(309\) 1.00000 0.0568880
\(310\) 6.00000 0.340777
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 3.00000 0.169842
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 24.0000 1.35440
\(315\) 3.00000 0.169031
\(316\) 4.00000 0.225018
\(317\) −33.0000 −1.85346 −0.926732 0.375722i \(-0.877395\pi\)
−0.926732 + 0.375722i \(0.877395\pi\)
\(318\) 12.0000 0.672927
\(319\) 4.00000 0.223957
\(320\) 3.00000 0.167705
\(321\) −4.00000 −0.223258
\(322\) 1.00000 0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 12.0000 0.665640
\(326\) 12.0000 0.664619
\(327\) 1.00000 0.0553001
\(328\) −5.00000 −0.276079
\(329\) −3.00000 −0.165395
\(330\) 12.0000 0.660578
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) −5.00000 −0.273998
\(334\) 12.0000 0.656611
\(335\) −36.0000 −1.96689
\(336\) −1.00000 −0.0545545
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 4.00000 0.217571
\(339\) −7.00000 −0.380188
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 7.00000 0.377415
\(345\) 3.00000 0.161515
\(346\) −2.00000 −0.107521
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −4.00000 −0.213809
\(351\) −3.00000 −0.160128
\(352\) −4.00000 −0.213201
\(353\) −17.0000 −0.904819 −0.452409 0.891810i \(-0.649435\pi\)
−0.452409 + 0.891810i \(0.649435\pi\)
\(354\) −2.00000 −0.106299
\(355\) 30.0000 1.59223
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) −3.00000 −0.158114
\(361\) −19.0000 −1.00000
\(362\) −18.0000 −0.946059
\(363\) −5.00000 −0.262432
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 21.0000 1.09619 0.548096 0.836416i \(-0.315353\pi\)
0.548096 + 0.836416i \(0.315353\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 5.00000 0.260290
\(370\) 15.0000 0.779813
\(371\) 12.0000 0.623009
\(372\) 2.00000 0.103695
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 3.00000 0.154713
\(377\) 3.00000 0.154508
\(378\) 1.00000 0.0514344
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 20.0000 1.02329
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 12.0000 0.611577
\(386\) 1.00000 0.0508987
\(387\) −7.00000 −0.355830
\(388\) 19.0000 0.964579
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 9.00000 0.455733
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 18.0000 0.907980
\(394\) 21.0000 1.05796
\(395\) 12.0000 0.603786
\(396\) 4.00000 0.201008
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 23.0000 1.15289
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −12.0000 −0.598506
\(403\) −6.00000 −0.298881
\(404\) 14.0000 0.696526
\(405\) 3.00000 0.149071
\(406\) −1.00000 −0.0496292
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) −15.0000 −0.740797
\(411\) 17.0000 0.838548
\(412\) −1.00000 −0.0492665
\(413\) −2.00000 −0.0984136
\(414\) 1.00000 0.0491473
\(415\) 12.0000 0.589057
\(416\) −3.00000 −0.147087
\(417\) −7.00000 −0.342791
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) −3.00000 −0.146385
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) −6.00000 −0.290360
\(428\) 4.00000 0.193347
\(429\) −12.0000 −0.579365
\(430\) 21.0000 1.01271
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −27.0000 −1.29754 −0.648769 0.760986i \(-0.724716\pi\)
−0.648769 + 0.760986i \(0.724716\pi\)
\(434\) 2.00000 0.0960031
\(435\) −3.00000 −0.143839
\(436\) −1.00000 −0.0478913
\(437\) 0 0
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) −12.0000 −0.572078
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −29.0000 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) 5.00000 0.237289
\(445\) 30.0000 1.42214
\(446\) −2.00000 −0.0947027
\(447\) −12.0000 −0.567581
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −4.00000 −0.188562
\(451\) 20.0000 0.941763
\(452\) 7.00000 0.329252
\(453\) 17.0000 0.798730
\(454\) 5.00000 0.234662
\(455\) 9.00000 0.421927
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 4.00000 0.186097
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) 1.00000 0.0464238
\(465\) 6.00000 0.278243
\(466\) 10.0000 0.463241
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 3.00000 0.138675
\(469\) −12.0000 −0.554109
\(470\) 9.00000 0.415139
\(471\) 24.0000 1.10586
\(472\) 2.00000 0.0920575
\(473\) −28.0000 −1.28744
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) −24.0000 −1.09773
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 3.00000 0.136931
\(481\) −15.0000 −0.683941
\(482\) −9.00000 −0.409939
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) 57.0000 2.58824
\(486\) 1.00000 0.0453609
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) 6.00000 0.271607
\(489\) 12.0000 0.542659
\(490\) −3.00000 −0.135526
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −5.00000 −0.225417
\(493\) 0 0
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) −2.00000 −0.0898027
\(497\) 10.0000 0.448561
\(498\) 4.00000 0.179244
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) −3.00000 −0.134164
\(501\) 12.0000 0.536120
\(502\) −19.0000 −0.848012
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 42.0000 1.86898
\(506\) 4.00000 0.177822
\(507\) 4.00000 0.177646
\(508\) 7.00000 0.310575
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −3.00000 −0.132196
\(516\) 7.00000 0.308158
\(517\) −12.0000 −0.527759
\(518\) 5.00000 0.219687
\(519\) −2.00000 −0.0877903
\(520\) −9.00000 −0.394676
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −18.0000 −0.786334
\(525\) −4.00000 −0.174574
\(526\) 21.0000 0.915644
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) −36.0000 −1.56374
\(531\) −2.00000 −0.0867926
\(532\) 0 0
\(533\) 15.0000 0.649722
\(534\) 10.0000 0.432742
\(535\) 12.0000 0.518805
\(536\) 12.0000 0.518321
\(537\) −9.00000 −0.388379
\(538\) 2.00000 0.0862261
\(539\) 4.00000 0.172292
\(540\) −3.00000 −0.129099
\(541\) −24.0000 −1.03184 −0.515920 0.856637i \(-0.672550\pi\)
−0.515920 + 0.856637i \(0.672550\pi\)
\(542\) 12.0000 0.515444
\(543\) −18.0000 −0.772454
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 3.00000 0.128388
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −17.0000 −0.726204
\(549\) −6.00000 −0.256074
\(550\) −16.0000 −0.682242
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 4.00000 0.170097
\(554\) 24.0000 1.01966
\(555\) 15.0000 0.636715
\(556\) 7.00000 0.296866
\(557\) 36.0000 1.52537 0.762684 0.646771i \(-0.223881\pi\)
0.762684 + 0.646771i \(0.223881\pi\)
\(558\) 2.00000 0.0846668
\(559\) −21.0000 −0.888205
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) −1.00000 −0.0421825
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 3.00000 0.126323
\(565\) 21.0000 0.883477
\(566\) −14.0000 −0.588464
\(567\) 1.00000 0.0419961
\(568\) −10.0000 −0.419591
\(569\) −19.0000 −0.796521 −0.398261 0.917272i \(-0.630386\pi\)
−0.398261 + 0.917272i \(0.630386\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 12.0000 0.501745
\(573\) 20.0000 0.835512
\(574\) −5.00000 −0.208696
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 17.0000 0.707107
\(579\) 1.00000 0.0415586
\(580\) 3.00000 0.124568
\(581\) 4.00000 0.165948
\(582\) 19.0000 0.787575
\(583\) 48.0000 1.98796
\(584\) 0 0
\(585\) 9.00000 0.372104
\(586\) −14.0000 −0.578335
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 6.00000 0.247016
\(591\) 21.0000 0.863825
\(592\) −5.00000 −0.205499
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 23.0000 0.941327
\(598\) 3.00000 0.122679
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 4.00000 0.163299
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 7.00000 0.285299
\(603\) −12.0000 −0.488678
\(604\) −17.0000 −0.691720
\(605\) 15.0000 0.609837
\(606\) 14.0000 0.568711
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 18.0000 0.728799
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) 21.0000 0.848182 0.424091 0.905620i \(-0.360594\pi\)
0.424091 + 0.905620i \(0.360594\pi\)
\(614\) −25.0000 −1.00892
\(615\) −15.0000 −0.604858
\(616\) −4.00000 −0.161165
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) −6.00000 −0.240966
\(621\) 1.00000 0.0401286
\(622\) 4.00000 0.160385
\(623\) 10.0000 0.400642
\(624\) −3.00000 −0.120096
\(625\) −29.0000 −1.16000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) −24.0000 −0.957704
\(629\) 0 0
\(630\) −3.00000 −0.119523
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 33.0000 1.31060
\(635\) 21.0000 0.833360
\(636\) −12.0000 −0.475831
\(637\) 3.00000 0.118864
\(638\) −4.00000 −0.158362
\(639\) 10.0000 0.395594
\(640\) −3.00000 −0.118585
\(641\) 1.00000 0.0394976 0.0197488 0.999805i \(-0.493713\pi\)
0.0197488 + 0.999805i \(0.493713\pi\)
\(642\) 4.00000 0.157867
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 21.0000 0.826874
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.00000 −0.314027
\(650\) −12.0000 −0.470679
\(651\) 2.00000 0.0783862
\(652\) −12.0000 −0.469956
\(653\) 13.0000 0.508729 0.254365 0.967108i \(-0.418134\pi\)
0.254365 + 0.967108i \(0.418134\pi\)
\(654\) −1.00000 −0.0391031
\(655\) −54.0000 −2.10995
\(656\) 5.00000 0.195217
\(657\) 0 0
\(658\) 3.00000 0.116952
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) −12.0000 −0.467099
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) −1.00000 −0.0387202
\(668\) −12.0000 −0.464294
\(669\) −2.00000 −0.0773245
\(670\) 36.0000 1.39080
\(671\) −24.0000 −0.926510
\(672\) 1.00000 0.0385758
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 20.0000 0.770371
\(675\) −4.00000 −0.153960
\(676\) −4.00000 −0.153846
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 7.00000 0.268833
\(679\) 19.0000 0.729153
\(680\) 0 0
\(681\) 5.00000 0.191600
\(682\) 8.00000 0.306336
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −51.0000 −1.94861
\(686\) −1.00000 −0.0381802
\(687\) −16.0000 −0.610438
\(688\) −7.00000 −0.266872
\(689\) 36.0000 1.37149
\(690\) −3.00000 −0.114208
\(691\) 21.0000 0.798878 0.399439 0.916760i \(-0.369205\pi\)
0.399439 + 0.916760i \(0.369205\pi\)
\(692\) 2.00000 0.0760286
\(693\) 4.00000 0.151947
\(694\) 3.00000 0.113878
\(695\) 21.0000 0.796575
\(696\) 1.00000 0.0379049
\(697\) 0 0
\(698\) −26.0000 −0.984115
\(699\) 10.0000 0.378235
\(700\) 4.00000 0.151186
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 3.00000 0.113228
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 9.00000 0.338960
\(706\) 17.0000 0.639803
\(707\) 14.0000 0.526524
\(708\) 2.00000 0.0751646
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −30.0000 −1.12588
\(711\) 4.00000 0.150012
\(712\) −10.0000 −0.374766
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) 9.00000 0.336346
\(717\) −24.0000 −0.896296
\(718\) −9.00000 −0.335877
\(719\) −43.0000 −1.60363 −0.801815 0.597573i \(-0.796132\pi\)
−0.801815 + 0.597573i \(0.796132\pi\)
\(720\) 3.00000 0.111803
\(721\) −1.00000 −0.0372419
\(722\) 19.0000 0.707107
\(723\) −9.00000 −0.334714
\(724\) 18.0000 0.668965
\(725\) 4.00000 0.148556
\(726\) 5.00000 0.185567
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −3.00000 −0.111187
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 6.00000 0.221766
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) −21.0000 −0.775124
\(735\) −3.00000 −0.110657
\(736\) 1.00000 0.0368605
\(737\) −48.0000 −1.76810
\(738\) −5.00000 −0.184053
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) −15.0000 −0.551411
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 36.0000 1.31894
\(746\) 2.00000 0.0732252
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) −3.00000 −0.109545
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) −3.00000 −0.109399
\(753\) −19.0000 −0.692398
\(754\) −3.00000 −0.109254
\(755\) −51.0000 −1.85608
\(756\) −1.00000 −0.0363696
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 5.00000 0.181608
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 7.00000 0.253583
\(763\) −1.00000 −0.0362024
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) −6.00000 −0.216647
\(768\) −1.00000 −0.0360844
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) −12.0000 −0.432450
\(771\) −14.0000 −0.504198
\(772\) −1.00000 −0.0359908
\(773\) 35.0000 1.25886 0.629431 0.777056i \(-0.283288\pi\)
0.629431 + 0.777056i \(0.283288\pi\)
\(774\) 7.00000 0.251610
\(775\) −8.00000 −0.287368
\(776\) −19.0000 −0.682060
\(777\) 5.00000 0.179374
\(778\) 14.0000 0.501924
\(779\) 0 0
\(780\) −9.00000 −0.322252
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 1.00000 0.0357143
\(785\) −72.0000 −2.56979
\(786\) −18.0000 −0.642039
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −21.0000 −0.748094
\(789\) 21.0000 0.747620
\(790\) −12.0000 −0.426941
\(791\) 7.00000 0.248891
\(792\) −4.00000 −0.142134
\(793\) −18.0000 −0.639199
\(794\) −6.00000 −0.212932
\(795\) −36.0000 −1.27679
\(796\) −23.0000 −0.815213
\(797\) 47.0000 1.66483 0.832413 0.554156i \(-0.186959\pi\)
0.832413 + 0.554156i \(0.186959\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) 10.0000 0.353333
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) −3.00000 −0.105736
\(806\) 6.00000 0.211341
\(807\) 2.00000 0.0704033
\(808\) −14.0000 −0.492518
\(809\) 48.0000 1.68759 0.843795 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) −3.00000 −0.105409
\(811\) 23.0000 0.807639 0.403820 0.914839i \(-0.367682\pi\)
0.403820 + 0.914839i \(0.367682\pi\)
\(812\) 1.00000 0.0350931
\(813\) 12.0000 0.420858
\(814\) 20.0000 0.701000
\(815\) −36.0000 −1.26102
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 3.00000 0.104828
\(820\) 15.0000 0.523823
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) −17.0000 −0.592943
\(823\) 31.0000 1.08059 0.540296 0.841475i \(-0.318312\pi\)
0.540296 + 0.841475i \(0.318312\pi\)
\(824\) 1.00000 0.0348367
\(825\) −16.0000 −0.557048
\(826\) 2.00000 0.0695889
\(827\) −42.0000 −1.46048 −0.730242 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −12.0000 −0.416526
\(831\) 24.0000 0.832551
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 7.00000 0.242390
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) −4.00000 −0.138178
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 3.00000 0.103510
\(841\) −28.0000 −0.965517
\(842\) 1.00000 0.0344623
\(843\) −1.00000 −0.0344418
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 3.00000 0.103142
\(847\) 5.00000 0.171802
\(848\) 12.0000 0.412082
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 5.00000 0.171398
\(852\) −10.0000 −0.342594
\(853\) 41.0000 1.40381 0.701907 0.712269i \(-0.252332\pi\)
0.701907 + 0.712269i \(0.252332\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) 12.0000 0.409673
\(859\) −29.0000 −0.989467 −0.494734 0.869045i \(-0.664734\pi\)
−0.494734 + 0.869045i \(0.664734\pi\)
\(860\) −21.0000 −0.716094
\(861\) −5.00000 −0.170400
\(862\) −3.00000 −0.102180
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.00000 0.204006
\(866\) 27.0000 0.917497
\(867\) 17.0000 0.577350
\(868\) −2.00000 −0.0678844
\(869\) 16.0000 0.542763
\(870\) 3.00000 0.101710
\(871\) −36.0000 −1.21981
\(872\) 1.00000 0.0338643
\(873\) 19.0000 0.643053
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) −14.0000 −0.472477
\(879\) −14.0000 −0.472208
\(880\) 12.0000 0.404520
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 6.00000 0.201916 0.100958 0.994891i \(-0.467809\pi\)
0.100958 + 0.994891i \(0.467809\pi\)
\(884\) 0 0
\(885\) 6.00000 0.201688
\(886\) 29.0000 0.974274
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) −5.00000 −0.167789
\(889\) 7.00000 0.234772
\(890\) −30.0000 −1.00560
\(891\) 4.00000 0.134005
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 12.0000 0.401340
\(895\) 27.0000 0.902510
\(896\) −1.00000 −0.0334077
\(897\) 3.00000 0.100167
\(898\) 18.0000 0.600668
\(899\) −2.00000 −0.0667037
\(900\) 4.00000 0.133333
\(901\) 0 0
\(902\) −20.0000 −0.665927
\(903\) 7.00000 0.232945
\(904\) −7.00000 −0.232817
\(905\) 54.0000 1.79502
\(906\) −17.0000 −0.564787
\(907\) −49.0000 −1.62702 −0.813509 0.581552i \(-0.802446\pi\)
−0.813509 + 0.581552i \(0.802446\pi\)
\(908\) −5.00000 −0.165931
\(909\) 14.0000 0.464351
\(910\) −9.00000 −0.298347
\(911\) −55.0000 −1.82223 −0.911116 0.412151i \(-0.864778\pi\)
−0.911116 + 0.412151i \(0.864778\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 4.00000 0.132308
\(915\) 18.0000 0.595062
\(916\) 16.0000 0.528655
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 60.0000 1.97922 0.989609 0.143787i \(-0.0459280\pi\)
0.989609 + 0.143787i \(0.0459280\pi\)
\(920\) 3.00000 0.0989071
\(921\) −25.0000 −0.823778
\(922\) 30.0000 0.987997
\(923\) 30.0000 0.987462
\(924\) −4.00000 −0.131590
\(925\) −20.0000 −0.657596
\(926\) −5.00000 −0.164310
\(927\) −1.00000 −0.0328443
\(928\) −1.00000 −0.0328266
\(929\) 21.0000 0.688988 0.344494 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) −6.00000 −0.196748
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 4.00000 0.130954
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) −53.0000 −1.73143 −0.865717 0.500533i \(-0.833137\pi\)
−0.865717 + 0.500533i \(0.833137\pi\)
\(938\) 12.0000 0.391814
\(939\) −14.0000 −0.456873
\(940\) −9.00000 −0.293548
\(941\) −21.0000 −0.684580 −0.342290 0.939594i \(-0.611203\pi\)
−0.342290 + 0.939594i \(0.611203\pi\)
\(942\) −24.0000 −0.781962
\(943\) −5.00000 −0.162822
\(944\) −2.00000 −0.0650945
\(945\) −3.00000 −0.0975900
\(946\) 28.0000 0.910359
\(947\) 3.00000 0.0974869 0.0487435 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) 0 0
\(951\) 33.0000 1.07010
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −12.0000 −0.388514
\(955\) −60.0000 −1.94155
\(956\) 24.0000 0.776215
\(957\) −4.00000 −0.129302
\(958\) 6.00000 0.193851
\(959\) −17.0000 −0.548959
\(960\) −3.00000 −0.0968246
\(961\) −27.0000 −0.870968
\(962\) 15.0000 0.483619
\(963\) 4.00000 0.128898
\(964\) 9.00000 0.289870
\(965\) −3.00000 −0.0965734
\(966\) −1.00000 −0.0321745
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) −57.0000 −1.83016
\(971\) −40.0000 −1.28366 −0.641831 0.766846i \(-0.721825\pi\)
−0.641831 + 0.766846i \(0.721825\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 7.00000 0.224410
\(974\) −13.0000 −0.416547
\(975\) −12.0000 −0.384308
\(976\) −6.00000 −0.192055
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) −12.0000 −0.383718
\(979\) 40.0000 1.27841
\(980\) 3.00000 0.0958315
\(981\) −1.00000 −0.0319275
\(982\) −28.0000 −0.893516
\(983\) 2.00000 0.0637901 0.0318950 0.999491i \(-0.489846\pi\)
0.0318950 + 0.999491i \(0.489846\pi\)
\(984\) 5.00000 0.159394
\(985\) −63.0000 −2.00735
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) 7.00000 0.222587
\(990\) −12.0000 −0.381385
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 2.00000 0.0635001
\(993\) −4.00000 −0.126936
\(994\) −10.0000 −0.317181
\(995\) −69.0000 −2.18745
\(996\) −4.00000 −0.126745
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 10.0000 0.316544
\(999\) 5.00000 0.158193
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 966.2.a.d.1.1 1
3.2 odd 2 2898.2.a.k.1.1 1
4.3 odd 2 7728.2.a.u.1.1 1
7.6 odd 2 6762.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.d.1.1 1 1.1 even 1 trivial
2898.2.a.k.1.1 1 3.2 odd 2
6762.2.a.l.1.1 1 7.6 odd 2
7728.2.a.u.1.1 1 4.3 odd 2