Properties

Label 5054.2.a.a.1.1
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +5.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{18} +1.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} -4.00000 q^{25} -5.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{35} -3.00000 q^{36} +4.00000 q^{37} -1.00000 q^{40} +2.00000 q^{41} -8.00000 q^{43} -2.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} +1.00000 q^{49} +4.00000 q^{50} +5.00000 q^{52} +2.00000 q^{53} -2.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} +7.00000 q^{59} -7.00000 q^{61} +4.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} -12.0000 q^{67} -1.00000 q^{70} +15.0000 q^{71} +3.00000 q^{72} -14.0000 q^{73} -4.00000 q^{74} -2.00000 q^{77} +4.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} -7.00000 q^{83} +8.00000 q^{86} +2.00000 q^{88} +3.00000 q^{90} +5.00000 q^{91} +1.00000 q^{92} -12.0000 q^{97} -1.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 3.00000 0.707107
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −3.00000 −0.500000
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −2.00000 −0.301511
\(45\) −3.00000 −0.447214
\(46\) −1.00000 −0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 7.00000 0.911322 0.455661 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 4.00000 0.508001
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 3.00000 0.353553
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 3.00000 0.316228
\(91\) 5.00000 0.524142
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −1.00000 −0.101015
\(99\) 6.00000 0.603023
\(100\) −4.00000 −0.400000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 5.00000 0.470360 0.235180 0.971952i \(-0.424432\pi\)
0.235180 + 0.971952i \(0.424432\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) −6.00000 −0.557086
\(117\) −15.0000 −1.38675
\(118\) −7.00000 −0.644402
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 7.00000 0.633750
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −9.00000 −0.804984
\(126\) 3.00000 0.267261
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −5.00000 −0.438529
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −15.0000 −1.25877
\(143\) −10.0000 −0.836242
\(144\) −3.00000 −0.250000
\(145\) −6.00000 −0.498273
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) −9.00000 −0.707107
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 13.0000 0.988372 0.494186 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −3.00000 −0.223607
\(181\) −23.0000 −1.70958 −0.854788 0.518977i \(-0.826313\pi\)
−0.854788 + 0.518977i \(0.826313\pi\)
\(182\) −5.00000 −0.370625
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.0000 1.23008 0.615038 0.788497i \(-0.289140\pi\)
0.615038 + 0.788497i \(0.289140\pi\)
\(192\) 0 0
\(193\) 3.00000 0.215945 0.107972 0.994154i \(-0.465564\pi\)
0.107972 + 0.994154i \(0.465564\pi\)
\(194\) 12.0000 0.861550
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −6.00000 −0.426401
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 6.00000 0.418040
\(207\) −3.00000 −0.208514
\(208\) 5.00000 0.346688
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 12.0000 0.800000
\(226\) −5.00000 −0.332595
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) 27.0000 1.78421 0.892105 0.451828i \(-0.149228\pi\)
0.892105 + 0.451828i \(0.149228\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −11.0000 −0.720634 −0.360317 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) 15.0000 0.980581
\(235\) 0 0
\(236\) 7.00000 0.455661
\(237\) 0 0
\(238\) 0 0
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) 9.00000 0.568075 0.284037 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) −3.00000 −0.188982
\(253\) −2.00000 −0.125739
\(254\) 13.0000 0.815693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 5.00000 0.310087
\(261\) 18.0000 1.11417
\(262\) −15.0000 −0.926703
\(263\) −3.00000 −0.184988 −0.0924940 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 12.0000 0.719712
\(279\) 12.0000 0.718421
\(280\) −1.00000 −0.0597614
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) −7.00000 −0.416107 −0.208053 0.978117i \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 15.0000 0.890086
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) 2.00000 0.118056
\(288\) 3.00000 0.176777
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 7.00000 0.407556
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 19.0000 1.09333
\(303\) 0 0
\(304\) 0 0
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) −19.0000 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) −7.00000 −0.395033
\(315\) −3.00000 −0.169031
\(316\) 4.00000 0.225018
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) −20.0000 −1.10940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 18.0000 0.989369 0.494685 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) −7.00000 −0.384175
\(333\) −12.0000 −0.657596
\(334\) 2.00000 0.109435
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −13.0000 −0.698884
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 15.0000 0.796117
\(356\) 0 0
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 3.00000 0.158114
\(361\) 0 0
\(362\) 23.0000 1.20885
\(363\) 0 0
\(364\) 5.00000 0.262071
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) −4.00000 −0.207950
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −30.0000 −1.54508
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −17.0000 −0.869796
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) −3.00000 −0.152696
\(387\) 24.0000 1.21999
\(388\) −12.0000 −0.609208
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 4.00000 0.201262
\(396\) 6.00000 0.301511
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −18.0000 −0.895533
\(405\) 9.00000 0.447214
\(406\) 6.00000 0.297775
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −6.00000 −0.295599
\(413\) 7.00000 0.344447
\(414\) 3.00000 0.147442
\(415\) −7.00000 −0.343616
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) −7.00000 −0.338754
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 2.00000 0.0953463
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −37.0000 −1.74614 −0.873069 0.487597i \(-0.837874\pi\)
−0.873069 + 0.487597i \(0.837874\pi\)
\(450\) −12.0000 −0.565685
\(451\) −4.00000 −0.188353
\(452\) 5.00000 0.235180
\(453\) 0 0
\(454\) −3.00000 −0.140797
\(455\) 5.00000 0.234404
\(456\) 0 0
\(457\) −39.0000 −1.82434 −0.912172 0.409809i \(-0.865595\pi\)
−0.912172 + 0.409809i \(0.865595\pi\)
\(458\) −27.0000 −1.26163
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) −37.0000 −1.72326 −0.861631 0.507535i \(-0.830557\pi\)
−0.861631 + 0.507535i \(0.830557\pi\)
\(462\) 0 0
\(463\) −5.00000 −0.232370 −0.116185 0.993228i \(-0.537067\pi\)
−0.116185 + 0.993228i \(0.537067\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 11.0000 0.509565
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −15.0000 −0.693375
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) −7.00000 −0.322201
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 5.00000 0.228695
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 7.00000 0.316875
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) −4.00000 −0.179605
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) −9.00000 −0.401690
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 3.00000 0.133631
\(505\) −18.0000 −0.800989
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) −13.0000 −0.576782
\(509\) −35.0000 −1.55135 −0.775674 0.631134i \(-0.782590\pi\)
−0.775674 + 0.631134i \(0.782590\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) −6.00000 −0.264392
\(516\) 0 0
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) −5.00000 −0.219265
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) −18.0000 −0.787839
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) 3.00000 0.130806
\(527\) 0 0
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −2.00000 −0.0868744
\(531\) −21.0000 −0.911322
\(532\) 0 0
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 14.0000 0.603583
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 36.0000 1.54776 0.773880 0.633332i \(-0.218313\pi\)
0.773880 + 0.633332i \(0.218313\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −9.00000 −0.384461
\(549\) 21.0000 0.896258
\(550\) −8.00000 −0.341121
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −12.0000 −0.508001
\(559\) −40.0000 −1.69182
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) 5.00000 0.210352
\(566\) 7.00000 0.294232
\(567\) 9.00000 0.377964
\(568\) −15.0000 −0.629386
\(569\) −31.0000 −1.29959 −0.649794 0.760111i \(-0.725145\pi\)
−0.649794 + 0.760111i \(0.725145\pi\)
\(570\) 0 0
\(571\) −30.0000 −1.25546 −0.627730 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) −10.0000 −0.418121
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) −4.00000 −0.166812
\(576\) −3.00000 −0.125000
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) −7.00000 −0.290409
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 14.0000 0.579324
\(585\) −15.0000 −0.620174
\(586\) 21.0000 0.867502
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −7.00000 −0.288185
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) −5.00000 −0.204465
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 8.00000 0.326056
\(603\) 36.0000 1.46603
\(604\) −19.0000 −0.773099
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 7.00000 0.283422
\(611\) 0 0
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 19.0000 0.766778
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 13.0000 0.523360 0.261680 0.965155i \(-0.415723\pi\)
0.261680 + 0.965155i \(0.415723\pi\)
\(618\) 0 0
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) 7.00000 0.279330
\(629\) 0 0
\(630\) 3.00000 0.119523
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) −13.0000 −0.515889
\(636\) 0 0
\(637\) 5.00000 0.198107
\(638\) −12.0000 −0.475085
\(639\) −45.0000 −1.78017
\(640\) −1.00000 −0.0395285
\(641\) −39.0000 −1.54041 −0.770204 0.637798i \(-0.779845\pi\)
−0.770204 + 0.637798i \(0.779845\pi\)
\(642\) 0 0
\(643\) −9.00000 −0.354925 −0.177463 0.984128i \(-0.556789\pi\)
−0.177463 + 0.984128i \(0.556789\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −9.00000 −0.353553
\(649\) −14.0000 −0.549548
\(650\) 20.0000 0.784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) 15.0000 0.586098
\(656\) 2.00000 0.0780869
\(657\) 42.0000 1.63858
\(658\) 0 0
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) −18.0000 −0.699590
\(663\) 0 0
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) −6.00000 −0.232321
\(668\) −2.00000 −0.0773823
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) −23.0000 −0.886585 −0.443292 0.896377i \(-0.646190\pi\)
−0.443292 + 0.896377i \(0.646190\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −29.0000 −1.10321 −0.551606 0.834105i \(-0.685985\pi\)
−0.551606 + 0.834105i \(0.685985\pi\)
\(692\) 13.0000 0.494186
\(693\) 6.00000 0.227921
\(694\) −18.0000 −0.683271
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) 0 0
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −18.0000 −0.676960
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −15.0000 −0.562940
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) −10.0000 −0.373979
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) −3.00000 −0.111803
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) −23.0000 −0.854788
\(725\) 24.0000 0.891338
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −5.00000 −0.185312
\(729\) −27.0000 −1.00000
\(730\) 14.0000 0.518163
\(731\) 0 0
\(732\) 0 0
\(733\) 13.0000 0.480166 0.240083 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 24.0000 0.884051
\(738\) 6.00000 0.220863
\(739\) −38.0000 −1.39785 −0.698926 0.715194i \(-0.746338\pi\)
−0.698926 + 0.715194i \(0.746338\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 43.0000 1.57752 0.788759 0.614703i \(-0.210724\pi\)
0.788759 + 0.614703i \(0.210724\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) −4.00000 −0.146450
\(747\) 21.0000 0.768350
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 30.0000 1.09254
\(755\) −19.0000 −0.691481
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 17.0000 0.615038
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 35.0000 1.26378
\(768\) 0 0
\(769\) −12.0000 −0.432731 −0.216366 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(770\) 2.00000 0.0720750
\(771\) 0 0
\(772\) 3.00000 0.107972
\(773\) −17.0000 −0.611448 −0.305724 0.952120i \(-0.598898\pi\)
−0.305724 + 0.952120i \(0.598898\pi\)
\(774\) −24.0000 −0.862662
\(775\) 16.0000 0.574737
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) −25.0000 −0.891154 −0.445577 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 5.00000 0.177780
\(792\) −6.00000 −0.213201
\(793\) −35.0000 −1.24289
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −7.00000 −0.247953 −0.123976 0.992285i \(-0.539565\pi\)
−0.123976 + 0.992285i \(0.539565\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −25.0000 −0.882781
\(803\) 28.0000 0.988099
\(804\) 0 0
\(805\) 1.00000 0.0352454
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) 18.0000 0.633238
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) −9.00000 −0.316228
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) −4.00000 −0.139857
\(819\) −15.0000 −0.524142
\(820\) 2.00000 0.0698430
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 0 0
\(823\) 5.00000 0.174289 0.0871445 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −7.00000 −0.243561
\(827\) −6.00000 −0.208640 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) −3.00000 −0.104257
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) 7.00000 0.242974
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) 0 0
\(838\) 28.0000 0.967244
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 7.00000 0.239535
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −14.0000 −0.478231 −0.239115 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 13.0000 0.442013
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −60.0000 −2.03302
\(872\) −10.0000 −0.338643
\(873\) 36.0000 1.21842
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 3.00000 0.101015
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 37.0000 1.23471
\(899\) 24.0000 0.800445
\(900\) 12.0000 0.400000
\(901\) 0 0
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) −5.00000 −0.166298
\(905\) −23.0000 −0.764546
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 3.00000 0.0995585
\(909\) 54.0000 1.79107
\(910\) −5.00000 −0.165748
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 0 0
\(913\) 14.0000 0.463332
\(914\) 39.0000 1.29001
\(915\) 0 0
\(916\) 27.0000 0.892105
\(917\) 15.0000 0.495344
\(918\) 0 0
\(919\) 3.00000 0.0989609 0.0494804 0.998775i \(-0.484243\pi\)
0.0494804 + 0.998775i \(0.484243\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) 37.0000 1.21853
\(923\) 75.0000 2.46866
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) 5.00000 0.164310
\(927\) 18.0000 0.591198
\(928\) 6.00000 0.196960
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −11.0000 −0.360317
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 15.0000 0.490290
\(937\) 58.0000 1.89478 0.947389 0.320085i \(-0.103712\pi\)
0.947389 + 0.320085i \(0.103712\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) −17.0000 −0.554184 −0.277092 0.960843i \(-0.589371\pi\)
−0.277092 + 0.960843i \(0.589371\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) 7.00000 0.227831
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −18.0000 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 0 0
\(949\) −70.0000 −2.27230
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 6.00000 0.194257
\(955\) 17.0000 0.550107
\(956\) −5.00000 −0.161712
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −20.0000 −0.644826
\(963\) −54.0000 −1.74013
\(964\) 4.00000 0.128831
\(965\) 3.00000 0.0965734
\(966\) 0 0
\(967\) −57.0000 −1.83300 −0.916498 0.400039i \(-0.868997\pi\)
−0.916498 + 0.400039i \(0.868997\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 12.0000 0.385297
\(971\) 41.0000 1.31575 0.657876 0.753126i \(-0.271455\pi\)
0.657876 + 0.753126i \(0.271455\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 37.0000 1.18373 0.591867 0.806035i \(-0.298391\pi\)
0.591867 + 0.806035i \(0.298391\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −30.0000 −0.957826
\(982\) −20.0000 −0.638226
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) −6.00000 −0.190693
\(991\) 31.0000 0.984747 0.492374 0.870384i \(-0.336129\pi\)
0.492374 + 0.870384i \(0.336129\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) −15.0000 −0.475771
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 17.0000 0.538395 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) −24.0000 −0.759707
\(999\) 0 0
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 5054.2.a.a.1.1 1
19.8 odd 6 266.2.f.a.197.1 2
19.12 odd 6 266.2.f.a.239.1 yes 2
19.18 odd 2 5054.2.a.b.1.1 1
57.8 even 6 2394.2.o.i.1261.1 2
57.50 even 6 2394.2.o.i.505.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.f.a.197.1 2 19.8 odd 6
266.2.f.a.239.1 yes 2 19.12 odd 6
2394.2.o.i.505.1 2 57.50 even 6
2394.2.o.i.1261.1 2 57.8 even 6
5054.2.a.a.1.1 1 1.1 even 1 trivial
5054.2.a.b.1.1 1 19.18 odd 2