Properties

Label 7728.2.a.by.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1229.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.841083\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.841083 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.841083 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.45150 q^{11} -5.29258 q^{13} +0.841083 q^{15} -4.13366 q^{17} -2.13366 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.29258 q^{25} +1.00000 q^{27} +4.13366 q^{29} +0.451497 q^{31} +4.45150 q^{33} -0.841083 q^{35} +9.81583 q^{37} -5.29258 q^{39} +1.54850 q^{41} +7.29258 q^{43} +0.841083 q^{45} -11.2208 q^{47} +1.00000 q^{49} -4.13366 q^{51} +12.0619 q^{53} +3.74408 q^{55} -2.13366 q^{57} -5.42624 q^{59} +12.6569 q^{61} -1.00000 q^{63} -4.45150 q^{65} +13.4262 q^{67} -1.00000 q^{69} -0.974745 q^{71} +9.03666 q^{73} -4.29258 q^{75} -4.45150 q^{77} +1.86634 q^{79} +1.00000 q^{81} -4.45150 q^{83} -3.47675 q^{85} +4.13366 q^{87} +10.9747 q^{89} +5.29258 q^{91} +0.451497 q^{93} -1.79459 q^{95} +10.4515 q^{97} +4.45150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 2 q^{29} - 10 q^{31} + 2 q^{33} - q^{35} + 12 q^{37} - 3 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} + 2 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} + 19 q^{61} - 3 q^{63} - 2 q^{65} + 13 q^{67} - 3 q^{69} + 13 q^{71} - 10 q^{73} - 2 q^{77} + 20 q^{79} + 3 q^{81} - 2 q^{83} - 15 q^{85} - 2 q^{87} + 17 q^{89} + 3 q^{91} - 10 q^{93} - 13 q^{95} + 20 q^{97} + 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.841083 0.376144 0.188072 0.982155i \(-0.439776\pi\)
0.188072 + 0.982155i \(0.439776\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.45150 1.34218 0.671088 0.741377i \(-0.265827\pi\)
0.671088 + 0.741377i \(0.265827\pi\)
\(12\) 0 0
\(13\) −5.29258 −1.46790 −0.733949 0.679205i \(-0.762325\pi\)
−0.733949 + 0.679205i \(0.762325\pi\)
\(14\) 0 0
\(15\) 0.841083 0.217167
\(16\) 0 0
\(17\) −4.13366 −1.00256 −0.501280 0.865285i \(-0.667137\pi\)
−0.501280 + 0.865285i \(0.667137\pi\)
\(18\) 0 0
\(19\) −2.13366 −0.489496 −0.244748 0.969587i \(-0.578705\pi\)
−0.244748 + 0.969587i \(0.578705\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.29258 −0.858516
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.13366 0.767602 0.383801 0.923416i \(-0.374615\pi\)
0.383801 + 0.923416i \(0.374615\pi\)
\(30\) 0 0
\(31\) 0.451497 0.0810913 0.0405457 0.999178i \(-0.487090\pi\)
0.0405457 + 0.999178i \(0.487090\pi\)
\(32\) 0 0
\(33\) 4.45150 0.774906
\(34\) 0 0
\(35\) −0.841083 −0.142169
\(36\) 0 0
\(37\) 9.81583 1.61371 0.806856 0.590748i \(-0.201167\pi\)
0.806856 + 0.590748i \(0.201167\pi\)
\(38\) 0 0
\(39\) −5.29258 −0.847491
\(40\) 0 0
\(41\) 1.54850 0.241835 0.120918 0.992663i \(-0.461416\pi\)
0.120918 + 0.992663i \(0.461416\pi\)
\(42\) 0 0
\(43\) 7.29258 1.11211 0.556054 0.831146i \(-0.312315\pi\)
0.556054 + 0.831146i \(0.312315\pi\)
\(44\) 0 0
\(45\) 0.841083 0.125381
\(46\) 0 0
\(47\) −11.2208 −1.63673 −0.818363 0.574702i \(-0.805118\pi\)
−0.818363 + 0.574702i \(0.805118\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.13366 −0.578829
\(52\) 0 0
\(53\) 12.0619 1.65683 0.828416 0.560114i \(-0.189243\pi\)
0.828416 + 0.560114i \(0.189243\pi\)
\(54\) 0 0
\(55\) 3.74408 0.504851
\(56\) 0 0
\(57\) −2.13366 −0.282611
\(58\) 0 0
\(59\) −5.42624 −0.706437 −0.353218 0.935541i \(-0.614913\pi\)
−0.353218 + 0.935541i \(0.614913\pi\)
\(60\) 0 0
\(61\) 12.6569 1.62055 0.810276 0.586049i \(-0.199317\pi\)
0.810276 + 0.586049i \(0.199317\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −4.45150 −0.552140
\(66\) 0 0
\(67\) 13.4262 1.64028 0.820138 0.572165i \(-0.193896\pi\)
0.820138 + 0.572165i \(0.193896\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −0.974745 −0.115681 −0.0578405 0.998326i \(-0.518421\pi\)
−0.0578405 + 0.998326i \(0.518421\pi\)
\(72\) 0 0
\(73\) 9.03666 1.05766 0.528830 0.848728i \(-0.322631\pi\)
0.528830 + 0.848728i \(0.322631\pi\)
\(74\) 0 0
\(75\) −4.29258 −0.495664
\(76\) 0 0
\(77\) −4.45150 −0.507295
\(78\) 0 0
\(79\) 1.86634 0.209979 0.104990 0.994473i \(-0.466519\pi\)
0.104990 + 0.994473i \(0.466519\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.45150 −0.488615 −0.244308 0.969698i \(-0.578561\pi\)
−0.244308 + 0.969698i \(0.578561\pi\)
\(84\) 0 0
\(85\) −3.47675 −0.377107
\(86\) 0 0
\(87\) 4.13366 0.443175
\(88\) 0 0
\(89\) 10.9747 1.16332 0.581660 0.813432i \(-0.302403\pi\)
0.581660 + 0.813432i \(0.302403\pi\)
\(90\) 0 0
\(91\) 5.29258 0.554813
\(92\) 0 0
\(93\) 0.451497 0.0468181
\(94\) 0 0
\(95\) −1.79459 −0.184121
\(96\) 0 0
\(97\) 10.4515 1.06119 0.530594 0.847626i \(-0.321969\pi\)
0.530594 + 0.847626i \(0.321969\pi\)
\(98\) 0 0
\(99\) 4.45150 0.447392
\(100\) 0 0
\(101\) −3.42624 −0.340924 −0.170462 0.985364i \(-0.554526\pi\)
−0.170462 + 0.985364i \(0.554526\pi\)
\(102\) 0 0
\(103\) 3.36433 0.331497 0.165749 0.986168i \(-0.446996\pi\)
0.165749 + 0.986168i \(0.446996\pi\)
\(104\) 0 0
\(105\) −0.841083 −0.0820813
\(106\) 0 0
\(107\) 17.2421 1.66685 0.833427 0.552630i \(-0.186376\pi\)
0.833427 + 0.552630i \(0.186376\pi\)
\(108\) 0 0
\(109\) 9.74408 0.933313 0.466657 0.884439i \(-0.345458\pi\)
0.466657 + 0.884439i \(0.345458\pi\)
\(110\) 0 0
\(111\) 9.81583 0.931677
\(112\) 0 0
\(113\) 9.47675 0.891498 0.445749 0.895158i \(-0.352937\pi\)
0.445749 + 0.895158i \(0.352937\pi\)
\(114\) 0 0
\(115\) −0.841083 −0.0784314
\(116\) 0 0
\(117\) −5.29258 −0.489299
\(118\) 0 0
\(119\) 4.13366 0.378932
\(120\) 0 0
\(121\) 8.81583 0.801439
\(122\) 0 0
\(123\) 1.54850 0.139624
\(124\) 0 0
\(125\) −7.81583 −0.699069
\(126\) 0 0
\(127\) 5.61041 0.497844 0.248922 0.968524i \(-0.419924\pi\)
0.248922 + 0.968524i \(0.419924\pi\)
\(128\) 0 0
\(129\) 7.29258 0.642076
\(130\) 0 0
\(131\) −3.54850 −0.310034 −0.155017 0.987912i \(-0.549543\pi\)
−0.155017 + 0.987912i \(0.549543\pi\)
\(132\) 0 0
\(133\) 2.13366 0.185012
\(134\) 0 0
\(135\) 0.841083 0.0723889
\(136\) 0 0
\(137\) 10.4515 0.892932 0.446466 0.894801i \(-0.352682\pi\)
0.446466 + 0.894801i \(0.352682\pi\)
\(138\) 0 0
\(139\) 2.20541 0.187061 0.0935304 0.995616i \(-0.470185\pi\)
0.0935304 + 0.995616i \(0.470185\pi\)
\(140\) 0 0
\(141\) −11.2208 −0.944964
\(142\) 0 0
\(143\) −23.5599 −1.97018
\(144\) 0 0
\(145\) 3.47675 0.288729
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −4.31783 −0.353731 −0.176865 0.984235i \(-0.556596\pi\)
−0.176865 + 0.984235i \(0.556596\pi\)
\(150\) 0 0
\(151\) −11.2208 −0.913138 −0.456569 0.889688i \(-0.650922\pi\)
−0.456569 + 0.889688i \(0.650922\pi\)
\(152\) 0 0
\(153\) −4.13366 −0.334187
\(154\) 0 0
\(155\) 0.379747 0.0305020
\(156\) 0 0
\(157\) −21.7555 −1.73628 −0.868138 0.496323i \(-0.834683\pi\)
−0.868138 + 0.496323i \(0.834683\pi\)
\(158\) 0 0
\(159\) 12.0619 0.956572
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 3.47675 0.272320 0.136160 0.990687i \(-0.456524\pi\)
0.136160 + 0.990687i \(0.456524\pi\)
\(164\) 0 0
\(165\) 3.74408 0.291476
\(166\) 0 0
\(167\) 4.18417 0.323781 0.161890 0.986809i \(-0.448241\pi\)
0.161890 + 0.986809i \(0.448241\pi\)
\(168\) 0 0
\(169\) 15.0114 1.15472
\(170\) 0 0
\(171\) −2.13366 −0.163165
\(172\) 0 0
\(173\) 1.28118 0.0974061 0.0487031 0.998813i \(-0.484491\pi\)
0.0487031 + 0.998813i \(0.484491\pi\)
\(174\) 0 0
\(175\) 4.29258 0.324489
\(176\) 0 0
\(177\) −5.42624 −0.407861
\(178\) 0 0
\(179\) 17.6936 1.32248 0.661240 0.750175i \(-0.270031\pi\)
0.661240 + 0.750175i \(0.270031\pi\)
\(180\) 0 0
\(181\) −10.0832 −0.749475 −0.374737 0.927131i \(-0.622267\pi\)
−0.374737 + 0.927131i \(0.622267\pi\)
\(182\) 0 0
\(183\) 12.6569 0.935626
\(184\) 0 0
\(185\) 8.25592 0.606988
\(186\) 0 0
\(187\) −18.4010 −1.34561
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −22.5852 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(192\) 0 0
\(193\) −4.58516 −0.330047 −0.165024 0.986290i \(-0.552770\pi\)
−0.165024 + 0.986290i \(0.552770\pi\)
\(194\) 0 0
\(195\) −4.45150 −0.318778
\(196\) 0 0
\(197\) 18.1956 1.29638 0.648191 0.761478i \(-0.275526\pi\)
0.648191 + 0.761478i \(0.275526\pi\)
\(198\) 0 0
\(199\) −1.93809 −0.137387 −0.0686937 0.997638i \(-0.521883\pi\)
−0.0686937 + 0.997638i \(0.521883\pi\)
\(200\) 0 0
\(201\) 13.4262 0.947014
\(202\) 0 0
\(203\) −4.13366 −0.290126
\(204\) 0 0
\(205\) 1.30242 0.0909649
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −9.49799 −0.656990
\(210\) 0 0
\(211\) −10.1337 −0.697630 −0.348815 0.937192i \(-0.613416\pi\)
−0.348815 + 0.937192i \(0.613416\pi\)
\(212\) 0 0
\(213\) −0.974745 −0.0667884
\(214\) 0 0
\(215\) 6.13366 0.418312
\(216\) 0 0
\(217\) −0.451497 −0.0306496
\(218\) 0 0
\(219\) 9.03666 0.610641
\(220\) 0 0
\(221\) 21.8777 1.47166
\(222\) 0 0
\(223\) −5.69357 −0.381269 −0.190635 0.981661i \(-0.561055\pi\)
−0.190635 + 0.981661i \(0.561055\pi\)
\(224\) 0 0
\(225\) −4.29258 −0.286172
\(226\) 0 0
\(227\) −0.112421 −0.00746166 −0.00373083 0.999993i \(-0.501188\pi\)
−0.00373083 + 0.999993i \(0.501188\pi\)
\(228\) 0 0
\(229\) 5.47675 0.361914 0.180957 0.983491i \(-0.442081\pi\)
0.180957 + 0.983491i \(0.442081\pi\)
\(230\) 0 0
\(231\) −4.45150 −0.292887
\(232\) 0 0
\(233\) 9.47675 0.620843 0.310421 0.950599i \(-0.399530\pi\)
0.310421 + 0.950599i \(0.399530\pi\)
\(234\) 0 0
\(235\) −9.43764 −0.615644
\(236\) 0 0
\(237\) 1.86634 0.121232
\(238\) 0 0
\(239\) −12.7906 −0.827353 −0.413677 0.910424i \(-0.635756\pi\)
−0.413677 + 0.910424i \(0.635756\pi\)
\(240\) 0 0
\(241\) −1.81583 −0.116968 −0.0584839 0.998288i \(-0.518627\pi\)
−0.0584839 + 0.998288i \(0.518627\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.841083 0.0537348
\(246\) 0 0
\(247\) 11.2926 0.718530
\(248\) 0 0
\(249\) −4.45150 −0.282102
\(250\) 0 0
\(251\) 13.0465 0.823488 0.411744 0.911300i \(-0.364920\pi\)
0.411744 + 0.911300i \(0.364920\pi\)
\(252\) 0 0
\(253\) −4.45150 −0.279863
\(254\) 0 0
\(255\) −3.47675 −0.217723
\(256\) 0 0
\(257\) −10.1238 −0.631507 −0.315753 0.948841i \(-0.602257\pi\)
−0.315753 + 0.948841i \(0.602257\pi\)
\(258\) 0 0
\(259\) −9.81583 −0.609926
\(260\) 0 0
\(261\) 4.13366 0.255867
\(262\) 0 0
\(263\) −12.8623 −0.793125 −0.396562 0.918008i \(-0.629797\pi\)
−0.396562 + 0.918008i \(0.629797\pi\)
\(264\) 0 0
\(265\) 10.1451 0.623206
\(266\) 0 0
\(267\) 10.9747 0.671644
\(268\) 0 0
\(269\) 2.70742 0.165074 0.0825372 0.996588i \(-0.473698\pi\)
0.0825372 + 0.996588i \(0.473698\pi\)
\(270\) 0 0
\(271\) −28.5753 −1.73583 −0.867914 0.496715i \(-0.834539\pi\)
−0.867914 + 0.496715i \(0.834539\pi\)
\(272\) 0 0
\(273\) 5.29258 0.320322
\(274\) 0 0
\(275\) −19.1084 −1.15228
\(276\) 0 0
\(277\) −21.8272 −1.31147 −0.655736 0.754991i \(-0.727641\pi\)
−0.655736 + 0.754991i \(0.727641\pi\)
\(278\) 0 0
\(279\) 0.451497 0.0270304
\(280\) 0 0
\(281\) 13.2208 0.788689 0.394344 0.918963i \(-0.370972\pi\)
0.394344 + 0.918963i \(0.370972\pi\)
\(282\) 0 0
\(283\) 1.69357 0.100672 0.0503361 0.998732i \(-0.483971\pi\)
0.0503361 + 0.998732i \(0.483971\pi\)
\(284\) 0 0
\(285\) −1.79459 −0.106302
\(286\) 0 0
\(287\) −1.54850 −0.0914052
\(288\) 0 0
\(289\) 0.0871666 0.00512745
\(290\) 0 0
\(291\) 10.4515 0.612678
\(292\) 0 0
\(293\) 31.4376 1.83661 0.918303 0.395877i \(-0.129559\pi\)
0.918303 + 0.395877i \(0.129559\pi\)
\(294\) 0 0
\(295\) −4.56392 −0.265722
\(296\) 0 0
\(297\) 4.45150 0.258302
\(298\) 0 0
\(299\) 5.29258 0.306078
\(300\) 0 0
\(301\) −7.29258 −0.420337
\(302\) 0 0
\(303\) −3.42624 −0.196832
\(304\) 0 0
\(305\) 10.6455 0.609560
\(306\) 0 0
\(307\) −24.3505 −1.38976 −0.694878 0.719128i \(-0.744541\pi\)
−0.694878 + 0.719128i \(0.744541\pi\)
\(308\) 0 0
\(309\) 3.36433 0.191390
\(310\) 0 0
\(311\) −18.0212 −1.02189 −0.510945 0.859613i \(-0.670705\pi\)
−0.510945 + 0.859613i \(0.670705\pi\)
\(312\) 0 0
\(313\) −2.26733 −0.128157 −0.0640784 0.997945i \(-0.520411\pi\)
−0.0640784 + 0.997945i \(0.520411\pi\)
\(314\) 0 0
\(315\) −0.841083 −0.0473896
\(316\) 0 0
\(317\) −0.205413 −0.0115372 −0.00576858 0.999983i \(-0.501836\pi\)
−0.00576858 + 0.999983i \(0.501836\pi\)
\(318\) 0 0
\(319\) 18.4010 1.03026
\(320\) 0 0
\(321\) 17.2421 0.962359
\(322\) 0 0
\(323\) 8.81984 0.490749
\(324\) 0 0
\(325\) 22.7188 1.26021
\(326\) 0 0
\(327\) 9.74408 0.538849
\(328\) 0 0
\(329\) 11.2208 0.618624
\(330\) 0 0
\(331\) 12.9030 0.709213 0.354606 0.935016i \(-0.384615\pi\)
0.354606 + 0.935016i \(0.384615\pi\)
\(332\) 0 0
\(333\) 9.81583 0.537904
\(334\) 0 0
\(335\) 11.2926 0.616980
\(336\) 0 0
\(337\) 29.8272 1.62479 0.812396 0.583106i \(-0.198163\pi\)
0.812396 + 0.583106i \(0.198163\pi\)
\(338\) 0 0
\(339\) 9.47675 0.514707
\(340\) 0 0
\(341\) 2.00984 0.108839
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −0.841083 −0.0452824
\(346\) 0 0
\(347\) 22.2575 1.19484 0.597422 0.801927i \(-0.296192\pi\)
0.597422 + 0.801927i \(0.296192\pi\)
\(348\) 0 0
\(349\) 15.4262 0.825748 0.412874 0.910788i \(-0.364525\pi\)
0.412874 + 0.910788i \(0.364525\pi\)
\(350\) 0 0
\(351\) −5.29258 −0.282497
\(352\) 0 0
\(353\) 29.1605 1.55206 0.776028 0.630699i \(-0.217232\pi\)
0.776028 + 0.630699i \(0.217232\pi\)
\(354\) 0 0
\(355\) −0.819841 −0.0435127
\(356\) 0 0
\(357\) 4.13366 0.218777
\(358\) 0 0
\(359\) −27.3757 −1.44484 −0.722418 0.691457i \(-0.756969\pi\)
−0.722418 + 0.691457i \(0.756969\pi\)
\(360\) 0 0
\(361\) −14.4475 −0.760394
\(362\) 0 0
\(363\) 8.81583 0.462711
\(364\) 0 0
\(365\) 7.60058 0.397832
\(366\) 0 0
\(367\) −2.38959 −0.124735 −0.0623677 0.998053i \(-0.519865\pi\)
−0.0623677 + 0.998053i \(0.519865\pi\)
\(368\) 0 0
\(369\) 1.54850 0.0806118
\(370\) 0 0
\(371\) −12.0619 −0.626223
\(372\) 0 0
\(373\) −8.27716 −0.428575 −0.214288 0.976771i \(-0.568743\pi\)
−0.214288 + 0.976771i \(0.568743\pi\)
\(374\) 0 0
\(375\) −7.81583 −0.403608
\(376\) 0 0
\(377\) −21.8777 −1.12676
\(378\) 0 0
\(379\) 23.8990 1.22761 0.613804 0.789458i \(-0.289638\pi\)
0.613804 + 0.789458i \(0.289638\pi\)
\(380\) 0 0
\(381\) 5.61041 0.287430
\(382\) 0 0
\(383\) −2.91283 −0.148839 −0.0744194 0.997227i \(-0.523710\pi\)
−0.0744194 + 0.997227i \(0.523710\pi\)
\(384\) 0 0
\(385\) −3.74408 −0.190816
\(386\) 0 0
\(387\) 7.29258 0.370703
\(388\) 0 0
\(389\) 18.2070 0.923130 0.461565 0.887106i \(-0.347288\pi\)
0.461565 + 0.887106i \(0.347288\pi\)
\(390\) 0 0
\(391\) 4.13366 0.209048
\(392\) 0 0
\(393\) −3.54850 −0.178998
\(394\) 0 0
\(395\) 1.56974 0.0789824
\(396\) 0 0
\(397\) 5.48815 0.275443 0.137721 0.990471i \(-0.456022\pi\)
0.137721 + 0.990471i \(0.456022\pi\)
\(398\) 0 0
\(399\) 2.13366 0.106817
\(400\) 0 0
\(401\) −34.2070 −1.70821 −0.854107 0.520097i \(-0.825896\pi\)
−0.854107 + 0.520097i \(0.825896\pi\)
\(402\) 0 0
\(403\) −2.38959 −0.119034
\(404\) 0 0
\(405\) 0.841083 0.0417937
\(406\) 0 0
\(407\) 43.6951 2.16589
\(408\) 0 0
\(409\) 14.7188 0.727799 0.363899 0.931438i \(-0.381445\pi\)
0.363899 + 0.931438i \(0.381445\pi\)
\(410\) 0 0
\(411\) 10.4515 0.515534
\(412\) 0 0
\(413\) 5.42624 0.267008
\(414\) 0 0
\(415\) −3.74408 −0.183790
\(416\) 0 0
\(417\) 2.20541 0.108000
\(418\) 0 0
\(419\) 14.8411 0.725034 0.362517 0.931977i \(-0.381917\pi\)
0.362517 + 0.931977i \(0.381917\pi\)
\(420\) 0 0
\(421\) −34.8312 −1.69757 −0.848785 0.528737i \(-0.822666\pi\)
−0.848785 + 0.528737i \(0.822666\pi\)
\(422\) 0 0
\(423\) −11.2208 −0.545575
\(424\) 0 0
\(425\) 17.7441 0.860714
\(426\) 0 0
\(427\) −12.6569 −0.612511
\(428\) 0 0
\(429\) −23.5599 −1.13748
\(430\) 0 0
\(431\) 12.5639 0.605183 0.302591 0.953120i \(-0.402148\pi\)
0.302591 + 0.953120i \(0.402148\pi\)
\(432\) 0 0
\(433\) −22.3911 −1.07605 −0.538025 0.842929i \(-0.680829\pi\)
−0.538025 + 0.842929i \(0.680829\pi\)
\(434\) 0 0
\(435\) 3.47675 0.166697
\(436\) 0 0
\(437\) 2.13366 0.102067
\(438\) 0 0
\(439\) 15.1802 0.724509 0.362255 0.932079i \(-0.382007\pi\)
0.362255 + 0.932079i \(0.382007\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 0.635669 0.0302016 0.0151008 0.999886i \(-0.495193\pi\)
0.0151008 + 0.999886i \(0.495193\pi\)
\(444\) 0 0
\(445\) 9.23067 0.437576
\(446\) 0 0
\(447\) −4.31783 −0.204227
\(448\) 0 0
\(449\) −6.01140 −0.283696 −0.141848 0.989888i \(-0.545304\pi\)
−0.141848 + 0.989888i \(0.545304\pi\)
\(450\) 0 0
\(451\) 6.89316 0.324586
\(452\) 0 0
\(453\) −11.2208 −0.527201
\(454\) 0 0
\(455\) 4.45150 0.208689
\(456\) 0 0
\(457\) −11.3024 −0.528705 −0.264352 0.964426i \(-0.585158\pi\)
−0.264352 + 0.964426i \(0.585158\pi\)
\(458\) 0 0
\(459\) −4.13366 −0.192943
\(460\) 0 0
\(461\) −8.59656 −0.400382 −0.200191 0.979757i \(-0.564156\pi\)
−0.200191 + 0.979757i \(0.564156\pi\)
\(462\) 0 0
\(463\) 31.4278 1.46057 0.730287 0.683140i \(-0.239386\pi\)
0.730287 + 0.683140i \(0.239386\pi\)
\(464\) 0 0
\(465\) 0.379747 0.0176103
\(466\) 0 0
\(467\) 17.0872 0.790700 0.395350 0.918531i \(-0.370623\pi\)
0.395350 + 0.918531i \(0.370623\pi\)
\(468\) 0 0
\(469\) −13.4262 −0.619966
\(470\) 0 0
\(471\) −21.7555 −1.00244
\(472\) 0 0
\(473\) 32.4629 1.49265
\(474\) 0 0
\(475\) 9.15892 0.420240
\(476\) 0 0
\(477\) 12.0619 0.552277
\(478\) 0 0
\(479\) −34.0326 −1.55499 −0.777496 0.628888i \(-0.783510\pi\)
−0.777496 + 0.628888i \(0.783510\pi\)
\(480\) 0 0
\(481\) −51.9511 −2.36876
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 8.79057 0.399159
\(486\) 0 0
\(487\) 20.7188 0.938859 0.469430 0.882970i \(-0.344460\pi\)
0.469430 + 0.882970i \(0.344460\pi\)
\(488\) 0 0
\(489\) 3.47675 0.157224
\(490\) 0 0
\(491\) −23.1491 −1.04470 −0.522352 0.852730i \(-0.674945\pi\)
−0.522352 + 0.852730i \(0.674945\pi\)
\(492\) 0 0
\(493\) −17.0872 −0.769567
\(494\) 0 0
\(495\) 3.74408 0.168284
\(496\) 0 0
\(497\) 0.974745 0.0437233
\(498\) 0 0
\(499\) 9.01542 0.403585 0.201793 0.979428i \(-0.435323\pi\)
0.201793 + 0.979428i \(0.435323\pi\)
\(500\) 0 0
\(501\) 4.18417 0.186935
\(502\) 0 0
\(503\) 33.0579 1.47398 0.736989 0.675904i \(-0.236247\pi\)
0.736989 + 0.675904i \(0.236247\pi\)
\(504\) 0 0
\(505\) −2.88175 −0.128236
\(506\) 0 0
\(507\) 15.0114 0.666680
\(508\) 0 0
\(509\) 13.7752 0.610573 0.305287 0.952261i \(-0.401248\pi\)
0.305287 + 0.952261i \(0.401248\pi\)
\(510\) 0 0
\(511\) −9.03666 −0.399758
\(512\) 0 0
\(513\) −2.13366 −0.0942035
\(514\) 0 0
\(515\) 2.82968 0.124691
\(516\) 0 0
\(517\) −49.9495 −2.19678
\(518\) 0 0
\(519\) 1.28118 0.0562375
\(520\) 0 0
\(521\) 31.1703 1.36560 0.682798 0.730607i \(-0.260763\pi\)
0.682798 + 0.730607i \(0.260763\pi\)
\(522\) 0 0
\(523\) −28.9030 −1.26384 −0.631920 0.775034i \(-0.717733\pi\)
−0.631920 + 0.775034i \(0.717733\pi\)
\(524\) 0 0
\(525\) 4.29258 0.187344
\(526\) 0 0
\(527\) −1.86634 −0.0812989
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −5.42624 −0.235479
\(532\) 0 0
\(533\) −8.19557 −0.354990
\(534\) 0 0
\(535\) 14.5020 0.626976
\(536\) 0 0
\(537\) 17.6936 0.763534
\(538\) 0 0
\(539\) 4.45150 0.191740
\(540\) 0 0
\(541\) −19.0465 −0.818873 −0.409436 0.912339i \(-0.634275\pi\)
−0.409436 + 0.912339i \(0.634275\pi\)
\(542\) 0 0
\(543\) −10.0832 −0.432710
\(544\) 0 0
\(545\) 8.19557 0.351060
\(546\) 0 0
\(547\) −15.2322 −0.651283 −0.325642 0.945493i \(-0.605580\pi\)
−0.325642 + 0.945493i \(0.605580\pi\)
\(548\) 0 0
\(549\) 12.6569 0.540184
\(550\) 0 0
\(551\) −8.81984 −0.375738
\(552\) 0 0
\(553\) −1.86634 −0.0793647
\(554\) 0 0
\(555\) 8.25592 0.350444
\(556\) 0 0
\(557\) −17.1198 −0.725390 −0.362695 0.931908i \(-0.618143\pi\)
−0.362695 + 0.931908i \(0.618143\pi\)
\(558\) 0 0
\(559\) −38.5966 −1.63246
\(560\) 0 0
\(561\) −18.4010 −0.776890
\(562\) 0 0
\(563\) −13.3431 −0.562344 −0.281172 0.959657i \(-0.590723\pi\)
−0.281172 + 0.959657i \(0.590723\pi\)
\(564\) 0 0
\(565\) 7.97073 0.335331
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −12.0733 −0.506140 −0.253070 0.967448i \(-0.581440\pi\)
−0.253070 + 0.967448i \(0.581440\pi\)
\(570\) 0 0
\(571\) −12.8426 −0.537448 −0.268724 0.963217i \(-0.586602\pi\)
−0.268724 + 0.963217i \(0.586602\pi\)
\(572\) 0 0
\(573\) −22.5852 −0.943509
\(574\) 0 0
\(575\) 4.29258 0.179013
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) −4.58516 −0.190553
\(580\) 0 0
\(581\) 4.45150 0.184679
\(582\) 0 0
\(583\) 53.6936 2.22376
\(584\) 0 0
\(585\) −4.45150 −0.184047
\(586\) 0 0
\(587\) −6.10258 −0.251881 −0.125940 0.992038i \(-0.540195\pi\)
−0.125940 + 0.992038i \(0.540195\pi\)
\(588\) 0 0
\(589\) −0.963343 −0.0396939
\(590\) 0 0
\(591\) 18.1956 0.748466
\(592\) 0 0
\(593\) −1.75548 −0.0720889 −0.0360445 0.999350i \(-0.511476\pi\)
−0.0360445 + 0.999350i \(0.511476\pi\)
\(594\) 0 0
\(595\) 3.47675 0.142533
\(596\) 0 0
\(597\) −1.93809 −0.0793207
\(598\) 0 0
\(599\) −13.2421 −0.541056 −0.270528 0.962712i \(-0.587198\pi\)
−0.270528 + 0.962712i \(0.587198\pi\)
\(600\) 0 0
\(601\) 29.3757 1.19826 0.599131 0.800651i \(-0.295513\pi\)
0.599131 + 0.800651i \(0.295513\pi\)
\(602\) 0 0
\(603\) 13.4262 0.546759
\(604\) 0 0
\(605\) 7.41484 0.301456
\(606\) 0 0
\(607\) 27.1491 1.10195 0.550974 0.834523i \(-0.314257\pi\)
0.550974 + 0.834523i \(0.314257\pi\)
\(608\) 0 0
\(609\) −4.13366 −0.167504
\(610\) 0 0
\(611\) 59.3871 2.40255
\(612\) 0 0
\(613\) 10.5753 0.427133 0.213567 0.976929i \(-0.431492\pi\)
0.213567 + 0.976929i \(0.431492\pi\)
\(614\) 0 0
\(615\) 1.30242 0.0525186
\(616\) 0 0
\(617\) 8.14507 0.327908 0.163954 0.986468i \(-0.447575\pi\)
0.163954 + 0.986468i \(0.447575\pi\)
\(618\) 0 0
\(619\) −14.1026 −0.566831 −0.283415 0.958997i \(-0.591467\pi\)
−0.283415 + 0.958997i \(0.591467\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −10.9747 −0.439694
\(624\) 0 0
\(625\) 14.8891 0.595566
\(626\) 0 0
\(627\) −9.49799 −0.379313
\(628\) 0 0
\(629\) −40.5753 −1.61784
\(630\) 0 0
\(631\) −24.9861 −0.994683 −0.497341 0.867555i \(-0.665690\pi\)
−0.497341 + 0.867555i \(0.665690\pi\)
\(632\) 0 0
\(633\) −10.1337 −0.402777
\(634\) 0 0
\(635\) 4.71882 0.187261
\(636\) 0 0
\(637\) −5.29258 −0.209700
\(638\) 0 0
\(639\) −0.974745 −0.0385603
\(640\) 0 0
\(641\) 13.1491 0.519357 0.259679 0.965695i \(-0.416383\pi\)
0.259679 + 0.965695i \(0.416383\pi\)
\(642\) 0 0
\(643\) 23.0677 0.909703 0.454851 0.890567i \(-0.349692\pi\)
0.454851 + 0.890567i \(0.349692\pi\)
\(644\) 0 0
\(645\) 6.13366 0.241513
\(646\) 0 0
\(647\) −3.88758 −0.152836 −0.0764182 0.997076i \(-0.524348\pi\)
−0.0764182 + 0.997076i \(0.524348\pi\)
\(648\) 0 0
\(649\) −24.1549 −0.948163
\(650\) 0 0
\(651\) −0.451497 −0.0176956
\(652\) 0 0
\(653\) 7.56974 0.296227 0.148113 0.988970i \(-0.452680\pi\)
0.148113 + 0.988970i \(0.452680\pi\)
\(654\) 0 0
\(655\) −2.98458 −0.116617
\(656\) 0 0
\(657\) 9.03666 0.352554
\(658\) 0 0
\(659\) −25.2535 −0.983736 −0.491868 0.870670i \(-0.663686\pi\)
−0.491868 + 0.870670i \(0.663686\pi\)
\(660\) 0 0
\(661\) 25.8158 1.00412 0.502060 0.864833i \(-0.332576\pi\)
0.502060 + 0.864833i \(0.332576\pi\)
\(662\) 0 0
\(663\) 21.8777 0.849661
\(664\) 0 0
\(665\) 1.79459 0.0695911
\(666\) 0 0
\(667\) −4.13366 −0.160056
\(668\) 0 0
\(669\) −5.69357 −0.220126
\(670\) 0 0
\(671\) 56.3422 2.17507
\(672\) 0 0
\(673\) −46.9861 −1.81118 −0.905591 0.424151i \(-0.860573\pi\)
−0.905591 + 0.424151i \(0.860573\pi\)
\(674\) 0 0
\(675\) −4.29258 −0.165221
\(676\) 0 0
\(677\) 28.9046 1.11089 0.555446 0.831552i \(-0.312547\pi\)
0.555446 + 0.831552i \(0.312547\pi\)
\(678\) 0 0
\(679\) −10.4515 −0.401092
\(680\) 0 0
\(681\) −0.112421 −0.00430799
\(682\) 0 0
\(683\) 39.4475 1.50942 0.754708 0.656061i \(-0.227779\pi\)
0.754708 + 0.656061i \(0.227779\pi\)
\(684\) 0 0
\(685\) 8.79057 0.335871
\(686\) 0 0
\(687\) 5.47675 0.208951
\(688\) 0 0
\(689\) −63.8386 −2.43206
\(690\) 0 0
\(691\) −44.3194 −1.68599 −0.842995 0.537922i \(-0.819210\pi\)
−0.842995 + 0.537922i \(0.819210\pi\)
\(692\) 0 0
\(693\) −4.45150 −0.169098
\(694\) 0 0
\(695\) 1.85493 0.0703617
\(696\) 0 0
\(697\) −6.40099 −0.242455
\(698\) 0 0
\(699\) 9.47675 0.358444
\(700\) 0 0
\(701\) −19.4262 −0.733719 −0.366860 0.930276i \(-0.619567\pi\)
−0.366860 + 0.930276i \(0.619567\pi\)
\(702\) 0 0
\(703\) −20.9437 −0.789905
\(704\) 0 0
\(705\) −9.43764 −0.355442
\(706\) 0 0
\(707\) 3.42624 0.128857
\(708\) 0 0
\(709\) −47.2421 −1.77421 −0.887107 0.461565i \(-0.847288\pi\)
−0.887107 + 0.461565i \(0.847288\pi\)
\(710\) 0 0
\(711\) 1.86634 0.0699931
\(712\) 0 0
\(713\) −0.451497 −0.0169087
\(714\) 0 0
\(715\) −19.8158 −0.741070
\(716\) 0 0
\(717\) −12.7906 −0.477673
\(718\) 0 0
\(719\) −11.1377 −0.415365 −0.207683 0.978196i \(-0.566592\pi\)
−0.207683 + 0.978196i \(0.566592\pi\)
\(720\) 0 0
\(721\) −3.36433 −0.125294
\(722\) 0 0
\(723\) −1.81583 −0.0675314
\(724\) 0 0
\(725\) −17.7441 −0.658998
\(726\) 0 0
\(727\) 39.2030 1.45396 0.726979 0.686660i \(-0.240924\pi\)
0.726979 + 0.686660i \(0.240924\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.1451 −1.11496
\(732\) 0 0
\(733\) 30.9258 1.14227 0.571135 0.820856i \(-0.306503\pi\)
0.571135 + 0.820856i \(0.306503\pi\)
\(734\) 0 0
\(735\) 0.841083 0.0310238
\(736\) 0 0
\(737\) 59.7669 2.20154
\(738\) 0 0
\(739\) −10.1337 −0.372773 −0.186386 0.982477i \(-0.559678\pi\)
−0.186386 + 0.982477i \(0.559678\pi\)
\(740\) 0 0
\(741\) 11.2926 0.414843
\(742\) 0 0
\(743\) −43.0887 −1.58077 −0.790386 0.612609i \(-0.790120\pi\)
−0.790386 + 0.612609i \(0.790120\pi\)
\(744\) 0 0
\(745\) −3.63166 −0.133054
\(746\) 0 0
\(747\) −4.45150 −0.162872
\(748\) 0 0
\(749\) −17.2421 −0.630012
\(750\) 0 0
\(751\) 46.4531 1.69510 0.847548 0.530719i \(-0.178078\pi\)
0.847548 + 0.530719i \(0.178078\pi\)
\(752\) 0 0
\(753\) 13.0465 0.475441
\(754\) 0 0
\(755\) −9.43764 −0.343471
\(756\) 0 0
\(757\) −21.2812 −0.773478 −0.386739 0.922189i \(-0.626398\pi\)
−0.386739 + 0.922189i \(0.626398\pi\)
\(758\) 0 0
\(759\) −4.45150 −0.161579
\(760\) 0 0
\(761\) 12.9960 0.471104 0.235552 0.971862i \(-0.424310\pi\)
0.235552 + 0.971862i \(0.424310\pi\)
\(762\) 0 0
\(763\) −9.74408 −0.352759
\(764\) 0 0
\(765\) −3.47675 −0.125702
\(766\) 0 0
\(767\) 28.7188 1.03698
\(768\) 0 0
\(769\) 23.5615 0.849648 0.424824 0.905276i \(-0.360336\pi\)
0.424824 + 0.905276i \(0.360336\pi\)
\(770\) 0 0
\(771\) −10.1238 −0.364601
\(772\) 0 0
\(773\) 54.6991 1.96739 0.983696 0.179841i \(-0.0575582\pi\)
0.983696 + 0.179841i \(0.0575582\pi\)
\(774\) 0 0
\(775\) −1.93809 −0.0696182
\(776\) 0 0
\(777\) −9.81583 −0.352141
\(778\) 0 0
\(779\) −3.30398 −0.118377
\(780\) 0 0
\(781\) −4.33908 −0.155264
\(782\) 0 0
\(783\) 4.13366 0.147725
\(784\) 0 0
\(785\) −18.2982 −0.653089
\(786\) 0 0
\(787\) −6.47274 −0.230728 −0.115364 0.993323i \(-0.536803\pi\)
−0.115364 + 0.993323i \(0.536803\pi\)
\(788\) 0 0
\(789\) −12.8623 −0.457911
\(790\) 0 0
\(791\) −9.47675 −0.336955
\(792\) 0 0
\(793\) −66.9877 −2.37880
\(794\) 0 0
\(795\) 10.1451 0.359808
\(796\) 0 0
\(797\) −10.5118 −0.372349 −0.186174 0.982517i \(-0.559609\pi\)
−0.186174 + 0.982517i \(0.559609\pi\)
\(798\) 0 0
\(799\) 46.3831 1.64092
\(800\) 0 0
\(801\) 10.9747 0.387774
\(802\) 0 0
\(803\) 40.2267 1.41957
\(804\) 0 0
\(805\) 0.841083 0.0296443
\(806\) 0 0
\(807\) 2.70742 0.0953057
\(808\) 0 0
\(809\) −30.9976 −1.08982 −0.544908 0.838496i \(-0.683435\pi\)
−0.544908 + 0.838496i \(0.683435\pi\)
\(810\) 0 0
\(811\) −17.2307 −0.605051 −0.302525 0.953141i \(-0.597830\pi\)
−0.302525 + 0.953141i \(0.597830\pi\)
\(812\) 0 0
\(813\) −28.5753 −1.00218
\(814\) 0 0
\(815\) 2.92424 0.102432
\(816\) 0 0
\(817\) −15.5599 −0.544372
\(818\) 0 0
\(819\) 5.29258 0.184938
\(820\) 0 0
\(821\) 10.1238 0.353324 0.176662 0.984272i \(-0.443470\pi\)
0.176662 + 0.984272i \(0.443470\pi\)
\(822\) 0 0
\(823\) 30.5966 1.06653 0.533265 0.845949i \(-0.320965\pi\)
0.533265 + 0.845949i \(0.320965\pi\)
\(824\) 0 0
\(825\) −19.1084 −0.665269
\(826\) 0 0
\(827\) −20.3797 −0.708673 −0.354337 0.935118i \(-0.615293\pi\)
−0.354337 + 0.935118i \(0.615293\pi\)
\(828\) 0 0
\(829\) 19.9088 0.691462 0.345731 0.938334i \(-0.387631\pi\)
0.345731 + 0.938334i \(0.387631\pi\)
\(830\) 0 0
\(831\) −21.8272 −0.757178
\(832\) 0 0
\(833\) −4.13366 −0.143223
\(834\) 0 0
\(835\) 3.51923 0.121788
\(836\) 0 0
\(837\) 0.451497 0.0156060
\(838\) 0 0
\(839\) −18.7807 −0.648383 −0.324191 0.945991i \(-0.605092\pi\)
−0.324191 + 0.945991i \(0.605092\pi\)
\(840\) 0 0
\(841\) −11.9128 −0.410787
\(842\) 0 0
\(843\) 13.2208 0.455350
\(844\) 0 0
\(845\) 12.6258 0.434342
\(846\) 0 0
\(847\) −8.81583 −0.302915
\(848\) 0 0
\(849\) 1.69357 0.0581231
\(850\) 0 0
\(851\) −9.81583 −0.336482
\(852\) 0 0
\(853\) −50.3911 −1.72536 −0.862680 0.505750i \(-0.831216\pi\)
−0.862680 + 0.505750i \(0.831216\pi\)
\(854\) 0 0
\(855\) −1.79459 −0.0613736
\(856\) 0 0
\(857\) 5.50783 0.188144 0.0940720 0.995565i \(-0.470012\pi\)
0.0940720 + 0.995565i \(0.470012\pi\)
\(858\) 0 0
\(859\) 51.3447 1.75186 0.875928 0.482441i \(-0.160250\pi\)
0.875928 + 0.482441i \(0.160250\pi\)
\(860\) 0 0
\(861\) −1.54850 −0.0527728
\(862\) 0 0
\(863\) 28.4108 0.967116 0.483558 0.875312i \(-0.339344\pi\)
0.483558 + 0.875312i \(0.339344\pi\)
\(864\) 0 0
\(865\) 1.07758 0.0366387
\(866\) 0 0
\(867\) 0.0871666 0.00296033
\(868\) 0 0
\(869\) 8.30800 0.281829
\(870\) 0 0
\(871\) −71.0595 −2.40776
\(872\) 0 0
\(873\) 10.4515 0.353730
\(874\) 0 0
\(875\) 7.81583 0.264223
\(876\) 0 0
\(877\) 55.1506 1.86230 0.931152 0.364630i \(-0.118805\pi\)
0.931152 + 0.364630i \(0.118805\pi\)
\(878\) 0 0
\(879\) 31.4376 1.06037
\(880\) 0 0
\(881\) 54.2901 1.82908 0.914540 0.404494i \(-0.132552\pi\)
0.914540 + 0.404494i \(0.132552\pi\)
\(882\) 0 0
\(883\) −48.2184 −1.62268 −0.811339 0.584576i \(-0.801261\pi\)
−0.811339 + 0.584576i \(0.801261\pi\)
\(884\) 0 0
\(885\) −4.56392 −0.153414
\(886\) 0 0
\(887\) 29.2421 0.981853 0.490926 0.871201i \(-0.336658\pi\)
0.490926 + 0.871201i \(0.336658\pi\)
\(888\) 0 0
\(889\) −5.61041 −0.188167
\(890\) 0 0
\(891\) 4.45150 0.149131
\(892\) 0 0
\(893\) 23.9415 0.801171
\(894\) 0 0
\(895\) 14.8818 0.497442
\(896\) 0 0
\(897\) 5.29258 0.176714
\(898\) 0 0
\(899\) 1.86634 0.0622458
\(900\) 0 0
\(901\) −49.8599 −1.66107
\(902\) 0 0
\(903\) −7.29258 −0.242682
\(904\) 0 0
\(905\) −8.48077 −0.281910
\(906\) 0 0
\(907\) 36.5461 1.21349 0.606746 0.794896i \(-0.292475\pi\)
0.606746 + 0.794896i \(0.292475\pi\)
\(908\) 0 0
\(909\) −3.42624 −0.113641
\(910\) 0 0
\(911\) −33.6625 −1.11529 −0.557644 0.830080i \(-0.688295\pi\)
−0.557644 + 0.830080i \(0.688295\pi\)
\(912\) 0 0
\(913\) −19.8158 −0.655808
\(914\) 0 0
\(915\) 10.6455 0.351930
\(916\) 0 0
\(917\) 3.54850 0.117182
\(918\) 0 0
\(919\) −22.8932 −0.755176 −0.377588 0.925974i \(-0.623246\pi\)
−0.377588 + 0.925974i \(0.623246\pi\)
\(920\) 0 0
\(921\) −24.3505 −0.802376
\(922\) 0 0
\(923\) 5.15892 0.169808
\(924\) 0 0
\(925\) −42.1352 −1.38540
\(926\) 0 0
\(927\) 3.36433 0.110499
\(928\) 0 0
\(929\) 58.3212 1.91346 0.956728 0.290982i \(-0.0939821\pi\)
0.956728 + 0.290982i \(0.0939821\pi\)
\(930\) 0 0
\(931\) −2.13366 −0.0699280
\(932\) 0 0
\(933\) −18.0212 −0.589989
\(934\) 0 0
\(935\) −15.4768 −0.506144
\(936\) 0 0
\(937\) 35.4980 1.15967 0.579834 0.814734i \(-0.303117\pi\)
0.579834 + 0.814734i \(0.303117\pi\)
\(938\) 0 0
\(939\) −2.26733 −0.0739914
\(940\) 0 0
\(941\) −54.5150 −1.77714 −0.888569 0.458744i \(-0.848300\pi\)
−0.888569 + 0.458744i \(0.848300\pi\)
\(942\) 0 0
\(943\) −1.54850 −0.0504262
\(944\) 0 0
\(945\) −0.841083 −0.0273604
\(946\) 0 0
\(947\) −41.3366 −1.34326 −0.671630 0.740887i \(-0.734405\pi\)
−0.671630 + 0.740887i \(0.734405\pi\)
\(948\) 0 0
\(949\) −47.8272 −1.55254
\(950\) 0 0
\(951\) −0.205413 −0.00666098
\(952\) 0 0
\(953\) −51.2649 −1.66063 −0.830316 0.557293i \(-0.811840\pi\)
−0.830316 + 0.557293i \(0.811840\pi\)
\(954\) 0 0
\(955\) −18.9960 −0.614696
\(956\) 0 0
\(957\) 18.4010 0.594819
\(958\) 0 0
\(959\) −10.4515 −0.337496
\(960\) 0 0
\(961\) −30.7962 −0.993424
\(962\) 0 0
\(963\) 17.2421 0.555618
\(964\) 0 0
\(965\) −3.85650 −0.124145
\(966\) 0 0
\(967\) 39.6317 1.27447 0.637234 0.770670i \(-0.280078\pi\)
0.637234 + 0.770670i \(0.280078\pi\)
\(968\) 0 0
\(969\) 8.81984 0.283334
\(970\) 0 0
\(971\) 28.3194 0.908813 0.454406 0.890795i \(-0.349851\pi\)
0.454406 + 0.890795i \(0.349851\pi\)
\(972\) 0 0
\(973\) −2.20541 −0.0707023
\(974\) 0 0
\(975\) 22.7188 0.727585
\(976\) 0 0
\(977\) 26.6064 0.851214 0.425607 0.904908i \(-0.360061\pi\)
0.425607 + 0.904908i \(0.360061\pi\)
\(978\) 0 0
\(979\) 48.8540 1.56138
\(980\) 0 0
\(981\) 9.74408 0.311104
\(982\) 0 0
\(983\) 31.7148 1.01155 0.505773 0.862667i \(-0.331207\pi\)
0.505773 + 0.862667i \(0.331207\pi\)
\(984\) 0 0
\(985\) 15.3040 0.487625
\(986\) 0 0
\(987\) 11.2208 0.357163
\(988\) 0 0
\(989\) −7.29258 −0.231891
\(990\) 0 0
\(991\) 3.94793 0.125410 0.0627050 0.998032i \(-0.480027\pi\)
0.0627050 + 0.998032i \(0.480027\pi\)
\(992\) 0 0
\(993\) 12.9030 0.409464
\(994\) 0 0
\(995\) −1.63009 −0.0516774
\(996\) 0 0
\(997\) 2.35048 0.0744404 0.0372202 0.999307i \(-0.488150\pi\)
0.0372202 + 0.999307i \(0.488150\pi\)
\(998\) 0 0
\(999\) 9.81583 0.310559
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 7728.2.a.by.1.2 3
4.3 odd 2 3864.2.a.n.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.n.1.2 3 4.3 odd 2
7728.2.a.by.1.2 3 1.1 even 1 trivial