Properties

Label 7728.2.a.bx.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
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 1.3
Root \(2.52892\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.52892 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.52892 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.86651 q^{11} -2.66241 q^{13} +2.52892 q^{15} -4.92434 q^{17} -2.13349 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.39543 q^{25} +1.00000 q^{27} +5.00000 q^{29} -3.86651 q^{31} -1.86651 q^{33} -2.52892 q^{35} +2.05784 q^{37} -2.66241 q^{39} +7.79085 q^{41} -5.72025 q^{43} +2.52892 q^{45} -5.92434 q^{47} +1.00000 q^{49} -4.92434 q^{51} -8.45326 q^{53} -4.72025 q^{55} -2.13349 q^{57} -14.1863 q^{59} -1.26193 q^{61} -1.00000 q^{63} -6.73302 q^{65} +11.2441 q^{67} +1.00000 q^{69} +1.26193 q^{71} +14.7730 q^{73} +1.39543 q^{75} +1.86651 q^{77} -15.9822 q^{79} +1.00000 q^{81} +17.8309 q^{83} -12.4533 q^{85} +5.00000 q^{87} -1.26193 q^{89} +2.66241 q^{91} -3.86651 q^{93} -5.39543 q^{95} -12.3248 q^{97} -1.86651 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(5\) 2.52892 1.13097 0.565483 0.824760i \(-0.308690\pi\)
0.565483 + 0.824760i \(0.308690\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.86651 −0.562773 −0.281387 0.959594i \(-0.590794\pi\)
−0.281387 + 0.959594i \(0.590794\pi\)
\(12\) 0 0
\(13\) −2.66241 −0.738420 −0.369210 0.929346i \(-0.620372\pi\)
−0.369210 + 0.929346i \(0.620372\pi\)
\(14\) 0 0
\(15\) 2.52892 0.652964
\(16\) 0 0
\(17\) −4.92434 −1.19433 −0.597164 0.802119i \(-0.703706\pi\)
−0.597164 + 0.802119i \(0.703706\pi\)
\(18\) 0 0
\(19\) −2.13349 −0.489457 −0.244728 0.969592i \(-0.578699\pi\)
−0.244728 + 0.969592i \(0.578699\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.39543 0.279085
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.86651 −0.694445 −0.347223 0.937783i \(-0.612875\pi\)
−0.347223 + 0.937783i \(0.612875\pi\)
\(32\) 0 0
\(33\) −1.86651 −0.324917
\(34\) 0 0
\(35\) −2.52892 −0.427465
\(36\) 0 0
\(37\) 2.05784 0.338306 0.169153 0.985590i \(-0.445897\pi\)
0.169153 + 0.985590i \(0.445897\pi\)
\(38\) 0 0
\(39\) −2.66241 −0.426327
\(40\) 0 0
\(41\) 7.79085 1.21673 0.608363 0.793659i \(-0.291826\pi\)
0.608363 + 0.793659i \(0.291826\pi\)
\(42\) 0 0
\(43\) −5.72025 −0.872329 −0.436165 0.899867i \(-0.643663\pi\)
−0.436165 + 0.899867i \(0.643663\pi\)
\(44\) 0 0
\(45\) 2.52892 0.376989
\(46\) 0 0
\(47\) −5.92434 −0.864154 −0.432077 0.901837i \(-0.642219\pi\)
−0.432077 + 0.901837i \(0.642219\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.92434 −0.689546
\(52\) 0 0
\(53\) −8.45326 −1.16114 −0.580572 0.814209i \(-0.697171\pi\)
−0.580572 + 0.814209i \(0.697171\pi\)
\(54\) 0 0
\(55\) −4.72025 −0.636478
\(56\) 0 0
\(57\) −2.13349 −0.282588
\(58\) 0 0
\(59\) −14.1863 −1.84690 −0.923448 0.383723i \(-0.874642\pi\)
−0.923448 + 0.383723i \(0.874642\pi\)
\(60\) 0 0
\(61\) −1.26193 −0.161574 −0.0807871 0.996731i \(-0.525743\pi\)
−0.0807871 + 0.996731i \(0.525743\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −6.73302 −0.835128
\(66\) 0 0
\(67\) 11.2441 1.37369 0.686844 0.726805i \(-0.258996\pi\)
0.686844 + 0.726805i \(0.258996\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 1.26193 0.149764 0.0748820 0.997192i \(-0.476142\pi\)
0.0748820 + 0.997192i \(0.476142\pi\)
\(72\) 0 0
\(73\) 14.7730 1.72905 0.864526 0.502588i \(-0.167619\pi\)
0.864526 + 0.502588i \(0.167619\pi\)
\(74\) 0 0
\(75\) 1.39543 0.161130
\(76\) 0 0
\(77\) 1.86651 0.212708
\(78\) 0 0
\(79\) −15.9822 −1.79814 −0.899068 0.437809i \(-0.855755\pi\)
−0.899068 + 0.437809i \(0.855755\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 17.8309 1.95719 0.978596 0.205791i \(-0.0659766\pi\)
0.978596 + 0.205791i \(0.0659766\pi\)
\(84\) 0 0
\(85\) −12.4533 −1.35075
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) −1.26193 −0.133765 −0.0668824 0.997761i \(-0.521305\pi\)
−0.0668824 + 0.997761i \(0.521305\pi\)
\(90\) 0 0
\(91\) 2.66241 0.279096
\(92\) 0 0
\(93\) −3.86651 −0.400938
\(94\) 0 0
\(95\) −5.39543 −0.553559
\(96\) 0 0
\(97\) −12.3248 −1.25140 −0.625698 0.780065i \(-0.715186\pi\)
−0.625698 + 0.780065i \(0.715186\pi\)
\(98\) 0 0
\(99\) −1.86651 −0.187591
\(100\) 0 0
\(101\) 7.66241 0.762438 0.381219 0.924485i \(-0.375504\pi\)
0.381219 + 0.924485i \(0.375504\pi\)
\(102\) 0 0
\(103\) −15.2492 −1.50254 −0.751272 0.659992i \(-0.770560\pi\)
−0.751272 + 0.659992i \(0.770560\pi\)
\(104\) 0 0
\(105\) −2.52892 −0.246797
\(106\) 0 0
\(107\) −4.47108 −0.432236 −0.216118 0.976367i \(-0.569340\pi\)
−0.216118 + 0.976367i \(0.569340\pi\)
\(108\) 0 0
\(109\) −18.3776 −1.76026 −0.880128 0.474737i \(-0.842543\pi\)
−0.880128 + 0.474737i \(0.842543\pi\)
\(110\) 0 0
\(111\) 2.05784 0.195321
\(112\) 0 0
\(113\) 5.85374 0.550673 0.275337 0.961348i \(-0.411211\pi\)
0.275337 + 0.961348i \(0.411211\pi\)
\(114\) 0 0
\(115\) 2.52892 0.235823
\(116\) 0 0
\(117\) −2.66241 −0.246140
\(118\) 0 0
\(119\) 4.92434 0.451414
\(120\) 0 0
\(121\) −7.51615 −0.683286
\(122\) 0 0
\(123\) 7.79085 0.702477
\(124\) 0 0
\(125\) −9.11567 −0.815330
\(126\) 0 0
\(127\) −13.5689 −1.20405 −0.602024 0.798478i \(-0.705639\pi\)
−0.602024 + 0.798478i \(0.705639\pi\)
\(128\) 0 0
\(129\) −5.72025 −0.503640
\(130\) 0 0
\(131\) −18.9243 −1.65343 −0.826714 0.562623i \(-0.809792\pi\)
−0.826714 + 0.562623i \(0.809792\pi\)
\(132\) 0 0
\(133\) 2.13349 0.184997
\(134\) 0 0
\(135\) 2.52892 0.217655
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) −19.1685 −1.62585 −0.812924 0.582370i \(-0.802125\pi\)
−0.812924 + 0.582370i \(0.802125\pi\)
\(140\) 0 0
\(141\) −5.92434 −0.498920
\(142\) 0 0
\(143\) 4.96941 0.415563
\(144\) 0 0
\(145\) 12.6446 1.05008
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 3.05784 0.250508 0.125254 0.992125i \(-0.460025\pi\)
0.125254 + 0.992125i \(0.460025\pi\)
\(150\) 0 0
\(151\) 8.04002 0.654287 0.327144 0.944975i \(-0.393914\pi\)
0.327144 + 0.944975i \(0.393914\pi\)
\(152\) 0 0
\(153\) −4.92434 −0.398110
\(154\) 0 0
\(155\) −9.77808 −0.785394
\(156\) 0 0
\(157\) 4.67518 0.373120 0.186560 0.982444i \(-0.440266\pi\)
0.186560 + 0.982444i \(0.440266\pi\)
\(158\) 0 0
\(159\) −8.45326 −0.670387
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −13.6624 −1.07012 −0.535061 0.844813i \(-0.679711\pi\)
−0.535061 + 0.844813i \(0.679711\pi\)
\(164\) 0 0
\(165\) −4.72025 −0.367471
\(166\) 0 0
\(167\) 11.3070 0.874962 0.437481 0.899228i \(-0.355871\pi\)
0.437481 + 0.899228i \(0.355871\pi\)
\(168\) 0 0
\(169\) −5.91157 −0.454736
\(170\) 0 0
\(171\) −2.13349 −0.163152
\(172\) 0 0
\(173\) −18.8887 −1.43608 −0.718041 0.696001i \(-0.754961\pi\)
−0.718041 + 0.696001i \(0.754961\pi\)
\(174\) 0 0
\(175\) −1.39543 −0.105484
\(176\) 0 0
\(177\) −14.1863 −1.06631
\(178\) 0 0
\(179\) −15.5868 −1.16501 −0.582504 0.812828i \(-0.697927\pi\)
−0.582504 + 0.812828i \(0.697927\pi\)
\(180\) 0 0
\(181\) 21.5740 1.60358 0.801791 0.597605i \(-0.203881\pi\)
0.801791 + 0.597605i \(0.203881\pi\)
\(182\) 0 0
\(183\) −1.26193 −0.0932849
\(184\) 0 0
\(185\) 5.20410 0.382613
\(186\) 0 0
\(187\) 9.19133 0.672136
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −18.5239 −1.34034 −0.670170 0.742208i \(-0.733779\pi\)
−0.670170 + 0.742208i \(0.733779\pi\)
\(192\) 0 0
\(193\) −4.07566 −0.293372 −0.146686 0.989183i \(-0.546861\pi\)
−0.146686 + 0.989183i \(0.546861\pi\)
\(194\) 0 0
\(195\) −6.73302 −0.482161
\(196\) 0 0
\(197\) 17.9771 1.28082 0.640409 0.768034i \(-0.278765\pi\)
0.640409 + 0.768034i \(0.278765\pi\)
\(198\) 0 0
\(199\) 25.2841 1.79234 0.896172 0.443706i \(-0.146337\pi\)
0.896172 + 0.443706i \(0.146337\pi\)
\(200\) 0 0
\(201\) 11.2441 0.793099
\(202\) 0 0
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 19.7024 1.37608
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 3.98218 0.275453
\(210\) 0 0
\(211\) 9.19133 0.632757 0.316379 0.948633i \(-0.397533\pi\)
0.316379 + 0.948633i \(0.397533\pi\)
\(212\) 0 0
\(213\) 1.26193 0.0864663
\(214\) 0 0
\(215\) −14.4660 −0.986575
\(216\) 0 0
\(217\) 3.86651 0.262476
\(218\) 0 0
\(219\) 14.7730 0.998269
\(220\) 0 0
\(221\) 13.1106 0.881916
\(222\) 0 0
\(223\) 9.51110 0.636910 0.318455 0.947938i \(-0.396836\pi\)
0.318455 + 0.947938i \(0.396836\pi\)
\(224\) 0 0
\(225\) 1.39543 0.0930284
\(226\) 0 0
\(227\) −4.68023 −0.310638 −0.155319 0.987864i \(-0.549641\pi\)
−0.155319 + 0.987864i \(0.549641\pi\)
\(228\) 0 0
\(229\) −4.45326 −0.294280 −0.147140 0.989116i \(-0.547007\pi\)
−0.147140 + 0.989116i \(0.547007\pi\)
\(230\) 0 0
\(231\) 1.86651 0.122807
\(232\) 0 0
\(233\) 6.72025 0.440258 0.220129 0.975471i \(-0.429352\pi\)
0.220129 + 0.975471i \(0.429352\pi\)
\(234\) 0 0
\(235\) −14.9822 −0.977330
\(236\) 0 0
\(237\) −15.9822 −1.03815
\(238\) 0 0
\(239\) 10.5689 0.683647 0.341824 0.939764i \(-0.388955\pi\)
0.341824 + 0.939764i \(0.388955\pi\)
\(240\) 0 0
\(241\) 2.74312 0.176700 0.0883498 0.996090i \(-0.471841\pi\)
0.0883498 + 0.996090i \(0.471841\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.52892 0.161567
\(246\) 0 0
\(247\) 5.68023 0.361424
\(248\) 0 0
\(249\) 17.8309 1.12999
\(250\) 0 0
\(251\) −27.9243 −1.76257 −0.881284 0.472586i \(-0.843321\pi\)
−0.881284 + 0.472586i \(0.843321\pi\)
\(252\) 0 0
\(253\) −1.86651 −0.117346
\(254\) 0 0
\(255\) −12.4533 −0.779854
\(256\) 0 0
\(257\) 22.9065 1.42887 0.714435 0.699702i \(-0.246684\pi\)
0.714435 + 0.699702i \(0.246684\pi\)
\(258\) 0 0
\(259\) −2.05784 −0.127868
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 0 0
\(263\) −10.5918 −0.653119 −0.326559 0.945177i \(-0.605889\pi\)
−0.326559 + 0.945177i \(0.605889\pi\)
\(264\) 0 0
\(265\) −21.3776 −1.31322
\(266\) 0 0
\(267\) −1.26193 −0.0772291
\(268\) 0 0
\(269\) 30.0272 1.83079 0.915397 0.402553i \(-0.131877\pi\)
0.915397 + 0.402553i \(0.131877\pi\)
\(270\) 0 0
\(271\) 2.80867 0.170615 0.0853073 0.996355i \(-0.472813\pi\)
0.0853073 + 0.996355i \(0.472813\pi\)
\(272\) 0 0
\(273\) 2.66241 0.161136
\(274\) 0 0
\(275\) −2.60457 −0.157062
\(276\) 0 0
\(277\) 11.2619 0.676664 0.338332 0.941027i \(-0.390137\pi\)
0.338332 + 0.941027i \(0.390137\pi\)
\(278\) 0 0
\(279\) −3.86651 −0.231482
\(280\) 0 0
\(281\) −7.39038 −0.440873 −0.220436 0.975401i \(-0.570748\pi\)
−0.220436 + 0.975401i \(0.570748\pi\)
\(282\) 0 0
\(283\) 0.337590 0.0200676 0.0100338 0.999950i \(-0.496806\pi\)
0.0100338 + 0.999950i \(0.496806\pi\)
\(284\) 0 0
\(285\) −5.39543 −0.319597
\(286\) 0 0
\(287\) −7.79085 −0.459879
\(288\) 0 0
\(289\) 7.24916 0.426421
\(290\) 0 0
\(291\) −12.3248 −0.722494
\(292\) 0 0
\(293\) −19.3147 −1.12838 −0.564189 0.825646i \(-0.690811\pi\)
−0.564189 + 0.825646i \(0.690811\pi\)
\(294\) 0 0
\(295\) −35.8759 −2.08878
\(296\) 0 0
\(297\) −1.86651 −0.108306
\(298\) 0 0
\(299\) −2.66241 −0.153971
\(300\) 0 0
\(301\) 5.72025 0.329709
\(302\) 0 0
\(303\) 7.66241 0.440194
\(304\) 0 0
\(305\) −3.19133 −0.182735
\(306\) 0 0
\(307\) 21.0800 1.20310 0.601550 0.798835i \(-0.294550\pi\)
0.601550 + 0.798835i \(0.294550\pi\)
\(308\) 0 0
\(309\) −15.2492 −0.867495
\(310\) 0 0
\(311\) 6.72797 0.381508 0.190754 0.981638i \(-0.438907\pi\)
0.190754 + 0.981638i \(0.438907\pi\)
\(312\) 0 0
\(313\) −24.8964 −1.40723 −0.703615 0.710582i \(-0.748432\pi\)
−0.703615 + 0.710582i \(0.748432\pi\)
\(314\) 0 0
\(315\) −2.52892 −0.142488
\(316\) 0 0
\(317\) 6.26193 0.351705 0.175853 0.984417i \(-0.443732\pi\)
0.175853 + 0.984417i \(0.443732\pi\)
\(318\) 0 0
\(319\) −9.33254 −0.522522
\(320\) 0 0
\(321\) −4.47108 −0.249551
\(322\) 0 0
\(323\) 10.5060 0.584572
\(324\) 0 0
\(325\) −3.71520 −0.206082
\(326\) 0 0
\(327\) −18.3776 −1.01628
\(328\) 0 0
\(329\) 5.92434 0.326620
\(330\) 0 0
\(331\) 0.266984 0.0146748 0.00733738 0.999973i \(-0.497664\pi\)
0.00733738 + 0.999973i \(0.497664\pi\)
\(332\) 0 0
\(333\) 2.05784 0.112769
\(334\) 0 0
\(335\) 28.4354 1.55359
\(336\) 0 0
\(337\) −5.00505 −0.272642 −0.136321 0.990665i \(-0.543528\pi\)
−0.136321 + 0.990665i \(0.543528\pi\)
\(338\) 0 0
\(339\) 5.85374 0.317931
\(340\) 0 0
\(341\) 7.21687 0.390815
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.52892 0.136152
\(346\) 0 0
\(347\) −8.85879 −0.475565 −0.237782 0.971318i \(-0.576420\pi\)
−0.237782 + 0.971318i \(0.576420\pi\)
\(348\) 0 0
\(349\) −10.8615 −0.581401 −0.290700 0.956814i \(-0.593888\pi\)
−0.290700 + 0.956814i \(0.593888\pi\)
\(350\) 0 0
\(351\) −2.66241 −0.142109
\(352\) 0 0
\(353\) 20.7075 1.10215 0.551074 0.834456i \(-0.314218\pi\)
0.551074 + 0.834456i \(0.314218\pi\)
\(354\) 0 0
\(355\) 3.19133 0.169378
\(356\) 0 0
\(357\) 4.92434 0.260624
\(358\) 0 0
\(359\) 30.4933 1.60937 0.804687 0.593700i \(-0.202333\pi\)
0.804687 + 0.593700i \(0.202333\pi\)
\(360\) 0 0
\(361\) −14.4482 −0.760432
\(362\) 0 0
\(363\) −7.51615 −0.394495
\(364\) 0 0
\(365\) 37.3598 1.95550
\(366\) 0 0
\(367\) −17.4533 −0.911053 −0.455526 0.890222i \(-0.650549\pi\)
−0.455526 + 0.890222i \(0.650549\pi\)
\(368\) 0 0
\(369\) 7.79085 0.405576
\(370\) 0 0
\(371\) 8.45326 0.438871
\(372\) 0 0
\(373\) −7.99228 −0.413825 −0.206912 0.978360i \(-0.566341\pi\)
−0.206912 + 0.978360i \(0.566341\pi\)
\(374\) 0 0
\(375\) −9.11567 −0.470731
\(376\) 0 0
\(377\) −13.3120 −0.685605
\(378\) 0 0
\(379\) 26.5639 1.36450 0.682248 0.731121i \(-0.261003\pi\)
0.682248 + 0.731121i \(0.261003\pi\)
\(380\) 0 0
\(381\) −13.5689 −0.695158
\(382\) 0 0
\(383\) −29.8309 −1.52429 −0.762143 0.647409i \(-0.775853\pi\)
−0.762143 + 0.647409i \(0.775853\pi\)
\(384\) 0 0
\(385\) 4.72025 0.240566
\(386\) 0 0
\(387\) −5.72025 −0.290776
\(388\) 0 0
\(389\) 12.5417 0.635889 0.317944 0.948109i \(-0.397007\pi\)
0.317944 + 0.948109i \(0.397007\pi\)
\(390\) 0 0
\(391\) −4.92434 −0.249035
\(392\) 0 0
\(393\) −18.9243 −0.954607
\(394\) 0 0
\(395\) −40.4176 −2.03363
\(396\) 0 0
\(397\) −11.4405 −0.574182 −0.287091 0.957903i \(-0.592688\pi\)
−0.287091 + 0.957903i \(0.592688\pi\)
\(398\) 0 0
\(399\) 2.13349 0.106808
\(400\) 0 0
\(401\) 6.92434 0.345785 0.172893 0.984941i \(-0.444689\pi\)
0.172893 + 0.984941i \(0.444689\pi\)
\(402\) 0 0
\(403\) 10.2942 0.512792
\(404\) 0 0
\(405\) 2.52892 0.125663
\(406\) 0 0
\(407\) −3.84097 −0.190390
\(408\) 0 0
\(409\) 31.1456 1.54005 0.770025 0.638014i \(-0.220244\pi\)
0.770025 + 0.638014i \(0.220244\pi\)
\(410\) 0 0
\(411\) −13.0000 −0.641243
\(412\) 0 0
\(413\) 14.1863 0.698061
\(414\) 0 0
\(415\) 45.0928 2.21352
\(416\) 0 0
\(417\) −19.1685 −0.938683
\(418\) 0 0
\(419\) −0.987230 −0.0482293 −0.0241147 0.999709i \(-0.507677\pi\)
−0.0241147 + 0.999709i \(0.507677\pi\)
\(420\) 0 0
\(421\) −5.08071 −0.247618 −0.123809 0.992306i \(-0.539511\pi\)
−0.123809 + 0.992306i \(0.539511\pi\)
\(422\) 0 0
\(423\) −5.92434 −0.288051
\(424\) 0 0
\(425\) −6.87156 −0.333320
\(426\) 0 0
\(427\) 1.26193 0.0610693
\(428\) 0 0
\(429\) 4.96941 0.239925
\(430\) 0 0
\(431\) 38.2186 1.84092 0.920462 0.390832i \(-0.127813\pi\)
0.920462 + 0.390832i \(0.127813\pi\)
\(432\) 0 0
\(433\) 21.8887 1.05190 0.525952 0.850514i \(-0.323709\pi\)
0.525952 + 0.850514i \(0.323709\pi\)
\(434\) 0 0
\(435\) 12.6446 0.606262
\(436\) 0 0
\(437\) −2.13349 −0.102059
\(438\) 0 0
\(439\) 1.71520 0.0818618 0.0409309 0.999162i \(-0.486968\pi\)
0.0409309 + 0.999162i \(0.486968\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −10.7152 −0.509094 −0.254547 0.967060i \(-0.581926\pi\)
−0.254547 + 0.967060i \(0.581926\pi\)
\(444\) 0 0
\(445\) −3.19133 −0.151283
\(446\) 0 0
\(447\) 3.05784 0.144631
\(448\) 0 0
\(449\) 0.977130 0.0461136 0.0230568 0.999734i \(-0.492660\pi\)
0.0230568 + 0.999734i \(0.492660\pi\)
\(450\) 0 0
\(451\) −14.5417 −0.684741
\(452\) 0 0
\(453\) 8.04002 0.377753
\(454\) 0 0
\(455\) 6.73302 0.315649
\(456\) 0 0
\(457\) −12.1863 −0.570050 −0.285025 0.958520i \(-0.592002\pi\)
−0.285025 + 0.958520i \(0.592002\pi\)
\(458\) 0 0
\(459\) −4.92434 −0.229849
\(460\) 0 0
\(461\) 8.71015 0.405672 0.202836 0.979213i \(-0.434984\pi\)
0.202836 + 0.979213i \(0.434984\pi\)
\(462\) 0 0
\(463\) 33.3827 1.55142 0.775712 0.631087i \(-0.217391\pi\)
0.775712 + 0.631087i \(0.217391\pi\)
\(464\) 0 0
\(465\) −9.77808 −0.453448
\(466\) 0 0
\(467\) −8.46603 −0.391761 −0.195881 0.980628i \(-0.562757\pi\)
−0.195881 + 0.980628i \(0.562757\pi\)
\(468\) 0 0
\(469\) −11.2441 −0.519205
\(470\) 0 0
\(471\) 4.67518 0.215421
\(472\) 0 0
\(473\) 10.6769 0.490924
\(474\) 0 0
\(475\) −2.97713 −0.136600
\(476\) 0 0
\(477\) −8.45326 −0.387048
\(478\) 0 0
\(479\) 30.5861 1.39751 0.698757 0.715359i \(-0.253737\pi\)
0.698757 + 0.715359i \(0.253737\pi\)
\(480\) 0 0
\(481\) −5.47880 −0.249812
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −31.1685 −1.41529
\(486\) 0 0
\(487\) −14.0477 −0.636564 −0.318282 0.947996i \(-0.603106\pi\)
−0.318282 + 0.947996i \(0.603106\pi\)
\(488\) 0 0
\(489\) −13.6624 −0.617836
\(490\) 0 0
\(491\) −35.9015 −1.62021 −0.810105 0.586284i \(-0.800590\pi\)
−0.810105 + 0.586284i \(0.800590\pi\)
\(492\) 0 0
\(493\) −24.6217 −1.10891
\(494\) 0 0
\(495\) −4.72025 −0.212159
\(496\) 0 0
\(497\) −1.26193 −0.0566055
\(498\) 0 0
\(499\) 2.75589 0.123370 0.0616852 0.998096i \(-0.480353\pi\)
0.0616852 + 0.998096i \(0.480353\pi\)
\(500\) 0 0
\(501\) 11.3070 0.505159
\(502\) 0 0
\(503\) −28.6745 −1.27853 −0.639267 0.768985i \(-0.720762\pi\)
−0.639267 + 0.768985i \(0.720762\pi\)
\(504\) 0 0
\(505\) 19.3776 0.862292
\(506\) 0 0
\(507\) −5.91157 −0.262542
\(508\) 0 0
\(509\) −2.01010 −0.0890961 −0.0445480 0.999007i \(-0.514185\pi\)
−0.0445480 + 0.999007i \(0.514185\pi\)
\(510\) 0 0
\(511\) −14.7730 −0.653520
\(512\) 0 0
\(513\) −2.13349 −0.0941960
\(514\) 0 0
\(515\) −38.5639 −1.69933
\(516\) 0 0
\(517\) 11.0578 0.486323
\(518\) 0 0
\(519\) −18.8887 −0.829122
\(520\) 0 0
\(521\) −3.58170 −0.156917 −0.0784587 0.996917i \(-0.525000\pi\)
−0.0784587 + 0.996917i \(0.525000\pi\)
\(522\) 0 0
\(523\) 2.53397 0.110803 0.0554013 0.998464i \(-0.482356\pi\)
0.0554013 + 0.998464i \(0.482356\pi\)
\(524\) 0 0
\(525\) −1.39543 −0.0609014
\(526\) 0 0
\(527\) 19.0400 0.829396
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −14.1863 −0.615632
\(532\) 0 0
\(533\) −20.7424 −0.898455
\(534\) 0 0
\(535\) −11.3070 −0.488844
\(536\) 0 0
\(537\) −15.5868 −0.672618
\(538\) 0 0
\(539\) −1.86651 −0.0803962
\(540\) 0 0
\(541\) 3.28918 0.141413 0.0707064 0.997497i \(-0.477475\pi\)
0.0707064 + 0.997497i \(0.477475\pi\)
\(542\) 0 0
\(543\) 21.5740 0.925828
\(544\) 0 0
\(545\) −46.4755 −1.99079
\(546\) 0 0
\(547\) 4.64021 0.198401 0.0992006 0.995067i \(-0.468371\pi\)
0.0992006 + 0.995067i \(0.468371\pi\)
\(548\) 0 0
\(549\) −1.26193 −0.0538580
\(550\) 0 0
\(551\) −10.6675 −0.454449
\(552\) 0 0
\(553\) 15.9822 0.679631
\(554\) 0 0
\(555\) 5.20410 0.220902
\(556\) 0 0
\(557\) −11.0222 −0.467025 −0.233513 0.972354i \(-0.575022\pi\)
−0.233513 + 0.972354i \(0.575022\pi\)
\(558\) 0 0
\(559\) 15.2296 0.644145
\(560\) 0 0
\(561\) 9.19133 0.388058
\(562\) 0 0
\(563\) −22.8837 −0.964431 −0.482216 0.876053i \(-0.660168\pi\)
−0.482216 + 0.876053i \(0.660168\pi\)
\(564\) 0 0
\(565\) 14.8036 0.622793
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 11.0979 0.465246 0.232623 0.972567i \(-0.425269\pi\)
0.232623 + 0.972567i \(0.425269\pi\)
\(570\) 0 0
\(571\) 39.9109 1.67022 0.835110 0.550084i \(-0.185404\pi\)
0.835110 + 0.550084i \(0.185404\pi\)
\(572\) 0 0
\(573\) −18.5239 −0.773846
\(574\) 0 0
\(575\) 1.39543 0.0581933
\(576\) 0 0
\(577\) 37.1634 1.54713 0.773566 0.633715i \(-0.218471\pi\)
0.773566 + 0.633715i \(0.218471\pi\)
\(578\) 0 0
\(579\) −4.07566 −0.169378
\(580\) 0 0
\(581\) −17.8309 −0.739749
\(582\) 0 0
\(583\) 15.7781 0.653461
\(584\) 0 0
\(585\) −6.73302 −0.278376
\(586\) 0 0
\(587\) 3.29757 0.136105 0.0680527 0.997682i \(-0.478321\pi\)
0.0680527 + 0.997682i \(0.478321\pi\)
\(588\) 0 0
\(589\) 8.24916 0.339901
\(590\) 0 0
\(591\) 17.9771 0.739480
\(592\) 0 0
\(593\) −29.7374 −1.22117 −0.610584 0.791951i \(-0.709065\pi\)
−0.610584 + 0.791951i \(0.709065\pi\)
\(594\) 0 0
\(595\) 12.4533 0.510534
\(596\) 0 0
\(597\) 25.2841 1.03481
\(598\) 0 0
\(599\) 1.14626 0.0468350 0.0234175 0.999726i \(-0.492545\pi\)
0.0234175 + 0.999726i \(0.492545\pi\)
\(600\) 0 0
\(601\) −28.9516 −1.18096 −0.590480 0.807052i \(-0.701062\pi\)
−0.590480 + 0.807052i \(0.701062\pi\)
\(602\) 0 0
\(603\) 11.2441 0.457896
\(604\) 0 0
\(605\) −19.0077 −0.772774
\(606\) 0 0
\(607\) 31.7858 1.29015 0.645073 0.764121i \(-0.276827\pi\)
0.645073 + 0.764121i \(0.276827\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) 15.7730 0.638109
\(612\) 0 0
\(613\) 26.0477 1.05206 0.526029 0.850467i \(-0.323680\pi\)
0.526029 + 0.850467i \(0.323680\pi\)
\(614\) 0 0
\(615\) 19.7024 0.794478
\(616\) 0 0
\(617\) −24.5511 −0.988391 −0.494195 0.869351i \(-0.664537\pi\)
−0.494195 + 0.869351i \(0.664537\pi\)
\(618\) 0 0
\(619\) 8.72797 0.350807 0.175403 0.984497i \(-0.443877\pi\)
0.175403 + 0.984497i \(0.443877\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 1.26193 0.0505583
\(624\) 0 0
\(625\) −30.0299 −1.20120
\(626\) 0 0
\(627\) 3.98218 0.159033
\(628\) 0 0
\(629\) −10.1335 −0.404049
\(630\) 0 0
\(631\) 0.621720 0.0247503 0.0123751 0.999923i \(-0.496061\pi\)
0.0123751 + 0.999923i \(0.496061\pi\)
\(632\) 0 0
\(633\) 9.19133 0.365322
\(634\) 0 0
\(635\) −34.3147 −1.36174
\(636\) 0 0
\(637\) −2.66241 −0.105489
\(638\) 0 0
\(639\) 1.26193 0.0499213
\(640\) 0 0
\(641\) 25.0094 0.987813 0.493906 0.869515i \(-0.335569\pi\)
0.493906 + 0.869515i \(0.335569\pi\)
\(642\) 0 0
\(643\) 29.4933 1.16310 0.581551 0.813510i \(-0.302446\pi\)
0.581551 + 0.813510i \(0.302446\pi\)
\(644\) 0 0
\(645\) −14.4660 −0.569599
\(646\) 0 0
\(647\) 31.7424 1.24792 0.623962 0.781455i \(-0.285522\pi\)
0.623962 + 0.781455i \(0.285522\pi\)
\(648\) 0 0
\(649\) 26.4788 1.03938
\(650\) 0 0
\(651\) 3.86651 0.151540
\(652\) 0 0
\(653\) −13.4253 −0.525374 −0.262687 0.964881i \(-0.584609\pi\)
−0.262687 + 0.964881i \(0.584609\pi\)
\(654\) 0 0
\(655\) −47.8581 −1.86997
\(656\) 0 0
\(657\) 14.7730 0.576351
\(658\) 0 0
\(659\) 17.3171 0.674578 0.337289 0.941401i \(-0.390490\pi\)
0.337289 + 0.941401i \(0.390490\pi\)
\(660\) 0 0
\(661\) 37.5740 1.46146 0.730729 0.682667i \(-0.239180\pi\)
0.730729 + 0.682667i \(0.239180\pi\)
\(662\) 0 0
\(663\) 13.1106 0.509174
\(664\) 0 0
\(665\) 5.39543 0.209226
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 0 0
\(669\) 9.51110 0.367720
\(670\) 0 0
\(671\) 2.35541 0.0909296
\(672\) 0 0
\(673\) −36.9644 −1.42487 −0.712436 0.701737i \(-0.752408\pi\)
−0.712436 + 0.701737i \(0.752408\pi\)
\(674\) 0 0
\(675\) 1.39543 0.0537100
\(676\) 0 0
\(677\) 45.4576 1.74708 0.873539 0.486753i \(-0.161819\pi\)
0.873539 + 0.486753i \(0.161819\pi\)
\(678\) 0 0
\(679\) 12.3248 0.472983
\(680\) 0 0
\(681\) −4.68023 −0.179347
\(682\) 0 0
\(683\) −41.9566 −1.60543 −0.802713 0.596365i \(-0.796611\pi\)
−0.802713 + 0.596365i \(0.796611\pi\)
\(684\) 0 0
\(685\) −32.8759 −1.25612
\(686\) 0 0
\(687\) −4.45326 −0.169903
\(688\) 0 0
\(689\) 22.5060 0.857412
\(690\) 0 0
\(691\) 19.3120 0.734665 0.367332 0.930090i \(-0.380271\pi\)
0.367332 + 0.930090i \(0.380271\pi\)
\(692\) 0 0
\(693\) 1.86651 0.0709028
\(694\) 0 0
\(695\) −48.4755 −1.83878
\(696\) 0 0
\(697\) −38.3648 −1.45317
\(698\) 0 0
\(699\) 6.72025 0.254183
\(700\) 0 0
\(701\) 20.4533 0.772509 0.386255 0.922392i \(-0.373769\pi\)
0.386255 + 0.922392i \(0.373769\pi\)
\(702\) 0 0
\(703\) −4.39038 −0.165586
\(704\) 0 0
\(705\) −14.9822 −0.564262
\(706\) 0 0
\(707\) −7.66241 −0.288175
\(708\) 0 0
\(709\) −47.2008 −1.77266 −0.886331 0.463053i \(-0.846754\pi\)
−0.886331 + 0.463053i \(0.846754\pi\)
\(710\) 0 0
\(711\) −15.9822 −0.599379
\(712\) 0 0
\(713\) −3.86651 −0.144802
\(714\) 0 0
\(715\) 12.5672 0.469988
\(716\) 0 0
\(717\) 10.5689 0.394704
\(718\) 0 0
\(719\) −3.38266 −0.126152 −0.0630759 0.998009i \(-0.520091\pi\)
−0.0630759 + 0.998009i \(0.520091\pi\)
\(720\) 0 0
\(721\) 15.2492 0.567909
\(722\) 0 0
\(723\) 2.74312 0.102018
\(724\) 0 0
\(725\) 6.97713 0.259124
\(726\) 0 0
\(727\) −0.516148 −0.0191429 −0.00957143 0.999954i \(-0.503047\pi\)
−0.00957143 + 0.999954i \(0.503047\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.1685 1.04185
\(732\) 0 0
\(733\) −45.4405 −1.67838 −0.839191 0.543836i \(-0.816971\pi\)
−0.839191 + 0.543836i \(0.816971\pi\)
\(734\) 0 0
\(735\) 2.52892 0.0932805
\(736\) 0 0
\(737\) −20.9872 −0.773075
\(738\) 0 0
\(739\) 22.9243 0.843286 0.421643 0.906762i \(-0.361454\pi\)
0.421643 + 0.906762i \(0.361454\pi\)
\(740\) 0 0
\(741\) 5.68023 0.208668
\(742\) 0 0
\(743\) 15.7781 0.578842 0.289421 0.957202i \(-0.406537\pi\)
0.289421 + 0.957202i \(0.406537\pi\)
\(744\) 0 0
\(745\) 7.73302 0.283316
\(746\) 0 0
\(747\) 17.8309 0.652397
\(748\) 0 0
\(749\) 4.47108 0.163370
\(750\) 0 0
\(751\) 49.7680 1.81606 0.908030 0.418906i \(-0.137586\pi\)
0.908030 + 0.418906i \(0.137586\pi\)
\(752\) 0 0
\(753\) −27.9243 −1.01762
\(754\) 0 0
\(755\) 20.3325 0.739977
\(756\) 0 0
\(757\) 21.5740 0.784120 0.392060 0.919940i \(-0.371763\pi\)
0.392060 + 0.919940i \(0.371763\pi\)
\(758\) 0 0
\(759\) −1.86651 −0.0677500
\(760\) 0 0
\(761\) −30.1957 −1.09459 −0.547297 0.836939i \(-0.684343\pi\)
−0.547297 + 0.836939i \(0.684343\pi\)
\(762\) 0 0
\(763\) 18.3776 0.665314
\(764\) 0 0
\(765\) −12.4533 −0.450249
\(766\) 0 0
\(767\) 37.7697 1.36378
\(768\) 0 0
\(769\) −15.5417 −0.560448 −0.280224 0.959935i \(-0.590409\pi\)
−0.280224 + 0.959935i \(0.590409\pi\)
\(770\) 0 0
\(771\) 22.9065 0.824958
\(772\) 0 0
\(773\) −30.5663 −1.09939 −0.549696 0.835365i \(-0.685256\pi\)
−0.549696 + 0.835365i \(0.685256\pi\)
\(774\) 0 0
\(775\) −5.39543 −0.193809
\(776\) 0 0
\(777\) −2.05784 −0.0738245
\(778\) 0 0
\(779\) −16.6217 −0.595535
\(780\) 0 0
\(781\) −2.35541 −0.0842832
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) 11.8231 0.421986
\(786\) 0 0
\(787\) −22.9415 −0.817776 −0.408888 0.912585i \(-0.634083\pi\)
−0.408888 + 0.912585i \(0.634083\pi\)
\(788\) 0 0
\(789\) −10.5918 −0.377078
\(790\) 0 0
\(791\) −5.85374 −0.208135
\(792\) 0 0
\(793\) 3.35979 0.119309
\(794\) 0 0
\(795\) −21.3776 −0.758186
\(796\) 0 0
\(797\) −49.0878 −1.73878 −0.869389 0.494129i \(-0.835487\pi\)
−0.869389 + 0.494129i \(0.835487\pi\)
\(798\) 0 0
\(799\) 29.1735 1.03208
\(800\) 0 0
\(801\) −1.26193 −0.0445882
\(802\) 0 0
\(803\) −27.5740 −0.973065
\(804\) 0 0
\(805\) −2.52892 −0.0891326
\(806\) 0 0
\(807\) 30.0272 1.05701
\(808\) 0 0
\(809\) −28.5767 −1.00470 −0.502351 0.864664i \(-0.667531\pi\)
−0.502351 + 0.864664i \(0.667531\pi\)
\(810\) 0 0
\(811\) −23.3369 −0.819470 −0.409735 0.912205i \(-0.634379\pi\)
−0.409735 + 0.912205i \(0.634379\pi\)
\(812\) 0 0
\(813\) 2.80867 0.0985044
\(814\) 0 0
\(815\) −34.5511 −1.21027
\(816\) 0 0
\(817\) 12.2041 0.426967
\(818\) 0 0
\(819\) 2.66241 0.0930321
\(820\) 0 0
\(821\) 3.74312 0.130636 0.0653178 0.997865i \(-0.479194\pi\)
0.0653178 + 0.997865i \(0.479194\pi\)
\(822\) 0 0
\(823\) 26.6089 0.927530 0.463765 0.885958i \(-0.346498\pi\)
0.463765 + 0.885958i \(0.346498\pi\)
\(824\) 0 0
\(825\) −2.60457 −0.0906796
\(826\) 0 0
\(827\) 56.6389 1.96953 0.984763 0.173901i \(-0.0556372\pi\)
0.984763 + 0.173901i \(0.0556372\pi\)
\(828\) 0 0
\(829\) −33.9465 −1.17901 −0.589506 0.807764i \(-0.700678\pi\)
−0.589506 + 0.807764i \(0.700678\pi\)
\(830\) 0 0
\(831\) 11.2619 0.390672
\(832\) 0 0
\(833\) −4.92434 −0.170618
\(834\) 0 0
\(835\) 28.5945 0.989553
\(836\) 0 0
\(837\) −3.86651 −0.133646
\(838\) 0 0
\(839\) 7.64459 0.263921 0.131960 0.991255i \(-0.457873\pi\)
0.131960 + 0.991255i \(0.457873\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −7.39038 −0.254538
\(844\) 0 0
\(845\) −14.9499 −0.514292
\(846\) 0 0
\(847\) 7.51615 0.258258
\(848\) 0 0
\(849\) 0.337590 0.0115861
\(850\) 0 0
\(851\) 2.05784 0.0705417
\(852\) 0 0
\(853\) 49.8887 1.70816 0.854078 0.520144i \(-0.174122\pi\)
0.854078 + 0.520144i \(0.174122\pi\)
\(854\) 0 0
\(855\) −5.39543 −0.184520
\(856\) 0 0
\(857\) −27.6217 −0.943540 −0.471770 0.881722i \(-0.656385\pi\)
−0.471770 + 0.881722i \(0.656385\pi\)
\(858\) 0 0
\(859\) −6.44821 −0.220010 −0.110005 0.993931i \(-0.535087\pi\)
−0.110005 + 0.993931i \(0.535087\pi\)
\(860\) 0 0
\(861\) −7.79085 −0.265512
\(862\) 0 0
\(863\) −50.7196 −1.72651 −0.863257 0.504764i \(-0.831579\pi\)
−0.863257 + 0.504764i \(0.831579\pi\)
\(864\) 0 0
\(865\) −47.7680 −1.62416
\(866\) 0 0
\(867\) 7.24916 0.246195
\(868\) 0 0
\(869\) 29.8309 1.01194
\(870\) 0 0
\(871\) −29.9364 −1.01436
\(872\) 0 0
\(873\) −12.3248 −0.417132
\(874\) 0 0
\(875\) 9.11567 0.308166
\(876\) 0 0
\(877\) −25.2993 −0.854296 −0.427148 0.904182i \(-0.640482\pi\)
−0.427148 + 0.904182i \(0.640482\pi\)
\(878\) 0 0
\(879\) −19.3147 −0.651469
\(880\) 0 0
\(881\) −47.7075 −1.60731 −0.803653 0.595098i \(-0.797113\pi\)
−0.803653 + 0.595098i \(0.797113\pi\)
\(882\) 0 0
\(883\) 33.0851 1.11340 0.556701 0.830713i \(-0.312067\pi\)
0.556701 + 0.830713i \(0.312067\pi\)
\(884\) 0 0
\(885\) −35.8759 −1.20596
\(886\) 0 0
\(887\) −53.7347 −1.80424 −0.902118 0.431490i \(-0.857988\pi\)
−0.902118 + 0.431490i \(0.857988\pi\)
\(888\) 0 0
\(889\) 13.5689 0.455087
\(890\) 0 0
\(891\) −1.86651 −0.0625304
\(892\) 0 0
\(893\) 12.6395 0.422966
\(894\) 0 0
\(895\) −39.4176 −1.31759
\(896\) 0 0
\(897\) −2.66241 −0.0888953
\(898\) 0 0
\(899\) −19.3325 −0.644776
\(900\) 0 0
\(901\) 41.6268 1.38679
\(902\) 0 0
\(903\) 5.72025 0.190358
\(904\) 0 0
\(905\) 54.5588 1.81360
\(906\) 0 0
\(907\) 20.1207 0.668098 0.334049 0.942556i \(-0.391585\pi\)
0.334049 + 0.942556i \(0.391585\pi\)
\(908\) 0 0
\(909\) 7.66241 0.254146
\(910\) 0 0
\(911\) −34.9721 −1.15868 −0.579338 0.815087i \(-0.696689\pi\)
−0.579338 + 0.815087i \(0.696689\pi\)
\(912\) 0 0
\(913\) −33.2815 −1.10146
\(914\) 0 0
\(915\) −3.19133 −0.105502
\(916\) 0 0
\(917\) 18.9243 0.624937
\(918\) 0 0
\(919\) −3.30700 −0.109088 −0.0545439 0.998511i \(-0.517370\pi\)
−0.0545439 + 0.998511i \(0.517370\pi\)
\(920\) 0 0
\(921\) 21.0800 0.694611
\(922\) 0 0
\(923\) −3.35979 −0.110589
\(924\) 0 0
\(925\) 2.87156 0.0944162
\(926\) 0 0
\(927\) −15.2492 −0.500848
\(928\) 0 0
\(929\) −22.7959 −0.747909 −0.373955 0.927447i \(-0.621998\pi\)
−0.373955 + 0.927447i \(0.621998\pi\)
\(930\) 0 0
\(931\) −2.13349 −0.0699224
\(932\) 0 0
\(933\) 6.72797 0.220264
\(934\) 0 0
\(935\) 23.2441 0.760164
\(936\) 0 0
\(937\) 27.7653 0.907053 0.453527 0.891243i \(-0.350166\pi\)
0.453527 + 0.891243i \(0.350166\pi\)
\(938\) 0 0
\(939\) −24.8964 −0.812464
\(940\) 0 0
\(941\) 56.8309 1.85263 0.926317 0.376746i \(-0.122957\pi\)
0.926317 + 0.376746i \(0.122957\pi\)
\(942\) 0 0
\(943\) 7.79085 0.253705
\(944\) 0 0
\(945\) −2.52892 −0.0822657
\(946\) 0 0
\(947\) 6.59952 0.214456 0.107228 0.994234i \(-0.465803\pi\)
0.107228 + 0.994234i \(0.465803\pi\)
\(948\) 0 0
\(949\) −39.3319 −1.27677
\(950\) 0 0
\(951\) 6.26193 0.203057
\(952\) 0 0
\(953\) −34.4354 −1.11547 −0.557737 0.830018i \(-0.688330\pi\)
−0.557737 + 0.830018i \(0.688330\pi\)
\(954\) 0 0
\(955\) −46.8453 −1.51588
\(956\) 0 0
\(957\) −9.33254 −0.301678
\(958\) 0 0
\(959\) 13.0000 0.419792
\(960\) 0 0
\(961\) −16.0501 −0.517746
\(962\) 0 0
\(963\) −4.47108 −0.144079
\(964\) 0 0
\(965\) −10.3070 −0.331794
\(966\) 0 0
\(967\) −5.03230 −0.161828 −0.0809139 0.996721i \(-0.525784\pi\)
−0.0809139 + 0.996721i \(0.525784\pi\)
\(968\) 0 0
\(969\) 10.5060 0.337503
\(970\) 0 0
\(971\) −24.5969 −0.789351 −0.394675 0.918821i \(-0.629143\pi\)
−0.394675 + 0.918821i \(0.629143\pi\)
\(972\) 0 0
\(973\) 19.1685 0.614513
\(974\) 0 0
\(975\) −3.71520 −0.118981
\(976\) 0 0
\(977\) −37.4176 −1.19710 −0.598548 0.801087i \(-0.704255\pi\)
−0.598548 + 0.801087i \(0.704255\pi\)
\(978\) 0 0
\(979\) 2.35541 0.0752792
\(980\) 0 0
\(981\) −18.3776 −0.586752
\(982\) 0 0
\(983\) −41.6796 −1.32937 −0.664686 0.747123i \(-0.731435\pi\)
−0.664686 + 0.747123i \(0.731435\pi\)
\(984\) 0 0
\(985\) 45.4627 1.44856
\(986\) 0 0
\(987\) 5.92434 0.188574
\(988\) 0 0
\(989\) −5.72025 −0.181893
\(990\) 0 0
\(991\) −5.44889 −0.173090 −0.0865448 0.996248i \(-0.527583\pi\)
−0.0865448 + 0.996248i \(0.527583\pi\)
\(992\) 0 0
\(993\) 0.266984 0.00847248
\(994\) 0 0
\(995\) 63.9415 2.02708
\(996\) 0 0
\(997\) 47.2815 1.49742 0.748709 0.662898i \(-0.230674\pi\)
0.748709 + 0.662898i \(0.230674\pi\)
\(998\) 0 0
\(999\) 2.05784 0.0651070
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.bx.1.3 3
4.3 odd 2 3864.2.a.m.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.m.1.3 3 4.3 odd 2
7728.2.a.bx.1.3 3 1.1 even 1 trivial