Properties

Label 4864.2.a.bo.1.8
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 68 x^{3} - 62 x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.681212\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.95163 q^{3} -2.13486 q^{5} -3.29464 q^{7} +5.71210 q^{9} +O(q^{10})\) \(q+2.95163 q^{3} -2.13486 q^{5} -3.29464 q^{7} +5.71210 q^{9} +3.71210 q^{11} +2.32843 q^{13} -6.30130 q^{15} -6.48822 q^{17} -1.00000 q^{19} -9.72456 q^{21} -7.32651 q^{23} -0.442384 q^{25} +8.00510 q^{27} -2.59857 q^{29} -1.34204 q^{31} +10.9567 q^{33} +7.03360 q^{35} +3.72986 q^{37} +6.87267 q^{39} -6.52385 q^{41} +1.97202 q^{43} -12.1945 q^{45} +5.45991 q^{47} +3.85468 q^{49} -19.1508 q^{51} -4.98640 q^{53} -7.92480 q^{55} -2.95163 q^{57} -9.67136 q^{59} +8.15570 q^{61} -18.8193 q^{63} -4.97088 q^{65} -0.524986 q^{67} -21.6251 q^{69} -7.17489 q^{71} +6.33130 q^{73} -1.30575 q^{75} -12.2300 q^{77} -8.75644 q^{79} +6.49178 q^{81} -7.74008 q^{83} +13.8514 q^{85} -7.67001 q^{87} +1.04368 q^{89} -7.67136 q^{91} -3.96119 q^{93} +2.13486 q^{95} -0.117594 q^{97} +21.2039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 4q^{7} + 12q^{9} - 4q^{11} - 8q^{13} - 4q^{17} - 8q^{19} - 16q^{21} + 12q^{25} - 28q^{29} + 8q^{31} + 12q^{35} - 4q^{37} - 4q^{39} - 8q^{41} + 4q^{43} - 24q^{45} + 12q^{47} + 12q^{49} - 12q^{51} - 32q^{53} - 8q^{55} - 12q^{59} - 8q^{61} - 16q^{63} + 8q^{65} + 4q^{67} - 28q^{69} - 24q^{71} - 24q^{77} - 24q^{79} - 8q^{81} - 40q^{83} - 24q^{85} + 24q^{87} + 8q^{89} + 4q^{91} - 32q^{93} + 8q^{95} + 16q^{97} + 76q^{99} + O(q^{100}) \)

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
\(3\) 2.95163 1.70412 0.852061 0.523442i \(-0.175352\pi\)
0.852061 + 0.523442i \(0.175352\pi\)
\(4\) 0 0
\(5\) −2.13486 −0.954737 −0.477369 0.878703i \(-0.658409\pi\)
−0.477369 + 0.878703i \(0.658409\pi\)
\(6\) 0 0
\(7\) −3.29464 −1.24526 −0.622629 0.782517i \(-0.713936\pi\)
−0.622629 + 0.782517i \(0.713936\pi\)
\(8\) 0 0
\(9\) 5.71210 1.90403
\(10\) 0 0
\(11\) 3.71210 1.11924 0.559620 0.828749i \(-0.310947\pi\)
0.559620 + 0.828749i \(0.310947\pi\)
\(12\) 0 0
\(13\) 2.32843 0.645792 0.322896 0.946435i \(-0.395344\pi\)
0.322896 + 0.946435i \(0.395344\pi\)
\(14\) 0 0
\(15\) −6.30130 −1.62699
\(16\) 0 0
\(17\) −6.48822 −1.57362 −0.786812 0.617192i \(-0.788270\pi\)
−0.786812 + 0.617192i \(0.788270\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −9.72456 −2.12207
\(22\) 0 0
\(23\) −7.32651 −1.52768 −0.763842 0.645404i \(-0.776689\pi\)
−0.763842 + 0.645404i \(0.776689\pi\)
\(24\) 0 0
\(25\) −0.442384 −0.0884769
\(26\) 0 0
\(27\) 8.00510 1.54058
\(28\) 0 0
\(29\) −2.59857 −0.482542 −0.241271 0.970458i \(-0.577564\pi\)
−0.241271 + 0.970458i \(0.577564\pi\)
\(30\) 0 0
\(31\) −1.34204 −0.241037 −0.120518 0.992711i \(-0.538456\pi\)
−0.120518 + 0.992711i \(0.538456\pi\)
\(32\) 0 0
\(33\) 10.9567 1.90732
\(34\) 0 0
\(35\) 7.03360 1.18889
\(36\) 0 0
\(37\) 3.72986 0.613186 0.306593 0.951841i \(-0.400811\pi\)
0.306593 + 0.951841i \(0.400811\pi\)
\(38\) 0 0
\(39\) 6.87267 1.10051
\(40\) 0 0
\(41\) −6.52385 −1.01885 −0.509427 0.860514i \(-0.670143\pi\)
−0.509427 + 0.860514i \(0.670143\pi\)
\(42\) 0 0
\(43\) 1.97202 0.300729 0.150365 0.988631i \(-0.451955\pi\)
0.150365 + 0.988631i \(0.451955\pi\)
\(44\) 0 0
\(45\) −12.1945 −1.81785
\(46\) 0 0
\(47\) 5.45991 0.796410 0.398205 0.917297i \(-0.369633\pi\)
0.398205 + 0.917297i \(0.369633\pi\)
\(48\) 0 0
\(49\) 3.85468 0.550669
\(50\) 0 0
\(51\) −19.1508 −2.68165
\(52\) 0 0
\(53\) −4.98640 −0.684935 −0.342467 0.939530i \(-0.611263\pi\)
−0.342467 + 0.939530i \(0.611263\pi\)
\(54\) 0 0
\(55\) −7.92480 −1.06858
\(56\) 0 0
\(57\) −2.95163 −0.390952
\(58\) 0 0
\(59\) −9.67136 −1.25910 −0.629552 0.776958i \(-0.716762\pi\)
−0.629552 + 0.776958i \(0.716762\pi\)
\(60\) 0 0
\(61\) 8.15570 1.04423 0.522115 0.852875i \(-0.325143\pi\)
0.522115 + 0.852875i \(0.325143\pi\)
\(62\) 0 0
\(63\) −18.8193 −2.37101
\(64\) 0 0
\(65\) −4.97088 −0.616561
\(66\) 0 0
\(67\) −0.524986 −0.0641372 −0.0320686 0.999486i \(-0.510210\pi\)
−0.0320686 + 0.999486i \(0.510210\pi\)
\(68\) 0 0
\(69\) −21.6251 −2.60336
\(70\) 0 0
\(71\) −7.17489 −0.851503 −0.425751 0.904840i \(-0.639990\pi\)
−0.425751 + 0.904840i \(0.639990\pi\)
\(72\) 0 0
\(73\) 6.33130 0.741022 0.370511 0.928828i \(-0.379183\pi\)
0.370511 + 0.928828i \(0.379183\pi\)
\(74\) 0 0
\(75\) −1.30575 −0.150775
\(76\) 0 0
\(77\) −12.2300 −1.39374
\(78\) 0 0
\(79\) −8.75644 −0.985176 −0.492588 0.870263i \(-0.663949\pi\)
−0.492588 + 0.870263i \(0.663949\pi\)
\(80\) 0 0
\(81\) 6.49178 0.721309
\(82\) 0 0
\(83\) −7.74008 −0.849585 −0.424792 0.905291i \(-0.639653\pi\)
−0.424792 + 0.905291i \(0.639653\pi\)
\(84\) 0 0
\(85\) 13.8514 1.50240
\(86\) 0 0
\(87\) −7.67001 −0.822311
\(88\) 0 0
\(89\) 1.04368 0.110630 0.0553148 0.998469i \(-0.482384\pi\)
0.0553148 + 0.998469i \(0.482384\pi\)
\(90\) 0 0
\(91\) −7.67136 −0.804178
\(92\) 0 0
\(93\) −3.96119 −0.410756
\(94\) 0 0
\(95\) 2.13486 0.219032
\(96\) 0 0
\(97\) −0.117594 −0.0119398 −0.00596992 0.999982i \(-0.501900\pi\)
−0.00596992 + 0.999982i \(0.501900\pi\)
\(98\) 0 0
\(99\) 21.2039 2.13107
\(100\) 0 0
\(101\) 10.6142 1.05615 0.528077 0.849197i \(-0.322913\pi\)
0.528077 + 0.849197i \(0.322913\pi\)
\(102\) 0 0
\(103\) −12.4190 −1.22368 −0.611840 0.790982i \(-0.709570\pi\)
−0.611840 + 0.790982i \(0.709570\pi\)
\(104\) 0 0
\(105\) 20.7605 2.02602
\(106\) 0 0
\(107\) 16.5439 1.59936 0.799678 0.600430i \(-0.205004\pi\)
0.799678 + 0.600430i \(0.205004\pi\)
\(108\) 0 0
\(109\) −17.4437 −1.67080 −0.835401 0.549641i \(-0.814765\pi\)
−0.835401 + 0.549641i \(0.814765\pi\)
\(110\) 0 0
\(111\) 11.0092 1.04494
\(112\) 0 0
\(113\) −14.3193 −1.34705 −0.673523 0.739166i \(-0.735220\pi\)
−0.673523 + 0.739166i \(0.735220\pi\)
\(114\) 0 0
\(115\) 15.6411 1.45854
\(116\) 0 0
\(117\) 13.3002 1.22961
\(118\) 0 0
\(119\) 21.3764 1.95957
\(120\) 0 0
\(121\) 2.77968 0.252698
\(122\) 0 0
\(123\) −19.2560 −1.73625
\(124\) 0 0
\(125\) 11.6187 1.03921
\(126\) 0 0
\(127\) 2.63985 0.234248 0.117124 0.993117i \(-0.462632\pi\)
0.117124 + 0.993117i \(0.462632\pi\)
\(128\) 0 0
\(129\) 5.82065 0.512480
\(130\) 0 0
\(131\) −15.3670 −1.34263 −0.671313 0.741174i \(-0.734269\pi\)
−0.671313 + 0.741174i \(0.734269\pi\)
\(132\) 0 0
\(133\) 3.29464 0.285682
\(134\) 0 0
\(135\) −17.0898 −1.47085
\(136\) 0 0
\(137\) 4.91853 0.420218 0.210109 0.977678i \(-0.432618\pi\)
0.210109 + 0.977678i \(0.432618\pi\)
\(138\) 0 0
\(139\) −9.80377 −0.831545 −0.415773 0.909469i \(-0.636489\pi\)
−0.415773 + 0.909469i \(0.636489\pi\)
\(140\) 0 0
\(141\) 16.1156 1.35718
\(142\) 0 0
\(143\) 8.64338 0.722796
\(144\) 0 0
\(145\) 5.54758 0.460701
\(146\) 0 0
\(147\) 11.3776 0.938407
\(148\) 0 0
\(149\) 0.724147 0.0593244 0.0296622 0.999560i \(-0.490557\pi\)
0.0296622 + 0.999560i \(0.490557\pi\)
\(150\) 0 0
\(151\) 15.7252 1.27970 0.639850 0.768500i \(-0.278997\pi\)
0.639850 + 0.768500i \(0.278997\pi\)
\(152\) 0 0
\(153\) −37.0614 −2.99623
\(154\) 0 0
\(155\) 2.86505 0.230127
\(156\) 0 0
\(157\) 0.141127 0.0112632 0.00563160 0.999984i \(-0.498207\pi\)
0.00563160 + 0.999984i \(0.498207\pi\)
\(158\) 0 0
\(159\) −14.7180 −1.16721
\(160\) 0 0
\(161\) 24.1383 1.90236
\(162\) 0 0
\(163\) 8.41859 0.659395 0.329697 0.944087i \(-0.393053\pi\)
0.329697 + 0.944087i \(0.393053\pi\)
\(164\) 0 0
\(165\) −23.3911 −1.82099
\(166\) 0 0
\(167\) −5.16538 −0.399709 −0.199854 0.979826i \(-0.564047\pi\)
−0.199854 + 0.979826i \(0.564047\pi\)
\(168\) 0 0
\(169\) −7.57839 −0.582953
\(170\) 0 0
\(171\) −5.71210 −0.436815
\(172\) 0 0
\(173\) −3.13988 −0.238721 −0.119360 0.992851i \(-0.538084\pi\)
−0.119360 + 0.992851i \(0.538084\pi\)
\(174\) 0 0
\(175\) 1.45750 0.110177
\(176\) 0 0
\(177\) −28.5463 −2.14567
\(178\) 0 0
\(179\) −18.1898 −1.35957 −0.679786 0.733410i \(-0.737927\pi\)
−0.679786 + 0.733410i \(0.737927\pi\)
\(180\) 0 0
\(181\) 14.0798 1.04654 0.523271 0.852166i \(-0.324712\pi\)
0.523271 + 0.852166i \(0.324712\pi\)
\(182\) 0 0
\(183\) 24.0726 1.77950
\(184\) 0 0
\(185\) −7.96273 −0.585431
\(186\) 0 0
\(187\) −24.0849 −1.76126
\(188\) 0 0
\(189\) −26.3740 −1.91842
\(190\) 0 0
\(191\) 4.87409 0.352677 0.176338 0.984330i \(-0.443575\pi\)
0.176338 + 0.984330i \(0.443575\pi\)
\(192\) 0 0
\(193\) −9.99845 −0.719704 −0.359852 0.933009i \(-0.617173\pi\)
−0.359852 + 0.933009i \(0.617173\pi\)
\(194\) 0 0
\(195\) −14.6722 −1.05070
\(196\) 0 0
\(197\) −19.2209 −1.36943 −0.684716 0.728810i \(-0.740074\pi\)
−0.684716 + 0.728810i \(0.740074\pi\)
\(198\) 0 0
\(199\) 11.7323 0.831683 0.415842 0.909437i \(-0.363487\pi\)
0.415842 + 0.909437i \(0.363487\pi\)
\(200\) 0 0
\(201\) −1.54956 −0.109298
\(202\) 0 0
\(203\) 8.56137 0.600890
\(204\) 0 0
\(205\) 13.9275 0.972737
\(206\) 0 0
\(207\) −41.8498 −2.90876
\(208\) 0 0
\(209\) −3.71210 −0.256771
\(210\) 0 0
\(211\) 0.399383 0.0274947 0.0137473 0.999906i \(-0.495624\pi\)
0.0137473 + 0.999906i \(0.495624\pi\)
\(212\) 0 0
\(213\) −21.1776 −1.45106
\(214\) 0 0
\(215\) −4.20997 −0.287118
\(216\) 0 0
\(217\) 4.42153 0.300153
\(218\) 0 0
\(219\) 18.6876 1.26279
\(220\) 0 0
\(221\) −15.1074 −1.01623
\(222\) 0 0
\(223\) −19.2548 −1.28939 −0.644697 0.764438i \(-0.723017\pi\)
−0.644697 + 0.764438i \(0.723017\pi\)
\(224\) 0 0
\(225\) −2.52694 −0.168463
\(226\) 0 0
\(227\) 20.0414 1.33019 0.665096 0.746758i \(-0.268391\pi\)
0.665096 + 0.746758i \(0.268391\pi\)
\(228\) 0 0
\(229\) −18.2013 −1.20277 −0.601387 0.798958i \(-0.705385\pi\)
−0.601387 + 0.798958i \(0.705385\pi\)
\(230\) 0 0
\(231\) −36.0985 −2.37511
\(232\) 0 0
\(233\) −14.2376 −0.932739 −0.466369 0.884590i \(-0.654438\pi\)
−0.466369 + 0.884590i \(0.654438\pi\)
\(234\) 0 0
\(235\) −11.6561 −0.760362
\(236\) 0 0
\(237\) −25.8457 −1.67886
\(238\) 0 0
\(239\) 8.31368 0.537767 0.268884 0.963173i \(-0.413345\pi\)
0.268884 + 0.963173i \(0.413345\pi\)
\(240\) 0 0
\(241\) −26.1776 −1.68625 −0.843125 0.537718i \(-0.819287\pi\)
−0.843125 + 0.537718i \(0.819287\pi\)
\(242\) 0 0
\(243\) −4.85401 −0.311385
\(244\) 0 0
\(245\) −8.22920 −0.525744
\(246\) 0 0
\(247\) −2.32843 −0.148155
\(248\) 0 0
\(249\) −22.8458 −1.44780
\(250\) 0 0
\(251\) 29.4740 1.86038 0.930191 0.367075i \(-0.119641\pi\)
0.930191 + 0.367075i \(0.119641\pi\)
\(252\) 0 0
\(253\) −27.1967 −1.70984
\(254\) 0 0
\(255\) 40.8842 2.56027
\(256\) 0 0
\(257\) 1.28195 0.0799657 0.0399828 0.999200i \(-0.487270\pi\)
0.0399828 + 0.999200i \(0.487270\pi\)
\(258\) 0 0
\(259\) −12.2886 −0.763575
\(260\) 0 0
\(261\) −14.8433 −0.918777
\(262\) 0 0
\(263\) 29.2736 1.80509 0.902544 0.430597i \(-0.141697\pi\)
0.902544 + 0.430597i \(0.141697\pi\)
\(264\) 0 0
\(265\) 10.6453 0.653933
\(266\) 0 0
\(267\) 3.08055 0.188526
\(268\) 0 0
\(269\) −14.1850 −0.864875 −0.432437 0.901664i \(-0.642346\pi\)
−0.432437 + 0.901664i \(0.642346\pi\)
\(270\) 0 0
\(271\) 26.3185 1.59874 0.799369 0.600840i \(-0.205167\pi\)
0.799369 + 0.600840i \(0.205167\pi\)
\(272\) 0 0
\(273\) −22.6430 −1.37042
\(274\) 0 0
\(275\) −1.64217 −0.0990269
\(276\) 0 0
\(277\) −3.58863 −0.215620 −0.107810 0.994172i \(-0.534384\pi\)
−0.107810 + 0.994172i \(0.534384\pi\)
\(278\) 0 0
\(279\) −7.66584 −0.458942
\(280\) 0 0
\(281\) 4.42597 0.264031 0.132016 0.991248i \(-0.457855\pi\)
0.132016 + 0.991248i \(0.457855\pi\)
\(282\) 0 0
\(283\) −0.493151 −0.0293148 −0.0146574 0.999893i \(-0.504666\pi\)
−0.0146574 + 0.999893i \(0.504666\pi\)
\(284\) 0 0
\(285\) 6.30130 0.373257
\(286\) 0 0
\(287\) 21.4938 1.26874
\(288\) 0 0
\(289\) 25.0970 1.47630
\(290\) 0 0
\(291\) −0.347093 −0.0203469
\(292\) 0 0
\(293\) −12.7228 −0.743275 −0.371637 0.928378i \(-0.621204\pi\)
−0.371637 + 0.928378i \(0.621204\pi\)
\(294\) 0 0
\(295\) 20.6470 1.20211
\(296\) 0 0
\(297\) 29.7157 1.72428
\(298\) 0 0
\(299\) −17.0593 −0.986565
\(300\) 0 0
\(301\) −6.49709 −0.374486
\(302\) 0 0
\(303\) 31.3292 1.79981
\(304\) 0 0
\(305\) −17.4113 −0.996966
\(306\) 0 0
\(307\) −10.2166 −0.583092 −0.291546 0.956557i \(-0.594170\pi\)
−0.291546 + 0.956557i \(0.594170\pi\)
\(308\) 0 0
\(309\) −36.6562 −2.08530
\(310\) 0 0
\(311\) −0.184934 −0.0104867 −0.00524333 0.999986i \(-0.501669\pi\)
−0.00524333 + 0.999986i \(0.501669\pi\)
\(312\) 0 0
\(313\) 19.0733 1.07809 0.539044 0.842278i \(-0.318786\pi\)
0.539044 + 0.842278i \(0.318786\pi\)
\(314\) 0 0
\(315\) 40.1766 2.26369
\(316\) 0 0
\(317\) 0.911635 0.0512025 0.0256013 0.999672i \(-0.491850\pi\)
0.0256013 + 0.999672i \(0.491850\pi\)
\(318\) 0 0
\(319\) −9.64615 −0.540081
\(320\) 0 0
\(321\) 48.8313 2.72550
\(322\) 0 0
\(323\) 6.48822 0.361014
\(324\) 0 0
\(325\) −1.03006 −0.0571376
\(326\) 0 0
\(327\) −51.4872 −2.84725
\(328\) 0 0
\(329\) −17.9885 −0.991736
\(330\) 0 0
\(331\) −2.45230 −0.134791 −0.0673953 0.997726i \(-0.521469\pi\)
−0.0673953 + 0.997726i \(0.521469\pi\)
\(332\) 0 0
\(333\) 21.3054 1.16753
\(334\) 0 0
\(335\) 1.12077 0.0612342
\(336\) 0 0
\(337\) 35.1467 1.91456 0.957282 0.289156i \(-0.0933746\pi\)
0.957282 + 0.289156i \(0.0933746\pi\)
\(338\) 0 0
\(339\) −42.2652 −2.29553
\(340\) 0 0
\(341\) −4.98177 −0.269778
\(342\) 0 0
\(343\) 10.3627 0.559533
\(344\) 0 0
\(345\) 46.1666 2.48552
\(346\) 0 0
\(347\) −5.30643 −0.284864 −0.142432 0.989805i \(-0.545492\pi\)
−0.142432 + 0.989805i \(0.545492\pi\)
\(348\) 0 0
\(349\) −26.8913 −1.43946 −0.719729 0.694255i \(-0.755734\pi\)
−0.719729 + 0.694255i \(0.755734\pi\)
\(350\) 0 0
\(351\) 18.6394 0.994895
\(352\) 0 0
\(353\) −3.21383 −0.171055 −0.0855275 0.996336i \(-0.527258\pi\)
−0.0855275 + 0.996336i \(0.527258\pi\)
\(354\) 0 0
\(355\) 15.3174 0.812961
\(356\) 0 0
\(357\) 63.0951 3.33935
\(358\) 0 0
\(359\) 15.6827 0.827699 0.413850 0.910345i \(-0.364184\pi\)
0.413850 + 0.910345i \(0.364184\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 8.20457 0.430628
\(364\) 0 0
\(365\) −13.5164 −0.707481
\(366\) 0 0
\(367\) 14.3509 0.749111 0.374555 0.927205i \(-0.377795\pi\)
0.374555 + 0.927205i \(0.377795\pi\)
\(368\) 0 0
\(369\) −37.2649 −1.93993
\(370\) 0 0
\(371\) 16.4284 0.852921
\(372\) 0 0
\(373\) 9.08701 0.470508 0.235254 0.971934i \(-0.424408\pi\)
0.235254 + 0.971934i \(0.424408\pi\)
\(374\) 0 0
\(375\) 34.2941 1.77094
\(376\) 0 0
\(377\) −6.05060 −0.311622
\(378\) 0 0
\(379\) 18.2006 0.934900 0.467450 0.884020i \(-0.345173\pi\)
0.467450 + 0.884020i \(0.345173\pi\)
\(380\) 0 0
\(381\) 7.79184 0.399188
\(382\) 0 0
\(383\) −22.4154 −1.14537 −0.572686 0.819775i \(-0.694099\pi\)
−0.572686 + 0.819775i \(0.694099\pi\)
\(384\) 0 0
\(385\) 26.1094 1.33066
\(386\) 0 0
\(387\) 11.2643 0.572599
\(388\) 0 0
\(389\) −34.9213 −1.77058 −0.885291 0.465038i \(-0.846041\pi\)
−0.885291 + 0.465038i \(0.846041\pi\)
\(390\) 0 0
\(391\) 47.5360 2.40400
\(392\) 0 0
\(393\) −45.3578 −2.28800
\(394\) 0 0
\(395\) 18.6937 0.940584
\(396\) 0 0
\(397\) 31.3443 1.57313 0.786563 0.617510i \(-0.211858\pi\)
0.786563 + 0.617510i \(0.211858\pi\)
\(398\) 0 0
\(399\) 9.72456 0.486837
\(400\) 0 0
\(401\) 1.64321 0.0820579 0.0410290 0.999158i \(-0.486936\pi\)
0.0410290 + 0.999158i \(0.486936\pi\)
\(402\) 0 0
\(403\) −3.12484 −0.155659
\(404\) 0 0
\(405\) −13.8590 −0.688660
\(406\) 0 0
\(407\) 13.8456 0.686302
\(408\) 0 0
\(409\) 2.46645 0.121958 0.0609791 0.998139i \(-0.480578\pi\)
0.0609791 + 0.998139i \(0.480578\pi\)
\(410\) 0 0
\(411\) 14.5177 0.716104
\(412\) 0 0
\(413\) 31.8637 1.56791
\(414\) 0 0
\(415\) 16.5240 0.811130
\(416\) 0 0
\(417\) −28.9371 −1.41705
\(418\) 0 0
\(419\) 4.25063 0.207657 0.103828 0.994595i \(-0.466891\pi\)
0.103828 + 0.994595i \(0.466891\pi\)
\(420\) 0 0
\(421\) −31.4900 −1.53473 −0.767364 0.641212i \(-0.778432\pi\)
−0.767364 + 0.641212i \(0.778432\pi\)
\(422\) 0 0
\(423\) 31.1875 1.51639
\(424\) 0 0
\(425\) 2.87029 0.139229
\(426\) 0 0
\(427\) −26.8701 −1.30034
\(428\) 0 0
\(429\) 25.5120 1.23173
\(430\) 0 0
\(431\) −21.0564 −1.01425 −0.507125 0.861872i \(-0.669292\pi\)
−0.507125 + 0.861872i \(0.669292\pi\)
\(432\) 0 0
\(433\) −7.53125 −0.361929 −0.180964 0.983490i \(-0.557922\pi\)
−0.180964 + 0.983490i \(0.557922\pi\)
\(434\) 0 0
\(435\) 16.3744 0.785091
\(436\) 0 0
\(437\) 7.32651 0.350475
\(438\) 0 0
\(439\) −0.823995 −0.0393271 −0.0196636 0.999807i \(-0.506260\pi\)
−0.0196636 + 0.999807i \(0.506260\pi\)
\(440\) 0 0
\(441\) 22.0183 1.04849
\(442\) 0 0
\(443\) 8.58526 0.407898 0.203949 0.978982i \(-0.434622\pi\)
0.203949 + 0.978982i \(0.434622\pi\)
\(444\) 0 0
\(445\) −2.22810 −0.105622
\(446\) 0 0
\(447\) 2.13741 0.101096
\(448\) 0 0
\(449\) 2.49451 0.117723 0.0588615 0.998266i \(-0.481253\pi\)
0.0588615 + 0.998266i \(0.481253\pi\)
\(450\) 0 0
\(451\) −24.2172 −1.14034
\(452\) 0 0
\(453\) 46.4150 2.18076
\(454\) 0 0
\(455\) 16.3773 0.767778
\(456\) 0 0
\(457\) −14.5004 −0.678301 −0.339150 0.940732i \(-0.610140\pi\)
−0.339150 + 0.940732i \(0.610140\pi\)
\(458\) 0 0
\(459\) −51.9389 −2.42430
\(460\) 0 0
\(461\) −23.1930 −1.08020 −0.540102 0.841600i \(-0.681614\pi\)
−0.540102 + 0.841600i \(0.681614\pi\)
\(462\) 0 0
\(463\) 26.0637 1.21128 0.605642 0.795737i \(-0.292916\pi\)
0.605642 + 0.795737i \(0.292916\pi\)
\(464\) 0 0
\(465\) 8.45657 0.392164
\(466\) 0 0
\(467\) 12.1953 0.564332 0.282166 0.959366i \(-0.408947\pi\)
0.282166 + 0.959366i \(0.408947\pi\)
\(468\) 0 0
\(469\) 1.72964 0.0798674
\(470\) 0 0
\(471\) 0.416556 0.0191939
\(472\) 0 0
\(473\) 7.32031 0.336588
\(474\) 0 0
\(475\) 0.442384 0.0202980
\(476\) 0 0
\(477\) −28.4828 −1.30414
\(478\) 0 0
\(479\) 18.1883 0.831045 0.415523 0.909583i \(-0.363599\pi\)
0.415523 + 0.909583i \(0.363599\pi\)
\(480\) 0 0
\(481\) 8.68475 0.395990
\(482\) 0 0
\(483\) 71.2471 3.24186
\(484\) 0 0
\(485\) 0.251046 0.0113994
\(486\) 0 0
\(487\) 2.13693 0.0968334 0.0484167 0.998827i \(-0.484582\pi\)
0.0484167 + 0.998827i \(0.484582\pi\)
\(488\) 0 0
\(489\) 24.8485 1.12369
\(490\) 0 0
\(491\) 2.32773 0.105049 0.0525244 0.998620i \(-0.483273\pi\)
0.0525244 + 0.998620i \(0.483273\pi\)
\(492\) 0 0
\(493\) 16.8601 0.759341
\(494\) 0 0
\(495\) −45.2673 −2.03461
\(496\) 0 0
\(497\) 23.6387 1.06034
\(498\) 0 0
\(499\) 16.9382 0.758260 0.379130 0.925343i \(-0.376223\pi\)
0.379130 + 0.925343i \(0.376223\pi\)
\(500\) 0 0
\(501\) −15.2463 −0.681153
\(502\) 0 0
\(503\) −8.22413 −0.366696 −0.183348 0.983048i \(-0.558693\pi\)
−0.183348 + 0.983048i \(0.558693\pi\)
\(504\) 0 0
\(505\) −22.6598 −1.00835
\(506\) 0 0
\(507\) −22.3686 −0.993424
\(508\) 0 0
\(509\) 19.8145 0.878261 0.439131 0.898423i \(-0.355287\pi\)
0.439131 + 0.898423i \(0.355287\pi\)
\(510\) 0 0
\(511\) −20.8594 −0.922764
\(512\) 0 0
\(513\) −8.00510 −0.353434
\(514\) 0 0
\(515\) 26.5128 1.16829
\(516\) 0 0
\(517\) 20.2677 0.891374
\(518\) 0 0
\(519\) −9.26775 −0.406809
\(520\) 0 0
\(521\) 36.0978 1.58147 0.790737 0.612156i \(-0.209698\pi\)
0.790737 + 0.612156i \(0.209698\pi\)
\(522\) 0 0
\(523\) −4.19475 −0.183424 −0.0917119 0.995786i \(-0.529234\pi\)
−0.0917119 + 0.995786i \(0.529234\pi\)
\(524\) 0 0
\(525\) 4.30199 0.187754
\(526\) 0 0
\(527\) 8.70743 0.379301
\(528\) 0 0
\(529\) 30.6778 1.33382
\(530\) 0 0
\(531\) −55.2438 −2.39738
\(532\) 0 0
\(533\) −15.1903 −0.657967
\(534\) 0 0
\(535\) −35.3188 −1.52696
\(536\) 0 0
\(537\) −53.6896 −2.31688
\(538\) 0 0
\(539\) 14.3090 0.616331
\(540\) 0 0
\(541\) −6.12211 −0.263210 −0.131605 0.991302i \(-0.542013\pi\)
−0.131605 + 0.991302i \(0.542013\pi\)
\(542\) 0 0
\(543\) 41.5582 1.78343
\(544\) 0 0
\(545\) 37.2398 1.59518
\(546\) 0 0
\(547\) −4.88879 −0.209029 −0.104515 0.994523i \(-0.533329\pi\)
−0.104515 + 0.994523i \(0.533329\pi\)
\(548\) 0 0
\(549\) 46.5862 1.98825
\(550\) 0 0
\(551\) 2.59857 0.110703
\(552\) 0 0
\(553\) 28.8493 1.22680
\(554\) 0 0
\(555\) −23.5030 −0.997647
\(556\) 0 0
\(557\) 28.5987 1.21176 0.605882 0.795554i \(-0.292820\pi\)
0.605882 + 0.795554i \(0.292820\pi\)
\(558\) 0 0
\(559\) 4.59171 0.194209
\(560\) 0 0
\(561\) −71.0897 −3.00141
\(562\) 0 0
\(563\) 9.83345 0.414431 0.207215 0.978295i \(-0.433560\pi\)
0.207215 + 0.978295i \(0.433560\pi\)
\(564\) 0 0
\(565\) 30.5697 1.28608
\(566\) 0 0
\(567\) −21.3881 −0.898216
\(568\) 0 0
\(569\) 10.3543 0.434075 0.217037 0.976163i \(-0.430361\pi\)
0.217037 + 0.976163i \(0.430361\pi\)
\(570\) 0 0
\(571\) 36.7988 1.53998 0.769991 0.638055i \(-0.220261\pi\)
0.769991 + 0.638055i \(0.220261\pi\)
\(572\) 0 0
\(573\) 14.3865 0.601005
\(574\) 0 0
\(575\) 3.24113 0.135165
\(576\) 0 0
\(577\) −8.22871 −0.342566 −0.171283 0.985222i \(-0.554791\pi\)
−0.171283 + 0.985222i \(0.554791\pi\)
\(578\) 0 0
\(579\) −29.5117 −1.22646
\(580\) 0 0
\(581\) 25.5008 1.05795
\(582\) 0 0
\(583\) −18.5100 −0.766606
\(584\) 0 0
\(585\) −28.3941 −1.17395
\(586\) 0 0
\(587\) 7.18047 0.296370 0.148185 0.988960i \(-0.452657\pi\)
0.148185 + 0.988960i \(0.452657\pi\)
\(588\) 0 0
\(589\) 1.34204 0.0552976
\(590\) 0 0
\(591\) −56.7329 −2.33368
\(592\) 0 0
\(593\) −24.7687 −1.01713 −0.508564 0.861024i \(-0.669823\pi\)
−0.508564 + 0.861024i \(0.669823\pi\)
\(594\) 0 0
\(595\) −45.6355 −1.87087
\(596\) 0 0
\(597\) 34.6295 1.41729
\(598\) 0 0
\(599\) −41.9307 −1.71324 −0.856621 0.515946i \(-0.827440\pi\)
−0.856621 + 0.515946i \(0.827440\pi\)
\(600\) 0 0
\(601\) −2.10027 −0.0856718 −0.0428359 0.999082i \(-0.513639\pi\)
−0.0428359 + 0.999082i \(0.513639\pi\)
\(602\) 0 0
\(603\) −2.99877 −0.122119
\(604\) 0 0
\(605\) −5.93422 −0.241260
\(606\) 0 0
\(607\) −11.2877 −0.458152 −0.229076 0.973409i \(-0.573570\pi\)
−0.229076 + 0.973409i \(0.573570\pi\)
\(608\) 0 0
\(609\) 25.2700 1.02399
\(610\) 0 0
\(611\) 12.7130 0.514315
\(612\) 0 0
\(613\) 31.4983 1.27220 0.636102 0.771605i \(-0.280546\pi\)
0.636102 + 0.771605i \(0.280546\pi\)
\(614\) 0 0
\(615\) 41.1087 1.65766
\(616\) 0 0
\(617\) 11.4563 0.461214 0.230607 0.973047i \(-0.425929\pi\)
0.230607 + 0.973047i \(0.425929\pi\)
\(618\) 0 0
\(619\) −37.6816 −1.51455 −0.757276 0.653095i \(-0.773470\pi\)
−0.757276 + 0.653095i \(0.773470\pi\)
\(620\) 0 0
\(621\) −58.6495 −2.35352
\(622\) 0 0
\(623\) −3.43855 −0.137763
\(624\) 0 0
\(625\) −22.5924 −0.903695
\(626\) 0 0
\(627\) −10.9567 −0.437570
\(628\) 0 0
\(629\) −24.2002 −0.964925
\(630\) 0 0
\(631\) −6.99670 −0.278534 −0.139267 0.990255i \(-0.544475\pi\)
−0.139267 + 0.990255i \(0.544475\pi\)
\(632\) 0 0
\(633\) 1.17883 0.0468543
\(634\) 0 0
\(635\) −5.63569 −0.223646
\(636\) 0 0
\(637\) 8.97538 0.355617
\(638\) 0 0
\(639\) −40.9837 −1.62129
\(640\) 0 0
\(641\) −6.20749 −0.245181 −0.122591 0.992457i \(-0.539120\pi\)
−0.122591 + 0.992457i \(0.539120\pi\)
\(642\) 0 0
\(643\) 29.4008 1.15945 0.579726 0.814811i \(-0.303159\pi\)
0.579726 + 0.814811i \(0.303159\pi\)
\(644\) 0 0
\(645\) −12.4263 −0.489284
\(646\) 0 0
\(647\) −5.62285 −0.221057 −0.110529 0.993873i \(-0.535254\pi\)
−0.110529 + 0.993873i \(0.535254\pi\)
\(648\) 0 0
\(649\) −35.9011 −1.40924
\(650\) 0 0
\(651\) 13.0507 0.511498
\(652\) 0 0
\(653\) −34.9447 −1.36749 −0.683745 0.729721i \(-0.739650\pi\)
−0.683745 + 0.729721i \(0.739650\pi\)
\(654\) 0 0
\(655\) 32.8064 1.28185
\(656\) 0 0
\(657\) 36.1650 1.41093
\(658\) 0 0
\(659\) 37.8768 1.47547 0.737736 0.675090i \(-0.235895\pi\)
0.737736 + 0.675090i \(0.235895\pi\)
\(660\) 0 0
\(661\) 6.77264 0.263425 0.131713 0.991288i \(-0.457952\pi\)
0.131713 + 0.991288i \(0.457952\pi\)
\(662\) 0 0
\(663\) −44.5914 −1.73179
\(664\) 0 0
\(665\) −7.03360 −0.272751
\(666\) 0 0
\(667\) 19.0385 0.737172
\(668\) 0 0
\(669\) −56.8329 −2.19729
\(670\) 0 0
\(671\) 30.2748 1.16874
\(672\) 0 0
\(673\) 24.6355 0.949630 0.474815 0.880086i \(-0.342515\pi\)
0.474815 + 0.880086i \(0.342515\pi\)
\(674\) 0 0
\(675\) −3.54133 −0.136306
\(676\) 0 0
\(677\) 28.2956 1.08749 0.543744 0.839251i \(-0.317006\pi\)
0.543744 + 0.839251i \(0.317006\pi\)
\(678\) 0 0
\(679\) 0.387430 0.0148682
\(680\) 0 0
\(681\) 59.1546 2.26681
\(682\) 0 0
\(683\) −34.4333 −1.31755 −0.658777 0.752339i \(-0.728926\pi\)
−0.658777 + 0.752339i \(0.728926\pi\)
\(684\) 0 0
\(685\) −10.5004 −0.401198
\(686\) 0 0
\(687\) −53.7234 −2.04967
\(688\) 0 0
\(689\) −11.6105 −0.442325
\(690\) 0 0
\(691\) 2.19928 0.0836646 0.0418323 0.999125i \(-0.486680\pi\)
0.0418323 + 0.999125i \(0.486680\pi\)
\(692\) 0 0
\(693\) −69.8592 −2.65373
\(694\) 0 0
\(695\) 20.9297 0.793907
\(696\) 0 0
\(697\) 42.3282 1.60329
\(698\) 0 0
\(699\) −42.0242 −1.58950
\(700\) 0 0
\(701\) 36.0608 1.36200 0.680999 0.732285i \(-0.261546\pi\)
0.680999 + 0.732285i \(0.261546\pi\)
\(702\) 0 0
\(703\) −3.72986 −0.140675
\(704\) 0 0
\(705\) −34.4045 −1.29575
\(706\) 0 0
\(707\) −34.9701 −1.31518
\(708\) 0 0
\(709\) 10.7299 0.402970 0.201485 0.979492i \(-0.435423\pi\)
0.201485 + 0.979492i \(0.435423\pi\)
\(710\) 0 0
\(711\) −50.0176 −1.87581
\(712\) 0 0
\(713\) 9.83244 0.368228
\(714\) 0 0
\(715\) −18.4524 −0.690080
\(716\) 0 0
\(717\) 24.5389 0.916421
\(718\) 0 0
\(719\) 29.1037 1.08539 0.542693 0.839931i \(-0.317405\pi\)
0.542693 + 0.839931i \(0.317405\pi\)
\(720\) 0 0
\(721\) 40.9162 1.52380
\(722\) 0 0
\(723\) −77.2666 −2.87358
\(724\) 0 0
\(725\) 1.14957 0.0426938
\(726\) 0 0
\(727\) 40.5850 1.50521 0.752607 0.658470i \(-0.228796\pi\)
0.752607 + 0.658470i \(0.228796\pi\)
\(728\) 0 0
\(729\) −33.8025 −1.25195
\(730\) 0 0
\(731\) −12.7949 −0.473235
\(732\) 0 0
\(733\) 18.6206 0.687766 0.343883 0.939012i \(-0.388258\pi\)
0.343883 + 0.939012i \(0.388258\pi\)
\(734\) 0 0
\(735\) −24.2895 −0.895933
\(736\) 0 0
\(737\) −1.94880 −0.0717849
\(738\) 0 0
\(739\) 7.89939 0.290584 0.145292 0.989389i \(-0.453588\pi\)
0.145292 + 0.989389i \(0.453588\pi\)
\(740\) 0 0
\(741\) −6.87267 −0.252474
\(742\) 0 0
\(743\) 14.4064 0.528521 0.264261 0.964451i \(-0.414872\pi\)
0.264261 + 0.964451i \(0.414872\pi\)
\(744\) 0 0
\(745\) −1.54595 −0.0566392
\(746\) 0 0
\(747\) −44.2121 −1.61764
\(748\) 0 0
\(749\) −54.5061 −1.99161
\(750\) 0 0
\(751\) 14.0914 0.514204 0.257102 0.966384i \(-0.417232\pi\)
0.257102 + 0.966384i \(0.417232\pi\)
\(752\) 0 0
\(753\) 86.9962 3.17032
\(754\) 0 0
\(755\) −33.5711 −1.22178
\(756\) 0 0
\(757\) −50.2311 −1.82568 −0.912840 0.408318i \(-0.866116\pi\)
−0.912840 + 0.408318i \(0.866116\pi\)
\(758\) 0 0
\(759\) −80.2746 −2.91378
\(760\) 0 0
\(761\) −23.7483 −0.860875 −0.430437 0.902620i \(-0.641641\pi\)
−0.430437 + 0.902620i \(0.641641\pi\)
\(762\) 0 0
\(763\) 57.4707 2.08058
\(764\) 0 0
\(765\) 79.1207 2.86062
\(766\) 0 0
\(767\) −22.5191 −0.813119
\(768\) 0 0
\(769\) −47.4035 −1.70942 −0.854708 0.519109i \(-0.826264\pi\)
−0.854708 + 0.519109i \(0.826264\pi\)
\(770\) 0 0
\(771\) 3.78383 0.136271
\(772\) 0 0
\(773\) −11.8508 −0.426242 −0.213121 0.977026i \(-0.568363\pi\)
−0.213121 + 0.977026i \(0.568363\pi\)
\(774\) 0 0
\(775\) 0.593696 0.0213262
\(776\) 0 0
\(777\) −36.2713 −1.30123
\(778\) 0 0
\(779\) 6.52385 0.233741
\(780\) 0 0
\(781\) −26.6339 −0.953036
\(782\) 0 0
\(783\) −20.8018 −0.743396
\(784\) 0 0
\(785\) −0.301287 −0.0107534
\(786\) 0 0
\(787\) 26.3903 0.940713 0.470356 0.882477i \(-0.344125\pi\)
0.470356 + 0.882477i \(0.344125\pi\)
\(788\) 0 0
\(789\) 86.4048 3.07609
\(790\) 0 0
\(791\) 47.1770 1.67742
\(792\) 0 0
\(793\) 18.9900 0.674356
\(794\) 0 0
\(795\) 31.4208 1.11438
\(796\) 0 0
\(797\) 14.1893 0.502610 0.251305 0.967908i \(-0.419140\pi\)
0.251305 + 0.967908i \(0.419140\pi\)
\(798\) 0 0
\(799\) −35.4251 −1.25325
\(800\) 0 0
\(801\) 5.96159 0.210643
\(802\) 0 0
\(803\) 23.5024 0.829382
\(804\) 0 0
\(805\) −51.5317 −1.81625
\(806\) 0 0
\(807\) −41.8688 −1.47385
\(808\) 0 0
\(809\) 29.9998 1.05474 0.527368 0.849637i \(-0.323179\pi\)
0.527368 + 0.849637i \(0.323179\pi\)
\(810\) 0 0
\(811\) 39.8833 1.40049 0.700246 0.713901i \(-0.253073\pi\)
0.700246 + 0.713901i \(0.253073\pi\)
\(812\) 0 0
\(813\) 77.6825 2.72445
\(814\) 0 0
\(815\) −17.9725 −0.629549
\(816\) 0 0
\(817\) −1.97202 −0.0689921
\(818\) 0 0
\(819\) −43.8196 −1.53118
\(820\) 0 0
\(821\) −56.4395 −1.96975 −0.984877 0.173256i \(-0.944571\pi\)
−0.984877 + 0.173256i \(0.944571\pi\)
\(822\) 0 0
\(823\) 41.7360 1.45482 0.727412 0.686201i \(-0.240723\pi\)
0.727412 + 0.686201i \(0.240723\pi\)
\(824\) 0 0
\(825\) −4.84709 −0.168754
\(826\) 0 0
\(827\) −17.4248 −0.605921 −0.302960 0.953003i \(-0.597975\pi\)
−0.302960 + 0.953003i \(0.597975\pi\)
\(828\) 0 0
\(829\) −52.2723 −1.81549 −0.907747 0.419519i \(-0.862199\pi\)
−0.907747 + 0.419519i \(0.862199\pi\)
\(830\) 0 0
\(831\) −10.5923 −0.367443
\(832\) 0 0
\(833\) −25.0100 −0.866547
\(834\) 0 0
\(835\) 11.0273 0.381617
\(836\) 0 0
\(837\) −10.7431 −0.371337
\(838\) 0 0
\(839\) −40.8429 −1.41005 −0.705027 0.709180i \(-0.749065\pi\)
−0.705027 + 0.709180i \(0.749065\pi\)
\(840\) 0 0
\(841\) −22.2474 −0.767153
\(842\) 0 0
\(843\) 13.0638 0.449941
\(844\) 0 0
\(845\) 16.1788 0.556567
\(846\) 0 0
\(847\) −9.15805 −0.314674
\(848\) 0 0
\(849\) −1.45560 −0.0499560
\(850\) 0 0
\(851\) −27.3269 −0.936754
\(852\) 0 0
\(853\) 42.8153 1.46597 0.732984 0.680245i \(-0.238127\pi\)
0.732984 + 0.680245i \(0.238127\pi\)
\(854\) 0 0
\(855\) 12.1945 0.417044
\(856\) 0 0
\(857\) −23.2947 −0.795732 −0.397866 0.917444i \(-0.630249\pi\)
−0.397866 + 0.917444i \(0.630249\pi\)
\(858\) 0 0
\(859\) −17.5879 −0.600091 −0.300045 0.953925i \(-0.597002\pi\)
−0.300045 + 0.953925i \(0.597002\pi\)
\(860\) 0 0
\(861\) 63.4415 2.16208
\(862\) 0 0
\(863\) −11.8886 −0.404692 −0.202346 0.979314i \(-0.564857\pi\)
−0.202346 + 0.979314i \(0.564857\pi\)
\(864\) 0 0
\(865\) 6.70319 0.227915
\(866\) 0 0
\(867\) 74.0770 2.51579
\(868\) 0 0
\(869\) −32.5048 −1.10265
\(870\) 0 0
\(871\) −1.22239 −0.0414193
\(872\) 0 0
\(873\) −0.671707 −0.0227338
\(874\) 0 0
\(875\) −38.2795 −1.29408
\(876\) 0 0
\(877\) 28.6698 0.968111 0.484056 0.875037i \(-0.339163\pi\)
0.484056 + 0.875037i \(0.339163\pi\)
\(878\) 0 0
\(879\) −37.5530 −1.26663
\(880\) 0 0
\(881\) −44.2297 −1.49014 −0.745069 0.666988i \(-0.767583\pi\)
−0.745069 + 0.666988i \(0.767583\pi\)
\(882\) 0 0
\(883\) −26.5869 −0.894719 −0.447359 0.894354i \(-0.647636\pi\)
−0.447359 + 0.894354i \(0.647636\pi\)
\(884\) 0 0
\(885\) 60.9422 2.04855
\(886\) 0 0
\(887\) −16.4124 −0.551074 −0.275537 0.961290i \(-0.588856\pi\)
−0.275537 + 0.961290i \(0.588856\pi\)
\(888\) 0 0
\(889\) −8.69735 −0.291700
\(890\) 0 0
\(891\) 24.0981 0.807317
\(892\) 0 0
\(893\) −5.45991 −0.182709
\(894\) 0 0
\(895\) 38.8327 1.29803
\(896\) 0 0
\(897\) −50.3527 −1.68123
\(898\) 0 0
\(899\) 3.48737 0.116310
\(900\) 0 0
\(901\) 32.3529 1.07783
\(902\) 0 0
\(903\) −19.1770 −0.638170
\(904\) 0 0
\(905\) −30.0583 −0.999172
\(906\) 0 0
\(907\) 47.3380 1.57183 0.785916 0.618333i \(-0.212192\pi\)
0.785916 + 0.618333i \(0.212192\pi\)
\(908\) 0 0
\(909\) 60.6294 2.01095
\(910\) 0 0
\(911\) 5.30896 0.175894 0.0879469 0.996125i \(-0.471969\pi\)
0.0879469 + 0.996125i \(0.471969\pi\)
\(912\) 0 0
\(913\) −28.7320 −0.950889
\(914\) 0 0
\(915\) −51.3915 −1.69895
\(916\) 0 0
\(917\) 50.6290 1.67192
\(918\) 0 0
\(919\) −49.2513 −1.62465 −0.812326 0.583204i \(-0.801799\pi\)
−0.812326 + 0.583204i \(0.801799\pi\)
\(920\) 0 0
\(921\) −30.1556 −0.993661
\(922\) 0 0
\(923\) −16.7063 −0.549893
\(924\) 0 0
\(925\) −1.65003 −0.0542528
\(926\) 0 0
\(927\) −70.9385 −2.32993
\(928\) 0 0
\(929\) −29.2990 −0.961270 −0.480635 0.876921i \(-0.659594\pi\)
−0.480635 + 0.876921i \(0.659594\pi\)
\(930\) 0 0
\(931\) −3.85468 −0.126332
\(932\) 0 0
\(933\) −0.545857 −0.0178706
\(934\) 0 0
\(935\) 51.4179 1.68154
\(936\) 0 0
\(937\) 6.35345 0.207558 0.103779 0.994600i \(-0.466907\pi\)
0.103779 + 0.994600i \(0.466907\pi\)
\(938\) 0 0
\(939\) 56.2973 1.83719
\(940\) 0 0
\(941\) 2.07523 0.0676505 0.0338253 0.999428i \(-0.489231\pi\)
0.0338253 + 0.999428i \(0.489231\pi\)
\(942\) 0 0
\(943\) 47.7970 1.55649
\(944\) 0 0
\(945\) 56.3047 1.83159
\(946\) 0 0
\(947\) 13.3618 0.434201 0.217101 0.976149i \(-0.430340\pi\)
0.217101 + 0.976149i \(0.430340\pi\)
\(948\) 0 0
\(949\) 14.7420 0.478546
\(950\) 0 0
\(951\) 2.69081 0.0872554
\(952\) 0 0
\(953\) 52.6796 1.70646 0.853231 0.521534i \(-0.174640\pi\)
0.853231 + 0.521534i \(0.174640\pi\)
\(954\) 0 0
\(955\) −10.4055 −0.336714
\(956\) 0 0
\(957\) −28.4718 −0.920364
\(958\) 0 0
\(959\) −16.2048 −0.523281
\(960\) 0 0
\(961\) −29.1989 −0.941901
\(962\) 0 0
\(963\) 94.5002 3.04522
\(964\) 0 0
\(965\) 21.3453 0.687128
\(966\) 0 0
\(967\) 7.17519 0.230739 0.115369 0.993323i \(-0.463195\pi\)
0.115369 + 0.993323i \(0.463195\pi\)
\(968\) 0 0
\(969\) 19.1508 0.615213
\(970\) 0 0
\(971\) −14.6991 −0.471718 −0.235859 0.971787i \(-0.575790\pi\)
−0.235859 + 0.971787i \(0.575790\pi\)
\(972\) 0 0
\(973\) 32.3000 1.03549
\(974\) 0 0
\(975\) −3.04036 −0.0973695
\(976\) 0 0
\(977\) 19.5470 0.625365 0.312682 0.949858i \(-0.398772\pi\)
0.312682 + 0.949858i \(0.398772\pi\)
\(978\) 0 0
\(979\) 3.87424 0.123821
\(980\) 0 0
\(981\) −99.6401 −3.18126
\(982\) 0 0
\(983\) −24.0339 −0.766562 −0.383281 0.923632i \(-0.625206\pi\)
−0.383281 + 0.923632i \(0.625206\pi\)
\(984\) 0 0
\(985\) 41.0339 1.30745
\(986\) 0 0
\(987\) −53.0952 −1.69004
\(988\) 0 0
\(989\) −14.4480 −0.459419
\(990\) 0 0
\(991\) 50.5419 1.60552 0.802758 0.596305i \(-0.203365\pi\)
0.802758 + 0.596305i \(0.203365\pi\)
\(992\) 0 0
\(993\) −7.23827 −0.229700
\(994\) 0 0
\(995\) −25.0469 −0.794039
\(996\) 0 0
\(997\) 21.9225 0.694293 0.347147 0.937811i \(-0.387151\pi\)
0.347147 + 0.937811i \(0.387151\pi\)
\(998\) 0 0
\(999\) 29.8580 0.944664
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 4864.2.a.bo.1.8 8
4.3 odd 2 4864.2.a.bn.1.1 8
8.3 odd 2 4864.2.a.bp.1.8 8
8.5 even 2 4864.2.a.bq.1.1 8
16.3 odd 4 608.2.c.b.305.2 16
16.5 even 4 152.2.c.b.77.3 16
16.11 odd 4 608.2.c.b.305.15 16
16.13 even 4 152.2.c.b.77.4 yes 16
48.5 odd 4 1368.2.g.b.685.14 16
48.11 even 4 5472.2.g.b.2737.13 16
48.29 odd 4 1368.2.g.b.685.13 16
48.35 even 4 5472.2.g.b.2737.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.3 16 16.5 even 4
152.2.c.b.77.4 yes 16 16.13 even 4
608.2.c.b.305.2 16 16.3 odd 4
608.2.c.b.305.15 16 16.11 odd 4
1368.2.g.b.685.13 16 48.29 odd 4
1368.2.g.b.685.14 16 48.5 odd 4
4864.2.a.bn.1.1 8 4.3 odd 2
4864.2.a.bo.1.8 8 1.1 even 1 trivial
4864.2.a.bp.1.8 8 8.3 odd 2
4864.2.a.bq.1.1 8 8.5 even 2
5472.2.g.b.2737.4 16 48.35 even 4
5472.2.g.b.2737.13 16 48.11 even 4