Properties

Label 8016.2.a.w.1.2
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.47270\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.01562 q^{5} +1.84677 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.01562 q^{5} +1.84677 q^{7} +1.00000 q^{9} +0.0702134 q^{11} +1.68809 q^{13} -3.01562 q^{15} -2.17362 q^{17} +2.90752 q^{19} +1.84677 q^{21} -1.67315 q^{23} +4.09394 q^{25} +1.00000 q^{27} -4.24297 q^{29} +4.28787 q^{31} +0.0702134 q^{33} -5.56914 q^{35} -10.8489 q^{37} +1.68809 q^{39} -7.20331 q^{41} +4.33921 q^{43} -3.01562 q^{45} -11.5944 q^{47} -3.58945 q^{49} -2.17362 q^{51} +3.88527 q^{53} -0.211737 q^{55} +2.90752 q^{57} -5.40850 q^{59} +2.01512 q^{61} +1.84677 q^{63} -5.09062 q^{65} +6.98849 q^{67} -1.67315 q^{69} -2.15920 q^{71} -8.46808 q^{73} +4.09394 q^{75} +0.129668 q^{77} +10.8442 q^{79} +1.00000 q^{81} +9.35160 q^{83} +6.55479 q^{85} -4.24297 q^{87} +16.2378 q^{89} +3.11750 q^{91} +4.28787 q^{93} -8.76795 q^{95} +13.2480 q^{97} +0.0702134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} + O(q^{10}) \) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} - q^{11} - 2q^{13} - 3q^{15} + 11q^{17} - 2q^{19} - 8q^{21} - 17q^{23} + 4q^{25} + 7q^{27} - 7q^{29} - 10q^{31} - q^{33} - 10q^{35} - 21q^{37} - 2q^{39} + 8q^{41} + 12q^{43} - 3q^{45} - 25q^{47} - 7q^{49} + 11q^{51} - 7q^{53} - 15q^{55} - 2q^{57} - 3q^{59} - 14q^{61} - 8q^{63} + 4q^{65} - 4q^{67} - 17q^{69} - 27q^{71} - 12q^{73} + 4q^{75} + 16q^{77} - 8q^{79} + 7q^{81} - 15q^{83} - 3q^{85} - 7q^{87} + 14q^{89} + 3q^{91} - 10q^{93} - 37q^{95} + 3q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) −3.01562 −1.34862 −0.674312 0.738446i \(-0.735560\pi\)
−0.674312 + 0.738446i \(0.735560\pi\)
\(6\) 0 0
\(7\) 1.84677 0.698012 0.349006 0.937120i \(-0.386519\pi\)
0.349006 + 0.937120i \(0.386519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.0702134 0.0211701 0.0105851 0.999944i \(-0.496631\pi\)
0.0105851 + 0.999944i \(0.496631\pi\)
\(12\) 0 0
\(13\) 1.68809 0.468191 0.234096 0.972214i \(-0.424787\pi\)
0.234096 + 0.972214i \(0.424787\pi\)
\(14\) 0 0
\(15\) −3.01562 −0.778629
\(16\) 0 0
\(17\) −2.17362 −0.527179 −0.263590 0.964635i \(-0.584907\pi\)
−0.263590 + 0.964635i \(0.584907\pi\)
\(18\) 0 0
\(19\) 2.90752 0.667030 0.333515 0.942745i \(-0.391765\pi\)
0.333515 + 0.942745i \(0.391765\pi\)
\(20\) 0 0
\(21\) 1.84677 0.402998
\(22\) 0 0
\(23\) −1.67315 −0.348876 −0.174438 0.984668i \(-0.555811\pi\)
−0.174438 + 0.984668i \(0.555811\pi\)
\(24\) 0 0
\(25\) 4.09394 0.818788
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.24297 −0.787899 −0.393950 0.919132i \(-0.628892\pi\)
−0.393950 + 0.919132i \(0.628892\pi\)
\(30\) 0 0
\(31\) 4.28787 0.770124 0.385062 0.922891i \(-0.374180\pi\)
0.385062 + 0.922891i \(0.374180\pi\)
\(32\) 0 0
\(33\) 0.0702134 0.0122226
\(34\) 0 0
\(35\) −5.56914 −0.941356
\(36\) 0 0
\(37\) −10.8489 −1.78355 −0.891776 0.452477i \(-0.850540\pi\)
−0.891776 + 0.452477i \(0.850540\pi\)
\(38\) 0 0
\(39\) 1.68809 0.270310
\(40\) 0 0
\(41\) −7.20331 −1.12497 −0.562484 0.826808i \(-0.690154\pi\)
−0.562484 + 0.826808i \(0.690154\pi\)
\(42\) 0 0
\(43\) 4.33921 0.661724 0.330862 0.943679i \(-0.392661\pi\)
0.330862 + 0.943679i \(0.392661\pi\)
\(44\) 0 0
\(45\) −3.01562 −0.449541
\(46\) 0 0
\(47\) −11.5944 −1.69122 −0.845609 0.533803i \(-0.820762\pi\)
−0.845609 + 0.533803i \(0.820762\pi\)
\(48\) 0 0
\(49\) −3.58945 −0.512779
\(50\) 0 0
\(51\) −2.17362 −0.304367
\(52\) 0 0
\(53\) 3.88527 0.533683 0.266841 0.963740i \(-0.414020\pi\)
0.266841 + 0.963740i \(0.414020\pi\)
\(54\) 0 0
\(55\) −0.211737 −0.0285505
\(56\) 0 0
\(57\) 2.90752 0.385110
\(58\) 0 0
\(59\) −5.40850 −0.704127 −0.352063 0.935976i \(-0.614520\pi\)
−0.352063 + 0.935976i \(0.614520\pi\)
\(60\) 0 0
\(61\) 2.01512 0.258010 0.129005 0.991644i \(-0.458822\pi\)
0.129005 + 0.991644i \(0.458822\pi\)
\(62\) 0 0
\(63\) 1.84677 0.232671
\(64\) 0 0
\(65\) −5.09062 −0.631414
\(66\) 0 0
\(67\) 6.98849 0.853780 0.426890 0.904304i \(-0.359609\pi\)
0.426890 + 0.904304i \(0.359609\pi\)
\(68\) 0 0
\(69\) −1.67315 −0.201424
\(70\) 0 0
\(71\) −2.15920 −0.256250 −0.128125 0.991758i \(-0.540896\pi\)
−0.128125 + 0.991758i \(0.540896\pi\)
\(72\) 0 0
\(73\) −8.46808 −0.991114 −0.495557 0.868575i \(-0.665036\pi\)
−0.495557 + 0.868575i \(0.665036\pi\)
\(74\) 0 0
\(75\) 4.09394 0.472727
\(76\) 0 0
\(77\) 0.129668 0.0147770
\(78\) 0 0
\(79\) 10.8442 1.22007 0.610035 0.792374i \(-0.291155\pi\)
0.610035 + 0.792374i \(0.291155\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.35160 1.02647 0.513236 0.858248i \(-0.328447\pi\)
0.513236 + 0.858248i \(0.328447\pi\)
\(84\) 0 0
\(85\) 6.55479 0.710967
\(86\) 0 0
\(87\) −4.24297 −0.454894
\(88\) 0 0
\(89\) 16.2378 1.72120 0.860601 0.509279i \(-0.170088\pi\)
0.860601 + 0.509279i \(0.170088\pi\)
\(90\) 0 0
\(91\) 3.11750 0.326803
\(92\) 0 0
\(93\) 4.28787 0.444631
\(94\) 0 0
\(95\) −8.76795 −0.899573
\(96\) 0 0
\(97\) 13.2480 1.34513 0.672563 0.740040i \(-0.265193\pi\)
0.672563 + 0.740040i \(0.265193\pi\)
\(98\) 0 0
\(99\) 0.0702134 0.00705671
\(100\) 0 0
\(101\) −5.67392 −0.564577 −0.282288 0.959330i \(-0.591093\pi\)
−0.282288 + 0.959330i \(0.591093\pi\)
\(102\) 0 0
\(103\) −17.5307 −1.72735 −0.863677 0.504046i \(-0.831844\pi\)
−0.863677 + 0.504046i \(0.831844\pi\)
\(104\) 0 0
\(105\) −5.56914 −0.543492
\(106\) 0 0
\(107\) −2.56293 −0.247768 −0.123884 0.992297i \(-0.539535\pi\)
−0.123884 + 0.992297i \(0.539535\pi\)
\(108\) 0 0
\(109\) 18.3817 1.76065 0.880325 0.474370i \(-0.157324\pi\)
0.880325 + 0.474370i \(0.157324\pi\)
\(110\) 0 0
\(111\) −10.8489 −1.02973
\(112\) 0 0
\(113\) 8.64947 0.813674 0.406837 0.913501i \(-0.366632\pi\)
0.406837 + 0.913501i \(0.366632\pi\)
\(114\) 0 0
\(115\) 5.04558 0.470503
\(116\) 0 0
\(117\) 1.68809 0.156064
\(118\) 0 0
\(119\) −4.01416 −0.367978
\(120\) 0 0
\(121\) −10.9951 −0.999552
\(122\) 0 0
\(123\) −7.20331 −0.649500
\(124\) 0 0
\(125\) 2.73233 0.244387
\(126\) 0 0
\(127\) 0.114006 0.0101164 0.00505819 0.999987i \(-0.498390\pi\)
0.00505819 + 0.999987i \(0.498390\pi\)
\(128\) 0 0
\(129\) 4.33921 0.382046
\(130\) 0 0
\(131\) −12.5617 −1.09752 −0.548759 0.835981i \(-0.684899\pi\)
−0.548759 + 0.835981i \(0.684899\pi\)
\(132\) 0 0
\(133\) 5.36950 0.465595
\(134\) 0 0
\(135\) −3.01562 −0.259543
\(136\) 0 0
\(137\) −6.59365 −0.563333 −0.281667 0.959512i \(-0.590887\pi\)
−0.281667 + 0.959512i \(0.590887\pi\)
\(138\) 0 0
\(139\) −5.80181 −0.492103 −0.246051 0.969257i \(-0.579133\pi\)
−0.246051 + 0.969257i \(0.579133\pi\)
\(140\) 0 0
\(141\) −11.5944 −0.976425
\(142\) 0 0
\(143\) 0.118526 0.00991167
\(144\) 0 0
\(145\) 12.7952 1.06258
\(146\) 0 0
\(147\) −3.58945 −0.296053
\(148\) 0 0
\(149\) 4.34532 0.355983 0.177991 0.984032i \(-0.443040\pi\)
0.177991 + 0.984032i \(0.443040\pi\)
\(150\) 0 0
\(151\) −17.8907 −1.45592 −0.727961 0.685619i \(-0.759532\pi\)
−0.727961 + 0.685619i \(0.759532\pi\)
\(152\) 0 0
\(153\) −2.17362 −0.175726
\(154\) 0 0
\(155\) −12.9306 −1.03861
\(156\) 0 0
\(157\) −17.9620 −1.43352 −0.716760 0.697320i \(-0.754376\pi\)
−0.716760 + 0.697320i \(0.754376\pi\)
\(158\) 0 0
\(159\) 3.88527 0.308122
\(160\) 0 0
\(161\) −3.08992 −0.243520
\(162\) 0 0
\(163\) 7.95764 0.623291 0.311645 0.950198i \(-0.399120\pi\)
0.311645 + 0.950198i \(0.399120\pi\)
\(164\) 0 0
\(165\) −0.211737 −0.0164837
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −10.1504 −0.780797
\(170\) 0 0
\(171\) 2.90752 0.222343
\(172\) 0 0
\(173\) −13.0823 −0.994632 −0.497316 0.867570i \(-0.665681\pi\)
−0.497316 + 0.867570i \(0.665681\pi\)
\(174\) 0 0
\(175\) 7.56055 0.571524
\(176\) 0 0
\(177\) −5.40850 −0.406528
\(178\) 0 0
\(179\) −10.4300 −0.779575 −0.389788 0.920905i \(-0.627452\pi\)
−0.389788 + 0.920905i \(0.627452\pi\)
\(180\) 0 0
\(181\) −4.31004 −0.320362 −0.160181 0.987088i \(-0.551208\pi\)
−0.160181 + 0.987088i \(0.551208\pi\)
\(182\) 0 0
\(183\) 2.01512 0.148962
\(184\) 0 0
\(185\) 32.7162 2.40534
\(186\) 0 0
\(187\) −0.152617 −0.0111605
\(188\) 0 0
\(189\) 1.84677 0.134333
\(190\) 0 0
\(191\) −18.7287 −1.35516 −0.677581 0.735448i \(-0.736972\pi\)
−0.677581 + 0.735448i \(0.736972\pi\)
\(192\) 0 0
\(193\) 15.3962 1.10824 0.554121 0.832436i \(-0.313054\pi\)
0.554121 + 0.832436i \(0.313054\pi\)
\(194\) 0 0
\(195\) −5.09062 −0.364547
\(196\) 0 0
\(197\) 0.0789986 0.00562842 0.00281421 0.999996i \(-0.499104\pi\)
0.00281421 + 0.999996i \(0.499104\pi\)
\(198\) 0 0
\(199\) −11.4542 −0.811967 −0.405983 0.913880i \(-0.633071\pi\)
−0.405983 + 0.913880i \(0.633071\pi\)
\(200\) 0 0
\(201\) 6.98849 0.492930
\(202\) 0 0
\(203\) −7.83577 −0.549963
\(204\) 0 0
\(205\) 21.7224 1.51716
\(206\) 0 0
\(207\) −1.67315 −0.116292
\(208\) 0 0
\(209\) 0.204146 0.0141211
\(210\) 0 0
\(211\) −15.3500 −1.05674 −0.528370 0.849014i \(-0.677197\pi\)
−0.528370 + 0.849014i \(0.677197\pi\)
\(212\) 0 0
\(213\) −2.15920 −0.147946
\(214\) 0 0
\(215\) −13.0854 −0.892417
\(216\) 0 0
\(217\) 7.91869 0.537556
\(218\) 0 0
\(219\) −8.46808 −0.572220
\(220\) 0 0
\(221\) −3.66926 −0.246821
\(222\) 0 0
\(223\) −24.6893 −1.65332 −0.826658 0.562705i \(-0.809761\pi\)
−0.826658 + 0.562705i \(0.809761\pi\)
\(224\) 0 0
\(225\) 4.09394 0.272929
\(226\) 0 0
\(227\) −24.5728 −1.63096 −0.815478 0.578788i \(-0.803526\pi\)
−0.815478 + 0.578788i \(0.803526\pi\)
\(228\) 0 0
\(229\) −8.98726 −0.593895 −0.296947 0.954894i \(-0.595969\pi\)
−0.296947 + 0.954894i \(0.595969\pi\)
\(230\) 0 0
\(231\) 0.129668 0.00853151
\(232\) 0 0
\(233\) −22.6629 −1.48470 −0.742348 0.670014i \(-0.766288\pi\)
−0.742348 + 0.670014i \(0.766288\pi\)
\(234\) 0 0
\(235\) 34.9643 2.28082
\(236\) 0 0
\(237\) 10.8442 0.704408
\(238\) 0 0
\(239\) −7.63603 −0.493934 −0.246967 0.969024i \(-0.579434\pi\)
−0.246967 + 0.969024i \(0.579434\pi\)
\(240\) 0 0
\(241\) −16.3487 −1.05311 −0.526555 0.850141i \(-0.676516\pi\)
−0.526555 + 0.850141i \(0.676516\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 10.8244 0.691546
\(246\) 0 0
\(247\) 4.90814 0.312298
\(248\) 0 0
\(249\) 9.35160 0.592633
\(250\) 0 0
\(251\) 25.3237 1.59842 0.799209 0.601053i \(-0.205252\pi\)
0.799209 + 0.601053i \(0.205252\pi\)
\(252\) 0 0
\(253\) −0.117478 −0.00738575
\(254\) 0 0
\(255\) 6.55479 0.410477
\(256\) 0 0
\(257\) 15.8241 0.987082 0.493541 0.869723i \(-0.335702\pi\)
0.493541 + 0.869723i \(0.335702\pi\)
\(258\) 0 0
\(259\) −20.0354 −1.24494
\(260\) 0 0
\(261\) −4.24297 −0.262633
\(262\) 0 0
\(263\) −19.1324 −1.17975 −0.589877 0.807493i \(-0.700824\pi\)
−0.589877 + 0.807493i \(0.700824\pi\)
\(264\) 0 0
\(265\) −11.7165 −0.719738
\(266\) 0 0
\(267\) 16.2378 0.993737
\(268\) 0 0
\(269\) 7.78513 0.474668 0.237334 0.971428i \(-0.423726\pi\)
0.237334 + 0.971428i \(0.423726\pi\)
\(270\) 0 0
\(271\) 11.7908 0.716237 0.358119 0.933676i \(-0.383418\pi\)
0.358119 + 0.933676i \(0.383418\pi\)
\(272\) 0 0
\(273\) 3.11750 0.188680
\(274\) 0 0
\(275\) 0.287449 0.0173338
\(276\) 0 0
\(277\) 2.54189 0.152728 0.0763638 0.997080i \(-0.475669\pi\)
0.0763638 + 0.997080i \(0.475669\pi\)
\(278\) 0 0
\(279\) 4.28787 0.256708
\(280\) 0 0
\(281\) −7.37140 −0.439741 −0.219870 0.975529i \(-0.570563\pi\)
−0.219870 + 0.975529i \(0.570563\pi\)
\(282\) 0 0
\(283\) −8.42659 −0.500909 −0.250455 0.968128i \(-0.580580\pi\)
−0.250455 + 0.968128i \(0.580580\pi\)
\(284\) 0 0
\(285\) −8.76795 −0.519369
\(286\) 0 0
\(287\) −13.3028 −0.785241
\(288\) 0 0
\(289\) −12.2754 −0.722082
\(290\) 0 0
\(291\) 13.2480 0.776609
\(292\) 0 0
\(293\) 17.3212 1.01192 0.505958 0.862558i \(-0.331139\pi\)
0.505958 + 0.862558i \(0.331139\pi\)
\(294\) 0 0
\(295\) 16.3100 0.949603
\(296\) 0 0
\(297\) 0.0702134 0.00407419
\(298\) 0 0
\(299\) −2.82442 −0.163341
\(300\) 0 0
\(301\) 8.01351 0.461891
\(302\) 0 0
\(303\) −5.67392 −0.325958
\(304\) 0 0
\(305\) −6.07684 −0.347959
\(306\) 0 0
\(307\) −6.40036 −0.365288 −0.182644 0.983179i \(-0.558466\pi\)
−0.182644 + 0.983179i \(0.558466\pi\)
\(308\) 0 0
\(309\) −17.5307 −0.997288
\(310\) 0 0
\(311\) −17.9862 −1.01991 −0.509953 0.860202i \(-0.670337\pi\)
−0.509953 + 0.860202i \(0.670337\pi\)
\(312\) 0 0
\(313\) −21.4712 −1.21363 −0.606813 0.794844i \(-0.707552\pi\)
−0.606813 + 0.794844i \(0.707552\pi\)
\(314\) 0 0
\(315\) −5.56914 −0.313785
\(316\) 0 0
\(317\) −22.8128 −1.28130 −0.640648 0.767834i \(-0.721334\pi\)
−0.640648 + 0.767834i \(0.721334\pi\)
\(318\) 0 0
\(319\) −0.297913 −0.0166799
\(320\) 0 0
\(321\) −2.56293 −0.143049
\(322\) 0 0
\(323\) −6.31982 −0.351644
\(324\) 0 0
\(325\) 6.91093 0.383349
\(326\) 0 0
\(327\) 18.3817 1.01651
\(328\) 0 0
\(329\) −21.4122 −1.18049
\(330\) 0 0
\(331\) 22.1607 1.21806 0.609031 0.793147i \(-0.291559\pi\)
0.609031 + 0.793147i \(0.291559\pi\)
\(332\) 0 0
\(333\) −10.8489 −0.594517
\(334\) 0 0
\(335\) −21.0746 −1.15143
\(336\) 0 0
\(337\) 32.6497 1.77854 0.889271 0.457380i \(-0.151212\pi\)
0.889271 + 0.457380i \(0.151212\pi\)
\(338\) 0 0
\(339\) 8.64947 0.469775
\(340\) 0 0
\(341\) 0.301066 0.0163036
\(342\) 0 0
\(343\) −19.5562 −1.05594
\(344\) 0 0
\(345\) 5.04558 0.271645
\(346\) 0 0
\(347\) 21.5493 1.15683 0.578414 0.815744i \(-0.303672\pi\)
0.578414 + 0.815744i \(0.303672\pi\)
\(348\) 0 0
\(349\) −1.96612 −0.105244 −0.0526219 0.998615i \(-0.516758\pi\)
−0.0526219 + 0.998615i \(0.516758\pi\)
\(350\) 0 0
\(351\) 1.68809 0.0901035
\(352\) 0 0
\(353\) 0.411409 0.0218971 0.0109485 0.999940i \(-0.496515\pi\)
0.0109485 + 0.999940i \(0.496515\pi\)
\(354\) 0 0
\(355\) 6.51131 0.345585
\(356\) 0 0
\(357\) −4.01416 −0.212452
\(358\) 0 0
\(359\) −17.1536 −0.905334 −0.452667 0.891680i \(-0.649527\pi\)
−0.452667 + 0.891680i \(0.649527\pi\)
\(360\) 0 0
\(361\) −10.5464 −0.555071
\(362\) 0 0
\(363\) −10.9951 −0.577092
\(364\) 0 0
\(365\) 25.5365 1.33664
\(366\) 0 0
\(367\) 23.3616 1.21947 0.609733 0.792607i \(-0.291277\pi\)
0.609733 + 0.792607i \(0.291277\pi\)
\(368\) 0 0
\(369\) −7.20331 −0.374989
\(370\) 0 0
\(371\) 7.17519 0.372517
\(372\) 0 0
\(373\) 10.3568 0.536257 0.268129 0.963383i \(-0.413595\pi\)
0.268129 + 0.963383i \(0.413595\pi\)
\(374\) 0 0
\(375\) 2.73233 0.141097
\(376\) 0 0
\(377\) −7.16250 −0.368888
\(378\) 0 0
\(379\) −1.75887 −0.0903471 −0.0451735 0.998979i \(-0.514384\pi\)
−0.0451735 + 0.998979i \(0.514384\pi\)
\(380\) 0 0
\(381\) 0.114006 0.00584069
\(382\) 0 0
\(383\) −2.23977 −0.114447 −0.0572233 0.998361i \(-0.518225\pi\)
−0.0572233 + 0.998361i \(0.518225\pi\)
\(384\) 0 0
\(385\) −0.391028 −0.0199286
\(386\) 0 0
\(387\) 4.33921 0.220575
\(388\) 0 0
\(389\) 10.8084 0.548008 0.274004 0.961729i \(-0.411652\pi\)
0.274004 + 0.961729i \(0.411652\pi\)
\(390\) 0 0
\(391\) 3.63679 0.183920
\(392\) 0 0
\(393\) −12.5617 −0.633652
\(394\) 0 0
\(395\) −32.7020 −1.64542
\(396\) 0 0
\(397\) −11.3168 −0.567975 −0.283987 0.958828i \(-0.591657\pi\)
−0.283987 + 0.958828i \(0.591657\pi\)
\(398\) 0 0
\(399\) 5.36950 0.268811
\(400\) 0 0
\(401\) −15.5327 −0.775667 −0.387834 0.921729i \(-0.626776\pi\)
−0.387834 + 0.921729i \(0.626776\pi\)
\(402\) 0 0
\(403\) 7.23830 0.360565
\(404\) 0 0
\(405\) −3.01562 −0.149847
\(406\) 0 0
\(407\) −0.761739 −0.0377580
\(408\) 0 0
\(409\) −16.0150 −0.791890 −0.395945 0.918274i \(-0.629583\pi\)
−0.395945 + 0.918274i \(0.629583\pi\)
\(410\) 0 0
\(411\) −6.59365 −0.325241
\(412\) 0 0
\(413\) −9.98824 −0.491489
\(414\) 0 0
\(415\) −28.2008 −1.38432
\(416\) 0 0
\(417\) −5.80181 −0.284116
\(418\) 0 0
\(419\) 4.60543 0.224990 0.112495 0.993652i \(-0.464116\pi\)
0.112495 + 0.993652i \(0.464116\pi\)
\(420\) 0 0
\(421\) −23.7515 −1.15758 −0.578789 0.815478i \(-0.696474\pi\)
−0.578789 + 0.815478i \(0.696474\pi\)
\(422\) 0 0
\(423\) −11.5944 −0.563739
\(424\) 0 0
\(425\) −8.89865 −0.431648
\(426\) 0 0
\(427\) 3.72146 0.180094
\(428\) 0 0
\(429\) 0.118526 0.00572250
\(430\) 0 0
\(431\) 8.50119 0.409488 0.204744 0.978816i \(-0.434364\pi\)
0.204744 + 0.978816i \(0.434364\pi\)
\(432\) 0 0
\(433\) 27.1642 1.30543 0.652715 0.757604i \(-0.273630\pi\)
0.652715 + 0.757604i \(0.273630\pi\)
\(434\) 0 0
\(435\) 12.7952 0.613481
\(436\) 0 0
\(437\) −4.86471 −0.232711
\(438\) 0 0
\(439\) −31.9205 −1.52348 −0.761740 0.647882i \(-0.775655\pi\)
−0.761740 + 0.647882i \(0.775655\pi\)
\(440\) 0 0
\(441\) −3.58945 −0.170926
\(442\) 0 0
\(443\) −4.34160 −0.206276 −0.103138 0.994667i \(-0.532888\pi\)
−0.103138 + 0.994667i \(0.532888\pi\)
\(444\) 0 0
\(445\) −48.9669 −2.32126
\(446\) 0 0
\(447\) 4.34532 0.205527
\(448\) 0 0
\(449\) −10.9358 −0.516093 −0.258046 0.966133i \(-0.583079\pi\)
−0.258046 + 0.966133i \(0.583079\pi\)
\(450\) 0 0
\(451\) −0.505768 −0.0238157
\(452\) 0 0
\(453\) −17.8907 −0.840577
\(454\) 0 0
\(455\) −9.40120 −0.440735
\(456\) 0 0
\(457\) 23.9782 1.12165 0.560826 0.827933i \(-0.310484\pi\)
0.560826 + 0.827933i \(0.310484\pi\)
\(458\) 0 0
\(459\) −2.17362 −0.101456
\(460\) 0 0
\(461\) 6.39210 0.297710 0.148855 0.988859i \(-0.452441\pi\)
0.148855 + 0.988859i \(0.452441\pi\)
\(462\) 0 0
\(463\) −19.8982 −0.924748 −0.462374 0.886685i \(-0.653002\pi\)
−0.462374 + 0.886685i \(0.653002\pi\)
\(464\) 0 0
\(465\) −12.9306 −0.599640
\(466\) 0 0
\(467\) −6.50454 −0.300994 −0.150497 0.988610i \(-0.548087\pi\)
−0.150497 + 0.988610i \(0.548087\pi\)
\(468\) 0 0
\(469\) 12.9061 0.595949
\(470\) 0 0
\(471\) −17.9620 −0.827643
\(472\) 0 0
\(473\) 0.304671 0.0140088
\(474\) 0 0
\(475\) 11.9032 0.546156
\(476\) 0 0
\(477\) 3.88527 0.177894
\(478\) 0 0
\(479\) −1.93092 −0.0882259 −0.0441130 0.999027i \(-0.514046\pi\)
−0.0441130 + 0.999027i \(0.514046\pi\)
\(480\) 0 0
\(481\) −18.3139 −0.835044
\(482\) 0 0
\(483\) −3.08992 −0.140596
\(484\) 0 0
\(485\) −39.9508 −1.81407
\(486\) 0 0
\(487\) −22.3754 −1.01392 −0.506962 0.861968i \(-0.669232\pi\)
−0.506962 + 0.861968i \(0.669232\pi\)
\(488\) 0 0
\(489\) 7.95764 0.359857
\(490\) 0 0
\(491\) −0.000619001 0 −2.79351e−5 0 −1.39676e−5 1.00000i \(-0.500004\pi\)
−1.39676e−5 1.00000i \(0.500004\pi\)
\(492\) 0 0
\(493\) 9.22259 0.415364
\(494\) 0 0
\(495\) −0.211737 −0.00951685
\(496\) 0 0
\(497\) −3.98754 −0.178865
\(498\) 0 0
\(499\) 38.6573 1.73054 0.865269 0.501309i \(-0.167148\pi\)
0.865269 + 0.501309i \(0.167148\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −23.2786 −1.03794 −0.518971 0.854792i \(-0.673685\pi\)
−0.518971 + 0.854792i \(0.673685\pi\)
\(504\) 0 0
\(505\) 17.1104 0.761402
\(506\) 0 0
\(507\) −10.1504 −0.450793
\(508\) 0 0
\(509\) 14.8713 0.659157 0.329578 0.944128i \(-0.393093\pi\)
0.329578 + 0.944128i \(0.393093\pi\)
\(510\) 0 0
\(511\) −15.6386 −0.691810
\(512\) 0 0
\(513\) 2.90752 0.128370
\(514\) 0 0
\(515\) 52.8659 2.32955
\(516\) 0 0
\(517\) −0.814082 −0.0358033
\(518\) 0 0
\(519\) −13.0823 −0.574251
\(520\) 0 0
\(521\) 10.4692 0.458663 0.229331 0.973348i \(-0.426346\pi\)
0.229331 + 0.973348i \(0.426346\pi\)
\(522\) 0 0
\(523\) 6.32882 0.276740 0.138370 0.990381i \(-0.455814\pi\)
0.138370 + 0.990381i \(0.455814\pi\)
\(524\) 0 0
\(525\) 7.56055 0.329969
\(526\) 0 0
\(527\) −9.32018 −0.405993
\(528\) 0 0
\(529\) −20.2006 −0.878286
\(530\) 0 0
\(531\) −5.40850 −0.234709
\(532\) 0 0
\(533\) −12.1598 −0.526700
\(534\) 0 0
\(535\) 7.72882 0.334146
\(536\) 0 0
\(537\) −10.4300 −0.450088
\(538\) 0 0
\(539\) −0.252027 −0.0108556
\(540\) 0 0
\(541\) 11.5446 0.496340 0.248170 0.968716i \(-0.420171\pi\)
0.248170 + 0.968716i \(0.420171\pi\)
\(542\) 0 0
\(543\) −4.31004 −0.184961
\(544\) 0 0
\(545\) −55.4322 −2.37446
\(546\) 0 0
\(547\) 20.7041 0.885243 0.442622 0.896708i \(-0.354048\pi\)
0.442622 + 0.896708i \(0.354048\pi\)
\(548\) 0 0
\(549\) 2.01512 0.0860034
\(550\) 0 0
\(551\) −12.3365 −0.525552
\(552\) 0 0
\(553\) 20.0268 0.851625
\(554\) 0 0
\(555\) 32.7162 1.38872
\(556\) 0 0
\(557\) 34.1956 1.44891 0.724457 0.689320i \(-0.242091\pi\)
0.724457 + 0.689320i \(0.242091\pi\)
\(558\) 0 0
\(559\) 7.32497 0.309813
\(560\) 0 0
\(561\) −0.152617 −0.00644349
\(562\) 0 0
\(563\) −13.4440 −0.566598 −0.283299 0.959032i \(-0.591429\pi\)
−0.283299 + 0.959032i \(0.591429\pi\)
\(564\) 0 0
\(565\) −26.0835 −1.09734
\(566\) 0 0
\(567\) 1.84677 0.0775569
\(568\) 0 0
\(569\) 1.42022 0.0595386 0.0297693 0.999557i \(-0.490523\pi\)
0.0297693 + 0.999557i \(0.490523\pi\)
\(570\) 0 0
\(571\) 19.0276 0.796280 0.398140 0.917325i \(-0.369656\pi\)
0.398140 + 0.917325i \(0.369656\pi\)
\(572\) 0 0
\(573\) −18.7287 −0.782404
\(574\) 0 0
\(575\) −6.84977 −0.285655
\(576\) 0 0
\(577\) 30.5919 1.27356 0.636778 0.771047i \(-0.280267\pi\)
0.636778 + 0.771047i \(0.280267\pi\)
\(578\) 0 0
\(579\) 15.3962 0.639844
\(580\) 0 0
\(581\) 17.2702 0.716490
\(582\) 0 0
\(583\) 0.272798 0.0112981
\(584\) 0 0
\(585\) −5.09062 −0.210471
\(586\) 0 0
\(587\) −32.6169 −1.34624 −0.673121 0.739532i \(-0.735047\pi\)
−0.673121 + 0.739532i \(0.735047\pi\)
\(588\) 0 0
\(589\) 12.4670 0.513696
\(590\) 0 0
\(591\) 0.0789986 0.00324957
\(592\) 0 0
\(593\) 16.2145 0.665849 0.332924 0.942953i \(-0.391965\pi\)
0.332924 + 0.942953i \(0.391965\pi\)
\(594\) 0 0
\(595\) 12.1052 0.496264
\(596\) 0 0
\(597\) −11.4542 −0.468789
\(598\) 0 0
\(599\) −20.5160 −0.838263 −0.419131 0.907926i \(-0.637665\pi\)
−0.419131 + 0.907926i \(0.637665\pi\)
\(600\) 0 0
\(601\) 21.1781 0.863874 0.431937 0.901904i \(-0.357830\pi\)
0.431937 + 0.901904i \(0.357830\pi\)
\(602\) 0 0
\(603\) 6.98849 0.284593
\(604\) 0 0
\(605\) 33.1569 1.34802
\(606\) 0 0
\(607\) 0.963530 0.0391085 0.0195542 0.999809i \(-0.493775\pi\)
0.0195542 + 0.999809i \(0.493775\pi\)
\(608\) 0 0
\(609\) −7.83577 −0.317522
\(610\) 0 0
\(611\) −19.5724 −0.791813
\(612\) 0 0
\(613\) −37.3703 −1.50937 −0.754685 0.656087i \(-0.772210\pi\)
−0.754685 + 0.656087i \(0.772210\pi\)
\(614\) 0 0
\(615\) 21.7224 0.875932
\(616\) 0 0
\(617\) −17.9204 −0.721448 −0.360724 0.932673i \(-0.617470\pi\)
−0.360724 + 0.932673i \(0.617470\pi\)
\(618\) 0 0
\(619\) −0.709375 −0.0285122 −0.0142561 0.999898i \(-0.504538\pi\)
−0.0142561 + 0.999898i \(0.504538\pi\)
\(620\) 0 0
\(621\) −1.67315 −0.0671412
\(622\) 0 0
\(623\) 29.9874 1.20142
\(624\) 0 0
\(625\) −28.7094 −1.14837
\(626\) 0 0
\(627\) 0.204146 0.00815282
\(628\) 0 0
\(629\) 23.5814 0.940252
\(630\) 0 0
\(631\) −12.1195 −0.482472 −0.241236 0.970467i \(-0.577553\pi\)
−0.241236 + 0.970467i \(0.577553\pi\)
\(632\) 0 0
\(633\) −15.3500 −0.610109
\(634\) 0 0
\(635\) −0.343798 −0.0136432
\(636\) 0 0
\(637\) −6.05931 −0.240079
\(638\) 0 0
\(639\) −2.15920 −0.0854166
\(640\) 0 0
\(641\) 15.8998 0.628003 0.314002 0.949422i \(-0.398330\pi\)
0.314002 + 0.949422i \(0.398330\pi\)
\(642\) 0 0
\(643\) 23.8562 0.940798 0.470399 0.882454i \(-0.344110\pi\)
0.470399 + 0.882454i \(0.344110\pi\)
\(644\) 0 0
\(645\) −13.0854 −0.515237
\(646\) 0 0
\(647\) −2.97952 −0.117137 −0.0585685 0.998283i \(-0.518654\pi\)
−0.0585685 + 0.998283i \(0.518654\pi\)
\(648\) 0 0
\(649\) −0.379749 −0.0149065
\(650\) 0 0
\(651\) 7.91869 0.310358
\(652\) 0 0
\(653\) 7.97720 0.312172 0.156086 0.987743i \(-0.450112\pi\)
0.156086 + 0.987743i \(0.450112\pi\)
\(654\) 0 0
\(655\) 37.8811 1.48014
\(656\) 0 0
\(657\) −8.46808 −0.330371
\(658\) 0 0
\(659\) 29.9289 1.16586 0.582931 0.812521i \(-0.301906\pi\)
0.582931 + 0.812521i \(0.301906\pi\)
\(660\) 0 0
\(661\) 15.2290 0.592339 0.296170 0.955135i \(-0.404291\pi\)
0.296170 + 0.955135i \(0.404291\pi\)
\(662\) 0 0
\(663\) −3.66926 −0.142502
\(664\) 0 0
\(665\) −16.1924 −0.627913
\(666\) 0 0
\(667\) 7.09912 0.274879
\(668\) 0 0
\(669\) −24.6893 −0.954542
\(670\) 0 0
\(671\) 0.141489 0.00546211
\(672\) 0 0
\(673\) 29.3603 1.13176 0.565878 0.824489i \(-0.308537\pi\)
0.565878 + 0.824489i \(0.308537\pi\)
\(674\) 0 0
\(675\) 4.09394 0.157576
\(676\) 0 0
\(677\) 0.142354 0.00547111 0.00273556 0.999996i \(-0.499129\pi\)
0.00273556 + 0.999996i \(0.499129\pi\)
\(678\) 0 0
\(679\) 24.4659 0.938915
\(680\) 0 0
\(681\) −24.5728 −0.941633
\(682\) 0 0
\(683\) 37.2505 1.42535 0.712676 0.701494i \(-0.247483\pi\)
0.712676 + 0.701494i \(0.247483\pi\)
\(684\) 0 0
\(685\) 19.8839 0.759725
\(686\) 0 0
\(687\) −8.98726 −0.342885
\(688\) 0 0
\(689\) 6.55868 0.249866
\(690\) 0 0
\(691\) −36.4586 −1.38695 −0.693475 0.720481i \(-0.743921\pi\)
−0.693475 + 0.720481i \(0.743921\pi\)
\(692\) 0 0
\(693\) 0.129668 0.00492567
\(694\) 0 0
\(695\) 17.4960 0.663662
\(696\) 0 0
\(697\) 15.6572 0.593060
\(698\) 0 0
\(699\) −22.6629 −0.857190
\(700\) 0 0
\(701\) −35.2147 −1.33004 −0.665020 0.746826i \(-0.731577\pi\)
−0.665020 + 0.746826i \(0.731577\pi\)
\(702\) 0 0
\(703\) −31.5434 −1.18968
\(704\) 0 0
\(705\) 34.9643 1.31683
\(706\) 0 0
\(707\) −10.4784 −0.394081
\(708\) 0 0
\(709\) −38.7522 −1.45537 −0.727684 0.685912i \(-0.759403\pi\)
−0.727684 + 0.685912i \(0.759403\pi\)
\(710\) 0 0
\(711\) 10.8442 0.406690
\(712\) 0 0
\(713\) −7.17425 −0.268678
\(714\) 0 0
\(715\) −0.357430 −0.0133671
\(716\) 0 0
\(717\) −7.63603 −0.285173
\(718\) 0 0
\(719\) 28.2855 1.05487 0.527435 0.849596i \(-0.323154\pi\)
0.527435 + 0.849596i \(0.323154\pi\)
\(720\) 0 0
\(721\) −32.3752 −1.20571
\(722\) 0 0
\(723\) −16.3487 −0.608013
\(724\) 0 0
\(725\) −17.3705 −0.645122
\(726\) 0 0
\(727\) −22.0460 −0.817642 −0.408821 0.912615i \(-0.634060\pi\)
−0.408821 + 0.912615i \(0.634060\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −9.43179 −0.348847
\(732\) 0 0
\(733\) 13.1625 0.486167 0.243084 0.970005i \(-0.421841\pi\)
0.243084 + 0.970005i \(0.421841\pi\)
\(734\) 0 0
\(735\) 10.8244 0.399264
\(736\) 0 0
\(737\) 0.490685 0.0180746
\(738\) 0 0
\(739\) 9.86634 0.362939 0.181470 0.983397i \(-0.441915\pi\)
0.181470 + 0.983397i \(0.441915\pi\)
\(740\) 0 0
\(741\) 4.90814 0.180305
\(742\) 0 0
\(743\) −5.70369 −0.209248 −0.104624 0.994512i \(-0.533364\pi\)
−0.104624 + 0.994512i \(0.533364\pi\)
\(744\) 0 0
\(745\) −13.1038 −0.480087
\(746\) 0 0
\(747\) 9.35160 0.342157
\(748\) 0 0
\(749\) −4.73314 −0.172945
\(750\) 0 0
\(751\) 25.2515 0.921439 0.460719 0.887546i \(-0.347591\pi\)
0.460719 + 0.887546i \(0.347591\pi\)
\(752\) 0 0
\(753\) 25.3237 0.922847
\(754\) 0 0
\(755\) 53.9514 1.96349
\(756\) 0 0
\(757\) 26.6032 0.966910 0.483455 0.875369i \(-0.339382\pi\)
0.483455 + 0.875369i \(0.339382\pi\)
\(758\) 0 0
\(759\) −0.117478 −0.00426416
\(760\) 0 0
\(761\) −38.0804 −1.38041 −0.690207 0.723612i \(-0.742481\pi\)
−0.690207 + 0.723612i \(0.742481\pi\)
\(762\) 0 0
\(763\) 33.9468 1.22896
\(764\) 0 0
\(765\) 6.55479 0.236989
\(766\) 0 0
\(767\) −9.13003 −0.329666
\(768\) 0 0
\(769\) 5.50542 0.198531 0.0992653 0.995061i \(-0.468351\pi\)
0.0992653 + 0.995061i \(0.468351\pi\)
\(770\) 0 0
\(771\) 15.8241 0.569892
\(772\) 0 0
\(773\) −17.0250 −0.612345 −0.306173 0.951976i \(-0.599048\pi\)
−0.306173 + 0.951976i \(0.599048\pi\)
\(774\) 0 0
\(775\) 17.5543 0.630568
\(776\) 0 0
\(777\) −20.0354 −0.718767
\(778\) 0 0
\(779\) −20.9437 −0.750387
\(780\) 0 0
\(781\) −0.151605 −0.00542484
\(782\) 0 0
\(783\) −4.24297 −0.151631
\(784\) 0 0
\(785\) 54.1664 1.93328
\(786\) 0 0
\(787\) −43.9181 −1.56551 −0.782756 0.622329i \(-0.786187\pi\)
−0.782756 + 0.622329i \(0.786187\pi\)
\(788\) 0 0
\(789\) −19.1324 −0.681131
\(790\) 0 0
\(791\) 15.9736 0.567955
\(792\) 0 0
\(793\) 3.40170 0.120798
\(794\) 0 0
\(795\) −11.7165 −0.415541
\(796\) 0 0
\(797\) 55.5931 1.96921 0.984605 0.174795i \(-0.0559264\pi\)
0.984605 + 0.174795i \(0.0559264\pi\)
\(798\) 0 0
\(799\) 25.2018 0.891575
\(800\) 0 0
\(801\) 16.2378 0.573734
\(802\) 0 0
\(803\) −0.594572 −0.0209820
\(804\) 0 0
\(805\) 9.31801 0.328417
\(806\) 0 0
\(807\) 7.78513 0.274050
\(808\) 0 0
\(809\) 39.9154 1.40335 0.701675 0.712497i \(-0.252436\pi\)
0.701675 + 0.712497i \(0.252436\pi\)
\(810\) 0 0
\(811\) 40.6843 1.42862 0.714309 0.699830i \(-0.246741\pi\)
0.714309 + 0.699830i \(0.246741\pi\)
\(812\) 0 0
\(813\) 11.7908 0.413520
\(814\) 0 0
\(815\) −23.9972 −0.840585
\(816\) 0 0
\(817\) 12.6163 0.441390
\(818\) 0 0
\(819\) 3.11750 0.108934
\(820\) 0 0
\(821\) −23.4731 −0.819217 −0.409608 0.912261i \(-0.634335\pi\)
−0.409608 + 0.912261i \(0.634335\pi\)
\(822\) 0 0
\(823\) 6.51601 0.227134 0.113567 0.993530i \(-0.463772\pi\)
0.113567 + 0.993530i \(0.463772\pi\)
\(824\) 0 0
\(825\) 0.287449 0.0100077
\(826\) 0 0
\(827\) −45.4145 −1.57922 −0.789609 0.613611i \(-0.789717\pi\)
−0.789609 + 0.613611i \(0.789717\pi\)
\(828\) 0 0
\(829\) 32.9723 1.14518 0.572588 0.819843i \(-0.305939\pi\)
0.572588 + 0.819843i \(0.305939\pi\)
\(830\) 0 0
\(831\) 2.54189 0.0881773
\(832\) 0 0
\(833\) 7.80209 0.270326
\(834\) 0 0
\(835\) −3.01562 −0.104360
\(836\) 0 0
\(837\) 4.28787 0.148210
\(838\) 0 0
\(839\) 27.4949 0.949230 0.474615 0.880193i \(-0.342587\pi\)
0.474615 + 0.880193i \(0.342587\pi\)
\(840\) 0 0
\(841\) −10.9972 −0.379215
\(842\) 0 0
\(843\) −7.37140 −0.253884
\(844\) 0 0
\(845\) 30.6096 1.05300
\(846\) 0 0
\(847\) −20.3053 −0.697699
\(848\) 0 0
\(849\) −8.42659 −0.289200
\(850\) 0 0
\(851\) 18.1519 0.622238
\(852\) 0 0
\(853\) 1.91679 0.0656298 0.0328149 0.999461i \(-0.489553\pi\)
0.0328149 + 0.999461i \(0.489553\pi\)
\(854\) 0 0
\(855\) −8.76795 −0.299858
\(856\) 0 0
\(857\) −0.386484 −0.0132021 −0.00660103 0.999978i \(-0.502101\pi\)
−0.00660103 + 0.999978i \(0.502101\pi\)
\(858\) 0 0
\(859\) 21.7433 0.741873 0.370937 0.928658i \(-0.379037\pi\)
0.370937 + 0.928658i \(0.379037\pi\)
\(860\) 0 0
\(861\) −13.3028 −0.453359
\(862\) 0 0
\(863\) 40.4063 1.37545 0.687723 0.725973i \(-0.258610\pi\)
0.687723 + 0.725973i \(0.258610\pi\)
\(864\) 0 0
\(865\) 39.4513 1.34138
\(866\) 0 0
\(867\) −12.2754 −0.416894
\(868\) 0 0
\(869\) 0.761410 0.0258291
\(870\) 0 0
\(871\) 11.7972 0.399732
\(872\) 0 0
\(873\) 13.2480 0.448376
\(874\) 0 0
\(875\) 5.04598 0.170585
\(876\) 0 0
\(877\) −47.3266 −1.59811 −0.799053 0.601261i \(-0.794665\pi\)
−0.799053 + 0.601261i \(0.794665\pi\)
\(878\) 0 0
\(879\) 17.3212 0.584229
\(880\) 0 0
\(881\) 40.0078 1.34790 0.673949 0.738778i \(-0.264597\pi\)
0.673949 + 0.738778i \(0.264597\pi\)
\(882\) 0 0
\(883\) −14.2412 −0.479255 −0.239627 0.970865i \(-0.577025\pi\)
−0.239627 + 0.970865i \(0.577025\pi\)
\(884\) 0 0
\(885\) 16.3100 0.548253
\(886\) 0 0
\(887\) −16.2299 −0.544948 −0.272474 0.962163i \(-0.587842\pi\)
−0.272474 + 0.962163i \(0.587842\pi\)
\(888\) 0 0
\(889\) 0.210542 0.00706136
\(890\) 0 0
\(891\) 0.0702134 0.00235224
\(892\) 0 0
\(893\) −33.7109 −1.12809
\(894\) 0 0
\(895\) 31.4529 1.05135
\(896\) 0 0
\(897\) −2.82442 −0.0943048
\(898\) 0 0
\(899\) −18.1933 −0.606780
\(900\) 0 0
\(901\) −8.44509 −0.281347
\(902\) 0 0
\(903\) 8.01351 0.266673
\(904\) 0 0
\(905\) 12.9974 0.432049
\(906\) 0 0
\(907\) −14.4413 −0.479516 −0.239758 0.970833i \(-0.577068\pi\)
−0.239758 + 0.970833i \(0.577068\pi\)
\(908\) 0 0
\(909\) −5.67392 −0.188192
\(910\) 0 0
\(911\) 53.7207 1.77984 0.889922 0.456112i \(-0.150758\pi\)
0.889922 + 0.456112i \(0.150758\pi\)
\(912\) 0 0
\(913\) 0.656607 0.0217305
\(914\) 0 0
\(915\) −6.07684 −0.200894
\(916\) 0 0
\(917\) −23.1985 −0.766081
\(918\) 0 0
\(919\) −29.4559 −0.971661 −0.485831 0.874053i \(-0.661483\pi\)
−0.485831 + 0.874053i \(0.661483\pi\)
\(920\) 0 0
\(921\) −6.40036 −0.210899
\(922\) 0 0
\(923\) −3.64492 −0.119974
\(924\) 0 0
\(925\) −44.4148 −1.46035
\(926\) 0 0
\(927\) −17.5307 −0.575785
\(928\) 0 0
\(929\) −34.9105 −1.14538 −0.572688 0.819773i \(-0.694100\pi\)
−0.572688 + 0.819773i \(0.694100\pi\)
\(930\) 0 0
\(931\) −10.4364 −0.342039
\(932\) 0 0
\(933\) −17.9862 −0.588843
\(934\) 0 0
\(935\) 0.460234 0.0150513
\(936\) 0 0
\(937\) 42.0982 1.37529 0.687644 0.726048i \(-0.258645\pi\)
0.687644 + 0.726048i \(0.258645\pi\)
\(938\) 0 0
\(939\) −21.4712 −0.700688
\(940\) 0 0
\(941\) 16.7714 0.546732 0.273366 0.961910i \(-0.411863\pi\)
0.273366 + 0.961910i \(0.411863\pi\)
\(942\) 0 0
\(943\) 12.0522 0.392474
\(944\) 0 0
\(945\) −5.56914 −0.181164
\(946\) 0 0
\(947\) 32.4958 1.05597 0.527985 0.849254i \(-0.322948\pi\)
0.527985 + 0.849254i \(0.322948\pi\)
\(948\) 0 0
\(949\) −14.2949 −0.464031
\(950\) 0 0
\(951\) −22.8128 −0.739757
\(952\) 0 0
\(953\) 37.5661 1.21689 0.608443 0.793597i \(-0.291794\pi\)
0.608443 + 0.793597i \(0.291794\pi\)
\(954\) 0 0
\(955\) 56.4786 1.82761
\(956\) 0 0
\(957\) −0.297913 −0.00963016
\(958\) 0 0
\(959\) −12.1769 −0.393214
\(960\) 0 0
\(961\) −12.6142 −0.406909
\(962\) 0 0
\(963\) −2.56293 −0.0825894
\(964\) 0 0
\(965\) −46.4290 −1.49460
\(966\) 0 0
\(967\) 5.46025 0.175590 0.0877949 0.996139i \(-0.472018\pi\)
0.0877949 + 0.996139i \(0.472018\pi\)
\(968\) 0 0
\(969\) −6.31982 −0.203022
\(970\) 0 0
\(971\) 34.4319 1.10497 0.552487 0.833522i \(-0.313679\pi\)
0.552487 + 0.833522i \(0.313679\pi\)
\(972\) 0 0
\(973\) −10.7146 −0.343494
\(974\) 0 0
\(975\) 6.91093 0.221327
\(976\) 0 0
\(977\) 13.1807 0.421689 0.210844 0.977520i \(-0.432379\pi\)
0.210844 + 0.977520i \(0.432379\pi\)
\(978\) 0 0
\(979\) 1.14011 0.0364381
\(980\) 0 0
\(981\) 18.3817 0.586884
\(982\) 0 0
\(983\) −10.2408 −0.326632 −0.163316 0.986574i \(-0.552219\pi\)
−0.163316 + 0.986574i \(0.552219\pi\)
\(984\) 0 0
\(985\) −0.238229 −0.00759062
\(986\) 0 0
\(987\) −21.4122 −0.681557
\(988\) 0 0
\(989\) −7.26015 −0.230859
\(990\) 0 0
\(991\) −27.8098 −0.883409 −0.441704 0.897161i \(-0.645626\pi\)
−0.441704 + 0.897161i \(0.645626\pi\)
\(992\) 0 0
\(993\) 22.1607 0.703248
\(994\) 0 0
\(995\) 34.5415 1.09504
\(996\) 0 0
\(997\) 11.4323 0.362065 0.181033 0.983477i \(-0.442056\pi\)
0.181033 + 0.983477i \(0.442056\pi\)
\(998\) 0 0
\(999\) −10.8489 −0.343245
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 8016.2.a.w.1.2 7
4.3 odd 2 4008.2.a.g.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.g.1.2 7 4.3 odd 2
8016.2.a.w.1.2 7 1.1 even 1 trivial