Properties

Label 3856.2.a.j.1.5
Level $3856$
Weight $2$
Character 3856.1
Self dual yes
Analytic conductor $30.790$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3856 = 2^{4} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7903150194\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.48734\) of defining polynomial
Character \(\chi\) \(=\) 3856.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.815004 q^{3} +0.961999 q^{5} +4.61392 q^{7} -2.33577 q^{9} +O(q^{10})\) \(q+0.815004 q^{3} +0.961999 q^{5} +4.61392 q^{7} -2.33577 q^{9} -1.93974 q^{11} -3.85571 q^{13} +0.784033 q^{15} +5.40289 q^{17} +4.17145 q^{19} +3.76036 q^{21} +1.42545 q^{23} -4.07456 q^{25} -4.34867 q^{27} -4.85744 q^{29} +7.24699 q^{31} -1.58090 q^{33} +4.43859 q^{35} +7.12597 q^{37} -3.14242 q^{39} +9.18955 q^{41} -2.93624 q^{43} -2.24701 q^{45} -2.48170 q^{47} +14.2883 q^{49} +4.40338 q^{51} +5.64997 q^{53} -1.86603 q^{55} +3.39975 q^{57} +11.9783 q^{59} -13.9214 q^{61} -10.7771 q^{63} -3.70919 q^{65} +7.30682 q^{67} +1.16175 q^{69} +14.8844 q^{71} +0.240264 q^{73} -3.32078 q^{75} -8.94983 q^{77} +3.15128 q^{79} +3.46313 q^{81} +2.46821 q^{83} +5.19758 q^{85} -3.95883 q^{87} +5.55181 q^{89} -17.7900 q^{91} +5.90632 q^{93} +4.01293 q^{95} +5.27964 q^{97} +4.53079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + O(q^{10}) \) \( 7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + 18q^{11} - q^{13} + 11q^{15} - 2q^{17} + 6q^{19} - 2q^{21} + 22q^{23} + 5q^{25} - 3q^{27} - 16q^{29} + 18q^{31} + 4q^{33} - 7q^{35} + 8q^{37} + 9q^{39} - 15q^{41} - 14q^{43} + 3q^{45} + 10q^{47} + 6q^{49} - 13q^{51} + 15q^{53} - 29q^{55} + 14q^{57} + 18q^{59} + 4q^{61} + 16q^{63} - 7q^{65} - 18q^{67} + 26q^{69} + 50q^{71} - 16q^{75} + 17q^{77} + 15q^{79} - 9q^{81} + 24q^{83} - 2q^{85} - 12q^{87} - 13q^{89} + 12q^{91} + 14q^{93} + 41q^{95} + q^{97} + 20q^{99} + O(q^{100}) \)

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
\(3\) 0.815004 0.470543 0.235271 0.971930i \(-0.424402\pi\)
0.235271 + 0.971930i \(0.424402\pi\)
\(4\) 0 0
\(5\) 0.961999 0.430219 0.215110 0.976590i \(-0.430989\pi\)
0.215110 + 0.976590i \(0.430989\pi\)
\(6\) 0 0
\(7\) 4.61392 1.74390 0.871950 0.489596i \(-0.162856\pi\)
0.871950 + 0.489596i \(0.162856\pi\)
\(8\) 0 0
\(9\) −2.33577 −0.778590
\(10\) 0 0
\(11\) −1.93974 −0.584855 −0.292427 0.956288i \(-0.594463\pi\)
−0.292427 + 0.956288i \(0.594463\pi\)
\(12\) 0 0
\(13\) −3.85571 −1.06938 −0.534691 0.845048i \(-0.679572\pi\)
−0.534691 + 0.845048i \(0.679572\pi\)
\(14\) 0 0
\(15\) 0.784033 0.202436
\(16\) 0 0
\(17\) 5.40289 1.31039 0.655197 0.755458i \(-0.272586\pi\)
0.655197 + 0.755458i \(0.272586\pi\)
\(18\) 0 0
\(19\) 4.17145 0.956997 0.478498 0.878088i \(-0.341181\pi\)
0.478498 + 0.878088i \(0.341181\pi\)
\(20\) 0 0
\(21\) 3.76036 0.820579
\(22\) 0 0
\(23\) 1.42545 0.297227 0.148614 0.988895i \(-0.452519\pi\)
0.148614 + 0.988895i \(0.452519\pi\)
\(24\) 0 0
\(25\) −4.07456 −0.814911
\(26\) 0 0
\(27\) −4.34867 −0.836902
\(28\) 0 0
\(29\) −4.85744 −0.902003 −0.451002 0.892523i \(-0.648933\pi\)
−0.451002 + 0.892523i \(0.648933\pi\)
\(30\) 0 0
\(31\) 7.24699 1.30160 0.650799 0.759250i \(-0.274434\pi\)
0.650799 + 0.759250i \(0.274434\pi\)
\(32\) 0 0
\(33\) −1.58090 −0.275199
\(34\) 0 0
\(35\) 4.43859 0.750259
\(36\) 0 0
\(37\) 7.12597 1.17150 0.585751 0.810491i \(-0.300800\pi\)
0.585751 + 0.810491i \(0.300800\pi\)
\(38\) 0 0
\(39\) −3.14242 −0.503190
\(40\) 0 0
\(41\) 9.18955 1.43517 0.717583 0.696473i \(-0.245248\pi\)
0.717583 + 0.696473i \(0.245248\pi\)
\(42\) 0 0
\(43\) −2.93624 −0.447772 −0.223886 0.974615i \(-0.571874\pi\)
−0.223886 + 0.974615i \(0.571874\pi\)
\(44\) 0 0
\(45\) −2.24701 −0.334964
\(46\) 0 0
\(47\) −2.48170 −0.361994 −0.180997 0.983484i \(-0.557932\pi\)
−0.180997 + 0.983484i \(0.557932\pi\)
\(48\) 0 0
\(49\) 14.2883 2.04118
\(50\) 0 0
\(51\) 4.40338 0.616596
\(52\) 0 0
\(53\) 5.64997 0.776083 0.388041 0.921642i \(-0.373152\pi\)
0.388041 + 0.921642i \(0.373152\pi\)
\(54\) 0 0
\(55\) −1.86603 −0.251616
\(56\) 0 0
\(57\) 3.39975 0.450308
\(58\) 0 0
\(59\) 11.9783 1.55944 0.779718 0.626131i \(-0.215362\pi\)
0.779718 + 0.626131i \(0.215362\pi\)
\(60\) 0 0
\(61\) −13.9214 −1.78245 −0.891223 0.453565i \(-0.850152\pi\)
−0.891223 + 0.453565i \(0.850152\pi\)
\(62\) 0 0
\(63\) −10.7771 −1.35778
\(64\) 0 0
\(65\) −3.70919 −0.460069
\(66\) 0 0
\(67\) 7.30682 0.892670 0.446335 0.894866i \(-0.352729\pi\)
0.446335 + 0.894866i \(0.352729\pi\)
\(68\) 0 0
\(69\) 1.16175 0.139858
\(70\) 0 0
\(71\) 14.8844 1.76645 0.883226 0.468948i \(-0.155367\pi\)
0.883226 + 0.468948i \(0.155367\pi\)
\(72\) 0 0
\(73\) 0.240264 0.0281208 0.0140604 0.999901i \(-0.495524\pi\)
0.0140604 + 0.999901i \(0.495524\pi\)
\(74\) 0 0
\(75\) −3.32078 −0.383450
\(76\) 0 0
\(77\) −8.94983 −1.01993
\(78\) 0 0
\(79\) 3.15128 0.354546 0.177273 0.984162i \(-0.443272\pi\)
0.177273 + 0.984162i \(0.443272\pi\)
\(80\) 0 0
\(81\) 3.46313 0.384792
\(82\) 0 0
\(83\) 2.46821 0.270922 0.135461 0.990783i \(-0.456749\pi\)
0.135461 + 0.990783i \(0.456749\pi\)
\(84\) 0 0
\(85\) 5.19758 0.563757
\(86\) 0 0
\(87\) −3.95883 −0.424431
\(88\) 0 0
\(89\) 5.55181 0.588491 0.294246 0.955730i \(-0.404932\pi\)
0.294246 + 0.955730i \(0.404932\pi\)
\(90\) 0 0
\(91\) −17.7900 −1.86489
\(92\) 0 0
\(93\) 5.90632 0.612457
\(94\) 0 0
\(95\) 4.01293 0.411718
\(96\) 0 0
\(97\) 5.27964 0.536067 0.268033 0.963410i \(-0.413626\pi\)
0.268033 + 0.963410i \(0.413626\pi\)
\(98\) 0 0
\(99\) 4.53079 0.455362
\(100\) 0 0
\(101\) −12.4239 −1.23623 −0.618113 0.786089i \(-0.712103\pi\)
−0.618113 + 0.786089i \(0.712103\pi\)
\(102\) 0 0
\(103\) 9.80533 0.966148 0.483074 0.875580i \(-0.339520\pi\)
0.483074 + 0.875580i \(0.339520\pi\)
\(104\) 0 0
\(105\) 3.61747 0.353029
\(106\) 0 0
\(107\) 9.15031 0.884593 0.442297 0.896869i \(-0.354164\pi\)
0.442297 + 0.896869i \(0.354164\pi\)
\(108\) 0 0
\(109\) 3.24534 0.310847 0.155424 0.987848i \(-0.450326\pi\)
0.155424 + 0.987848i \(0.450326\pi\)
\(110\) 0 0
\(111\) 5.80769 0.551241
\(112\) 0 0
\(113\) −0.842874 −0.0792909 −0.0396455 0.999214i \(-0.512623\pi\)
−0.0396455 + 0.999214i \(0.512623\pi\)
\(114\) 0 0
\(115\) 1.37128 0.127873
\(116\) 0 0
\(117\) 9.00605 0.832610
\(118\) 0 0
\(119\) 24.9285 2.28520
\(120\) 0 0
\(121\) −7.23740 −0.657945
\(122\) 0 0
\(123\) 7.48952 0.675307
\(124\) 0 0
\(125\) −8.72972 −0.780810
\(126\) 0 0
\(127\) −11.3416 −1.00641 −0.503203 0.864168i \(-0.667845\pi\)
−0.503203 + 0.864168i \(0.667845\pi\)
\(128\) 0 0
\(129\) −2.39305 −0.210696
\(130\) 0 0
\(131\) −14.1261 −1.23421 −0.617103 0.786882i \(-0.711694\pi\)
−0.617103 + 0.786882i \(0.711694\pi\)
\(132\) 0 0
\(133\) 19.2468 1.66891
\(134\) 0 0
\(135\) −4.18342 −0.360051
\(136\) 0 0
\(137\) −12.9555 −1.10686 −0.553430 0.832896i \(-0.686681\pi\)
−0.553430 + 0.832896i \(0.686681\pi\)
\(138\) 0 0
\(139\) −8.23606 −0.698574 −0.349287 0.937016i \(-0.613576\pi\)
−0.349287 + 0.937016i \(0.613576\pi\)
\(140\) 0 0
\(141\) −2.02260 −0.170333
\(142\) 0 0
\(143\) 7.47909 0.625433
\(144\) 0 0
\(145\) −4.67285 −0.388059
\(146\) 0 0
\(147\) 11.6450 0.960464
\(148\) 0 0
\(149\) 4.48606 0.367513 0.183756 0.982972i \(-0.441174\pi\)
0.183756 + 0.982972i \(0.441174\pi\)
\(150\) 0 0
\(151\) −0.0933624 −0.00759773 −0.00379886 0.999993i \(-0.501209\pi\)
−0.00379886 + 0.999993i \(0.501209\pi\)
\(152\) 0 0
\(153\) −12.6199 −1.02026
\(154\) 0 0
\(155\) 6.97160 0.559972
\(156\) 0 0
\(157\) −10.1337 −0.808760 −0.404380 0.914591i \(-0.632513\pi\)
−0.404380 + 0.914591i \(0.632513\pi\)
\(158\) 0 0
\(159\) 4.60474 0.365180
\(160\) 0 0
\(161\) 6.57693 0.518334
\(162\) 0 0
\(163\) −7.49804 −0.587292 −0.293646 0.955914i \(-0.594869\pi\)
−0.293646 + 0.955914i \(0.594869\pi\)
\(164\) 0 0
\(165\) −1.52082 −0.118396
\(166\) 0 0
\(167\) −17.7922 −1.37680 −0.688399 0.725332i \(-0.741686\pi\)
−0.688399 + 0.725332i \(0.741686\pi\)
\(168\) 0 0
\(169\) 1.86650 0.143577
\(170\) 0 0
\(171\) −9.74355 −0.745108
\(172\) 0 0
\(173\) 18.4927 1.40597 0.702987 0.711203i \(-0.251849\pi\)
0.702987 + 0.711203i \(0.251849\pi\)
\(174\) 0 0
\(175\) −18.7997 −1.42112
\(176\) 0 0
\(177\) 9.76232 0.733781
\(178\) 0 0
\(179\) 17.3387 1.29595 0.647977 0.761660i \(-0.275615\pi\)
0.647977 + 0.761660i \(0.275615\pi\)
\(180\) 0 0
\(181\) 17.7124 1.31655 0.658275 0.752778i \(-0.271287\pi\)
0.658275 + 0.752778i \(0.271287\pi\)
\(182\) 0 0
\(183\) −11.3460 −0.838717
\(184\) 0 0
\(185\) 6.85518 0.504003
\(186\) 0 0
\(187\) −10.4802 −0.766390
\(188\) 0 0
\(189\) −20.0644 −1.45947
\(190\) 0 0
\(191\) −13.2440 −0.958303 −0.479151 0.877732i \(-0.659056\pi\)
−0.479151 + 0.877732i \(0.659056\pi\)
\(192\) 0 0
\(193\) −25.5030 −1.83574 −0.917872 0.396877i \(-0.870094\pi\)
−0.917872 + 0.396877i \(0.870094\pi\)
\(194\) 0 0
\(195\) −3.02300 −0.216482
\(196\) 0 0
\(197\) −15.0303 −1.07087 −0.535433 0.844578i \(-0.679851\pi\)
−0.535433 + 0.844578i \(0.679851\pi\)
\(198\) 0 0
\(199\) 19.3801 1.37382 0.686911 0.726742i \(-0.258966\pi\)
0.686911 + 0.726742i \(0.258966\pi\)
\(200\) 0 0
\(201\) 5.95508 0.420039
\(202\) 0 0
\(203\) −22.4118 −1.57300
\(204\) 0 0
\(205\) 8.84034 0.617436
\(206\) 0 0
\(207\) −3.32953 −0.231418
\(208\) 0 0
\(209\) −8.09154 −0.559704
\(210\) 0 0
\(211\) −27.9383 −1.92335 −0.961675 0.274192i \(-0.911590\pi\)
−0.961675 + 0.274192i \(0.911590\pi\)
\(212\) 0 0
\(213\) 12.1308 0.831191
\(214\) 0 0
\(215\) −2.82466 −0.192640
\(216\) 0 0
\(217\) 33.4370 2.26985
\(218\) 0 0
\(219\) 0.195816 0.0132320
\(220\) 0 0
\(221\) −20.8320 −1.40131
\(222\) 0 0
\(223\) 0.387500 0.0259489 0.0129745 0.999916i \(-0.495870\pi\)
0.0129745 + 0.999916i \(0.495870\pi\)
\(224\) 0 0
\(225\) 9.51722 0.634482
\(226\) 0 0
\(227\) −10.2778 −0.682161 −0.341081 0.940034i \(-0.610793\pi\)
−0.341081 + 0.940034i \(0.610793\pi\)
\(228\) 0 0
\(229\) −10.5346 −0.696147 −0.348074 0.937467i \(-0.613164\pi\)
−0.348074 + 0.937467i \(0.613164\pi\)
\(230\) 0 0
\(231\) −7.29414 −0.479919
\(232\) 0 0
\(233\) 25.6725 1.68186 0.840932 0.541141i \(-0.182007\pi\)
0.840932 + 0.541141i \(0.182007\pi\)
\(234\) 0 0
\(235\) −2.38740 −0.155737
\(236\) 0 0
\(237\) 2.56830 0.166829
\(238\) 0 0
\(239\) 25.1312 1.62560 0.812801 0.582542i \(-0.197942\pi\)
0.812801 + 0.582542i \(0.197942\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 0 0
\(243\) 15.8685 1.01796
\(244\) 0 0
\(245\) 13.7453 0.878157
\(246\) 0 0
\(247\) −16.0839 −1.02339
\(248\) 0 0
\(249\) 2.01160 0.127480
\(250\) 0 0
\(251\) −2.52238 −0.159211 −0.0796055 0.996826i \(-0.525366\pi\)
−0.0796055 + 0.996826i \(0.525366\pi\)
\(252\) 0 0
\(253\) −2.76501 −0.173835
\(254\) 0 0
\(255\) 4.23605 0.265272
\(256\) 0 0
\(257\) 3.92348 0.244740 0.122370 0.992485i \(-0.460951\pi\)
0.122370 + 0.992485i \(0.460951\pi\)
\(258\) 0 0
\(259\) 32.8787 2.04298
\(260\) 0 0
\(261\) 11.3458 0.702290
\(262\) 0 0
\(263\) −6.24984 −0.385382 −0.192691 0.981260i \(-0.561721\pi\)
−0.192691 + 0.981260i \(0.561721\pi\)
\(264\) 0 0
\(265\) 5.43527 0.333886
\(266\) 0 0
\(267\) 4.52475 0.276910
\(268\) 0 0
\(269\) 20.4943 1.24956 0.624780 0.780801i \(-0.285189\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(270\) 0 0
\(271\) −15.3179 −0.930499 −0.465249 0.885180i \(-0.654035\pi\)
−0.465249 + 0.885180i \(0.654035\pi\)
\(272\) 0 0
\(273\) −14.4989 −0.877512
\(274\) 0 0
\(275\) 7.90359 0.476605
\(276\) 0 0
\(277\) 3.26388 0.196108 0.0980539 0.995181i \(-0.468738\pi\)
0.0980539 + 0.995181i \(0.468738\pi\)
\(278\) 0 0
\(279\) −16.9273 −1.01341
\(280\) 0 0
\(281\) −12.4791 −0.744442 −0.372221 0.928144i \(-0.621404\pi\)
−0.372221 + 0.928144i \(0.621404\pi\)
\(282\) 0 0
\(283\) 15.8115 0.939899 0.469949 0.882693i \(-0.344272\pi\)
0.469949 + 0.882693i \(0.344272\pi\)
\(284\) 0 0
\(285\) 3.27056 0.193731
\(286\) 0 0
\(287\) 42.3999 2.50279
\(288\) 0 0
\(289\) 12.1913 0.717133
\(290\) 0 0
\(291\) 4.30293 0.252242
\(292\) 0 0
\(293\) 18.7147 1.09333 0.546663 0.837353i \(-0.315898\pi\)
0.546663 + 0.837353i \(0.315898\pi\)
\(294\) 0 0
\(295\) 11.5231 0.670899
\(296\) 0 0
\(297\) 8.43530 0.489466
\(298\) 0 0
\(299\) −5.49613 −0.317849
\(300\) 0 0
\(301\) −13.5476 −0.780870
\(302\) 0 0
\(303\) −10.1255 −0.581697
\(304\) 0 0
\(305\) −13.3923 −0.766843
\(306\) 0 0
\(307\) −5.23477 −0.298764 −0.149382 0.988780i \(-0.547728\pi\)
−0.149382 + 0.988780i \(0.547728\pi\)
\(308\) 0 0
\(309\) 7.99138 0.454614
\(310\) 0 0
\(311\) 26.6878 1.51332 0.756662 0.653806i \(-0.226829\pi\)
0.756662 + 0.653806i \(0.226829\pi\)
\(312\) 0 0
\(313\) 6.24141 0.352786 0.176393 0.984320i \(-0.443557\pi\)
0.176393 + 0.984320i \(0.443557\pi\)
\(314\) 0 0
\(315\) −10.3675 −0.584144
\(316\) 0 0
\(317\) 6.89526 0.387276 0.193638 0.981073i \(-0.437971\pi\)
0.193638 + 0.981073i \(0.437971\pi\)
\(318\) 0 0
\(319\) 9.42218 0.527541
\(320\) 0 0
\(321\) 7.45753 0.416239
\(322\) 0 0
\(323\) 22.5379 1.25404
\(324\) 0 0
\(325\) 15.7103 0.871451
\(326\) 0 0
\(327\) 2.64497 0.146267
\(328\) 0 0
\(329\) −11.4504 −0.631281
\(330\) 0 0
\(331\) 13.5697 0.745860 0.372930 0.927859i \(-0.378353\pi\)
0.372930 + 0.927859i \(0.378353\pi\)
\(332\) 0 0
\(333\) −16.6446 −0.912119
\(334\) 0 0
\(335\) 7.02916 0.384044
\(336\) 0 0
\(337\) −20.2288 −1.10193 −0.550966 0.834528i \(-0.685740\pi\)
−0.550966 + 0.834528i \(0.685740\pi\)
\(338\) 0 0
\(339\) −0.686946 −0.0373098
\(340\) 0 0
\(341\) −14.0573 −0.761245
\(342\) 0 0
\(343\) 33.6276 1.81572
\(344\) 0 0
\(345\) 1.11760 0.0601696
\(346\) 0 0
\(347\) −1.52458 −0.0818437 −0.0409219 0.999162i \(-0.513029\pi\)
−0.0409219 + 0.999162i \(0.513029\pi\)
\(348\) 0 0
\(349\) 16.5585 0.886358 0.443179 0.896433i \(-0.353851\pi\)
0.443179 + 0.896433i \(0.353851\pi\)
\(350\) 0 0
\(351\) 16.7672 0.894968
\(352\) 0 0
\(353\) 34.6802 1.84584 0.922920 0.384992i \(-0.125796\pi\)
0.922920 + 0.384992i \(0.125796\pi\)
\(354\) 0 0
\(355\) 14.3188 0.759961
\(356\) 0 0
\(357\) 20.3168 1.07528
\(358\) 0 0
\(359\) −18.1341 −0.957084 −0.478542 0.878065i \(-0.658835\pi\)
−0.478542 + 0.878065i \(0.658835\pi\)
\(360\) 0 0
\(361\) −1.59899 −0.0841573
\(362\) 0 0
\(363\) −5.89850 −0.309591
\(364\) 0 0
\(365\) 0.231134 0.0120981
\(366\) 0 0
\(367\) 10.9879 0.573566 0.286783 0.957996i \(-0.407414\pi\)
0.286783 + 0.957996i \(0.407414\pi\)
\(368\) 0 0
\(369\) −21.4647 −1.11741
\(370\) 0 0
\(371\) 26.0685 1.35341
\(372\) 0 0
\(373\) 5.23646 0.271134 0.135567 0.990768i \(-0.456714\pi\)
0.135567 + 0.990768i \(0.456714\pi\)
\(374\) 0 0
\(375\) −7.11475 −0.367404
\(376\) 0 0
\(377\) 18.7289 0.964586
\(378\) 0 0
\(379\) −8.07101 −0.414580 −0.207290 0.978280i \(-0.566464\pi\)
−0.207290 + 0.978280i \(0.566464\pi\)
\(380\) 0 0
\(381\) −9.24346 −0.473557
\(382\) 0 0
\(383\) −2.46503 −0.125957 −0.0629785 0.998015i \(-0.520060\pi\)
−0.0629785 + 0.998015i \(0.520060\pi\)
\(384\) 0 0
\(385\) −8.60973 −0.438792
\(386\) 0 0
\(387\) 6.85838 0.348631
\(388\) 0 0
\(389\) −28.7159 −1.45595 −0.727977 0.685601i \(-0.759539\pi\)
−0.727977 + 0.685601i \(0.759539\pi\)
\(390\) 0 0
\(391\) 7.70157 0.389485
\(392\) 0 0
\(393\) −11.5128 −0.580746
\(394\) 0 0
\(395\) 3.03153 0.152533
\(396\) 0 0
\(397\) 12.0605 0.605301 0.302651 0.953102i \(-0.402129\pi\)
0.302651 + 0.953102i \(0.402129\pi\)
\(398\) 0 0
\(399\) 15.6862 0.785291
\(400\) 0 0
\(401\) −31.7082 −1.58343 −0.791716 0.610890i \(-0.790812\pi\)
−0.791716 + 0.610890i \(0.790812\pi\)
\(402\) 0 0
\(403\) −27.9423 −1.39190
\(404\) 0 0
\(405\) 3.33152 0.165545
\(406\) 0 0
\(407\) −13.8225 −0.685158
\(408\) 0 0
\(409\) −14.1893 −0.701617 −0.350809 0.936447i \(-0.614093\pi\)
−0.350809 + 0.936447i \(0.614093\pi\)
\(410\) 0 0
\(411\) −10.5587 −0.520825
\(412\) 0 0
\(413\) 55.2668 2.71950
\(414\) 0 0
\(415\) 2.37442 0.116556
\(416\) 0 0
\(417\) −6.71242 −0.328709
\(418\) 0 0
\(419\) −31.1014 −1.51941 −0.759703 0.650271i \(-0.774655\pi\)
−0.759703 + 0.650271i \(0.774655\pi\)
\(420\) 0 0
\(421\) −27.4415 −1.33742 −0.668709 0.743524i \(-0.733153\pi\)
−0.668709 + 0.743524i \(0.733153\pi\)
\(422\) 0 0
\(423\) 5.79669 0.281845
\(424\) 0 0
\(425\) −22.0144 −1.06786
\(426\) 0 0
\(427\) −64.2321 −3.10841
\(428\) 0 0
\(429\) 6.09548 0.294293
\(430\) 0 0
\(431\) 14.1005 0.679199 0.339599 0.940570i \(-0.389709\pi\)
0.339599 + 0.940570i \(0.389709\pi\)
\(432\) 0 0
\(433\) −17.7403 −0.852546 −0.426273 0.904595i \(-0.640174\pi\)
−0.426273 + 0.904595i \(0.640174\pi\)
\(434\) 0 0
\(435\) −3.80839 −0.182598
\(436\) 0 0
\(437\) 5.94621 0.284446
\(438\) 0 0
\(439\) 2.52447 0.120486 0.0602431 0.998184i \(-0.480812\pi\)
0.0602431 + 0.998184i \(0.480812\pi\)
\(440\) 0 0
\(441\) −33.3741 −1.58924
\(442\) 0 0
\(443\) 16.8382 0.800008 0.400004 0.916513i \(-0.369009\pi\)
0.400004 + 0.916513i \(0.369009\pi\)
\(444\) 0 0
\(445\) 5.34084 0.253180
\(446\) 0 0
\(447\) 3.65616 0.172930
\(448\) 0 0
\(449\) −30.5688 −1.44263 −0.721314 0.692608i \(-0.756462\pi\)
−0.721314 + 0.692608i \(0.756462\pi\)
\(450\) 0 0
\(451\) −17.8254 −0.839364
\(452\) 0 0
\(453\) −0.0760907 −0.00357505
\(454\) 0 0
\(455\) −17.1139 −0.802313
\(456\) 0 0
\(457\) −32.7043 −1.52984 −0.764922 0.644123i \(-0.777223\pi\)
−0.764922 + 0.644123i \(0.777223\pi\)
\(458\) 0 0
\(459\) −23.4954 −1.09667
\(460\) 0 0
\(461\) 12.3932 0.577210 0.288605 0.957448i \(-0.406809\pi\)
0.288605 + 0.957448i \(0.406809\pi\)
\(462\) 0 0
\(463\) −12.1379 −0.564094 −0.282047 0.959401i \(-0.591013\pi\)
−0.282047 + 0.959401i \(0.591013\pi\)
\(464\) 0 0
\(465\) 5.68188 0.263491
\(466\) 0 0
\(467\) −5.38203 −0.249051 −0.124525 0.992216i \(-0.539741\pi\)
−0.124525 + 0.992216i \(0.539741\pi\)
\(468\) 0 0
\(469\) 33.7131 1.55673
\(470\) 0 0
\(471\) −8.25903 −0.380556
\(472\) 0 0
\(473\) 5.69555 0.261882
\(474\) 0 0
\(475\) −16.9968 −0.779868
\(476\) 0 0
\(477\) −13.1970 −0.604250
\(478\) 0 0
\(479\) 25.3752 1.15942 0.579711 0.814822i \(-0.303166\pi\)
0.579711 + 0.814822i \(0.303166\pi\)
\(480\) 0 0
\(481\) −27.4757 −1.25278
\(482\) 0 0
\(483\) 5.36022 0.243898
\(484\) 0 0
\(485\) 5.07901 0.230626
\(486\) 0 0
\(487\) 10.4243 0.472369 0.236185 0.971708i \(-0.424103\pi\)
0.236185 + 0.971708i \(0.424103\pi\)
\(488\) 0 0
\(489\) −6.11093 −0.276346
\(490\) 0 0
\(491\) −34.2858 −1.54730 −0.773649 0.633614i \(-0.781571\pi\)
−0.773649 + 0.633614i \(0.781571\pi\)
\(492\) 0 0
\(493\) −26.2442 −1.18198
\(494\) 0 0
\(495\) 4.35862 0.195905
\(496\) 0 0
\(497\) 68.6754 3.08051
\(498\) 0 0
\(499\) −15.9672 −0.714792 −0.357396 0.933953i \(-0.616335\pi\)
−0.357396 + 0.933953i \(0.616335\pi\)
\(500\) 0 0
\(501\) −14.5007 −0.647842
\(502\) 0 0
\(503\) 4.53174 0.202060 0.101030 0.994883i \(-0.467786\pi\)
0.101030 + 0.994883i \(0.467786\pi\)
\(504\) 0 0
\(505\) −11.9518 −0.531849
\(506\) 0 0
\(507\) 1.52121 0.0675591
\(508\) 0 0
\(509\) −4.08798 −0.181197 −0.0905983 0.995888i \(-0.528878\pi\)
−0.0905983 + 0.995888i \(0.528878\pi\)
\(510\) 0 0
\(511\) 1.10856 0.0490398
\(512\) 0 0
\(513\) −18.1403 −0.800913
\(514\) 0 0
\(515\) 9.43272 0.415655
\(516\) 0 0
\(517\) 4.81387 0.211714
\(518\) 0 0
\(519\) 15.0716 0.661571
\(520\) 0 0
\(521\) 34.2884 1.50220 0.751102 0.660186i \(-0.229523\pi\)
0.751102 + 0.660186i \(0.229523\pi\)
\(522\) 0 0
\(523\) −33.7724 −1.47676 −0.738382 0.674382i \(-0.764410\pi\)
−0.738382 + 0.674382i \(0.764410\pi\)
\(524\) 0 0
\(525\) −15.3218 −0.668699
\(526\) 0 0
\(527\) 39.1547 1.70561
\(528\) 0 0
\(529\) −20.9681 −0.911656
\(530\) 0 0
\(531\) −27.9784 −1.21416
\(532\) 0 0
\(533\) −35.4322 −1.53474
\(534\) 0 0
\(535\) 8.80259 0.380569
\(536\) 0 0
\(537\) 14.1311 0.609801
\(538\) 0 0
\(539\) −27.7156 −1.19380
\(540\) 0 0
\(541\) −31.0330 −1.33421 −0.667106 0.744963i \(-0.732467\pi\)
−0.667106 + 0.744963i \(0.732467\pi\)
\(542\) 0 0
\(543\) 14.4356 0.619492
\(544\) 0 0
\(545\) 3.12202 0.133733
\(546\) 0 0
\(547\) 33.0104 1.41142 0.705712 0.708499i \(-0.250627\pi\)
0.705712 + 0.708499i \(0.250627\pi\)
\(548\) 0 0
\(549\) 32.5171 1.38779
\(550\) 0 0
\(551\) −20.2626 −0.863214
\(552\) 0 0
\(553\) 14.5398 0.618293
\(554\) 0 0
\(555\) 5.58699 0.237155
\(556\) 0 0
\(557\) 15.4826 0.656019 0.328010 0.944674i \(-0.393622\pi\)
0.328010 + 0.944674i \(0.393622\pi\)
\(558\) 0 0
\(559\) 11.3213 0.478840
\(560\) 0 0
\(561\) −8.54142 −0.360619
\(562\) 0 0
\(563\) −6.21072 −0.261751 −0.130875 0.991399i \(-0.541779\pi\)
−0.130875 + 0.991399i \(0.541779\pi\)
\(564\) 0 0
\(565\) −0.810845 −0.0341125
\(566\) 0 0
\(567\) 15.9786 0.671038
\(568\) 0 0
\(569\) −20.3014 −0.851078 −0.425539 0.904940i \(-0.639916\pi\)
−0.425539 + 0.904940i \(0.639916\pi\)
\(570\) 0 0
\(571\) 38.5209 1.61205 0.806024 0.591882i \(-0.201615\pi\)
0.806024 + 0.591882i \(0.201615\pi\)
\(572\) 0 0
\(573\) −10.7939 −0.450922
\(574\) 0 0
\(575\) −5.80809 −0.242214
\(576\) 0 0
\(577\) 19.3481 0.805472 0.402736 0.915316i \(-0.368059\pi\)
0.402736 + 0.915316i \(0.368059\pi\)
\(578\) 0 0
\(579\) −20.7850 −0.863795
\(580\) 0 0
\(581\) 11.3881 0.472460
\(582\) 0 0
\(583\) −10.9595 −0.453896
\(584\) 0 0
\(585\) 8.66381 0.358205
\(586\) 0 0
\(587\) 20.0183 0.826243 0.413122 0.910676i \(-0.364438\pi\)
0.413122 + 0.910676i \(0.364438\pi\)
\(588\) 0 0
\(589\) 30.2305 1.24562
\(590\) 0 0
\(591\) −12.2498 −0.503888
\(592\) 0 0
\(593\) 12.5904 0.517027 0.258513 0.966008i \(-0.416767\pi\)
0.258513 + 0.966008i \(0.416767\pi\)
\(594\) 0 0
\(595\) 23.9812 0.983135
\(596\) 0 0
\(597\) 15.7949 0.646441
\(598\) 0 0
\(599\) −24.4282 −0.998107 −0.499054 0.866571i \(-0.666319\pi\)
−0.499054 + 0.866571i \(0.666319\pi\)
\(600\) 0 0
\(601\) 9.02256 0.368038 0.184019 0.982923i \(-0.441089\pi\)
0.184019 + 0.982923i \(0.441089\pi\)
\(602\) 0 0
\(603\) −17.0670 −0.695024
\(604\) 0 0
\(605\) −6.96237 −0.283061
\(606\) 0 0
\(607\) 19.0044 0.771364 0.385682 0.922632i \(-0.373966\pi\)
0.385682 + 0.922632i \(0.373966\pi\)
\(608\) 0 0
\(609\) −18.2657 −0.740165
\(610\) 0 0
\(611\) 9.56873 0.387110
\(612\) 0 0
\(613\) 3.38617 0.136766 0.0683831 0.997659i \(-0.478216\pi\)
0.0683831 + 0.997659i \(0.478216\pi\)
\(614\) 0 0
\(615\) 7.20491 0.290530
\(616\) 0 0
\(617\) −33.9920 −1.36847 −0.684234 0.729263i \(-0.739863\pi\)
−0.684234 + 0.729263i \(0.739863\pi\)
\(618\) 0 0
\(619\) −29.6477 −1.19164 −0.595822 0.803117i \(-0.703174\pi\)
−0.595822 + 0.803117i \(0.703174\pi\)
\(620\) 0 0
\(621\) −6.19882 −0.248750
\(622\) 0 0
\(623\) 25.6156 1.02627
\(624\) 0 0
\(625\) 11.9748 0.478992
\(626\) 0 0
\(627\) −6.59464 −0.263364
\(628\) 0 0
\(629\) 38.5008 1.53513
\(630\) 0 0
\(631\) −14.6241 −0.582176 −0.291088 0.956696i \(-0.594017\pi\)
−0.291088 + 0.956696i \(0.594017\pi\)
\(632\) 0 0
\(633\) −22.7698 −0.905018
\(634\) 0 0
\(635\) −10.9106 −0.432975
\(636\) 0 0
\(637\) −55.0915 −2.18280
\(638\) 0 0
\(639\) −34.7665 −1.37534
\(640\) 0 0
\(641\) 4.17547 0.164921 0.0824606 0.996594i \(-0.473722\pi\)
0.0824606 + 0.996594i \(0.473722\pi\)
\(642\) 0 0
\(643\) −23.1928 −0.914635 −0.457317 0.889304i \(-0.651190\pi\)
−0.457317 + 0.889304i \(0.651190\pi\)
\(644\) 0 0
\(645\) −2.30211 −0.0906455
\(646\) 0 0
\(647\) 31.0975 1.22257 0.611284 0.791411i \(-0.290653\pi\)
0.611284 + 0.791411i \(0.290653\pi\)
\(648\) 0 0
\(649\) −23.2347 −0.912043
\(650\) 0 0
\(651\) 27.2513 1.06806
\(652\) 0 0
\(653\) −31.5439 −1.23441 −0.617204 0.786803i \(-0.711735\pi\)
−0.617204 + 0.786803i \(0.711735\pi\)
\(654\) 0 0
\(655\) −13.5893 −0.530979
\(656\) 0 0
\(657\) −0.561202 −0.0218946
\(658\) 0 0
\(659\) −25.2754 −0.984590 −0.492295 0.870428i \(-0.663842\pi\)
−0.492295 + 0.870428i \(0.663842\pi\)
\(660\) 0 0
\(661\) −21.7628 −0.846476 −0.423238 0.906018i \(-0.639107\pi\)
−0.423238 + 0.906018i \(0.639107\pi\)
\(662\) 0 0
\(663\) −16.9781 −0.659377
\(664\) 0 0
\(665\) 18.5154 0.717995
\(666\) 0 0
\(667\) −6.92404 −0.268100
\(668\) 0 0
\(669\) 0.315814 0.0122101
\(670\) 0 0
\(671\) 27.0038 1.04247
\(672\) 0 0
\(673\) −37.2710 −1.43669 −0.718346 0.695686i \(-0.755101\pi\)
−0.718346 + 0.695686i \(0.755101\pi\)
\(674\) 0 0
\(675\) 17.7189 0.682001
\(676\) 0 0
\(677\) −33.5515 −1.28949 −0.644745 0.764398i \(-0.723036\pi\)
−0.644745 + 0.764398i \(0.723036\pi\)
\(678\) 0 0
\(679\) 24.3599 0.934846
\(680\) 0 0
\(681\) −8.37644 −0.320986
\(682\) 0 0
\(683\) −8.94737 −0.342362 −0.171181 0.985240i \(-0.554758\pi\)
−0.171181 + 0.985240i \(0.554758\pi\)
\(684\) 0 0
\(685\) −12.4631 −0.476192
\(686\) 0 0
\(687\) −8.58575 −0.327567
\(688\) 0 0
\(689\) −21.7846 −0.829929
\(690\) 0 0
\(691\) −7.04969 −0.268183 −0.134092 0.990969i \(-0.542812\pi\)
−0.134092 + 0.990969i \(0.542812\pi\)
\(692\) 0 0
\(693\) 20.9047 0.794105
\(694\) 0 0
\(695\) −7.92309 −0.300540
\(696\) 0 0
\(697\) 49.6502 1.88063
\(698\) 0 0
\(699\) 20.9232 0.791389
\(700\) 0 0
\(701\) −5.19179 −0.196091 −0.0980456 0.995182i \(-0.531259\pi\)
−0.0980456 + 0.995182i \(0.531259\pi\)
\(702\) 0 0
\(703\) 29.7256 1.12112
\(704\) 0 0
\(705\) −1.94574 −0.0732807
\(706\) 0 0
\(707\) −57.3230 −2.15586
\(708\) 0 0
\(709\) −16.3067 −0.612410 −0.306205 0.951966i \(-0.599059\pi\)
−0.306205 + 0.951966i \(0.599059\pi\)
\(710\) 0 0
\(711\) −7.36066 −0.276046
\(712\) 0 0
\(713\) 10.3302 0.386870
\(714\) 0 0
\(715\) 7.19488 0.269073
\(716\) 0 0
\(717\) 20.4820 0.764915
\(718\) 0 0
\(719\) 27.4968 1.02546 0.512729 0.858551i \(-0.328635\pi\)
0.512729 + 0.858551i \(0.328635\pi\)
\(720\) 0 0
\(721\) 45.2410 1.68486
\(722\) 0 0
\(723\) −0.815004 −0.0303103
\(724\) 0 0
\(725\) 19.7919 0.735053
\(726\) 0 0
\(727\) 15.2319 0.564919 0.282460 0.959279i \(-0.408850\pi\)
0.282460 + 0.959279i \(0.408850\pi\)
\(728\) 0 0
\(729\) 2.54349 0.0942032
\(730\) 0 0
\(731\) −15.8642 −0.586758
\(732\) 0 0
\(733\) 6.76475 0.249862 0.124931 0.992165i \(-0.460129\pi\)
0.124931 + 0.992165i \(0.460129\pi\)
\(734\) 0 0
\(735\) 11.2025 0.413210
\(736\) 0 0
\(737\) −14.1734 −0.522082
\(738\) 0 0
\(739\) −17.9519 −0.660372 −0.330186 0.943916i \(-0.607111\pi\)
−0.330186 + 0.943916i \(0.607111\pi\)
\(740\) 0 0
\(741\) −13.1084 −0.481551
\(742\) 0 0
\(743\) 17.2096 0.631357 0.315679 0.948866i \(-0.397768\pi\)
0.315679 + 0.948866i \(0.397768\pi\)
\(744\) 0 0
\(745\) 4.31559 0.158111
\(746\) 0 0
\(747\) −5.76518 −0.210937
\(748\) 0 0
\(749\) 42.2188 1.54264
\(750\) 0 0
\(751\) −4.98853 −0.182034 −0.0910170 0.995849i \(-0.529012\pi\)
−0.0910170 + 0.995849i \(0.529012\pi\)
\(752\) 0 0
\(753\) −2.05575 −0.0749156
\(754\) 0 0
\(755\) −0.0898146 −0.00326869
\(756\) 0 0
\(757\) −19.5585 −0.710864 −0.355432 0.934702i \(-0.615666\pi\)
−0.355432 + 0.934702i \(0.615666\pi\)
\(758\) 0 0
\(759\) −2.25349 −0.0817966
\(760\) 0 0
\(761\) −41.0235 −1.48710 −0.743550 0.668681i \(-0.766859\pi\)
−0.743550 + 0.668681i \(0.766859\pi\)
\(762\) 0 0
\(763\) 14.9738 0.542087
\(764\) 0 0
\(765\) −12.1403 −0.438935
\(766\) 0 0
\(767\) −46.1847 −1.66763
\(768\) 0 0
\(769\) 11.7068 0.422159 0.211079 0.977469i \(-0.432302\pi\)
0.211079 + 0.977469i \(0.432302\pi\)
\(770\) 0 0
\(771\) 3.19765 0.115161
\(772\) 0 0
\(773\) −6.66278 −0.239643 −0.119822 0.992795i \(-0.538232\pi\)
−0.119822 + 0.992795i \(0.538232\pi\)
\(774\) 0 0
\(775\) −29.5283 −1.06069
\(776\) 0 0
\(777\) 26.7962 0.961309
\(778\) 0 0
\(779\) 38.3338 1.37345
\(780\) 0 0
\(781\) −28.8719 −1.03312
\(782\) 0 0
\(783\) 21.1234 0.754888
\(784\) 0 0
\(785\) −9.74865 −0.347944
\(786\) 0 0
\(787\) −2.88890 −0.102978 −0.0514891 0.998674i \(-0.516397\pi\)
−0.0514891 + 0.998674i \(0.516397\pi\)
\(788\) 0 0
\(789\) −5.09364 −0.181338
\(790\) 0 0
\(791\) −3.88896 −0.138275
\(792\) 0 0
\(793\) 53.6767 1.90612
\(794\) 0 0
\(795\) 4.42976 0.157107
\(796\) 0 0
\(797\) 18.0912 0.640824 0.320412 0.947278i \(-0.396179\pi\)
0.320412 + 0.947278i \(0.396179\pi\)
\(798\) 0 0
\(799\) −13.4084 −0.474355
\(800\) 0 0
\(801\) −12.9678 −0.458193
\(802\) 0 0
\(803\) −0.466051 −0.0164466
\(804\) 0 0
\(805\) 6.32700 0.222997
\(806\) 0 0
\(807\) 16.7029 0.587971
\(808\) 0 0
\(809\) 8.04121 0.282714 0.141357 0.989959i \(-0.454853\pi\)
0.141357 + 0.989959i \(0.454853\pi\)
\(810\) 0 0
\(811\) −24.5111 −0.860701 −0.430350 0.902662i \(-0.641610\pi\)
−0.430350 + 0.902662i \(0.641610\pi\)
\(812\) 0 0
\(813\) −12.4842 −0.437839
\(814\) 0 0
\(815\) −7.21311 −0.252664
\(816\) 0 0
\(817\) −12.2484 −0.428517
\(818\) 0 0
\(819\) 41.5532 1.45199
\(820\) 0 0
\(821\) −10.6539 −0.371823 −0.185912 0.982566i \(-0.559524\pi\)
−0.185912 + 0.982566i \(0.559524\pi\)
\(822\) 0 0
\(823\) −25.6473 −0.894009 −0.447004 0.894532i \(-0.647509\pi\)
−0.447004 + 0.894532i \(0.647509\pi\)
\(824\) 0 0
\(825\) 6.44146 0.224263
\(826\) 0 0
\(827\) −18.0150 −0.626442 −0.313221 0.949680i \(-0.601408\pi\)
−0.313221 + 0.949680i \(0.601408\pi\)
\(828\) 0 0
\(829\) 10.1189 0.351444 0.175722 0.984440i \(-0.443774\pi\)
0.175722 + 0.984440i \(0.443774\pi\)
\(830\) 0 0
\(831\) 2.66008 0.0922771
\(832\) 0 0
\(833\) 77.1981 2.67476
\(834\) 0 0
\(835\) −17.1160 −0.592325
\(836\) 0 0
\(837\) −31.5148 −1.08931
\(838\) 0 0
\(839\) 46.0365 1.58936 0.794679 0.607030i \(-0.207639\pi\)
0.794679 + 0.607030i \(0.207639\pi\)
\(840\) 0 0
\(841\) −5.40531 −0.186390
\(842\) 0 0
\(843\) −10.1705 −0.350292
\(844\) 0 0
\(845\) 1.79557 0.0617696
\(846\) 0 0
\(847\) −33.3928 −1.14739
\(848\) 0 0
\(849\) 12.8865 0.442262
\(850\) 0 0
\(851\) 10.1577 0.348202
\(852\) 0 0
\(853\) −55.0001 −1.88317 −0.941584 0.336780i \(-0.890662\pi\)
−0.941584 + 0.336780i \(0.890662\pi\)
\(854\) 0 0
\(855\) −9.37329 −0.320560
\(856\) 0 0
\(857\) −31.5202 −1.07671 −0.538354 0.842719i \(-0.680954\pi\)
−0.538354 + 0.842719i \(0.680954\pi\)
\(858\) 0 0
\(859\) 37.2208 1.26996 0.634979 0.772529i \(-0.281009\pi\)
0.634979 + 0.772529i \(0.281009\pi\)
\(860\) 0 0
\(861\) 34.5561 1.17767
\(862\) 0 0
\(863\) −30.3224 −1.03218 −0.516092 0.856533i \(-0.672614\pi\)
−0.516092 + 0.856533i \(0.672614\pi\)
\(864\) 0 0
\(865\) 17.7900 0.604877
\(866\) 0 0
\(867\) 9.93592 0.337442
\(868\) 0 0
\(869\) −6.11267 −0.207358
\(870\) 0 0
\(871\) −28.1730 −0.954605
\(872\) 0 0
\(873\) −12.3320 −0.417376
\(874\) 0 0
\(875\) −40.2783 −1.36165
\(876\) 0 0
\(877\) 27.0037 0.911849 0.455925 0.890018i \(-0.349309\pi\)
0.455925 + 0.890018i \(0.349309\pi\)
\(878\) 0 0
\(879\) 15.2526 0.514457
\(880\) 0 0
\(881\) −0.565506 −0.0190524 −0.00952619 0.999955i \(-0.503032\pi\)
−0.00952619 + 0.999955i \(0.503032\pi\)
\(882\) 0 0
\(883\) −23.8838 −0.803752 −0.401876 0.915694i \(-0.631642\pi\)
−0.401876 + 0.915694i \(0.631642\pi\)
\(884\) 0 0
\(885\) 9.39135 0.315687
\(886\) 0 0
\(887\) 10.1717 0.341533 0.170767 0.985311i \(-0.445376\pi\)
0.170767 + 0.985311i \(0.445376\pi\)
\(888\) 0 0
\(889\) −52.3294 −1.75507
\(890\) 0 0
\(891\) −6.71757 −0.225047
\(892\) 0 0
\(893\) −10.3523 −0.346427
\(894\) 0 0
\(895\) 16.6798 0.557544
\(896\) 0 0
\(897\) −4.47937 −0.149562
\(898\) 0 0
\(899\) −35.2018 −1.17405
\(900\) 0 0
\(901\) 30.5262 1.01697
\(902\) 0 0
\(903\) −11.0413 −0.367432
\(904\) 0 0
\(905\) 17.0393 0.566405
\(906\) 0 0
\(907\) 44.1661 1.46651 0.733255 0.679954i \(-0.238000\pi\)
0.733255 + 0.679954i \(0.238000\pi\)
\(908\) 0 0
\(909\) 29.0194 0.962514
\(910\) 0 0
\(911\) 7.07316 0.234344 0.117172 0.993112i \(-0.462617\pi\)
0.117172 + 0.993112i \(0.462617\pi\)
\(912\) 0 0
\(913\) −4.78770 −0.158450
\(914\) 0 0
\(915\) −10.9148 −0.360832
\(916\) 0 0
\(917\) −65.1769 −2.15233
\(918\) 0 0
\(919\) −23.6016 −0.778544 −0.389272 0.921123i \(-0.627273\pi\)
−0.389272 + 0.921123i \(0.627273\pi\)
\(920\) 0 0
\(921\) −4.26636 −0.140581
\(922\) 0 0
\(923\) −57.3899 −1.88901
\(924\) 0 0
\(925\) −29.0352 −0.954670
\(926\) 0 0
\(927\) −22.9030 −0.752233
\(928\) 0 0
\(929\) 31.8496 1.04495 0.522475 0.852654i \(-0.325009\pi\)
0.522475 + 0.852654i \(0.325009\pi\)
\(930\) 0 0
\(931\) 59.6029 1.95341
\(932\) 0 0
\(933\) 21.7506 0.712083
\(934\) 0 0
\(935\) −10.0820 −0.329716
\(936\) 0 0
\(937\) 44.1729 1.44307 0.721533 0.692380i \(-0.243438\pi\)
0.721533 + 0.692380i \(0.243438\pi\)
\(938\) 0 0
\(939\) 5.08677 0.166001
\(940\) 0 0
\(941\) −34.5303 −1.12566 −0.562828 0.826574i \(-0.690287\pi\)
−0.562828 + 0.826574i \(0.690287\pi\)
\(942\) 0 0
\(943\) 13.0993 0.426571
\(944\) 0 0
\(945\) −19.3020 −0.627893
\(946\) 0 0
\(947\) 3.46191 0.112497 0.0562485 0.998417i \(-0.482086\pi\)
0.0562485 + 0.998417i \(0.482086\pi\)
\(948\) 0 0
\(949\) −0.926389 −0.0300719
\(950\) 0 0
\(951\) 5.61966 0.182230
\(952\) 0 0
\(953\) 60.2596 1.95200 0.976000 0.217771i \(-0.0698785\pi\)
0.976000 + 0.217771i \(0.0698785\pi\)
\(954\) 0 0
\(955\) −12.7407 −0.412280
\(956\) 0 0
\(957\) 7.67911 0.248230
\(958\) 0 0
\(959\) −59.7755 −1.93025
\(960\) 0 0
\(961\) 21.5188 0.694156
\(962\) 0 0
\(963\) −21.3730 −0.688735
\(964\) 0 0
\(965\) −24.5338 −0.789772
\(966\) 0 0
\(967\) 39.9168 1.28364 0.641819 0.766856i \(-0.278180\pi\)
0.641819 + 0.766856i \(0.278180\pi\)
\(968\) 0 0
\(969\) 18.3685 0.590081
\(970\) 0 0
\(971\) −43.7104 −1.40273 −0.701366 0.712801i \(-0.747426\pi\)
−0.701366 + 0.712801i \(0.747426\pi\)
\(972\) 0 0
\(973\) −38.0006 −1.21824
\(974\) 0 0
\(975\) 12.8040 0.410055
\(976\) 0 0
\(977\) −0.955388 −0.0305656 −0.0152828 0.999883i \(-0.504865\pi\)
−0.0152828 + 0.999883i \(0.504865\pi\)
\(978\) 0 0
\(979\) −10.7691 −0.344182
\(980\) 0 0
\(981\) −7.58037 −0.242023
\(982\) 0 0
\(983\) 48.7488 1.55485 0.777423 0.628978i \(-0.216526\pi\)
0.777423 + 0.628978i \(0.216526\pi\)
\(984\) 0 0
\(985\) −14.4591 −0.460707
\(986\) 0 0
\(987\) −9.33211 −0.297044
\(988\) 0 0
\(989\) −4.18547 −0.133090
\(990\) 0 0
\(991\) −0.488986 −0.0155332 −0.00776658 0.999970i \(-0.502472\pi\)
−0.00776658 + 0.999970i \(0.502472\pi\)
\(992\) 0 0
\(993\) 11.0594 0.350959
\(994\) 0 0
\(995\) 18.6437 0.591044
\(996\) 0 0
\(997\) 39.5339 1.25205 0.626026 0.779802i \(-0.284680\pi\)
0.626026 + 0.779802i \(0.284680\pi\)
\(998\) 0 0
\(999\) −30.9885 −0.980432
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 3856.2.a.j.1.5 7
4.3 odd 2 241.2.a.a.1.6 7
12.11 even 2 2169.2.a.e.1.2 7
20.19 odd 2 6025.2.a.f.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.6 7 4.3 odd 2
2169.2.a.e.1.2 7 12.11 even 2
3856.2.a.j.1.5 7 1.1 even 1 trivial
6025.2.a.f.1.2 7 20.19 odd 2