Properties

Label 3800.2.a.bb.1.4
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.471016\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.471016 q^{3} +0.567324 q^{7} -2.77814 q^{9} +O(q^{10})\) \(q+0.471016 q^{3} +0.567324 q^{7} -2.77814 q^{9} +4.37804 q^{11} -0.165457 q^{13} -7.94693 q^{17} +1.00000 q^{19} +0.267219 q^{21} +3.87445 q^{23} -2.72160 q^{27} +3.53231 q^{29} +3.20380 q^{31} +2.06213 q^{33} +10.1779 q^{37} -0.0779330 q^{39} -5.97409 q^{41} +12.0904 q^{43} -5.46140 q^{47} -6.67814 q^{49} -3.74313 q^{51} -2.00333 q^{53} +0.471016 q^{57} +8.32164 q^{59} +11.8181 q^{61} -1.57611 q^{63} +8.79599 q^{67} +1.82493 q^{69} +0.720031 q^{71} +4.54017 q^{73} +2.48377 q^{77} +11.7123 q^{79} +7.05251 q^{81} -6.72351 q^{83} +1.66378 q^{87} +8.11941 q^{89} -0.0938678 q^{91} +1.50904 q^{93} -13.6321 q^{97} -12.1628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.471016 0.271941 0.135971 0.990713i \(-0.456585\pi\)
0.135971 + 0.990713i \(0.456585\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.567324 0.214428 0.107214 0.994236i \(-0.465807\pi\)
0.107214 + 0.994236i \(0.465807\pi\)
\(8\) 0 0
\(9\) −2.77814 −0.926048
\(10\) 0 0
\(11\) 4.37804 1.32003 0.660014 0.751253i \(-0.270550\pi\)
0.660014 + 0.751253i \(0.270550\pi\)
\(12\) 0 0
\(13\) −0.165457 −0.0458895 −0.0229448 0.999737i \(-0.507304\pi\)
−0.0229448 + 0.999737i \(0.507304\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.94693 −1.92741 −0.963707 0.266963i \(-0.913980\pi\)
−0.963707 + 0.266963i \(0.913980\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.267219 0.0583119
\(22\) 0 0
\(23\) 3.87445 0.807879 0.403939 0.914786i \(-0.367641\pi\)
0.403939 + 0.914786i \(0.367641\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.72160 −0.523772
\(28\) 0 0
\(29\) 3.53231 0.655934 0.327967 0.944689i \(-0.393636\pi\)
0.327967 + 0.944689i \(0.393636\pi\)
\(30\) 0 0
\(31\) 3.20380 0.575419 0.287709 0.957718i \(-0.407106\pi\)
0.287709 + 0.957718i \(0.407106\pi\)
\(32\) 0 0
\(33\) 2.06213 0.358970
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.1779 1.67323 0.836617 0.547788i \(-0.184530\pi\)
0.836617 + 0.547788i \(0.184530\pi\)
\(38\) 0 0
\(39\) −0.0779330 −0.0124793
\(40\) 0 0
\(41\) −5.97409 −0.932996 −0.466498 0.884522i \(-0.654485\pi\)
−0.466498 + 0.884522i \(0.654485\pi\)
\(42\) 0 0
\(43\) 12.0904 1.84376 0.921881 0.387472i \(-0.126652\pi\)
0.921881 + 0.387472i \(0.126652\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.46140 −0.796627 −0.398314 0.917249i \(-0.630404\pi\)
−0.398314 + 0.917249i \(0.630404\pi\)
\(48\) 0 0
\(49\) −6.67814 −0.954020
\(50\) 0 0
\(51\) −3.74313 −0.524143
\(52\) 0 0
\(53\) −2.00333 −0.275179 −0.137589 0.990489i \(-0.543935\pi\)
−0.137589 + 0.990489i \(0.543935\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.471016 0.0623876
\(58\) 0 0
\(59\) 8.32164 1.08339 0.541693 0.840577i \(-0.317784\pi\)
0.541693 + 0.840577i \(0.317784\pi\)
\(60\) 0 0
\(61\) 11.8181 1.51315 0.756573 0.653909i \(-0.226872\pi\)
0.756573 + 0.653909i \(0.226872\pi\)
\(62\) 0 0
\(63\) −1.57611 −0.198571
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.79599 1.07460 0.537300 0.843391i \(-0.319444\pi\)
0.537300 + 0.843391i \(0.319444\pi\)
\(68\) 0 0
\(69\) 1.82493 0.219696
\(70\) 0 0
\(71\) 0.720031 0.0854520 0.0427260 0.999087i \(-0.486396\pi\)
0.0427260 + 0.999087i \(0.486396\pi\)
\(72\) 0 0
\(73\) 4.54017 0.531386 0.265693 0.964058i \(-0.414399\pi\)
0.265693 + 0.964058i \(0.414399\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.48377 0.283051
\(78\) 0 0
\(79\) 11.7123 1.31774 0.658870 0.752257i \(-0.271035\pi\)
0.658870 + 0.752257i \(0.271035\pi\)
\(80\) 0 0
\(81\) 7.05251 0.783613
\(82\) 0 0
\(83\) −6.72351 −0.738001 −0.369000 0.929429i \(-0.620300\pi\)
−0.369000 + 0.929429i \(0.620300\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.66378 0.178376
\(88\) 0 0
\(89\) 8.11941 0.860656 0.430328 0.902673i \(-0.358398\pi\)
0.430328 + 0.902673i \(0.358398\pi\)
\(90\) 0 0
\(91\) −0.0938678 −0.00984002
\(92\) 0 0
\(93\) 1.50904 0.156480
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.6321 −1.38413 −0.692065 0.721835i \(-0.743299\pi\)
−0.692065 + 0.721835i \(0.743299\pi\)
\(98\) 0 0
\(99\) −12.1628 −1.22241
\(100\) 0 0
\(101\) 19.2686 1.91730 0.958649 0.284590i \(-0.0918575\pi\)
0.958649 + 0.284590i \(0.0918575\pi\)
\(102\) 0 0
\(103\) −19.2069 −1.89251 −0.946257 0.323417i \(-0.895168\pi\)
−0.946257 + 0.323417i \(0.895168\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0091 −1.16096 −0.580480 0.814275i \(-0.697135\pi\)
−0.580480 + 0.814275i \(0.697135\pi\)
\(108\) 0 0
\(109\) −15.0032 −1.43704 −0.718521 0.695505i \(-0.755180\pi\)
−0.718521 + 0.695505i \(0.755180\pi\)
\(110\) 0 0
\(111\) 4.79395 0.455022
\(112\) 0 0
\(113\) 8.13049 0.764852 0.382426 0.923986i \(-0.375089\pi\)
0.382426 + 0.923986i \(0.375089\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.459664 0.0424959
\(118\) 0 0
\(119\) −4.50848 −0.413292
\(120\) 0 0
\(121\) 8.16722 0.742474
\(122\) 0 0
\(123\) −2.81389 −0.253720
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.37051 −0.742762 −0.371381 0.928480i \(-0.621116\pi\)
−0.371381 + 0.928480i \(0.621116\pi\)
\(128\) 0 0
\(129\) 5.69476 0.501395
\(130\) 0 0
\(131\) −5.03206 −0.439653 −0.219826 0.975539i \(-0.570549\pi\)
−0.219826 + 0.975539i \(0.570549\pi\)
\(132\) 0 0
\(133\) 0.567324 0.0491932
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.6498 1.76423 0.882115 0.471033i \(-0.156119\pi\)
0.882115 + 0.471033i \(0.156119\pi\)
\(138\) 0 0
\(139\) 11.4726 0.973092 0.486546 0.873655i \(-0.338257\pi\)
0.486546 + 0.873655i \(0.338257\pi\)
\(140\) 0 0
\(141\) −2.57241 −0.216636
\(142\) 0 0
\(143\) −0.724378 −0.0605755
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.14551 −0.259438
\(148\) 0 0
\(149\) 13.0611 1.07001 0.535003 0.844850i \(-0.320311\pi\)
0.535003 + 0.844850i \(0.320311\pi\)
\(150\) 0 0
\(151\) 17.8177 1.44998 0.724992 0.688758i \(-0.241844\pi\)
0.724992 + 0.688758i \(0.241844\pi\)
\(152\) 0 0
\(153\) 22.0777 1.78488
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.3959 −0.909490 −0.454745 0.890622i \(-0.650270\pi\)
−0.454745 + 0.890622i \(0.650270\pi\)
\(158\) 0 0
\(159\) −0.943601 −0.0748324
\(160\) 0 0
\(161\) 2.19807 0.173232
\(162\) 0 0
\(163\) 16.5741 1.29819 0.649093 0.760709i \(-0.275149\pi\)
0.649093 + 0.760709i \(0.275149\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.3378 −1.49640 −0.748200 0.663473i \(-0.769082\pi\)
−0.748200 + 0.663473i \(0.769082\pi\)
\(168\) 0 0
\(169\) −12.9726 −0.997894
\(170\) 0 0
\(171\) −2.77814 −0.212450
\(172\) 0 0
\(173\) 9.31290 0.708046 0.354023 0.935237i \(-0.384813\pi\)
0.354023 + 0.935237i \(0.384813\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.91963 0.294617
\(178\) 0 0
\(179\) 24.1703 1.80658 0.903288 0.429035i \(-0.141146\pi\)
0.903288 + 0.429035i \(0.141146\pi\)
\(180\) 0 0
\(181\) 18.6981 1.38982 0.694910 0.719097i \(-0.255444\pi\)
0.694910 + 0.719097i \(0.255444\pi\)
\(182\) 0 0
\(183\) 5.56649 0.411487
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −34.7920 −2.54424
\(188\) 0 0
\(189\) −1.54403 −0.112312
\(190\) 0 0
\(191\) 6.27452 0.454008 0.227004 0.973894i \(-0.427107\pi\)
0.227004 + 0.973894i \(0.427107\pi\)
\(192\) 0 0
\(193\) 8.49096 0.611193 0.305596 0.952161i \(-0.401144\pi\)
0.305596 + 0.952161i \(0.401144\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.97358 0.425600 0.212800 0.977096i \(-0.431742\pi\)
0.212800 + 0.977096i \(0.431742\pi\)
\(198\) 0 0
\(199\) 19.8763 1.40899 0.704497 0.709707i \(-0.251173\pi\)
0.704497 + 0.709707i \(0.251173\pi\)
\(200\) 0 0
\(201\) 4.14305 0.292228
\(202\) 0 0
\(203\) 2.00397 0.140651
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −10.7638 −0.748135
\(208\) 0 0
\(209\) 4.37804 0.302835
\(210\) 0 0
\(211\) −2.84905 −0.196137 −0.0980685 0.995180i \(-0.531266\pi\)
−0.0980685 + 0.995180i \(0.531266\pi\)
\(212\) 0 0
\(213\) 0.339146 0.0232379
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.81759 0.123386
\(218\) 0 0
\(219\) 2.13849 0.144506
\(220\) 0 0
\(221\) 1.31488 0.0884481
\(222\) 0 0
\(223\) −0.883914 −0.0591912 −0.0295956 0.999562i \(-0.509422\pi\)
−0.0295956 + 0.999562i \(0.509422\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.02836 0.0682549 0.0341275 0.999417i \(-0.489135\pi\)
0.0341275 + 0.999417i \(0.489135\pi\)
\(228\) 0 0
\(229\) −24.5878 −1.62481 −0.812404 0.583095i \(-0.801842\pi\)
−0.812404 + 0.583095i \(0.801842\pi\)
\(230\) 0 0
\(231\) 1.16989 0.0769734
\(232\) 0 0
\(233\) −5.23585 −0.343012 −0.171506 0.985183i \(-0.554863\pi\)
−0.171506 + 0.985183i \(0.554863\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.51669 0.358348
\(238\) 0 0
\(239\) −12.8124 −0.828767 −0.414384 0.910102i \(-0.636003\pi\)
−0.414384 + 0.910102i \(0.636003\pi\)
\(240\) 0 0
\(241\) −11.9996 −0.772962 −0.386481 0.922297i \(-0.626309\pi\)
−0.386481 + 0.922297i \(0.626309\pi\)
\(242\) 0 0
\(243\) 11.4866 0.736869
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.165457 −0.0105278
\(248\) 0 0
\(249\) −3.16688 −0.200693
\(250\) 0 0
\(251\) 23.7189 1.49713 0.748563 0.663064i \(-0.230744\pi\)
0.748563 + 0.663064i \(0.230744\pi\)
\(252\) 0 0
\(253\) 16.9625 1.06642
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.11678 −0.132041 −0.0660206 0.997818i \(-0.521030\pi\)
−0.0660206 + 0.997818i \(0.521030\pi\)
\(258\) 0 0
\(259\) 5.77416 0.358789
\(260\) 0 0
\(261\) −9.81327 −0.607426
\(262\) 0 0
\(263\) −1.95674 −0.120658 −0.0603288 0.998179i \(-0.519215\pi\)
−0.0603288 + 0.998179i \(0.519215\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.82437 0.234048
\(268\) 0 0
\(269\) −22.8278 −1.39184 −0.695919 0.718120i \(-0.745003\pi\)
−0.695919 + 0.718120i \(0.745003\pi\)
\(270\) 0 0
\(271\) −15.8203 −0.961014 −0.480507 0.876991i \(-0.659547\pi\)
−0.480507 + 0.876991i \(0.659547\pi\)
\(272\) 0 0
\(273\) −0.0442133 −0.00267591
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9.47924 0.569553 0.284776 0.958594i \(-0.408081\pi\)
0.284776 + 0.958594i \(0.408081\pi\)
\(278\) 0 0
\(279\) −8.90061 −0.532866
\(280\) 0 0
\(281\) 13.3160 0.794367 0.397183 0.917739i \(-0.369988\pi\)
0.397183 + 0.917739i \(0.369988\pi\)
\(282\) 0 0
\(283\) −9.20676 −0.547285 −0.273643 0.961831i \(-0.588229\pi\)
−0.273643 + 0.961831i \(0.588229\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.38924 −0.200061
\(288\) 0 0
\(289\) 46.1537 2.71492
\(290\) 0 0
\(291\) −6.42094 −0.376402
\(292\) 0 0
\(293\) 10.3435 0.604273 0.302137 0.953265i \(-0.402300\pi\)
0.302137 + 0.953265i \(0.402300\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −11.9153 −0.691394
\(298\) 0 0
\(299\) −0.641056 −0.0370732
\(300\) 0 0
\(301\) 6.85915 0.395355
\(302\) 0 0
\(303\) 9.07583 0.521393
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0642 1.25927 0.629636 0.776891i \(-0.283204\pi\)
0.629636 + 0.776891i \(0.283204\pi\)
\(308\) 0 0
\(309\) −9.04677 −0.514653
\(310\) 0 0
\(311\) −31.4064 −1.78089 −0.890447 0.455087i \(-0.849608\pi\)
−0.890447 + 0.455087i \(0.849608\pi\)
\(312\) 0 0
\(313\) −15.1428 −0.855924 −0.427962 0.903797i \(-0.640768\pi\)
−0.427962 + 0.903797i \(0.640768\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.5791 −1.04350 −0.521752 0.853097i \(-0.674721\pi\)
−0.521752 + 0.853097i \(0.674721\pi\)
\(318\) 0 0
\(319\) 15.4646 0.865852
\(320\) 0 0
\(321\) −5.65646 −0.315713
\(322\) 0 0
\(323\) −7.94693 −0.442179
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −7.06673 −0.390791
\(328\) 0 0
\(329\) −3.09839 −0.170820
\(330\) 0 0
\(331\) 22.8001 1.25320 0.626602 0.779339i \(-0.284445\pi\)
0.626602 + 0.779339i \(0.284445\pi\)
\(332\) 0 0
\(333\) −28.2756 −1.54950
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 28.9306 1.57595 0.787976 0.615706i \(-0.211129\pi\)
0.787976 + 0.615706i \(0.211129\pi\)
\(338\) 0 0
\(339\) 3.82959 0.207995
\(340\) 0 0
\(341\) 14.0263 0.759569
\(342\) 0 0
\(343\) −7.75994 −0.418997
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.2803 −1.08870 −0.544352 0.838857i \(-0.683224\pi\)
−0.544352 + 0.838857i \(0.683224\pi\)
\(348\) 0 0
\(349\) 20.4355 1.09389 0.546943 0.837170i \(-0.315792\pi\)
0.546943 + 0.837170i \(0.315792\pi\)
\(350\) 0 0
\(351\) 0.450308 0.0240357
\(352\) 0 0
\(353\) 7.50588 0.399498 0.199749 0.979847i \(-0.435987\pi\)
0.199749 + 0.979847i \(0.435987\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.12357 −0.112391
\(358\) 0 0
\(359\) −18.4383 −0.973135 −0.486567 0.873643i \(-0.661751\pi\)
−0.486567 + 0.873643i \(0.661751\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 3.84689 0.201909
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −12.0105 −0.626943 −0.313471 0.949598i \(-0.601492\pi\)
−0.313471 + 0.949598i \(0.601492\pi\)
\(368\) 0 0
\(369\) 16.5969 0.863999
\(370\) 0 0
\(371\) −1.13654 −0.0590061
\(372\) 0 0
\(373\) −15.0719 −0.780395 −0.390197 0.920731i \(-0.627593\pi\)
−0.390197 + 0.920731i \(0.627593\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.584446 −0.0301005
\(378\) 0 0
\(379\) −8.50216 −0.436727 −0.218363 0.975868i \(-0.570072\pi\)
−0.218363 + 0.975868i \(0.570072\pi\)
\(380\) 0 0
\(381\) −3.94264 −0.201988
\(382\) 0 0
\(383\) −34.9232 −1.78449 −0.892247 0.451547i \(-0.850872\pi\)
−0.892247 + 0.451547i \(0.850872\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −33.5888 −1.70741
\(388\) 0 0
\(389\) −10.7455 −0.544820 −0.272410 0.962181i \(-0.587821\pi\)
−0.272410 + 0.962181i \(0.587821\pi\)
\(390\) 0 0
\(391\) −30.7900 −1.55712
\(392\) 0 0
\(393\) −2.37018 −0.119560
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.45462 0.0730051 0.0365025 0.999334i \(-0.488378\pi\)
0.0365025 + 0.999334i \(0.488378\pi\)
\(398\) 0 0
\(399\) 0.267219 0.0133777
\(400\) 0 0
\(401\) 11.5379 0.576176 0.288088 0.957604i \(-0.406980\pi\)
0.288088 + 0.957604i \(0.406980\pi\)
\(402\) 0 0
\(403\) −0.530091 −0.0264057
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 44.5592 2.20872
\(408\) 0 0
\(409\) −20.5596 −1.01661 −0.508304 0.861178i \(-0.669727\pi\)
−0.508304 + 0.861178i \(0.669727\pi\)
\(410\) 0 0
\(411\) 9.72639 0.479767
\(412\) 0 0
\(413\) 4.72107 0.232308
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.40377 0.264624
\(418\) 0 0
\(419\) −11.0760 −0.541096 −0.270548 0.962706i \(-0.587205\pi\)
−0.270548 + 0.962706i \(0.587205\pi\)
\(420\) 0 0
\(421\) 18.9178 0.921999 0.461000 0.887400i \(-0.347491\pi\)
0.461000 + 0.887400i \(0.347491\pi\)
\(422\) 0 0
\(423\) 15.1726 0.737715
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.70467 0.324461
\(428\) 0 0
\(429\) −0.341194 −0.0164730
\(430\) 0 0
\(431\) 5.23923 0.252365 0.126182 0.992007i \(-0.459728\pi\)
0.126182 + 0.992007i \(0.459728\pi\)
\(432\) 0 0
\(433\) −8.26963 −0.397413 −0.198707 0.980059i \(-0.563674\pi\)
−0.198707 + 0.980059i \(0.563674\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.87445 0.185340
\(438\) 0 0
\(439\) 26.2786 1.25421 0.627104 0.778936i \(-0.284240\pi\)
0.627104 + 0.778936i \(0.284240\pi\)
\(440\) 0 0
\(441\) 18.5528 0.883469
\(442\) 0 0
\(443\) −9.03372 −0.429205 −0.214602 0.976701i \(-0.568846\pi\)
−0.214602 + 0.976701i \(0.568846\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.15198 0.290979
\(448\) 0 0
\(449\) −10.5553 −0.498135 −0.249067 0.968486i \(-0.580124\pi\)
−0.249067 + 0.968486i \(0.580124\pi\)
\(450\) 0 0
\(451\) −26.1548 −1.23158
\(452\) 0 0
\(453\) 8.39242 0.394310
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13.7694 0.644105 0.322053 0.946722i \(-0.395627\pi\)
0.322053 + 0.946722i \(0.395627\pi\)
\(458\) 0 0
\(459\) 21.6284 1.00953
\(460\) 0 0
\(461\) −0.0771895 −0.00359507 −0.00179754 0.999998i \(-0.500572\pi\)
−0.00179754 + 0.999998i \(0.500572\pi\)
\(462\) 0 0
\(463\) 7.73598 0.359521 0.179761 0.983710i \(-0.442468\pi\)
0.179761 + 0.983710i \(0.442468\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.11290 −0.421695 −0.210847 0.977519i \(-0.567622\pi\)
−0.210847 + 0.977519i \(0.567622\pi\)
\(468\) 0 0
\(469\) 4.99017 0.230425
\(470\) 0 0
\(471\) −5.36764 −0.247328
\(472\) 0 0
\(473\) 52.9321 2.43382
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5.56554 0.254828
\(478\) 0 0
\(479\) 14.8868 0.680197 0.340099 0.940390i \(-0.389540\pi\)
0.340099 + 0.940390i \(0.389540\pi\)
\(480\) 0 0
\(481\) −1.68400 −0.0767840
\(482\) 0 0
\(483\) 1.03533 0.0471090
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 37.1917 1.68532 0.842658 0.538449i \(-0.180989\pi\)
0.842658 + 0.538449i \(0.180989\pi\)
\(488\) 0 0
\(489\) 7.80668 0.353030
\(490\) 0 0
\(491\) −6.94872 −0.313591 −0.156796 0.987631i \(-0.550116\pi\)
−0.156796 + 0.987631i \(0.550116\pi\)
\(492\) 0 0
\(493\) −28.0711 −1.26426
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.408491 0.0183233
\(498\) 0 0
\(499\) 12.2261 0.547314 0.273657 0.961827i \(-0.411767\pi\)
0.273657 + 0.961827i \(0.411767\pi\)
\(500\) 0 0
\(501\) −9.10840 −0.406933
\(502\) 0 0
\(503\) 28.0625 1.25125 0.625623 0.780126i \(-0.284845\pi\)
0.625623 + 0.780126i \(0.284845\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.11032 −0.271369
\(508\) 0 0
\(509\) 3.23221 0.143265 0.0716326 0.997431i \(-0.477179\pi\)
0.0716326 + 0.997431i \(0.477179\pi\)
\(510\) 0 0
\(511\) 2.57575 0.113944
\(512\) 0 0
\(513\) −2.72160 −0.120162
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −23.9102 −1.05157
\(518\) 0 0
\(519\) 4.38653 0.192547
\(520\) 0 0
\(521\) −27.8354 −1.21949 −0.609745 0.792598i \(-0.708728\pi\)
−0.609745 + 0.792598i \(0.708728\pi\)
\(522\) 0 0
\(523\) 10.1695 0.444682 0.222341 0.974969i \(-0.428630\pi\)
0.222341 + 0.974969i \(0.428630\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −25.4604 −1.10907
\(528\) 0 0
\(529\) −7.98863 −0.347332
\(530\) 0 0
\(531\) −23.1187 −1.00327
\(532\) 0 0
\(533\) 0.988456 0.0428148
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 11.3846 0.491283
\(538\) 0 0
\(539\) −29.2372 −1.25933
\(540\) 0 0
\(541\) 27.4801 1.18146 0.590731 0.806869i \(-0.298840\pi\)
0.590731 + 0.806869i \(0.298840\pi\)
\(542\) 0 0
\(543\) 8.80711 0.377949
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.84680 −0.249991 −0.124995 0.992157i \(-0.539892\pi\)
−0.124995 + 0.992157i \(0.539892\pi\)
\(548\) 0 0
\(549\) −32.8322 −1.40125
\(550\) 0 0
\(551\) 3.53231 0.150482
\(552\) 0 0
\(553\) 6.64468 0.282561
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.2691 0.519860 0.259930 0.965628i \(-0.416301\pi\)
0.259930 + 0.965628i \(0.416301\pi\)
\(558\) 0 0
\(559\) −2.00044 −0.0846095
\(560\) 0 0
\(561\) −16.3876 −0.691884
\(562\) 0 0
\(563\) −17.3234 −0.730096 −0.365048 0.930989i \(-0.618947\pi\)
−0.365048 + 0.930989i \(0.618947\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.00106 0.168029
\(568\) 0 0
\(569\) −10.3824 −0.435252 −0.217626 0.976032i \(-0.569831\pi\)
−0.217626 + 0.976032i \(0.569831\pi\)
\(570\) 0 0
\(571\) 10.3852 0.434607 0.217304 0.976104i \(-0.430274\pi\)
0.217304 + 0.976104i \(0.430274\pi\)
\(572\) 0 0
\(573\) 2.95540 0.123464
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 16.6524 0.693248 0.346624 0.938004i \(-0.387328\pi\)
0.346624 + 0.938004i \(0.387328\pi\)
\(578\) 0 0
\(579\) 3.99938 0.166209
\(580\) 0 0
\(581\) −3.81441 −0.158248
\(582\) 0 0
\(583\) −8.77065 −0.363243
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.4077 −0.553393 −0.276697 0.960957i \(-0.589240\pi\)
−0.276697 + 0.960957i \(0.589240\pi\)
\(588\) 0 0
\(589\) 3.20380 0.132010
\(590\) 0 0
\(591\) 2.81365 0.115738
\(592\) 0 0
\(593\) 20.8050 0.854361 0.427180 0.904166i \(-0.359507\pi\)
0.427180 + 0.904166i \(0.359507\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 9.36207 0.383164
\(598\) 0 0
\(599\) 9.75788 0.398696 0.199348 0.979929i \(-0.436118\pi\)
0.199348 + 0.979929i \(0.436118\pi\)
\(600\) 0 0
\(601\) 3.27717 0.133679 0.0668393 0.997764i \(-0.478709\pi\)
0.0668393 + 0.997764i \(0.478709\pi\)
\(602\) 0 0
\(603\) −24.4365 −0.995132
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −31.1984 −1.26630 −0.633151 0.774028i \(-0.718239\pi\)
−0.633151 + 0.774028i \(0.718239\pi\)
\(608\) 0 0
\(609\) 0.943901 0.0382488
\(610\) 0 0
\(611\) 0.903628 0.0365569
\(612\) 0 0
\(613\) −30.3455 −1.22564 −0.612821 0.790222i \(-0.709965\pi\)
−0.612821 + 0.790222i \(0.709965\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.7447 1.64032 0.820159 0.572135i \(-0.193885\pi\)
0.820159 + 0.572135i \(0.193885\pi\)
\(618\) 0 0
\(619\) −13.7115 −0.551113 −0.275557 0.961285i \(-0.588862\pi\)
−0.275557 + 0.961285i \(0.588862\pi\)
\(620\) 0 0
\(621\) −10.5447 −0.423144
\(622\) 0 0
\(623\) 4.60634 0.184549
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.06213 0.0823534
\(628\) 0 0
\(629\) −80.8830 −3.22501
\(630\) 0 0
\(631\) −32.3748 −1.28882 −0.644411 0.764680i \(-0.722897\pi\)
−0.644411 + 0.764680i \(0.722897\pi\)
\(632\) 0 0
\(633\) −1.34195 −0.0533378
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.10495 0.0437796
\(638\) 0 0
\(639\) −2.00035 −0.0791326
\(640\) 0 0
\(641\) 0.701397 0.0277035 0.0138518 0.999904i \(-0.495591\pi\)
0.0138518 + 0.999904i \(0.495591\pi\)
\(642\) 0 0
\(643\) −12.5736 −0.495856 −0.247928 0.968778i \(-0.579750\pi\)
−0.247928 + 0.968778i \(0.579750\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −36.4494 −1.43297 −0.716487 0.697600i \(-0.754251\pi\)
−0.716487 + 0.697600i \(0.754251\pi\)
\(648\) 0 0
\(649\) 36.4325 1.43010
\(650\) 0 0
\(651\) 0.856115 0.0335538
\(652\) 0 0
\(653\) −44.2535 −1.73177 −0.865887 0.500240i \(-0.833245\pi\)
−0.865887 + 0.500240i \(0.833245\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −12.6132 −0.492089
\(658\) 0 0
\(659\) −18.4958 −0.720493 −0.360247 0.932857i \(-0.617307\pi\)
−0.360247 + 0.932857i \(0.617307\pi\)
\(660\) 0 0
\(661\) 15.3505 0.597067 0.298533 0.954399i \(-0.403503\pi\)
0.298533 + 0.954399i \(0.403503\pi\)
\(662\) 0 0
\(663\) 0.619328 0.0240527
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.6858 0.529915
\(668\) 0 0
\(669\) −0.416338 −0.0160965
\(670\) 0 0
\(671\) 51.7399 1.99740
\(672\) 0 0
\(673\) −34.1219 −1.31530 −0.657651 0.753322i \(-0.728450\pi\)
−0.657651 + 0.753322i \(0.728450\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.7808 0.529639 0.264819 0.964298i \(-0.414688\pi\)
0.264819 + 0.964298i \(0.414688\pi\)
\(678\) 0 0
\(679\) −7.73382 −0.296797
\(680\) 0 0
\(681\) 0.484376 0.0185613
\(682\) 0 0
\(683\) −35.8693 −1.37250 −0.686250 0.727366i \(-0.740744\pi\)
−0.686250 + 0.727366i \(0.740744\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −11.5813 −0.441853
\(688\) 0 0
\(689\) 0.331465 0.0126278
\(690\) 0 0
\(691\) 17.5888 0.669111 0.334555 0.942376i \(-0.391414\pi\)
0.334555 + 0.942376i \(0.391414\pi\)
\(692\) 0 0
\(693\) −6.90026 −0.262119
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 47.4757 1.79827
\(698\) 0 0
\(699\) −2.46617 −0.0932792
\(700\) 0 0
\(701\) −40.3585 −1.52432 −0.762159 0.647389i \(-0.775861\pi\)
−0.762159 + 0.647389i \(0.775861\pi\)
\(702\) 0 0
\(703\) 10.1779 0.383866
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.9315 0.411123
\(708\) 0 0
\(709\) 20.5258 0.770862 0.385431 0.922737i \(-0.374053\pi\)
0.385431 + 0.922737i \(0.374053\pi\)
\(710\) 0 0
\(711\) −32.5385 −1.22029
\(712\) 0 0
\(713\) 12.4130 0.464869
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.03486 −0.225376
\(718\) 0 0
\(719\) −29.8673 −1.11386 −0.556930 0.830559i \(-0.688021\pi\)
−0.556930 + 0.830559i \(0.688021\pi\)
\(720\) 0 0
\(721\) −10.8965 −0.405808
\(722\) 0 0
\(723\) −5.65200 −0.210200
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 26.4630 0.981458 0.490729 0.871312i \(-0.336731\pi\)
0.490729 + 0.871312i \(0.336731\pi\)
\(728\) 0 0
\(729\) −15.7471 −0.583228
\(730\) 0 0
\(731\) −96.0813 −3.55369
\(732\) 0 0
\(733\) −30.5488 −1.12834 −0.564172 0.825657i \(-0.690804\pi\)
−0.564172 + 0.825657i \(0.690804\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 38.5092 1.41850
\(738\) 0 0
\(739\) −34.4001 −1.26543 −0.632714 0.774386i \(-0.718059\pi\)
−0.632714 + 0.774386i \(0.718059\pi\)
\(740\) 0 0
\(741\) −0.0779330 −0.00286294
\(742\) 0 0
\(743\) 21.9370 0.804790 0.402395 0.915466i \(-0.368178\pi\)
0.402395 + 0.915466i \(0.368178\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 18.6789 0.683424
\(748\) 0 0
\(749\) −6.81303 −0.248943
\(750\) 0 0
\(751\) −45.4750 −1.65941 −0.829703 0.558206i \(-0.811490\pi\)
−0.829703 + 0.558206i \(0.811490\pi\)
\(752\) 0 0
\(753\) 11.1720 0.407130
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.91122 0.323884 0.161942 0.986800i \(-0.448224\pi\)
0.161942 + 0.986800i \(0.448224\pi\)
\(758\) 0 0
\(759\) 7.98961 0.290005
\(760\) 0 0
\(761\) −48.1068 −1.74387 −0.871934 0.489623i \(-0.837134\pi\)
−0.871934 + 0.489623i \(0.837134\pi\)
\(762\) 0 0
\(763\) −8.51165 −0.308142
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.37687 −0.0497161
\(768\) 0 0
\(769\) −11.2615 −0.406099 −0.203049 0.979169i \(-0.565085\pi\)
−0.203049 + 0.979169i \(0.565085\pi\)
\(770\) 0 0
\(771\) −0.997039 −0.0359075
\(772\) 0 0
\(773\) 20.4897 0.736964 0.368482 0.929635i \(-0.379878\pi\)
0.368482 + 0.929635i \(0.379878\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.71972 0.0975695
\(778\) 0 0
\(779\) −5.97409 −0.214044
\(780\) 0 0
\(781\) 3.15232 0.112799
\(782\) 0 0
\(783\) −9.61354 −0.343560
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.45356 0.194398 0.0971992 0.995265i \(-0.469012\pi\)
0.0971992 + 0.995265i \(0.469012\pi\)
\(788\) 0 0
\(789\) −0.921655 −0.0328118
\(790\) 0 0
\(791\) 4.61262 0.164006
\(792\) 0 0
\(793\) −1.95538 −0.0694376
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.96228 −0.175773 −0.0878864 0.996131i \(-0.528011\pi\)
−0.0878864 + 0.996131i \(0.528011\pi\)
\(798\) 0 0
\(799\) 43.4014 1.53543
\(800\) 0 0
\(801\) −22.5569 −0.797008
\(802\) 0 0
\(803\) 19.8770 0.701445
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.7523 −0.378498
\(808\) 0 0
\(809\) −18.4521 −0.648741 −0.324370 0.945930i \(-0.605152\pi\)
−0.324370 + 0.945930i \(0.605152\pi\)
\(810\) 0 0
\(811\) 31.3590 1.10116 0.550582 0.834781i \(-0.314406\pi\)
0.550582 + 0.834781i \(0.314406\pi\)
\(812\) 0 0
\(813\) −7.45161 −0.261339
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 12.0904 0.422988
\(818\) 0 0
\(819\) 0.260778 0.00911233
\(820\) 0 0
\(821\) −52.9504 −1.84798 −0.923991 0.382413i \(-0.875093\pi\)
−0.923991 + 0.382413i \(0.875093\pi\)
\(822\) 0 0
\(823\) −46.0815 −1.60630 −0.803150 0.595776i \(-0.796845\pi\)
−0.803150 + 0.595776i \(0.796845\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.9170 −0.379622 −0.189811 0.981821i \(-0.560788\pi\)
−0.189811 + 0.981821i \(0.560788\pi\)
\(828\) 0 0
\(829\) 48.6141 1.68844 0.844219 0.535998i \(-0.180064\pi\)
0.844219 + 0.535998i \(0.180064\pi\)
\(830\) 0 0
\(831\) 4.46488 0.154885
\(832\) 0 0
\(833\) 53.0707 1.83879
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.71945 −0.301388
\(838\) 0 0
\(839\) −35.2171 −1.21583 −0.607915 0.794002i \(-0.707994\pi\)
−0.607915 + 0.794002i \(0.707994\pi\)
\(840\) 0 0
\(841\) −16.5228 −0.569750
\(842\) 0 0
\(843\) 6.27206 0.216021
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.63346 0.159208
\(848\) 0 0
\(849\) −4.33653 −0.148829
\(850\) 0 0
\(851\) 39.4337 1.35177
\(852\) 0 0
\(853\) 7.77006 0.266042 0.133021 0.991113i \(-0.457532\pi\)
0.133021 + 0.991113i \(0.457532\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −20.8792 −0.713219 −0.356610 0.934253i \(-0.616067\pi\)
−0.356610 + 0.934253i \(0.616067\pi\)
\(858\) 0 0
\(859\) 5.48065 0.186997 0.0934987 0.995619i \(-0.470195\pi\)
0.0934987 + 0.995619i \(0.470195\pi\)
\(860\) 0 0
\(861\) −1.59639 −0.0544048
\(862\) 0 0
\(863\) −23.9562 −0.815479 −0.407740 0.913098i \(-0.633683\pi\)
−0.407740 + 0.913098i \(0.633683\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21.7391 0.738300
\(868\) 0 0
\(869\) 51.2770 1.73945
\(870\) 0 0
\(871\) −1.45536 −0.0493129
\(872\) 0 0
\(873\) 37.8719 1.28177
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 18.0925 0.610941 0.305471 0.952201i \(-0.401186\pi\)
0.305471 + 0.952201i \(0.401186\pi\)
\(878\) 0 0
\(879\) 4.87195 0.164327
\(880\) 0 0
\(881\) 6.10173 0.205572 0.102786 0.994703i \(-0.467224\pi\)
0.102786 + 0.994703i \(0.467224\pi\)
\(882\) 0 0
\(883\) −11.8538 −0.398914 −0.199457 0.979907i \(-0.563918\pi\)
−0.199457 + 0.979907i \(0.563918\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 46.7513 1.56976 0.784878 0.619651i \(-0.212726\pi\)
0.784878 + 0.619651i \(0.212726\pi\)
\(888\) 0 0
\(889\) −4.74879 −0.159269
\(890\) 0 0
\(891\) 30.8762 1.03439
\(892\) 0 0
\(893\) −5.46140 −0.182759
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −0.301948 −0.0100817
\(898\) 0 0
\(899\) 11.3168 0.377437
\(900\) 0 0
\(901\) 15.9203 0.530383
\(902\) 0 0
\(903\) 3.23077 0.107513
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 36.8753 1.22442 0.612212 0.790693i \(-0.290280\pi\)
0.612212 + 0.790693i \(0.290280\pi\)
\(908\) 0 0
\(909\) −53.5310 −1.77551
\(910\) 0 0
\(911\) −24.9642 −0.827099 −0.413550 0.910482i \(-0.635711\pi\)
−0.413550 + 0.910482i \(0.635711\pi\)
\(912\) 0 0
\(913\) −29.4358 −0.974182
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.85481 −0.0942740
\(918\) 0 0
\(919\) −8.45204 −0.278807 −0.139404 0.990236i \(-0.544519\pi\)
−0.139404 + 0.990236i \(0.544519\pi\)
\(920\) 0 0
\(921\) 10.3926 0.342448
\(922\) 0 0
\(923\) −0.119134 −0.00392135
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 53.3596 1.75256
\(928\) 0 0
\(929\) 34.3786 1.12793 0.563963 0.825800i \(-0.309276\pi\)
0.563963 + 0.825800i \(0.309276\pi\)
\(930\) 0 0
\(931\) −6.67814 −0.218867
\(932\) 0 0
\(933\) −14.7929 −0.484299
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.23838 −0.171130 −0.0855652 0.996333i \(-0.527270\pi\)
−0.0855652 + 0.996333i \(0.527270\pi\)
\(938\) 0 0
\(939\) −7.13252 −0.232761
\(940\) 0 0
\(941\) 14.3720 0.468514 0.234257 0.972175i \(-0.424734\pi\)
0.234257 + 0.972175i \(0.424734\pi\)
\(942\) 0 0
\(943\) −23.1463 −0.753748
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.8865 0.906190 0.453095 0.891462i \(-0.350320\pi\)
0.453095 + 0.891462i \(0.350320\pi\)
\(948\) 0 0
\(949\) −0.751203 −0.0243851
\(950\) 0 0
\(951\) −8.75104 −0.283772
\(952\) 0 0
\(953\) −18.8741 −0.611392 −0.305696 0.952129i \(-0.598889\pi\)
−0.305696 + 0.952129i \(0.598889\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 7.28408 0.235461
\(958\) 0 0
\(959\) 11.7151 0.378301
\(960\) 0 0
\(961\) −20.7357 −0.668893
\(962\) 0 0
\(963\) 33.3629 1.07510
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.87426 −0.0924301 −0.0462150 0.998932i \(-0.514716\pi\)
−0.0462150 + 0.998932i \(0.514716\pi\)
\(968\) 0 0
\(969\) −3.74313 −0.120247
\(970\) 0 0
\(971\) −50.4156 −1.61791 −0.808956 0.587869i \(-0.799967\pi\)
−0.808956 + 0.587869i \(0.799967\pi\)
\(972\) 0 0
\(973\) 6.50867 0.208658
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −48.5411 −1.55297 −0.776484 0.630137i \(-0.782999\pi\)
−0.776484 + 0.630137i \(0.782999\pi\)
\(978\) 0 0
\(979\) 35.5471 1.13609
\(980\) 0 0
\(981\) 41.6809 1.33077
\(982\) 0 0
\(983\) 5.00431 0.159613 0.0798063 0.996810i \(-0.474570\pi\)
0.0798063 + 0.996810i \(0.474570\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.45939 −0.0464529
\(988\) 0 0
\(989\) 46.8435 1.48954
\(990\) 0 0
\(991\) −43.9761 −1.39695 −0.698474 0.715635i \(-0.746137\pi\)
−0.698474 + 0.715635i \(0.746137\pi\)
\(992\) 0 0
\(993\) 10.7392 0.340798
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −11.4138 −0.361478 −0.180739 0.983531i \(-0.557849\pi\)
−0.180739 + 0.983531i \(0.557849\pi\)
\(998\) 0 0
\(999\) −27.7001 −0.876393
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 3800.2.a.bb.1.4 6
4.3 odd 2 7600.2.a.cm.1.3 6
5.2 odd 4 3800.2.d.p.3649.6 12
5.3 odd 4 3800.2.d.p.3649.7 12
5.4 even 2 3800.2.a.bd.1.3 yes 6
20.19 odd 2 7600.2.a.ci.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.bb.1.4 6 1.1 even 1 trivial
3800.2.a.bd.1.3 yes 6 5.4 even 2
3800.2.d.p.3649.6 12 5.2 odd 4
3800.2.d.p.3649.7 12 5.3 odd 4
7600.2.a.ci.1.4 6 20.19 odd 2
7600.2.a.cm.1.3 6 4.3 odd 2