Properties

Label 4840.2.a.o.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{3} +1.00000 q^{5} +0.414214 q^{7} -2.82843 q^{9} +O(q^{10})\) \(q-0.414214 q^{3} +1.00000 q^{5} +0.414214 q^{7} -2.82843 q^{9} -2.00000 q^{13} -0.414214 q^{15} +5.65685 q^{17} +0.828427 q^{19} -0.171573 q^{21} -7.65685 q^{23} +1.00000 q^{25} +2.41421 q^{27} -2.00000 q^{29} +5.65685 q^{31} +0.414214 q^{35} -4.00000 q^{37} +0.828427 q^{39} -7.48528 q^{41} -6.41421 q^{43} -2.82843 q^{45} +10.4142 q^{47} -6.82843 q^{49} -2.34315 q^{51} +3.65685 q^{53} -0.343146 q^{57} +11.3137 q^{59} -1.00000 q^{61} -1.17157 q^{63} -2.00000 q^{65} -3.58579 q^{67} +3.17157 q^{69} -4.00000 q^{71} -4.00000 q^{73} -0.414214 q^{75} -2.48528 q^{79} +7.48528 q^{81} -9.31371 q^{83} +5.65685 q^{85} +0.828427 q^{87} -8.65685 q^{89} -0.828427 q^{91} -2.34315 q^{93} +0.828427 q^{95} +5.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{13} + 2 q^{15} - 4 q^{19} - 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 2 q^{35} - 8 q^{37} - 4 q^{39} + 2 q^{41} - 10 q^{43} + 18 q^{47} - 8 q^{49} - 16 q^{51} - 4 q^{53} - 12 q^{57} - 2 q^{61} - 8 q^{63} - 4 q^{65} - 10 q^{67} + 12 q^{69} - 8 q^{71} - 8 q^{73} + 2 q^{75} + 12 q^{79} - 2 q^{81} + 4 q^{83} - 4 q^{87} - 6 q^{89} + 4 q^{91} - 16 q^{93} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) 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
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.414214 0.156558 0.0782790 0.996931i \(-0.475058\pi\)
0.0782790 + 0.996931i \(0.475058\pi\)
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −0.414214 −0.106949
\(16\) 0 0
\(17\) 5.65685 1.37199 0.685994 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) 0 0
\(19\) 0.828427 0.190054 0.0950271 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) 0 0
\(21\) −0.171573 −0.0374403
\(22\) 0 0
\(23\) −7.65685 −1.59656 −0.798282 0.602284i \(-0.794258\pi\)
−0.798282 + 0.602284i \(0.794258\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.414214 0.0700149
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0.828427 0.132655
\(40\) 0 0
\(41\) −7.48528 −1.16900 −0.584502 0.811392i \(-0.698710\pi\)
−0.584502 + 0.811392i \(0.698710\pi\)
\(42\) 0 0
\(43\) −6.41421 −0.978158 −0.489079 0.872239i \(-0.662667\pi\)
−0.489079 + 0.872239i \(0.662667\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 10.4142 1.51907 0.759535 0.650467i \(-0.225427\pi\)
0.759535 + 0.650467i \(0.225427\pi\)
\(48\) 0 0
\(49\) −6.82843 −0.975490
\(50\) 0 0
\(51\) −2.34315 −0.328106
\(52\) 0 0
\(53\) 3.65685 0.502308 0.251154 0.967947i \(-0.419190\pi\)
0.251154 + 0.967947i \(0.419190\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.343146 −0.0454508
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −1.17157 −0.147604
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −3.58579 −0.438074 −0.219037 0.975717i \(-0.570291\pi\)
−0.219037 + 0.975717i \(0.570291\pi\)
\(68\) 0 0
\(69\) 3.17157 0.381813
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −0.414214 −0.0478293
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.48528 −0.279616 −0.139808 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) −9.31371 −1.02231 −0.511156 0.859488i \(-0.670783\pi\)
−0.511156 + 0.859488i \(0.670783\pi\)
\(84\) 0 0
\(85\) 5.65685 0.613572
\(86\) 0 0
\(87\) 0.828427 0.0888167
\(88\) 0 0
\(89\) −8.65685 −0.917625 −0.458812 0.888533i \(-0.651725\pi\)
−0.458812 + 0.888533i \(0.651725\pi\)
\(90\) 0 0
\(91\) −0.828427 −0.0868428
\(92\) 0 0
\(93\) −2.34315 −0.242973
\(94\) 0 0
\(95\) 0.828427 0.0849948
\(96\) 0 0
\(97\) 5.31371 0.539525 0.269763 0.962927i \(-0.413055\pi\)
0.269763 + 0.962927i \(0.413055\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.8284 −1.57499 −0.787494 0.616323i \(-0.788622\pi\)
−0.787494 + 0.616323i \(0.788622\pi\)
\(102\) 0 0
\(103\) −9.31371 −0.917707 −0.458853 0.888512i \(-0.651740\pi\)
−0.458853 + 0.888512i \(0.651740\pi\)
\(104\) 0 0
\(105\) −0.171573 −0.0167438
\(106\) 0 0
\(107\) −6.75736 −0.653259 −0.326629 0.945153i \(-0.605913\pi\)
−0.326629 + 0.945153i \(0.605913\pi\)
\(108\) 0 0
\(109\) −5.34315 −0.511781 −0.255890 0.966706i \(-0.582369\pi\)
−0.255890 + 0.966706i \(0.582369\pi\)
\(110\) 0 0
\(111\) 1.65685 0.157262
\(112\) 0 0
\(113\) −5.31371 −0.499872 −0.249936 0.968262i \(-0.580410\pi\)
−0.249936 + 0.968262i \(0.580410\pi\)
\(114\) 0 0
\(115\) −7.65685 −0.714005
\(116\) 0 0
\(117\) 5.65685 0.522976
\(118\) 0 0
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 3.10051 0.279563
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.58579 0.140716 0.0703579 0.997522i \(-0.477586\pi\)
0.0703579 + 0.997522i \(0.477586\pi\)
\(128\) 0 0
\(129\) 2.65685 0.233923
\(130\) 0 0
\(131\) −14.4853 −1.26558 −0.632792 0.774321i \(-0.718091\pi\)
−0.632792 + 0.774321i \(0.718091\pi\)
\(132\) 0 0
\(133\) 0.343146 0.0297545
\(134\) 0 0
\(135\) 2.41421 0.207782
\(136\) 0 0
\(137\) −9.31371 −0.795724 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −4.31371 −0.363280
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 2.82843 0.233285
\(148\) 0 0
\(149\) 17.0000 1.39269 0.696347 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 0 0
\(151\) 5.65685 0.460348 0.230174 0.973149i \(-0.426070\pi\)
0.230174 + 0.973149i \(0.426070\pi\)
\(152\) 0 0
\(153\) −16.0000 −1.29352
\(154\) 0 0
\(155\) 5.65685 0.454369
\(156\) 0 0
\(157\) 6.34315 0.506238 0.253119 0.967435i \(-0.418544\pi\)
0.253119 + 0.967435i \(0.418544\pi\)
\(158\) 0 0
\(159\) −1.51472 −0.120125
\(160\) 0 0
\(161\) −3.17157 −0.249955
\(162\) 0 0
\(163\) 10.4142 0.815704 0.407852 0.913048i \(-0.366278\pi\)
0.407852 + 0.913048i \(0.366278\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.58579 −0.277476 −0.138738 0.990329i \(-0.544305\pi\)
−0.138738 + 0.990329i \(0.544305\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.34315 −0.179185
\(172\) 0 0
\(173\) −7.65685 −0.582140 −0.291070 0.956702i \(-0.594011\pi\)
−0.291070 + 0.956702i \(0.594011\pi\)
\(174\) 0 0
\(175\) 0.414214 0.0313116
\(176\) 0 0
\(177\) −4.68629 −0.352243
\(178\) 0 0
\(179\) 21.7990 1.62933 0.814667 0.579930i \(-0.196920\pi\)
0.814667 + 0.579930i \(0.196920\pi\)
\(180\) 0 0
\(181\) −20.1716 −1.49934 −0.749671 0.661811i \(-0.769788\pi\)
−0.749671 + 0.661811i \(0.769788\pi\)
\(182\) 0 0
\(183\) 0.414214 0.0306195
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 11.1716 0.808347 0.404173 0.914682i \(-0.367559\pi\)
0.404173 + 0.914682i \(0.367559\pi\)
\(192\) 0 0
\(193\) 21.3137 1.53419 0.767097 0.641531i \(-0.221700\pi\)
0.767097 + 0.641531i \(0.221700\pi\)
\(194\) 0 0
\(195\) 0.828427 0.0593249
\(196\) 0 0
\(197\) −26.9706 −1.92157 −0.960787 0.277289i \(-0.910564\pi\)
−0.960787 + 0.277289i \(0.910564\pi\)
\(198\) 0 0
\(199\) −12.8284 −0.909383 −0.454692 0.890649i \(-0.650250\pi\)
−0.454692 + 0.890649i \(0.650250\pi\)
\(200\) 0 0
\(201\) 1.48528 0.104764
\(202\) 0 0
\(203\) −0.828427 −0.0581442
\(204\) 0 0
\(205\) −7.48528 −0.522795
\(206\) 0 0
\(207\) 21.6569 1.50526
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 1.65685 0.113526
\(214\) 0 0
\(215\) −6.41421 −0.437446
\(216\) 0 0
\(217\) 2.34315 0.159063
\(218\) 0 0
\(219\) 1.65685 0.111960
\(220\) 0 0
\(221\) −11.3137 −0.761042
\(222\) 0 0
\(223\) 22.0711 1.47799 0.738994 0.673712i \(-0.235301\pi\)
0.738994 + 0.673712i \(0.235301\pi\)
\(224\) 0 0
\(225\) −2.82843 −0.188562
\(226\) 0 0
\(227\) −11.9289 −0.791751 −0.395876 0.918304i \(-0.629559\pi\)
−0.395876 + 0.918304i \(0.629559\pi\)
\(228\) 0 0
\(229\) −5.48528 −0.362478 −0.181239 0.983439i \(-0.558011\pi\)
−0.181239 + 0.983439i \(0.558011\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.6569 −1.02571 −0.512857 0.858474i \(-0.671413\pi\)
−0.512857 + 0.858474i \(0.671413\pi\)
\(234\) 0 0
\(235\) 10.4142 0.679348
\(236\) 0 0
\(237\) 1.02944 0.0668691
\(238\) 0 0
\(239\) −4.14214 −0.267932 −0.133966 0.990986i \(-0.542771\pi\)
−0.133966 + 0.990986i \(0.542771\pi\)
\(240\) 0 0
\(241\) −21.8284 −1.40609 −0.703046 0.711144i \(-0.748177\pi\)
−0.703046 + 0.711144i \(0.748177\pi\)
\(242\) 0 0
\(243\) −10.3431 −0.663513
\(244\) 0 0
\(245\) −6.82843 −0.436252
\(246\) 0 0
\(247\) −1.65685 −0.105423
\(248\) 0 0
\(249\) 3.85786 0.244482
\(250\) 0 0
\(251\) 23.3137 1.47155 0.735774 0.677227i \(-0.236819\pi\)
0.735774 + 0.677227i \(0.236819\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −2.34315 −0.146733
\(256\) 0 0
\(257\) −20.9706 −1.30811 −0.654054 0.756448i \(-0.726933\pi\)
−0.654054 + 0.756448i \(0.726933\pi\)
\(258\) 0 0
\(259\) −1.65685 −0.102952
\(260\) 0 0
\(261\) 5.65685 0.350150
\(262\) 0 0
\(263\) 20.6274 1.27194 0.635971 0.771713i \(-0.280600\pi\)
0.635971 + 0.771713i \(0.280600\pi\)
\(264\) 0 0
\(265\) 3.65685 0.224639
\(266\) 0 0
\(267\) 3.58579 0.219447
\(268\) 0 0
\(269\) 6.31371 0.384954 0.192477 0.981302i \(-0.438348\pi\)
0.192477 + 0.981302i \(0.438348\pi\)
\(270\) 0 0
\(271\) −11.3137 −0.687259 −0.343629 0.939105i \(-0.611656\pi\)
−0.343629 + 0.939105i \(0.611656\pi\)
\(272\) 0 0
\(273\) 0.343146 0.0207681
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 16.6274 0.999045 0.499522 0.866301i \(-0.333509\pi\)
0.499522 + 0.866301i \(0.333509\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) −17.3137 −1.03285 −0.516425 0.856333i \(-0.672737\pi\)
−0.516425 + 0.856333i \(0.672737\pi\)
\(282\) 0 0
\(283\) −22.2132 −1.32044 −0.660219 0.751073i \(-0.729537\pi\)
−0.660219 + 0.751073i \(0.729537\pi\)
\(284\) 0 0
\(285\) −0.343146 −0.0203262
\(286\) 0 0
\(287\) −3.10051 −0.183017
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) −2.20101 −0.129025
\(292\) 0 0
\(293\) −3.31371 −0.193589 −0.0967945 0.995304i \(-0.530859\pi\)
−0.0967945 + 0.995304i \(0.530859\pi\)
\(294\) 0 0
\(295\) 11.3137 0.658710
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.3137 0.885615
\(300\) 0 0
\(301\) −2.65685 −0.153139
\(302\) 0 0
\(303\) 6.55635 0.376652
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 2.68629 0.153315 0.0766574 0.997057i \(-0.475575\pi\)
0.0766574 + 0.997057i \(0.475575\pi\)
\(308\) 0 0
\(309\) 3.85786 0.219466
\(310\) 0 0
\(311\) 7.17157 0.406663 0.203331 0.979110i \(-0.434823\pi\)
0.203331 + 0.979110i \(0.434823\pi\)
\(312\) 0 0
\(313\) 1.31371 0.0742552 0.0371276 0.999311i \(-0.488179\pi\)
0.0371276 + 0.999311i \(0.488179\pi\)
\(314\) 0 0
\(315\) −1.17157 −0.0660107
\(316\) 0 0
\(317\) −25.3137 −1.42176 −0.710880 0.703314i \(-0.751703\pi\)
−0.710880 + 0.703314i \(0.751703\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 2.79899 0.156224
\(322\) 0 0
\(323\) 4.68629 0.260752
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 2.21320 0.122390
\(328\) 0 0
\(329\) 4.31371 0.237822
\(330\) 0 0
\(331\) −0.828427 −0.0455345 −0.0227672 0.999741i \(-0.507248\pi\)
−0.0227672 + 0.999741i \(0.507248\pi\)
\(332\) 0 0
\(333\) 11.3137 0.619987
\(334\) 0 0
\(335\) −3.58579 −0.195912
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 2.20101 0.119542
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −5.72792 −0.309279
\(344\) 0 0
\(345\) 3.17157 0.170752
\(346\) 0 0
\(347\) 35.3848 1.89955 0.949777 0.312927i \(-0.101310\pi\)
0.949777 + 0.312927i \(0.101310\pi\)
\(348\) 0 0
\(349\) −17.3137 −0.926782 −0.463391 0.886154i \(-0.653367\pi\)
−0.463391 + 0.886154i \(0.653367\pi\)
\(350\) 0 0
\(351\) −4.82843 −0.257722
\(352\) 0 0
\(353\) 10.6274 0.565640 0.282820 0.959173i \(-0.408730\pi\)
0.282820 + 0.959173i \(0.408730\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) −0.970563 −0.0513676
\(358\) 0 0
\(359\) 8.82843 0.465947 0.232973 0.972483i \(-0.425155\pi\)
0.232973 + 0.972483i \(0.425155\pi\)
\(360\) 0 0
\(361\) −18.3137 −0.963879
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −3.24264 −0.169264 −0.0846322 0.996412i \(-0.526972\pi\)
−0.0846322 + 0.996412i \(0.526972\pi\)
\(368\) 0 0
\(369\) 21.1716 1.10215
\(370\) 0 0
\(371\) 1.51472 0.0786403
\(372\) 0 0
\(373\) −12.9706 −0.671590 −0.335795 0.941935i \(-0.609005\pi\)
−0.335795 + 0.941935i \(0.609005\pi\)
\(374\) 0 0
\(375\) −0.414214 −0.0213899
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 8.14214 0.418233 0.209117 0.977891i \(-0.432941\pi\)
0.209117 + 0.977891i \(0.432941\pi\)
\(380\) 0 0
\(381\) −0.656854 −0.0336517
\(382\) 0 0
\(383\) −2.97056 −0.151789 −0.0758943 0.997116i \(-0.524181\pi\)
−0.0758943 + 0.997116i \(0.524181\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 18.1421 0.922217
\(388\) 0 0
\(389\) 3.68629 0.186902 0.0934512 0.995624i \(-0.470210\pi\)
0.0934512 + 0.995624i \(0.470210\pi\)
\(390\) 0 0
\(391\) −43.3137 −2.19047
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) −2.48528 −0.125048
\(396\) 0 0
\(397\) −24.0000 −1.20453 −0.602263 0.798298i \(-0.705734\pi\)
−0.602263 + 0.798298i \(0.705734\pi\)
\(398\) 0 0
\(399\) −0.142136 −0.00711568
\(400\) 0 0
\(401\) −16.3137 −0.814668 −0.407334 0.913279i \(-0.633541\pi\)
−0.407334 + 0.913279i \(0.633541\pi\)
\(402\) 0 0
\(403\) −11.3137 −0.563576
\(404\) 0 0
\(405\) 7.48528 0.371947
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −33.7696 −1.66980 −0.834898 0.550404i \(-0.814474\pi\)
−0.834898 + 0.550404i \(0.814474\pi\)
\(410\) 0 0
\(411\) 3.85786 0.190294
\(412\) 0 0
\(413\) 4.68629 0.230597
\(414\) 0 0
\(415\) −9.31371 −0.457192
\(416\) 0 0
\(417\) 4.97056 0.243410
\(418\) 0 0
\(419\) −34.4853 −1.68472 −0.842358 0.538918i \(-0.818833\pi\)
−0.842358 + 0.538918i \(0.818833\pi\)
\(420\) 0 0
\(421\) −30.3137 −1.47740 −0.738700 0.674034i \(-0.764560\pi\)
−0.738700 + 0.674034i \(0.764560\pi\)
\(422\) 0 0
\(423\) −29.4558 −1.43219
\(424\) 0 0
\(425\) 5.65685 0.274398
\(426\) 0 0
\(427\) −0.414214 −0.0200452
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −34.4853 −1.66110 −0.830549 0.556946i \(-0.811973\pi\)
−0.830549 + 0.556946i \(0.811973\pi\)
\(432\) 0 0
\(433\) 4.97056 0.238870 0.119435 0.992842i \(-0.461892\pi\)
0.119435 + 0.992842i \(0.461892\pi\)
\(434\) 0 0
\(435\) 0.828427 0.0397200
\(436\) 0 0
\(437\) −6.34315 −0.303434
\(438\) 0 0
\(439\) −32.2843 −1.54084 −0.770422 0.637534i \(-0.779954\pi\)
−0.770422 + 0.637534i \(0.779954\pi\)
\(440\) 0 0
\(441\) 19.3137 0.919700
\(442\) 0 0
\(443\) 0.757359 0.0359832 0.0179916 0.999838i \(-0.494273\pi\)
0.0179916 + 0.999838i \(0.494273\pi\)
\(444\) 0 0
\(445\) −8.65685 −0.410374
\(446\) 0 0
\(447\) −7.04163 −0.333058
\(448\) 0 0
\(449\) 10.4558 0.493442 0.246721 0.969087i \(-0.420647\pi\)
0.246721 + 0.969087i \(0.420647\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.34315 −0.110091
\(454\) 0 0
\(455\) −0.828427 −0.0388373
\(456\) 0 0
\(457\) 19.3137 0.903457 0.451729 0.892155i \(-0.350808\pi\)
0.451729 + 0.892155i \(0.350808\pi\)
\(458\) 0 0
\(459\) 13.6569 0.637447
\(460\) 0 0
\(461\) −19.1421 −0.891538 −0.445769 0.895148i \(-0.647070\pi\)
−0.445769 + 0.895148i \(0.647070\pi\)
\(462\) 0 0
\(463\) 6.75736 0.314041 0.157021 0.987595i \(-0.449811\pi\)
0.157021 + 0.987595i \(0.449811\pi\)
\(464\) 0 0
\(465\) −2.34315 −0.108661
\(466\) 0 0
\(467\) 11.4437 0.529549 0.264775 0.964310i \(-0.414702\pi\)
0.264775 + 0.964310i \(0.414702\pi\)
\(468\) 0 0
\(469\) −1.48528 −0.0685839
\(470\) 0 0
\(471\) −2.62742 −0.121065
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.828427 0.0380108
\(476\) 0 0
\(477\) −10.3431 −0.473580
\(478\) 0 0
\(479\) −37.7990 −1.72708 −0.863540 0.504280i \(-0.831758\pi\)
−0.863540 + 0.504280i \(0.831758\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 1.31371 0.0597758
\(484\) 0 0
\(485\) 5.31371 0.241283
\(486\) 0 0
\(487\) 41.3137 1.87210 0.936051 0.351863i \(-0.114452\pi\)
0.936051 + 0.351863i \(0.114452\pi\)
\(488\) 0 0
\(489\) −4.31371 −0.195073
\(490\) 0 0
\(491\) −34.4853 −1.55630 −0.778149 0.628079i \(-0.783841\pi\)
−0.778149 + 0.628079i \(0.783841\pi\)
\(492\) 0 0
\(493\) −11.3137 −0.509544
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.65685 −0.0743201
\(498\) 0 0
\(499\) 1.79899 0.0805338 0.0402669 0.999189i \(-0.487179\pi\)
0.0402669 + 0.999189i \(0.487179\pi\)
\(500\) 0 0
\(501\) 1.48528 0.0663575
\(502\) 0 0
\(503\) 27.3848 1.22103 0.610513 0.792006i \(-0.290963\pi\)
0.610513 + 0.792006i \(0.290963\pi\)
\(504\) 0 0
\(505\) −15.8284 −0.704356
\(506\) 0 0
\(507\) 3.72792 0.165563
\(508\) 0 0
\(509\) −5.48528 −0.243131 −0.121565 0.992583i \(-0.538791\pi\)
−0.121565 + 0.992583i \(0.538791\pi\)
\(510\) 0 0
\(511\) −1.65685 −0.0732949
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) −9.31371 −0.410411
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 3.17157 0.139217
\(520\) 0 0
\(521\) 39.9706 1.75114 0.875571 0.483089i \(-0.160485\pi\)
0.875571 + 0.483089i \(0.160485\pi\)
\(522\) 0 0
\(523\) −9.31371 −0.407260 −0.203630 0.979048i \(-0.565274\pi\)
−0.203630 + 0.979048i \(0.565274\pi\)
\(524\) 0 0
\(525\) −0.171573 −0.00748805
\(526\) 0 0
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) 35.6274 1.54902
\(530\) 0 0
\(531\) −32.0000 −1.38868
\(532\) 0 0
\(533\) 14.9706 0.648447
\(534\) 0 0
\(535\) −6.75736 −0.292146
\(536\) 0 0
\(537\) −9.02944 −0.389649
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23.1421 −0.994958 −0.497479 0.867476i \(-0.665741\pi\)
−0.497479 + 0.867476i \(0.665741\pi\)
\(542\) 0 0
\(543\) 8.35534 0.358562
\(544\) 0 0
\(545\) −5.34315 −0.228875
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 0 0
\(549\) 2.82843 0.120714
\(550\) 0 0
\(551\) −1.65685 −0.0705844
\(552\) 0 0
\(553\) −1.02944 −0.0437761
\(554\) 0 0
\(555\) 1.65685 0.0703295
\(556\) 0 0
\(557\) 22.6274 0.958754 0.479377 0.877609i \(-0.340863\pi\)
0.479377 + 0.877609i \(0.340863\pi\)
\(558\) 0 0
\(559\) 12.8284 0.542585
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.8995 0.712229 0.356114 0.934442i \(-0.384101\pi\)
0.356114 + 0.934442i \(0.384101\pi\)
\(564\) 0 0
\(565\) −5.31371 −0.223549
\(566\) 0 0
\(567\) 3.10051 0.130209
\(568\) 0 0
\(569\) −34.4558 −1.44446 −0.722232 0.691651i \(-0.756884\pi\)
−0.722232 + 0.691651i \(0.756884\pi\)
\(570\) 0 0
\(571\) 27.4558 1.14899 0.574496 0.818508i \(-0.305198\pi\)
0.574496 + 0.818508i \(0.305198\pi\)
\(572\) 0 0
\(573\) −4.62742 −0.193313
\(574\) 0 0
\(575\) −7.65685 −0.319313
\(576\) 0 0
\(577\) 33.9411 1.41299 0.706494 0.707719i \(-0.250276\pi\)
0.706494 + 0.707719i \(0.250276\pi\)
\(578\) 0 0
\(579\) −8.82843 −0.366897
\(580\) 0 0
\(581\) −3.85786 −0.160051
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 5.65685 0.233882
\(586\) 0 0
\(587\) −22.6985 −0.936867 −0.468433 0.883499i \(-0.655181\pi\)
−0.468433 + 0.883499i \(0.655181\pi\)
\(588\) 0 0
\(589\) 4.68629 0.193095
\(590\) 0 0
\(591\) 11.1716 0.459537
\(592\) 0 0
\(593\) 12.3431 0.506872 0.253436 0.967352i \(-0.418439\pi\)
0.253436 + 0.967352i \(0.418439\pi\)
\(594\) 0 0
\(595\) 2.34315 0.0960596
\(596\) 0 0
\(597\) 5.31371 0.217476
\(598\) 0 0
\(599\) 44.1421 1.80360 0.901799 0.432155i \(-0.142247\pi\)
0.901799 + 0.432155i \(0.142247\pi\)
\(600\) 0 0
\(601\) 27.9411 1.13974 0.569871 0.821734i \(-0.306993\pi\)
0.569871 + 0.821734i \(0.306993\pi\)
\(602\) 0 0
\(603\) 10.1421 0.413020
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 45.5980 1.85076 0.925382 0.379035i \(-0.123744\pi\)
0.925382 + 0.379035i \(0.123744\pi\)
\(608\) 0 0
\(609\) 0.343146 0.0139050
\(610\) 0 0
\(611\) −20.8284 −0.842628
\(612\) 0 0
\(613\) 18.9706 0.766214 0.383107 0.923704i \(-0.374854\pi\)
0.383107 + 0.923704i \(0.374854\pi\)
\(614\) 0 0
\(615\) 3.10051 0.125024
\(616\) 0 0
\(617\) −40.0000 −1.61034 −0.805170 0.593045i \(-0.797926\pi\)
−0.805170 + 0.593045i \(0.797926\pi\)
\(618\) 0 0
\(619\) −15.3137 −0.615510 −0.307755 0.951466i \(-0.599578\pi\)
−0.307755 + 0.951466i \(0.599578\pi\)
\(620\) 0 0
\(621\) −18.4853 −0.741789
\(622\) 0 0
\(623\) −3.58579 −0.143662
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −22.6274 −0.902214
\(630\) 0 0
\(631\) 45.6569 1.81757 0.908785 0.417264i \(-0.137011\pi\)
0.908785 + 0.417264i \(0.137011\pi\)
\(632\) 0 0
\(633\) −3.31371 −0.131708
\(634\) 0 0
\(635\) 1.58579 0.0629300
\(636\) 0 0
\(637\) 13.6569 0.541104
\(638\) 0 0
\(639\) 11.3137 0.447563
\(640\) 0 0
\(641\) −3.37258 −0.133209 −0.0666045 0.997779i \(-0.521217\pi\)
−0.0666045 + 0.997779i \(0.521217\pi\)
\(642\) 0 0
\(643\) 25.7279 1.01461 0.507305 0.861767i \(-0.330642\pi\)
0.507305 + 0.861767i \(0.330642\pi\)
\(644\) 0 0
\(645\) 2.65685 0.104614
\(646\) 0 0
\(647\) 23.8701 0.938429 0.469214 0.883084i \(-0.344537\pi\)
0.469214 + 0.883084i \(0.344537\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.970563 −0.0380394
\(652\) 0 0
\(653\) 3.65685 0.143104 0.0715519 0.997437i \(-0.477205\pi\)
0.0715519 + 0.997437i \(0.477205\pi\)
\(654\) 0 0
\(655\) −14.4853 −0.565987
\(656\) 0 0
\(657\) 11.3137 0.441390
\(658\) 0 0
\(659\) 28.8284 1.12300 0.561498 0.827478i \(-0.310225\pi\)
0.561498 + 0.827478i \(0.310225\pi\)
\(660\) 0 0
\(661\) 46.6569 1.81474 0.907371 0.420331i \(-0.138086\pi\)
0.907371 + 0.420331i \(0.138086\pi\)
\(662\) 0 0
\(663\) 4.68629 0.182000
\(664\) 0 0
\(665\) 0.343146 0.0133066
\(666\) 0 0
\(667\) 15.3137 0.592949
\(668\) 0 0
\(669\) −9.14214 −0.353455
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 16.9706 0.654167 0.327084 0.944995i \(-0.393934\pi\)
0.327084 + 0.944995i \(0.393934\pi\)
\(674\) 0 0
\(675\) 2.41421 0.0929231
\(676\) 0 0
\(677\) 31.3137 1.20348 0.601742 0.798691i \(-0.294474\pi\)
0.601742 + 0.798691i \(0.294474\pi\)
\(678\) 0 0
\(679\) 2.20101 0.0844670
\(680\) 0 0
\(681\) 4.94113 0.189344
\(682\) 0 0
\(683\) −33.8701 −1.29600 −0.648001 0.761640i \(-0.724395\pi\)
−0.648001 + 0.761640i \(0.724395\pi\)
\(684\) 0 0
\(685\) −9.31371 −0.355859
\(686\) 0 0
\(687\) 2.27208 0.0866852
\(688\) 0 0
\(689\) −7.31371 −0.278630
\(690\) 0 0
\(691\) 2.48528 0.0945446 0.0472723 0.998882i \(-0.484947\pi\)
0.0472723 + 0.998882i \(0.484947\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −42.3431 −1.60386
\(698\) 0 0
\(699\) 6.48528 0.245296
\(700\) 0 0
\(701\) 27.9411 1.05532 0.527661 0.849455i \(-0.323069\pi\)
0.527661 + 0.849455i \(0.323069\pi\)
\(702\) 0 0
\(703\) −3.31371 −0.124979
\(704\) 0 0
\(705\) −4.31371 −0.162464
\(706\) 0 0
\(707\) −6.55635 −0.246577
\(708\) 0 0
\(709\) −6.85786 −0.257553 −0.128776 0.991674i \(-0.541105\pi\)
−0.128776 + 0.991674i \(0.541105\pi\)
\(710\) 0 0
\(711\) 7.02944 0.263624
\(712\) 0 0
\(713\) −43.3137 −1.62211
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.71573 0.0640751
\(718\) 0 0
\(719\) −36.8284 −1.37347 −0.686734 0.726909i \(-0.740956\pi\)
−0.686734 + 0.726909i \(0.740956\pi\)
\(720\) 0 0
\(721\) −3.85786 −0.143674
\(722\) 0 0
\(723\) 9.04163 0.336262
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 37.9289 1.40671 0.703353 0.710841i \(-0.251685\pi\)
0.703353 + 0.710841i \(0.251685\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) −36.2843 −1.34202
\(732\) 0 0
\(733\) 16.2843 0.601473 0.300737 0.953707i \(-0.402767\pi\)
0.300737 + 0.953707i \(0.402767\pi\)
\(734\) 0 0
\(735\) 2.82843 0.104328
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 50.4853 1.85713 0.928566 0.371168i \(-0.121043\pi\)
0.928566 + 0.371168i \(0.121043\pi\)
\(740\) 0 0
\(741\) 0.686292 0.0252115
\(742\) 0 0
\(743\) −29.5858 −1.08540 −0.542699 0.839928i \(-0.682597\pi\)
−0.542699 + 0.839928i \(0.682597\pi\)
\(744\) 0 0
\(745\) 17.0000 0.622832
\(746\) 0 0
\(747\) 26.3431 0.963845
\(748\) 0 0
\(749\) −2.79899 −0.102273
\(750\) 0 0
\(751\) −0.828427 −0.0302297 −0.0151149 0.999886i \(-0.504811\pi\)
−0.0151149 + 0.999886i \(0.504811\pi\)
\(752\) 0 0
\(753\) −9.65685 −0.351915
\(754\) 0 0
\(755\) 5.65685 0.205874
\(756\) 0 0
\(757\) 2.97056 0.107967 0.0539835 0.998542i \(-0.482808\pi\)
0.0539835 + 0.998542i \(0.482808\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −2.21320 −0.0801233
\(764\) 0 0
\(765\) −16.0000 −0.578481
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 8.68629 0.312829
\(772\) 0 0
\(773\) 4.97056 0.178779 0.0893894 0.995997i \(-0.471508\pi\)
0.0893894 + 0.995997i \(0.471508\pi\)
\(774\) 0 0
\(775\) 5.65685 0.203200
\(776\) 0 0
\(777\) 0.686292 0.0246206
\(778\) 0 0
\(779\) −6.20101 −0.222174
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −4.82843 −0.172554
\(784\) 0 0
\(785\) 6.34315 0.226397
\(786\) 0 0
\(787\) 42.2132 1.50474 0.752369 0.658742i \(-0.228911\pi\)
0.752369 + 0.658742i \(0.228911\pi\)
\(788\) 0 0
\(789\) −8.54416 −0.304180
\(790\) 0 0
\(791\) −2.20101 −0.0782589
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) −1.51472 −0.0537215
\(796\) 0 0
\(797\) 45.6569 1.61725 0.808624 0.588325i \(-0.200213\pi\)
0.808624 + 0.588325i \(0.200213\pi\)
\(798\) 0 0
\(799\) 58.9117 2.08415
\(800\) 0 0
\(801\) 24.4853 0.865145
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −3.17157 −0.111783
\(806\) 0 0
\(807\) −2.61522 −0.0920602
\(808\) 0 0
\(809\) 20.6274 0.725221 0.362611 0.931941i \(-0.381885\pi\)
0.362611 + 0.931941i \(0.381885\pi\)
\(810\) 0 0
\(811\) −5.65685 −0.198639 −0.0993195 0.995056i \(-0.531667\pi\)
−0.0993195 + 0.995056i \(0.531667\pi\)
\(812\) 0 0
\(813\) 4.68629 0.164355
\(814\) 0 0
\(815\) 10.4142 0.364794
\(816\) 0 0
\(817\) −5.31371 −0.185903
\(818\) 0 0
\(819\) 2.34315 0.0818761
\(820\) 0 0
\(821\) −29.9706 −1.04598 −0.522990 0.852339i \(-0.675183\pi\)
−0.522990 + 0.852339i \(0.675183\pi\)
\(822\) 0 0
\(823\) −16.0711 −0.560202 −0.280101 0.959971i \(-0.590368\pi\)
−0.280101 + 0.959971i \(0.590368\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38.3553 −1.33375 −0.666873 0.745171i \(-0.732368\pi\)
−0.666873 + 0.745171i \(0.732368\pi\)
\(828\) 0 0
\(829\) −34.9411 −1.21356 −0.606778 0.794872i \(-0.707538\pi\)
−0.606778 + 0.794872i \(0.707538\pi\)
\(830\) 0 0
\(831\) −6.88730 −0.238918
\(832\) 0 0
\(833\) −38.6274 −1.33836
\(834\) 0 0
\(835\) −3.58579 −0.124091
\(836\) 0 0
\(837\) 13.6569 0.472050
\(838\) 0 0
\(839\) 2.20101 0.0759873 0.0379937 0.999278i \(-0.487903\pi\)
0.0379937 + 0.999278i \(0.487903\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 7.17157 0.247002
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 9.20101 0.315778
\(850\) 0 0
\(851\) 30.6274 1.04989
\(852\) 0 0
\(853\) 18.3431 0.628057 0.314029 0.949413i \(-0.398321\pi\)
0.314029 + 0.949413i \(0.398321\pi\)
\(854\) 0 0
\(855\) −2.34315 −0.0801339
\(856\) 0 0
\(857\) −22.9706 −0.784659 −0.392330 0.919825i \(-0.628331\pi\)
−0.392330 + 0.919825i \(0.628331\pi\)
\(858\) 0 0
\(859\) −36.1421 −1.23315 −0.616577 0.787295i \(-0.711481\pi\)
−0.616577 + 0.787295i \(0.711481\pi\)
\(860\) 0 0
\(861\) 1.28427 0.0437678
\(862\) 0 0
\(863\) 45.8701 1.56143 0.780717 0.624884i \(-0.214854\pi\)
0.780717 + 0.624884i \(0.214854\pi\)
\(864\) 0 0
\(865\) −7.65685 −0.260341
\(866\) 0 0
\(867\) −6.21320 −0.211011
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 7.17157 0.242999
\(872\) 0 0
\(873\) −15.0294 −0.508669
\(874\) 0 0
\(875\) 0.414214 0.0140030
\(876\) 0 0
\(877\) 24.6863 0.833597 0.416798 0.908999i \(-0.363152\pi\)
0.416798 + 0.908999i \(0.363152\pi\)
\(878\) 0 0
\(879\) 1.37258 0.0462961
\(880\) 0 0
\(881\) 19.4853 0.656476 0.328238 0.944595i \(-0.393545\pi\)
0.328238 + 0.944595i \(0.393545\pi\)
\(882\) 0 0
\(883\) −43.6569 −1.46917 −0.734585 0.678517i \(-0.762623\pi\)
−0.734585 + 0.678517i \(0.762623\pi\)
\(884\) 0 0
\(885\) −4.68629 −0.157528
\(886\) 0 0
\(887\) 37.3848 1.25526 0.627629 0.778513i \(-0.284026\pi\)
0.627629 + 0.778513i \(0.284026\pi\)
\(888\) 0 0
\(889\) 0.656854 0.0220302
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.62742 0.288706
\(894\) 0 0
\(895\) 21.7990 0.728660
\(896\) 0 0
\(897\) −6.34315 −0.211791
\(898\) 0 0
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) 20.6863 0.689160
\(902\) 0 0
\(903\) 1.10051 0.0366225
\(904\) 0 0
\(905\) −20.1716 −0.670526
\(906\) 0 0
\(907\) −18.6985 −0.620873 −0.310436 0.950594i \(-0.600475\pi\)
−0.310436 + 0.950594i \(0.600475\pi\)
\(908\) 0 0
\(909\) 44.7696 1.48491
\(910\) 0 0
\(911\) −32.8284 −1.08765 −0.543827 0.839197i \(-0.683025\pi\)
−0.543827 + 0.839197i \(0.683025\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0.414214 0.0136935
\(916\) 0 0
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) 52.4264 1.72939 0.864694 0.502299i \(-0.167512\pi\)
0.864694 + 0.502299i \(0.167512\pi\)
\(920\) 0 0
\(921\) −1.11270 −0.0366647
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 26.3431 0.865222
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) −5.65685 −0.185396
\(932\) 0 0
\(933\) −2.97056 −0.0972519
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) −0.544156 −0.0177579
\(940\) 0 0
\(941\) −46.1716 −1.50515 −0.752575 0.658506i \(-0.771189\pi\)
−0.752575 + 0.658506i \(0.771189\pi\)
\(942\) 0 0
\(943\) 57.3137 1.86639
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 10.0000 0.324956 0.162478 0.986712i \(-0.448051\pi\)
0.162478 + 0.986712i \(0.448051\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 10.4853 0.340009
\(952\) 0 0
\(953\) 32.2843 1.04579 0.522895 0.852397i \(-0.324852\pi\)
0.522895 + 0.852397i \(0.324852\pi\)
\(954\) 0 0
\(955\) 11.1716 0.361504
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.85786 −0.124577
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 19.1127 0.615898
\(964\) 0 0
\(965\) 21.3137 0.686113
\(966\) 0 0
\(967\) 37.5980 1.20907 0.604535 0.796579i \(-0.293359\pi\)
0.604535 + 0.796579i \(0.293359\pi\)
\(968\) 0 0
\(969\) −1.94113 −0.0623579
\(970\) 0 0
\(971\) −60.2843 −1.93461 −0.967307 0.253608i \(-0.918383\pi\)
−0.967307 + 0.253608i \(0.918383\pi\)
\(972\) 0 0
\(973\) −4.97056 −0.159349
\(974\) 0 0
\(975\) 0.828427 0.0265309
\(976\) 0 0
\(977\) −42.6274 −1.36377 −0.681886 0.731459i \(-0.738840\pi\)
−0.681886 + 0.731459i \(0.738840\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 15.1127 0.482511
\(982\) 0 0
\(983\) −20.0711 −0.640168 −0.320084 0.947389i \(-0.603711\pi\)
−0.320084 + 0.947389i \(0.603711\pi\)
\(984\) 0 0
\(985\) −26.9706 −0.859354
\(986\) 0 0
\(987\) −1.78680 −0.0568744
\(988\) 0 0
\(989\) 49.1127 1.56169
\(990\) 0 0
\(991\) 41.5147 1.31876 0.659379 0.751810i \(-0.270819\pi\)
0.659379 + 0.751810i \(0.270819\pi\)
\(992\) 0 0
\(993\) 0.343146 0.0108894
\(994\) 0 0
\(995\) −12.8284 −0.406688
\(996\) 0 0
\(997\) 6.68629 0.211757 0.105878 0.994379i \(-0.466235\pi\)
0.105878 + 0.994379i \(0.466235\pi\)
\(998\) 0 0
\(999\) −9.65685 −0.305529
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 4840.2.a.o.1.1 2
4.3 odd 2 9680.2.a.bk.1.2 2
11.10 odd 2 4840.2.a.p.1.1 yes 2
44.43 even 2 9680.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.o.1.1 2 1.1 even 1 trivial
4840.2.a.p.1.1 yes 2 11.10 odd 2
9680.2.a.bj.1.2 2 44.43 even 2
9680.2.a.bk.1.2 2 4.3 odd 2