Properties

Label 4840.2.a.s.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.788.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) 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 \(-1.87740\) 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)
 
\(f(q)\) \(=\) \(q-1.87740 q^{3} -1.00000 q^{5} +4.15686 q^{7} +0.524645 q^{9} +O(q^{10})\) \(q-1.87740 q^{3} -1.00000 q^{5} +4.15686 q^{7} +0.524645 q^{9} -2.27945 q^{13} +1.87740 q^{15} -2.80410 q^{17} +4.27945 q^{19} -7.80410 q^{21} -0.524645 q^{23} +1.00000 q^{25} +4.64724 q^{27} +3.75481 q^{29} -10.2795 q^{31} -4.15686 q^{35} -5.75481 q^{37} +4.27945 q^{39} +1.27945 q^{41} +6.96095 q^{43} -0.524645 q^{45} -10.4363 q^{47} +10.2795 q^{49} +5.26442 q^{51} -10.8384 q^{53} -8.03426 q^{57} -0.524645 q^{59} +8.47536 q^{61} +2.18087 q^{63} +2.27945 q^{65} +11.6322 q^{67} +0.984970 q^{69} -13.6425 q^{71} -4.55890 q^{73} -1.87740 q^{75} +0.804097 q^{79} -10.2987 q^{81} +4.03426 q^{83} +2.80410 q^{85} -7.04929 q^{87} +7.80410 q^{89} -9.47536 q^{91} +19.2987 q^{93} -4.27945 q^{95} +2.27945 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} + 2 q^{13} - q^{15} - 4 q^{17} + 4 q^{19} - 19 q^{21} - 6 q^{23} + 3 q^{25} + 25 q^{27} - 2 q^{29} - 22 q^{31} + 3 q^{35} - 4 q^{37} + 4 q^{39} - 5 q^{41} + q^{43} - 6 q^{45} - 7 q^{47} + 22 q^{49} - 24 q^{51} - 6 q^{53} - 2 q^{57} - 6 q^{59} + 21 q^{61} - 20 q^{63} - 2 q^{65} + 15 q^{67} - 28 q^{69} - 10 q^{71} + 4 q^{73} + q^{75} - 2 q^{79} + 31 q^{81} - 10 q^{83} + 4 q^{85} - 30 q^{87} + 19 q^{89} - 24 q^{91} - 4 q^{93} - 4 q^{95} - 2 q^{97}+O(q^{100}) \) 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
\(3\) −1.87740 −1.08392 −0.541960 0.840404i \(-0.682317\pi\)
−0.541960 + 0.840404i \(0.682317\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.15686 1.57114 0.785572 0.618770i \(-0.212369\pi\)
0.785572 + 0.618770i \(0.212369\pi\)
\(8\) 0 0
\(9\) 0.524645 0.174882
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.27945 −0.632206 −0.316103 0.948725i \(-0.602375\pi\)
−0.316103 + 0.948725i \(0.602375\pi\)
\(14\) 0 0
\(15\) 1.87740 0.484744
\(16\) 0 0
\(17\) −2.80410 −0.680093 −0.340047 0.940409i \(-0.610443\pi\)
−0.340047 + 0.940409i \(0.610443\pi\)
\(18\) 0 0
\(19\) 4.27945 0.981774 0.490887 0.871223i \(-0.336673\pi\)
0.490887 + 0.871223i \(0.336673\pi\)
\(20\) 0 0
\(21\) −7.80410 −1.70299
\(22\) 0 0
\(23\) −0.524645 −0.109396 −0.0546980 0.998503i \(-0.517420\pi\)
−0.0546980 + 0.998503i \(0.517420\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.64724 0.894362
\(28\) 0 0
\(29\) 3.75481 0.697250 0.348625 0.937262i \(-0.386649\pi\)
0.348625 + 0.937262i \(0.386649\pi\)
\(30\) 0 0
\(31\) −10.2795 −1.84624 −0.923122 0.384507i \(-0.874371\pi\)
−0.923122 + 0.384507i \(0.874371\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.15686 −0.702637
\(36\) 0 0
\(37\) −5.75481 −0.946085 −0.473042 0.881040i \(-0.656844\pi\)
−0.473042 + 0.881040i \(0.656844\pi\)
\(38\) 0 0
\(39\) 4.27945 0.685261
\(40\) 0 0
\(41\) 1.27945 0.199817 0.0999084 0.994997i \(-0.468145\pi\)
0.0999084 + 0.994997i \(0.468145\pi\)
\(42\) 0 0
\(43\) 6.96095 1.06154 0.530768 0.847517i \(-0.321904\pi\)
0.530768 + 0.847517i \(0.321904\pi\)
\(44\) 0 0
\(45\) −0.524645 −0.0782094
\(46\) 0 0
\(47\) −10.4363 −1.52229 −0.761146 0.648581i \(-0.775363\pi\)
−0.761146 + 0.648581i \(0.775363\pi\)
\(48\) 0 0
\(49\) 10.2795 1.46849
\(50\) 0 0
\(51\) 5.26442 0.737167
\(52\) 0 0
\(53\) −10.8384 −1.48876 −0.744381 0.667755i \(-0.767256\pi\)
−0.744381 + 0.667755i \(0.767256\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.03426 −1.06416
\(58\) 0 0
\(59\) −0.524645 −0.0683029 −0.0341515 0.999417i \(-0.510873\pi\)
−0.0341515 + 0.999417i \(0.510873\pi\)
\(60\) 0 0
\(61\) 8.47536 1.08516 0.542579 0.840005i \(-0.317448\pi\)
0.542579 + 0.840005i \(0.317448\pi\)
\(62\) 0 0
\(63\) 2.18087 0.274764
\(64\) 0 0
\(65\) 2.27945 0.282731
\(66\) 0 0
\(67\) 11.6322 1.42110 0.710550 0.703646i \(-0.248446\pi\)
0.710550 + 0.703646i \(0.248446\pi\)
\(68\) 0 0
\(69\) 0.984970 0.118576
\(70\) 0 0
\(71\) −13.6425 −1.61906 −0.809531 0.587078i \(-0.800278\pi\)
−0.809531 + 0.587078i \(0.800278\pi\)
\(72\) 0 0
\(73\) −4.55890 −0.533579 −0.266790 0.963755i \(-0.585963\pi\)
−0.266790 + 0.963755i \(0.585963\pi\)
\(74\) 0 0
\(75\) −1.87740 −0.216784
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.804097 0.0904680 0.0452340 0.998976i \(-0.485597\pi\)
0.0452340 + 0.998976i \(0.485597\pi\)
\(80\) 0 0
\(81\) −10.2987 −1.14430
\(82\) 0 0
\(83\) 4.03426 0.442818 0.221409 0.975181i \(-0.428934\pi\)
0.221409 + 0.975181i \(0.428934\pi\)
\(84\) 0 0
\(85\) 2.80410 0.304147
\(86\) 0 0
\(87\) −7.04929 −0.755763
\(88\) 0 0
\(89\) 7.80410 0.827233 0.413616 0.910451i \(-0.364266\pi\)
0.413616 + 0.910451i \(0.364266\pi\)
\(90\) 0 0
\(91\) −9.47536 −0.993287
\(92\) 0 0
\(93\) 19.2987 2.00118
\(94\) 0 0
\(95\) −4.27945 −0.439063
\(96\) 0 0
\(97\) 2.27945 0.231443 0.115722 0.993282i \(-0.463082\pi\)
0.115722 + 0.993282i \(0.463082\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.8726 1.18137 0.590685 0.806902i \(-0.298858\pi\)
0.590685 + 0.806902i \(0.298858\pi\)
\(102\) 0 0
\(103\) 9.64245 0.950099 0.475050 0.879959i \(-0.342430\pi\)
0.475050 + 0.879959i \(0.342430\pi\)
\(104\) 0 0
\(105\) 7.80410 0.761602
\(106\) 0 0
\(107\) 0.681501 0.0658832 0.0329416 0.999457i \(-0.489512\pi\)
0.0329416 + 0.999457i \(0.489512\pi\)
\(108\) 0 0
\(109\) −19.9850 −1.91421 −0.957106 0.289736i \(-0.906432\pi\)
−0.957106 + 0.289736i \(0.906432\pi\)
\(110\) 0 0
\(111\) 10.8041 1.02548
\(112\) 0 0
\(113\) 14.6274 1.37603 0.688016 0.725695i \(-0.258482\pi\)
0.688016 + 0.725695i \(0.258482\pi\)
\(114\) 0 0
\(115\) 0.524645 0.0489234
\(116\) 0 0
\(117\) −1.19590 −0.110561
\(118\) 0 0
\(119\) −11.6562 −1.06852
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −2.40205 −0.216585
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.07331 −0.627654 −0.313827 0.949480i \(-0.601611\pi\)
−0.313827 + 0.949480i \(0.601611\pi\)
\(128\) 0 0
\(129\) −13.0685 −1.15062
\(130\) 0 0
\(131\) −12.3480 −1.07885 −0.539424 0.842035i \(-0.681358\pi\)
−0.539424 + 0.842035i \(0.681358\pi\)
\(132\) 0 0
\(133\) 17.7891 1.54251
\(134\) 0 0
\(135\) −4.64724 −0.399971
\(136\) 0 0
\(137\) −13.1521 −1.12366 −0.561829 0.827254i \(-0.689902\pi\)
−0.561829 + 0.827254i \(0.689902\pi\)
\(138\) 0 0
\(139\) −2.76984 −0.234935 −0.117467 0.993077i \(-0.537478\pi\)
−0.117467 + 0.993077i \(0.537478\pi\)
\(140\) 0 0
\(141\) 19.5932 1.65004
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −3.75481 −0.311820
\(146\) 0 0
\(147\) −19.2987 −1.59173
\(148\) 0 0
\(149\) 9.83836 0.805990 0.402995 0.915202i \(-0.367969\pi\)
0.402995 + 0.915202i \(0.367969\pi\)
\(150\) 0 0
\(151\) −17.1521 −1.39582 −0.697908 0.716188i \(-0.745885\pi\)
−0.697908 + 0.716188i \(0.745885\pi\)
\(152\) 0 0
\(153\) −1.47116 −0.118936
\(154\) 0 0
\(155\) 10.2795 0.825665
\(156\) 0 0
\(157\) 16.0685 1.28241 0.641204 0.767371i \(-0.278435\pi\)
0.641204 + 0.767371i \(0.278435\pi\)
\(158\) 0 0
\(159\) 20.3480 1.61370
\(160\) 0 0
\(161\) −2.18087 −0.171877
\(162\) 0 0
\(163\) −18.1569 −1.42215 −0.711077 0.703114i \(-0.751792\pi\)
−0.711077 + 0.703114i \(0.751792\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0583 −1.24263 −0.621313 0.783562i \(-0.713401\pi\)
−0.621313 + 0.783562i \(0.713401\pi\)
\(168\) 0 0
\(169\) −7.80410 −0.600315
\(170\) 0 0
\(171\) 2.24519 0.171694
\(172\) 0 0
\(173\) 5.78907 0.440135 0.220067 0.975485i \(-0.429372\pi\)
0.220067 + 0.975485i \(0.429372\pi\)
\(174\) 0 0
\(175\) 4.15686 0.314229
\(176\) 0 0
\(177\) 0.984970 0.0740349
\(178\) 0 0
\(179\) −1.44110 −0.107713 −0.0538563 0.998549i \(-0.517151\pi\)
−0.0538563 + 0.998549i \(0.517151\pi\)
\(180\) 0 0
\(181\) 20.9219 1.55511 0.777557 0.628813i \(-0.216459\pi\)
0.777557 + 0.628813i \(0.216459\pi\)
\(182\) 0 0
\(183\) −15.9117 −1.17622
\(184\) 0 0
\(185\) 5.75481 0.423102
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 19.3179 1.40517
\(190\) 0 0
\(191\) −3.08355 −0.223118 −0.111559 0.993758i \(-0.535584\pi\)
−0.111559 + 0.993758i \(0.535584\pi\)
\(192\) 0 0
\(193\) 15.7548 1.13406 0.567028 0.823699i \(-0.308093\pi\)
0.567028 + 0.823699i \(0.308093\pi\)
\(194\) 0 0
\(195\) −4.27945 −0.306458
\(196\) 0 0
\(197\) −19.2644 −1.37253 −0.686267 0.727350i \(-0.740752\pi\)
−0.686267 + 0.727350i \(0.740752\pi\)
\(198\) 0 0
\(199\) −14.0685 −0.997291 −0.498645 0.866806i \(-0.666169\pi\)
−0.498645 + 0.866806i \(0.666169\pi\)
\(200\) 0 0
\(201\) −21.8384 −1.54036
\(202\) 0 0
\(203\) 15.6082 1.09548
\(204\) 0 0
\(205\) −1.27945 −0.0893608
\(206\) 0 0
\(207\) −0.275252 −0.0191313
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 13.2302 0.910802 0.455401 0.890286i \(-0.349496\pi\)
0.455401 + 0.890286i \(0.349496\pi\)
\(212\) 0 0
\(213\) 25.6124 1.75493
\(214\) 0 0
\(215\) −6.96095 −0.474733
\(216\) 0 0
\(217\) −42.7302 −2.90072
\(218\) 0 0
\(219\) 8.55890 0.578357
\(220\) 0 0
\(221\) 6.39181 0.429959
\(222\) 0 0
\(223\) −21.3528 −1.42989 −0.714943 0.699182i \(-0.753547\pi\)
−0.714943 + 0.699182i \(0.753547\pi\)
\(224\) 0 0
\(225\) 0.524645 0.0349763
\(226\) 0 0
\(227\) −28.7158 −1.90593 −0.952966 0.303077i \(-0.901986\pi\)
−0.952966 + 0.303077i \(0.901986\pi\)
\(228\) 0 0
\(229\) −7.31371 −0.483304 −0.241652 0.970363i \(-0.577689\pi\)
−0.241652 + 0.970363i \(0.577689\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −29.6425 −1.94194 −0.970971 0.239198i \(-0.923116\pi\)
−0.970971 + 0.239198i \(0.923116\pi\)
\(234\) 0 0
\(235\) 10.4363 0.680790
\(236\) 0 0
\(237\) −1.50961 −0.0980600
\(238\) 0 0
\(239\) 17.9562 1.16149 0.580744 0.814086i \(-0.302762\pi\)
0.580744 + 0.814086i \(0.302762\pi\)
\(240\) 0 0
\(241\) 6.88765 0.443672 0.221836 0.975084i \(-0.428795\pi\)
0.221836 + 0.975084i \(0.428795\pi\)
\(242\) 0 0
\(243\) 5.39306 0.345965
\(244\) 0 0
\(245\) −10.2795 −0.656730
\(246\) 0 0
\(247\) −9.75481 −0.620684
\(248\) 0 0
\(249\) −7.57393 −0.479979
\(250\) 0 0
\(251\) −4.03426 −0.254640 −0.127320 0.991862i \(-0.540638\pi\)
−0.127320 + 0.991862i \(0.540638\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −5.26442 −0.329671
\(256\) 0 0
\(257\) −7.72055 −0.481595 −0.240797 0.970575i \(-0.577409\pi\)
−0.240797 + 0.970575i \(0.577409\pi\)
\(258\) 0 0
\(259\) −23.9219 −1.48643
\(260\) 0 0
\(261\) 1.96994 0.121936
\(262\) 0 0
\(263\) −18.8384 −1.16162 −0.580811 0.814038i \(-0.697265\pi\)
−0.580811 + 0.814038i \(0.697265\pi\)
\(264\) 0 0
\(265\) 10.8384 0.665795
\(266\) 0 0
\(267\) −14.6514 −0.896654
\(268\) 0 0
\(269\) −14.1521 −0.862867 −0.431433 0.902145i \(-0.641992\pi\)
−0.431433 + 0.902145i \(0.641992\pi\)
\(270\) 0 0
\(271\) 18.8726 1.14643 0.573215 0.819405i \(-0.305696\pi\)
0.573215 + 0.819405i \(0.305696\pi\)
\(272\) 0 0
\(273\) 17.7891 1.07664
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −27.7548 −1.66762 −0.833812 0.552048i \(-0.813847\pi\)
−0.833812 + 0.552048i \(0.813847\pi\)
\(278\) 0 0
\(279\) −5.39306 −0.322874
\(280\) 0 0
\(281\) 25.0192 1.49252 0.746261 0.665653i \(-0.231847\pi\)
0.746261 + 0.665653i \(0.231847\pi\)
\(282\) 0 0
\(283\) 2.04450 0.121533 0.0607665 0.998152i \(-0.480645\pi\)
0.0607665 + 0.998152i \(0.480645\pi\)
\(284\) 0 0
\(285\) 8.03426 0.475908
\(286\) 0 0
\(287\) 5.31850 0.313941
\(288\) 0 0
\(289\) −9.13704 −0.537473
\(290\) 0 0
\(291\) −4.27945 −0.250866
\(292\) 0 0
\(293\) −22.5932 −1.31991 −0.659953 0.751307i \(-0.729424\pi\)
−0.659953 + 0.751307i \(0.729424\pi\)
\(294\) 0 0
\(295\) 0.524645 0.0305460
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.19590 0.0691608
\(300\) 0 0
\(301\) 28.9357 1.66782
\(302\) 0 0
\(303\) −22.2897 −1.28051
\(304\) 0 0
\(305\) −8.47536 −0.485297
\(306\) 0 0
\(307\) −7.96574 −0.454629 −0.227314 0.973821i \(-0.572995\pi\)
−0.227314 + 0.973821i \(0.572995\pi\)
\(308\) 0 0
\(309\) −18.1028 −1.02983
\(310\) 0 0
\(311\) −18.6713 −1.05875 −0.529375 0.848388i \(-0.677574\pi\)
−0.529375 + 0.848388i \(0.677574\pi\)
\(312\) 0 0
\(313\) 4.81787 0.272322 0.136161 0.990687i \(-0.456524\pi\)
0.136161 + 0.990687i \(0.456524\pi\)
\(314\) 0 0
\(315\) −2.18087 −0.122878
\(316\) 0 0
\(317\) 8.39181 0.471331 0.235665 0.971834i \(-0.424273\pi\)
0.235665 + 0.971834i \(0.424273\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.27945 −0.0714121
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −2.27945 −0.126441
\(326\) 0 0
\(327\) 37.5199 2.07485
\(328\) 0 0
\(329\) −43.3822 −2.39174
\(330\) 0 0
\(331\) −11.7110 −0.643693 −0.321847 0.946792i \(-0.604304\pi\)
−0.321847 + 0.946792i \(0.604304\pi\)
\(332\) 0 0
\(333\) −3.01923 −0.165453
\(334\) 0 0
\(335\) −11.6322 −0.635536
\(336\) 0 0
\(337\) −12.7398 −0.693980 −0.346990 0.937869i \(-0.612796\pi\)
−0.346990 + 0.937869i \(0.612796\pi\)
\(338\) 0 0
\(339\) −27.4616 −1.49151
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.6322 0.736070
\(344\) 0 0
\(345\) −0.984970 −0.0530290
\(346\) 0 0
\(347\) −30.0583 −1.61361 −0.806806 0.590816i \(-0.798806\pi\)
−0.806806 + 0.590816i \(0.798806\pi\)
\(348\) 0 0
\(349\) 28.3137 1.51560 0.757799 0.652488i \(-0.226275\pi\)
0.757799 + 0.652488i \(0.226275\pi\)
\(350\) 0 0
\(351\) −10.5932 −0.565421
\(352\) 0 0
\(353\) −26.1028 −1.38931 −0.694655 0.719343i \(-0.744443\pi\)
−0.694655 + 0.719343i \(0.744443\pi\)
\(354\) 0 0
\(355\) 13.6425 0.724066
\(356\) 0 0
\(357\) 21.8834 1.15819
\(358\) 0 0
\(359\) −24.4946 −1.29277 −0.646387 0.763009i \(-0.723721\pi\)
−0.646387 + 0.763009i \(0.723721\pi\)
\(360\) 0 0
\(361\) −0.686288 −0.0361204
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.55890 0.238624
\(366\) 0 0
\(367\) −33.7993 −1.76431 −0.882155 0.470960i \(-0.843908\pi\)
−0.882155 + 0.470960i \(0.843908\pi\)
\(368\) 0 0
\(369\) 0.671258 0.0349443
\(370\) 0 0
\(371\) −45.0535 −2.33906
\(372\) 0 0
\(373\) −22.1028 −1.14444 −0.572219 0.820101i \(-0.693917\pi\)
−0.572219 + 0.820101i \(0.693917\pi\)
\(374\) 0 0
\(375\) 1.87740 0.0969487
\(376\) 0 0
\(377\) −8.55890 −0.440806
\(378\) 0 0
\(379\) 29.1658 1.49815 0.749074 0.662486i \(-0.230499\pi\)
0.749074 + 0.662486i \(0.230499\pi\)
\(380\) 0 0
\(381\) 13.2795 0.680327
\(382\) 0 0
\(383\) 0.378032 0.0193165 0.00965826 0.999953i \(-0.496926\pi\)
0.00965826 + 0.999953i \(0.496926\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.65203 0.185643
\(388\) 0 0
\(389\) 28.0055 1.41993 0.709966 0.704235i \(-0.248710\pi\)
0.709966 + 0.704235i \(0.248710\pi\)
\(390\) 0 0
\(391\) 1.47116 0.0743995
\(392\) 0 0
\(393\) 23.1821 1.16938
\(394\) 0 0
\(395\) −0.804097 −0.0404585
\(396\) 0 0
\(397\) 0.0685196 0.00343890 0.00171945 0.999999i \(-0.499453\pi\)
0.00171945 + 0.999999i \(0.499453\pi\)
\(398\) 0 0
\(399\) −33.3973 −1.67195
\(400\) 0 0
\(401\) −24.0192 −1.19946 −0.599732 0.800201i \(-0.704726\pi\)
−0.599732 + 0.800201i \(0.704726\pi\)
\(402\) 0 0
\(403\) 23.4315 1.16721
\(404\) 0 0
\(405\) 10.2987 0.511746
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.83836 −0.0909009 −0.0454505 0.998967i \(-0.514472\pi\)
−0.0454505 + 0.998967i \(0.514472\pi\)
\(410\) 0 0
\(411\) 24.6917 1.21795
\(412\) 0 0
\(413\) −2.18087 −0.107314
\(414\) 0 0
\(415\) −4.03426 −0.198034
\(416\) 0 0
\(417\) 5.20010 0.254650
\(418\) 0 0
\(419\) −0.984970 −0.0481189 −0.0240595 0.999711i \(-0.507659\pi\)
−0.0240595 + 0.999711i \(0.507659\pi\)
\(420\) 0 0
\(421\) −26.9562 −1.31376 −0.656882 0.753994i \(-0.728125\pi\)
−0.656882 + 0.753994i \(0.728125\pi\)
\(422\) 0 0
\(423\) −5.47536 −0.266221
\(424\) 0 0
\(425\) −2.80410 −0.136019
\(426\) 0 0
\(427\) 35.2308 1.70494
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.41229 −0.308869 −0.154435 0.988003i \(-0.549356\pi\)
−0.154435 + 0.988003i \(0.549356\pi\)
\(432\) 0 0
\(433\) 22.7398 1.09280 0.546402 0.837523i \(-0.315997\pi\)
0.546402 + 0.837523i \(0.315997\pi\)
\(434\) 0 0
\(435\) 7.04929 0.337988
\(436\) 0 0
\(437\) −2.24519 −0.107402
\(438\) 0 0
\(439\) 12.2151 0.582996 0.291498 0.956571i \(-0.405846\pi\)
0.291498 + 0.956571i \(0.405846\pi\)
\(440\) 0 0
\(441\) 5.39306 0.256812
\(442\) 0 0
\(443\) 1.98976 0.0945362 0.0472681 0.998882i \(-0.484948\pi\)
0.0472681 + 0.998882i \(0.484948\pi\)
\(444\) 0 0
\(445\) −7.80410 −0.369950
\(446\) 0 0
\(447\) −18.4706 −0.873628
\(448\) 0 0
\(449\) 5.45613 0.257490 0.128745 0.991678i \(-0.458905\pi\)
0.128745 + 0.991678i \(0.458905\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 32.2014 1.51295
\(454\) 0 0
\(455\) 9.47536 0.444212
\(456\) 0 0
\(457\) 6.37803 0.298352 0.149176 0.988811i \(-0.452338\pi\)
0.149176 + 0.988811i \(0.452338\pi\)
\(458\) 0 0
\(459\) −13.0313 −0.608250
\(460\) 0 0
\(461\) −4.19590 −0.195423 −0.0977113 0.995215i \(-0.531152\pi\)
−0.0977113 + 0.995215i \(0.531152\pi\)
\(462\) 0 0
\(463\) −30.4706 −1.41609 −0.708044 0.706169i \(-0.750422\pi\)
−0.708044 + 0.706169i \(0.750422\pi\)
\(464\) 0 0
\(465\) −19.2987 −0.894955
\(466\) 0 0
\(467\) −12.9952 −0.601347 −0.300673 0.953727i \(-0.597211\pi\)
−0.300673 + 0.953727i \(0.597211\pi\)
\(468\) 0 0
\(469\) 48.3534 2.23275
\(470\) 0 0
\(471\) −30.1671 −1.39003
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.27945 0.196355
\(476\) 0 0
\(477\) −5.68629 −0.260357
\(478\) 0 0
\(479\) 6.97120 0.318522 0.159261 0.987237i \(-0.449089\pi\)
0.159261 + 0.987237i \(0.449089\pi\)
\(480\) 0 0
\(481\) 13.1178 0.598121
\(482\) 0 0
\(483\) 4.09438 0.186301
\(484\) 0 0
\(485\) −2.27945 −0.103505
\(486\) 0 0
\(487\) 23.6124 1.06998 0.534990 0.844858i \(-0.320315\pi\)
0.534990 + 0.844858i \(0.320315\pi\)
\(488\) 0 0
\(489\) 34.0877 1.54150
\(490\) 0 0
\(491\) −34.4808 −1.55610 −0.778049 0.628204i \(-0.783790\pi\)
−0.778049 + 0.628204i \(0.783790\pi\)
\(492\) 0 0
\(493\) −10.5288 −0.474195
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −56.7097 −2.54378
\(498\) 0 0
\(499\) 15.7452 0.704853 0.352427 0.935839i \(-0.385357\pi\)
0.352427 + 0.935839i \(0.385357\pi\)
\(500\) 0 0
\(501\) 30.1479 1.34691
\(502\) 0 0
\(503\) −28.0144 −1.24910 −0.624551 0.780984i \(-0.714718\pi\)
−0.624551 + 0.780984i \(0.714718\pi\)
\(504\) 0 0
\(505\) −11.8726 −0.528325
\(506\) 0 0
\(507\) 14.6514 0.650693
\(508\) 0 0
\(509\) 36.0397 1.59743 0.798716 0.601708i \(-0.205513\pi\)
0.798716 + 0.601708i \(0.205513\pi\)
\(510\) 0 0
\(511\) −18.9507 −0.838330
\(512\) 0 0
\(513\) 19.8876 0.878061
\(514\) 0 0
\(515\) −9.64245 −0.424897
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −10.8684 −0.477071
\(520\) 0 0
\(521\) 15.1466 0.663585 0.331793 0.943352i \(-0.392347\pi\)
0.331793 + 0.943352i \(0.392347\pi\)
\(522\) 0 0
\(523\) 11.3287 0.495371 0.247686 0.968840i \(-0.420330\pi\)
0.247686 + 0.968840i \(0.420330\pi\)
\(524\) 0 0
\(525\) −7.80410 −0.340599
\(526\) 0 0
\(527\) 28.8246 1.25562
\(528\) 0 0
\(529\) −22.7247 −0.988033
\(530\) 0 0
\(531\) −0.275252 −0.0119449
\(532\) 0 0
\(533\) −2.91645 −0.126325
\(534\) 0 0
\(535\) −0.681501 −0.0294639
\(536\) 0 0
\(537\) 2.70552 0.116752
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.62742 −0.327929 −0.163964 0.986466i \(-0.552428\pi\)
−0.163964 + 0.986466i \(0.552428\pi\)
\(542\) 0 0
\(543\) −39.2789 −1.68562
\(544\) 0 0
\(545\) 19.9850 0.856062
\(546\) 0 0
\(547\) −28.1713 −1.20452 −0.602259 0.798301i \(-0.705733\pi\)
−0.602259 + 0.798301i \(0.705733\pi\)
\(548\) 0 0
\(549\) 4.44655 0.189774
\(550\) 0 0
\(551\) 16.0685 0.684542
\(552\) 0 0
\(553\) 3.34252 0.142138
\(554\) 0 0
\(555\) −10.8041 −0.458608
\(556\) 0 0
\(557\) −14.8726 −0.630173 −0.315086 0.949063i \(-0.602034\pi\)
−0.315086 + 0.949063i \(0.602034\pi\)
\(558\) 0 0
\(559\) −15.8672 −0.671109
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.0487 1.60356 0.801781 0.597618i \(-0.203886\pi\)
0.801781 + 0.597618i \(0.203886\pi\)
\(564\) 0 0
\(565\) −14.6274 −0.615380
\(566\) 0 0
\(567\) −42.8101 −1.79786
\(568\) 0 0
\(569\) −12.1521 −0.509441 −0.254721 0.967015i \(-0.581984\pi\)
−0.254721 + 0.967015i \(0.581984\pi\)
\(570\) 0 0
\(571\) 15.8233 0.662186 0.331093 0.943598i \(-0.392583\pi\)
0.331093 + 0.943598i \(0.392583\pi\)
\(572\) 0 0
\(573\) 5.78907 0.241842
\(574\) 0 0
\(575\) −0.524645 −0.0218792
\(576\) 0 0
\(577\) 44.2014 1.84013 0.920063 0.391770i \(-0.128137\pi\)
0.920063 + 0.391770i \(0.128137\pi\)
\(578\) 0 0
\(579\) −29.5781 −1.22923
\(580\) 0 0
\(581\) 16.7698 0.695730
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.19590 0.0494445
\(586\) 0 0
\(587\) −11.5980 −0.478699 −0.239349 0.970934i \(-0.576934\pi\)
−0.239349 + 0.970934i \(0.576934\pi\)
\(588\) 0 0
\(589\) −43.9904 −1.81259
\(590\) 0 0
\(591\) 36.1671 1.48772
\(592\) 0 0
\(593\) 27.3973 1.12507 0.562535 0.826773i \(-0.309826\pi\)
0.562535 + 0.826773i \(0.309826\pi\)
\(594\) 0 0
\(595\) 11.6562 0.477859
\(596\) 0 0
\(597\) 26.4123 1.08098
\(598\) 0 0
\(599\) 28.4165 1.16107 0.580533 0.814237i \(-0.302844\pi\)
0.580533 + 0.814237i \(0.302844\pi\)
\(600\) 0 0
\(601\) −27.0397 −1.10297 −0.551487 0.834184i \(-0.685939\pi\)
−0.551487 + 0.834184i \(0.685939\pi\)
\(602\) 0 0
\(603\) 6.10278 0.248524
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 31.4658 1.27716 0.638578 0.769557i \(-0.279523\pi\)
0.638578 + 0.769557i \(0.279523\pi\)
\(608\) 0 0
\(609\) −29.3029 −1.18741
\(610\) 0 0
\(611\) 23.7891 0.962403
\(612\) 0 0
\(613\) 9.09732 0.367437 0.183719 0.982979i \(-0.441186\pi\)
0.183719 + 0.982979i \(0.441186\pi\)
\(614\) 0 0
\(615\) 2.40205 0.0968599
\(616\) 0 0
\(617\) −19.3768 −0.780080 −0.390040 0.920798i \(-0.627539\pi\)
−0.390040 + 0.920798i \(0.627539\pi\)
\(618\) 0 0
\(619\) 28.8521 1.15967 0.579833 0.814736i \(-0.303118\pi\)
0.579833 + 0.814736i \(0.303118\pi\)
\(620\) 0 0
\(621\) −2.43815 −0.0978396
\(622\) 0 0
\(623\) 32.4405 1.29970
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.1370 0.643426
\(630\) 0 0
\(631\) 44.6959 1.77932 0.889659 0.456626i \(-0.150942\pi\)
0.889659 + 0.456626i \(0.150942\pi\)
\(632\) 0 0
\(633\) −24.8384 −0.987236
\(634\) 0 0
\(635\) 7.07331 0.280696
\(636\) 0 0
\(637\) −23.4315 −0.928391
\(638\) 0 0
\(639\) −7.15744 −0.283144
\(640\) 0 0
\(641\) 28.3041 1.11795 0.558973 0.829186i \(-0.311195\pi\)
0.558973 + 0.829186i \(0.311195\pi\)
\(642\) 0 0
\(643\) −3.98018 −0.156963 −0.0784816 0.996916i \(-0.525007\pi\)
−0.0784816 + 0.996916i \(0.525007\pi\)
\(644\) 0 0
\(645\) 13.0685 0.514572
\(646\) 0 0
\(647\) −9.37324 −0.368500 −0.184250 0.982879i \(-0.558986\pi\)
−0.184250 + 0.982879i \(0.558986\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 80.2218 3.14414
\(652\) 0 0
\(653\) −19.5301 −0.764272 −0.382136 0.924106i \(-0.624811\pi\)
−0.382136 + 0.924106i \(0.624811\pi\)
\(654\) 0 0
\(655\) 12.3480 0.482475
\(656\) 0 0
\(657\) −2.39181 −0.0933132
\(658\) 0 0
\(659\) 13.0973 0.510199 0.255100 0.966915i \(-0.417892\pi\)
0.255100 + 0.966915i \(0.417892\pi\)
\(660\) 0 0
\(661\) 45.0932 1.75392 0.876961 0.480561i \(-0.159567\pi\)
0.876961 + 0.480561i \(0.159567\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −17.7891 −0.689830
\(666\) 0 0
\(667\) −1.96994 −0.0762764
\(668\) 0 0
\(669\) 40.0877 1.54988
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.57268 −0.330452 −0.165226 0.986256i \(-0.552835\pi\)
−0.165226 + 0.986256i \(0.552835\pi\)
\(674\) 0 0
\(675\) 4.64724 0.178872
\(676\) 0 0
\(677\) −30.7260 −1.18090 −0.590448 0.807076i \(-0.701049\pi\)
−0.590448 + 0.807076i \(0.701049\pi\)
\(678\) 0 0
\(679\) 9.47536 0.363631
\(680\) 0 0
\(681\) 53.9111 2.06588
\(682\) 0 0
\(683\) 35.1761 1.34598 0.672988 0.739654i \(-0.265011\pi\)
0.672988 + 0.739654i \(0.265011\pi\)
\(684\) 0 0
\(685\) 13.1521 0.502515
\(686\) 0 0
\(687\) 13.7308 0.523862
\(688\) 0 0
\(689\) 24.7055 0.941205
\(690\) 0 0
\(691\) −1.54387 −0.0587318 −0.0293659 0.999569i \(-0.509349\pi\)
−0.0293659 + 0.999569i \(0.509349\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.76984 0.105066
\(696\) 0 0
\(697\) −3.58771 −0.135894
\(698\) 0 0
\(699\) 55.6509 2.10491
\(700\) 0 0
\(701\) 27.5301 1.03980 0.519899 0.854228i \(-0.325970\pi\)
0.519899 + 0.854228i \(0.325970\pi\)
\(702\) 0 0
\(703\) −24.6274 −0.928841
\(704\) 0 0
\(705\) −19.5932 −0.737921
\(706\) 0 0
\(707\) 49.3528 1.85610
\(708\) 0 0
\(709\) −36.0877 −1.35530 −0.677652 0.735383i \(-0.737002\pi\)
−0.677652 + 0.735383i \(0.737002\pi\)
\(710\) 0 0
\(711\) 0.421865 0.0158212
\(712\) 0 0
\(713\) 5.39306 0.201972
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −33.7110 −1.25896
\(718\) 0 0
\(719\) 0.132839 0.00495406 0.00247703 0.999997i \(-0.499212\pi\)
0.00247703 + 0.999997i \(0.499212\pi\)
\(720\) 0 0
\(721\) 40.0823 1.49274
\(722\) 0 0
\(723\) −12.9309 −0.480905
\(724\) 0 0
\(725\) 3.75481 0.139450
\(726\) 0 0
\(727\) 30.3925 1.12719 0.563597 0.826050i \(-0.309417\pi\)
0.563597 + 0.826050i \(0.309417\pi\)
\(728\) 0 0
\(729\) 20.7711 0.769300
\(730\) 0 0
\(731\) −19.5192 −0.721943
\(732\) 0 0
\(733\) 6.60694 0.244033 0.122016 0.992528i \(-0.461064\pi\)
0.122016 + 0.992528i \(0.461064\pi\)
\(734\) 0 0
\(735\) 19.2987 0.711843
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 13.2945 0.489045 0.244523 0.969644i \(-0.421369\pi\)
0.244523 + 0.969644i \(0.421369\pi\)
\(740\) 0 0
\(741\) 18.3137 0.672771
\(742\) 0 0
\(743\) 8.60341 0.315628 0.157814 0.987469i \(-0.449555\pi\)
0.157814 + 0.987469i \(0.449555\pi\)
\(744\) 0 0
\(745\) −9.83836 −0.360450
\(746\) 0 0
\(747\) 2.11655 0.0774406
\(748\) 0 0
\(749\) 2.83290 0.103512
\(750\) 0 0
\(751\) −7.27400 −0.265432 −0.132716 0.991154i \(-0.542370\pi\)
−0.132716 + 0.991154i \(0.542370\pi\)
\(752\) 0 0
\(753\) 7.57393 0.276010
\(754\) 0 0
\(755\) 17.1521 0.624228
\(756\) 0 0
\(757\) −34.4946 −1.25373 −0.626864 0.779129i \(-0.715662\pi\)
−0.626864 + 0.779129i \(0.715662\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.6274 −0.530244 −0.265122 0.964215i \(-0.585412\pi\)
−0.265122 + 0.964215i \(0.585412\pi\)
\(762\) 0 0
\(763\) −83.0746 −3.00750
\(764\) 0 0
\(765\) 1.47116 0.0531897
\(766\) 0 0
\(767\) 1.19590 0.0431815
\(768\) 0 0
\(769\) −37.5686 −1.35476 −0.677378 0.735635i \(-0.736884\pi\)
−0.677378 + 0.735635i \(0.736884\pi\)
\(770\) 0 0
\(771\) 14.4946 0.522010
\(772\) 0 0
\(773\) 1.76858 0.0636115 0.0318057 0.999494i \(-0.489874\pi\)
0.0318057 + 0.999494i \(0.489874\pi\)
\(774\) 0 0
\(775\) −10.2795 −0.369249
\(776\) 0 0
\(777\) 44.9111 1.61118
\(778\) 0 0
\(779\) 5.47536 0.196175
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 17.4495 0.623594
\(784\) 0 0
\(785\) −16.0685 −0.573510
\(786\) 0 0
\(787\) −26.2692 −0.936396 −0.468198 0.883624i \(-0.655097\pi\)
−0.468198 + 0.883624i \(0.655097\pi\)
\(788\) 0 0
\(789\) 35.3672 1.25911
\(790\) 0 0
\(791\) 60.8041 2.16194
\(792\) 0 0
\(793\) −19.3192 −0.686044
\(794\) 0 0
\(795\) −20.3480 −0.721668
\(796\) 0 0
\(797\) −37.5439 −1.32987 −0.664936 0.746900i \(-0.731541\pi\)
−0.664936 + 0.746900i \(0.731541\pi\)
\(798\) 0 0
\(799\) 29.2644 1.03530
\(800\) 0 0
\(801\) 4.09438 0.144668
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 2.18087 0.0768657
\(806\) 0 0
\(807\) 26.5691 0.935278
\(808\) 0 0
\(809\) −23.1274 −0.813115 −0.406558 0.913625i \(-0.633271\pi\)
−0.406558 + 0.913625i \(0.633271\pi\)
\(810\) 0 0
\(811\) 5.56436 0.195391 0.0976956 0.995216i \(-0.468853\pi\)
0.0976956 + 0.995216i \(0.468853\pi\)
\(812\) 0 0
\(813\) −35.4315 −1.24264
\(814\) 0 0
\(815\) 18.1569 0.636007
\(816\) 0 0
\(817\) 29.7891 1.04219
\(818\) 0 0
\(819\) −4.97120 −0.173708
\(820\) 0 0
\(821\) 21.6713 0.756332 0.378166 0.925738i \(-0.376555\pi\)
0.378166 + 0.925738i \(0.376555\pi\)
\(822\) 0 0
\(823\) −37.4075 −1.30394 −0.651972 0.758243i \(-0.726058\pi\)
−0.651972 + 0.758243i \(0.726058\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.0157 −1.21762 −0.608808 0.793318i \(-0.708352\pi\)
−0.608808 + 0.793318i \(0.708352\pi\)
\(828\) 0 0
\(829\) −20.4658 −0.710806 −0.355403 0.934713i \(-0.615656\pi\)
−0.355403 + 0.934713i \(0.615656\pi\)
\(830\) 0 0
\(831\) 52.1070 1.80757
\(832\) 0 0
\(833\) −28.8246 −0.998713
\(834\) 0 0
\(835\) 16.0583 0.555720
\(836\) 0 0
\(837\) −47.7711 −1.65121
\(838\) 0 0
\(839\) 44.3041 1.52955 0.764774 0.644298i \(-0.222850\pi\)
0.764774 + 0.644298i \(0.222850\pi\)
\(840\) 0 0
\(841\) −14.9014 −0.513842
\(842\) 0 0
\(843\) −46.9712 −1.61777
\(844\) 0 0
\(845\) 7.80410 0.268469
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3.83836 −0.131732
\(850\) 0 0
\(851\) 3.01923 0.103498
\(852\) 0 0
\(853\) 12.6274 0.432355 0.216177 0.976354i \(-0.430641\pi\)
0.216177 + 0.976354i \(0.430641\pi\)
\(854\) 0 0
\(855\) −2.24519 −0.0767840
\(856\) 0 0
\(857\) 53.9904 1.84428 0.922139 0.386859i \(-0.126440\pi\)
0.922139 + 0.386859i \(0.126440\pi\)
\(858\) 0 0
\(859\) −12.8479 −0.438366 −0.219183 0.975684i \(-0.570339\pi\)
−0.219183 + 0.975684i \(0.570339\pi\)
\(860\) 0 0
\(861\) −9.98497 −0.340287
\(862\) 0 0
\(863\) −44.9952 −1.53165 −0.765827 0.643046i \(-0.777670\pi\)
−0.765827 + 0.643046i \(0.777670\pi\)
\(864\) 0 0
\(865\) −5.78907 −0.196834
\(866\) 0 0
\(867\) 17.1539 0.582577
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −26.5151 −0.898429
\(872\) 0 0
\(873\) 1.19590 0.0404752
\(874\) 0 0
\(875\) −4.15686 −0.140527
\(876\) 0 0
\(877\) 37.6124 1.27008 0.635040 0.772479i \(-0.280984\pi\)
0.635040 + 0.772479i \(0.280984\pi\)
\(878\) 0 0
\(879\) 42.4165 1.43067
\(880\) 0 0
\(881\) −31.8288 −1.07234 −0.536169 0.844110i \(-0.680129\pi\)
−0.536169 + 0.844110i \(0.680129\pi\)
\(882\) 0 0
\(883\) 26.9850 0.908117 0.454058 0.890972i \(-0.349976\pi\)
0.454058 + 0.890972i \(0.349976\pi\)
\(884\) 0 0
\(885\) −0.984970 −0.0331094
\(886\) 0 0
\(887\) 32.2939 1.08432 0.542161 0.840274i \(-0.317606\pi\)
0.542161 + 0.840274i \(0.317606\pi\)
\(888\) 0 0
\(889\) −29.4027 −0.986135
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −44.6617 −1.49455
\(894\) 0 0
\(895\) 1.44110 0.0481705
\(896\) 0 0
\(897\) −2.24519 −0.0749648
\(898\) 0 0
\(899\) −38.5974 −1.28729
\(900\) 0 0
\(901\) 30.3918 1.01250
\(902\) 0 0
\(903\) −54.3240 −1.80779
\(904\) 0 0
\(905\) −20.9219 −0.695468
\(906\) 0 0
\(907\) 21.6665 0.719423 0.359712 0.933064i \(-0.382875\pi\)
0.359712 + 0.933064i \(0.382875\pi\)
\(908\) 0 0
\(909\) 6.22891 0.206600
\(910\) 0 0
\(911\) 17.9219 0.593779 0.296890 0.954912i \(-0.404051\pi\)
0.296890 + 0.954912i \(0.404051\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 15.9117 0.526023
\(916\) 0 0
\(917\) −51.3287 −1.69502
\(918\) 0 0
\(919\) −25.5439 −0.842615 −0.421307 0.906918i \(-0.638429\pi\)
−0.421307 + 0.906918i \(0.638429\pi\)
\(920\) 0 0
\(921\) 14.9549 0.492781
\(922\) 0 0
\(923\) 31.0973 1.02358
\(924\) 0 0
\(925\) −5.75481 −0.189217
\(926\) 0 0
\(927\) 5.05886 0.166155
\(928\) 0 0
\(929\) 4.79452 0.157303 0.0786516 0.996902i \(-0.474939\pi\)
0.0786516 + 0.996902i \(0.474939\pi\)
\(930\) 0 0
\(931\) 43.9904 1.44173
\(932\) 0 0
\(933\) 35.0535 1.14760
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 27.5301 0.899369 0.449685 0.893187i \(-0.351536\pi\)
0.449685 + 0.893187i \(0.351536\pi\)
\(938\) 0 0
\(939\) −9.04509 −0.295175
\(940\) 0 0
\(941\) −29.5974 −0.964847 −0.482423 0.875938i \(-0.660243\pi\)
−0.482423 + 0.875938i \(0.660243\pi\)
\(942\) 0 0
\(943\) −0.671258 −0.0218592
\(944\) 0 0
\(945\) −19.3179 −0.628412
\(946\) 0 0
\(947\) 17.1616 0.557678 0.278839 0.960338i \(-0.410050\pi\)
0.278839 + 0.960338i \(0.410050\pi\)
\(948\) 0 0
\(949\) 10.3918 0.337332
\(950\) 0 0
\(951\) −15.7548 −0.510885
\(952\) 0 0
\(953\) 45.6671 1.47930 0.739652 0.672990i \(-0.234990\pi\)
0.739652 + 0.672990i \(0.234990\pi\)
\(954\) 0 0
\(955\) 3.08355 0.0997813
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −54.6713 −1.76543
\(960\) 0 0
\(961\) 74.6671 2.40862
\(962\) 0 0
\(963\) 0.357546 0.0115218
\(964\) 0 0
\(965\) −15.7548 −0.507165
\(966\) 0 0
\(967\) −26.9850 −0.867778 −0.433889 0.900966i \(-0.642859\pi\)
−0.433889 + 0.900966i \(0.642859\pi\)
\(968\) 0 0
\(969\) 22.5288 0.723731
\(970\) 0 0
\(971\) −16.5493 −0.531093 −0.265547 0.964098i \(-0.585552\pi\)
−0.265547 + 0.964098i \(0.585552\pi\)
\(972\) 0 0
\(973\) −11.5138 −0.369116
\(974\) 0 0
\(975\) 4.27945 0.137052
\(976\) 0 0
\(977\) −7.92191 −0.253444 −0.126722 0.991938i \(-0.540446\pi\)
−0.126722 + 0.991938i \(0.540446\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −10.4850 −0.334761
\(982\) 0 0
\(983\) 0.388274 0.0123840 0.00619202 0.999981i \(-0.498029\pi\)
0.00619202 + 0.999981i \(0.498029\pi\)
\(984\) 0 0
\(985\) 19.2644 0.613816
\(986\) 0 0
\(987\) 81.4460 2.59245
\(988\) 0 0
\(989\) −3.65203 −0.116128
\(990\) 0 0
\(991\) 7.98623 0.253691 0.126845 0.991922i \(-0.459515\pi\)
0.126845 + 0.991922i \(0.459515\pi\)
\(992\) 0 0
\(993\) 21.9862 0.697712
\(994\) 0 0
\(995\) 14.0685 0.446002
\(996\) 0 0
\(997\) −48.1713 −1.52560 −0.762800 0.646634i \(-0.776176\pi\)
−0.762800 + 0.646634i \(0.776176\pi\)
\(998\) 0 0
\(999\) −26.7440 −0.846142
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.s.1.1 3
4.3 odd 2 9680.2.a.cd.1.3 3
11.10 odd 2 4840.2.a.v.1.1 yes 3
44.43 even 2 9680.2.a.by.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.s.1.1 3 1.1 even 1 trivial
4840.2.a.v.1.1 yes 3 11.10 odd 2
9680.2.a.by.1.3 3 44.43 even 2
9680.2.a.cd.1.3 3 4.3 odd 2