Properties

Label 4788.2.a.o.1.2
Level $4788$
Weight $2$
Character 4788.1
Self dual yes
Analytic conductor $38.232$
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] = [4788,2,Mod(1,4788)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4788, 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("4788.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4788.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: \(38.2323724878\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.69639\) of defining polynomial
Character \(\chi\) \(=\) 4788.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.42586 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-1.42586 q^{5} -1.00000 q^{7} +3.27053 q^{11} +3.42586 q^{13} -5.27053 q^{17} -1.00000 q^{19} -0.574141 q^{23} -2.96693 q^{25} +0.122252 q^{29} +2.12225 q^{31} +1.42586 q^{35} -3.96693 q^{37} +4.66332 q^{41} -11.9339 q^{43} -3.11521 q^{47} +1.00000 q^{49} +6.08918 q^{53} -4.66332 q^{55} +1.72242 q^{59} +12.2114 q^{61} -4.88479 q^{65} -5.27053 q^{67} +6.81864 q^{71} -11.5481 q^{73} -3.27053 q^{77} +11.0821 q^{79} -5.23746 q^{83} +7.51504 q^{85} +1.45893 q^{89} -3.42586 q^{91} +1.42586 q^{95} +17.6563 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{5} - 3 q^{7} + 3 q^{11} + 8 q^{13} - 9 q^{17} - 3 q^{19} - 4 q^{23} + 7 q^{25} - 11 q^{29} - 5 q^{31} + 2 q^{35} + 4 q^{37} - 11 q^{41} - 4 q^{43} + 2 q^{47} + 3 q^{49} - 9 q^{53} + 11 q^{55} + 12 q^{59} - 2 q^{61} - 26 q^{65} - 9 q^{67} - 21 q^{73} - 3 q^{77} + 6 q^{79} + 7 q^{83} - 7 q^{85} + 18 q^{89} - 8 q^{91} + 2 q^{95} + 28 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) −1.42586 −0.637663 −0.318832 0.947811i \(-0.603290\pi\)
−0.318832 + 0.947811i \(0.603290\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.27053 0.986103 0.493052 0.870000i \(-0.335881\pi\)
0.493052 + 0.870000i \(0.335881\pi\)
\(12\) 0 0
\(13\) 3.42586 0.950162 0.475081 0.879942i \(-0.342419\pi\)
0.475081 + 0.879942i \(0.342419\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.27053 −1.27829 −0.639146 0.769085i \(-0.720712\pi\)
−0.639146 + 0.769085i \(0.720712\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.574141 −0.119717 −0.0598584 0.998207i \(-0.519065\pi\)
−0.0598584 + 0.998207i \(0.519065\pi\)
\(24\) 0 0
\(25\) −2.96693 −0.593385
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.122252 0.0227015 0.0113508 0.999936i \(-0.496387\pi\)
0.0113508 + 0.999936i \(0.496387\pi\)
\(30\) 0 0
\(31\) 2.12225 0.381168 0.190584 0.981671i \(-0.438962\pi\)
0.190584 + 0.981671i \(0.438962\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.42586 0.241014
\(36\) 0 0
\(37\) −3.96693 −0.652159 −0.326079 0.945342i \(-0.605728\pi\)
−0.326079 + 0.945342i \(0.605728\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.66332 0.728288 0.364144 0.931343i \(-0.381362\pi\)
0.364144 + 0.931343i \(0.381362\pi\)
\(42\) 0 0
\(43\) −11.9339 −1.81990 −0.909948 0.414723i \(-0.863879\pi\)
−0.909948 + 0.414723i \(0.863879\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.11521 −0.454400 −0.227200 0.973848i \(-0.572957\pi\)
−0.227200 + 0.973848i \(0.572957\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.08918 0.836413 0.418206 0.908352i \(-0.362659\pi\)
0.418206 + 0.908352i \(0.362659\pi\)
\(54\) 0 0
\(55\) −4.66332 −0.628802
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.72242 0.224240 0.112120 0.993695i \(-0.464236\pi\)
0.112120 + 0.993695i \(0.464236\pi\)
\(60\) 0 0
\(61\) 12.2114 1.56351 0.781757 0.623584i \(-0.214324\pi\)
0.781757 + 0.623584i \(0.214324\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.88479 −0.605884
\(66\) 0 0
\(67\) −5.27053 −0.643898 −0.321949 0.946757i \(-0.604338\pi\)
−0.321949 + 0.946757i \(0.604338\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.81864 0.809224 0.404612 0.914488i \(-0.367407\pi\)
0.404612 + 0.914488i \(0.367407\pi\)
\(72\) 0 0
\(73\) −11.5481 −1.35160 −0.675802 0.737083i \(-0.736203\pi\)
−0.675802 + 0.737083i \(0.736203\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.27053 −0.372712
\(78\) 0 0
\(79\) 11.0821 1.24684 0.623419 0.781888i \(-0.285743\pi\)
0.623419 + 0.781888i \(0.285743\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.23746 −0.574886 −0.287443 0.957798i \(-0.592805\pi\)
−0.287443 + 0.957798i \(0.592805\pi\)
\(84\) 0 0
\(85\) 7.51504 0.815120
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.45893 0.154646 0.0773232 0.997006i \(-0.475363\pi\)
0.0773232 + 0.997006i \(0.475363\pi\)
\(90\) 0 0
\(91\) −3.42586 −0.359128
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.42586 0.146290
\(96\) 0 0
\(97\) 17.6563 1.79272 0.896362 0.443324i \(-0.146201\pi\)
0.896362 + 0.443324i \(0.146201\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −17.0491 −1.69645 −0.848223 0.529640i \(-0.822327\pi\)
−0.848223 + 0.529640i \(0.822327\pi\)
\(102\) 0 0
\(103\) −11.3597 −1.11931 −0.559653 0.828727i \(-0.689066\pi\)
−0.559653 + 0.828727i \(0.689066\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.8186 −1.04588 −0.522939 0.852370i \(-0.675164\pi\)
−0.522939 + 0.852370i \(0.675164\pi\)
\(108\) 0 0
\(109\) −12.2776 −1.17598 −0.587989 0.808869i \(-0.700080\pi\)
−0.587989 + 0.808869i \(0.700080\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.696393 0.0655111 0.0327556 0.999463i \(-0.489572\pi\)
0.0327556 + 0.999463i \(0.489572\pi\)
\(114\) 0 0
\(115\) 0.818644 0.0763390
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.27053 0.483149
\(120\) 0 0
\(121\) −0.303607 −0.0276007
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3597 1.01604
\(126\) 0 0
\(127\) −5.68935 −0.504848 −0.252424 0.967617i \(-0.581228\pi\)
−0.252424 + 0.967617i \(0.581228\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.2044 1.32841 0.664207 0.747549i \(-0.268769\pi\)
0.664207 + 0.747549i \(0.268769\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.70344 −0.316406 −0.158203 0.987407i \(-0.550570\pi\)
−0.158203 + 0.987407i \(0.550570\pi\)
\(138\) 0 0
\(139\) −11.6704 −0.989867 −0.494934 0.868931i \(-0.664808\pi\)
−0.494934 + 0.868931i \(0.664808\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.2044 0.936958
\(144\) 0 0
\(145\) −0.174313 −0.0144759
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.8186 −1.54168 −0.770842 0.637027i \(-0.780164\pi\)
−0.770842 + 0.637027i \(0.780164\pi\)
\(150\) 0 0
\(151\) −4.43290 −0.360744 −0.180372 0.983598i \(-0.557730\pi\)
−0.180372 + 0.983598i \(0.557730\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.02603 −0.243057
\(156\) 0 0
\(157\) −9.82569 −0.784175 −0.392088 0.919928i \(-0.628247\pi\)
−0.392088 + 0.919928i \(0.628247\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.574141 0.0452487
\(162\) 0 0
\(163\) −10.9409 −0.856957 −0.428479 0.903552i \(-0.640950\pi\)
−0.428479 + 0.903552i \(0.640950\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.7525 0.986818 0.493409 0.869797i \(-0.335751\pi\)
0.493409 + 0.869797i \(0.335751\pi\)
\(168\) 0 0
\(169\) −1.26349 −0.0971917
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.5411 −1.71377 −0.856883 0.515511i \(-0.827602\pi\)
−0.856883 + 0.515511i \(0.827602\pi\)
\(174\) 0 0
\(175\) 2.96693 0.224279
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.41882 −0.479765 −0.239882 0.970802i \(-0.577109\pi\)
−0.239882 + 0.970802i \(0.577109\pi\)
\(180\) 0 0
\(181\) 9.81160 0.729291 0.364645 0.931146i \(-0.381190\pi\)
0.364645 + 0.931146i \(0.381190\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.65628 0.415858
\(186\) 0 0
\(187\) −17.2375 −1.26053
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.2515 −1.10356 −0.551782 0.833989i \(-0.686052\pi\)
−0.551782 + 0.833989i \(0.686052\pi\)
\(192\) 0 0
\(193\) −15.7455 −1.13338 −0.566691 0.823930i \(-0.691777\pi\)
−0.566691 + 0.823930i \(0.691777\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.8086 −1.76754 −0.883770 0.467922i \(-0.845003\pi\)
−0.883770 + 0.467922i \(0.845003\pi\)
\(198\) 0 0
\(199\) −2.58823 −0.183474 −0.0917372 0.995783i \(-0.529242\pi\)
−0.0917372 + 0.995783i \(0.529242\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.122252 −0.00858038
\(204\) 0 0
\(205\) −6.64924 −0.464403
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.27053 −0.226228
\(210\) 0 0
\(211\) 21.7785 1.49930 0.749648 0.661837i \(-0.230223\pi\)
0.749648 + 0.661837i \(0.230223\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 17.0160 1.16048
\(216\) 0 0
\(217\) −2.12225 −0.144068
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −18.0561 −1.21459
\(222\) 0 0
\(223\) −9.90078 −0.663005 −0.331503 0.943454i \(-0.607556\pi\)
−0.331503 + 0.943454i \(0.607556\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.71538 −0.180226 −0.0901131 0.995932i \(-0.528723\pi\)
−0.0901131 + 0.995932i \(0.528723\pi\)
\(228\) 0 0
\(229\) −8.29656 −0.548252 −0.274126 0.961694i \(-0.588389\pi\)
−0.274126 + 0.961694i \(0.588389\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.7295 −0.964959 −0.482480 0.875907i \(-0.660264\pi\)
−0.482480 + 0.875907i \(0.660264\pi\)
\(234\) 0 0
\(235\) 4.44185 0.289754
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −28.9970 −1.87566 −0.937830 0.347095i \(-0.887168\pi\)
−0.937830 + 0.347095i \(0.887168\pi\)
\(240\) 0 0
\(241\) −4.31065 −0.277673 −0.138837 0.990315i \(-0.544336\pi\)
−0.138837 + 0.990315i \(0.544336\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.42586 −0.0910948
\(246\) 0 0
\(247\) −3.42586 −0.217982
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.4329 0.658519 0.329259 0.944239i \(-0.393201\pi\)
0.329259 + 0.944239i \(0.393201\pi\)
\(252\) 0 0
\(253\) −1.87775 −0.118053
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −23.7124 −1.47914 −0.739569 0.673081i \(-0.764971\pi\)
−0.739569 + 0.673081i \(0.764971\pi\)
\(258\) 0 0
\(259\) 3.96693 0.246493
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −30.9409 −1.90790 −0.953949 0.299970i \(-0.903023\pi\)
−0.953949 + 0.299970i \(0.903023\pi\)
\(264\) 0 0
\(265\) −8.68231 −0.533350
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.17131 0.559185 0.279592 0.960119i \(-0.409801\pi\)
0.279592 + 0.960119i \(0.409801\pi\)
\(270\) 0 0
\(271\) 20.0420 1.21747 0.608733 0.793375i \(-0.291678\pi\)
0.608733 + 0.793375i \(0.291678\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.70344 −0.585139
\(276\) 0 0
\(277\) −7.45893 −0.448164 −0.224082 0.974570i \(-0.571938\pi\)
−0.224082 + 0.974570i \(0.571938\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.2114 −0.728473 −0.364236 0.931307i \(-0.618670\pi\)
−0.364236 + 0.931307i \(0.618670\pi\)
\(282\) 0 0
\(283\) −8.02303 −0.476920 −0.238460 0.971152i \(-0.576643\pi\)
−0.238460 + 0.971152i \(0.576643\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.66332 −0.275267
\(288\) 0 0
\(289\) 10.7785 0.634031
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −14.0331 −0.819821 −0.409910 0.912126i \(-0.634440\pi\)
−0.409910 + 0.912126i \(0.634440\pi\)
\(294\) 0 0
\(295\) −2.45593 −0.142990
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.96693 −0.113750
\(300\) 0 0
\(301\) 11.9339 0.687856
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −17.4118 −0.996995
\(306\) 0 0
\(307\) 16.0702 0.917174 0.458587 0.888649i \(-0.348356\pi\)
0.458587 + 0.888649i \(0.348356\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.56710 0.0888620 0.0444310 0.999012i \(-0.485853\pi\)
0.0444310 + 0.999012i \(0.485853\pi\)
\(312\) 0 0
\(313\) 29.6232 1.67440 0.837201 0.546895i \(-0.184190\pi\)
0.837201 + 0.546895i \(0.184190\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.4890 1.26311 0.631554 0.775332i \(-0.282417\pi\)
0.631554 + 0.775332i \(0.282417\pi\)
\(318\) 0 0
\(319\) 0.399828 0.0223861
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.27053 0.293260
\(324\) 0 0
\(325\) −10.1643 −0.563812
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.11521 0.171747
\(330\) 0 0
\(331\) 28.3857 1.56022 0.780111 0.625641i \(-0.215163\pi\)
0.780111 + 0.625641i \(0.215163\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.51504 0.410590
\(336\) 0 0
\(337\) 29.6272 1.61390 0.806950 0.590620i \(-0.201117\pi\)
0.806950 + 0.590620i \(0.201117\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.94090 0.375871
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.8607 1.76405 0.882026 0.471200i \(-0.156179\pi\)
0.882026 + 0.471200i \(0.156179\pi\)
\(348\) 0 0
\(349\) −32.5641 −1.74312 −0.871558 0.490292i \(-0.836890\pi\)
−0.871558 + 0.490292i \(0.836890\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.6633 1.63204 0.816022 0.578021i \(-0.196175\pi\)
0.816022 + 0.578021i \(0.196175\pi\)
\(354\) 0 0
\(355\) −9.72242 −0.516013
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.67740 0.352420 0.176210 0.984353i \(-0.443616\pi\)
0.176210 + 0.984353i \(0.443616\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.4660 0.861868
\(366\) 0 0
\(367\) 5.19544 0.271200 0.135600 0.990764i \(-0.456704\pi\)
0.135600 + 0.990764i \(0.456704\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.08918 −0.316134
\(372\) 0 0
\(373\) −1.31769 −0.0682275 −0.0341137 0.999418i \(-0.510861\pi\)
−0.0341137 + 0.999418i \(0.510861\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.418817 0.0215701
\(378\) 0 0
\(379\) −15.6563 −0.804209 −0.402104 0.915594i \(-0.631721\pi\)
−0.402104 + 0.915594i \(0.631721\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −29.0631 −1.48506 −0.742529 0.669814i \(-0.766374\pi\)
−0.742529 + 0.669814i \(0.766374\pi\)
\(384\) 0 0
\(385\) 4.66332 0.237665
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.7295 −0.544006 −0.272003 0.962296i \(-0.587686\pi\)
−0.272003 + 0.962296i \(0.587686\pi\)
\(390\) 0 0
\(391\) 3.02603 0.153033
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −15.8016 −0.795063
\(396\) 0 0
\(397\) 10.7997 0.542019 0.271010 0.962577i \(-0.412642\pi\)
0.271010 + 0.962577i \(0.412642\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.66332 0.332750 0.166375 0.986063i \(-0.446794\pi\)
0.166375 + 0.986063i \(0.446794\pi\)
\(402\) 0 0
\(403\) 7.27053 0.362171
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.9740 −0.643096
\(408\) 0 0
\(409\) −25.5481 −1.26327 −0.631636 0.775265i \(-0.717616\pi\)
−0.631636 + 0.775265i \(0.717616\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.72242 −0.0847549
\(414\) 0 0
\(415\) 7.46788 0.366584
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −33.2415 −1.62395 −0.811977 0.583690i \(-0.801608\pi\)
−0.811977 + 0.583690i \(0.801608\pi\)
\(420\) 0 0
\(421\) 2.95094 0.143820 0.0719099 0.997411i \(-0.477091\pi\)
0.0719099 + 0.997411i \(0.477091\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 15.6373 0.758520
\(426\) 0 0
\(427\) −12.2114 −0.590953
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.29656 −0.206958 −0.103479 0.994632i \(-0.532997\pi\)
−0.103479 + 0.994632i \(0.532997\pi\)
\(432\) 0 0
\(433\) 5.96693 0.286752 0.143376 0.989668i \(-0.454204\pi\)
0.143376 + 0.989668i \(0.454204\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.574141 0.0274649
\(438\) 0 0
\(439\) 37.5711 1.79317 0.896586 0.442869i \(-0.146039\pi\)
0.896586 + 0.442869i \(0.146039\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.2515 0.724623 0.362311 0.932057i \(-0.381988\pi\)
0.362311 + 0.932057i \(0.381988\pi\)
\(444\) 0 0
\(445\) −2.08023 −0.0986124
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 35.7926 1.68916 0.844579 0.535431i \(-0.179851\pi\)
0.844579 + 0.535431i \(0.179851\pi\)
\(450\) 0 0
\(451\) 15.2515 0.718167
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.88479 0.229003
\(456\) 0 0
\(457\) 18.6443 0.872145 0.436073 0.899912i \(-0.356369\pi\)
0.436073 + 0.899912i \(0.356369\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.8306 0.551006 0.275503 0.961300i \(-0.411156\pi\)
0.275503 + 0.961300i \(0.411156\pi\)
\(462\) 0 0
\(463\) 0.752498 0.0349715 0.0174858 0.999847i \(-0.494434\pi\)
0.0174858 + 0.999847i \(0.494434\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.8588 0.641307 0.320653 0.947197i \(-0.396098\pi\)
0.320653 + 0.947197i \(0.396098\pi\)
\(468\) 0 0
\(469\) 5.27053 0.243371
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −39.0301 −1.79460
\(474\) 0 0
\(475\) 2.96693 0.136132
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.58118 0.346393 0.173197 0.984887i \(-0.444590\pi\)
0.173197 + 0.984887i \(0.444590\pi\)
\(480\) 0 0
\(481\) −13.5901 −0.619657
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −25.1754 −1.14315
\(486\) 0 0
\(487\) −10.6864 −0.484245 −0.242122 0.970246i \(-0.577844\pi\)
−0.242122 + 0.970246i \(0.577844\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.1643 0.639225 0.319612 0.947548i \(-0.396447\pi\)
0.319612 + 0.947548i \(0.396447\pi\)
\(492\) 0 0
\(493\) −0.644331 −0.0290192
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.81864 −0.305858
\(498\) 0 0
\(499\) −23.9108 −1.07040 −0.535198 0.844727i \(-0.679763\pi\)
−0.535198 + 0.844727i \(0.679763\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 37.4388 1.66932 0.834658 0.550769i \(-0.185665\pi\)
0.834658 + 0.550769i \(0.185665\pi\)
\(504\) 0 0
\(505\) 24.3096 1.08176
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −31.3126 −1.38790 −0.693952 0.720021i \(-0.744132\pi\)
−0.693952 + 0.720021i \(0.744132\pi\)
\(510\) 0 0
\(511\) 11.5481 0.510858
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.1973 0.713740
\(516\) 0 0
\(517\) −10.1884 −0.448085
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −21.8346 −0.956593 −0.478296 0.878199i \(-0.658746\pi\)
−0.478296 + 0.878199i \(0.658746\pi\)
\(522\) 0 0
\(523\) 9.37380 0.409888 0.204944 0.978774i \(-0.434299\pi\)
0.204944 + 0.978774i \(0.434299\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.1854 −0.487244
\(528\) 0 0
\(529\) −22.6704 −0.985668
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.9759 0.691992
\(534\) 0 0
\(535\) 15.4259 0.666918
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.27053 0.140872
\(540\) 0 0
\(541\) −30.4608 −1.30961 −0.654807 0.755796i \(-0.727250\pi\)
−0.654807 + 0.755796i \(0.727250\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 17.5061 0.749878
\(546\) 0 0
\(547\) 34.2675 1.46517 0.732587 0.680673i \(-0.238313\pi\)
0.732587 + 0.680673i \(0.238313\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.122252 −0.00520809
\(552\) 0 0
\(553\) −11.0821 −0.471260
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.44889 0.0613915 0.0306957 0.999529i \(-0.490228\pi\)
0.0306957 + 0.999529i \(0.490228\pi\)
\(558\) 0 0
\(559\) −40.8837 −1.72920
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.8707 0.458146 0.229073 0.973409i \(-0.426431\pi\)
0.229073 + 0.973409i \(0.426431\pi\)
\(564\) 0 0
\(565\) −0.992958 −0.0417740
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −25.0772 −1.05129 −0.525646 0.850703i \(-0.676176\pi\)
−0.525646 + 0.850703i \(0.676176\pi\)
\(570\) 0 0
\(571\) −11.6042 −0.485621 −0.242811 0.970074i \(-0.578069\pi\)
−0.242811 + 0.970074i \(0.578069\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.70344 0.0710382
\(576\) 0 0
\(577\) 26.5782 1.10646 0.553232 0.833027i \(-0.313394\pi\)
0.553232 + 0.833027i \(0.313394\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.23746 0.217286
\(582\) 0 0
\(583\) 19.9149 0.824789
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.5552 0.683304 0.341652 0.939826i \(-0.389014\pi\)
0.341652 + 0.939826i \(0.389014\pi\)
\(588\) 0 0
\(589\) −2.12225 −0.0874459
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26.6864 −1.09588 −0.547939 0.836519i \(-0.684587\pi\)
−0.547939 + 0.836519i \(0.684587\pi\)
\(594\) 0 0
\(595\) −7.51504 −0.308086
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 19.6603 0.803299 0.401649 0.915793i \(-0.368437\pi\)
0.401649 + 0.915793i \(0.368437\pi\)
\(600\) 0 0
\(601\) −23.1854 −0.945752 −0.472876 0.881129i \(-0.656784\pi\)
−0.472876 + 0.881129i \(0.656784\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.432901 0.0175999
\(606\) 0 0
\(607\) 0.441848 0.0179341 0.00896704 0.999960i \(-0.497146\pi\)
0.00896704 + 0.999960i \(0.497146\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.6723 −0.431754
\(612\) 0 0
\(613\) −28.3478 −1.14496 −0.572478 0.819920i \(-0.694018\pi\)
−0.572478 + 0.819920i \(0.694018\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.7455 −1.27802 −0.639012 0.769197i \(-0.720657\pi\)
−0.639012 + 0.769197i \(0.720657\pi\)
\(618\) 0 0
\(619\) 22.2535 0.894442 0.447221 0.894424i \(-0.352414\pi\)
0.447221 + 0.894424i \(0.352414\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.45893 −0.0584509
\(624\) 0 0
\(625\) −1.36271 −0.0545085
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.9078 0.833649
\(630\) 0 0
\(631\) −19.4719 −0.775165 −0.387583 0.921835i \(-0.626690\pi\)
−0.387583 + 0.921835i \(0.626690\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.11221 0.321923
\(636\) 0 0
\(637\) 3.42586 0.135737
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −40.2345 −1.58917 −0.794583 0.607156i \(-0.792310\pi\)
−0.794583 + 0.607156i \(0.792310\pi\)
\(642\) 0 0
\(643\) −17.1011 −0.674403 −0.337201 0.941433i \(-0.609480\pi\)
−0.337201 + 0.941433i \(0.609480\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.37380 0.211266 0.105633 0.994405i \(-0.466313\pi\)
0.105633 + 0.994405i \(0.466313\pi\)
\(648\) 0 0
\(649\) 5.63325 0.221124
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.6183 0.454659 0.227330 0.973818i \(-0.427001\pi\)
0.227330 + 0.973818i \(0.427001\pi\)
\(654\) 0 0
\(655\) −21.6793 −0.847081
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 41.8718 1.63109 0.815546 0.578692i \(-0.196437\pi\)
0.815546 + 0.578692i \(0.196437\pi\)
\(660\) 0 0
\(661\) 6.71942 0.261355 0.130678 0.991425i \(-0.458285\pi\)
0.130678 + 0.991425i \(0.458285\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.42586 −0.0552924
\(666\) 0 0
\(667\) −0.0701896 −0.00271775
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 39.9379 1.54179
\(672\) 0 0
\(673\) 43.5280 1.67788 0.838941 0.544222i \(-0.183175\pi\)
0.838941 + 0.544222i \(0.183175\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.1033 −0.695765 −0.347882 0.937538i \(-0.613099\pi\)
−0.347882 + 0.937538i \(0.613099\pi\)
\(678\) 0 0
\(679\) −17.6563 −0.677586
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −27.3456 −1.04635 −0.523176 0.852225i \(-0.675253\pi\)
−0.523176 + 0.852225i \(0.675253\pi\)
\(684\) 0 0
\(685\) 5.28058 0.201760
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 20.8607 0.794728
\(690\) 0 0
\(691\) −30.8316 −1.17289 −0.586445 0.809989i \(-0.699473\pi\)
−0.586445 + 0.809989i \(0.699473\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.6403 0.631202
\(696\) 0 0
\(697\) −24.5782 −0.930965
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.23042 0.310859 0.155429 0.987847i \(-0.450324\pi\)
0.155429 + 0.987847i \(0.450324\pi\)
\(702\) 0 0
\(703\) 3.96693 0.149615
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.0491 0.641196
\(708\) 0 0
\(709\) −15.6183 −0.586558 −0.293279 0.956027i \(-0.594746\pi\)
−0.293279 + 0.956027i \(0.594746\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.21847 −0.0456321
\(714\) 0 0
\(715\) −15.9759 −0.597464
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.3567 −0.610002 −0.305001 0.952352i \(-0.598657\pi\)
−0.305001 + 0.952352i \(0.598657\pi\)
\(720\) 0 0
\(721\) 11.3597 0.423058
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.362711 −0.0134708
\(726\) 0 0
\(727\) −17.6183 −0.653427 −0.326713 0.945123i \(-0.605941\pi\)
−0.326713 + 0.945123i \(0.605941\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 62.8978 2.32636
\(732\) 0 0
\(733\) −42.8837 −1.58395 −0.791973 0.610556i \(-0.790946\pi\)
−0.791973 + 0.610556i \(0.790946\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −17.2375 −0.634950
\(738\) 0 0
\(739\) 19.4259 0.714592 0.357296 0.933991i \(-0.383699\pi\)
0.357296 + 0.933991i \(0.383699\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −51.7305 −1.89781 −0.948904 0.315564i \(-0.897806\pi\)
−0.948904 + 0.315564i \(0.897806\pi\)
\(744\) 0 0
\(745\) 26.8327 0.983075
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.8186 0.395305
\(750\) 0 0
\(751\) 23.9428 0.873685 0.436843 0.899538i \(-0.356097\pi\)
0.436843 + 0.899538i \(0.356097\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.32069 0.230033
\(756\) 0 0
\(757\) −2.48901 −0.0904645 −0.0452322 0.998976i \(-0.514403\pi\)
−0.0452322 + 0.998976i \(0.514403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.14528 0.150266 0.0751332 0.997174i \(-0.476062\pi\)
0.0751332 + 0.997174i \(0.476062\pi\)
\(762\) 0 0
\(763\) 12.2776 0.444478
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.90078 0.213065
\(768\) 0 0
\(769\) −11.0442 −0.398263 −0.199131 0.979973i \(-0.563812\pi\)
−0.199131 + 0.979973i \(0.563812\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.3426 1.52296 0.761479 0.648189i \(-0.224473\pi\)
0.761479 + 0.648189i \(0.224473\pi\)
\(774\) 0 0
\(775\) −6.29656 −0.226179
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.66332 −0.167081
\(780\) 0 0
\(781\) 22.3006 0.797979
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0100 0.500040
\(786\) 0 0
\(787\) 49.5711 1.76702 0.883510 0.468412i \(-0.155174\pi\)
0.883510 + 0.468412i \(0.155174\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.696393 −0.0247609
\(792\) 0 0
\(793\) 41.8346 1.48559
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −38.8276 −1.37534 −0.687672 0.726022i \(-0.741367\pi\)
−0.687672 + 0.726022i \(0.741367\pi\)
\(798\) 0 0
\(799\) 16.4188 0.580856
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −37.7685 −1.33282
\(804\) 0 0
\(805\) −0.818644 −0.0288534
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −13.4920 −0.474354 −0.237177 0.971466i \(-0.576222\pi\)
−0.237177 + 0.971466i \(0.576222\pi\)
\(810\) 0 0
\(811\) 24.5552 0.862248 0.431124 0.902293i \(-0.358117\pi\)
0.431124 + 0.902293i \(0.358117\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.6002 0.546450
\(816\) 0 0
\(817\) 11.9339 0.417513
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 37.9339 1.32390 0.661950 0.749548i \(-0.269729\pi\)
0.661950 + 0.749548i \(0.269729\pi\)
\(822\) 0 0
\(823\) −10.0141 −0.349069 −0.174535 0.984651i \(-0.555842\pi\)
−0.174535 + 0.984651i \(0.555842\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.7685 1.24379 0.621896 0.783100i \(-0.286363\pi\)
0.621896 + 0.783100i \(0.286363\pi\)
\(828\) 0 0
\(829\) 25.6753 0.891739 0.445869 0.895098i \(-0.352895\pi\)
0.445869 + 0.895098i \(0.352895\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.27053 −0.182613
\(834\) 0 0
\(835\) −18.1833 −0.629258
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 42.7856 1.47712 0.738561 0.674187i \(-0.235506\pi\)
0.738561 + 0.674187i \(0.235506\pi\)
\(840\) 0 0
\(841\) −28.9851 −0.999485
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.80156 0.0619756
\(846\) 0 0
\(847\) 0.303607 0.0104321
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.27758 0.0780743
\(852\) 0 0
\(853\) 17.8447 0.610990 0.305495 0.952194i \(-0.401178\pi\)
0.305495 + 0.952194i \(0.401178\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −37.8116 −1.29162 −0.645810 0.763498i \(-0.723480\pi\)
−0.645810 + 0.763498i \(0.723480\pi\)
\(858\) 0 0
\(859\) 22.8558 0.779828 0.389914 0.920851i \(-0.372505\pi\)
0.389914 + 0.920851i \(0.372505\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13.3177 −0.453339 −0.226670 0.973972i \(-0.572784\pi\)
−0.226670 + 0.973972i \(0.572784\pi\)
\(864\) 0 0
\(865\) 32.1404 1.09281
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 36.2445 1.22951
\(870\) 0 0
\(871\) −18.0561 −0.611808
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.3597 −0.384028
\(876\) 0 0
\(877\) −33.5571 −1.13314 −0.566571 0.824013i \(-0.691730\pi\)
−0.566571 + 0.824013i \(0.691730\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 52.5641 1.77093 0.885465 0.464707i \(-0.153840\pi\)
0.885465 + 0.464707i \(0.153840\pi\)
\(882\) 0 0
\(883\) −5.68445 −0.191297 −0.0956484 0.995415i \(-0.530492\pi\)
−0.0956484 + 0.995415i \(0.530492\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26.3898 −0.886082 −0.443041 0.896501i \(-0.646100\pi\)
−0.443041 + 0.896501i \(0.646100\pi\)
\(888\) 0 0
\(889\) 5.68935 0.190815
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.11521 0.104247
\(894\) 0 0
\(895\) 9.15233 0.305929
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.259449 0.00865309
\(900\) 0 0
\(901\) −32.0932 −1.06918
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13.9900 −0.465042
\(906\) 0 0
\(907\) 0.211430 0.00702041 0.00351021 0.999994i \(-0.498883\pi\)
0.00351021 + 0.999994i \(0.498883\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21.3787 0.708308 0.354154 0.935187i \(-0.384769\pi\)
0.354154 + 0.935187i \(0.384769\pi\)
\(912\) 0 0
\(913\) −17.1293 −0.566897
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.2044 −0.502093
\(918\) 0 0
\(919\) −16.8186 −0.554796 −0.277398 0.960755i \(-0.589472\pi\)
−0.277398 + 0.960755i \(0.589472\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 23.3597 0.768894
\(924\) 0 0
\(925\) 11.7696 0.386981
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.3497 1.15979 0.579893 0.814693i \(-0.303095\pi\)
0.579893 + 0.814693i \(0.303095\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.5782 0.803793
\(936\) 0 0
\(937\) 4.78153 0.156206 0.0781029 0.996945i \(-0.475114\pi\)
0.0781029 + 0.996945i \(0.475114\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 13.6373 0.444563 0.222281 0.974983i \(-0.428650\pi\)
0.222281 + 0.974983i \(0.428650\pi\)
\(942\) 0 0
\(943\) −2.67740 −0.0871883
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 59.3638 1.92906 0.964531 0.263968i \(-0.0850313\pi\)
0.964531 + 0.263968i \(0.0850313\pi\)
\(948\) 0 0
\(949\) −39.5622 −1.28424
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 52.4790 1.69996 0.849980 0.526815i \(-0.176614\pi\)
0.849980 + 0.526815i \(0.176614\pi\)
\(954\) 0 0
\(955\) 21.7465 0.703702
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.70344 0.119590
\(960\) 0 0
\(961\) −26.4960 −0.854711
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 22.4508 0.722717
\(966\) 0 0
\(967\) 21.9900 0.707149 0.353575 0.935406i \(-0.384966\pi\)
0.353575 + 0.935406i \(0.384966\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.9309 1.31353 0.656767 0.754094i \(-0.271924\pi\)
0.656767 + 0.754094i \(0.271924\pi\)
\(972\) 0 0
\(973\) 11.6704 0.374135
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.56924 −0.274154 −0.137077 0.990560i \(-0.543771\pi\)
−0.137077 + 0.990560i \(0.543771\pi\)
\(978\) 0 0
\(979\) 4.77149 0.152497
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −35.2654 −1.12479 −0.562396 0.826868i \(-0.690120\pi\)
−0.562396 + 0.826868i \(0.690120\pi\)
\(984\) 0 0
\(985\) 35.3736 1.12710
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.85172 0.217872
\(990\) 0 0
\(991\) 1.78367 0.0566600 0.0283300 0.999599i \(-0.490981\pi\)
0.0283300 + 0.999599i \(0.490981\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.69044 0.116995
\(996\) 0 0
\(997\) −37.4299 −1.18542 −0.592708 0.805417i \(-0.701941\pi\)
−0.592708 + 0.805417i \(0.701941\pi\)
\(998\) 0 0
\(999\) 0 0
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 4788.2.a.o.1.2 3
3.2 odd 2 532.2.a.e.1.3 3
12.11 even 2 2128.2.a.r.1.1 3
21.20 even 2 3724.2.a.i.1.1 3
24.5 odd 2 8512.2.a.bn.1.1 3
24.11 even 2 8512.2.a.bl.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.e.1.3 3 3.2 odd 2
2128.2.a.r.1.1 3 12.11 even 2
3724.2.a.i.1.1 3 21.20 even 2
4788.2.a.o.1.2 3 1.1 even 1 trivial
8512.2.a.bl.1.3 3 24.11 even 2
8512.2.a.bn.1.1 3 24.5 odd 2