Properties

Label 363.2.f.g.215.1
Level $363$
Weight $2$
Character 363.215
Analytic conductor $2.899$
Analytic rank $0$
Dimension $16$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(161,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.f (of order \(10\), degree \(4\), not minimal)

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: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 11^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label 215.1
Root \(-1.83730 + 0.596975i\) of defining polynomial
Character \(\chi\) \(=\) 363.215
Dual form 363.2.f.g.233.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.40126 - 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(-2.79802 - 3.85115i) q^{7} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\) \(q+(-1.40126 - 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(-2.79802 - 3.85115i) q^{7} +(0.927051 + 2.85317i) q^{9} -3.46410 q^{12} +(-3.31421 + 1.07685i) q^{13} +(1.23607 - 3.80423i) q^{16} +(-0.749728 + 1.03191i) q^{19} +8.24504i q^{21} +(-4.04508 - 2.93893i) q^{25} +(1.60570 - 4.94183i) q^{27} +(-9.05459 - 2.94201i) q^{28} +(-0.535233 - 1.64728i) q^{31} +(4.85410 + 3.52671i) q^{36} +(4.20378 - 3.05422i) q^{37} +(5.74038 + 1.86516i) q^{39} -13.0053i q^{43} +(-5.60503 + 4.07230i) q^{48} +(-4.83928 + 14.8938i) q^{49} +(-4.09659 + 5.63847i) q^{52} +(2.10112 - 0.682697i) q^{57} +(11.1557 + 3.62471i) q^{61} +(8.39406 - 11.5534i) q^{63} +(-2.47214 - 7.60845i) q^{64} +12.1244 q^{67} +(6.34577 + 8.73420i) q^{73} +(2.67617 + 8.23639i) q^{75} +2.55103i q^{76} +(16.8961 - 5.48987i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(9.69263 + 13.3408i) q^{84} +(13.4203 + 9.75045i) q^{91} +(-0.927051 + 2.85317i) q^{93} +(-1.54508 - 4.75528i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{10}\right)\)

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 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(3\) −1.40126 1.01807i −0.809017 0.587785i
\(4\) 1.61803 1.17557i 0.809017 0.587785i
\(5\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(6\) 0 0
\(7\) −2.79802 3.85115i −1.05755 1.45560i −0.882073 0.471114i \(-0.843852\pi\)
−0.175480 0.984483i \(-0.556148\pi\)
\(8\) 0 0
\(9\) 0.927051 + 2.85317i 0.309017 + 0.951057i
\(10\) 0 0
\(11\) 0 0
\(12\) −3.46410 −1.00000
\(13\) −3.31421 + 1.07685i −0.919196 + 0.298665i −0.730137 0.683301i \(-0.760544\pi\)
−0.189059 + 0.981966i \(0.560544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.23607 3.80423i 0.309017 0.951057i
\(17\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(18\) 0 0
\(19\) −0.749728 + 1.03191i −0.171999 + 0.236737i −0.886310 0.463091i \(-0.846740\pi\)
0.714311 + 0.699828i \(0.246740\pi\)
\(20\) 0 0
\(21\) 8.24504i 1.79922i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −4.04508 2.93893i −0.809017 0.587785i
\(26\) 0 0
\(27\) 1.60570 4.94183i 0.309017 0.951057i
\(28\) −9.05459 2.94201i −1.71116 0.555988i
\(29\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(30\) 0 0
\(31\) −0.535233 1.64728i −0.0961307 0.295860i 0.891416 0.453186i \(-0.149713\pi\)
−0.987547 + 0.157326i \(0.949713\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 4.85410 + 3.52671i 0.809017 + 0.587785i
\(37\) 4.20378 3.05422i 0.691096 0.502111i −0.185924 0.982564i \(-0.559528\pi\)
0.877020 + 0.480453i \(0.159528\pi\)
\(38\) 0 0
\(39\) 5.74038 + 1.86516i 0.919196 + 0.298665i
\(40\) 0 0
\(41\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(42\) 0 0
\(43\) 13.0053i 1.98329i −0.128984 0.991647i \(-0.541172\pi\)
0.128984 0.991647i \(-0.458828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(48\) −5.60503 + 4.07230i −0.809017 + 0.587785i
\(49\) −4.83928 + 14.8938i −0.691326 + 2.12768i
\(50\) 0 0
\(51\) 0 0
\(52\) −4.09659 + 5.63847i −0.568095 + 0.781915i
\(53\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.10112 0.682697i 0.278301 0.0904254i
\(58\) 0 0
\(59\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(60\) 0 0
\(61\) 11.1557 + 3.62471i 1.42834 + 0.464097i 0.918243 0.396018i \(-0.129608\pi\)
0.510100 + 0.860115i \(0.329608\pi\)
\(62\) 0 0
\(63\) 8.39406 11.5534i 1.05755 1.45560i
\(64\) −2.47214 7.60845i −0.309017 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) 12.1244 1.48123 0.740613 0.671932i \(-0.234535\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(72\) 0 0
\(73\) 6.34577 + 8.73420i 0.742716 + 1.02226i 0.998458 + 0.0555161i \(0.0176804\pi\)
−0.255741 + 0.966745i \(0.582320\pi\)
\(74\) 0 0
\(75\) 2.67617 + 8.23639i 0.309017 + 0.951057i
\(76\) 2.55103i 0.292623i
\(77\) 0 0
\(78\) 0 0
\(79\) 16.8961 5.48987i 1.90096 0.617659i 0.939716 0.341956i \(-0.111089\pi\)
0.961243 0.275703i \(-0.0889106\pi\)
\(80\) 0 0
\(81\) −7.28115 + 5.29007i −0.809017 + 0.587785i
\(82\) 0 0
\(83\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(84\) 9.69263 + 13.3408i 1.05755 + 1.45560i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 13.4203 + 9.75045i 1.40683 + 1.02212i
\(92\) 0 0
\(93\) −0.927051 + 2.85317i −0.0961307 + 0.295860i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.54508 4.75528i −0.156880 0.482826i 0.841467 0.540309i \(-0.181693\pi\)
−0.998346 + 0.0574829i \(0.981693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −10.0000 −1.00000
\(101\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(102\) 0 0
\(103\) −5.66312 + 4.11450i −0.558004 + 0.405413i −0.830728 0.556679i \(-0.812075\pi\)
0.272724 + 0.962092i \(0.412075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(108\) −3.21140 9.88367i −0.309017 0.951057i
\(109\) 7.31130i 0.700296i −0.936694 0.350148i \(-0.886131\pi\)
0.936694 0.350148i \(-0.113869\pi\)
\(110\) 0 0
\(111\) −9.00000 −0.854242
\(112\) −18.1092 + 5.88403i −1.71116 + 0.555988i
\(113\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.14488 8.45770i −0.568095 0.781915i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −2.80252 2.03615i −0.251673 0.182851i
\(125\) 0 0
\(126\) 0 0
\(127\) −14.4699 4.70156i −1.28400 0.417196i −0.414011 0.910272i \(-0.635873\pi\)
−0.869987 + 0.493075i \(0.835873\pi\)
\(128\) 0 0
\(129\) −13.2404 + 18.2238i −1.16575 + 1.60452i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 6.07180 0.526492
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(138\) 0 0
\(139\) 0.548839 + 0.755412i 0.0465519 + 0.0640732i 0.831658 0.555288i \(-0.187392\pi\)
−0.785106 + 0.619361i \(0.787392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 12.0000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 21.9441 15.9433i 1.80992 1.31498i
\(148\) 3.21140 9.88367i 0.263975 0.812433i
\(149\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(150\) 0 0
\(151\) 11.7409 16.1600i 0.955463 1.31508i 0.00640530 0.999979i \(-0.497961\pi\)
0.949058 0.315102i \(-0.102039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 11.4808 3.73032i 0.919196 0.298665i
\(157\) −18.2164 13.2350i −1.45382 1.05626i −0.984919 0.173016i \(-0.944649\pi\)
−0.468905 0.883249i \(-0.655351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.25329 + 16.1680i 0.411469 + 1.26637i 0.915371 + 0.402611i \(0.131897\pi\)
−0.503902 + 0.863761i \(0.668103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(168\) 0 0
\(169\) −0.692847 + 0.503383i −0.0532960 + 0.0387218i
\(170\) 0 0
\(171\) −3.63925 1.18247i −0.278301 0.0904254i
\(172\) −15.2887 21.0431i −1.16575 1.60452i
\(173\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(174\) 0 0
\(175\) 23.8014i 1.79922i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(180\) 0 0
\(181\) 5.87132 18.0701i 0.436412 1.34314i −0.455221 0.890379i \(-0.650440\pi\)
0.891633 0.452759i \(-0.149560\pi\)
\(182\) 0 0
\(183\) −11.9418 16.4365i −0.882764 1.21502i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −23.5245 + 7.64358i −1.71116 + 0.555988i
\(190\) 0 0
\(191\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(192\) −4.28187 + 13.1782i −0.309017 + 0.951057i
\(193\) 25.9507 + 8.43189i 1.86797 + 0.606941i 0.992272 + 0.124083i \(0.0395989\pi\)
0.875699 + 0.482857i \(0.160401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 9.67857 + 29.7876i 0.691326 + 2.12768i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −25.9808 −1.84173 −0.920864 0.389885i \(-0.872515\pi\)
−0.920864 + 0.389885i \(0.872515\pi\)
\(200\) 0 0
\(201\) −16.9894 12.3435i −1.19834 0.870643i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 13.9391i 0.966500i
\(209\) 0 0
\(210\) 0 0
\(211\) −8.72954 + 2.83640i −0.600967 + 0.195266i −0.593671 0.804708i \(-0.702322\pi\)
−0.00729508 + 0.999973i \(0.502322\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.84632 + 6.67038i −0.328989 + 0.452815i
\(218\) 0 0
\(219\) 18.6993i 1.26358i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 18.6074 + 13.5191i 1.24604 + 0.905303i 0.997986 0.0634420i \(-0.0202078\pi\)
0.248058 + 0.968745i \(0.420208\pi\)
\(224\) 0 0
\(225\) 4.63525 14.2658i 0.309017 0.951057i
\(226\) 0 0
\(227\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(228\) 2.59713 3.57465i 0.171999 0.236737i
\(229\) −6.42280 19.7673i −0.424430 1.30626i −0.903539 0.428507i \(-0.859040\pi\)
0.479108 0.877756i \(-0.340960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −29.2649 9.50874i −1.90096 0.617659i
\(238\) 0 0
\(239\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(240\) 0 0
\(241\) 29.4954i 1.89997i 0.312301 + 0.949983i \(0.398900\pi\)
−0.312301 + 0.949983i \(0.601100\pi\)
\(242\) 0 0
\(243\) 15.5885 1.00000
\(244\) 22.3114 7.24942i 1.42834 0.464097i
\(245\) 0 0
\(246\) 0 0
\(247\) 1.37354 4.22732i 0.0873962 0.268978i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(252\) 28.5617i 1.79922i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −12.9443 9.40456i −0.809017 0.587785i
\(257\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(258\) 0 0
\(259\) −23.5245 7.64358i −1.46174 0.474949i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 19.6176 14.2530i 1.19834 0.870643i
\(269\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(270\) 0 0
\(271\) 18.8364 + 25.9261i 1.14423 + 1.57490i 0.757673 + 0.652634i \(0.226336\pi\)
0.386558 + 0.922265i \(0.373664\pi\)
\(272\) 0 0
\(273\) −8.87869 27.3258i −0.537363 1.65383i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −12.6938 + 4.12448i −0.762699 + 0.247816i −0.664437 0.747345i \(-0.731328\pi\)
−0.0982624 + 0.995161i \(0.531328\pi\)
\(278\) 0 0
\(279\) 4.20378 3.05422i 0.251673 0.182851i
\(280\) 0 0
\(281\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(282\) 0 0
\(283\) −16.5872 + 22.8304i −0.986009 + 1.35713i −0.0524806 + 0.998622i \(0.516713\pi\)
−0.933528 + 0.358503i \(0.883287\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.7533 + 9.99235i 0.809017 + 0.587785i
\(290\) 0 0
\(291\) −2.67617 + 8.23639i −0.156880 + 0.482826i
\(292\) 20.5353 + 6.67234i 1.20174 + 0.390469i
\(293\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 14.0126 + 10.1807i 0.809017 + 0.587785i
\(301\) −50.0854 + 36.3892i −2.88687 + 2.09744i
\(302\) 0 0
\(303\) 0 0
\(304\) 2.99891 + 4.12765i 0.171999 + 0.236737i
\(305\) 0 0
\(306\) 0 0
\(307\) 34.2557i 1.95508i −0.210760 0.977538i \(-0.567594\pi\)
0.210760 0.977538i \(-0.432406\pi\)
\(308\) 0 0
\(309\) 12.1244 0.689730
\(310\) 0 0
\(311\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(312\) 0 0
\(313\) −8.56373 + 26.3565i −0.484051 + 1.48975i 0.349300 + 0.937011i \(0.386419\pi\)
−0.833351 + 0.552744i \(0.813581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 20.8847 28.7453i 1.17486 1.61705i
\(317\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −5.56231 + 17.1190i −0.309017 + 0.951057i
\(325\) 16.5710 + 5.38426i 0.919196 + 0.298665i
\(326\) 0 0
\(327\) −7.44345 + 10.2450i −0.411624 + 0.566551i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −31.0000 −1.70391 −0.851957 0.523612i \(-0.824584\pi\)
−0.851957 + 0.523612i \(0.824584\pi\)
\(332\) 0 0
\(333\) 12.6113 + 9.16267i 0.691096 + 0.502111i
\(334\) 0 0
\(335\) 0 0
\(336\) 31.3660 + 10.1914i 1.71116 + 0.555988i
\(337\) 13.0395 + 17.9473i 0.710306 + 0.977653i 0.999790 + 0.0204700i \(0.00651627\pi\)
−0.289484 + 0.957183i \(0.593484\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 39.2075 12.7393i 2.11701 0.687857i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(348\) 0 0
\(349\) 17.5379 24.1388i 0.938780 1.29212i −0.0175547 0.999846i \(-0.505588\pi\)
0.956335 0.292274i \(-0.0944119\pi\)
\(350\) 0 0
\(351\) 18.1074i 0.966500i
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(360\) 0 0
\(361\) 5.36857 + 16.5228i 0.282556 + 0.869619i
\(362\) 0 0
\(363\) 0 0
\(364\) 33.1769 1.73894
\(365\) 0 0
\(366\) 0 0
\(367\) 3.23607 2.35114i 0.168921 0.122729i −0.500113 0.865960i \(-0.666708\pi\)
0.669034 + 0.743232i \(0.266708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.85410 + 5.70634i 0.0961307 + 0.295860i
\(373\) 3.14299i 0.162738i −0.996684 0.0813690i \(-0.974071\pi\)
0.996684 0.0813690i \(-0.0259292\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.47214 7.60845i 0.126985 0.390820i −0.867272 0.497834i \(-0.834129\pi\)
0.994258 + 0.107014i \(0.0341289\pi\)
\(380\) 0 0
\(381\) 15.4896 + 21.3196i 0.793554 + 1.09223i
\(382\) 0 0
\(383\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 37.1064 12.0566i 1.88622 0.612871i
\(388\) −8.09017 5.87785i −0.410716 0.298403i
\(389\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 35.0000 1.75660 0.878300 0.478110i \(-0.158678\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) 0 0
\(399\) −8.50816 6.18154i −0.425941 0.309464i
\(400\) −16.1803 + 11.7557i −0.809017 + 0.587785i
\(401\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(402\) 0 0
\(403\) 3.54775 + 4.88306i 0.176726 + 0.243242i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −38.3195 + 12.4508i −1.89478 + 0.615650i −0.920273 + 0.391277i \(0.872033\pi\)
−0.974503 + 0.224373i \(0.927967\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.32624 + 13.3148i −0.213138 + 0.655973i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.61729i 0.0791988i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 28.0252 + 20.3615i 1.36586 + 0.992358i 0.998048 + 0.0624590i \(0.0198943\pi\)
0.367816 + 0.929899i \(0.380106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −17.2546 53.1043i −0.835010 2.56990i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(432\) −16.8151 12.2169i −0.809017 0.587785i
\(433\) 29.9336 21.7481i 1.43852 1.04514i 0.450169 0.892943i \(-0.351364\pi\)
0.988350 0.152201i \(-0.0486362\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.59495 11.8299i −0.411624 0.566551i
\(437\) 0 0
\(438\) 0 0
\(439\) 22.8677i 1.09141i −0.837976 0.545707i \(-0.816261\pi\)
0.837976 0.545707i \(-0.183739\pi\)
\(440\) 0 0
\(441\) −46.9808 −2.23718
\(442\) 0 0
\(443\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(444\) −14.5623 + 10.5801i −0.691096 + 0.502111i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −22.3842 + 30.8092i −1.05755 + 1.45560i
\(449\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −32.9041 + 10.6912i −1.54597 + 0.502317i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −34.6802 11.2683i −1.62227 0.527108i −0.649796 0.760109i \(-0.725146\pi\)
−0.972477 + 0.233001i \(0.925146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(468\) −19.8853 6.46111i −0.919196 0.298665i
\(469\) −33.9242 46.6927i −1.56647 2.15607i
\(470\) 0 0
\(471\) 12.0517 + 37.0912i 0.555311 + 1.70907i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.06542 1.97078i 0.278301 0.0904254i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(480\) 0 0
\(481\) −10.6432 + 14.6492i −0.485290 + 0.667945i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.80252 2.03615i −0.126994 0.0922667i 0.522475 0.852655i \(-0.325009\pi\)
−0.649469 + 0.760388i \(0.725009\pi\)
\(488\) 0 0
\(489\) 9.09896 28.0037i 0.411469 1.26637i
\(490\) 0 0
\(491\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −6.92820 −0.311086
\(497\) 0 0
\(498\) 0 0
\(499\) 35.0315 25.4518i 1.56822 1.13938i 0.639391 0.768882i \(-0.279187\pi\)
0.928833 0.370499i \(-0.120813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.48334 0.0658774
\(508\) −28.9398 + 9.40313i −1.28400 + 0.417196i
\(509\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(510\) 0 0
\(511\) 15.8811 48.8770i 0.702538 2.16219i
\(512\) 0 0
\(513\) 3.89570 + 5.36197i 0.171999 + 0.236737i
\(514\) 0 0
\(515\) 0 0
\(516\) 45.0518i 1.98329i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(522\) 0 0
\(523\) −18.4342 5.98964i −0.806072 0.261909i −0.123139 0.992389i \(-0.539296\pi\)
−0.682933 + 0.730481i \(0.739296\pi\)
\(524\) 0 0
\(525\) 24.2316 33.3519i 1.05755 1.45560i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 9.82437 7.13783i 0.425941 0.309464i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 26.2757 8.53750i 1.12968 0.367056i 0.316227 0.948684i \(-0.397584\pi\)
0.813455 + 0.581628i \(0.197584\pi\)
\(542\) 0 0
\(543\) −26.6239 + 19.3434i −1.14254 + 0.830105i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −6.54666 + 9.01070i −0.279915 + 0.385270i −0.925706 0.378245i \(-0.876528\pi\)
0.645791 + 0.763514i \(0.276528\pi\)
\(548\) 0 0
\(549\) 35.1894i 1.50185i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −68.4179 49.7085i −2.90943 2.11382i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.77608 + 0.577083i 0.0753225 + 0.0244738i
\(557\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(558\) 0 0
\(559\) 14.0048 + 43.1024i 0.592340 + 1.82304i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 40.7456 + 13.2391i 1.71116 + 0.555988i
\(568\) 0 0
\(569\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(570\) 0 0
\(571\) 39.3577i 1.64707i 0.567265 + 0.823535i \(0.308001\pi\)
−0.567265 + 0.823535i \(0.691999\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 19.4164 14.1068i 0.809017 0.587785i
\(577\) −14.4513 + 44.4765i −0.601615 + 1.85158i −0.0830461 + 0.996546i \(0.526465\pi\)
−0.518569 + 0.855036i \(0.673535\pi\)
\(578\) 0 0
\(579\) −27.7793 38.2350i −1.15447 1.58899i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(588\) 16.7638 51.5936i 0.691326 2.12768i
\(589\) 2.10112 + 0.682697i 0.0865753 + 0.0281300i
\(590\) 0 0
\(591\) 0 0
\(592\) −6.42280 19.7673i −0.263975 0.812433i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 36.4058 + 26.4503i 1.48999 + 1.08254i
\(598\) 0 0
\(599\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(600\) 0 0
\(601\) −21.0856 29.0218i −0.860100 1.18383i −0.981546 0.191228i \(-0.938753\pi\)
0.121446 0.992598i \(-0.461247\pi\)
\(602\) 0 0
\(603\) 11.2399 + 34.5928i 0.457724 + 1.40873i
\(604\) 39.9497i 1.62553i
\(605\) 0 0
\(606\) 0 0
\(607\) −43.7348 + 14.2103i −1.77514 + 0.576778i −0.998580 0.0532640i \(-0.983038\pi\)
−0.776561 + 0.630042i \(0.783038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.70034 2.34032i 0.0686763 0.0945248i −0.773297 0.634044i \(-0.781394\pi\)
0.841973 + 0.539519i \(0.181394\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −25.8885 18.8091i −1.04055 0.756003i −0.0701559 0.997536i \(-0.522350\pi\)
−0.970393 + 0.241533i \(0.922350\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 14.1910 19.5322i 0.568095 0.781915i
\(625\) 7.72542 + 23.7764i 0.309017 + 0.951057i
\(626\) 0 0
\(627\) 0 0
\(628\) −45.0333 −1.79703
\(629\) 0 0
\(630\) 0 0
\(631\) 19.6176 14.2530i 0.780965 0.567404i −0.124303 0.992244i \(-0.539670\pi\)
0.905269 + 0.424840i \(0.139670\pi\)
\(632\) 0 0
\(633\) 15.1200 + 4.91279i 0.600967 + 0.195266i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 54.5723i 2.16223i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(642\) 0 0
\(643\) −14.5238 + 44.6997i −0.572763 + 1.76278i 0.0709114 + 0.997483i \(0.477409\pi\)
−0.643674 + 0.765300i \(0.722591\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 13.5819 4.41302i 0.532316 0.172960i
\(652\) 27.5066 + 19.9847i 1.07724 + 0.782661i
\(653\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −19.0373 + 26.2026i −0.742716 + 1.02226i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 50.2295 1.95370 0.976850 0.213925i \(-0.0686249\pi\)
0.976850 + 0.213925i \(0.0686249\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −12.3104 37.8874i −0.475946 1.46481i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 42.5217 13.8161i 1.63909 0.532573i 0.662757 0.748835i \(-0.269386\pi\)
0.976335 + 0.216262i \(0.0693864\pi\)
\(674\) 0 0
\(675\) −21.0189 + 15.2711i −0.809017 + 0.587785i
\(676\) −0.529288 + 1.62898i −0.0203572 + 0.0626531i
\(677\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(678\) 0 0
\(679\) −13.9901 + 19.2557i −0.536891 + 0.738967i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −7.27851 + 2.36493i −0.278301 + 0.0904254i
\(685\) 0 0
\(686\) 0 0
\(687\) −11.1246 + 34.2380i −0.424430 + 1.30626i
\(688\) −49.4752 16.0755i −1.88622 0.612871i
\(689\) 0 0
\(690\) 0 0
\(691\) −15.1418 46.6018i −0.576022 1.77281i −0.632671 0.774421i \(-0.718041\pi\)
0.0566486 0.998394i \(-0.481959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 27.9802 + 38.5115i 1.05755 + 1.45560i
\(701\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(702\) 0 0
\(703\) 6.62776i 0.249971i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14.9865 46.1238i 0.562831 1.73222i −0.111479 0.993767i \(-0.535559\pi\)
0.674310 0.738448i \(-0.264441\pi\)
\(710\) 0 0
\(711\) 31.3271 + 43.1180i 1.17486 + 1.61705i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(720\) 0 0
\(721\) 31.6911 + 10.2970i 1.18024 + 0.383482i
\(722\) 0 0
\(723\) 30.0285 41.3307i 1.11677 1.53711i
\(724\) −11.7426 36.1401i −0.436412 1.34314i
\(725\) 0 0
\(726\) 0 0
\(727\) 31.1769 1.15629 0.578144 0.815935i \(-0.303777\pi\)
0.578144 + 0.815935i \(0.303777\pi\)
\(728\) 0 0
\(729\) −21.8435 15.8702i −0.809017 0.587785i
\(730\) 0 0
\(731\) 0 0
\(732\) −38.6445 12.5564i −1.42834 0.464097i
\(733\) 12.1427 + 16.7130i 0.448501 + 0.617309i 0.972075 0.234672i \(-0.0754015\pi\)
−0.523574 + 0.851980i \(0.675402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −27.4888 + 8.93166i −1.01119 + 0.328556i −0.767329 0.641253i \(-0.778415\pi\)
−0.243863 + 0.969810i \(0.578415\pi\)
\(740\) 0 0
\(741\) −6.22840 + 4.52520i −0.228806 + 0.166237i
\(742\) 0 0
\(743\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 43.4390 + 31.5603i 1.58511 + 1.15165i 0.910523 + 0.413458i \(0.135679\pi\)
0.674589 + 0.738194i \(0.264321\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −29.0779 + 40.0223i −1.05755 + 1.45560i
\(757\) −0.535233 1.64728i −0.0194534 0.0598713i 0.940859 0.338800i \(-0.110021\pi\)
−0.960312 + 0.278928i \(0.910021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(762\) 0 0
\(763\) −28.1569 + 20.4572i −1.01935 + 0.740600i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 8.56373 + 26.3565i 0.309017 + 0.951057i
\(769\) 12.4134i 0.447637i 0.974631 + 0.223819i \(0.0718523\pi\)
−0.974631 + 0.223819i \(0.928148\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 51.9014 16.8638i 1.86797 0.606941i
\(773\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(774\) 0 0
\(775\) −2.67617 + 8.23639i −0.0961307 + 0.295860i
\(776\) 0 0
\(777\) 25.1822 + 34.6603i 0.903406 + 1.24343i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 50.6776 + 36.8195i 1.80992 + 1.31498i
\(785\) 0 0
\(786\) 0 0
\(787\) 35.3303 + 11.4795i 1.25939 + 0.409201i 0.861277 0.508136i \(-0.169665\pi\)
0.398113 + 0.917336i \(0.369665\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −40.8756 −1.45154
\(794\) 0 0
\(795\) 0 0
\(796\) −42.0378 + 30.5422i −1.48999 + 1.08254i
\(797\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −42.0000 −1.48123
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(810\) 0 0
\(811\) 33.3754 45.9373i 1.17197 1.61308i 0.522695 0.852520i \(-0.324927\pi\)
0.649273 0.760555i \(-0.275073\pi\)
\(812\) 0 0
\(813\) 55.5061i 1.94668i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 13.4203 + 9.75045i 0.469518 + 0.341125i
\(818\) 0 0
\(819\) −15.3783 + 47.3297i −0.537363 + 1.65383i
\(820\) 0 0
\(821\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(822\) 0 0
\(823\) −1.54508 4.75528i −0.0538583 0.165759i 0.920509 0.390721i \(-0.127774\pi\)
−0.974368 + 0.224962i \(0.927774\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(828\) 0 0
\(829\) −5.66312 + 4.11450i −0.196688 + 0.142902i −0.681770 0.731566i \(-0.738790\pi\)
0.485082 + 0.874469i \(0.338790\pi\)
\(830\) 0 0
\(831\) 21.9864 + 7.14381i 0.762699 + 0.247816i
\(832\) 16.3864 + 22.5539i 0.568095 + 0.781915i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −9.00000 −0.311086
\(838\) 0 0
\(839\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(840\) 0 0
\(841\) −8.96149 + 27.5806i −0.309017 + 0.951057i
\(842\) 0 0
\(843\) 0 0
\(844\) −10.7903 + 14.8516i −0.371418 + 0.511213i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 46.4860 15.1042i 1.59540 0.518376i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 51.5763 + 16.7582i 1.76594 + 0.573788i 0.997788 0.0664795i \(-0.0211767\pi\)
0.768152 + 0.640268i \(0.221177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −9.09896 28.0037i −0.309017 0.951057i
\(868\) 16.4901i 0.559710i
\(869\) 0 0
\(870\) 0 0
\(871\) −40.1827 + 13.0561i −1.36154 + 0.442390i
\(872\) 0 0
\(873\) 12.1353 8.81678i 0.410716 0.298403i
\(874\) 0 0
\(875\) 0 0
\(876\) −21.9824 30.2562i −0.742716 1.02226i
\(877\) 26.6817 36.7241i 0.900975 1.24009i −0.0691806 0.997604i \(-0.522038\pi\)
0.970156 0.242482i \(-0.0779615\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −18.2164 13.2350i −0.613029 0.445392i 0.237450 0.971400i \(-0.423688\pi\)
−0.850480 + 0.526008i \(0.823688\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(888\) 0 0
\(889\) 22.3807 + 68.8809i 0.750626 + 2.31019i
\(890\) 0 0
\(891\) 0 0
\(892\) 46.0000 1.54019
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −9.27051 28.5317i −0.309017 0.951057i
\(901\) 0 0
\(902\) 0 0
\(903\) 107.229 3.56837
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.87132 18.0701i 0.194954 0.600007i −0.805023 0.593244i \(-0.797847\pi\)
0.999977 0.00676342i \(-0.00215288\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(912\) 8.83701i 0.292623i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −33.6302 24.4338i −1.11117 0.807315i
\(917\) 0 0
\(918\) 0 0
\(919\) −40.0956 13.0278i −1.32263 0.429749i −0.439233 0.898373i \(-0.644750\pi\)
−0.883397 + 0.468625i \(0.844750\pi\)
\(920\) 0 0
\(921\) −34.8748 + 48.0011i −1.14916 + 1.58169i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −25.9808 −0.854242
\(926\) 0 0
\(927\) −16.9894 12.3435i −0.558004 0.405413i
\(928\) 0 0
\(929\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(930\) 0 0
\(931\) −11.7409 16.1600i −0.384793 0.529623i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 57.3167 18.6233i 1.87245 0.608397i 0.881871 0.471491i \(-0.156284\pi\)
0.990584 0.136906i \(-0.0437159\pi\)
\(938\) 0 0
\(939\) 38.8328 28.2137i 1.26726 0.920719i
\(940\) 0 0
\(941\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −58.5298 + 19.0175i −1.90096 + 0.617659i
\(949\) −30.4367 22.1135i −0.988016 0.717835i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 22.6525 16.4580i 0.730725 0.530903i
\(962\) 0 0
\(963\) 0 0
\(964\) 34.6739 + 47.7246i 1.11677 + 1.53711i
\(965\) 0 0
\(966\) 0 0
\(967\) 4.07673i 0.131099i 0.997849 + 0.0655495i \(0.0208800\pi\)
−0.997849 + 0.0655495i \(0.979120\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(972\) 25.2227 18.3253i 0.809017 0.587785i
\(973\) 1.37354 4.22732i 0.0440336 0.135522i
\(974\) 0 0
\(975\) −17.7387 24.4153i −0.568095 0.781915i
\(976\) 27.5784 37.9585i 0.882764 1.21502i
\(977\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 20.8604 6.77795i 0.666021 0.216403i
\(982\) 0 0
\(983\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −2.74708 8.45463i −0.0873962 0.268978i
\(989\) 0 0
\(990\) 0 0
\(991\) −45.0333 −1.43053 −0.715265 0.698853i \(-0.753694\pi\)
−0.715265 + 0.698853i \(0.753694\pi\)
\(992\) 0 0
\(993\) 43.4390 + 31.5603i 1.37850 + 1.00154i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −36.9231 50.8203i −1.16937 1.60950i −0.667687 0.744442i \(-0.732715\pi\)
−0.501680 0.865053i \(-0.667285\pi\)
\(998\) 0 0
\(999\) −8.34346 25.6785i −0.263975 0.812433i
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 363.2.f.g.215.1 16
3.2 odd 2 CM 363.2.f.g.215.1 16
11.2 odd 10 inner 363.2.f.g.233.1 16
11.3 even 5 363.2.d.d.362.4 yes 4
11.4 even 5 inner 363.2.f.g.239.4 16
11.5 even 5 inner 363.2.f.g.161.3 16
11.6 odd 10 inner 363.2.f.g.161.4 16
11.7 odd 10 inner 363.2.f.g.239.3 16
11.8 odd 10 363.2.d.d.362.3 4
11.9 even 5 inner 363.2.f.g.233.2 16
11.10 odd 2 inner 363.2.f.g.215.2 16
33.2 even 10 inner 363.2.f.g.233.1 16
33.5 odd 10 inner 363.2.f.g.161.3 16
33.8 even 10 363.2.d.d.362.3 4
33.14 odd 10 363.2.d.d.362.4 yes 4
33.17 even 10 inner 363.2.f.g.161.4 16
33.20 odd 10 inner 363.2.f.g.233.2 16
33.26 odd 10 inner 363.2.f.g.239.4 16
33.29 even 10 inner 363.2.f.g.239.3 16
33.32 even 2 inner 363.2.f.g.215.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.d.d.362.3 4 11.8 odd 10
363.2.d.d.362.3 4 33.8 even 10
363.2.d.d.362.4 yes 4 11.3 even 5
363.2.d.d.362.4 yes 4 33.14 odd 10
363.2.f.g.161.3 16 11.5 even 5 inner
363.2.f.g.161.3 16 33.5 odd 10 inner
363.2.f.g.161.4 16 11.6 odd 10 inner
363.2.f.g.161.4 16 33.17 even 10 inner
363.2.f.g.215.1 16 1.1 even 1 trivial
363.2.f.g.215.1 16 3.2 odd 2 CM
363.2.f.g.215.2 16 11.10 odd 2 inner
363.2.f.g.215.2 16 33.32 even 2 inner
363.2.f.g.233.1 16 11.2 odd 10 inner
363.2.f.g.233.1 16 33.2 even 10 inner
363.2.f.g.233.2 16 11.9 even 5 inner
363.2.f.g.233.2 16 33.20 odd 10 inner
363.2.f.g.239.3 16 11.7 odd 10 inner
363.2.f.g.239.3 16 33.29 even 10 inner
363.2.f.g.239.4 16 11.4 even 5 inner
363.2.f.g.239.4 16 33.26 odd 10 inner