Properties

Label 363.2.f.g.233.2
Level $363$
Weight $2$
Character 363.233
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 233.2
Root \(1.83730 + 0.596975i\) of defining polynomial
Character \(\chi\) \(=\) 363.233
Dual form 363.2.f.g.215.2

$q$-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 \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

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

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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(3\) −1.40126 + 1.01807i −0.809017 + 0.587785i
\(4\) 1.61803 + 1.17557i 0.809017 + 0.587785i
\(5\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(6\) 0 0
\(7\) 2.79802 3.85115i 1.05755 1.45560i 0.175480 0.984483i \(-0.443852\pi\)
0.882073 0.471114i \(-0.156148\pi\)
\(8\) 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 \(-0.239456\pi\)
0.189059 + 0.981966i \(0.439456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.23607 + 3.80423i 0.309017 + 0.951057i
\(17\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(18\) 0 0
\(19\) 0.749728 + 1.03191i 0.171999 + 0.236737i 0.886310 0.463091i \(-0.153260\pi\)
−0.714311 + 0.699828i \(0.753260\pi\)
\(20\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(30\) 0 0
\(31\) −0.535233 + 1.64728i −0.0961307 + 0.295860i −0.987547 0.157326i \(-0.949713\pi\)
0.891416 + 0.453186i \(0.149713\pi\)
\(32\) 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 0.877020 0.480453i \(-0.159528\pi\)
−0.185924 + 0.982564i \(0.559528\pi\)
\(38\) 0 0
\(39\) −5.74038 + 1.86516i −0.919196 + 0.298665i
\(40\) 0 0
\(41\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(42\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(48\) −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 \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(54\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(60\) 0 0
\(61\) −11.1557 + 3.62471i −1.42834 + 0.464097i −0.918243 0.396018i \(-0.870392\pi\)
−0.510100 + 0.860115i \(0.670392\pi\)
\(62\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(72\) 0 0
\(73\) −6.34577 + 8.73420i −0.742716 + 1.02226i 0.255741 + 0.966745i \(0.417680\pi\)
−0.998458 + 0.0555161i \(0.982320\pi\)
\(74\) 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 −0.961243 0.275703i \(-0.911089\pi\)
−0.939716 0.341956i \(-0.888911\pi\)
\(80\) 0 0
\(81\) −7.28115 5.29007i −0.809017 0.587785i
\(82\) 0 0
\(83\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(84\) −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 −0.998346 0.0574829i \(-0.981693\pi\)
0.841467 + 0.540309i \(0.181693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −10.0000 −1.00000
\(101\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(102\) 0 0
\(103\) −5.66312 4.11450i −0.558004 0.405413i 0.272724 0.962092i \(-0.412075\pi\)
−0.830728 + 0.556679i \(0.812075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(108\) −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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(114\) 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 \(-0.364127\pi\)
0.869987 + 0.493075i \(0.164127\pi\)
\(128\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(138\) 0 0
\(139\) −0.548839 + 0.755412i −0.0465519 + 0.0640732i −0.831658 0.555288i \(-0.812608\pi\)
0.785106 + 0.619361i \(0.212608\pi\)
\(140\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(150\) 0 0
\(151\) −11.7409 16.1600i −0.955463 1.31508i −0.949058 0.315102i \(-0.897961\pi\)
−0.00640530 0.999979i \(-0.502039\pi\)
\(152\) 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 −0.468905 + 0.883249i \(0.655351\pi\)
−0.984919 + 0.173016i \(0.944649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.25329 16.1680i 0.411469 1.26637i −0.503902 0.863761i \(-0.668103\pi\)
0.915371 0.402611i \(-0.131897\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(168\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(174\) 0 0
\(175\) 23.8014i 1.79922i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(180\) 0 0
\(181\) 5.87132 + 18.0701i 0.436412 + 1.34314i 0.891633 + 0.452759i \(0.149560\pi\)
−0.455221 + 0.890379i \(0.650440\pi\)
\(182\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(192\) −4.28187 13.1782i −0.309017 0.951057i
\(193\) −25.9507 + 8.43189i −1.86797 + 0.606941i −0.875699 + 0.482857i \(0.839599\pi\)
−0.992272 + 0.124083i \(0.960401\pi\)
\(194\) 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 \(-0.297678\pi\)
0.00729508 + 0.999973i \(0.497678\pi\)
\(212\) 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 0.248058 0.968745i \(-0.420208\pi\)
0.997986 + 0.0634420i \(0.0202078\pi\)
\(224\) 0 0
\(225\) 4.63525 + 14.2658i 0.309017 + 0.951057i
\(226\) 0 0
\(227\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(228\) −2.59713 3.57465i −0.171999 0.236737i
\(229\) −6.42280 + 19.7673i −0.424430 + 1.30626i 0.479108 + 0.877756i \(0.340960\pi\)
−0.903539 + 0.428507i \(0.859040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(234\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(240\) 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 \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(252\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(258\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(270\) 0 0
\(271\) −18.8364 + 25.9261i −1.14423 + 1.57490i −0.386558 + 0.922265i \(0.626336\pi\)
−0.757673 + 0.652634i \(0.773664\pi\)
\(272\) 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 \(-0.268672\pi\)
0.0982624 + 0.995161i \(0.468672\pi\)
\(278\) 0 0
\(279\) 4.20378 + 3.05422i 0.251673 + 0.182851i
\(280\) 0 0
\(281\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(282\) 0 0
\(283\) 16.5872 + 22.8304i 0.986009 + 1.35713i 0.933528 + 0.358503i \(0.116713\pi\)
0.0524806 + 0.998622i \(0.483287\pi\)
\(284\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(294\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(312\) 0 0
\(313\) −8.56373 26.3565i −0.484051 1.48975i −0.833351 0.552744i \(-0.813581\pi\)
0.349300 0.937011i \(-0.386419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −20.8847 28.7453i −1.17486 1.61705i
\(317\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(318\) 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 0.289484 + 0.957183i \(0.406516\pi\)
−0.999790 + 0.0204700i \(0.993484\pi\)
\(338\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(348\) 0 0
\(349\) −17.5379 24.1388i −0.938780 1.29212i −0.956335 0.292274i \(-0.905588\pi\)
0.0175547 0.999846i \(-0.494412\pi\)
\(350\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(360\) 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 0.669034 0.743232i \(-0.266708\pi\)
−0.500113 + 0.865960i \(0.666708\pi\)
\(368\) 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 0.994258 0.107014i \(-0.0341289\pi\)
−0.867272 + 0.497834i \(0.834129\pi\)
\(380\) 0 0
\(381\) −15.4896 + 21.3196i −0.793554 + 1.09223i
\(382\) 0 0
\(383\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(384\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(390\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(402\) 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 0.974503 + 0.224373i \(0.0720334\pi\)
0.920273 + 0.391277i \(0.127967\pi\)
\(410\) 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 0.367816 0.929899i \(-0.380106\pi\)
0.998048 0.0624590i \(-0.0198943\pi\)
\(422\) 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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(432\) −16.8151 + 12.2169i −0.809017 + 0.587785i
\(433\) 29.9336 + 21.7481i 1.43852 + 1.04514i 0.988350 + 0.152201i \(0.0486362\pi\)
0.450169 + 0.892943i \(0.351364\pi\)
\(434\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(444\) −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 \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(450\) 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 \(-0.274854\pi\)
0.972477 + 0.233001i \(0.0748544\pi\)
\(458\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(468\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(480\) 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 −0.649469 0.760388i \(-0.725009\pi\)
0.522475 + 0.852655i \(0.325009\pi\)
\(488\) 0 0
\(489\) 9.09896 + 28.0037i 0.411469 + 1.26637i
\(490\) 0 0
\(491\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(492\) 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 0.928833 + 0.370499i \(0.120813\pi\)
0.639391 + 0.768882i \(0.279187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(504\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(510\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(522\) 0 0
\(523\) 18.4342 5.98964i 0.806072 0.261909i 0.123139 0.992389i \(-0.460704\pi\)
0.682933 + 0.730481i \(0.260704\pi\)
\(524\) 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 \(-0.602416\pi\)
−0.813455 + 0.581628i \(0.802416\pi\)
\(542\) 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 \(-0.123472\pi\)
−0.645791 + 0.763514i \(0.723472\pi\)
\(548\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(558\) 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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(564\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(570\) 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 −0.518569 0.855036i \(-0.673535\pi\)
−0.0830461 0.996546i \(-0.526465\pi\)
\(578\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(588\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(600\) 0 0
\(601\) 21.0856 29.0218i 0.860100 1.18383i −0.121446 0.992598i \(-0.538753\pi\)
0.981546 0.191228i \(-0.0612469\pi\)
\(602\) 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 \(-0.0169625\pi\)
0.776561 + 0.630042i \(0.216962\pi\)
\(608\) 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 \(-0.218606\pi\)
−0.841973 + 0.539519i \(0.818606\pi\)
\(614\) 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 −0.970393 0.241533i \(-0.922350\pi\)
−0.0701559 + 0.997536i \(0.522350\pi\)
\(620\) 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 0.905269 0.424840i \(-0.139670\pi\)
−0.124303 + 0.992244i \(0.539670\pi\)
\(632\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(642\) 0 0
\(643\) −14.5238 44.6997i −0.572763 1.76278i −0.643674 0.765300i \(-0.722591\pi\)
0.0709114 0.997483i \(-0.477409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(648\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(654\) 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 \(-0.730614\pi\)
−0.976335 + 0.216262i \(0.930614\pi\)
\(674\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(678\) 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 0.0566486 + 0.998394i \(0.481959\pi\)
−0.632671 + 0.774421i \(0.718041\pi\)
\(692\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(702\) 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 0.674310 + 0.738448i \(0.264441\pi\)
−0.111479 + 0.993767i \(0.535559\pi\)
\(710\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(720\) 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 \(-0.924598\pi\)
0.523574 + 0.851980i \(0.324598\pi\)
\(734\) 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 \(-0.221585\pi\)
0.243863 + 0.969810i \(0.421585\pi\)
\(740\) 0 0
\(741\) −6.22840 4.52520i −0.228806 0.166237i
\(742\) 0 0
\(743\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(744\) 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 0.674589 0.738194i \(-0.264321\pi\)
0.910523 0.413458i \(-0.135679\pi\)
\(752\) 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 −0.960312 0.278928i \(-0.910021\pi\)
0.940859 + 0.338800i \(0.110021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(762\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(774\) 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 \(-0.830335\pi\)
−0.398113 + 0.917336i \(0.630335\pi\)
\(788\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(798\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(810\) 0 0
\(811\) −33.3754 45.9373i −1.17197 1.61308i −0.649273 0.760555i \(-0.724927\pi\)
−0.522695 0.852520i \(-0.675073\pi\)
\(812\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(822\) 0 0
\(823\) −1.54508 + 4.75528i −0.0538583 + 0.165759i −0.974368 0.224962i \(-0.927774\pi\)
0.920509 + 0.390721i \(0.127774\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(828\) 0 0
\(829\) −5.66312 4.11450i −0.196688 0.142902i 0.485082 0.874469i \(-0.338790\pi\)
−0.681770 + 0.731566i \(0.738790\pi\)
\(830\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(840\) 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 \(-0.978823\pi\)
−0.768152 + 0.640268i \(0.778823\pi\)
\(854\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(864\) 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 −0.970156 0.242482i \(-0.922038\pi\)
0.0691806 0.997604i \(-0.477962\pi\)
\(878\) 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 −0.850480 0.526008i \(-0.823688\pi\)
0.237450 + 0.971400i \(0.423688\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(888\) 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 0.999977 + 0.00676342i \(0.00215288\pi\)
−0.805023 + 0.593244i \(0.797847\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(912\) 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 \(-0.355250\pi\)
0.883397 + 0.468625i \(0.155250\pi\)
\(920\) 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 \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(930\) 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 −0.990584 0.136906i \(-0.956284\pi\)
−0.881871 0.471491i \(-0.843716\pi\)
\(938\) 0 0
\(939\) 38.8328 + 28.2137i 1.26726 + 0.920719i
\(940\) 0 0
\(941\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(942\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(954\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(972\) 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 \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(978\) 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 \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(984\) 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 0.501680 0.865053i \(-0.332715\pi\)
0.667687 0.744442i \(-0.267285\pi\)
\(998\) 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.233.2 16
3.2 odd 2 CM 363.2.f.g.233.2 16
11.2 odd 10 inner 363.2.f.g.239.3 16
11.3 even 5 inner 363.2.f.g.161.3 16
11.4 even 5 363.2.d.d.362.4 yes 4
11.5 even 5 inner 363.2.f.g.215.1 16
11.6 odd 10 inner 363.2.f.g.215.2 16
11.7 odd 10 363.2.d.d.362.3 4
11.8 odd 10 inner 363.2.f.g.161.4 16
11.9 even 5 inner 363.2.f.g.239.4 16
11.10 odd 2 inner 363.2.f.g.233.1 16
33.2 even 10 inner 363.2.f.g.239.3 16
33.5 odd 10 inner 363.2.f.g.215.1 16
33.8 even 10 inner 363.2.f.g.161.4 16
33.14 odd 10 inner 363.2.f.g.161.3 16
33.17 even 10 inner 363.2.f.g.215.2 16
33.20 odd 10 inner 363.2.f.g.239.4 16
33.26 odd 10 363.2.d.d.362.4 yes 4
33.29 even 10 363.2.d.d.362.3 4
33.32 even 2 inner 363.2.f.g.233.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.d.d.362.3 4 11.7 odd 10
363.2.d.d.362.3 4 33.29 even 10
363.2.d.d.362.4 yes 4 11.4 even 5
363.2.d.d.362.4 yes 4 33.26 odd 10
363.2.f.g.161.3 16 11.3 even 5 inner
363.2.f.g.161.3 16 33.14 odd 10 inner
363.2.f.g.161.4 16 11.8 odd 10 inner
363.2.f.g.161.4 16 33.8 even 10 inner
363.2.f.g.215.1 16 11.5 even 5 inner
363.2.f.g.215.1 16 33.5 odd 10 inner
363.2.f.g.215.2 16 11.6 odd 10 inner
363.2.f.g.215.2 16 33.17 even 10 inner
363.2.f.g.233.1 16 11.10 odd 2 inner
363.2.f.g.233.1 16 33.32 even 2 inner
363.2.f.g.233.2 16 1.1 even 1 trivial
363.2.f.g.233.2 16 3.2 odd 2 CM
363.2.f.g.239.3 16 11.2 odd 10 inner
363.2.f.g.239.3 16 33.2 even 10 inner
363.2.f.g.239.4 16 11.9 even 5 inner
363.2.f.g.239.4 16 33.20 odd 10 inner