Properties

Label 4140.2.i.a.1241.15
Level $4140$
Weight $2$
Character 4140.1241
Analytic conductor $33.058$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,2,Mod(1241,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.1241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.i (of order \(2\), degree \(1\), 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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.15
Root \(1.41255i\) of defining polynomial
Character \(\chi\) \(=\) 4140.1241
Dual form 4140.2.i.a.1241.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +4.35549i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +4.35549i q^{7} -0.0533304 q^{11} -4.90540 q^{13} -1.18157 q^{17} -5.30903i q^{19} +(1.90776 - 4.40005i) q^{23} +1.00000 q^{25} +3.23248i q^{29} -6.80149 q^{31} -4.35549i q^{35} -4.86073i q^{37} -3.15866i q^{41} +10.5277i q^{43} -11.6729i q^{47} -11.9703 q^{49} +6.20332 q^{53} +0.0533304 q^{55} +4.11933i q^{59} -0.265744i q^{61} +4.90540 q^{65} -12.6537i q^{67} +1.13167i q^{71} +9.67564 q^{73} -0.232280i q^{77} -3.25652i q^{79} +17.8978 q^{83} +1.18157 q^{85} +12.1425 q^{89} -21.3654i q^{91} +5.30903i q^{95} -6.81245i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{5} - 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} + 4 q^{53} + 8 q^{73} + 20 q^{83} + 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.35549i 1.64622i 0.567881 + 0.823111i \(0.307763\pi\)
−0.567881 + 0.823111i \(0.692237\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.0533304 −0.0160797 −0.00803986 0.999968i \(-0.502559\pi\)
−0.00803986 + 0.999968i \(0.502559\pi\)
\(12\) 0 0
\(13\) −4.90540 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.18157 −0.286572 −0.143286 0.989681i \(-0.545767\pi\)
−0.143286 + 0.989681i \(0.545767\pi\)
\(18\) 0 0
\(19\) 5.30903i 1.21797i −0.793180 0.608987i \(-0.791576\pi\)
0.793180 0.608987i \(-0.208424\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.90776 4.40005i 0.397795 0.917474i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.23248i 0.600256i 0.953899 + 0.300128i \(0.0970293\pi\)
−0.953899 + 0.300128i \(0.902971\pi\)
\(30\) 0 0
\(31\) −6.80149 −1.22158 −0.610792 0.791791i \(-0.709149\pi\)
−0.610792 + 0.791791i \(0.709149\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.35549i 0.736213i
\(36\) 0 0
\(37\) 4.86073i 0.799100i −0.916711 0.399550i \(-0.869166\pi\)
0.916711 0.399550i \(-0.130834\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.15866i 0.493299i −0.969105 0.246650i \(-0.920670\pi\)
0.969105 0.246650i \(-0.0793297\pi\)
\(42\) 0 0
\(43\) 10.5277i 1.60546i 0.596340 + 0.802732i \(0.296621\pi\)
−0.596340 + 0.802732i \(0.703379\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.6729i 1.70266i −0.524629 0.851331i \(-0.675796\pi\)
0.524629 0.851331i \(-0.324204\pi\)
\(48\) 0 0
\(49\) −11.9703 −1.71005
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.20332 0.852091 0.426046 0.904702i \(-0.359906\pi\)
0.426046 + 0.904702i \(0.359906\pi\)
\(54\) 0 0
\(55\) 0.0533304 0.00719107
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.11933i 0.536291i 0.963378 + 0.268145i \(0.0864108\pi\)
−0.963378 + 0.268145i \(0.913589\pi\)
\(60\) 0 0
\(61\) 0.265744i 0.0340250i −0.999855 0.0170125i \(-0.994584\pi\)
0.999855 0.0170125i \(-0.00541551\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.90540 0.608440
\(66\) 0 0
\(67\) 12.6537i 1.54589i −0.634470 0.772947i \(-0.718782\pi\)
0.634470 0.772947i \(-0.281218\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.13167i 0.134304i 0.997743 + 0.0671520i \(0.0213912\pi\)
−0.997743 + 0.0671520i \(0.978609\pi\)
\(72\) 0 0
\(73\) 9.67564 1.13245 0.566224 0.824251i \(-0.308404\pi\)
0.566224 + 0.824251i \(0.308404\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.232280i 0.0264708i
\(78\) 0 0
\(79\) 3.25652i 0.366387i −0.983077 0.183193i \(-0.941357\pi\)
0.983077 0.183193i \(-0.0586434\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.8978 1.96454 0.982268 0.187484i \(-0.0600334\pi\)
0.982268 + 0.187484i \(0.0600334\pi\)
\(84\) 0 0
\(85\) 1.18157 0.128159
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.1425 1.28710 0.643552 0.765402i \(-0.277460\pi\)
0.643552 + 0.765402i \(0.277460\pi\)
\(90\) 0 0
\(91\) 21.3654i 2.23971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.30903i 0.544695i
\(96\) 0 0
\(97\) 6.81245i 0.691699i −0.938290 0.345850i \(-0.887591\pi\)
0.938290 0.345850i \(-0.112409\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.69296i 0.765478i 0.923857 + 0.382739i \(0.125019\pi\)
−0.923857 + 0.382739i \(0.874981\pi\)
\(102\) 0 0
\(103\) 3.66586i 0.361208i 0.983556 + 0.180604i \(0.0578053\pi\)
−0.983556 + 0.180604i \(0.942195\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.44772 −0.429977 −0.214988 0.976617i \(-0.568971\pi\)
−0.214988 + 0.976617i \(0.568971\pi\)
\(108\) 0 0
\(109\) 2.59386i 0.248447i 0.992254 + 0.124224i \(0.0396440\pi\)
−0.992254 + 0.124224i \(0.960356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.3149 1.25256 0.626278 0.779600i \(-0.284577\pi\)
0.626278 + 0.779600i \(0.284577\pi\)
\(114\) 0 0
\(115\) −1.90776 + 4.40005i −0.177899 + 0.410307i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.14631i 0.471762i
\(120\) 0 0
\(121\) −10.9972 −0.999741
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 21.5250 1.91004 0.955019 0.296544i \(-0.0958343\pi\)
0.955019 + 0.296544i \(0.0958343\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.36869i 0.818546i 0.912412 + 0.409273i \(0.134218\pi\)
−0.912412 + 0.409273i \(0.865782\pi\)
\(132\) 0 0
\(133\) 23.1234 2.00506
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.29045 0.281122 0.140561 0.990072i \(-0.455109\pi\)
0.140561 + 0.990072i \(0.455109\pi\)
\(138\) 0 0
\(139\) 2.68501 0.227740 0.113870 0.993496i \(-0.463675\pi\)
0.113870 + 0.993496i \(0.463675\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.261607 0.0218767
\(144\) 0 0
\(145\) 3.23248i 0.268442i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.517635 −0.0424063 −0.0212032 0.999775i \(-0.506750\pi\)
−0.0212032 + 0.999775i \(0.506750\pi\)
\(150\) 0 0
\(151\) −19.1978 −1.56230 −0.781149 0.624345i \(-0.785366\pi\)
−0.781149 + 0.624345i \(0.785366\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.80149 0.546309
\(156\) 0 0
\(157\) 13.7067i 1.09391i −0.837161 0.546956i \(-0.815786\pi\)
0.837161 0.546956i \(-0.184214\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 19.1644 + 8.30922i 1.51037 + 0.654858i
\(162\) 0 0
\(163\) −15.0108 −1.17574 −0.587869 0.808956i \(-0.700033\pi\)
−0.587869 + 0.808956i \(0.700033\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.6143i 0.976126i −0.872808 0.488063i \(-0.837704\pi\)
0.872808 0.488063i \(-0.162296\pi\)
\(168\) 0 0
\(169\) 11.0630 0.850997
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.49207i 0.341526i −0.985312 0.170763i \(-0.945377\pi\)
0.985312 0.170763i \(-0.0546232\pi\)
\(174\) 0 0
\(175\) 4.35549i 0.329244i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.66095i 0.722093i −0.932548 0.361047i \(-0.882420\pi\)
0.932548 0.361047i \(-0.117580\pi\)
\(180\) 0 0
\(181\) 0.575728i 0.0427935i −0.999771 0.0213967i \(-0.993189\pi\)
0.999771 0.0213967i \(-0.00681132\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.86073i 0.357368i
\(186\) 0 0
\(187\) 0.0630135 0.00460800
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.8620 −1.29245 −0.646224 0.763148i \(-0.723653\pi\)
−0.646224 + 0.763148i \(0.723653\pi\)
\(192\) 0 0
\(193\) 0.760186 0.0547194 0.0273597 0.999626i \(-0.491290\pi\)
0.0273597 + 0.999626i \(0.491290\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.2534i 1.65674i 0.560184 + 0.828368i \(0.310731\pi\)
−0.560184 + 0.828368i \(0.689269\pi\)
\(198\) 0 0
\(199\) 14.2715i 1.01168i −0.862628 0.505839i \(-0.831183\pi\)
0.862628 0.505839i \(-0.168817\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −14.0790 −0.988154
\(204\) 0 0
\(205\) 3.15866i 0.220610i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.283132i 0.0195847i
\(210\) 0 0
\(211\) 2.56124 0.176323 0.0881616 0.996106i \(-0.471901\pi\)
0.0881616 + 0.996106i \(0.471901\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.5277i 0.717985i
\(216\) 0 0
\(217\) 29.6238i 2.01100i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.79607 0.389886
\(222\) 0 0
\(223\) −18.2514 −1.22221 −0.611104 0.791551i \(-0.709274\pi\)
−0.611104 + 0.791551i \(0.709274\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.6784 1.04061 0.520304 0.853981i \(-0.325819\pi\)
0.520304 + 0.853981i \(0.325819\pi\)
\(228\) 0 0
\(229\) 2.23945i 0.147987i −0.997259 0.0739936i \(-0.976426\pi\)
0.997259 0.0739936i \(-0.0235744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.5141i 1.27841i −0.769035 0.639206i \(-0.779263\pi\)
0.769035 0.639206i \(-0.220737\pi\)
\(234\) 0 0
\(235\) 11.6729i 0.761454i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.8368i 1.73593i −0.496626 0.867965i \(-0.665428\pi\)
0.496626 0.867965i \(-0.334572\pi\)
\(240\) 0 0
\(241\) 14.2600i 0.918566i −0.888290 0.459283i \(-0.848106\pi\)
0.888290 0.459283i \(-0.151894\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 11.9703 0.764756
\(246\) 0 0
\(247\) 26.0429i 1.65707i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.23056 −0.267031 −0.133515 0.991047i \(-0.542627\pi\)
−0.133515 + 0.991047i \(0.542627\pi\)
\(252\) 0 0
\(253\) −0.101741 + 0.234657i −0.00639643 + 0.0147527i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.1291i 1.69226i −0.532973 0.846132i \(-0.678925\pi\)
0.532973 0.846132i \(-0.321075\pi\)
\(258\) 0 0
\(259\) 21.1709 1.31550
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.78709 −0.541835 −0.270918 0.962603i \(-0.587327\pi\)
−0.270918 + 0.962603i \(0.587327\pi\)
\(264\) 0 0
\(265\) −6.20332 −0.381067
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.9826i 1.52322i −0.648037 0.761609i \(-0.724410\pi\)
0.648037 0.761609i \(-0.275590\pi\)
\(270\) 0 0
\(271\) −5.35815 −0.325484 −0.162742 0.986669i \(-0.552034\pi\)
−0.162742 + 0.986669i \(0.552034\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.0533304 −0.00321594
\(276\) 0 0
\(277\) −11.2055 −0.673274 −0.336637 0.941634i \(-0.609290\pi\)
−0.336637 + 0.941634i \(0.609290\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.8269 1.42139 0.710696 0.703499i \(-0.248380\pi\)
0.710696 + 0.703499i \(0.248380\pi\)
\(282\) 0 0
\(283\) 8.56706i 0.509259i 0.967039 + 0.254630i \(0.0819535\pi\)
−0.967039 + 0.254630i \(0.918046\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 13.7575 0.812080
\(288\) 0 0
\(289\) −15.6039 −0.917876
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.8009 1.44888 0.724442 0.689336i \(-0.242097\pi\)
0.724442 + 0.689336i \(0.242097\pi\)
\(294\) 0 0
\(295\) 4.11933i 0.239837i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.35831 + 21.5840i −0.541205 + 1.24824i
\(300\) 0 0
\(301\) −45.8534 −2.64295
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.265744i 0.0152164i
\(306\) 0 0
\(307\) 10.8960 0.621866 0.310933 0.950432i \(-0.399358\pi\)
0.310933 + 0.950432i \(0.399358\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.6067i 1.22520i −0.790392 0.612602i \(-0.790123\pi\)
0.790392 0.612602i \(-0.209877\pi\)
\(312\) 0 0
\(313\) 27.6548i 1.56314i −0.623816 0.781572i \(-0.714418\pi\)
0.623816 0.781572i \(-0.285582\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.6630i 0.598895i −0.954113 0.299447i \(-0.903198\pi\)
0.954113 0.299447i \(-0.0968023\pi\)
\(318\) 0 0
\(319\) 0.172389i 0.00965194i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.27298i 0.349038i
\(324\) 0 0
\(325\) −4.90540 −0.272103
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 50.8411 2.80296
\(330\) 0 0
\(331\) 19.0311 1.04604 0.523022 0.852319i \(-0.324805\pi\)
0.523022 + 0.852319i \(0.324805\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.6537i 0.691345i
\(336\) 0 0
\(337\) 9.01836i 0.491261i −0.969364 0.245631i \(-0.921005\pi\)
0.969364 0.245631i \(-0.0789950\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.362726 0.0196427
\(342\) 0 0
\(343\) 21.6482i 1.16889i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.8249i 1.22530i 0.790353 + 0.612652i \(0.209897\pi\)
−0.790353 + 0.612652i \(0.790103\pi\)
\(348\) 0 0
\(349\) 14.0814 0.753761 0.376880 0.926262i \(-0.376997\pi\)
0.376880 + 0.926262i \(0.376997\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.2950i 1.13342i −0.823917 0.566710i \(-0.808216\pi\)
0.823917 0.566710i \(-0.191784\pi\)
\(354\) 0 0
\(355\) 1.13167i 0.0600626i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −35.4491 −1.87093 −0.935465 0.353418i \(-0.885019\pi\)
−0.935465 + 0.353418i \(0.885019\pi\)
\(360\) 0 0
\(361\) −9.18576 −0.483461
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.67564 −0.506446
\(366\) 0 0
\(367\) 20.0679i 1.04753i 0.851861 + 0.523767i \(0.175474\pi\)
−0.851861 + 0.523767i \(0.824526\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 27.0185i 1.40273i
\(372\) 0 0
\(373\) 11.0961i 0.574535i 0.957850 + 0.287267i \(0.0927468\pi\)
−0.957850 + 0.287267i \(0.907253\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.8566i 0.816656i
\(378\) 0 0
\(379\) 0.876305i 0.0450128i 0.999747 + 0.0225064i \(0.00716461\pi\)
−0.999747 + 0.0225064i \(0.992835\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.0864 −0.719781 −0.359890 0.932995i \(-0.617186\pi\)
−0.359890 + 0.932995i \(0.617186\pi\)
\(384\) 0 0
\(385\) 0.232280i 0.0118381i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −16.8276 −0.853193 −0.426596 0.904442i \(-0.640287\pi\)
−0.426596 + 0.904442i \(0.640287\pi\)
\(390\) 0 0
\(391\) −2.25415 + 5.19896i −0.113997 + 0.262923i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.25652i 0.163853i
\(396\) 0 0
\(397\) −36.4052 −1.82713 −0.913563 0.406697i \(-0.866680\pi\)
−0.913563 + 0.406697i \(0.866680\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.8784 −1.14249 −0.571246 0.820779i \(-0.693540\pi\)
−0.571246 + 0.820779i \(0.693540\pi\)
\(402\) 0 0
\(403\) 33.3640 1.66198
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.259225i 0.0128493i
\(408\) 0 0
\(409\) −7.58108 −0.374860 −0.187430 0.982278i \(-0.560016\pi\)
−0.187430 + 0.982278i \(0.560016\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −17.9417 −0.882854
\(414\) 0 0
\(415\) −17.8978 −0.878567
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.6066 1.64179 0.820895 0.571079i \(-0.193475\pi\)
0.820895 + 0.571079i \(0.193475\pi\)
\(420\) 0 0
\(421\) 31.2672i 1.52387i 0.647655 + 0.761934i \(0.275750\pi\)
−0.647655 + 0.761934i \(0.724250\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.18157 −0.0573145
\(426\) 0 0
\(427\) 1.15744 0.0560127
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 31.9116 1.53713 0.768564 0.639772i \(-0.220971\pi\)
0.768564 + 0.639772i \(0.220971\pi\)
\(432\) 0 0
\(433\) 5.65026i 0.271534i −0.990741 0.135767i \(-0.956650\pi\)
0.990741 0.135767i \(-0.0433499\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −23.3600 10.1283i −1.11746 0.484504i
\(438\) 0 0
\(439\) −20.1206 −0.960305 −0.480152 0.877185i \(-0.659419\pi\)
−0.480152 + 0.877185i \(0.659419\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.5062i 1.35437i 0.735813 + 0.677185i \(0.236800\pi\)
−0.735813 + 0.677185i \(0.763200\pi\)
\(444\) 0 0
\(445\) −12.1425 −0.575611
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.4642i 0.541029i 0.962716 + 0.270515i \(0.0871938\pi\)
−0.962716 + 0.270515i \(0.912806\pi\)
\(450\) 0 0
\(451\) 0.168452i 0.00793211i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 21.3654i 1.00163i
\(456\) 0 0
\(457\) 6.50050i 0.304080i −0.988374 0.152040i \(-0.951416\pi\)
0.988374 0.152040i \(-0.0485843\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0113i 0.838872i −0.907785 0.419436i \(-0.862228\pi\)
0.907785 0.419436i \(-0.137772\pi\)
\(462\) 0 0
\(463\) −14.5620 −0.676753 −0.338376 0.941011i \(-0.609878\pi\)
−0.338376 + 0.941011i \(0.609878\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.1466 −0.747174 −0.373587 0.927595i \(-0.621872\pi\)
−0.373587 + 0.927595i \(0.621872\pi\)
\(468\) 0 0
\(469\) 55.1131 2.54488
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.561448i 0.0258154i
\(474\) 0 0
\(475\) 5.30903i 0.243595i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.0188 −1.37160 −0.685798 0.727792i \(-0.740547\pi\)
−0.685798 + 0.727792i \(0.740547\pi\)
\(480\) 0 0
\(481\) 23.8438i 1.08719i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.81245i 0.309337i
\(486\) 0 0
\(487\) 13.2977 0.602576 0.301288 0.953533i \(-0.402583\pi\)
0.301288 + 0.953533i \(0.402583\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.53612i 0.159583i 0.996812 + 0.0797915i \(0.0254255\pi\)
−0.996812 + 0.0797915i \(0.974575\pi\)
\(492\) 0 0
\(493\) 3.81939i 0.172017i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.92896 −0.221094
\(498\) 0 0
\(499\) 27.9990 1.25341 0.626705 0.779257i \(-0.284403\pi\)
0.626705 + 0.779257i \(0.284403\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.91684 0.174643 0.0873217 0.996180i \(-0.472169\pi\)
0.0873217 + 0.996180i \(0.472169\pi\)
\(504\) 0 0
\(505\) 7.69296i 0.342332i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.1501i 0.715839i −0.933752 0.357920i \(-0.883486\pi\)
0.933752 0.357920i \(-0.116514\pi\)
\(510\) 0 0
\(511\) 42.1422i 1.86426i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.66586i 0.161537i
\(516\) 0 0
\(517\) 0.622519i 0.0273783i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.1551 1.05825 0.529126 0.848543i \(-0.322520\pi\)
0.529126 + 0.848543i \(0.322520\pi\)
\(522\) 0 0
\(523\) 1.30455i 0.0570439i 0.999593 + 0.0285220i \(0.00908005\pi\)
−0.999593 + 0.0285220i \(0.990920\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.03642 0.350072
\(528\) 0 0
\(529\) −15.7209 16.7885i −0.683519 0.729933i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.4945i 0.671140i
\(534\) 0 0
\(535\) 4.44772 0.192292
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.638382 0.0274971
\(540\) 0 0
\(541\) −15.5323 −0.667784 −0.333892 0.942611i \(-0.608362\pi\)
−0.333892 + 0.942611i \(0.608362\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.59386i 0.111109i
\(546\) 0 0
\(547\) 14.6332 0.625671 0.312836 0.949807i \(-0.398721\pi\)
0.312836 + 0.949807i \(0.398721\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 17.1613 0.731096
\(552\) 0 0
\(553\) 14.1837 0.603154
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.5916 −0.491154 −0.245577 0.969377i \(-0.578977\pi\)
−0.245577 + 0.969377i \(0.578977\pi\)
\(558\) 0 0
\(559\) 51.6427i 2.18425i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.21549 −0.219807 −0.109903 0.993942i \(-0.535054\pi\)
−0.109903 + 0.993942i \(0.535054\pi\)
\(564\) 0 0
\(565\) −13.3149 −0.560160
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.94892 0.375158 0.187579 0.982250i \(-0.439936\pi\)
0.187579 + 0.982250i \(0.439936\pi\)
\(570\) 0 0
\(571\) 2.90991i 0.121776i −0.998145 0.0608879i \(-0.980607\pi\)
0.998145 0.0608879i \(-0.0193932\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.90776 4.40005i 0.0795590 0.183495i
\(576\) 0 0
\(577\) 36.2025 1.50713 0.753565 0.657373i \(-0.228332\pi\)
0.753565 + 0.657373i \(0.228332\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 77.9536i 3.23406i
\(582\) 0 0
\(583\) −0.330825 −0.0137014
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.5893i 0.891088i −0.895260 0.445544i \(-0.853010\pi\)
0.895260 0.445544i \(-0.146990\pi\)
\(588\) 0 0
\(589\) 36.1093i 1.48786i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.2330i 0.543412i −0.962380 0.271706i \(-0.912412\pi\)
0.962380 0.271706i \(-0.0875879\pi\)
\(594\) 0 0
\(595\) 5.14631i 0.210978i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28.9912i 1.18455i 0.805736 + 0.592275i \(0.201770\pi\)
−0.805736 + 0.592275i \(0.798230\pi\)
\(600\) 0 0
\(601\) −2.30240 −0.0939170 −0.0469585 0.998897i \(-0.514953\pi\)
−0.0469585 + 0.998897i \(0.514953\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.9972 0.447098
\(606\) 0 0
\(607\) 47.3449 1.92167 0.960836 0.277118i \(-0.0893792\pi\)
0.960836 + 0.277118i \(0.0893792\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 57.2601i 2.31650i
\(612\) 0 0
\(613\) 2.90409i 0.117295i 0.998279 + 0.0586475i \(0.0186788\pi\)
−0.998279 + 0.0586475i \(0.981321\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.3417 −0.537118 −0.268559 0.963263i \(-0.586547\pi\)
−0.268559 + 0.963263i \(0.586547\pi\)
\(618\) 0 0
\(619\) 21.1961i 0.851943i 0.904737 + 0.425971i \(0.140068\pi\)
−0.904737 + 0.425971i \(0.859932\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 52.8867i 2.11886i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.74329i 0.229000i
\(630\) 0 0
\(631\) 27.0046i 1.07503i 0.843253 + 0.537517i \(0.180638\pi\)
−0.843253 + 0.537517i \(0.819362\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −21.5250 −0.854195
\(636\) 0 0
\(637\) 58.7192 2.32654
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.03440 −0.0408565 −0.0204282 0.999791i \(-0.506503\pi\)
−0.0204282 + 0.999791i \(0.506503\pi\)
\(642\) 0 0
\(643\) 23.0638i 0.909546i −0.890608 0.454773i \(-0.849720\pi\)
0.890608 0.454773i \(-0.150280\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0526i 0.945604i −0.881169 0.472802i \(-0.843243\pi\)
0.881169 0.472802i \(-0.156757\pi\)
\(648\) 0 0
\(649\) 0.219685i 0.00862341i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 44.7405i 1.75083i −0.483370 0.875416i \(-0.660587\pi\)
0.483370 0.875416i \(-0.339413\pi\)
\(654\) 0 0
\(655\) 9.36869i 0.366065i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.7570 −0.847533 −0.423767 0.905771i \(-0.639292\pi\)
−0.423767 + 0.905771i \(0.639292\pi\)
\(660\) 0 0
\(661\) 15.8400i 0.616104i 0.951370 + 0.308052i \(0.0996770\pi\)
−0.951370 + 0.308052i \(0.900323\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −23.1234 −0.896688
\(666\) 0 0
\(667\) 14.2231 + 6.16678i 0.550719 + 0.238779i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.0141722i 0.000547112i
\(672\) 0 0
\(673\) −29.9845 −1.15582 −0.577909 0.816101i \(-0.696131\pi\)
−0.577909 + 0.816101i \(0.696131\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 46.0816 1.77106 0.885530 0.464582i \(-0.153795\pi\)
0.885530 + 0.464582i \(0.153795\pi\)
\(678\) 0 0
\(679\) 29.6716 1.13869
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.1398i 0.387988i −0.981003 0.193994i \(-0.937856\pi\)
0.981003 0.193994i \(-0.0621443\pi\)
\(684\) 0 0
\(685\) −3.29045 −0.125722
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −30.4298 −1.15928
\(690\) 0 0
\(691\) −46.2807 −1.76060 −0.880301 0.474416i \(-0.842659\pi\)
−0.880301 + 0.474416i \(0.842659\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.68501 −0.101848
\(696\) 0 0
\(697\) 3.73217i 0.141366i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.8970 0.562653 0.281327 0.959612i \(-0.409226\pi\)
0.281327 + 0.959612i \(0.409226\pi\)
\(702\) 0 0
\(703\) −25.8058 −0.973283
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −33.5066 −1.26015
\(708\) 0 0
\(709\) 37.5138i 1.40886i −0.709773 0.704431i \(-0.751202\pi\)
0.709773 0.704431i \(-0.248798\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.9756 + 29.9269i −0.485939 + 1.12077i
\(714\) 0 0
\(715\) −0.261607 −0.00978355
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.5872i 0.693186i −0.938016 0.346593i \(-0.887339\pi\)
0.938016 0.346593i \(-0.112661\pi\)
\(720\) 0 0
\(721\) −15.9666 −0.594629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.23248i 0.120051i
\(726\) 0 0
\(727\) 0.682442i 0.0253104i 0.999920 + 0.0126552i \(0.00402838\pi\)
−0.999920 + 0.0126552i \(0.995972\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.4392i 0.460082i
\(732\) 0 0
\(733\) 23.5325i 0.869191i −0.900626 0.434596i \(-0.856891\pi\)
0.900626 0.434596i \(-0.143109\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.674826i 0.0248575i
\(738\) 0 0
\(739\) −35.6626 −1.31187 −0.655935 0.754817i \(-0.727726\pi\)
−0.655935 + 0.754817i \(0.727726\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −53.4011 −1.95910 −0.979548 0.201209i \(-0.935513\pi\)
−0.979548 + 0.201209i \(0.935513\pi\)
\(744\) 0 0
\(745\) 0.517635 0.0189647
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.3720i 0.707837i
\(750\) 0 0
\(751\) 27.4014i 0.999891i 0.866057 + 0.499945i \(0.166647\pi\)
−0.866057 + 0.499945i \(0.833353\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 19.1978 0.698681
\(756\) 0 0
\(757\) 17.3599i 0.630955i −0.948933 0.315478i \(-0.897835\pi\)
0.948933 0.315478i \(-0.102165\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36.9950i 1.34107i −0.741879 0.670534i \(-0.766065\pi\)
0.741879 0.670534i \(-0.233935\pi\)
\(762\) 0 0
\(763\) −11.2976 −0.408999
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.2070i 0.729631i
\(768\) 0 0
\(769\) 2.63844i 0.0951447i 0.998868 + 0.0475724i \(0.0151485\pi\)
−0.998868 + 0.0475724i \(0.984852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.74882 −0.0629009 −0.0314504 0.999505i \(-0.510013\pi\)
−0.0314504 + 0.999505i \(0.510013\pi\)
\(774\) 0 0
\(775\) −6.80149 −0.244317
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.7694 −0.600826
\(780\) 0 0
\(781\) 0.0603522i 0.00215957i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 13.7067i 0.489213i
\(786\) 0 0
\(787\) 45.2802i 1.61407i 0.590506 + 0.807033i \(0.298928\pi\)
−0.590506 + 0.807033i \(0.701072\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 57.9928i 2.06199i
\(792\) 0 0
\(793\) 1.30358i 0.0462915i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0454 0.639202 0.319601 0.947552i \(-0.396451\pi\)
0.319601 + 0.947552i \(0.396451\pi\)
\(798\) 0 0
\(799\) 13.7923i 0.487936i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.516006 −0.0182094
\(804\) 0 0
\(805\) −19.1644 8.30922i −0.675456 0.292862i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.7188i 0.552645i −0.961065 0.276322i \(-0.910884\pi\)
0.961065 0.276322i \(-0.0891158\pi\)
\(810\) 0 0
\(811\) −14.6749 −0.515306 −0.257653 0.966238i \(-0.582949\pi\)
−0.257653 + 0.966238i \(0.582949\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.0108 0.525806
\(816\) 0 0
\(817\) 55.8920 1.95541
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 41.8173i 1.45943i −0.683750 0.729717i \(-0.739652\pi\)
0.683750 0.729717i \(-0.260348\pi\)
\(822\) 0 0
\(823\) 48.6855 1.69707 0.848534 0.529140i \(-0.177485\pi\)
0.848534 + 0.529140i \(0.177485\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −44.5025 −1.54750 −0.773752 0.633489i \(-0.781622\pi\)
−0.773752 + 0.633489i \(0.781622\pi\)
\(828\) 0 0
\(829\) −25.2852 −0.878191 −0.439096 0.898440i \(-0.644701\pi\)
−0.439096 + 0.898440i \(0.644701\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14.1438 0.490052
\(834\) 0 0
\(835\) 12.6143i 0.436537i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −33.0296 −1.14031 −0.570154 0.821538i \(-0.693117\pi\)
−0.570154 + 0.821538i \(0.693117\pi\)
\(840\) 0 0
\(841\) 18.5511 0.639693
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −11.0630 −0.380577
\(846\) 0 0
\(847\) 47.8980i 1.64580i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −21.3875 9.27310i −0.733153 0.317878i
\(852\) 0 0
\(853\) 11.6648 0.399394 0.199697 0.979858i \(-0.436004\pi\)
0.199697 + 0.979858i \(0.436004\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 30.5236i 1.04267i −0.853353 0.521333i \(-0.825435\pi\)
0.853353 0.521333i \(-0.174565\pi\)
\(858\) 0 0
\(859\) 33.7320 1.15092 0.575460 0.817830i \(-0.304823\pi\)
0.575460 + 0.817830i \(0.304823\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.442368i 0.0150584i −0.999972 0.00752920i \(-0.997603\pi\)
0.999972 0.00752920i \(-0.00239664\pi\)
\(864\) 0 0
\(865\) 4.49207i 0.152735i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.173671i 0.00589139i
\(870\) 0 0
\(871\) 62.0714i 2.10321i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.35549i 0.147243i
\(876\) 0 0
\(877\) 23.3839 0.789618 0.394809 0.918763i \(-0.370811\pi\)
0.394809 + 0.918763i \(0.370811\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.6795 0.663019 0.331509 0.943452i \(-0.392442\pi\)
0.331509 + 0.943452i \(0.392442\pi\)
\(882\) 0 0
\(883\) −24.1161 −0.811571 −0.405785 0.913968i \(-0.633002\pi\)
−0.405785 + 0.913968i \(0.633002\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.3880i 0.483103i 0.970388 + 0.241552i \(0.0776563\pi\)
−0.970388 + 0.241552i \(0.922344\pi\)
\(888\) 0 0
\(889\) 93.7522i 3.14435i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −61.9716 −2.07380
\(894\) 0 0
\(895\) 9.66095i 0.322930i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 21.9856i 0.733262i
\(900\) 0 0
\(901\) −7.32964 −0.244186
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.575728i 0.0191378i
\(906\) 0 0
\(907\) 0.153589i 0.00509983i 0.999997 + 0.00254992i \(0.000811665\pi\)
−0.999997 + 0.00254992i \(0.999188\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.89188 −0.0626809 −0.0313405 0.999509i \(-0.509978\pi\)
−0.0313405 + 0.999509i \(0.509978\pi\)
\(912\) 0 0
\(913\) −0.954495 −0.0315892
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −40.8053 −1.34751
\(918\) 0 0
\(919\) 38.3096i 1.26372i 0.775084 + 0.631859i \(0.217708\pi\)
−0.775084 + 0.631859i \(0.782292\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.55127i 0.182722i
\(924\) 0 0
\(925\) 4.86073i 0.159820i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.4363i 0.736113i 0.929804 + 0.368056i \(0.119977\pi\)
−0.929804 + 0.368056i \(0.880023\pi\)
\(930\) 0 0
\(931\) 63.5508i 2.08279i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.0630135 −0.00206076
\(936\) 0 0
\(937\) 17.3629i 0.567221i 0.958940 + 0.283610i \(0.0915322\pi\)
−0.958940 + 0.283610i \(0.908468\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.9128 −0.420946 −0.210473 0.977600i \(-0.567500\pi\)
−0.210473 + 0.977600i \(0.567500\pi\)
\(942\) 0 0
\(943\) −13.8983 6.02595i −0.452589 0.196232i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 43.8747i 1.42574i −0.701297 0.712869i \(-0.747395\pi\)
0.701297 0.712869i \(-0.252605\pi\)
\(948\) 0 0
\(949\) −47.4629 −1.54071
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −48.0731 −1.55724 −0.778621 0.627495i \(-0.784080\pi\)
−0.778621 + 0.627495i \(0.784080\pi\)
\(954\) 0 0
\(955\) 17.8620 0.578001
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 14.3315i 0.462790i
\(960\) 0 0
\(961\) 15.2602 0.492265
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.760186 −0.0244713
\(966\) 0 0
\(967\) 10.9967 0.353630 0.176815 0.984244i \(-0.443421\pi\)
0.176815 + 0.984244i \(0.443421\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −47.4613 −1.52311 −0.761553 0.648103i \(-0.775563\pi\)
−0.761553 + 0.648103i \(0.775563\pi\)
\(972\) 0 0
\(973\) 11.6945i 0.374910i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −10.8494 −0.347102 −0.173551 0.984825i \(-0.555524\pi\)
−0.173551 + 0.984825i \(0.555524\pi\)
\(978\) 0 0
\(979\) −0.647566 −0.0206963
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −53.0065 −1.69064 −0.845322 0.534257i \(-0.820591\pi\)
−0.845322 + 0.534257i \(0.820591\pi\)
\(984\) 0 0
\(985\) 23.2534i 0.740915i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.3226 + 20.0843i 1.47297 + 0.638645i
\(990\) 0 0
\(991\) −36.1662 −1.14886 −0.574429 0.818555i \(-0.694776\pi\)
−0.574429 + 0.818555i \(0.694776\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.2715i 0.452436i
\(996\) 0 0
\(997\) −15.7675 −0.499363 −0.249681 0.968328i \(-0.580326\pi\)
−0.249681 + 0.968328i \(0.580326\pi\)
\(998\) 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 4140.2.i.a.1241.15 yes 16
3.2 odd 2 4140.2.i.b.1241.15 yes 16
23.22 odd 2 4140.2.i.b.1241.2 yes 16
69.68 even 2 inner 4140.2.i.a.1241.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.i.a.1241.2 16 69.68 even 2 inner
4140.2.i.a.1241.15 yes 16 1.1 even 1 trivial
4140.2.i.b.1241.2 yes 16 23.22 odd 2
4140.2.i.b.1241.15 yes 16 3.2 odd 2