Properties

Label 1520.2.g.f.1519.3
Level $1520$
Weight $2$
Character 1520.1519
Analytic conductor $12.137$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(1519,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.1519");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.g (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: \(12.1372611072\)
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: \( x^{16} - 32x^{14} + 380x^{12} - 1752x^{10} + 1904x^{8} + 7824x^{6} + 7352x^{4} + 2992x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1519.3
Root \(3.52761 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1520.1519
Dual form 1520.2.g.f.1519.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.59462i q^{3} +(-1.73205 + 1.41421i) q^{5} -3.38587 q^{7} -3.73205 q^{9} +O(q^{10})\) \(q-2.59462i q^{3} +(-1.73205 + 1.41421i) q^{5} -3.38587 q^{7} -3.73205 q^{9} +1.75265i q^{11} +6.21196 q^{13} +(3.66935 + 4.49401i) q^{15} +6.69213i q^{17} +(-1.34307 - 4.14682i) q^{19} +8.78504i q^{21} +5.86450 q^{23} +(1.00000 - 4.89898i) q^{25} +1.89939i q^{27} +6.43110i q^{29} -6.35549 q^{31} +4.54747 q^{33} +(5.86450 - 4.78834i) q^{35} +1.66449 q^{37} -16.1177i q^{39} -8.78504i q^{41} -0.907241 q^{43} +(6.46410 - 5.27792i) q^{45} +10.1576 q^{47} +4.46410 q^{49} +17.3635 q^{51} +10.7594 q^{53} +(-2.47863 - 3.03569i) q^{55} +(-10.7594 + 3.48477i) q^{57} +6.35549 q^{59} -2.19615 q^{61} +12.6362 q^{63} +(-10.7594 + 8.78504i) q^{65} +4.49401i q^{67} -15.2161i q^{69} -1.70295 q^{71} -3.58630i q^{73} +(-12.7110 - 2.59462i) q^{75} -5.93426i q^{77} +16.3803 q^{79} -6.26795 q^{81} -1.57139 q^{83} +(-9.46410 - 11.5911i) q^{85} +16.6862 q^{87} -6.43110i q^{89} -21.0329 q^{91} +16.4901i q^{93} +(8.19077 + 5.28312i) q^{95} -1.66449 q^{97} -6.54099i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{9} + 16 q^{25} + 48 q^{45} + 16 q^{49} + 48 q^{61} - 128 q^{81} - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 2.59462i 1.49800i −0.662568 0.749002i \(-0.730533\pi\)
0.662568 0.749002i \(-0.269467\pi\)
\(4\) 0 0
\(5\) −1.73205 + 1.41421i −0.774597 + 0.632456i
\(6\) 0 0
\(7\) −3.38587 −1.27974 −0.639869 0.768484i \(-0.721011\pi\)
−0.639869 + 0.768484i \(0.721011\pi\)
\(8\) 0 0
\(9\) −3.73205 −1.24402
\(10\) 0 0
\(11\) 1.75265i 0.528445i 0.964462 + 0.264223i \(0.0851153\pi\)
−0.964462 + 0.264223i \(0.914885\pi\)
\(12\) 0 0
\(13\) 6.21196 1.72289 0.861444 0.507853i \(-0.169561\pi\)
0.861444 + 0.507853i \(0.169561\pi\)
\(14\) 0 0
\(15\) 3.66935 + 4.49401i 0.947421 + 1.16035i
\(16\) 0 0
\(17\) 6.69213i 1.62308i 0.584297 + 0.811540i \(0.301370\pi\)
−0.584297 + 0.811540i \(0.698630\pi\)
\(18\) 0 0
\(19\) −1.34307 4.14682i −0.308122 0.951347i
\(20\) 0 0
\(21\) 8.78504i 1.91705i
\(22\) 0 0
\(23\) 5.86450 1.22283 0.611416 0.791309i \(-0.290600\pi\)
0.611416 + 0.791309i \(0.290600\pi\)
\(24\) 0 0
\(25\) 1.00000 4.89898i 0.200000 0.979796i
\(26\) 0 0
\(27\) 1.89939i 0.365538i
\(28\) 0 0
\(29\) 6.43110i 1.19422i 0.802158 + 0.597112i \(0.203685\pi\)
−0.802158 + 0.597112i \(0.796315\pi\)
\(30\) 0 0
\(31\) −6.35549 −1.14148 −0.570740 0.821131i \(-0.693344\pi\)
−0.570740 + 0.821131i \(0.693344\pi\)
\(32\) 0 0
\(33\) 4.54747 0.791613
\(34\) 0 0
\(35\) 5.86450 4.78834i 0.991281 0.809377i
\(36\) 0 0
\(37\) 1.66449 0.273640 0.136820 0.990596i \(-0.456312\pi\)
0.136820 + 0.990596i \(0.456312\pi\)
\(38\) 0 0
\(39\) 16.1177i 2.58089i
\(40\) 0 0
\(41\) 8.78504i 1.37199i −0.727605 0.685996i \(-0.759367\pi\)
0.727605 0.685996i \(-0.240633\pi\)
\(42\) 0 0
\(43\) −0.907241 −0.138353 −0.0691764 0.997604i \(-0.522037\pi\)
−0.0691764 + 0.997604i \(0.522037\pi\)
\(44\) 0 0
\(45\) 6.46410 5.27792i 0.963611 0.786785i
\(46\) 0 0
\(47\) 10.1576 1.48164 0.740819 0.671704i \(-0.234437\pi\)
0.740819 + 0.671704i \(0.234437\pi\)
\(48\) 0 0
\(49\) 4.46410 0.637729
\(50\) 0 0
\(51\) 17.3635 2.43138
\(52\) 0 0
\(53\) 10.7594 1.47792 0.738961 0.673748i \(-0.235317\pi\)
0.738961 + 0.673748i \(0.235317\pi\)
\(54\) 0 0
\(55\) −2.47863 3.03569i −0.334218 0.409332i
\(56\) 0 0
\(57\) −10.7594 + 3.48477i −1.42512 + 0.461569i
\(58\) 0 0
\(59\) 6.35549 0.827415 0.413707 0.910410i \(-0.364234\pi\)
0.413707 + 0.910410i \(0.364234\pi\)
\(60\) 0 0
\(61\) −2.19615 −0.281189 −0.140594 0.990067i \(-0.544901\pi\)
−0.140594 + 0.990067i \(0.544901\pi\)
\(62\) 0 0
\(63\) 12.6362 1.59202
\(64\) 0 0
\(65\) −10.7594 + 8.78504i −1.33454 + 1.08965i
\(66\) 0 0
\(67\) 4.49401i 0.549031i 0.961583 + 0.274516i \(0.0885175\pi\)
−0.961583 + 0.274516i \(0.911483\pi\)
\(68\) 0 0
\(69\) 15.2161i 1.83181i
\(70\) 0 0
\(71\) −1.70295 −0.202103 −0.101051 0.994881i \(-0.532221\pi\)
−0.101051 + 0.994881i \(0.532221\pi\)
\(72\) 0 0
\(73\) 3.58630i 0.419745i −0.977729 0.209872i \(-0.932695\pi\)
0.977729 0.209872i \(-0.0673049\pi\)
\(74\) 0 0
\(75\) −12.7110 2.59462i −1.46774 0.299601i
\(76\) 0 0
\(77\) 5.93426i 0.676271i
\(78\) 0 0
\(79\) 16.3803 1.84293 0.921466 0.388459i \(-0.126993\pi\)
0.921466 + 0.388459i \(0.126993\pi\)
\(80\) 0 0
\(81\) −6.26795 −0.696439
\(82\) 0 0
\(83\) −1.57139 −0.172482 −0.0862411 0.996274i \(-0.527486\pi\)
−0.0862411 + 0.996274i \(0.527486\pi\)
\(84\) 0 0
\(85\) −9.46410 11.5911i −1.02653 1.25723i
\(86\) 0 0
\(87\) 16.6862 1.78895
\(88\) 0 0
\(89\) 6.43110i 0.681695i −0.940119 0.340847i \(-0.889286\pi\)
0.940119 0.340847i \(-0.110714\pi\)
\(90\) 0 0
\(91\) −21.0329 −2.20484
\(92\) 0 0
\(93\) 16.4901i 1.70994i
\(94\) 0 0
\(95\) 8.19077 + 5.28312i 0.840355 + 0.542036i
\(96\) 0 0
\(97\) −1.66449 −0.169003 −0.0845017 0.996423i \(-0.526930\pi\)
−0.0845017 + 0.996423i \(0.526930\pi\)
\(98\) 0 0
\(99\) 6.54099i 0.657395i
\(100\) 0 0
\(101\) 8.19615 0.815548 0.407774 0.913083i \(-0.366305\pi\)
0.407774 + 0.913083i \(0.366305\pi\)
\(102\) 0 0
\(103\) 13.4820i 1.32842i 0.747544 + 0.664212i \(0.231233\pi\)
−0.747544 + 0.664212i \(0.768767\pi\)
\(104\) 0 0
\(105\) −12.4239 15.2161i −1.21245 1.48494i
\(106\) 0 0
\(107\) 9.68325i 0.936115i 0.883698 + 0.468058i \(0.155046\pi\)
−0.883698 + 0.468058i \(0.844954\pi\)
\(108\) 0 0
\(109\) 15.2161i 1.45744i 0.684811 + 0.728721i \(0.259885\pi\)
−0.684811 + 0.728721i \(0.740115\pi\)
\(110\) 0 0
\(111\) 4.31872i 0.409915i
\(112\) 0 0
\(113\) 2.88298 0.271208 0.135604 0.990763i \(-0.456703\pi\)
0.135604 + 0.990763i \(0.456703\pi\)
\(114\) 0 0
\(115\) −10.1576 + 8.29365i −0.947201 + 0.773387i
\(116\) 0 0
\(117\) −23.1834 −2.14330
\(118\) 0 0
\(119\) 22.6587i 2.07712i
\(120\) 0 0
\(121\) 7.92820 0.720746
\(122\) 0 0
\(123\) −22.7938 −2.05525
\(124\) 0 0
\(125\) 5.19615 + 9.89949i 0.464758 + 0.885438i
\(126\) 0 0
\(127\) 16.7719i 1.48826i −0.668033 0.744132i \(-0.732863\pi\)
0.668033 0.744132i \(-0.267137\pi\)
\(128\) 0 0
\(129\) 2.35394i 0.207253i
\(130\) 0 0
\(131\) 13.0820i 1.14298i −0.820609 0.571489i \(-0.806366\pi\)
0.820609 0.571489i \(-0.193634\pi\)
\(132\) 0 0
\(133\) 4.54747 + 14.0406i 0.394316 + 1.21747i
\(134\) 0 0
\(135\) −2.68615 3.28985i −0.231187 0.283145i
\(136\) 0 0
\(137\) 0.277401i 0.0237000i 0.999930 + 0.0118500i \(0.00377206\pi\)
−0.999930 + 0.0118500i \(0.996228\pi\)
\(138\) 0 0
\(139\) 17.4007i 1.47591i 0.674851 + 0.737954i \(0.264208\pi\)
−0.674851 + 0.737954i \(0.735792\pi\)
\(140\) 0 0
\(141\) 26.3551i 2.21950i
\(142\) 0 0
\(143\) 10.8874i 0.910452i
\(144\) 0 0
\(145\) −9.09494 11.1390i −0.755294 0.925042i
\(146\) 0 0
\(147\) 11.5826i 0.955320i
\(148\) 0 0
\(149\) −11.6603 −0.955245 −0.477623 0.878565i \(-0.658501\pi\)
−0.477623 + 0.878565i \(0.658501\pi\)
\(150\) 0 0
\(151\) 17.3635 1.41302 0.706512 0.707701i \(-0.250268\pi\)
0.706512 + 0.707701i \(0.250268\pi\)
\(152\) 0 0
\(153\) 24.9754i 2.01914i
\(154\) 0 0
\(155\) 11.0080 8.98803i 0.884187 0.721936i
\(156\) 0 0
\(157\) 1.79315i 0.143109i 0.997437 + 0.0715545i \(0.0227960\pi\)
−0.997437 + 0.0715545i \(0.977204\pi\)
\(158\) 0 0
\(159\) 27.9166i 2.21393i
\(160\) 0 0
\(161\) −19.8564 −1.56490
\(162\) 0 0
\(163\) 12.6362 0.989746 0.494873 0.868965i \(-0.335215\pi\)
0.494873 + 0.868965i \(0.335215\pi\)
\(164\) 0 0
\(165\) −7.87645 + 6.43110i −0.613181 + 0.500660i
\(166\) 0 0
\(167\) 10.1922i 0.788696i 0.918961 + 0.394348i \(0.129029\pi\)
−0.918961 + 0.394348i \(0.870971\pi\)
\(168\) 0 0
\(169\) 25.5885 1.96834
\(170\) 0 0
\(171\) 5.01242 + 15.4762i 0.383309 + 1.18349i
\(172\) 0 0
\(173\) 18.6359 1.41686 0.708430 0.705781i \(-0.249404\pi\)
0.708430 + 0.705781i \(0.249404\pi\)
\(174\) 0 0
\(175\) −3.38587 + 16.5873i −0.255948 + 1.25388i
\(176\) 0 0
\(177\) 16.4901i 1.23947i
\(178\) 0 0
\(179\) −15.6606 −1.17053 −0.585263 0.810843i \(-0.699009\pi\)
−0.585263 + 0.810843i \(0.699009\pi\)
\(180\) 0 0
\(181\) 11.1390i 0.827954i 0.910287 + 0.413977i \(0.135861\pi\)
−0.910287 + 0.413977i \(0.864139\pi\)
\(182\) 0 0
\(183\) 5.69818i 0.421222i
\(184\) 0 0
\(185\) −2.88298 + 2.35394i −0.211961 + 0.173065i
\(186\) 0 0
\(187\) −11.7290 −0.857709
\(188\) 0 0
\(189\) 6.43110i 0.467793i
\(190\) 0 0
\(191\) 1.28303i 0.0928369i −0.998922 0.0464185i \(-0.985219\pi\)
0.998922 0.0464185i \(-0.0147808\pi\)
\(192\) 0 0
\(193\) 14.0884 1.01411 0.507053 0.861915i \(-0.330735\pi\)
0.507053 + 0.861915i \(0.330735\pi\)
\(194\) 0 0
\(195\) 22.7938 + 27.9166i 1.63230 + 1.99915i
\(196\) 0 0
\(197\) 1.79315i 0.127757i 0.997958 + 0.0638784i \(0.0203470\pi\)
−0.997958 + 0.0638784i \(0.979653\pi\)
\(198\) 0 0
\(199\) 2.22228i 0.157533i −0.996893 0.0787665i \(-0.974902\pi\)
0.996893 0.0787665i \(-0.0250982\pi\)
\(200\) 0 0
\(201\) 11.6603 0.822451
\(202\) 0 0
\(203\) 21.7748i 1.52829i
\(204\) 0 0
\(205\) 12.4239 + 15.2161i 0.867724 + 1.06274i
\(206\) 0 0
\(207\) −21.8866 −1.52122
\(208\) 0 0
\(209\) 7.26795 2.35394i 0.502735 0.162826i
\(210\) 0 0
\(211\) 2.68615 0.184922 0.0924610 0.995716i \(-0.470527\pi\)
0.0924610 + 0.995716i \(0.470527\pi\)
\(212\) 0 0
\(213\) 4.41851i 0.302751i
\(214\) 0 0
\(215\) 1.57139 1.28303i 0.107168 0.0875021i
\(216\) 0 0
\(217\) 21.5189 1.46080
\(218\) 0 0
\(219\) −9.30509 −0.628780
\(220\) 0 0
\(221\) 41.5713i 2.79639i
\(222\) 0 0
\(223\) 19.1802i 1.28440i −0.766536 0.642201i \(-0.778021\pi\)
0.766536 0.642201i \(-0.221979\pi\)
\(224\) 0 0
\(225\) −3.73205 + 18.2832i −0.248803 + 1.21888i
\(226\) 0 0
\(227\) 11.0737i 0.734988i 0.930026 + 0.367494i \(0.119784\pi\)
−0.930026 + 0.367494i \(0.880216\pi\)
\(228\) 0 0
\(229\) −15.1244 −0.999446 −0.499723 0.866185i \(-0.666565\pi\)
−0.499723 + 0.866185i \(0.666565\pi\)
\(230\) 0 0
\(231\) −15.3971 −1.01306
\(232\) 0 0
\(233\) 8.76268i 0.574062i −0.957921 0.287031i \(-0.907332\pi\)
0.957921 0.287031i \(-0.0926683\pi\)
\(234\) 0 0
\(235\) −17.5935 + 14.3650i −1.14767 + 0.937071i
\(236\) 0 0
\(237\) 42.5007i 2.76072i
\(238\) 0 0
\(239\) 7.01062i 0.453479i −0.973955 0.226740i \(-0.927193\pi\)
0.973955 0.226740i \(-0.0728066\pi\)
\(240\) 0 0
\(241\) 15.2161i 0.980157i 0.871678 + 0.490079i \(0.163032\pi\)
−0.871678 + 0.490079i \(0.836968\pi\)
\(242\) 0 0
\(243\) 21.9611i 1.40881i
\(244\) 0 0
\(245\) −7.73205 + 6.31319i −0.493983 + 0.403335i
\(246\) 0 0
\(247\) −8.34312 25.7599i −0.530860 1.63906i
\(248\) 0 0
\(249\) 4.07715i 0.258379i
\(250\) 0 0
\(251\) 7.35440i 0.464206i 0.972691 + 0.232103i \(0.0745606\pi\)
−0.972691 + 0.232103i \(0.925439\pi\)
\(252\) 0 0
\(253\) 10.2784i 0.646200i
\(254\) 0 0
\(255\) −30.0745 + 24.5557i −1.88334 + 1.53774i
\(256\) 0 0
\(257\) 26.5123 1.65379 0.826897 0.562353i \(-0.190104\pi\)
0.826897 + 0.562353i \(0.190104\pi\)
\(258\) 0 0
\(259\) −5.63574 −0.350188
\(260\) 0 0
\(261\) 24.0012i 1.48564i
\(262\) 0 0
\(263\) −14.4507 −0.891069 −0.445535 0.895265i \(-0.646986\pi\)
−0.445535 + 0.895265i \(0.646986\pi\)
\(264\) 0 0
\(265\) −18.6359 + 15.2161i −1.14479 + 0.934720i
\(266\) 0 0
\(267\) −16.6862 −1.02118
\(268\) 0 0
\(269\) 8.78504i 0.535633i 0.963470 + 0.267817i \(0.0863021\pi\)
−0.963470 + 0.267817i \(0.913698\pi\)
\(270\) 0 0
\(271\) 17.4007i 1.05702i 0.848928 + 0.528509i \(0.177249\pi\)
−0.848928 + 0.528509i \(0.822751\pi\)
\(272\) 0 0
\(273\) 54.5723i 3.30287i
\(274\) 0 0
\(275\) 8.58622 + 1.75265i 0.517768 + 0.105689i
\(276\) 0 0
\(277\) 18.7637i 1.12740i 0.825979 + 0.563701i \(0.190623\pi\)
−0.825979 + 0.563701i \(0.809377\pi\)
\(278\) 0 0
\(279\) 23.7190 1.42002
\(280\) 0 0
\(281\) 2.35394i 0.140425i −0.997532 0.0702123i \(-0.977632\pi\)
0.997532 0.0702123i \(-0.0223677\pi\)
\(282\) 0 0
\(283\) −6.52864 −0.388087 −0.194044 0.980993i \(-0.562160\pi\)
−0.194044 + 0.980993i \(0.562160\pi\)
\(284\) 0 0
\(285\) 13.7077 21.2519i 0.811973 1.25886i
\(286\) 0 0
\(287\) 29.7450i 1.75579i
\(288\) 0 0
\(289\) −27.7846 −1.63439
\(290\) 0 0
\(291\) 4.31872i 0.253168i
\(292\) 0 0
\(293\) −10.7594 −0.628573 −0.314286 0.949328i \(-0.601765\pi\)
−0.314286 + 0.949328i \(0.601765\pi\)
\(294\) 0 0
\(295\) −11.0080 + 8.98803i −0.640913 + 0.523303i
\(296\) 0 0
\(297\) −3.32898 −0.193167
\(298\) 0 0
\(299\) 36.4300 2.10680
\(300\) 0 0
\(301\) 3.07180 0.177055
\(302\) 0 0
\(303\) 21.2659i 1.22169i
\(304\) 0 0
\(305\) 3.80385 3.10583i 0.217808 0.177839i
\(306\) 0 0
\(307\) 15.8904i 0.906911i −0.891279 0.453456i \(-0.850191\pi\)
0.891279 0.453456i \(-0.149809\pi\)
\(308\) 0 0
\(309\) 34.9808 1.98999
\(310\) 0 0
\(311\) 25.3506i 1.43750i 0.695269 + 0.718749i \(0.255285\pi\)
−0.695269 + 0.718749i \(0.744715\pi\)
\(312\) 0 0
\(313\) 8.00481i 0.452459i −0.974074 0.226229i \(-0.927360\pi\)
0.974074 0.226229i \(-0.0726399\pi\)
\(314\) 0 0
\(315\) −21.8866 + 17.8703i −1.23317 + 1.00688i
\(316\) 0 0
\(317\) 18.6359 1.04670 0.523348 0.852119i \(-0.324683\pi\)
0.523348 + 0.852119i \(0.324683\pi\)
\(318\) 0 0
\(319\) −11.2715 −0.631082
\(320\) 0 0
\(321\) 25.1244 1.40230
\(322\) 0 0
\(323\) 27.7511 8.98803i 1.54411 0.500107i
\(324\) 0 0
\(325\) 6.21196 30.4323i 0.344578 1.68808i
\(326\) 0 0
\(327\) 39.4801 2.18325
\(328\) 0 0
\(329\) −34.3923 −1.89611
\(330\) 0 0
\(331\) −9.30509 −0.511454 −0.255727 0.966749i \(-0.582315\pi\)
−0.255727 + 0.966749i \(0.582315\pi\)
\(332\) 0 0
\(333\) −6.21196 −0.340413
\(334\) 0 0
\(335\) −6.35549 7.78386i −0.347238 0.425278i
\(336\) 0 0
\(337\) 15.3069 0.833820 0.416910 0.908948i \(-0.363113\pi\)
0.416910 + 0.908948i \(0.363113\pi\)
\(338\) 0 0
\(339\) 7.48024i 0.406271i
\(340\) 0 0
\(341\) 11.1390i 0.603210i
\(342\) 0 0
\(343\) 8.58622 0.463612
\(344\) 0 0
\(345\) 21.5189 + 26.3551i 1.15854 + 1.41891i
\(346\) 0 0
\(347\) −5.86450 −0.314823 −0.157411 0.987533i \(-0.550315\pi\)
−0.157411 + 0.987533i \(0.550315\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 11.7990i 0.629782i
\(352\) 0 0
\(353\) 31.9449i 1.70026i −0.526576 0.850128i \(-0.676525\pi\)
0.526576 0.850128i \(-0.323475\pi\)
\(354\) 0 0
\(355\) 2.94960 2.40833i 0.156548 0.127821i
\(356\) 0 0
\(357\) −58.7906 −3.11153
\(358\) 0 0
\(359\) 14.8346i 0.782943i −0.920190 0.391471i \(-0.871966\pi\)
0.920190 0.391471i \(-0.128034\pi\)
\(360\) 0 0
\(361\) −15.3923 + 11.1390i −0.810121 + 0.586262i
\(362\) 0 0
\(363\) 20.5707i 1.07968i
\(364\) 0 0
\(365\) 5.07180 + 6.21166i 0.265470 + 0.325133i
\(366\) 0 0
\(367\) −24.3652 −1.27185 −0.635927 0.771749i \(-0.719382\pi\)
−0.635927 + 0.771749i \(0.719382\pi\)
\(368\) 0 0
\(369\) 32.7862i 1.70678i
\(370\) 0 0
\(371\) −36.4300 −1.89135
\(372\) 0 0
\(373\) −1.66449 −0.0861840 −0.0430920 0.999071i \(-0.513721\pi\)
−0.0430920 + 0.999071i \(0.513721\pi\)
\(374\) 0 0
\(375\) 25.6854 13.4820i 1.32639 0.696209i
\(376\) 0 0
\(377\) 39.9497i 2.05751i
\(378\) 0 0
\(379\) 8.05844 0.413934 0.206967 0.978348i \(-0.433641\pi\)
0.206967 + 0.978348i \(0.433641\pi\)
\(380\) 0 0
\(381\) −43.5167 −2.22943
\(382\) 0 0
\(383\) 9.68325i 0.494791i −0.968915 0.247396i \(-0.920425\pi\)
0.968915 0.247396i \(-0.0795747\pi\)
\(384\) 0 0
\(385\) 8.39230 + 10.2784i 0.427711 + 0.523837i
\(386\) 0 0
\(387\) 3.38587 0.172113
\(388\) 0 0
\(389\) 21.4641 1.08827 0.544137 0.838997i \(-0.316857\pi\)
0.544137 + 0.838997i \(0.316857\pi\)
\(390\) 0 0
\(391\) 39.2460i 1.98475i
\(392\) 0 0
\(393\) −33.9428 −1.71219
\(394\) 0 0
\(395\) −28.3716 + 23.1653i −1.42753 + 1.16557i
\(396\) 0 0
\(397\) 1.31268i 0.0658814i 0.999457 + 0.0329407i \(0.0104872\pi\)
−0.999457 + 0.0329407i \(0.989513\pi\)
\(398\) 0 0
\(399\) 36.4300 11.7990i 1.82378 0.590687i
\(400\) 0 0
\(401\) 12.8622i 0.642307i −0.947027 0.321154i \(-0.895929\pi\)
0.947027 0.321154i \(-0.104071\pi\)
\(402\) 0 0
\(403\) −39.4801 −1.96664
\(404\) 0 0
\(405\) 10.8564 8.86422i 0.539459 0.440467i
\(406\) 0 0
\(407\) 2.91728i 0.144604i
\(408\) 0 0
\(409\) 26.3551i 1.30318i −0.758573 0.651588i \(-0.774103\pi\)
0.758573 0.651588i \(-0.225897\pi\)
\(410\) 0 0
\(411\) 0.719751 0.0355027
\(412\) 0 0
\(413\) −21.5189 −1.05887
\(414\) 0 0
\(415\) 2.72172 2.22228i 0.133604 0.109087i
\(416\) 0 0
\(417\) 45.1482 2.21092
\(418\) 0 0
\(419\) 20.0926i 0.981588i −0.871276 0.490794i \(-0.836707\pi\)
0.871276 0.490794i \(-0.163293\pi\)
\(420\) 0 0
\(421\) 26.3551i 1.28447i −0.766508 0.642235i \(-0.778007\pi\)
0.766508 0.642235i \(-0.221993\pi\)
\(422\) 0 0
\(423\) −37.9087 −1.84318
\(424\) 0 0
\(425\) 32.7846 + 6.69213i 1.59029 + 0.324616i
\(426\) 0 0
\(427\) 7.43588 0.359848
\(428\) 0 0
\(429\) 28.2487 1.36386
\(430\) 0 0
\(431\) −20.7694 −1.00043 −0.500214 0.865902i \(-0.666745\pi\)
−0.500214 + 0.865902i \(0.666745\pi\)
\(432\) 0 0
\(433\) −15.3069 −0.735603 −0.367801 0.929904i \(-0.619889\pi\)
−0.367801 + 0.929904i \(0.619889\pi\)
\(434\) 0 0
\(435\) −28.9014 + 23.5979i −1.38572 + 1.13143i
\(436\) 0 0
\(437\) −7.87645 24.3190i −0.376782 1.16334i
\(438\) 0 0
\(439\) −38.3964 −1.83256 −0.916280 0.400537i \(-0.868823\pi\)
−0.916280 + 0.400537i \(0.868823\pi\)
\(440\) 0 0
\(441\) −16.6603 −0.793345
\(442\) 0 0
\(443\) 5.86450 0.278631 0.139315 0.990248i \(-0.455510\pi\)
0.139315 + 0.990248i \(0.455510\pi\)
\(444\) 0 0
\(445\) 9.09494 + 11.1390i 0.431142 + 0.528038i
\(446\) 0 0
\(447\) 30.2539i 1.43096i
\(448\) 0 0
\(449\) 2.35394i 0.111089i 0.998456 + 0.0555447i \(0.0176895\pi\)
−0.998456 + 0.0555447i \(0.982310\pi\)
\(450\) 0 0
\(451\) 15.3971 0.725023
\(452\) 0 0
\(453\) 45.0518i 2.11672i
\(454\) 0 0
\(455\) 36.4300 29.7450i 1.70787 1.39447i
\(456\) 0 0
\(457\) 25.4558i 1.19077i −0.803439 0.595387i \(-0.796999\pi\)
0.803439 0.595387i \(-0.203001\pi\)
\(458\) 0 0
\(459\) −12.7110 −0.593298
\(460\) 0 0
\(461\) 24.2487 1.12938 0.564688 0.825305i \(-0.308997\pi\)
0.564688 + 0.825305i \(0.308997\pi\)
\(462\) 0 0
\(463\) −9.49346 −0.441198 −0.220599 0.975365i \(-0.570801\pi\)
−0.220599 + 0.975365i \(0.570801\pi\)
\(464\) 0 0
\(465\) −23.3205 28.5617i −1.08146 1.32452i
\(466\) 0 0
\(467\) −17.5935 −0.814129 −0.407065 0.913399i \(-0.633448\pi\)
−0.407065 + 0.913399i \(0.633448\pi\)
\(468\) 0 0
\(469\) 15.2161i 0.702616i
\(470\) 0 0
\(471\) 4.65254 0.214378
\(472\) 0 0
\(473\) 1.59008i 0.0731119i
\(474\) 0 0
\(475\) −21.6583 + 2.43287i −0.993750 + 0.111628i
\(476\) 0 0
\(477\) −40.1547 −1.83856
\(478\) 0 0
\(479\) 1.75265i 0.0800808i −0.999198 0.0400404i \(-0.987251\pi\)
0.999198 0.0400404i \(-0.0127487\pi\)
\(480\) 0 0
\(481\) 10.3397 0.471452
\(482\) 0 0
\(483\) 51.5198i 2.34423i
\(484\) 0 0
\(485\) 2.88298 2.35394i 0.130909 0.106887i
\(486\) 0 0
\(487\) 26.6414i 1.20724i 0.797273 + 0.603619i \(0.206275\pi\)
−0.797273 + 0.603619i \(0.793725\pi\)
\(488\) 0 0
\(489\) 32.7862i 1.48264i
\(490\) 0 0
\(491\) 37.9629i 1.71324i 0.515945 + 0.856622i \(0.327441\pi\)
−0.515945 + 0.856622i \(0.672559\pi\)
\(492\) 0 0
\(493\) −43.0377 −1.93832
\(494\) 0 0
\(495\) 9.25036 + 11.3293i 0.415773 + 0.509216i
\(496\) 0 0
\(497\) 5.76596 0.258639
\(498\) 0 0
\(499\) 17.4007i 0.778963i 0.921034 + 0.389481i \(0.127346\pi\)
−0.921034 + 0.389481i \(0.872654\pi\)
\(500\) 0 0
\(501\) 26.4449 1.18147
\(502\) 0 0
\(503\) −14.4507 −0.644325 −0.322163 0.946684i \(-0.604410\pi\)
−0.322163 + 0.946684i \(0.604410\pi\)
\(504\) 0 0
\(505\) −14.1962 + 11.5911i −0.631720 + 0.515798i
\(506\) 0 0
\(507\) 66.3923i 2.94859i
\(508\) 0 0
\(509\) 35.1402i 1.55756i −0.627297 0.778780i \(-0.715839\pi\)
0.627297 0.778780i \(-0.284161\pi\)
\(510\) 0 0
\(511\) 12.1427i 0.537163i
\(512\) 0 0
\(513\) 7.87645 2.55103i 0.347754 0.112631i
\(514\) 0 0
\(515\) −19.0665 23.3516i −0.840170 1.02899i
\(516\) 0 0
\(517\) 17.8028i 0.782965i
\(518\) 0 0
\(519\) 48.3530i 2.12246i
\(520\) 0 0
\(521\) 6.43110i 0.281751i −0.990027 0.140876i \(-0.955008\pi\)
0.990027 0.140876i \(-0.0449918\pi\)
\(522\) 0 0
\(523\) 16.7719i 0.733383i 0.930343 + 0.366692i \(0.119510\pi\)
−0.930343 + 0.366692i \(0.880490\pi\)
\(524\) 0 0
\(525\) 43.0377 + 8.78504i 1.87832 + 0.383411i
\(526\) 0 0
\(527\) 42.5318i 1.85271i
\(528\) 0 0
\(529\) 11.3923 0.495318
\(530\) 0 0
\(531\) −23.7190 −1.02932
\(532\) 0 0
\(533\) 54.5723i 2.36379i
\(534\) 0 0
\(535\) −13.6942 16.7719i −0.592051 0.725112i
\(536\) 0 0
\(537\) 40.6333i 1.75345i
\(538\) 0 0
\(539\) 7.82403i 0.337005i
\(540\) 0 0
\(541\) 12.1962 0.524354 0.262177 0.965020i \(-0.415560\pi\)
0.262177 + 0.965020i \(0.415560\pi\)
\(542\) 0 0
\(543\) 28.9014 1.24028
\(544\) 0 0
\(545\) −21.5189 26.3551i −0.921767 1.12893i
\(546\) 0 0
\(547\) 3.61250i 0.154459i 0.997013 + 0.0772297i \(0.0246075\pi\)
−0.997013 + 0.0772297i \(0.975393\pi\)
\(548\) 0 0
\(549\) 8.19615 0.349803
\(550\) 0 0
\(551\) 26.6686 8.63744i 1.13612 0.367967i
\(552\) 0 0
\(553\) −55.4616 −2.35847
\(554\) 0 0
\(555\) 6.10759 + 7.48024i 0.259253 + 0.317518i
\(556\) 0 0
\(557\) 25.7332i 1.09035i 0.838321 + 0.545176i \(0.183537\pi\)
−0.838321 + 0.545176i \(0.816463\pi\)
\(558\) 0 0
\(559\) −5.63574 −0.238367
\(560\) 0 0
\(561\) 30.4323i 1.28485i
\(562\) 0 0
\(563\) 10.1922i 0.429550i 0.976664 + 0.214775i \(0.0689018\pi\)
−0.976664 + 0.214775i \(0.931098\pi\)
\(564\) 0 0
\(565\) −4.99347 + 4.07715i −0.210077 + 0.171527i
\(566\) 0 0
\(567\) 21.2224 0.891259
\(568\) 0 0
\(569\) 32.7862i 1.37447i 0.726435 + 0.687235i \(0.241176\pi\)
−0.726435 + 0.687235i \(0.758824\pi\)
\(570\) 0 0
\(571\) 23.4721i 0.982276i 0.871082 + 0.491138i \(0.163419\pi\)
−0.871082 + 0.491138i \(0.836581\pi\)
\(572\) 0 0
\(573\) −3.32898 −0.139070
\(574\) 0 0
\(575\) 5.86450 28.7300i 0.244566 1.19813i
\(576\) 0 0
\(577\) 16.4901i 0.686491i 0.939246 + 0.343246i \(0.111526\pi\)
−0.939246 + 0.343246i \(0.888474\pi\)
\(578\) 0 0
\(579\) 36.5541i 1.51914i
\(580\) 0 0
\(581\) 5.32051 0.220732
\(582\) 0 0
\(583\) 18.8576i 0.781000i
\(584\) 0 0
\(585\) 40.1547 32.7862i 1.66019 1.35554i
\(586\) 0 0
\(587\) 33.6156 1.38746 0.693732 0.720233i \(-0.255965\pi\)
0.693732 + 0.720233i \(0.255965\pi\)
\(588\) 0 0
\(589\) 8.53590 + 26.3551i 0.351716 + 1.08594i
\(590\) 0 0
\(591\) 4.65254 0.191380
\(592\) 0 0
\(593\) 17.5254i 0.719681i −0.933014 0.359840i \(-0.882831\pi\)
0.933014 0.359840i \(-0.117169\pi\)
\(594\) 0 0
\(595\) 32.0442 + 39.2460i 1.31368 + 1.60893i
\(596\) 0 0
\(597\) −5.76596 −0.235985
\(598\) 0 0
\(599\) 1.70295 0.0695806 0.0347903 0.999395i \(-0.488924\pi\)
0.0347903 + 0.999395i \(0.488924\pi\)
\(600\) 0 0
\(601\) 15.2161i 0.620679i −0.950626 0.310340i \(-0.899557\pi\)
0.950626 0.310340i \(-0.100443\pi\)
\(602\) 0 0
\(603\) 16.7719i 0.683004i
\(604\) 0 0
\(605\) −13.7321 + 11.2122i −0.558287 + 0.455840i
\(606\) 0 0
\(607\) 17.6534i 0.716529i 0.933620 + 0.358265i \(0.116631\pi\)
−0.933620 + 0.358265i \(0.883369\pi\)
\(608\) 0 0
\(609\) −56.4974 −2.28939
\(610\) 0 0
\(611\) 63.0986 2.55270
\(612\) 0 0
\(613\) 38.8401i 1.56874i 0.620295 + 0.784369i \(0.287013\pi\)
−0.620295 + 0.784369i \(0.712987\pi\)
\(614\) 0 0
\(615\) 39.4801 32.2354i 1.59199 1.29985i
\(616\) 0 0
\(617\) 20.9086i 0.841748i 0.907119 + 0.420874i \(0.138277\pi\)
−0.907119 + 0.420874i \(0.861723\pi\)
\(618\) 0 0
\(619\) 21.8453i 0.878035i −0.898478 0.439018i \(-0.855327\pi\)
0.898478 0.439018i \(-0.144673\pi\)
\(620\) 0 0
\(621\) 11.1390i 0.446992i
\(622\) 0 0
\(623\) 21.7748i 0.872390i
\(624\) 0 0
\(625\) −23.0000 9.79796i −0.920000 0.391918i
\(626\) 0 0
\(627\) −6.10759 18.8576i −0.243914 0.753099i
\(628\) 0 0
\(629\) 11.1390i 0.444140i
\(630\) 0 0
\(631\) 27.9166i 1.11134i −0.831402 0.555672i \(-0.812461\pi\)
0.831402 0.555672i \(-0.187539\pi\)
\(632\) 0 0
\(633\) 6.96953i 0.277014i
\(634\) 0 0
\(635\) 23.7190 + 29.0498i 0.941261 + 1.15280i
\(636\) 0 0
\(637\) 27.7308 1.09874
\(638\) 0 0
\(639\) 6.35549 0.251419
\(640\) 0 0
\(641\) 28.0783i 1.10903i 0.832175 + 0.554514i \(0.187096\pi\)
−0.832175 + 0.554514i \(0.812904\pi\)
\(642\) 0 0
\(643\) −24.3652 −0.960871 −0.480435 0.877030i \(-0.659521\pi\)
−0.480435 + 0.877030i \(0.659521\pi\)
\(644\) 0 0
\(645\) −3.32898 4.07715i −0.131078 0.160538i
\(646\) 0 0
\(647\) −27.3300 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(648\) 0 0
\(649\) 11.1390i 0.437243i
\(650\) 0 0
\(651\) 55.8333i 2.18828i
\(652\) 0 0
\(653\) 5.85993i 0.229317i 0.993405 + 0.114658i \(0.0365773\pi\)
−0.993405 + 0.114658i \(0.963423\pi\)
\(654\) 0 0
\(655\) 18.5007 + 22.6587i 0.722883 + 0.885347i
\(656\) 0 0
\(657\) 13.3843i 0.522170i
\(658\) 0 0
\(659\) 33.0241 1.28644 0.643218 0.765683i \(-0.277599\pi\)
0.643218 + 0.765683i \(0.277599\pi\)
\(660\) 0 0
\(661\) 4.07715i 0.158583i −0.996851 0.0792914i \(-0.974734\pi\)
0.996851 0.0792914i \(-0.0252658\pi\)
\(662\) 0 0
\(663\) 107.862 4.18900
\(664\) 0 0
\(665\) −27.7329 17.8879i −1.07543 0.693664i
\(666\) 0 0
\(667\) 37.7151i 1.46034i
\(668\) 0 0
\(669\) −49.7654 −1.92404
\(670\) 0 0
\(671\) 3.84910i 0.148593i
\(672\) 0 0
\(673\) 15.3069 0.590038 0.295019 0.955491i \(-0.404674\pi\)
0.295019 + 0.955491i \(0.404674\pi\)
\(674\) 0 0
\(675\) 9.30509 + 1.89939i 0.358153 + 0.0731077i
\(676\) 0 0
\(677\) 4.99347 0.191915 0.0959573 0.995385i \(-0.469409\pi\)
0.0959573 + 0.995385i \(0.469409\pi\)
\(678\) 0 0
\(679\) 5.63574 0.216280
\(680\) 0 0
\(681\) 28.7321 1.10101
\(682\) 0 0
\(683\) 0.186285i 0.00712801i −0.999994 0.00356400i \(-0.998866\pi\)
0.999994 0.00356400i \(-0.00113446\pi\)
\(684\) 0 0
\(685\) −0.392305 0.480473i −0.0149892 0.0183579i
\(686\) 0 0
\(687\) 39.2419i 1.49717i
\(688\) 0 0
\(689\) 66.8372 2.54629
\(690\) 0 0
\(691\) 40.0594i 1.52393i −0.647618 0.761965i \(-0.724235\pi\)
0.647618 0.761965i \(-0.275765\pi\)
\(692\) 0 0
\(693\) 22.1469i 0.841293i
\(694\) 0 0
\(695\) −24.6083 30.1389i −0.933447 1.14323i
\(696\) 0 0
\(697\) 58.7906 2.22685
\(698\) 0 0
\(699\) −22.7358 −0.859948
\(700\) 0 0
\(701\) −37.5167 −1.41698 −0.708492 0.705718i \(-0.750624\pi\)
−0.708492 + 0.705718i \(0.750624\pi\)
\(702\) 0 0
\(703\) −2.23553 6.90235i −0.0843147 0.260327i
\(704\) 0 0
\(705\) 37.2718 + 45.6484i 1.40374 + 1.71922i
\(706\) 0 0
\(707\) −27.7511 −1.04369
\(708\) 0 0
\(709\) −28.3923 −1.06630 −0.533148 0.846022i \(-0.678991\pi\)
−0.533148 + 0.846022i \(0.678991\pi\)
\(710\) 0 0
\(711\) −61.1322 −2.29264
\(712\) 0 0
\(713\) −37.2718 −1.39584
\(714\) 0 0
\(715\) −15.3971 18.8576i −0.575820 0.705233i
\(716\) 0 0
\(717\) −18.1899 −0.679314
\(718\) 0 0
\(719\) 47.0700i 1.75541i −0.479197 0.877707i \(-0.659072\pi\)
0.479197 0.877707i \(-0.340928\pi\)
\(720\) 0 0
\(721\) 45.6484i 1.70004i
\(722\) 0 0
\(723\) 39.4801 1.46828
\(724\) 0 0
\(725\) 31.5058 + 6.43110i 1.17010 + 0.238845i
\(726\) 0 0
\(727\) 18.0797 0.670538 0.335269 0.942122i \(-0.391173\pi\)
0.335269 + 0.942122i \(0.391173\pi\)
\(728\) 0 0
\(729\) 38.1769 1.41396
\(730\) 0 0
\(731\) 6.07137i 0.224558i
\(732\) 0 0
\(733\) 5.37945i 0.198695i 0.995053 + 0.0993473i \(0.0316755\pi\)
−0.995053 + 0.0993473i \(0.968325\pi\)
\(734\) 0 0
\(735\) 16.3803 + 20.0617i 0.604198 + 0.739988i
\(736\) 0 0
\(737\) −7.87645 −0.290133
\(738\) 0 0
\(739\) 3.84910i 0.141591i −0.997491 0.0707956i \(-0.977446\pi\)
0.997491 0.0707956i \(-0.0225538\pi\)
\(740\) 0 0
\(741\) −66.8372 + 21.6472i −2.45532 + 0.795231i
\(742\) 0 0
\(743\) 7.41129i 0.271894i 0.990716 + 0.135947i \(0.0434077\pi\)
−0.990716 + 0.135947i \(0.956592\pi\)
\(744\) 0 0
\(745\) 20.1962 16.4901i 0.739930 0.604150i
\(746\) 0 0
\(747\) 5.86450 0.214571
\(748\) 0 0
\(749\) 32.7862i 1.19798i
\(750\) 0 0
\(751\) −26.6686 −0.973152 −0.486576 0.873638i \(-0.661754\pi\)
−0.486576 + 0.873638i \(0.661754\pi\)
\(752\) 0 0
\(753\) 19.0819 0.695382
\(754\) 0 0
\(755\) −30.0745 + 24.5557i −1.09452 + 0.893675i
\(756\) 0 0
\(757\) 49.4703i 1.79803i −0.437920 0.899014i \(-0.644285\pi\)
0.437920 0.899014i \(-0.355715\pi\)
\(758\) 0 0
\(759\) 26.6686 0.968010
\(760\) 0 0
\(761\) −26.4449 −0.958626 −0.479313 0.877644i \(-0.659114\pi\)
−0.479313 + 0.877644i \(0.659114\pi\)
\(762\) 0 0
\(763\) 51.5198i 1.86514i
\(764\) 0 0
\(765\) 35.3205 + 43.2586i 1.27702 + 1.56402i
\(766\) 0 0
\(767\) 39.4801 1.42554
\(768\) 0 0
\(769\) 24.1962 0.872536 0.436268 0.899817i \(-0.356300\pi\)
0.436268 + 0.899817i \(0.356300\pi\)
\(770\) 0 0
\(771\) 68.7894i 2.47739i
\(772\) 0 0
\(773\) 2.88298 0.103694 0.0518468 0.998655i \(-0.483489\pi\)
0.0518468 + 0.998655i \(0.483489\pi\)
\(774\) 0 0
\(775\) −6.35549 + 31.1354i −0.228296 + 1.11842i
\(776\) 0 0
\(777\) 14.6226i 0.524583i
\(778\) 0 0
\(779\) −36.4300 + 11.7990i −1.30524 + 0.422742i
\(780\) 0 0
\(781\) 2.98468i 0.106800i
\(782\) 0 0
\(783\) −12.2152 −0.436535
\(784\) 0 0
\(785\) −2.53590 3.10583i −0.0905101 0.110852i
\(786\) 0 0
\(787\) 45.4990i 1.62186i −0.585141 0.810932i \(-0.698961\pi\)
0.585141 0.810932i \(-0.301039\pi\)
\(788\) 0 0
\(789\) 37.4941i 1.33483i
\(790\) 0 0
\(791\) −9.76139 −0.347075
\(792\) 0 0
\(793\) −13.6424 −0.484456
\(794\) 0 0
\(795\) 39.4801 + 48.3530i 1.40021 + 1.71491i
\(796\) 0 0
\(797\) 38.0443 1.34760 0.673798 0.738915i \(-0.264662\pi\)
0.673798 + 0.738915i \(0.264662\pi\)
\(798\) 0 0
\(799\) 67.9760i 2.40482i
\(800\) 0 0
\(801\) 24.0012i 0.848040i
\(802\) 0 0
\(803\) 6.28555 0.221812
\(804\) 0 0
\(805\) 34.3923 28.0812i 1.21217 0.989732i
\(806\) 0 0
\(807\) 22.7938 0.802381
\(808\) 0 0
\(809\) −2.53590 −0.0891574 −0.0445787 0.999006i \(-0.514195\pi\)
−0.0445787 + 0.999006i \(0.514195\pi\)
\(810\) 0 0
\(811\) 43.7687 1.53693 0.768464 0.639894i \(-0.221022\pi\)
0.768464 + 0.639894i \(0.221022\pi\)
\(812\) 0 0
\(813\) 45.1482 1.58342
\(814\) 0 0
\(815\) −21.8866 + 17.8703i −0.766654 + 0.625970i
\(816\) 0 0
\(817\) 1.21849 + 3.76217i 0.0426296 + 0.131622i
\(818\) 0 0
\(819\) 78.4958 2.74286
\(820\) 0 0
\(821\) 0.679492 0.0237144 0.0118572 0.999930i \(-0.496226\pi\)
0.0118572 + 0.999930i \(0.496226\pi\)
\(822\) 0 0
\(823\) −12.6362 −0.440471 −0.220236 0.975447i \(-0.570683\pi\)
−0.220236 + 0.975447i \(0.570683\pi\)
\(824\) 0 0
\(825\) 4.54747 22.2780i 0.158323 0.775619i
\(826\) 0 0
\(827\) 1.20417i 0.0418730i 0.999781 + 0.0209365i \(0.00666478\pi\)
−0.999781 + 0.0209365i \(0.993335\pi\)
\(828\) 0 0
\(829\) 7.06183i 0.245268i −0.992452 0.122634i \(-0.960866\pi\)
0.992452 0.122634i \(-0.0391341\pi\)
\(830\) 0 0
\(831\) 48.6847 1.68885
\(832\) 0 0
\(833\) 29.8744i 1.03508i
\(834\) 0 0
\(835\) −14.4139 17.6534i −0.498815 0.610921i
\(836\) 0 0
\(837\) 12.0716i 0.417255i
\(838\) 0 0
\(839\) 6.35549 0.219416 0.109708 0.993964i \(-0.465008\pi\)
0.109708 + 0.993964i \(0.465008\pi\)
\(840\) 0 0
\(841\) −12.3590 −0.426172
\(842\) 0 0
\(843\) −6.10759 −0.210357
\(844\) 0 0
\(845\) −44.3205 + 36.1875i −1.52467 + 1.24489i
\(846\) 0 0
\(847\) −26.8438 −0.922366
\(848\) 0 0
\(849\) 16.9393i 0.581357i
\(850\) 0 0
\(851\) 9.76139 0.334616
\(852\) 0 0
\(853\) 9.79796i 0.335476i 0.985832 + 0.167738i \(0.0536462\pi\)
−0.985832 + 0.167738i \(0.946354\pi\)
\(854\) 0 0
\(855\) −30.5684 19.7169i −1.04542 0.674302i
\(856\) 0 0
\(857\) 10.7594 0.367535 0.183768 0.982970i \(-0.441171\pi\)
0.183768 + 0.982970i \(0.441171\pi\)
\(858\) 0 0
\(859\) 12.1427i 0.414305i −0.978309 0.207153i \(-0.933580\pi\)
0.978309 0.207153i \(-0.0664196\pi\)
\(860\) 0 0
\(861\) 77.1769 2.63018
\(862\) 0 0
\(863\) 3.98507i 0.135653i 0.997697 + 0.0678267i \(0.0216065\pi\)
−0.997697 + 0.0678267i \(0.978393\pi\)
\(864\) 0 0
\(865\) −32.2783 + 26.3551i −1.09749 + 0.896101i
\(866\) 0 0
\(867\) 72.0905i 2.44832i
\(868\) 0 0
\(869\) 28.7091i 0.973888i
\(870\) 0 0
\(871\) 27.9166i 0.945919i
\(872\) 0 0
\(873\) 6.21196 0.210243
\(874\) 0 0
\(875\) −17.5935 33.5184i −0.594768 1.13313i
\(876\) 0 0
\(877\) −19.8544 −0.670435 −0.335217 0.942141i \(-0.608810\pi\)
−0.335217 + 0.942141i \(0.608810\pi\)
\(878\) 0 0
\(879\) 27.9166i 0.941605i
\(880\) 0 0
\(881\) 34.3013 1.15564 0.577820 0.816165i \(-0.303904\pi\)
0.577820 + 0.816165i \(0.303904\pi\)
\(882\) 0 0
\(883\) −11.9721 −0.402893 −0.201446 0.979500i \(-0.564564\pi\)
−0.201446 + 0.979500i \(0.564564\pi\)
\(884\) 0 0
\(885\) 23.3205 + 28.5617i 0.783910 + 0.960090i
\(886\) 0 0
\(887\) 29.4223i 0.987905i −0.869489 0.493953i \(-0.835552\pi\)
0.869489 0.493953i \(-0.164448\pi\)
\(888\) 0 0
\(889\) 56.7874i 1.90459i
\(890\) 0 0
\(891\) 10.9855i 0.368030i
\(892\) 0 0
\(893\) −13.6424 42.1218i −0.456526 1.40955i
\(894\) 0 0
\(895\) 27.1249 22.1474i 0.906686 0.740306i
\(896\) 0 0
\(897\) 94.5220i 3.15600i
\(898\) 0 0
\(899\) 40.8728i 1.36318i
\(900\) 0 0
\(901\) 72.0035i 2.39879i
\(902\) 0 0
\(903\) 7.97014i 0.265230i
\(904\) 0 0
\(905\) −15.7529 19.2933i −0.523644 0.641331i
\(906\) 0 0
\(907\) 13.4820i 0.447664i 0.974628 + 0.223832i \(0.0718567\pi\)
−0.974628 + 0.223832i \(0.928143\pi\)
\(908\) 0 0
\(909\) −30.5885 −1.01456
\(910\) 0 0
\(911\) −20.7694 −0.688122 −0.344061 0.938947i \(-0.611803\pi\)
−0.344061 + 0.938947i \(0.611803\pi\)
\(912\) 0 0
\(913\) 2.75410i 0.0911473i
\(914\) 0 0
\(915\) −8.05844 9.86954i −0.266404 0.326277i
\(916\) 0 0
\(917\) 44.2939i 1.46271i
\(918\) 0 0
\(919\) 28.7300i 0.947717i 0.880601 + 0.473858i \(0.157139\pi\)
−0.880601 + 0.473858i \(0.842861\pi\)
\(920\) 0 0
\(921\) −41.2295 −1.35856
\(922\) 0 0
\(923\) −10.5787 −0.348201
\(924\) 0 0
\(925\) 1.66449 8.15430i 0.0547281 0.268112i
\(926\) 0 0
\(927\) 50.3157i 1.65258i
\(928\) 0 0
\(929\) 12.2487 0.401867 0.200934 0.979605i \(-0.435602\pi\)
0.200934 + 0.979605i \(0.435602\pi\)
\(930\) 0 0
\(931\) −5.99562 18.5118i −0.196498 0.606701i
\(932\) 0 0
\(933\) 65.7751 2.15338
\(934\) 0 0
\(935\) 20.3152 16.5873i 0.664378 0.542463i
\(936\) 0 0
\(937\) 55.8107i 1.82326i −0.411017 0.911628i \(-0.634826\pi\)
0.411017 0.911628i \(-0.365174\pi\)
\(938\) 0 0
\(939\) −20.7694 −0.677785
\(940\) 0 0
\(941\) 48.0023i 1.56483i −0.622756 0.782416i \(-0.713987\pi\)
0.622756 0.782416i \(-0.286013\pi\)
\(942\) 0 0
\(943\) 51.5198i 1.67772i
\(944\) 0 0
\(945\) 9.09494 + 11.1390i 0.295858 + 0.362351i
\(946\) 0 0
\(947\) 52.7805 1.71513 0.857567 0.514372i \(-0.171975\pi\)
0.857567 + 0.514372i \(0.171975\pi\)
\(948\) 0 0
\(949\) 22.2780i 0.723173i
\(950\) 0 0
\(951\) 48.3530i 1.56795i
\(952\) 0 0
\(953\) −10.7594 −0.348532 −0.174266 0.984699i \(-0.555755\pi\)
−0.174266 + 0.984699i \(0.555755\pi\)
\(954\) 0 0
\(955\) 1.81448 + 2.22228i 0.0587152 + 0.0719112i
\(956\) 0 0
\(957\) 29.2452i 0.945364i
\(958\) 0 0
\(959\) 0.939245i 0.0303298i
\(960\) 0 0
\(961\) 9.39230 0.302978
\(962\) 0 0
\(963\) 36.1384i 1.16454i
\(964\) 0 0
\(965\) −24.4018 + 19.9240i −0.785523 + 0.641377i
\(966\) 0 0
\(967\) −52.6025 −1.69158 −0.845791 0.533514i \(-0.820871\pi\)
−0.845791 + 0.533514i \(0.820871\pi\)
\(968\) 0 0
\(969\) −23.3205 72.0035i −0.749163 2.31309i
\(970\) 0 0
\(971\) −14.4139 −0.462565 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(972\) 0 0
\(973\) 58.9165i 1.88878i
\(974\) 0 0
\(975\) −78.9602 16.1177i −2.52875 0.516179i
\(976\) 0 0
\(977\) −18.6359 −0.596215 −0.298107 0.954532i \(-0.596355\pi\)
−0.298107 + 0.954532i \(0.596355\pi\)
\(978\) 0 0
\(979\) 11.2715 0.360238
\(980\) 0 0
\(981\) 56.7874i 1.81308i
\(982\) 0 0
\(983\) 20.0617i 0.639870i 0.947440 + 0.319935i \(0.103661\pi\)
−0.947440 + 0.319935i \(0.896339\pi\)
\(984\) 0 0
\(985\) −2.53590 3.10583i −0.0808004 0.0989599i
\(986\) 0 0
\(987\) 89.2349i 2.84038i
\(988\) 0 0
\(989\) −5.32051 −0.169182
\(990\) 0 0
\(991\) 13.4307 0.426641 0.213321 0.976982i \(-0.431572\pi\)
0.213321 + 0.976982i \(0.431572\pi\)
\(992\) 0 0
\(993\) 24.1432i 0.766160i
\(994\) 0 0
\(995\) 3.14277 + 3.84910i 0.0996326 + 0.122025i
\(996\) 0 0
\(997\) 26.2880i 0.832551i 0.909239 + 0.416275i \(0.136665\pi\)
−0.909239 + 0.416275i \(0.863335\pi\)
\(998\) 0 0
\(999\) 3.16152i 0.100026i
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 1520.2.g.f.1519.3 yes 16
4.3 odd 2 inner 1520.2.g.f.1519.16 yes 16
5.4 even 2 inner 1520.2.g.f.1519.14 yes 16
19.18 odd 2 inner 1520.2.g.f.1519.15 yes 16
20.19 odd 2 inner 1520.2.g.f.1519.1 16
76.75 even 2 inner 1520.2.g.f.1519.4 yes 16
95.94 odd 2 inner 1520.2.g.f.1519.2 yes 16
380.379 even 2 inner 1520.2.g.f.1519.13 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1520.2.g.f.1519.1 16 20.19 odd 2 inner
1520.2.g.f.1519.2 yes 16 95.94 odd 2 inner
1520.2.g.f.1519.3 yes 16 1.1 even 1 trivial
1520.2.g.f.1519.4 yes 16 76.75 even 2 inner
1520.2.g.f.1519.13 yes 16 380.379 even 2 inner
1520.2.g.f.1519.14 yes 16 5.4 even 2 inner
1520.2.g.f.1519.15 yes 16 19.18 odd 2 inner
1520.2.g.f.1519.16 yes 16 4.3 odd 2 inner