Properties

Label 164.2.o.a.7.1
Level $164$
Weight $2$
Character 164.7
Analytic conductor $1.310$
Analytic rank $0$
Dimension $16$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,2,Mod(7,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 39]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 164.o (of order \(40\), degree \(16\), 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: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{40}]$

Embedding invariants

Embedding label 7.1
Root \(-0.156434 - 0.987688i\) of defining polynomial
Character \(\chi\) \(=\) 164.7
Dual form 164.2.o.a.47.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.642040 - 1.26007i) q^{2} +(-1.17557 + 1.61803i) q^{4} +(-0.490920 - 3.09954i) q^{5} +(2.79360 + 0.442463i) q^{8} +(-2.12132 - 2.12132i) q^{9} +O(q^{10})\) \(q+(-0.642040 - 1.26007i) q^{2} +(-1.17557 + 1.61803i) q^{4} +(-0.490920 - 3.09954i) q^{5} +(2.79360 + 0.442463i) q^{8} +(-2.12132 - 2.12132i) q^{9} +(-3.59046 + 2.60863i) q^{10} +(-5.14643 - 4.39547i) q^{13} +(-1.23607 - 3.80423i) q^{16} +(5.72667 + 3.50931i) q^{17} +(-1.31105 + 4.03499i) q^{18} +(5.59228 + 2.84941i) q^{20} +(-4.61089 + 1.49817i) q^{25} +(-2.23440 + 9.30694i) q^{26} +(6.41386 - 3.93042i) q^{29} +(-4.00000 + 4.00000i) q^{32} +(0.745237 - 9.46914i) q^{34} +(5.92613 - 0.938607i) q^{36} +(9.33556 + 6.78268i) q^{37} -8.87612i q^{40} +(5.34931 - 3.51921i) q^{41} +(-5.53373 + 7.61653i) q^{45} +(6.91382 + 1.09504i) q^{49} +(4.84818 + 4.84818i) q^{50} +(13.1620 - 3.15992i) q^{52} +(-2.65748 - 4.33661i) q^{53} +(-9.07057 - 5.55845i) q^{58} +(-10.4730 - 5.33625i) q^{61} +(7.60845 + 2.47214i) q^{64} +(-11.0975 + 18.1094i) q^{65} +(-12.4103 + 5.14051i) q^{68} +(-4.98752 - 6.86474i) q^{72} +(-11.6467 + 11.6467i) q^{73} +(2.55288 - 16.1182i) q^{74} +(-11.1846 + 5.69882i) q^{80} +9.00000i q^{81} +(-7.86894 - 4.48105i) q^{82} +(8.06592 - 19.4729i) q^{85} +(3.10626 + 0.244468i) q^{89} +(13.1503 + 2.08280i) q^{90} +(-15.7778 - 3.78791i) q^{97} +(-3.05911 - 9.41498i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 8 q^{8} - 8 q^{13} + 16 q^{16} - 4 q^{17} - 20 q^{26} + 20 q^{29} - 64 q^{32} - 48 q^{34} - 20 q^{41} + 56 q^{50} + 96 q^{52} + 28 q^{53} + 12 q^{58} - 44 q^{61} - 112 q^{65} + 32 q^{68} - 36 q^{82} + 56 q^{85} + 32 q^{89} + 48 q^{90} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(e\left(\frac{39}{40}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.642040 1.26007i −0.453990 0.891007i
\(3\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(4\) −1.17557 + 1.61803i −0.587785 + 0.809017i
\(5\) −0.490920 3.09954i −0.219546 1.38616i −0.813456 0.581627i \(-0.802416\pi\)
0.593910 0.804532i \(-0.297584\pi\)
\(6\) 0 0
\(7\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(8\) 2.79360 + 0.442463i 0.987688 + 0.156434i
\(9\) −2.12132 2.12132i −0.707107 0.707107i
\(10\) −3.59046 + 2.60863i −1.13540 + 0.824920i
\(11\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(12\) 0 0
\(13\) −5.14643 4.39547i −1.42736 1.21908i −0.937829 0.347098i \(-0.887167\pi\)
−0.489534 0.871984i \(-0.662833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.23607 3.80423i −0.309017 0.951057i
\(17\) 5.72667 + 3.50931i 1.38892 + 0.851132i 0.997608 0.0691254i \(-0.0220209\pi\)
0.391313 + 0.920258i \(0.372021\pi\)
\(18\) −1.31105 + 4.03499i −0.309017 + 0.951057i
\(19\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(20\) 5.59228 + 2.84941i 1.25047 + 0.637147i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(24\) 0 0
\(25\) −4.61089 + 1.49817i −0.922179 + 0.299634i
\(26\) −2.23440 + 9.30694i −0.438202 + 1.82524i
\(27\) 0 0
\(28\) 0 0
\(29\) 6.41386 3.93042i 1.19102 0.729860i 0.221572 0.975144i \(-0.428881\pi\)
0.969451 + 0.245284i \(0.0788811\pi\)
\(30\) 0 0
\(31\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(32\) −4.00000 + 4.00000i −0.707107 + 0.707107i
\(33\) 0 0
\(34\) 0.745237 9.46914i 0.127807 1.62394i
\(35\) 0 0
\(36\) 5.92613 0.938607i 0.987688 0.156434i
\(37\) 9.33556 + 6.78268i 1.53476 + 1.11507i 0.953519 + 0.301333i \(0.0974315\pi\)
0.581238 + 0.813733i \(0.302568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 8.87612i 1.40344i
\(41\) 5.34931 3.51921i 0.835422 0.549609i
\(42\) 0 0
\(43\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(44\) 0 0
\(45\) −5.53373 + 7.61653i −0.824920 + 1.13540i
\(46\) 0 0
\(47\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(48\) 0 0
\(49\) 6.91382 + 1.09504i 0.987688 + 0.156434i
\(50\) 4.84818 + 4.84818i 0.685636 + 0.685636i
\(51\) 0 0
\(52\) 13.1620 3.15992i 1.82524 0.438202i
\(53\) −2.65748 4.33661i −0.365033 0.595679i 0.616412 0.787424i \(-0.288586\pi\)
−0.981445 + 0.191744i \(0.938586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −9.07057 5.55845i −1.19102 0.729860i
\(59\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(60\) 0 0
\(61\) −10.4730 5.33625i −1.34093 0.683237i −0.371458 0.928450i \(-0.621142\pi\)
−0.969469 + 0.245213i \(0.921142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.60845 + 2.47214i 0.951057 + 0.309017i
\(65\) −11.0975 + 18.1094i −1.37647 + 2.24620i
\(66\) 0 0
\(67\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(68\) −12.4103 + 5.14051i −1.50497 + 0.623378i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(72\) −4.98752 6.86474i −0.587785 0.809017i
\(73\) −11.6467 + 11.6467i −1.36315 + 1.36315i −0.493275 + 0.869874i \(0.664200\pi\)
−0.869874 + 0.493275i \(0.835800\pi\)
\(74\) 2.55288 16.1182i 0.296766 1.87371i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(80\) −11.1846 + 5.69882i −1.25047 + 0.637147i
\(81\) 9.00000i 1.00000i
\(82\) −7.86894 4.48105i −0.868979 0.494849i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 8.06592 19.4729i 0.874872 2.11213i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.10626 + 0.244468i 0.329262 + 0.0259135i 0.242013 0.970273i \(-0.422192\pi\)
0.0872493 + 0.996187i \(0.472192\pi\)
\(90\) 13.1503 + 2.08280i 1.38616 + 0.219546i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.7778 3.78791i −1.60199 0.384604i −0.668644 0.743583i \(-0.733125\pi\)
−0.933346 + 0.358979i \(0.883125\pi\)
\(98\) −3.05911 9.41498i −0.309017 0.951057i
\(99\) 0 0
\(100\) 2.99634 9.22179i 0.299634 0.922179i
\(101\) 10.5767 9.03332i 1.05242 0.898849i 0.0573794 0.998352i \(-0.481726\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) 0 0
\(103\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(104\) −12.4323 14.5563i −1.21908 1.42736i
\(105\) 0 0
\(106\) −3.75824 + 6.13289i −0.365033 + 0.595679i
\(107\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(108\) 0 0
\(109\) 4.07645 1.68852i 0.390453 0.161731i −0.178815 0.983883i \(-0.557226\pi\)
0.569269 + 0.822152i \(0.307226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.65648 + 9.16186i 0.626189 + 0.861876i 0.997785 0.0665190i \(-0.0211893\pi\)
−0.371596 + 0.928395i \(0.621189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.18039 + 14.9983i −0.109597 + 1.39256i
\(117\) 1.59303 + 20.2414i 0.147276 + 1.87132i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.80107 4.99390i 0.891007 0.453990i
\(122\) 16.6228i 1.50496i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.216299 0.424510i −0.0193463 0.0379693i
\(126\) 0 0
\(127\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(128\) −1.76985 11.1744i −0.156434 0.987688i
\(129\) 0 0
\(130\) 29.9442 + 2.35666i 2.62628 + 0.206693i
\(131\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 14.4453 + 12.3375i 1.23867 + 1.05793i
\(137\) −1.01149 2.44196i −0.0864178 0.208631i 0.874763 0.484552i \(-0.161017\pi\)
−0.961180 + 0.275921i \(0.911017\pi\)
\(138\) 0 0
\(139\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −5.44789 + 10.6921i −0.453990 + 0.891007i
\(145\) −15.3312 17.9505i −1.27319 1.49071i
\(146\) 22.1534 + 7.19808i 1.83343 + 0.595718i
\(147\) 0 0
\(148\) −21.9492 + 7.13174i −1.80422 + 0.586225i
\(149\) 3.77695 15.7321i 0.309420 1.28883i −0.573462 0.819232i \(-0.694400\pi\)
0.882882 0.469595i \(-0.155600\pi\)
\(150\) 0 0
\(151\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(152\) 0 0
\(153\) −4.70373 19.5925i −0.380274 1.58396i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.979805 12.4496i 0.0781970 0.993587i −0.825217 0.564816i \(-0.808947\pi\)
0.903414 0.428770i \(-0.141053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 14.3619 + 10.4345i 1.13540 + 0.824920i
\(161\) 0 0
\(162\) 11.3407 5.77836i 0.891007 0.453990i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) −0.594283 + 12.7925i −0.0464057 + 0.998923i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(168\) 0 0
\(169\) 5.13196 + 32.4019i 0.394766 + 2.49246i
\(170\) −29.7159 + 2.33869i −2.27910 + 0.179369i
\(171\) 0 0
\(172\) 0 0
\(173\) 10.8667 + 10.8667i 0.826177 + 0.826177i 0.986986 0.160809i \(-0.0514102\pi\)
−0.160809 + 0.986986i \(0.551410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.68629 4.07107i −0.126393 0.305139i
\(179\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(180\) −5.81851 17.9075i −0.433686 1.33475i
\(181\) 21.9960 + 13.4792i 1.63495 + 1.00190i 0.965984 + 0.258603i \(0.0832623\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.4402 32.2658i 1.20871 2.37223i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(192\) 0 0
\(193\) −23.3129 + 14.2862i −1.67810 + 1.02834i −0.746132 + 0.665798i \(0.768091\pi\)
−0.931967 + 0.362543i \(0.881909\pi\)
\(194\) 5.35691 + 22.3131i 0.384604 + 1.60199i
\(195\) 0 0
\(196\) −9.89949 + 9.89949i −0.707107 + 0.707107i
\(197\) −1.70979 + 10.7952i −0.121817 + 0.769123i 0.848839 + 0.528651i \(0.177302\pi\)
−0.970656 + 0.240472i \(0.922698\pi\)
\(198\) 0 0
\(199\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(200\) −13.5439 + 2.14514i −0.957698 + 0.151684i
\(201\) 0 0
\(202\) −18.1733 7.52762i −1.27867 0.529641i
\(203\) 0 0
\(204\) 0 0
\(205\) −13.5340 14.8528i −0.945259 1.03736i
\(206\) 0 0
\(207\) 0 0
\(208\) −10.3600 + 25.0113i −0.718337 + 1.73422i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(212\) 10.1408 + 0.798101i 0.696475 + 0.0548138i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −4.74491 4.05253i −0.321366 0.274472i
\(219\) 0 0
\(220\) 0 0
\(221\) −14.0469 43.2318i −0.944894 2.90808i
\(222\) 0 0
\(223\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(224\) 0 0
\(225\) 12.9593 + 6.60308i 0.863952 + 0.440206i
\(226\) 7.27090 14.2699i 0.483653 0.949222i
\(227\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(228\) 0 0
\(229\) −5.69149 + 9.28767i −0.376104 + 0.613747i −0.983555 0.180611i \(-0.942192\pi\)
0.607450 + 0.794358i \(0.292192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 19.6569 8.14214i 1.29054 0.534557i
\(233\) −9.03430 + 10.5778i −0.591857 + 0.692975i −0.972806 0.231621i \(-0.925597\pi\)
0.380949 + 0.924596i \(0.375597\pi\)
\(234\) 24.4829 15.0031i 1.60050 0.980785i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(240\) 0 0
\(241\) 6.41174 1.01552i 0.413016 0.0654154i 0.0535313 0.998566i \(-0.482952\pi\)
0.359485 + 0.933151i \(0.382952\pi\)
\(242\) −12.5854 9.14379i −0.809017 0.587785i
\(243\) 0 0
\(244\) 20.9460 10.6725i 1.34093 0.683237i
\(245\) 21.9673i 1.40344i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.396041 + 0.545104i −0.0250479 + 0.0344754i
\(251\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −12.9443 + 9.40456i −0.809017 + 0.587785i
\(257\) 28.0031 6.72294i 1.74678 0.419366i 0.770776 0.637106i \(-0.219869\pi\)
0.976007 + 0.217740i \(0.0698685\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −16.2558 39.2449i −1.00814 2.43387i
\(261\) −21.9435 5.26817i −1.35827 0.326092i
\(262\) 0 0
\(263\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(264\) 0 0
\(265\) −12.1369 + 10.3659i −0.745564 + 0.636772i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −31.1853 10.1327i −1.90140 0.617802i −0.959159 0.282867i \(-0.908714\pi\)
−0.942241 0.334935i \(-0.891286\pi\)
\(270\) 0 0
\(271\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(272\) 6.27165 26.1233i 0.380274 1.58396i
\(273\) 0 0
\(274\) −2.42763 + 2.84239i −0.146659 + 0.171715i
\(275\) 0 0
\(276\) 0 0
\(277\) 19.4675 + 26.7947i 1.16969 + 1.60993i 0.665689 + 0.746229i \(0.268138\pi\)
0.503997 + 0.863706i \(0.331862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.73077 + 21.9915i 0.103249 + 1.31190i 0.802339 + 0.596869i \(0.203589\pi\)
−0.699090 + 0.715034i \(0.746411\pi\)
\(282\) 0 0
\(283\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 16.9706 1.00000
\(289\) 12.7617 + 25.0462i 0.750686 + 1.47330i
\(290\) −12.7757 + 30.8434i −0.750218 + 1.81119i
\(291\) 0 0
\(292\) −5.15326 32.5364i −0.301572 1.90405i
\(293\) −33.4778 + 2.63476i −1.95580 + 0.153924i −0.993151 0.116841i \(-0.962723\pi\)
−0.962645 + 0.270766i \(0.912723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 23.0788 + 23.0788i 1.34143 + 1.34143i
\(297\) 0 0
\(298\) −22.2486 + 5.34141i −1.28883 + 0.309420i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.3986 + 35.0811i −0.652679 + 2.00874i
\(306\) −21.6680 + 18.5062i −1.23867 + 1.05793i
\(307\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(312\) 0 0
\(313\) 7.75162 32.2878i 0.438147 1.82501i −0.116834 0.993152i \(-0.537274\pi\)
0.554981 0.831863i \(-0.312726\pi\)
\(314\) −16.3165 + 6.75851i −0.920793 + 0.381405i
\(315\) 0 0
\(316\) 0 0
\(317\) 8.25670 + 34.3916i 0.463743 + 1.93163i 0.344597 + 0.938751i \(0.388015\pi\)
0.119145 + 0.992877i \(0.461985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 3.92736 24.7964i 0.219546 1.38616i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −14.5623 10.5801i −0.809017 0.587785i
\(325\) 30.3148 + 12.5568i 1.68156 + 0.696526i
\(326\) 0 0
\(327\) 0 0
\(328\) 16.5010 7.46442i 0.911114 0.412154i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(332\) 0 0
\(333\) −5.41548 34.1920i −0.296766 1.87371i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 17.5868 + 17.5868i 0.958014 + 0.958014i 0.999153 0.0411389i \(-0.0130986\pi\)
−0.0411389 + 0.999153i \(0.513099\pi\)
\(338\) 37.5339 27.2700i 2.04158 1.48329i
\(339\) 0 0
\(340\) 22.0257 + 35.9426i 1.19451 + 1.94926i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 6.71597 20.6696i 0.361053 1.11121i
\(347\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(348\) 0 0
\(349\) −14.1408 + 27.7530i −0.756942 + 1.48558i 0.113621 + 0.993524i \(0.463755\pi\)
−0.870563 + 0.492057i \(0.836245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 35.7198 11.6061i 1.90118 0.617729i 0.940887 0.338719i \(-0.109994\pi\)
0.960288 0.279010i \(-0.0900062\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.04718 + 4.73864i −0.214500 + 0.251147i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(360\) −18.8291 + 18.8291i −0.992380 + 0.992380i
\(361\) 2.97225 18.7661i 0.156434 0.987688i
\(362\) 2.86244 36.3707i 0.150446 1.91160i
\(363\) 0 0
\(364\) 0 0
\(365\) 41.8172 + 30.3820i 2.18881 + 1.59027i
\(366\) 0 0
\(367\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(368\) 0 0
\(369\) −18.8130 3.88222i −0.979365 0.202100i
\(370\) −51.2125 −2.66241
\(371\) 0 0
\(372\) 0 0
\(373\) −19.0951 + 26.2821i −0.988706 + 1.36084i −0.0567016 + 0.998391i \(0.518058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −50.2845 7.96428i −2.58978 0.410181i
\(378\) 0 0
\(379\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 32.9694 + 20.2037i 1.67810 + 1.02834i
\(387\) 0 0
\(388\) 24.6768 21.0760i 1.25278 1.06997i
\(389\) −2.12456 1.08252i −0.107720 0.0548859i 0.399300 0.916820i \(-0.369253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 18.8300 + 6.11822i 0.951057 + 0.309017i
\(393\) 0 0
\(394\) 14.7004 4.77646i 0.740598 0.240635i
\(395\) 0 0
\(396\) 0 0
\(397\) 13.7047 16.0461i 0.687818 0.805332i −0.301131 0.953583i \(-0.597364\pi\)
0.988950 + 0.148251i \(0.0473643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 11.3988 + 15.6890i 0.569938 + 0.784452i
\(401\) 26.4629 26.4629i 1.32150 1.32150i 0.408929 0.912566i \(-0.365902\pi\)
0.912566 0.408929i \(-0.134098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 2.18261 + 27.7327i 0.108589 + 1.37975i
\(405\) 27.8959 4.41828i 1.38616 0.219546i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 12.1835i 0.602436i −0.953555 0.301218i \(-0.902607\pi\)
0.953555 0.301218i \(-0.0973931\pi\)
\(410\) −10.0262 + 26.5900i −0.495159 + 1.31318i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 38.1676 3.00385i 1.87132 0.147276i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(420\) 0 0
\(421\) −39.6482 + 9.51869i −1.93233 + 0.463913i −0.939842 + 0.341609i \(0.889028\pi\)
−0.992493 + 0.122304i \(0.960972\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −5.50515 13.2906i −0.267354 0.645449i
\(425\) −31.6626 7.60152i −1.53586 0.368728i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(432\) 0 0
\(433\) 11.7159 + 3.80671i 0.563028 + 0.182939i 0.576683 0.816968i \(-0.304347\pi\)
−0.0136552 + 0.999907i \(0.504347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.06007 + 8.58082i −0.0986596 + 0.410947i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(440\) 0 0
\(441\) −12.3435 16.9894i −0.587785 0.809017i
\(442\) −45.4566 + 45.4566i −2.16215 + 2.16215i
\(443\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(444\) 0 0
\(445\) −0.767184 9.74799i −0.0363680 0.462099i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 37.4188 19.0658i 1.76590 0.899773i 0.822045 0.569422i \(-0.192833\pi\)
0.943858 0.330350i \(-0.107167\pi\)
\(450\) 20.5691i 0.969636i
\(451\) 0 0
\(452\) −22.6494 −1.06534
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −31.5334 + 2.48173i −1.47507 + 0.116091i −0.790323 0.612690i \(-0.790087\pi\)
−0.684747 + 0.728781i \(0.740087\pi\)
\(458\) 15.3573 + 1.20865i 0.717600 + 0.0564764i
\(459\) 0 0
\(460\) 0 0
\(461\) 27.8330 20.2219i 1.29631 0.941825i 0.296399 0.955064i \(-0.404214\pi\)
0.999912 + 0.0132386i \(0.00421411\pi\)
\(462\) 0 0
\(463\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(464\) −22.8802 19.5415i −1.06219 0.907192i
\(465\) 0 0
\(466\) 19.1292 + 4.59251i 0.886143 + 0.212744i
\(467\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(468\) −34.6240 21.2176i −1.60050 0.980785i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.56198 + 14.8367i −0.163092 + 0.679326i
\(478\) 0 0
\(479\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(480\) 0 0
\(481\) −18.2318 75.9407i −0.831297 3.46260i
\(482\) −5.39622 7.42726i −0.245791 0.338302i
\(483\) 0 0
\(484\) −3.44156 + 21.7291i −0.156434 + 0.987688i
\(485\) −3.99517 + 50.7635i −0.181411 + 2.30505i
\(486\) 0 0
\(487\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(488\) −26.8963 19.5413i −1.21754 0.884592i
\(489\) 0 0
\(490\) −27.6804 + 14.1039i −1.25047 + 0.637147i
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 50.5231 2.27545
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(500\) 0.941145 + 0.149063i 0.0420893 + 0.00666629i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(504\) 0 0
\(505\) −33.1915 28.3482i −1.47700 1.26148i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −36.7315 22.5091i −1.62810 0.997699i −0.969464 0.245235i \(-0.921135\pi\)
−0.658633 0.752464i \(-0.728865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 20.1612 + 10.2726i 0.891007 + 0.453990i
\(513\) 0 0
\(514\) −26.4505 30.9695i −1.16668 1.36601i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −39.0147 + 45.6803i −1.71091 + 2.00321i
\(521\) 10.2780 6.29837i 0.450287 0.275936i −0.278810 0.960346i \(-0.589940\pi\)
0.729098 + 0.684410i \(0.239940\pi\)
\(522\) 7.45032 + 31.0328i 0.326092 + 1.35827i
\(523\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 18.6074 + 13.5191i 0.809017 + 0.587785i
\(530\) 20.8542 + 8.63808i 0.905847 + 0.375214i
\(531\) 0 0
\(532\) 0 0
\(533\) −42.9984 5.40132i −1.86247 0.233957i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 7.25422 + 45.8014i 0.312752 + 1.97464i
\(539\) 0 0
\(540\) 0 0
\(541\) −5.96802 0.945242i −0.256585 0.0406391i 0.0268165 0.999640i \(-0.491463\pi\)
−0.283402 + 0.959001i \(0.591463\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −36.9439 + 8.86945i −1.58396 + 0.380274i
\(545\) −7.23486 11.8062i −0.309907 0.505723i
\(546\) 0 0
\(547\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(548\) 5.14026 + 1.23407i 0.219581 + 0.0527168i
\(549\) 10.8966 + 33.5364i 0.465058 + 1.43130i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 21.2644 41.7336i 0.903436 1.77309i
\(555\) 0 0
\(556\) 0 0
\(557\) −12.7449 + 20.7978i −0.540020 + 0.881233i −0.999978 0.00670815i \(-0.997865\pi\)
0.459957 + 0.887941i \(0.347865\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 26.5997 16.3003i 1.12204 0.687587i
\(563\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(564\) 0 0
\(565\) 25.1298 25.1298i 1.05722 1.05722i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16.8680 + 2.67163i −0.707143 + 0.112001i −0.499639 0.866234i \(-0.666534\pi\)
−0.207504 + 0.978234i \(0.566534\pi\)
\(570\) 0 0
\(571\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −10.8958 21.3842i −0.453990 0.891007i
\(577\) −2.20421 + 5.32143i −0.0917624 + 0.221534i −0.963097 0.269156i \(-0.913255\pi\)
0.871334 + 0.490690i \(0.163255\pi\)
\(578\) 23.3665 32.1613i 0.971919 1.33773i
\(579\) 0 0
\(580\) 47.0675 3.70429i 1.95437 0.153812i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −37.6897 + 27.3831i −1.55961 + 1.13312i
\(585\) 61.9571 14.8746i 2.56161 0.614988i
\(586\) 24.8141 + 40.4929i 1.02506 + 1.67275i
\(587\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 14.2635 43.8985i 0.586225 1.80422i
\(593\) −7.90266 + 6.74951i −0.324524 + 0.277169i −0.796751 0.604307i \(-0.793450\pi\)
0.472228 + 0.881477i \(0.343450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 21.0151 + 24.6055i 0.860810 + 1.00788i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(600\) 0 0
\(601\) 1.36889 0.567014i 0.0558383 0.0231290i −0.354590 0.935022i \(-0.615379\pi\)
0.410428 + 0.911893i \(0.365379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −20.2903 27.9273i −0.824920 1.13540i
\(606\) 0 0
\(607\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 51.5231 8.16046i 2.08611 0.330407i
\(611\) 0 0
\(612\) 37.2308 + 15.4215i 1.50497 + 0.623378i
\(613\) 43.7567 22.2952i 1.76732 0.900493i 0.825474 0.564440i \(-0.190908\pi\)
0.941843 0.336053i \(-0.109092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.11219 + 10.0332i 0.205809 + 0.403923i 0.970720 0.240213i \(-0.0772172\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.8209 + 15.1273i −0.832836 + 0.605091i
\(626\) −45.6619 + 10.9624i −1.82501 + 0.438147i
\(627\) 0 0
\(628\) 18.9921 + 16.2207i 0.757865 + 0.647278i
\(629\) 29.6592 + 71.6036i 1.18259 + 2.85502i
\(630\) 0 0
\(631\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 38.0349 32.4849i 1.51056 1.29014i
\(635\) 0 0
\(636\) 0 0
\(637\) −30.7683 36.0250i −1.21908 1.42736i
\(638\) 0 0
\(639\) 0 0
\(640\) −33.7668 + 10.9715i −1.33475 + 0.433686i
\(641\) −6.16043 + 25.6600i −0.243322 + 1.01351i 0.708220 + 0.705992i \(0.249498\pi\)
−0.951542 + 0.307518i \(0.900502\pi\)
\(642\) 0 0
\(643\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) −3.98217 + 25.1424i −0.156434 + 0.987688i
\(649\) 0 0
\(650\) −3.64081 46.2608i −0.142804 1.81450i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.758775 0.314295i −0.0296932 0.0122993i 0.367787 0.929910i \(-0.380115\pi\)
−0.397481 + 0.917611i \(0.630115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −20.0000 16.0000i −0.780869 0.624695i
\(657\) 49.4130 1.92778
\(658\) 0 0
\(659\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(660\) 0 0
\(661\) 5.22451 + 32.9862i 0.203210 + 1.28302i 0.852601 + 0.522562i \(0.175024\pi\)
−0.649391 + 0.760454i \(0.724976\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −39.6074 + 28.7765i −1.53476 + 1.11507i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 33.3607 + 20.4435i 1.28596 + 0.788037i 0.986133 0.165957i \(-0.0530712\pi\)
0.299827 + 0.953994i \(0.403071\pi\)
\(674\) 10.8692 33.4521i 0.418668 1.28853i
\(675\) 0 0
\(676\) −58.4604 29.7871i −2.24848 1.14566i
\(677\) −14.6107 + 28.6751i −0.561535 + 1.10208i 0.419411 + 0.907796i \(0.362237\pi\)
−0.980946 + 0.194279i \(0.937763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 31.1490 50.8306i 1.19451 1.94926i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(684\) 0 0
\(685\) −7.07241 + 4.33398i −0.270223 + 0.165593i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.38490 + 33.9989i −0.205148 + 1.29526i
\(690\) 0 0
\(691\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(692\) −30.3572 + 4.80810i −1.15401 + 0.182777i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 42.9837 1.38100i 1.62813 0.0523091i
\(698\) 44.0497 1.66731
\(699\) 0 0
\(700\) 0 0
\(701\) −29.9020 + 41.1565i −1.12938 + 1.55446i −0.340142 + 0.940374i \(0.610475\pi\)
−0.789239 + 0.614086i \(0.789525\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −37.5581 37.5581i −1.41352 1.41352i
\(707\) 0 0
\(708\) 0 0
\(709\) −3.01144 4.91422i −0.113097 0.184557i 0.791083 0.611708i \(-0.209517\pi\)
−0.904180 + 0.427151i \(0.859517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 8.56948 + 2.05735i 0.321155 + 0.0771025i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(720\) 35.8150 + 11.6370i 1.33475 + 0.433686i
\(721\) 0 0
\(722\) −25.5549 + 8.30330i −0.951057 + 0.309017i
\(723\) 0 0
\(724\) −47.6676 + 19.7445i −1.77155 + 0.733800i
\(725\) −23.6852 + 27.7318i −0.879646 + 1.02993i
\(726\) 0 0
\(727\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(728\) 0 0
\(729\) 19.0919 19.0919i 0.707107 0.707107i
\(730\) 11.4352 72.1992i 0.423237 2.67221i
\(731\) 0 0
\(732\) 0 0
\(733\) −12.2456 + 1.93951i −0.452301 + 0.0716374i −0.378429 0.925630i \(-0.623536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 7.18680 + 26.1983i 0.264550 + 0.964372i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 32.8804 + 64.5315i 1.20871 + 2.37223i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(744\) 0 0
\(745\) −50.6166 3.98361i −1.85445 0.145948i
\(746\) 45.3772 + 7.18705i 1.66138 + 0.263136i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 22.2490 + 68.4755i 0.810263 + 2.49373i
\(755\) 0 0
\(756\) 0 0
\(757\) 40.4931 34.5844i 1.47175 1.25699i 0.572239 0.820087i \(-0.306075\pi\)
0.899508 0.436904i \(-0.143925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.9303 4.85114i −0.541222 0.175854i 0.0256326 0.999671i \(-0.491840\pi\)
−0.566855 + 0.823818i \(0.691840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −58.4186 + 24.1978i −2.11213 + 0.874872i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −28.6873 39.4846i −1.03449 1.42385i −0.901523 0.432731i \(-0.857550\pi\)
−0.132966 0.991121i \(-0.542450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.29046 54.5155i 0.154417 1.96205i
\(773\) −3.78565 48.1012i −0.136160 1.73008i −0.563609 0.826042i \(-0.690588\pi\)
0.427449 0.904040i \(-0.359412\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −42.4008 17.5630i −1.52210 0.630475i
\(777\) 0 0
\(778\) 3.37213i 0.120897i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −4.38017 27.6553i −0.156434 0.987688i
\(785\) −39.0691 + 3.07481i −1.39444 + 0.109745i
\(786\) 0 0
\(787\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(788\) −15.4570 15.4570i −0.550631 0.550631i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 30.4431 + 73.4962i 1.08107 + 2.60993i
\(794\) −29.0182 6.96666i −1.02982 0.247238i
\(795\) 0 0
\(796\) 0 0
\(797\) −17.3832 + 53.5001i −0.615746 + 1.89507i −0.226106 + 0.974103i \(0.572599\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 12.4509 24.4363i 0.440206 0.863952i
\(801\) −6.07077 7.10796i −0.214500 0.251147i
\(802\) −50.3355 16.3550i −1.77741 0.577515i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 33.5439 20.5557i 1.18007 0.723148i
\(809\) −13.2567 55.2181i −0.466081 1.94137i −0.290290 0.956939i \(-0.593752\pi\)
−0.175791 0.984428i \(-0.556248\pi\)
\(810\) −23.4776 32.3142i −0.824920 1.13540i
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −15.3521 + 7.82229i −0.536774 + 0.273500i
\(819\) 0 0
\(820\) 39.9425 4.43806i 1.39485 0.154984i
\(821\) −35.4770 −1.23816 −0.619078 0.785330i \(-0.712493\pi\)
−0.619078 + 0.785330i \(0.712493\pi\)
\(822\) 0 0
\(823\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(828\) 0 0
\(829\) −11.0026 11.0026i −0.382137 0.382137i 0.489735 0.871872i \(-0.337094\pi\)
−0.871872 + 0.489735i \(0.837094\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −28.2902 46.1654i −0.980785 1.60050i
\(833\) 35.7503 + 30.5337i 1.23867 + 1.05793i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(840\) 0 0
\(841\) 12.5237 24.5791i 0.431851 0.847556i
\(842\) 37.4500 + 43.8483i 1.29061 + 1.51111i
\(843\) 0 0
\(844\) 0 0
\(845\) 97.9119 31.8135i 3.36827 1.09442i
\(846\) 0 0
\(847\) 0 0
\(848\) −13.2126 + 15.4700i −0.453723 + 0.531242i
\(849\) 0 0
\(850\) 10.7502 + 44.7777i 0.368728 + 1.53586i
\(851\) 0 0
\(852\) 0 0
\(853\) 1.10616 6.98401i 0.0378742 0.239128i −0.961488 0.274848i \(-0.911372\pi\)
0.999362 + 0.0357200i \(0.0113725\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −44.7470 32.5106i −1.52853 1.11054i −0.957050 0.289922i \(-0.906370\pi\)
−0.571477 0.820618i \(-0.693630\pi\)
\(858\) 0 0
\(859\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(864\) 0 0
\(865\) 28.3470 39.0164i 0.963828 1.32660i
\(866\) −2.72530 17.2069i −0.0926096 0.584714i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 12.1351 2.91338i 0.410947 0.0986596i
\(873\) 25.4343 + 41.5051i 0.860822 + 1.40473i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.01180 + 6.19167i 0.0679335 + 0.209078i 0.979260 0.202606i \(-0.0649409\pi\)
−0.911327 + 0.411683i \(0.864941\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −11.3407 5.77836i −0.382077 0.194678i 0.252394 0.967624i \(-0.418782\pi\)
−0.634471 + 0.772947i \(0.718782\pi\)
\(882\) −13.4828 + 26.4615i −0.453990 + 0.891007i
\(883\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(884\) 86.4635 + 28.0937i 2.90808 + 0.944894i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −11.7906 + 7.22530i −0.395223 + 0.242193i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −48.0487 34.9094i −1.60341 1.16494i
\(899\) 0 0
\(900\) −25.9186 + 13.2062i −0.863952 + 0.440206i
\(901\) 34.1602i 1.13804i
\(902\) 0 0
\(903\) 0 0
\(904\) 14.5418 + 28.5399i 0.483653 + 0.949222i
\(905\) 30.9810 74.7947i 1.02984 2.48626i
\(906\) 0 0
\(907\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(908\) 0 0
\(909\) −41.5990 3.27392i −1.37975 0.108589i
\(910\) 0 0
\(911\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 23.3728 + 38.1410i 0.773105 + 1.26159i
\(915\) 0 0
\(916\) −8.33702 20.1273i −0.275463 0.665026i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −43.3509 22.0884i −1.42769 0.727442i
\(923\) 0 0
\(924\) 0 0
\(925\) −53.2069 17.2880i −1.74943 0.568425i
\(926\) 0 0
\(927\) 0 0
\(928\) −9.93377 + 41.3771i −0.326092 + 1.35827i
\(929\) 16.9269 7.01134i 0.555352 0.230034i −0.0873137 0.996181i \(-0.527828\pi\)
0.642666 + 0.766146i \(0.277828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.49480 27.0528i −0.212744 0.886143i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −4.50578 + 57.2514i −0.147276 + 1.87132i
\(937\) 4.62031 + 58.7067i 0.150939 + 1.91786i 0.348040 + 0.937480i \(0.386847\pi\)
−0.197101 + 0.980383i \(0.563153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.51646 4.84888i 0.310228 0.158069i −0.291944 0.956435i \(-0.594302\pi\)
0.602172 + 0.798366i \(0.294302\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(948\) 0 0
\(949\) 111.132 8.74628i 3.60750 0.283916i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −17.1618 + 12.4688i −0.555927 + 0.403904i −0.829966 0.557814i \(-0.811640\pi\)
0.274039 + 0.961719i \(0.411640\pi\)
\(954\) 20.9823 5.03740i 0.679326 0.163092i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9.57953 + 29.4828i −0.309017 + 0.951057i
\(962\) −83.9854 + 71.7303i −2.70780 + 2.31268i
\(963\) 0 0
\(964\) −5.89431 + 11.5682i −0.189843 + 0.372588i
\(965\) 55.7254 + 65.2460i 1.79386 + 2.10034i
\(966\) 0 0
\(967\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(968\) 29.5899 9.61435i 0.951057 0.309017i
\(969\) 0 0
\(970\) 66.5307 27.5579i 2.13617 0.884832i
\(971\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −7.35498 + 46.4375i −0.235427 + 1.48643i
\(977\) 1.28166 16.2850i 0.0410038 0.521003i −0.941778 0.336234i \(-0.890847\pi\)
0.982782 0.184768i \(-0.0591534\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 35.5438 + 25.8241i 1.13540 + 0.824920i
\(981\) −12.2294 5.06557i −0.390453 0.161731i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 34.2994 1.09287
\(986\) −32.4378 63.6628i −1.03303 2.02744i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −47.7158 40.7531i −1.51117 1.29066i −0.828589 0.559857i \(-0.810856\pi\)
−0.682585 0.730807i \(-0.739144\pi\)
\(998\) 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 164.2.o.a.7.1 16
4.3 odd 2 CM 164.2.o.a.7.1 16
41.6 odd 40 inner 164.2.o.a.47.1 yes 16
164.47 even 40 inner 164.2.o.a.47.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.2.o.a.7.1 16 1.1 even 1 trivial
164.2.o.a.7.1 16 4.3 odd 2 CM
164.2.o.a.47.1 yes 16 41.6 odd 40 inner
164.2.o.a.47.1 yes 16 164.47 even 40 inner