Properties

Label 8046.2.a.g.1.4
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.a (trivial)

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: yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.67390\) of defining polynomial
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.102080 q^{5} +0.350570 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.102080 q^{5} +0.350570 q^{7} -1.00000 q^{8} +0.102080 q^{10} -2.02675 q^{11} -0.0298835 q^{13} -0.350570 q^{14} +1.00000 q^{16} +3.57487 q^{17} -2.36593 q^{19} -0.102080 q^{20} +2.02675 q^{22} +3.59145 q^{23} -4.98958 q^{25} +0.0298835 q^{26} +0.350570 q^{28} +4.55651 q^{29} +0.839401 q^{31} -1.00000 q^{32} -3.57487 q^{34} -0.0357860 q^{35} -3.75798 q^{37} +2.36593 q^{38} +0.102080 q^{40} -1.43684 q^{41} -5.01492 q^{43} -2.02675 q^{44} -3.59145 q^{46} -6.28480 q^{47} -6.87710 q^{49} +4.98958 q^{50} -0.0298835 q^{52} +9.52893 q^{53} +0.206890 q^{55} -0.350570 q^{56} -4.55651 q^{58} +9.78689 q^{59} -7.40365 q^{61} -0.839401 q^{62} +1.00000 q^{64} +0.00305050 q^{65} -4.06800 q^{67} +3.57487 q^{68} +0.0357860 q^{70} -9.49501 q^{71} +7.96509 q^{73} +3.75798 q^{74} -2.36593 q^{76} -0.710518 q^{77} +9.24244 q^{79} -0.102080 q^{80} +1.43684 q^{82} +9.62022 q^{83} -0.364921 q^{85} +5.01492 q^{86} +2.02675 q^{88} +0.871304 q^{89} -0.0104763 q^{91} +3.59145 q^{92} +6.28480 q^{94} +0.241512 q^{95} -12.9918 q^{97} +6.87710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.102080 −0.0456513 −0.0228257 0.999739i \(-0.507266\pi\)
−0.0228257 + 0.999739i \(0.507266\pi\)
\(6\) 0 0
\(7\) 0.350570 0.132503 0.0662515 0.997803i \(-0.478896\pi\)
0.0662515 + 0.997803i \(0.478896\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.102080 0.0322804
\(11\) −2.02675 −0.611089 −0.305544 0.952178i \(-0.598838\pi\)
−0.305544 + 0.952178i \(0.598838\pi\)
\(12\) 0 0
\(13\) −0.0298835 −0.00828821 −0.00414410 0.999991i \(-0.501319\pi\)
−0.00414410 + 0.999991i \(0.501319\pi\)
\(14\) −0.350570 −0.0936937
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.57487 0.867033 0.433516 0.901146i \(-0.357273\pi\)
0.433516 + 0.901146i \(0.357273\pi\)
\(18\) 0 0
\(19\) −2.36593 −0.542780 −0.271390 0.962469i \(-0.587483\pi\)
−0.271390 + 0.962469i \(0.587483\pi\)
\(20\) −0.102080 −0.0228257
\(21\) 0 0
\(22\) 2.02675 0.432105
\(23\) 3.59145 0.748869 0.374435 0.927253i \(-0.377837\pi\)
0.374435 + 0.927253i \(0.377837\pi\)
\(24\) 0 0
\(25\) −4.98958 −0.997916
\(26\) 0.0298835 0.00586065
\(27\) 0 0
\(28\) 0.350570 0.0662515
\(29\) 4.55651 0.846122 0.423061 0.906101i \(-0.360956\pi\)
0.423061 + 0.906101i \(0.360956\pi\)
\(30\) 0 0
\(31\) 0.839401 0.150761 0.0753805 0.997155i \(-0.475983\pi\)
0.0753805 + 0.997155i \(0.475983\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.57487 −0.613085
\(35\) −0.0357860 −0.00604894
\(36\) 0 0
\(37\) −3.75798 −0.617808 −0.308904 0.951093i \(-0.599962\pi\)
−0.308904 + 0.951093i \(0.599962\pi\)
\(38\) 2.36593 0.383804
\(39\) 0 0
\(40\) 0.102080 0.0161402
\(41\) −1.43684 −0.224396 −0.112198 0.993686i \(-0.535789\pi\)
−0.112198 + 0.993686i \(0.535789\pi\)
\(42\) 0 0
\(43\) −5.01492 −0.764768 −0.382384 0.924003i \(-0.624897\pi\)
−0.382384 + 0.924003i \(0.624897\pi\)
\(44\) −2.02675 −0.305544
\(45\) 0 0
\(46\) −3.59145 −0.529531
\(47\) −6.28480 −0.916732 −0.458366 0.888764i \(-0.651565\pi\)
−0.458366 + 0.888764i \(0.651565\pi\)
\(48\) 0 0
\(49\) −6.87710 −0.982443
\(50\) 4.98958 0.705633
\(51\) 0 0
\(52\) −0.0298835 −0.00414410
\(53\) 9.52893 1.30890 0.654449 0.756106i \(-0.272900\pi\)
0.654449 + 0.756106i \(0.272900\pi\)
\(54\) 0 0
\(55\) 0.206890 0.0278970
\(56\) −0.350570 −0.0468469
\(57\) 0 0
\(58\) −4.55651 −0.598299
\(59\) 9.78689 1.27414 0.637072 0.770804i \(-0.280145\pi\)
0.637072 + 0.770804i \(0.280145\pi\)
\(60\) 0 0
\(61\) −7.40365 −0.947941 −0.473970 0.880541i \(-0.657180\pi\)
−0.473970 + 0.880541i \(0.657180\pi\)
\(62\) −0.839401 −0.106604
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.00305050 0.000378368 0
\(66\) 0 0
\(67\) −4.06800 −0.496985 −0.248493 0.968634i \(-0.579935\pi\)
−0.248493 + 0.968634i \(0.579935\pi\)
\(68\) 3.57487 0.433516
\(69\) 0 0
\(70\) 0.0357860 0.00427724
\(71\) −9.49501 −1.12685 −0.563425 0.826167i \(-0.690517\pi\)
−0.563425 + 0.826167i \(0.690517\pi\)
\(72\) 0 0
\(73\) 7.96509 0.932243 0.466122 0.884721i \(-0.345651\pi\)
0.466122 + 0.884721i \(0.345651\pi\)
\(74\) 3.75798 0.436856
\(75\) 0 0
\(76\) −2.36593 −0.271390
\(77\) −0.710518 −0.0809710
\(78\) 0 0
\(79\) 9.24244 1.03986 0.519928 0.854210i \(-0.325959\pi\)
0.519928 + 0.854210i \(0.325959\pi\)
\(80\) −0.102080 −0.0114128
\(81\) 0 0
\(82\) 1.43684 0.158672
\(83\) 9.62022 1.05596 0.527978 0.849258i \(-0.322950\pi\)
0.527978 + 0.849258i \(0.322950\pi\)
\(84\) 0 0
\(85\) −0.364921 −0.0395812
\(86\) 5.01492 0.540773
\(87\) 0 0
\(88\) 2.02675 0.216052
\(89\) 0.871304 0.0923580 0.0461790 0.998933i \(-0.485296\pi\)
0.0461790 + 0.998933i \(0.485296\pi\)
\(90\) 0 0
\(91\) −0.0104763 −0.00109821
\(92\) 3.59145 0.374435
\(93\) 0 0
\(94\) 6.28480 0.648227
\(95\) 0.241512 0.0247787
\(96\) 0 0
\(97\) −12.9918 −1.31912 −0.659559 0.751653i \(-0.729257\pi\)
−0.659559 + 0.751653i \(0.729257\pi\)
\(98\) 6.87710 0.694692
\(99\) 0 0
\(100\) −4.98958 −0.498958
\(101\) −6.60527 −0.657249 −0.328624 0.944461i \(-0.606585\pi\)
−0.328624 + 0.944461i \(0.606585\pi\)
\(102\) 0 0
\(103\) 5.27651 0.519910 0.259955 0.965621i \(-0.416292\pi\)
0.259955 + 0.965621i \(0.416292\pi\)
\(104\) 0.0298835 0.00293032
\(105\) 0 0
\(106\) −9.52893 −0.925531
\(107\) −8.42424 −0.814402 −0.407201 0.913339i \(-0.633495\pi\)
−0.407201 + 0.913339i \(0.633495\pi\)
\(108\) 0 0
\(109\) 3.75345 0.359516 0.179758 0.983711i \(-0.442469\pi\)
0.179758 + 0.983711i \(0.442469\pi\)
\(110\) −0.206890 −0.0197262
\(111\) 0 0
\(112\) 0.350570 0.0331257
\(113\) −4.46292 −0.419836 −0.209918 0.977719i \(-0.567320\pi\)
−0.209918 + 0.977719i \(0.567320\pi\)
\(114\) 0 0
\(115\) −0.366614 −0.0341869
\(116\) 4.55651 0.423061
\(117\) 0 0
\(118\) −9.78689 −0.900956
\(119\) 1.25324 0.114884
\(120\) 0 0
\(121\) −6.89228 −0.626571
\(122\) 7.40365 0.670295
\(123\) 0 0
\(124\) 0.839401 0.0753805
\(125\) 1.01973 0.0912075
\(126\) 0 0
\(127\) −1.36720 −0.121320 −0.0606598 0.998158i \(-0.519320\pi\)
−0.0606598 + 0.998158i \(0.519320\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −0.00305050 −0.000267546 0
\(131\) 8.75704 0.765106 0.382553 0.923934i \(-0.375045\pi\)
0.382553 + 0.923934i \(0.375045\pi\)
\(132\) 0 0
\(133\) −0.829422 −0.0719200
\(134\) 4.06800 0.351422
\(135\) 0 0
\(136\) −3.57487 −0.306542
\(137\) 7.96214 0.680251 0.340126 0.940380i \(-0.389530\pi\)
0.340126 + 0.940380i \(0.389530\pi\)
\(138\) 0 0
\(139\) 3.31018 0.280766 0.140383 0.990097i \(-0.455167\pi\)
0.140383 + 0.990097i \(0.455167\pi\)
\(140\) −0.0357860 −0.00302447
\(141\) 0 0
\(142\) 9.49501 0.796804
\(143\) 0.0605665 0.00506483
\(144\) 0 0
\(145\) −0.465126 −0.0386266
\(146\) −7.96509 −0.659196
\(147\) 0 0
\(148\) −3.75798 −0.308904
\(149\) 1.00000 0.0819232
\(150\) 0 0
\(151\) 6.90585 0.561990 0.280995 0.959709i \(-0.409336\pi\)
0.280995 + 0.959709i \(0.409336\pi\)
\(152\) 2.36593 0.191902
\(153\) 0 0
\(154\) 0.710518 0.0572552
\(155\) −0.0856857 −0.00688244
\(156\) 0 0
\(157\) −14.0313 −1.11982 −0.559909 0.828554i \(-0.689164\pi\)
−0.559909 + 0.828554i \(0.689164\pi\)
\(158\) −9.24244 −0.735289
\(159\) 0 0
\(160\) 0.102080 0.00807009
\(161\) 1.25905 0.0992274
\(162\) 0 0
\(163\) −20.7001 −1.62135 −0.810677 0.585494i \(-0.800900\pi\)
−0.810677 + 0.585494i \(0.800900\pi\)
\(164\) −1.43684 −0.112198
\(165\) 0 0
\(166\) −9.62022 −0.746674
\(167\) −13.5549 −1.04891 −0.524455 0.851438i \(-0.675731\pi\)
−0.524455 + 0.851438i \(0.675731\pi\)
\(168\) 0 0
\(169\) −12.9991 −0.999931
\(170\) 0.364921 0.0279881
\(171\) 0 0
\(172\) −5.01492 −0.382384
\(173\) −24.3064 −1.84798 −0.923991 0.382415i \(-0.875093\pi\)
−0.923991 + 0.382415i \(0.875093\pi\)
\(174\) 0 0
\(175\) −1.74920 −0.132227
\(176\) −2.02675 −0.152772
\(177\) 0 0
\(178\) −0.871304 −0.0653070
\(179\) 19.8845 1.48623 0.743117 0.669162i \(-0.233347\pi\)
0.743117 + 0.669162i \(0.233347\pi\)
\(180\) 0 0
\(181\) 1.35829 0.100961 0.0504803 0.998725i \(-0.483925\pi\)
0.0504803 + 0.998725i \(0.483925\pi\)
\(182\) 0.0104763 0.000776553 0
\(183\) 0 0
\(184\) −3.59145 −0.264765
\(185\) 0.383613 0.0282038
\(186\) 0 0
\(187\) −7.24537 −0.529834
\(188\) −6.28480 −0.458366
\(189\) 0 0
\(190\) −0.241512 −0.0175212
\(191\) −12.0574 −0.872440 −0.436220 0.899840i \(-0.643683\pi\)
−0.436220 + 0.899840i \(0.643683\pi\)
\(192\) 0 0
\(193\) 17.1757 1.23634 0.618168 0.786046i \(-0.287875\pi\)
0.618168 + 0.786046i \(0.287875\pi\)
\(194\) 12.9918 0.932758
\(195\) 0 0
\(196\) −6.87710 −0.491221
\(197\) −15.6315 −1.11370 −0.556848 0.830614i \(-0.687989\pi\)
−0.556848 + 0.830614i \(0.687989\pi\)
\(198\) 0 0
\(199\) −15.7562 −1.11692 −0.558462 0.829530i \(-0.688609\pi\)
−0.558462 + 0.829530i \(0.688609\pi\)
\(200\) 4.98958 0.352817
\(201\) 0 0
\(202\) 6.60527 0.464745
\(203\) 1.59737 0.112114
\(204\) 0 0
\(205\) 0.146672 0.0102440
\(206\) −5.27651 −0.367632
\(207\) 0 0
\(208\) −0.0298835 −0.00207205
\(209\) 4.79514 0.331687
\(210\) 0 0
\(211\) 8.83252 0.608056 0.304028 0.952663i \(-0.401668\pi\)
0.304028 + 0.952663i \(0.401668\pi\)
\(212\) 9.52893 0.654449
\(213\) 0 0
\(214\) 8.42424 0.575869
\(215\) 0.511921 0.0349127
\(216\) 0 0
\(217\) 0.294269 0.0199763
\(218\) −3.75345 −0.254216
\(219\) 0 0
\(220\) 0.206890 0.0139485
\(221\) −0.106830 −0.00718614
\(222\) 0 0
\(223\) 20.9511 1.40299 0.701493 0.712676i \(-0.252517\pi\)
0.701493 + 0.712676i \(0.252517\pi\)
\(224\) −0.350570 −0.0234234
\(225\) 0 0
\(226\) 4.46292 0.296869
\(227\) −4.91723 −0.326368 −0.163184 0.986596i \(-0.552176\pi\)
−0.163184 + 0.986596i \(0.552176\pi\)
\(228\) 0 0
\(229\) −7.53842 −0.498153 −0.249077 0.968484i \(-0.580127\pi\)
−0.249077 + 0.968484i \(0.580127\pi\)
\(230\) 0.366614 0.0241738
\(231\) 0 0
\(232\) −4.55651 −0.299149
\(233\) −11.9221 −0.781041 −0.390521 0.920594i \(-0.627705\pi\)
−0.390521 + 0.920594i \(0.627705\pi\)
\(234\) 0 0
\(235\) 0.641549 0.0418500
\(236\) 9.78689 0.637072
\(237\) 0 0
\(238\) −1.25324 −0.0812355
\(239\) 16.1316 1.04347 0.521734 0.853108i \(-0.325285\pi\)
0.521734 + 0.853108i \(0.325285\pi\)
\(240\) 0 0
\(241\) 20.6366 1.32932 0.664662 0.747144i \(-0.268576\pi\)
0.664662 + 0.747144i \(0.268576\pi\)
\(242\) 6.89228 0.443052
\(243\) 0 0
\(244\) −7.40365 −0.473970
\(245\) 0.702011 0.0448498
\(246\) 0 0
\(247\) 0.0707022 0.00449868
\(248\) −0.839401 −0.0533020
\(249\) 0 0
\(250\) −1.01973 −0.0644935
\(251\) −4.82748 −0.304708 −0.152354 0.988326i \(-0.548685\pi\)
−0.152354 + 0.988326i \(0.548685\pi\)
\(252\) 0 0
\(253\) −7.27898 −0.457626
\(254\) 1.36720 0.0857859
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.38502 0.211152 0.105576 0.994411i \(-0.466331\pi\)
0.105576 + 0.994411i \(0.466331\pi\)
\(258\) 0 0
\(259\) −1.31743 −0.0818614
\(260\) 0.00305050 0.000189184 0
\(261\) 0 0
\(262\) −8.75704 −0.541012
\(263\) −15.5141 −0.956642 −0.478321 0.878185i \(-0.658754\pi\)
−0.478321 + 0.878185i \(0.658754\pi\)
\(264\) 0 0
\(265\) −0.972708 −0.0597530
\(266\) 0.829422 0.0508551
\(267\) 0 0
\(268\) −4.06800 −0.248493
\(269\) 16.6115 1.01282 0.506410 0.862293i \(-0.330972\pi\)
0.506410 + 0.862293i \(0.330972\pi\)
\(270\) 0 0
\(271\) −23.4563 −1.42487 −0.712435 0.701738i \(-0.752408\pi\)
−0.712435 + 0.701738i \(0.752408\pi\)
\(272\) 3.57487 0.216758
\(273\) 0 0
\(274\) −7.96214 −0.481010
\(275\) 10.1126 0.609815
\(276\) 0 0
\(277\) −13.5781 −0.815826 −0.407913 0.913021i \(-0.633743\pi\)
−0.407913 + 0.913021i \(0.633743\pi\)
\(278\) −3.31018 −0.198532
\(279\) 0 0
\(280\) 0.0357860 0.00213862
\(281\) −8.60505 −0.513334 −0.256667 0.966500i \(-0.582624\pi\)
−0.256667 + 0.966500i \(0.582624\pi\)
\(282\) 0 0
\(283\) −18.6623 −1.10936 −0.554680 0.832063i \(-0.687159\pi\)
−0.554680 + 0.832063i \(0.687159\pi\)
\(284\) −9.49501 −0.563425
\(285\) 0 0
\(286\) −0.0605665 −0.00358137
\(287\) −0.503712 −0.0297332
\(288\) 0 0
\(289\) −4.22032 −0.248254
\(290\) 0.465126 0.0273131
\(291\) 0 0
\(292\) 7.96509 0.466122
\(293\) 8.92222 0.521241 0.260621 0.965441i \(-0.416073\pi\)
0.260621 + 0.965441i \(0.416073\pi\)
\(294\) 0 0
\(295\) −0.999041 −0.0581664
\(296\) 3.75798 0.218428
\(297\) 0 0
\(298\) −1.00000 −0.0579284
\(299\) −0.107325 −0.00620678
\(300\) 0 0
\(301\) −1.75808 −0.101334
\(302\) −6.90585 −0.397387
\(303\) 0 0
\(304\) −2.36593 −0.135695
\(305\) 0.755761 0.0432748
\(306\) 0 0
\(307\) 15.6841 0.895139 0.447570 0.894249i \(-0.352290\pi\)
0.447570 + 0.894249i \(0.352290\pi\)
\(308\) −0.710518 −0.0404855
\(309\) 0 0
\(310\) 0.0856857 0.00486662
\(311\) 3.64245 0.206544 0.103272 0.994653i \(-0.467069\pi\)
0.103272 + 0.994653i \(0.467069\pi\)
\(312\) 0 0
\(313\) −31.7926 −1.79703 −0.898513 0.438947i \(-0.855352\pi\)
−0.898513 + 0.438947i \(0.855352\pi\)
\(314\) 14.0313 0.791830
\(315\) 0 0
\(316\) 9.24244 0.519928
\(317\) 1.50799 0.0846972 0.0423486 0.999103i \(-0.486516\pi\)
0.0423486 + 0.999103i \(0.486516\pi\)
\(318\) 0 0
\(319\) −9.23491 −0.517056
\(320\) −0.102080 −0.00570642
\(321\) 0 0
\(322\) −1.25905 −0.0701644
\(323\) −8.45787 −0.470608
\(324\) 0 0
\(325\) 0.149106 0.00827093
\(326\) 20.7001 1.14647
\(327\) 0 0
\(328\) 1.43684 0.0793360
\(329\) −2.20326 −0.121470
\(330\) 0 0
\(331\) 1.26926 0.0697647 0.0348824 0.999391i \(-0.488894\pi\)
0.0348824 + 0.999391i \(0.488894\pi\)
\(332\) 9.62022 0.527978
\(333\) 0 0
\(334\) 13.5549 0.741692
\(335\) 0.415260 0.0226881
\(336\) 0 0
\(337\) 14.2539 0.776457 0.388229 0.921563i \(-0.373087\pi\)
0.388229 + 0.921563i \(0.373087\pi\)
\(338\) 12.9991 0.707058
\(339\) 0 0
\(340\) −0.364921 −0.0197906
\(341\) −1.70126 −0.0921283
\(342\) 0 0
\(343\) −4.86489 −0.262680
\(344\) 5.01492 0.270386
\(345\) 0 0
\(346\) 24.3064 1.30672
\(347\) −17.9926 −0.965891 −0.482945 0.875650i \(-0.660433\pi\)
−0.482945 + 0.875650i \(0.660433\pi\)
\(348\) 0 0
\(349\) −30.2747 −1.62057 −0.810284 0.586037i \(-0.800687\pi\)
−0.810284 + 0.586037i \(0.800687\pi\)
\(350\) 1.74920 0.0934985
\(351\) 0 0
\(352\) 2.02675 0.108026
\(353\) 16.5740 0.882146 0.441073 0.897471i \(-0.354598\pi\)
0.441073 + 0.897471i \(0.354598\pi\)
\(354\) 0 0
\(355\) 0.969246 0.0514422
\(356\) 0.871304 0.0461790
\(357\) 0 0
\(358\) −19.8845 −1.05093
\(359\) 23.0600 1.21706 0.608531 0.793530i \(-0.291759\pi\)
0.608531 + 0.793530i \(0.291759\pi\)
\(360\) 0 0
\(361\) −13.4024 −0.705389
\(362\) −1.35829 −0.0713899
\(363\) 0 0
\(364\) −0.0104763 −0.000549106 0
\(365\) −0.813073 −0.0425582
\(366\) 0 0
\(367\) −10.3939 −0.542558 −0.271279 0.962501i \(-0.587447\pi\)
−0.271279 + 0.962501i \(0.587447\pi\)
\(368\) 3.59145 0.187217
\(369\) 0 0
\(370\) −0.383613 −0.0199431
\(371\) 3.34055 0.173433
\(372\) 0 0
\(373\) 17.6768 0.915269 0.457634 0.889140i \(-0.348697\pi\)
0.457634 + 0.889140i \(0.348697\pi\)
\(374\) 7.24537 0.374649
\(375\) 0 0
\(376\) 6.28480 0.324114
\(377\) −0.136165 −0.00701284
\(378\) 0 0
\(379\) −6.04973 −0.310754 −0.155377 0.987855i \(-0.549659\pi\)
−0.155377 + 0.987855i \(0.549659\pi\)
\(380\) 0.241512 0.0123893
\(381\) 0 0
\(382\) 12.0574 0.616908
\(383\) 20.8782 1.06682 0.533412 0.845855i \(-0.320909\pi\)
0.533412 + 0.845855i \(0.320909\pi\)
\(384\) 0 0
\(385\) 0.0725293 0.00369644
\(386\) −17.1757 −0.874222
\(387\) 0 0
\(388\) −12.9918 −0.659559
\(389\) −14.4959 −0.734971 −0.367486 0.930029i \(-0.619781\pi\)
−0.367486 + 0.930029i \(0.619781\pi\)
\(390\) 0 0
\(391\) 12.8390 0.649294
\(392\) 6.87710 0.347346
\(393\) 0 0
\(394\) 15.6315 0.787502
\(395\) −0.943464 −0.0474708
\(396\) 0 0
\(397\) −6.17054 −0.309690 −0.154845 0.987939i \(-0.549488\pi\)
−0.154845 + 0.987939i \(0.549488\pi\)
\(398\) 15.7562 0.789785
\(399\) 0 0
\(400\) −4.98958 −0.249479
\(401\) −10.3024 −0.514480 −0.257240 0.966348i \(-0.582813\pi\)
−0.257240 + 0.966348i \(0.582813\pi\)
\(402\) 0 0
\(403\) −0.0250843 −0.00124954
\(404\) −6.60527 −0.328624
\(405\) 0 0
\(406\) −1.59737 −0.0792764
\(407\) 7.61649 0.377536
\(408\) 0 0
\(409\) 9.18360 0.454100 0.227050 0.973883i \(-0.427092\pi\)
0.227050 + 0.973883i \(0.427092\pi\)
\(410\) −0.146672 −0.00724359
\(411\) 0 0
\(412\) 5.27651 0.259955
\(413\) 3.43099 0.168828
\(414\) 0 0
\(415\) −0.982027 −0.0482058
\(416\) 0.0298835 0.00146516
\(417\) 0 0
\(418\) −4.79514 −0.234538
\(419\) −23.3723 −1.14181 −0.570906 0.821015i \(-0.693408\pi\)
−0.570906 + 0.821015i \(0.693408\pi\)
\(420\) 0 0
\(421\) −22.5192 −1.09752 −0.548760 0.835980i \(-0.684900\pi\)
−0.548760 + 0.835980i \(0.684900\pi\)
\(422\) −8.83252 −0.429960
\(423\) 0 0
\(424\) −9.52893 −0.462766
\(425\) −17.8371 −0.865226
\(426\) 0 0
\(427\) −2.59550 −0.125605
\(428\) −8.42424 −0.407201
\(429\) 0 0
\(430\) −0.511921 −0.0246870
\(431\) −4.03344 −0.194284 −0.0971421 0.995271i \(-0.530970\pi\)
−0.0971421 + 0.995271i \(0.530970\pi\)
\(432\) 0 0
\(433\) 4.71921 0.226791 0.113395 0.993550i \(-0.463827\pi\)
0.113395 + 0.993550i \(0.463827\pi\)
\(434\) −0.294269 −0.0141254
\(435\) 0 0
\(436\) 3.75345 0.179758
\(437\) −8.49711 −0.406472
\(438\) 0 0
\(439\) −32.7549 −1.56331 −0.781653 0.623713i \(-0.785623\pi\)
−0.781653 + 0.623713i \(0.785623\pi\)
\(440\) −0.206890 −0.00986309
\(441\) 0 0
\(442\) 0.106830 0.00508137
\(443\) 35.0469 1.66513 0.832565 0.553927i \(-0.186871\pi\)
0.832565 + 0.553927i \(0.186871\pi\)
\(444\) 0 0
\(445\) −0.0889423 −0.00421627
\(446\) −20.9511 −0.992061
\(447\) 0 0
\(448\) 0.350570 0.0165629
\(449\) −32.0551 −1.51278 −0.756388 0.654123i \(-0.773038\pi\)
−0.756388 + 0.654123i \(0.773038\pi\)
\(450\) 0 0
\(451\) 2.91211 0.137126
\(452\) −4.46292 −0.209918
\(453\) 0 0
\(454\) 4.91723 0.230777
\(455\) 0.00106941 5.01348e−5 0
\(456\) 0 0
\(457\) 24.0443 1.12475 0.562373 0.826884i \(-0.309888\pi\)
0.562373 + 0.826884i \(0.309888\pi\)
\(458\) 7.53842 0.352247
\(459\) 0 0
\(460\) −0.366614 −0.0170934
\(461\) −7.50713 −0.349642 −0.174821 0.984600i \(-0.555935\pi\)
−0.174821 + 0.984600i \(0.555935\pi\)
\(462\) 0 0
\(463\) −37.7034 −1.75223 −0.876114 0.482105i \(-0.839872\pi\)
−0.876114 + 0.482105i \(0.839872\pi\)
\(464\) 4.55651 0.211531
\(465\) 0 0
\(466\) 11.9221 0.552280
\(467\) 11.1921 0.517909 0.258954 0.965890i \(-0.416622\pi\)
0.258954 + 0.965890i \(0.416622\pi\)
\(468\) 0 0
\(469\) −1.42612 −0.0658520
\(470\) −0.641549 −0.0295925
\(471\) 0 0
\(472\) −9.78689 −0.450478
\(473\) 10.1640 0.467341
\(474\) 0 0
\(475\) 11.8050 0.541649
\(476\) 1.25324 0.0574422
\(477\) 0 0
\(478\) −16.1316 −0.737843
\(479\) −37.4425 −1.71079 −0.855396 0.517975i \(-0.826686\pi\)
−0.855396 + 0.517975i \(0.826686\pi\)
\(480\) 0 0
\(481\) 0.112302 0.00512052
\(482\) −20.6366 −0.939974
\(483\) 0 0
\(484\) −6.89228 −0.313285
\(485\) 1.32620 0.0602195
\(486\) 0 0
\(487\) −9.80226 −0.444183 −0.222091 0.975026i \(-0.571288\pi\)
−0.222091 + 0.975026i \(0.571288\pi\)
\(488\) 7.40365 0.335148
\(489\) 0 0
\(490\) −0.702011 −0.0317136
\(491\) 32.0256 1.44529 0.722647 0.691217i \(-0.242925\pi\)
0.722647 + 0.691217i \(0.242925\pi\)
\(492\) 0 0
\(493\) 16.2889 0.733616
\(494\) −0.0707022 −0.00318104
\(495\) 0 0
\(496\) 0.839401 0.0376902
\(497\) −3.32866 −0.149311
\(498\) 0 0
\(499\) 13.3658 0.598336 0.299168 0.954201i \(-0.403291\pi\)
0.299168 + 0.954201i \(0.403291\pi\)
\(500\) 1.01973 0.0456038
\(501\) 0 0
\(502\) 4.82748 0.215461
\(503\) −0.515588 −0.0229889 −0.0114945 0.999934i \(-0.503659\pi\)
−0.0114945 + 0.999934i \(0.503659\pi\)
\(504\) 0 0
\(505\) 0.674262 0.0300043
\(506\) 7.27898 0.323590
\(507\) 0 0
\(508\) −1.36720 −0.0606598
\(509\) −16.3872 −0.726351 −0.363176 0.931721i \(-0.618307\pi\)
−0.363176 + 0.931721i \(0.618307\pi\)
\(510\) 0 0
\(511\) 2.79232 0.123525
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −3.38502 −0.149307
\(515\) −0.538624 −0.0237346
\(516\) 0 0
\(517\) 12.7377 0.560205
\(518\) 1.31743 0.0578847
\(519\) 0 0
\(520\) −0.00305050 −0.000133773 0
\(521\) 31.8411 1.39499 0.697493 0.716592i \(-0.254299\pi\)
0.697493 + 0.716592i \(0.254299\pi\)
\(522\) 0 0
\(523\) −7.22133 −0.315766 −0.157883 0.987458i \(-0.550467\pi\)
−0.157883 + 0.987458i \(0.550467\pi\)
\(524\) 8.75704 0.382553
\(525\) 0 0
\(526\) 15.5141 0.676448
\(527\) 3.00075 0.130715
\(528\) 0 0
\(529\) −10.1015 −0.439195
\(530\) 0.972708 0.0422517
\(531\) 0 0
\(532\) −0.829422 −0.0359600
\(533\) 0.0429378 0.00185984
\(534\) 0 0
\(535\) 0.859942 0.0371786
\(536\) 4.06800 0.175711
\(537\) 0 0
\(538\) −16.6115 −0.716172
\(539\) 13.9382 0.600360
\(540\) 0 0
\(541\) −24.9022 −1.07063 −0.535314 0.844653i \(-0.679807\pi\)
−0.535314 + 0.844653i \(0.679807\pi\)
\(542\) 23.4563 1.00754
\(543\) 0 0
\(544\) −3.57487 −0.153271
\(545\) −0.383151 −0.0164124
\(546\) 0 0
\(547\) 31.6658 1.35393 0.676966 0.736014i \(-0.263294\pi\)
0.676966 + 0.736014i \(0.263294\pi\)
\(548\) 7.96214 0.340126
\(549\) 0 0
\(550\) −10.1126 −0.431204
\(551\) −10.7804 −0.459259
\(552\) 0 0
\(553\) 3.24012 0.137784
\(554\) 13.5781 0.576876
\(555\) 0 0
\(556\) 3.31018 0.140383
\(557\) 28.4674 1.20620 0.603101 0.797665i \(-0.293932\pi\)
0.603101 + 0.797665i \(0.293932\pi\)
\(558\) 0 0
\(559\) 0.149864 0.00633856
\(560\) −0.0357860 −0.00151223
\(561\) 0 0
\(562\) 8.60505 0.362982
\(563\) −18.4847 −0.779037 −0.389518 0.921019i \(-0.627358\pi\)
−0.389518 + 0.921019i \(0.627358\pi\)
\(564\) 0 0
\(565\) 0.455572 0.0191661
\(566\) 18.6623 0.784437
\(567\) 0 0
\(568\) 9.49501 0.398402
\(569\) −30.2299 −1.26730 −0.633651 0.773619i \(-0.718444\pi\)
−0.633651 + 0.773619i \(0.718444\pi\)
\(570\) 0 0
\(571\) −5.28489 −0.221166 −0.110583 0.993867i \(-0.535272\pi\)
−0.110583 + 0.993867i \(0.535272\pi\)
\(572\) 0.0605665 0.00253241
\(573\) 0 0
\(574\) 0.503712 0.0210245
\(575\) −17.9198 −0.747309
\(576\) 0 0
\(577\) −41.7286 −1.73718 −0.868592 0.495527i \(-0.834975\pi\)
−0.868592 + 0.495527i \(0.834975\pi\)
\(578\) 4.22032 0.175542
\(579\) 0 0
\(580\) −0.465126 −0.0193133
\(581\) 3.37256 0.139917
\(582\) 0 0
\(583\) −19.3128 −0.799853
\(584\) −7.96509 −0.329598
\(585\) 0 0
\(586\) −8.92222 −0.368573
\(587\) −6.32683 −0.261136 −0.130568 0.991439i \(-0.541680\pi\)
−0.130568 + 0.991439i \(0.541680\pi\)
\(588\) 0 0
\(589\) −1.98596 −0.0818301
\(590\) 0.999041 0.0411299
\(591\) 0 0
\(592\) −3.75798 −0.154452
\(593\) −29.1140 −1.19557 −0.597784 0.801657i \(-0.703952\pi\)
−0.597784 + 0.801657i \(0.703952\pi\)
\(594\) 0 0
\(595\) −0.127930 −0.00524463
\(596\) 1.00000 0.0409616
\(597\) 0 0
\(598\) 0.107325 0.00438886
\(599\) 23.2937 0.951755 0.475878 0.879511i \(-0.342130\pi\)
0.475878 + 0.879511i \(0.342130\pi\)
\(600\) 0 0
\(601\) −5.33860 −0.217766 −0.108883 0.994055i \(-0.534727\pi\)
−0.108883 + 0.994055i \(0.534727\pi\)
\(602\) 1.75808 0.0716540
\(603\) 0 0
\(604\) 6.90585 0.280995
\(605\) 0.703560 0.0286038
\(606\) 0 0
\(607\) 21.5161 0.873310 0.436655 0.899629i \(-0.356163\pi\)
0.436655 + 0.899629i \(0.356163\pi\)
\(608\) 2.36593 0.0959509
\(609\) 0 0
\(610\) −0.755761 −0.0305999
\(611\) 0.187812 0.00759806
\(612\) 0 0
\(613\) 12.2745 0.495762 0.247881 0.968791i \(-0.420266\pi\)
0.247881 + 0.968791i \(0.420266\pi\)
\(614\) −15.6841 −0.632959
\(615\) 0 0
\(616\) 0.710518 0.0286276
\(617\) 4.83318 0.194577 0.0972883 0.995256i \(-0.468983\pi\)
0.0972883 + 0.995256i \(0.468983\pi\)
\(618\) 0 0
\(619\) −4.13677 −0.166271 −0.0831353 0.996538i \(-0.526493\pi\)
−0.0831353 + 0.996538i \(0.526493\pi\)
\(620\) −0.0856857 −0.00344122
\(621\) 0 0
\(622\) −3.64245 −0.146049
\(623\) 0.305453 0.0122377
\(624\) 0 0
\(625\) 24.8438 0.993752
\(626\) 31.7926 1.27069
\(627\) 0 0
\(628\) −14.0313 −0.559909
\(629\) −13.4343 −0.535660
\(630\) 0 0
\(631\) 10.8814 0.433182 0.216591 0.976262i \(-0.430506\pi\)
0.216591 + 0.976262i \(0.430506\pi\)
\(632\) −9.24244 −0.367645
\(633\) 0 0
\(634\) −1.50799 −0.0598899
\(635\) 0.139563 0.00553840
\(636\) 0 0
\(637\) 0.205512 0.00814269
\(638\) 9.23491 0.365614
\(639\) 0 0
\(640\) 0.102080 0.00403505
\(641\) −46.5892 −1.84016 −0.920081 0.391728i \(-0.871878\pi\)
−0.920081 + 0.391728i \(0.871878\pi\)
\(642\) 0 0
\(643\) 18.1726 0.716655 0.358328 0.933596i \(-0.383347\pi\)
0.358328 + 0.933596i \(0.383347\pi\)
\(644\) 1.25905 0.0496137
\(645\) 0 0
\(646\) 8.45787 0.332770
\(647\) −23.8592 −0.938002 −0.469001 0.883198i \(-0.655386\pi\)
−0.469001 + 0.883198i \(0.655386\pi\)
\(648\) 0 0
\(649\) −19.8356 −0.778615
\(650\) −0.149106 −0.00584843
\(651\) 0 0
\(652\) −20.7001 −0.810677
\(653\) 7.62071 0.298222 0.149111 0.988820i \(-0.452359\pi\)
0.149111 + 0.988820i \(0.452359\pi\)
\(654\) 0 0
\(655\) −0.893914 −0.0349281
\(656\) −1.43684 −0.0560991
\(657\) 0 0
\(658\) 2.20326 0.0858920
\(659\) −12.4087 −0.483376 −0.241688 0.970354i \(-0.577701\pi\)
−0.241688 + 0.970354i \(0.577701\pi\)
\(660\) 0 0
\(661\) 25.9018 1.00746 0.503732 0.863860i \(-0.331960\pi\)
0.503732 + 0.863860i \(0.331960\pi\)
\(662\) −1.26926 −0.0493311
\(663\) 0 0
\(664\) −9.62022 −0.373337
\(665\) 0.0846670 0.00328324
\(666\) 0 0
\(667\) 16.3645 0.633635
\(668\) −13.5549 −0.524455
\(669\) 0 0
\(670\) −0.415260 −0.0160429
\(671\) 15.0054 0.579276
\(672\) 0 0
\(673\) −7.27877 −0.280576 −0.140288 0.990111i \(-0.544803\pi\)
−0.140288 + 0.990111i \(0.544803\pi\)
\(674\) −14.2539 −0.549038
\(675\) 0 0
\(676\) −12.9991 −0.499966
\(677\) −50.8122 −1.95287 −0.976435 0.215812i \(-0.930760\pi\)
−0.976435 + 0.215812i \(0.930760\pi\)
\(678\) 0 0
\(679\) −4.55454 −0.174787
\(680\) 0.364921 0.0139941
\(681\) 0 0
\(682\) 1.70126 0.0651445
\(683\) −42.7539 −1.63593 −0.817965 0.575267i \(-0.804898\pi\)
−0.817965 + 0.575267i \(0.804898\pi\)
\(684\) 0 0
\(685\) −0.812771 −0.0310544
\(686\) 4.86489 0.185742
\(687\) 0 0
\(688\) −5.01492 −0.191192
\(689\) −0.284758 −0.0108484
\(690\) 0 0
\(691\) 0.949122 0.0361063 0.0180532 0.999837i \(-0.494253\pi\)
0.0180532 + 0.999837i \(0.494253\pi\)
\(692\) −24.3064 −0.923991
\(693\) 0 0
\(694\) 17.9926 0.682988
\(695\) −0.337902 −0.0128174
\(696\) 0 0
\(697\) −5.13650 −0.194559
\(698\) 30.2747 1.14591
\(699\) 0 0
\(700\) −1.74920 −0.0661134
\(701\) −24.8882 −0.940013 −0.470006 0.882663i \(-0.655748\pi\)
−0.470006 + 0.882663i \(0.655748\pi\)
\(702\) 0 0
\(703\) 8.89110 0.335334
\(704\) −2.02675 −0.0763861
\(705\) 0 0
\(706\) −16.5740 −0.623771
\(707\) −2.31561 −0.0870874
\(708\) 0 0
\(709\) 15.5052 0.582311 0.291155 0.956676i \(-0.405960\pi\)
0.291155 + 0.956676i \(0.405960\pi\)
\(710\) −0.969246 −0.0363752
\(711\) 0 0
\(712\) −0.871304 −0.0326535
\(713\) 3.01467 0.112900
\(714\) 0 0
\(715\) −0.00618260 −0.000231216 0
\(716\) 19.8845 0.743117
\(717\) 0 0
\(718\) −23.0600 −0.860593
\(719\) −13.6341 −0.508465 −0.254233 0.967143i \(-0.581823\pi\)
−0.254233 + 0.967143i \(0.581823\pi\)
\(720\) 0 0
\(721\) 1.84979 0.0688896
\(722\) 13.4024 0.498786
\(723\) 0 0
\(724\) 1.35829 0.0504803
\(725\) −22.7351 −0.844359
\(726\) 0 0
\(727\) 9.59451 0.355841 0.177920 0.984045i \(-0.443063\pi\)
0.177920 + 0.984045i \(0.443063\pi\)
\(728\) 0.0104763 0.000388276 0
\(729\) 0 0
\(730\) 0.813073 0.0300932
\(731\) −17.9277 −0.663079
\(732\) 0 0
\(733\) 18.3649 0.678324 0.339162 0.940728i \(-0.389857\pi\)
0.339162 + 0.940728i \(0.389857\pi\)
\(734\) 10.3939 0.383646
\(735\) 0 0
\(736\) −3.59145 −0.132383
\(737\) 8.24483 0.303702
\(738\) 0 0
\(739\) −39.9117 −1.46817 −0.734087 0.679055i \(-0.762390\pi\)
−0.734087 + 0.679055i \(0.762390\pi\)
\(740\) 0.383613 0.0141019
\(741\) 0 0
\(742\) −3.34055 −0.122636
\(743\) 8.04457 0.295127 0.147563 0.989053i \(-0.452857\pi\)
0.147563 + 0.989053i \(0.452857\pi\)
\(744\) 0 0
\(745\) −0.102080 −0.00373990
\(746\) −17.6768 −0.647193
\(747\) 0 0
\(748\) −7.24537 −0.264917
\(749\) −2.95328 −0.107911
\(750\) 0 0
\(751\) 10.3350 0.377128 0.188564 0.982061i \(-0.439617\pi\)
0.188564 + 0.982061i \(0.439617\pi\)
\(752\) −6.28480 −0.229183
\(753\) 0 0
\(754\) 0.136165 0.00495882
\(755\) −0.704946 −0.0256556
\(756\) 0 0
\(757\) −49.2360 −1.78951 −0.894756 0.446556i \(-0.852651\pi\)
−0.894756 + 0.446556i \(0.852651\pi\)
\(758\) 6.04973 0.219736
\(759\) 0 0
\(760\) −0.241512 −0.00876058
\(761\) −8.72729 −0.316364 −0.158182 0.987410i \(-0.550563\pi\)
−0.158182 + 0.987410i \(0.550563\pi\)
\(762\) 0 0
\(763\) 1.31585 0.0476369
\(764\) −12.0574 −0.436220
\(765\) 0 0
\(766\) −20.8782 −0.754359
\(767\) −0.292467 −0.0105604
\(768\) 0 0
\(769\) −32.5167 −1.17258 −0.586290 0.810101i \(-0.699412\pi\)
−0.586290 + 0.810101i \(0.699412\pi\)
\(770\) −0.0725293 −0.00261378
\(771\) 0 0
\(772\) 17.1757 0.618168
\(773\) −30.6093 −1.10094 −0.550469 0.834855i \(-0.685551\pi\)
−0.550469 + 0.834855i \(0.685551\pi\)
\(774\) 0 0
\(775\) −4.18826 −0.150447
\(776\) 12.9918 0.466379
\(777\) 0 0
\(778\) 14.4959 0.519703
\(779\) 3.39945 0.121798
\(780\) 0 0
\(781\) 19.2440 0.688606
\(782\) −12.8390 −0.459120
\(783\) 0 0
\(784\) −6.87710 −0.245611
\(785\) 1.43231 0.0511212
\(786\) 0 0
\(787\) −24.9239 −0.888440 −0.444220 0.895918i \(-0.646519\pi\)
−0.444220 + 0.895918i \(0.646519\pi\)
\(788\) −15.6315 −0.556848
\(789\) 0 0
\(790\) 0.943464 0.0335669
\(791\) −1.56456 −0.0556295
\(792\) 0 0
\(793\) 0.221247 0.00785673
\(794\) 6.17054 0.218984
\(795\) 0 0
\(796\) −15.7562 −0.558462
\(797\) 21.6064 0.765338 0.382669 0.923886i \(-0.375005\pi\)
0.382669 + 0.923886i \(0.375005\pi\)
\(798\) 0 0
\(799\) −22.4673 −0.794837
\(800\) 4.98958 0.176408
\(801\) 0 0
\(802\) 10.3024 0.363792
\(803\) −16.1433 −0.569683
\(804\) 0 0
\(805\) −0.128524 −0.00452986
\(806\) 0.0250843 0.000883556 0
\(807\) 0 0
\(808\) 6.60527 0.232373
\(809\) 16.4003 0.576603 0.288302 0.957540i \(-0.406909\pi\)
0.288302 + 0.957540i \(0.406909\pi\)
\(810\) 0 0
\(811\) −11.7483 −0.412539 −0.206269 0.978495i \(-0.566132\pi\)
−0.206269 + 0.978495i \(0.566132\pi\)
\(812\) 1.59737 0.0560569
\(813\) 0 0
\(814\) −7.61649 −0.266958
\(815\) 2.11305 0.0740169
\(816\) 0 0
\(817\) 11.8649 0.415101
\(818\) −9.18360 −0.321097
\(819\) 0 0
\(820\) 0.146672 0.00512199
\(821\) 51.3053 1.79057 0.895284 0.445495i \(-0.146972\pi\)
0.895284 + 0.445495i \(0.146972\pi\)
\(822\) 0 0
\(823\) −0.892424 −0.0311079 −0.0155540 0.999879i \(-0.504951\pi\)
−0.0155540 + 0.999879i \(0.504951\pi\)
\(824\) −5.27651 −0.183816
\(825\) 0 0
\(826\) −3.43099 −0.119379
\(827\) 21.4142 0.744644 0.372322 0.928104i \(-0.378562\pi\)
0.372322 + 0.928104i \(0.378562\pi\)
\(828\) 0 0
\(829\) −10.1286 −0.351782 −0.175891 0.984410i \(-0.556281\pi\)
−0.175891 + 0.984410i \(0.556281\pi\)
\(830\) 0.982027 0.0340867
\(831\) 0 0
\(832\) −0.0298835 −0.00103603
\(833\) −24.5847 −0.851810
\(834\) 0 0
\(835\) 1.38368 0.0478842
\(836\) 4.79514 0.165843
\(837\) 0 0
\(838\) 23.3723 0.807384
\(839\) −13.1240 −0.453091 −0.226545 0.974001i \(-0.572743\pi\)
−0.226545 + 0.974001i \(0.572743\pi\)
\(840\) 0 0
\(841\) −8.23823 −0.284077
\(842\) 22.5192 0.776064
\(843\) 0 0
\(844\) 8.83252 0.304028
\(845\) 1.32694 0.0456482
\(846\) 0 0
\(847\) −2.41622 −0.0830224
\(848\) 9.52893 0.327225
\(849\) 0 0
\(850\) 17.8371 0.611807
\(851\) −13.4966 −0.462658
\(852\) 0 0
\(853\) 24.7181 0.846333 0.423167 0.906052i \(-0.360919\pi\)
0.423167 + 0.906052i \(0.360919\pi\)
\(854\) 2.59550 0.0888161
\(855\) 0 0
\(856\) 8.42424 0.287935
\(857\) −43.6328 −1.49047 −0.745234 0.666803i \(-0.767662\pi\)
−0.745234 + 0.666803i \(0.767662\pi\)
\(858\) 0 0
\(859\) 43.6531 1.48942 0.744712 0.667386i \(-0.232587\pi\)
0.744712 + 0.667386i \(0.232587\pi\)
\(860\) 0.511921 0.0174563
\(861\) 0 0
\(862\) 4.03344 0.137380
\(863\) −9.57447 −0.325919 −0.162959 0.986633i \(-0.552104\pi\)
−0.162959 + 0.986633i \(0.552104\pi\)
\(864\) 0 0
\(865\) 2.48119 0.0843628
\(866\) −4.71921 −0.160365
\(867\) 0 0
\(868\) 0.294269 0.00998813
\(869\) −18.7321 −0.635444
\(870\) 0 0
\(871\) 0.121566 0.00411912
\(872\) −3.75345 −0.127108
\(873\) 0 0
\(874\) 8.49711 0.287419
\(875\) 0.357487 0.0120853
\(876\) 0 0
\(877\) −17.1383 −0.578719 −0.289360 0.957220i \(-0.593442\pi\)
−0.289360 + 0.957220i \(0.593442\pi\)
\(878\) 32.7549 1.10542
\(879\) 0 0
\(880\) 0.206890 0.00697425
\(881\) 19.2478 0.648475 0.324237 0.945976i \(-0.394892\pi\)
0.324237 + 0.945976i \(0.394892\pi\)
\(882\) 0 0
\(883\) −30.4527 −1.02482 −0.512408 0.858742i \(-0.671246\pi\)
−0.512408 + 0.858742i \(0.671246\pi\)
\(884\) −0.106830 −0.00359307
\(885\) 0 0
\(886\) −35.0469 −1.17742
\(887\) 27.3879 0.919595 0.459797 0.888024i \(-0.347922\pi\)
0.459797 + 0.888024i \(0.347922\pi\)
\(888\) 0 0
\(889\) −0.479300 −0.0160752
\(890\) 0.0889423 0.00298135
\(891\) 0 0
\(892\) 20.9511 0.701493
\(893\) 14.8694 0.497584
\(894\) 0 0
\(895\) −2.02980 −0.0678486
\(896\) −0.350570 −0.0117117
\(897\) 0 0
\(898\) 32.0551 1.06969
\(899\) 3.82474 0.127562
\(900\) 0 0
\(901\) 34.0646 1.13486
\(902\) −2.91211 −0.0969627
\(903\) 0 0
\(904\) 4.46292 0.148434
\(905\) −0.138653 −0.00460899
\(906\) 0 0
\(907\) 23.6464 0.785166 0.392583 0.919717i \(-0.371582\pi\)
0.392583 + 0.919717i \(0.371582\pi\)
\(908\) −4.91723 −0.163184
\(909\) 0 0
\(910\) −0.00106941 −3.54507e−5 0
\(911\) −40.4265 −1.33939 −0.669695 0.742636i \(-0.733575\pi\)
−0.669695 + 0.742636i \(0.733575\pi\)
\(912\) 0 0
\(913\) −19.4978 −0.645283
\(914\) −24.0443 −0.795316
\(915\) 0 0
\(916\) −7.53842 −0.249077
\(917\) 3.06995 0.101379
\(918\) 0 0
\(919\) −22.4929 −0.741973 −0.370986 0.928638i \(-0.620980\pi\)
−0.370986 + 0.928638i \(0.620980\pi\)
\(920\) 0.366614 0.0120869
\(921\) 0 0
\(922\) 7.50713 0.247234
\(923\) 0.283745 0.00933957
\(924\) 0 0
\(925\) 18.7507 0.616521
\(926\) 37.7034 1.23901
\(927\) 0 0
\(928\) −4.55651 −0.149575
\(929\) 29.1055 0.954920 0.477460 0.878653i \(-0.341558\pi\)
0.477460 + 0.878653i \(0.341558\pi\)
\(930\) 0 0
\(931\) 16.2707 0.533251
\(932\) −11.9221 −0.390521
\(933\) 0 0
\(934\) −11.1921 −0.366217
\(935\) 0.739604 0.0241876
\(936\) 0 0
\(937\) −6.36306 −0.207872 −0.103936 0.994584i \(-0.533144\pi\)
−0.103936 + 0.994584i \(0.533144\pi\)
\(938\) 1.42612 0.0465644
\(939\) 0 0
\(940\) 0.641549 0.0209250
\(941\) −35.5770 −1.15978 −0.579889 0.814696i \(-0.696904\pi\)
−0.579889 + 0.814696i \(0.696904\pi\)
\(942\) 0 0
\(943\) −5.16033 −0.168043
\(944\) 9.78689 0.318536
\(945\) 0 0
\(946\) −10.1640 −0.330460
\(947\) −28.1479 −0.914684 −0.457342 0.889291i \(-0.651198\pi\)
−0.457342 + 0.889291i \(0.651198\pi\)
\(948\) 0 0
\(949\) −0.238025 −0.00772663
\(950\) −11.8050 −0.383004
\(951\) 0 0
\(952\) −1.25324 −0.0406178
\(953\) −47.8502 −1.55002 −0.775010 0.631949i \(-0.782255\pi\)
−0.775010 + 0.631949i \(0.782255\pi\)
\(954\) 0 0
\(955\) 1.23081 0.0398281
\(956\) 16.1316 0.521734
\(957\) 0 0
\(958\) 37.4425 1.20971
\(959\) 2.79128 0.0901353
\(960\) 0 0
\(961\) −30.2954 −0.977271
\(962\) −0.112302 −0.00362076
\(963\) 0 0
\(964\) 20.6366 0.664662
\(965\) −1.75329 −0.0564404
\(966\) 0 0
\(967\) 11.0594 0.355646 0.177823 0.984063i \(-0.443095\pi\)
0.177823 + 0.984063i \(0.443095\pi\)
\(968\) 6.89228 0.221526
\(969\) 0 0
\(970\) −1.32620 −0.0425816
\(971\) 11.6714 0.374552 0.187276 0.982307i \(-0.440034\pi\)
0.187276 + 0.982307i \(0.440034\pi\)
\(972\) 0 0
\(973\) 1.16045 0.0372023
\(974\) 9.80226 0.314085
\(975\) 0 0
\(976\) −7.40365 −0.236985
\(977\) 29.4805 0.943163 0.471582 0.881822i \(-0.343683\pi\)
0.471582 + 0.881822i \(0.343683\pi\)
\(978\) 0 0
\(979\) −1.76592 −0.0564389
\(980\) 0.702011 0.0224249
\(981\) 0 0
\(982\) −32.0256 −1.02198
\(983\) 28.4486 0.907371 0.453685 0.891162i \(-0.350109\pi\)
0.453685 + 0.891162i \(0.350109\pi\)
\(984\) 0 0
\(985\) 1.59565 0.0508417
\(986\) −16.2889 −0.518745
\(987\) 0 0
\(988\) 0.0707022 0.00224934
\(989\) −18.0108 −0.572712
\(990\) 0 0
\(991\) −52.8403 −1.67853 −0.839264 0.543724i \(-0.817014\pi\)
−0.839264 + 0.543724i \(0.817014\pi\)
\(992\) −0.839401 −0.0266510
\(993\) 0 0
\(994\) 3.32866 0.105579
\(995\) 1.60838 0.0509891
\(996\) 0 0
\(997\) 21.3350 0.675685 0.337843 0.941203i \(-0.390303\pi\)
0.337843 + 0.941203i \(0.390303\pi\)
\(998\) −13.3658 −0.423087
\(999\) 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 8046.2.a.g.1.4 9
3.2 odd 2 8046.2.a.h.1.6 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.g.1.4 9 1.1 even 1 trivial
8046.2.a.h.1.6 yes 9 3.2 odd 2