Properties

Label 5586.2.a.bm.1.2
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
Dimension $2$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.267949 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.267949 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.267949 q^{10} -1.00000 q^{11} +1.00000 q^{12} -5.46410 q^{13} -0.267949 q^{15} +1.00000 q^{16} -4.73205 q^{17} +1.00000 q^{18} +1.00000 q^{19} -0.267949 q^{20} -1.00000 q^{22} -2.73205 q^{23} +1.00000 q^{24} -4.92820 q^{25} -5.46410 q^{26} +1.00000 q^{27} +1.00000 q^{29} -0.267949 q^{30} +2.46410 q^{31} +1.00000 q^{32} -1.00000 q^{33} -4.73205 q^{34} +1.00000 q^{36} +6.19615 q^{37} +1.00000 q^{38} -5.46410 q^{39} -0.267949 q^{40} -5.26795 q^{41} +0.196152 q^{43} -1.00000 q^{44} -0.267949 q^{45} -2.73205 q^{46} -11.4641 q^{47} +1.00000 q^{48} -4.92820 q^{50} -4.73205 q^{51} -5.46410 q^{52} +1.53590 q^{53} +1.00000 q^{54} +0.267949 q^{55} +1.00000 q^{57} +1.00000 q^{58} +1.73205 q^{59} -0.267949 q^{60} -4.53590 q^{61} +2.46410 q^{62} +1.00000 q^{64} +1.46410 q^{65} -1.00000 q^{66} -0.928203 q^{67} -4.73205 q^{68} -2.73205 q^{69} -12.1962 q^{71} +1.00000 q^{72} -6.92820 q^{73} +6.19615 q^{74} -4.92820 q^{75} +1.00000 q^{76} -5.46410 q^{78} -11.3923 q^{79} -0.267949 q^{80} +1.00000 q^{81} -5.26795 q^{82} +3.00000 q^{83} +1.26795 q^{85} +0.196152 q^{86} +1.00000 q^{87} -1.00000 q^{88} -0.339746 q^{89} -0.267949 q^{90} -2.73205 q^{92} +2.46410 q^{93} -11.4641 q^{94} -0.267949 q^{95} +1.00000 q^{96} -5.73205 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} - 4 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{23} + 2 q^{24} + 4 q^{25} - 4 q^{26} + 2 q^{27} + 2 q^{29} - 4 q^{30} - 2 q^{31} + 2 q^{32} - 2 q^{33} - 6 q^{34} + 2 q^{36} + 2 q^{37} + 2 q^{38} - 4 q^{39} - 4 q^{40} - 14 q^{41} - 10 q^{43} - 2 q^{44} - 4 q^{45} - 2 q^{46} - 16 q^{47} + 2 q^{48} + 4 q^{50} - 6 q^{51} - 4 q^{52} + 10 q^{53} + 2 q^{54} + 4 q^{55} + 2 q^{57} + 2 q^{58} - 4 q^{60} - 16 q^{61} - 2 q^{62} + 2 q^{64} - 4 q^{65} - 2 q^{66} + 12 q^{67} - 6 q^{68} - 2 q^{69} - 14 q^{71} + 2 q^{72} + 2 q^{74} + 4 q^{75} + 2 q^{76} - 4 q^{78} - 2 q^{79} - 4 q^{80} + 2 q^{81} - 14 q^{82} + 6 q^{83} + 6 q^{85} - 10 q^{86} + 2 q^{87} - 2 q^{88} - 18 q^{89} - 4 q^{90} - 2 q^{92} - 2 q^{93} - 16 q^{94} - 4 q^{95} + 2 q^{96} - 8 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −0.267949 −0.119831 −0.0599153 0.998203i \(-0.519083\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.267949 −0.0847330
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 0 0
\(15\) −0.267949 −0.0691842
\(16\) 1.00000 0.250000
\(17\) −4.73205 −1.14769 −0.573845 0.818964i \(-0.694549\pi\)
−0.573845 + 0.818964i \(0.694549\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) −0.267949 −0.0599153
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −2.73205 −0.569672 −0.284836 0.958576i \(-0.591939\pi\)
−0.284836 + 0.958576i \(0.591939\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.92820 −0.985641
\(26\) −5.46410 −1.07160
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) −0.267949 −0.0489206
\(31\) 2.46410 0.442566 0.221283 0.975210i \(-0.428976\pi\)
0.221283 + 0.975210i \(0.428976\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.73205 −0.811540
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.19615 1.01864 0.509321 0.860577i \(-0.329897\pi\)
0.509321 + 0.860577i \(0.329897\pi\)
\(38\) 1.00000 0.162221
\(39\) −5.46410 −0.874957
\(40\) −0.267949 −0.0423665
\(41\) −5.26795 −0.822715 −0.411358 0.911474i \(-0.634945\pi\)
−0.411358 + 0.911474i \(0.634945\pi\)
\(42\) 0 0
\(43\) 0.196152 0.0299130 0.0149565 0.999888i \(-0.495239\pi\)
0.0149565 + 0.999888i \(0.495239\pi\)
\(44\) −1.00000 −0.150756
\(45\) −0.267949 −0.0399435
\(46\) −2.73205 −0.402819
\(47\) −11.4641 −1.67221 −0.836106 0.548569i \(-0.815173\pi\)
−0.836106 + 0.548569i \(0.815173\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.92820 −0.696953
\(51\) −4.73205 −0.662620
\(52\) −5.46410 −0.757735
\(53\) 1.53590 0.210972 0.105486 0.994421i \(-0.466360\pi\)
0.105486 + 0.994421i \(0.466360\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.267949 0.0361303
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 1.00000 0.131306
\(59\) 1.73205 0.225494 0.112747 0.993624i \(-0.464035\pi\)
0.112747 + 0.993624i \(0.464035\pi\)
\(60\) −0.267949 −0.0345921
\(61\) −4.53590 −0.580762 −0.290381 0.956911i \(-0.593782\pi\)
−0.290381 + 0.956911i \(0.593782\pi\)
\(62\) 2.46410 0.312941
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.46410 0.181599
\(66\) −1.00000 −0.123091
\(67\) −0.928203 −0.113398 −0.0566990 0.998391i \(-0.518058\pi\)
−0.0566990 + 0.998391i \(0.518058\pi\)
\(68\) −4.73205 −0.573845
\(69\) −2.73205 −0.328900
\(70\) 0 0
\(71\) −12.1962 −1.44742 −0.723708 0.690106i \(-0.757564\pi\)
−0.723708 + 0.690106i \(0.757564\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.92820 −0.810885 −0.405442 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) 6.19615 0.720288
\(75\) −4.92820 −0.569060
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −5.46410 −0.618688
\(79\) −11.3923 −1.28173 −0.640867 0.767652i \(-0.721425\pi\)
−0.640867 + 0.767652i \(0.721425\pi\)
\(80\) −0.267949 −0.0299576
\(81\) 1.00000 0.111111
\(82\) −5.26795 −0.581748
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 1.26795 0.137528
\(86\) 0.196152 0.0211517
\(87\) 1.00000 0.107211
\(88\) −1.00000 −0.106600
\(89\) −0.339746 −0.0360130 −0.0180065 0.999838i \(-0.505732\pi\)
−0.0180065 + 0.999838i \(0.505732\pi\)
\(90\) −0.267949 −0.0282443
\(91\) 0 0
\(92\) −2.73205 −0.284836
\(93\) 2.46410 0.255515
\(94\) −11.4641 −1.18243
\(95\) −0.267949 −0.0274910
\(96\) 1.00000 0.102062
\(97\) −5.73205 −0.582002 −0.291001 0.956723i \(-0.593988\pi\)
−0.291001 + 0.956723i \(0.593988\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −4.92820 −0.492820
\(101\) 4.92820 0.490375 0.245187 0.969476i \(-0.421151\pi\)
0.245187 + 0.969476i \(0.421151\pi\)
\(102\) −4.73205 −0.468543
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −5.46410 −0.535799
\(105\) 0 0
\(106\) 1.53590 0.149180
\(107\) −1.33975 −0.129518 −0.0647591 0.997901i \(-0.520628\pi\)
−0.0647591 + 0.997901i \(0.520628\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.7321 1.02794 0.513972 0.857807i \(-0.328174\pi\)
0.513972 + 0.857807i \(0.328174\pi\)
\(110\) 0.267949 0.0255480
\(111\) 6.19615 0.588113
\(112\) 0 0
\(113\) −5.26795 −0.495567 −0.247783 0.968815i \(-0.579702\pi\)
−0.247783 + 0.968815i \(0.579702\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0.732051 0.0682641
\(116\) 1.00000 0.0928477
\(117\) −5.46410 −0.505156
\(118\) 1.73205 0.159448
\(119\) 0 0
\(120\) −0.267949 −0.0244603
\(121\) −10.0000 −0.909091
\(122\) −4.53590 −0.410661
\(123\) −5.26795 −0.474995
\(124\) 2.46410 0.221283
\(125\) 2.66025 0.237940
\(126\) 0 0
\(127\) 3.92820 0.348572 0.174286 0.984695i \(-0.444238\pi\)
0.174286 + 0.984695i \(0.444238\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.196152 0.0172703
\(130\) 1.46410 0.128410
\(131\) 0.464102 0.0405487 0.0202744 0.999794i \(-0.493546\pi\)
0.0202744 + 0.999794i \(0.493546\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −0.928203 −0.0801845
\(135\) −0.267949 −0.0230614
\(136\) −4.73205 −0.405770
\(137\) 11.8564 1.01296 0.506481 0.862251i \(-0.330946\pi\)
0.506481 + 0.862251i \(0.330946\pi\)
\(138\) −2.73205 −0.232568
\(139\) −8.19615 −0.695189 −0.347594 0.937645i \(-0.613001\pi\)
−0.347594 + 0.937645i \(0.613001\pi\)
\(140\) 0 0
\(141\) −11.4641 −0.965452
\(142\) −12.1962 −1.02348
\(143\) 5.46410 0.456931
\(144\) 1.00000 0.0833333
\(145\) −0.267949 −0.0222520
\(146\) −6.92820 −0.573382
\(147\) 0 0
\(148\) 6.19615 0.509321
\(149\) 1.60770 0.131708 0.0658538 0.997829i \(-0.479023\pi\)
0.0658538 + 0.997829i \(0.479023\pi\)
\(150\) −4.92820 −0.402386
\(151\) 15.3923 1.25261 0.626304 0.779579i \(-0.284567\pi\)
0.626304 + 0.779579i \(0.284567\pi\)
\(152\) 1.00000 0.0811107
\(153\) −4.73205 −0.382564
\(154\) 0 0
\(155\) −0.660254 −0.0530329
\(156\) −5.46410 −0.437478
\(157\) −1.46410 −0.116848 −0.0584240 0.998292i \(-0.518608\pi\)
−0.0584240 + 0.998292i \(0.518608\pi\)
\(158\) −11.3923 −0.906323
\(159\) 1.53590 0.121805
\(160\) −0.267949 −0.0211832
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 12.3923 0.970640 0.485320 0.874337i \(-0.338703\pi\)
0.485320 + 0.874337i \(0.338703\pi\)
\(164\) −5.26795 −0.411358
\(165\) 0.267949 0.0208598
\(166\) 3.00000 0.232845
\(167\) −20.1962 −1.56283 −0.781413 0.624015i \(-0.785501\pi\)
−0.781413 + 0.624015i \(0.785501\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 1.26795 0.0972473
\(171\) 1.00000 0.0764719
\(172\) 0.196152 0.0149565
\(173\) 20.7846 1.58022 0.790112 0.612962i \(-0.210022\pi\)
0.790112 + 0.612962i \(0.210022\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 1.73205 0.130189
\(178\) −0.339746 −0.0254650
\(179\) −9.46410 −0.707380 −0.353690 0.935363i \(-0.615073\pi\)
−0.353690 + 0.935363i \(0.615073\pi\)
\(180\) −0.267949 −0.0199718
\(181\) −1.46410 −0.108826 −0.0544129 0.998519i \(-0.517329\pi\)
−0.0544129 + 0.998519i \(0.517329\pi\)
\(182\) 0 0
\(183\) −4.53590 −0.335303
\(184\) −2.73205 −0.201409
\(185\) −1.66025 −0.122064
\(186\) 2.46410 0.180677
\(187\) 4.73205 0.346042
\(188\) −11.4641 −0.836106
\(189\) 0 0
\(190\) −0.267949 −0.0194391
\(191\) 13.4641 0.974228 0.487114 0.873338i \(-0.338050\pi\)
0.487114 + 0.873338i \(0.338050\pi\)
\(192\) 1.00000 0.0721688
\(193\) 27.5885 1.98586 0.992930 0.118699i \(-0.0378723\pi\)
0.992930 + 0.118699i \(0.0378723\pi\)
\(194\) −5.73205 −0.411537
\(195\) 1.46410 0.104846
\(196\) 0 0
\(197\) 16.7846 1.19585 0.597927 0.801551i \(-0.295991\pi\)
0.597927 + 0.801551i \(0.295991\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −12.3923 −0.878467 −0.439234 0.898373i \(-0.644750\pi\)
−0.439234 + 0.898373i \(0.644750\pi\)
\(200\) −4.92820 −0.348477
\(201\) −0.928203 −0.0654704
\(202\) 4.92820 0.346747
\(203\) 0 0
\(204\) −4.73205 −0.331310
\(205\) 1.41154 0.0985864
\(206\) −6.00000 −0.418040
\(207\) −2.73205 −0.189891
\(208\) −5.46410 −0.378867
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −7.26795 −0.500346 −0.250173 0.968201i \(-0.580488\pi\)
−0.250173 + 0.968201i \(0.580488\pi\)
\(212\) 1.53590 0.105486
\(213\) −12.1962 −0.835667
\(214\) −1.33975 −0.0915831
\(215\) −0.0525589 −0.00358449
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 10.7321 0.726866
\(219\) −6.92820 −0.468165
\(220\) 0.267949 0.0180651
\(221\) 25.8564 1.73929
\(222\) 6.19615 0.415859
\(223\) −10.3205 −0.691112 −0.345556 0.938398i \(-0.612310\pi\)
−0.345556 + 0.938398i \(0.612310\pi\)
\(224\) 0 0
\(225\) −4.92820 −0.328547
\(226\) −5.26795 −0.350419
\(227\) −18.6603 −1.23852 −0.619262 0.785184i \(-0.712568\pi\)
−0.619262 + 0.785184i \(0.712568\pi\)
\(228\) 1.00000 0.0662266
\(229\) 27.3205 1.80539 0.902695 0.430281i \(-0.141586\pi\)
0.902695 + 0.430281i \(0.141586\pi\)
\(230\) 0.732051 0.0482700
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 2.53590 0.166132 0.0830661 0.996544i \(-0.473529\pi\)
0.0830661 + 0.996544i \(0.473529\pi\)
\(234\) −5.46410 −0.357199
\(235\) 3.07180 0.200382
\(236\) 1.73205 0.112747
\(237\) −11.3923 −0.740010
\(238\) 0 0
\(239\) −18.9282 −1.22436 −0.612182 0.790717i \(-0.709708\pi\)
−0.612182 + 0.790717i \(0.709708\pi\)
\(240\) −0.267949 −0.0172960
\(241\) −23.9808 −1.54474 −0.772368 0.635175i \(-0.780928\pi\)
−0.772368 + 0.635175i \(0.780928\pi\)
\(242\) −10.0000 −0.642824
\(243\) 1.00000 0.0641500
\(244\) −4.53590 −0.290381
\(245\) 0 0
\(246\) −5.26795 −0.335872
\(247\) −5.46410 −0.347672
\(248\) 2.46410 0.156471
\(249\) 3.00000 0.190117
\(250\) 2.66025 0.168249
\(251\) 2.85641 0.180295 0.0901474 0.995928i \(-0.471266\pi\)
0.0901474 + 0.995928i \(0.471266\pi\)
\(252\) 0 0
\(253\) 2.73205 0.171763
\(254\) 3.92820 0.246477
\(255\) 1.26795 0.0794021
\(256\) 1.00000 0.0625000
\(257\) 4.92820 0.307413 0.153706 0.988117i \(-0.450879\pi\)
0.153706 + 0.988117i \(0.450879\pi\)
\(258\) 0.196152 0.0122119
\(259\) 0 0
\(260\) 1.46410 0.0907997
\(261\) 1.00000 0.0618984
\(262\) 0.464102 0.0286723
\(263\) 16.0526 0.989843 0.494922 0.868938i \(-0.335197\pi\)
0.494922 + 0.868938i \(0.335197\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −0.411543 −0.0252809
\(266\) 0 0
\(267\) −0.339746 −0.0207921
\(268\) −0.928203 −0.0566990
\(269\) −20.3205 −1.23896 −0.619482 0.785011i \(-0.712657\pi\)
−0.619482 + 0.785011i \(0.712657\pi\)
\(270\) −0.267949 −0.0163069
\(271\) −23.0526 −1.40034 −0.700172 0.713975i \(-0.746893\pi\)
−0.700172 + 0.713975i \(0.746893\pi\)
\(272\) −4.73205 −0.286923
\(273\) 0 0
\(274\) 11.8564 0.716272
\(275\) 4.92820 0.297182
\(276\) −2.73205 −0.164450
\(277\) 19.8038 1.18990 0.594949 0.803763i \(-0.297172\pi\)
0.594949 + 0.803763i \(0.297172\pi\)
\(278\) −8.19615 −0.491573
\(279\) 2.46410 0.147522
\(280\) 0 0
\(281\) −27.8564 −1.66177 −0.830887 0.556441i \(-0.812166\pi\)
−0.830887 + 0.556441i \(0.812166\pi\)
\(282\) −11.4641 −0.682677
\(283\) 26.4449 1.57198 0.785992 0.618237i \(-0.212153\pi\)
0.785992 + 0.618237i \(0.212153\pi\)
\(284\) −12.1962 −0.723708
\(285\) −0.267949 −0.0158719
\(286\) 5.46410 0.323099
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 5.39230 0.317194
\(290\) −0.267949 −0.0157345
\(291\) −5.73205 −0.336019
\(292\) −6.92820 −0.405442
\(293\) −30.4641 −1.77973 −0.889866 0.456222i \(-0.849202\pi\)
−0.889866 + 0.456222i \(0.849202\pi\)
\(294\) 0 0
\(295\) −0.464102 −0.0270210
\(296\) 6.19615 0.360144
\(297\) −1.00000 −0.0580259
\(298\) 1.60770 0.0931313
\(299\) 14.9282 0.863320
\(300\) −4.92820 −0.284530
\(301\) 0 0
\(302\) 15.3923 0.885728
\(303\) 4.92820 0.283118
\(304\) 1.00000 0.0573539
\(305\) 1.21539 0.0695930
\(306\) −4.73205 −0.270513
\(307\) 24.5885 1.40334 0.701669 0.712503i \(-0.252439\pi\)
0.701669 + 0.712503i \(0.252439\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) −0.660254 −0.0374999
\(311\) 35.1244 1.99172 0.995860 0.0909031i \(-0.0289754\pi\)
0.995860 + 0.0909031i \(0.0289754\pi\)
\(312\) −5.46410 −0.309344
\(313\) −31.7846 −1.79657 −0.898286 0.439411i \(-0.855187\pi\)
−0.898286 + 0.439411i \(0.855187\pi\)
\(314\) −1.46410 −0.0826240
\(315\) 0 0
\(316\) −11.3923 −0.640867
\(317\) −25.3923 −1.42617 −0.713087 0.701076i \(-0.752704\pi\)
−0.713087 + 0.701076i \(0.752704\pi\)
\(318\) 1.53590 0.0861289
\(319\) −1.00000 −0.0559893
\(320\) −0.267949 −0.0149788
\(321\) −1.33975 −0.0747773
\(322\) 0 0
\(323\) −4.73205 −0.263298
\(324\) 1.00000 0.0555556
\(325\) 26.9282 1.49371
\(326\) 12.3923 0.686346
\(327\) 10.7321 0.593484
\(328\) −5.26795 −0.290874
\(329\) 0 0
\(330\) 0.267949 0.0147501
\(331\) 15.5167 0.852873 0.426436 0.904518i \(-0.359769\pi\)
0.426436 + 0.904518i \(0.359769\pi\)
\(332\) 3.00000 0.164646
\(333\) 6.19615 0.339547
\(334\) −20.1962 −1.10508
\(335\) 0.248711 0.0135886
\(336\) 0 0
\(337\) −27.9808 −1.52421 −0.762105 0.647454i \(-0.775834\pi\)
−0.762105 + 0.647454i \(0.775834\pi\)
\(338\) 16.8564 0.916868
\(339\) −5.26795 −0.286116
\(340\) 1.26795 0.0687642
\(341\) −2.46410 −0.133439
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 0.196152 0.0105758
\(345\) 0.732051 0.0394123
\(346\) 20.7846 1.11739
\(347\) −6.39230 −0.343157 −0.171578 0.985170i \(-0.554887\pi\)
−0.171578 + 0.985170i \(0.554887\pi\)
\(348\) 1.00000 0.0536056
\(349\) 15.2679 0.817275 0.408637 0.912697i \(-0.366004\pi\)
0.408637 + 0.912697i \(0.366004\pi\)
\(350\) 0 0
\(351\) −5.46410 −0.291652
\(352\) −1.00000 −0.0533002
\(353\) 14.1962 0.755585 0.377792 0.925890i \(-0.376683\pi\)
0.377792 + 0.925890i \(0.376683\pi\)
\(354\) 1.73205 0.0920575
\(355\) 3.26795 0.173445
\(356\) −0.339746 −0.0180065
\(357\) 0 0
\(358\) −9.46410 −0.500193
\(359\) −0.928203 −0.0489887 −0.0244943 0.999700i \(-0.507798\pi\)
−0.0244943 + 0.999700i \(0.507798\pi\)
\(360\) −0.267949 −0.0141222
\(361\) 1.00000 0.0526316
\(362\) −1.46410 −0.0769515
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 1.85641 0.0971688
\(366\) −4.53590 −0.237095
\(367\) 24.5167 1.27976 0.639880 0.768475i \(-0.278984\pi\)
0.639880 + 0.768475i \(0.278984\pi\)
\(368\) −2.73205 −0.142418
\(369\) −5.26795 −0.274238
\(370\) −1.66025 −0.0863125
\(371\) 0 0
\(372\) 2.46410 0.127758
\(373\) −15.3205 −0.793266 −0.396633 0.917977i \(-0.629821\pi\)
−0.396633 + 0.917977i \(0.629821\pi\)
\(374\) 4.73205 0.244689
\(375\) 2.66025 0.137375
\(376\) −11.4641 −0.591216
\(377\) −5.46410 −0.281416
\(378\) 0 0
\(379\) 10.1436 0.521041 0.260521 0.965468i \(-0.416106\pi\)
0.260521 + 0.965468i \(0.416106\pi\)
\(380\) −0.267949 −0.0137455
\(381\) 3.92820 0.201248
\(382\) 13.4641 0.688883
\(383\) −6.78461 −0.346677 −0.173339 0.984862i \(-0.555456\pi\)
−0.173339 + 0.984862i \(0.555456\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 27.5885 1.40422
\(387\) 0.196152 0.00997099
\(388\) −5.73205 −0.291001
\(389\) 17.3205 0.878185 0.439092 0.898442i \(-0.355300\pi\)
0.439092 + 0.898442i \(0.355300\pi\)
\(390\) 1.46410 0.0741377
\(391\) 12.9282 0.653807
\(392\) 0 0
\(393\) 0.464102 0.0234108
\(394\) 16.7846 0.845596
\(395\) 3.05256 0.153591
\(396\) −1.00000 −0.0502519
\(397\) −19.3205 −0.969669 −0.484834 0.874606i \(-0.661120\pi\)
−0.484834 + 0.874606i \(0.661120\pi\)
\(398\) −12.3923 −0.621170
\(399\) 0 0
\(400\) −4.92820 −0.246410
\(401\) −0.392305 −0.0195908 −0.00979538 0.999952i \(-0.503118\pi\)
−0.00979538 + 0.999952i \(0.503118\pi\)
\(402\) −0.928203 −0.0462946
\(403\) −13.4641 −0.670695
\(404\) 4.92820 0.245187
\(405\) −0.267949 −0.0133145
\(406\) 0 0
\(407\) −6.19615 −0.307132
\(408\) −4.73205 −0.234271
\(409\) 10.6603 0.527116 0.263558 0.964644i \(-0.415104\pi\)
0.263558 + 0.964644i \(0.415104\pi\)
\(410\) 1.41154 0.0697111
\(411\) 11.8564 0.584833
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) −2.73205 −0.134273
\(415\) −0.803848 −0.0394593
\(416\) −5.46410 −0.267900
\(417\) −8.19615 −0.401367
\(418\) −1.00000 −0.0489116
\(419\) −31.1769 −1.52309 −0.761546 0.648111i \(-0.775559\pi\)
−0.761546 + 0.648111i \(0.775559\pi\)
\(420\) 0 0
\(421\) −16.2487 −0.791914 −0.395957 0.918269i \(-0.629587\pi\)
−0.395957 + 0.918269i \(0.629587\pi\)
\(422\) −7.26795 −0.353798
\(423\) −11.4641 −0.557404
\(424\) 1.53590 0.0745898
\(425\) 23.3205 1.13121
\(426\) −12.1962 −0.590906
\(427\) 0 0
\(428\) −1.33975 −0.0647591
\(429\) 5.46410 0.263809
\(430\) −0.0525589 −0.00253461
\(431\) −20.4449 −0.984794 −0.492397 0.870371i \(-0.663879\pi\)
−0.492397 + 0.870371i \(0.663879\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) −0.267949 −0.0128472
\(436\) 10.7321 0.513972
\(437\) −2.73205 −0.130692
\(438\) −6.92820 −0.331042
\(439\) −2.46410 −0.117605 −0.0588026 0.998270i \(-0.518728\pi\)
−0.0588026 + 0.998270i \(0.518728\pi\)
\(440\) 0.267949 0.0127740
\(441\) 0 0
\(442\) 25.8564 1.22986
\(443\) 7.00000 0.332580 0.166290 0.986077i \(-0.446821\pi\)
0.166290 + 0.986077i \(0.446821\pi\)
\(444\) 6.19615 0.294056
\(445\) 0.0910347 0.00431546
\(446\) −10.3205 −0.488690
\(447\) 1.60770 0.0760414
\(448\) 0 0
\(449\) −23.1244 −1.09131 −0.545653 0.838011i \(-0.683718\pi\)
−0.545653 + 0.838011i \(0.683718\pi\)
\(450\) −4.92820 −0.232318
\(451\) 5.26795 0.248058
\(452\) −5.26795 −0.247783
\(453\) 15.3923 0.723194
\(454\) −18.6603 −0.875769
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 23.9282 1.11931 0.559657 0.828724i \(-0.310933\pi\)
0.559657 + 0.828724i \(0.310933\pi\)
\(458\) 27.3205 1.27660
\(459\) −4.73205 −0.220873
\(460\) 0.732051 0.0341320
\(461\) −39.8564 −1.85630 −0.928149 0.372209i \(-0.878600\pi\)
−0.928149 + 0.372209i \(0.878600\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 1.00000 0.0464238
\(465\) −0.660254 −0.0306185
\(466\) 2.53590 0.117473
\(467\) −16.9282 −0.783344 −0.391672 0.920105i \(-0.628103\pi\)
−0.391672 + 0.920105i \(0.628103\pi\)
\(468\) −5.46410 −0.252578
\(469\) 0 0
\(470\) 3.07180 0.141691
\(471\) −1.46410 −0.0674622
\(472\) 1.73205 0.0797241
\(473\) −0.196152 −0.00901910
\(474\) −11.3923 −0.523266
\(475\) −4.92820 −0.226121
\(476\) 0 0
\(477\) 1.53590 0.0703240
\(478\) −18.9282 −0.865756
\(479\) −23.8564 −1.09003 −0.545014 0.838427i \(-0.683475\pi\)
−0.545014 + 0.838427i \(0.683475\pi\)
\(480\) −0.267949 −0.0122302
\(481\) −33.8564 −1.54372
\(482\) −23.9808 −1.09229
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 1.53590 0.0697416
\(486\) 1.00000 0.0453609
\(487\) 38.8564 1.76075 0.880376 0.474277i \(-0.157290\pi\)
0.880376 + 0.474277i \(0.157290\pi\)
\(488\) −4.53590 −0.205330
\(489\) 12.3923 0.560399
\(490\) 0 0
\(491\) −43.0000 −1.94056 −0.970281 0.241979i \(-0.922203\pi\)
−0.970281 + 0.241979i \(0.922203\pi\)
\(492\) −5.26795 −0.237497
\(493\) −4.73205 −0.213121
\(494\) −5.46410 −0.245842
\(495\) 0.267949 0.0120434
\(496\) 2.46410 0.110641
\(497\) 0 0
\(498\) 3.00000 0.134433
\(499\) −7.07180 −0.316577 −0.158289 0.987393i \(-0.550598\pi\)
−0.158289 + 0.987393i \(0.550598\pi\)
\(500\) 2.66025 0.118970
\(501\) −20.1962 −0.902298
\(502\) 2.85641 0.127488
\(503\) 41.6603 1.85754 0.928769 0.370658i \(-0.120868\pi\)
0.928769 + 0.370658i \(0.120868\pi\)
\(504\) 0 0
\(505\) −1.32051 −0.0587618
\(506\) 2.73205 0.121454
\(507\) 16.8564 0.748619
\(508\) 3.92820 0.174286
\(509\) 0.464102 0.0205709 0.0102855 0.999947i \(-0.496726\pi\)
0.0102855 + 0.999947i \(0.496726\pi\)
\(510\) 1.26795 0.0561457
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 4.92820 0.217374
\(515\) 1.60770 0.0708435
\(516\) 0.196152 0.00863513
\(517\) 11.4641 0.504191
\(518\) 0 0
\(519\) 20.7846 0.912343
\(520\) 1.46410 0.0642051
\(521\) 23.8564 1.04517 0.522584 0.852588i \(-0.324968\pi\)
0.522584 + 0.852588i \(0.324968\pi\)
\(522\) 1.00000 0.0437688
\(523\) 9.66025 0.422413 0.211207 0.977441i \(-0.432261\pi\)
0.211207 + 0.977441i \(0.432261\pi\)
\(524\) 0.464102 0.0202744
\(525\) 0 0
\(526\) 16.0526 0.699925
\(527\) −11.6603 −0.507929
\(528\) −1.00000 −0.0435194
\(529\) −15.5359 −0.675474
\(530\) −0.411543 −0.0178763
\(531\) 1.73205 0.0751646
\(532\) 0 0
\(533\) 28.7846 1.24680
\(534\) −0.339746 −0.0147022
\(535\) 0.358984 0.0155202
\(536\) −0.928203 −0.0400923
\(537\) −9.46410 −0.408406
\(538\) −20.3205 −0.876079
\(539\) 0 0
\(540\) −0.267949 −0.0115307
\(541\) 16.4449 0.707020 0.353510 0.935431i \(-0.384988\pi\)
0.353510 + 0.935431i \(0.384988\pi\)
\(542\) −23.0526 −0.990192
\(543\) −1.46410 −0.0628306
\(544\) −4.73205 −0.202885
\(545\) −2.87564 −0.123179
\(546\) 0 0
\(547\) 44.9808 1.92324 0.961619 0.274387i \(-0.0884750\pi\)
0.961619 + 0.274387i \(0.0884750\pi\)
\(548\) 11.8564 0.506481
\(549\) −4.53590 −0.193587
\(550\) 4.92820 0.210139
\(551\) 1.00000 0.0426014
\(552\) −2.73205 −0.116284
\(553\) 0 0
\(554\) 19.8038 0.841385
\(555\) −1.66025 −0.0704739
\(556\) −8.19615 −0.347594
\(557\) −0.267949 −0.0113534 −0.00567669 0.999984i \(-0.501807\pi\)
−0.00567669 + 0.999984i \(0.501807\pi\)
\(558\) 2.46410 0.104314
\(559\) −1.07180 −0.0453322
\(560\) 0 0
\(561\) 4.73205 0.199787
\(562\) −27.8564 −1.17505
\(563\) 11.8756 0.500499 0.250249 0.968181i \(-0.419487\pi\)
0.250249 + 0.968181i \(0.419487\pi\)
\(564\) −11.4641 −0.482726
\(565\) 1.41154 0.0593840
\(566\) 26.4449 1.11156
\(567\) 0 0
\(568\) −12.1962 −0.511739
\(569\) −42.6410 −1.78760 −0.893802 0.448461i \(-0.851972\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(570\) −0.267949 −0.0112232
\(571\) 18.0526 0.755476 0.377738 0.925913i \(-0.376702\pi\)
0.377738 + 0.925913i \(0.376702\pi\)
\(572\) 5.46410 0.228466
\(573\) 13.4641 0.562471
\(574\) 0 0
\(575\) 13.4641 0.561492
\(576\) 1.00000 0.0416667
\(577\) 20.6077 0.857910 0.428955 0.903326i \(-0.358882\pi\)
0.428955 + 0.903326i \(0.358882\pi\)
\(578\) 5.39230 0.224290
\(579\) 27.5885 1.14654
\(580\) −0.267949 −0.0111260
\(581\) 0 0
\(582\) −5.73205 −0.237601
\(583\) −1.53590 −0.0636104
\(584\) −6.92820 −0.286691
\(585\) 1.46410 0.0605332
\(586\) −30.4641 −1.25846
\(587\) 1.14359 0.0472012 0.0236006 0.999721i \(-0.492487\pi\)
0.0236006 + 0.999721i \(0.492487\pi\)
\(588\) 0 0
\(589\) 2.46410 0.101532
\(590\) −0.464102 −0.0191068
\(591\) 16.7846 0.690427
\(592\) 6.19615 0.254660
\(593\) 22.5359 0.925438 0.462719 0.886505i \(-0.346874\pi\)
0.462719 + 0.886505i \(0.346874\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 1.60770 0.0658538
\(597\) −12.3923 −0.507183
\(598\) 14.9282 0.610460
\(599\) 35.9090 1.46720 0.733600 0.679581i \(-0.237838\pi\)
0.733600 + 0.679581i \(0.237838\pi\)
\(600\) −4.92820 −0.201193
\(601\) 21.9808 0.896614 0.448307 0.893880i \(-0.352027\pi\)
0.448307 + 0.893880i \(0.352027\pi\)
\(602\) 0 0
\(603\) −0.928203 −0.0377994
\(604\) 15.3923 0.626304
\(605\) 2.67949 0.108937
\(606\) 4.92820 0.200195
\(607\) 14.1769 0.575423 0.287712 0.957717i \(-0.407106\pi\)
0.287712 + 0.957717i \(0.407106\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 1.21539 0.0492097
\(611\) 62.6410 2.53418
\(612\) −4.73205 −0.191282
\(613\) 19.8038 0.799870 0.399935 0.916544i \(-0.369033\pi\)
0.399935 + 0.916544i \(0.369033\pi\)
\(614\) 24.5885 0.992309
\(615\) 1.41154 0.0569189
\(616\) 0 0
\(617\) −12.5885 −0.506792 −0.253396 0.967363i \(-0.581548\pi\)
−0.253396 + 0.967363i \(0.581548\pi\)
\(618\) −6.00000 −0.241355
\(619\) −31.4641 −1.26465 −0.632325 0.774704i \(-0.717899\pi\)
−0.632325 + 0.774704i \(0.717899\pi\)
\(620\) −0.660254 −0.0265164
\(621\) −2.73205 −0.109633
\(622\) 35.1244 1.40836
\(623\) 0 0
\(624\) −5.46410 −0.218739
\(625\) 23.9282 0.957128
\(626\) −31.7846 −1.27037
\(627\) −1.00000 −0.0399362
\(628\) −1.46410 −0.0584240
\(629\) −29.3205 −1.16909
\(630\) 0 0
\(631\) −17.9808 −0.715803 −0.357901 0.933759i \(-0.616508\pi\)
−0.357901 + 0.933759i \(0.616508\pi\)
\(632\) −11.3923 −0.453162
\(633\) −7.26795 −0.288875
\(634\) −25.3923 −1.00846
\(635\) −1.05256 −0.0417695
\(636\) 1.53590 0.0609023
\(637\) 0 0
\(638\) −1.00000 −0.0395904
\(639\) −12.1962 −0.482472
\(640\) −0.267949 −0.0105916
\(641\) 34.9808 1.38166 0.690829 0.723019i \(-0.257246\pi\)
0.690829 + 0.723019i \(0.257246\pi\)
\(642\) −1.33975 −0.0528756
\(643\) 23.8038 0.938732 0.469366 0.883004i \(-0.344482\pi\)
0.469366 + 0.883004i \(0.344482\pi\)
\(644\) 0 0
\(645\) −0.0525589 −0.00206950
\(646\) −4.73205 −0.186180
\(647\) 46.5885 1.83158 0.915791 0.401656i \(-0.131565\pi\)
0.915791 + 0.401656i \(0.131565\pi\)
\(648\) 1.00000 0.0392837
\(649\) −1.73205 −0.0679889
\(650\) 26.9282 1.05621
\(651\) 0 0
\(652\) 12.3923 0.485320
\(653\) −42.1244 −1.64845 −0.824227 0.566260i \(-0.808390\pi\)
−0.824227 + 0.566260i \(0.808390\pi\)
\(654\) 10.7321 0.419656
\(655\) −0.124356 −0.00485898
\(656\) −5.26795 −0.205679
\(657\) −6.92820 −0.270295
\(658\) 0 0
\(659\) −19.4641 −0.758214 −0.379107 0.925353i \(-0.623769\pi\)
−0.379107 + 0.925353i \(0.623769\pi\)
\(660\) 0.267949 0.0104299
\(661\) −23.5167 −0.914692 −0.457346 0.889289i \(-0.651200\pi\)
−0.457346 + 0.889289i \(0.651200\pi\)
\(662\) 15.5167 0.603072
\(663\) 25.8564 1.00418
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) 6.19615 0.240096
\(667\) −2.73205 −0.105785
\(668\) −20.1962 −0.781413
\(669\) −10.3205 −0.399014
\(670\) 0.248711 0.00960856
\(671\) 4.53590 0.175106
\(672\) 0 0
\(673\) 27.9808 1.07858 0.539290 0.842120i \(-0.318693\pi\)
0.539290 + 0.842120i \(0.318693\pi\)
\(674\) −27.9808 −1.07778
\(675\) −4.92820 −0.189687
\(676\) 16.8564 0.648323
\(677\) 21.6410 0.831732 0.415866 0.909426i \(-0.363479\pi\)
0.415866 + 0.909426i \(0.363479\pi\)
\(678\) −5.26795 −0.202314
\(679\) 0 0
\(680\) 1.26795 0.0486236
\(681\) −18.6603 −0.715062
\(682\) −2.46410 −0.0943553
\(683\) −38.8038 −1.48479 −0.742394 0.669964i \(-0.766310\pi\)
−0.742394 + 0.669964i \(0.766310\pi\)
\(684\) 1.00000 0.0382360
\(685\) −3.17691 −0.121384
\(686\) 0 0
\(687\) 27.3205 1.04234
\(688\) 0.196152 0.00747824
\(689\) −8.39230 −0.319721
\(690\) 0.732051 0.0278687
\(691\) −18.5885 −0.707138 −0.353569 0.935408i \(-0.615032\pi\)
−0.353569 + 0.935408i \(0.615032\pi\)
\(692\) 20.7846 0.790112
\(693\) 0 0
\(694\) −6.39230 −0.242649
\(695\) 2.19615 0.0833048
\(696\) 1.00000 0.0379049
\(697\) 24.9282 0.944223
\(698\) 15.2679 0.577900
\(699\) 2.53590 0.0959165
\(700\) 0 0
\(701\) −4.12436 −0.155775 −0.0778874 0.996962i \(-0.524817\pi\)
−0.0778874 + 0.996962i \(0.524817\pi\)
\(702\) −5.46410 −0.206229
\(703\) 6.19615 0.233692
\(704\) −1.00000 −0.0376889
\(705\) 3.07180 0.115691
\(706\) 14.1962 0.534279
\(707\) 0 0
\(708\) 1.73205 0.0650945
\(709\) −21.1244 −0.793342 −0.396671 0.917961i \(-0.629835\pi\)
−0.396671 + 0.917961i \(0.629835\pi\)
\(710\) 3.26795 0.122644
\(711\) −11.3923 −0.427245
\(712\) −0.339746 −0.0127325
\(713\) −6.73205 −0.252117
\(714\) 0 0
\(715\) −1.46410 −0.0547543
\(716\) −9.46410 −0.353690
\(717\) −18.9282 −0.706887
\(718\) −0.928203 −0.0346402
\(719\) 40.5359 1.51173 0.755867 0.654725i \(-0.227216\pi\)
0.755867 + 0.654725i \(0.227216\pi\)
\(720\) −0.267949 −0.00998588
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −23.9808 −0.891854
\(724\) −1.46410 −0.0544129
\(725\) −4.92820 −0.183029
\(726\) −10.0000 −0.371135
\(727\) 0.267949 0.00993769 0.00496884 0.999988i \(-0.498418\pi\)
0.00496884 + 0.999988i \(0.498418\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.85641 0.0687087
\(731\) −0.928203 −0.0343308
\(732\) −4.53590 −0.167652
\(733\) 17.9090 0.661483 0.330741 0.943721i \(-0.392701\pi\)
0.330741 + 0.943721i \(0.392701\pi\)
\(734\) 24.5167 0.904926
\(735\) 0 0
\(736\) −2.73205 −0.100705
\(737\) 0.928203 0.0341908
\(738\) −5.26795 −0.193916
\(739\) −30.4449 −1.11993 −0.559966 0.828515i \(-0.689186\pi\)
−0.559966 + 0.828515i \(0.689186\pi\)
\(740\) −1.66025 −0.0610322
\(741\) −5.46410 −0.200729
\(742\) 0 0
\(743\) −1.21539 −0.0445883 −0.0222942 0.999751i \(-0.507097\pi\)
−0.0222942 + 0.999751i \(0.507097\pi\)
\(744\) 2.46410 0.0903383
\(745\) −0.430781 −0.0157826
\(746\) −15.3205 −0.560924
\(747\) 3.00000 0.109764
\(748\) 4.73205 0.173021
\(749\) 0 0
\(750\) 2.66025 0.0971387
\(751\) 6.60770 0.241118 0.120559 0.992706i \(-0.461531\pi\)
0.120559 + 0.992706i \(0.461531\pi\)
\(752\) −11.4641 −0.418053
\(753\) 2.85641 0.104093
\(754\) −5.46410 −0.198991
\(755\) −4.12436 −0.150101
\(756\) 0 0
\(757\) −17.1244 −0.622395 −0.311198 0.950345i \(-0.600730\pi\)
−0.311198 + 0.950345i \(0.600730\pi\)
\(758\) 10.1436 0.368432
\(759\) 2.73205 0.0991672
\(760\) −0.267949 −0.00971954
\(761\) 12.6795 0.459631 0.229816 0.973234i \(-0.426188\pi\)
0.229816 + 0.973234i \(0.426188\pi\)
\(762\) 3.92820 0.142304
\(763\) 0 0
\(764\) 13.4641 0.487114
\(765\) 1.26795 0.0458428
\(766\) −6.78461 −0.245138
\(767\) −9.46410 −0.341729
\(768\) 1.00000 0.0360844
\(769\) 12.6077 0.454645 0.227323 0.973820i \(-0.427003\pi\)
0.227323 + 0.973820i \(0.427003\pi\)
\(770\) 0 0
\(771\) 4.92820 0.177485
\(772\) 27.5885 0.992930
\(773\) 35.3205 1.27039 0.635195 0.772352i \(-0.280920\pi\)
0.635195 + 0.772352i \(0.280920\pi\)
\(774\) 0.196152 0.00705055
\(775\) −12.1436 −0.436211
\(776\) −5.73205 −0.205769
\(777\) 0 0
\(778\) 17.3205 0.620970
\(779\) −5.26795 −0.188744
\(780\) 1.46410 0.0524232
\(781\) 12.1962 0.436413
\(782\) 12.9282 0.462312
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 0.392305 0.0140020
\(786\) 0.464102 0.0165540
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 16.7846 0.597927
\(789\) 16.0526 0.571486
\(790\) 3.05256 0.108605
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 24.7846 0.880127
\(794\) −19.3205 −0.685659
\(795\) −0.411543 −0.0145959
\(796\) −12.3923 −0.439234
\(797\) −2.32051 −0.0821966 −0.0410983 0.999155i \(-0.513086\pi\)
−0.0410983 + 0.999155i \(0.513086\pi\)
\(798\) 0 0
\(799\) 54.2487 1.91918
\(800\) −4.92820 −0.174238
\(801\) −0.339746 −0.0120043
\(802\) −0.392305 −0.0138528
\(803\) 6.92820 0.244491
\(804\) −0.928203 −0.0327352
\(805\) 0 0
\(806\) −13.4641 −0.474253
\(807\) −20.3205 −0.715316
\(808\) 4.92820 0.173374
\(809\) −40.0526 −1.40817 −0.704086 0.710114i \(-0.748643\pi\)
−0.704086 + 0.710114i \(0.748643\pi\)
\(810\) −0.267949 −0.00941477
\(811\) 22.9808 0.806964 0.403482 0.914988i \(-0.367800\pi\)
0.403482 + 0.914988i \(0.367800\pi\)
\(812\) 0 0
\(813\) −23.0526 −0.808489
\(814\) −6.19615 −0.217175
\(815\) −3.32051 −0.116312
\(816\) −4.73205 −0.165655
\(817\) 0.196152 0.00686250
\(818\) 10.6603 0.372727
\(819\) 0 0
\(820\) 1.41154 0.0492932
\(821\) −7.73205 −0.269850 −0.134925 0.990856i \(-0.543079\pi\)
−0.134925 + 0.990856i \(0.543079\pi\)
\(822\) 11.8564 0.413540
\(823\) −33.0718 −1.15281 −0.576405 0.817164i \(-0.695545\pi\)
−0.576405 + 0.817164i \(0.695545\pi\)
\(824\) −6.00000 −0.209020
\(825\) 4.92820 0.171578
\(826\) 0 0
\(827\) 33.9808 1.18163 0.590813 0.806808i \(-0.298807\pi\)
0.590813 + 0.806808i \(0.298807\pi\)
\(828\) −2.73205 −0.0949453
\(829\) 10.1436 0.352302 0.176151 0.984363i \(-0.443635\pi\)
0.176151 + 0.984363i \(0.443635\pi\)
\(830\) −0.803848 −0.0279020
\(831\) 19.8038 0.686988
\(832\) −5.46410 −0.189434
\(833\) 0 0
\(834\) −8.19615 −0.283810
\(835\) 5.41154 0.187274
\(836\) −1.00000 −0.0345857
\(837\) 2.46410 0.0851718
\(838\) −31.1769 −1.07699
\(839\) 24.4449 0.843930 0.421965 0.906612i \(-0.361340\pi\)
0.421965 + 0.906612i \(0.361340\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −16.2487 −0.559968
\(843\) −27.8564 −0.959426
\(844\) −7.26795 −0.250173
\(845\) −4.51666 −0.155378
\(846\) −11.4641 −0.394144
\(847\) 0 0
\(848\) 1.53590 0.0527430
\(849\) 26.4449 0.907585
\(850\) 23.3205 0.799887
\(851\) −16.9282 −0.580291
\(852\) −12.1962 −0.417833
\(853\) −40.7321 −1.39464 −0.697319 0.716761i \(-0.745624\pi\)
−0.697319 + 0.716761i \(0.745624\pi\)
\(854\) 0 0
\(855\) −0.267949 −0.00916367
\(856\) −1.33975 −0.0457916
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 5.46410 0.186541
\(859\) 15.1769 0.517830 0.258915 0.965900i \(-0.416635\pi\)
0.258915 + 0.965900i \(0.416635\pi\)
\(860\) −0.0525589 −0.00179224
\(861\) 0 0
\(862\) −20.4449 −0.696355
\(863\) 15.1244 0.514839 0.257420 0.966300i \(-0.417128\pi\)
0.257420 + 0.966300i \(0.417128\pi\)
\(864\) 1.00000 0.0340207
\(865\) −5.56922 −0.189359
\(866\) 16.0000 0.543702
\(867\) 5.39230 0.183132
\(868\) 0 0
\(869\) 11.3923 0.386457
\(870\) −0.267949 −0.00908433
\(871\) 5.07180 0.171851
\(872\) 10.7321 0.363433
\(873\) −5.73205 −0.194001
\(874\) −2.73205 −0.0924130
\(875\) 0 0
\(876\) −6.92820 −0.234082
\(877\) 6.87564 0.232174 0.116087 0.993239i \(-0.462965\pi\)
0.116087 + 0.993239i \(0.462965\pi\)
\(878\) −2.46410 −0.0831594
\(879\) −30.4641 −1.02753
\(880\) 0.267949 0.00903257
\(881\) 9.21539 0.310474 0.155237 0.987877i \(-0.450386\pi\)
0.155237 + 0.987877i \(0.450386\pi\)
\(882\) 0 0
\(883\) 42.1962 1.42001 0.710007 0.704195i \(-0.248692\pi\)
0.710007 + 0.704195i \(0.248692\pi\)
\(884\) 25.8564 0.869645
\(885\) −0.464102 −0.0156006
\(886\) 7.00000 0.235170
\(887\) −26.9808 −0.905925 −0.452963 0.891529i \(-0.649633\pi\)
−0.452963 + 0.891529i \(0.649633\pi\)
\(888\) 6.19615 0.207929
\(889\) 0 0
\(890\) 0.0910347 0.00305149
\(891\) −1.00000 −0.0335013
\(892\) −10.3205 −0.345556
\(893\) −11.4641 −0.383632
\(894\) 1.60770 0.0537694
\(895\) 2.53590 0.0847657
\(896\) 0 0
\(897\) 14.9282 0.498438
\(898\) −23.1244 −0.771670
\(899\) 2.46410 0.0821824
\(900\) −4.92820 −0.164273
\(901\) −7.26795 −0.242130
\(902\) 5.26795 0.175404
\(903\) 0 0
\(904\) −5.26795 −0.175209
\(905\) 0.392305 0.0130407
\(906\) 15.3923 0.511375
\(907\) −31.8038 −1.05603 −0.528015 0.849235i \(-0.677063\pi\)
−0.528015 + 0.849235i \(0.677063\pi\)
\(908\) −18.6603 −0.619262
\(909\) 4.92820 0.163458
\(910\) 0 0
\(911\) −8.53590 −0.282807 −0.141403 0.989952i \(-0.545161\pi\)
−0.141403 + 0.989952i \(0.545161\pi\)
\(912\) 1.00000 0.0331133
\(913\) −3.00000 −0.0992855
\(914\) 23.9282 0.791475
\(915\) 1.21539 0.0401796
\(916\) 27.3205 0.902695
\(917\) 0 0
\(918\) −4.73205 −0.156181
\(919\) −47.1769 −1.55622 −0.778111 0.628126i \(-0.783822\pi\)
−0.778111 + 0.628126i \(0.783822\pi\)
\(920\) 0.732051 0.0241350
\(921\) 24.5885 0.810217
\(922\) −39.8564 −1.31260
\(923\) 66.6410 2.19352
\(924\) 0 0
\(925\) −30.5359 −1.00401
\(926\) 4.00000 0.131448
\(927\) −6.00000 −0.197066
\(928\) 1.00000 0.0328266
\(929\) 0.196152 0.00643555 0.00321778 0.999995i \(-0.498976\pi\)
0.00321778 + 0.999995i \(0.498976\pi\)
\(930\) −0.660254 −0.0216506
\(931\) 0 0
\(932\) 2.53590 0.0830661
\(933\) 35.1244 1.14992
\(934\) −16.9282 −0.553908
\(935\) −1.26795 −0.0414664
\(936\) −5.46410 −0.178600
\(937\) −27.3923 −0.894868 −0.447434 0.894317i \(-0.647662\pi\)
−0.447434 + 0.894317i \(0.647662\pi\)
\(938\) 0 0
\(939\) −31.7846 −1.03725
\(940\) 3.07180 0.100191
\(941\) 19.7846 0.644960 0.322480 0.946576i \(-0.395484\pi\)
0.322480 + 0.946576i \(0.395484\pi\)
\(942\) −1.46410 −0.0477030
\(943\) 14.3923 0.468678
\(944\) 1.73205 0.0563735
\(945\) 0 0
\(946\) −0.196152 −0.00637747
\(947\) 1.60770 0.0522431 0.0261215 0.999659i \(-0.491684\pi\)
0.0261215 + 0.999659i \(0.491684\pi\)
\(948\) −11.3923 −0.370005
\(949\) 37.8564 1.22887
\(950\) −4.92820 −0.159892
\(951\) −25.3923 −0.823402
\(952\) 0 0
\(953\) −52.2487 −1.69250 −0.846251 0.532785i \(-0.821146\pi\)
−0.846251 + 0.532785i \(0.821146\pi\)
\(954\) 1.53590 0.0497265
\(955\) −3.60770 −0.116742
\(956\) −18.9282 −0.612182
\(957\) −1.00000 −0.0323254
\(958\) −23.8564 −0.770766
\(959\) 0 0
\(960\) −0.267949 −0.00864802
\(961\) −24.9282 −0.804136
\(962\) −33.8564 −1.09157
\(963\) −1.33975 −0.0431727
\(964\) −23.9808 −0.772368
\(965\) −7.39230 −0.237967
\(966\) 0 0
\(967\) −9.58846 −0.308344 −0.154172 0.988044i \(-0.549271\pi\)
−0.154172 + 0.988044i \(0.549271\pi\)
\(968\) −10.0000 −0.321412
\(969\) −4.73205 −0.152015
\(970\) 1.53590 0.0493147
\(971\) −7.19615 −0.230936 −0.115468 0.993311i \(-0.536837\pi\)
−0.115468 + 0.993311i \(0.536837\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 38.8564 1.24504
\(975\) 26.9282 0.862393
\(976\) −4.53590 −0.145191
\(977\) 0.143594 0.00459396 0.00229698 0.999997i \(-0.499269\pi\)
0.00229698 + 0.999997i \(0.499269\pi\)
\(978\) 12.3923 0.396262
\(979\) 0.339746 0.0108583
\(980\) 0 0
\(981\) 10.7321 0.342648
\(982\) −43.0000 −1.37219
\(983\) −51.7128 −1.64938 −0.824691 0.565583i \(-0.808651\pi\)
−0.824691 + 0.565583i \(0.808651\pi\)
\(984\) −5.26795 −0.167936
\(985\) −4.49742 −0.143300
\(986\) −4.73205 −0.150699
\(987\) 0 0
\(988\) −5.46410 −0.173836
\(989\) −0.535898 −0.0170406
\(990\) 0.267949 0.00851598
\(991\) 58.0333 1.84349 0.921745 0.387797i \(-0.126764\pi\)
0.921745 + 0.387797i \(0.126764\pi\)
\(992\) 2.46410 0.0782353
\(993\) 15.5167 0.492406
\(994\) 0 0
\(995\) 3.32051 0.105267
\(996\) 3.00000 0.0950586
\(997\) 61.3205 1.94204 0.971020 0.238998i \(-0.0768190\pi\)
0.971020 + 0.238998i \(0.0768190\pi\)
\(998\) −7.07180 −0.223854
\(999\) 6.19615 0.196038
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 5586.2.a.bm.1.2 2
7.3 odd 6 798.2.j.g.457.2 4
7.5 odd 6 798.2.j.g.571.2 yes 4
7.6 odd 2 5586.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.g.457.2 4 7.3 odd 6
798.2.j.g.571.2 yes 4 7.5 odd 6
5586.2.a.bl.1.1 2 7.6 odd 2
5586.2.a.bm.1.2 2 1.1 even 1 trivial