Properties

Label 5586.2.a.bt.1.1
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) 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 1.1
Root \(2.86620\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.34889 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.34889 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.34889 q^{10} -2.21509 q^{11} +1.00000 q^{12} -5.73240 q^{13} -2.34889 q^{15} +1.00000 q^{16} +5.34889 q^{17} -1.00000 q^{18} +1.00000 q^{19} -2.34889 q^{20} +2.21509 q^{22} +4.83159 q^{23} -1.00000 q^{24} +0.517304 q^{25} +5.73240 q^{26} +1.00000 q^{27} +3.24970 q^{29} +2.34889 q^{30} +9.76700 q^{31} -1.00000 q^{32} -2.21509 q^{33} -5.34889 q^{34} +1.00000 q^{36} -8.11590 q^{37} -1.00000 q^{38} -5.73240 q^{39} +2.34889 q^{40} -1.86620 q^{41} -9.08129 q^{43} -2.21509 q^{44} -2.34889 q^{45} -4.83159 q^{46} -1.03461 q^{47} +1.00000 q^{48} -0.517304 q^{50} +5.34889 q^{51} -5.73240 q^{52} -4.73240 q^{53} -1.00000 q^{54} +5.20302 q^{55} +1.00000 q^{57} -3.24970 q^{58} -9.63320 q^{59} -2.34889 q^{60} +3.30221 q^{61} -9.76700 q^{62} +1.00000 q^{64} +13.4648 q^{65} +2.21509 q^{66} -10.6978 q^{67} +5.34889 q^{68} +4.83159 q^{69} -5.34889 q^{71} -1.00000 q^{72} -13.9821 q^{73} +8.11590 q^{74} +0.517304 q^{75} +1.00000 q^{76} +5.73240 q^{78} +1.58652 q^{79} -2.34889 q^{80} +1.00000 q^{81} +1.86620 q^{82} -2.30221 q^{83} -12.5640 q^{85} +9.08129 q^{86} +3.24970 q^{87} +2.21509 q^{88} +17.3310 q^{89} +2.34889 q^{90} +4.83159 q^{92} +9.76700 q^{93} +1.03461 q^{94} -2.34889 q^{95} -1.00000 q^{96} +8.33099 q^{97} -2.21509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} + q^{10} + 7 q^{11} + 3 q^{12} - 2 q^{13} - q^{15} + 3 q^{16} + 10 q^{17} - 3 q^{18} + 3 q^{19} - q^{20} - 7 q^{22} + 10 q^{23} - 3 q^{24} + 2 q^{26} + 3 q^{27} - 7 q^{29} + q^{30} + 11 q^{31} - 3 q^{32} + 7 q^{33} - 10 q^{34} + 3 q^{36} - 3 q^{38} - 2 q^{39} + q^{40} + 2 q^{41} - 6 q^{43} + 7 q^{44} - q^{45} - 10 q^{46} + 3 q^{48} + 10 q^{51} - 2 q^{52} + q^{53} - 3 q^{54} + 17 q^{55} + 3 q^{57} + 7 q^{58} - 3 q^{59} - q^{60} + 22 q^{61} - 11 q^{62} + 3 q^{64} + 10 q^{65} - 7 q^{66} - 20 q^{67} + 10 q^{68} + 10 q^{69} - 10 q^{71} - 3 q^{72} - 10 q^{73} + 3 q^{76} + 2 q^{78} - 3 q^{79} - q^{80} + 3 q^{81} - 2 q^{82} - 19 q^{83} - 18 q^{85} + 6 q^{86} - 7 q^{87} - 7 q^{88} + 14 q^{89} + q^{90} + 10 q^{92} + 11 q^{93} - q^{95} - 3 q^{96} - 13 q^{97} + 7 q^{99}+O(q^{100}) \) 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
\(5\) −2.34889 −1.05046 −0.525229 0.850961i \(-0.676020\pi\)
−0.525229 + 0.850961i \(0.676020\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.34889 0.742786
\(11\) −2.21509 −0.667876 −0.333938 0.942595i \(-0.608378\pi\)
−0.333938 + 0.942595i \(0.608378\pi\)
\(12\) 1.00000 0.288675
\(13\) −5.73240 −1.58988 −0.794940 0.606688i \(-0.792498\pi\)
−0.794940 + 0.606688i \(0.792498\pi\)
\(14\) 0 0
\(15\) −2.34889 −0.606482
\(16\) 1.00000 0.250000
\(17\) 5.34889 1.29730 0.648649 0.761088i \(-0.275334\pi\)
0.648649 + 0.761088i \(0.275334\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) −2.34889 −0.525229
\(21\) 0 0
\(22\) 2.21509 0.472259
\(23\) 4.83159 1.00746 0.503728 0.863862i \(-0.331961\pi\)
0.503728 + 0.863862i \(0.331961\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0.517304 0.103461
\(26\) 5.73240 1.12422
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.24970 0.603454 0.301727 0.953394i \(-0.402437\pi\)
0.301727 + 0.953394i \(0.402437\pi\)
\(30\) 2.34889 0.428847
\(31\) 9.76700 1.75421 0.877103 0.480302i \(-0.159473\pi\)
0.877103 + 0.480302i \(0.159473\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.21509 −0.385598
\(34\) −5.34889 −0.917328
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.11590 −1.33425 −0.667123 0.744948i \(-0.732474\pi\)
−0.667123 + 0.744948i \(0.732474\pi\)
\(38\) −1.00000 −0.162221
\(39\) −5.73240 −0.917918
\(40\) 2.34889 0.371393
\(41\) −1.86620 −0.291451 −0.145726 0.989325i \(-0.546552\pi\)
−0.145726 + 0.989325i \(0.546552\pi\)
\(42\) 0 0
\(43\) −9.08129 −1.38488 −0.692442 0.721474i \(-0.743465\pi\)
−0.692442 + 0.721474i \(0.743465\pi\)
\(44\) −2.21509 −0.333938
\(45\) −2.34889 −0.350152
\(46\) −4.83159 −0.712379
\(47\) −1.03461 −0.150913 −0.0754566 0.997149i \(-0.524041\pi\)
−0.0754566 + 0.997149i \(0.524041\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −0.517304 −0.0731578
\(51\) 5.34889 0.748995
\(52\) −5.73240 −0.794940
\(53\) −4.73240 −0.650045 −0.325022 0.945706i \(-0.605372\pi\)
−0.325022 + 0.945706i \(0.605372\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.20302 0.701575
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −3.24970 −0.426707
\(59\) −9.63320 −1.25414 −0.627068 0.778964i \(-0.715745\pi\)
−0.627068 + 0.778964i \(0.715745\pi\)
\(60\) −2.34889 −0.303241
\(61\) 3.30221 0.422805 0.211402 0.977399i \(-0.432197\pi\)
0.211402 + 0.977399i \(0.432197\pi\)
\(62\) −9.76700 −1.24041
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 13.4648 1.67010
\(66\) 2.21509 0.272659
\(67\) −10.6978 −1.30694 −0.653471 0.756951i \(-0.726688\pi\)
−0.653471 + 0.756951i \(0.726688\pi\)
\(68\) 5.34889 0.648649
\(69\) 4.83159 0.581655
\(70\) 0 0
\(71\) −5.34889 −0.634797 −0.317398 0.948292i \(-0.602809\pi\)
−0.317398 + 0.948292i \(0.602809\pi\)
\(72\) −1.00000 −0.117851
\(73\) −13.9821 −1.63648 −0.818240 0.574876i \(-0.805050\pi\)
−0.818240 + 0.574876i \(0.805050\pi\)
\(74\) 8.11590 0.943454
\(75\) 0.517304 0.0597331
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 5.73240 0.649066
\(79\) 1.58652 0.178497 0.0892487 0.996009i \(-0.471553\pi\)
0.0892487 + 0.996009i \(0.471553\pi\)
\(80\) −2.34889 −0.262614
\(81\) 1.00000 0.111111
\(82\) 1.86620 0.206087
\(83\) −2.30221 −0.252701 −0.126350 0.991986i \(-0.540326\pi\)
−0.126350 + 0.991986i \(0.540326\pi\)
\(84\) 0 0
\(85\) −12.5640 −1.36276
\(86\) 9.08129 0.979261
\(87\) 3.24970 0.348404
\(88\) 2.21509 0.236130
\(89\) 17.3310 1.83708 0.918541 0.395326i \(-0.129369\pi\)
0.918541 + 0.395326i \(0.129369\pi\)
\(90\) 2.34889 0.247595
\(91\) 0 0
\(92\) 4.83159 0.503728
\(93\) 9.76700 1.01279
\(94\) 1.03461 0.106712
\(95\) −2.34889 −0.240991
\(96\) −1.00000 −0.102062
\(97\) 8.33099 0.845884 0.422942 0.906157i \(-0.360997\pi\)
0.422942 + 0.906157i \(0.360997\pi\)
\(98\) 0 0
\(99\) −2.21509 −0.222625
\(100\) 0.517304 0.0517304
\(101\) 14.7670 1.46937 0.734686 0.678407i \(-0.237330\pi\)
0.734686 + 0.678407i \(0.237330\pi\)
\(102\) −5.34889 −0.529619
\(103\) 16.6107 1.63670 0.818349 0.574722i \(-0.194890\pi\)
0.818349 + 0.574722i \(0.194890\pi\)
\(104\) 5.73240 0.562108
\(105\) 0 0
\(106\) 4.73240 0.459651
\(107\) 0.884101 0.0854693 0.0427346 0.999086i \(-0.486393\pi\)
0.0427346 + 0.999086i \(0.486393\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.4769 1.19507 0.597534 0.801844i \(-0.296147\pi\)
0.597534 + 0.801844i \(0.296147\pi\)
\(110\) −5.20302 −0.496088
\(111\) −8.11590 −0.770327
\(112\) 0 0
\(113\) 10.9700 1.03197 0.515986 0.856597i \(-0.327425\pi\)
0.515986 + 0.856597i \(0.327425\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −11.3489 −1.05829
\(116\) 3.24970 0.301727
\(117\) −5.73240 −0.529960
\(118\) 9.63320 0.886808
\(119\) 0 0
\(120\) 2.34889 0.214424
\(121\) −6.09337 −0.553942
\(122\) −3.30221 −0.298968
\(123\) −1.86620 −0.168269
\(124\) 9.76700 0.877103
\(125\) 10.5294 0.941776
\(126\) 0 0
\(127\) −6.10382 −0.541627 −0.270813 0.962632i \(-0.587293\pi\)
−0.270813 + 0.962632i \(0.587293\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −9.08129 −0.799563
\(130\) −13.4648 −1.18094
\(131\) 16.1280 1.40911 0.704554 0.709651i \(-0.251147\pi\)
0.704554 + 0.709651i \(0.251147\pi\)
\(132\) −2.21509 −0.192799
\(133\) 0 0
\(134\) 10.6978 0.924148
\(135\) −2.34889 −0.202161
\(136\) −5.34889 −0.458664
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) −4.83159 −0.411292
\(139\) 17.1505 1.45469 0.727344 0.686274i \(-0.240755\pi\)
0.727344 + 0.686274i \(0.240755\pi\)
\(140\) 0 0
\(141\) −1.03461 −0.0871297
\(142\) 5.34889 0.448869
\(143\) 12.6978 1.06184
\(144\) 1.00000 0.0833333
\(145\) −7.63320 −0.633903
\(146\) 13.9821 1.15717
\(147\) 0 0
\(148\) −8.11590 −0.667123
\(149\) 16.6978 1.36794 0.683968 0.729512i \(-0.260253\pi\)
0.683968 + 0.729512i \(0.260253\pi\)
\(150\) −0.517304 −0.0422377
\(151\) −6.56982 −0.534644 −0.267322 0.963607i \(-0.586139\pi\)
−0.267322 + 0.963607i \(0.586139\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 5.34889 0.432432
\(154\) 0 0
\(155\) −22.9417 −1.84272
\(156\) −5.73240 −0.458959
\(157\) 2.36097 0.188426 0.0942129 0.995552i \(-0.469967\pi\)
0.0942129 + 0.995552i \(0.469967\pi\)
\(158\) −1.58652 −0.126217
\(159\) −4.73240 −0.375303
\(160\) 2.34889 0.185696
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 9.80161 0.767722 0.383861 0.923391i \(-0.374594\pi\)
0.383861 + 0.923391i \(0.374594\pi\)
\(164\) −1.86620 −0.145726
\(165\) 5.20302 0.405054
\(166\) 2.30221 0.178686
\(167\) 4.72032 0.365269 0.182635 0.983181i \(-0.441537\pi\)
0.182635 + 0.983181i \(0.441537\pi\)
\(168\) 0 0
\(169\) 19.8604 1.52772
\(170\) 12.5640 0.963614
\(171\) 1.00000 0.0764719
\(172\) −9.08129 −0.692442
\(173\) −10.3431 −0.786369 −0.393184 0.919460i \(-0.628627\pi\)
−0.393184 + 0.919460i \(0.628627\pi\)
\(174\) −3.24970 −0.246359
\(175\) 0 0
\(176\) −2.21509 −0.166969
\(177\) −9.63320 −0.724076
\(178\) −17.3310 −1.29901
\(179\) 16.8604 1.26020 0.630102 0.776513i \(-0.283013\pi\)
0.630102 + 0.776513i \(0.283013\pi\)
\(180\) −2.34889 −0.175076
\(181\) −24.6620 −1.83311 −0.916555 0.399908i \(-0.869042\pi\)
−0.916555 + 0.399908i \(0.869042\pi\)
\(182\) 0 0
\(183\) 3.30221 0.244107
\(184\) −4.83159 −0.356190
\(185\) 19.0634 1.40157
\(186\) −9.76700 −0.716152
\(187\) −11.8483 −0.866433
\(188\) −1.03461 −0.0754566
\(189\) 0 0
\(190\) 2.34889 0.170407
\(191\) 18.6978 1.35292 0.676462 0.736477i \(-0.263512\pi\)
0.676462 + 0.736477i \(0.263512\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.77908 0.344006 0.172003 0.985096i \(-0.444976\pi\)
0.172003 + 0.985096i \(0.444976\pi\)
\(194\) −8.33099 −0.598130
\(195\) 13.4648 0.964234
\(196\) 0 0
\(197\) 4.87827 0.347563 0.173781 0.984784i \(-0.444401\pi\)
0.173781 + 0.984784i \(0.444401\pi\)
\(198\) 2.21509 0.157420
\(199\) 2.06922 0.146683 0.0733414 0.997307i \(-0.476634\pi\)
0.0733414 + 0.997307i \(0.476634\pi\)
\(200\) −0.517304 −0.0365789
\(201\) −10.6978 −0.754564
\(202\) −14.7670 −1.03900
\(203\) 0 0
\(204\) 5.34889 0.374498
\(205\) 4.38350 0.306157
\(206\) −16.6107 −1.15732
\(207\) 4.83159 0.335819
\(208\) −5.73240 −0.397470
\(209\) −2.21509 −0.153221
\(210\) 0 0
\(211\) −8.06459 −0.555189 −0.277594 0.960698i \(-0.589537\pi\)
−0.277594 + 0.960698i \(0.589537\pi\)
\(212\) −4.73240 −0.325022
\(213\) −5.34889 −0.366500
\(214\) −0.884101 −0.0604359
\(215\) 21.3310 1.45476
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −12.4769 −0.845040
\(219\) −13.9821 −0.944823
\(220\) 5.20302 0.350787
\(221\) −30.6620 −2.06255
\(222\) 8.11590 0.544703
\(223\) −3.31892 −0.222251 −0.111126 0.993806i \(-0.535446\pi\)
−0.111126 + 0.993806i \(0.535446\pi\)
\(224\) 0 0
\(225\) 0.517304 0.0344869
\(226\) −10.9700 −0.729715
\(227\) −3.58189 −0.237738 −0.118869 0.992910i \(-0.537927\pi\)
−0.118869 + 0.992910i \(0.537927\pi\)
\(228\) 1.00000 0.0662266
\(229\) −7.39558 −0.488713 −0.244357 0.969685i \(-0.578577\pi\)
−0.244357 + 0.969685i \(0.578577\pi\)
\(230\) 11.3489 0.748324
\(231\) 0 0
\(232\) −3.24970 −0.213353
\(233\) 26.0934 1.70943 0.854717 0.519095i \(-0.173731\pi\)
0.854717 + 0.519095i \(0.173731\pi\)
\(234\) 5.73240 0.374738
\(235\) 2.43018 0.158528
\(236\) −9.63320 −0.627068
\(237\) 1.58652 0.103056
\(238\) 0 0
\(239\) −1.89498 −0.122576 −0.0612880 0.998120i \(-0.519521\pi\)
−0.0612880 + 0.998120i \(0.519521\pi\)
\(240\) −2.34889 −0.151620
\(241\) 4.36680 0.281290 0.140645 0.990060i \(-0.455082\pi\)
0.140645 + 0.990060i \(0.455082\pi\)
\(242\) 6.09337 0.391696
\(243\) 1.00000 0.0641500
\(244\) 3.30221 0.211402
\(245\) 0 0
\(246\) 1.86620 0.118984
\(247\) −5.73240 −0.364744
\(248\) −9.76700 −0.620205
\(249\) −2.30221 −0.145897
\(250\) −10.5294 −0.665936
\(251\) −4.98210 −0.314467 −0.157234 0.987561i \(-0.550258\pi\)
−0.157234 + 0.987561i \(0.550258\pi\)
\(252\) 0 0
\(253\) −10.7024 −0.672855
\(254\) 6.10382 0.382988
\(255\) −12.5640 −0.786787
\(256\) 1.00000 0.0625000
\(257\) 13.4648 0.839911 0.419955 0.907545i \(-0.362046\pi\)
0.419955 + 0.907545i \(0.362046\pi\)
\(258\) 9.08129 0.565376
\(259\) 0 0
\(260\) 13.4648 0.835051
\(261\) 3.24970 0.201151
\(262\) −16.1280 −0.996390
\(263\) 24.2272 1.49391 0.746956 0.664874i \(-0.231515\pi\)
0.746956 + 0.664874i \(0.231515\pi\)
\(264\) 2.21509 0.136330
\(265\) 11.1159 0.682844
\(266\) 0 0
\(267\) 17.3310 1.06064
\(268\) −10.6978 −0.653471
\(269\) 18.3777 1.12051 0.560253 0.828321i \(-0.310704\pi\)
0.560253 + 0.828321i \(0.310704\pi\)
\(270\) 2.34889 0.142949
\(271\) 24.4181 1.48329 0.741647 0.670790i \(-0.234045\pi\)
0.741647 + 0.670790i \(0.234045\pi\)
\(272\) 5.34889 0.324324
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) −1.14588 −0.0690989
\(276\) 4.83159 0.290828
\(277\) −5.33099 −0.320308 −0.160154 0.987092i \(-0.551199\pi\)
−0.160154 + 0.987092i \(0.551199\pi\)
\(278\) −17.1505 −1.02862
\(279\) 9.76700 0.584735
\(280\) 0 0
\(281\) −3.56982 −0.212957 −0.106479 0.994315i \(-0.533958\pi\)
−0.106479 + 0.994315i \(0.533958\pi\)
\(282\) 1.03461 0.0616100
\(283\) 12.0467 0.716101 0.358050 0.933702i \(-0.383442\pi\)
0.358050 + 0.933702i \(0.383442\pi\)
\(284\) −5.34889 −0.317398
\(285\) −2.34889 −0.139136
\(286\) −12.6978 −0.750836
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 11.6107 0.682981
\(290\) 7.63320 0.448237
\(291\) 8.33099 0.488371
\(292\) −13.9821 −0.818240
\(293\) 24.5761 1.43575 0.717874 0.696173i \(-0.245115\pi\)
0.717874 + 0.696173i \(0.245115\pi\)
\(294\) 0 0
\(295\) 22.6274 1.31742
\(296\) 8.11590 0.471727
\(297\) −2.21509 −0.128533
\(298\) −16.6978 −0.967277
\(299\) −27.6966 −1.60174
\(300\) 0.517304 0.0298666
\(301\) 0 0
\(302\) 6.56982 0.378050
\(303\) 14.7670 0.848342
\(304\) 1.00000 0.0573539
\(305\) −7.75655 −0.444139
\(306\) −5.34889 −0.305776
\(307\) 29.6227 1.69066 0.845330 0.534245i \(-0.179404\pi\)
0.845330 + 0.534245i \(0.179404\pi\)
\(308\) 0 0
\(309\) 16.6107 0.944948
\(310\) 22.9417 1.30300
\(311\) 21.8903 1.24129 0.620644 0.784093i \(-0.286871\pi\)
0.620644 + 0.784093i \(0.286871\pi\)
\(312\) 5.73240 0.324533
\(313\) 27.7491 1.56847 0.784236 0.620463i \(-0.213055\pi\)
0.784236 + 0.620463i \(0.213055\pi\)
\(314\) −2.36097 −0.133237
\(315\) 0 0
\(316\) 1.58652 0.0892487
\(317\) −11.9296 −0.670032 −0.335016 0.942212i \(-0.608742\pi\)
−0.335016 + 0.942212i \(0.608742\pi\)
\(318\) 4.73240 0.265380
\(319\) −7.19839 −0.403032
\(320\) −2.34889 −0.131307
\(321\) 0.884101 0.0493457
\(322\) 0 0
\(323\) 5.34889 0.297620
\(324\) 1.00000 0.0555556
\(325\) −2.96539 −0.164490
\(326\) −9.80161 −0.542861
\(327\) 12.4769 0.689972
\(328\) 1.86620 0.103044
\(329\) 0 0
\(330\) −5.20302 −0.286417
\(331\) −25.6678 −1.41083 −0.705415 0.708794i \(-0.749239\pi\)
−0.705415 + 0.708794i \(0.749239\pi\)
\(332\) −2.30221 −0.126350
\(333\) −8.11590 −0.444749
\(334\) −4.72032 −0.258285
\(335\) 25.1280 1.37289
\(336\) 0 0
\(337\) 13.7266 0.747734 0.373867 0.927482i \(-0.378032\pi\)
0.373867 + 0.927482i \(0.378032\pi\)
\(338\) −19.8604 −1.08026
\(339\) 10.9700 0.595810
\(340\) −12.5640 −0.681378
\(341\) −21.6348 −1.17159
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) 9.08129 0.489630
\(345\) −11.3489 −0.611004
\(346\) 10.3431 0.556047
\(347\) −12.8783 −0.691342 −0.345671 0.938356i \(-0.612349\pi\)
−0.345671 + 0.938356i \(0.612349\pi\)
\(348\) 3.24970 0.174202
\(349\) −2.76237 −0.147866 −0.0739332 0.997263i \(-0.523555\pi\)
−0.0739332 + 0.997263i \(0.523555\pi\)
\(350\) 0 0
\(351\) −5.73240 −0.305973
\(352\) 2.21509 0.118065
\(353\) −28.0109 −1.49087 −0.745434 0.666579i \(-0.767758\pi\)
−0.745434 + 0.666579i \(0.767758\pi\)
\(354\) 9.63320 0.511999
\(355\) 12.5640 0.666827
\(356\) 17.3310 0.918541
\(357\) 0 0
\(358\) −16.8604 −0.891098
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 2.34889 0.123798
\(361\) 1.00000 0.0526316
\(362\) 24.6620 1.29621
\(363\) −6.09337 −0.319819
\(364\) 0 0
\(365\) 32.8425 1.71905
\(366\) −3.30221 −0.172609
\(367\) −30.3131 −1.58233 −0.791165 0.611603i \(-0.790525\pi\)
−0.791165 + 0.611603i \(0.790525\pi\)
\(368\) 4.83159 0.251864
\(369\) −1.86620 −0.0971504
\(370\) −19.0634 −0.991058
\(371\) 0 0
\(372\) 9.76700 0.506396
\(373\) 18.5236 0.959113 0.479557 0.877511i \(-0.340798\pi\)
0.479557 + 0.877511i \(0.340798\pi\)
\(374\) 11.8483 0.612661
\(375\) 10.5294 0.543735
\(376\) 1.03461 0.0533558
\(377\) −18.6286 −0.959420
\(378\) 0 0
\(379\) −9.17304 −0.471187 −0.235594 0.971852i \(-0.575703\pi\)
−0.235594 + 0.971852i \(0.575703\pi\)
\(380\) −2.34889 −0.120496
\(381\) −6.10382 −0.312708
\(382\) −18.6978 −0.956662
\(383\) 6.43018 0.328567 0.164284 0.986413i \(-0.447469\pi\)
0.164284 + 0.986413i \(0.447469\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −4.77908 −0.243249
\(387\) −9.08129 −0.461628
\(388\) 8.33099 0.422942
\(389\) 23.9883 1.21626 0.608129 0.793838i \(-0.291920\pi\)
0.608129 + 0.793838i \(0.291920\pi\)
\(390\) −13.4648 −0.681816
\(391\) 25.8437 1.30697
\(392\) 0 0
\(393\) 16.1280 0.813549
\(394\) −4.87827 −0.245764
\(395\) −3.72657 −0.187504
\(396\) −2.21509 −0.111313
\(397\) 15.3598 0.770885 0.385442 0.922732i \(-0.374049\pi\)
0.385442 + 0.922732i \(0.374049\pi\)
\(398\) −2.06922 −0.103720
\(399\) 0 0
\(400\) 0.517304 0.0258652
\(401\) −30.9988 −1.54801 −0.774003 0.633182i \(-0.781749\pi\)
−0.774003 + 0.633182i \(0.781749\pi\)
\(402\) 10.6978 0.533557
\(403\) −55.9883 −2.78898
\(404\) 14.7670 0.734686
\(405\) −2.34889 −0.116717
\(406\) 0 0
\(407\) 17.9775 0.891110
\(408\) −5.34889 −0.264810
\(409\) 23.5189 1.16294 0.581468 0.813569i \(-0.302479\pi\)
0.581468 + 0.813569i \(0.302479\pi\)
\(410\) −4.38350 −0.216486
\(411\) 4.00000 0.197305
\(412\) 16.6107 0.818349
\(413\) 0 0
\(414\) −4.83159 −0.237460
\(415\) 5.40765 0.265451
\(416\) 5.73240 0.281054
\(417\) 17.1505 0.839864
\(418\) 2.21509 0.108344
\(419\) −23.5403 −1.15002 −0.575008 0.818148i \(-0.695001\pi\)
−0.575008 + 0.818148i \(0.695001\pi\)
\(420\) 0 0
\(421\) −9.05876 −0.441497 −0.220748 0.975331i \(-0.570850\pi\)
−0.220748 + 0.975331i \(0.570850\pi\)
\(422\) 8.06459 0.392578
\(423\) −1.03461 −0.0503044
\(424\) 4.73240 0.229825
\(425\) 2.76700 0.134219
\(426\) 5.34889 0.259155
\(427\) 0 0
\(428\) 0.884101 0.0427346
\(429\) 12.6978 0.613055
\(430\) −21.3310 −1.02867
\(431\) 8.09175 0.389766 0.194883 0.980827i \(-0.437567\pi\)
0.194883 + 0.980827i \(0.437567\pi\)
\(432\) 1.00000 0.0481125
\(433\) −34.7733 −1.67110 −0.835548 0.549418i \(-0.814850\pi\)
−0.835548 + 0.549418i \(0.814850\pi\)
\(434\) 0 0
\(435\) −7.63320 −0.365984
\(436\) 12.4769 0.597534
\(437\) 4.83159 0.231126
\(438\) 13.9821 0.668090
\(439\) −15.4815 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(440\) −5.20302 −0.248044
\(441\) 0 0
\(442\) 30.6620 1.45844
\(443\) −29.9024 −1.42071 −0.710353 0.703845i \(-0.751465\pi\)
−0.710353 + 0.703845i \(0.751465\pi\)
\(444\) −8.11590 −0.385164
\(445\) −40.7087 −1.92978
\(446\) 3.31892 0.157155
\(447\) 16.6978 0.789778
\(448\) 0 0
\(449\) 4.36560 0.206025 0.103013 0.994680i \(-0.467152\pi\)
0.103013 + 0.994680i \(0.467152\pi\)
\(450\) −0.517304 −0.0243859
\(451\) 4.13380 0.194653
\(452\) 10.9700 0.515986
\(453\) −6.56982 −0.308677
\(454\) 3.58189 0.168106
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −25.8425 −1.20886 −0.604430 0.796658i \(-0.706599\pi\)
−0.604430 + 0.796658i \(0.706599\pi\)
\(458\) 7.39558 0.345573
\(459\) 5.34889 0.249665
\(460\) −11.3489 −0.529145
\(461\) −24.7912 −1.15464 −0.577320 0.816518i \(-0.695901\pi\)
−0.577320 + 0.816518i \(0.695901\pi\)
\(462\) 0 0
\(463\) 11.9642 0.556023 0.278012 0.960578i \(-0.410325\pi\)
0.278012 + 0.960578i \(0.410325\pi\)
\(464\) 3.24970 0.150864
\(465\) −22.9417 −1.06389
\(466\) −26.0934 −1.20875
\(467\) 13.5068 0.625022 0.312511 0.949914i \(-0.398830\pi\)
0.312511 + 0.949914i \(0.398830\pi\)
\(468\) −5.73240 −0.264980
\(469\) 0 0
\(470\) −2.43018 −0.112096
\(471\) 2.36097 0.108788
\(472\) 9.63320 0.443404
\(473\) 20.1159 0.924930
\(474\) −1.58652 −0.0728713
\(475\) 0.517304 0.0237355
\(476\) 0 0
\(477\) −4.73240 −0.216682
\(478\) 1.89498 0.0866743
\(479\) −8.59277 −0.392614 −0.196307 0.980543i \(-0.562895\pi\)
−0.196307 + 0.980543i \(0.562895\pi\)
\(480\) 2.34889 0.107212
\(481\) 46.5236 2.12129
\(482\) −4.36680 −0.198902
\(483\) 0 0
\(484\) −6.09337 −0.276971
\(485\) −19.5686 −0.888565
\(486\) −1.00000 −0.0453609
\(487\) −18.4648 −0.836720 −0.418360 0.908281i \(-0.637395\pi\)
−0.418360 + 0.908281i \(0.637395\pi\)
\(488\) −3.30221 −0.149484
\(489\) 9.80161 0.443244
\(490\) 0 0
\(491\) −25.7912 −1.16394 −0.581969 0.813211i \(-0.697718\pi\)
−0.581969 + 0.813211i \(0.697718\pi\)
\(492\) −1.86620 −0.0841347
\(493\) 17.3823 0.782860
\(494\) 5.73240 0.257913
\(495\) 5.20302 0.233858
\(496\) 9.76700 0.438551
\(497\) 0 0
\(498\) 2.30221 0.103165
\(499\) −10.3610 −0.463821 −0.231910 0.972737i \(-0.574498\pi\)
−0.231910 + 0.972737i \(0.574498\pi\)
\(500\) 10.5294 0.470888
\(501\) 4.72032 0.210888
\(502\) 4.98210 0.222362
\(503\) −10.5881 −0.472102 −0.236051 0.971741i \(-0.575853\pi\)
−0.236051 + 0.971741i \(0.575853\pi\)
\(504\) 0 0
\(505\) −34.6861 −1.54351
\(506\) 10.7024 0.475781
\(507\) 19.8604 0.882030
\(508\) −6.10382 −0.270813
\(509\) 13.2497 0.587283 0.293641 0.955916i \(-0.405133\pi\)
0.293641 + 0.955916i \(0.405133\pi\)
\(510\) 12.5640 0.556343
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −13.4648 −0.593907
\(515\) −39.0167 −1.71928
\(516\) −9.08129 −0.399782
\(517\) 2.29175 0.100791
\(518\) 0 0
\(519\) −10.3431 −0.454010
\(520\) −13.4648 −0.590470
\(521\) 28.0934 1.23079 0.615396 0.788218i \(-0.288996\pi\)
0.615396 + 0.788218i \(0.288996\pi\)
\(522\) −3.24970 −0.142236
\(523\) 36.1672 1.58148 0.790741 0.612151i \(-0.209696\pi\)
0.790741 + 0.612151i \(0.209696\pi\)
\(524\) 16.1280 0.704554
\(525\) 0 0
\(526\) −24.2272 −1.05635
\(527\) 52.2427 2.27573
\(528\) −2.21509 −0.0963995
\(529\) 0.344264 0.0149680
\(530\) −11.1159 −0.482844
\(531\) −9.63320 −0.418045
\(532\) 0 0
\(533\) 10.6978 0.463373
\(534\) −17.3310 −0.749985
\(535\) −2.07666 −0.0897818
\(536\) 10.6978 0.462074
\(537\) 16.8604 0.727579
\(538\) −18.3777 −0.792318
\(539\) 0 0
\(540\) −2.34889 −0.101080
\(541\) 27.5986 1.18656 0.593278 0.804998i \(-0.297833\pi\)
0.593278 + 0.804998i \(0.297833\pi\)
\(542\) −24.4181 −1.04885
\(543\) −24.6620 −1.05835
\(544\) −5.34889 −0.229332
\(545\) −29.3068 −1.25537
\(546\) 0 0
\(547\) −27.3551 −1.16962 −0.584811 0.811170i \(-0.698831\pi\)
−0.584811 + 0.811170i \(0.698831\pi\)
\(548\) 4.00000 0.170872
\(549\) 3.30221 0.140935
\(550\) 1.14588 0.0488603
\(551\) 3.24970 0.138442
\(552\) −4.83159 −0.205646
\(553\) 0 0
\(554\) 5.33099 0.226492
\(555\) 19.0634 0.809196
\(556\) 17.1505 0.727344
\(557\) 28.4694 1.20629 0.603144 0.797632i \(-0.293914\pi\)
0.603144 + 0.797632i \(0.293914\pi\)
\(558\) −9.76700 −0.413470
\(559\) 52.0576 2.20180
\(560\) 0 0
\(561\) −11.8483 −0.500235
\(562\) 3.56982 0.150584
\(563\) −10.1233 −0.426648 −0.213324 0.976982i \(-0.568429\pi\)
−0.213324 + 0.976982i \(0.568429\pi\)
\(564\) −1.03461 −0.0435649
\(565\) −25.7674 −1.08404
\(566\) −12.0467 −0.506360
\(567\) 0 0
\(568\) 5.34889 0.224435
\(569\) 2.23180 0.0935618 0.0467809 0.998905i \(-0.485104\pi\)
0.0467809 + 0.998905i \(0.485104\pi\)
\(570\) 2.34889 0.0983844
\(571\) −7.08129 −0.296343 −0.148171 0.988962i \(-0.547339\pi\)
−0.148171 + 0.988962i \(0.547339\pi\)
\(572\) 12.6978 0.530921
\(573\) 18.6978 0.781111
\(574\) 0 0
\(575\) 2.49940 0.104232
\(576\) 1.00000 0.0416667
\(577\) 6.53401 0.272014 0.136007 0.990708i \(-0.456573\pi\)
0.136007 + 0.990708i \(0.456573\pi\)
\(578\) −11.6107 −0.482940
\(579\) 4.77908 0.198612
\(580\) −7.63320 −0.316951
\(581\) 0 0
\(582\) −8.33099 −0.345331
\(583\) 10.4827 0.434149
\(584\) 13.9821 0.578583
\(585\) 13.4648 0.556701
\(586\) −24.5761 −1.01523
\(587\) 24.8529 1.02579 0.512895 0.858451i \(-0.328573\pi\)
0.512895 + 0.858451i \(0.328573\pi\)
\(588\) 0 0
\(589\) 9.76700 0.402442
\(590\) −22.6274 −0.931554
\(591\) 4.87827 0.200665
\(592\) −8.11590 −0.333561
\(593\) −32.0576 −1.31645 −0.658223 0.752823i \(-0.728692\pi\)
−0.658223 + 0.752823i \(0.728692\pi\)
\(594\) 2.21509 0.0908863
\(595\) 0 0
\(596\) 16.6978 0.683968
\(597\) 2.06922 0.0846874
\(598\) 27.6966 1.13260
\(599\) 19.5565 0.799059 0.399529 0.916720i \(-0.369174\pi\)
0.399529 + 0.916720i \(0.369174\pi\)
\(600\) −0.517304 −0.0211188
\(601\) 46.9479 1.91505 0.957523 0.288358i \(-0.0931095\pi\)
0.957523 + 0.288358i \(0.0931095\pi\)
\(602\) 0 0
\(603\) −10.6978 −0.435648
\(604\) −6.56982 −0.267322
\(605\) 14.3127 0.581893
\(606\) −14.7670 −0.599869
\(607\) −4.17929 −0.169632 −0.0848160 0.996397i \(-0.527030\pi\)
−0.0848160 + 0.996397i \(0.527030\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 7.75655 0.314053
\(611\) 5.93078 0.239934
\(612\) 5.34889 0.216216
\(613\) 30.2606 1.22221 0.611107 0.791548i \(-0.290725\pi\)
0.611107 + 0.791548i \(0.290725\pi\)
\(614\) −29.6227 −1.19548
\(615\) 4.38350 0.176760
\(616\) 0 0
\(617\) −21.2889 −0.857060 −0.428530 0.903527i \(-0.640968\pi\)
−0.428530 + 0.903527i \(0.640968\pi\)
\(618\) −16.6107 −0.668179
\(619\) 7.44064 0.299065 0.149532 0.988757i \(-0.452223\pi\)
0.149532 + 0.988757i \(0.452223\pi\)
\(620\) −22.9417 −0.921359
\(621\) 4.83159 0.193885
\(622\) −21.8903 −0.877723
\(623\) 0 0
\(624\) −5.73240 −0.229480
\(625\) −27.3189 −1.09276
\(626\) −27.7491 −1.10908
\(627\) −2.21509 −0.0884623
\(628\) 2.36097 0.0942129
\(629\) −43.4111 −1.73091
\(630\) 0 0
\(631\) 24.6499 0.981297 0.490649 0.871357i \(-0.336760\pi\)
0.490649 + 0.871357i \(0.336760\pi\)
\(632\) −1.58652 −0.0631084
\(633\) −8.06459 −0.320538
\(634\) 11.9296 0.473784
\(635\) 14.3372 0.568956
\(636\) −4.73240 −0.187652
\(637\) 0 0
\(638\) 7.19839 0.284987
\(639\) −5.34889 −0.211599
\(640\) 2.34889 0.0928482
\(641\) 13.4602 0.531644 0.265822 0.964022i \(-0.414357\pi\)
0.265822 + 0.964022i \(0.414357\pi\)
\(642\) −0.884101 −0.0348927
\(643\) 12.6511 0.498911 0.249455 0.968386i \(-0.419748\pi\)
0.249455 + 0.968386i \(0.419748\pi\)
\(644\) 0 0
\(645\) 21.3310 0.839907
\(646\) −5.34889 −0.210449
\(647\) −8.02878 −0.315644 −0.157822 0.987468i \(-0.550447\pi\)
−0.157822 + 0.987468i \(0.550447\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 21.3384 0.837607
\(650\) 2.96539 0.116312
\(651\) 0 0
\(652\) 9.80161 0.383861
\(653\) −12.0392 −0.471132 −0.235566 0.971858i \(-0.575694\pi\)
−0.235566 + 0.971858i \(0.575694\pi\)
\(654\) −12.4769 −0.487884
\(655\) −37.8829 −1.48021
\(656\) −1.86620 −0.0728628
\(657\) −13.9821 −0.545494
\(658\) 0 0
\(659\) −44.8183 −1.74587 −0.872937 0.487833i \(-0.837787\pi\)
−0.872937 + 0.487833i \(0.837787\pi\)
\(660\) 5.20302 0.202527
\(661\) 47.7433 1.85700 0.928499 0.371335i \(-0.121100\pi\)
0.928499 + 0.371335i \(0.121100\pi\)
\(662\) 25.6678 0.997608
\(663\) −30.6620 −1.19081
\(664\) 2.30221 0.0893431
\(665\) 0 0
\(666\) 8.11590 0.314485
\(667\) 15.7012 0.607954
\(668\) 4.72032 0.182635
\(669\) −3.31892 −0.128317
\(670\) −25.1280 −0.970778
\(671\) −7.31470 −0.282381
\(672\) 0 0
\(673\) −30.4181 −1.17253 −0.586266 0.810119i \(-0.699403\pi\)
−0.586266 + 0.810119i \(0.699403\pi\)
\(674\) −13.7266 −0.528728
\(675\) 0.517304 0.0199110
\(676\) 19.8604 0.763860
\(677\) −2.03461 −0.0781963 −0.0390982 0.999235i \(-0.512449\pi\)
−0.0390982 + 0.999235i \(0.512449\pi\)
\(678\) −10.9700 −0.421301
\(679\) 0 0
\(680\) 12.5640 0.481807
\(681\) −3.58189 −0.137258
\(682\) 21.6348 0.828440
\(683\) −13.9521 −0.533863 −0.266932 0.963716i \(-0.586010\pi\)
−0.266932 + 0.963716i \(0.586010\pi\)
\(684\) 1.00000 0.0382360
\(685\) −9.39558 −0.358987
\(686\) 0 0
\(687\) −7.39558 −0.282159
\(688\) −9.08129 −0.346221
\(689\) 27.1280 1.03349
\(690\) 11.3489 0.432045
\(691\) 17.8841 0.680343 0.340172 0.940363i \(-0.389515\pi\)
0.340172 + 0.940363i \(0.389515\pi\)
\(692\) −10.3431 −0.393184
\(693\) 0 0
\(694\) 12.8783 0.488853
\(695\) −40.2847 −1.52809
\(696\) −3.24970 −0.123180
\(697\) −9.98210 −0.378099
\(698\) 2.76237 0.104557
\(699\) 26.0934 0.986942
\(700\) 0 0
\(701\) −35.5282 −1.34188 −0.670940 0.741511i \(-0.734109\pi\)
−0.670940 + 0.741511i \(0.734109\pi\)
\(702\) 5.73240 0.216355
\(703\) −8.11590 −0.306097
\(704\) −2.21509 −0.0834844
\(705\) 2.43018 0.0915261
\(706\) 28.0109 1.05420
\(707\) 0 0
\(708\) −9.63320 −0.362038
\(709\) 5.16841 0.194104 0.0970519 0.995279i \(-0.469059\pi\)
0.0970519 + 0.995279i \(0.469059\pi\)
\(710\) −12.5640 −0.471518
\(711\) 1.58652 0.0594992
\(712\) −17.3310 −0.649506
\(713\) 47.1902 1.76729
\(714\) 0 0
\(715\) −29.8258 −1.11542
\(716\) 16.8604 0.630102
\(717\) −1.89498 −0.0707693
\(718\) 18.0000 0.671754
\(719\) −14.9537 −0.557680 −0.278840 0.960338i \(-0.589950\pi\)
−0.278840 + 0.960338i \(0.589950\pi\)
\(720\) −2.34889 −0.0875381
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 4.36680 0.162403
\(724\) −24.6620 −0.916555
\(725\) 1.68108 0.0624339
\(726\) 6.09337 0.226146
\(727\) 0.511476 0.0189696 0.00948479 0.999955i \(-0.496981\pi\)
0.00948479 + 0.999955i \(0.496981\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −32.8425 −1.21555
\(731\) −48.5749 −1.79661
\(732\) 3.30221 0.122053
\(733\) 29.4360 1.08724 0.543622 0.839330i \(-0.317053\pi\)
0.543622 + 0.839330i \(0.317053\pi\)
\(734\) 30.3131 1.11888
\(735\) 0 0
\(736\) −4.83159 −0.178095
\(737\) 23.6966 0.872875
\(738\) 1.86620 0.0686957
\(739\) −24.4077 −0.897850 −0.448925 0.893569i \(-0.648193\pi\)
−0.448925 + 0.893569i \(0.648193\pi\)
\(740\) 19.0634 0.700784
\(741\) −5.73240 −0.210585
\(742\) 0 0
\(743\) 6.25595 0.229508 0.114754 0.993394i \(-0.463392\pi\)
0.114754 + 0.993394i \(0.463392\pi\)
\(744\) −9.76700 −0.358076
\(745\) −39.2213 −1.43696
\(746\) −18.5236 −0.678195
\(747\) −2.30221 −0.0842335
\(748\) −11.8483 −0.433217
\(749\) 0 0
\(750\) −10.5294 −0.384479
\(751\) −10.6211 −0.387570 −0.193785 0.981044i \(-0.562076\pi\)
−0.193785 + 0.981044i \(0.562076\pi\)
\(752\) −1.03461 −0.0377283
\(753\) −4.98210 −0.181558
\(754\) 18.6286 0.678413
\(755\) 15.4318 0.561621
\(756\) 0 0
\(757\) −21.9837 −0.799012 −0.399506 0.916731i \(-0.630818\pi\)
−0.399506 + 0.916731i \(0.630818\pi\)
\(758\) 9.17304 0.333180
\(759\) −10.7024 −0.388473
\(760\) 2.34889 0.0852033
\(761\) 16.0241 0.580875 0.290437 0.956894i \(-0.406199\pi\)
0.290437 + 0.956894i \(0.406199\pi\)
\(762\) 6.10382 0.221118
\(763\) 0 0
\(764\) 18.6978 0.676462
\(765\) −12.5640 −0.454252
\(766\) −6.43018 −0.232332
\(767\) 55.2213 1.99393
\(768\) 1.00000 0.0360844
\(769\) −40.4469 −1.45855 −0.729276 0.684220i \(-0.760143\pi\)
−0.729276 + 0.684220i \(0.760143\pi\)
\(770\) 0 0
\(771\) 13.4648 0.484923
\(772\) 4.77908 0.172003
\(773\) 20.0421 0.720863 0.360431 0.932786i \(-0.382630\pi\)
0.360431 + 0.932786i \(0.382630\pi\)
\(774\) 9.08129 0.326420
\(775\) 5.05251 0.181492
\(776\) −8.33099 −0.299065
\(777\) 0 0
\(778\) −23.9883 −0.860024
\(779\) −1.86620 −0.0668635
\(780\) 13.4648 0.482117
\(781\) 11.8483 0.423965
\(782\) −25.8437 −0.924168
\(783\) 3.24970 0.116135
\(784\) 0 0
\(785\) −5.54567 −0.197933
\(786\) −16.1280 −0.575266
\(787\) −25.0680 −0.893578 −0.446789 0.894639i \(-0.647433\pi\)
−0.446789 + 0.894639i \(0.647433\pi\)
\(788\) 4.87827 0.173781
\(789\) 24.2272 0.862510
\(790\) 3.72657 0.132585
\(791\) 0 0
\(792\) 2.21509 0.0787099
\(793\) −18.9296 −0.672209
\(794\) −15.3598 −0.545098
\(795\) 11.1159 0.394240
\(796\) 2.06922 0.0733414
\(797\) 43.9358 1.55629 0.778144 0.628087i \(-0.216162\pi\)
0.778144 + 0.628087i \(0.216162\pi\)
\(798\) 0 0
\(799\) −5.53401 −0.195779
\(800\) −0.517304 −0.0182895
\(801\) 17.3310 0.612360
\(802\) 30.9988 1.09461
\(803\) 30.9716 1.09297
\(804\) −10.6978 −0.377282
\(805\) 0 0
\(806\) 55.9883 1.97211
\(807\) 18.3777 0.646925
\(808\) −14.7670 −0.519501
\(809\) 8.29014 0.291466 0.145733 0.989324i \(-0.453446\pi\)
0.145733 + 0.989324i \(0.453446\pi\)
\(810\) 2.34889 0.0825317
\(811\) −7.26178 −0.254995 −0.127498 0.991839i \(-0.540695\pi\)
−0.127498 + 0.991839i \(0.540695\pi\)
\(812\) 0 0
\(813\) 24.4181 0.856381
\(814\) −17.9775 −0.630110
\(815\) −23.0230 −0.806459
\(816\) 5.34889 0.187249
\(817\) −9.08129 −0.317714
\(818\) −23.5189 −0.822320
\(819\) 0 0
\(820\) 4.38350 0.153079
\(821\) 54.6562 1.90751 0.953756 0.300580i \(-0.0971804\pi\)
0.953756 + 0.300580i \(0.0971804\pi\)
\(822\) −4.00000 −0.139516
\(823\) 35.8833 1.25081 0.625407 0.780299i \(-0.284933\pi\)
0.625407 + 0.780299i \(0.284933\pi\)
\(824\) −16.6107 −0.578660
\(825\) −1.14588 −0.0398943
\(826\) 0 0
\(827\) 44.5177 1.54803 0.774016 0.633166i \(-0.218245\pi\)
0.774016 + 0.633166i \(0.218245\pi\)
\(828\) 4.83159 0.167909
\(829\) −1.20885 −0.0419850 −0.0209925 0.999780i \(-0.506683\pi\)
−0.0209925 + 0.999780i \(0.506683\pi\)
\(830\) −5.40765 −0.187702
\(831\) −5.33099 −0.184930
\(832\) −5.73240 −0.198735
\(833\) 0 0
\(834\) −17.1505 −0.593873
\(835\) −11.0875 −0.383700
\(836\) −2.21509 −0.0766106
\(837\) 9.76700 0.337597
\(838\) 23.5403 0.813185
\(839\) −20.0708 −0.692922 −0.346461 0.938064i \(-0.612617\pi\)
−0.346461 + 0.938064i \(0.612617\pi\)
\(840\) 0 0
\(841\) −18.4394 −0.635843
\(842\) 9.05876 0.312185
\(843\) −3.56982 −0.122951
\(844\) −8.06459 −0.277594
\(845\) −46.6499 −1.60481
\(846\) 1.03461 0.0355706
\(847\) 0 0
\(848\) −4.73240 −0.162511
\(849\) 12.0467 0.413441
\(850\) −2.76700 −0.0949075
\(851\) −39.2127 −1.34419
\(852\) −5.34889 −0.183250
\(853\) −51.5628 −1.76548 −0.882738 0.469865i \(-0.844302\pi\)
−0.882738 + 0.469865i \(0.844302\pi\)
\(854\) 0 0
\(855\) −2.34889 −0.0803305
\(856\) −0.884101 −0.0302180
\(857\) −6.92032 −0.236394 −0.118197 0.992990i \(-0.537711\pi\)
−0.118197 + 0.992990i \(0.537711\pi\)
\(858\) −12.6978 −0.433495
\(859\) 37.2306 1.27029 0.635146 0.772392i \(-0.280940\pi\)
0.635146 + 0.772392i \(0.280940\pi\)
\(860\) 21.3310 0.727381
\(861\) 0 0
\(862\) −8.09175 −0.275606
\(863\) −20.6511 −0.702972 −0.351486 0.936193i \(-0.614323\pi\)
−0.351486 + 0.936193i \(0.614323\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 24.2948 0.826047
\(866\) 34.7733 1.18164
\(867\) 11.6107 0.394319
\(868\) 0 0
\(869\) −3.51429 −0.119214
\(870\) 7.63320 0.258790
\(871\) 61.3240 2.07788
\(872\) −12.4769 −0.422520
\(873\) 8.33099 0.281961
\(874\) −4.83159 −0.163431
\(875\) 0 0
\(876\) −13.9821 −0.472411
\(877\) 0.927971 0.0313353 0.0156677 0.999877i \(-0.495013\pi\)
0.0156677 + 0.999877i \(0.495013\pi\)
\(878\) 15.4815 0.522475
\(879\) 24.5761 0.828930
\(880\) 5.20302 0.175394
\(881\) −37.0346 −1.24773 −0.623864 0.781533i \(-0.714438\pi\)
−0.623864 + 0.781533i \(0.714438\pi\)
\(882\) 0 0
\(883\) 24.7087 0.831513 0.415756 0.909476i \(-0.363517\pi\)
0.415756 + 0.909476i \(0.363517\pi\)
\(884\) −30.6620 −1.03127
\(885\) 22.6274 0.760611
\(886\) 29.9024 1.00459
\(887\) 24.0801 0.808530 0.404265 0.914642i \(-0.367527\pi\)
0.404265 + 0.914642i \(0.367527\pi\)
\(888\) 8.11590 0.272352
\(889\) 0 0
\(890\) 40.7087 1.36456
\(891\) −2.21509 −0.0742084
\(892\) −3.31892 −0.111126
\(893\) −1.03461 −0.0346218
\(894\) −16.6978 −0.558458
\(895\) −39.6032 −1.32379
\(896\) 0 0
\(897\) −27.6966 −0.924762
\(898\) −4.36560 −0.145682
\(899\) 31.7398 1.05858
\(900\) 0.517304 0.0172435
\(901\) −25.3131 −0.843301
\(902\) −4.13380 −0.137641
\(903\) 0 0
\(904\) −10.9700 −0.364858
\(905\) 57.9284 1.92560
\(906\) 6.56982 0.218267
\(907\) 7.29758 0.242312 0.121156 0.992633i \(-0.461340\pi\)
0.121156 + 0.992633i \(0.461340\pi\)
\(908\) −3.58189 −0.118869
\(909\) 14.7670 0.489791
\(910\) 0 0
\(911\) 5.73240 0.189923 0.0949614 0.995481i \(-0.469727\pi\)
0.0949614 + 0.995481i \(0.469727\pi\)
\(912\) 1.00000 0.0331133
\(913\) 5.09961 0.168773
\(914\) 25.8425 0.854793
\(915\) −7.75655 −0.256423
\(916\) −7.39558 −0.244357
\(917\) 0 0
\(918\) −5.34889 −0.176540
\(919\) 29.4376 0.971058 0.485529 0.874221i \(-0.338627\pi\)
0.485529 + 0.874221i \(0.338627\pi\)
\(920\) 11.3489 0.374162
\(921\) 29.6227 0.976103
\(922\) 24.7912 0.816453
\(923\) 30.6620 1.00925
\(924\) 0 0
\(925\) −4.19839 −0.138042
\(926\) −11.9642 −0.393168
\(927\) 16.6107 0.545566
\(928\) −3.24970 −0.106677
\(929\) −38.0467 −1.24827 −0.624136 0.781316i \(-0.714549\pi\)
−0.624136 + 0.781316i \(0.714549\pi\)
\(930\) 22.9417 0.752287
\(931\) 0 0
\(932\) 26.0934 0.854717
\(933\) 21.8903 0.716658
\(934\) −13.5068 −0.441957
\(935\) 27.8304 0.910151
\(936\) 5.73240 0.187369
\(937\) −13.1533 −0.429700 −0.214850 0.976647i \(-0.568926\pi\)
−0.214850 + 0.976647i \(0.568926\pi\)
\(938\) 0 0
\(939\) 27.7491 0.905558
\(940\) 2.43018 0.0792639
\(941\) −8.95493 −0.291922 −0.145961 0.989290i \(-0.546627\pi\)
−0.145961 + 0.989290i \(0.546627\pi\)
\(942\) −2.36097 −0.0769245
\(943\) −9.01671 −0.293624
\(944\) −9.63320 −0.313534
\(945\) 0 0
\(946\) −20.1159 −0.654024
\(947\) 15.0859 0.490227 0.245113 0.969494i \(-0.421175\pi\)
0.245113 + 0.969494i \(0.421175\pi\)
\(948\) 1.58652 0.0515278
\(949\) 80.1509 2.60181
\(950\) −0.517304 −0.0167836
\(951\) −11.9296 −0.386843
\(952\) 0 0
\(953\) −47.2906 −1.53189 −0.765946 0.642905i \(-0.777729\pi\)
−0.765946 + 0.642905i \(0.777729\pi\)
\(954\) 4.73240 0.153217
\(955\) −43.9191 −1.42119
\(956\) −1.89498 −0.0612880
\(957\) −7.19839 −0.232691
\(958\) 8.59277 0.277620
\(959\) 0 0
\(960\) −2.34889 −0.0758102
\(961\) 64.3944 2.07724
\(962\) −46.5236 −1.49998
\(963\) 0.884101 0.0284898
\(964\) 4.36680 0.140645
\(965\) −11.2256 −0.361363
\(966\) 0 0
\(967\) 45.0863 1.44988 0.724939 0.688813i \(-0.241868\pi\)
0.724939 + 0.688813i \(0.241868\pi\)
\(968\) 6.09337 0.195848
\(969\) 5.34889 0.171831
\(970\) 19.5686 0.628310
\(971\) −45.8441 −1.47121 −0.735603 0.677413i \(-0.763101\pi\)
−0.735603 + 0.677413i \(0.763101\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 18.4648 0.591650
\(975\) −2.96539 −0.0949685
\(976\) 3.30221 0.105701
\(977\) 32.6169 1.04351 0.521754 0.853096i \(-0.325278\pi\)
0.521754 + 0.853096i \(0.325278\pi\)
\(978\) −9.80161 −0.313421
\(979\) −38.3897 −1.22694
\(980\) 0 0
\(981\) 12.4769 0.398356
\(982\) 25.7912 0.823029
\(983\) 21.1730 0.675315 0.337658 0.941269i \(-0.390365\pi\)
0.337658 + 0.941269i \(0.390365\pi\)
\(984\) 1.86620 0.0594922
\(985\) −11.4585 −0.365100
\(986\) −17.3823 −0.553565
\(987\) 0 0
\(988\) −5.73240 −0.182372
\(989\) −43.8771 −1.39521
\(990\) −5.20302 −0.165363
\(991\) −51.1718 −1.62553 −0.812764 0.582594i \(-0.802038\pi\)
−0.812764 + 0.582594i \(0.802038\pi\)
\(992\) −9.76700 −0.310103
\(993\) −25.6678 −0.814543
\(994\) 0 0
\(995\) −4.86037 −0.154084
\(996\) −2.30221 −0.0729484
\(997\) −21.7082 −0.687507 −0.343754 0.939060i \(-0.611698\pi\)
−0.343754 + 0.939060i \(0.611698\pi\)
\(998\) 10.3610 0.327971
\(999\) −8.11590 −0.256776
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.bt.1.1 3
7.3 odd 6 798.2.j.j.457.1 6
7.5 odd 6 798.2.j.j.571.1 yes 6
7.6 odd 2 5586.2.a.bs.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.j.457.1 6 7.3 odd 6
798.2.j.j.571.1 yes 6 7.5 odd 6
5586.2.a.bs.1.3 3 7.6 odd 2
5586.2.a.bt.1.1 3 1.1 even 1 trivial