Properties

Label 6018.2.a.y.1.9
Level $6018$
Weight $2$
Character 6018.1
Self dual yes
Analytic conductor $48.054$
Analytic rank $1$
Dimension $10$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, 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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 33x^{8} + 53x^{7} + 356x^{6} - 433x^{5} - 1296x^{4} + 1135x^{3} + 930x^{2} - 186x - 104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-3.46791\) of defining polynomial
Character \(\chi\) \(=\) 6018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.46791 q^{5} -1.00000 q^{6} -1.61634 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.46791 q^{5} -1.00000 q^{6} -1.61634 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.46791 q^{10} +4.46894 q^{11} +1.00000 q^{12} -4.75403 q^{13} +1.61634 q^{14} +3.46791 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.47086 q^{19} +3.46791 q^{20} -1.61634 q^{21} -4.46894 q^{22} -8.14330 q^{23} -1.00000 q^{24} +7.02642 q^{25} +4.75403 q^{26} +1.00000 q^{27} -1.61634 q^{28} +6.35448 q^{29} -3.46791 q^{30} -10.4107 q^{31} -1.00000 q^{32} +4.46894 q^{33} +1.00000 q^{34} -5.60533 q^{35} +1.00000 q^{36} -11.5338 q^{37} +4.47086 q^{38} -4.75403 q^{39} -3.46791 q^{40} -6.83512 q^{41} +1.61634 q^{42} -3.54907 q^{43} +4.46894 q^{44} +3.46791 q^{45} +8.14330 q^{46} -4.44449 q^{47} +1.00000 q^{48} -4.38744 q^{49} -7.02642 q^{50} -1.00000 q^{51} -4.75403 q^{52} +7.21748 q^{53} -1.00000 q^{54} +15.4979 q^{55} +1.61634 q^{56} -4.47086 q^{57} -6.35448 q^{58} +1.00000 q^{59} +3.46791 q^{60} -3.12863 q^{61} +10.4107 q^{62} -1.61634 q^{63} +1.00000 q^{64} -16.4866 q^{65} -4.46894 q^{66} -13.7911 q^{67} -1.00000 q^{68} -8.14330 q^{69} +5.60533 q^{70} -6.19584 q^{71} -1.00000 q^{72} -6.44163 q^{73} +11.5338 q^{74} +7.02642 q^{75} -4.47086 q^{76} -7.22334 q^{77} +4.75403 q^{78} +1.63136 q^{79} +3.46791 q^{80} +1.00000 q^{81} +6.83512 q^{82} +15.0535 q^{83} -1.61634 q^{84} -3.46791 q^{85} +3.54907 q^{86} +6.35448 q^{87} -4.46894 q^{88} +11.4660 q^{89} -3.46791 q^{90} +7.68415 q^{91} -8.14330 q^{92} -10.4107 q^{93} +4.44449 q^{94} -15.5046 q^{95} -1.00000 q^{96} +8.18102 q^{97} +4.38744 q^{98} +4.46894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{5} - 10 q^{6} - 6 q^{7} - 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{5} - 10 q^{6} - 6 q^{7} - 10 q^{8} + 10 q^{9} + 2 q^{10} - 3 q^{11} + 10 q^{12} - 10 q^{13} + 6 q^{14} - 2 q^{15} + 10 q^{16} - 10 q^{17} - 10 q^{18} + 8 q^{19} - 2 q^{20} - 6 q^{21} + 3 q^{22} - 9 q^{23} - 10 q^{24} + 20 q^{25} + 10 q^{26} + 10 q^{27} - 6 q^{28} - 24 q^{29} + 2 q^{30} - 7 q^{31} - 10 q^{32} - 3 q^{33} + 10 q^{34} - 22 q^{35} + 10 q^{36} - 4 q^{37} - 8 q^{38} - 10 q^{39} + 2 q^{40} - 9 q^{41} + 6 q^{42} - 11 q^{43} - 3 q^{44} - 2 q^{45} + 9 q^{46} - 18 q^{47} + 10 q^{48} + 6 q^{49} - 20 q^{50} - 10 q^{51} - 10 q^{52} - 9 q^{53} - 10 q^{54} + q^{55} + 6 q^{56} + 8 q^{57} + 24 q^{58} + 10 q^{59} - 2 q^{60} - 25 q^{61} + 7 q^{62} - 6 q^{63} + 10 q^{64} - 28 q^{65} + 3 q^{66} + 2 q^{67} - 10 q^{68} - 9 q^{69} + 22 q^{70} - 30 q^{71} - 10 q^{72} - 11 q^{73} + 4 q^{74} + 20 q^{75} + 8 q^{76} + 4 q^{77} + 10 q^{78} + 3 q^{79} - 2 q^{80} + 10 q^{81} + 9 q^{82} - q^{83} - 6 q^{84} + 2 q^{85} + 11 q^{86} - 24 q^{87} + 3 q^{88} - 14 q^{89} + 2 q^{90} - 13 q^{91} - 9 q^{92} - 7 q^{93} + 18 q^{94} - 35 q^{95} - 10 q^{96} - 10 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100}) \) 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
\(5\) 3.46791 1.55090 0.775449 0.631410i \(-0.217524\pi\)
0.775449 + 0.631410i \(0.217524\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.61634 −0.610920 −0.305460 0.952205i \(-0.598810\pi\)
−0.305460 + 0.952205i \(0.598810\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.46791 −1.09665
\(11\) 4.46894 1.34744 0.673719 0.738988i \(-0.264696\pi\)
0.673719 + 0.738988i \(0.264696\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.75403 −1.31853 −0.659266 0.751910i \(-0.729133\pi\)
−0.659266 + 0.751910i \(0.729133\pi\)
\(14\) 1.61634 0.431986
\(15\) 3.46791 0.895411
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −4.47086 −1.02569 −0.512843 0.858483i \(-0.671408\pi\)
−0.512843 + 0.858483i \(0.671408\pi\)
\(20\) 3.46791 0.775449
\(21\) −1.61634 −0.352715
\(22\) −4.46894 −0.952782
\(23\) −8.14330 −1.69800 −0.848998 0.528396i \(-0.822794\pi\)
−0.848998 + 0.528396i \(0.822794\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.02642 1.40528
\(26\) 4.75403 0.932343
\(27\) 1.00000 0.192450
\(28\) −1.61634 −0.305460
\(29\) 6.35448 1.18000 0.589998 0.807404i \(-0.299128\pi\)
0.589998 + 0.807404i \(0.299128\pi\)
\(30\) −3.46791 −0.633151
\(31\) −10.4107 −1.86982 −0.934912 0.354879i \(-0.884522\pi\)
−0.934912 + 0.354879i \(0.884522\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.46894 0.777943
\(34\) 1.00000 0.171499
\(35\) −5.60533 −0.947474
\(36\) 1.00000 0.166667
\(37\) −11.5338 −1.89615 −0.948073 0.318054i \(-0.896971\pi\)
−0.948073 + 0.318054i \(0.896971\pi\)
\(38\) 4.47086 0.725269
\(39\) −4.75403 −0.761255
\(40\) −3.46791 −0.548325
\(41\) −6.83512 −1.06747 −0.533733 0.845653i \(-0.679211\pi\)
−0.533733 + 0.845653i \(0.679211\pi\)
\(42\) 1.61634 0.249407
\(43\) −3.54907 −0.541228 −0.270614 0.962688i \(-0.587227\pi\)
−0.270614 + 0.962688i \(0.587227\pi\)
\(44\) 4.46894 0.673719
\(45\) 3.46791 0.516966
\(46\) 8.14330 1.20066
\(47\) −4.44449 −0.648295 −0.324148 0.946007i \(-0.605077\pi\)
−0.324148 + 0.946007i \(0.605077\pi\)
\(48\) 1.00000 0.144338
\(49\) −4.38744 −0.626777
\(50\) −7.02642 −0.993685
\(51\) −1.00000 −0.140028
\(52\) −4.75403 −0.659266
\(53\) 7.21748 0.991397 0.495698 0.868495i \(-0.334912\pi\)
0.495698 + 0.868495i \(0.334912\pi\)
\(54\) −1.00000 −0.136083
\(55\) 15.4979 2.08974
\(56\) 1.61634 0.215993
\(57\) −4.47086 −0.592180
\(58\) −6.35448 −0.834384
\(59\) 1.00000 0.130189
\(60\) 3.46791 0.447706
\(61\) −3.12863 −0.400580 −0.200290 0.979737i \(-0.564188\pi\)
−0.200290 + 0.979737i \(0.564188\pi\)
\(62\) 10.4107 1.32217
\(63\) −1.61634 −0.203640
\(64\) 1.00000 0.125000
\(65\) −16.4866 −2.04491
\(66\) −4.46894 −0.550089
\(67\) −13.7911 −1.68485 −0.842425 0.538814i \(-0.818873\pi\)
−0.842425 + 0.538814i \(0.818873\pi\)
\(68\) −1.00000 −0.121268
\(69\) −8.14330 −0.980338
\(70\) 5.60533 0.669965
\(71\) −6.19584 −0.735311 −0.367655 0.929962i \(-0.619839\pi\)
−0.367655 + 0.929962i \(0.619839\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.44163 −0.753936 −0.376968 0.926226i \(-0.623033\pi\)
−0.376968 + 0.926226i \(0.623033\pi\)
\(74\) 11.5338 1.34078
\(75\) 7.02642 0.811341
\(76\) −4.47086 −0.512843
\(77\) −7.22334 −0.823176
\(78\) 4.75403 0.538288
\(79\) 1.63136 0.183542 0.0917711 0.995780i \(-0.470747\pi\)
0.0917711 + 0.995780i \(0.470747\pi\)
\(80\) 3.46791 0.387724
\(81\) 1.00000 0.111111
\(82\) 6.83512 0.754813
\(83\) 15.0535 1.65233 0.826166 0.563426i \(-0.190517\pi\)
0.826166 + 0.563426i \(0.190517\pi\)
\(84\) −1.61634 −0.176357
\(85\) −3.46791 −0.376148
\(86\) 3.54907 0.382706
\(87\) 6.35448 0.681271
\(88\) −4.46894 −0.476391
\(89\) 11.4660 1.21540 0.607699 0.794167i \(-0.292093\pi\)
0.607699 + 0.794167i \(0.292093\pi\)
\(90\) −3.46791 −0.365550
\(91\) 7.68415 0.805517
\(92\) −8.14330 −0.848998
\(93\) −10.4107 −1.07954
\(94\) 4.44449 0.458414
\(95\) −15.5046 −1.59073
\(96\) −1.00000 −0.102062
\(97\) 8.18102 0.830657 0.415328 0.909672i \(-0.363667\pi\)
0.415328 + 0.909672i \(0.363667\pi\)
\(98\) 4.38744 0.443198
\(99\) 4.46894 0.449146
\(100\) 7.02642 0.702642
\(101\) 1.99314 0.198325 0.0991623 0.995071i \(-0.468384\pi\)
0.0991623 + 0.995071i \(0.468384\pi\)
\(102\) 1.00000 0.0990148
\(103\) −14.2936 −1.40839 −0.704195 0.710007i \(-0.748692\pi\)
−0.704195 + 0.710007i \(0.748692\pi\)
\(104\) 4.75403 0.466171
\(105\) −5.60533 −0.547025
\(106\) −7.21748 −0.701023
\(107\) 4.73424 0.457676 0.228838 0.973465i \(-0.426507\pi\)
0.228838 + 0.973465i \(0.426507\pi\)
\(108\) 1.00000 0.0962250
\(109\) −7.09575 −0.679650 −0.339825 0.940489i \(-0.610368\pi\)
−0.339825 + 0.940489i \(0.610368\pi\)
\(110\) −15.4979 −1.47767
\(111\) −11.5338 −1.09474
\(112\) −1.61634 −0.152730
\(113\) −2.12850 −0.200232 −0.100116 0.994976i \(-0.531921\pi\)
−0.100116 + 0.994976i \(0.531921\pi\)
\(114\) 4.47086 0.418734
\(115\) −28.2403 −2.63342
\(116\) 6.35448 0.589998
\(117\) −4.75403 −0.439511
\(118\) −1.00000 −0.0920575
\(119\) 1.61634 0.148170
\(120\) −3.46791 −0.316576
\(121\) 8.97147 0.815588
\(122\) 3.12863 0.283253
\(123\) −6.83512 −0.616302
\(124\) −10.4107 −0.934912
\(125\) 7.02743 0.628553
\(126\) 1.61634 0.143995
\(127\) 4.83842 0.429340 0.214670 0.976687i \(-0.431132\pi\)
0.214670 + 0.976687i \(0.431132\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.54907 −0.312478
\(130\) 16.4866 1.44597
\(131\) −15.6705 −1.36914 −0.684569 0.728948i \(-0.740010\pi\)
−0.684569 + 0.728948i \(0.740010\pi\)
\(132\) 4.46894 0.388972
\(133\) 7.22644 0.626612
\(134\) 13.7911 1.19137
\(135\) 3.46791 0.298470
\(136\) 1.00000 0.0857493
\(137\) 10.1634 0.868319 0.434159 0.900836i \(-0.357045\pi\)
0.434159 + 0.900836i \(0.357045\pi\)
\(138\) 8.14330 0.693204
\(139\) −8.26128 −0.700712 −0.350356 0.936617i \(-0.613939\pi\)
−0.350356 + 0.936617i \(0.613939\pi\)
\(140\) −5.60533 −0.473737
\(141\) −4.44449 −0.374293
\(142\) 6.19584 0.519943
\(143\) −21.2455 −1.77664
\(144\) 1.00000 0.0833333
\(145\) 22.0368 1.83005
\(146\) 6.44163 0.533113
\(147\) −4.38744 −0.361870
\(148\) −11.5338 −0.948073
\(149\) −2.73869 −0.224362 −0.112181 0.993688i \(-0.535784\pi\)
−0.112181 + 0.993688i \(0.535784\pi\)
\(150\) −7.02642 −0.573705
\(151\) −2.48875 −0.202532 −0.101266 0.994859i \(-0.532289\pi\)
−0.101266 + 0.994859i \(0.532289\pi\)
\(152\) 4.47086 0.362635
\(153\) −1.00000 −0.0808452
\(154\) 7.22334 0.582074
\(155\) −36.1035 −2.89991
\(156\) −4.75403 −0.380627
\(157\) 18.6301 1.48684 0.743422 0.668822i \(-0.233201\pi\)
0.743422 + 0.668822i \(0.233201\pi\)
\(158\) −1.63136 −0.129784
\(159\) 7.21748 0.572383
\(160\) −3.46791 −0.274163
\(161\) 13.1624 1.03734
\(162\) −1.00000 −0.0785674
\(163\) 16.2123 1.26985 0.634924 0.772575i \(-0.281032\pi\)
0.634924 + 0.772575i \(0.281032\pi\)
\(164\) −6.83512 −0.533733
\(165\) 15.4979 1.20651
\(166\) −15.0535 −1.16838
\(167\) 9.85713 0.762768 0.381384 0.924417i \(-0.375448\pi\)
0.381384 + 0.924417i \(0.375448\pi\)
\(168\) 1.61634 0.124704
\(169\) 9.60084 0.738526
\(170\) 3.46791 0.265977
\(171\) −4.47086 −0.341895
\(172\) −3.54907 −0.270614
\(173\) −7.57175 −0.575669 −0.287835 0.957680i \(-0.592935\pi\)
−0.287835 + 0.957680i \(0.592935\pi\)
\(174\) −6.35448 −0.481732
\(175\) −11.3571 −0.858516
\(176\) 4.46894 0.336859
\(177\) 1.00000 0.0751646
\(178\) −11.4660 −0.859416
\(179\) 14.1115 1.05474 0.527371 0.849635i \(-0.323178\pi\)
0.527371 + 0.849635i \(0.323178\pi\)
\(180\) 3.46791 0.258483
\(181\) 24.0460 1.78733 0.893664 0.448737i \(-0.148126\pi\)
0.893664 + 0.448737i \(0.148126\pi\)
\(182\) −7.68415 −0.569587
\(183\) −3.12863 −0.231275
\(184\) 8.14330 0.600332
\(185\) −39.9982 −2.94073
\(186\) 10.4107 0.763353
\(187\) −4.46894 −0.326802
\(188\) −4.44449 −0.324148
\(189\) −1.61634 −0.117572
\(190\) 15.5046 1.12482
\(191\) 27.3260 1.97724 0.988621 0.150430i \(-0.0480657\pi\)
0.988621 + 0.150430i \(0.0480657\pi\)
\(192\) 1.00000 0.0721688
\(193\) −15.3997 −1.10850 −0.554248 0.832352i \(-0.686994\pi\)
−0.554248 + 0.832352i \(0.686994\pi\)
\(194\) −8.18102 −0.587363
\(195\) −16.4866 −1.18063
\(196\) −4.38744 −0.313388
\(197\) −6.44083 −0.458890 −0.229445 0.973322i \(-0.573691\pi\)
−0.229445 + 0.973322i \(0.573691\pi\)
\(198\) −4.46894 −0.317594
\(199\) −22.0655 −1.56418 −0.782092 0.623163i \(-0.785847\pi\)
−0.782092 + 0.623163i \(0.785847\pi\)
\(200\) −7.02642 −0.496843
\(201\) −13.7911 −0.972748
\(202\) −1.99314 −0.140237
\(203\) −10.2710 −0.720884
\(204\) −1.00000 −0.0700140
\(205\) −23.7036 −1.65553
\(206\) 14.2936 0.995882
\(207\) −8.14330 −0.565999
\(208\) −4.75403 −0.329633
\(209\) −19.9800 −1.38205
\(210\) 5.60533 0.386805
\(211\) 12.2576 0.843846 0.421923 0.906632i \(-0.361355\pi\)
0.421923 + 0.906632i \(0.361355\pi\)
\(212\) 7.21748 0.495698
\(213\) −6.19584 −0.424532
\(214\) −4.73424 −0.323626
\(215\) −12.3079 −0.839389
\(216\) −1.00000 −0.0680414
\(217\) 16.8273 1.14231
\(218\) 7.09575 0.480585
\(219\) −6.44163 −0.435285
\(220\) 15.4979 1.04487
\(221\) 4.75403 0.319791
\(222\) 11.5338 0.774098
\(223\) 25.7757 1.72607 0.863033 0.505147i \(-0.168562\pi\)
0.863033 + 0.505147i \(0.168562\pi\)
\(224\) 1.61634 0.107996
\(225\) 7.02642 0.468428
\(226\) 2.12850 0.141586
\(227\) −18.7200 −1.24249 −0.621247 0.783615i \(-0.713373\pi\)
−0.621247 + 0.783615i \(0.713373\pi\)
\(228\) −4.47086 −0.296090
\(229\) 16.0744 1.06222 0.531112 0.847301i \(-0.321774\pi\)
0.531112 + 0.847301i \(0.321774\pi\)
\(230\) 28.2403 1.86211
\(231\) −7.22334 −0.475261
\(232\) −6.35448 −0.417192
\(233\) −4.16581 −0.272911 −0.136455 0.990646i \(-0.543571\pi\)
−0.136455 + 0.990646i \(0.543571\pi\)
\(234\) 4.75403 0.310781
\(235\) −15.4131 −1.00544
\(236\) 1.00000 0.0650945
\(237\) 1.63136 0.105968
\(238\) −1.61634 −0.104772
\(239\) 14.3722 0.929662 0.464831 0.885399i \(-0.346115\pi\)
0.464831 + 0.885399i \(0.346115\pi\)
\(240\) 3.46791 0.223853
\(241\) 25.4335 1.63832 0.819158 0.573568i \(-0.194441\pi\)
0.819158 + 0.573568i \(0.194441\pi\)
\(242\) −8.97147 −0.576708
\(243\) 1.00000 0.0641500
\(244\) −3.12863 −0.200290
\(245\) −15.2153 −0.972067
\(246\) 6.83512 0.435791
\(247\) 21.2546 1.35240
\(248\) 10.4107 0.661083
\(249\) 15.0535 0.953975
\(250\) −7.02743 −0.444454
\(251\) 19.8080 1.25027 0.625135 0.780516i \(-0.285044\pi\)
0.625135 + 0.780516i \(0.285044\pi\)
\(252\) −1.61634 −0.101820
\(253\) −36.3920 −2.28794
\(254\) −4.83842 −0.303589
\(255\) −3.46791 −0.217169
\(256\) 1.00000 0.0625000
\(257\) −3.19238 −0.199135 −0.0995676 0.995031i \(-0.531746\pi\)
−0.0995676 + 0.995031i \(0.531746\pi\)
\(258\) 3.54907 0.220955
\(259\) 18.6426 1.15839
\(260\) −16.4866 −1.02245
\(261\) 6.35448 0.393332
\(262\) 15.6705 0.968127
\(263\) −23.7811 −1.46641 −0.733204 0.680009i \(-0.761976\pi\)
−0.733204 + 0.680009i \(0.761976\pi\)
\(264\) −4.46894 −0.275045
\(265\) 25.0296 1.53756
\(266\) −7.22644 −0.443081
\(267\) 11.4660 0.701711
\(268\) −13.7911 −0.842425
\(269\) 9.46182 0.576897 0.288449 0.957495i \(-0.406861\pi\)
0.288449 + 0.957495i \(0.406861\pi\)
\(270\) −3.46791 −0.211050
\(271\) −9.12209 −0.554128 −0.277064 0.960852i \(-0.589361\pi\)
−0.277064 + 0.960852i \(0.589361\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 7.68415 0.465066
\(274\) −10.1634 −0.613994
\(275\) 31.4007 1.89353
\(276\) −8.14330 −0.490169
\(277\) −28.5229 −1.71377 −0.856887 0.515504i \(-0.827605\pi\)
−0.856887 + 0.515504i \(0.827605\pi\)
\(278\) 8.26128 0.495479
\(279\) −10.4107 −0.623275
\(280\) 5.60533 0.334983
\(281\) −2.93316 −0.174977 −0.0874887 0.996166i \(-0.527884\pi\)
−0.0874887 + 0.996166i \(0.527884\pi\)
\(282\) 4.44449 0.264665
\(283\) 2.59567 0.154297 0.0771483 0.997020i \(-0.475418\pi\)
0.0771483 + 0.997020i \(0.475418\pi\)
\(284\) −6.19584 −0.367655
\(285\) −15.5046 −0.918410
\(286\) 21.2455 1.25627
\(287\) 11.0479 0.652136
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −22.0368 −1.29404
\(291\) 8.18102 0.479580
\(292\) −6.44163 −0.376968
\(293\) 4.30235 0.251346 0.125673 0.992072i \(-0.459891\pi\)
0.125673 + 0.992072i \(0.459891\pi\)
\(294\) 4.38744 0.255881
\(295\) 3.46791 0.201910
\(296\) 11.5338 0.670389
\(297\) 4.46894 0.259314
\(298\) 2.73869 0.158648
\(299\) 38.7135 2.23886
\(300\) 7.02642 0.405670
\(301\) 5.73651 0.330647
\(302\) 2.48875 0.143212
\(303\) 1.99314 0.114503
\(304\) −4.47086 −0.256421
\(305\) −10.8498 −0.621259
\(306\) 1.00000 0.0571662
\(307\) 8.18122 0.466927 0.233463 0.972366i \(-0.424994\pi\)
0.233463 + 0.972366i \(0.424994\pi\)
\(308\) −7.22334 −0.411588
\(309\) −14.2936 −0.813134
\(310\) 36.1035 2.05054
\(311\) −28.1098 −1.59396 −0.796979 0.604006i \(-0.793570\pi\)
−0.796979 + 0.604006i \(0.793570\pi\)
\(312\) 4.75403 0.269144
\(313\) 9.95547 0.562716 0.281358 0.959603i \(-0.409215\pi\)
0.281358 + 0.959603i \(0.409215\pi\)
\(314\) −18.6301 −1.05136
\(315\) −5.60533 −0.315825
\(316\) 1.63136 0.0917711
\(317\) 3.36859 0.189199 0.0945995 0.995515i \(-0.469843\pi\)
0.0945995 + 0.995515i \(0.469843\pi\)
\(318\) −7.21748 −0.404736
\(319\) 28.3978 1.58997
\(320\) 3.46791 0.193862
\(321\) 4.73424 0.264239
\(322\) −13.1624 −0.733510
\(323\) 4.47086 0.248765
\(324\) 1.00000 0.0555556
\(325\) −33.4038 −1.85291
\(326\) −16.2123 −0.897918
\(327\) −7.09575 −0.392396
\(328\) 6.83512 0.377406
\(329\) 7.18381 0.396057
\(330\) −15.4979 −0.853132
\(331\) 26.7813 1.47203 0.736016 0.676964i \(-0.236705\pi\)
0.736016 + 0.676964i \(0.236705\pi\)
\(332\) 15.0535 0.826166
\(333\) −11.5338 −0.632048
\(334\) −9.85713 −0.539358
\(335\) −47.8263 −2.61303
\(336\) −1.61634 −0.0881787
\(337\) −13.9894 −0.762051 −0.381026 0.924564i \(-0.624429\pi\)
−0.381026 + 0.924564i \(0.624429\pi\)
\(338\) −9.60084 −0.522217
\(339\) −2.12850 −0.115604
\(340\) −3.46791 −0.188074
\(341\) −46.5250 −2.51947
\(342\) 4.47086 0.241756
\(343\) 18.4060 0.993830
\(344\) 3.54907 0.191353
\(345\) −28.2403 −1.52040
\(346\) 7.57175 0.407060
\(347\) −24.1939 −1.29879 −0.649397 0.760449i \(-0.724979\pi\)
−0.649397 + 0.760449i \(0.724979\pi\)
\(348\) 6.35448 0.340636
\(349\) −17.9313 −0.959838 −0.479919 0.877313i \(-0.659334\pi\)
−0.479919 + 0.877313i \(0.659334\pi\)
\(350\) 11.3571 0.607062
\(351\) −4.75403 −0.253752
\(352\) −4.46894 −0.238196
\(353\) 19.6448 1.04559 0.522793 0.852460i \(-0.324890\pi\)
0.522793 + 0.852460i \(0.324890\pi\)
\(354\) −1.00000 −0.0531494
\(355\) −21.4866 −1.14039
\(356\) 11.4660 0.607699
\(357\) 1.61634 0.0855459
\(358\) −14.1115 −0.745815
\(359\) 26.0866 1.37680 0.688400 0.725331i \(-0.258313\pi\)
0.688400 + 0.725331i \(0.258313\pi\)
\(360\) −3.46791 −0.182775
\(361\) 0.988588 0.0520309
\(362\) −24.0460 −1.26383
\(363\) 8.97147 0.470880
\(364\) 7.68415 0.402759
\(365\) −22.3390 −1.16928
\(366\) 3.12863 0.163536
\(367\) −4.27970 −0.223398 −0.111699 0.993742i \(-0.535629\pi\)
−0.111699 + 0.993742i \(0.535629\pi\)
\(368\) −8.14330 −0.424499
\(369\) −6.83512 −0.355822
\(370\) 39.9982 2.07941
\(371\) −11.6659 −0.605664
\(372\) −10.4107 −0.539772
\(373\) −6.48381 −0.335719 −0.167859 0.985811i \(-0.553685\pi\)
−0.167859 + 0.985811i \(0.553685\pi\)
\(374\) 4.46894 0.231084
\(375\) 7.02743 0.362895
\(376\) 4.44449 0.229207
\(377\) −30.2094 −1.55586
\(378\) 1.61634 0.0831357
\(379\) 16.7701 0.861423 0.430712 0.902490i \(-0.358263\pi\)
0.430712 + 0.902490i \(0.358263\pi\)
\(380\) −15.5046 −0.795367
\(381\) 4.83842 0.247880
\(382\) −27.3260 −1.39812
\(383\) 14.5264 0.742264 0.371132 0.928580i \(-0.378970\pi\)
0.371132 + 0.928580i \(0.378970\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −25.0499 −1.27666
\(386\) 15.3997 0.783824
\(387\) −3.54907 −0.180409
\(388\) 8.18102 0.415328
\(389\) −7.40238 −0.375315 −0.187658 0.982234i \(-0.560090\pi\)
−0.187658 + 0.982234i \(0.560090\pi\)
\(390\) 16.4866 0.834830
\(391\) 8.14330 0.411824
\(392\) 4.38744 0.221599
\(393\) −15.6705 −0.790473
\(394\) 6.44083 0.324484
\(395\) 5.65741 0.284655
\(396\) 4.46894 0.224573
\(397\) 23.5013 1.17949 0.589747 0.807588i \(-0.299227\pi\)
0.589747 + 0.807588i \(0.299227\pi\)
\(398\) 22.0655 1.10605
\(399\) 7.22644 0.361774
\(400\) 7.02642 0.351321
\(401\) 33.5228 1.67405 0.837025 0.547164i \(-0.184293\pi\)
0.837025 + 0.547164i \(0.184293\pi\)
\(402\) 13.7911 0.687837
\(403\) 49.4930 2.46542
\(404\) 1.99314 0.0991623
\(405\) 3.46791 0.172322
\(406\) 10.2710 0.509742
\(407\) −51.5439 −2.55494
\(408\) 1.00000 0.0495074
\(409\) 17.5661 0.868589 0.434295 0.900771i \(-0.356998\pi\)
0.434295 + 0.900771i \(0.356998\pi\)
\(410\) 23.7036 1.17064
\(411\) 10.1634 0.501324
\(412\) −14.2936 −0.704195
\(413\) −1.61634 −0.0795350
\(414\) 8.14330 0.400221
\(415\) 52.2041 2.56260
\(416\) 4.75403 0.233086
\(417\) −8.26128 −0.404557
\(418\) 19.9800 0.977255
\(419\) −6.89542 −0.336863 −0.168432 0.985713i \(-0.553870\pi\)
−0.168432 + 0.985713i \(0.553870\pi\)
\(420\) −5.60533 −0.273512
\(421\) 2.86944 0.139848 0.0699240 0.997552i \(-0.477724\pi\)
0.0699240 + 0.997552i \(0.477724\pi\)
\(422\) −12.2576 −0.596689
\(423\) −4.44449 −0.216098
\(424\) −7.21748 −0.350512
\(425\) −7.02642 −0.340831
\(426\) 6.19584 0.300189
\(427\) 5.05694 0.244722
\(428\) 4.73424 0.228838
\(429\) −21.2455 −1.02574
\(430\) 12.3079 0.593538
\(431\) 30.4374 1.46612 0.733060 0.680164i \(-0.238092\pi\)
0.733060 + 0.680164i \(0.238092\pi\)
\(432\) 1.00000 0.0481125
\(433\) 40.5363 1.94805 0.974025 0.226442i \(-0.0727092\pi\)
0.974025 + 0.226442i \(0.0727092\pi\)
\(434\) −16.8273 −0.807737
\(435\) 22.0368 1.05658
\(436\) −7.09575 −0.339825
\(437\) 36.4076 1.74161
\(438\) 6.44163 0.307793
\(439\) −10.1448 −0.484185 −0.242093 0.970253i \(-0.577834\pi\)
−0.242093 + 0.970253i \(0.577834\pi\)
\(440\) −15.4979 −0.738834
\(441\) −4.38744 −0.208926
\(442\) −4.75403 −0.226126
\(443\) 39.0984 1.85762 0.928811 0.370555i \(-0.120832\pi\)
0.928811 + 0.370555i \(0.120832\pi\)
\(444\) −11.5338 −0.547370
\(445\) 39.7632 1.88496
\(446\) −25.7757 −1.22051
\(447\) −2.73869 −0.129535
\(448\) −1.61634 −0.0763650
\(449\) 11.7537 0.554693 0.277347 0.960770i \(-0.410545\pi\)
0.277347 + 0.960770i \(0.410545\pi\)
\(450\) −7.02642 −0.331228
\(451\) −30.5458 −1.43834
\(452\) −2.12850 −0.100116
\(453\) −2.48875 −0.116932
\(454\) 18.7200 0.878575
\(455\) 26.6479 1.24927
\(456\) 4.47086 0.209367
\(457\) −17.9916 −0.841609 −0.420805 0.907151i \(-0.638252\pi\)
−0.420805 + 0.907151i \(0.638252\pi\)
\(458\) −16.0744 −0.751106
\(459\) −1.00000 −0.0466760
\(460\) −28.2403 −1.31671
\(461\) −5.57560 −0.259682 −0.129841 0.991535i \(-0.541447\pi\)
−0.129841 + 0.991535i \(0.541447\pi\)
\(462\) 7.22334 0.336060
\(463\) −29.3945 −1.36608 −0.683038 0.730383i \(-0.739342\pi\)
−0.683038 + 0.730383i \(0.739342\pi\)
\(464\) 6.35448 0.294999
\(465\) −36.1035 −1.67426
\(466\) 4.16581 0.192977
\(467\) −15.0331 −0.695651 −0.347825 0.937559i \(-0.613080\pi\)
−0.347825 + 0.937559i \(0.613080\pi\)
\(468\) −4.75403 −0.219755
\(469\) 22.2911 1.02931
\(470\) 15.4131 0.710953
\(471\) 18.6301 0.858430
\(472\) −1.00000 −0.0460287
\(473\) −15.8606 −0.729271
\(474\) −1.63136 −0.0749308
\(475\) −31.4141 −1.44138
\(476\) 1.61634 0.0740849
\(477\) 7.21748 0.330466
\(478\) −14.3722 −0.657371
\(479\) −37.9795 −1.73533 −0.867663 0.497153i \(-0.834379\pi\)
−0.867663 + 0.497153i \(0.834379\pi\)
\(480\) −3.46791 −0.158288
\(481\) 54.8321 2.50013
\(482\) −25.4335 −1.15846
\(483\) 13.1624 0.598908
\(484\) 8.97147 0.407794
\(485\) 28.3711 1.28826
\(486\) −1.00000 −0.0453609
\(487\) −0.239403 −0.0108484 −0.00542420 0.999985i \(-0.501727\pi\)
−0.00542420 + 0.999985i \(0.501727\pi\)
\(488\) 3.12863 0.141626
\(489\) 16.2123 0.733147
\(490\) 15.2153 0.687355
\(491\) −16.5815 −0.748312 −0.374156 0.927366i \(-0.622067\pi\)
−0.374156 + 0.927366i \(0.622067\pi\)
\(492\) −6.83512 −0.308151
\(493\) −6.35448 −0.286191
\(494\) −21.2546 −0.956291
\(495\) 15.4979 0.696579
\(496\) −10.4107 −0.467456
\(497\) 10.0146 0.449216
\(498\) −15.0535 −0.674562
\(499\) −10.1572 −0.454699 −0.227349 0.973813i \(-0.573006\pi\)
−0.227349 + 0.973813i \(0.573006\pi\)
\(500\) 7.02743 0.314276
\(501\) 9.85713 0.440384
\(502\) −19.8080 −0.884075
\(503\) −30.5565 −1.36245 −0.681225 0.732075i \(-0.738552\pi\)
−0.681225 + 0.732075i \(0.738552\pi\)
\(504\) 1.61634 0.0719976
\(505\) 6.91203 0.307581
\(506\) 36.3920 1.61782
\(507\) 9.60084 0.426388
\(508\) 4.83842 0.214670
\(509\) 11.7305 0.519947 0.259973 0.965616i \(-0.416286\pi\)
0.259973 + 0.965616i \(0.416286\pi\)
\(510\) 3.46791 0.153562
\(511\) 10.4119 0.460595
\(512\) −1.00000 −0.0441942
\(513\) −4.47086 −0.197393
\(514\) 3.19238 0.140810
\(515\) −49.5689 −2.18427
\(516\) −3.54907 −0.156239
\(517\) −19.8622 −0.873537
\(518\) −18.6426 −0.819107
\(519\) −7.57175 −0.332363
\(520\) 16.4866 0.722984
\(521\) 1.94656 0.0852804 0.0426402 0.999090i \(-0.486423\pi\)
0.0426402 + 0.999090i \(0.486423\pi\)
\(522\) −6.35448 −0.278128
\(523\) −31.0243 −1.35660 −0.678299 0.734786i \(-0.737283\pi\)
−0.678299 + 0.734786i \(0.737283\pi\)
\(524\) −15.6705 −0.684569
\(525\) −11.3571 −0.495664
\(526\) 23.7811 1.03691
\(527\) 10.4107 0.453499
\(528\) 4.46894 0.194486
\(529\) 43.3134 1.88319
\(530\) −25.0296 −1.08722
\(531\) 1.00000 0.0433963
\(532\) 7.22644 0.313306
\(533\) 32.4944 1.40749
\(534\) −11.4660 −0.496184
\(535\) 16.4179 0.709808
\(536\) 13.7911 0.595684
\(537\) 14.1115 0.608956
\(538\) −9.46182 −0.407928
\(539\) −19.6072 −0.844543
\(540\) 3.46791 0.149235
\(541\) −31.9444 −1.37340 −0.686698 0.726943i \(-0.740941\pi\)
−0.686698 + 0.726943i \(0.740941\pi\)
\(542\) 9.12209 0.391827
\(543\) 24.0460 1.03191
\(544\) 1.00000 0.0428746
\(545\) −24.6074 −1.05407
\(546\) −7.68415 −0.328851
\(547\) 13.4088 0.573318 0.286659 0.958033i \(-0.407455\pi\)
0.286659 + 0.958033i \(0.407455\pi\)
\(548\) 10.1634 0.434159
\(549\) −3.12863 −0.133527
\(550\) −31.4007 −1.33893
\(551\) −28.4100 −1.21031
\(552\) 8.14330 0.346602
\(553\) −2.63683 −0.112130
\(554\) 28.5229 1.21182
\(555\) −39.9982 −1.69783
\(556\) −8.26128 −0.350356
\(557\) 25.2577 1.07021 0.535103 0.844787i \(-0.320273\pi\)
0.535103 + 0.844787i \(0.320273\pi\)
\(558\) 10.4107 0.440722
\(559\) 16.8724 0.713626
\(560\) −5.60533 −0.236869
\(561\) −4.46894 −0.188679
\(562\) 2.93316 0.123728
\(563\) −46.3267 −1.95244 −0.976218 0.216792i \(-0.930441\pi\)
−0.976218 + 0.216792i \(0.930441\pi\)
\(564\) −4.44449 −0.187147
\(565\) −7.38145 −0.310540
\(566\) −2.59567 −0.109104
\(567\) −1.61634 −0.0678800
\(568\) 6.19584 0.259972
\(569\) −19.5129 −0.818024 −0.409012 0.912529i \(-0.634127\pi\)
−0.409012 + 0.912529i \(0.634127\pi\)
\(570\) 15.5046 0.649414
\(571\) 3.60657 0.150930 0.0754652 0.997148i \(-0.475956\pi\)
0.0754652 + 0.997148i \(0.475956\pi\)
\(572\) −21.2455 −0.888320
\(573\) 27.3260 1.14156
\(574\) −11.0479 −0.461130
\(575\) −57.2182 −2.38616
\(576\) 1.00000 0.0416667
\(577\) −5.13057 −0.213588 −0.106794 0.994281i \(-0.534059\pi\)
−0.106794 + 0.994281i \(0.534059\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −15.3997 −0.639990
\(580\) 22.0368 0.915027
\(581\) −24.3316 −1.00944
\(582\) −8.18102 −0.339114
\(583\) 32.2545 1.33585
\(584\) 6.44163 0.266557
\(585\) −16.4866 −0.681636
\(586\) −4.30235 −0.177728
\(587\) 7.56561 0.312266 0.156133 0.987736i \(-0.450097\pi\)
0.156133 + 0.987736i \(0.450097\pi\)
\(588\) −4.38744 −0.180935
\(589\) 46.5450 1.91785
\(590\) −3.46791 −0.142772
\(591\) −6.44083 −0.264940
\(592\) −11.5338 −0.474036
\(593\) −16.4343 −0.674876 −0.337438 0.941348i \(-0.609560\pi\)
−0.337438 + 0.941348i \(0.609560\pi\)
\(594\) −4.46894 −0.183363
\(595\) 5.60533 0.229796
\(596\) −2.73869 −0.112181
\(597\) −22.0655 −0.903082
\(598\) −38.7135 −1.58311
\(599\) −31.8269 −1.30041 −0.650206 0.759758i \(-0.725317\pi\)
−0.650206 + 0.759758i \(0.725317\pi\)
\(600\) −7.02642 −0.286852
\(601\) 28.2907 1.15400 0.577001 0.816744i \(-0.304223\pi\)
0.577001 + 0.816744i \(0.304223\pi\)
\(602\) −5.73651 −0.233803
\(603\) −13.7911 −0.561617
\(604\) −2.48875 −0.101266
\(605\) 31.1123 1.26489
\(606\) −1.99314 −0.0809657
\(607\) 7.17649 0.291285 0.145642 0.989337i \(-0.453475\pi\)
0.145642 + 0.989337i \(0.453475\pi\)
\(608\) 4.47086 0.181317
\(609\) −10.2710 −0.416202
\(610\) 10.8498 0.439296
\(611\) 21.1292 0.854798
\(612\) −1.00000 −0.0404226
\(613\) −18.8731 −0.762279 −0.381140 0.924517i \(-0.624468\pi\)
−0.381140 + 0.924517i \(0.624468\pi\)
\(614\) −8.18122 −0.330167
\(615\) −23.7036 −0.955821
\(616\) 7.22334 0.291037
\(617\) −33.3657 −1.34325 −0.671626 0.740891i \(-0.734404\pi\)
−0.671626 + 0.740891i \(0.734404\pi\)
\(618\) 14.2936 0.574973
\(619\) −12.0309 −0.483564 −0.241782 0.970331i \(-0.577732\pi\)
−0.241782 + 0.970331i \(0.577732\pi\)
\(620\) −36.1035 −1.44995
\(621\) −8.14330 −0.326779
\(622\) 28.1098 1.12710
\(623\) −18.5331 −0.742511
\(624\) −4.75403 −0.190314
\(625\) −10.7616 −0.430462
\(626\) −9.95547 −0.397900
\(627\) −19.9800 −0.797925
\(628\) 18.6301 0.743422
\(629\) 11.5338 0.459883
\(630\) 5.60533 0.223322
\(631\) −39.5712 −1.57530 −0.787652 0.616121i \(-0.788703\pi\)
−0.787652 + 0.616121i \(0.788703\pi\)
\(632\) −1.63136 −0.0648919
\(633\) 12.2576 0.487195
\(634\) −3.36859 −0.133784
\(635\) 16.7792 0.665862
\(636\) 7.21748 0.286192
\(637\) 20.8580 0.826425
\(638\) −28.3978 −1.12428
\(639\) −6.19584 −0.245104
\(640\) −3.46791 −0.137081
\(641\) −22.2416 −0.878491 −0.439245 0.898367i \(-0.644754\pi\)
−0.439245 + 0.898367i \(0.644754\pi\)
\(642\) −4.73424 −0.186845
\(643\) −43.9222 −1.73212 −0.866062 0.499936i \(-0.833357\pi\)
−0.866062 + 0.499936i \(0.833357\pi\)
\(644\) 13.1624 0.518670
\(645\) −12.3079 −0.484622
\(646\) −4.47086 −0.175904
\(647\) −17.1562 −0.674481 −0.337241 0.941418i \(-0.609494\pi\)
−0.337241 + 0.941418i \(0.609494\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.46894 0.175421
\(650\) 33.4038 1.31021
\(651\) 16.8273 0.659515
\(652\) 16.2123 0.634924
\(653\) −11.5343 −0.451371 −0.225686 0.974200i \(-0.572462\pi\)
−0.225686 + 0.974200i \(0.572462\pi\)
\(654\) 7.09575 0.277466
\(655\) −54.3440 −2.12339
\(656\) −6.83512 −0.266867
\(657\) −6.44163 −0.251312
\(658\) −7.18381 −0.280054
\(659\) 22.1531 0.862962 0.431481 0.902122i \(-0.357991\pi\)
0.431481 + 0.902122i \(0.357991\pi\)
\(660\) 15.4979 0.603255
\(661\) −38.4517 −1.49560 −0.747798 0.663926i \(-0.768889\pi\)
−0.747798 + 0.663926i \(0.768889\pi\)
\(662\) −26.7813 −1.04088
\(663\) 4.75403 0.184631
\(664\) −15.0535 −0.584188
\(665\) 25.0607 0.971811
\(666\) 11.5338 0.446926
\(667\) −51.7464 −2.00363
\(668\) 9.85713 0.381384
\(669\) 25.7757 0.996545
\(670\) 47.8263 1.84769
\(671\) −13.9817 −0.539757
\(672\) 1.61634 0.0623518
\(673\) −41.2596 −1.59044 −0.795220 0.606320i \(-0.792645\pi\)
−0.795220 + 0.606320i \(0.792645\pi\)
\(674\) 13.9894 0.538852
\(675\) 7.02642 0.270447
\(676\) 9.60084 0.369263
\(677\) 15.2627 0.586595 0.293297 0.956021i \(-0.405247\pi\)
0.293297 + 0.956021i \(0.405247\pi\)
\(678\) 2.12850 0.0817445
\(679\) −13.2233 −0.507465
\(680\) 3.46791 0.132988
\(681\) −18.7200 −0.717354
\(682\) 46.5250 1.78154
\(683\) 1.15361 0.0441418 0.0220709 0.999756i \(-0.492974\pi\)
0.0220709 + 0.999756i \(0.492974\pi\)
\(684\) −4.47086 −0.170948
\(685\) 35.2458 1.34667
\(686\) −18.4060 −0.702744
\(687\) 16.0744 0.613276
\(688\) −3.54907 −0.135307
\(689\) −34.3121 −1.30719
\(690\) 28.2403 1.07509
\(691\) −38.3330 −1.45826 −0.729128 0.684378i \(-0.760074\pi\)
−0.729128 + 0.684378i \(0.760074\pi\)
\(692\) −7.57175 −0.287835
\(693\) −7.22334 −0.274392
\(694\) 24.1939 0.918386
\(695\) −28.6494 −1.08673
\(696\) −6.35448 −0.240866
\(697\) 6.83512 0.258899
\(698\) 17.9313 0.678708
\(699\) −4.16581 −0.157565
\(700\) −11.3571 −0.429258
\(701\) −29.6260 −1.11896 −0.559480 0.828844i \(-0.688999\pi\)
−0.559480 + 0.828844i \(0.688999\pi\)
\(702\) 4.75403 0.179429
\(703\) 51.5660 1.94485
\(704\) 4.46894 0.168430
\(705\) −15.4131 −0.580491
\(706\) −19.6448 −0.739340
\(707\) −3.22159 −0.121160
\(708\) 1.00000 0.0375823
\(709\) −35.1146 −1.31876 −0.659379 0.751811i \(-0.729181\pi\)
−0.659379 + 0.751811i \(0.729181\pi\)
\(710\) 21.4866 0.806379
\(711\) 1.63136 0.0611807
\(712\) −11.4660 −0.429708
\(713\) 84.7778 3.17495
\(714\) −1.61634 −0.0604901
\(715\) −73.6776 −2.75539
\(716\) 14.1115 0.527371
\(717\) 14.3722 0.536741
\(718\) −26.0866 −0.973545
\(719\) 15.3282 0.571644 0.285822 0.958283i \(-0.407733\pi\)
0.285822 + 0.958283i \(0.407733\pi\)
\(720\) 3.46791 0.129241
\(721\) 23.1033 0.860413
\(722\) −0.988588 −0.0367914
\(723\) 25.4335 0.945882
\(724\) 24.0460 0.893664
\(725\) 44.6492 1.65823
\(726\) −8.97147 −0.332962
\(727\) −14.8629 −0.551233 −0.275617 0.961268i \(-0.588882\pi\)
−0.275617 + 0.961268i \(0.588882\pi\)
\(728\) −7.68415 −0.284793
\(729\) 1.00000 0.0370370
\(730\) 22.3390 0.826804
\(731\) 3.54907 0.131267
\(732\) −3.12863 −0.115637
\(733\) −49.4269 −1.82563 −0.912813 0.408379i \(-0.866094\pi\)
−0.912813 + 0.408379i \(0.866094\pi\)
\(734\) 4.27970 0.157966
\(735\) −15.2153 −0.561223
\(736\) 8.14330 0.300166
\(737\) −61.6316 −2.27023
\(738\) 6.83512 0.251604
\(739\) −42.0135 −1.54549 −0.772745 0.634716i \(-0.781117\pi\)
−0.772745 + 0.634716i \(0.781117\pi\)
\(740\) −39.9982 −1.47036
\(741\) 21.2546 0.780808
\(742\) 11.6659 0.428269
\(743\) −19.5197 −0.716109 −0.358055 0.933701i \(-0.616560\pi\)
−0.358055 + 0.933701i \(0.616560\pi\)
\(744\) 10.4107 0.381676
\(745\) −9.49752 −0.347962
\(746\) 6.48381 0.237389
\(747\) 15.0535 0.550778
\(748\) −4.46894 −0.163401
\(749\) −7.65215 −0.279603
\(750\) −7.02743 −0.256606
\(751\) −17.8153 −0.650089 −0.325044 0.945699i \(-0.605379\pi\)
−0.325044 + 0.945699i \(0.605379\pi\)
\(752\) −4.44449 −0.162074
\(753\) 19.8080 0.721844
\(754\) 30.2094 1.10016
\(755\) −8.63078 −0.314106
\(756\) −1.61634 −0.0587858
\(757\) −41.0173 −1.49080 −0.745399 0.666618i \(-0.767741\pi\)
−0.745399 + 0.666618i \(0.767741\pi\)
\(758\) −16.7701 −0.609118
\(759\) −36.3920 −1.32094
\(760\) 15.5046 0.562409
\(761\) −25.2129 −0.913967 −0.456984 0.889475i \(-0.651070\pi\)
−0.456984 + 0.889475i \(0.651070\pi\)
\(762\) −4.83842 −0.175277
\(763\) 11.4692 0.415212
\(764\) 27.3260 0.988621
\(765\) −3.46791 −0.125383
\(766\) −14.5264 −0.524860
\(767\) −4.75403 −0.171658
\(768\) 1.00000 0.0360844
\(769\) 23.8825 0.861227 0.430613 0.902537i \(-0.358297\pi\)
0.430613 + 0.902537i \(0.358297\pi\)
\(770\) 25.0499 0.902737
\(771\) −3.19238 −0.114971
\(772\) −15.3997 −0.554248
\(773\) 11.1622 0.401476 0.200738 0.979645i \(-0.435666\pi\)
0.200738 + 0.979645i \(0.435666\pi\)
\(774\) 3.54907 0.127569
\(775\) −73.1502 −2.62763
\(776\) −8.18102 −0.293682
\(777\) 18.6426 0.668798
\(778\) 7.40238 0.265388
\(779\) 30.5589 1.09488
\(780\) −16.4866 −0.590314
\(781\) −27.6889 −0.990785
\(782\) −8.14330 −0.291204
\(783\) 6.35448 0.227090
\(784\) −4.38744 −0.156694
\(785\) 64.6076 2.30594
\(786\) 15.6705 0.558949
\(787\) 15.2737 0.544450 0.272225 0.962234i \(-0.412241\pi\)
0.272225 + 0.962234i \(0.412241\pi\)
\(788\) −6.44083 −0.229445
\(789\) −23.7811 −0.846631
\(790\) −5.65741 −0.201281
\(791\) 3.44038 0.122326
\(792\) −4.46894 −0.158797
\(793\) 14.8736 0.528178
\(794\) −23.5013 −0.834029
\(795\) 25.0296 0.887708
\(796\) −22.0655 −0.782092
\(797\) −38.0890 −1.34918 −0.674590 0.738192i \(-0.735680\pi\)
−0.674590 + 0.738192i \(0.735680\pi\)
\(798\) −7.22644 −0.255813
\(799\) 4.44449 0.157235
\(800\) −7.02642 −0.248421
\(801\) 11.4660 0.405133
\(802\) −33.5228 −1.18373
\(803\) −28.7873 −1.01588
\(804\) −13.7911 −0.486374
\(805\) 45.6459 1.60881
\(806\) −49.4930 −1.74332
\(807\) 9.46182 0.333072
\(808\) −1.99314 −0.0701183
\(809\) 12.6651 0.445282 0.222641 0.974901i \(-0.428532\pi\)
0.222641 + 0.974901i \(0.428532\pi\)
\(810\) −3.46791 −0.121850
\(811\) 45.8034 1.60837 0.804187 0.594377i \(-0.202601\pi\)
0.804187 + 0.594377i \(0.202601\pi\)
\(812\) −10.2710 −0.360442
\(813\) −9.12209 −0.319926
\(814\) 51.5439 1.80661
\(815\) 56.2229 1.96940
\(816\) −1.00000 −0.0350070
\(817\) 15.8674 0.555130
\(818\) −17.5661 −0.614185
\(819\) 7.68415 0.268506
\(820\) −23.7036 −0.827765
\(821\) 28.2967 0.987563 0.493782 0.869586i \(-0.335614\pi\)
0.493782 + 0.869586i \(0.335614\pi\)
\(822\) −10.1634 −0.354490
\(823\) −49.7708 −1.73490 −0.867450 0.497525i \(-0.834242\pi\)
−0.867450 + 0.497525i \(0.834242\pi\)
\(824\) 14.2936 0.497941
\(825\) 31.4007 1.09323
\(826\) 1.61634 0.0562397
\(827\) 8.29893 0.288582 0.144291 0.989535i \(-0.453910\pi\)
0.144291 + 0.989535i \(0.453910\pi\)
\(828\) −8.14330 −0.282999
\(829\) 18.8724 0.655465 0.327732 0.944771i \(-0.393716\pi\)
0.327732 + 0.944771i \(0.393716\pi\)
\(830\) −52.2041 −1.81203
\(831\) −28.5229 −0.989448
\(832\) −4.75403 −0.164816
\(833\) 4.38744 0.152016
\(834\) 8.26128 0.286065
\(835\) 34.1837 1.18297
\(836\) −19.9800 −0.691024
\(837\) −10.4107 −0.359848
\(838\) 6.89542 0.238198
\(839\) 8.29043 0.286217 0.143109 0.989707i \(-0.454290\pi\)
0.143109 + 0.989707i \(0.454290\pi\)
\(840\) 5.60533 0.193402
\(841\) 11.3794 0.392392
\(842\) −2.86944 −0.0988875
\(843\) −2.93316 −0.101023
\(844\) 12.2576 0.421923
\(845\) 33.2949 1.14538
\(846\) 4.44449 0.152805
\(847\) −14.5010 −0.498259
\(848\) 7.21748 0.247849
\(849\) 2.59567 0.0890832
\(850\) 7.02642 0.241004
\(851\) 93.9232 3.21965
\(852\) −6.19584 −0.212266
\(853\) −20.9784 −0.718286 −0.359143 0.933283i \(-0.616931\pi\)
−0.359143 + 0.933283i \(0.616931\pi\)
\(854\) −5.05694 −0.173045
\(855\) −15.5046 −0.530244
\(856\) −4.73424 −0.161813
\(857\) −18.6774 −0.638009 −0.319005 0.947753i \(-0.603349\pi\)
−0.319005 + 0.947753i \(0.603349\pi\)
\(858\) 21.2455 0.725310
\(859\) −6.62451 −0.226026 −0.113013 0.993594i \(-0.536050\pi\)
−0.113013 + 0.993594i \(0.536050\pi\)
\(860\) −12.3079 −0.419695
\(861\) 11.0479 0.376511
\(862\) −30.4374 −1.03670
\(863\) −23.3004 −0.793154 −0.396577 0.918001i \(-0.629802\pi\)
−0.396577 + 0.918001i \(0.629802\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −26.2581 −0.892804
\(866\) −40.5363 −1.37748
\(867\) 1.00000 0.0339618
\(868\) 16.8273 0.571156
\(869\) 7.29045 0.247312
\(870\) −22.0368 −0.747116
\(871\) 65.5633 2.22153
\(872\) 7.09575 0.240292
\(873\) 8.18102 0.276886
\(874\) −36.4076 −1.23150
\(875\) −11.3587 −0.383995
\(876\) −6.44163 −0.217643
\(877\) 42.3728 1.43083 0.715414 0.698701i \(-0.246238\pi\)
0.715414 + 0.698701i \(0.246238\pi\)
\(878\) 10.1448 0.342371
\(879\) 4.30235 0.145115
\(880\) 15.4979 0.522434
\(881\) 22.4733 0.757145 0.378573 0.925572i \(-0.376415\pi\)
0.378573 + 0.925572i \(0.376415\pi\)
\(882\) 4.38744 0.147733
\(883\) −18.4621 −0.621301 −0.310650 0.950524i \(-0.600547\pi\)
−0.310650 + 0.950524i \(0.600547\pi\)
\(884\) 4.75403 0.159895
\(885\) 3.46791 0.116573
\(886\) −39.0984 −1.31354
\(887\) 33.3760 1.12066 0.560328 0.828271i \(-0.310675\pi\)
0.560328 + 0.828271i \(0.310675\pi\)
\(888\) 11.5338 0.387049
\(889\) −7.82054 −0.262292
\(890\) −39.7632 −1.33287
\(891\) 4.46894 0.149715
\(892\) 25.7757 0.863033
\(893\) 19.8707 0.664947
\(894\) 2.73869 0.0915954
\(895\) 48.9374 1.63580
\(896\) 1.61634 0.0539982
\(897\) 38.7135 1.29261
\(898\) −11.7537 −0.392228
\(899\) −66.1548 −2.20639
\(900\) 7.02642 0.234214
\(901\) −7.21748 −0.240449
\(902\) 30.5458 1.01706
\(903\) 5.73651 0.190899
\(904\) 2.12850 0.0707928
\(905\) 83.3895 2.77196
\(906\) 2.48875 0.0826833
\(907\) −22.0740 −0.732955 −0.366477 0.930427i \(-0.619436\pi\)
−0.366477 + 0.930427i \(0.619436\pi\)
\(908\) −18.7200 −0.621247
\(909\) 1.99314 0.0661082
\(910\) −26.6479 −0.883371
\(911\) −14.6962 −0.486908 −0.243454 0.969912i \(-0.578280\pi\)
−0.243454 + 0.969912i \(0.578280\pi\)
\(912\) −4.47086 −0.148045
\(913\) 67.2731 2.22641
\(914\) 17.9916 0.595108
\(915\) −10.8498 −0.358684
\(916\) 16.0744 0.531112
\(917\) 25.3289 0.836434
\(918\) 1.00000 0.0330049
\(919\) −42.1922 −1.39179 −0.695896 0.718142i \(-0.744993\pi\)
−0.695896 + 0.718142i \(0.744993\pi\)
\(920\) 28.2403 0.931054
\(921\) 8.18122 0.269580
\(922\) 5.57560 0.183623
\(923\) 29.4552 0.969531
\(924\) −7.22334 −0.237631
\(925\) −81.0413 −2.66462
\(926\) 29.3945 0.965962
\(927\) −14.2936 −0.469463
\(928\) −6.35448 −0.208596
\(929\) 37.2670 1.22269 0.611345 0.791365i \(-0.290629\pi\)
0.611345 + 0.791365i \(0.290629\pi\)
\(930\) 36.1035 1.18388
\(931\) 19.6156 0.642876
\(932\) −4.16581 −0.136455
\(933\) −28.1098 −0.920273
\(934\) 15.0331 0.491899
\(935\) −15.4979 −0.506836
\(936\) 4.75403 0.155390
\(937\) −16.7330 −0.546644 −0.273322 0.961923i \(-0.588122\pi\)
−0.273322 + 0.961923i \(0.588122\pi\)
\(938\) −22.2911 −0.727831
\(939\) 9.95547 0.324884
\(940\) −15.4131 −0.502720
\(941\) 12.2415 0.399061 0.199531 0.979892i \(-0.436058\pi\)
0.199531 + 0.979892i \(0.436058\pi\)
\(942\) −18.6301 −0.607002
\(943\) 55.6604 1.81255
\(944\) 1.00000 0.0325472
\(945\) −5.60533 −0.182342
\(946\) 15.8606 0.515672
\(947\) −51.3523 −1.66873 −0.834363 0.551215i \(-0.814165\pi\)
−0.834363 + 0.551215i \(0.814165\pi\)
\(948\) 1.63136 0.0529840
\(949\) 30.6237 0.994089
\(950\) 31.4141 1.01921
\(951\) 3.36859 0.109234
\(952\) −1.61634 −0.0523860
\(953\) 0.954361 0.0309148 0.0154574 0.999881i \(-0.495080\pi\)
0.0154574 + 0.999881i \(0.495080\pi\)
\(954\) −7.21748 −0.233674
\(955\) 94.7642 3.06650
\(956\) 14.3722 0.464831
\(957\) 28.3978 0.917971
\(958\) 37.9795 1.22706
\(959\) −16.4275 −0.530473
\(960\) 3.46791 0.111926
\(961\) 77.3835 2.49624
\(962\) −54.8321 −1.76786
\(963\) 4.73424 0.152559
\(964\) 25.4335 0.819158
\(965\) −53.4048 −1.71916
\(966\) −13.1624 −0.423492
\(967\) −16.5773 −0.533090 −0.266545 0.963822i \(-0.585882\pi\)
−0.266545 + 0.963822i \(0.585882\pi\)
\(968\) −8.97147 −0.288354
\(969\) 4.47086 0.143625
\(970\) −28.3711 −0.910940
\(971\) 21.6566 0.694994 0.347497 0.937681i \(-0.387032\pi\)
0.347497 + 0.937681i \(0.387032\pi\)
\(972\) 1.00000 0.0320750
\(973\) 13.3531 0.428079
\(974\) 0.239403 0.00767098
\(975\) −33.4038 −1.06978
\(976\) −3.12863 −0.100145
\(977\) 38.3897 1.22819 0.614097 0.789230i \(-0.289520\pi\)
0.614097 + 0.789230i \(0.289520\pi\)
\(978\) −16.2123 −0.518413
\(979\) 51.2411 1.63767
\(980\) −15.2153 −0.486033
\(981\) −7.09575 −0.226550
\(982\) 16.5815 0.529136
\(983\) 0.501750 0.0160033 0.00800167 0.999968i \(-0.497453\pi\)
0.00800167 + 0.999968i \(0.497453\pi\)
\(984\) 6.83512 0.217896
\(985\) −22.3362 −0.711692
\(986\) 6.35448 0.202368
\(987\) 7.18381 0.228663
\(988\) 21.2546 0.676200
\(989\) 28.9011 0.919003
\(990\) −15.4979 −0.492556
\(991\) −24.5139 −0.778711 −0.389356 0.921088i \(-0.627302\pi\)
−0.389356 + 0.921088i \(0.627302\pi\)
\(992\) 10.4107 0.330541
\(993\) 26.7813 0.849878
\(994\) −10.0146 −0.317644
\(995\) −76.5213 −2.42589
\(996\) 15.0535 0.476987
\(997\) 5.89342 0.186647 0.0933233 0.995636i \(-0.470251\pi\)
0.0933233 + 0.995636i \(0.470251\pi\)
\(998\) 10.1572 0.321521
\(999\) −11.5338 −0.364913
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 6018.2.a.y.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6018.2.a.y.1.9 10 1.1 even 1 trivial