Properties

Label 6042.2.a.bc.1.9
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 20x^{7} + 69x^{6} + 27x^{5} - 185x^{4} + 8x^{3} + 109x^{2} - 8x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-4.36993\) of defining polynomial
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.36993 q^{5} -1.00000 q^{6} -4.18318 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.36993 q^{5} -1.00000 q^{6} -4.18318 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.36993 q^{10} -0.462917 q^{11} -1.00000 q^{12} -5.66621 q^{13} -4.18318 q^{14} -4.36993 q^{15} +1.00000 q^{16} -7.64277 q^{17} +1.00000 q^{18} +1.00000 q^{19} +4.36993 q^{20} +4.18318 q^{21} -0.462917 q^{22} +5.05356 q^{23} -1.00000 q^{24} +14.0963 q^{25} -5.66621 q^{26} -1.00000 q^{27} -4.18318 q^{28} +0.972185 q^{29} -4.36993 q^{30} +6.32253 q^{31} +1.00000 q^{32} +0.462917 q^{33} -7.64277 q^{34} -18.2802 q^{35} +1.00000 q^{36} -8.67622 q^{37} +1.00000 q^{38} +5.66621 q^{39} +4.36993 q^{40} -3.87511 q^{41} +4.18318 q^{42} -7.90866 q^{43} -0.462917 q^{44} +4.36993 q^{45} +5.05356 q^{46} +5.36140 q^{47} -1.00000 q^{48} +10.4990 q^{49} +14.0963 q^{50} +7.64277 q^{51} -5.66621 q^{52} -1.00000 q^{53} -1.00000 q^{54} -2.02292 q^{55} -4.18318 q^{56} -1.00000 q^{57} +0.972185 q^{58} +11.8679 q^{59} -4.36993 q^{60} -2.69186 q^{61} +6.32253 q^{62} -4.18318 q^{63} +1.00000 q^{64} -24.7609 q^{65} +0.462917 q^{66} -8.85931 q^{67} -7.64277 q^{68} -5.05356 q^{69} -18.2802 q^{70} -9.10783 q^{71} +1.00000 q^{72} -8.12473 q^{73} -8.67622 q^{74} -14.0963 q^{75} +1.00000 q^{76} +1.93647 q^{77} +5.66621 q^{78} -3.22329 q^{79} +4.36993 q^{80} +1.00000 q^{81} -3.87511 q^{82} +10.8534 q^{83} +4.18318 q^{84} -33.3984 q^{85} -7.90866 q^{86} -0.972185 q^{87} -0.462917 q^{88} -17.1661 q^{89} +4.36993 q^{90} +23.7028 q^{91} +5.05356 q^{92} -6.32253 q^{93} +5.36140 q^{94} +4.36993 q^{95} -1.00000 q^{96} -12.7574 q^{97} +10.4990 q^{98} -0.462917 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 3 q^{5} - 9 q^{6} - 7 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 3 q^{5} - 9 q^{6} - 7 q^{7} + 9 q^{8} + 9 q^{9} - 3 q^{10} + 4 q^{11} - 9 q^{12} - 5 q^{13} - 7 q^{14} + 3 q^{15} + 9 q^{16} - 28 q^{17} + 9 q^{18} + 9 q^{19} - 3 q^{20} + 7 q^{21} + 4 q^{22} - 10 q^{23} - 9 q^{24} + 4 q^{25} - 5 q^{26} - 9 q^{27} - 7 q^{28} + 3 q^{30} + 5 q^{31} + 9 q^{32} - 4 q^{33} - 28 q^{34} - 10 q^{35} + 9 q^{36} - 25 q^{37} + 9 q^{38} + 5 q^{39} - 3 q^{40} - 7 q^{41} + 7 q^{42} - 16 q^{43} + 4 q^{44} - 3 q^{45} - 10 q^{46} - 9 q^{47} - 9 q^{48} + 44 q^{49} + 4 q^{50} + 28 q^{51} - 5 q^{52} - 9 q^{53} - 9 q^{54} - 31 q^{55} - 7 q^{56} - 9 q^{57} - 3 q^{59} + 3 q^{60} - 16 q^{61} + 5 q^{62} - 7 q^{63} + 9 q^{64} - 33 q^{65} - 4 q^{66} - 13 q^{67} - 28 q^{68} + 10 q^{69} - 10 q^{70} - 4 q^{71} + 9 q^{72} - 29 q^{73} - 25 q^{74} - 4 q^{75} + 9 q^{76} - 33 q^{77} + 5 q^{78} + 13 q^{79} - 3 q^{80} + 9 q^{81} - 7 q^{82} - 35 q^{83} + 7 q^{84} + 3 q^{85} - 16 q^{86} + 4 q^{88} - 19 q^{89} - 3 q^{90} - 10 q^{92} - 5 q^{93} - 9 q^{94} - 3 q^{95} - 9 q^{96} - 12 q^{97} + 44 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(5\) 4.36993 1.95429 0.977146 0.212569i \(-0.0681830\pi\)
0.977146 + 0.212569i \(0.0681830\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.18318 −1.58110 −0.790548 0.612401i \(-0.790204\pi\)
−0.790548 + 0.612401i \(0.790204\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.36993 1.38189
\(11\) −0.462917 −0.139575 −0.0697874 0.997562i \(-0.522232\pi\)
−0.0697874 + 0.997562i \(0.522232\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.66621 −1.57152 −0.785762 0.618529i \(-0.787729\pi\)
−0.785762 + 0.618529i \(0.787729\pi\)
\(14\) −4.18318 −1.11800
\(15\) −4.36993 −1.12831
\(16\) 1.00000 0.250000
\(17\) −7.64277 −1.85364 −0.926822 0.375500i \(-0.877471\pi\)
−0.926822 + 0.375500i \(0.877471\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 4.36993 0.977146
\(21\) 4.18318 0.912846
\(22\) −0.462917 −0.0986943
\(23\) 5.05356 1.05374 0.526870 0.849946i \(-0.323366\pi\)
0.526870 + 0.849946i \(0.323366\pi\)
\(24\) −1.00000 −0.204124
\(25\) 14.0963 2.81926
\(26\) −5.66621 −1.11124
\(27\) −1.00000 −0.192450
\(28\) −4.18318 −0.790548
\(29\) 0.972185 0.180530 0.0902651 0.995918i \(-0.471229\pi\)
0.0902651 + 0.995918i \(0.471229\pi\)
\(30\) −4.36993 −0.797836
\(31\) 6.32253 1.13556 0.567780 0.823180i \(-0.307802\pi\)
0.567780 + 0.823180i \(0.307802\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.462917 0.0805835
\(34\) −7.64277 −1.31072
\(35\) −18.2802 −3.08992
\(36\) 1.00000 0.166667
\(37\) −8.67622 −1.42636 −0.713181 0.700980i \(-0.752746\pi\)
−0.713181 + 0.700980i \(0.752746\pi\)
\(38\) 1.00000 0.162221
\(39\) 5.66621 0.907320
\(40\) 4.36993 0.690947
\(41\) −3.87511 −0.605190 −0.302595 0.953119i \(-0.597853\pi\)
−0.302595 + 0.953119i \(0.597853\pi\)
\(42\) 4.18318 0.645479
\(43\) −7.90866 −1.20606 −0.603029 0.797719i \(-0.706040\pi\)
−0.603029 + 0.797719i \(0.706040\pi\)
\(44\) −0.462917 −0.0697874
\(45\) 4.36993 0.651431
\(46\) 5.05356 0.745106
\(47\) 5.36140 0.782040 0.391020 0.920382i \(-0.372122\pi\)
0.391020 + 0.920382i \(0.372122\pi\)
\(48\) −1.00000 −0.144338
\(49\) 10.4990 1.49986
\(50\) 14.0963 1.99352
\(51\) 7.64277 1.07020
\(52\) −5.66621 −0.785762
\(53\) −1.00000 −0.137361
\(54\) −1.00000 −0.136083
\(55\) −2.02292 −0.272770
\(56\) −4.18318 −0.559002
\(57\) −1.00000 −0.132453
\(58\) 0.972185 0.127654
\(59\) 11.8679 1.54506 0.772532 0.634976i \(-0.218990\pi\)
0.772532 + 0.634976i \(0.218990\pi\)
\(60\) −4.36993 −0.564156
\(61\) −2.69186 −0.344657 −0.172329 0.985040i \(-0.555129\pi\)
−0.172329 + 0.985040i \(0.555129\pi\)
\(62\) 6.32253 0.802962
\(63\) −4.18318 −0.527032
\(64\) 1.00000 0.125000
\(65\) −24.7609 −3.07122
\(66\) 0.462917 0.0569812
\(67\) −8.85931 −1.08234 −0.541168 0.840914i \(-0.682018\pi\)
−0.541168 + 0.840914i \(0.682018\pi\)
\(68\) −7.64277 −0.926822
\(69\) −5.05356 −0.608377
\(70\) −18.2802 −2.18490
\(71\) −9.10783 −1.08090 −0.540450 0.841376i \(-0.681746\pi\)
−0.540450 + 0.841376i \(0.681746\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.12473 −0.950928 −0.475464 0.879735i \(-0.657720\pi\)
−0.475464 + 0.879735i \(0.657720\pi\)
\(74\) −8.67622 −1.00859
\(75\) −14.0963 −1.62770
\(76\) 1.00000 0.114708
\(77\) 1.93647 0.220681
\(78\) 5.66621 0.641572
\(79\) −3.22329 −0.362648 −0.181324 0.983423i \(-0.558038\pi\)
−0.181324 + 0.983423i \(0.558038\pi\)
\(80\) 4.36993 0.488573
\(81\) 1.00000 0.111111
\(82\) −3.87511 −0.427934
\(83\) 10.8534 1.19132 0.595658 0.803238i \(-0.296891\pi\)
0.595658 + 0.803238i \(0.296891\pi\)
\(84\) 4.18318 0.456423
\(85\) −33.3984 −3.62256
\(86\) −7.90866 −0.852812
\(87\) −0.972185 −0.104229
\(88\) −0.462917 −0.0493471
\(89\) −17.1661 −1.81960 −0.909800 0.415047i \(-0.863765\pi\)
−0.909800 + 0.415047i \(0.863765\pi\)
\(90\) 4.36993 0.460631
\(91\) 23.7028 2.48473
\(92\) 5.05356 0.526870
\(93\) −6.32253 −0.655616
\(94\) 5.36140 0.552986
\(95\) 4.36993 0.448345
\(96\) −1.00000 −0.102062
\(97\) −12.7574 −1.29532 −0.647658 0.761932i \(-0.724251\pi\)
−0.647658 + 0.761932i \(0.724251\pi\)
\(98\) 10.4990 1.06056
\(99\) −0.462917 −0.0465249
\(100\) 14.0963 1.40963
\(101\) −14.9630 −1.48887 −0.744437 0.667693i \(-0.767282\pi\)
−0.744437 + 0.667693i \(0.767282\pi\)
\(102\) 7.64277 0.756747
\(103\) −11.2134 −1.10489 −0.552444 0.833550i \(-0.686305\pi\)
−0.552444 + 0.833550i \(0.686305\pi\)
\(104\) −5.66621 −0.555618
\(105\) 18.2802 1.78397
\(106\) −1.00000 −0.0971286
\(107\) −15.8051 −1.52794 −0.763968 0.645254i \(-0.776751\pi\)
−0.763968 + 0.645254i \(0.776751\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −13.5273 −1.29568 −0.647840 0.761776i \(-0.724328\pi\)
−0.647840 + 0.761776i \(0.724328\pi\)
\(110\) −2.02292 −0.192877
\(111\) 8.67622 0.823511
\(112\) −4.18318 −0.395274
\(113\) −0.380231 −0.0357691 −0.0178846 0.999840i \(-0.505693\pi\)
−0.0178846 + 0.999840i \(0.505693\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 22.0837 2.05931
\(116\) 0.972185 0.0902651
\(117\) −5.66621 −0.523841
\(118\) 11.8679 1.09252
\(119\) 31.9711 2.93079
\(120\) −4.36993 −0.398918
\(121\) −10.7857 −0.980519
\(122\) −2.69186 −0.243709
\(123\) 3.87511 0.349407
\(124\) 6.32253 0.567780
\(125\) 39.7501 3.55536
\(126\) −4.18318 −0.372668
\(127\) −8.23191 −0.730464 −0.365232 0.930916i \(-0.619010\pi\)
−0.365232 + 0.930916i \(0.619010\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.90866 0.696318
\(130\) −24.7609 −2.17168
\(131\) 9.88981 0.864076 0.432038 0.901855i \(-0.357795\pi\)
0.432038 + 0.901855i \(0.357795\pi\)
\(132\) 0.462917 0.0402918
\(133\) −4.18318 −0.362728
\(134\) −8.85931 −0.765328
\(135\) −4.36993 −0.376104
\(136\) −7.64277 −0.655362
\(137\) −9.88872 −0.844850 −0.422425 0.906398i \(-0.638821\pi\)
−0.422425 + 0.906398i \(0.638821\pi\)
\(138\) −5.05356 −0.430187
\(139\) −5.37706 −0.456076 −0.228038 0.973652i \(-0.573231\pi\)
−0.228038 + 0.973652i \(0.573231\pi\)
\(140\) −18.2802 −1.54496
\(141\) −5.36140 −0.451511
\(142\) −9.10783 −0.764312
\(143\) 2.62299 0.219345
\(144\) 1.00000 0.0833333
\(145\) 4.24838 0.352809
\(146\) −8.12473 −0.672408
\(147\) −10.4990 −0.865946
\(148\) −8.67622 −0.713181
\(149\) 6.00567 0.492004 0.246002 0.969269i \(-0.420883\pi\)
0.246002 + 0.969269i \(0.420883\pi\)
\(150\) −14.0963 −1.15096
\(151\) −5.75130 −0.468034 −0.234017 0.972233i \(-0.575187\pi\)
−0.234017 + 0.972233i \(0.575187\pi\)
\(152\) 1.00000 0.0811107
\(153\) −7.64277 −0.617882
\(154\) 1.93647 0.156045
\(155\) 27.6290 2.21922
\(156\) 5.66621 0.453660
\(157\) 0.329748 0.0263168 0.0131584 0.999913i \(-0.495811\pi\)
0.0131584 + 0.999913i \(0.495811\pi\)
\(158\) −3.22329 −0.256431
\(159\) 1.00000 0.0793052
\(160\) 4.36993 0.345473
\(161\) −21.1400 −1.66606
\(162\) 1.00000 0.0785674
\(163\) −18.6873 −1.46370 −0.731850 0.681466i \(-0.761343\pi\)
−0.731850 + 0.681466i \(0.761343\pi\)
\(164\) −3.87511 −0.302595
\(165\) 2.02292 0.157484
\(166\) 10.8534 0.842388
\(167\) 16.5992 1.28448 0.642242 0.766502i \(-0.278004\pi\)
0.642242 + 0.766502i \(0.278004\pi\)
\(168\) 4.18318 0.322740
\(169\) 19.1059 1.46969
\(170\) −33.3984 −2.56154
\(171\) 1.00000 0.0764719
\(172\) −7.90866 −0.603029
\(173\) −4.31430 −0.328010 −0.164005 0.986459i \(-0.552441\pi\)
−0.164005 + 0.986459i \(0.552441\pi\)
\(174\) −0.972185 −0.0737012
\(175\) −58.9674 −4.45751
\(176\) −0.462917 −0.0348937
\(177\) −11.8679 −0.892043
\(178\) −17.1661 −1.28665
\(179\) 5.74119 0.429117 0.214558 0.976711i \(-0.431169\pi\)
0.214558 + 0.976711i \(0.431169\pi\)
\(180\) 4.36993 0.325715
\(181\) −17.6145 −1.30927 −0.654637 0.755944i \(-0.727178\pi\)
−0.654637 + 0.755944i \(0.727178\pi\)
\(182\) 23.7028 1.75697
\(183\) 2.69186 0.198988
\(184\) 5.05356 0.372553
\(185\) −37.9145 −2.78753
\(186\) −6.32253 −0.463590
\(187\) 3.53797 0.258722
\(188\) 5.36140 0.391020
\(189\) 4.18318 0.304282
\(190\) 4.36993 0.317028
\(191\) 17.8500 1.29158 0.645790 0.763515i \(-0.276528\pi\)
0.645790 + 0.763515i \(0.276528\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.80102 0.633512 0.316756 0.948507i \(-0.397406\pi\)
0.316756 + 0.948507i \(0.397406\pi\)
\(194\) −12.7574 −0.915926
\(195\) 24.7609 1.77317
\(196\) 10.4990 0.749931
\(197\) 11.0422 0.786721 0.393361 0.919384i \(-0.371312\pi\)
0.393361 + 0.919384i \(0.371312\pi\)
\(198\) −0.462917 −0.0328981
\(199\) 5.15706 0.365574 0.182787 0.983153i \(-0.441488\pi\)
0.182787 + 0.983153i \(0.441488\pi\)
\(200\) 14.0963 0.996758
\(201\) 8.85931 0.624887
\(202\) −14.9630 −1.05279
\(203\) −4.06683 −0.285436
\(204\) 7.64277 0.535101
\(205\) −16.9339 −1.18272
\(206\) −11.2134 −0.781274
\(207\) 5.05356 0.351247
\(208\) −5.66621 −0.392881
\(209\) −0.462917 −0.0320207
\(210\) 18.2802 1.26146
\(211\) 8.38109 0.576978 0.288489 0.957483i \(-0.406847\pi\)
0.288489 + 0.957483i \(0.406847\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 9.10783 0.624058
\(214\) −15.8051 −1.08041
\(215\) −34.5603 −2.35699
\(216\) −1.00000 −0.0680414
\(217\) −26.4483 −1.79543
\(218\) −13.5273 −0.916184
\(219\) 8.12473 0.549019
\(220\) −2.02292 −0.136385
\(221\) 43.3056 2.91305
\(222\) 8.67622 0.582310
\(223\) 13.4343 0.899628 0.449814 0.893122i \(-0.351490\pi\)
0.449814 + 0.893122i \(0.351490\pi\)
\(224\) −4.18318 −0.279501
\(225\) 14.0963 0.939753
\(226\) −0.380231 −0.0252926
\(227\) 21.2391 1.40969 0.704844 0.709362i \(-0.251017\pi\)
0.704844 + 0.709362i \(0.251017\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 14.2131 0.939226 0.469613 0.882872i \(-0.344394\pi\)
0.469613 + 0.882872i \(0.344394\pi\)
\(230\) 22.0837 1.45616
\(231\) −1.93647 −0.127410
\(232\) 0.972185 0.0638271
\(233\) −7.85524 −0.514614 −0.257307 0.966330i \(-0.582835\pi\)
−0.257307 + 0.966330i \(0.582835\pi\)
\(234\) −5.66621 −0.370412
\(235\) 23.4289 1.52834
\(236\) 11.8679 0.772532
\(237\) 3.22329 0.209375
\(238\) 31.9711 2.07238
\(239\) 14.4480 0.934564 0.467282 0.884108i \(-0.345233\pi\)
0.467282 + 0.884108i \(0.345233\pi\)
\(240\) −4.36993 −0.282078
\(241\) −13.1851 −0.849330 −0.424665 0.905351i \(-0.639608\pi\)
−0.424665 + 0.905351i \(0.639608\pi\)
\(242\) −10.7857 −0.693332
\(243\) −1.00000 −0.0641500
\(244\) −2.69186 −0.172329
\(245\) 45.8800 2.93117
\(246\) 3.87511 0.247068
\(247\) −5.66621 −0.360532
\(248\) 6.32253 0.401481
\(249\) −10.8534 −0.687807
\(250\) 39.7501 2.51402
\(251\) −8.60144 −0.542918 −0.271459 0.962450i \(-0.587506\pi\)
−0.271459 + 0.962450i \(0.587506\pi\)
\(252\) −4.18318 −0.263516
\(253\) −2.33938 −0.147075
\(254\) −8.23191 −0.516516
\(255\) 33.3984 2.09149
\(256\) 1.00000 0.0625000
\(257\) 23.5374 1.46822 0.734111 0.679030i \(-0.237599\pi\)
0.734111 + 0.679030i \(0.237599\pi\)
\(258\) 7.90866 0.492371
\(259\) 36.2943 2.25522
\(260\) −24.7609 −1.53561
\(261\) 0.972185 0.0601767
\(262\) 9.88981 0.610994
\(263\) −3.24994 −0.200400 −0.100200 0.994967i \(-0.531948\pi\)
−0.100200 + 0.994967i \(0.531948\pi\)
\(264\) 0.462917 0.0284906
\(265\) −4.36993 −0.268443
\(266\) −4.18318 −0.256488
\(267\) 17.1661 1.05055
\(268\) −8.85931 −0.541168
\(269\) 15.4797 0.943816 0.471908 0.881648i \(-0.343566\pi\)
0.471908 + 0.881648i \(0.343566\pi\)
\(270\) −4.36993 −0.265945
\(271\) −30.8138 −1.87180 −0.935902 0.352261i \(-0.885413\pi\)
−0.935902 + 0.352261i \(0.885413\pi\)
\(272\) −7.64277 −0.463411
\(273\) −23.7028 −1.43456
\(274\) −9.88872 −0.597399
\(275\) −6.52541 −0.393497
\(276\) −5.05356 −0.304188
\(277\) 0.982640 0.0590411 0.0295206 0.999564i \(-0.490602\pi\)
0.0295206 + 0.999564i \(0.490602\pi\)
\(278\) −5.37706 −0.322495
\(279\) 6.32253 0.378520
\(280\) −18.2802 −1.09245
\(281\) 13.3044 0.793675 0.396837 0.917889i \(-0.370108\pi\)
0.396837 + 0.917889i \(0.370108\pi\)
\(282\) −5.36140 −0.319267
\(283\) −5.49847 −0.326850 −0.163425 0.986556i \(-0.552254\pi\)
−0.163425 + 0.986556i \(0.552254\pi\)
\(284\) −9.10783 −0.540450
\(285\) −4.36993 −0.258852
\(286\) 2.62299 0.155100
\(287\) 16.2103 0.956863
\(288\) 1.00000 0.0589256
\(289\) 41.4120 2.43600
\(290\) 4.24838 0.249474
\(291\) 12.7574 0.747850
\(292\) −8.12473 −0.475464
\(293\) 15.9482 0.931705 0.465853 0.884862i \(-0.345748\pi\)
0.465853 + 0.884862i \(0.345748\pi\)
\(294\) −10.4990 −0.612316
\(295\) 51.8617 3.01951
\(296\) −8.67622 −0.504295
\(297\) 0.462917 0.0268612
\(298\) 6.00567 0.347899
\(299\) −28.6345 −1.65598
\(300\) −14.0963 −0.813850
\(301\) 33.0834 1.90689
\(302\) −5.75130 −0.330950
\(303\) 14.9630 0.859602
\(304\) 1.00000 0.0573539
\(305\) −11.7632 −0.673561
\(306\) −7.64277 −0.436908
\(307\) −13.1936 −0.753001 −0.376501 0.926416i \(-0.622873\pi\)
−0.376501 + 0.926416i \(0.622873\pi\)
\(308\) 1.93647 0.110341
\(309\) 11.2134 0.637907
\(310\) 27.6290 1.56922
\(311\) −11.4478 −0.649147 −0.324573 0.945861i \(-0.605221\pi\)
−0.324573 + 0.945861i \(0.605221\pi\)
\(312\) 5.66621 0.320786
\(313\) 7.86254 0.444417 0.222209 0.974999i \(-0.428673\pi\)
0.222209 + 0.974999i \(0.428673\pi\)
\(314\) 0.329748 0.0186088
\(315\) −18.2802 −1.02997
\(316\) −3.22329 −0.181324
\(317\) 13.7246 0.770850 0.385425 0.922739i \(-0.374055\pi\)
0.385425 + 0.922739i \(0.374055\pi\)
\(318\) 1.00000 0.0560772
\(319\) −0.450041 −0.0251975
\(320\) 4.36993 0.244287
\(321\) 15.8051 0.882154
\(322\) −21.1400 −1.17808
\(323\) −7.64277 −0.425255
\(324\) 1.00000 0.0555556
\(325\) −79.8725 −4.43053
\(326\) −18.6873 −1.03499
\(327\) 13.5273 0.748061
\(328\) −3.87511 −0.213967
\(329\) −22.4277 −1.23648
\(330\) 2.02292 0.111358
\(331\) −8.55029 −0.469966 −0.234983 0.971999i \(-0.575503\pi\)
−0.234983 + 0.971999i \(0.575503\pi\)
\(332\) 10.8534 0.595658
\(333\) −8.67622 −0.475454
\(334\) 16.5992 0.908267
\(335\) −38.7146 −2.11520
\(336\) 4.18318 0.228211
\(337\) 1.35714 0.0739279 0.0369640 0.999317i \(-0.488231\pi\)
0.0369640 + 0.999317i \(0.488231\pi\)
\(338\) 19.1059 1.03923
\(339\) 0.380231 0.0206513
\(340\) −33.3984 −1.81128
\(341\) −2.92681 −0.158496
\(342\) 1.00000 0.0540738
\(343\) −14.6371 −0.790330
\(344\) −7.90866 −0.426406
\(345\) −22.0837 −1.18895
\(346\) −4.31430 −0.231938
\(347\) 14.8543 0.797422 0.398711 0.917077i \(-0.369458\pi\)
0.398711 + 0.917077i \(0.369458\pi\)
\(348\) −0.972185 −0.0521146
\(349\) 8.57010 0.458747 0.229373 0.973338i \(-0.426332\pi\)
0.229373 + 0.973338i \(0.426332\pi\)
\(350\) −58.9674 −3.15194
\(351\) 5.66621 0.302440
\(352\) −0.462917 −0.0246736
\(353\) −27.4357 −1.46025 −0.730126 0.683312i \(-0.760539\pi\)
−0.730126 + 0.683312i \(0.760539\pi\)
\(354\) −11.8679 −0.630770
\(355\) −39.8006 −2.11240
\(356\) −17.1661 −0.909800
\(357\) −31.9711 −1.69209
\(358\) 5.74119 0.303431
\(359\) −22.3571 −1.17996 −0.589980 0.807418i \(-0.700864\pi\)
−0.589980 + 0.807418i \(0.700864\pi\)
\(360\) 4.36993 0.230316
\(361\) 1.00000 0.0526316
\(362\) −17.6145 −0.925796
\(363\) 10.7857 0.566103
\(364\) 23.7028 1.24236
\(365\) −35.5045 −1.85839
\(366\) 2.69186 0.140706
\(367\) −34.2891 −1.78987 −0.894937 0.446192i \(-0.852780\pi\)
−0.894937 + 0.446192i \(0.852780\pi\)
\(368\) 5.05356 0.263435
\(369\) −3.87511 −0.201730
\(370\) −37.9145 −1.97108
\(371\) 4.18318 0.217180
\(372\) −6.32253 −0.327808
\(373\) 15.6803 0.811894 0.405947 0.913897i \(-0.366942\pi\)
0.405947 + 0.913897i \(0.366942\pi\)
\(374\) 3.53797 0.182944
\(375\) −39.7501 −2.05269
\(376\) 5.36140 0.276493
\(377\) −5.50860 −0.283708
\(378\) 4.18318 0.215160
\(379\) 0.666461 0.0342338 0.0171169 0.999853i \(-0.494551\pi\)
0.0171169 + 0.999853i \(0.494551\pi\)
\(380\) 4.36993 0.224173
\(381\) 8.23191 0.421734
\(382\) 17.8500 0.913285
\(383\) 11.4556 0.585354 0.292677 0.956211i \(-0.405454\pi\)
0.292677 + 0.956211i \(0.405454\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.46223 0.431275
\(386\) 8.80102 0.447960
\(387\) −7.90866 −0.402020
\(388\) −12.7574 −0.647658
\(389\) 21.0287 1.06620 0.533099 0.846053i \(-0.321027\pi\)
0.533099 + 0.846053i \(0.321027\pi\)
\(390\) 24.7609 1.25382
\(391\) −38.6232 −1.95326
\(392\) 10.4990 0.530281
\(393\) −9.88981 −0.498875
\(394\) 11.0422 0.556296
\(395\) −14.0855 −0.708720
\(396\) −0.462917 −0.0232625
\(397\) −32.8051 −1.64644 −0.823222 0.567720i \(-0.807826\pi\)
−0.823222 + 0.567720i \(0.807826\pi\)
\(398\) 5.15706 0.258500
\(399\) 4.18318 0.209421
\(400\) 14.0963 0.704814
\(401\) 36.2376 1.80962 0.904810 0.425815i \(-0.140013\pi\)
0.904810 + 0.425815i \(0.140013\pi\)
\(402\) 8.85931 0.441862
\(403\) −35.8248 −1.78456
\(404\) −14.9630 −0.744437
\(405\) 4.36993 0.217144
\(406\) −4.06683 −0.201833
\(407\) 4.01637 0.199084
\(408\) 7.64277 0.378374
\(409\) −3.38851 −0.167551 −0.0837755 0.996485i \(-0.526698\pi\)
−0.0837755 + 0.996485i \(0.526698\pi\)
\(410\) −16.9339 −0.836308
\(411\) 9.88872 0.487775
\(412\) −11.2134 −0.552444
\(413\) −49.6454 −2.44289
\(414\) 5.05356 0.248369
\(415\) 47.4287 2.32818
\(416\) −5.66621 −0.277809
\(417\) 5.37706 0.263316
\(418\) −0.462917 −0.0226420
\(419\) 23.9914 1.17206 0.586029 0.810290i \(-0.300690\pi\)
0.586029 + 0.810290i \(0.300690\pi\)
\(420\) 18.2802 0.891984
\(421\) −4.33701 −0.211373 −0.105686 0.994400i \(-0.533704\pi\)
−0.105686 + 0.994400i \(0.533704\pi\)
\(422\) 8.38109 0.407985
\(423\) 5.36140 0.260680
\(424\) −1.00000 −0.0485643
\(425\) −107.735 −5.22590
\(426\) 9.10783 0.441276
\(427\) 11.2605 0.544936
\(428\) −15.8051 −0.763968
\(429\) −2.62299 −0.126639
\(430\) −34.5603 −1.66664
\(431\) 20.6639 0.995347 0.497673 0.867365i \(-0.334188\pi\)
0.497673 + 0.867365i \(0.334188\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 4.41555 0.212198 0.106099 0.994356i \(-0.466164\pi\)
0.106099 + 0.994356i \(0.466164\pi\)
\(434\) −26.4483 −1.26956
\(435\) −4.24838 −0.203694
\(436\) −13.5273 −0.647840
\(437\) 5.05356 0.241744
\(438\) 8.12473 0.388215
\(439\) 30.7827 1.46918 0.734589 0.678512i \(-0.237375\pi\)
0.734589 + 0.678512i \(0.237375\pi\)
\(440\) −2.02292 −0.0964387
\(441\) 10.4990 0.499954
\(442\) 43.3056 2.05984
\(443\) 12.9311 0.614377 0.307188 0.951649i \(-0.400612\pi\)
0.307188 + 0.951649i \(0.400612\pi\)
\(444\) 8.67622 0.411755
\(445\) −75.0145 −3.55603
\(446\) 13.4343 0.636133
\(447\) −6.00567 −0.284059
\(448\) −4.18318 −0.197637
\(449\) −23.5987 −1.11369 −0.556846 0.830616i \(-0.687988\pi\)
−0.556846 + 0.830616i \(0.687988\pi\)
\(450\) 14.0963 0.664505
\(451\) 1.79385 0.0844693
\(452\) −0.380231 −0.0178846
\(453\) 5.75130 0.270220
\(454\) 21.2391 0.996800
\(455\) 103.580 4.85589
\(456\) −1.00000 −0.0468293
\(457\) −7.65871 −0.358260 −0.179130 0.983825i \(-0.557328\pi\)
−0.179130 + 0.983825i \(0.557328\pi\)
\(458\) 14.2131 0.664133
\(459\) 7.64277 0.356734
\(460\) 22.0837 1.02966
\(461\) −10.5593 −0.491795 −0.245898 0.969296i \(-0.579083\pi\)
−0.245898 + 0.969296i \(0.579083\pi\)
\(462\) −1.93647 −0.0900927
\(463\) 14.7577 0.685846 0.342923 0.939363i \(-0.388583\pi\)
0.342923 + 0.939363i \(0.388583\pi\)
\(464\) 0.972185 0.0451326
\(465\) −27.6290 −1.28126
\(466\) −7.85524 −0.363887
\(467\) −18.3402 −0.848682 −0.424341 0.905502i \(-0.639494\pi\)
−0.424341 + 0.905502i \(0.639494\pi\)
\(468\) −5.66621 −0.261921
\(469\) 37.0601 1.71128
\(470\) 23.4289 1.08070
\(471\) −0.329748 −0.0151940
\(472\) 11.8679 0.546262
\(473\) 3.66105 0.168335
\(474\) 3.22329 0.148050
\(475\) 14.0963 0.646782
\(476\) 31.9711 1.46539
\(477\) −1.00000 −0.0457869
\(478\) 14.4480 0.660836
\(479\) −20.7969 −0.950237 −0.475118 0.879922i \(-0.657595\pi\)
−0.475118 + 0.879922i \(0.657595\pi\)
\(480\) −4.36993 −0.199459
\(481\) 49.1613 2.24156
\(482\) −13.1851 −0.600567
\(483\) 21.1400 0.961902
\(484\) −10.7857 −0.490259
\(485\) −55.7488 −2.53142
\(486\) −1.00000 −0.0453609
\(487\) 4.73868 0.214730 0.107365 0.994220i \(-0.465759\pi\)
0.107365 + 0.994220i \(0.465759\pi\)
\(488\) −2.69186 −0.121855
\(489\) 18.6873 0.845068
\(490\) 45.8800 2.07265
\(491\) 7.66928 0.346110 0.173055 0.984912i \(-0.444636\pi\)
0.173055 + 0.984912i \(0.444636\pi\)
\(492\) 3.87511 0.174703
\(493\) −7.43019 −0.334639
\(494\) −5.66621 −0.254935
\(495\) −2.02292 −0.0909233
\(496\) 6.32253 0.283890
\(497\) 38.0997 1.70901
\(498\) −10.8534 −0.486353
\(499\) −16.1706 −0.723895 −0.361948 0.932198i \(-0.617888\pi\)
−0.361948 + 0.932198i \(0.617888\pi\)
\(500\) 39.7501 1.77768
\(501\) −16.5992 −0.741597
\(502\) −8.60144 −0.383901
\(503\) −4.14454 −0.184796 −0.0923980 0.995722i \(-0.529453\pi\)
−0.0923980 + 0.995722i \(0.529453\pi\)
\(504\) −4.18318 −0.186334
\(505\) −65.3873 −2.90970
\(506\) −2.33938 −0.103998
\(507\) −19.1059 −0.848524
\(508\) −8.23191 −0.365232
\(509\) −8.16352 −0.361842 −0.180921 0.983498i \(-0.557908\pi\)
−0.180921 + 0.983498i \(0.557908\pi\)
\(510\) 33.3984 1.47891
\(511\) 33.9873 1.50351
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 23.5374 1.03819
\(515\) −49.0017 −2.15927
\(516\) 7.90866 0.348159
\(517\) −2.48188 −0.109153
\(518\) 36.2943 1.59468
\(519\) 4.31430 0.189377
\(520\) −24.7609 −1.08584
\(521\) −33.1616 −1.45283 −0.726417 0.687254i \(-0.758816\pi\)
−0.726417 + 0.687254i \(0.758816\pi\)
\(522\) 0.972185 0.0425514
\(523\) 3.91437 0.171163 0.0855816 0.996331i \(-0.472725\pi\)
0.0855816 + 0.996331i \(0.472725\pi\)
\(524\) 9.88981 0.432038
\(525\) 58.9674 2.57355
\(526\) −3.24994 −0.141704
\(527\) −48.3217 −2.10492
\(528\) 0.462917 0.0201459
\(529\) 2.53844 0.110367
\(530\) −4.36993 −0.189818
\(531\) 11.8679 0.515021
\(532\) −4.18318 −0.181364
\(533\) 21.9572 0.951070
\(534\) 17.1661 0.742849
\(535\) −69.0671 −2.98603
\(536\) −8.85931 −0.382664
\(537\) −5.74119 −0.247751
\(538\) 15.4797 0.667378
\(539\) −4.86018 −0.209343
\(540\) −4.36993 −0.188052
\(541\) 39.4448 1.69586 0.847932 0.530105i \(-0.177847\pi\)
0.847932 + 0.530105i \(0.177847\pi\)
\(542\) −30.8138 −1.32356
\(543\) 17.6145 0.755909
\(544\) −7.64277 −0.327681
\(545\) −59.1134 −2.53214
\(546\) −23.7028 −1.01439
\(547\) 7.37946 0.315523 0.157761 0.987477i \(-0.449572\pi\)
0.157761 + 0.987477i \(0.449572\pi\)
\(548\) −9.88872 −0.422425
\(549\) −2.69186 −0.114886
\(550\) −6.52541 −0.278245
\(551\) 0.972185 0.0414165
\(552\) −5.05356 −0.215094
\(553\) 13.4836 0.573381
\(554\) 0.982640 0.0417484
\(555\) 37.9145 1.60938
\(556\) −5.37706 −0.228038
\(557\) 8.53921 0.361818 0.180909 0.983500i \(-0.442096\pi\)
0.180909 + 0.983500i \(0.442096\pi\)
\(558\) 6.32253 0.267654
\(559\) 44.8121 1.89535
\(560\) −18.2802 −0.772480
\(561\) −3.53797 −0.149373
\(562\) 13.3044 0.561213
\(563\) −41.6161 −1.75391 −0.876956 0.480572i \(-0.840429\pi\)
−0.876956 + 0.480572i \(0.840429\pi\)
\(564\) −5.36140 −0.225756
\(565\) −1.66158 −0.0699033
\(566\) −5.49847 −0.231118
\(567\) −4.18318 −0.175677
\(568\) −9.10783 −0.382156
\(569\) 23.3829 0.980264 0.490132 0.871648i \(-0.336949\pi\)
0.490132 + 0.871648i \(0.336949\pi\)
\(570\) −4.36993 −0.183036
\(571\) 6.61125 0.276672 0.138336 0.990385i \(-0.455825\pi\)
0.138336 + 0.990385i \(0.455825\pi\)
\(572\) 2.62299 0.109673
\(573\) −17.8500 −0.745694
\(574\) 16.2103 0.676604
\(575\) 71.2364 2.97076
\(576\) 1.00000 0.0416667
\(577\) −4.40595 −0.183422 −0.0917110 0.995786i \(-0.529234\pi\)
−0.0917110 + 0.995786i \(0.529234\pi\)
\(578\) 41.4120 1.72251
\(579\) −8.80102 −0.365758
\(580\) 4.24838 0.176404
\(581\) −45.4018 −1.88359
\(582\) 12.7574 0.528810
\(583\) 0.462917 0.0191721
\(584\) −8.12473 −0.336204
\(585\) −24.7609 −1.02374
\(586\) 15.9482 0.658815
\(587\) 17.3677 0.716840 0.358420 0.933560i \(-0.383316\pi\)
0.358420 + 0.933560i \(0.383316\pi\)
\(588\) −10.4990 −0.432973
\(589\) 6.32253 0.260515
\(590\) 51.8617 2.13511
\(591\) −11.0422 −0.454214
\(592\) −8.67622 −0.356591
\(593\) −4.65942 −0.191339 −0.0956696 0.995413i \(-0.530499\pi\)
−0.0956696 + 0.995413i \(0.530499\pi\)
\(594\) 0.462917 0.0189937
\(595\) 139.712 5.72762
\(596\) 6.00567 0.246002
\(597\) −5.15706 −0.211064
\(598\) −28.6345 −1.17095
\(599\) 22.4463 0.917132 0.458566 0.888660i \(-0.348363\pi\)
0.458566 + 0.888660i \(0.348363\pi\)
\(600\) −14.0963 −0.575479
\(601\) 23.2169 0.947037 0.473518 0.880784i \(-0.342984\pi\)
0.473518 + 0.880784i \(0.342984\pi\)
\(602\) 33.0834 1.34838
\(603\) −8.85931 −0.360779
\(604\) −5.75130 −0.234017
\(605\) −47.1328 −1.91622
\(606\) 14.9630 0.607830
\(607\) −4.61777 −0.187430 −0.0937148 0.995599i \(-0.529874\pi\)
−0.0937148 + 0.995599i \(0.529874\pi\)
\(608\) 1.00000 0.0405554
\(609\) 4.06683 0.164796
\(610\) −11.7632 −0.476280
\(611\) −30.3788 −1.22900
\(612\) −7.64277 −0.308941
\(613\) 40.5344 1.63717 0.818584 0.574386i \(-0.194759\pi\)
0.818584 + 0.574386i \(0.194759\pi\)
\(614\) −13.1936 −0.532452
\(615\) 16.9339 0.682843
\(616\) 1.93647 0.0780225
\(617\) 12.4860 0.502669 0.251334 0.967900i \(-0.419131\pi\)
0.251334 + 0.967900i \(0.419131\pi\)
\(618\) 11.2134 0.451069
\(619\) −40.5165 −1.62850 −0.814248 0.580517i \(-0.802850\pi\)
−0.814248 + 0.580517i \(0.802850\pi\)
\(620\) 27.6290 1.10961
\(621\) −5.05356 −0.202792
\(622\) −11.4478 −0.459016
\(623\) 71.8089 2.87696
\(624\) 5.66621 0.226830
\(625\) 103.224 4.12896
\(626\) 7.86254 0.314250
\(627\) 0.462917 0.0184871
\(628\) 0.329748 0.0131584
\(629\) 66.3104 2.64397
\(630\) −18.2802 −0.728302
\(631\) 6.88667 0.274154 0.137077 0.990560i \(-0.456229\pi\)
0.137077 + 0.990560i \(0.456229\pi\)
\(632\) −3.22329 −0.128215
\(633\) −8.38109 −0.333119
\(634\) 13.7246 0.545073
\(635\) −35.9729 −1.42754
\(636\) 1.00000 0.0396526
\(637\) −59.4897 −2.35707
\(638\) −0.450041 −0.0178173
\(639\) −9.10783 −0.360300
\(640\) 4.36993 0.172737
\(641\) 6.80835 0.268914 0.134457 0.990919i \(-0.457071\pi\)
0.134457 + 0.990919i \(0.457071\pi\)
\(642\) 15.8051 0.623777
\(643\) 32.7345 1.29092 0.645461 0.763793i \(-0.276665\pi\)
0.645461 + 0.763793i \(0.276665\pi\)
\(644\) −21.1400 −0.833031
\(645\) 34.5603 1.36081
\(646\) −7.64277 −0.300701
\(647\) 35.5943 1.39936 0.699678 0.714458i \(-0.253327\pi\)
0.699678 + 0.714458i \(0.253327\pi\)
\(648\) 1.00000 0.0392837
\(649\) −5.49384 −0.215652
\(650\) −79.8725 −3.13286
\(651\) 26.4483 1.03659
\(652\) −18.6873 −0.731850
\(653\) −9.12422 −0.357058 −0.178529 0.983935i \(-0.557134\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(654\) 13.5273 0.528959
\(655\) 43.2178 1.68866
\(656\) −3.87511 −0.151297
\(657\) −8.12473 −0.316976
\(658\) −22.4277 −0.874324
\(659\) −1.10571 −0.0430722 −0.0215361 0.999768i \(-0.506856\pi\)
−0.0215361 + 0.999768i \(0.506856\pi\)
\(660\) 2.02292 0.0787419
\(661\) −33.3931 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(662\) −8.55029 −0.332316
\(663\) −43.3056 −1.68185
\(664\) 10.8534 0.421194
\(665\) −18.2802 −0.708877
\(666\) −8.67622 −0.336197
\(667\) 4.91299 0.190232
\(668\) 16.5992 0.642242
\(669\) −13.4343 −0.519401
\(670\) −38.7146 −1.49567
\(671\) 1.24611 0.0481055
\(672\) 4.18318 0.161370
\(673\) −48.1559 −1.85627 −0.928136 0.372240i \(-0.878590\pi\)
−0.928136 + 0.372240i \(0.878590\pi\)
\(674\) 1.35714 0.0522749
\(675\) −14.0963 −0.542566
\(676\) 19.1059 0.734843
\(677\) −39.1433 −1.50440 −0.752200 0.658934i \(-0.771007\pi\)
−0.752200 + 0.658934i \(0.771007\pi\)
\(678\) 0.380231 0.0146027
\(679\) 53.3665 2.04802
\(680\) −33.3984 −1.28077
\(681\) −21.2391 −0.813884
\(682\) −2.92681 −0.112073
\(683\) 14.7869 0.565805 0.282902 0.959149i \(-0.408703\pi\)
0.282902 + 0.959149i \(0.408703\pi\)
\(684\) 1.00000 0.0382360
\(685\) −43.2130 −1.65108
\(686\) −14.6371 −0.558848
\(687\) −14.2131 −0.542262
\(688\) −7.90866 −0.301515
\(689\) 5.66621 0.215865
\(690\) −22.0837 −0.840712
\(691\) −2.93764 −0.111753 −0.0558765 0.998438i \(-0.517795\pi\)
−0.0558765 + 0.998438i \(0.517795\pi\)
\(692\) −4.31430 −0.164005
\(693\) 1.93647 0.0735603
\(694\) 14.8543 0.563862
\(695\) −23.4974 −0.891306
\(696\) −0.972185 −0.0368506
\(697\) 29.6166 1.12181
\(698\) 8.57010 0.324383
\(699\) 7.85524 0.297113
\(700\) −58.9674 −2.22876
\(701\) −19.3762 −0.731828 −0.365914 0.930649i \(-0.619244\pi\)
−0.365914 + 0.930649i \(0.619244\pi\)
\(702\) 5.66621 0.213857
\(703\) −8.67622 −0.327230
\(704\) −0.462917 −0.0174468
\(705\) −23.4289 −0.882385
\(706\) −27.4357 −1.03255
\(707\) 62.5930 2.35405
\(708\) −11.8679 −0.446021
\(709\) 1.81717 0.0682452 0.0341226 0.999418i \(-0.489136\pi\)
0.0341226 + 0.999418i \(0.489136\pi\)
\(710\) −39.8006 −1.49369
\(711\) −3.22329 −0.120883
\(712\) −17.1661 −0.643326
\(713\) 31.9513 1.19658
\(714\) −31.9711 −1.19649
\(715\) 11.4623 0.428664
\(716\) 5.74119 0.214558
\(717\) −14.4480 −0.539571
\(718\) −22.3571 −0.834358
\(719\) −42.8677 −1.59870 −0.799348 0.600869i \(-0.794821\pi\)
−0.799348 + 0.600869i \(0.794821\pi\)
\(720\) 4.36993 0.162858
\(721\) 46.9077 1.74693
\(722\) 1.00000 0.0372161
\(723\) 13.1851 0.490361
\(724\) −17.6145 −0.654637
\(725\) 13.7042 0.508961
\(726\) 10.7857 0.400295
\(727\) 46.7541 1.73401 0.867006 0.498297i \(-0.166041\pi\)
0.867006 + 0.498297i \(0.166041\pi\)
\(728\) 23.7028 0.878484
\(729\) 1.00000 0.0370370
\(730\) −35.5045 −1.31408
\(731\) 60.4441 2.23560
\(732\) 2.69186 0.0994940
\(733\) 42.7273 1.57817 0.789084 0.614286i \(-0.210556\pi\)
0.789084 + 0.614286i \(0.210556\pi\)
\(734\) −34.2891 −1.26563
\(735\) −45.8800 −1.69231
\(736\) 5.05356 0.186277
\(737\) 4.10113 0.151067
\(738\) −3.87511 −0.142645
\(739\) −16.6710 −0.613252 −0.306626 0.951830i \(-0.599200\pi\)
−0.306626 + 0.951830i \(0.599200\pi\)
\(740\) −37.9145 −1.39376
\(741\) 5.66621 0.208153
\(742\) 4.18318 0.153570
\(743\) 1.56657 0.0574719 0.0287359 0.999587i \(-0.490852\pi\)
0.0287359 + 0.999587i \(0.490852\pi\)
\(744\) −6.32253 −0.231795
\(745\) 26.2444 0.961520
\(746\) 15.6803 0.574096
\(747\) 10.8534 0.397106
\(748\) 3.53797 0.129361
\(749\) 66.1156 2.41581
\(750\) −39.7501 −1.45147
\(751\) 13.8383 0.504966 0.252483 0.967601i \(-0.418753\pi\)
0.252483 + 0.967601i \(0.418753\pi\)
\(752\) 5.36140 0.195510
\(753\) 8.60144 0.313454
\(754\) −5.50860 −0.200612
\(755\) −25.1328 −0.914675
\(756\) 4.18318 0.152141
\(757\) 13.8095 0.501914 0.250957 0.967998i \(-0.419255\pi\)
0.250957 + 0.967998i \(0.419255\pi\)
\(758\) 0.666461 0.0242070
\(759\) 2.33938 0.0849141
\(760\) 4.36993 0.158514
\(761\) 46.7409 1.69436 0.847179 0.531308i \(-0.178299\pi\)
0.847179 + 0.531308i \(0.178299\pi\)
\(762\) 8.23191 0.298211
\(763\) 56.5872 2.04859
\(764\) 17.8500 0.645790
\(765\) −33.3984 −1.20752
\(766\) 11.4556 0.413908
\(767\) −67.2458 −2.42810
\(768\) −1.00000 −0.0360844
\(769\) 17.6905 0.637935 0.318968 0.947766i \(-0.396664\pi\)
0.318968 + 0.947766i \(0.396664\pi\)
\(770\) 8.46223 0.304958
\(771\) −23.5374 −0.847678
\(772\) 8.80102 0.316756
\(773\) 20.2522 0.728421 0.364211 0.931317i \(-0.381339\pi\)
0.364211 + 0.931317i \(0.381339\pi\)
\(774\) −7.90866 −0.284271
\(775\) 89.1242 3.20144
\(776\) −12.7574 −0.457963
\(777\) −36.2943 −1.30205
\(778\) 21.0287 0.753916
\(779\) −3.87511 −0.138840
\(780\) 24.7609 0.886584
\(781\) 4.21617 0.150866
\(782\) −38.6232 −1.38116
\(783\) −0.972185 −0.0347431
\(784\) 10.4990 0.374966
\(785\) 1.44098 0.0514307
\(786\) −9.88981 −0.352758
\(787\) −20.7858 −0.740934 −0.370467 0.928846i \(-0.620802\pi\)
−0.370467 + 0.928846i \(0.620802\pi\)
\(788\) 11.0422 0.393361
\(789\) 3.24994 0.115701
\(790\) −14.0855 −0.501141
\(791\) 1.59058 0.0565544
\(792\) −0.462917 −0.0164490
\(793\) 15.2526 0.541637
\(794\) −32.8051 −1.16421
\(795\) 4.36993 0.154985
\(796\) 5.15706 0.182787
\(797\) −26.5141 −0.939177 −0.469589 0.882885i \(-0.655598\pi\)
−0.469589 + 0.882885i \(0.655598\pi\)
\(798\) 4.18318 0.148083
\(799\) −40.9760 −1.44963
\(800\) 14.0963 0.498379
\(801\) −17.1661 −0.606533
\(802\) 36.2376 1.27959
\(803\) 3.76108 0.132726
\(804\) 8.85931 0.312444
\(805\) −92.3802 −3.25597
\(806\) −35.8248 −1.26187
\(807\) −15.4797 −0.544912
\(808\) −14.9630 −0.526397
\(809\) 45.9823 1.61665 0.808326 0.588736i \(-0.200374\pi\)
0.808326 + 0.588736i \(0.200374\pi\)
\(810\) 4.36993 0.153544
\(811\) 12.3784 0.434666 0.217333 0.976098i \(-0.430264\pi\)
0.217333 + 0.976098i \(0.430264\pi\)
\(812\) −4.06683 −0.142718
\(813\) 30.8138 1.08069
\(814\) 4.01637 0.140774
\(815\) −81.6621 −2.86050
\(816\) 7.64277 0.267551
\(817\) −7.90866 −0.276689
\(818\) −3.38851 −0.118476
\(819\) 23.7028 0.828243
\(820\) −16.9339 −0.591359
\(821\) −29.1984 −1.01903 −0.509516 0.860461i \(-0.670175\pi\)
−0.509516 + 0.860461i \(0.670175\pi\)
\(822\) 9.88872 0.344909
\(823\) 8.25456 0.287736 0.143868 0.989597i \(-0.454046\pi\)
0.143868 + 0.989597i \(0.454046\pi\)
\(824\) −11.2134 −0.390637
\(825\) 6.52541 0.227186
\(826\) −49.6454 −1.72739
\(827\) 19.0354 0.661925 0.330962 0.943644i \(-0.392627\pi\)
0.330962 + 0.943644i \(0.392627\pi\)
\(828\) 5.05356 0.175623
\(829\) −18.5500 −0.644269 −0.322134 0.946694i \(-0.604400\pi\)
−0.322134 + 0.946694i \(0.604400\pi\)
\(830\) 47.4287 1.64627
\(831\) −0.982640 −0.0340874
\(832\) −5.66621 −0.196440
\(833\) −80.2418 −2.78021
\(834\) 5.37706 0.186192
\(835\) 72.5373 2.51026
\(836\) −0.462917 −0.0160103
\(837\) −6.32253 −0.218539
\(838\) 23.9914 0.828770
\(839\) −41.8878 −1.44613 −0.723064 0.690781i \(-0.757267\pi\)
−0.723064 + 0.690781i \(0.757267\pi\)
\(840\) 18.2802 0.630728
\(841\) −28.0549 −0.967409
\(842\) −4.33701 −0.149463
\(843\) −13.3044 −0.458228
\(844\) 8.38109 0.288489
\(845\) 83.4916 2.87220
\(846\) 5.36140 0.184329
\(847\) 45.1186 1.55029
\(848\) −1.00000 −0.0343401
\(849\) 5.49847 0.188707
\(850\) −107.735 −3.69527
\(851\) −43.8458 −1.50301
\(852\) 9.10783 0.312029
\(853\) 17.3721 0.594808 0.297404 0.954752i \(-0.403879\pi\)
0.297404 + 0.954752i \(0.403879\pi\)
\(854\) 11.2605 0.385328
\(855\) 4.36993 0.149448
\(856\) −15.8051 −0.540207
\(857\) 8.41806 0.287556 0.143778 0.989610i \(-0.454075\pi\)
0.143778 + 0.989610i \(0.454075\pi\)
\(858\) −2.62299 −0.0895473
\(859\) −10.5012 −0.358295 −0.179148 0.983822i \(-0.557334\pi\)
−0.179148 + 0.983822i \(0.557334\pi\)
\(860\) −34.5603 −1.17850
\(861\) −16.2103 −0.552445
\(862\) 20.6639 0.703816
\(863\) −48.4929 −1.65072 −0.825358 0.564609i \(-0.809027\pi\)
−0.825358 + 0.564609i \(0.809027\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.8532 −0.641028
\(866\) 4.41555 0.150047
\(867\) −41.4120 −1.40643
\(868\) −26.4483 −0.897714
\(869\) 1.49211 0.0506165
\(870\) −4.24838 −0.144034
\(871\) 50.1987 1.70092
\(872\) −13.5273 −0.458092
\(873\) −12.7574 −0.431772
\(874\) 5.05356 0.170939
\(875\) −166.282 −5.62136
\(876\) 8.12473 0.274509
\(877\) 14.4814 0.489001 0.244501 0.969649i \(-0.421376\pi\)
0.244501 + 0.969649i \(0.421376\pi\)
\(878\) 30.7827 1.03887
\(879\) −15.9482 −0.537920
\(880\) −2.02292 −0.0681925
\(881\) 28.9785 0.976310 0.488155 0.872757i \(-0.337670\pi\)
0.488155 + 0.872757i \(0.337670\pi\)
\(882\) 10.4990 0.353521
\(883\) 33.1809 1.11663 0.558313 0.829631i \(-0.311449\pi\)
0.558313 + 0.829631i \(0.311449\pi\)
\(884\) 43.3056 1.45652
\(885\) −51.8617 −1.74331
\(886\) 12.9311 0.434430
\(887\) 23.8738 0.801602 0.400801 0.916165i \(-0.368732\pi\)
0.400801 + 0.916165i \(0.368732\pi\)
\(888\) 8.67622 0.291155
\(889\) 34.4356 1.15493
\(890\) −75.0145 −2.51449
\(891\) −0.462917 −0.0155083
\(892\) 13.4343 0.449814
\(893\) 5.36140 0.179412
\(894\) −6.00567 −0.200860
\(895\) 25.0886 0.838620
\(896\) −4.18318 −0.139750
\(897\) 28.6345 0.956079
\(898\) −23.5987 −0.787499
\(899\) 6.14667 0.205003
\(900\) 14.0963 0.469876
\(901\) 7.64277 0.254618
\(902\) 1.79385 0.0597288
\(903\) −33.0834 −1.10095
\(904\) −0.380231 −0.0126463
\(905\) −76.9740 −2.55870
\(906\) 5.75130 0.191074
\(907\) −8.02094 −0.266331 −0.133165 0.991094i \(-0.542514\pi\)
−0.133165 + 0.991094i \(0.542514\pi\)
\(908\) 21.2391 0.704844
\(909\) −14.9630 −0.496291
\(910\) 103.580 3.43363
\(911\) −37.5553 −1.24426 −0.622131 0.782913i \(-0.713733\pi\)
−0.622131 + 0.782913i \(0.713733\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −5.02423 −0.166278
\(914\) −7.65871 −0.253328
\(915\) 11.7632 0.388881
\(916\) 14.2131 0.469613
\(917\) −41.3709 −1.36619
\(918\) 7.64277 0.252249
\(919\) 51.3959 1.69539 0.847697 0.530480i \(-0.177988\pi\)
0.847697 + 0.530480i \(0.177988\pi\)
\(920\) 22.0837 0.728078
\(921\) 13.1936 0.434745
\(922\) −10.5593 −0.347752
\(923\) 51.6069 1.69866
\(924\) −1.93647 −0.0637051
\(925\) −122.303 −4.02128
\(926\) 14.7577 0.484967
\(927\) −11.2134 −0.368296
\(928\) 0.972185 0.0319135
\(929\) −37.0142 −1.21440 −0.607198 0.794551i \(-0.707706\pi\)
−0.607198 + 0.794551i \(0.707706\pi\)
\(930\) −27.6290 −0.905991
\(931\) 10.4990 0.344092
\(932\) −7.85524 −0.257307
\(933\) 11.4478 0.374785
\(934\) −18.3402 −0.600109
\(935\) 15.4607 0.505619
\(936\) −5.66621 −0.185206
\(937\) −32.6951 −1.06810 −0.534051 0.845452i \(-0.679331\pi\)
−0.534051 + 0.845452i \(0.679331\pi\)
\(938\) 37.0601 1.21006
\(939\) −7.86254 −0.256584
\(940\) 23.4289 0.764168
\(941\) −46.4722 −1.51495 −0.757475 0.652865i \(-0.773567\pi\)
−0.757475 + 0.652865i \(0.773567\pi\)
\(942\) −0.329748 −0.0107438
\(943\) −19.5831 −0.637713
\(944\) 11.8679 0.386266
\(945\) 18.2802 0.594656
\(946\) 3.66105 0.119031
\(947\) 56.0273 1.82064 0.910321 0.413903i \(-0.135835\pi\)
0.910321 + 0.413903i \(0.135835\pi\)
\(948\) 3.22329 0.104687
\(949\) 46.0364 1.49441
\(950\) 14.0963 0.457344
\(951\) −13.7246 −0.445050
\(952\) 31.9711 1.03619
\(953\) −2.71220 −0.0878567 −0.0439283 0.999035i \(-0.513987\pi\)
−0.0439283 + 0.999035i \(0.513987\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 78.0032 2.52413
\(956\) 14.4480 0.467282
\(957\) 0.450041 0.0145478
\(958\) −20.7969 −0.671919
\(959\) 41.3663 1.33579
\(960\) −4.36993 −0.141039
\(961\) 8.97439 0.289496
\(962\) 49.1613 1.58502
\(963\) −15.8051 −0.509312
\(964\) −13.1851 −0.424665
\(965\) 38.4599 1.23807
\(966\) 21.1400 0.680167
\(967\) 9.67385 0.311090 0.155545 0.987829i \(-0.450287\pi\)
0.155545 + 0.987829i \(0.450287\pi\)
\(968\) −10.7857 −0.346666
\(969\) 7.64277 0.245521
\(970\) −55.7488 −1.78999
\(971\) 3.05353 0.0979924 0.0489962 0.998799i \(-0.484398\pi\)
0.0489962 + 0.998799i \(0.484398\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 22.4932 0.721100
\(974\) 4.73868 0.151837
\(975\) 79.8725 2.55797
\(976\) −2.69186 −0.0861643
\(977\) −29.2886 −0.937027 −0.468513 0.883456i \(-0.655210\pi\)
−0.468513 + 0.883456i \(0.655210\pi\)
\(978\) 18.6873 0.597553
\(979\) 7.94647 0.253970
\(980\) 45.8800 1.46558
\(981\) −13.5273 −0.431893
\(982\) 7.66928 0.244737
\(983\) −14.3083 −0.456365 −0.228183 0.973618i \(-0.573278\pi\)
−0.228183 + 0.973618i \(0.573278\pi\)
\(984\) 3.87511 0.123534
\(985\) 48.2535 1.53748
\(986\) −7.43019 −0.236625
\(987\) 22.4277 0.713882
\(988\) −5.66621 −0.180266
\(989\) −39.9668 −1.27087
\(990\) −2.02292 −0.0642925
\(991\) −45.3779 −1.44148 −0.720739 0.693207i \(-0.756197\pi\)
−0.720739 + 0.693207i \(0.756197\pi\)
\(992\) 6.32253 0.200741
\(993\) 8.55029 0.271335
\(994\) 38.0997 1.20845
\(995\) 22.5360 0.714438
\(996\) −10.8534 −0.343904
\(997\) −48.0528 −1.52185 −0.760923 0.648842i \(-0.775254\pi\)
−0.760923 + 0.648842i \(0.775254\pi\)
\(998\) −16.1706 −0.511871
\(999\) 8.67622 0.274504
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 6042.2.a.bc.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.bc.1.9 9 1.1 even 1 trivial