Properties

Label 6046.2.a.g.1.16
Level $6046$
Weight $2$
Character 6046.1
Self dual yes
Analytic conductor $48.278$
Analytic rank $0$
Dimension $69$
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] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(0\)
Dimension: \(69\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Character \(\chi\) \(=\) 6046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.10025 q^{3} +1.00000 q^{4} +4.19656 q^{5} +2.10025 q^{6} +2.72635 q^{7} -1.00000 q^{8} +1.41104 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.10025 q^{3} +1.00000 q^{4} +4.19656 q^{5} +2.10025 q^{6} +2.72635 q^{7} -1.00000 q^{8} +1.41104 q^{9} -4.19656 q^{10} +2.34795 q^{11} -2.10025 q^{12} +4.28809 q^{13} -2.72635 q^{14} -8.81381 q^{15} +1.00000 q^{16} +4.62658 q^{17} -1.41104 q^{18} +1.86144 q^{19} +4.19656 q^{20} -5.72601 q^{21} -2.34795 q^{22} -4.16667 q^{23} +2.10025 q^{24} +12.6111 q^{25} -4.28809 q^{26} +3.33720 q^{27} +2.72635 q^{28} -4.64041 q^{29} +8.81381 q^{30} -2.65007 q^{31} -1.00000 q^{32} -4.93128 q^{33} -4.62658 q^{34} +11.4413 q^{35} +1.41104 q^{36} +8.00548 q^{37} -1.86144 q^{38} -9.00605 q^{39} -4.19656 q^{40} +1.70811 q^{41} +5.72601 q^{42} -5.72273 q^{43} +2.34795 q^{44} +5.92153 q^{45} +4.16667 q^{46} +5.21811 q^{47} -2.10025 q^{48} +0.432985 q^{49} -12.6111 q^{50} -9.71696 q^{51} +4.28809 q^{52} +13.6464 q^{53} -3.33720 q^{54} +9.85331 q^{55} -2.72635 q^{56} -3.90948 q^{57} +4.64041 q^{58} -4.37714 q^{59} -8.81381 q^{60} -12.1769 q^{61} +2.65007 q^{62} +3.84700 q^{63} +1.00000 q^{64} +17.9952 q^{65} +4.93128 q^{66} +10.3490 q^{67} +4.62658 q^{68} +8.75103 q^{69} -11.4413 q^{70} +7.91164 q^{71} -1.41104 q^{72} -3.12735 q^{73} -8.00548 q^{74} -26.4864 q^{75} +1.86144 q^{76} +6.40134 q^{77} +9.00605 q^{78} +0.147439 q^{79} +4.19656 q^{80} -11.2421 q^{81} -1.70811 q^{82} -8.32890 q^{83} -5.72601 q^{84} +19.4157 q^{85} +5.72273 q^{86} +9.74602 q^{87} -2.34795 q^{88} +3.25753 q^{89} -5.92153 q^{90} +11.6908 q^{91} -4.16667 q^{92} +5.56580 q^{93} -5.21811 q^{94} +7.81162 q^{95} +2.10025 q^{96} +3.43929 q^{97} -0.432985 q^{98} +3.31306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9} - 13 q^{10} + 42 q^{11} - 5 q^{13} + 27 q^{14} + 18 q^{15} + 69 q^{16} + 24 q^{17} - 99 q^{18} + q^{19} + 13 q^{20} + 7 q^{21} - 42 q^{22} + 25 q^{23} + 100 q^{25} + 5 q^{26} + 15 q^{27} - 27 q^{28} + 87 q^{29} - 18 q^{30} + 5 q^{31} - 69 q^{32} + 28 q^{33} - 24 q^{34} + 33 q^{35} + 99 q^{36} - 5 q^{37} - q^{38} + 22 q^{39} - 13 q^{40} + 47 q^{41} - 7 q^{42} - 23 q^{43} + 42 q^{44} + 14 q^{45} - 25 q^{46} + 13 q^{47} + 106 q^{49} - 100 q^{50} + 2 q^{51} - 5 q^{52} + 51 q^{53} - 15 q^{54} - 11 q^{55} + 27 q^{56} + 52 q^{57} - 87 q^{58} + 73 q^{59} + 18 q^{60} + 4 q^{61} - 5 q^{62} - 86 q^{63} + 69 q^{64} + 70 q^{65} - 28 q^{66} - 24 q^{67} + 24 q^{68} + 56 q^{69} - 33 q^{70} + 84 q^{71} - 99 q^{72} + 27 q^{73} + 5 q^{74} + 27 q^{75} + q^{76} + 45 q^{77} - 22 q^{78} + 42 q^{79} + 13 q^{80} + 205 q^{81} - 47 q^{82} + q^{83} + 7 q^{84} - 18 q^{85} + 23 q^{86} - q^{87} - 42 q^{88} + 94 q^{89} - 14 q^{90} + 6 q^{91} + 25 q^{92} - 13 q^{93} - 13 q^{94} + 86 q^{95} + 35 q^{97} - 106 q^{98} + 83 q^{99}+O(q^{100}) \) 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
\(3\) −2.10025 −1.21258 −0.606290 0.795244i \(-0.707343\pi\)
−0.606290 + 0.795244i \(0.707343\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.19656 1.87676 0.938379 0.345609i \(-0.112328\pi\)
0.938379 + 0.345609i \(0.112328\pi\)
\(6\) 2.10025 0.857423
\(7\) 2.72635 1.03046 0.515232 0.857051i \(-0.327706\pi\)
0.515232 + 0.857051i \(0.327706\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.41104 0.470348
\(10\) −4.19656 −1.32707
\(11\) 2.34795 0.707934 0.353967 0.935258i \(-0.384833\pi\)
0.353967 + 0.935258i \(0.384833\pi\)
\(12\) −2.10025 −0.606290
\(13\) 4.28809 1.18930 0.594651 0.803984i \(-0.297290\pi\)
0.594651 + 0.803984i \(0.297290\pi\)
\(14\) −2.72635 −0.728648
\(15\) −8.81381 −2.27572
\(16\) 1.00000 0.250000
\(17\) 4.62658 1.12211 0.561055 0.827779i \(-0.310396\pi\)
0.561055 + 0.827779i \(0.310396\pi\)
\(18\) −1.41104 −0.332586
\(19\) 1.86144 0.427043 0.213521 0.976938i \(-0.431507\pi\)
0.213521 + 0.976938i \(0.431507\pi\)
\(20\) 4.19656 0.938379
\(21\) −5.72601 −1.24952
\(22\) −2.34795 −0.500585
\(23\) −4.16667 −0.868810 −0.434405 0.900718i \(-0.643041\pi\)
−0.434405 + 0.900718i \(0.643041\pi\)
\(24\) 2.10025 0.428711
\(25\) 12.6111 2.52222
\(26\) −4.28809 −0.840963
\(27\) 3.33720 0.642245
\(28\) 2.72635 0.515232
\(29\) −4.64041 −0.861703 −0.430851 0.902423i \(-0.641787\pi\)
−0.430851 + 0.902423i \(0.641787\pi\)
\(30\) 8.81381 1.60917
\(31\) −2.65007 −0.475966 −0.237983 0.971269i \(-0.576486\pi\)
−0.237983 + 0.971269i \(0.576486\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.93128 −0.858426
\(34\) −4.62658 −0.793451
\(35\) 11.4413 1.93393
\(36\) 1.41104 0.235174
\(37\) 8.00548 1.31609 0.658046 0.752977i \(-0.271383\pi\)
0.658046 + 0.752977i \(0.271383\pi\)
\(38\) −1.86144 −0.301965
\(39\) −9.00605 −1.44212
\(40\) −4.19656 −0.663534
\(41\) 1.70811 0.266762 0.133381 0.991065i \(-0.457417\pi\)
0.133381 + 0.991065i \(0.457417\pi\)
\(42\) 5.72601 0.883543
\(43\) −5.72273 −0.872708 −0.436354 0.899775i \(-0.643730\pi\)
−0.436354 + 0.899775i \(0.643730\pi\)
\(44\) 2.34795 0.353967
\(45\) 5.92153 0.882729
\(46\) 4.16667 0.614341
\(47\) 5.21811 0.761139 0.380570 0.924752i \(-0.375728\pi\)
0.380570 + 0.924752i \(0.375728\pi\)
\(48\) −2.10025 −0.303145
\(49\) 0.432985 0.0618551
\(50\) −12.6111 −1.78348
\(51\) −9.71696 −1.36065
\(52\) 4.28809 0.594651
\(53\) 13.6464 1.87448 0.937240 0.348685i \(-0.113372\pi\)
0.937240 + 0.348685i \(0.113372\pi\)
\(54\) −3.33720 −0.454136
\(55\) 9.85331 1.32862
\(56\) −2.72635 −0.364324
\(57\) −3.90948 −0.517823
\(58\) 4.64041 0.609316
\(59\) −4.37714 −0.569855 −0.284928 0.958549i \(-0.591970\pi\)
−0.284928 + 0.958549i \(0.591970\pi\)
\(60\) −8.81381 −1.13786
\(61\) −12.1769 −1.55909 −0.779545 0.626347i \(-0.784549\pi\)
−0.779545 + 0.626347i \(0.784549\pi\)
\(62\) 2.65007 0.336559
\(63\) 3.84700 0.484676
\(64\) 1.00000 0.125000
\(65\) 17.9952 2.23203
\(66\) 4.93128 0.606999
\(67\) 10.3490 1.26433 0.632165 0.774834i \(-0.282167\pi\)
0.632165 + 0.774834i \(0.282167\pi\)
\(68\) 4.62658 0.561055
\(69\) 8.75103 1.05350
\(70\) −11.4413 −1.36749
\(71\) 7.91164 0.938939 0.469469 0.882949i \(-0.344445\pi\)
0.469469 + 0.882949i \(0.344445\pi\)
\(72\) −1.41104 −0.166293
\(73\) −3.12735 −0.366029 −0.183014 0.983110i \(-0.558586\pi\)
−0.183014 + 0.983110i \(0.558586\pi\)
\(74\) −8.00548 −0.930618
\(75\) −26.4864 −3.05839
\(76\) 1.86144 0.213521
\(77\) 6.40134 0.729500
\(78\) 9.00605 1.01973
\(79\) 0.147439 0.0165882 0.00829411 0.999966i \(-0.497360\pi\)
0.00829411 + 0.999966i \(0.497360\pi\)
\(80\) 4.19656 0.469189
\(81\) −11.2421 −1.24912
\(82\) −1.70811 −0.188629
\(83\) −8.32890 −0.914215 −0.457108 0.889411i \(-0.651115\pi\)
−0.457108 + 0.889411i \(0.651115\pi\)
\(84\) −5.72601 −0.624759
\(85\) 19.4157 2.10593
\(86\) 5.72273 0.617098
\(87\) 9.74602 1.04488
\(88\) −2.34795 −0.250292
\(89\) 3.25753 0.345297 0.172649 0.984983i \(-0.444767\pi\)
0.172649 + 0.984983i \(0.444767\pi\)
\(90\) −5.92153 −0.624184
\(91\) 11.6908 1.22553
\(92\) −4.16667 −0.434405
\(93\) 5.56580 0.577147
\(94\) −5.21811 −0.538207
\(95\) 7.81162 0.801456
\(96\) 2.10025 0.214356
\(97\) 3.43929 0.349207 0.174604 0.984639i \(-0.444136\pi\)
0.174604 + 0.984639i \(0.444136\pi\)
\(98\) −0.432985 −0.0437381
\(99\) 3.31306 0.332975
\(100\) 12.6111 1.26111
\(101\) 17.9553 1.78662 0.893311 0.449439i \(-0.148376\pi\)
0.893311 + 0.449439i \(0.148376\pi\)
\(102\) 9.71696 0.962122
\(103\) 11.6765 1.15052 0.575258 0.817972i \(-0.304902\pi\)
0.575258 + 0.817972i \(0.304902\pi\)
\(104\) −4.28809 −0.420482
\(105\) −24.0295 −2.34504
\(106\) −13.6464 −1.32546
\(107\) −8.37290 −0.809439 −0.404720 0.914441i \(-0.632631\pi\)
−0.404720 + 0.914441i \(0.632631\pi\)
\(108\) 3.33720 0.321123
\(109\) −7.43302 −0.711954 −0.355977 0.934495i \(-0.615852\pi\)
−0.355977 + 0.934495i \(0.615852\pi\)
\(110\) −9.85331 −0.939476
\(111\) −16.8135 −1.59587
\(112\) 2.72635 0.257616
\(113\) 9.43626 0.887689 0.443844 0.896104i \(-0.353614\pi\)
0.443844 + 0.896104i \(0.353614\pi\)
\(114\) 3.90948 0.366156
\(115\) −17.4856 −1.63054
\(116\) −4.64041 −0.430851
\(117\) 6.05068 0.559385
\(118\) 4.37714 0.402949
\(119\) 12.6137 1.15629
\(120\) 8.81381 0.804587
\(121\) −5.48713 −0.498830
\(122\) 12.1769 1.10244
\(123\) −3.58746 −0.323470
\(124\) −2.65007 −0.237983
\(125\) 31.9404 2.85683
\(126\) −3.84700 −0.342718
\(127\) −21.3319 −1.89290 −0.946452 0.322846i \(-0.895361\pi\)
−0.946452 + 0.322846i \(0.895361\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0192 1.05823
\(130\) −17.9952 −1.57828
\(131\) −10.5976 −0.925919 −0.462959 0.886379i \(-0.653212\pi\)
−0.462959 + 0.886379i \(0.653212\pi\)
\(132\) −4.93128 −0.429213
\(133\) 5.07493 0.440052
\(134\) −10.3490 −0.894016
\(135\) 14.0048 1.20534
\(136\) −4.62658 −0.396726
\(137\) 6.57723 0.561931 0.280965 0.959718i \(-0.409345\pi\)
0.280965 + 0.959718i \(0.409345\pi\)
\(138\) −8.75103 −0.744937
\(139\) 20.8114 1.76520 0.882600 0.470124i \(-0.155791\pi\)
0.882600 + 0.470124i \(0.155791\pi\)
\(140\) 11.4413 0.966965
\(141\) −10.9593 −0.922941
\(142\) −7.91164 −0.663930
\(143\) 10.0682 0.841947
\(144\) 1.41104 0.117587
\(145\) −19.4737 −1.61721
\(146\) 3.12735 0.258821
\(147\) −0.909377 −0.0750041
\(148\) 8.00548 0.658046
\(149\) 2.64225 0.216462 0.108231 0.994126i \(-0.465481\pi\)
0.108231 + 0.994126i \(0.465481\pi\)
\(150\) 26.4864 2.16261
\(151\) −17.0437 −1.38700 −0.693499 0.720458i \(-0.743932\pi\)
−0.693499 + 0.720458i \(0.743932\pi\)
\(152\) −1.86144 −0.150982
\(153\) 6.52830 0.527782
\(154\) −6.40134 −0.515834
\(155\) −11.1212 −0.893273
\(156\) −9.00605 −0.721061
\(157\) −20.1146 −1.60532 −0.802660 0.596437i \(-0.796583\pi\)
−0.802660 + 0.596437i \(0.796583\pi\)
\(158\) −0.147439 −0.0117296
\(159\) −28.6609 −2.27296
\(160\) −4.19656 −0.331767
\(161\) −11.3598 −0.895277
\(162\) 11.2421 0.883262
\(163\) 5.28295 0.413793 0.206896 0.978363i \(-0.433664\pi\)
0.206896 + 0.978363i \(0.433664\pi\)
\(164\) 1.70811 0.133381
\(165\) −20.6944 −1.61106
\(166\) 8.32890 0.646448
\(167\) −18.7836 −1.45352 −0.726760 0.686891i \(-0.758975\pi\)
−0.726760 + 0.686891i \(0.758975\pi\)
\(168\) 5.72601 0.441771
\(169\) 5.38769 0.414438
\(170\) −19.4157 −1.48912
\(171\) 2.62657 0.200859
\(172\) −5.72273 −0.436354
\(173\) −15.7375 −1.19650 −0.598251 0.801308i \(-0.704138\pi\)
−0.598251 + 0.801308i \(0.704138\pi\)
\(174\) −9.74602 −0.738844
\(175\) 34.3822 2.59905
\(176\) 2.34795 0.176983
\(177\) 9.19309 0.690995
\(178\) −3.25753 −0.244162
\(179\) 19.4081 1.45063 0.725316 0.688417i \(-0.241694\pi\)
0.725316 + 0.688417i \(0.241694\pi\)
\(180\) 5.92153 0.441364
\(181\) 11.5761 0.860441 0.430221 0.902724i \(-0.358436\pi\)
0.430221 + 0.902724i \(0.358436\pi\)
\(182\) −11.6908 −0.866582
\(183\) 25.5745 1.89052
\(184\) 4.16667 0.307171
\(185\) 33.5954 2.46999
\(186\) −5.56580 −0.408104
\(187\) 10.8630 0.794379
\(188\) 5.21811 0.380570
\(189\) 9.09838 0.661810
\(190\) −7.81162 −0.566715
\(191\) −17.4749 −1.26444 −0.632221 0.774788i \(-0.717857\pi\)
−0.632221 + 0.774788i \(0.717857\pi\)
\(192\) −2.10025 −0.151572
\(193\) −4.30096 −0.309590 −0.154795 0.987947i \(-0.549472\pi\)
−0.154795 + 0.987947i \(0.549472\pi\)
\(194\) −3.43929 −0.246927
\(195\) −37.7944 −2.70651
\(196\) 0.432985 0.0309275
\(197\) −24.4924 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(198\) −3.31306 −0.235449
\(199\) −1.02679 −0.0727875 −0.0363937 0.999338i \(-0.511587\pi\)
−0.0363937 + 0.999338i \(0.511587\pi\)
\(200\) −12.6111 −0.891739
\(201\) −21.7354 −1.53310
\(202\) −17.9553 −1.26333
\(203\) −12.6514 −0.887953
\(204\) −9.71696 −0.680323
\(205\) 7.16818 0.500648
\(206\) −11.6765 −0.813538
\(207\) −5.87935 −0.408643
\(208\) 4.28809 0.297325
\(209\) 4.37056 0.302318
\(210\) 24.0295 1.65820
\(211\) −13.9129 −0.957805 −0.478902 0.877868i \(-0.658965\pi\)
−0.478902 + 0.877868i \(0.658965\pi\)
\(212\) 13.6464 0.937240
\(213\) −16.6164 −1.13854
\(214\) 8.37290 0.572360
\(215\) −24.0158 −1.63786
\(216\) −3.33720 −0.227068
\(217\) −7.22501 −0.490466
\(218\) 7.43302 0.503428
\(219\) 6.56822 0.443839
\(220\) 9.85331 0.664310
\(221\) 19.8392 1.33453
\(222\) 16.8135 1.12845
\(223\) −14.2628 −0.955105 −0.477552 0.878603i \(-0.658476\pi\)
−0.477552 + 0.878603i \(0.658476\pi\)
\(224\) −2.72635 −0.182162
\(225\) 17.7948 1.18632
\(226\) −9.43626 −0.627691
\(227\) −27.4971 −1.82505 −0.912523 0.409026i \(-0.865868\pi\)
−0.912523 + 0.409026i \(0.865868\pi\)
\(228\) −3.90948 −0.258912
\(229\) 1.83928 0.121543 0.0607716 0.998152i \(-0.480644\pi\)
0.0607716 + 0.998152i \(0.480644\pi\)
\(230\) 17.4856 1.15297
\(231\) −13.4444 −0.884576
\(232\) 4.64041 0.304658
\(233\) −9.08508 −0.595183 −0.297592 0.954693i \(-0.596183\pi\)
−0.297592 + 0.954693i \(0.596183\pi\)
\(234\) −6.05068 −0.395545
\(235\) 21.8981 1.42847
\(236\) −4.37714 −0.284928
\(237\) −0.309659 −0.0201145
\(238\) −12.6137 −0.817622
\(239\) 4.75495 0.307572 0.153786 0.988104i \(-0.450853\pi\)
0.153786 + 0.988104i \(0.450853\pi\)
\(240\) −8.81381 −0.568929
\(241\) −3.30036 −0.212595 −0.106297 0.994334i \(-0.533900\pi\)
−0.106297 + 0.994334i \(0.533900\pi\)
\(242\) 5.48713 0.352726
\(243\) 13.5996 0.872413
\(244\) −12.1769 −0.779545
\(245\) 1.81705 0.116087
\(246\) 3.58746 0.228728
\(247\) 7.98200 0.507883
\(248\) 2.65007 0.168279
\(249\) 17.4928 1.10856
\(250\) −31.9404 −2.02009
\(251\) −24.0193 −1.51608 −0.758041 0.652207i \(-0.773843\pi\)
−0.758041 + 0.652207i \(0.773843\pi\)
\(252\) 3.84700 0.242338
\(253\) −9.78312 −0.615060
\(254\) 21.3319 1.33848
\(255\) −40.7778 −2.55360
\(256\) 1.00000 0.0625000
\(257\) −16.3421 −1.01939 −0.509696 0.860354i \(-0.670242\pi\)
−0.509696 + 0.860354i \(0.670242\pi\)
\(258\) −12.0192 −0.748280
\(259\) 21.8257 1.35619
\(260\) 17.9952 1.11602
\(261\) −6.54782 −0.405300
\(262\) 10.5976 0.654724
\(263\) −27.7339 −1.71015 −0.855073 0.518507i \(-0.826488\pi\)
−0.855073 + 0.518507i \(0.826488\pi\)
\(264\) 4.93128 0.303499
\(265\) 57.2680 3.51794
\(266\) −5.07493 −0.311164
\(267\) −6.84162 −0.418700
\(268\) 10.3490 0.632165
\(269\) 25.0318 1.52621 0.763107 0.646272i \(-0.223673\pi\)
0.763107 + 0.646272i \(0.223673\pi\)
\(270\) −14.0048 −0.852303
\(271\) 18.7328 1.13794 0.568969 0.822359i \(-0.307342\pi\)
0.568969 + 0.822359i \(0.307342\pi\)
\(272\) 4.62658 0.280527
\(273\) −24.5536 −1.48605
\(274\) −6.57723 −0.397345
\(275\) 29.6102 1.78556
\(276\) 8.75103 0.526750
\(277\) 1.97682 0.118775 0.0593877 0.998235i \(-0.481085\pi\)
0.0593877 + 0.998235i \(0.481085\pi\)
\(278\) −20.8114 −1.24819
\(279\) −3.73936 −0.223870
\(280\) −11.4413 −0.683747
\(281\) −6.89170 −0.411124 −0.205562 0.978644i \(-0.565902\pi\)
−0.205562 + 0.978644i \(0.565902\pi\)
\(282\) 10.9593 0.652618
\(283\) −22.1297 −1.31547 −0.657737 0.753247i \(-0.728486\pi\)
−0.657737 + 0.753247i \(0.728486\pi\)
\(284\) 7.91164 0.469469
\(285\) −16.4063 −0.971828
\(286\) −10.0682 −0.595346
\(287\) 4.65691 0.274889
\(288\) −1.41104 −0.0831465
\(289\) 4.40520 0.259129
\(290\) 19.4737 1.14354
\(291\) −7.22337 −0.423441
\(292\) −3.12735 −0.183014
\(293\) 25.5694 1.49378 0.746889 0.664948i \(-0.231546\pi\)
0.746889 + 0.664948i \(0.231546\pi\)
\(294\) 0.909377 0.0530359
\(295\) −18.3689 −1.06948
\(296\) −8.00548 −0.465309
\(297\) 7.83559 0.454667
\(298\) −2.64225 −0.153062
\(299\) −17.8670 −1.03328
\(300\) −26.4864 −1.52919
\(301\) −15.6022 −0.899294
\(302\) 17.0437 0.980755
\(303\) −37.7106 −2.16642
\(304\) 1.86144 0.106761
\(305\) −51.1009 −2.92603
\(306\) −6.52830 −0.373198
\(307\) 20.1772 1.15157 0.575786 0.817600i \(-0.304696\pi\)
0.575786 + 0.817600i \(0.304696\pi\)
\(308\) 6.40134 0.364750
\(309\) −24.5235 −1.39509
\(310\) 11.1212 0.631639
\(311\) −3.56956 −0.202411 −0.101205 0.994866i \(-0.532270\pi\)
−0.101205 + 0.994866i \(0.532270\pi\)
\(312\) 9.00605 0.509867
\(313\) −13.4653 −0.761102 −0.380551 0.924760i \(-0.624266\pi\)
−0.380551 + 0.924760i \(0.624266\pi\)
\(314\) 20.1146 1.13513
\(315\) 16.1442 0.909620
\(316\) 0.147439 0.00829411
\(317\) 24.2502 1.36203 0.681014 0.732271i \(-0.261539\pi\)
0.681014 + 0.732271i \(0.261539\pi\)
\(318\) 28.6609 1.60722
\(319\) −10.8955 −0.610028
\(320\) 4.19656 0.234595
\(321\) 17.5852 0.981509
\(322\) 11.3598 0.633056
\(323\) 8.61207 0.479189
\(324\) −11.2421 −0.624560
\(325\) 54.0775 2.99968
\(326\) −5.28295 −0.292596
\(327\) 15.6112 0.863301
\(328\) −1.70811 −0.0943146
\(329\) 14.2264 0.784326
\(330\) 20.6944 1.13919
\(331\) 18.6103 1.02292 0.511458 0.859308i \(-0.329106\pi\)
0.511458 + 0.859308i \(0.329106\pi\)
\(332\) −8.32890 −0.457108
\(333\) 11.2961 0.619021
\(334\) 18.7836 1.02779
\(335\) 43.4301 2.37284
\(336\) −5.72601 −0.312380
\(337\) 25.7371 1.40199 0.700993 0.713168i \(-0.252740\pi\)
0.700993 + 0.713168i \(0.252740\pi\)
\(338\) −5.38769 −0.293052
\(339\) −19.8185 −1.07639
\(340\) 19.4157 1.05296
\(341\) −6.22223 −0.336952
\(342\) −2.62657 −0.142029
\(343\) −17.9040 −0.966724
\(344\) 5.72273 0.308549
\(345\) 36.7242 1.97716
\(346\) 15.7375 0.846055
\(347\) −6.68163 −0.358688 −0.179344 0.983786i \(-0.557398\pi\)
−0.179344 + 0.983786i \(0.557398\pi\)
\(348\) 9.74602 0.522441
\(349\) −14.7538 −0.789755 −0.394878 0.918734i \(-0.629213\pi\)
−0.394878 + 0.918734i \(0.629213\pi\)
\(350\) −34.3822 −1.83781
\(351\) 14.3102 0.763823
\(352\) −2.34795 −0.125146
\(353\) 21.0243 1.11901 0.559505 0.828827i \(-0.310991\pi\)
0.559505 + 0.828827i \(0.310991\pi\)
\(354\) −9.19309 −0.488607
\(355\) 33.2016 1.76216
\(356\) 3.25753 0.172649
\(357\) −26.4918 −1.40210
\(358\) −19.4081 −1.02575
\(359\) −8.26515 −0.436218 −0.218109 0.975924i \(-0.569989\pi\)
−0.218109 + 0.975924i \(0.569989\pi\)
\(360\) −5.92153 −0.312092
\(361\) −15.5351 −0.817635
\(362\) −11.5761 −0.608424
\(363\) 11.5243 0.604871
\(364\) 11.6908 0.612766
\(365\) −13.1241 −0.686947
\(366\) −25.5745 −1.33680
\(367\) −11.9739 −0.625032 −0.312516 0.949913i \(-0.601172\pi\)
−0.312516 + 0.949913i \(0.601172\pi\)
\(368\) −4.16667 −0.217202
\(369\) 2.41022 0.125471
\(370\) −33.5954 −1.74654
\(371\) 37.2049 1.93158
\(372\) 5.56580 0.288573
\(373\) −26.4342 −1.36871 −0.684354 0.729149i \(-0.739916\pi\)
−0.684354 + 0.729149i \(0.739916\pi\)
\(374\) −10.8630 −0.561711
\(375\) −67.0827 −3.46414
\(376\) −5.21811 −0.269103
\(377\) −19.8985 −1.02482
\(378\) −9.09838 −0.467970
\(379\) 16.0303 0.823423 0.411712 0.911314i \(-0.364931\pi\)
0.411712 + 0.911314i \(0.364931\pi\)
\(380\) 7.81162 0.400728
\(381\) 44.8024 2.29529
\(382\) 17.4749 0.894096
\(383\) 12.0712 0.616808 0.308404 0.951255i \(-0.400205\pi\)
0.308404 + 0.951255i \(0.400205\pi\)
\(384\) 2.10025 0.107178
\(385\) 26.8636 1.36909
\(386\) 4.30096 0.218913
\(387\) −8.07502 −0.410476
\(388\) 3.43929 0.174604
\(389\) 19.4306 0.985171 0.492585 0.870264i \(-0.336052\pi\)
0.492585 + 0.870264i \(0.336052\pi\)
\(390\) 37.7944 1.91379
\(391\) −19.2774 −0.974899
\(392\) −0.432985 −0.0218691
\(393\) 22.2577 1.12275
\(394\) 24.4924 1.23391
\(395\) 0.618737 0.0311321
\(396\) 3.31306 0.166488
\(397\) −33.4445 −1.67853 −0.839266 0.543721i \(-0.817015\pi\)
−0.839266 + 0.543721i \(0.817015\pi\)
\(398\) 1.02679 0.0514685
\(399\) −10.6586 −0.533598
\(400\) 12.6111 0.630554
\(401\) 10.5349 0.526090 0.263045 0.964784i \(-0.415273\pi\)
0.263045 + 0.964784i \(0.415273\pi\)
\(402\) 21.7354 1.08406
\(403\) −11.3637 −0.566067
\(404\) 17.9553 0.893311
\(405\) −47.1781 −2.34430
\(406\) 12.6514 0.627878
\(407\) 18.7965 0.931706
\(408\) 9.71696 0.481061
\(409\) 18.5593 0.917699 0.458849 0.888514i \(-0.348262\pi\)
0.458849 + 0.888514i \(0.348262\pi\)
\(410\) −7.16818 −0.354011
\(411\) −13.8138 −0.681385
\(412\) 11.6765 0.575258
\(413\) −11.9336 −0.587215
\(414\) 5.87935 0.288954
\(415\) −34.9527 −1.71576
\(416\) −4.28809 −0.210241
\(417\) −43.7091 −2.14045
\(418\) −4.37056 −0.213771
\(419\) 5.76487 0.281633 0.140816 0.990036i \(-0.455027\pi\)
0.140816 + 0.990036i \(0.455027\pi\)
\(420\) −24.0295 −1.17252
\(421\) −4.18467 −0.203948 −0.101974 0.994787i \(-0.532516\pi\)
−0.101974 + 0.994787i \(0.532516\pi\)
\(422\) 13.9129 0.677270
\(423\) 7.36298 0.358000
\(424\) −13.6464 −0.662729
\(425\) 58.3462 2.83020
\(426\) 16.6164 0.805068
\(427\) −33.1984 −1.60658
\(428\) −8.37290 −0.404720
\(429\) −21.1458 −1.02093
\(430\) 24.0158 1.15814
\(431\) −5.80236 −0.279490 −0.139745 0.990188i \(-0.544628\pi\)
−0.139745 + 0.990188i \(0.544628\pi\)
\(432\) 3.33720 0.160561
\(433\) 32.6204 1.56764 0.783819 0.620989i \(-0.213269\pi\)
0.783819 + 0.620989i \(0.213269\pi\)
\(434\) 7.22501 0.346812
\(435\) 40.8997 1.96099
\(436\) −7.43302 −0.355977
\(437\) −7.75598 −0.371019
\(438\) −6.56822 −0.313842
\(439\) 22.7808 1.08727 0.543635 0.839322i \(-0.317048\pi\)
0.543635 + 0.839322i \(0.317048\pi\)
\(440\) −9.85331 −0.469738
\(441\) 0.610961 0.0290934
\(442\) −19.8392 −0.943653
\(443\) 22.9851 1.09206 0.546028 0.837767i \(-0.316139\pi\)
0.546028 + 0.837767i \(0.316139\pi\)
\(444\) −16.8135 −0.797933
\(445\) 13.6704 0.648039
\(446\) 14.2628 0.675361
\(447\) −5.54939 −0.262477
\(448\) 2.72635 0.128808
\(449\) −18.3512 −0.866047 −0.433024 0.901383i \(-0.642553\pi\)
−0.433024 + 0.901383i \(0.642553\pi\)
\(450\) −17.7948 −0.838855
\(451\) 4.01056 0.188850
\(452\) 9.43626 0.443844
\(453\) 35.7960 1.68184
\(454\) 27.4971 1.29050
\(455\) 49.0612 2.30003
\(456\) 3.90948 0.183078
\(457\) −11.9075 −0.557009 −0.278504 0.960435i \(-0.589839\pi\)
−0.278504 + 0.960435i \(0.589839\pi\)
\(458\) −1.83928 −0.0859440
\(459\) 15.4398 0.720669
\(460\) −17.4856 −0.815272
\(461\) 0.191347 0.00891190 0.00445595 0.999990i \(-0.498582\pi\)
0.00445595 + 0.999990i \(0.498582\pi\)
\(462\) 13.4444 0.625490
\(463\) 0.344055 0.0159896 0.00799479 0.999968i \(-0.497455\pi\)
0.00799479 + 0.999968i \(0.497455\pi\)
\(464\) −4.64041 −0.215426
\(465\) 23.3572 1.08316
\(466\) 9.08508 0.420858
\(467\) −0.403713 −0.0186816 −0.00934082 0.999956i \(-0.502973\pi\)
−0.00934082 + 0.999956i \(0.502973\pi\)
\(468\) 6.05068 0.279693
\(469\) 28.2149 1.30285
\(470\) −21.8981 −1.01008
\(471\) 42.2457 1.94658
\(472\) 4.37714 0.201474
\(473\) −13.4367 −0.617819
\(474\) 0.309659 0.0142231
\(475\) 23.4747 1.07709
\(476\) 12.6137 0.578146
\(477\) 19.2557 0.881658
\(478\) −4.75495 −0.217487
\(479\) 12.3751 0.565431 0.282716 0.959204i \(-0.408765\pi\)
0.282716 + 0.959204i \(0.408765\pi\)
\(480\) 8.81381 0.402294
\(481\) 34.3282 1.56523
\(482\) 3.30036 0.150327
\(483\) 23.8584 1.08559
\(484\) −5.48713 −0.249415
\(485\) 14.4332 0.655377
\(486\) −13.5996 −0.616889
\(487\) −35.6262 −1.61438 −0.807188 0.590294i \(-0.799012\pi\)
−0.807188 + 0.590294i \(0.799012\pi\)
\(488\) 12.1769 0.551221
\(489\) −11.0955 −0.501756
\(490\) −1.81705 −0.0820859
\(491\) 21.8556 0.986327 0.493164 0.869937i \(-0.335840\pi\)
0.493164 + 0.869937i \(0.335840\pi\)
\(492\) −3.58746 −0.161735
\(493\) −21.4692 −0.966925
\(494\) −7.98200 −0.359127
\(495\) 13.9034 0.624914
\(496\) −2.65007 −0.118992
\(497\) 21.5699 0.967542
\(498\) −17.4928 −0.783869
\(499\) 2.53687 0.113566 0.0567830 0.998387i \(-0.481916\pi\)
0.0567830 + 0.998387i \(0.481916\pi\)
\(500\) 31.9404 1.42842
\(501\) 39.4503 1.76251
\(502\) 24.0193 1.07203
\(503\) −20.7392 −0.924716 −0.462358 0.886693i \(-0.652997\pi\)
−0.462358 + 0.886693i \(0.652997\pi\)
\(504\) −3.84700 −0.171359
\(505\) 75.3505 3.35306
\(506\) 9.78312 0.434913
\(507\) −11.3155 −0.502539
\(508\) −21.3319 −0.946452
\(509\) −27.1126 −1.20175 −0.600873 0.799345i \(-0.705180\pi\)
−0.600873 + 0.799345i \(0.705180\pi\)
\(510\) 40.7778 1.80567
\(511\) −8.52626 −0.377179
\(512\) −1.00000 −0.0441942
\(513\) 6.21199 0.274266
\(514\) 16.3421 0.720820
\(515\) 49.0010 2.15924
\(516\) 12.0192 0.529114
\(517\) 12.2519 0.538836
\(518\) −21.8257 −0.958968
\(519\) 33.0527 1.45085
\(520\) −17.9952 −0.789142
\(521\) 9.40566 0.412069 0.206035 0.978545i \(-0.433944\pi\)
0.206035 + 0.978545i \(0.433944\pi\)
\(522\) 6.54782 0.286590
\(523\) −7.41472 −0.324223 −0.162112 0.986772i \(-0.551830\pi\)
−0.162112 + 0.986772i \(0.551830\pi\)
\(524\) −10.5976 −0.462959
\(525\) −72.2113 −3.15156
\(526\) 27.7339 1.20926
\(527\) −12.2607 −0.534086
\(528\) −4.93128 −0.214606
\(529\) −5.63890 −0.245170
\(530\) −57.2680 −2.48756
\(531\) −6.17634 −0.268030
\(532\) 5.07493 0.220026
\(533\) 7.32453 0.317261
\(534\) 6.84162 0.296066
\(535\) −35.1374 −1.51912
\(536\) −10.3490 −0.447008
\(537\) −40.7619 −1.75900
\(538\) −25.0318 −1.07920
\(539\) 1.01663 0.0437893
\(540\) 14.0048 0.602669
\(541\) 22.0332 0.947282 0.473641 0.880718i \(-0.342939\pi\)
0.473641 + 0.880718i \(0.342939\pi\)
\(542\) −18.7328 −0.804644
\(543\) −24.3126 −1.04335
\(544\) −4.62658 −0.198363
\(545\) −31.1931 −1.33617
\(546\) 24.5536 1.05080
\(547\) 8.33944 0.356569 0.178284 0.983979i \(-0.442945\pi\)
0.178284 + 0.983979i \(0.442945\pi\)
\(548\) 6.57723 0.280965
\(549\) −17.1821 −0.733314
\(550\) −29.6102 −1.26258
\(551\) −8.63783 −0.367984
\(552\) −8.75103 −0.372469
\(553\) 0.401971 0.0170936
\(554\) −1.97682 −0.0839869
\(555\) −70.5588 −2.99505
\(556\) 20.8114 0.882600
\(557\) 2.79343 0.118361 0.0591807 0.998247i \(-0.481151\pi\)
0.0591807 + 0.998247i \(0.481151\pi\)
\(558\) 3.73936 0.158300
\(559\) −24.5396 −1.03791
\(560\) 11.4413 0.483482
\(561\) −22.8149 −0.963247
\(562\) 6.89170 0.290709
\(563\) 0.482130 0.0203194 0.0101597 0.999948i \(-0.496766\pi\)
0.0101597 + 0.999948i \(0.496766\pi\)
\(564\) −10.9593 −0.461471
\(565\) 39.5998 1.66598
\(566\) 22.1297 0.930181
\(567\) −30.6499 −1.28717
\(568\) −7.91164 −0.331965
\(569\) 5.31075 0.222638 0.111319 0.993785i \(-0.464492\pi\)
0.111319 + 0.993785i \(0.464492\pi\)
\(570\) 16.4063 0.687186
\(571\) −6.78172 −0.283806 −0.141903 0.989881i \(-0.545322\pi\)
−0.141903 + 0.989881i \(0.545322\pi\)
\(572\) 10.0682 0.420973
\(573\) 36.7017 1.53324
\(574\) −4.65691 −0.194376
\(575\) −52.5462 −2.19133
\(576\) 1.41104 0.0587935
\(577\) 6.01415 0.250373 0.125186 0.992133i \(-0.460047\pi\)
0.125186 + 0.992133i \(0.460047\pi\)
\(578\) −4.40520 −0.183232
\(579\) 9.03309 0.375403
\(580\) −19.4737 −0.808603
\(581\) −22.7075 −0.942065
\(582\) 7.22337 0.299418
\(583\) 32.0411 1.32701
\(584\) 3.12735 0.129411
\(585\) 25.3920 1.04983
\(586\) −25.5694 −1.05626
\(587\) 30.9975 1.27940 0.639701 0.768623i \(-0.279058\pi\)
0.639701 + 0.768623i \(0.279058\pi\)
\(588\) −0.909377 −0.0375021
\(589\) −4.93293 −0.203258
\(590\) 18.3689 0.756237
\(591\) 51.4402 2.11597
\(592\) 8.00548 0.329023
\(593\) 27.4988 1.12924 0.564620 0.825351i \(-0.309023\pi\)
0.564620 + 0.825351i \(0.309023\pi\)
\(594\) −7.83559 −0.321498
\(595\) 52.9340 2.17008
\(596\) 2.64225 0.108231
\(597\) 2.15652 0.0882606
\(598\) 17.8670 0.730637
\(599\) 11.5512 0.471968 0.235984 0.971757i \(-0.424169\pi\)
0.235984 + 0.971757i \(0.424169\pi\)
\(600\) 26.4864 1.08130
\(601\) 31.1016 1.26866 0.634330 0.773062i \(-0.281276\pi\)
0.634330 + 0.773062i \(0.281276\pi\)
\(602\) 15.6022 0.635897
\(603\) 14.6029 0.594675
\(604\) −17.0437 −0.693499
\(605\) −23.0270 −0.936183
\(606\) 37.7106 1.53189
\(607\) 3.08921 0.125387 0.0626936 0.998033i \(-0.480031\pi\)
0.0626936 + 0.998033i \(0.480031\pi\)
\(608\) −1.86144 −0.0754912
\(609\) 26.5711 1.07671
\(610\) 51.1009 2.06902
\(611\) 22.3757 0.905224
\(612\) 6.52830 0.263891
\(613\) 36.3876 1.46968 0.734840 0.678240i \(-0.237257\pi\)
0.734840 + 0.678240i \(0.237257\pi\)
\(614\) −20.1772 −0.814284
\(615\) −15.0550 −0.607075
\(616\) −6.40134 −0.257917
\(617\) 7.35811 0.296226 0.148113 0.988970i \(-0.452680\pi\)
0.148113 + 0.988970i \(0.452680\pi\)
\(618\) 24.5235 0.986479
\(619\) 16.5856 0.666631 0.333315 0.942815i \(-0.391833\pi\)
0.333315 + 0.942815i \(0.391833\pi\)
\(620\) −11.1212 −0.446636
\(621\) −13.9050 −0.557989
\(622\) 3.56956 0.143126
\(623\) 8.88116 0.355816
\(624\) −9.00605 −0.360531
\(625\) 70.9841 2.83937
\(626\) 13.4653 0.538181
\(627\) −9.17926 −0.366584
\(628\) −20.1146 −0.802660
\(629\) 37.0379 1.47680
\(630\) −16.1442 −0.643198
\(631\) 5.82222 0.231779 0.115889 0.993262i \(-0.463028\pi\)
0.115889 + 0.993262i \(0.463028\pi\)
\(632\) −0.147439 −0.00586482
\(633\) 29.2206 1.16141
\(634\) −24.2502 −0.963099
\(635\) −89.5207 −3.55252
\(636\) −28.6609 −1.13648
\(637\) 1.85668 0.0735643
\(638\) 10.8955 0.431355
\(639\) 11.1637 0.441628
\(640\) −4.19656 −0.165883
\(641\) −39.3312 −1.55349 −0.776744 0.629817i \(-0.783130\pi\)
−0.776744 + 0.629817i \(0.783130\pi\)
\(642\) −17.5852 −0.694032
\(643\) 4.80588 0.189525 0.0947627 0.995500i \(-0.469791\pi\)
0.0947627 + 0.995500i \(0.469791\pi\)
\(644\) −11.3598 −0.447638
\(645\) 50.4391 1.98604
\(646\) −8.61207 −0.338838
\(647\) −14.5215 −0.570900 −0.285450 0.958394i \(-0.592143\pi\)
−0.285450 + 0.958394i \(0.592143\pi\)
\(648\) 11.2421 0.441631
\(649\) −10.2773 −0.403420
\(650\) −54.0775 −2.12109
\(651\) 15.1743 0.594728
\(652\) 5.28295 0.206896
\(653\) −32.8222 −1.28443 −0.642216 0.766524i \(-0.721985\pi\)
−0.642216 + 0.766524i \(0.721985\pi\)
\(654\) −15.6112 −0.610446
\(655\) −44.4735 −1.73773
\(656\) 1.70811 0.0666905
\(657\) −4.41283 −0.172161
\(658\) −14.2264 −0.554602
\(659\) 38.2114 1.48850 0.744252 0.667899i \(-0.232806\pi\)
0.744252 + 0.667899i \(0.232806\pi\)
\(660\) −20.6944 −0.805528
\(661\) 16.0617 0.624726 0.312363 0.949963i \(-0.398879\pi\)
0.312363 + 0.949963i \(0.398879\pi\)
\(662\) −18.6103 −0.723310
\(663\) −41.6672 −1.61822
\(664\) 8.32890 0.323224
\(665\) 21.2972 0.825871
\(666\) −11.2961 −0.437714
\(667\) 19.3350 0.748656
\(668\) −18.7836 −0.726760
\(669\) 29.9553 1.15814
\(670\) −43.4301 −1.67785
\(671\) −28.5907 −1.10373
\(672\) 5.72601 0.220886
\(673\) −36.6337 −1.41212 −0.706062 0.708150i \(-0.749530\pi\)
−0.706062 + 0.708150i \(0.749530\pi\)
\(674\) −25.7371 −0.991355
\(675\) 42.0858 1.61988
\(676\) 5.38769 0.207219
\(677\) 33.5506 1.28946 0.644728 0.764412i \(-0.276971\pi\)
0.644728 + 0.764412i \(0.276971\pi\)
\(678\) 19.8185 0.761125
\(679\) 9.37671 0.359845
\(680\) −19.4157 −0.744558
\(681\) 57.7507 2.21301
\(682\) 6.22223 0.238261
\(683\) 26.9587 1.03154 0.515772 0.856726i \(-0.327505\pi\)
0.515772 + 0.856726i \(0.327505\pi\)
\(684\) 2.62657 0.100429
\(685\) 27.6017 1.05461
\(686\) 17.9040 0.683577
\(687\) −3.86295 −0.147381
\(688\) −5.72273 −0.218177
\(689\) 58.5171 2.22932
\(690\) −36.7242 −1.39807
\(691\) −25.7477 −0.979487 −0.489743 0.871867i \(-0.662910\pi\)
−0.489743 + 0.871867i \(0.662910\pi\)
\(692\) −15.7375 −0.598251
\(693\) 9.03256 0.343119
\(694\) 6.68163 0.253631
\(695\) 87.3363 3.31285
\(696\) −9.74602 −0.369422
\(697\) 7.90270 0.299336
\(698\) 14.7538 0.558441
\(699\) 19.0809 0.721706
\(700\) 34.3822 1.29953
\(701\) −29.3989 −1.11038 −0.555191 0.831723i \(-0.687355\pi\)
−0.555191 + 0.831723i \(0.687355\pi\)
\(702\) −14.3102 −0.540104
\(703\) 14.9017 0.562028
\(704\) 2.34795 0.0884917
\(705\) −45.9914 −1.73214
\(706\) −21.0243 −0.791260
\(707\) 48.9525 1.84105
\(708\) 9.19309 0.345497
\(709\) −31.8895 −1.19764 −0.598818 0.800885i \(-0.704363\pi\)
−0.598818 + 0.800885i \(0.704363\pi\)
\(710\) −33.2016 −1.24604
\(711\) 0.208043 0.00780223
\(712\) −3.25753 −0.122081
\(713\) 11.0419 0.413524
\(714\) 26.4918 0.991432
\(715\) 42.2518 1.58013
\(716\) 19.4081 0.725316
\(717\) −9.98658 −0.372956
\(718\) 8.26515 0.308453
\(719\) 12.7218 0.474444 0.237222 0.971455i \(-0.423763\pi\)
0.237222 + 0.971455i \(0.423763\pi\)
\(720\) 5.92153 0.220682
\(721\) 31.8341 1.18557
\(722\) 15.5351 0.578155
\(723\) 6.93157 0.257788
\(724\) 11.5761 0.430221
\(725\) −58.5206 −2.17340
\(726\) −11.5243 −0.427708
\(727\) 45.0305 1.67009 0.835044 0.550184i \(-0.185442\pi\)
0.835044 + 0.550184i \(0.185442\pi\)
\(728\) −11.6908 −0.433291
\(729\) 5.16379 0.191252
\(730\) 13.1241 0.485745
\(731\) −26.4766 −0.979274
\(732\) 25.5745 0.945259
\(733\) −24.6880 −0.911873 −0.455937 0.890012i \(-0.650696\pi\)
−0.455937 + 0.890012i \(0.650696\pi\)
\(734\) 11.9739 0.441964
\(735\) −3.81625 −0.140765
\(736\) 4.16667 0.153585
\(737\) 24.2989 0.895061
\(738\) −2.41022 −0.0887214
\(739\) −48.2684 −1.77558 −0.887792 0.460245i \(-0.847761\pi\)
−0.887792 + 0.460245i \(0.847761\pi\)
\(740\) 33.5954 1.23499
\(741\) −16.7642 −0.615848
\(742\) −37.2049 −1.36584
\(743\) 47.6956 1.74978 0.874890 0.484321i \(-0.160933\pi\)
0.874890 + 0.484321i \(0.160933\pi\)
\(744\) −5.56580 −0.204052
\(745\) 11.0884 0.406246
\(746\) 26.4342 0.967823
\(747\) −11.7524 −0.429999
\(748\) 10.8630 0.397190
\(749\) −22.8275 −0.834098
\(750\) 67.0827 2.44951
\(751\) −23.5969 −0.861064 −0.430532 0.902575i \(-0.641674\pi\)
−0.430532 + 0.902575i \(0.641674\pi\)
\(752\) 5.21811 0.190285
\(753\) 50.4464 1.83837
\(754\) 19.8985 0.724660
\(755\) −71.5249 −2.60306
\(756\) 9.09838 0.330905
\(757\) −43.7178 −1.58895 −0.794475 0.607297i \(-0.792254\pi\)
−0.794475 + 0.607297i \(0.792254\pi\)
\(758\) −16.0303 −0.582248
\(759\) 20.5470 0.745808
\(760\) −7.81162 −0.283357
\(761\) −25.2780 −0.916328 −0.458164 0.888868i \(-0.651493\pi\)
−0.458164 + 0.888868i \(0.651493\pi\)
\(762\) −44.8024 −1.62302
\(763\) −20.2650 −0.733643
\(764\) −17.4749 −0.632221
\(765\) 27.3964 0.990518
\(766\) −12.0712 −0.436149
\(767\) −18.7696 −0.677730
\(768\) −2.10025 −0.0757862
\(769\) −4.76921 −0.171982 −0.0859910 0.996296i \(-0.527406\pi\)
−0.0859910 + 0.996296i \(0.527406\pi\)
\(770\) −26.8636 −0.968096
\(771\) 34.3225 1.23609
\(772\) −4.30096 −0.154795
\(773\) 31.9420 1.14887 0.574437 0.818549i \(-0.305221\pi\)
0.574437 + 0.818549i \(0.305221\pi\)
\(774\) 8.07502 0.290251
\(775\) −33.4202 −1.20049
\(776\) −3.43929 −0.123463
\(777\) −45.8395 −1.64448
\(778\) −19.4306 −0.696621
\(779\) 3.17954 0.113919
\(780\) −37.7944 −1.35326
\(781\) 18.5761 0.664707
\(782\) 19.2774 0.689358
\(783\) −15.4860 −0.553424
\(784\) 0.432985 0.0154638
\(785\) −84.4121 −3.01280
\(786\) −22.2577 −0.793904
\(787\) 30.9627 1.10370 0.551850 0.833943i \(-0.313922\pi\)
0.551850 + 0.833943i \(0.313922\pi\)
\(788\) −24.4924 −0.872506
\(789\) 58.2481 2.07369
\(790\) −0.618737 −0.0220137
\(791\) 25.7265 0.914731
\(792\) −3.31306 −0.117724
\(793\) −52.2155 −1.85423
\(794\) 33.4445 1.18690
\(795\) −120.277 −4.26579
\(796\) −1.02679 −0.0363937
\(797\) 11.4900 0.406997 0.203498 0.979075i \(-0.434769\pi\)
0.203498 + 0.979075i \(0.434769\pi\)
\(798\) 10.6586 0.377311
\(799\) 24.1420 0.854081
\(800\) −12.6111 −0.445869
\(801\) 4.59651 0.162410
\(802\) −10.5349 −0.372001
\(803\) −7.34287 −0.259124
\(804\) −21.7354 −0.766550
\(805\) −47.6720 −1.68022
\(806\) 11.3637 0.400270
\(807\) −52.5729 −1.85065
\(808\) −17.9553 −0.631666
\(809\) 34.9288 1.22803 0.614016 0.789293i \(-0.289553\pi\)
0.614016 + 0.789293i \(0.289553\pi\)
\(810\) 47.1781 1.65767
\(811\) −36.0547 −1.26605 −0.633026 0.774131i \(-0.718187\pi\)
−0.633026 + 0.774131i \(0.718187\pi\)
\(812\) −12.6514 −0.443977
\(813\) −39.3436 −1.37984
\(814\) −18.7965 −0.658816
\(815\) 22.1702 0.776588
\(816\) −9.71696 −0.340162
\(817\) −10.6525 −0.372684
\(818\) −18.5593 −0.648911
\(819\) 16.4963 0.576426
\(820\) 7.16818 0.250324
\(821\) −1.28425 −0.0448207 −0.0224103 0.999749i \(-0.507134\pi\)
−0.0224103 + 0.999749i \(0.507134\pi\)
\(822\) 13.8138 0.481812
\(823\) 6.04580 0.210743 0.105372 0.994433i \(-0.466397\pi\)
0.105372 + 0.994433i \(0.466397\pi\)
\(824\) −11.6765 −0.406769
\(825\) −62.1888 −2.16514
\(826\) 11.9336 0.415224
\(827\) 16.8031 0.584301 0.292150 0.956372i \(-0.405629\pi\)
0.292150 + 0.956372i \(0.405629\pi\)
\(828\) −5.87935 −0.204321
\(829\) −19.7056 −0.684403 −0.342202 0.939627i \(-0.611173\pi\)
−0.342202 + 0.939627i \(0.611173\pi\)
\(830\) 34.9527 1.21323
\(831\) −4.15181 −0.144025
\(832\) 4.28809 0.148663
\(833\) 2.00324 0.0694081
\(834\) 43.7091 1.51352
\(835\) −78.8265 −2.72790
\(836\) 4.37056 0.151159
\(837\) −8.84381 −0.305687
\(838\) −5.76487 −0.199144
\(839\) −27.7956 −0.959611 −0.479806 0.877375i \(-0.659293\pi\)
−0.479806 + 0.877375i \(0.659293\pi\)
\(840\) 24.0295 0.829098
\(841\) −7.46659 −0.257469
\(842\) 4.18467 0.144213
\(843\) 14.4743 0.498521
\(844\) −13.9129 −0.478902
\(845\) 22.6098 0.777800
\(846\) −7.36298 −0.253144
\(847\) −14.9598 −0.514026
\(848\) 13.6464 0.468620
\(849\) 46.4779 1.59512
\(850\) −58.3462 −2.00126
\(851\) −33.3561 −1.14343
\(852\) −16.6164 −0.569269
\(853\) −45.8249 −1.56901 −0.784507 0.620121i \(-0.787084\pi\)
−0.784507 + 0.620121i \(0.787084\pi\)
\(854\) 33.1984 1.13603
\(855\) 11.0225 0.376963
\(856\) 8.37290 0.286180
\(857\) 33.4509 1.14266 0.571330 0.820721i \(-0.306428\pi\)
0.571330 + 0.820721i \(0.306428\pi\)
\(858\) 21.1458 0.721904
\(859\) 30.8890 1.05392 0.526959 0.849891i \(-0.323332\pi\)
0.526959 + 0.849891i \(0.323332\pi\)
\(860\) −24.0158 −0.818930
\(861\) −9.78066 −0.333324
\(862\) 5.80236 0.197629
\(863\) −32.6839 −1.11257 −0.556287 0.830990i \(-0.687774\pi\)
−0.556287 + 0.830990i \(0.687774\pi\)
\(864\) −3.33720 −0.113534
\(865\) −66.0435 −2.24555
\(866\) −32.6204 −1.10849
\(867\) −9.25201 −0.314215
\(868\) −7.22501 −0.245233
\(869\) 0.346180 0.0117434
\(870\) −40.8997 −1.38663
\(871\) 44.3773 1.50367
\(872\) 7.43302 0.251714
\(873\) 4.85299 0.164249
\(874\) 7.75598 0.262350
\(875\) 87.0806 2.94386
\(876\) 6.56822 0.221919
\(877\) −30.1155 −1.01693 −0.508463 0.861084i \(-0.669786\pi\)
−0.508463 + 0.861084i \(0.669786\pi\)
\(878\) −22.7808 −0.768816
\(879\) −53.7020 −1.81132
\(880\) 9.85331 0.332155
\(881\) −3.62693 −0.122194 −0.0610971 0.998132i \(-0.519460\pi\)
−0.0610971 + 0.998132i \(0.519460\pi\)
\(882\) −0.610961 −0.0205721
\(883\) −5.41120 −0.182101 −0.0910507 0.995846i \(-0.529023\pi\)
−0.0910507 + 0.995846i \(0.529023\pi\)
\(884\) 19.8392 0.667263
\(885\) 38.5793 1.29683
\(886\) −22.9851 −0.772200
\(887\) −4.22032 −0.141705 −0.0708523 0.997487i \(-0.522572\pi\)
−0.0708523 + 0.997487i \(0.522572\pi\)
\(888\) 16.8135 0.564224
\(889\) −58.1583 −1.95057
\(890\) −13.6704 −0.458233
\(891\) −26.3959 −0.884295
\(892\) −14.2628 −0.477552
\(893\) 9.71317 0.325039
\(894\) 5.54939 0.185599
\(895\) 81.4473 2.72248
\(896\) −2.72635 −0.0910810
\(897\) 37.5252 1.25293
\(898\) 18.3512 0.612388
\(899\) 12.2974 0.410141
\(900\) 17.7948 0.593160
\(901\) 63.1362 2.10337
\(902\) −4.01056 −0.133537
\(903\) 32.7684 1.09046
\(904\) −9.43626 −0.313845
\(905\) 48.5796 1.61484
\(906\) −35.7960 −1.18924
\(907\) 10.9447 0.363414 0.181707 0.983353i \(-0.441838\pi\)
0.181707 + 0.983353i \(0.441838\pi\)
\(908\) −27.4971 −0.912523
\(909\) 25.3358 0.840334
\(910\) −49.0612 −1.62636
\(911\) −24.2244 −0.802591 −0.401296 0.915949i \(-0.631440\pi\)
−0.401296 + 0.915949i \(0.631440\pi\)
\(912\) −3.90948 −0.129456
\(913\) −19.5558 −0.647204
\(914\) 11.9075 0.393865
\(915\) 107.325 3.54804
\(916\) 1.83928 0.0607716
\(917\) −28.8928 −0.954126
\(918\) −15.4398 −0.509590
\(919\) −2.68405 −0.0885388 −0.0442694 0.999020i \(-0.514096\pi\)
−0.0442694 + 0.999020i \(0.514096\pi\)
\(920\) 17.4856 0.576485
\(921\) −42.3771 −1.39637
\(922\) −0.191347 −0.00630167
\(923\) 33.9258 1.11668
\(924\) −13.4444 −0.442288
\(925\) 100.958 3.31947
\(926\) −0.344055 −0.0113063
\(927\) 16.4760 0.541143
\(928\) 4.64041 0.152329
\(929\) 37.4174 1.22762 0.613812 0.789452i \(-0.289635\pi\)
0.613812 + 0.789452i \(0.289635\pi\)
\(930\) −23.3572 −0.765913
\(931\) 0.805975 0.0264148
\(932\) −9.08508 −0.297592
\(933\) 7.49696 0.245439
\(934\) 0.403713 0.0132099
\(935\) 45.5871 1.49086
\(936\) −6.05068 −0.197773
\(937\) −35.7845 −1.16903 −0.584514 0.811384i \(-0.698715\pi\)
−0.584514 + 0.811384i \(0.698715\pi\)
\(938\) −28.2149 −0.921251
\(939\) 28.2804 0.922897
\(940\) 21.8981 0.714237
\(941\) 55.5362 1.81043 0.905215 0.424954i \(-0.139710\pi\)
0.905215 + 0.424954i \(0.139710\pi\)
\(942\) −42.2457 −1.37644
\(943\) −7.11712 −0.231765
\(944\) −4.37714 −0.142464
\(945\) 38.1819 1.24206
\(946\) 13.4367 0.436864
\(947\) 31.4831 1.02306 0.511531 0.859265i \(-0.329078\pi\)
0.511531 + 0.859265i \(0.329078\pi\)
\(948\) −0.309659 −0.0100573
\(949\) −13.4104 −0.435319
\(950\) −23.4747 −0.761621
\(951\) −50.9315 −1.65157
\(952\) −12.6137 −0.408811
\(953\) −34.6687 −1.12303 −0.561515 0.827466i \(-0.689781\pi\)
−0.561515 + 0.827466i \(0.689781\pi\)
\(954\) −19.2557 −0.623426
\(955\) −73.3346 −2.37305
\(956\) 4.75495 0.153786
\(957\) 22.8832 0.739708
\(958\) −12.3751 −0.399820
\(959\) 17.9318 0.579049
\(960\) −8.81381 −0.284465
\(961\) −23.9771 −0.773456
\(962\) −34.3282 −1.10679
\(963\) −11.8145 −0.380718
\(964\) −3.30036 −0.106297
\(965\) −18.0492 −0.581026
\(966\) −23.8584 −0.767631
\(967\) −39.4697 −1.26926 −0.634629 0.772817i \(-0.718847\pi\)
−0.634629 + 0.772817i \(0.718847\pi\)
\(968\) 5.48713 0.176363
\(969\) −18.0875 −0.581054
\(970\) −14.4332 −0.463422
\(971\) 45.5251 1.46097 0.730485 0.682928i \(-0.239294\pi\)
0.730485 + 0.682928i \(0.239294\pi\)
\(972\) 13.5996 0.436206
\(973\) 56.7392 1.81897
\(974\) 35.6262 1.14154
\(975\) −113.576 −3.63735
\(976\) −12.1769 −0.389772
\(977\) −1.48194 −0.0474113 −0.0237057 0.999719i \(-0.507546\pi\)
−0.0237057 + 0.999719i \(0.507546\pi\)
\(978\) 11.0955 0.354795
\(979\) 7.64851 0.244448
\(980\) 1.81705 0.0580435
\(981\) −10.4883 −0.334866
\(982\) −21.8556 −0.697439
\(983\) −35.7828 −1.14129 −0.570647 0.821195i \(-0.693308\pi\)
−0.570647 + 0.821195i \(0.693308\pi\)
\(984\) 3.58746 0.114364
\(985\) −102.784 −3.27497
\(986\) 21.4692 0.683719
\(987\) −29.8790 −0.951057
\(988\) 7.98200 0.253941
\(989\) 23.8447 0.758217
\(990\) −13.9034 −0.441881
\(991\) 38.0912 1.21001 0.605004 0.796223i \(-0.293172\pi\)
0.605004 + 0.796223i \(0.293172\pi\)
\(992\) 2.65007 0.0841397
\(993\) −39.0863 −1.24037
\(994\) −21.5699 −0.684156
\(995\) −4.30900 −0.136604
\(996\) 17.4928 0.554279
\(997\) 23.5263 0.745087 0.372543 0.928015i \(-0.378486\pi\)
0.372543 + 0.928015i \(0.378486\pi\)
\(998\) −2.53687 −0.0803032
\(999\) 26.7159 0.845254
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 6046.2.a.g.1.16 69
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6046.2.a.g.1.16 69 1.1 even 1 trivial