Properties

Label 6031.2.a.d.1.12
Level $6031$
Weight $2$
Character 6031.1
Self dual yes
Analytic conductor $48.158$
Analytic rank $0$
Dimension $133$
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] = [6031,2,Mod(1,6031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6031, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6031 = 37 \cdot 163 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6031.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: \(48.1577774590\)
Analytic rank: \(0\)
Dimension: \(133\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 6031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.38561 q^{2} -1.27766 q^{3} +3.69114 q^{4} +1.05833 q^{5} +3.04800 q^{6} +3.63452 q^{7} -4.03440 q^{8} -1.36759 q^{9} +O(q^{10})\) \(q-2.38561 q^{2} -1.27766 q^{3} +3.69114 q^{4} +1.05833 q^{5} +3.04800 q^{6} +3.63452 q^{7} -4.03440 q^{8} -1.36759 q^{9} -2.52477 q^{10} +6.37491 q^{11} -4.71602 q^{12} +3.34884 q^{13} -8.67055 q^{14} -1.35219 q^{15} +2.24223 q^{16} -1.62321 q^{17} +3.26253 q^{18} -1.80216 q^{19} +3.90645 q^{20} -4.64368 q^{21} -15.2081 q^{22} +2.14602 q^{23} +5.15459 q^{24} -3.87993 q^{25} -7.98904 q^{26} +5.58029 q^{27} +13.4155 q^{28} -8.47687 q^{29} +3.22580 q^{30} -0.456835 q^{31} +2.71971 q^{32} -8.14497 q^{33} +3.87235 q^{34} +3.84653 q^{35} -5.04795 q^{36} -1.00000 q^{37} +4.29926 q^{38} -4.27868 q^{39} -4.26974 q^{40} -3.27551 q^{41} +11.0780 q^{42} -1.63618 q^{43} +23.5307 q^{44} -1.44736 q^{45} -5.11957 q^{46} -0.868467 q^{47} -2.86481 q^{48} +6.20974 q^{49} +9.25601 q^{50} +2.07391 q^{51} +12.3610 q^{52} +5.15215 q^{53} -13.3124 q^{54} +6.74678 q^{55} -14.6631 q^{56} +2.30255 q^{57} +20.2225 q^{58} +12.0518 q^{59} -4.99112 q^{60} +0.335052 q^{61} +1.08983 q^{62} -4.97052 q^{63} -10.9726 q^{64} +3.54419 q^{65} +19.4307 q^{66} -2.75376 q^{67} -5.99150 q^{68} -2.74189 q^{69} -9.17633 q^{70} -7.94862 q^{71} +5.51739 q^{72} -4.92432 q^{73} +2.38561 q^{74} +4.95723 q^{75} -6.65203 q^{76} +23.1698 q^{77} +10.2073 q^{78} +11.9529 q^{79} +2.37303 q^{80} -3.02695 q^{81} +7.81410 q^{82} +9.43378 q^{83} -17.1405 q^{84} -1.71790 q^{85} +3.90328 q^{86} +10.8306 q^{87} -25.7190 q^{88} +10.3818 q^{89} +3.45284 q^{90} +12.1714 q^{91} +7.92127 q^{92} +0.583680 q^{93} +2.07182 q^{94} -1.90729 q^{95} -3.47487 q^{96} -10.0161 q^{97} -14.8140 q^{98} -8.71824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 133 q + 14 q^{2} + 8 q^{3} + 142 q^{4} + 34 q^{5} + 20 q^{6} + 8 q^{7} + 42 q^{8} + 177 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 133 q + 14 q^{2} + 8 q^{3} + 142 q^{4} + 34 q^{5} + 20 q^{6} + 8 q^{7} + 42 q^{8} + 177 q^{9} + 9 q^{10} + 23 q^{11} + 24 q^{12} + 23 q^{13} + 31 q^{14} + 9 q^{15} + 168 q^{16} + 98 q^{17} + 38 q^{18} + 29 q^{19} + 83 q^{20} + 26 q^{21} + 2 q^{22} + 34 q^{23} + 75 q^{24} + 177 q^{25} + 67 q^{26} + 32 q^{27} + 32 q^{28} + 91 q^{29} + 12 q^{30} + 24 q^{31} + 88 q^{32} + 27 q^{33} + 23 q^{34} + 66 q^{35} + 232 q^{36} - 133 q^{37} + 26 q^{38} + 28 q^{39} + 41 q^{40} + 132 q^{41} + 13 q^{42} + 11 q^{43} + 65 q^{44} + 107 q^{45} + 20 q^{46} + 10 q^{47} + 27 q^{48} + 229 q^{49} + 78 q^{50} + 19 q^{51} + 71 q^{52} + 7 q^{53} + 43 q^{54} + 41 q^{55} + 67 q^{56} + 45 q^{57} + 25 q^{58} + 97 q^{59} - 42 q^{60} + 65 q^{61} + 24 q^{62} + 39 q^{63} + 200 q^{64} + 60 q^{65} + 35 q^{66} + 25 q^{67} + 227 q^{68} + 120 q^{69} + 37 q^{70} + 26 q^{71} + 93 q^{72} + 55 q^{73} - 14 q^{74} + 5 q^{75} + 34 q^{76} + 21 q^{77} - 2 q^{78} + 50 q^{79} + 162 q^{80} + 341 q^{81} + 66 q^{82} + 30 q^{83} - 89 q^{84} + 30 q^{85} - 12 q^{86} + 80 q^{87} - 85 q^{88} + 225 q^{89} - 86 q^{90} + q^{91} + 82 q^{92} + 42 q^{93} - 17 q^{94} + 70 q^{95} + 55 q^{96} + 12 q^{97} + 90 q^{98} + 17 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38561 −1.68688 −0.843441 0.537222i \(-0.819474\pi\)
−0.843441 + 0.537222i \(0.819474\pi\)
\(3\) −1.27766 −0.737657 −0.368829 0.929497i \(-0.620241\pi\)
−0.368829 + 0.929497i \(0.620241\pi\)
\(4\) 3.69114 1.84557
\(5\) 1.05833 0.473301 0.236650 0.971595i \(-0.423950\pi\)
0.236650 + 0.971595i \(0.423950\pi\)
\(6\) 3.04800 1.24434
\(7\) 3.63452 1.37372 0.686860 0.726790i \(-0.258989\pi\)
0.686860 + 0.726790i \(0.258989\pi\)
\(8\) −4.03440 −1.42638
\(9\) −1.36759 −0.455862
\(10\) −2.52477 −0.798403
\(11\) 6.37491 1.92211 0.961055 0.276359i \(-0.0891278\pi\)
0.961055 + 0.276359i \(0.0891278\pi\)
\(12\) −4.71602 −1.36140
\(13\) 3.34884 0.928802 0.464401 0.885625i \(-0.346270\pi\)
0.464401 + 0.885625i \(0.346270\pi\)
\(14\) −8.67055 −2.31730
\(15\) −1.35219 −0.349134
\(16\) 2.24223 0.560557
\(17\) −1.62321 −0.393687 −0.196843 0.980435i \(-0.563069\pi\)
−0.196843 + 0.980435i \(0.563069\pi\)
\(18\) 3.26253 0.768985
\(19\) −1.80216 −0.413444 −0.206722 0.978400i \(-0.566280\pi\)
−0.206722 + 0.978400i \(0.566280\pi\)
\(20\) 3.90645 0.873510
\(21\) −4.64368 −1.01333
\(22\) −15.2081 −3.24237
\(23\) 2.14602 0.447477 0.223738 0.974649i \(-0.428174\pi\)
0.223738 + 0.974649i \(0.428174\pi\)
\(24\) 5.15459 1.05218
\(25\) −3.87993 −0.775986
\(26\) −7.98904 −1.56678
\(27\) 5.58029 1.07393
\(28\) 13.4155 2.53530
\(29\) −8.47687 −1.57412 −0.787058 0.616879i \(-0.788397\pi\)
−0.787058 + 0.616879i \(0.788397\pi\)
\(30\) 3.22580 0.588947
\(31\) −0.456835 −0.0820501 −0.0410250 0.999158i \(-0.513062\pi\)
−0.0410250 + 0.999158i \(0.513062\pi\)
\(32\) 2.71971 0.480782
\(33\) −8.14497 −1.41786
\(34\) 3.87235 0.664103
\(35\) 3.84653 0.650183
\(36\) −5.04795 −0.841325
\(37\) −1.00000 −0.164399
\(38\) 4.29926 0.697432
\(39\) −4.27868 −0.685138
\(40\) −4.26974 −0.675105
\(41\) −3.27551 −0.511549 −0.255775 0.966736i \(-0.582331\pi\)
−0.255775 + 0.966736i \(0.582331\pi\)
\(42\) 11.0780 1.70937
\(43\) −1.63618 −0.249515 −0.124757 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(44\) 23.5307 3.54739
\(45\) −1.44736 −0.215760
\(46\) −5.11957 −0.754840
\(47\) −0.868467 −0.126679 −0.0633395 0.997992i \(-0.520175\pi\)
−0.0633395 + 0.997992i \(0.520175\pi\)
\(48\) −2.86481 −0.413499
\(49\) 6.20974 0.887106
\(50\) 9.25601 1.30900
\(51\) 2.07391 0.290406
\(52\) 12.3610 1.71417
\(53\) 5.15215 0.707703 0.353851 0.935302i \(-0.384872\pi\)
0.353851 + 0.935302i \(0.384872\pi\)
\(54\) −13.3124 −1.81159
\(55\) 6.74678 0.909736
\(56\) −14.6631 −1.95944
\(57\) 2.30255 0.304980
\(58\) 20.2225 2.65535
\(59\) 12.0518 1.56901 0.784506 0.620121i \(-0.212917\pi\)
0.784506 + 0.620121i \(0.212917\pi\)
\(60\) −4.99112 −0.644351
\(61\) 0.335052 0.0428991 0.0214495 0.999770i \(-0.493172\pi\)
0.0214495 + 0.999770i \(0.493172\pi\)
\(62\) 1.08983 0.138409
\(63\) −4.97052 −0.626227
\(64\) −10.9726 −1.37158
\(65\) 3.54419 0.439603
\(66\) 19.4307 2.39176
\(67\) −2.75376 −0.336425 −0.168213 0.985751i \(-0.553800\pi\)
−0.168213 + 0.985751i \(0.553800\pi\)
\(68\) −5.99150 −0.726577
\(69\) −2.74189 −0.330084
\(70\) −9.17633 −1.09678
\(71\) −7.94862 −0.943328 −0.471664 0.881778i \(-0.656346\pi\)
−0.471664 + 0.881778i \(0.656346\pi\)
\(72\) 5.51739 0.650230
\(73\) −4.92432 −0.576348 −0.288174 0.957578i \(-0.593048\pi\)
−0.288174 + 0.957578i \(0.593048\pi\)
\(74\) 2.38561 0.277322
\(75\) 4.95723 0.572412
\(76\) −6.65203 −0.763041
\(77\) 23.1698 2.64044
\(78\) 10.2073 1.15575
\(79\) 11.9529 1.34480 0.672401 0.740187i \(-0.265263\pi\)
0.672401 + 0.740187i \(0.265263\pi\)
\(80\) 2.37303 0.265312
\(81\) −3.02695 −0.336328
\(82\) 7.81410 0.862923
\(83\) 9.43378 1.03549 0.517746 0.855535i \(-0.326771\pi\)
0.517746 + 0.855535i \(0.326771\pi\)
\(84\) −17.1405 −1.87018
\(85\) −1.71790 −0.186332
\(86\) 3.90328 0.420902
\(87\) 10.8306 1.16116
\(88\) −25.7190 −2.74165
\(89\) 10.3818 1.10047 0.550237 0.835009i \(-0.314537\pi\)
0.550237 + 0.835009i \(0.314537\pi\)
\(90\) 3.45284 0.363961
\(91\) 12.1714 1.27591
\(92\) 7.92127 0.825849
\(93\) 0.583680 0.0605248
\(94\) 2.07182 0.213692
\(95\) −1.90729 −0.195684
\(96\) −3.47487 −0.354652
\(97\) −10.0161 −1.01698 −0.508490 0.861068i \(-0.669796\pi\)
−0.508490 + 0.861068i \(0.669796\pi\)
\(98\) −14.8140 −1.49644
\(99\) −8.71824 −0.876216
\(100\) −14.3214 −1.43214
\(101\) −11.1360 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(102\) −4.94755 −0.489880
\(103\) 4.02708 0.396800 0.198400 0.980121i \(-0.436425\pi\)
0.198400 + 0.980121i \(0.436425\pi\)
\(104\) −13.5106 −1.32482
\(105\) −4.91456 −0.479612
\(106\) −12.2910 −1.19381
\(107\) 18.0569 1.74563 0.872814 0.488053i \(-0.162293\pi\)
0.872814 + 0.488053i \(0.162293\pi\)
\(108\) 20.5976 1.98201
\(109\) 9.94049 0.952126 0.476063 0.879411i \(-0.342063\pi\)
0.476063 + 0.879411i \(0.342063\pi\)
\(110\) −16.0952 −1.53462
\(111\) 1.27766 0.121270
\(112\) 8.14943 0.770049
\(113\) 6.41720 0.603680 0.301840 0.953359i \(-0.402399\pi\)
0.301840 + 0.953359i \(0.402399\pi\)
\(114\) −5.49299 −0.514466
\(115\) 2.27121 0.211791
\(116\) −31.2893 −2.90514
\(117\) −4.57983 −0.423406
\(118\) −28.7509 −2.64674
\(119\) −5.89960 −0.540816
\(120\) 5.45527 0.497996
\(121\) 29.6395 2.69450
\(122\) −0.799305 −0.0723657
\(123\) 4.18499 0.377348
\(124\) −1.68624 −0.151429
\(125\) −9.39792 −0.840576
\(126\) 11.8577 1.05637
\(127\) −10.0174 −0.888897 −0.444448 0.895804i \(-0.646600\pi\)
−0.444448 + 0.895804i \(0.646600\pi\)
\(128\) 20.7370 1.83291
\(129\) 2.09048 0.184056
\(130\) −8.45506 −0.741558
\(131\) 17.0598 1.49052 0.745261 0.666773i \(-0.232325\pi\)
0.745261 + 0.666773i \(0.232325\pi\)
\(132\) −30.0642 −2.61675
\(133\) −6.55000 −0.567957
\(134\) 6.56940 0.567510
\(135\) 5.90580 0.508291
\(136\) 6.54869 0.561545
\(137\) 8.84841 0.755971 0.377985 0.925812i \(-0.376617\pi\)
0.377985 + 0.925812i \(0.376617\pi\)
\(138\) 6.54107 0.556813
\(139\) 20.9195 1.77437 0.887186 0.461412i \(-0.152657\pi\)
0.887186 + 0.461412i \(0.152657\pi\)
\(140\) 14.1981 1.19996
\(141\) 1.10961 0.0934456
\(142\) 18.9623 1.59128
\(143\) 21.3486 1.78526
\(144\) −3.06644 −0.255537
\(145\) −8.97135 −0.745030
\(146\) 11.7475 0.972231
\(147\) −7.93394 −0.654380
\(148\) −3.69114 −0.303410
\(149\) −2.23035 −0.182718 −0.0913588 0.995818i \(-0.529121\pi\)
−0.0913588 + 0.995818i \(0.529121\pi\)
\(150\) −11.8260 −0.965591
\(151\) 1.12941 0.0919099 0.0459549 0.998944i \(-0.485367\pi\)
0.0459549 + 0.998944i \(0.485367\pi\)
\(152\) 7.27064 0.589727
\(153\) 2.21988 0.179467
\(154\) −55.2740 −4.45411
\(155\) −0.483484 −0.0388344
\(156\) −15.7932 −1.26447
\(157\) −2.00470 −0.159992 −0.0799962 0.996795i \(-0.525491\pi\)
−0.0799962 + 0.996795i \(0.525491\pi\)
\(158\) −28.5149 −2.26852
\(159\) −6.58270 −0.522042
\(160\) 2.87836 0.227554
\(161\) 7.79976 0.614707
\(162\) 7.22113 0.567345
\(163\) 1.00000 0.0783260
\(164\) −12.0904 −0.944100
\(165\) −8.62009 −0.671073
\(166\) −22.5053 −1.74675
\(167\) 3.43488 0.265799 0.132899 0.991130i \(-0.457571\pi\)
0.132899 + 0.991130i \(0.457571\pi\)
\(168\) 18.7345 1.44540
\(169\) −1.78524 −0.137326
\(170\) 4.09824 0.314321
\(171\) 2.46461 0.188474
\(172\) −6.03936 −0.460497
\(173\) −6.87158 −0.522437 −0.261218 0.965280i \(-0.584124\pi\)
−0.261218 + 0.965280i \(0.584124\pi\)
\(174\) −25.8375 −1.95874
\(175\) −14.1017 −1.06599
\(176\) 14.2940 1.07745
\(177\) −15.3981 −1.15739
\(178\) −24.7670 −1.85637
\(179\) 9.23808 0.690486 0.345243 0.938513i \(-0.387796\pi\)
0.345243 + 0.938513i \(0.387796\pi\)
\(180\) −5.34241 −0.398200
\(181\) −16.1734 −1.20216 −0.601081 0.799188i \(-0.705263\pi\)
−0.601081 + 0.799188i \(0.705263\pi\)
\(182\) −29.0363 −2.15232
\(183\) −0.428083 −0.0316448
\(184\) −8.65791 −0.638270
\(185\) −1.05833 −0.0778102
\(186\) −1.39243 −0.102098
\(187\) −10.3478 −0.756709
\(188\) −3.20563 −0.233795
\(189\) 20.2817 1.47527
\(190\) 4.55005 0.330095
\(191\) 6.49253 0.469783 0.234891 0.972022i \(-0.424527\pi\)
0.234891 + 0.972022i \(0.424527\pi\)
\(192\) 14.0193 1.01176
\(193\) −17.1509 −1.23455 −0.617273 0.786749i \(-0.711763\pi\)
−0.617273 + 0.786749i \(0.711763\pi\)
\(194\) 23.8945 1.71553
\(195\) −4.52827 −0.324276
\(196\) 22.9210 1.63722
\(197\) 18.7610 1.33667 0.668334 0.743862i \(-0.267008\pi\)
0.668334 + 0.743862i \(0.267008\pi\)
\(198\) 20.7983 1.47807
\(199\) 7.77477 0.551139 0.275569 0.961281i \(-0.411134\pi\)
0.275569 + 0.961281i \(0.411134\pi\)
\(200\) 15.6532 1.10685
\(201\) 3.51837 0.248166
\(202\) 26.5662 1.86919
\(203\) −30.8094 −2.16239
\(204\) 7.65510 0.535964
\(205\) −3.46659 −0.242117
\(206\) −9.60705 −0.669355
\(207\) −2.93487 −0.203988
\(208\) 7.50888 0.520647
\(209\) −11.4886 −0.794685
\(210\) 11.7242 0.809049
\(211\) 7.92778 0.545771 0.272885 0.962047i \(-0.412022\pi\)
0.272885 + 0.962047i \(0.412022\pi\)
\(212\) 19.0173 1.30611
\(213\) 10.1556 0.695852
\(214\) −43.0768 −2.94467
\(215\) −1.73162 −0.118096
\(216\) −22.5131 −1.53182
\(217\) −1.66038 −0.112714
\(218\) −23.7141 −1.60612
\(219\) 6.29160 0.425147
\(220\) 24.9033 1.67898
\(221\) −5.43589 −0.365657
\(222\) −3.04800 −0.204568
\(223\) 22.1897 1.48593 0.742967 0.669328i \(-0.233418\pi\)
0.742967 + 0.669328i \(0.233418\pi\)
\(224\) 9.88485 0.660459
\(225\) 5.30614 0.353743
\(226\) −15.3089 −1.01834
\(227\) 1.90344 0.126336 0.0631680 0.998003i \(-0.479880\pi\)
0.0631680 + 0.998003i \(0.479880\pi\)
\(228\) 8.49903 0.562862
\(229\) 7.79373 0.515024 0.257512 0.966275i \(-0.417097\pi\)
0.257512 + 0.966275i \(0.417097\pi\)
\(230\) −5.41821 −0.357266
\(231\) −29.6031 −1.94774
\(232\) 34.1991 2.24528
\(233\) −2.40280 −0.157412 −0.0787062 0.996898i \(-0.525079\pi\)
−0.0787062 + 0.996898i \(0.525079\pi\)
\(234\) 10.9257 0.714235
\(235\) −0.919128 −0.0599573
\(236\) 44.4849 2.89572
\(237\) −15.2717 −0.992002
\(238\) 14.0742 0.912292
\(239\) −0.587973 −0.0380328 −0.0190164 0.999819i \(-0.506053\pi\)
−0.0190164 + 0.999819i \(0.506053\pi\)
\(240\) −3.03192 −0.195710
\(241\) −15.4995 −0.998410 −0.499205 0.866484i \(-0.666374\pi\)
−0.499205 + 0.866484i \(0.666374\pi\)
\(242\) −70.7084 −4.54531
\(243\) −12.8735 −0.825832
\(244\) 1.23673 0.0791732
\(245\) 6.57198 0.419868
\(246\) −9.98376 −0.636542
\(247\) −6.03516 −0.384008
\(248\) 1.84306 0.117034
\(249\) −12.0532 −0.763838
\(250\) 22.4198 1.41795
\(251\) 29.3174 1.85050 0.925250 0.379358i \(-0.123855\pi\)
0.925250 + 0.379358i \(0.123855\pi\)
\(252\) −18.3469 −1.15574
\(253\) 13.6807 0.860099
\(254\) 23.8975 1.49946
\(255\) 2.19489 0.137449
\(256\) −27.5252 −1.72032
\(257\) −15.6563 −0.976612 −0.488306 0.872673i \(-0.662385\pi\)
−0.488306 + 0.872673i \(0.662385\pi\)
\(258\) −4.98707 −0.310481
\(259\) −3.63452 −0.225838
\(260\) 13.0821 0.811318
\(261\) 11.5929 0.717579
\(262\) −40.6980 −2.51433
\(263\) −6.19871 −0.382229 −0.191114 0.981568i \(-0.561210\pi\)
−0.191114 + 0.981568i \(0.561210\pi\)
\(264\) 32.8601 2.02240
\(265\) 5.45269 0.334956
\(266\) 15.6257 0.958076
\(267\) −13.2645 −0.811772
\(268\) −10.1645 −0.620896
\(269\) −27.5181 −1.67781 −0.838903 0.544281i \(-0.816802\pi\)
−0.838903 + 0.544281i \(0.816802\pi\)
\(270\) −14.0889 −0.857426
\(271\) 7.35320 0.446675 0.223337 0.974741i \(-0.428305\pi\)
0.223337 + 0.974741i \(0.428305\pi\)
\(272\) −3.63962 −0.220684
\(273\) −15.5510 −0.941187
\(274\) −21.1089 −1.27523
\(275\) −24.7342 −1.49153
\(276\) −10.1207 −0.609193
\(277\) −12.0892 −0.726372 −0.363186 0.931717i \(-0.618311\pi\)
−0.363186 + 0.931717i \(0.618311\pi\)
\(278\) −49.9059 −2.99316
\(279\) 0.624762 0.0374035
\(280\) −15.5185 −0.927405
\(281\) 24.9391 1.48774 0.743872 0.668322i \(-0.232987\pi\)
0.743872 + 0.668322i \(0.232987\pi\)
\(282\) −2.64709 −0.157632
\(283\) 9.67230 0.574958 0.287479 0.957787i \(-0.407183\pi\)
0.287479 + 0.957787i \(0.407183\pi\)
\(284\) −29.3395 −1.74098
\(285\) 2.43687 0.144347
\(286\) −50.9294 −3.01152
\(287\) −11.9049 −0.702726
\(288\) −3.71944 −0.219170
\(289\) −14.3652 −0.845011
\(290\) 21.4022 1.25678
\(291\) 12.7972 0.750183
\(292\) −18.1764 −1.06369
\(293\) −3.19113 −0.186428 −0.0932138 0.995646i \(-0.529714\pi\)
−0.0932138 + 0.995646i \(0.529714\pi\)
\(294\) 18.9273 1.10386
\(295\) 12.7548 0.742615
\(296\) 4.03440 0.234495
\(297\) 35.5739 2.06420
\(298\) 5.32075 0.308223
\(299\) 7.18669 0.415617
\(300\) 18.2978 1.05643
\(301\) −5.94672 −0.342763
\(302\) −2.69433 −0.155041
\(303\) 14.2280 0.817379
\(304\) −4.04086 −0.231759
\(305\) 0.354597 0.0203042
\(306\) −5.29578 −0.302739
\(307\) −0.199609 −0.0113923 −0.00569614 0.999984i \(-0.501813\pi\)
−0.00569614 + 0.999984i \(0.501813\pi\)
\(308\) 85.5228 4.87311
\(309\) −5.14524 −0.292702
\(310\) 1.15340 0.0655090
\(311\) 24.1056 1.36690 0.683451 0.729996i \(-0.260478\pi\)
0.683451 + 0.729996i \(0.260478\pi\)
\(312\) 17.2619 0.977264
\(313\) −6.78884 −0.383728 −0.191864 0.981422i \(-0.561453\pi\)
−0.191864 + 0.981422i \(0.561453\pi\)
\(314\) 4.78243 0.269888
\(315\) −5.26047 −0.296394
\(316\) 44.1197 2.48192
\(317\) −13.6757 −0.768105 −0.384053 0.923311i \(-0.625472\pi\)
−0.384053 + 0.923311i \(0.625472\pi\)
\(318\) 15.7038 0.880623
\(319\) −54.0393 −3.02562
\(320\) −11.6127 −0.649170
\(321\) −23.0706 −1.28767
\(322\) −18.6072 −1.03694
\(323\) 2.92529 0.162768
\(324\) −11.1729 −0.620716
\(325\) −12.9933 −0.720738
\(326\) −2.38561 −0.132127
\(327\) −12.7006 −0.702343
\(328\) 13.2147 0.729662
\(329\) −3.15646 −0.174021
\(330\) 20.5642 1.13202
\(331\) 22.7086 1.24818 0.624088 0.781354i \(-0.285471\pi\)
0.624088 + 0.781354i \(0.285471\pi\)
\(332\) 34.8214 1.91107
\(333\) 1.36759 0.0749432
\(334\) −8.19428 −0.448371
\(335\) −2.91439 −0.159230
\(336\) −10.4122 −0.568032
\(337\) −27.7135 −1.50965 −0.754826 0.655926i \(-0.772278\pi\)
−0.754826 + 0.655926i \(0.772278\pi\)
\(338\) 4.25890 0.231653
\(339\) −8.19900 −0.445309
\(340\) −6.34101 −0.343889
\(341\) −2.91229 −0.157709
\(342\) −5.87961 −0.317933
\(343\) −2.87220 −0.155084
\(344\) 6.60100 0.355902
\(345\) −2.90183 −0.156229
\(346\) 16.3929 0.881289
\(347\) 14.0521 0.754357 0.377179 0.926140i \(-0.376894\pi\)
0.377179 + 0.926140i \(0.376894\pi\)
\(348\) 39.9771 2.14300
\(349\) −19.1006 −1.02243 −0.511215 0.859453i \(-0.670804\pi\)
−0.511215 + 0.859453i \(0.670804\pi\)
\(350\) 33.6411 1.79819
\(351\) 18.6875 0.997466
\(352\) 17.3379 0.924115
\(353\) 18.0643 0.961463 0.480732 0.876868i \(-0.340371\pi\)
0.480732 + 0.876868i \(0.340371\pi\)
\(354\) 36.7339 1.95238
\(355\) −8.41229 −0.446478
\(356\) 38.3208 2.03100
\(357\) 7.53768 0.398936
\(358\) −22.0385 −1.16477
\(359\) −14.9244 −0.787679 −0.393839 0.919179i \(-0.628853\pi\)
−0.393839 + 0.919179i \(0.628853\pi\)
\(360\) 5.83924 0.307755
\(361\) −15.7522 −0.829064
\(362\) 38.5835 2.02790
\(363\) −37.8692 −1.98762
\(364\) 44.9265 2.35479
\(365\) −5.21157 −0.272786
\(366\) 1.02124 0.0533810
\(367\) −33.3017 −1.73834 −0.869168 0.494516i \(-0.835345\pi\)
−0.869168 + 0.494516i \(0.835345\pi\)
\(368\) 4.81187 0.250836
\(369\) 4.47955 0.233196
\(370\) 2.52477 0.131257
\(371\) 18.7256 0.972185
\(372\) 2.15444 0.111703
\(373\) −8.21808 −0.425516 −0.212758 0.977105i \(-0.568245\pi\)
−0.212758 + 0.977105i \(0.568245\pi\)
\(374\) 24.6859 1.27648
\(375\) 12.0073 0.620057
\(376\) 3.50374 0.180692
\(377\) −28.3877 −1.46204
\(378\) −48.3842 −2.48861
\(379\) 17.9178 0.920377 0.460189 0.887821i \(-0.347782\pi\)
0.460189 + 0.887821i \(0.347782\pi\)
\(380\) −7.04007 −0.361148
\(381\) 12.7988 0.655701
\(382\) −15.4886 −0.792468
\(383\) −6.54536 −0.334453 −0.167226 0.985919i \(-0.553481\pi\)
−0.167226 + 0.985919i \(0.553481\pi\)
\(384\) −26.4948 −1.35206
\(385\) 24.5213 1.24972
\(386\) 40.9153 2.08253
\(387\) 2.23761 0.113744
\(388\) −36.9708 −1.87691
\(389\) −16.9083 −0.857287 −0.428643 0.903474i \(-0.641008\pi\)
−0.428643 + 0.903474i \(0.641008\pi\)
\(390\) 10.8027 0.547016
\(391\) −3.48345 −0.176166
\(392\) −25.0526 −1.26535
\(393\) −21.7966 −1.09949
\(394\) −44.7565 −2.25480
\(395\) 12.6501 0.636496
\(396\) −32.1803 −1.61712
\(397\) 7.76674 0.389802 0.194901 0.980823i \(-0.437562\pi\)
0.194901 + 0.980823i \(0.437562\pi\)
\(398\) −18.5476 −0.929706
\(399\) 8.36867 0.418957
\(400\) −8.69970 −0.434985
\(401\) 20.5245 1.02494 0.512471 0.858704i \(-0.328730\pi\)
0.512471 + 0.858704i \(0.328730\pi\)
\(402\) −8.39345 −0.418627
\(403\) −1.52987 −0.0762083
\(404\) −41.1046 −2.04503
\(405\) −3.20352 −0.159184
\(406\) 73.4992 3.64770
\(407\) −6.37491 −0.315993
\(408\) −8.36700 −0.414228
\(409\) 10.6232 0.525285 0.262642 0.964893i \(-0.415406\pi\)
0.262642 + 0.964893i \(0.415406\pi\)
\(410\) 8.26992 0.408422
\(411\) −11.3053 −0.557647
\(412\) 14.8645 0.732322
\(413\) 43.8026 2.15538
\(414\) 7.00146 0.344103
\(415\) 9.98408 0.490099
\(416\) 9.10789 0.446551
\(417\) −26.7281 −1.30888
\(418\) 27.4074 1.34054
\(419\) 29.3565 1.43416 0.717079 0.696992i \(-0.245479\pi\)
0.717079 + 0.696992i \(0.245479\pi\)
\(420\) −18.1403 −0.885157
\(421\) 19.2212 0.936785 0.468393 0.883520i \(-0.344833\pi\)
0.468393 + 0.883520i \(0.344833\pi\)
\(422\) −18.9126 −0.920651
\(423\) 1.18770 0.0577481
\(424\) −20.7858 −1.00945
\(425\) 6.29795 0.305496
\(426\) −24.2274 −1.17382
\(427\) 1.21776 0.0589313
\(428\) 66.6506 3.22168
\(429\) −27.2762 −1.31691
\(430\) 4.13097 0.199213
\(431\) −8.03518 −0.387041 −0.193520 0.981096i \(-0.561991\pi\)
−0.193520 + 0.981096i \(0.561991\pi\)
\(432\) 12.5123 0.601998
\(433\) 32.7921 1.57589 0.787943 0.615749i \(-0.211146\pi\)
0.787943 + 0.615749i \(0.211146\pi\)
\(434\) 3.96102 0.190135
\(435\) 11.4623 0.549577
\(436\) 36.6917 1.75722
\(437\) −3.86748 −0.185007
\(438\) −15.0093 −0.717173
\(439\) 13.1832 0.629199 0.314599 0.949225i \(-0.398130\pi\)
0.314599 + 0.949225i \(0.398130\pi\)
\(440\) −27.2192 −1.29763
\(441\) −8.49236 −0.404398
\(442\) 12.9679 0.616821
\(443\) 20.7843 0.987493 0.493746 0.869606i \(-0.335627\pi\)
0.493746 + 0.869606i \(0.335627\pi\)
\(444\) 4.71602 0.223812
\(445\) 10.9875 0.520855
\(446\) −52.9361 −2.50660
\(447\) 2.84963 0.134783
\(448\) −39.8803 −1.88417
\(449\) 30.9573 1.46097 0.730483 0.682931i \(-0.239295\pi\)
0.730483 + 0.682931i \(0.239295\pi\)
\(450\) −12.6584 −0.596722
\(451\) −20.8811 −0.983254
\(452\) 23.6868 1.11413
\(453\) −1.44300 −0.0677980
\(454\) −4.54088 −0.213114
\(455\) 12.8814 0.603891
\(456\) −9.28941 −0.435016
\(457\) 7.77568 0.363731 0.181865 0.983323i \(-0.441786\pi\)
0.181865 + 0.983323i \(0.441786\pi\)
\(458\) −18.5928 −0.868785
\(459\) −9.05799 −0.422791
\(460\) 8.38334 0.390875
\(461\) 26.6159 1.23963 0.619813 0.784749i \(-0.287208\pi\)
0.619813 + 0.784749i \(0.287208\pi\)
\(462\) 70.6214 3.28560
\(463\) 9.44524 0.438958 0.219479 0.975617i \(-0.429564\pi\)
0.219479 + 0.975617i \(0.429564\pi\)
\(464\) −19.0071 −0.882382
\(465\) 0.617728 0.0286465
\(466\) 5.73213 0.265536
\(467\) −28.0646 −1.29868 −0.649338 0.760500i \(-0.724954\pi\)
−0.649338 + 0.760500i \(0.724954\pi\)
\(468\) −16.9048 −0.781425
\(469\) −10.0086 −0.462154
\(470\) 2.19268 0.101141
\(471\) 2.56132 0.118019
\(472\) −48.6218 −2.23800
\(473\) −10.4305 −0.479595
\(474\) 36.4323 1.67339
\(475\) 6.99227 0.320827
\(476\) −21.7762 −0.998113
\(477\) −7.04601 −0.322615
\(478\) 1.40267 0.0641568
\(479\) −20.8581 −0.953033 −0.476517 0.879166i \(-0.658101\pi\)
−0.476517 + 0.879166i \(0.658101\pi\)
\(480\) −3.67757 −0.167857
\(481\) −3.34884 −0.152694
\(482\) 36.9757 1.68420
\(483\) −9.96544 −0.453443
\(484\) 109.404 4.97289
\(485\) −10.6004 −0.481338
\(486\) 30.7110 1.39308
\(487\) −42.9596 −1.94668 −0.973342 0.229360i \(-0.926337\pi\)
−0.973342 + 0.229360i \(0.926337\pi\)
\(488\) −1.35174 −0.0611902
\(489\) −1.27766 −0.0577778
\(490\) −15.6782 −0.708268
\(491\) 23.4348 1.05760 0.528798 0.848748i \(-0.322643\pi\)
0.528798 + 0.848748i \(0.322643\pi\)
\(492\) 15.4474 0.696422
\(493\) 13.7598 0.619709
\(494\) 14.3975 0.647776
\(495\) −9.22681 −0.414714
\(496\) −1.02433 −0.0459938
\(497\) −28.8894 −1.29587
\(498\) 28.7541 1.28850
\(499\) 24.7700 1.10886 0.554428 0.832232i \(-0.312937\pi\)
0.554428 + 0.832232i \(0.312937\pi\)
\(500\) −34.6890 −1.55134
\(501\) −4.38860 −0.196068
\(502\) −69.9400 −3.12157
\(503\) 11.3696 0.506944 0.253472 0.967343i \(-0.418427\pi\)
0.253472 + 0.967343i \(0.418427\pi\)
\(504\) 20.0531 0.893235
\(505\) −11.7856 −0.524453
\(506\) −32.6368 −1.45088
\(507\) 2.28093 0.101300
\(508\) −36.9755 −1.64052
\(509\) −25.1393 −1.11428 −0.557139 0.830419i \(-0.688101\pi\)
−0.557139 + 0.830419i \(0.688101\pi\)
\(510\) −5.23616 −0.231861
\(511\) −17.8975 −0.791741
\(512\) 24.1903 1.06907
\(513\) −10.0566 −0.444009
\(514\) 37.3498 1.64743
\(515\) 4.26199 0.187806
\(516\) 7.71625 0.339689
\(517\) −5.53640 −0.243491
\(518\) 8.67055 0.380962
\(519\) 8.77955 0.385379
\(520\) −14.2987 −0.627039
\(521\) 10.1344 0.443997 0.221998 0.975047i \(-0.428742\pi\)
0.221998 + 0.975047i \(0.428742\pi\)
\(522\) −27.6560 −1.21047
\(523\) 3.89363 0.170257 0.0851283 0.996370i \(-0.472870\pi\)
0.0851283 + 0.996370i \(0.472870\pi\)
\(524\) 62.9701 2.75086
\(525\) 18.0172 0.786333
\(526\) 14.7877 0.644775
\(527\) 0.741541 0.0323020
\(528\) −18.2629 −0.794791
\(529\) −18.3946 −0.799765
\(530\) −13.0080 −0.565032
\(531\) −16.4819 −0.715253
\(532\) −24.1770 −1.04820
\(533\) −10.9692 −0.475128
\(534\) 31.6439 1.36936
\(535\) 19.1102 0.826207
\(536\) 11.1098 0.479869
\(537\) −11.8031 −0.509342
\(538\) 65.6474 2.83026
\(539\) 39.5866 1.70511
\(540\) 21.7991 0.938086
\(541\) 17.9159 0.770263 0.385132 0.922862i \(-0.374156\pi\)
0.385132 + 0.922862i \(0.374156\pi\)
\(542\) −17.5419 −0.753488
\(543\) 20.6641 0.886783
\(544\) −4.41467 −0.189277
\(545\) 10.5204 0.450642
\(546\) 37.0985 1.58767
\(547\) −14.3505 −0.613582 −0.306791 0.951777i \(-0.599255\pi\)
−0.306791 + 0.951777i \(0.599255\pi\)
\(548\) 32.6607 1.39520
\(549\) −0.458213 −0.0195561
\(550\) 59.0062 2.51603
\(551\) 15.2767 0.650809
\(552\) 11.0619 0.470824
\(553\) 43.4429 1.84738
\(554\) 28.8402 1.22530
\(555\) 1.35219 0.0573972
\(556\) 77.2169 3.27473
\(557\) 25.2670 1.07060 0.535298 0.844663i \(-0.320199\pi\)
0.535298 + 0.844663i \(0.320199\pi\)
\(558\) −1.49044 −0.0630953
\(559\) −5.47931 −0.231750
\(560\) 8.62481 0.364465
\(561\) 13.2210 0.558192
\(562\) −59.4951 −2.50965
\(563\) −21.0305 −0.886331 −0.443165 0.896440i \(-0.646145\pi\)
−0.443165 + 0.896440i \(0.646145\pi\)
\(564\) 4.09571 0.172460
\(565\) 6.79154 0.285722
\(566\) −23.0743 −0.969887
\(567\) −11.0015 −0.462020
\(568\) 32.0679 1.34554
\(569\) −14.2962 −0.599329 −0.299664 0.954045i \(-0.596875\pi\)
−0.299664 + 0.954045i \(0.596875\pi\)
\(570\) −5.81341 −0.243497
\(571\) 46.6942 1.95409 0.977045 0.213034i \(-0.0683345\pi\)
0.977045 + 0.213034i \(0.0683345\pi\)
\(572\) 78.8006 3.29482
\(573\) −8.29524 −0.346539
\(574\) 28.4005 1.18541
\(575\) −8.32642 −0.347236
\(576\) 15.0060 0.625251
\(577\) −1.17680 −0.0489909 −0.0244954 0.999700i \(-0.507798\pi\)
−0.0244954 + 0.999700i \(0.507798\pi\)
\(578\) 34.2697 1.42543
\(579\) 21.9130 0.910672
\(580\) −33.1145 −1.37501
\(581\) 34.2873 1.42248
\(582\) −30.5291 −1.26547
\(583\) 32.8445 1.36028
\(584\) 19.8667 0.822089
\(585\) −4.84699 −0.200398
\(586\) 7.61279 0.314481
\(587\) −3.93425 −0.162384 −0.0811919 0.996698i \(-0.525873\pi\)
−0.0811919 + 0.996698i \(0.525873\pi\)
\(588\) −29.2853 −1.20770
\(589\) 0.823292 0.0339231
\(590\) −30.4281 −1.25270
\(591\) −23.9702 −0.986002
\(592\) −2.24223 −0.0921551
\(593\) 25.2795 1.03810 0.519051 0.854743i \(-0.326285\pi\)
0.519051 + 0.854743i \(0.326285\pi\)
\(594\) −84.8654 −3.48207
\(595\) −6.24374 −0.255968
\(596\) −8.23254 −0.337218
\(597\) −9.93351 −0.406552
\(598\) −17.1447 −0.701097
\(599\) 5.51051 0.225153 0.112577 0.993643i \(-0.464090\pi\)
0.112577 + 0.993643i \(0.464090\pi\)
\(600\) −19.9995 −0.816474
\(601\) −31.2876 −1.27625 −0.638124 0.769933i \(-0.720289\pi\)
−0.638124 + 0.769933i \(0.720289\pi\)
\(602\) 14.1866 0.578201
\(603\) 3.76600 0.153363
\(604\) 4.16880 0.169626
\(605\) 31.3685 1.27531
\(606\) −33.9426 −1.37882
\(607\) −32.5715 −1.32204 −0.661018 0.750370i \(-0.729875\pi\)
−0.661018 + 0.750370i \(0.729875\pi\)
\(608\) −4.90136 −0.198777
\(609\) 39.3639 1.59511
\(610\) −0.845931 −0.0342507
\(611\) −2.90836 −0.117660
\(612\) 8.19390 0.331219
\(613\) 37.3315 1.50780 0.753902 0.656987i \(-0.228169\pi\)
0.753902 + 0.656987i \(0.228169\pi\)
\(614\) 0.476189 0.0192174
\(615\) 4.42912 0.178599
\(616\) −93.4761 −3.76626
\(617\) 24.4064 0.982565 0.491283 0.871000i \(-0.336528\pi\)
0.491283 + 0.871000i \(0.336528\pi\)
\(618\) 12.2745 0.493754
\(619\) 2.15033 0.0864291 0.0432145 0.999066i \(-0.486240\pi\)
0.0432145 + 0.999066i \(0.486240\pi\)
\(620\) −1.78461 −0.0716715
\(621\) 11.9754 0.480557
\(622\) −57.5065 −2.30580
\(623\) 37.7330 1.51174
\(624\) −9.59379 −0.384059
\(625\) 9.45352 0.378141
\(626\) 16.1955 0.647304
\(627\) 14.6786 0.586205
\(628\) −7.39962 −0.295277
\(629\) 1.62321 0.0647217
\(630\) 12.5494 0.499981
\(631\) −5.62209 −0.223812 −0.111906 0.993719i \(-0.535696\pi\)
−0.111906 + 0.993719i \(0.535696\pi\)
\(632\) −48.2226 −1.91819
\(633\) −10.1290 −0.402592
\(634\) 32.6250 1.29570
\(635\) −10.6017 −0.420716
\(636\) −24.2977 −0.963465
\(637\) 20.7955 0.823946
\(638\) 128.917 5.10387
\(639\) 10.8704 0.430027
\(640\) 21.9467 0.867518
\(641\) 20.9671 0.828149 0.414075 0.910243i \(-0.364105\pi\)
0.414075 + 0.910243i \(0.364105\pi\)
\(642\) 55.0374 2.17215
\(643\) −19.7292 −0.778044 −0.389022 0.921229i \(-0.627187\pi\)
−0.389022 + 0.921229i \(0.627187\pi\)
\(644\) 28.7900 1.13449
\(645\) 2.21242 0.0871141
\(646\) −6.97861 −0.274570
\(647\) 30.9230 1.21571 0.607854 0.794048i \(-0.292030\pi\)
0.607854 + 0.794048i \(0.292030\pi\)
\(648\) 12.2119 0.479730
\(649\) 76.8293 3.01581
\(650\) 30.9969 1.21580
\(651\) 2.12140 0.0831441
\(652\) 3.69114 0.144556
\(653\) 10.4491 0.408906 0.204453 0.978876i \(-0.434459\pi\)
0.204453 + 0.978876i \(0.434459\pi\)
\(654\) 30.2986 1.18477
\(655\) 18.0550 0.705465
\(656\) −7.34446 −0.286753
\(657\) 6.73443 0.262735
\(658\) 7.53009 0.293553
\(659\) −28.2674 −1.10114 −0.550571 0.834788i \(-0.685590\pi\)
−0.550571 + 0.834788i \(0.685590\pi\)
\(660\) −31.8180 −1.23851
\(661\) 18.2170 0.708559 0.354280 0.935140i \(-0.384726\pi\)
0.354280 + 0.935140i \(0.384726\pi\)
\(662\) −54.1738 −2.10553
\(663\) 6.94521 0.269730
\(664\) −38.0596 −1.47700
\(665\) −6.93208 −0.268815
\(666\) −3.26253 −0.126420
\(667\) −18.1916 −0.704380
\(668\) 12.6786 0.490550
\(669\) −28.3509 −1.09611
\(670\) 6.95261 0.268603
\(671\) 2.13593 0.0824567
\(672\) −12.6295 −0.487193
\(673\) −28.8117 −1.11061 −0.555305 0.831647i \(-0.687398\pi\)
−0.555305 + 0.831647i \(0.687398\pi\)
\(674\) 66.1136 2.54660
\(675\) −21.6511 −0.833353
\(676\) −6.58958 −0.253445
\(677\) 14.8273 0.569860 0.284930 0.958548i \(-0.408030\pi\)
0.284930 + 0.958548i \(0.408030\pi\)
\(678\) 19.5596 0.751183
\(679\) −36.4037 −1.39705
\(680\) 6.93069 0.265780
\(681\) −2.43195 −0.0931927
\(682\) 6.94758 0.266037
\(683\) 9.54038 0.365053 0.182526 0.983201i \(-0.441573\pi\)
0.182526 + 0.983201i \(0.441573\pi\)
\(684\) 9.09723 0.347841
\(685\) 9.36457 0.357802
\(686\) 6.85196 0.261609
\(687\) −9.95774 −0.379911
\(688\) −3.66869 −0.139867
\(689\) 17.2538 0.657316
\(690\) 6.92263 0.263540
\(691\) 22.9783 0.874137 0.437068 0.899428i \(-0.356017\pi\)
0.437068 + 0.899428i \(0.356017\pi\)
\(692\) −25.3640 −0.964194
\(693\) −31.6866 −1.20368
\(694\) −33.5229 −1.27251
\(695\) 22.1398 0.839812
\(696\) −43.6948 −1.65625
\(697\) 5.31686 0.201390
\(698\) 45.5665 1.72472
\(699\) 3.06995 0.116116
\(700\) −52.0513 −1.96735
\(701\) 38.7262 1.46267 0.731334 0.682019i \(-0.238898\pi\)
0.731334 + 0.682019i \(0.238898\pi\)
\(702\) −44.5811 −1.68261
\(703\) 1.80216 0.0679699
\(704\) −69.9496 −2.63632
\(705\) 1.17433 0.0442279
\(706\) −43.0943 −1.62187
\(707\) −40.4741 −1.52218
\(708\) −56.8366 −2.13605
\(709\) −11.4267 −0.429140 −0.214570 0.976709i \(-0.568835\pi\)
−0.214570 + 0.976709i \(0.568835\pi\)
\(710\) 20.0684 0.753155
\(711\) −16.3466 −0.613044
\(712\) −41.8845 −1.56969
\(713\) −0.980379 −0.0367155
\(714\) −17.9820 −0.672958
\(715\) 22.5939 0.844965
\(716\) 34.0990 1.27434
\(717\) 0.751229 0.0280552
\(718\) 35.6038 1.32872
\(719\) −11.6324 −0.433815 −0.216907 0.976192i \(-0.569597\pi\)
−0.216907 + 0.976192i \(0.569597\pi\)
\(720\) −3.24532 −0.120946
\(721\) 14.6365 0.545092
\(722\) 37.5786 1.39853
\(723\) 19.8031 0.736484
\(724\) −59.6984 −2.21867
\(725\) 32.8897 1.22149
\(726\) 90.3413 3.35288
\(727\) 29.9366 1.11029 0.555144 0.831755i \(-0.312663\pi\)
0.555144 + 0.831755i \(0.312663\pi\)
\(728\) −49.1045 −1.81993
\(729\) 25.5287 0.945509
\(730\) 12.4328 0.460158
\(731\) 2.65587 0.0982307
\(732\) −1.58011 −0.0584027
\(733\) 42.5192 1.57048 0.785241 0.619191i \(-0.212539\pi\)
0.785241 + 0.619191i \(0.212539\pi\)
\(734\) 79.4450 2.93237
\(735\) −8.39675 −0.309719
\(736\) 5.83656 0.215139
\(737\) −17.5550 −0.646646
\(738\) −10.6865 −0.393374
\(739\) −13.7936 −0.507407 −0.253703 0.967282i \(-0.581649\pi\)
−0.253703 + 0.967282i \(0.581649\pi\)
\(740\) −3.90645 −0.143604
\(741\) 7.71088 0.283266
\(742\) −44.6720 −1.63996
\(743\) −9.17497 −0.336597 −0.168299 0.985736i \(-0.553827\pi\)
−0.168299 + 0.985736i \(0.553827\pi\)
\(744\) −2.35480 −0.0863311
\(745\) −2.36046 −0.0864804
\(746\) 19.6051 0.717795
\(747\) −12.9015 −0.472041
\(748\) −38.1953 −1.39656
\(749\) 65.6282 2.39800
\(750\) −28.6449 −1.04596
\(751\) −31.8375 −1.16177 −0.580884 0.813986i \(-0.697293\pi\)
−0.580884 + 0.813986i \(0.697293\pi\)
\(752\) −1.94730 −0.0710108
\(753\) −37.4577 −1.36503
\(754\) 67.7221 2.46629
\(755\) 1.19529 0.0435010
\(756\) 74.8625 2.72272
\(757\) 28.5958 1.03933 0.519666 0.854370i \(-0.326057\pi\)
0.519666 + 0.854370i \(0.326057\pi\)
\(758\) −42.7450 −1.55257
\(759\) −17.4793 −0.634458
\(760\) 7.69476 0.279118
\(761\) −25.7172 −0.932249 −0.466124 0.884719i \(-0.654350\pi\)
−0.466124 + 0.884719i \(0.654350\pi\)
\(762\) −30.5329 −1.10609
\(763\) 36.1289 1.30795
\(764\) 23.9648 0.867017
\(765\) 2.34938 0.0849418
\(766\) 15.6147 0.564182
\(767\) 40.3596 1.45730
\(768\) 35.1678 1.26901
\(769\) 16.4920 0.594718 0.297359 0.954766i \(-0.403894\pi\)
0.297359 + 0.954766i \(0.403894\pi\)
\(770\) −58.4983 −2.10813
\(771\) 20.0034 0.720405
\(772\) −63.3063 −2.27844
\(773\) 25.1581 0.904872 0.452436 0.891797i \(-0.350555\pi\)
0.452436 + 0.891797i \(0.350555\pi\)
\(774\) −5.33808 −0.191873
\(775\) 1.77249 0.0636697
\(776\) 40.4089 1.45060
\(777\) 4.64368 0.166591
\(778\) 40.3367 1.44614
\(779\) 5.90301 0.211497
\(780\) −16.7145 −0.598474
\(781\) −50.6718 −1.81318
\(782\) 8.31016 0.297171
\(783\) −47.3034 −1.69049
\(784\) 13.9237 0.497274
\(785\) −2.12164 −0.0757245
\(786\) 51.9982 1.85472
\(787\) −28.3678 −1.01120 −0.505602 0.862767i \(-0.668730\pi\)
−0.505602 + 0.862767i \(0.668730\pi\)
\(788\) 69.2495 2.46691
\(789\) 7.91984 0.281954
\(790\) −30.1782 −1.07369
\(791\) 23.3235 0.829287
\(792\) 35.1729 1.24981
\(793\) 1.12204 0.0398448
\(794\) −18.5284 −0.657549
\(795\) −6.96669 −0.247083
\(796\) 28.6978 1.01717
\(797\) 27.9988 0.991767 0.495883 0.868389i \(-0.334844\pi\)
0.495883 + 0.868389i \(0.334844\pi\)
\(798\) −19.9644 −0.706732
\(799\) 1.40971 0.0498718
\(800\) −10.5523 −0.373080
\(801\) −14.1981 −0.501664
\(802\) −48.9634 −1.72896
\(803\) −31.3921 −1.10780
\(804\) 12.9868 0.458008
\(805\) 8.25475 0.290942
\(806\) 3.64968 0.128554
\(807\) 35.1587 1.23765
\(808\) 44.9271 1.58053
\(809\) 17.2548 0.606646 0.303323 0.952888i \(-0.401904\pi\)
0.303323 + 0.952888i \(0.401904\pi\)
\(810\) 7.64236 0.268525
\(811\) 29.1635 1.02407 0.512035 0.858965i \(-0.328892\pi\)
0.512035 + 0.858965i \(0.328892\pi\)
\(812\) −113.722 −3.99085
\(813\) −9.39488 −0.329493
\(814\) 15.2081 0.533042
\(815\) 1.05833 0.0370718
\(816\) 4.65019 0.162789
\(817\) 2.94866 0.103161
\(818\) −25.3429 −0.886093
\(819\) −16.6455 −0.581641
\(820\) −12.7956 −0.446843
\(821\) −7.24680 −0.252915 −0.126457 0.991972i \(-0.540361\pi\)
−0.126457 + 0.991972i \(0.540361\pi\)
\(822\) 26.9700 0.940685
\(823\) −32.0667 −1.11777 −0.558887 0.829244i \(-0.688772\pi\)
−0.558887 + 0.829244i \(0.688772\pi\)
\(824\) −16.2469 −0.565986
\(825\) 31.6019 1.10024
\(826\) −104.496 −3.63588
\(827\) −39.4316 −1.37117 −0.685585 0.727993i \(-0.740453\pi\)
−0.685585 + 0.727993i \(0.740453\pi\)
\(828\) −10.8330 −0.376473
\(829\) −36.3763 −1.26340 −0.631701 0.775212i \(-0.717643\pi\)
−0.631701 + 0.775212i \(0.717643\pi\)
\(830\) −23.8181 −0.826739
\(831\) 15.4459 0.535813
\(832\) −36.7456 −1.27393
\(833\) −10.0797 −0.349242
\(834\) 63.7627 2.20792
\(835\) 3.63524 0.125803
\(836\) −42.4061 −1.46665
\(837\) −2.54927 −0.0881158
\(838\) −70.0331 −2.41925
\(839\) −38.4451 −1.32727 −0.663636 0.748055i \(-0.730988\pi\)
−0.663636 + 0.748055i \(0.730988\pi\)
\(840\) 19.8273 0.684107
\(841\) 42.8574 1.47784
\(842\) −45.8544 −1.58025
\(843\) −31.8637 −1.09744
\(844\) 29.2625 1.00726
\(845\) −1.88938 −0.0649967
\(846\) −2.83340 −0.0974142
\(847\) 107.726 3.70149
\(848\) 11.5523 0.396708
\(849\) −12.3579 −0.424122
\(850\) −15.0245 −0.515335
\(851\) −2.14602 −0.0735647
\(852\) 37.4859 1.28424
\(853\) −29.1175 −0.996965 −0.498483 0.866900i \(-0.666109\pi\)
−0.498483 + 0.866900i \(0.666109\pi\)
\(854\) −2.90509 −0.0994101
\(855\) 2.60838 0.0892047
\(856\) −72.8488 −2.48992
\(857\) −28.5512 −0.975290 −0.487645 0.873042i \(-0.662144\pi\)
−0.487645 + 0.873042i \(0.662144\pi\)
\(858\) 65.0705 2.22147
\(859\) −8.84258 −0.301705 −0.150853 0.988556i \(-0.548202\pi\)
−0.150853 + 0.988556i \(0.548202\pi\)
\(860\) −6.39166 −0.217954
\(861\) 15.2104 0.518371
\(862\) 19.1688 0.652892
\(863\) 34.5391 1.17572 0.587862 0.808961i \(-0.299970\pi\)
0.587862 + 0.808961i \(0.299970\pi\)
\(864\) 15.1768 0.516324
\(865\) −7.27243 −0.247270
\(866\) −78.2291 −2.65833
\(867\) 18.3538 0.623328
\(868\) −6.12869 −0.208021
\(869\) 76.1984 2.58486
\(870\) −27.3447 −0.927071
\(871\) −9.22191 −0.312472
\(872\) −40.1039 −1.35809
\(873\) 13.6979 0.463603
\(874\) 9.22630 0.312084
\(875\) −34.1570 −1.15472
\(876\) 23.2232 0.784639
\(877\) −57.3543 −1.93672 −0.968358 0.249564i \(-0.919713\pi\)
−0.968358 + 0.249564i \(0.919713\pi\)
\(878\) −31.4499 −1.06138
\(879\) 4.07717 0.137520
\(880\) 15.1278 0.509959
\(881\) 20.3193 0.684574 0.342287 0.939595i \(-0.388798\pi\)
0.342287 + 0.939595i \(0.388798\pi\)
\(882\) 20.2595 0.682172
\(883\) 13.0358 0.438689 0.219344 0.975648i \(-0.429608\pi\)
0.219344 + 0.975648i \(0.429608\pi\)
\(884\) −20.0646 −0.674846
\(885\) −16.2963 −0.547795
\(886\) −49.5833 −1.66578
\(887\) −24.5973 −0.825896 −0.412948 0.910755i \(-0.635501\pi\)
−0.412948 + 0.910755i \(0.635501\pi\)
\(888\) −5.15459 −0.172977
\(889\) −36.4083 −1.22110
\(890\) −26.2118 −0.878621
\(891\) −19.2966 −0.646459
\(892\) 81.9054 2.74239
\(893\) 1.56512 0.0523747
\(894\) −6.79811 −0.227363
\(895\) 9.77696 0.326808
\(896\) 75.3691 2.51791
\(897\) −9.18215 −0.306583
\(898\) −73.8521 −2.46448
\(899\) 3.87254 0.129156
\(900\) 19.5857 0.652857
\(901\) −8.36304 −0.278613
\(902\) 49.8142 1.65863
\(903\) 7.59789 0.252842
\(904\) −25.8896 −0.861074
\(905\) −17.1169 −0.568984
\(906\) 3.44243 0.114367
\(907\) −38.6492 −1.28333 −0.641663 0.766987i \(-0.721755\pi\)
−0.641663 + 0.766987i \(0.721755\pi\)
\(908\) 7.02588 0.233162
\(909\) 15.2295 0.505129
\(910\) −30.7301 −1.01869
\(911\) −37.0511 −1.22756 −0.613779 0.789478i \(-0.710351\pi\)
−0.613779 + 0.789478i \(0.710351\pi\)
\(912\) 5.16285 0.170959
\(913\) 60.1395 1.99033
\(914\) −18.5497 −0.613571
\(915\) −0.453054 −0.0149775
\(916\) 28.7678 0.950514
\(917\) 62.0042 2.04756
\(918\) 21.6088 0.713198
\(919\) −7.27746 −0.240061 −0.120031 0.992770i \(-0.538299\pi\)
−0.120031 + 0.992770i \(0.538299\pi\)
\(920\) −9.16295 −0.302094
\(921\) 0.255032 0.00840360
\(922\) −63.4952 −2.09110
\(923\) −26.6187 −0.876165
\(924\) −109.269 −3.59469
\(925\) 3.87993 0.127571
\(926\) −22.5327 −0.740470
\(927\) −5.50738 −0.180886
\(928\) −23.0546 −0.756806
\(929\) −44.2105 −1.45050 −0.725249 0.688486i \(-0.758275\pi\)
−0.725249 + 0.688486i \(0.758275\pi\)
\(930\) −1.47366 −0.0483232
\(931\) −11.1910 −0.366769
\(932\) −8.86905 −0.290515
\(933\) −30.7987 −1.00831
\(934\) 66.9513 2.19071
\(935\) −10.9515 −0.358151
\(936\) 18.4769 0.603936
\(937\) −21.1418 −0.690673 −0.345336 0.938479i \(-0.612235\pi\)
−0.345336 + 0.938479i \(0.612235\pi\)
\(938\) 23.8766 0.779599
\(939\) 8.67383 0.283060
\(940\) −3.39263 −0.110655
\(941\) 11.4252 0.372450 0.186225 0.982507i \(-0.440375\pi\)
0.186225 + 0.982507i \(0.440375\pi\)
\(942\) −6.11032 −0.199085
\(943\) −7.02933 −0.228906
\(944\) 27.0229 0.879521
\(945\) 21.4648 0.698249
\(946\) 24.8831 0.809019
\(947\) 5.31208 0.172619 0.0863097 0.996268i \(-0.472493\pi\)
0.0863097 + 0.996268i \(0.472493\pi\)
\(948\) −56.3699 −1.83081
\(949\) −16.4908 −0.535313
\(950\) −16.6808 −0.541198
\(951\) 17.4729 0.566598
\(952\) 23.8013 0.771406
\(953\) −30.4192 −0.985376 −0.492688 0.870206i \(-0.663986\pi\)
−0.492688 + 0.870206i \(0.663986\pi\)
\(954\) 16.8090 0.544213
\(955\) 6.87126 0.222349
\(956\) −2.17029 −0.0701922
\(957\) 69.0439 2.23187
\(958\) 49.7594 1.60765
\(959\) 32.1597 1.03849
\(960\) 14.8371 0.478865
\(961\) −30.7913 −0.993268
\(962\) 7.98904 0.257577
\(963\) −24.6944 −0.795765
\(964\) −57.2108 −1.84263
\(965\) −18.1513 −0.584312
\(966\) 23.7737 0.764905
\(967\) 9.93886 0.319612 0.159806 0.987148i \(-0.448913\pi\)
0.159806 + 0.987148i \(0.448913\pi\)
\(968\) −119.578 −3.84337
\(969\) −3.73753 −0.120067
\(970\) 25.2884 0.811960
\(971\) 34.1939 1.09733 0.548667 0.836041i \(-0.315135\pi\)
0.548667 + 0.836041i \(0.315135\pi\)
\(972\) −47.5177 −1.52413
\(973\) 76.0325 2.43749
\(974\) 102.485 3.28382
\(975\) 16.6010 0.531657
\(976\) 0.751265 0.0240474
\(977\) −12.2301 −0.391276 −0.195638 0.980676i \(-0.562678\pi\)
−0.195638 + 0.980676i \(0.562678\pi\)
\(978\) 3.04800 0.0974642
\(979\) 66.1834 2.11523
\(980\) 24.2581 0.774896
\(981\) −13.5945 −0.434038
\(982\) −55.9062 −1.78404
\(983\) −42.8678 −1.36727 −0.683636 0.729823i \(-0.739602\pi\)
−0.683636 + 0.729823i \(0.739602\pi\)
\(984\) −16.8839 −0.538240
\(985\) 19.8554 0.632646
\(986\) −32.8254 −1.04538
\(987\) 4.03288 0.128368
\(988\) −22.2766 −0.708714
\(989\) −3.51127 −0.111652
\(990\) 22.0116 0.699574
\(991\) 23.9459 0.760665 0.380333 0.924850i \(-0.375810\pi\)
0.380333 + 0.924850i \(0.375810\pi\)
\(992\) −1.24246 −0.0394482
\(993\) −29.0138 −0.920726
\(994\) 68.9189 2.18598
\(995\) 8.22830 0.260855
\(996\) −44.4899 −1.40972
\(997\) −45.8007 −1.45052 −0.725262 0.688473i \(-0.758281\pi\)
−0.725262 + 0.688473i \(0.758281\pi\)
\(998\) −59.0915 −1.87051
\(999\) −5.58029 −0.176553
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 6031.2.a.d.1.12 133
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6031.2.a.d.1.12 133 1.1 even 1 trivial