Properties

Label 5054.2.a.bf.1.5
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $8$
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] = [5054,2,Mod(1,5054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5054, 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("5054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.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: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.236286\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.0295377 q^{3} +1.00000 q^{4} -2.02989 q^{5} -0.0295377 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.99913 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.0295377 q^{3} +1.00000 q^{4} -2.02989 q^{5} -0.0295377 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.99913 q^{9} +2.02989 q^{10} -2.94417 q^{11} +0.0295377 q^{12} +1.18331 q^{13} +1.00000 q^{14} -0.0599583 q^{15} +1.00000 q^{16} +4.65490 q^{17} +2.99913 q^{18} -2.02989 q^{20} -0.0295377 q^{21} +2.94417 q^{22} +6.80976 q^{23} -0.0295377 q^{24} -0.879544 q^{25} -1.18331 q^{26} -0.177200 q^{27} -1.00000 q^{28} -0.396447 q^{29} +0.0599583 q^{30} +0.265419 q^{31} -1.00000 q^{32} -0.0869639 q^{33} -4.65490 q^{34} +2.02989 q^{35} -2.99913 q^{36} +2.79771 q^{37} +0.0349521 q^{39} +2.02989 q^{40} +5.15394 q^{41} +0.0295377 q^{42} -0.743718 q^{43} -2.94417 q^{44} +6.08790 q^{45} -6.80976 q^{46} -2.97389 q^{47} +0.0295377 q^{48} +1.00000 q^{49} +0.879544 q^{50} +0.137495 q^{51} +1.18331 q^{52} -4.60071 q^{53} +0.177200 q^{54} +5.97634 q^{55} +1.00000 q^{56} +0.396447 q^{58} +7.11953 q^{59} -0.0599583 q^{60} +9.03552 q^{61} -0.265419 q^{62} +2.99913 q^{63} +1.00000 q^{64} -2.40198 q^{65} +0.0869639 q^{66} -11.1770 q^{67} +4.65490 q^{68} +0.201145 q^{69} -2.02989 q^{70} +13.9312 q^{71} +2.99913 q^{72} +4.21053 q^{73} -2.79771 q^{74} -0.0259797 q^{75} +2.94417 q^{77} -0.0349521 q^{78} -2.21703 q^{79} -2.02989 q^{80} +8.99215 q^{81} -5.15394 q^{82} +12.5400 q^{83} -0.0295377 q^{84} -9.44895 q^{85} +0.743718 q^{86} -0.0117101 q^{87} +2.94417 q^{88} -16.7143 q^{89} -6.08790 q^{90} -1.18331 q^{91} +6.80976 q^{92} +0.00783987 q^{93} +2.97389 q^{94} -0.0295377 q^{96} -2.11838 q^{97} -1.00000 q^{98} +8.82994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 0.0295377 0.0170536 0.00852679 0.999964i \(-0.497286\pi\)
0.00852679 + 0.999964i \(0.497286\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.02989 −0.907795 −0.453897 0.891054i \(-0.649967\pi\)
−0.453897 + 0.891054i \(0.649967\pi\)
\(6\) −0.0295377 −0.0120587
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.99913 −0.999709
\(10\) 2.02989 0.641908
\(11\) −2.94417 −0.887700 −0.443850 0.896101i \(-0.646388\pi\)
−0.443850 + 0.896101i \(0.646388\pi\)
\(12\) 0.0295377 0.00852679
\(13\) 1.18331 0.328190 0.164095 0.986445i \(-0.447530\pi\)
0.164095 + 0.986445i \(0.447530\pi\)
\(14\) 1.00000 0.267261
\(15\) −0.0599583 −0.0154812
\(16\) 1.00000 0.250000
\(17\) 4.65490 1.12898 0.564490 0.825440i \(-0.309073\pi\)
0.564490 + 0.825440i \(0.309073\pi\)
\(18\) 2.99913 0.706901
\(19\) 0 0
\(20\) −2.02989 −0.453897
\(21\) −0.0295377 −0.00644565
\(22\) 2.94417 0.627699
\(23\) 6.80976 1.41993 0.709967 0.704235i \(-0.248710\pi\)
0.709967 + 0.704235i \(0.248710\pi\)
\(24\) −0.0295377 −0.00602935
\(25\) −0.879544 −0.175909
\(26\) −1.18331 −0.232066
\(27\) −0.177200 −0.0341022
\(28\) −1.00000 −0.188982
\(29\) −0.396447 −0.0736184 −0.0368092 0.999322i \(-0.511719\pi\)
−0.0368092 + 0.999322i \(0.511719\pi\)
\(30\) 0.0599583 0.0109468
\(31\) 0.265419 0.0476707 0.0238354 0.999716i \(-0.492412\pi\)
0.0238354 + 0.999716i \(0.492412\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.0869639 −0.0151385
\(34\) −4.65490 −0.798309
\(35\) 2.02989 0.343114
\(36\) −2.99913 −0.499855
\(37\) 2.79771 0.459941 0.229971 0.973198i \(-0.426137\pi\)
0.229971 + 0.973198i \(0.426137\pi\)
\(38\) 0 0
\(39\) 0.0349521 0.00559682
\(40\) 2.02989 0.320954
\(41\) 5.15394 0.804911 0.402455 0.915440i \(-0.368157\pi\)
0.402455 + 0.915440i \(0.368157\pi\)
\(42\) 0.0295377 0.00455776
\(43\) −0.743718 −0.113416 −0.0567080 0.998391i \(-0.518060\pi\)
−0.0567080 + 0.998391i \(0.518060\pi\)
\(44\) −2.94417 −0.443850
\(45\) 6.08790 0.907531
\(46\) −6.80976 −1.00405
\(47\) −2.97389 −0.433787 −0.216893 0.976195i \(-0.569592\pi\)
−0.216893 + 0.976195i \(0.569592\pi\)
\(48\) 0.0295377 0.00426340
\(49\) 1.00000 0.142857
\(50\) 0.879544 0.124386
\(51\) 0.137495 0.0192532
\(52\) 1.18331 0.164095
\(53\) −4.60071 −0.631957 −0.315978 0.948766i \(-0.602333\pi\)
−0.315978 + 0.948766i \(0.602333\pi\)
\(54\) 0.177200 0.0241139
\(55\) 5.97634 0.805850
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 0.396447 0.0520560
\(59\) 7.11953 0.926884 0.463442 0.886127i \(-0.346614\pi\)
0.463442 + 0.886127i \(0.346614\pi\)
\(60\) −0.0599583 −0.00774058
\(61\) 9.03552 1.15688 0.578440 0.815725i \(-0.303662\pi\)
0.578440 + 0.815725i \(0.303662\pi\)
\(62\) −0.265419 −0.0337083
\(63\) 2.99913 0.377855
\(64\) 1.00000 0.125000
\(65\) −2.40198 −0.297929
\(66\) 0.0869639 0.0107045
\(67\) −11.1770 −1.36549 −0.682743 0.730659i \(-0.739213\pi\)
−0.682743 + 0.730659i \(0.739213\pi\)
\(68\) 4.65490 0.564490
\(69\) 0.201145 0.0242150
\(70\) −2.02989 −0.242618
\(71\) 13.9312 1.65333 0.826663 0.562697i \(-0.190236\pi\)
0.826663 + 0.562697i \(0.190236\pi\)
\(72\) 2.99913 0.353451
\(73\) 4.21053 0.492805 0.246403 0.969168i \(-0.420751\pi\)
0.246403 + 0.969168i \(0.420751\pi\)
\(74\) −2.79771 −0.325228
\(75\) −0.0259797 −0.00299988
\(76\) 0 0
\(77\) 2.94417 0.335519
\(78\) −0.0349521 −0.00395755
\(79\) −2.21703 −0.249435 −0.124718 0.992192i \(-0.539803\pi\)
−0.124718 + 0.992192i \(0.539803\pi\)
\(80\) −2.02989 −0.226949
\(81\) 8.99215 0.999128
\(82\) −5.15394 −0.569158
\(83\) 12.5400 1.37645 0.688224 0.725498i \(-0.258391\pi\)
0.688224 + 0.725498i \(0.258391\pi\)
\(84\) −0.0295377 −0.00322282
\(85\) −9.44895 −1.02488
\(86\) 0.743718 0.0801972
\(87\) −0.0117101 −0.00125546
\(88\) 2.94417 0.313849
\(89\) −16.7143 −1.77172 −0.885859 0.463955i \(-0.846430\pi\)
−0.885859 + 0.463955i \(0.846430\pi\)
\(90\) −6.08790 −0.641721
\(91\) −1.18331 −0.124044
\(92\) 6.80976 0.709967
\(93\) 0.00783987 0.000812957 0
\(94\) 2.97389 0.306734
\(95\) 0 0
\(96\) −0.0295377 −0.00301468
\(97\) −2.11838 −0.215089 −0.107545 0.994200i \(-0.534299\pi\)
−0.107545 + 0.994200i \(0.534299\pi\)
\(98\) −1.00000 −0.101015
\(99\) 8.82994 0.887442
\(100\) −0.879544 −0.0879544
\(101\) −13.3618 −1.32955 −0.664775 0.747043i \(-0.731473\pi\)
−0.664775 + 0.747043i \(0.731473\pi\)
\(102\) −0.137495 −0.0136140
\(103\) −13.3318 −1.31362 −0.656811 0.754055i \(-0.728095\pi\)
−0.656811 + 0.754055i \(0.728095\pi\)
\(104\) −1.18331 −0.116033
\(105\) 0.0599583 0.00585133
\(106\) 4.60071 0.446861
\(107\) −9.44616 −0.913195 −0.456597 0.889673i \(-0.650932\pi\)
−0.456597 + 0.889673i \(0.650932\pi\)
\(108\) −0.177200 −0.0170511
\(109\) −9.67812 −0.926996 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(110\) −5.97634 −0.569822
\(111\) 0.0826380 0.00784365
\(112\) −1.00000 −0.0944911
\(113\) −8.76641 −0.824675 −0.412337 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(114\) 0 0
\(115\) −13.8231 −1.28901
\(116\) −0.396447 −0.0368092
\(117\) −3.54889 −0.328095
\(118\) −7.11953 −0.655406
\(119\) −4.65490 −0.426714
\(120\) 0.0599583 0.00547341
\(121\) −2.33187 −0.211988
\(122\) −9.03552 −0.818038
\(123\) 0.152236 0.0137266
\(124\) 0.265419 0.0238354
\(125\) 11.9348 1.06748
\(126\) −2.99913 −0.267184
\(127\) −17.1303 −1.52007 −0.760033 0.649884i \(-0.774818\pi\)
−0.760033 + 0.649884i \(0.774818\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.0219677 −0.00193415
\(130\) 2.40198 0.210668
\(131\) 6.67961 0.583600 0.291800 0.956479i \(-0.405746\pi\)
0.291800 + 0.956479i \(0.405746\pi\)
\(132\) −0.0869639 −0.00756924
\(133\) 0 0
\(134\) 11.1770 0.965544
\(135\) 0.359697 0.0309578
\(136\) −4.65490 −0.399155
\(137\) 7.13250 0.609371 0.304685 0.952453i \(-0.401449\pi\)
0.304685 + 0.952453i \(0.401449\pi\)
\(138\) −0.201145 −0.0171226
\(139\) 17.3500 1.47161 0.735803 0.677195i \(-0.236805\pi\)
0.735803 + 0.677195i \(0.236805\pi\)
\(140\) 2.02989 0.171557
\(141\) −0.0878419 −0.00739762
\(142\) −13.9312 −1.16908
\(143\) −3.48386 −0.291335
\(144\) −2.99913 −0.249927
\(145\) 0.804744 0.0668304
\(146\) −4.21053 −0.348466
\(147\) 0.0295377 0.00243623
\(148\) 2.79771 0.229971
\(149\) 10.6622 0.873480 0.436740 0.899588i \(-0.356133\pi\)
0.436740 + 0.899588i \(0.356133\pi\)
\(150\) 0.0259797 0.00212123
\(151\) 4.50841 0.366889 0.183445 0.983030i \(-0.441275\pi\)
0.183445 + 0.983030i \(0.441275\pi\)
\(152\) 0 0
\(153\) −13.9607 −1.12865
\(154\) −2.94417 −0.237248
\(155\) −0.538772 −0.0432752
\(156\) 0.0349521 0.00279841
\(157\) −1.30678 −0.104293 −0.0521463 0.998639i \(-0.516606\pi\)
−0.0521463 + 0.998639i \(0.516606\pi\)
\(158\) 2.21703 0.176377
\(159\) −0.135894 −0.0107771
\(160\) 2.02989 0.160477
\(161\) −6.80976 −0.536685
\(162\) −8.99215 −0.706490
\(163\) −12.8874 −1.00942 −0.504710 0.863289i \(-0.668401\pi\)
−0.504710 + 0.863289i \(0.668401\pi\)
\(164\) 5.15394 0.402455
\(165\) 0.176527 0.0137426
\(166\) −12.5400 −0.973296
\(167\) −17.2129 −1.33198 −0.665988 0.745963i \(-0.731990\pi\)
−0.665988 + 0.745963i \(0.731990\pi\)
\(168\) 0.0295377 0.00227888
\(169\) −11.5998 −0.892291
\(170\) 9.44895 0.724701
\(171\) 0 0
\(172\) −0.743718 −0.0567080
\(173\) −12.2229 −0.929291 −0.464646 0.885497i \(-0.653818\pi\)
−0.464646 + 0.885497i \(0.653818\pi\)
\(174\) 0.0117101 0.000887742 0
\(175\) 0.879544 0.0664873
\(176\) −2.94417 −0.221925
\(177\) 0.210294 0.0158067
\(178\) 16.7143 1.25279
\(179\) 10.7092 0.800445 0.400222 0.916418i \(-0.368933\pi\)
0.400222 + 0.916418i \(0.368933\pi\)
\(180\) 6.08790 0.453765
\(181\) −21.8663 −1.62531 −0.812653 0.582748i \(-0.801978\pi\)
−0.812653 + 0.582748i \(0.801978\pi\)
\(182\) 1.18331 0.0877126
\(183\) 0.266888 0.0197290
\(184\) −6.80976 −0.502023
\(185\) −5.67905 −0.417532
\(186\) −0.00783987 −0.000574847 0
\(187\) −13.7048 −1.00220
\(188\) −2.97389 −0.216893
\(189\) 0.177200 0.0128894
\(190\) 0 0
\(191\) −19.1654 −1.38676 −0.693380 0.720572i \(-0.743879\pi\)
−0.693380 + 0.720572i \(0.743879\pi\)
\(192\) 0.0295377 0.00213170
\(193\) 12.9081 0.929145 0.464572 0.885535i \(-0.346208\pi\)
0.464572 + 0.885535i \(0.346208\pi\)
\(194\) 2.11838 0.152091
\(195\) −0.0709490 −0.00508077
\(196\) 1.00000 0.0714286
\(197\) 23.6787 1.68704 0.843520 0.537098i \(-0.180479\pi\)
0.843520 + 0.537098i \(0.180479\pi\)
\(198\) −8.82994 −0.627516
\(199\) −6.56540 −0.465409 −0.232705 0.972547i \(-0.574758\pi\)
−0.232705 + 0.972547i \(0.574758\pi\)
\(200\) 0.879544 0.0621932
\(201\) −0.330142 −0.0232864
\(202\) 13.3618 0.940134
\(203\) 0.396447 0.0278251
\(204\) 0.137495 0.00962658
\(205\) −10.4619 −0.730694
\(206\) 13.3318 0.928871
\(207\) −20.4234 −1.41952
\(208\) 1.18331 0.0820476
\(209\) 0 0
\(210\) −0.0599583 −0.00413751
\(211\) 11.9730 0.824255 0.412128 0.911126i \(-0.364786\pi\)
0.412128 + 0.911126i \(0.364786\pi\)
\(212\) −4.60071 −0.315978
\(213\) 0.411495 0.0281951
\(214\) 9.44616 0.645726
\(215\) 1.50967 0.102958
\(216\) 0.177200 0.0120570
\(217\) −0.265419 −0.0180178
\(218\) 9.67812 0.655485
\(219\) 0.124369 0.00840410
\(220\) 5.97634 0.402925
\(221\) 5.50818 0.370520
\(222\) −0.0826380 −0.00554630
\(223\) −8.47530 −0.567548 −0.283774 0.958891i \(-0.591587\pi\)
−0.283774 + 0.958891i \(0.591587\pi\)
\(224\) 1.00000 0.0668153
\(225\) 2.63787 0.175858
\(226\) 8.76641 0.583133
\(227\) −2.19728 −0.145839 −0.0729193 0.997338i \(-0.523232\pi\)
−0.0729193 + 0.997338i \(0.523232\pi\)
\(228\) 0 0
\(229\) −14.5591 −0.962089 −0.481045 0.876696i \(-0.659743\pi\)
−0.481045 + 0.876696i \(0.659743\pi\)
\(230\) 13.8231 0.911467
\(231\) 0.0869639 0.00572181
\(232\) 0.396447 0.0260280
\(233\) −17.7732 −1.16436 −0.582179 0.813060i \(-0.697800\pi\)
−0.582179 + 0.813060i \(0.697800\pi\)
\(234\) 3.54889 0.231998
\(235\) 6.03668 0.393789
\(236\) 7.11953 0.463442
\(237\) −0.0654859 −0.00425377
\(238\) 4.65490 0.301733
\(239\) −18.9372 −1.22495 −0.612473 0.790492i \(-0.709825\pi\)
−0.612473 + 0.790492i \(0.709825\pi\)
\(240\) −0.0599583 −0.00387029
\(241\) 19.6449 1.26544 0.632720 0.774380i \(-0.281938\pi\)
0.632720 + 0.774380i \(0.281938\pi\)
\(242\) 2.33187 0.149898
\(243\) 0.797208 0.0511409
\(244\) 9.03552 0.578440
\(245\) −2.02989 −0.129685
\(246\) −0.152236 −0.00970618
\(247\) 0 0
\(248\) −0.265419 −0.0168541
\(249\) 0.370404 0.0234734
\(250\) −11.9348 −0.754825
\(251\) 23.0399 1.45426 0.727132 0.686497i \(-0.240853\pi\)
0.727132 + 0.686497i \(0.240853\pi\)
\(252\) 2.99913 0.188927
\(253\) −20.0491 −1.26048
\(254\) 17.1303 1.07485
\(255\) −0.279100 −0.0174779
\(256\) 1.00000 0.0625000
\(257\) −19.0390 −1.18762 −0.593811 0.804604i \(-0.702377\pi\)
−0.593811 + 0.804604i \(0.702377\pi\)
\(258\) 0.0219677 0.00136765
\(259\) −2.79771 −0.173842
\(260\) −2.40198 −0.148965
\(261\) 1.18900 0.0735970
\(262\) −6.67961 −0.412668
\(263\) 15.3994 0.949569 0.474785 0.880102i \(-0.342526\pi\)
0.474785 + 0.880102i \(0.342526\pi\)
\(264\) 0.0869639 0.00535226
\(265\) 9.33895 0.573687
\(266\) 0 0
\(267\) −0.493703 −0.0302141
\(268\) −11.1770 −0.682743
\(269\) 15.8634 0.967207 0.483603 0.875287i \(-0.339328\pi\)
0.483603 + 0.875287i \(0.339328\pi\)
\(270\) −0.359697 −0.0218905
\(271\) −23.1622 −1.40701 −0.703503 0.710692i \(-0.748382\pi\)
−0.703503 + 0.710692i \(0.748382\pi\)
\(272\) 4.65490 0.282245
\(273\) −0.0349521 −0.00211540
\(274\) −7.13250 −0.430890
\(275\) 2.58953 0.156154
\(276\) 0.201145 0.0121075
\(277\) 0.153840 0.00924332 0.00462166 0.999989i \(-0.498529\pi\)
0.00462166 + 0.999989i \(0.498529\pi\)
\(278\) −17.3500 −1.04058
\(279\) −0.796026 −0.0476569
\(280\) −2.02989 −0.121309
\(281\) −12.5517 −0.748770 −0.374385 0.927273i \(-0.622146\pi\)
−0.374385 + 0.927273i \(0.622146\pi\)
\(282\) 0.0878419 0.00523091
\(283\) −0.132478 −0.00787500 −0.00393750 0.999992i \(-0.501253\pi\)
−0.00393750 + 0.999992i \(0.501253\pi\)
\(284\) 13.9312 0.826663
\(285\) 0 0
\(286\) 3.48386 0.206005
\(287\) −5.15394 −0.304228
\(288\) 2.99913 0.176725
\(289\) 4.66813 0.274596
\(290\) −0.804744 −0.0472562
\(291\) −0.0625721 −0.00366804
\(292\) 4.21053 0.246403
\(293\) 31.7197 1.85308 0.926542 0.376191i \(-0.122766\pi\)
0.926542 + 0.376191i \(0.122766\pi\)
\(294\) −0.0295377 −0.00172267
\(295\) −14.4519 −0.841421
\(296\) −2.79771 −0.162614
\(297\) 0.521708 0.0302725
\(298\) −10.6622 −0.617644
\(299\) 8.05804 0.466009
\(300\) −0.0259797 −0.00149994
\(301\) 0.743718 0.0428672
\(302\) −4.50841 −0.259430
\(303\) −0.394677 −0.0226736
\(304\) 0 0
\(305\) −18.3411 −1.05021
\(306\) 13.9607 0.798077
\(307\) −6.49152 −0.370491 −0.185245 0.982692i \(-0.559308\pi\)
−0.185245 + 0.982692i \(0.559308\pi\)
\(308\) 2.94417 0.167760
\(309\) −0.393790 −0.0224020
\(310\) 0.538772 0.0306002
\(311\) 15.9813 0.906218 0.453109 0.891455i \(-0.350315\pi\)
0.453109 + 0.891455i \(0.350315\pi\)
\(312\) −0.0349521 −0.00197878
\(313\) 2.85203 0.161207 0.0806033 0.996746i \(-0.474315\pi\)
0.0806033 + 0.996746i \(0.474315\pi\)
\(314\) 1.30678 0.0737460
\(315\) −6.08790 −0.343014
\(316\) −2.21703 −0.124718
\(317\) 16.9449 0.951720 0.475860 0.879521i \(-0.342137\pi\)
0.475860 + 0.879521i \(0.342137\pi\)
\(318\) 0.135894 0.00762058
\(319\) 1.16721 0.0653510
\(320\) −2.02989 −0.113474
\(321\) −0.279018 −0.0155732
\(322\) 6.80976 0.379493
\(323\) 0 0
\(324\) 8.99215 0.499564
\(325\) −1.04077 −0.0577316
\(326\) 12.8874 0.713767
\(327\) −0.285869 −0.0158086
\(328\) −5.15394 −0.284579
\(329\) 2.97389 0.163956
\(330\) −0.176527 −0.00971750
\(331\) −21.8016 −1.19832 −0.599162 0.800628i \(-0.704499\pi\)
−0.599162 + 0.800628i \(0.704499\pi\)
\(332\) 12.5400 0.688224
\(333\) −8.39070 −0.459808
\(334\) 17.2129 0.941849
\(335\) 22.6880 1.23958
\(336\) −0.0295377 −0.00161141
\(337\) 24.5605 1.33790 0.668949 0.743309i \(-0.266745\pi\)
0.668949 + 0.743309i \(0.266745\pi\)
\(338\) 11.5998 0.630945
\(339\) −0.258939 −0.0140637
\(340\) −9.44895 −0.512441
\(341\) −0.781439 −0.0423173
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0.743718 0.0400986
\(345\) −0.408302 −0.0219822
\(346\) 12.2229 0.657108
\(347\) −22.6149 −1.21403 −0.607015 0.794690i \(-0.707633\pi\)
−0.607015 + 0.794690i \(0.707633\pi\)
\(348\) −0.0117101 −0.000627729 0
\(349\) −28.8834 −1.54609 −0.773046 0.634350i \(-0.781268\pi\)
−0.773046 + 0.634350i \(0.781268\pi\)
\(350\) −0.879544 −0.0470136
\(351\) −0.209682 −0.0111920
\(352\) 2.94417 0.156925
\(353\) −21.9422 −1.16787 −0.583933 0.811802i \(-0.698487\pi\)
−0.583933 + 0.811802i \(0.698487\pi\)
\(354\) −0.210294 −0.0111770
\(355\) −28.2788 −1.50088
\(356\) −16.7143 −0.885859
\(357\) −0.137495 −0.00727701
\(358\) −10.7092 −0.566000
\(359\) −14.3290 −0.756257 −0.378128 0.925753i \(-0.623432\pi\)
−0.378128 + 0.925753i \(0.623432\pi\)
\(360\) −6.08790 −0.320861
\(361\) 0 0
\(362\) 21.8663 1.14927
\(363\) −0.0688780 −0.00361516
\(364\) −1.18331 −0.0620221
\(365\) −8.54692 −0.447366
\(366\) −0.266888 −0.0139505
\(367\) −4.90373 −0.255973 −0.127986 0.991776i \(-0.540851\pi\)
−0.127986 + 0.991776i \(0.540851\pi\)
\(368\) 6.80976 0.354984
\(369\) −15.4573 −0.804677
\(370\) 5.67905 0.295240
\(371\) 4.60071 0.238857
\(372\) 0.00783987 0.000406478 0
\(373\) −27.4511 −1.42137 −0.710683 0.703512i \(-0.751614\pi\)
−0.710683 + 0.703512i \(0.751614\pi\)
\(374\) 13.7048 0.708660
\(375\) 0.352527 0.0182044
\(376\) 2.97389 0.153367
\(377\) −0.469119 −0.0241608
\(378\) −0.177200 −0.00911420
\(379\) −3.03127 −0.155706 −0.0778528 0.996965i \(-0.524806\pi\)
−0.0778528 + 0.996965i \(0.524806\pi\)
\(380\) 0 0
\(381\) −0.505989 −0.0259226
\(382\) 19.1654 0.980588
\(383\) 1.79284 0.0916098 0.0458049 0.998950i \(-0.485415\pi\)
0.0458049 + 0.998950i \(0.485415\pi\)
\(384\) −0.0295377 −0.00150734
\(385\) −5.97634 −0.304583
\(386\) −12.9081 −0.657005
\(387\) 2.23051 0.113383
\(388\) −2.11838 −0.107545
\(389\) −30.3079 −1.53667 −0.768335 0.640048i \(-0.778915\pi\)
−0.768335 + 0.640048i \(0.778915\pi\)
\(390\) 0.0709490 0.00359264
\(391\) 31.6988 1.60308
\(392\) −1.00000 −0.0505076
\(393\) 0.197300 0.00995247
\(394\) −23.6787 −1.19292
\(395\) 4.50033 0.226436
\(396\) 8.82994 0.443721
\(397\) 39.1611 1.96544 0.982719 0.185102i \(-0.0592614\pi\)
0.982719 + 0.185102i \(0.0592614\pi\)
\(398\) 6.56540 0.329094
\(399\) 0 0
\(400\) −0.879544 −0.0439772
\(401\) 0.123230 0.00615384 0.00307692 0.999995i \(-0.499021\pi\)
0.00307692 + 0.999995i \(0.499021\pi\)
\(402\) 0.330142 0.0164660
\(403\) 0.314073 0.0156451
\(404\) −13.3618 −0.664775
\(405\) −18.2531 −0.907003
\(406\) −0.396447 −0.0196753
\(407\) −8.23694 −0.408290
\(408\) −0.137495 −0.00680702
\(409\) −29.4542 −1.45642 −0.728208 0.685356i \(-0.759647\pi\)
−0.728208 + 0.685356i \(0.759647\pi\)
\(410\) 10.4619 0.516678
\(411\) 0.210678 0.0103920
\(412\) −13.3318 −0.656811
\(413\) −7.11953 −0.350329
\(414\) 20.4234 1.00375
\(415\) −25.4549 −1.24953
\(416\) −1.18331 −0.0580164
\(417\) 0.512478 0.0250962
\(418\) 0 0
\(419\) 16.1094 0.786996 0.393498 0.919325i \(-0.371265\pi\)
0.393498 + 0.919325i \(0.371265\pi\)
\(420\) 0.0599583 0.00292566
\(421\) −28.9033 −1.40866 −0.704329 0.709873i \(-0.748752\pi\)
−0.704329 + 0.709873i \(0.748752\pi\)
\(422\) −11.9730 −0.582837
\(423\) 8.91908 0.433661
\(424\) 4.60071 0.223430
\(425\) −4.09419 −0.198598
\(426\) −0.411495 −0.0199370
\(427\) −9.03552 −0.437260
\(428\) −9.44616 −0.456597
\(429\) −0.102905 −0.00496830
\(430\) −1.50967 −0.0728026
\(431\) 39.7770 1.91599 0.957996 0.286781i \(-0.0925852\pi\)
0.957996 + 0.286781i \(0.0925852\pi\)
\(432\) −0.177200 −0.00852555
\(433\) −26.9114 −1.29328 −0.646641 0.762795i \(-0.723827\pi\)
−0.646641 + 0.762795i \(0.723827\pi\)
\(434\) 0.265419 0.0127405
\(435\) 0.0237703 0.00113970
\(436\) −9.67812 −0.463498
\(437\) 0 0
\(438\) −0.124369 −0.00594260
\(439\) 0.705775 0.0336848 0.0168424 0.999858i \(-0.494639\pi\)
0.0168424 + 0.999858i \(0.494639\pi\)
\(440\) −5.97634 −0.284911
\(441\) −2.99913 −0.142816
\(442\) −5.50818 −0.261997
\(443\) 15.2248 0.723350 0.361675 0.932304i \(-0.382205\pi\)
0.361675 + 0.932304i \(0.382205\pi\)
\(444\) 0.0826380 0.00392183
\(445\) 33.9283 1.60836
\(446\) 8.47530 0.401317
\(447\) 0.314936 0.0148960
\(448\) −1.00000 −0.0472456
\(449\) 16.0822 0.758968 0.379484 0.925198i \(-0.376102\pi\)
0.379484 + 0.925198i \(0.376102\pi\)
\(450\) −2.63787 −0.124350
\(451\) −15.1741 −0.714520
\(452\) −8.76641 −0.412337
\(453\) 0.133168 0.00625678
\(454\) 2.19728 0.103124
\(455\) 2.40198 0.112607
\(456\) 0 0
\(457\) −1.39979 −0.0654794 −0.0327397 0.999464i \(-0.510423\pi\)
−0.0327397 + 0.999464i \(0.510423\pi\)
\(458\) 14.5591 0.680300
\(459\) −0.824850 −0.0385007
\(460\) −13.8231 −0.644504
\(461\) 15.4887 0.721380 0.360690 0.932686i \(-0.382541\pi\)
0.360690 + 0.932686i \(0.382541\pi\)
\(462\) −0.0869639 −0.00404593
\(463\) −1.80796 −0.0840229 −0.0420115 0.999117i \(-0.513377\pi\)
−0.0420115 + 0.999117i \(0.513377\pi\)
\(464\) −0.396447 −0.0184046
\(465\) −0.0159141 −0.000737998 0
\(466\) 17.7732 0.823326
\(467\) 11.8566 0.548659 0.274330 0.961636i \(-0.411544\pi\)
0.274330 + 0.961636i \(0.411544\pi\)
\(468\) −3.54889 −0.164047
\(469\) 11.1770 0.516105
\(470\) −6.03668 −0.278451
\(471\) −0.0385993 −0.00177856
\(472\) −7.11953 −0.327703
\(473\) 2.18963 0.100679
\(474\) 0.0654859 0.00300787
\(475\) 0 0
\(476\) −4.65490 −0.213357
\(477\) 13.7981 0.631773
\(478\) 18.9372 0.866167
\(479\) 2.75396 0.125831 0.0629157 0.998019i \(-0.479960\pi\)
0.0629157 + 0.998019i \(0.479960\pi\)
\(480\) 0.0599583 0.00273671
\(481\) 3.31056 0.150948
\(482\) −19.6449 −0.894802
\(483\) −0.201145 −0.00915240
\(484\) −2.33187 −0.105994
\(485\) 4.30008 0.195257
\(486\) −0.797208 −0.0361621
\(487\) −11.6214 −0.526615 −0.263307 0.964712i \(-0.584813\pi\)
−0.263307 + 0.964712i \(0.584813\pi\)
\(488\) −9.03552 −0.409019
\(489\) −0.380664 −0.0172142
\(490\) 2.02989 0.0917011
\(491\) −20.5701 −0.928317 −0.464158 0.885752i \(-0.653643\pi\)
−0.464158 + 0.885752i \(0.653643\pi\)
\(492\) 0.152236 0.00686331
\(493\) −1.84542 −0.0831137
\(494\) 0 0
\(495\) −17.9238 −0.805615
\(496\) 0.265419 0.0119177
\(497\) −13.9312 −0.624899
\(498\) −0.370404 −0.0165982
\(499\) −13.7051 −0.613522 −0.306761 0.951787i \(-0.599245\pi\)
−0.306761 + 0.951787i \(0.599245\pi\)
\(500\) 11.9348 0.533742
\(501\) −0.508429 −0.0227150
\(502\) −23.0399 −1.02832
\(503\) −3.42219 −0.152588 −0.0762939 0.997085i \(-0.524309\pi\)
−0.0762939 + 0.997085i \(0.524309\pi\)
\(504\) −2.99913 −0.133592
\(505\) 27.1230 1.20696
\(506\) 20.0491 0.891291
\(507\) −0.342631 −0.0152168
\(508\) −17.1303 −0.760033
\(509\) −29.8042 −1.32105 −0.660524 0.750805i \(-0.729666\pi\)
−0.660524 + 0.750805i \(0.729666\pi\)
\(510\) 0.279100 0.0123588
\(511\) −4.21053 −0.186263
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 19.0390 0.839776
\(515\) 27.0621 1.19250
\(516\) −0.0219677 −0.000967074 0
\(517\) 8.75564 0.385073
\(518\) 2.79771 0.122925
\(519\) −0.361037 −0.0158477
\(520\) 2.40198 0.105334
\(521\) −19.6327 −0.860125 −0.430062 0.902799i \(-0.641508\pi\)
−0.430062 + 0.902799i \(0.641508\pi\)
\(522\) −1.18900 −0.0520409
\(523\) −25.6599 −1.12203 −0.561016 0.827805i \(-0.689589\pi\)
−0.561016 + 0.827805i \(0.689589\pi\)
\(524\) 6.67961 0.291800
\(525\) 0.0259797 0.00113385
\(526\) −15.3994 −0.671447
\(527\) 1.23550 0.0538193
\(528\) −0.0869639 −0.00378462
\(529\) 23.3729 1.01621
\(530\) −9.33895 −0.405658
\(531\) −21.3524 −0.926615
\(532\) 0 0
\(533\) 6.09870 0.264164
\(534\) 0.493703 0.0213646
\(535\) 19.1747 0.828993
\(536\) 11.1770 0.482772
\(537\) 0.316326 0.0136505
\(538\) −15.8634 −0.683919
\(539\) −2.94417 −0.126814
\(540\) 0.359697 0.0154789
\(541\) −8.99925 −0.386908 −0.193454 0.981109i \(-0.561969\pi\)
−0.193454 + 0.981109i \(0.561969\pi\)
\(542\) 23.1622 0.994903
\(543\) −0.645879 −0.0277173
\(544\) −4.65490 −0.199577
\(545\) 19.6455 0.841522
\(546\) 0.0349521 0.00149581
\(547\) 21.8354 0.933616 0.466808 0.884359i \(-0.345404\pi\)
0.466808 + 0.884359i \(0.345404\pi\)
\(548\) 7.13250 0.304685
\(549\) −27.0987 −1.15654
\(550\) −2.58953 −0.110418
\(551\) 0 0
\(552\) −0.201145 −0.00856128
\(553\) 2.21703 0.0942777
\(554\) −0.153840 −0.00653602
\(555\) −0.167746 −0.00712043
\(556\) 17.3500 0.735803
\(557\) −14.1210 −0.598324 −0.299162 0.954202i \(-0.596707\pi\)
−0.299162 + 0.954202i \(0.596707\pi\)
\(558\) 0.796026 0.0336985
\(559\) −0.880047 −0.0372220
\(560\) 2.02989 0.0857785
\(561\) −0.404809 −0.0170910
\(562\) 12.5517 0.529461
\(563\) −25.8400 −1.08903 −0.544513 0.838752i \(-0.683286\pi\)
−0.544513 + 0.838752i \(0.683286\pi\)
\(564\) −0.0878419 −0.00369881
\(565\) 17.7949 0.748635
\(566\) 0.132478 0.00556847
\(567\) −8.99215 −0.377635
\(568\) −13.9312 −0.584539
\(569\) 27.5260 1.15395 0.576976 0.816761i \(-0.304233\pi\)
0.576976 + 0.816761i \(0.304233\pi\)
\(570\) 0 0
\(571\) 21.9105 0.916928 0.458464 0.888713i \(-0.348400\pi\)
0.458464 + 0.888713i \(0.348400\pi\)
\(572\) −3.48386 −0.145667
\(573\) −0.566102 −0.0236492
\(574\) 5.15394 0.215121
\(575\) −5.98949 −0.249779
\(576\) −2.99913 −0.124964
\(577\) −46.9112 −1.95294 −0.976469 0.215659i \(-0.930810\pi\)
−0.976469 + 0.215659i \(0.930810\pi\)
\(578\) −4.66813 −0.194169
\(579\) 0.381275 0.0158452
\(580\) 0.804744 0.0334152
\(581\) −12.5400 −0.520249
\(582\) 0.0625721 0.00259370
\(583\) 13.5453 0.560988
\(584\) −4.21053 −0.174233
\(585\) 7.20386 0.297843
\(586\) −31.7197 −1.31033
\(587\) 20.2603 0.836234 0.418117 0.908393i \(-0.362690\pi\)
0.418117 + 0.908393i \(0.362690\pi\)
\(588\) 0.0295377 0.00121811
\(589\) 0 0
\(590\) 14.4519 0.594974
\(591\) 0.699415 0.0287701
\(592\) 2.79771 0.114985
\(593\) −30.8939 −1.26866 −0.634330 0.773062i \(-0.718724\pi\)
−0.634330 + 0.773062i \(0.718724\pi\)
\(594\) −0.521708 −0.0214059
\(595\) 9.44895 0.387369
\(596\) 10.6622 0.436740
\(597\) −0.193927 −0.00793689
\(598\) −8.05804 −0.329518
\(599\) 30.1902 1.23354 0.616769 0.787144i \(-0.288441\pi\)
0.616769 + 0.787144i \(0.288441\pi\)
\(600\) 0.0259797 0.00106062
\(601\) −42.7945 −1.74562 −0.872812 0.488057i \(-0.837706\pi\)
−0.872812 + 0.488057i \(0.837706\pi\)
\(602\) −0.743718 −0.0303117
\(603\) 33.5212 1.36509
\(604\) 4.50841 0.183445
\(605\) 4.73344 0.192442
\(606\) 0.394677 0.0160327
\(607\) −25.8352 −1.04862 −0.524309 0.851528i \(-0.675676\pi\)
−0.524309 + 0.851528i \(0.675676\pi\)
\(608\) 0 0
\(609\) 0.0117101 0.000474518 0
\(610\) 18.3411 0.742610
\(611\) −3.51903 −0.142365
\(612\) −13.9607 −0.564326
\(613\) 8.26177 0.333690 0.166845 0.985983i \(-0.446642\pi\)
0.166845 + 0.985983i \(0.446642\pi\)
\(614\) 6.49152 0.261977
\(615\) −0.309021 −0.0124609
\(616\) −2.94417 −0.118624
\(617\) 14.5351 0.585163 0.292581 0.956241i \(-0.405486\pi\)
0.292581 + 0.956241i \(0.405486\pi\)
\(618\) 0.393790 0.0158406
\(619\) −42.9334 −1.72564 −0.862819 0.505513i \(-0.831303\pi\)
−0.862819 + 0.505513i \(0.831303\pi\)
\(620\) −0.538772 −0.0216376
\(621\) −1.20669 −0.0484229
\(622\) −15.9813 −0.640793
\(623\) 16.7143 0.669646
\(624\) 0.0349521 0.00139921
\(625\) −19.8287 −0.793147
\(626\) −2.85203 −0.113990
\(627\) 0 0
\(628\) −1.30678 −0.0521463
\(629\) 13.0231 0.519265
\(630\) 6.08790 0.242548
\(631\) 11.4310 0.455062 0.227531 0.973771i \(-0.426935\pi\)
0.227531 + 0.973771i \(0.426935\pi\)
\(632\) 2.21703 0.0881887
\(633\) 0.353655 0.0140565
\(634\) −16.9449 −0.672968
\(635\) 34.7726 1.37991
\(636\) −0.135894 −0.00538856
\(637\) 1.18331 0.0468843
\(638\) −1.16721 −0.0462102
\(639\) −41.7814 −1.65285
\(640\) 2.02989 0.0802385
\(641\) 15.4233 0.609184 0.304592 0.952483i \(-0.401480\pi\)
0.304592 + 0.952483i \(0.401480\pi\)
\(642\) 0.279018 0.0110119
\(643\) 25.3895 1.00126 0.500632 0.865660i \(-0.333101\pi\)
0.500632 + 0.865660i \(0.333101\pi\)
\(644\) −6.80976 −0.268342
\(645\) 0.0445920 0.00175581
\(646\) 0 0
\(647\) 18.4239 0.724318 0.362159 0.932116i \(-0.382040\pi\)
0.362159 + 0.932116i \(0.382040\pi\)
\(648\) −8.99215 −0.353245
\(649\) −20.9611 −0.822796
\(650\) 1.04077 0.0408224
\(651\) −0.00783987 −0.000307269 0
\(652\) −12.8874 −0.504710
\(653\) 25.5335 0.999205 0.499602 0.866255i \(-0.333479\pi\)
0.499602 + 0.866255i \(0.333479\pi\)
\(654\) 0.285869 0.0111784
\(655\) −13.5589 −0.529789
\(656\) 5.15394 0.201228
\(657\) −12.6279 −0.492662
\(658\) −2.97389 −0.115934
\(659\) 4.27377 0.166482 0.0832412 0.996529i \(-0.473473\pi\)
0.0832412 + 0.996529i \(0.473473\pi\)
\(660\) 0.176527 0.00687131
\(661\) −18.0211 −0.700941 −0.350470 0.936574i \(-0.613978\pi\)
−0.350470 + 0.936574i \(0.613978\pi\)
\(662\) 21.8016 0.847343
\(663\) 0.162699 0.00631870
\(664\) −12.5400 −0.486648
\(665\) 0 0
\(666\) 8.39070 0.325133
\(667\) −2.69971 −0.104533
\(668\) −17.2129 −0.665988
\(669\) −0.250341 −0.00967873
\(670\) −22.6880 −0.876516
\(671\) −26.6021 −1.02696
\(672\) 0.0295377 0.00113944
\(673\) 12.0784 0.465587 0.232793 0.972526i \(-0.425213\pi\)
0.232793 + 0.972526i \(0.425213\pi\)
\(674\) −24.5605 −0.946036
\(675\) 0.155856 0.00599888
\(676\) −11.5998 −0.446146
\(677\) 18.5172 0.711674 0.355837 0.934548i \(-0.384196\pi\)
0.355837 + 0.934548i \(0.384196\pi\)
\(678\) 0.258939 0.00994451
\(679\) 2.11838 0.0812960
\(680\) 9.44895 0.362351
\(681\) −0.0649026 −0.00248707
\(682\) 0.781439 0.0299229
\(683\) −40.1450 −1.53611 −0.768053 0.640386i \(-0.778774\pi\)
−0.768053 + 0.640386i \(0.778774\pi\)
\(684\) 0 0
\(685\) −14.4782 −0.553184
\(686\) 1.00000 0.0381802
\(687\) −0.430041 −0.0164071
\(688\) −0.743718 −0.0283540
\(689\) −5.44406 −0.207402
\(690\) 0.408302 0.0155438
\(691\) −37.9320 −1.44300 −0.721501 0.692414i \(-0.756547\pi\)
−0.721501 + 0.692414i \(0.756547\pi\)
\(692\) −12.2229 −0.464646
\(693\) −8.82994 −0.335422
\(694\) 22.6149 0.858449
\(695\) −35.2186 −1.33592
\(696\) 0.0117101 0.000443871 0
\(697\) 23.9911 0.908728
\(698\) 28.8834 1.09325
\(699\) −0.524978 −0.0198565
\(700\) 0.879544 0.0332436
\(701\) −9.19991 −0.347476 −0.173738 0.984792i \(-0.555585\pi\)
−0.173738 + 0.984792i \(0.555585\pi\)
\(702\) 0.209682 0.00791395
\(703\) 0 0
\(704\) −2.94417 −0.110963
\(705\) 0.178309 0.00671552
\(706\) 21.9422 0.825806
\(707\) 13.3618 0.502523
\(708\) 0.210294 0.00790335
\(709\) 40.5184 1.52170 0.760850 0.648928i \(-0.224782\pi\)
0.760850 + 0.648928i \(0.224782\pi\)
\(710\) 28.2788 1.06128
\(711\) 6.64915 0.249363
\(712\) 16.7143 0.626397
\(713\) 1.80744 0.0676893
\(714\) 0.137495 0.00514562
\(715\) 7.07185 0.264472
\(716\) 10.7092 0.400222
\(717\) −0.559361 −0.0208897
\(718\) 14.3290 0.534754
\(719\) −47.7082 −1.77921 −0.889607 0.456727i \(-0.849022\pi\)
−0.889607 + 0.456727i \(0.849022\pi\)
\(720\) 6.08790 0.226883
\(721\) 13.3318 0.496502
\(722\) 0 0
\(723\) 0.580265 0.0215803
\(724\) −21.8663 −0.812653
\(725\) 0.348693 0.0129501
\(726\) 0.0688780 0.00255630
\(727\) 1.93893 0.0719108 0.0359554 0.999353i \(-0.488553\pi\)
0.0359554 + 0.999353i \(0.488553\pi\)
\(728\) 1.18331 0.0438563
\(729\) −26.9529 −0.998255
\(730\) 8.54692 0.316336
\(731\) −3.46194 −0.128044
\(732\) 0.266888 0.00986448
\(733\) 43.0842 1.59135 0.795676 0.605723i \(-0.207116\pi\)
0.795676 + 0.605723i \(0.207116\pi\)
\(734\) 4.90373 0.181000
\(735\) −0.0599583 −0.00221159
\(736\) −6.80976 −0.251011
\(737\) 32.9069 1.21214
\(738\) 15.4573 0.568992
\(739\) 23.6580 0.870273 0.435137 0.900364i \(-0.356700\pi\)
0.435137 + 0.900364i \(0.356700\pi\)
\(740\) −5.67905 −0.208766
\(741\) 0 0
\(742\) −4.60071 −0.168898
\(743\) −25.8480 −0.948270 −0.474135 0.880452i \(-0.657239\pi\)
−0.474135 + 0.880452i \(0.657239\pi\)
\(744\) −0.00783987 −0.000287424 0
\(745\) −21.6431 −0.792941
\(746\) 27.4511 1.00506
\(747\) −37.6092 −1.37605
\(748\) −13.7048 −0.501098
\(749\) 9.44616 0.345155
\(750\) −0.352527 −0.0128725
\(751\) 23.4711 0.856472 0.428236 0.903667i \(-0.359135\pi\)
0.428236 + 0.903667i \(0.359135\pi\)
\(752\) −2.97389 −0.108447
\(753\) 0.680545 0.0248004
\(754\) 0.469119 0.0170843
\(755\) −9.15158 −0.333060
\(756\) 0.177200 0.00644471
\(757\) −21.0205 −0.764005 −0.382002 0.924161i \(-0.624765\pi\)
−0.382002 + 0.924161i \(0.624765\pi\)
\(758\) 3.03127 0.110101
\(759\) −0.592204 −0.0214956
\(760\) 0 0
\(761\) 34.0214 1.23327 0.616637 0.787248i \(-0.288495\pi\)
0.616637 + 0.787248i \(0.288495\pi\)
\(762\) 0.505989 0.0183300
\(763\) 9.67812 0.350371
\(764\) −19.1654 −0.693380
\(765\) 28.3386 1.02458
\(766\) −1.79284 −0.0647779
\(767\) 8.42460 0.304194
\(768\) 0.0295377 0.00106585
\(769\) 37.4225 1.34949 0.674745 0.738051i \(-0.264254\pi\)
0.674745 + 0.738051i \(0.264254\pi\)
\(770\) 5.97634 0.215372
\(771\) −0.562369 −0.0202532
\(772\) 12.9081 0.464572
\(773\) 20.7862 0.747627 0.373814 0.927504i \(-0.378050\pi\)
0.373814 + 0.927504i \(0.378050\pi\)
\(774\) −2.23051 −0.0801739
\(775\) −0.233448 −0.00838570
\(776\) 2.11838 0.0760455
\(777\) −0.0826380 −0.00296462
\(778\) 30.3079 1.08659
\(779\) 0 0
\(780\) −0.0709490 −0.00254038
\(781\) −41.0157 −1.46766
\(782\) −31.6988 −1.13355
\(783\) 0.0702505 0.00251055
\(784\) 1.00000 0.0357143
\(785\) 2.65263 0.0946763
\(786\) −0.197300 −0.00703746
\(787\) 33.4364 1.19188 0.595939 0.803029i \(-0.296780\pi\)
0.595939 + 0.803029i \(0.296780\pi\)
\(788\) 23.6787 0.843520
\(789\) 0.454863 0.0161936
\(790\) −4.50033 −0.160114
\(791\) 8.76641 0.311698
\(792\) −8.82994 −0.313758
\(793\) 10.6918 0.379677
\(794\) −39.1611 −1.38978
\(795\) 0.275851 0.00978342
\(796\) −6.56540 −0.232705
\(797\) −26.0945 −0.924316 −0.462158 0.886798i \(-0.652925\pi\)
−0.462158 + 0.886798i \(0.652925\pi\)
\(798\) 0 0
\(799\) −13.8432 −0.489737
\(800\) 0.879544 0.0310966
\(801\) 50.1285 1.77120
\(802\) −0.123230 −0.00435142
\(803\) −12.3965 −0.437464
\(804\) −0.330142 −0.0116432
\(805\) 13.8231 0.487199
\(806\) −0.314073 −0.0110627
\(807\) 0.468567 0.0164943
\(808\) 13.3618 0.470067
\(809\) −40.7851 −1.43393 −0.716964 0.697110i \(-0.754469\pi\)
−0.716964 + 0.697110i \(0.754469\pi\)
\(810\) 18.2531 0.641348
\(811\) −14.5813 −0.512017 −0.256009 0.966674i \(-0.582408\pi\)
−0.256009 + 0.966674i \(0.582408\pi\)
\(812\) 0.396447 0.0139126
\(813\) −0.684159 −0.0239945
\(814\) 8.23694 0.288705
\(815\) 26.1600 0.916346
\(816\) 0.137495 0.00481329
\(817\) 0 0
\(818\) 29.4542 1.02984
\(819\) 3.54889 0.124008
\(820\) −10.4619 −0.365347
\(821\) −20.1502 −0.703246 −0.351623 0.936142i \(-0.614370\pi\)
−0.351623 + 0.936142i \(0.614370\pi\)
\(822\) −0.210678 −0.00734822
\(823\) 49.4504 1.72373 0.861866 0.507136i \(-0.169296\pi\)
0.861866 + 0.507136i \(0.169296\pi\)
\(824\) 13.3318 0.464435
\(825\) 0.0764886 0.00266299
\(826\) 7.11953 0.247720
\(827\) −35.7637 −1.24363 −0.621813 0.783166i \(-0.713603\pi\)
−0.621813 + 0.783166i \(0.713603\pi\)
\(828\) −20.4234 −0.709761
\(829\) −26.6486 −0.925544 −0.462772 0.886477i \(-0.653145\pi\)
−0.462772 + 0.886477i \(0.653145\pi\)
\(830\) 25.4549 0.883553
\(831\) 0.00454406 0.000157632 0
\(832\) 1.18331 0.0410238
\(833\) 4.65490 0.161283
\(834\) −0.512478 −0.0177457
\(835\) 34.9403 1.20916
\(836\) 0 0
\(837\) −0.0470324 −0.00162568
\(838\) −16.1094 −0.556491
\(839\) 13.0277 0.449766 0.224883 0.974386i \(-0.427800\pi\)
0.224883 + 0.974386i \(0.427800\pi\)
\(840\) −0.0599583 −0.00206876
\(841\) −28.8428 −0.994580
\(842\) 28.9033 0.996072
\(843\) −0.370747 −0.0127692
\(844\) 11.9730 0.412128
\(845\) 23.5463 0.810017
\(846\) −8.91908 −0.306644
\(847\) 2.33187 0.0801240
\(848\) −4.60071 −0.157989
\(849\) −0.00391309 −0.000134297 0
\(850\) 4.09419 0.140430
\(851\) 19.0518 0.653087
\(852\) 0.411495 0.0140976
\(853\) 25.8812 0.886156 0.443078 0.896483i \(-0.353886\pi\)
0.443078 + 0.896483i \(0.353886\pi\)
\(854\) 9.03552 0.309189
\(855\) 0 0
\(856\) 9.44616 0.322863
\(857\) −30.2522 −1.03340 −0.516698 0.856168i \(-0.672839\pi\)
−0.516698 + 0.856168i \(0.672839\pi\)
\(858\) 0.102905 0.00351312
\(859\) −7.90397 −0.269680 −0.134840 0.990867i \(-0.543052\pi\)
−0.134840 + 0.990867i \(0.543052\pi\)
\(860\) 1.50967 0.0514792
\(861\) −0.152236 −0.00518817
\(862\) −39.7770 −1.35481
\(863\) 52.7257 1.79480 0.897402 0.441214i \(-0.145452\pi\)
0.897402 + 0.441214i \(0.145452\pi\)
\(864\) 0.177200 0.00602848
\(865\) 24.8112 0.843606
\(866\) 26.9114 0.914488
\(867\) 0.137886 0.00468284
\(868\) −0.265419 −0.00900892
\(869\) 6.52731 0.221424
\(870\) −0.0237703 −0.000805888 0
\(871\) −13.2258 −0.448139
\(872\) 9.67812 0.327742
\(873\) 6.35330 0.215027
\(874\) 0 0
\(875\) −11.9348 −0.403471
\(876\) 0.124369 0.00420205
\(877\) −19.6747 −0.664369 −0.332184 0.943214i \(-0.607786\pi\)
−0.332184 + 0.943214i \(0.607786\pi\)
\(878\) −0.705775 −0.0238188
\(879\) 0.936926 0.0316017
\(880\) 5.97634 0.201462
\(881\) −45.4201 −1.53024 −0.765121 0.643887i \(-0.777321\pi\)
−0.765121 + 0.643887i \(0.777321\pi\)
\(882\) 2.99913 0.100986
\(883\) 23.3127 0.784534 0.392267 0.919851i \(-0.371691\pi\)
0.392267 + 0.919851i \(0.371691\pi\)
\(884\) 5.50818 0.185260
\(885\) −0.426875 −0.0143492
\(886\) −15.2248 −0.511486
\(887\) −54.4206 −1.82727 −0.913633 0.406541i \(-0.866735\pi\)
−0.913633 + 0.406541i \(0.866735\pi\)
\(888\) −0.0826380 −0.00277315
\(889\) 17.1303 0.574531
\(890\) −33.9283 −1.13728
\(891\) −26.4744 −0.886926
\(892\) −8.47530 −0.283774
\(893\) 0 0
\(894\) −0.314936 −0.0105330
\(895\) −21.7386 −0.726640
\(896\) 1.00000 0.0334077
\(897\) 0.238016 0.00794712
\(898\) −16.0822 −0.536671
\(899\) −0.105225 −0.00350944
\(900\) 2.63787 0.0879288
\(901\) −21.4159 −0.713467
\(902\) 15.1741 0.505242
\(903\) 0.0219677 0.000731040 0
\(904\) 8.76641 0.291566
\(905\) 44.3861 1.47544
\(906\) −0.133168 −0.00442421
\(907\) −40.9915 −1.36110 −0.680550 0.732702i \(-0.738259\pi\)
−0.680550 + 0.732702i \(0.738259\pi\)
\(908\) −2.19728 −0.0729193
\(909\) 40.0738 1.32916
\(910\) −2.40198 −0.0796250
\(911\) −25.5209 −0.845544 −0.422772 0.906236i \(-0.638943\pi\)
−0.422772 + 0.906236i \(0.638943\pi\)
\(912\) 0 0
\(913\) −36.9200 −1.22187
\(914\) 1.39979 0.0463009
\(915\) −0.541754 −0.0179098
\(916\) −14.5591 −0.481045
\(917\) −6.67961 −0.220580
\(918\) 0.824850 0.0272241
\(919\) −6.80969 −0.224631 −0.112315 0.993673i \(-0.535827\pi\)
−0.112315 + 0.993673i \(0.535827\pi\)
\(920\) 13.8231 0.455733
\(921\) −0.191745 −0.00631820
\(922\) −15.4887 −0.510093
\(923\) 16.4849 0.542606
\(924\) 0.0869639 0.00286090
\(925\) −2.46071 −0.0809078
\(926\) 1.80796 0.0594132
\(927\) 39.9838 1.31324
\(928\) 0.396447 0.0130140
\(929\) −11.7047 −0.384018 −0.192009 0.981393i \(-0.561500\pi\)
−0.192009 + 0.981393i \(0.561500\pi\)
\(930\) 0.0159141 0.000521843 0
\(931\) 0 0
\(932\) −17.7732 −0.582179
\(933\) 0.472051 0.0154543
\(934\) −11.8566 −0.387961
\(935\) 27.8193 0.909788
\(936\) 3.54889 0.115999
\(937\) −9.26878 −0.302798 −0.151399 0.988473i \(-0.548378\pi\)
−0.151399 + 0.988473i \(0.548378\pi\)
\(938\) −11.1770 −0.364941
\(939\) 0.0842425 0.00274915
\(940\) 6.03668 0.196895
\(941\) 14.0020 0.456451 0.228226 0.973608i \(-0.426708\pi\)
0.228226 + 0.973608i \(0.426708\pi\)
\(942\) 0.0385993 0.00125763
\(943\) 35.0971 1.14292
\(944\) 7.11953 0.231721
\(945\) −0.359697 −0.0117010
\(946\) −2.18963 −0.0711911
\(947\) −32.5246 −1.05691 −0.528453 0.848962i \(-0.677228\pi\)
−0.528453 + 0.848962i \(0.677228\pi\)
\(948\) −0.0654859 −0.00212688
\(949\) 4.98235 0.161734
\(950\) 0 0
\(951\) 0.500513 0.0162302
\(952\) 4.65490 0.150866
\(953\) −5.75300 −0.186358 −0.0931790 0.995649i \(-0.529703\pi\)
−0.0931790 + 0.995649i \(0.529703\pi\)
\(954\) −13.7981 −0.446731
\(955\) 38.9037 1.25889
\(956\) −18.9372 −0.612473
\(957\) 0.0344766 0.00111447
\(958\) −2.75396 −0.0889763
\(959\) −7.13250 −0.230321
\(960\) −0.0599583 −0.00193514
\(961\) −30.9296 −0.997728
\(962\) −3.31056 −0.106737
\(963\) 28.3302 0.912929
\(964\) 19.6449 0.632720
\(965\) −26.2020 −0.843473
\(966\) 0.201145 0.00647172
\(967\) 54.9793 1.76802 0.884008 0.467471i \(-0.154835\pi\)
0.884008 + 0.467471i \(0.154835\pi\)
\(968\) 2.33187 0.0749491
\(969\) 0 0
\(970\) −4.30008 −0.138067
\(971\) −33.7559 −1.08328 −0.541639 0.840611i \(-0.682196\pi\)
−0.541639 + 0.840611i \(0.682196\pi\)
\(972\) 0.797208 0.0255705
\(973\) −17.3500 −0.556215
\(974\) 11.6214 0.372373
\(975\) −0.0307420 −0.000984531 0
\(976\) 9.03552 0.289220
\(977\) −21.6915 −0.693973 −0.346987 0.937870i \(-0.612795\pi\)
−0.346987 + 0.937870i \(0.612795\pi\)
\(978\) 0.380664 0.0121723
\(979\) 49.2099 1.57275
\(980\) −2.02989 −0.0648425
\(981\) 29.0259 0.926726
\(982\) 20.5701 0.656419
\(983\) −8.62213 −0.275003 −0.137502 0.990502i \(-0.543907\pi\)
−0.137502 + 0.990502i \(0.543907\pi\)
\(984\) −0.152236 −0.00485309
\(985\) −48.0652 −1.53149
\(986\) 1.84542 0.0587702
\(987\) 0.0878419 0.00279604
\(988\) 0 0
\(989\) −5.06455 −0.161043
\(990\) 17.9238 0.569656
\(991\) −10.2995 −0.327176 −0.163588 0.986529i \(-0.552307\pi\)
−0.163588 + 0.986529i \(0.552307\pi\)
\(992\) −0.265419 −0.00842707
\(993\) −0.643969 −0.0204357
\(994\) 13.9312 0.441870
\(995\) 13.3270 0.422496
\(996\) 0.370404 0.0117367
\(997\) 22.2647 0.705129 0.352565 0.935787i \(-0.385310\pi\)
0.352565 + 0.935787i \(0.385310\pi\)
\(998\) 13.7051 0.433826
\(999\) −0.495756 −0.0156850
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 5054.2.a.bf.1.5 8
19.18 odd 2 5054.2.a.bi.1.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.bf.1.5 8 1.1 even 1 trivial
5054.2.a.bi.1.4 yes 8 19.18 odd 2