Properties

Label 731.2.a.e.1.19
Level $731$
Weight $2$
Character 731.1
Self dual yes
Analytic conductor $5.837$
Analytic rank $0$
Dimension $19$
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] = [731,2,Mod(1,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, 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("731.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 2 x^{18} - 30 x^{17} + 62 x^{16} + 365 x^{15} - 786 x^{14} - 2295 x^{13} + 5233 x^{12} + 7816 x^{11} - 19517 x^{10} - 13527 x^{9} + 40173 x^{8} + 8942 x^{7} - 41911 x^{6} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Root \(2.66743\) of defining polynomial
Character \(\chi\) \(=\) 731.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.66743 q^{2} -1.22120 q^{3} +5.11519 q^{4} -1.15779 q^{5} -3.25748 q^{6} +2.36958 q^{7} +8.30956 q^{8} -1.50866 q^{9} +O(q^{10})\) \(q+2.66743 q^{2} -1.22120 q^{3} +5.11519 q^{4} -1.15779 q^{5} -3.25748 q^{6} +2.36958 q^{7} +8.30956 q^{8} -1.50866 q^{9} -3.08834 q^{10} +6.51509 q^{11} -6.24669 q^{12} -3.92858 q^{13} +6.32068 q^{14} +1.41390 q^{15} +11.9348 q^{16} +1.00000 q^{17} -4.02426 q^{18} +0.308012 q^{19} -5.92234 q^{20} -2.89373 q^{21} +17.3786 q^{22} +1.28661 q^{23} -10.1477 q^{24} -3.65951 q^{25} -10.4792 q^{26} +5.50599 q^{27} +12.1208 q^{28} -2.29611 q^{29} +3.77148 q^{30} +5.07718 q^{31} +15.2162 q^{32} -7.95625 q^{33} +2.66743 q^{34} -2.74348 q^{35} -7.71711 q^{36} -5.52176 q^{37} +0.821602 q^{38} +4.79759 q^{39} -9.62076 q^{40} -0.358377 q^{41} -7.71884 q^{42} -1.00000 q^{43} +33.3260 q^{44} +1.74672 q^{45} +3.43196 q^{46} -7.52351 q^{47} -14.5748 q^{48} -1.38511 q^{49} -9.76150 q^{50} -1.22120 q^{51} -20.0954 q^{52} -0.0422801 q^{53} +14.6869 q^{54} -7.54313 q^{55} +19.6901 q^{56} -0.376146 q^{57} -6.12471 q^{58} -5.52410 q^{59} +7.23237 q^{60} -4.20526 q^{61} +13.5430 q^{62} -3.57489 q^{63} +16.7185 q^{64} +4.54848 q^{65} -21.2228 q^{66} +7.62209 q^{67} +5.11519 q^{68} -1.57122 q^{69} -7.31805 q^{70} +6.07143 q^{71} -12.5363 q^{72} -7.32893 q^{73} -14.7289 q^{74} +4.46901 q^{75} +1.57554 q^{76} +15.4380 q^{77} +12.7973 q^{78} -11.0101 q^{79} -13.8180 q^{80} -2.19794 q^{81} -0.955947 q^{82} -7.59813 q^{83} -14.8020 q^{84} -1.15779 q^{85} -2.66743 q^{86} +2.80401 q^{87} +54.1376 q^{88} -12.4245 q^{89} +4.65926 q^{90} -9.30907 q^{91} +6.58128 q^{92} -6.20027 q^{93} -20.0685 q^{94} -0.356615 q^{95} -18.5820 q^{96} -17.8513 q^{97} -3.69468 q^{98} -9.82908 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9} - 2 q^{10} + 4 q^{11} + 9 q^{12} + 14 q^{13} + 5 q^{14} - 7 q^{15} + 32 q^{16} + 19 q^{17} + 12 q^{18} + 12 q^{19} + 23 q^{20} + 16 q^{21} + 36 q^{22} - q^{23} - 13 q^{24} + 30 q^{25} - 21 q^{26} + 8 q^{27} + 5 q^{28} + 41 q^{29} - 26 q^{30} - 8 q^{31} - 20 q^{32} - 14 q^{33} + 2 q^{34} + 3 q^{35} - 5 q^{36} + 50 q^{37} - 29 q^{38} + 17 q^{39} - 15 q^{40} + 6 q^{41} - q^{42} - 19 q^{43} + 16 q^{44} + 24 q^{45} + 38 q^{46} - 21 q^{47} - 2 q^{48} + 46 q^{49} - 36 q^{50} + 5 q^{51} + 39 q^{52} - 9 q^{53} + 53 q^{54} + 10 q^{55} - 12 q^{56} - 5 q^{57} - 45 q^{58} - 4 q^{59} - 7 q^{60} + 68 q^{61} - 25 q^{62} + 61 q^{63} - 14 q^{64} + 22 q^{65} - 17 q^{66} + 26 q^{68} - 9 q^{69} - 37 q^{70} + 23 q^{71} - 4 q^{72} - q^{73} - 30 q^{74} - 25 q^{75} + 47 q^{76} - 19 q^{77} + 12 q^{78} + 16 q^{79} + 28 q^{80} - 21 q^{81} - 13 q^{82} - 32 q^{83} - 47 q^{84} + 11 q^{85} - 2 q^{86} - 8 q^{87} + 108 q^{88} + 11 q^{89} + 5 q^{90} + 52 q^{91} - 23 q^{92} - 23 q^{93} + 47 q^{94} - 25 q^{95} - 103 q^{96} + 36 q^{97} - 100 q^{98} - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66743 1.88616 0.943080 0.332567i \(-0.107915\pi\)
0.943080 + 0.332567i \(0.107915\pi\)
\(3\) −1.22120 −0.705062 −0.352531 0.935800i \(-0.614679\pi\)
−0.352531 + 0.935800i \(0.614679\pi\)
\(4\) 5.11519 2.55760
\(5\) −1.15779 −0.517781 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(6\) −3.25748 −1.32986
\(7\) 2.36958 0.895616 0.447808 0.894130i \(-0.352205\pi\)
0.447808 + 0.894130i \(0.352205\pi\)
\(8\) 8.30956 2.93787
\(9\) −1.50866 −0.502888
\(10\) −3.08834 −0.976617
\(11\) 6.51509 1.96437 0.982187 0.187905i \(-0.0601698\pi\)
0.982187 + 0.187905i \(0.0601698\pi\)
\(12\) −6.24669 −1.80326
\(13\) −3.92858 −1.08959 −0.544796 0.838569i \(-0.683393\pi\)
−0.544796 + 0.838569i \(0.683393\pi\)
\(14\) 6.32068 1.68927
\(15\) 1.41390 0.365068
\(16\) 11.9348 2.98370
\(17\) 1.00000 0.242536
\(18\) −4.02426 −0.948527
\(19\) 0.308012 0.0706629 0.0353315 0.999376i \(-0.488751\pi\)
0.0353315 + 0.999376i \(0.488751\pi\)
\(20\) −5.92234 −1.32427
\(21\) −2.89373 −0.631464
\(22\) 17.3786 3.70512
\(23\) 1.28661 0.268278 0.134139 0.990963i \(-0.457173\pi\)
0.134139 + 0.990963i \(0.457173\pi\)
\(24\) −10.1477 −2.07138
\(25\) −3.65951 −0.731903
\(26\) −10.4792 −2.05514
\(27\) 5.50599 1.05963
\(28\) 12.1208 2.29062
\(29\) −2.29611 −0.426377 −0.213188 0.977011i \(-0.568385\pi\)
−0.213188 + 0.977011i \(0.568385\pi\)
\(30\) 3.77148 0.688576
\(31\) 5.07718 0.911889 0.455945 0.890008i \(-0.349301\pi\)
0.455945 + 0.890008i \(0.349301\pi\)
\(32\) 15.2162 2.68986
\(33\) −7.95625 −1.38501
\(34\) 2.66743 0.457461
\(35\) −2.74348 −0.463733
\(36\) −7.71711 −1.28618
\(37\) −5.52176 −0.907771 −0.453886 0.891060i \(-0.649963\pi\)
−0.453886 + 0.891060i \(0.649963\pi\)
\(38\) 0.821602 0.133281
\(39\) 4.79759 0.768230
\(40\) −9.62076 −1.52118
\(41\) −0.358377 −0.0559691 −0.0279846 0.999608i \(-0.508909\pi\)
−0.0279846 + 0.999608i \(0.508909\pi\)
\(42\) −7.71884 −1.19104
\(43\) −1.00000 −0.152499
\(44\) 33.3260 5.02408
\(45\) 1.74672 0.260386
\(46\) 3.43196 0.506014
\(47\) −7.52351 −1.09742 −0.548709 0.836014i \(-0.684880\pi\)
−0.548709 + 0.836014i \(0.684880\pi\)
\(48\) −14.5748 −2.10369
\(49\) −1.38511 −0.197873
\(50\) −9.76150 −1.38049
\(51\) −1.22120 −0.171003
\(52\) −20.0954 −2.78674
\(53\) −0.0422801 −0.00580761 −0.00290381 0.999996i \(-0.500924\pi\)
−0.00290381 + 0.999996i \(0.500924\pi\)
\(54\) 14.6869 1.99863
\(55\) −7.54313 −1.01712
\(56\) 19.6901 2.63121
\(57\) −0.376146 −0.0498217
\(58\) −6.12471 −0.804214
\(59\) −5.52410 −0.719176 −0.359588 0.933111i \(-0.617083\pi\)
−0.359588 + 0.933111i \(0.617083\pi\)
\(60\) 7.23237 0.933695
\(61\) −4.20526 −0.538428 −0.269214 0.963080i \(-0.586764\pi\)
−0.269214 + 0.963080i \(0.586764\pi\)
\(62\) 13.5430 1.71997
\(63\) −3.57489 −0.450394
\(64\) 16.7185 2.08981
\(65\) 4.54848 0.564170
\(66\) −21.2228 −2.61234
\(67\) 7.62209 0.931186 0.465593 0.884999i \(-0.345841\pi\)
0.465593 + 0.884999i \(0.345841\pi\)
\(68\) 5.11519 0.620308
\(69\) −1.57122 −0.189152
\(70\) −7.31805 −0.874674
\(71\) 6.07143 0.720546 0.360273 0.932847i \(-0.382684\pi\)
0.360273 + 0.932847i \(0.382684\pi\)
\(72\) −12.5363 −1.47742
\(73\) −7.32893 −0.857786 −0.428893 0.903355i \(-0.641096\pi\)
−0.428893 + 0.903355i \(0.641096\pi\)
\(74\) −14.7289 −1.71220
\(75\) 4.46901 0.516037
\(76\) 1.57554 0.180727
\(77\) 15.4380 1.75932
\(78\) 12.7973 1.44900
\(79\) −11.0101 −1.23874 −0.619368 0.785101i \(-0.712611\pi\)
−0.619368 + 0.785101i \(0.712611\pi\)
\(80\) −13.8180 −1.54490
\(81\) −2.19794 −0.244216
\(82\) −0.955947 −0.105567
\(83\) −7.59813 −0.834003 −0.417002 0.908906i \(-0.636919\pi\)
−0.417002 + 0.908906i \(0.636919\pi\)
\(84\) −14.8020 −1.61503
\(85\) −1.15779 −0.125580
\(86\) −2.66743 −0.287637
\(87\) 2.80401 0.300622
\(88\) 54.1376 5.77109
\(89\) −12.4245 −1.31699 −0.658497 0.752583i \(-0.728808\pi\)
−0.658497 + 0.752583i \(0.728808\pi\)
\(90\) 4.65926 0.491129
\(91\) −9.30907 −0.975856
\(92\) 6.58128 0.686146
\(93\) −6.20027 −0.642938
\(94\) −20.0685 −2.06990
\(95\) −0.356615 −0.0365879
\(96\) −18.5820 −1.89652
\(97\) −17.8513 −1.81252 −0.906261 0.422718i \(-0.861076\pi\)
−0.906261 + 0.422718i \(0.861076\pi\)
\(98\) −3.69468 −0.373219
\(99\) −9.82908 −0.987860
\(100\) −18.7191 −1.87191
\(101\) 5.07769 0.505249 0.252624 0.967564i \(-0.418706\pi\)
0.252624 + 0.967564i \(0.418706\pi\)
\(102\) −3.25748 −0.322538
\(103\) 0.113534 0.0111868 0.00559342 0.999984i \(-0.498220\pi\)
0.00559342 + 0.999984i \(0.498220\pi\)
\(104\) −32.6448 −3.20108
\(105\) 3.35035 0.326960
\(106\) −0.112779 −0.0109541
\(107\) −11.0063 −1.06402 −0.532012 0.846737i \(-0.678564\pi\)
−0.532012 + 0.846737i \(0.678564\pi\)
\(108\) 28.1642 2.71010
\(109\) 15.2804 1.46360 0.731798 0.681522i \(-0.238682\pi\)
0.731798 + 0.681522i \(0.238682\pi\)
\(110\) −20.1208 −1.91844
\(111\) 6.74319 0.640035
\(112\) 28.2804 2.67225
\(113\) 16.0296 1.50794 0.753969 0.656910i \(-0.228137\pi\)
0.753969 + 0.656910i \(0.228137\pi\)
\(114\) −1.00334 −0.0939717
\(115\) −1.48963 −0.138909
\(116\) −11.7450 −1.09050
\(117\) 5.92691 0.547943
\(118\) −14.7352 −1.35648
\(119\) 2.36958 0.217219
\(120\) 11.7489 1.07252
\(121\) 31.4464 2.85877
\(122\) −11.2172 −1.01556
\(123\) 0.437651 0.0394617
\(124\) 25.9708 2.33224
\(125\) 10.0259 0.896746
\(126\) −9.53579 −0.849515
\(127\) 17.2139 1.52749 0.763743 0.645521i \(-0.223360\pi\)
0.763743 + 0.645521i \(0.223360\pi\)
\(128\) 14.1630 1.25185
\(129\) 1.22120 0.107521
\(130\) 12.1328 1.06411
\(131\) −2.11340 −0.184649 −0.0923243 0.995729i \(-0.529430\pi\)
−0.0923243 + 0.995729i \(0.529430\pi\)
\(132\) −40.6977 −3.54228
\(133\) 0.729859 0.0632868
\(134\) 20.3314 1.75637
\(135\) −6.37480 −0.548656
\(136\) 8.30956 0.712539
\(137\) −13.1346 −1.12217 −0.561084 0.827759i \(-0.689616\pi\)
−0.561084 + 0.827759i \(0.689616\pi\)
\(138\) −4.19111 −0.356771
\(139\) 21.0890 1.78875 0.894375 0.447318i \(-0.147621\pi\)
0.894375 + 0.447318i \(0.147621\pi\)
\(140\) −14.0334 −1.18604
\(141\) 9.18774 0.773747
\(142\) 16.1951 1.35906
\(143\) −25.5951 −2.14037
\(144\) −18.0056 −1.50047
\(145\) 2.65842 0.220770
\(146\) −19.5494 −1.61792
\(147\) 1.69150 0.139512
\(148\) −28.2449 −2.32171
\(149\) −18.6209 −1.52548 −0.762741 0.646704i \(-0.776147\pi\)
−0.762741 + 0.646704i \(0.776147\pi\)
\(150\) 11.9208 0.973327
\(151\) 7.82065 0.636436 0.318218 0.948018i \(-0.396916\pi\)
0.318218 + 0.948018i \(0.396916\pi\)
\(152\) 2.55945 0.207599
\(153\) −1.50866 −0.121968
\(154\) 41.1798 3.31837
\(155\) −5.87833 −0.472159
\(156\) 24.5406 1.96482
\(157\) 19.0355 1.51920 0.759600 0.650391i \(-0.225395\pi\)
0.759600 + 0.650391i \(0.225395\pi\)
\(158\) −29.3688 −2.33645
\(159\) 0.0516325 0.00409472
\(160\) −17.6172 −1.39276
\(161\) 3.04873 0.240274
\(162\) −5.86286 −0.460630
\(163\) −9.68110 −0.758282 −0.379141 0.925339i \(-0.623781\pi\)
−0.379141 + 0.925339i \(0.623781\pi\)
\(164\) −1.83317 −0.143146
\(165\) 9.21169 0.717129
\(166\) −20.2675 −1.57306
\(167\) −6.67564 −0.516576 −0.258288 0.966068i \(-0.583158\pi\)
−0.258288 + 0.966068i \(0.583158\pi\)
\(168\) −24.0457 −1.85516
\(169\) 2.43374 0.187211
\(170\) −3.08834 −0.236865
\(171\) −0.464687 −0.0355355
\(172\) −5.11519 −0.390030
\(173\) −2.47305 −0.188023 −0.0940113 0.995571i \(-0.529969\pi\)
−0.0940113 + 0.995571i \(0.529969\pi\)
\(174\) 7.47952 0.567021
\(175\) −8.67150 −0.655504
\(176\) 77.7564 5.86111
\(177\) 6.74604 0.507064
\(178\) −33.1415 −2.48406
\(179\) 18.3300 1.37005 0.685025 0.728520i \(-0.259791\pi\)
0.685025 + 0.728520i \(0.259791\pi\)
\(180\) 8.93482 0.665962
\(181\) 12.4942 0.928690 0.464345 0.885654i \(-0.346290\pi\)
0.464345 + 0.885654i \(0.346290\pi\)
\(182\) −24.8313 −1.84062
\(183\) 5.13547 0.379625
\(184\) 10.6912 0.788166
\(185\) 6.39305 0.470027
\(186\) −16.5388 −1.21268
\(187\) 6.51509 0.476431
\(188\) −38.4842 −2.80675
\(189\) 13.0469 0.949020
\(190\) −0.951246 −0.0690106
\(191\) 12.0780 0.873936 0.436968 0.899477i \(-0.356052\pi\)
0.436968 + 0.899477i \(0.356052\pi\)
\(192\) −20.4166 −1.47344
\(193\) −22.5069 −1.62008 −0.810042 0.586372i \(-0.800556\pi\)
−0.810042 + 0.586372i \(0.800556\pi\)
\(194\) −47.6171 −3.41871
\(195\) −5.55462 −0.397775
\(196\) −7.08510 −0.506078
\(197\) 8.66711 0.617506 0.308753 0.951142i \(-0.400088\pi\)
0.308753 + 0.951142i \(0.400088\pi\)
\(198\) −26.2184 −1.86326
\(199\) −9.76275 −0.692063 −0.346032 0.938223i \(-0.612471\pi\)
−0.346032 + 0.938223i \(0.612471\pi\)
\(200\) −30.4090 −2.15024
\(201\) −9.30812 −0.656544
\(202\) 13.5444 0.952980
\(203\) −5.44080 −0.381870
\(204\) −6.24669 −0.437356
\(205\) 0.414927 0.0289798
\(206\) 0.302844 0.0211002
\(207\) −1.94107 −0.134914
\(208\) −46.8869 −3.25102
\(209\) 2.00673 0.138808
\(210\) 8.93682 0.616699
\(211\) −0.635111 −0.0437229 −0.0218614 0.999761i \(-0.506959\pi\)
−0.0218614 + 0.999761i \(0.506959\pi\)
\(212\) −0.216271 −0.0148535
\(213\) −7.41445 −0.508029
\(214\) −29.3587 −2.00692
\(215\) 1.15779 0.0789609
\(216\) 45.7524 3.11306
\(217\) 12.0308 0.816702
\(218\) 40.7594 2.76057
\(219\) 8.95010 0.604792
\(220\) −38.5846 −2.60137
\(221\) −3.92858 −0.264265
\(222\) 17.9870 1.20721
\(223\) 23.6836 1.58597 0.792987 0.609239i \(-0.208525\pi\)
0.792987 + 0.609239i \(0.208525\pi\)
\(224\) 36.0559 2.40908
\(225\) 5.52098 0.368065
\(226\) 42.7579 2.84421
\(227\) 28.1558 1.86876 0.934381 0.356274i \(-0.115953\pi\)
0.934381 + 0.356274i \(0.115953\pi\)
\(228\) −1.92406 −0.127424
\(229\) −0.785206 −0.0518879 −0.0259439 0.999663i \(-0.508259\pi\)
−0.0259439 + 0.999663i \(0.508259\pi\)
\(230\) −3.97350 −0.262005
\(231\) −18.8529 −1.24043
\(232\) −19.0797 −1.25264
\(233\) −0.995090 −0.0651905 −0.0325953 0.999469i \(-0.510377\pi\)
−0.0325953 + 0.999469i \(0.510377\pi\)
\(234\) 15.8096 1.03351
\(235\) 8.71068 0.568222
\(236\) −28.2568 −1.83936
\(237\) 13.4456 0.873385
\(238\) 6.32068 0.409709
\(239\) −7.10271 −0.459436 −0.229718 0.973257i \(-0.573780\pi\)
−0.229718 + 0.973257i \(0.573780\pi\)
\(240\) 16.8746 1.08925
\(241\) 29.9977 1.93232 0.966161 0.257940i \(-0.0830438\pi\)
0.966161 + 0.257940i \(0.0830438\pi\)
\(242\) 83.8812 5.39209
\(243\) −13.8338 −0.887442
\(244\) −21.5107 −1.37708
\(245\) 1.60367 0.102455
\(246\) 1.16741 0.0744310
\(247\) −1.21005 −0.0769937
\(248\) 42.1892 2.67902
\(249\) 9.27886 0.588024
\(250\) 26.7435 1.69141
\(251\) −15.0432 −0.949517 −0.474759 0.880116i \(-0.657465\pi\)
−0.474759 + 0.880116i \(0.657465\pi\)
\(252\) −18.2863 −1.15193
\(253\) 8.38241 0.526998
\(254\) 45.9169 2.88108
\(255\) 1.41390 0.0885419
\(256\) 4.34199 0.271375
\(257\) −13.1966 −0.823181 −0.411590 0.911369i \(-0.635027\pi\)
−0.411590 + 0.911369i \(0.635027\pi\)
\(258\) 3.25748 0.202802
\(259\) −13.0842 −0.813014
\(260\) 23.2664 1.44292
\(261\) 3.46406 0.214420
\(262\) −5.63735 −0.348277
\(263\) −9.40757 −0.580095 −0.290048 0.957012i \(-0.593671\pi\)
−0.290048 + 0.957012i \(0.593671\pi\)
\(264\) −66.1130 −4.06897
\(265\) 0.0489516 0.00300707
\(266\) 1.94685 0.119369
\(267\) 15.1728 0.928562
\(268\) 38.9885 2.38160
\(269\) 23.7915 1.45059 0.725297 0.688436i \(-0.241702\pi\)
0.725297 + 0.688436i \(0.241702\pi\)
\(270\) −17.0044 −1.03485
\(271\) 10.4516 0.634889 0.317445 0.948277i \(-0.397175\pi\)
0.317445 + 0.948277i \(0.397175\pi\)
\(272\) 11.9348 0.723654
\(273\) 11.3683 0.688039
\(274\) −35.0358 −2.11659
\(275\) −23.8421 −1.43773
\(276\) −8.03708 −0.483775
\(277\) 28.7979 1.73030 0.865149 0.501515i \(-0.167224\pi\)
0.865149 + 0.501515i \(0.167224\pi\)
\(278\) 56.2536 3.37387
\(279\) −7.65977 −0.458578
\(280\) −22.7971 −1.36239
\(281\) −22.0010 −1.31247 −0.656234 0.754558i \(-0.727851\pi\)
−0.656234 + 0.754558i \(0.727851\pi\)
\(282\) 24.5077 1.45941
\(283\) 1.40000 0.0832211 0.0416106 0.999134i \(-0.486751\pi\)
0.0416106 + 0.999134i \(0.486751\pi\)
\(284\) 31.0565 1.84287
\(285\) 0.435499 0.0257967
\(286\) −68.2731 −4.03707
\(287\) −0.849202 −0.0501268
\(288\) −22.9561 −1.35270
\(289\) 1.00000 0.0588235
\(290\) 7.09115 0.416407
\(291\) 21.8000 1.27794
\(292\) −37.4889 −2.19387
\(293\) −26.4316 −1.54415 −0.772074 0.635533i \(-0.780781\pi\)
−0.772074 + 0.635533i \(0.780781\pi\)
\(294\) 4.51196 0.263143
\(295\) 6.39576 0.372376
\(296\) −45.8834 −2.66692
\(297\) 35.8721 2.08151
\(298\) −49.6699 −2.87730
\(299\) −5.05457 −0.292313
\(300\) 22.8598 1.31981
\(301\) −2.36958 −0.136580
\(302\) 20.8611 1.20042
\(303\) −6.20089 −0.356232
\(304\) 3.67607 0.210837
\(305\) 4.86882 0.278788
\(306\) −4.02426 −0.230052
\(307\) 4.69458 0.267934 0.133967 0.990986i \(-0.457228\pi\)
0.133967 + 0.990986i \(0.457228\pi\)
\(308\) 78.9684 4.49964
\(309\) −0.138648 −0.00788741
\(310\) −15.6800 −0.890567
\(311\) −9.45381 −0.536076 −0.268038 0.963408i \(-0.586375\pi\)
−0.268038 + 0.963408i \(0.586375\pi\)
\(312\) 39.8659 2.25696
\(313\) 0.488917 0.0276352 0.0138176 0.999905i \(-0.495602\pi\)
0.0138176 + 0.999905i \(0.495602\pi\)
\(314\) 50.7760 2.86545
\(315\) 4.13899 0.233206
\(316\) −56.3189 −3.16819
\(317\) 2.98536 0.167675 0.0838373 0.996479i \(-0.473282\pi\)
0.0838373 + 0.996479i \(0.473282\pi\)
\(318\) 0.137726 0.00772330
\(319\) −14.9594 −0.837563
\(320\) −19.3565 −1.08206
\(321\) 13.4410 0.750202
\(322\) 8.13228 0.453194
\(323\) 0.308012 0.0171383
\(324\) −11.2429 −0.624605
\(325\) 14.3767 0.797476
\(326\) −25.8237 −1.43024
\(327\) −18.6604 −1.03193
\(328\) −2.97796 −0.164430
\(329\) −17.8275 −0.982864
\(330\) 24.5716 1.35262
\(331\) −6.43306 −0.353593 −0.176796 0.984247i \(-0.556573\pi\)
−0.176796 + 0.984247i \(0.556573\pi\)
\(332\) −38.8659 −2.13304
\(333\) 8.33048 0.456507
\(334\) −17.8068 −0.974345
\(335\) −8.82481 −0.482151
\(336\) −34.5362 −1.88410
\(337\) 31.9329 1.73950 0.869749 0.493494i \(-0.164281\pi\)
0.869749 + 0.493494i \(0.164281\pi\)
\(338\) 6.49184 0.353110
\(339\) −19.5754 −1.06319
\(340\) −5.92234 −0.321184
\(341\) 33.0783 1.79129
\(342\) −1.23952 −0.0670257
\(343\) −19.8692 −1.07283
\(344\) −8.30956 −0.448022
\(345\) 1.81914 0.0979395
\(346\) −6.59669 −0.354640
\(347\) −21.9876 −1.18036 −0.590179 0.807273i \(-0.700943\pi\)
−0.590179 + 0.807273i \(0.700943\pi\)
\(348\) 14.3431 0.768869
\(349\) 21.2791 1.13905 0.569523 0.821976i \(-0.307128\pi\)
0.569523 + 0.821976i \(0.307128\pi\)
\(350\) −23.1306 −1.23638
\(351\) −21.6307 −1.15456
\(352\) 99.1347 5.28390
\(353\) 29.6150 1.57625 0.788124 0.615517i \(-0.211053\pi\)
0.788124 + 0.615517i \(0.211053\pi\)
\(354\) 17.9946 0.956403
\(355\) −7.02946 −0.373085
\(356\) −63.5537 −3.36834
\(357\) −2.89373 −0.153153
\(358\) 48.8940 2.58413
\(359\) 33.6193 1.77436 0.887180 0.461423i \(-0.152661\pi\)
0.887180 + 0.461423i \(0.152661\pi\)
\(360\) 14.5145 0.764981
\(361\) −18.9051 −0.995007
\(362\) 33.3276 1.75166
\(363\) −38.4025 −2.01561
\(364\) −47.6177 −2.49584
\(365\) 8.48538 0.444145
\(366\) 13.6985 0.716033
\(367\) −8.84424 −0.461666 −0.230833 0.972993i \(-0.574145\pi\)
−0.230833 + 0.972993i \(0.574145\pi\)
\(368\) 15.3555 0.800461
\(369\) 0.540671 0.0281462
\(370\) 17.0530 0.886545
\(371\) −0.100186 −0.00520139
\(372\) −31.7156 −1.64438
\(373\) −36.6255 −1.89640 −0.948198 0.317679i \(-0.897097\pi\)
−0.948198 + 0.317679i \(0.897097\pi\)
\(374\) 17.3786 0.898624
\(375\) −12.2437 −0.632262
\(376\) −62.5171 −3.22407
\(377\) 9.02045 0.464577
\(378\) 34.8016 1.79000
\(379\) −9.39461 −0.482569 −0.241284 0.970454i \(-0.577569\pi\)
−0.241284 + 0.970454i \(0.577569\pi\)
\(380\) −1.82415 −0.0935771
\(381\) −21.0216 −1.07697
\(382\) 32.2173 1.64838
\(383\) 25.7441 1.31546 0.657730 0.753253i \(-0.271517\pi\)
0.657730 + 0.753253i \(0.271517\pi\)
\(384\) −17.2959 −0.882629
\(385\) −17.8740 −0.910945
\(386\) −60.0357 −3.05574
\(387\) 1.50866 0.0766897
\(388\) −91.3127 −4.63570
\(389\) −31.9152 −1.61817 −0.809083 0.587695i \(-0.800036\pi\)
−0.809083 + 0.587695i \(0.800036\pi\)
\(390\) −14.8166 −0.750266
\(391\) 1.28661 0.0650669
\(392\) −11.5096 −0.581325
\(393\) 2.58089 0.130189
\(394\) 23.1189 1.16471
\(395\) 12.7475 0.641394
\(396\) −50.2777 −2.52655
\(397\) 5.89272 0.295747 0.147874 0.989006i \(-0.452757\pi\)
0.147874 + 0.989006i \(0.452757\pi\)
\(398\) −26.0415 −1.30534
\(399\) −0.891306 −0.0446211
\(400\) −43.6756 −2.18378
\(401\) −20.7143 −1.03442 −0.517210 0.855858i \(-0.673029\pi\)
−0.517210 + 0.855858i \(0.673029\pi\)
\(402\) −24.8288 −1.23835
\(403\) −19.9461 −0.993587
\(404\) 25.9733 1.29222
\(405\) 2.54476 0.126450
\(406\) −14.5130 −0.720267
\(407\) −35.9748 −1.78320
\(408\) −10.1477 −0.502384
\(409\) −36.5799 −1.80876 −0.904379 0.426731i \(-0.859665\pi\)
−0.904379 + 0.426731i \(0.859665\pi\)
\(410\) 1.10679 0.0546604
\(411\) 16.0401 0.791198
\(412\) 0.580749 0.0286114
\(413\) −13.0898 −0.644105
\(414\) −5.17767 −0.254468
\(415\) 8.79707 0.431831
\(416\) −59.7779 −2.93085
\(417\) −25.7540 −1.26118
\(418\) 5.35282 0.261815
\(419\) 7.11670 0.347674 0.173837 0.984774i \(-0.444383\pi\)
0.173837 + 0.984774i \(0.444383\pi\)
\(420\) 17.1377 0.836232
\(421\) 35.3632 1.72350 0.861748 0.507337i \(-0.169370\pi\)
0.861748 + 0.507337i \(0.169370\pi\)
\(422\) −1.69412 −0.0824683
\(423\) 11.3505 0.551878
\(424\) −0.351329 −0.0170620
\(425\) −3.65951 −0.177513
\(426\) −19.7775 −0.958224
\(427\) −9.96468 −0.482225
\(428\) −56.2996 −2.72134
\(429\) 31.2568 1.50909
\(430\) 3.08834 0.148933
\(431\) 8.88389 0.427922 0.213961 0.976842i \(-0.431363\pi\)
0.213961 + 0.976842i \(0.431363\pi\)
\(432\) 65.7130 3.16162
\(433\) 5.10237 0.245204 0.122602 0.992456i \(-0.460876\pi\)
0.122602 + 0.992456i \(0.460876\pi\)
\(434\) 32.0913 1.54043
\(435\) −3.24647 −0.155656
\(436\) 78.1621 3.74329
\(437\) 0.396293 0.0189573
\(438\) 23.8738 1.14073
\(439\) 24.0015 1.14553 0.572764 0.819720i \(-0.305871\pi\)
0.572764 + 0.819720i \(0.305871\pi\)
\(440\) −62.6801 −2.98816
\(441\) 2.08966 0.0995078
\(442\) −10.4792 −0.498446
\(443\) 2.14142 0.101742 0.0508710 0.998705i \(-0.483800\pi\)
0.0508710 + 0.998705i \(0.483800\pi\)
\(444\) 34.4927 1.63695
\(445\) 14.3850 0.681915
\(446\) 63.1745 2.99140
\(447\) 22.7399 1.07556
\(448\) 39.6157 1.87166
\(449\) 13.9735 0.659448 0.329724 0.944077i \(-0.393044\pi\)
0.329724 + 0.944077i \(0.393044\pi\)
\(450\) 14.7268 0.694229
\(451\) −2.33486 −0.109944
\(452\) 81.9945 3.85670
\(453\) −9.55060 −0.448727
\(454\) 75.1035 3.52478
\(455\) 10.7780 0.505280
\(456\) −3.12561 −0.146370
\(457\) 0.782294 0.0365942 0.0182971 0.999833i \(-0.494176\pi\)
0.0182971 + 0.999833i \(0.494176\pi\)
\(458\) −2.09448 −0.0978688
\(459\) 5.50599 0.256998
\(460\) −7.61976 −0.355273
\(461\) −38.4583 −1.79118 −0.895592 0.444877i \(-0.853247\pi\)
−0.895592 + 0.444877i \(0.853247\pi\)
\(462\) −50.2889 −2.33965
\(463\) −4.07968 −0.189599 −0.0947994 0.995496i \(-0.530221\pi\)
−0.0947994 + 0.995496i \(0.530221\pi\)
\(464\) −27.4036 −1.27218
\(465\) 7.17863 0.332901
\(466\) −2.65433 −0.122960
\(467\) −21.9870 −1.01744 −0.508719 0.860932i \(-0.669881\pi\)
−0.508719 + 0.860932i \(0.669881\pi\)
\(468\) 30.3173 1.40142
\(469\) 18.0611 0.833985
\(470\) 23.2351 1.07176
\(471\) −23.2462 −1.07113
\(472\) −45.9028 −2.11285
\(473\) −6.51509 −0.299564
\(474\) 35.8652 1.64734
\(475\) −1.12718 −0.0517184
\(476\) 12.1208 0.555558
\(477\) 0.0637864 0.00292058
\(478\) −18.9460 −0.866569
\(479\) −10.1140 −0.462121 −0.231061 0.972939i \(-0.574220\pi\)
−0.231061 + 0.972939i \(0.574220\pi\)
\(480\) 21.5141 0.981982
\(481\) 21.6927 0.989100
\(482\) 80.0168 3.64467
\(483\) −3.72312 −0.169408
\(484\) 160.855 7.31157
\(485\) 20.6681 0.938490
\(486\) −36.9008 −1.67386
\(487\) −34.1263 −1.54641 −0.773205 0.634157i \(-0.781347\pi\)
−0.773205 + 0.634157i \(0.781347\pi\)
\(488\) −34.9439 −1.58183
\(489\) 11.8226 0.534636
\(490\) 4.27768 0.193246
\(491\) 8.07034 0.364209 0.182105 0.983279i \(-0.441709\pi\)
0.182105 + 0.983279i \(0.441709\pi\)
\(492\) 2.23867 0.100927
\(493\) −2.29611 −0.103412
\(494\) −3.22773 −0.145222
\(495\) 11.3801 0.511495
\(496\) 60.5952 2.72081
\(497\) 14.3867 0.645332
\(498\) 24.7507 1.10911
\(499\) 15.3883 0.688875 0.344437 0.938809i \(-0.388070\pi\)
0.344437 + 0.938809i \(0.388070\pi\)
\(500\) 51.2846 2.29352
\(501\) 8.15231 0.364218
\(502\) −40.1267 −1.79094
\(503\) 26.7497 1.19271 0.596356 0.802720i \(-0.296615\pi\)
0.596356 + 0.802720i \(0.296615\pi\)
\(504\) −29.7058 −1.32320
\(505\) −5.87891 −0.261608
\(506\) 22.3595 0.994001
\(507\) −2.97209 −0.131995
\(508\) 88.0524 3.90669
\(509\) 29.6718 1.31518 0.657589 0.753377i \(-0.271576\pi\)
0.657589 + 0.753377i \(0.271576\pi\)
\(510\) 3.77148 0.167004
\(511\) −17.3664 −0.768246
\(512\) −16.7441 −0.739991
\(513\) 1.69591 0.0748765
\(514\) −35.2010 −1.55265
\(515\) −0.131449 −0.00579233
\(516\) 6.24669 0.274995
\(517\) −49.0164 −2.15574
\(518\) −34.9013 −1.53347
\(519\) 3.02010 0.132567
\(520\) 37.7959 1.65746
\(521\) −30.3472 −1.32953 −0.664767 0.747051i \(-0.731469\pi\)
−0.664767 + 0.747051i \(0.731469\pi\)
\(522\) 9.24013 0.404430
\(523\) 27.5156 1.20317 0.601587 0.798807i \(-0.294535\pi\)
0.601587 + 0.798807i \(0.294535\pi\)
\(524\) −10.8105 −0.472257
\(525\) 10.5897 0.462171
\(526\) −25.0940 −1.09415
\(527\) 5.07718 0.221166
\(528\) −94.9563 −4.13244
\(529\) −21.3446 −0.928027
\(530\) 0.130575 0.00567181
\(531\) 8.33400 0.361665
\(532\) 3.73337 0.161862
\(533\) 1.40791 0.0609835
\(534\) 40.4725 1.75142
\(535\) 12.7431 0.550931
\(536\) 63.3362 2.73571
\(537\) −22.3847 −0.965969
\(538\) 63.4622 2.73605
\(539\) −9.02411 −0.388696
\(540\) −32.6083 −1.40324
\(541\) 19.6595 0.845226 0.422613 0.906310i \(-0.361113\pi\)
0.422613 + 0.906310i \(0.361113\pi\)
\(542\) 27.8789 1.19750
\(543\) −15.2580 −0.654784
\(544\) 15.2162 0.652388
\(545\) −17.6915 −0.757822
\(546\) 30.3241 1.29775
\(547\) −9.16183 −0.391732 −0.195866 0.980631i \(-0.562752\pi\)
−0.195866 + 0.980631i \(0.562752\pi\)
\(548\) −67.1862 −2.87005
\(549\) 6.34432 0.270769
\(550\) −63.5971 −2.71179
\(551\) −0.707230 −0.0301290
\(552\) −13.0561 −0.555706
\(553\) −26.0893 −1.10943
\(554\) 76.8164 3.26362
\(555\) −7.80722 −0.331398
\(556\) 107.875 4.57490
\(557\) 25.8790 1.09653 0.548263 0.836306i \(-0.315289\pi\)
0.548263 + 0.836306i \(0.315289\pi\)
\(558\) −20.4319 −0.864951
\(559\) 3.92858 0.166161
\(560\) −32.7429 −1.38364
\(561\) −7.95625 −0.335913
\(562\) −58.6861 −2.47552
\(563\) 36.5929 1.54221 0.771103 0.636711i \(-0.219706\pi\)
0.771103 + 0.636711i \(0.219706\pi\)
\(564\) 46.9970 1.97893
\(565\) −18.5590 −0.780782
\(566\) 3.73439 0.156968
\(567\) −5.20819 −0.218723
\(568\) 50.4509 2.11687
\(569\) −22.5560 −0.945596 −0.472798 0.881171i \(-0.656756\pi\)
−0.472798 + 0.881171i \(0.656756\pi\)
\(570\) 1.16166 0.0486568
\(571\) 28.1823 1.17939 0.589696 0.807626i \(-0.299248\pi\)
0.589696 + 0.807626i \(0.299248\pi\)
\(572\) −130.924 −5.47419
\(573\) −14.7497 −0.616179
\(574\) −2.26519 −0.0945472
\(575\) −4.70838 −0.196353
\(576\) −25.2225 −1.05094
\(577\) 7.01637 0.292095 0.146048 0.989278i \(-0.453345\pi\)
0.146048 + 0.989278i \(0.453345\pi\)
\(578\) 2.66743 0.110951
\(579\) 27.4855 1.14226
\(580\) 13.5983 0.564640
\(581\) −18.0043 −0.746946
\(582\) 58.1501 2.41040
\(583\) −0.275458 −0.0114083
\(584\) −60.9002 −2.52007
\(585\) −6.86213 −0.283714
\(586\) −70.5044 −2.91251
\(587\) 3.53126 0.145751 0.0728753 0.997341i \(-0.476782\pi\)
0.0728753 + 0.997341i \(0.476782\pi\)
\(588\) 8.65234 0.356816
\(589\) 1.56384 0.0644368
\(590\) 17.0603 0.702360
\(591\) −10.5843 −0.435380
\(592\) −65.9011 −2.70852
\(593\) −16.2807 −0.668569 −0.334284 0.942472i \(-0.608495\pi\)
−0.334284 + 0.942472i \(0.608495\pi\)
\(594\) 95.6863 3.92605
\(595\) −2.74348 −0.112472
\(596\) −95.2494 −3.90157
\(597\) 11.9223 0.487947
\(598\) −13.4827 −0.551349
\(599\) −43.2707 −1.76799 −0.883996 0.467495i \(-0.845157\pi\)
−0.883996 + 0.467495i \(0.845157\pi\)
\(600\) 37.1355 1.51605
\(601\) 42.0708 1.71611 0.858053 0.513561i \(-0.171674\pi\)
0.858053 + 0.513561i \(0.171674\pi\)
\(602\) −6.32068 −0.257612
\(603\) −11.4992 −0.468282
\(604\) 40.0042 1.62775
\(605\) −36.4085 −1.48021
\(606\) −16.5404 −0.671909
\(607\) −1.60950 −0.0653275 −0.0326638 0.999466i \(-0.510399\pi\)
−0.0326638 + 0.999466i \(0.510399\pi\)
\(608\) 4.68677 0.190074
\(609\) 6.64432 0.269242
\(610\) 12.9872 0.525838
\(611\) 29.5567 1.19574
\(612\) −7.71711 −0.311946
\(613\) −29.3932 −1.18718 −0.593590 0.804767i \(-0.702290\pi\)
−0.593590 + 0.804767i \(0.702290\pi\)
\(614\) 12.5225 0.505366
\(615\) −0.506710 −0.0204325
\(616\) 128.283 5.16867
\(617\) −33.5535 −1.35081 −0.675406 0.737446i \(-0.736032\pi\)
−0.675406 + 0.737446i \(0.736032\pi\)
\(618\) −0.369834 −0.0148769
\(619\) 21.9301 0.881445 0.440723 0.897643i \(-0.354722\pi\)
0.440723 + 0.897643i \(0.354722\pi\)
\(620\) −30.0688 −1.20759
\(621\) 7.08409 0.284275
\(622\) −25.2174 −1.01113
\(623\) −29.4408 −1.17952
\(624\) 57.2584 2.29217
\(625\) 6.68961 0.267585
\(626\) 1.30415 0.0521245
\(627\) −2.45062 −0.0978685
\(628\) 97.3704 3.88550
\(629\) −5.52176 −0.220167
\(630\) 11.0405 0.439863
\(631\) −2.89353 −0.115190 −0.0575949 0.998340i \(-0.518343\pi\)
−0.0575949 + 0.998340i \(0.518343\pi\)
\(632\) −91.4894 −3.63925
\(633\) 0.775599 0.0308273
\(634\) 7.96325 0.316261
\(635\) −19.9301 −0.790903
\(636\) 0.264110 0.0104727
\(637\) 5.44151 0.215600
\(638\) −39.9031 −1.57978
\(639\) −9.15975 −0.362354
\(640\) −16.3979 −0.648183
\(641\) −6.49838 −0.256671 −0.128335 0.991731i \(-0.540963\pi\)
−0.128335 + 0.991731i \(0.540963\pi\)
\(642\) 35.8529 1.41500
\(643\) −34.3274 −1.35374 −0.676869 0.736103i \(-0.736664\pi\)
−0.676869 + 0.736103i \(0.736664\pi\)
\(644\) 15.5948 0.614523
\(645\) −1.41390 −0.0556723
\(646\) 0.821602 0.0323255
\(647\) −0.516741 −0.0203152 −0.0101576 0.999948i \(-0.503233\pi\)
−0.0101576 + 0.999948i \(0.503233\pi\)
\(648\) −18.2639 −0.717475
\(649\) −35.9900 −1.41273
\(650\) 38.3489 1.50417
\(651\) −14.6920 −0.575826
\(652\) −49.5207 −1.93938
\(653\) 3.02652 0.118437 0.0592184 0.998245i \(-0.481139\pi\)
0.0592184 + 0.998245i \(0.481139\pi\)
\(654\) −49.7755 −1.94638
\(655\) 2.44688 0.0956076
\(656\) −4.27717 −0.166995
\(657\) 11.0569 0.431370
\(658\) −47.5537 −1.85384
\(659\) 24.7953 0.965887 0.482943 0.875652i \(-0.339568\pi\)
0.482943 + 0.875652i \(0.339568\pi\)
\(660\) 47.1196 1.83413
\(661\) 1.16506 0.0453156 0.0226578 0.999743i \(-0.492787\pi\)
0.0226578 + 0.999743i \(0.492787\pi\)
\(662\) −17.1597 −0.666932
\(663\) 4.79759 0.186323
\(664\) −63.1371 −2.45020
\(665\) −0.845026 −0.0327687
\(666\) 22.2210 0.861045
\(667\) −2.95421 −0.114387
\(668\) −34.1472 −1.32119
\(669\) −28.9225 −1.11821
\(670\) −23.5396 −0.909413
\(671\) −27.3976 −1.05767
\(672\) −44.0315 −1.69855
\(673\) −15.3866 −0.593109 −0.296554 0.955016i \(-0.595838\pi\)
−0.296554 + 0.955016i \(0.595838\pi\)
\(674\) 85.1789 3.28097
\(675\) −20.1493 −0.775545
\(676\) 12.4491 0.478810
\(677\) −28.6663 −1.10174 −0.550868 0.834592i \(-0.685703\pi\)
−0.550868 + 0.834592i \(0.685703\pi\)
\(678\) −52.2160 −2.00534
\(679\) −42.3000 −1.62332
\(680\) −9.62076 −0.368939
\(681\) −34.3839 −1.31759
\(682\) 88.2342 3.37866
\(683\) 14.2794 0.546385 0.273192 0.961959i \(-0.411920\pi\)
0.273192 + 0.961959i \(0.411920\pi\)
\(684\) −2.37697 −0.0908855
\(685\) 15.2072 0.581037
\(686\) −52.9996 −2.02353
\(687\) 0.958896 0.0365842
\(688\) −11.9348 −0.455010
\(689\) 0.166101 0.00632793
\(690\) 4.85244 0.184729
\(691\) 36.2344 1.37842 0.689211 0.724561i \(-0.257957\pi\)
0.689211 + 0.724561i \(0.257957\pi\)
\(692\) −12.6501 −0.480886
\(693\) −23.2908 −0.884743
\(694\) −58.6505 −2.22634
\(695\) −24.4168 −0.926181
\(696\) 23.3001 0.883189
\(697\) −0.358377 −0.0135745
\(698\) 56.7606 2.14842
\(699\) 1.21521 0.0459633
\(700\) −44.3564 −1.67651
\(701\) 5.14395 0.194284 0.0971422 0.995271i \(-0.469030\pi\)
0.0971422 + 0.995271i \(0.469030\pi\)
\(702\) −57.6985 −2.17769
\(703\) −1.70077 −0.0641458
\(704\) 108.922 4.10516
\(705\) −10.6375 −0.400631
\(706\) 78.9960 2.97305
\(707\) 12.0320 0.452509
\(708\) 34.5073 1.29686
\(709\) 7.73810 0.290610 0.145305 0.989387i \(-0.453584\pi\)
0.145305 + 0.989387i \(0.453584\pi\)
\(710\) −18.7506 −0.703698
\(711\) 16.6106 0.622946
\(712\) −103.242 −3.86916
\(713\) 6.53238 0.244639
\(714\) −7.71884 −0.288870
\(715\) 29.6338 1.10824
\(716\) 93.7615 3.50403
\(717\) 8.67384 0.323931
\(718\) 89.6773 3.34673
\(719\) −6.85306 −0.255576 −0.127788 0.991802i \(-0.540788\pi\)
−0.127788 + 0.991802i \(0.540788\pi\)
\(720\) 20.8468 0.776914
\(721\) 0.269028 0.0100191
\(722\) −50.4281 −1.87674
\(723\) −36.6333 −1.36241
\(724\) 63.9105 2.37521
\(725\) 8.40264 0.312066
\(726\) −102.436 −3.80175
\(727\) −25.9649 −0.962983 −0.481492 0.876451i \(-0.659905\pi\)
−0.481492 + 0.876451i \(0.659905\pi\)
\(728\) −77.3543 −2.86694
\(729\) 23.4878 0.869917
\(730\) 22.6342 0.837729
\(731\) −1.00000 −0.0369863
\(732\) 26.2689 0.970927
\(733\) 32.5717 1.20306 0.601531 0.798850i \(-0.294558\pi\)
0.601531 + 0.798850i \(0.294558\pi\)
\(734\) −23.5914 −0.870775
\(735\) −1.95841 −0.0722369
\(736\) 19.5773 0.721630
\(737\) 49.6586 1.82920
\(738\) 1.44220 0.0530882
\(739\) −22.9751 −0.845151 −0.422576 0.906328i \(-0.638874\pi\)
−0.422576 + 0.906328i \(0.638874\pi\)
\(740\) 32.7017 1.20214
\(741\) 1.47772 0.0542853
\(742\) −0.267239 −0.00981065
\(743\) −2.51329 −0.0922038 −0.0461019 0.998937i \(-0.514680\pi\)
−0.0461019 + 0.998937i \(0.514680\pi\)
\(744\) −51.5216 −1.88887
\(745\) 21.5591 0.789866
\(746\) −97.6960 −3.57691
\(747\) 11.4630 0.419410
\(748\) 33.3260 1.21852
\(749\) −26.0804 −0.952956
\(750\) −32.6592 −1.19255
\(751\) −33.9701 −1.23959 −0.619794 0.784765i \(-0.712784\pi\)
−0.619794 + 0.784765i \(0.712784\pi\)
\(752\) −89.7917 −3.27437
\(753\) 18.3708 0.669468
\(754\) 24.0614 0.876265
\(755\) −9.05470 −0.329534
\(756\) 66.7372 2.42721
\(757\) 5.93628 0.215758 0.107879 0.994164i \(-0.465594\pi\)
0.107879 + 0.994164i \(0.465594\pi\)
\(758\) −25.0595 −0.910201
\(759\) −10.2366 −0.371566
\(760\) −2.96331 −0.107491
\(761\) −8.61012 −0.312116 −0.156058 0.987748i \(-0.549879\pi\)
−0.156058 + 0.987748i \(0.549879\pi\)
\(762\) −56.0738 −2.03134
\(763\) 36.2080 1.31082
\(764\) 61.7815 2.23518
\(765\) 1.74672 0.0631528
\(766\) 68.6706 2.48117
\(767\) 21.7019 0.783609
\(768\) −5.30245 −0.191336
\(769\) 38.3439 1.38271 0.691357 0.722513i \(-0.257013\pi\)
0.691357 + 0.722513i \(0.257013\pi\)
\(770\) −47.6777 −1.71819
\(771\) 16.1157 0.580393
\(772\) −115.127 −4.14352
\(773\) −8.12687 −0.292303 −0.146152 0.989262i \(-0.546689\pi\)
−0.146152 + 0.989262i \(0.546689\pi\)
\(774\) 4.02426 0.144649
\(775\) −18.5800 −0.667414
\(776\) −148.336 −5.32496
\(777\) 15.9785 0.573225
\(778\) −85.1316 −3.05212
\(779\) −0.110385 −0.00395494
\(780\) −28.4130 −1.01735
\(781\) 39.5559 1.41542
\(782\) 3.43196 0.122726
\(783\) −12.6424 −0.451801
\(784\) −16.5310 −0.590393
\(785\) −22.0392 −0.786613
\(786\) 6.88435 0.245557
\(787\) 35.5439 1.26700 0.633502 0.773741i \(-0.281617\pi\)
0.633502 + 0.773741i \(0.281617\pi\)
\(788\) 44.3339 1.57933
\(789\) 11.4885 0.409003
\(790\) 34.0030 1.20977
\(791\) 37.9834 1.35053
\(792\) −81.6754 −2.90221
\(793\) 16.5207 0.586667
\(794\) 15.7184 0.557826
\(795\) −0.0597798 −0.00212017
\(796\) −49.9384 −1.77002
\(797\) 0.405256 0.0143549 0.00717746 0.999974i \(-0.497715\pi\)
0.00717746 + 0.999974i \(0.497715\pi\)
\(798\) −2.37750 −0.0841625
\(799\) −7.52351 −0.266163
\(800\) −55.6838 −1.96872
\(801\) 18.7444 0.662301
\(802\) −55.2539 −1.95108
\(803\) −47.7486 −1.68501
\(804\) −47.6128 −1.67917
\(805\) −3.52980 −0.124409
\(806\) −53.2049 −1.87406
\(807\) −29.0543 −1.02276
\(808\) 42.1934 1.48436
\(809\) −31.3337 −1.10163 −0.550817 0.834626i \(-0.685684\pi\)
−0.550817 + 0.834626i \(0.685684\pi\)
\(810\) 6.78798 0.238505
\(811\) 4.94465 0.173630 0.0868151 0.996224i \(-0.472331\pi\)
0.0868151 + 0.996224i \(0.472331\pi\)
\(812\) −27.8308 −0.976668
\(813\) −12.7635 −0.447636
\(814\) −95.9602 −3.36340
\(815\) 11.2087 0.392624
\(816\) −14.5748 −0.510221
\(817\) −0.308012 −0.0107760
\(818\) −97.5743 −3.41160
\(819\) 14.0443 0.490746
\(820\) 2.12243 0.0741185
\(821\) −5.52402 −0.192790 −0.0963948 0.995343i \(-0.530731\pi\)
−0.0963948 + 0.995343i \(0.530731\pi\)
\(822\) 42.7858 1.49233
\(823\) −3.26294 −0.113739 −0.0568694 0.998382i \(-0.518112\pi\)
−0.0568694 + 0.998382i \(0.518112\pi\)
\(824\) 0.943418 0.0328655
\(825\) 29.1160 1.01369
\(826\) −34.9161 −1.21489
\(827\) −20.4127 −0.709818 −0.354909 0.934901i \(-0.615488\pi\)
−0.354909 + 0.934901i \(0.615488\pi\)
\(828\) −9.92894 −0.345054
\(829\) 25.9529 0.901381 0.450691 0.892680i \(-0.351178\pi\)
0.450691 + 0.892680i \(0.351178\pi\)
\(830\) 23.4656 0.814502
\(831\) −35.1681 −1.21997
\(832\) −65.6798 −2.27704
\(833\) −1.38511 −0.0479912
\(834\) −68.6971 −2.37878
\(835\) 7.72901 0.267473
\(836\) 10.2648 0.355016
\(837\) 27.9549 0.966264
\(838\) 18.9833 0.655768
\(839\) 13.3088 0.459472 0.229736 0.973253i \(-0.426214\pi\)
0.229736 + 0.973253i \(0.426214\pi\)
\(840\) 27.8399 0.960568
\(841\) −23.7279 −0.818203
\(842\) 94.3288 3.25079
\(843\) 26.8676 0.925370
\(844\) −3.24872 −0.111825
\(845\) −2.81777 −0.0969343
\(846\) 30.2766 1.04093
\(847\) 74.5147 2.56036
\(848\) −0.504604 −0.0173282
\(849\) −1.70968 −0.0586760
\(850\) −9.76150 −0.334817
\(851\) −7.10437 −0.243535
\(852\) −37.9263 −1.29933
\(853\) −46.9946 −1.60906 −0.804532 0.593909i \(-0.797584\pi\)
−0.804532 + 0.593909i \(0.797584\pi\)
\(854\) −26.5801 −0.909552
\(855\) 0.538012 0.0183996
\(856\) −91.4579 −3.12597
\(857\) 2.55569 0.0873008 0.0436504 0.999047i \(-0.486101\pi\)
0.0436504 + 0.999047i \(0.486101\pi\)
\(858\) 83.3753 2.84639
\(859\) 6.56470 0.223985 0.111992 0.993709i \(-0.464277\pi\)
0.111992 + 0.993709i \(0.464277\pi\)
\(860\) 5.92234 0.201950
\(861\) 1.03705 0.0353425
\(862\) 23.6972 0.807129
\(863\) 10.0905 0.343484 0.171742 0.985142i \(-0.445060\pi\)
0.171742 + 0.985142i \(0.445060\pi\)
\(864\) 83.7801 2.85026
\(865\) 2.86328 0.0973545
\(866\) 13.6102 0.462494
\(867\) −1.22120 −0.0414742
\(868\) 61.5397 2.08879
\(869\) −71.7320 −2.43334
\(870\) −8.65974 −0.293593
\(871\) −29.9440 −1.01461
\(872\) 126.973 4.29986
\(873\) 26.9316 0.911496
\(874\) 1.05709 0.0357564
\(875\) 23.7572 0.803140
\(876\) 45.7815 1.54681
\(877\) −55.3115 −1.86774 −0.933868 0.357619i \(-0.883589\pi\)
−0.933868 + 0.357619i \(0.883589\pi\)
\(878\) 64.0223 2.16065
\(879\) 32.2783 1.08872
\(880\) −90.0258 −3.03477
\(881\) 8.63268 0.290842 0.145421 0.989370i \(-0.453546\pi\)
0.145421 + 0.989370i \(0.453546\pi\)
\(882\) 5.57403 0.187687
\(883\) 20.1129 0.676854 0.338427 0.940993i \(-0.390105\pi\)
0.338427 + 0.940993i \(0.390105\pi\)
\(884\) −20.0954 −0.675883
\(885\) −7.81052 −0.262548
\(886\) 5.71210 0.191902
\(887\) −27.7739 −0.932557 −0.466279 0.884638i \(-0.654406\pi\)
−0.466279 + 0.884638i \(0.654406\pi\)
\(888\) 56.0329 1.88034
\(889\) 40.7896 1.36804
\(890\) 38.3710 1.28620
\(891\) −14.3198 −0.479731
\(892\) 121.146 4.05628
\(893\) −2.31734 −0.0775467
\(894\) 60.6571 2.02868
\(895\) −21.2224 −0.709385
\(896\) 33.5604 1.12117
\(897\) 6.17265 0.206099
\(898\) 37.2732 1.24382
\(899\) −11.6578 −0.388808
\(900\) 28.2409 0.941362
\(901\) −0.0422801 −0.00140855
\(902\) −6.22808 −0.207372
\(903\) 2.89373 0.0962974
\(904\) 133.199 4.43013
\(905\) −14.4658 −0.480858
\(906\) −25.4756 −0.846370
\(907\) 24.3607 0.808883 0.404442 0.914564i \(-0.367466\pi\)
0.404442 + 0.914564i \(0.367466\pi\)
\(908\) 144.022 4.77954
\(909\) −7.66052 −0.254084
\(910\) 28.7495 0.953038
\(911\) −3.04070 −0.100743 −0.0503714 0.998731i \(-0.516041\pi\)
−0.0503714 + 0.998731i \(0.516041\pi\)
\(912\) −4.48923 −0.148653
\(913\) −49.5025 −1.63829
\(914\) 2.08672 0.0690224
\(915\) −5.94582 −0.196563
\(916\) −4.01648 −0.132708
\(917\) −5.00786 −0.165374
\(918\) 14.6869 0.484739
\(919\) 32.3552 1.06730 0.533649 0.845706i \(-0.320820\pi\)
0.533649 + 0.845706i \(0.320820\pi\)
\(920\) −12.3782 −0.408097
\(921\) −5.73303 −0.188910
\(922\) −102.585 −3.37846
\(923\) −23.8521 −0.785101
\(924\) −96.4364 −3.17252
\(925\) 20.2069 0.664400
\(926\) −10.8823 −0.357614
\(927\) −0.171285 −0.00562573
\(928\) −34.9380 −1.14689
\(929\) 1.68345 0.0552321 0.0276161 0.999619i \(-0.491208\pi\)
0.0276161 + 0.999619i \(0.491208\pi\)
\(930\) 19.1485 0.627905
\(931\) −0.426631 −0.0139823
\(932\) −5.09008 −0.166731
\(933\) 11.5450 0.377967
\(934\) −58.6489 −1.91905
\(935\) −7.54313 −0.246687
\(936\) 49.2500 1.60979
\(937\) −20.5243 −0.670500 −0.335250 0.942129i \(-0.608821\pi\)
−0.335250 + 0.942129i \(0.608821\pi\)
\(938\) 48.1768 1.57303
\(939\) −0.597067 −0.0194845
\(940\) 44.5568 1.45328
\(941\) 8.41695 0.274385 0.137192 0.990544i \(-0.456192\pi\)
0.137192 + 0.990544i \(0.456192\pi\)
\(942\) −62.0077 −2.02032
\(943\) −0.461093 −0.0150153
\(944\) −65.9290 −2.14581
\(945\) −15.1056 −0.491385
\(946\) −17.3786 −0.565026
\(947\) 0.843099 0.0273970 0.0136985 0.999906i \(-0.495639\pi\)
0.0136985 + 0.999906i \(0.495639\pi\)
\(948\) 68.7768 2.23377
\(949\) 28.7923 0.934637
\(950\) −3.00667 −0.0975491
\(951\) −3.64573 −0.118221
\(952\) 19.6901 0.638161
\(953\) 10.6277 0.344266 0.172133 0.985074i \(-0.444934\pi\)
0.172133 + 0.985074i \(0.444934\pi\)
\(954\) 0.170146 0.00550867
\(955\) −13.9839 −0.452507
\(956\) −36.3317 −1.17505
\(957\) 18.2684 0.590534
\(958\) −26.9784 −0.871634
\(959\) −31.1235 −1.00503
\(960\) 23.6382 0.762921
\(961\) −5.22220 −0.168458
\(962\) 57.8637 1.86560
\(963\) 16.6049 0.535085
\(964\) 153.444 4.94210
\(965\) 26.0584 0.838848
\(966\) −9.93116 −0.319530
\(967\) −41.9531 −1.34912 −0.674561 0.738219i \(-0.735667\pi\)
−0.674561 + 0.738219i \(0.735667\pi\)
\(968\) 261.306 8.39870
\(969\) −0.376146 −0.0120835
\(970\) 55.1307 1.77014
\(971\) −33.1186 −1.06283 −0.531414 0.847112i \(-0.678339\pi\)
−0.531414 + 0.847112i \(0.678339\pi\)
\(972\) −70.7628 −2.26972
\(973\) 49.9721 1.60203
\(974\) −91.0295 −2.91677
\(975\) −17.5569 −0.562269
\(976\) −50.1889 −1.60651
\(977\) −53.0473 −1.69713 −0.848566 0.529089i \(-0.822534\pi\)
−0.848566 + 0.529089i \(0.822534\pi\)
\(978\) 31.5359 1.00841
\(979\) −80.9468 −2.58707
\(980\) 8.20308 0.262038
\(981\) −23.0530 −0.736025
\(982\) 21.5271 0.686957
\(983\) 36.1599 1.15332 0.576661 0.816984i \(-0.304356\pi\)
0.576661 + 0.816984i \(0.304356\pi\)
\(984\) 3.63669 0.115933
\(985\) −10.0347 −0.319733
\(986\) −6.12471 −0.195051
\(987\) 21.7710 0.692980
\(988\) −6.18965 −0.196919
\(989\) −1.28661 −0.0409120
\(990\) 30.3555 0.964761
\(991\) 53.5511 1.70111 0.850554 0.525888i \(-0.176267\pi\)
0.850554 + 0.525888i \(0.176267\pi\)
\(992\) 77.2553 2.45286
\(993\) 7.85607 0.249305
\(994\) 38.3756 1.21720
\(995\) 11.3033 0.358337
\(996\) 47.4631 1.50393
\(997\) −23.1974 −0.734668 −0.367334 0.930089i \(-0.619729\pi\)
−0.367334 + 0.930089i \(0.619729\pi\)
\(998\) 41.0472 1.29933
\(999\) −30.4028 −0.961901
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 731.2.a.e.1.19 19
3.2 odd 2 6579.2.a.t.1.1 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.a.e.1.19 19 1.1 even 1 trivial
6579.2.a.t.1.1 19 3.2 odd 2