Properties

Label 103.2.a.a.1.2
Level $103$
Weight $2$
Character 103.1
Self dual yes
Analytic conductor $0.822$
Analytic rank $1$
Dimension $2$
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] = [103,2,Mod(1,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(0.822459140819\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 103.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} -2.61803 q^{5} +0.381966 q^{6} -1.00000 q^{7} +1.47214 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} -2.61803 q^{5} +0.381966 q^{6} -1.00000 q^{7} +1.47214 q^{8} -2.00000 q^{9} +1.00000 q^{10} -0.381966 q^{11} +1.85410 q^{12} +1.85410 q^{13} +0.381966 q^{14} +2.61803 q^{15} +3.14590 q^{16} -3.38197 q^{17} +0.763932 q^{18} -0.854102 q^{19} +4.85410 q^{20} +1.00000 q^{21} +0.145898 q^{22} -4.47214 q^{23} -1.47214 q^{24} +1.85410 q^{25} -0.708204 q^{26} +5.00000 q^{27} +1.85410 q^{28} -0.763932 q^{29} -1.00000 q^{30} +6.70820 q^{31} -4.14590 q^{32} +0.381966 q^{33} +1.29180 q^{34} +2.61803 q^{35} +3.70820 q^{36} -6.70820 q^{37} +0.326238 q^{38} -1.85410 q^{39} -3.85410 q^{40} -8.94427 q^{41} -0.381966 q^{42} +4.70820 q^{43} +0.708204 q^{44} +5.23607 q^{45} +1.70820 q^{46} -7.09017 q^{47} -3.14590 q^{48} -6.00000 q^{49} -0.708204 q^{50} +3.38197 q^{51} -3.43769 q^{52} -10.0902 q^{53} -1.90983 q^{54} +1.00000 q^{55} -1.47214 q^{56} +0.854102 q^{57} +0.291796 q^{58} +8.61803 q^{59} -4.85410 q^{60} +10.8541 q^{61} -2.56231 q^{62} +2.00000 q^{63} -4.70820 q^{64} -4.85410 q^{65} -0.145898 q^{66} -12.4164 q^{67} +6.27051 q^{68} +4.47214 q^{69} -1.00000 q^{70} +7.09017 q^{71} -2.94427 q^{72} -4.14590 q^{73} +2.56231 q^{74} -1.85410 q^{75} +1.58359 q^{76} +0.381966 q^{77} +0.708204 q^{78} +13.5623 q^{79} -8.23607 q^{80} +1.00000 q^{81} +3.41641 q^{82} +9.32624 q^{83} -1.85410 q^{84} +8.85410 q^{85} -1.79837 q^{86} +0.763932 q^{87} -0.562306 q^{88} -15.7082 q^{89} -2.00000 q^{90} -1.85410 q^{91} +8.29180 q^{92} -6.70820 q^{93} +2.70820 q^{94} +2.23607 q^{95} +4.14590 q^{96} +11.7082 q^{97} +2.29180 q^{98} +0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 2 q^{10} - 3 q^{11} - 3 q^{12} - 3 q^{13} + 3 q^{14} + 3 q^{15} + 13 q^{16} - 9 q^{17} + 6 q^{18} + 5 q^{19} + 3 q^{20} + 2 q^{21} + 7 q^{22} + 6 q^{24} - 3 q^{25} + 12 q^{26} + 10 q^{27} - 3 q^{28} - 6 q^{29} - 2 q^{30} - 15 q^{32} + 3 q^{33} + 16 q^{34} + 3 q^{35} - 6 q^{36} - 15 q^{38} + 3 q^{39} - q^{40} - 3 q^{42} - 4 q^{43} - 12 q^{44} + 6 q^{45} - 10 q^{46} - 3 q^{47} - 13 q^{48} - 12 q^{49} + 12 q^{50} + 9 q^{51} - 27 q^{52} - 9 q^{53} - 15 q^{54} + 2 q^{55} + 6 q^{56} - 5 q^{57} + 14 q^{58} + 15 q^{59} - 3 q^{60} + 15 q^{61} + 15 q^{62} + 4 q^{63} + 4 q^{64} - 3 q^{65} - 7 q^{66} + 2 q^{67} - 21 q^{68} - 2 q^{70} + 3 q^{71} + 12 q^{72} - 15 q^{73} - 15 q^{74} + 3 q^{75} + 30 q^{76} + 3 q^{77} - 12 q^{78} + 7 q^{79} - 12 q^{80} + 2 q^{81} - 20 q^{82} + 3 q^{83} + 3 q^{84} + 11 q^{85} + 21 q^{86} + 6 q^{87} + 19 q^{88} - 18 q^{89} - 4 q^{90} + 3 q^{91} + 30 q^{92} - 8 q^{94} + 15 q^{96} + 10 q^{97} + 18 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.381966 −0.270091 −0.135045 0.990839i \(-0.543118\pi\)
−0.135045 + 0.990839i \(0.543118\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −1.85410 −0.927051
\(5\) −2.61803 −1.17082 −0.585410 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) 0.381966 0.155937
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.47214 0.520479
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) −0.381966 −0.115167 −0.0575835 0.998341i \(-0.518340\pi\)
−0.0575835 + 0.998341i \(0.518340\pi\)
\(12\) 1.85410 0.535233
\(13\) 1.85410 0.514235 0.257118 0.966380i \(-0.417227\pi\)
0.257118 + 0.966380i \(0.417227\pi\)
\(14\) 0.381966 0.102085
\(15\) 2.61803 0.675973
\(16\) 3.14590 0.786475
\(17\) −3.38197 −0.820247 −0.410124 0.912030i \(-0.634514\pi\)
−0.410124 + 0.912030i \(0.634514\pi\)
\(18\) 0.763932 0.180061
\(19\) −0.854102 −0.195944 −0.0979722 0.995189i \(-0.531236\pi\)
−0.0979722 + 0.995189i \(0.531236\pi\)
\(20\) 4.85410 1.08541
\(21\) 1.00000 0.218218
\(22\) 0.145898 0.0311056
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) −1.47214 −0.300498
\(25\) 1.85410 0.370820
\(26\) −0.708204 −0.138890
\(27\) 5.00000 0.962250
\(28\) 1.85410 0.350392
\(29\) −0.763932 −0.141859 −0.0709293 0.997481i \(-0.522596\pi\)
−0.0709293 + 0.997481i \(0.522596\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) −4.14590 −0.732898
\(33\) 0.381966 0.0664917
\(34\) 1.29180 0.221541
\(35\) 2.61803 0.442529
\(36\) 3.70820 0.618034
\(37\) −6.70820 −1.10282 −0.551411 0.834234i \(-0.685910\pi\)
−0.551411 + 0.834234i \(0.685910\pi\)
\(38\) 0.326238 0.0529228
\(39\) −1.85410 −0.296894
\(40\) −3.85410 −0.609387
\(41\) −8.94427 −1.39686 −0.698430 0.715678i \(-0.746118\pi\)
−0.698430 + 0.715678i \(0.746118\pi\)
\(42\) −0.381966 −0.0589386
\(43\) 4.70820 0.717994 0.358997 0.933339i \(-0.383119\pi\)
0.358997 + 0.933339i \(0.383119\pi\)
\(44\) 0.708204 0.106766
\(45\) 5.23607 0.780547
\(46\) 1.70820 0.251861
\(47\) −7.09017 −1.03421 −0.517104 0.855923i \(-0.672990\pi\)
−0.517104 + 0.855923i \(0.672990\pi\)
\(48\) −3.14590 −0.454071
\(49\) −6.00000 −0.857143
\(50\) −0.708204 −0.100155
\(51\) 3.38197 0.473570
\(52\) −3.43769 −0.476722
\(53\) −10.0902 −1.38599 −0.692996 0.720942i \(-0.743710\pi\)
−0.692996 + 0.720942i \(0.743710\pi\)
\(54\) −1.90983 −0.259895
\(55\) 1.00000 0.134840
\(56\) −1.47214 −0.196722
\(57\) 0.854102 0.113129
\(58\) 0.291796 0.0383147
\(59\) 8.61803 1.12197 0.560986 0.827825i \(-0.310422\pi\)
0.560986 + 0.827825i \(0.310422\pi\)
\(60\) −4.85410 −0.626662
\(61\) 10.8541 1.38973 0.694863 0.719142i \(-0.255465\pi\)
0.694863 + 0.719142i \(0.255465\pi\)
\(62\) −2.56231 −0.325413
\(63\) 2.00000 0.251976
\(64\) −4.70820 −0.588525
\(65\) −4.85410 −0.602077
\(66\) −0.145898 −0.0179588
\(67\) −12.4164 −1.51691 −0.758453 0.651728i \(-0.774044\pi\)
−0.758453 + 0.651728i \(0.774044\pi\)
\(68\) 6.27051 0.760411
\(69\) 4.47214 0.538382
\(70\) −1.00000 −0.119523
\(71\) 7.09017 0.841448 0.420724 0.907189i \(-0.361776\pi\)
0.420724 + 0.907189i \(0.361776\pi\)
\(72\) −2.94427 −0.346986
\(73\) −4.14590 −0.485241 −0.242620 0.970121i \(-0.578007\pi\)
−0.242620 + 0.970121i \(0.578007\pi\)
\(74\) 2.56231 0.297862
\(75\) −1.85410 −0.214093
\(76\) 1.58359 0.181650
\(77\) 0.381966 0.0435291
\(78\) 0.708204 0.0801883
\(79\) 13.5623 1.52588 0.762939 0.646470i \(-0.223755\pi\)
0.762939 + 0.646470i \(0.223755\pi\)
\(80\) −8.23607 −0.920820
\(81\) 1.00000 0.111111
\(82\) 3.41641 0.377279
\(83\) 9.32624 1.02369 0.511844 0.859079i \(-0.328963\pi\)
0.511844 + 0.859079i \(0.328963\pi\)
\(84\) −1.85410 −0.202299
\(85\) 8.85410 0.960362
\(86\) −1.79837 −0.193924
\(87\) 0.763932 0.0819021
\(88\) −0.562306 −0.0599420
\(89\) −15.7082 −1.66507 −0.832533 0.553975i \(-0.813110\pi\)
−0.832533 + 0.553975i \(0.813110\pi\)
\(90\) −2.00000 −0.210819
\(91\) −1.85410 −0.194363
\(92\) 8.29180 0.864479
\(93\) −6.70820 −0.695608
\(94\) 2.70820 0.279330
\(95\) 2.23607 0.229416
\(96\) 4.14590 0.423139
\(97\) 11.7082 1.18879 0.594394 0.804174i \(-0.297392\pi\)
0.594394 + 0.804174i \(0.297392\pi\)
\(98\) 2.29180 0.231506
\(99\) 0.763932 0.0767781
\(100\) −3.43769 −0.343769
\(101\) 13.0902 1.30252 0.651260 0.758854i \(-0.274241\pi\)
0.651260 + 0.758854i \(0.274241\pi\)
\(102\) −1.29180 −0.127907
\(103\) −1.00000 −0.0985329
\(104\) 2.72949 0.267649
\(105\) −2.61803 −0.255494
\(106\) 3.85410 0.374343
\(107\) 4.09017 0.395412 0.197706 0.980261i \(-0.436651\pi\)
0.197706 + 0.980261i \(0.436651\pi\)
\(108\) −9.27051 −0.892055
\(109\) 10.5623 1.01169 0.505843 0.862626i \(-0.331182\pi\)
0.505843 + 0.862626i \(0.331182\pi\)
\(110\) −0.381966 −0.0364190
\(111\) 6.70820 0.636715
\(112\) −3.14590 −0.297259
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) −0.326238 −0.0305550
\(115\) 11.7082 1.09180
\(116\) 1.41641 0.131510
\(117\) −3.70820 −0.342824
\(118\) −3.29180 −0.303034
\(119\) 3.38197 0.310024
\(120\) 3.85410 0.351830
\(121\) −10.8541 −0.986737
\(122\) −4.14590 −0.375352
\(123\) 8.94427 0.806478
\(124\) −12.4377 −1.11694
\(125\) 8.23607 0.736656
\(126\) −0.763932 −0.0680565
\(127\) −18.2705 −1.62125 −0.810623 0.585569i \(-0.800871\pi\)
−0.810623 + 0.585569i \(0.800871\pi\)
\(128\) 10.0902 0.891853
\(129\) −4.70820 −0.414534
\(130\) 1.85410 0.162615
\(131\) 2.23607 0.195366 0.0976831 0.995218i \(-0.468857\pi\)
0.0976831 + 0.995218i \(0.468857\pi\)
\(132\) −0.708204 −0.0616412
\(133\) 0.854102 0.0740600
\(134\) 4.74265 0.409702
\(135\) −13.0902 −1.12662
\(136\) −4.97871 −0.426921
\(137\) 0.708204 0.0605059 0.0302530 0.999542i \(-0.490369\pi\)
0.0302530 + 0.999542i \(0.490369\pi\)
\(138\) −1.70820 −0.145412
\(139\) −17.8541 −1.51437 −0.757183 0.653203i \(-0.773425\pi\)
−0.757183 + 0.653203i \(0.773425\pi\)
\(140\) −4.85410 −0.410246
\(141\) 7.09017 0.597100
\(142\) −2.70820 −0.227267
\(143\) −0.708204 −0.0592230
\(144\) −6.29180 −0.524316
\(145\) 2.00000 0.166091
\(146\) 1.58359 0.131059
\(147\) 6.00000 0.494872
\(148\) 12.4377 1.02237
\(149\) 1.47214 0.120602 0.0603010 0.998180i \(-0.480794\pi\)
0.0603010 + 0.998180i \(0.480794\pi\)
\(150\) 0.708204 0.0578246
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) −1.25735 −0.101985
\(153\) 6.76393 0.546831
\(154\) −0.145898 −0.0117568
\(155\) −17.5623 −1.41064
\(156\) 3.43769 0.275236
\(157\) 3.29180 0.262714 0.131357 0.991335i \(-0.458067\pi\)
0.131357 + 0.991335i \(0.458067\pi\)
\(158\) −5.18034 −0.412126
\(159\) 10.0902 0.800203
\(160\) 10.8541 0.858092
\(161\) 4.47214 0.352454
\(162\) −0.381966 −0.0300101
\(163\) −10.7082 −0.838731 −0.419366 0.907817i \(-0.637747\pi\)
−0.419366 + 0.907817i \(0.637747\pi\)
\(164\) 16.5836 1.29496
\(165\) −1.00000 −0.0778499
\(166\) −3.56231 −0.276489
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 1.47214 0.113578
\(169\) −9.56231 −0.735562
\(170\) −3.38197 −0.259385
\(171\) 1.70820 0.130630
\(172\) −8.72949 −0.665617
\(173\) −13.0344 −0.990990 −0.495495 0.868611i \(-0.665013\pi\)
−0.495495 + 0.868611i \(0.665013\pi\)
\(174\) −0.291796 −0.0221210
\(175\) −1.85410 −0.140157
\(176\) −1.20163 −0.0905760
\(177\) −8.61803 −0.647771
\(178\) 6.00000 0.449719
\(179\) −1.14590 −0.0856484 −0.0428242 0.999083i \(-0.513636\pi\)
−0.0428242 + 0.999083i \(0.513636\pi\)
\(180\) −9.70820 −0.723607
\(181\) −2.85410 −0.212144 −0.106072 0.994358i \(-0.533827\pi\)
−0.106072 + 0.994358i \(0.533827\pi\)
\(182\) 0.708204 0.0524956
\(183\) −10.8541 −0.802358
\(184\) −6.58359 −0.485349
\(185\) 17.5623 1.29121
\(186\) 2.56231 0.187877
\(187\) 1.29180 0.0944655
\(188\) 13.1459 0.958763
\(189\) −5.00000 −0.363696
\(190\) −0.854102 −0.0619631
\(191\) 3.38197 0.244710 0.122355 0.992486i \(-0.460955\pi\)
0.122355 + 0.992486i \(0.460955\pi\)
\(192\) 4.70820 0.339785
\(193\) −20.1246 −1.44860 −0.724301 0.689484i \(-0.757837\pi\)
−0.724301 + 0.689484i \(0.757837\pi\)
\(194\) −4.47214 −0.321081
\(195\) 4.85410 0.347609
\(196\) 11.1246 0.794615
\(197\) 10.4164 0.742138 0.371069 0.928605i \(-0.378991\pi\)
0.371069 + 0.928605i \(0.378991\pi\)
\(198\) −0.291796 −0.0207370
\(199\) 3.41641 0.242183 0.121091 0.992641i \(-0.461361\pi\)
0.121091 + 0.992641i \(0.461361\pi\)
\(200\) 2.72949 0.193004
\(201\) 12.4164 0.875786
\(202\) −5.00000 −0.351799
\(203\) 0.763932 0.0536175
\(204\) −6.27051 −0.439024
\(205\) 23.4164 1.63547
\(206\) 0.381966 0.0266128
\(207\) 8.94427 0.621670
\(208\) 5.83282 0.404433
\(209\) 0.326238 0.0225663
\(210\) 1.00000 0.0690066
\(211\) 8.14590 0.560787 0.280393 0.959885i \(-0.409535\pi\)
0.280393 + 0.959885i \(0.409535\pi\)
\(212\) 18.7082 1.28488
\(213\) −7.09017 −0.485810
\(214\) −1.56231 −0.106797
\(215\) −12.3262 −0.840642
\(216\) 7.36068 0.500831
\(217\) −6.70820 −0.455383
\(218\) −4.03444 −0.273247
\(219\) 4.14590 0.280154
\(220\) −1.85410 −0.125004
\(221\) −6.27051 −0.421800
\(222\) −2.56231 −0.171971
\(223\) −5.70820 −0.382250 −0.191125 0.981566i \(-0.561214\pi\)
−0.191125 + 0.981566i \(0.561214\pi\)
\(224\) 4.14590 0.277009
\(225\) −3.70820 −0.247214
\(226\) 5.72949 0.381120
\(227\) −14.9443 −0.991886 −0.495943 0.868355i \(-0.665178\pi\)
−0.495943 + 0.868355i \(0.665178\pi\)
\(228\) −1.58359 −0.104876
\(229\) −6.70820 −0.443291 −0.221645 0.975127i \(-0.571143\pi\)
−0.221645 + 0.975127i \(0.571143\pi\)
\(230\) −4.47214 −0.294884
\(231\) −0.381966 −0.0251315
\(232\) −1.12461 −0.0738344
\(233\) 8.88854 0.582308 0.291154 0.956676i \(-0.405961\pi\)
0.291154 + 0.956676i \(0.405961\pi\)
\(234\) 1.41641 0.0925935
\(235\) 18.5623 1.21087
\(236\) −15.9787 −1.04013
\(237\) −13.5623 −0.880966
\(238\) −1.29180 −0.0837347
\(239\) 24.3262 1.57353 0.786767 0.617250i \(-0.211753\pi\)
0.786767 + 0.617250i \(0.211753\pi\)
\(240\) 8.23607 0.531636
\(241\) 25.2705 1.62782 0.813908 0.580993i \(-0.197336\pi\)
0.813908 + 0.580993i \(0.197336\pi\)
\(242\) 4.14590 0.266508
\(243\) −16.0000 −1.02640
\(244\) −20.1246 −1.28835
\(245\) 15.7082 1.00356
\(246\) −3.41641 −0.217822
\(247\) −1.58359 −0.100762
\(248\) 9.87539 0.627088
\(249\) −9.32624 −0.591026
\(250\) −3.14590 −0.198964
\(251\) 6.76393 0.426936 0.213468 0.976950i \(-0.431524\pi\)
0.213468 + 0.976950i \(0.431524\pi\)
\(252\) −3.70820 −0.233595
\(253\) 1.70820 0.107394
\(254\) 6.97871 0.437883
\(255\) −8.85410 −0.554465
\(256\) 5.56231 0.347644
\(257\) −4.52786 −0.282440 −0.141220 0.989978i \(-0.545103\pi\)
−0.141220 + 0.989978i \(0.545103\pi\)
\(258\) 1.79837 0.111962
\(259\) 6.70820 0.416828
\(260\) 9.00000 0.558156
\(261\) 1.52786 0.0945724
\(262\) −0.854102 −0.0527666
\(263\) −18.3820 −1.13348 −0.566740 0.823896i \(-0.691796\pi\)
−0.566740 + 0.823896i \(0.691796\pi\)
\(264\) 0.562306 0.0346075
\(265\) 26.4164 1.62275
\(266\) −0.326238 −0.0200029
\(267\) 15.7082 0.961326
\(268\) 23.0213 1.40625
\(269\) −3.32624 −0.202804 −0.101402 0.994846i \(-0.532333\pi\)
−0.101402 + 0.994846i \(0.532333\pi\)
\(270\) 5.00000 0.304290
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) −10.6393 −0.645104
\(273\) 1.85410 0.112215
\(274\) −0.270510 −0.0163421
\(275\) −0.708204 −0.0427063
\(276\) −8.29180 −0.499107
\(277\) −8.70820 −0.523225 −0.261613 0.965173i \(-0.584254\pi\)
−0.261613 + 0.965173i \(0.584254\pi\)
\(278\) 6.81966 0.409016
\(279\) −13.4164 −0.803219
\(280\) 3.85410 0.230327
\(281\) −22.5279 −1.34390 −0.671950 0.740597i \(-0.734543\pi\)
−0.671950 + 0.740597i \(0.734543\pi\)
\(282\) −2.70820 −0.161271
\(283\) −6.29180 −0.374008 −0.187004 0.982359i \(-0.559878\pi\)
−0.187004 + 0.982359i \(0.559878\pi\)
\(284\) −13.1459 −0.780066
\(285\) −2.23607 −0.132453
\(286\) 0.270510 0.0159956
\(287\) 8.94427 0.527964
\(288\) 8.29180 0.488599
\(289\) −5.56231 −0.327194
\(290\) −0.763932 −0.0448596
\(291\) −11.7082 −0.686347
\(292\) 7.68692 0.449843
\(293\) 9.65248 0.563904 0.281952 0.959429i \(-0.409018\pi\)
0.281952 + 0.959429i \(0.409018\pi\)
\(294\) −2.29180 −0.133660
\(295\) −22.5623 −1.31363
\(296\) −9.87539 −0.573995
\(297\) −1.90983 −0.110820
\(298\) −0.562306 −0.0325735
\(299\) −8.29180 −0.479527
\(300\) 3.43769 0.198475
\(301\) −4.70820 −0.271376
\(302\) 7.25735 0.417614
\(303\) −13.0902 −0.752011
\(304\) −2.68692 −0.154105
\(305\) −28.4164 −1.62712
\(306\) −2.58359 −0.147694
\(307\) 3.85410 0.219965 0.109983 0.993934i \(-0.464920\pi\)
0.109983 + 0.993934i \(0.464920\pi\)
\(308\) −0.708204 −0.0403537
\(309\) 1.00000 0.0568880
\(310\) 6.70820 0.381000
\(311\) −32.8885 −1.86494 −0.932469 0.361250i \(-0.882350\pi\)
−0.932469 + 0.361250i \(0.882350\pi\)
\(312\) −2.72949 −0.154527
\(313\) 29.7082 1.67921 0.839603 0.543200i \(-0.182787\pi\)
0.839603 + 0.543200i \(0.182787\pi\)
\(314\) −1.25735 −0.0709566
\(315\) −5.23607 −0.295019
\(316\) −25.1459 −1.41457
\(317\) 1.58359 0.0889434 0.0444717 0.999011i \(-0.485840\pi\)
0.0444717 + 0.999011i \(0.485840\pi\)
\(318\) −3.85410 −0.216127
\(319\) 0.291796 0.0163374
\(320\) 12.3262 0.689058
\(321\) −4.09017 −0.228291
\(322\) −1.70820 −0.0951945
\(323\) 2.88854 0.160723
\(324\) −1.85410 −0.103006
\(325\) 3.43769 0.190689
\(326\) 4.09017 0.226534
\(327\) −10.5623 −0.584097
\(328\) −13.1672 −0.727036
\(329\) 7.09017 0.390894
\(330\) 0.381966 0.0210265
\(331\) −22.8541 −1.25618 −0.628088 0.778143i \(-0.716162\pi\)
−0.628088 + 0.778143i \(0.716162\pi\)
\(332\) −17.2918 −0.949011
\(333\) 13.4164 0.735215
\(334\) −3.43769 −0.188102
\(335\) 32.5066 1.77602
\(336\) 3.14590 0.171623
\(337\) −22.5623 −1.22905 −0.614524 0.788898i \(-0.710652\pi\)
−0.614524 + 0.788898i \(0.710652\pi\)
\(338\) 3.65248 0.198668
\(339\) 15.0000 0.814688
\(340\) −16.4164 −0.890305
\(341\) −2.56231 −0.138757
\(342\) −0.652476 −0.0352819
\(343\) 13.0000 0.701934
\(344\) 6.93112 0.373701
\(345\) −11.7082 −0.630349
\(346\) 4.97871 0.267657
\(347\) 7.47214 0.401125 0.200563 0.979681i \(-0.435723\pi\)
0.200563 + 0.979681i \(0.435723\pi\)
\(348\) −1.41641 −0.0759274
\(349\) −11.4164 −0.611106 −0.305553 0.952175i \(-0.598841\pi\)
−0.305553 + 0.952175i \(0.598841\pi\)
\(350\) 0.708204 0.0378551
\(351\) 9.27051 0.494823
\(352\) 1.58359 0.0844057
\(353\) 4.03444 0.214732 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(354\) 3.29180 0.174957
\(355\) −18.5623 −0.985185
\(356\) 29.1246 1.54360
\(357\) −3.38197 −0.178993
\(358\) 0.437694 0.0231329
\(359\) 30.3262 1.60056 0.800279 0.599628i \(-0.204685\pi\)
0.800279 + 0.599628i \(0.204685\pi\)
\(360\) 7.70820 0.406258
\(361\) −18.2705 −0.961606
\(362\) 1.09017 0.0572981
\(363\) 10.8541 0.569693
\(364\) 3.43769 0.180184
\(365\) 10.8541 0.568130
\(366\) 4.14590 0.216710
\(367\) 36.5623 1.90854 0.954268 0.298951i \(-0.0966368\pi\)
0.954268 + 0.298951i \(0.0966368\pi\)
\(368\) −14.0689 −0.733391
\(369\) 17.8885 0.931240
\(370\) −6.70820 −0.348743
\(371\) 10.0902 0.523856
\(372\) 12.4377 0.644864
\(373\) 37.6869 1.95135 0.975677 0.219212i \(-0.0703485\pi\)
0.975677 + 0.219212i \(0.0703485\pi\)
\(374\) −0.493422 −0.0255143
\(375\) −8.23607 −0.425309
\(376\) −10.4377 −0.538283
\(377\) −1.41641 −0.0729487
\(378\) 1.90983 0.0982311
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) −4.14590 −0.212680
\(381\) 18.2705 0.936027
\(382\) −1.29180 −0.0660940
\(383\) 23.1803 1.18446 0.592230 0.805769i \(-0.298248\pi\)
0.592230 + 0.805769i \(0.298248\pi\)
\(384\) −10.0902 −0.514912
\(385\) −1.00000 −0.0509647
\(386\) 7.68692 0.391254
\(387\) −9.41641 −0.478663
\(388\) −21.7082 −1.10207
\(389\) −19.4164 −0.984451 −0.492225 0.870468i \(-0.663816\pi\)
−0.492225 + 0.870468i \(0.663816\pi\)
\(390\) −1.85410 −0.0938861
\(391\) 15.1246 0.764884
\(392\) −8.83282 −0.446125
\(393\) −2.23607 −0.112795
\(394\) −3.97871 −0.200445
\(395\) −35.5066 −1.78653
\(396\) −1.41641 −0.0711772
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) −1.30495 −0.0654113
\(399\) −0.854102 −0.0427586
\(400\) 5.83282 0.291641
\(401\) 11.8885 0.593686 0.296843 0.954926i \(-0.404066\pi\)
0.296843 + 0.954926i \(0.404066\pi\)
\(402\) −4.74265 −0.236542
\(403\) 12.4377 0.619566
\(404\) −24.2705 −1.20750
\(405\) −2.61803 −0.130091
\(406\) −0.291796 −0.0144816
\(407\) 2.56231 0.127009
\(408\) 4.97871 0.246483
\(409\) 23.2918 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(410\) −8.94427 −0.441726
\(411\) −0.708204 −0.0349331
\(412\) 1.85410 0.0913450
\(413\) −8.61803 −0.424066
\(414\) −3.41641 −0.167907
\(415\) −24.4164 −1.19855
\(416\) −7.68692 −0.376882
\(417\) 17.8541 0.874319
\(418\) −0.124612 −0.00609496
\(419\) 7.09017 0.346377 0.173189 0.984889i \(-0.444593\pi\)
0.173189 + 0.984889i \(0.444593\pi\)
\(420\) 4.85410 0.236856
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) −3.11146 −0.151463
\(423\) 14.1803 0.689472
\(424\) −14.8541 −0.721379
\(425\) −6.27051 −0.304164
\(426\) 2.70820 0.131213
\(427\) −10.8541 −0.525267
\(428\) −7.58359 −0.366567
\(429\) 0.708204 0.0341924
\(430\) 4.70820 0.227050
\(431\) −10.3607 −0.499056 −0.249528 0.968368i \(-0.580276\pi\)
−0.249528 + 0.968368i \(0.580276\pi\)
\(432\) 15.7295 0.756785
\(433\) −12.4164 −0.596694 −0.298347 0.954457i \(-0.596435\pi\)
−0.298347 + 0.954457i \(0.596435\pi\)
\(434\) 2.56231 0.122995
\(435\) −2.00000 −0.0958927
\(436\) −19.5836 −0.937884
\(437\) 3.81966 0.182719
\(438\) −1.58359 −0.0756670
\(439\) 9.43769 0.450437 0.225218 0.974308i \(-0.427690\pi\)
0.225218 + 0.974308i \(0.427690\pi\)
\(440\) 1.47214 0.0701813
\(441\) 12.0000 0.571429
\(442\) 2.39512 0.113924
\(443\) −35.5623 −1.68962 −0.844808 0.535069i \(-0.820285\pi\)
−0.844808 + 0.535069i \(0.820285\pi\)
\(444\) −12.4377 −0.590267
\(445\) 41.1246 1.94949
\(446\) 2.18034 0.103242
\(447\) −1.47214 −0.0696296
\(448\) 4.70820 0.222442
\(449\) 31.3607 1.48000 0.740001 0.672606i \(-0.234825\pi\)
0.740001 + 0.672606i \(0.234825\pi\)
\(450\) 1.41641 0.0667701
\(451\) 3.41641 0.160872
\(452\) 27.8115 1.30814
\(453\) 19.0000 0.892698
\(454\) 5.70820 0.267899
\(455\) 4.85410 0.227564
\(456\) 1.25735 0.0588810
\(457\) −3.14590 −0.147159 −0.0735795 0.997289i \(-0.523442\pi\)
−0.0735795 + 0.997289i \(0.523442\pi\)
\(458\) 2.56231 0.119729
\(459\) −16.9098 −0.789283
\(460\) −21.7082 −1.01215
\(461\) −39.2148 −1.82641 −0.913207 0.407495i \(-0.866402\pi\)
−0.913207 + 0.407495i \(0.866402\pi\)
\(462\) 0.145898 0.00678779
\(463\) 5.41641 0.251722 0.125861 0.992048i \(-0.459831\pi\)
0.125861 + 0.992048i \(0.459831\pi\)
\(464\) −2.40325 −0.111568
\(465\) 17.5623 0.814432
\(466\) −3.39512 −0.157276
\(467\) 27.6525 1.27960 0.639802 0.768540i \(-0.279016\pi\)
0.639802 + 0.768540i \(0.279016\pi\)
\(468\) 6.87539 0.317815
\(469\) 12.4164 0.573336
\(470\) −7.09017 −0.327045
\(471\) −3.29180 −0.151678
\(472\) 12.6869 0.583963
\(473\) −1.79837 −0.0826893
\(474\) 5.18034 0.237941
\(475\) −1.58359 −0.0726602
\(476\) −6.27051 −0.287408
\(477\) 20.1803 0.923994
\(478\) −9.29180 −0.424997
\(479\) −14.1803 −0.647916 −0.323958 0.946071i \(-0.605014\pi\)
−0.323958 + 0.946071i \(0.605014\pi\)
\(480\) −10.8541 −0.495420
\(481\) −12.4377 −0.567110
\(482\) −9.65248 −0.439658
\(483\) −4.47214 −0.203489
\(484\) 20.1246 0.914755
\(485\) −30.6525 −1.39186
\(486\) 6.11146 0.277221
\(487\) −23.0000 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) 15.9787 0.723322
\(489\) 10.7082 0.484242
\(490\) −6.00000 −0.271052
\(491\) −35.7771 −1.61460 −0.807299 0.590143i \(-0.799071\pi\)
−0.807299 + 0.590143i \(0.799071\pi\)
\(492\) −16.5836 −0.747646
\(493\) 2.58359 0.116359
\(494\) 0.604878 0.0272148
\(495\) −2.00000 −0.0898933
\(496\) 21.1033 0.947567
\(497\) −7.09017 −0.318038
\(498\) 3.56231 0.159631
\(499\) −20.2705 −0.907433 −0.453716 0.891146i \(-0.649902\pi\)
−0.453716 + 0.891146i \(0.649902\pi\)
\(500\) −15.2705 −0.682918
\(501\) −9.00000 −0.402090
\(502\) −2.58359 −0.115311
\(503\) −31.3607 −1.39830 −0.699152 0.714973i \(-0.746439\pi\)
−0.699152 + 0.714973i \(0.746439\pi\)
\(504\) 2.94427 0.131148
\(505\) −34.2705 −1.52502
\(506\) −0.652476 −0.0290061
\(507\) 9.56231 0.424677
\(508\) 33.8754 1.50298
\(509\) 24.3820 1.08071 0.540356 0.841437i \(-0.318290\pi\)
0.540356 + 0.841437i \(0.318290\pi\)
\(510\) 3.38197 0.149756
\(511\) 4.14590 0.183404
\(512\) −22.3050 −0.985749
\(513\) −4.27051 −0.188548
\(514\) 1.72949 0.0762845
\(515\) 2.61803 0.115364
\(516\) 8.72949 0.384294
\(517\) 2.70820 0.119107
\(518\) −2.56231 −0.112581
\(519\) 13.0344 0.572148
\(520\) −7.14590 −0.313368
\(521\) −5.18034 −0.226955 −0.113477 0.993541i \(-0.536199\pi\)
−0.113477 + 0.993541i \(0.536199\pi\)
\(522\) −0.583592 −0.0255431
\(523\) −37.4164 −1.63611 −0.818053 0.575143i \(-0.804946\pi\)
−0.818053 + 0.575143i \(0.804946\pi\)
\(524\) −4.14590 −0.181114
\(525\) 1.85410 0.0809196
\(526\) 7.02129 0.306143
\(527\) −22.6869 −0.988258
\(528\) 1.20163 0.0522941
\(529\) −3.00000 −0.130435
\(530\) −10.0902 −0.438289
\(531\) −17.2361 −0.747982
\(532\) −1.58359 −0.0686574
\(533\) −16.5836 −0.718315
\(534\) −6.00000 −0.259645
\(535\) −10.7082 −0.462956
\(536\) −18.2786 −0.789517
\(537\) 1.14590 0.0494492
\(538\) 1.27051 0.0547756
\(539\) 2.29180 0.0987146
\(540\) 24.2705 1.04444
\(541\) 15.8541 0.681621 0.340811 0.940132i \(-0.389299\pi\)
0.340811 + 0.940132i \(0.389299\pi\)
\(542\) −0.381966 −0.0164068
\(543\) 2.85410 0.122481
\(544\) 14.0213 0.601158
\(545\) −27.6525 −1.18450
\(546\) −0.708204 −0.0303083
\(547\) 31.2705 1.33703 0.668515 0.743698i \(-0.266930\pi\)
0.668515 + 0.743698i \(0.266930\pi\)
\(548\) −1.31308 −0.0560921
\(549\) −21.7082 −0.926484
\(550\) 0.270510 0.0115346
\(551\) 0.652476 0.0277964
\(552\) 6.58359 0.280216
\(553\) −13.5623 −0.576728
\(554\) 3.32624 0.141318
\(555\) −17.5623 −0.745478
\(556\) 33.1033 1.40389
\(557\) 28.6869 1.21550 0.607752 0.794127i \(-0.292072\pi\)
0.607752 + 0.794127i \(0.292072\pi\)
\(558\) 5.12461 0.216942
\(559\) 8.72949 0.369218
\(560\) 8.23607 0.348037
\(561\) −1.29180 −0.0545397
\(562\) 8.60488 0.362975
\(563\) −25.7984 −1.08727 −0.543636 0.839321i \(-0.682953\pi\)
−0.543636 + 0.839321i \(0.682953\pi\)
\(564\) −13.1459 −0.553542
\(565\) 39.2705 1.65212
\(566\) 2.40325 0.101016
\(567\) −1.00000 −0.0419961
\(568\) 10.4377 0.437956
\(569\) 42.1591 1.76740 0.883700 0.468054i \(-0.155045\pi\)
0.883700 + 0.468054i \(0.155045\pi\)
\(570\) 0.854102 0.0357744
\(571\) −2.43769 −0.102014 −0.0510072 0.998698i \(-0.516243\pi\)
−0.0510072 + 0.998698i \(0.516243\pi\)
\(572\) 1.31308 0.0549027
\(573\) −3.38197 −0.141284
\(574\) −3.41641 −0.142598
\(575\) −8.29180 −0.345792
\(576\) 9.41641 0.392350
\(577\) 16.8328 0.700759 0.350380 0.936608i \(-0.386053\pi\)
0.350380 + 0.936608i \(0.386053\pi\)
\(578\) 2.12461 0.0883722
\(579\) 20.1246 0.836350
\(580\) −3.70820 −0.153975
\(581\) −9.32624 −0.386918
\(582\) 4.47214 0.185376
\(583\) 3.85410 0.159621
\(584\) −6.10333 −0.252557
\(585\) 9.70820 0.401385
\(586\) −3.68692 −0.152305
\(587\) 11.0689 0.456862 0.228431 0.973560i \(-0.426641\pi\)
0.228431 + 0.973560i \(0.426641\pi\)
\(588\) −11.1246 −0.458771
\(589\) −5.72949 −0.236080
\(590\) 8.61803 0.354799
\(591\) −10.4164 −0.428474
\(592\) −21.1033 −0.867341
\(593\) −20.1803 −0.828707 −0.414354 0.910116i \(-0.635992\pi\)
−0.414354 + 0.910116i \(0.635992\pi\)
\(594\) 0.729490 0.0299313
\(595\) −8.85410 −0.362983
\(596\) −2.72949 −0.111804
\(597\) −3.41641 −0.139824
\(598\) 3.16718 0.129516
\(599\) 20.4508 0.835599 0.417800 0.908539i \(-0.362801\pi\)
0.417800 + 0.908539i \(0.362801\pi\)
\(600\) −2.72949 −0.111431
\(601\) −16.5623 −0.675591 −0.337795 0.941220i \(-0.609681\pi\)
−0.337795 + 0.941220i \(0.609681\pi\)
\(602\) 1.79837 0.0732962
\(603\) 24.8328 1.01127
\(604\) 35.2279 1.43340
\(605\) 28.4164 1.15529
\(606\) 5.00000 0.203111
\(607\) −7.70820 −0.312866 −0.156433 0.987689i \(-0.550000\pi\)
−0.156433 + 0.987689i \(0.550000\pi\)
\(608\) 3.54102 0.143607
\(609\) −0.763932 −0.0309561
\(610\) 10.8541 0.439470
\(611\) −13.1459 −0.531826
\(612\) −12.5410 −0.506941
\(613\) −2.41641 −0.0975978 −0.0487989 0.998809i \(-0.515539\pi\)
−0.0487989 + 0.998809i \(0.515539\pi\)
\(614\) −1.47214 −0.0594106
\(615\) −23.4164 −0.944241
\(616\) 0.562306 0.0226560
\(617\) −30.2705 −1.21864 −0.609322 0.792923i \(-0.708558\pi\)
−0.609322 + 0.792923i \(0.708558\pi\)
\(618\) −0.381966 −0.0153649
\(619\) 31.6869 1.27360 0.636802 0.771027i \(-0.280257\pi\)
0.636802 + 0.771027i \(0.280257\pi\)
\(620\) 32.5623 1.30773
\(621\) −22.3607 −0.897303
\(622\) 12.5623 0.503703
\(623\) 15.7082 0.629336
\(624\) −5.83282 −0.233500
\(625\) −30.8328 −1.23331
\(626\) −11.3475 −0.453538
\(627\) −0.326238 −0.0130287
\(628\) −6.10333 −0.243549
\(629\) 22.6869 0.904587
\(630\) 2.00000 0.0796819
\(631\) 8.72949 0.347516 0.173758 0.984788i \(-0.444409\pi\)
0.173758 + 0.984788i \(0.444409\pi\)
\(632\) 19.9656 0.794187
\(633\) −8.14590 −0.323770
\(634\) −0.604878 −0.0240228
\(635\) 47.8328 1.89819
\(636\) −18.7082 −0.741829
\(637\) −11.1246 −0.440773
\(638\) −0.111456 −0.00441259
\(639\) −14.1803 −0.560966
\(640\) −26.4164 −1.04420
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 1.56231 0.0616593
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) −8.29180 −0.326743
\(645\) 12.3262 0.485345
\(646\) −1.10333 −0.0434098
\(647\) 43.7426 1.71970 0.859850 0.510546i \(-0.170557\pi\)
0.859850 + 0.510546i \(0.170557\pi\)
\(648\) 1.47214 0.0578310
\(649\) −3.29180 −0.129214
\(650\) −1.31308 −0.0515033
\(651\) 6.70820 0.262915
\(652\) 19.8541 0.777547
\(653\) −0.763932 −0.0298950 −0.0149475 0.999888i \(-0.504758\pi\)
−0.0149475 + 0.999888i \(0.504758\pi\)
\(654\) 4.03444 0.157759
\(655\) −5.85410 −0.228739
\(656\) −28.1378 −1.09860
\(657\) 8.29180 0.323494
\(658\) −2.70820 −0.105577
\(659\) −12.5967 −0.490700 −0.245350 0.969435i \(-0.578903\pi\)
−0.245350 + 0.969435i \(0.578903\pi\)
\(660\) 1.85410 0.0721708
\(661\) −11.4377 −0.444875 −0.222437 0.974947i \(-0.571401\pi\)
−0.222437 + 0.974947i \(0.571401\pi\)
\(662\) 8.72949 0.339281
\(663\) 6.27051 0.243526
\(664\) 13.7295 0.532808
\(665\) −2.23607 −0.0867110
\(666\) −5.12461 −0.198575
\(667\) 3.41641 0.132284
\(668\) −16.6869 −0.645636
\(669\) 5.70820 0.220692
\(670\) −12.4164 −0.479688
\(671\) −4.14590 −0.160051
\(672\) −4.14590 −0.159931
\(673\) −37.7082 −1.45354 −0.726772 0.686879i \(-0.758980\pi\)
−0.726772 + 0.686879i \(0.758980\pi\)
\(674\) 8.61803 0.331954
\(675\) 9.27051 0.356822
\(676\) 17.7295 0.681903
\(677\) 10.9656 0.421441 0.210720 0.977546i \(-0.432419\pi\)
0.210720 + 0.977546i \(0.432419\pi\)
\(678\) −5.72949 −0.220040
\(679\) −11.7082 −0.449320
\(680\) 13.0344 0.499848
\(681\) 14.9443 0.572666
\(682\) 0.978714 0.0374769
\(683\) 42.7639 1.63632 0.818158 0.574993i \(-0.194995\pi\)
0.818158 + 0.574993i \(0.194995\pi\)
\(684\) −3.16718 −0.121100
\(685\) −1.85410 −0.0708416
\(686\) −4.96556 −0.189586
\(687\) 6.70820 0.255934
\(688\) 14.8115 0.564684
\(689\) −18.7082 −0.712726
\(690\) 4.47214 0.170251
\(691\) 1.14590 0.0435920 0.0217960 0.999762i \(-0.493062\pi\)
0.0217960 + 0.999762i \(0.493062\pi\)
\(692\) 24.1672 0.918698
\(693\) −0.763932 −0.0290194
\(694\) −2.85410 −0.108340
\(695\) 46.7426 1.77305
\(696\) 1.12461 0.0426283
\(697\) 30.2492 1.14577
\(698\) 4.36068 0.165054
\(699\) −8.88854 −0.336196
\(700\) 3.43769 0.129933
\(701\) −1.20163 −0.0453848 −0.0226924 0.999742i \(-0.507224\pi\)
−0.0226924 + 0.999742i \(0.507224\pi\)
\(702\) −3.54102 −0.133647
\(703\) 5.72949 0.216092
\(704\) 1.79837 0.0677788
\(705\) −18.5623 −0.699097
\(706\) −1.54102 −0.0579970
\(707\) −13.0902 −0.492307
\(708\) 15.9787 0.600517
\(709\) −47.9787 −1.80188 −0.900939 0.433945i \(-0.857121\pi\)
−0.900939 + 0.433945i \(0.857121\pi\)
\(710\) 7.09017 0.266089
\(711\) −27.1246 −1.01725
\(712\) −23.1246 −0.866631
\(713\) −30.0000 −1.12351
\(714\) 1.29180 0.0483443
\(715\) 1.85410 0.0693395
\(716\) 2.12461 0.0794005
\(717\) −24.3262 −0.908480
\(718\) −11.5836 −0.432296
\(719\) 23.6738 0.882882 0.441441 0.897290i \(-0.354467\pi\)
0.441441 + 0.897290i \(0.354467\pi\)
\(720\) 16.4721 0.613880
\(721\) 1.00000 0.0372419
\(722\) 6.97871 0.259721
\(723\) −25.2705 −0.939820
\(724\) 5.29180 0.196668
\(725\) −1.41641 −0.0526041
\(726\) −4.14590 −0.153869
\(727\) −38.2705 −1.41937 −0.709687 0.704517i \(-0.751164\pi\)
−0.709687 + 0.704517i \(0.751164\pi\)
\(728\) −2.72949 −0.101162
\(729\) 13.0000 0.481481
\(730\) −4.14590 −0.153447
\(731\) −15.9230 −0.588933
\(732\) 20.1246 0.743827
\(733\) 1.29180 0.0477136 0.0238568 0.999715i \(-0.492405\pi\)
0.0238568 + 0.999715i \(0.492405\pi\)
\(734\) −13.9656 −0.515478
\(735\) −15.7082 −0.579406
\(736\) 18.5410 0.683431
\(737\) 4.74265 0.174698
\(738\) −6.83282 −0.251519
\(739\) 36.8328 1.35492 0.677459 0.735561i \(-0.263081\pi\)
0.677459 + 0.735561i \(0.263081\pi\)
\(740\) −32.5623 −1.19701
\(741\) 1.58359 0.0581747
\(742\) −3.85410 −0.141489
\(743\) 17.2918 0.634374 0.317187 0.948363i \(-0.397262\pi\)
0.317187 + 0.948363i \(0.397262\pi\)
\(744\) −9.87539 −0.362049
\(745\) −3.85410 −0.141203
\(746\) −14.3951 −0.527043
\(747\) −18.6525 −0.682458
\(748\) −2.39512 −0.0875743
\(749\) −4.09017 −0.149452
\(750\) 3.14590 0.114872
\(751\) −3.87539 −0.141415 −0.0707075 0.997497i \(-0.522526\pi\)
−0.0707075 + 0.997497i \(0.522526\pi\)
\(752\) −22.3050 −0.813378
\(753\) −6.76393 −0.246491
\(754\) 0.541020 0.0197028
\(755\) 49.7426 1.81032
\(756\) 9.27051 0.337165
\(757\) −34.7082 −1.26149 −0.630746 0.775990i \(-0.717251\pi\)
−0.630746 + 0.775990i \(0.717251\pi\)
\(758\) −1.90983 −0.0693682
\(759\) −1.70820 −0.0620039
\(760\) 3.29180 0.119406
\(761\) 1.47214 0.0533649 0.0266824 0.999644i \(-0.491506\pi\)
0.0266824 + 0.999644i \(0.491506\pi\)
\(762\) −6.97871 −0.252812
\(763\) −10.5623 −0.382381
\(764\) −6.27051 −0.226859
\(765\) −17.7082 −0.640241
\(766\) −8.85410 −0.319912
\(767\) 15.9787 0.576958
\(768\) −5.56231 −0.200712
\(769\) −42.3951 −1.52881 −0.764404 0.644738i \(-0.776966\pi\)
−0.764404 + 0.644738i \(0.776966\pi\)
\(770\) 0.381966 0.0137651
\(771\) 4.52786 0.163067
\(772\) 37.3131 1.34293
\(773\) −25.4721 −0.916169 −0.458085 0.888909i \(-0.651464\pi\)
−0.458085 + 0.888909i \(0.651464\pi\)
\(774\) 3.59675 0.129282
\(775\) 12.4377 0.446775
\(776\) 17.2361 0.618739
\(777\) −6.70820 −0.240655
\(778\) 7.41641 0.265891
\(779\) 7.63932 0.273707
\(780\) −9.00000 −0.322252
\(781\) −2.70820 −0.0969072
\(782\) −5.77709 −0.206588
\(783\) −3.81966 −0.136504
\(784\) −18.8754 −0.674121
\(785\) −8.61803 −0.307591
\(786\) 0.854102 0.0304648
\(787\) 16.5836 0.591141 0.295571 0.955321i \(-0.404490\pi\)
0.295571 + 0.955321i \(0.404490\pi\)
\(788\) −19.3131 −0.688000
\(789\) 18.3820 0.654415
\(790\) 13.5623 0.482525
\(791\) 15.0000 0.533339
\(792\) 1.12461 0.0399613
\(793\) 20.1246 0.714646
\(794\) −7.63932 −0.271109
\(795\) −26.4164 −0.936893
\(796\) −6.33437 −0.224516
\(797\) −9.87539 −0.349804 −0.174902 0.984586i \(-0.555961\pi\)
−0.174902 + 0.984586i \(0.555961\pi\)
\(798\) 0.326238 0.0115487
\(799\) 23.9787 0.848306
\(800\) −7.68692 −0.271774
\(801\) 31.4164 1.11004
\(802\) −4.54102 −0.160349
\(803\) 1.58359 0.0558838
\(804\) −23.0213 −0.811898
\(805\) −11.7082 −0.412660
\(806\) −4.75078 −0.167339
\(807\) 3.32624 0.117089
\(808\) 19.2705 0.677934
\(809\) −14.9443 −0.525413 −0.262706 0.964876i \(-0.584615\pi\)
−0.262706 + 0.964876i \(0.584615\pi\)
\(810\) 1.00000 0.0351364
\(811\) −45.5410 −1.59916 −0.799581 0.600559i \(-0.794945\pi\)
−0.799581 + 0.600559i \(0.794945\pi\)
\(812\) −1.41641 −0.0497062
\(813\) −1.00000 −0.0350715
\(814\) −0.978714 −0.0343039
\(815\) 28.0344 0.982004
\(816\) 10.6393 0.372451
\(817\) −4.02129 −0.140687
\(818\) −8.89667 −0.311065
\(819\) 3.70820 0.129575
\(820\) −43.4164 −1.51617
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0.270510 0.00943511
\(823\) 23.5623 0.821330 0.410665 0.911786i \(-0.365297\pi\)
0.410665 + 0.911786i \(0.365297\pi\)
\(824\) −1.47214 −0.0512843
\(825\) 0.708204 0.0246565
\(826\) 3.29180 0.114536
\(827\) −6.70820 −0.233267 −0.116634 0.993175i \(-0.537210\pi\)
−0.116634 + 0.993175i \(0.537210\pi\)
\(828\) −16.5836 −0.576320
\(829\) −19.2705 −0.669292 −0.334646 0.942344i \(-0.608617\pi\)
−0.334646 + 0.942344i \(0.608617\pi\)
\(830\) 9.32624 0.323718
\(831\) 8.70820 0.302084
\(832\) −8.72949 −0.302641
\(833\) 20.2918 0.703069
\(834\) −6.81966 −0.236146
\(835\) −23.5623 −0.815407
\(836\) −0.604878 −0.0209202
\(837\) 33.5410 1.15935
\(838\) −2.70820 −0.0935534
\(839\) 21.3820 0.738187 0.369094 0.929392i \(-0.379668\pi\)
0.369094 + 0.929392i \(0.379668\pi\)
\(840\) −3.85410 −0.132979
\(841\) −28.4164 −0.979876
\(842\) −1.14590 −0.0394903
\(843\) 22.5279 0.775901
\(844\) −15.1033 −0.519878
\(845\) 25.0344 0.861211
\(846\) −5.41641 −0.186220
\(847\) 10.8541 0.372951
\(848\) −31.7426 −1.09005
\(849\) 6.29180 0.215934
\(850\) 2.39512 0.0821520
\(851\) 30.0000 1.02839
\(852\) 13.1459 0.450371
\(853\) −10.7295 −0.367371 −0.183685 0.982985i \(-0.558803\pi\)
−0.183685 + 0.982985i \(0.558803\pi\)
\(854\) 4.14590 0.141870
\(855\) −4.47214 −0.152944
\(856\) 6.02129 0.205803
\(857\) 3.76393 0.128573 0.0642867 0.997931i \(-0.479523\pi\)
0.0642867 + 0.997931i \(0.479523\pi\)
\(858\) −0.270510 −0.00923505
\(859\) −9.56231 −0.326262 −0.163131 0.986604i \(-0.552159\pi\)
−0.163131 + 0.986604i \(0.552159\pi\)
\(860\) 22.8541 0.779318
\(861\) −8.94427 −0.304820
\(862\) 3.95743 0.134791
\(863\) −8.29180 −0.282256 −0.141128 0.989991i \(-0.545073\pi\)
−0.141128 + 0.989991i \(0.545073\pi\)
\(864\) −20.7295 −0.705232
\(865\) 34.1246 1.16027
\(866\) 4.74265 0.161162
\(867\) 5.56231 0.188906
\(868\) 12.4377 0.422163
\(869\) −5.18034 −0.175731
\(870\) 0.763932 0.0258997
\(871\) −23.0213 −0.780047
\(872\) 15.5492 0.526561
\(873\) −23.4164 −0.792525
\(874\) −1.45898 −0.0493507
\(875\) −8.23607 −0.278430
\(876\) −7.68692 −0.259717
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −3.60488 −0.121659
\(879\) −9.65248 −0.325570
\(880\) 3.14590 0.106048
\(881\) −5.88854 −0.198390 −0.0991950 0.995068i \(-0.531627\pi\)
−0.0991950 + 0.995068i \(0.531627\pi\)
\(882\) −4.58359 −0.154338
\(883\) 46.1246 1.55222 0.776108 0.630600i \(-0.217191\pi\)
0.776108 + 0.630600i \(0.217191\pi\)
\(884\) 11.6262 0.391030
\(885\) 22.5623 0.758424
\(886\) 13.5836 0.456350
\(887\) −58.1935 −1.95395 −0.976973 0.213362i \(-0.931559\pi\)
−0.976973 + 0.213362i \(0.931559\pi\)
\(888\) 9.87539 0.331396
\(889\) 18.2705 0.612773
\(890\) −15.7082 −0.526540
\(891\) −0.381966 −0.0127963
\(892\) 10.5836 0.354365
\(893\) 6.05573 0.202647
\(894\) 0.562306 0.0188063
\(895\) 3.00000 0.100279
\(896\) −10.0902 −0.337089
\(897\) 8.29180 0.276855
\(898\) −11.9787 −0.399735
\(899\) −5.12461 −0.170915
\(900\) 6.87539 0.229180
\(901\) 34.1246 1.13686
\(902\) −1.30495 −0.0434501
\(903\) 4.70820 0.156679
\(904\) −22.0820 −0.734438
\(905\) 7.47214 0.248382
\(906\) −7.25735 −0.241109
\(907\) 33.1246 1.09988 0.549942 0.835203i \(-0.314650\pi\)
0.549942 + 0.835203i \(0.314650\pi\)
\(908\) 27.7082 0.919529
\(909\) −26.1803 −0.868347
\(910\) −1.85410 −0.0614629
\(911\) 19.0344 0.630639 0.315320 0.948986i \(-0.397888\pi\)
0.315320 + 0.948986i \(0.397888\pi\)
\(912\) 2.68692 0.0889727
\(913\) −3.56231 −0.117895
\(914\) 1.20163 0.0397463
\(915\) 28.4164 0.939417
\(916\) 12.4377 0.410953
\(917\) −2.23607 −0.0738415
\(918\) 6.45898 0.213178
\(919\) 18.9787 0.626050 0.313025 0.949745i \(-0.398658\pi\)
0.313025 + 0.949745i \(0.398658\pi\)
\(920\) 17.2361 0.568256
\(921\) −3.85410 −0.126997
\(922\) 14.9787 0.493298
\(923\) 13.1459 0.432703
\(924\) 0.708204 0.0232982
\(925\) −12.4377 −0.408949
\(926\) −2.06888 −0.0679877
\(927\) 2.00000 0.0656886
\(928\) 3.16718 0.103968
\(929\) −2.94427 −0.0965984 −0.0482992 0.998833i \(-0.515380\pi\)
−0.0482992 + 0.998833i \(0.515380\pi\)
\(930\) −6.70820 −0.219971
\(931\) 5.12461 0.167952
\(932\) −16.4803 −0.539829
\(933\) 32.8885 1.07672
\(934\) −10.5623 −0.345609
\(935\) −3.38197 −0.110602
\(936\) −5.45898 −0.178432
\(937\) −11.0000 −0.359354 −0.179677 0.983726i \(-0.557505\pi\)
−0.179677 + 0.983726i \(0.557505\pi\)
\(938\) −4.74265 −0.154853
\(939\) −29.7082 −0.969491
\(940\) −34.4164 −1.12254
\(941\) 50.3951 1.64283 0.821417 0.570328i \(-0.193184\pi\)
0.821417 + 0.570328i \(0.193184\pi\)
\(942\) 1.25735 0.0409668
\(943\) 40.0000 1.30258
\(944\) 27.1115 0.882403
\(945\) 13.0902 0.425823
\(946\) 0.686918 0.0223336
\(947\) 35.0132 1.13777 0.568887 0.822415i \(-0.307374\pi\)
0.568887 + 0.822415i \(0.307374\pi\)
\(948\) 25.1459 0.816701
\(949\) −7.68692 −0.249528
\(950\) 0.604878 0.0196248
\(951\) −1.58359 −0.0513515
\(952\) 4.97871 0.161361
\(953\) −31.3607 −1.01587 −0.507936 0.861395i \(-0.669591\pi\)
−0.507936 + 0.861395i \(0.669591\pi\)
\(954\) −7.70820 −0.249562
\(955\) −8.85410 −0.286512
\(956\) −45.1033 −1.45875
\(957\) −0.291796 −0.00943243
\(958\) 5.41641 0.174996
\(959\) −0.708204 −0.0228691
\(960\) −12.3262 −0.397828
\(961\) 14.0000 0.451613
\(962\) 4.75078 0.153171
\(963\) −8.18034 −0.263608
\(964\) −46.8541 −1.50907
\(965\) 52.6869 1.69605
\(966\) 1.70820 0.0549606
\(967\) 12.4164 0.399285 0.199642 0.979869i \(-0.436022\pi\)
0.199642 + 0.979869i \(0.436022\pi\)
\(968\) −15.9787 −0.513575
\(969\) −2.88854 −0.0927934
\(970\) 11.7082 0.375928
\(971\) 23.0132 0.738527 0.369264 0.929325i \(-0.379610\pi\)
0.369264 + 0.929325i \(0.379610\pi\)
\(972\) 29.6656 0.951526
\(973\) 17.8541 0.572376
\(974\) 8.78522 0.281497
\(975\) −3.43769 −0.110094
\(976\) 34.1459 1.09298
\(977\) 4.25735 0.136205 0.0681024 0.997678i \(-0.478306\pi\)
0.0681024 + 0.997678i \(0.478306\pi\)
\(978\) −4.09017 −0.130789
\(979\) 6.00000 0.191761
\(980\) −29.1246 −0.930352
\(981\) −21.1246 −0.674457
\(982\) 13.6656 0.436088
\(983\) −12.6525 −0.403551 −0.201776 0.979432i \(-0.564671\pi\)
−0.201776 + 0.979432i \(0.564671\pi\)
\(984\) 13.1672 0.419755
\(985\) −27.2705 −0.868911
\(986\) −0.986844 −0.0314275
\(987\) −7.09017 −0.225683
\(988\) 2.93614 0.0934111
\(989\) −21.0557 −0.669533
\(990\) 0.763932 0.0242794
\(991\) −9.27051 −0.294487 −0.147244 0.989100i \(-0.547040\pi\)
−0.147244 + 0.989100i \(0.547040\pi\)
\(992\) −27.8115 −0.883017
\(993\) 22.8541 0.725253
\(994\) 2.70820 0.0858990
\(995\) −8.94427 −0.283552
\(996\) 17.2918 0.547912
\(997\) −52.7082 −1.66929 −0.834643 0.550792i \(-0.814326\pi\)
−0.834643 + 0.550792i \(0.814326\pi\)
\(998\) 7.74265 0.245089
\(999\) −33.5410 −1.06119
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 103.2.a.a.1.2 2
3.2 odd 2 927.2.a.b.1.1 2
4.3 odd 2 1648.2.a.f.1.1 2
5.4 even 2 2575.2.a.g.1.1 2
7.6 odd 2 5047.2.a.a.1.2 2
8.3 odd 2 6592.2.a.h.1.2 2
8.5 even 2 6592.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.2 2 1.1 even 1 trivial
927.2.a.b.1.1 2 3.2 odd 2
1648.2.a.f.1.1 2 4.3 odd 2
2575.2.a.g.1.1 2 5.4 even 2
5047.2.a.a.1.2 2 7.6 odd 2
6592.2.a.h.1.2 2 8.3 odd 2
6592.2.a.t.1.2 2 8.5 even 2