Properties

Label 5047.2.a.a.1.2
Level $5047$
Weight $2$
Character 5047.1
Self dual yes
Analytic conductor $40.300$
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] = [5047,2,Mod(1,5047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5047, 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("5047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5047 = 7^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5047.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: \(40.3004979001\)
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: no (minimal twist has level 103)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5047.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} +2.61803 q^{5} -0.381966 q^{6} +1.47214 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} +2.61803 q^{5} -0.381966 q^{6} +1.47214 q^{8} -2.00000 q^{9} -1.00000 q^{10} -0.381966 q^{11} -1.85410 q^{12} -1.85410 q^{13} +2.61803 q^{15} +3.14590 q^{16} +3.38197 q^{17} +0.763932 q^{18} +0.854102 q^{19} -4.85410 q^{20} +0.145898 q^{22} -4.47214 q^{23} +1.47214 q^{24} +1.85410 q^{25} +0.708204 q^{26} -5.00000 q^{27} -0.763932 q^{29} -1.00000 q^{30} -6.70820 q^{31} -4.14590 q^{32} -0.381966 q^{33} -1.29180 q^{34} +3.70820 q^{36} -6.70820 q^{37} -0.326238 q^{38} -1.85410 q^{39} +3.85410 q^{40} +8.94427 q^{41} +4.70820 q^{43} +0.708204 q^{44} -5.23607 q^{45} +1.70820 q^{46} +7.09017 q^{47} +3.14590 q^{48} -0.708204 q^{50} +3.38197 q^{51} +3.43769 q^{52} -10.0902 q^{53} +1.90983 q^{54} -1.00000 q^{55} +0.854102 q^{57} +0.291796 q^{58} -8.61803 q^{59} -4.85410 q^{60} -10.8541 q^{61} +2.56231 q^{62} -4.70820 q^{64} -4.85410 q^{65} +0.145898 q^{66} -12.4164 q^{67} -6.27051 q^{68} -4.47214 q^{69} +7.09017 q^{71} -2.94427 q^{72} +4.14590 q^{73} +2.56231 q^{74} +1.85410 q^{75} -1.58359 q^{76} +0.708204 q^{78} +13.5623 q^{79} +8.23607 q^{80} +1.00000 q^{81} -3.41641 q^{82} -9.32624 q^{83} +8.85410 q^{85} -1.79837 q^{86} -0.763932 q^{87} -0.562306 q^{88} +15.7082 q^{89} +2.00000 q^{90} +8.29180 q^{92} -6.70820 q^{93} -2.70820 q^{94} +2.23607 q^{95} -4.14590 q^{96} -11.7082 q^{97} +0.763932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 4 q^{9} - 2 q^{10} - 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{15} + 13 q^{16} + 9 q^{17} + 6 q^{18} - 5 q^{19} - 3 q^{20} + 7 q^{22} - 6 q^{24} - 3 q^{25} - 12 q^{26} - 10 q^{27} - 6 q^{29} - 2 q^{30} - 15 q^{32} - 3 q^{33} - 16 q^{34} - 6 q^{36} + 15 q^{38} + 3 q^{39} + q^{40} - 4 q^{43} - 12 q^{44} - 6 q^{45} - 10 q^{46} + 3 q^{47} + 13 q^{48} + 12 q^{50} + 9 q^{51} + 27 q^{52} - 9 q^{53} + 15 q^{54} - 2 q^{55} - 5 q^{57} + 14 q^{58} - 15 q^{59} - 3 q^{60} - 15 q^{61} - 15 q^{62} + 4 q^{64} - 3 q^{65} + 7 q^{66} + 2 q^{67} + 21 q^{68} + 3 q^{71} + 12 q^{72} + 15 q^{73} - 15 q^{74} - 3 q^{75} - 30 q^{76} - 12 q^{78} + 7 q^{79} + 12 q^{80} + 2 q^{81} + 20 q^{82} - 3 q^{83} + 11 q^{85} + 21 q^{86} - 6 q^{87} + 19 q^{88} + 18 q^{89} + 4 q^{90} + 30 q^{92} + 8 q^{94} - 15 q^{96} - 10 q^{97} + 6 q^{99}+O(q^{100}) \) 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 \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) −1.85410 −0.927051
\(5\) 2.61803 1.17082 0.585410 0.810737i \(-0.300933\pi\)
0.585410 + 0.810737i \(0.300933\pi\)
\(6\) −0.381966 −0.155937
\(7\) 0 0
\(8\) 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 \(-0.582773\pi\)
−0.257118 + 0.966380i \(0.582773\pi\)
\(14\) 0 0
\(15\) 2.61803 0.675973
\(16\) 3.14590 0.786475
\(17\) 3.38197 0.820247 0.410124 0.912030i \(-0.365486\pi\)
0.410124 + 0.912030i \(0.365486\pi\)
\(18\) 0.763932 0.180061
\(19\) 0.854102 0.195944 0.0979722 0.995189i \(-0.468764\pi\)
0.0979722 + 0.995189i \(0.468764\pi\)
\(20\) −4.85410 −1.08541
\(21\) 0 0
\(22\) 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
\(28\) 0 0
\(29\) −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 \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) −4.14590 −0.732898
\(33\) −0.381966 −0.0664917
\(34\) −1.29180 −0.221541
\(35\) 0 0
\(36\) 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 \(-0.253882\pi\)
0.698430 + 0.715678i \(0.253882\pi\)
\(42\) 0 0
\(43\) 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 \(-0.327010\pi\)
0.517104 + 0.855923i \(0.327010\pi\)
\(48\) 3.14590 0.454071
\(49\) 0 0
\(50\) −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
\(56\) 0 0
\(57\) 0.854102 0.113129
\(58\) 0.291796 0.0383147
\(59\) −8.61803 −1.12197 −0.560986 0.827825i \(-0.689578\pi\)
−0.560986 + 0.827825i \(0.689578\pi\)
\(60\) −4.85410 −0.626662
\(61\) −10.8541 −1.38973 −0.694863 0.719142i \(-0.744535\pi\)
−0.694863 + 0.719142i \(0.744535\pi\)
\(62\) 2.56231 0.325413
\(63\) 0 0
\(64\) −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
\(70\) 0 0
\(71\) 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 \(-0.421993\pi\)
0.242620 + 0.970121i \(0.421993\pi\)
\(74\) 2.56231 0.297862
\(75\) 1.85410 0.214093
\(76\) −1.58359 −0.181650
\(77\) 0 0
\(78\) 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 \(-0.671037\pi\)
−0.511844 + 0.859079i \(0.671037\pi\)
\(84\) 0 0
\(85\) 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 \(-0.186890\pi\)
0.832533 + 0.553975i \(0.186890\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 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 \(-0.702608\pi\)
−0.594394 + 0.804174i \(0.702608\pi\)
\(98\) 0 0
\(99\) 0.763932 0.0767781
\(100\) −3.43769 −0.343769
\(101\) −13.0902 −1.30252 −0.651260 0.758854i \(-0.725759\pi\)
−0.651260 + 0.758854i \(0.725759\pi\)
\(102\) −1.29180 −0.127907
\(103\) 1.00000 0.0985329
\(104\) −2.72949 −0.267649
\(105\) 0 0
\(106\) 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
\(112\) 0 0
\(113\) −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
\(119\) 0 0
\(120\) 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 0
\(127\) −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 \(-0.531143\pi\)
−0.0976831 + 0.995218i \(0.531143\pi\)
\(132\) 0.708204 0.0616412
\(133\) 0 0
\(134\) 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 \(-0.226575\pi\)
0.757183 + 0.653203i \(0.226575\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) 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 0
\(155\) −17.5623 −1.41064
\(156\) 3.43769 0.275236
\(157\) −3.29180 −0.262714 −0.131357 0.991335i \(-0.541933\pi\)
−0.131357 + 0.991335i \(0.541933\pi\)
\(158\) −5.18034 −0.412126
\(159\) −10.0902 −0.800203
\(160\) −10.8541 −0.858092
\(161\) 0 0
\(162\) −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 \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) −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 \(-0.334987\pi\)
0.495495 + 0.868611i \(0.334987\pi\)
\(174\) 0.291796 0.0221210
\(175\) 0 0
\(176\) −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 \(-0.466173\pi\)
0.106072 + 0.994358i \(0.466173\pi\)
\(182\) 0 0
\(183\) −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
\(189\) 0 0
\(190\) −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
\(196\) 0 0
\(197\) 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 \(-0.538639\pi\)
−0.121091 + 0.992641i \(0.538639\pi\)
\(200\) 2.72949 0.193004
\(201\) −12.4164 −0.875786
\(202\) 5.00000 0.351799
\(203\) 0 0
\(204\) −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
\(210\) 0 0
\(211\) 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
\(217\) 0 0
\(218\) −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 \(-0.438786\pi\)
0.191125 + 0.981566i \(0.438786\pi\)
\(224\) 0 0
\(225\) −3.70820 −0.247214
\(226\) 5.72949 0.381120
\(227\) 14.9443 0.991886 0.495943 0.868355i \(-0.334822\pi\)
0.495943 + 0.868355i \(0.334822\pi\)
\(228\) −1.58359 −0.104876
\(229\) 6.70820 0.443291 0.221645 0.975127i \(-0.428857\pi\)
0.221645 + 0.975127i \(0.428857\pi\)
\(230\) 4.47214 0.294884
\(231\) 0 0
\(232\) −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
\(238\) 0 0
\(239\) 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 \(-0.802664\pi\)
−0.813908 + 0.580993i \(0.802664\pi\)
\(242\) 4.14590 0.266508
\(243\) 16.0000 1.02640
\(244\) 20.1246 1.28835
\(245\) 0 0
\(246\) −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 \(-0.568476\pi\)
−0.213468 + 0.976950i \(0.568476\pi\)
\(252\) 0 0
\(253\) 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 \(-0.454897\pi\)
0.141220 + 0.989978i \(0.454897\pi\)
\(258\) −1.79837 −0.111962
\(259\) 0 0
\(260\) 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 0
\(267\) 15.7082 0.961326
\(268\) 23.0213 1.40625
\(269\) 3.32624 0.202804 0.101402 0.994846i \(-0.467667\pi\)
0.101402 + 0.994846i \(0.467667\pi\)
\(270\) 5.00000 0.304290
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 10.6393 0.645104
\(273\) 0 0
\(274\) −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
\(280\) 0 0
\(281\) −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 \(-0.440122\pi\)
0.187004 + 0.982359i \(0.440122\pi\)
\(284\) −13.1459 −0.780066
\(285\) 2.23607 0.132453
\(286\) −0.270510 −0.0159956
\(287\) 0 0
\(288\) 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 \(-0.590982\pi\)
−0.281952 + 0.959429i \(0.590982\pi\)
\(294\) 0 0
\(295\) −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
\(301\) 0 0
\(302\) 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 \(-0.535080\pi\)
−0.109983 + 0.993934i \(0.535080\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 6.70820 0.381000
\(311\) 32.8885 1.86494 0.932469 0.361250i \(-0.117650\pi\)
0.932469 + 0.361250i \(0.117650\pi\)
\(312\) −2.72949 −0.154527
\(313\) −29.7082 −1.67921 −0.839603 0.543200i \(-0.817213\pi\)
−0.839603 + 0.543200i \(0.817213\pi\)
\(314\) 1.25735 0.0709566
\(315\) 0 0
\(316\) −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
\(322\) 0 0
\(323\) 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
\(329\) 0 0
\(330\) 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
\(336\) 0 0
\(337\) −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
\(343\) 0 0
\(344\) 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 \(-0.401159\pi\)
0.305553 + 0.952175i \(0.401159\pi\)
\(350\) 0 0
\(351\) 9.27051 0.494823
\(352\) 1.58359 0.0844057
\(353\) −4.03444 −0.214732 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(354\) 3.29180 0.174957
\(355\) 18.5623 0.985185
\(356\) −29.1246 −1.54360
\(357\) 0 0
\(358\) 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
\(364\) 0 0
\(365\) 10.8541 0.568130
\(366\) 4.14590 0.216710
\(367\) −36.5623 −1.90854 −0.954268 0.298951i \(-0.903363\pi\)
−0.954268 + 0.298951i \(0.903363\pi\)
\(368\) −14.0689 −0.733391
\(369\) −17.8885 −0.931240
\(370\) 6.70820 0.348743
\(371\) 0 0
\(372\) 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
\(378\) 0 0
\(379\) 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 \(-0.701752\pi\)
−0.592230 + 0.805769i \(0.701752\pi\)
\(384\) 10.0902 0.514912
\(385\) 0 0
\(386\) 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
\(392\) 0 0
\(393\) −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 \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 1.30495 0.0654113
\(399\) 0 0
\(400\) 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 0
\(407\) 2.56231 0.127009
\(408\) 4.97871 0.246483
\(409\) −23.2918 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(410\) −8.94427 −0.441726
\(411\) 0.708204 0.0349331
\(412\) −1.85410 −0.0913450
\(413\) 0 0
\(414\) −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 \(-0.555407\pi\)
−0.173189 + 0.984889i \(0.555407\pi\)
\(420\) 0 0
\(421\) 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
\(427\) 0 0
\(428\) −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 \(-0.403565\pi\)
0.298347 + 0.954457i \(0.403565\pi\)
\(434\) 0 0
\(435\) −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 \(-0.572310\pi\)
−0.225218 + 0.974308i \(0.572310\pi\)
\(440\) −1.47214 −0.0701813
\(441\) 0 0
\(442\) 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
\(448\) 0 0
\(449\) 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
\(455\) 0 0
\(456\) 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 \(-0.133598\pi\)
0.913207 + 0.407495i \(0.133598\pi\)
\(462\) 0 0
\(463\) 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 \(-0.720984\pi\)
−0.639802 + 0.768540i \(0.720984\pi\)
\(468\) −6.87539 −0.317815
\(469\) 0 0
\(470\) −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
\(476\) 0 0
\(477\) 20.1803 0.923994
\(478\) −9.29180 −0.424997
\(479\) 14.1803 0.647916 0.323958 0.946071i \(-0.394986\pi\)
0.323958 + 0.946071i \(0.394986\pi\)
\(480\) −10.8541 −0.495420
\(481\) 12.4377 0.567110
\(482\) 9.65248 0.439658
\(483\) 0 0
\(484\) 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
\(490\) 0 0
\(491\) −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
\(497\) 0 0
\(498\) 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 \(-0.253561\pi\)
0.699152 + 0.714973i \(0.253561\pi\)
\(504\) 0 0
\(505\) −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 \(-0.681710\pi\)
−0.540356 + 0.841437i \(0.681710\pi\)
\(510\) −3.38197 −0.149756
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) 13.0344 0.572148
\(520\) −7.14590 −0.313368
\(521\) 5.18034 0.226955 0.113477 0.993541i \(-0.463801\pi\)
0.113477 + 0.993541i \(0.463801\pi\)
\(522\) −0.583592 −0.0255431
\(523\) 37.4164 1.63611 0.818053 0.575143i \(-0.195054\pi\)
0.818053 + 0.575143i \(0.195054\pi\)
\(524\) 4.14590 0.181114
\(525\) 0 0
\(526\) 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
\(532\) 0 0
\(533\) −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
\(539\) 0 0
\(540\) 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 0
\(547\) 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
\(553\) 0 0
\(554\) 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
\(560\) 0 0
\(561\) −1.29180 −0.0545397
\(562\) 8.60488 0.362975
\(563\) 25.7984 1.08727 0.543636 0.839321i \(-0.317047\pi\)
0.543636 + 0.839321i \(0.317047\pi\)
\(564\) −13.1459 −0.553542
\(565\) −39.2705 −1.65212
\(566\) −2.40325 −0.101016
\(567\) 0 0
\(568\) 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
\(574\) 0 0
\(575\) −8.29180 −0.345792
\(576\) 9.41641 0.392350
\(577\) −16.8328 −0.700759 −0.350380 0.936608i \(-0.613947\pi\)
−0.350380 + 0.936608i \(0.613947\pi\)
\(578\) 2.12461 0.0883722
\(579\) −20.1246 −0.836350
\(580\) 3.70820 0.153975
\(581\) 0 0
\(582\) 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 \(-0.573359\pi\)
−0.228431 + 0.973560i \(0.573359\pi\)
\(588\) 0 0
\(589\) −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 \(-0.364008\pi\)
0.414354 + 0.910116i \(0.364008\pi\)
\(594\) −0.729490 −0.0299313
\(595\) 0 0
\(596\) −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 \(-0.390319\pi\)
0.337795 + 0.941220i \(0.390319\pi\)
\(602\) 0 0
\(603\) 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 \(-0.450000\pi\)
0.156433 + 0.987689i \(0.450000\pi\)
\(608\) −3.54102 −0.143607
\(609\) 0 0
\(610\) 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 0
\(617\) −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 \(-0.719743\pi\)
−0.636802 + 0.771027i \(0.719743\pi\)
\(620\) 32.5623 1.30773
\(621\) 22.3607 0.897303
\(622\) −12.5623 −0.503703
\(623\) 0 0
\(624\) −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
\(630\) 0 0
\(631\) 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
\(637\) 0 0
\(638\) −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 \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) 12.3262 0.485345
\(646\) −1.10333 −0.0434098
\(647\) −43.7426 −1.71970 −0.859850 0.510546i \(-0.829443\pi\)
−0.859850 + 0.510546i \(0.829443\pi\)
\(648\) 1.47214 0.0578310
\(649\) 3.29180 0.129214
\(650\) 1.31308 0.0515033
\(651\) 0 0
\(652\) 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
\(658\) 0 0
\(659\) −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 \(-0.428599\pi\)
0.222437 + 0.974947i \(0.428599\pi\)
\(662\) 8.72949 0.339281
\(663\) −6.27051 −0.243526
\(664\) −13.7295 −0.532808
\(665\) 0 0
\(666\) −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
\(672\) 0 0
\(673\) −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 \(-0.567581\pi\)
−0.210720 + 0.977546i \(0.567581\pi\)
\(678\) 5.72949 0.220040
\(679\) 0 0
\(680\) 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
\(686\) 0 0
\(687\) 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 \(-0.506938\pi\)
−0.0217960 + 0.999762i \(0.506938\pi\)
\(692\) −24.1672 −0.918698
\(693\) 0 0
\(694\) −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
\(700\) 0 0
\(701\) −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
\(707\) 0 0
\(708\) 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
\(714\) 0 0
\(715\) 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 \(-0.645533\pi\)
−0.441441 + 0.897290i \(0.645533\pi\)
\(720\) −16.4721 −0.613880
\(721\) 0 0
\(722\) 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 \(-0.248836\pi\)
0.709687 + 0.704517i \(0.248836\pi\)
\(728\) 0 0
\(729\) 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 \(-0.507595\pi\)
−0.0238568 + 0.999715i \(0.507595\pi\)
\(734\) 13.9656 0.515478
\(735\) 0 0
\(736\) 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
\(742\) 0 0
\(743\) 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
\(749\) 0 0
\(750\) 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
\(756\) 0 0
\(757\) −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 \(-0.508494\pi\)
−0.0266824 + 0.999644i \(0.508494\pi\)
\(762\) 6.97871 0.252812
\(763\) 0 0
\(764\) −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 \(-0.223034\pi\)
0.764404 + 0.644738i \(0.223034\pi\)
\(770\) 0 0
\(771\) 4.52786 0.163067
\(772\) 37.3131 1.34293
\(773\) 25.4721 0.916169 0.458085 0.888909i \(-0.348536\pi\)
0.458085 + 0.888909i \(0.348536\pi\)
\(774\) 3.59675 0.129282
\(775\) −12.4377 −0.446775
\(776\) −17.2361 −0.618739
\(777\) 0 0
\(778\) 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
\(784\) 0 0
\(785\) −8.61803 −0.307591
\(786\) 0.854102 0.0304648
\(787\) −16.5836 −0.591141 −0.295571 0.955321i \(-0.595510\pi\)
−0.295571 + 0.955321i \(0.595510\pi\)
\(788\) −19.3131 −0.688000
\(789\) −18.3820 −0.654415
\(790\) −13.5623 −0.482525
\(791\) 0 0
\(792\) 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 \(-0.444039\pi\)
0.174902 + 0.984586i \(0.444039\pi\)
\(798\) 0 0
\(799\) 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
\(805\) 0 0
\(806\) −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 \(-0.205055\pi\)
0.799581 + 0.600559i \(0.205055\pi\)
\(812\) 0 0
\(813\) −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
\(819\) 0 0
\(820\) −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
\(826\) 0 0
\(827\) −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 \(-0.391383\pi\)
0.334646 + 0.942344i \(0.391383\pi\)
\(830\) 9.32624 0.323718
\(831\) −8.70820 −0.302084
\(832\) 8.72949 0.302641
\(833\) 0 0
\(834\) −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 \(-0.620332\pi\)
−0.369094 + 0.929392i \(0.620332\pi\)
\(840\) 0 0
\(841\) −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
\(847\) 0 0
\(848\) −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 \(-0.441197\pi\)
0.183685 + 0.982985i \(0.441197\pi\)
\(854\) 0 0
\(855\) −4.47214 −0.152944
\(856\) 6.02129 0.205803
\(857\) −3.76393 −0.128573 −0.0642867 0.997931i \(-0.520477\pi\)
−0.0642867 + 0.997931i \(0.520477\pi\)
\(858\) −0.270510 −0.00923505
\(859\) 9.56231 0.326262 0.163131 0.986604i \(-0.447841\pi\)
0.163131 + 0.986604i \(0.447841\pi\)
\(860\) −22.8541 −0.779318
\(861\) 0 0
\(862\) 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
\(868\) 0 0
\(869\) −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
\(875\) 0 0
\(876\) −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 \(-0.468373\pi\)
0.0991950 + 0.995068i \(0.468373\pi\)
\(882\) 0 0
\(883\) 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 \(-0.0684414\pi\)
0.976973 + 0.213362i \(0.0684414\pi\)
\(888\) −9.87539 −0.331396
\(889\) 0 0
\(890\) −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
\(896\) 0 0
\(897\) 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
\(903\) 0 0
\(904\) −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
\(910\) 0 0
\(911\) 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
\(917\) 0 0
\(918\) 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 0
\(925\) −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 \(-0.484620\pi\)
0.0482992 + 0.998833i \(0.484620\pi\)
\(930\) 6.70820 0.219971
\(931\) 0 0
\(932\) −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 \(-0.442495\pi\)
0.179677 + 0.983726i \(0.442495\pi\)
\(938\) 0 0
\(939\) −29.7082 −0.969491
\(940\) −34.4164 −1.12254
\(941\) −50.3951 −1.64283 −0.821417 0.570328i \(-0.806816\pi\)
−0.821417 + 0.570328i \(0.806816\pi\)
\(942\) 1.25735 0.0409668
\(943\) −40.0000 −1.30258
\(944\) −27.1115 −0.882403
\(945\) 0 0
\(946\) 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
\(952\) 0 0
\(953\) −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 0
\(960\) −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
\(966\) 0 0
\(967\) 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 \(-0.620390\pi\)
−0.369264 + 0.929325i \(0.620390\pi\)
\(972\) −29.6656 −0.951526
\(973\) 0 0
\(974\) 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
\(980\) 0 0
\(981\) −21.1246 −0.674457
\(982\) 13.6656 0.436088
\(983\) 12.6525 0.403551 0.201776 0.979432i \(-0.435329\pi\)
0.201776 + 0.979432i \(0.435329\pi\)
\(984\) 13.1672 0.419755
\(985\) 27.2705 0.868911
\(986\) 0.986844 0.0314275
\(987\) 0 0
\(988\) 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
\(994\) 0 0
\(995\) −8.94427 −0.283552
\(996\) 17.2918 0.547912
\(997\) 52.7082 1.66929 0.834643 0.550792i \(-0.185674\pi\)
0.834643 + 0.550792i \(0.185674\pi\)
\(998\) 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 5047.2.a.a.1.2 2
7.6 odd 2 103.2.a.a.1.2 2
21.20 even 2 927.2.a.b.1.1 2
28.27 even 2 1648.2.a.f.1.1 2
35.34 odd 2 2575.2.a.g.1.1 2
56.13 odd 2 6592.2.a.t.1.2 2
56.27 even 2 6592.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.2 2 7.6 odd 2
927.2.a.b.1.1 2 21.20 even 2
1648.2.a.f.1.1 2 28.27 even 2
2575.2.a.g.1.1 2 35.34 odd 2
5047.2.a.a.1.2 2 1.1 even 1 trivial
6592.2.a.h.1.2 2 56.27 even 2
6592.2.a.t.1.2 2 56.13 odd 2