Properties

Label 927.2.a.b.1.1
Level $927$
Weight $2$
Character 927.1
Self dual yes
Analytic conductor $7.402$
Analytic rank $0$
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] = [927,2,Mod(1,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, 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("927.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.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: \(7.40213226737\)
Analytic rank: \(0\)
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 927.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.381966 q^{2} -1.85410 q^{4} +2.61803 q^{5} -1.00000 q^{7} -1.47214 q^{8} +O(q^{10})\) \(q+0.381966 q^{2} -1.85410 q^{4} +2.61803 q^{5} -1.00000 q^{7} -1.47214 q^{8} +1.00000 q^{10} +0.381966 q^{11} +1.85410 q^{13} -0.381966 q^{14} +3.14590 q^{16} +3.38197 q^{17} -0.854102 q^{19} -4.85410 q^{20} +0.145898 q^{22} +4.47214 q^{23} +1.85410 q^{25} +0.708204 q^{26} +1.85410 q^{28} +0.763932 q^{29} +6.70820 q^{31} +4.14590 q^{32} +1.29180 q^{34} -2.61803 q^{35} -6.70820 q^{37} -0.326238 q^{38} -3.85410 q^{40} +8.94427 q^{41} +4.70820 q^{43} -0.708204 q^{44} +1.70820 q^{46} +7.09017 q^{47} -6.00000 q^{49} +0.708204 q^{50} -3.43769 q^{52} +10.0902 q^{53} +1.00000 q^{55} +1.47214 q^{56} +0.291796 q^{58} -8.61803 q^{59} +10.8541 q^{61} +2.56231 q^{62} -4.70820 q^{64} +4.85410 q^{65} -12.4164 q^{67} -6.27051 q^{68} -1.00000 q^{70} -7.09017 q^{71} -4.14590 q^{73} -2.56231 q^{74} +1.58359 q^{76} -0.381966 q^{77} +13.5623 q^{79} +8.23607 q^{80} +3.41641 q^{82} -9.32624 q^{83} +8.85410 q^{85} +1.79837 q^{86} -0.562306 q^{88} +15.7082 q^{89} -1.85410 q^{91} -8.29180 q^{92} +2.70820 q^{94} -2.23607 q^{95} +11.7082 q^{97} -2.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} + 6 q^{8} + 2 q^{10} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 13 q^{16} + 9 q^{17} + 5 q^{19} - 3 q^{20} + 7 q^{22} - 3 q^{25} - 12 q^{26} - 3 q^{28} + 6 q^{29} + 15 q^{32} + 16 q^{34} - 3 q^{35} + 15 q^{38} - q^{40} - 4 q^{43} + 12 q^{44} - 10 q^{46} + 3 q^{47} - 12 q^{49} - 12 q^{50} - 27 q^{52} + 9 q^{53} + 2 q^{55} - 6 q^{56} + 14 q^{58} - 15 q^{59} + 15 q^{61} - 15 q^{62} + 4 q^{64} + 3 q^{65} + 2 q^{67} + 21 q^{68} - 2 q^{70} - 3 q^{71} - 15 q^{73} + 15 q^{74} + 30 q^{76} - 3 q^{77} + 7 q^{79} + 12 q^{80} - 20 q^{82} - 3 q^{83} + 11 q^{85} - 21 q^{86} + 19 q^{88} + 18 q^{89} + 3 q^{91} - 30 q^{92} - 8 q^{94} + 10 q^{97} - 18 q^{98}+O(q^{100}) \) 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 \(-0.456882\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) 0 0
\(4\) −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 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.47214 −0.520479
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0.381966 0.115167 0.0575835 0.998341i \(-0.481660\pi\)
0.0575835 + 0.998341i \(0.481660\pi\)
\(12\) 0 0
\(13\) 1.85410 0.514235 0.257118 0.966380i \(-0.417227\pi\)
0.257118 + 0.966380i \(0.417227\pi\)
\(14\) −0.381966 −0.102085
\(15\) 0 0
\(16\) 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 0
\(19\) −0.854102 −0.195944 −0.0979722 0.995189i \(-0.531236\pi\)
−0.0979722 + 0.995189i \(0.531236\pi\)
\(20\) −4.85410 −1.08541
\(21\) 0 0
\(22\) 0.145898 0.0311056
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 0 0
\(25\) 1.85410 0.370820
\(26\) 0.708204 0.138890
\(27\) 0 0
\(28\) 1.85410 0.350392
\(29\) 0.763932 0.141859 0.0709293 0.997481i \(-0.477404\pi\)
0.0709293 + 0.997481i \(0.477404\pi\)
\(30\) 0 0
\(31\) 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 0
\(34\) 1.29180 0.221541
\(35\) −2.61803 −0.442529
\(36\) 0 0
\(37\) −6.70820 −1.10282 −0.551411 0.834234i \(-0.685910\pi\)
−0.551411 + 0.834234i \(0.685910\pi\)
\(38\) −0.326238 −0.0529228
\(39\) 0 0
\(40\) −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
\(45\) 0 0
\(46\) 1.70820 0.251861
\(47\) 7.09017 1.03421 0.517104 0.855923i \(-0.327010\pi\)
0.517104 + 0.855923i \(0.327010\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0.708204 0.100155
\(51\) 0 0
\(52\) −3.43769 −0.476722
\(53\) 10.0902 1.38599 0.692996 0.720942i \(-0.256290\pi\)
0.692996 + 0.720942i \(0.256290\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.47214 0.196722
\(57\) 0 0
\(58\) 0.291796 0.0383147
\(59\) −8.61803 −1.12197 −0.560986 0.827825i \(-0.689578\pi\)
−0.560986 + 0.827825i \(0.689578\pi\)
\(60\) 0 0
\(61\) 10.8541 1.38973 0.694863 0.719142i \(-0.255465\pi\)
0.694863 + 0.719142i \(0.255465\pi\)
\(62\) 2.56231 0.325413
\(63\) 0 0
\(64\) −4.70820 −0.588525
\(65\) 4.85410 0.602077
\(66\) 0 0
\(67\) −12.4164 −1.51691 −0.758453 0.651728i \(-0.774044\pi\)
−0.758453 + 0.651728i \(0.774044\pi\)
\(68\) −6.27051 −0.760411
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −7.09017 −0.841448 −0.420724 0.907189i \(-0.638224\pi\)
−0.420724 + 0.907189i \(0.638224\pi\)
\(72\) 0 0
\(73\) −4.14590 −0.485241 −0.242620 0.970121i \(-0.578007\pi\)
−0.242620 + 0.970121i \(0.578007\pi\)
\(74\) −2.56231 −0.297862
\(75\) 0 0
\(76\) 1.58359 0.181650
\(77\) −0.381966 −0.0435291
\(78\) 0 0
\(79\) 13.5623 1.52588 0.762939 0.646470i \(-0.223755\pi\)
0.762939 + 0.646470i \(0.223755\pi\)
\(80\) 8.23607 0.920820
\(81\) 0 0
\(82\) 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 0
\(88\) −0.562306 −0.0599420
\(89\) 15.7082 1.66507 0.832533 0.553975i \(-0.186890\pi\)
0.832533 + 0.553975i \(0.186890\pi\)
\(90\) 0 0
\(91\) −1.85410 −0.194363
\(92\) −8.29180 −0.864479
\(93\) 0 0
\(94\) 2.70820 0.279330
\(95\) −2.23607 −0.229416
\(96\) 0 0
\(97\) 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 0
\(100\) −3.43769 −0.343769
\(101\) −13.0902 −1.30252 −0.651260 0.758854i \(-0.725759\pi\)
−0.651260 + 0.758854i \(0.725759\pi\)
\(102\) 0 0
\(103\) −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 \(-0.563349\pi\)
−0.197706 + 0.980261i \(0.563349\pi\)
\(108\) 0 0
\(109\) 10.5623 1.01169 0.505843 0.862626i \(-0.331182\pi\)
0.505843 + 0.862626i \(0.331182\pi\)
\(110\) 0.381966 0.0364190
\(111\) 0 0
\(112\) −3.14590 −0.297259
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) 11.7082 1.09180
\(116\) −1.41641 −0.131510
\(117\) 0 0
\(118\) −3.29180 −0.303034
\(119\) −3.38197 −0.310024
\(120\) 0 0
\(121\) −10.8541 −0.986737
\(122\) 4.14590 0.375352
\(123\) 0 0
\(124\) −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
\(129\) 0 0
\(130\) 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 0
\(133\) 0.854102 0.0740600
\(134\) −4.74265 −0.409702
\(135\) 0 0
\(136\) −4.97871 −0.426921
\(137\) −0.708204 −0.0605059 −0.0302530 0.999542i \(-0.509631\pi\)
−0.0302530 + 0.999542i \(0.509631\pi\)
\(138\) 0 0
\(139\) −17.8541 −1.51437 −0.757183 0.653203i \(-0.773425\pi\)
−0.757183 + 0.653203i \(0.773425\pi\)
\(140\) 4.85410 0.410246
\(141\) 0 0
\(142\) −2.70820 −0.227267
\(143\) 0.708204 0.0592230
\(144\) 0 0
\(145\) 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 \(-0.519206\pi\)
−0.0603010 + 0.998180i \(0.519206\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 1.25735 0.101985
\(153\) 0 0
\(154\) −0.145898 −0.0117568
\(155\) 17.5623 1.41064
\(156\) 0 0
\(157\) 3.29180 0.262714 0.131357 0.991335i \(-0.458067\pi\)
0.131357 + 0.991335i \(0.458067\pi\)
\(158\) 5.18034 0.412126
\(159\) 0 0
\(160\) 10.8541 0.858092
\(161\) −4.47214 −0.352454
\(162\) 0 0
\(163\) −10.7082 −0.838731 −0.419366 0.907817i \(-0.637747\pi\)
−0.419366 + 0.907817i \(0.637747\pi\)
\(164\) −16.5836 −1.29496
\(165\) 0 0
\(166\) −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
\(171\) 0 0
\(172\) −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 0
\(175\) −1.85410 −0.140157
\(176\) 1.20163 0.0905760
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 1.14590 0.0856484 0.0428242 0.999083i \(-0.486364\pi\)
0.0428242 + 0.999083i \(0.486364\pi\)
\(180\) 0 0
\(181\) −2.85410 −0.212144 −0.106072 0.994358i \(-0.533827\pi\)
−0.106072 + 0.994358i \(0.533827\pi\)
\(182\) −0.708204 −0.0524956
\(183\) 0 0
\(184\) −6.58359 −0.485349
\(185\) −17.5623 −1.29121
\(186\) 0 0
\(187\) 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 \(-0.539045\pi\)
−0.122355 + 0.992486i \(0.539045\pi\)
\(192\) 0 0
\(193\) −20.1246 −1.44860 −0.724301 0.689484i \(-0.757837\pi\)
−0.724301 + 0.689484i \(0.757837\pi\)
\(194\) 4.47214 0.321081
\(195\) 0 0
\(196\) 11.1246 0.794615
\(197\) −10.4164 −0.742138 −0.371069 0.928605i \(-0.621009\pi\)
−0.371069 + 0.928605i \(0.621009\pi\)
\(198\) 0 0
\(199\) 3.41641 0.242183 0.121091 0.992641i \(-0.461361\pi\)
0.121091 + 0.992641i \(0.461361\pi\)
\(200\) −2.72949 −0.193004
\(201\) 0 0
\(202\) −5.00000 −0.351799
\(203\) −0.763932 −0.0536175
\(204\) 0 0
\(205\) 23.4164 1.63547
\(206\) −0.381966 −0.0266128
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) −1.56231 −0.106797
\(215\) 12.3262 0.840642
\(216\) 0 0
\(217\) −6.70820 −0.455383
\(218\) 4.03444 0.273247
\(219\) 0 0
\(220\) −1.85410 −0.125004
\(221\) 6.27051 0.421800
\(222\) 0 0
\(223\) −5.70820 −0.382250 −0.191125 0.981566i \(-0.561214\pi\)
−0.191125 + 0.981566i \(0.561214\pi\)
\(224\) −4.14590 −0.277009
\(225\) 0 0
\(226\) 5.72949 0.381120
\(227\) 14.9443 0.991886 0.495943 0.868355i \(-0.334822\pi\)
0.495943 + 0.868355i \(0.334822\pi\)
\(228\) 0 0
\(229\) −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 0
\(232\) −1.12461 −0.0738344
\(233\) −8.88854 −0.582308 −0.291154 0.956676i \(-0.594039\pi\)
−0.291154 + 0.956676i \(0.594039\pi\)
\(234\) 0 0
\(235\) 18.5623 1.21087
\(236\) 15.9787 1.04013
\(237\) 0 0
\(238\) −1.29180 −0.0837347
\(239\) −24.3262 −1.57353 −0.786767 0.617250i \(-0.788247\pi\)
−0.786767 + 0.617250i \(0.788247\pi\)
\(240\) 0 0
\(241\) 25.2705 1.62782 0.813908 0.580993i \(-0.197336\pi\)
0.813908 + 0.580993i \(0.197336\pi\)
\(242\) −4.14590 −0.266508
\(243\) 0 0
\(244\) −20.1246 −1.28835
\(245\) −15.7082 −1.00356
\(246\) 0 0
\(247\) −1.58359 −0.100762
\(248\) −9.87539 −0.627088
\(249\) 0 0
\(250\) −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
\(255\) 0 0
\(256\) 5.56231 0.347644
\(257\) 4.52786 0.282440 0.141220 0.989978i \(-0.454897\pi\)
0.141220 + 0.989978i \(0.454897\pi\)
\(258\) 0 0
\(259\) 6.70820 0.416828
\(260\) −9.00000 −0.558156
\(261\) 0 0
\(262\) −0.854102 −0.0527666
\(263\) 18.3820 1.13348 0.566740 0.823896i \(-0.308204\pi\)
0.566740 + 0.823896i \(0.308204\pi\)
\(264\) 0 0
\(265\) 26.4164 1.62275
\(266\) 0.326238 0.0200029
\(267\) 0 0
\(268\) 23.0213 1.40625
\(269\) 3.32624 0.202804 0.101402 0.994846i \(-0.467667\pi\)
0.101402 + 0.994846i \(0.467667\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) 10.6393 0.645104
\(273\) 0 0
\(274\) −0.270510 −0.0163421
\(275\) 0.708204 0.0427063
\(276\) 0 0
\(277\) −8.70820 −0.523225 −0.261613 0.965173i \(-0.584254\pi\)
−0.261613 + 0.965173i \(0.584254\pi\)
\(278\) −6.81966 −0.409016
\(279\) 0 0
\(280\) 3.85410 0.230327
\(281\) 22.5279 1.34390 0.671950 0.740597i \(-0.265457\pi\)
0.671950 + 0.740597i \(0.265457\pi\)
\(282\) 0 0
\(283\) −6.29180 −0.374008 −0.187004 0.982359i \(-0.559878\pi\)
−0.187004 + 0.982359i \(0.559878\pi\)
\(284\) 13.1459 0.780066
\(285\) 0 0
\(286\) 0.270510 0.0159956
\(287\) −8.94427 −0.527964
\(288\) 0 0
\(289\) −5.56231 −0.327194
\(290\) 0.763932 0.0448596
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) −0.562306 −0.0325735
\(299\) 8.29180 0.479527
\(300\) 0 0
\(301\) −4.70820 −0.271376
\(302\) −7.25735 −0.417614
\(303\) 0 0
\(304\) −2.68692 −0.154105
\(305\) 28.4164 1.62712
\(306\) 0 0
\(307\) 3.85410 0.219965 0.109983 0.993934i \(-0.464920\pi\)
0.109983 + 0.993934i \(0.464920\pi\)
\(308\) 0.708204 0.0403537
\(309\) 0 0
\(310\) 6.70820 0.381000
\(311\) 32.8885 1.86494 0.932469 0.361250i \(-0.117650\pi\)
0.932469 + 0.361250i \(0.117650\pi\)
\(312\) 0 0
\(313\) 29.7082 1.67921 0.839603 0.543200i \(-0.182787\pi\)
0.839603 + 0.543200i \(0.182787\pi\)
\(314\) 1.25735 0.0709566
\(315\) 0 0
\(316\) −25.1459 −1.41457
\(317\) −1.58359 −0.0889434 −0.0444717 0.999011i \(-0.514160\pi\)
−0.0444717 + 0.999011i \(0.514160\pi\)
\(318\) 0 0
\(319\) 0.291796 0.0163374
\(320\) −12.3262 −0.689058
\(321\) 0 0
\(322\) −1.70820 −0.0951945
\(323\) −2.88854 −0.160723
\(324\) 0 0
\(325\) 3.43769 0.190689
\(326\) −4.09017 −0.226534
\(327\) 0 0
\(328\) −13.1672 −0.727036
\(329\) −7.09017 −0.390894
\(330\) 0 0
\(331\) −22.8541 −1.25618 −0.628088 0.778143i \(-0.716162\pi\)
−0.628088 + 0.778143i \(0.716162\pi\)
\(332\) 17.2918 0.949011
\(333\) 0 0
\(334\) −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
\(339\) 0 0
\(340\) −16.4164 −0.890305
\(341\) 2.56231 0.138757
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −6.93112 −0.373701
\(345\) 0 0
\(346\) 4.97871 0.267657
\(347\) −7.47214 −0.401125 −0.200563 0.979681i \(-0.564277\pi\)
−0.200563 + 0.979681i \(0.564277\pi\)
\(348\) 0 0
\(349\) −11.4164 −0.611106 −0.305553 0.952175i \(-0.598841\pi\)
−0.305553 + 0.952175i \(0.598841\pi\)
\(350\) −0.708204 −0.0378551
\(351\) 0 0
\(352\) 1.58359 0.0844057
\(353\) −4.03444 −0.214732 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(354\) 0 0
\(355\) −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 \(-0.795315\pi\)
−0.800279 + 0.599628i \(0.795315\pi\)
\(360\) 0 0
\(361\) −18.2705 −0.961606
\(362\) −1.09017 −0.0572981
\(363\) 0 0
\(364\) 3.43769 0.180184
\(365\) −10.8541 −0.568130
\(366\) 0 0
\(367\) 36.5623 1.90854 0.954268 0.298951i \(-0.0966368\pi\)
0.954268 + 0.298951i \(0.0966368\pi\)
\(368\) 14.0689 0.733391
\(369\) 0 0
\(370\) −6.70820 −0.348743
\(371\) −10.0902 −0.523856
\(372\) 0 0
\(373\) 37.6869 1.95135 0.975677 0.219212i \(-0.0703485\pi\)
0.975677 + 0.219212i \(0.0703485\pi\)
\(374\) 0.493422 0.0255143
\(375\) 0 0
\(376\) −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
\(381\) 0 0
\(382\) −1.29180 −0.0660940
\(383\) −23.1803 −1.18446 −0.592230 0.805769i \(-0.701752\pi\)
−0.592230 + 0.805769i \(0.701752\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −7.68692 −0.391254
\(387\) 0 0
\(388\) −21.7082 −1.10207
\(389\) 19.4164 0.984451 0.492225 0.870468i \(-0.336184\pi\)
0.492225 + 0.870468i \(0.336184\pi\)
\(390\) 0 0
\(391\) 15.1246 0.764884
\(392\) 8.83282 0.446125
\(393\) 0 0
\(394\) −3.97871 −0.200445
\(395\) 35.5066 1.78653
\(396\) 0 0
\(397\) 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 0
\(400\) 5.83282 0.291641
\(401\) −11.8885 −0.593686 −0.296843 0.954926i \(-0.595934\pi\)
−0.296843 + 0.954926i \(0.595934\pi\)
\(402\) 0 0
\(403\) 12.4377 0.619566
\(404\) 24.2705 1.20750
\(405\) 0 0
\(406\) −0.291796 −0.0144816
\(407\) −2.56231 −0.127009
\(408\) 0 0
\(409\) 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 0
\(412\) 1.85410 0.0913450
\(413\) 8.61803 0.424066
\(414\) 0 0
\(415\) −24.4164 −1.19855
\(416\) 7.68692 0.376882
\(417\) 0 0
\(418\) −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
\(423\) 0 0
\(424\) −14.8541 −0.721379
\(425\) 6.27051 0.304164
\(426\) 0 0
\(427\) −10.8541 −0.525267
\(428\) 7.58359 0.366567
\(429\) 0 0
\(430\) 4.70820 0.227050
\(431\) 10.3607 0.499056 0.249528 0.968368i \(-0.419724\pi\)
0.249528 + 0.968368i \(0.419724\pi\)
\(432\) 0 0
\(433\) −12.4164 −0.596694 −0.298347 0.954457i \(-0.596435\pi\)
−0.298347 + 0.954457i \(0.596435\pi\)
\(434\) −2.56231 −0.122995
\(435\) 0 0
\(436\) −19.5836 −0.937884
\(437\) −3.81966 −0.182719
\(438\) 0 0
\(439\) 9.43769 0.450437 0.225218 0.974308i \(-0.427690\pi\)
0.225218 + 0.974308i \(0.427690\pi\)
\(440\) −1.47214 −0.0701813
\(441\) 0 0
\(442\) 2.39512 0.113924
\(443\) 35.5623 1.68962 0.844808 0.535069i \(-0.179715\pi\)
0.844808 + 0.535069i \(0.179715\pi\)
\(444\) 0 0
\(445\) 41.1246 1.94949
\(446\) −2.18034 −0.103242
\(447\) 0 0
\(448\) 4.70820 0.222442
\(449\) −31.3607 −1.48000 −0.740001 0.672606i \(-0.765175\pi\)
−0.740001 + 0.672606i \(0.765175\pi\)
\(450\) 0 0
\(451\) 3.41641 0.160872
\(452\) −27.8115 −1.30814
\(453\) 0 0
\(454\) 5.70820 0.267899
\(455\) −4.85410 −0.227564
\(456\) 0 0
\(457\) −3.14590 −0.147159 −0.0735795 0.997289i \(-0.523442\pi\)
−0.0735795 + 0.997289i \(0.523442\pi\)
\(458\) −2.56231 −0.119729
\(459\) 0 0
\(460\) −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
\(465\) 0 0
\(466\) −3.39512 −0.157276
\(467\) −27.6525 −1.27960 −0.639802 0.768540i \(-0.720984\pi\)
−0.639802 + 0.768540i \(0.720984\pi\)
\(468\) 0 0
\(469\) 12.4164 0.573336
\(470\) 7.09017 0.327045
\(471\) 0 0
\(472\) 12.6869 0.583963
\(473\) 1.79837 0.0826893
\(474\) 0 0
\(475\) −1.58359 −0.0726602
\(476\) 6.27051 0.287408
\(477\) 0 0
\(478\) −9.29180 −0.424997
\(479\) 14.1803 0.647916 0.323958 0.946071i \(-0.394986\pi\)
0.323958 + 0.946071i \(0.394986\pi\)
\(480\) 0 0
\(481\) −12.4377 −0.567110
\(482\) 9.65248 0.439658
\(483\) 0 0
\(484\) 20.1246 0.914755
\(485\) 30.6525 1.39186
\(486\) 0 0
\(487\) −23.0000 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) −15.9787 −0.723322
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) 35.7771 1.61460 0.807299 0.590143i \(-0.200929\pi\)
0.807299 + 0.590143i \(0.200929\pi\)
\(492\) 0 0
\(493\) 2.58359 0.116359
\(494\) −0.604878 −0.0272148
\(495\) 0 0
\(496\) 21.1033 0.947567
\(497\) 7.09017 0.318038
\(498\) 0 0
\(499\) −20.2705 −0.907433 −0.453716 0.891146i \(-0.649902\pi\)
−0.453716 + 0.891146i \(0.649902\pi\)
\(500\) 15.2705 0.682918
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) 33.8754 1.50298
\(509\) −24.3820 −1.08071 −0.540356 0.841437i \(-0.681710\pi\)
−0.540356 + 0.841437i \(0.681710\pi\)
\(510\) 0 0
\(511\) 4.14590 0.183404
\(512\) 22.3050 0.985749
\(513\) 0 0
\(514\) 1.72949 0.0762845
\(515\) −2.61803 −0.115364
\(516\) 0 0
\(517\) 2.70820 0.119107
\(518\) 2.56231 0.112581
\(519\) 0 0
\(520\) −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 0
\(523\) −37.4164 −1.63611 −0.818053 0.575143i \(-0.804946\pi\)
−0.818053 + 0.575143i \(0.804946\pi\)
\(524\) 4.14590 0.181114
\(525\) 0 0
\(526\) 7.02129 0.306143
\(527\) 22.6869 0.988258
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) 10.0902 0.438289
\(531\) 0 0
\(532\) −1.58359 −0.0686574
\(533\) 16.5836 0.718315
\(534\) 0 0
\(535\) −10.7082 −0.462956
\(536\) 18.2786 0.789517
\(537\) 0 0
\(538\) 1.27051 0.0547756
\(539\) −2.29180 −0.0987146
\(540\) 0 0
\(541\) 15.8541 0.681621 0.340811 0.940132i \(-0.389299\pi\)
0.340811 + 0.940132i \(0.389299\pi\)
\(542\) 0.381966 0.0164068
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0.270510 0.0115346
\(551\) −0.652476 −0.0277964
\(552\) 0 0
\(553\) −13.5623 −0.576728
\(554\) −3.32624 −0.141318
\(555\) 0 0
\(556\) 33.1033 1.40389
\(557\) −28.6869 −1.21550 −0.607752 0.794127i \(-0.707928\pi\)
−0.607752 + 0.794127i \(0.707928\pi\)
\(558\) 0 0
\(559\) 8.72949 0.369218
\(560\) −8.23607 −0.348037
\(561\) 0 0
\(562\) 8.60488 0.362975
\(563\) 25.7984 1.08727 0.543636 0.839321i \(-0.317047\pi\)
0.543636 + 0.839321i \(0.317047\pi\)
\(564\) 0 0
\(565\) 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 \(-0.844955\pi\)
−0.883700 + 0.468054i \(0.844955\pi\)
\(570\) 0 0
\(571\) −2.43769 −0.102014 −0.0510072 0.998698i \(-0.516243\pi\)
−0.0510072 + 0.998698i \(0.516243\pi\)
\(572\) −1.31308 −0.0549027
\(573\) 0 0
\(574\) −3.41641 −0.142598
\(575\) 8.29180 0.345792
\(576\) 0 0
\(577\) 16.8328 0.700759 0.350380 0.936608i \(-0.386053\pi\)
0.350380 + 0.936608i \(0.386053\pi\)
\(578\) −2.12461 −0.0883722
\(579\) 0 0
\(580\) −3.70820 −0.153975
\(581\) 9.32624 0.386918
\(582\) 0 0
\(583\) 3.85410 0.159621
\(584\) 6.10333 0.252557
\(585\) 0 0
\(586\) −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
\(591\) 0 0
\(592\) −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 0
\(595\) −8.85410 −0.362983
\(596\) 2.72949 0.111804
\(597\) 0 0
\(598\) 3.16718 0.129516
\(599\) −20.4508 −0.835599 −0.417800 0.908539i \(-0.637199\pi\)
−0.417800 + 0.908539i \(0.637199\pi\)
\(600\) 0 0
\(601\) −16.5623 −0.675591 −0.337795 0.941220i \(-0.609681\pi\)
−0.337795 + 0.941220i \(0.609681\pi\)
\(602\) −1.79837 −0.0732962
\(603\) 0 0
\(604\) 35.2279 1.43340
\(605\) −28.4164 −1.15529
\(606\) 0 0
\(607\) −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 0
\(610\) 10.8541 0.439470
\(611\) 13.1459 0.531826
\(612\) 0 0
\(613\) −2.41641 −0.0975978 −0.0487989 0.998809i \(-0.515539\pi\)
−0.0487989 + 0.998809i \(0.515539\pi\)
\(614\) 1.47214 0.0594106
\(615\) 0 0
\(616\) 0.562306 0.0226560
\(617\) 30.2705 1.21864 0.609322 0.792923i \(-0.291442\pi\)
0.609322 + 0.792923i \(0.291442\pi\)
\(618\) 0 0
\(619\) 31.6869 1.27360 0.636802 0.771027i \(-0.280257\pi\)
0.636802 + 0.771027i \(0.280257\pi\)
\(620\) −32.5623 −1.30773
\(621\) 0 0
\(622\) 12.5623 0.503703
\(623\) −15.7082 −0.629336
\(624\) 0 0
\(625\) −30.8328 −1.23331
\(626\) 11.3475 0.453538
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) −0.604878 −0.0240228
\(635\) −47.8328 −1.89819
\(636\) 0 0
\(637\) −11.1246 −0.440773
\(638\) 0.111456 0.00441259
\(639\) 0 0
\(640\) −26.4164 −1.04420
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) 8.29180 0.326743
\(645\) 0 0
\(646\) −1.10333 −0.0434098
\(647\) −43.7426 −1.71970 −0.859850 0.510546i \(-0.829443\pi\)
−0.859850 + 0.510546i \(0.829443\pi\)
\(648\) 0 0
\(649\) −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 \(-0.495242\pi\)
0.0149475 + 0.999888i \(0.495242\pi\)
\(654\) 0 0
\(655\) −5.85410 −0.228739
\(656\) 28.1378 1.09860
\(657\) 0 0
\(658\) −2.70820 −0.105577
\(659\) 12.5967 0.490700 0.245350 0.969435i \(-0.421097\pi\)
0.245350 + 0.969435i \(0.421097\pi\)
\(660\) 0 0
\(661\) −11.4377 −0.444875 −0.222437 0.974947i \(-0.571401\pi\)
−0.222437 + 0.974947i \(0.571401\pi\)
\(662\) −8.72949 −0.339281
\(663\) 0 0
\(664\) 13.7295 0.532808
\(665\) 2.23607 0.0867110
\(666\) 0 0
\(667\) 3.41641 0.132284
\(668\) 16.6869 0.645636
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 17.7295 0.681903
\(677\) −10.9656 −0.421441 −0.210720 0.977546i \(-0.567581\pi\)
−0.210720 + 0.977546i \(0.567581\pi\)
\(678\) 0 0
\(679\) −11.7082 −0.449320
\(680\) −13.0344 −0.499848
\(681\) 0 0
\(682\) 0.978714 0.0374769
\(683\) −42.7639 −1.63632 −0.818158 0.574993i \(-0.805005\pi\)
−0.818158 + 0.574993i \(0.805005\pi\)
\(684\) 0 0
\(685\) −1.85410 −0.0708416
\(686\) 4.96556 0.189586
\(687\) 0 0
\(688\) 14.8115 0.564684
\(689\) 18.7082 0.712726
\(690\) 0 0
\(691\) 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 0
\(694\) −2.85410 −0.108340
\(695\) −46.7426 −1.77305
\(696\) 0 0
\(697\) 30.2492 1.14577
\(698\) −4.36068 −0.165054
\(699\) 0 0
\(700\) 3.43769 0.129933
\(701\) 1.20163 0.0453848 0.0226924 0.999742i \(-0.492776\pi\)
0.0226924 + 0.999742i \(0.492776\pi\)
\(702\) 0 0
\(703\) 5.72949 0.216092
\(704\) −1.79837 −0.0677788
\(705\) 0 0
\(706\) −1.54102 −0.0579970
\(707\) 13.0902 0.492307
\(708\) 0 0
\(709\) −47.9787 −1.80188 −0.900939 0.433945i \(-0.857121\pi\)
−0.900939 + 0.433945i \(0.857121\pi\)
\(710\) −7.09017 −0.266089
\(711\) 0 0
\(712\) −23.1246 −0.866631
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 1.85410 0.0693395
\(716\) −2.12461 −0.0794005
\(717\) 0 0
\(718\) −11.5836 −0.432296
\(719\) −23.6738 −0.882882 −0.441441 0.897290i \(-0.645533\pi\)
−0.441441 + 0.897290i \(0.645533\pi\)
\(720\) 0 0
\(721\) 1.00000 0.0372419
\(722\) −6.97871 −0.259721
\(723\) 0 0
\(724\) 5.29180 0.196668
\(725\) 1.41641 0.0526041
\(726\) 0 0
\(727\) −38.2705 −1.41937 −0.709687 0.704517i \(-0.751164\pi\)
−0.709687 + 0.704517i \(0.751164\pi\)
\(728\) 2.72949 0.101162
\(729\) 0 0
\(730\) −4.14590 −0.153447
\(731\) 15.9230 0.588933
\(732\) 0 0
\(733\) 1.29180 0.0477136 0.0238568 0.999715i \(-0.492405\pi\)
0.0238568 + 0.999715i \(0.492405\pi\)
\(734\) 13.9656 0.515478
\(735\) 0 0
\(736\) 18.5410 0.683431
\(737\) −4.74265 −0.174698
\(738\) 0 0
\(739\) 36.8328 1.35492 0.677459 0.735561i \(-0.263081\pi\)
0.677459 + 0.735561i \(0.263081\pi\)
\(740\) 32.5623 1.19701
\(741\) 0 0
\(742\) −3.85410 −0.141489
\(743\) −17.2918 −0.634374 −0.317187 0.948363i \(-0.602738\pi\)
−0.317187 + 0.948363i \(0.602738\pi\)
\(744\) 0 0
\(745\) −3.85410 −0.141203
\(746\) 14.3951 0.527043
\(747\) 0 0
\(748\) −2.39512 −0.0875743
\(749\) 4.09017 0.149452
\(750\) 0 0
\(751\) −3.87539 −0.141415 −0.0707075 0.997497i \(-0.522526\pi\)
−0.0707075 + 0.997497i \(0.522526\pi\)
\(752\) 22.3050 0.813378
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) 3.29180 0.119406
\(761\) −1.47214 −0.0533649 −0.0266824 0.999644i \(-0.508494\pi\)
−0.0266824 + 0.999644i \(0.508494\pi\)
\(762\) 0 0
\(763\) −10.5623 −0.382381
\(764\) 6.27051 0.226859
\(765\) 0 0
\(766\) −8.85410 −0.319912
\(767\) −15.9787 −0.576958
\(768\) 0 0
\(769\) −42.3951 −1.52881 −0.764404 0.644738i \(-0.776966\pi\)
−0.764404 + 0.644738i \(0.776966\pi\)
\(770\) −0.381966 −0.0137651
\(771\) 0 0
\(772\) 37.3131 1.34293
\(773\) 25.4721 0.916169 0.458085 0.888909i \(-0.348536\pi\)
0.458085 + 0.888909i \(0.348536\pi\)
\(774\) 0 0
\(775\) 12.4377 0.446775
\(776\) −17.2361 −0.618739
\(777\) 0 0
\(778\) 7.41641 0.265891
\(779\) −7.63932 −0.273707
\(780\) 0 0
\(781\) −2.70820 −0.0969072
\(782\) 5.77709 0.206588
\(783\) 0 0
\(784\) −18.8754 −0.674121
\(785\) 8.61803 0.307591
\(786\) 0 0
\(787\) 16.5836 0.591141 0.295571 0.955321i \(-0.404490\pi\)
0.295571 + 0.955321i \(0.404490\pi\)
\(788\) 19.3131 0.688000
\(789\) 0 0
\(790\) 13.5623 0.482525
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) 20.1246 0.714646
\(794\) 7.63932 0.271109
\(795\) 0 0
\(796\) −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
\(801\) 0 0
\(802\) −4.54102 −0.160349
\(803\) −1.58359 −0.0558838
\(804\) 0 0
\(805\) −11.7082 −0.412660
\(806\) 4.75078 0.167339
\(807\) 0 0
\(808\) 19.2705 0.677934
\(809\) 14.9443 0.525413 0.262706 0.964876i \(-0.415385\pi\)
0.262706 + 0.964876i \(0.415385\pi\)
\(810\) 0 0
\(811\) −45.5410 −1.59916 −0.799581 0.600559i \(-0.794945\pi\)
−0.799581 + 0.600559i \(0.794945\pi\)
\(812\) 1.41641 0.0497062
\(813\) 0 0
\(814\) −0.978714 −0.0343039
\(815\) −28.0344 −0.982004
\(816\) 0 0
\(817\) −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 \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 0 0
\(823\) 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 0
\(826\) 3.29180 0.114536
\(827\) 6.70820 0.233267 0.116634 0.993175i \(-0.462790\pi\)
0.116634 + 0.993175i \(0.462790\pi\)
\(828\) 0 0
\(829\) −19.2705 −0.669292 −0.334646 0.942344i \(-0.608617\pi\)
−0.334646 + 0.942344i \(0.608617\pi\)
\(830\) −9.32624 −0.323718
\(831\) 0 0
\(832\) −8.72949 −0.302641
\(833\) −20.2918 −0.703069
\(834\) 0 0
\(835\) −23.5623 −0.815407
\(836\) 0.604878 0.0209202
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) −15.1033 −0.519878
\(845\) −25.0344 −0.861211
\(846\) 0 0
\(847\) 10.8541 0.372951
\(848\) 31.7426 1.09005
\(849\) 0 0
\(850\) 2.39512 0.0821520
\(851\) −30.0000 −1.02839
\(852\) 0 0
\(853\) −10.7295 −0.367371 −0.183685 0.982985i \(-0.558803\pi\)
−0.183685 + 0.982985i \(0.558803\pi\)
\(854\) −4.14590 −0.141870
\(855\) 0 0
\(856\) 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 0
\(859\) −9.56231 −0.326262 −0.163131 0.986604i \(-0.552159\pi\)
−0.163131 + 0.986604i \(0.552159\pi\)
\(860\) −22.8541 −0.779318
\(861\) 0 0
\(862\) 3.95743 0.134791
\(863\) 8.29180 0.282256 0.141128 0.989991i \(-0.454927\pi\)
0.141128 + 0.989991i \(0.454927\pi\)
\(864\) 0 0
\(865\) 34.1246 1.16027
\(866\) −4.74265 −0.161162
\(867\) 0 0
\(868\) 12.4377 0.422163
\(869\) 5.18034 0.175731
\(870\) 0 0
\(871\) −23.0213 −0.780047
\(872\) −15.5492 −0.526561
\(873\) 0 0
\(874\) −1.45898 −0.0493507
\(875\) 8.23607 0.278430
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 3.60488 0.121659
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) 13.5836 0.456350
\(887\) 58.1935 1.95395 0.976973 0.213362i \(-0.0684414\pi\)
0.976973 + 0.213362i \(0.0684414\pi\)
\(888\) 0 0
\(889\) 18.2705 0.612773
\(890\) 15.7082 0.526540
\(891\) 0 0
\(892\) 10.5836 0.354365
\(893\) −6.05573 −0.202647
\(894\) 0 0
\(895\) 3.00000 0.100279
\(896\) 10.0902 0.337089
\(897\) 0 0
\(898\) −11.9787 −0.399735
\(899\) 5.12461 0.170915
\(900\) 0 0
\(901\) 34.1246 1.13686
\(902\) 1.30495 0.0434501
\(903\) 0 0
\(904\) −22.0820 −0.734438
\(905\) −7.47214 −0.248382
\(906\) 0 0
\(907\) 33.1246 1.09988 0.549942 0.835203i \(-0.314650\pi\)
0.549942 + 0.835203i \(0.314650\pi\)
\(908\) −27.7082 −0.919529
\(909\) 0 0
\(910\) −1.85410 −0.0614629
\(911\) −19.0344 −0.630639 −0.315320 0.948986i \(-0.602112\pi\)
−0.315320 + 0.948986i \(0.602112\pi\)
\(912\) 0 0
\(913\) −3.56231 −0.117895
\(914\) −1.20163 −0.0397463
\(915\) 0 0
\(916\) 12.4377 0.410953
\(917\) 2.23607 0.0738415
\(918\) 0 0
\(919\) 18.9787 0.626050 0.313025 0.949745i \(-0.398658\pi\)
0.313025 + 0.949745i \(0.398658\pi\)
\(920\) −17.2361 −0.568256
\(921\) 0 0
\(922\) 14.9787 0.493298
\(923\) −13.1459 −0.432703
\(924\) 0 0
\(925\) −12.4377 −0.408949
\(926\) 2.06888 0.0679877
\(927\) 0 0
\(928\) 3.16718 0.103968
\(929\) 2.94427 0.0965984 0.0482992 0.998833i \(-0.484620\pi\)
0.0482992 + 0.998833i \(0.484620\pi\)
\(930\) 0 0
\(931\) 5.12461 0.167952
\(932\) 16.4803 0.539829
\(933\) 0 0
\(934\) −10.5623 −0.345609
\(935\) 3.38197 0.110602
\(936\) 0 0
\(937\) −11.0000 −0.359354 −0.179677 0.983726i \(-0.557505\pi\)
−0.179677 + 0.983726i \(0.557505\pi\)
\(938\) 4.74265 0.154853
\(939\) 0 0
\(940\) −34.4164 −1.12254
\(941\) −50.3951 −1.64283 −0.821417 0.570328i \(-0.806816\pi\)
−0.821417 + 0.570328i \(0.806816\pi\)
\(942\) 0 0
\(943\) 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 \(-0.692626\pi\)
−0.568887 + 0.822415i \(0.692626\pi\)
\(948\) 0 0
\(949\) −7.68692 −0.249528
\(950\) −0.604878 −0.0196248
\(951\) 0 0
\(952\) 4.97871 0.161361
\(953\) 31.3607 1.01587 0.507936 0.861395i \(-0.330409\pi\)
0.507936 + 0.861395i \(0.330409\pi\)
\(954\) 0 0
\(955\) −8.85410 −0.286512
\(956\) 45.1033 1.45875
\(957\) 0 0
\(958\) 5.41641 0.174996
\(959\) 0.708204 0.0228691
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) −4.75078 −0.153171
\(963\) 0 0
\(964\) −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
\(969\) 0 0
\(970\) 11.7082 0.375928
\(971\) −23.0132 −0.738527 −0.369264 0.929325i \(-0.620390\pi\)
−0.369264 + 0.929325i \(0.620390\pi\)
\(972\) 0 0
\(973\) 17.8541 0.572376
\(974\) −8.78522 −0.281497
\(975\) 0 0
\(976\) 34.1459 1.09298
\(977\) −4.25735 −0.136205 −0.0681024 0.997678i \(-0.521694\pi\)
−0.0681024 + 0.997678i \(0.521694\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 29.1246 0.930352
\(981\) 0 0
\(982\) 13.6656 0.436088
\(983\) 12.6525 0.403551 0.201776 0.979432i \(-0.435329\pi\)
0.201776 + 0.979432i \(0.435329\pi\)
\(984\) 0 0
\(985\) −27.2705 −0.868911
\(986\) 0.986844 0.0314275
\(987\) 0 0
\(988\) 2.93614 0.0934111
\(989\) 21.0557 0.669533
\(990\) 0 0
\(991\) −9.27051 −0.294487 −0.147244 0.989100i \(-0.547040\pi\)
−0.147244 + 0.989100i \(0.547040\pi\)
\(992\) 27.8115 0.883017
\(993\) 0 0
\(994\) 2.70820 0.0858990
\(995\) 8.94427 0.283552
\(996\) 0 0
\(997\) −52.7082 −1.66929 −0.834643 0.550792i \(-0.814326\pi\)
−0.834643 + 0.550792i \(0.814326\pi\)
\(998\) −7.74265 −0.245089
\(999\) 0 0
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 927.2.a.b.1.1 2
3.2 odd 2 103.2.a.a.1.2 2
12.11 even 2 1648.2.a.f.1.1 2
15.14 odd 2 2575.2.a.g.1.1 2
21.20 even 2 5047.2.a.a.1.2 2
24.5 odd 2 6592.2.a.t.1.2 2
24.11 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 3.2 odd 2
927.2.a.b.1.1 2 1.1 even 1 trivial
1648.2.a.f.1.1 2 12.11 even 2
2575.2.a.g.1.1 2 15.14 odd 2
5047.2.a.a.1.2 2 21.20 even 2
6592.2.a.h.1.2 2 24.11 even 2
6592.2.a.t.1.2 2 24.5 odd 2