Properties

Label 2738.2.a.f.1.2
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
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] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, 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("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.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: \(21.8630400734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.73205 q^{5} -0.732051 q^{6} +4.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.73205 q^{5} -0.732051 q^{6} +4.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} -1.73205 q^{10} -4.73205 q^{11} +0.732051 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.26795 q^{15} +1.00000 q^{16} -1.73205 q^{17} +2.46410 q^{18} +1.26795 q^{19} +1.73205 q^{20} +2.92820 q^{21} +4.73205 q^{22} +1.26795 q^{23} -0.732051 q^{24} -2.00000 q^{25} -6.00000 q^{26} -4.00000 q^{27} +4.00000 q^{28} +4.26795 q^{29} -1.26795 q^{30} +1.26795 q^{31} -1.00000 q^{32} -3.46410 q^{33} +1.73205 q^{34} +6.92820 q^{35} -2.46410 q^{36} -1.26795 q^{38} +4.39230 q^{39} -1.73205 q^{40} -0.464102 q^{41} -2.92820 q^{42} +9.46410 q^{43} -4.73205 q^{44} -4.26795 q^{45} -1.26795 q^{46} +11.6603 q^{47} +0.732051 q^{48} +9.00000 q^{49} +2.00000 q^{50} -1.26795 q^{51} +6.00000 q^{52} -2.53590 q^{53} +4.00000 q^{54} -8.19615 q^{55} -4.00000 q^{56} +0.928203 q^{57} -4.26795 q^{58} -2.53590 q^{59} +1.26795 q^{60} +14.6603 q^{61} -1.26795 q^{62} -9.85641 q^{63} +1.00000 q^{64} +10.3923 q^{65} +3.46410 q^{66} -6.19615 q^{67} -1.73205 q^{68} +0.928203 q^{69} -6.92820 q^{70} +2.53590 q^{71} +2.46410 q^{72} +12.3923 q^{73} -1.46410 q^{75} +1.26795 q^{76} -18.9282 q^{77} -4.39230 q^{78} +8.19615 q^{79} +1.73205 q^{80} +4.46410 q^{81} +0.464102 q^{82} +11.6603 q^{83} +2.92820 q^{84} -3.00000 q^{85} -9.46410 q^{86} +3.12436 q^{87} +4.73205 q^{88} -5.19615 q^{89} +4.26795 q^{90} +24.0000 q^{91} +1.26795 q^{92} +0.928203 q^{93} -11.6603 q^{94} +2.19615 q^{95} -0.732051 q^{96} +5.19615 q^{97} -9.00000 q^{98} +11.6603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 q^{7} - 2 q^{8} + 2 q^{9} - 6 q^{11} - 2 q^{12} + 12 q^{13} - 8 q^{14} + 6 q^{15} + 2 q^{16} - 2 q^{18} + 6 q^{19} - 8 q^{21} + 6 q^{22} + 6 q^{23} + 2 q^{24} - 4 q^{25} - 12 q^{26} - 8 q^{27} + 8 q^{28} + 12 q^{29} - 6 q^{30} + 6 q^{31} - 2 q^{32} + 2 q^{36} - 6 q^{38} - 12 q^{39} + 6 q^{41} + 8 q^{42} + 12 q^{43} - 6 q^{44} - 12 q^{45} - 6 q^{46} + 6 q^{47} - 2 q^{48} + 18 q^{49} + 4 q^{50} - 6 q^{51} + 12 q^{52} - 12 q^{53} + 8 q^{54} - 6 q^{55} - 8 q^{56} - 12 q^{57} - 12 q^{58} - 12 q^{59} + 6 q^{60} + 12 q^{61} - 6 q^{62} + 8 q^{63} + 2 q^{64} - 2 q^{67} - 12 q^{69} + 12 q^{71} - 2 q^{72} + 4 q^{73} + 4 q^{75} + 6 q^{76} - 24 q^{77} + 12 q^{78} + 6 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{83} - 8 q^{84} - 6 q^{85} - 12 q^{86} - 18 q^{87} + 6 q^{88} + 12 q^{90} + 48 q^{91} + 6 q^{92} - 12 q^{93} - 6 q^{94} - 6 q^{95} + 2 q^{96} - 18 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.73205 0.774597 0.387298 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) −0.732051 −0.298858
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.46410 −0.821367
\(10\) −1.73205 −0.547723
\(11\) −4.73205 −1.42677 −0.713384 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0.732051 0.211325
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 1.26795 0.327383
\(16\) 1.00000 0.250000
\(17\) −1.73205 −0.420084 −0.210042 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 2.46410 0.580794
\(19\) 1.26795 0.290887 0.145444 0.989367i \(-0.453539\pi\)
0.145444 + 0.989367i \(0.453539\pi\)
\(20\) 1.73205 0.387298
\(21\) 2.92820 0.638986
\(22\) 4.73205 1.00888
\(23\) 1.26795 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(24\) −0.732051 −0.149429
\(25\) −2.00000 −0.400000
\(26\) −6.00000 −1.17670
\(27\) −4.00000 −0.769800
\(28\) 4.00000 0.755929
\(29\) 4.26795 0.792538 0.396269 0.918134i \(-0.370305\pi\)
0.396269 + 0.918134i \(0.370305\pi\)
\(30\) −1.26795 −0.231495
\(31\) 1.26795 0.227730 0.113865 0.993496i \(-0.463677\pi\)
0.113865 + 0.993496i \(0.463677\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.46410 −0.603023
\(34\) 1.73205 0.297044
\(35\) 6.92820 1.17108
\(36\) −2.46410 −0.410684
\(37\) 0 0
\(38\) −1.26795 −0.205689
\(39\) 4.39230 0.703332
\(40\) −1.73205 −0.273861
\(41\) −0.464102 −0.0724805 −0.0362402 0.999343i \(-0.511538\pi\)
−0.0362402 + 0.999343i \(0.511538\pi\)
\(42\) −2.92820 −0.451832
\(43\) 9.46410 1.44326 0.721631 0.692278i \(-0.243393\pi\)
0.721631 + 0.692278i \(0.243393\pi\)
\(44\) −4.73205 −0.713384
\(45\) −4.26795 −0.636228
\(46\) −1.26795 −0.186949
\(47\) 11.6603 1.70082 0.850411 0.526118i \(-0.176353\pi\)
0.850411 + 0.526118i \(0.176353\pi\)
\(48\) 0.732051 0.105662
\(49\) 9.00000 1.28571
\(50\) 2.00000 0.282843
\(51\) −1.26795 −0.177548
\(52\) 6.00000 0.832050
\(53\) −2.53590 −0.348332 −0.174166 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 4.00000 0.544331
\(55\) −8.19615 −1.10517
\(56\) −4.00000 −0.534522
\(57\) 0.928203 0.122944
\(58\) −4.26795 −0.560409
\(59\) −2.53590 −0.330146 −0.165073 0.986281i \(-0.552786\pi\)
−0.165073 + 0.986281i \(0.552786\pi\)
\(60\) 1.26795 0.163692
\(61\) 14.6603 1.87705 0.938527 0.345207i \(-0.112191\pi\)
0.938527 + 0.345207i \(0.112191\pi\)
\(62\) −1.26795 −0.161030
\(63\) −9.85641 −1.24179
\(64\) 1.00000 0.125000
\(65\) 10.3923 1.28901
\(66\) 3.46410 0.426401
\(67\) −6.19615 −0.756980 −0.378490 0.925605i \(-0.623557\pi\)
−0.378490 + 0.925605i \(0.623557\pi\)
\(68\) −1.73205 −0.210042
\(69\) 0.928203 0.111743
\(70\) −6.92820 −0.828079
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 2.46410 0.290397
\(73\) 12.3923 1.45041 0.725205 0.688533i \(-0.241745\pi\)
0.725205 + 0.688533i \(0.241745\pi\)
\(74\) 0 0
\(75\) −1.46410 −0.169060
\(76\) 1.26795 0.145444
\(77\) −18.9282 −2.15707
\(78\) −4.39230 −0.497331
\(79\) 8.19615 0.922139 0.461070 0.887364i \(-0.347466\pi\)
0.461070 + 0.887364i \(0.347466\pi\)
\(80\) 1.73205 0.193649
\(81\) 4.46410 0.496011
\(82\) 0.464102 0.0512514
\(83\) 11.6603 1.27988 0.639940 0.768425i \(-0.278959\pi\)
0.639940 + 0.768425i \(0.278959\pi\)
\(84\) 2.92820 0.319493
\(85\) −3.00000 −0.325396
\(86\) −9.46410 −1.02054
\(87\) 3.12436 0.334966
\(88\) 4.73205 0.504438
\(89\) −5.19615 −0.550791 −0.275396 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 4.26795 0.449881
\(91\) 24.0000 2.51588
\(92\) 1.26795 0.132193
\(93\) 0.928203 0.0962502
\(94\) −11.6603 −1.20266
\(95\) 2.19615 0.225320
\(96\) −0.732051 −0.0747146
\(97\) 5.19615 0.527589 0.263795 0.964579i \(-0.415026\pi\)
0.263795 + 0.964579i \(0.415026\pi\)
\(98\) −9.00000 −0.909137
\(99\) 11.6603 1.17190
\(100\) −2.00000 −0.200000
\(101\) 0.464102 0.0461798 0.0230899 0.999733i \(-0.492650\pi\)
0.0230899 + 0.999733i \(0.492650\pi\)
\(102\) 1.26795 0.125546
\(103\) −6.92820 −0.682656 −0.341328 0.939944i \(-0.610877\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(104\) −6.00000 −0.588348
\(105\) 5.07180 0.494957
\(106\) 2.53590 0.246308
\(107\) −16.3923 −1.58470 −0.792352 0.610064i \(-0.791144\pi\)
−0.792352 + 0.610064i \(0.791144\pi\)
\(108\) −4.00000 −0.384900
\(109\) −7.73205 −0.740596 −0.370298 0.928913i \(-0.620744\pi\)
−0.370298 + 0.928913i \(0.620744\pi\)
\(110\) 8.19615 0.781472
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) −0.928203 −0.0869342
\(115\) 2.19615 0.204792
\(116\) 4.26795 0.396269
\(117\) −14.7846 −1.36684
\(118\) 2.53590 0.233448
\(119\) −6.92820 −0.635107
\(120\) −1.26795 −0.115747
\(121\) 11.3923 1.03566
\(122\) −14.6603 −1.32728
\(123\) −0.339746 −0.0306339
\(124\) 1.26795 0.113865
\(125\) −12.1244 −1.08444
\(126\) 9.85641 0.878078
\(127\) −10.5885 −0.939574 −0.469787 0.882780i \(-0.655669\pi\)
−0.469787 + 0.882780i \(0.655669\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.92820 0.609994
\(130\) −10.3923 −0.911465
\(131\) 8.19615 0.716101 0.358051 0.933702i \(-0.383442\pi\)
0.358051 + 0.933702i \(0.383442\pi\)
\(132\) −3.46410 −0.301511
\(133\) 5.07180 0.439781
\(134\) 6.19615 0.535266
\(135\) −6.92820 −0.596285
\(136\) 1.73205 0.148522
\(137\) −13.3923 −1.14418 −0.572091 0.820190i \(-0.693868\pi\)
−0.572091 + 0.820190i \(0.693868\pi\)
\(138\) −0.928203 −0.0790139
\(139\) −1.80385 −0.153000 −0.0765002 0.997070i \(-0.524375\pi\)
−0.0765002 + 0.997070i \(0.524375\pi\)
\(140\) 6.92820 0.585540
\(141\) 8.53590 0.718852
\(142\) −2.53590 −0.212808
\(143\) −28.3923 −2.37428
\(144\) −2.46410 −0.205342
\(145\) 7.39230 0.613898
\(146\) −12.3923 −1.02559
\(147\) 6.58846 0.543407
\(148\) 0 0
\(149\) 3.92820 0.321811 0.160905 0.986970i \(-0.448559\pi\)
0.160905 + 0.986970i \(0.448559\pi\)
\(150\) 1.46410 0.119543
\(151\) 1.80385 0.146795 0.0733975 0.997303i \(-0.476616\pi\)
0.0733975 + 0.997303i \(0.476616\pi\)
\(152\) −1.26795 −0.102844
\(153\) 4.26795 0.345043
\(154\) 18.9282 1.52528
\(155\) 2.19615 0.176399
\(156\) 4.39230 0.351666
\(157\) −9.39230 −0.749588 −0.374794 0.927108i \(-0.622286\pi\)
−0.374794 + 0.927108i \(0.622286\pi\)
\(158\) −8.19615 −0.652051
\(159\) −1.85641 −0.147223
\(160\) −1.73205 −0.136931
\(161\) 5.07180 0.399714
\(162\) −4.46410 −0.350733
\(163\) 2.53590 0.198627 0.0993134 0.995056i \(-0.468335\pi\)
0.0993134 + 0.995056i \(0.468335\pi\)
\(164\) −0.464102 −0.0362402
\(165\) −6.00000 −0.467099
\(166\) −11.6603 −0.905011
\(167\) 6.92820 0.536120 0.268060 0.963402i \(-0.413617\pi\)
0.268060 + 0.963402i \(0.413617\pi\)
\(168\) −2.92820 −0.225916
\(169\) 23.0000 1.76923
\(170\) 3.00000 0.230089
\(171\) −3.12436 −0.238925
\(172\) 9.46410 0.721631
\(173\) 9.92820 0.754827 0.377414 0.926045i \(-0.376813\pi\)
0.377414 + 0.926045i \(0.376813\pi\)
\(174\) −3.12436 −0.236857
\(175\) −8.00000 −0.604743
\(176\) −4.73205 −0.356692
\(177\) −1.85641 −0.139536
\(178\) 5.19615 0.389468
\(179\) 11.3205 0.846135 0.423067 0.906098i \(-0.360953\pi\)
0.423067 + 0.906098i \(0.360953\pi\)
\(180\) −4.26795 −0.318114
\(181\) −15.3923 −1.14410 −0.572051 0.820218i \(-0.693852\pi\)
−0.572051 + 0.820218i \(0.693852\pi\)
\(182\) −24.0000 −1.77900
\(183\) 10.7321 0.793336
\(184\) −1.26795 −0.0934745
\(185\) 0 0
\(186\) −0.928203 −0.0680592
\(187\) 8.19615 0.599362
\(188\) 11.6603 0.850411
\(189\) −16.0000 −1.16383
\(190\) −2.19615 −0.159326
\(191\) 24.5885 1.77916 0.889579 0.456781i \(-0.150998\pi\)
0.889579 + 0.456781i \(0.150998\pi\)
\(192\) 0.732051 0.0528312
\(193\) −12.1244 −0.872730 −0.436365 0.899770i \(-0.643734\pi\)
−0.436365 + 0.899770i \(0.643734\pi\)
\(194\) −5.19615 −0.373062
\(195\) 7.60770 0.544798
\(196\) 9.00000 0.642857
\(197\) 26.3205 1.87526 0.937629 0.347637i \(-0.113016\pi\)
0.937629 + 0.347637i \(0.113016\pi\)
\(198\) −11.6603 −0.828658
\(199\) 5.07180 0.359530 0.179765 0.983710i \(-0.442466\pi\)
0.179765 + 0.983710i \(0.442466\pi\)
\(200\) 2.00000 0.141421
\(201\) −4.53590 −0.319938
\(202\) −0.464102 −0.0326541
\(203\) 17.0718 1.19821
\(204\) −1.26795 −0.0887742
\(205\) −0.803848 −0.0561432
\(206\) 6.92820 0.482711
\(207\) −3.12436 −0.217158
\(208\) 6.00000 0.416025
\(209\) −6.00000 −0.415029
\(210\) −5.07180 −0.349987
\(211\) 12.3923 0.853121 0.426561 0.904459i \(-0.359725\pi\)
0.426561 + 0.904459i \(0.359725\pi\)
\(212\) −2.53590 −0.174166
\(213\) 1.85641 0.127199
\(214\) 16.3923 1.12055
\(215\) 16.3923 1.11795
\(216\) 4.00000 0.272166
\(217\) 5.07180 0.344296
\(218\) 7.73205 0.523681
\(219\) 9.07180 0.613015
\(220\) −8.19615 −0.552584
\(221\) −10.3923 −0.699062
\(222\) 0 0
\(223\) −22.5885 −1.51263 −0.756317 0.654205i \(-0.773003\pi\)
−0.756317 + 0.654205i \(0.773003\pi\)
\(224\) −4.00000 −0.267261
\(225\) 4.92820 0.328547
\(226\) 19.8564 1.32083
\(227\) 1.26795 0.0841567 0.0420784 0.999114i \(-0.486602\pi\)
0.0420784 + 0.999114i \(0.486602\pi\)
\(228\) 0.928203 0.0614718
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) −2.19615 −0.144810
\(231\) −13.8564 −0.911685
\(232\) −4.26795 −0.280205
\(233\) 4.60770 0.301860 0.150930 0.988544i \(-0.451773\pi\)
0.150930 + 0.988544i \(0.451773\pi\)
\(234\) 14.7846 0.966500
\(235\) 20.1962 1.31745
\(236\) −2.53590 −0.165073
\(237\) 6.00000 0.389742
\(238\) 6.92820 0.449089
\(239\) −5.07180 −0.328067 −0.164034 0.986455i \(-0.552451\pi\)
−0.164034 + 0.986455i \(0.552451\pi\)
\(240\) 1.26795 0.0818458
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −11.3923 −0.732325
\(243\) 15.2679 0.979439
\(244\) 14.6603 0.938527
\(245\) 15.5885 0.995910
\(246\) 0.339746 0.0216614
\(247\) 7.60770 0.484066
\(248\) −1.26795 −0.0805149
\(249\) 8.53590 0.540941
\(250\) 12.1244 0.766812
\(251\) 5.07180 0.320129 0.160064 0.987107i \(-0.448830\pi\)
0.160064 + 0.987107i \(0.448830\pi\)
\(252\) −9.85641 −0.620895
\(253\) −6.00000 −0.377217
\(254\) 10.5885 0.664379
\(255\) −2.19615 −0.137528
\(256\) 1.00000 0.0625000
\(257\) 11.1962 0.698397 0.349198 0.937049i \(-0.386454\pi\)
0.349198 + 0.937049i \(0.386454\pi\)
\(258\) −6.92820 −0.431331
\(259\) 0 0
\(260\) 10.3923 0.644503
\(261\) −10.5167 −0.650965
\(262\) −8.19615 −0.506360
\(263\) −23.3205 −1.43800 −0.719002 0.695008i \(-0.755401\pi\)
−0.719002 + 0.695008i \(0.755401\pi\)
\(264\) 3.46410 0.213201
\(265\) −4.39230 −0.269817
\(266\) −5.07180 −0.310972
\(267\) −3.80385 −0.232792
\(268\) −6.19615 −0.378490
\(269\) −7.60770 −0.463849 −0.231925 0.972734i \(-0.574502\pi\)
−0.231925 + 0.972734i \(0.574502\pi\)
\(270\) 6.92820 0.421637
\(271\) −14.5885 −0.886186 −0.443093 0.896476i \(-0.646119\pi\)
−0.443093 + 0.896476i \(0.646119\pi\)
\(272\) −1.73205 −0.105021
\(273\) 17.5692 1.06334
\(274\) 13.3923 0.809059
\(275\) 9.46410 0.570707
\(276\) 0.928203 0.0558713
\(277\) −6.80385 −0.408804 −0.204402 0.978887i \(-0.565525\pi\)
−0.204402 + 0.978887i \(0.565525\pi\)
\(278\) 1.80385 0.108188
\(279\) −3.12436 −0.187050
\(280\) −6.92820 −0.414039
\(281\) −11.1962 −0.667906 −0.333953 0.942590i \(-0.608383\pi\)
−0.333953 + 0.942590i \(0.608383\pi\)
\(282\) −8.53590 −0.508305
\(283\) 2.53590 0.150744 0.0753718 0.997156i \(-0.475986\pi\)
0.0753718 + 0.997156i \(0.475986\pi\)
\(284\) 2.53590 0.150478
\(285\) 1.60770 0.0952316
\(286\) 28.3923 1.67887
\(287\) −1.85641 −0.109580
\(288\) 2.46410 0.145199
\(289\) −14.0000 −0.823529
\(290\) −7.39230 −0.434091
\(291\) 3.80385 0.222985
\(292\) 12.3923 0.725205
\(293\) 25.3923 1.48343 0.741717 0.670713i \(-0.234012\pi\)
0.741717 + 0.670713i \(0.234012\pi\)
\(294\) −6.58846 −0.384247
\(295\) −4.39230 −0.255730
\(296\) 0 0
\(297\) 18.9282 1.09833
\(298\) −3.92820 −0.227555
\(299\) 7.60770 0.439964
\(300\) −1.46410 −0.0845299
\(301\) 37.8564 2.18201
\(302\) −1.80385 −0.103800
\(303\) 0.339746 0.0195179
\(304\) 1.26795 0.0727219
\(305\) 25.3923 1.45396
\(306\) −4.26795 −0.243982
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −18.9282 −1.07853
\(309\) −5.07180 −0.288524
\(310\) −2.19615 −0.124733
\(311\) −16.3923 −0.929522 −0.464761 0.885436i \(-0.653860\pi\)
−0.464761 + 0.885436i \(0.653860\pi\)
\(312\) −4.39230 −0.248665
\(313\) 21.5885 1.22025 0.610126 0.792304i \(-0.291119\pi\)
0.610126 + 0.792304i \(0.291119\pi\)
\(314\) 9.39230 0.530038
\(315\) −17.0718 −0.961887
\(316\) 8.19615 0.461070
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) 1.85641 0.104102
\(319\) −20.1962 −1.13077
\(320\) 1.73205 0.0968246
\(321\) −12.0000 −0.669775
\(322\) −5.07180 −0.282640
\(323\) −2.19615 −0.122197
\(324\) 4.46410 0.248006
\(325\) −12.0000 −0.665640
\(326\) −2.53590 −0.140450
\(327\) −5.66025 −0.313013
\(328\) 0.464102 0.0256257
\(329\) 46.6410 2.57140
\(330\) 6.00000 0.330289
\(331\) −30.9282 −1.69997 −0.849984 0.526809i \(-0.823388\pi\)
−0.849984 + 0.526809i \(0.823388\pi\)
\(332\) 11.6603 0.639940
\(333\) 0 0
\(334\) −6.92820 −0.379094
\(335\) −10.7321 −0.586355
\(336\) 2.92820 0.159747
\(337\) 7.00000 0.381314 0.190657 0.981657i \(-0.438938\pi\)
0.190657 + 0.981657i \(0.438938\pi\)
\(338\) −23.0000 −1.25104
\(339\) −14.5359 −0.789482
\(340\) −3.00000 −0.162698
\(341\) −6.00000 −0.324918
\(342\) 3.12436 0.168946
\(343\) 8.00000 0.431959
\(344\) −9.46410 −0.510270
\(345\) 1.60770 0.0865554
\(346\) −9.92820 −0.533744
\(347\) −10.0526 −0.539650 −0.269825 0.962909i \(-0.586966\pi\)
−0.269825 + 0.962909i \(0.586966\pi\)
\(348\) 3.12436 0.167483
\(349\) −6.60770 −0.353702 −0.176851 0.984238i \(-0.556591\pi\)
−0.176851 + 0.984238i \(0.556591\pi\)
\(350\) 8.00000 0.427618
\(351\) −24.0000 −1.28103
\(352\) 4.73205 0.252219
\(353\) −30.1244 −1.60336 −0.801679 0.597755i \(-0.796060\pi\)
−0.801679 + 0.597755i \(0.796060\pi\)
\(354\) 1.85641 0.0986669
\(355\) 4.39230 0.233119
\(356\) −5.19615 −0.275396
\(357\) −5.07180 −0.268428
\(358\) −11.3205 −0.598307
\(359\) 33.4641 1.76617 0.883084 0.469215i \(-0.155463\pi\)
0.883084 + 0.469215i \(0.155463\pi\)
\(360\) 4.26795 0.224941
\(361\) −17.3923 −0.915384
\(362\) 15.3923 0.809002
\(363\) 8.33975 0.437723
\(364\) 24.0000 1.25794
\(365\) 21.4641 1.12348
\(366\) −10.7321 −0.560973
\(367\) 0.392305 0.0204781 0.0102391 0.999948i \(-0.496741\pi\)
0.0102391 + 0.999948i \(0.496741\pi\)
\(368\) 1.26795 0.0660964
\(369\) 1.14359 0.0595331
\(370\) 0 0
\(371\) −10.1436 −0.526629
\(372\) 0.928203 0.0481251
\(373\) 19.7846 1.02441 0.512204 0.858864i \(-0.328829\pi\)
0.512204 + 0.858864i \(0.328829\pi\)
\(374\) −8.19615 −0.423813
\(375\) −8.87564 −0.458336
\(376\) −11.6603 −0.601332
\(377\) 25.6077 1.31886
\(378\) 16.0000 0.822951
\(379\) 24.7846 1.27310 0.636550 0.771235i \(-0.280361\pi\)
0.636550 + 0.771235i \(0.280361\pi\)
\(380\) 2.19615 0.112660
\(381\) −7.75129 −0.397111
\(382\) −24.5885 −1.25805
\(383\) −25.8564 −1.32120 −0.660600 0.750738i \(-0.729698\pi\)
−0.660600 + 0.750738i \(0.729698\pi\)
\(384\) −0.732051 −0.0373573
\(385\) −32.7846 −1.67086
\(386\) 12.1244 0.617113
\(387\) −23.3205 −1.18545
\(388\) 5.19615 0.263795
\(389\) −0.803848 −0.0407567 −0.0203783 0.999792i \(-0.506487\pi\)
−0.0203783 + 0.999792i \(0.506487\pi\)
\(390\) −7.60770 −0.385231
\(391\) −2.19615 −0.111064
\(392\) −9.00000 −0.454569
\(393\) 6.00000 0.302660
\(394\) −26.3205 −1.32601
\(395\) 14.1962 0.714286
\(396\) 11.6603 0.585950
\(397\) 5.00000 0.250943 0.125471 0.992097i \(-0.459956\pi\)
0.125471 + 0.992097i \(0.459956\pi\)
\(398\) −5.07180 −0.254226
\(399\) 3.71281 0.185873
\(400\) −2.00000 −0.100000
\(401\) 14.7846 0.738308 0.369154 0.929368i \(-0.379647\pi\)
0.369154 + 0.929368i \(0.379647\pi\)
\(402\) 4.53590 0.226230
\(403\) 7.60770 0.378966
\(404\) 0.464102 0.0230899
\(405\) 7.73205 0.384209
\(406\) −17.0718 −0.847259
\(407\) 0 0
\(408\) 1.26795 0.0627728
\(409\) 33.5885 1.66084 0.830421 0.557136i \(-0.188100\pi\)
0.830421 + 0.557136i \(0.188100\pi\)
\(410\) 0.803848 0.0396992
\(411\) −9.80385 −0.483588
\(412\) −6.92820 −0.341328
\(413\) −10.1436 −0.499134
\(414\) 3.12436 0.153554
\(415\) 20.1962 0.991390
\(416\) −6.00000 −0.294174
\(417\) −1.32051 −0.0646656
\(418\) 6.00000 0.293470
\(419\) 10.1436 0.495547 0.247773 0.968818i \(-0.420301\pi\)
0.247773 + 0.968818i \(0.420301\pi\)
\(420\) 5.07180 0.247478
\(421\) −3.33975 −0.162769 −0.0813846 0.996683i \(-0.525934\pi\)
−0.0813846 + 0.996683i \(0.525934\pi\)
\(422\) −12.3923 −0.603248
\(423\) −28.7321 −1.39700
\(424\) 2.53590 0.123154
\(425\) 3.46410 0.168034
\(426\) −1.85641 −0.0899432
\(427\) 58.6410 2.83784
\(428\) −16.3923 −0.792352
\(429\) −20.7846 −1.00349
\(430\) −16.3923 −0.790507
\(431\) 22.0526 1.06223 0.531117 0.847298i \(-0.321772\pi\)
0.531117 + 0.847298i \(0.321772\pi\)
\(432\) −4.00000 −0.192450
\(433\) −12.6077 −0.605887 −0.302944 0.953008i \(-0.597969\pi\)
−0.302944 + 0.953008i \(0.597969\pi\)
\(434\) −5.07180 −0.243454
\(435\) 5.41154 0.259464
\(436\) −7.73205 −0.370298
\(437\) 1.60770 0.0769065
\(438\) −9.07180 −0.433467
\(439\) 2.53590 0.121032 0.0605159 0.998167i \(-0.480725\pi\)
0.0605159 + 0.998167i \(0.480725\pi\)
\(440\) 8.19615 0.390736
\(441\) −22.1769 −1.05604
\(442\) 10.3923 0.494312
\(443\) 9.46410 0.449653 0.224827 0.974399i \(-0.427818\pi\)
0.224827 + 0.974399i \(0.427818\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 22.5885 1.06959
\(447\) 2.87564 0.136013
\(448\) 4.00000 0.188982
\(449\) −11.0718 −0.522510 −0.261255 0.965270i \(-0.584136\pi\)
−0.261255 + 0.965270i \(0.584136\pi\)
\(450\) −4.92820 −0.232318
\(451\) 2.19615 0.103413
\(452\) −19.8564 −0.933967
\(453\) 1.32051 0.0620429
\(454\) −1.26795 −0.0595078
\(455\) 41.5692 1.94880
\(456\) −0.928203 −0.0434671
\(457\) −31.9808 −1.49600 −0.747998 0.663700i \(-0.768985\pi\)
−0.747998 + 0.663700i \(0.768985\pi\)
\(458\) 19.0000 0.887812
\(459\) 6.92820 0.323381
\(460\) 2.19615 0.102396
\(461\) −21.7128 −1.01127 −0.505633 0.862749i \(-0.668741\pi\)
−0.505633 + 0.862749i \(0.668741\pi\)
\(462\) 13.8564 0.644658
\(463\) −21.4641 −0.997521 −0.498761 0.866740i \(-0.666211\pi\)
−0.498761 + 0.866740i \(0.666211\pi\)
\(464\) 4.26795 0.198135
\(465\) 1.60770 0.0745551
\(466\) −4.60770 −0.213447
\(467\) −18.2487 −0.844450 −0.422225 0.906491i \(-0.638751\pi\)
−0.422225 + 0.906491i \(0.638751\pi\)
\(468\) −14.7846 −0.683419
\(469\) −24.7846 −1.14445
\(470\) −20.1962 −0.931579
\(471\) −6.87564 −0.316813
\(472\) 2.53590 0.116724
\(473\) −44.7846 −2.05920
\(474\) −6.00000 −0.275589
\(475\) −2.53590 −0.116355
\(476\) −6.92820 −0.317554
\(477\) 6.24871 0.286109
\(478\) 5.07180 0.231979
\(479\) 2.53590 0.115868 0.0579341 0.998320i \(-0.481549\pi\)
0.0579341 + 0.998320i \(0.481549\pi\)
\(480\) −1.26795 −0.0578737
\(481\) 0 0
\(482\) −18.0000 −0.819878
\(483\) 3.71281 0.168939
\(484\) 11.3923 0.517832
\(485\) 9.00000 0.408669
\(486\) −15.2679 −0.692568
\(487\) −10.0526 −0.455525 −0.227762 0.973717i \(-0.573141\pi\)
−0.227762 + 0.973717i \(0.573141\pi\)
\(488\) −14.6603 −0.663639
\(489\) 1.85641 0.0839496
\(490\) −15.5885 −0.704215
\(491\) −22.9808 −1.03711 −0.518554 0.855045i \(-0.673529\pi\)
−0.518554 + 0.855045i \(0.673529\pi\)
\(492\) −0.339746 −0.0153169
\(493\) −7.39230 −0.332933
\(494\) −7.60770 −0.342286
\(495\) 20.1962 0.907750
\(496\) 1.26795 0.0569326
\(497\) 10.1436 0.455002
\(498\) −8.53590 −0.382503
\(499\) 8.19615 0.366910 0.183455 0.983028i \(-0.441272\pi\)
0.183455 + 0.983028i \(0.441272\pi\)
\(500\) −12.1244 −0.542218
\(501\) 5.07180 0.226591
\(502\) −5.07180 −0.226365
\(503\) 12.5885 0.561292 0.280646 0.959811i \(-0.409451\pi\)
0.280646 + 0.959811i \(0.409451\pi\)
\(504\) 9.85641 0.439039
\(505\) 0.803848 0.0357707
\(506\) 6.00000 0.266733
\(507\) 16.8372 0.747765
\(508\) −10.5885 −0.469787
\(509\) −31.6410 −1.40246 −0.701232 0.712933i \(-0.747366\pi\)
−0.701232 + 0.712933i \(0.747366\pi\)
\(510\) 2.19615 0.0972473
\(511\) 49.5692 2.19281
\(512\) −1.00000 −0.0441942
\(513\) −5.07180 −0.223925
\(514\) −11.1962 −0.493841
\(515\) −12.0000 −0.528783
\(516\) 6.92820 0.304997
\(517\) −55.1769 −2.42668
\(518\) 0 0
\(519\) 7.26795 0.319028
\(520\) −10.3923 −0.455733
\(521\) −26.5359 −1.16256 −0.581279 0.813704i \(-0.697448\pi\)
−0.581279 + 0.813704i \(0.697448\pi\)
\(522\) 10.5167 0.460302
\(523\) −12.5885 −0.550455 −0.275227 0.961379i \(-0.588753\pi\)
−0.275227 + 0.961379i \(0.588753\pi\)
\(524\) 8.19615 0.358051
\(525\) −5.85641 −0.255595
\(526\) 23.3205 1.01682
\(527\) −2.19615 −0.0956659
\(528\) −3.46410 −0.150756
\(529\) −21.3923 −0.930100
\(530\) 4.39230 0.190790
\(531\) 6.24871 0.271171
\(532\) 5.07180 0.219890
\(533\) −2.78461 −0.120615
\(534\) 3.80385 0.164609
\(535\) −28.3923 −1.22751
\(536\) 6.19615 0.267633
\(537\) 8.28719 0.357619
\(538\) 7.60770 0.327991
\(539\) −42.5885 −1.83441
\(540\) −6.92820 −0.298142
\(541\) 3.58846 0.154280 0.0771399 0.997020i \(-0.475421\pi\)
0.0771399 + 0.997020i \(0.475421\pi\)
\(542\) 14.5885 0.626628
\(543\) −11.2679 −0.483554
\(544\) 1.73205 0.0742611
\(545\) −13.3923 −0.573663
\(546\) −17.5692 −0.751893
\(547\) 12.5885 0.538244 0.269122 0.963106i \(-0.413267\pi\)
0.269122 + 0.963106i \(0.413267\pi\)
\(548\) −13.3923 −0.572091
\(549\) −36.1244 −1.54175
\(550\) −9.46410 −0.403551
\(551\) 5.41154 0.230539
\(552\) −0.928203 −0.0395070
\(553\) 32.7846 1.39414
\(554\) 6.80385 0.289068
\(555\) 0 0
\(556\) −1.80385 −0.0765002
\(557\) −10.5167 −0.445605 −0.222803 0.974864i \(-0.571521\pi\)
−0.222803 + 0.974864i \(0.571521\pi\)
\(558\) 3.12436 0.132265
\(559\) 56.7846 2.40173
\(560\) 6.92820 0.292770
\(561\) 6.00000 0.253320
\(562\) 11.1962 0.472281
\(563\) 11.4115 0.480939 0.240470 0.970657i \(-0.422699\pi\)
0.240470 + 0.970657i \(0.422699\pi\)
\(564\) 8.53590 0.359426
\(565\) −34.3923 −1.44690
\(566\) −2.53590 −0.106592
\(567\) 17.8564 0.749899
\(568\) −2.53590 −0.106404
\(569\) 28.5167 1.19548 0.597740 0.801690i \(-0.296065\pi\)
0.597740 + 0.801690i \(0.296065\pi\)
\(570\) −1.60770 −0.0673389
\(571\) 14.9808 0.626925 0.313463 0.949601i \(-0.398511\pi\)
0.313463 + 0.949601i \(0.398511\pi\)
\(572\) −28.3923 −1.18714
\(573\) 18.0000 0.751961
\(574\) 1.85641 0.0774849
\(575\) −2.53590 −0.105754
\(576\) −2.46410 −0.102671
\(577\) 4.14359 0.172500 0.0862500 0.996274i \(-0.472512\pi\)
0.0862500 + 0.996274i \(0.472512\pi\)
\(578\) 14.0000 0.582323
\(579\) −8.87564 −0.368859
\(580\) 7.39230 0.306949
\(581\) 46.6410 1.93500
\(582\) −3.80385 −0.157675
\(583\) 12.0000 0.496989
\(584\) −12.3923 −0.512797
\(585\) −25.6077 −1.05875
\(586\) −25.3923 −1.04895
\(587\) 22.0526 0.910207 0.455103 0.890439i \(-0.349602\pi\)
0.455103 + 0.890439i \(0.349602\pi\)
\(588\) 6.58846 0.271703
\(589\) 1.60770 0.0662439
\(590\) 4.39230 0.180828
\(591\) 19.2679 0.792578
\(592\) 0 0
\(593\) 23.5359 0.966504 0.483252 0.875481i \(-0.339456\pi\)
0.483252 + 0.875481i \(0.339456\pi\)
\(594\) −18.9282 −0.776634
\(595\) −12.0000 −0.491952
\(596\) 3.92820 0.160905
\(597\) 3.71281 0.151955
\(598\) −7.60770 −0.311102
\(599\) 4.73205 0.193346 0.0966732 0.995316i \(-0.469180\pi\)
0.0966732 + 0.995316i \(0.469180\pi\)
\(600\) 1.46410 0.0597717
\(601\) −25.7846 −1.05178 −0.525888 0.850554i \(-0.676267\pi\)
−0.525888 + 0.850554i \(0.676267\pi\)
\(602\) −37.8564 −1.54291
\(603\) 15.2679 0.621759
\(604\) 1.80385 0.0733975
\(605\) 19.7321 0.802222
\(606\) −0.339746 −0.0138012
\(607\) −1.94744 −0.0790442 −0.0395221 0.999219i \(-0.512584\pi\)
−0.0395221 + 0.999219i \(0.512584\pi\)
\(608\) −1.26795 −0.0514221
\(609\) 12.4974 0.506421
\(610\) −25.3923 −1.02810
\(611\) 69.9615 2.83034
\(612\) 4.26795 0.172522
\(613\) −12.6077 −0.509220 −0.254610 0.967044i \(-0.581947\pi\)
−0.254610 + 0.967044i \(0.581947\pi\)
\(614\) −4.00000 −0.161427
\(615\) −0.588457 −0.0237289
\(616\) 18.9282 0.762639
\(617\) −23.3205 −0.938848 −0.469424 0.882973i \(-0.655538\pi\)
−0.469424 + 0.882973i \(0.655538\pi\)
\(618\) 5.07180 0.204018
\(619\) −21.1769 −0.851172 −0.425586 0.904918i \(-0.639932\pi\)
−0.425586 + 0.904918i \(0.639932\pi\)
\(620\) 2.19615 0.0881996
\(621\) −5.07180 −0.203524
\(622\) 16.3923 0.657272
\(623\) −20.7846 −0.832718
\(624\) 4.39230 0.175833
\(625\) −11.0000 −0.440000
\(626\) −21.5885 −0.862848
\(627\) −4.39230 −0.175412
\(628\) −9.39230 −0.374794
\(629\) 0 0
\(630\) 17.0718 0.680157
\(631\) 29.6603 1.18076 0.590378 0.807127i \(-0.298979\pi\)
0.590378 + 0.807127i \(0.298979\pi\)
\(632\) −8.19615 −0.326025
\(633\) 9.07180 0.360572
\(634\) 9.00000 0.357436
\(635\) −18.3397 −0.727791
\(636\) −1.85641 −0.0736113
\(637\) 54.0000 2.13956
\(638\) 20.1962 0.799573
\(639\) −6.24871 −0.247195
\(640\) −1.73205 −0.0684653
\(641\) −19.1436 −0.756126 −0.378063 0.925780i \(-0.623410\pi\)
−0.378063 + 0.925780i \(0.623410\pi\)
\(642\) 12.0000 0.473602
\(643\) −15.1244 −0.596446 −0.298223 0.954496i \(-0.596394\pi\)
−0.298223 + 0.954496i \(0.596394\pi\)
\(644\) 5.07180 0.199857
\(645\) 12.0000 0.472500
\(646\) 2.19615 0.0864065
\(647\) 2.53590 0.0996965 0.0498482 0.998757i \(-0.484126\pi\)
0.0498482 + 0.998757i \(0.484126\pi\)
\(648\) −4.46410 −0.175366
\(649\) 12.0000 0.471041
\(650\) 12.0000 0.470679
\(651\) 3.71281 0.145517
\(652\) 2.53590 0.0993134
\(653\) −17.1962 −0.672937 −0.336469 0.941695i \(-0.609233\pi\)
−0.336469 + 0.941695i \(0.609233\pi\)
\(654\) 5.66025 0.221333
\(655\) 14.1962 0.554690
\(656\) −0.464102 −0.0181201
\(657\) −30.5359 −1.19132
\(658\) −46.6410 −1.81826
\(659\) −21.4641 −0.836123 −0.418061 0.908419i \(-0.637290\pi\)
−0.418061 + 0.908419i \(0.637290\pi\)
\(660\) −6.00000 −0.233550
\(661\) −18.8038 −0.731385 −0.365692 0.930736i \(-0.619168\pi\)
−0.365692 + 0.930736i \(0.619168\pi\)
\(662\) 30.9282 1.20206
\(663\) −7.60770 −0.295458
\(664\) −11.6603 −0.452506
\(665\) 8.78461 0.340653
\(666\) 0 0
\(667\) 5.41154 0.209536
\(668\) 6.92820 0.268060
\(669\) −16.5359 −0.639315
\(670\) 10.7321 0.414615
\(671\) −69.3731 −2.67812
\(672\) −2.92820 −0.112958
\(673\) 8.39230 0.323500 0.161750 0.986832i \(-0.448286\pi\)
0.161750 + 0.986832i \(0.448286\pi\)
\(674\) −7.00000 −0.269630
\(675\) 8.00000 0.307920
\(676\) 23.0000 0.884615
\(677\) −34.8564 −1.33964 −0.669820 0.742523i \(-0.733629\pi\)
−0.669820 + 0.742523i \(0.733629\pi\)
\(678\) 14.5359 0.558248
\(679\) 20.7846 0.797640
\(680\) 3.00000 0.115045
\(681\) 0.928203 0.0355688
\(682\) 6.00000 0.229752
\(683\) −17.0718 −0.653234 −0.326617 0.945157i \(-0.605909\pi\)
−0.326617 + 0.945157i \(0.605909\pi\)
\(684\) −3.12436 −0.119463
\(685\) −23.1962 −0.886279
\(686\) −8.00000 −0.305441
\(687\) −13.9090 −0.530660
\(688\) 9.46410 0.360815
\(689\) −15.2154 −0.579660
\(690\) −1.60770 −0.0612039
\(691\) 22.1962 0.844381 0.422191 0.906507i \(-0.361261\pi\)
0.422191 + 0.906507i \(0.361261\pi\)
\(692\) 9.92820 0.377414
\(693\) 46.6410 1.77175
\(694\) 10.0526 0.381590
\(695\) −3.12436 −0.118514
\(696\) −3.12436 −0.118428
\(697\) 0.803848 0.0304479
\(698\) 6.60770 0.250105
\(699\) 3.37307 0.127581
\(700\) −8.00000 −0.302372
\(701\) 1.85641 0.0701155 0.0350578 0.999385i \(-0.488838\pi\)
0.0350578 + 0.999385i \(0.488838\pi\)
\(702\) 24.0000 0.905822
\(703\) 0 0
\(704\) −4.73205 −0.178346
\(705\) 14.7846 0.556821
\(706\) 30.1244 1.13375
\(707\) 1.85641 0.0698174
\(708\) −1.85641 −0.0697680
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) −4.39230 −0.164840
\(711\) −20.1962 −0.757415
\(712\) 5.19615 0.194734
\(713\) 1.60770 0.0602087
\(714\) 5.07180 0.189807
\(715\) −49.1769 −1.83911
\(716\) 11.3205 0.423067
\(717\) −3.71281 −0.138658
\(718\) −33.4641 −1.24887
\(719\) −32.4449 −1.20999 −0.604995 0.796230i \(-0.706825\pi\)
−0.604995 + 0.796230i \(0.706825\pi\)
\(720\) −4.26795 −0.159057
\(721\) −27.7128 −1.03208
\(722\) 17.3923 0.647275
\(723\) 13.1769 0.490055
\(724\) −15.3923 −0.572051
\(725\) −8.53590 −0.317015
\(726\) −8.33975 −0.309517
\(727\) −1.85641 −0.0688503 −0.0344252 0.999407i \(-0.510960\pi\)
−0.0344252 + 0.999407i \(0.510960\pi\)
\(728\) −24.0000 −0.889499
\(729\) −2.21539 −0.0820515
\(730\) −21.4641 −0.794422
\(731\) −16.3923 −0.606291
\(732\) 10.7321 0.396668
\(733\) −15.6077 −0.576483 −0.288242 0.957558i \(-0.593071\pi\)
−0.288242 + 0.957558i \(0.593071\pi\)
\(734\) −0.392305 −0.0144802
\(735\) 11.4115 0.420921
\(736\) −1.26795 −0.0467372
\(737\) 29.3205 1.08003
\(738\) −1.14359 −0.0420963
\(739\) −6.98076 −0.256791 −0.128396 0.991723i \(-0.540983\pi\)
−0.128396 + 0.991723i \(0.540983\pi\)
\(740\) 0 0
\(741\) 5.56922 0.204590
\(742\) 10.1436 0.372383
\(743\) 41.9090 1.53749 0.768745 0.639555i \(-0.220881\pi\)
0.768745 + 0.639555i \(0.220881\pi\)
\(744\) −0.928203 −0.0340296
\(745\) 6.80385 0.249274
\(746\) −19.7846 −0.724366
\(747\) −28.7321 −1.05125
\(748\) 8.19615 0.299681
\(749\) −65.5692 −2.39585
\(750\) 8.87564 0.324093
\(751\) 36.3923 1.32797 0.663987 0.747744i \(-0.268863\pi\)
0.663987 + 0.747744i \(0.268863\pi\)
\(752\) 11.6603 0.425206
\(753\) 3.71281 0.135302
\(754\) −25.6077 −0.932577
\(755\) 3.12436 0.113707
\(756\) −16.0000 −0.581914
\(757\) 4.26795 0.155121 0.0775606 0.996988i \(-0.475287\pi\)
0.0775606 + 0.996988i \(0.475287\pi\)
\(758\) −24.7846 −0.900218
\(759\) −4.39230 −0.159431
\(760\) −2.19615 −0.0796628
\(761\) −32.3205 −1.17162 −0.585809 0.810449i \(-0.699223\pi\)
−0.585809 + 0.810449i \(0.699223\pi\)
\(762\) 7.75129 0.280800
\(763\) −30.9282 −1.11968
\(764\) 24.5885 0.889579
\(765\) 7.39230 0.267269
\(766\) 25.8564 0.934230
\(767\) −15.2154 −0.549396
\(768\) 0.732051 0.0264156
\(769\) −21.7128 −0.782984 −0.391492 0.920181i \(-0.628041\pi\)
−0.391492 + 0.920181i \(0.628041\pi\)
\(770\) 32.7846 1.18148
\(771\) 8.19615 0.295177
\(772\) −12.1244 −0.436365
\(773\) −21.9282 −0.788703 −0.394351 0.918960i \(-0.629031\pi\)
−0.394351 + 0.918960i \(0.629031\pi\)
\(774\) 23.3205 0.838238
\(775\) −2.53590 −0.0910922
\(776\) −5.19615 −0.186531
\(777\) 0 0
\(778\) 0.803848 0.0288193
\(779\) −0.588457 −0.0210837
\(780\) 7.60770 0.272399
\(781\) −12.0000 −0.429394
\(782\) 2.19615 0.0785343
\(783\) −17.0718 −0.610096
\(784\) 9.00000 0.321429
\(785\) −16.2679 −0.580628
\(786\) −6.00000 −0.214013
\(787\) 33.1769 1.18263 0.591315 0.806441i \(-0.298609\pi\)
0.591315 + 0.806441i \(0.298609\pi\)
\(788\) 26.3205 0.937629
\(789\) −17.0718 −0.607772
\(790\) −14.1962 −0.505076
\(791\) −79.4256 −2.82405
\(792\) −11.6603 −0.414329
\(793\) 87.9615 3.12361
\(794\) −5.00000 −0.177443
\(795\) −3.21539 −0.114038
\(796\) 5.07180 0.179765
\(797\) −20.7846 −0.736229 −0.368114 0.929781i \(-0.619996\pi\)
−0.368114 + 0.929781i \(0.619996\pi\)
\(798\) −3.71281 −0.131432
\(799\) −20.1962 −0.714489
\(800\) 2.00000 0.0707107
\(801\) 12.8038 0.452402
\(802\) −14.7846 −0.522063
\(803\) −58.6410 −2.06940
\(804\) −4.53590 −0.159969
\(805\) 8.78461 0.309617
\(806\) −7.60770 −0.267970
\(807\) −5.56922 −0.196046
\(808\) −0.464102 −0.0163270
\(809\) 29.0718 1.02211 0.511055 0.859548i \(-0.329255\pi\)
0.511055 + 0.859548i \(0.329255\pi\)
\(810\) −7.73205 −0.271677
\(811\) −36.3923 −1.27791 −0.638953 0.769246i \(-0.720632\pi\)
−0.638953 + 0.769246i \(0.720632\pi\)
\(812\) 17.0718 0.599103
\(813\) −10.6795 −0.374546
\(814\) 0 0
\(815\) 4.39230 0.153856
\(816\) −1.26795 −0.0443871
\(817\) 12.0000 0.419827
\(818\) −33.5885 −1.17439
\(819\) −59.1384 −2.06646
\(820\) −0.803848 −0.0280716
\(821\) −38.5359 −1.34491 −0.672456 0.740137i \(-0.734761\pi\)
−0.672456 + 0.740137i \(0.734761\pi\)
\(822\) 9.80385 0.341948
\(823\) 38.5885 1.34511 0.672555 0.740048i \(-0.265197\pi\)
0.672555 + 0.740048i \(0.265197\pi\)
\(824\) 6.92820 0.241355
\(825\) 6.92820 0.241209
\(826\) 10.1436 0.352941
\(827\) −46.6410 −1.62187 −0.810934 0.585138i \(-0.801040\pi\)
−0.810934 + 0.585138i \(0.801040\pi\)
\(828\) −3.12436 −0.108579
\(829\) −23.0718 −0.801317 −0.400658 0.916228i \(-0.631219\pi\)
−0.400658 + 0.916228i \(0.631219\pi\)
\(830\) −20.1962 −0.701019
\(831\) −4.98076 −0.172781
\(832\) 6.00000 0.208013
\(833\) −15.5885 −0.540108
\(834\) 1.32051 0.0457255
\(835\) 12.0000 0.415277
\(836\) −6.00000 −0.207514
\(837\) −5.07180 −0.175307
\(838\) −10.1436 −0.350405
\(839\) −9.80385 −0.338466 −0.169233 0.985576i \(-0.554129\pi\)
−0.169233 + 0.985576i \(0.554129\pi\)
\(840\) −5.07180 −0.174994
\(841\) −10.7846 −0.371883
\(842\) 3.33975 0.115095
\(843\) −8.19615 −0.282290
\(844\) 12.3923 0.426561
\(845\) 39.8372 1.37044
\(846\) 28.7321 0.987828
\(847\) 45.5692 1.56578
\(848\) −2.53590 −0.0870831
\(849\) 1.85641 0.0637117
\(850\) −3.46410 −0.118818
\(851\) 0 0
\(852\) 1.85641 0.0635994
\(853\) 25.7321 0.881049 0.440524 0.897741i \(-0.354793\pi\)
0.440524 + 0.897741i \(0.354793\pi\)
\(854\) −58.6410 −2.00665
\(855\) −5.41154 −0.185071
\(856\) 16.3923 0.560277
\(857\) −1.05256 −0.0359547 −0.0179774 0.999838i \(-0.505723\pi\)
−0.0179774 + 0.999838i \(0.505723\pi\)
\(858\) 20.7846 0.709575
\(859\) 14.5359 0.495958 0.247979 0.968765i \(-0.420234\pi\)
0.247979 + 0.968765i \(0.420234\pi\)
\(860\) 16.3923 0.558973
\(861\) −1.35898 −0.0463140
\(862\) −22.0526 −0.751113
\(863\) 4.39230 0.149516 0.0747579 0.997202i \(-0.476182\pi\)
0.0747579 + 0.997202i \(0.476182\pi\)
\(864\) 4.00000 0.136083
\(865\) 17.1962 0.584687
\(866\) 12.6077 0.428427
\(867\) −10.2487 −0.348064
\(868\) 5.07180 0.172148
\(869\) −38.7846 −1.31568
\(870\) −5.41154 −0.183468
\(871\) −37.1769 −1.25969
\(872\) 7.73205 0.261840
\(873\) −12.8038 −0.433345
\(874\) −1.60770 −0.0543811
\(875\) −48.4974 −1.63951
\(876\) 9.07180 0.306508
\(877\) −18.1769 −0.613791 −0.306895 0.951743i \(-0.599290\pi\)
−0.306895 + 0.951743i \(0.599290\pi\)
\(878\) −2.53590 −0.0855824
\(879\) 18.5885 0.626973
\(880\) −8.19615 −0.276292
\(881\) −54.9615 −1.85170 −0.925850 0.377890i \(-0.876650\pi\)
−0.925850 + 0.377890i \(0.876650\pi\)
\(882\) 22.1769 0.746736
\(883\) −18.3397 −0.617182 −0.308591 0.951195i \(-0.599857\pi\)
−0.308591 + 0.951195i \(0.599857\pi\)
\(884\) −10.3923 −0.349531
\(885\) −3.21539 −0.108084
\(886\) −9.46410 −0.317953
\(887\) −2.87564 −0.0965547 −0.0482773 0.998834i \(-0.515373\pi\)
−0.0482773 + 0.998834i \(0.515373\pi\)
\(888\) 0 0
\(889\) −42.3538 −1.42050
\(890\) 9.00000 0.301681
\(891\) −21.1244 −0.707693
\(892\) −22.5885 −0.756317
\(893\) 14.7846 0.494748
\(894\) −2.87564 −0.0961759
\(895\) 19.6077 0.655413
\(896\) −4.00000 −0.133631
\(897\) 5.56922 0.185951
\(898\) 11.0718 0.369471
\(899\) 5.41154 0.180485
\(900\) 4.92820 0.164273
\(901\) 4.39230 0.146329
\(902\) −2.19615 −0.0731239
\(903\) 27.7128 0.922225
\(904\) 19.8564 0.660414
\(905\) −26.6603 −0.886217
\(906\) −1.32051 −0.0438709
\(907\) −40.3013 −1.33818 −0.669091 0.743181i \(-0.733316\pi\)
−0.669091 + 0.743181i \(0.733316\pi\)
\(908\) 1.26795 0.0420784
\(909\) −1.14359 −0.0379306
\(910\) −41.5692 −1.37801
\(911\) 27.1244 0.898670 0.449335 0.893363i \(-0.351661\pi\)
0.449335 + 0.893363i \(0.351661\pi\)
\(912\) 0.928203 0.0307359
\(913\) −55.1769 −1.82609
\(914\) 31.9808 1.05783
\(915\) 18.5885 0.614515
\(916\) −19.0000 −0.627778
\(917\) 32.7846 1.08264
\(918\) −6.92820 −0.228665
\(919\) −10.0526 −0.331603 −0.165802 0.986159i \(-0.553021\pi\)
−0.165802 + 0.986159i \(0.553021\pi\)
\(920\) −2.19615 −0.0724050
\(921\) 2.92820 0.0964876
\(922\) 21.7128 0.715073
\(923\) 15.2154 0.500821
\(924\) −13.8564 −0.455842
\(925\) 0 0
\(926\) 21.4641 0.705354
\(927\) 17.0718 0.560711
\(928\) −4.26795 −0.140102
\(929\) 16.6077 0.544881 0.272440 0.962173i \(-0.412169\pi\)
0.272440 + 0.962173i \(0.412169\pi\)
\(930\) −1.60770 −0.0527184
\(931\) 11.4115 0.373998
\(932\) 4.60770 0.150930
\(933\) −12.0000 −0.392862
\(934\) 18.2487 0.597116
\(935\) 14.1962 0.464264
\(936\) 14.7846 0.483250
\(937\) −26.1769 −0.855163 −0.427581 0.903977i \(-0.640634\pi\)
−0.427581 + 0.903977i \(0.640634\pi\)
\(938\) 24.7846 0.809246
\(939\) 15.8038 0.515739
\(940\) 20.1962 0.658726
\(941\) 18.2154 0.593805 0.296902 0.954908i \(-0.404046\pi\)
0.296902 + 0.954908i \(0.404046\pi\)
\(942\) 6.87564 0.224021
\(943\) −0.588457 −0.0191628
\(944\) −2.53590 −0.0825365
\(945\) −27.7128 −0.901498
\(946\) 44.7846 1.45607
\(947\) 31.6077 1.02711 0.513556 0.858056i \(-0.328328\pi\)
0.513556 + 0.858056i \(0.328328\pi\)
\(948\) 6.00000 0.194871
\(949\) 74.3538 2.41363
\(950\) 2.53590 0.0822754
\(951\) −6.58846 −0.213645
\(952\) 6.92820 0.224544
\(953\) 19.8564 0.643212 0.321606 0.946874i \(-0.395777\pi\)
0.321606 + 0.946874i \(0.395777\pi\)
\(954\) −6.24871 −0.202309
\(955\) 42.5885 1.37813
\(956\) −5.07180 −0.164034
\(957\) −14.7846 −0.477919
\(958\) −2.53590 −0.0819312
\(959\) −53.5692 −1.72984
\(960\) 1.26795 0.0409229
\(961\) −29.3923 −0.948139
\(962\) 0 0
\(963\) 40.3923 1.30162
\(964\) 18.0000 0.579741
\(965\) −21.0000 −0.676014
\(966\) −3.71281 −0.119458
\(967\) 58.6410 1.88577 0.942884 0.333121i \(-0.108102\pi\)
0.942884 + 0.333121i \(0.108102\pi\)
\(968\) −11.3923 −0.366163
\(969\) −1.60770 −0.0516466
\(970\) −9.00000 −0.288973
\(971\) 18.5885 0.596532 0.298266 0.954483i \(-0.403592\pi\)
0.298266 + 0.954483i \(0.403592\pi\)
\(972\) 15.2679 0.489720
\(973\) −7.21539 −0.231315
\(974\) 10.0526 0.322105
\(975\) −8.78461 −0.281333
\(976\) 14.6603 0.469263
\(977\) −2.28719 −0.0731736 −0.0365868 0.999330i \(-0.511649\pi\)
−0.0365868 + 0.999330i \(0.511649\pi\)
\(978\) −1.85641 −0.0593613
\(979\) 24.5885 0.785851
\(980\) 15.5885 0.497955
\(981\) 19.0526 0.608301
\(982\) 22.9808 0.733346
\(983\) 25.5167 0.813855 0.406928 0.913460i \(-0.366600\pi\)
0.406928 + 0.913460i \(0.366600\pi\)
\(984\) 0.339746 0.0108307
\(985\) 45.5885 1.45257
\(986\) 7.39230 0.235419
\(987\) 34.1436 1.08680
\(988\) 7.60770 0.242033
\(989\) 12.0000 0.381578
\(990\) −20.1962 −0.641876
\(991\) 32.1051 1.01985 0.509926 0.860218i \(-0.329673\pi\)
0.509926 + 0.860218i \(0.329673\pi\)
\(992\) −1.26795 −0.0402574
\(993\) −22.6410 −0.718491
\(994\) −10.1436 −0.321735
\(995\) 8.78461 0.278491
\(996\) 8.53590 0.270470
\(997\) −59.5692 −1.88658 −0.943288 0.331975i \(-0.892285\pi\)
−0.943288 + 0.331975i \(0.892285\pi\)
\(998\) −8.19615 −0.259445
\(999\) 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 2738.2.a.f.1.2 2
37.23 odd 12 74.2.e.a.11.2 4
37.29 odd 12 74.2.e.a.27.2 yes 4
37.36 even 2 2738.2.a.j.1.2 2
111.23 even 12 666.2.s.e.307.1 4
111.29 even 12 666.2.s.e.397.1 4
148.23 even 12 592.2.w.d.529.1 4
148.103 even 12 592.2.w.d.545.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.2 4 37.23 odd 12
74.2.e.a.27.2 yes 4 37.29 odd 12
592.2.w.d.529.1 4 148.23 even 12
592.2.w.d.545.1 4 148.103 even 12
666.2.s.e.307.1 4 111.23 even 12
666.2.s.e.397.1 4 111.29 even 12
2738.2.a.f.1.2 2 1.1 even 1 trivial
2738.2.a.j.1.2 2 37.36 even 2