Properties

Label 145.2.a.b.1.1
Level $145$
Weight $2$
Character 145.1
Self dual yes
Analytic conductor $1.158$
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] = [145,2,Mod(1,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, 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("145.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.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: \(1.15783082931\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 145.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} -2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -4.82843 q^{11} -7.65685 q^{12} -2.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} +3.00000 q^{16} -2.82843 q^{17} -2.41421 q^{18} +0.828427 q^{19} +3.82843 q^{20} -1.65685 q^{21} +11.6569 q^{22} -8.82843 q^{23} +8.82843 q^{24} +1.00000 q^{25} +4.82843 q^{26} +4.00000 q^{27} +3.17157 q^{28} +1.00000 q^{29} +4.82843 q^{30} -10.4853 q^{31} +1.58579 q^{32} +9.65685 q^{33} +6.82843 q^{34} +0.828427 q^{35} +3.82843 q^{36} +8.48528 q^{37} -2.00000 q^{38} +4.00000 q^{39} -4.41421 q^{40} -6.00000 q^{41} +4.00000 q^{42} -6.00000 q^{43} -18.4853 q^{44} +1.00000 q^{45} +21.3137 q^{46} -0.343146 q^{47} -6.00000 q^{48} -6.31371 q^{49} -2.41421 q^{50} +5.65685 q^{51} -7.65685 q^{52} +7.65685 q^{53} -9.65685 q^{54} -4.82843 q^{55} -3.65685 q^{56} -1.65685 q^{57} -2.41421 q^{58} -7.65685 q^{60} +7.65685 q^{61} +25.3137 q^{62} +0.828427 q^{63} -9.82843 q^{64} -2.00000 q^{65} -23.3137 q^{66} -10.4853 q^{67} -10.8284 q^{68} +17.6569 q^{69} -2.00000 q^{70} +7.31371 q^{71} -4.41421 q^{72} -8.48528 q^{73} -20.4853 q^{74} -2.00000 q^{75} +3.17157 q^{76} -4.00000 q^{77} -9.65685 q^{78} +14.4853 q^{79} +3.00000 q^{80} -11.0000 q^{81} +14.4853 q^{82} +12.8284 q^{83} -6.34315 q^{84} -2.82843 q^{85} +14.4853 q^{86} -2.00000 q^{87} +21.3137 q^{88} +3.65685 q^{89} -2.41421 q^{90} -1.65685 q^{91} -33.7990 q^{92} +20.9706 q^{93} +0.828427 q^{94} +0.828427 q^{95} -3.17157 q^{96} +4.48528 q^{97} +15.2426 q^{98} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21} + 12 q^{22} - 12 q^{23} + 12 q^{24} + 2 q^{25} + 4 q^{26} + 8 q^{27} + 12 q^{28} + 2 q^{29} + 4 q^{30} - 4 q^{31} + 6 q^{32} + 8 q^{33} + 8 q^{34} - 4 q^{35} + 2 q^{36} - 4 q^{38} + 8 q^{39} - 6 q^{40} - 12 q^{41} + 8 q^{42} - 12 q^{43} - 20 q^{44} + 2 q^{45} + 20 q^{46} - 12 q^{47} - 12 q^{48} + 10 q^{49} - 2 q^{50} - 4 q^{52} + 4 q^{53} - 8 q^{54} - 4 q^{55} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 4 q^{60} + 4 q^{61} + 28 q^{62} - 4 q^{63} - 14 q^{64} - 4 q^{65} - 24 q^{66} - 4 q^{67} - 16 q^{68} + 24 q^{69} - 4 q^{70} - 8 q^{71} - 6 q^{72} - 24 q^{74} - 4 q^{75} + 12 q^{76} - 8 q^{77} - 8 q^{78} + 12 q^{79} + 6 q^{80} - 22 q^{81} + 12 q^{82} + 20 q^{83} - 24 q^{84} + 12 q^{86} - 4 q^{87} + 20 q^{88} - 4 q^{89} - 2 q^{90} + 8 q^{91} - 28 q^{92} + 8 q^{93} - 4 q^{94} - 4 q^{95} - 12 q^{96} - 8 q^{97} + 22 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 4.82843 1.97120
\(7\) 0.828427 0.313116 0.156558 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) −2.41421 −0.763441
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) −7.65685 −2.21034
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) −2.00000 −0.516398
\(16\) 3.00000 0.750000
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) −2.41421 −0.569036
\(19\) 0.828427 0.190054 0.0950271 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) 3.82843 0.856062
\(21\) −1.65685 −0.361555
\(22\) 11.6569 2.48525
\(23\) −8.82843 −1.84085 −0.920427 0.390914i \(-0.872159\pi\)
−0.920427 + 0.390914i \(0.872159\pi\)
\(24\) 8.82843 1.80210
\(25\) 1.00000 0.200000
\(26\) 4.82843 0.946932
\(27\) 4.00000 0.769800
\(28\) 3.17157 0.599371
\(29\) 1.00000 0.185695
\(30\) 4.82843 0.881546
\(31\) −10.4853 −1.88321 −0.941606 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) 1.58579 0.280330
\(33\) 9.65685 1.68104
\(34\) 6.82843 1.17107
\(35\) 0.828427 0.140030
\(36\) 3.82843 0.638071
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) −2.00000 −0.324443
\(39\) 4.00000 0.640513
\(40\) −4.41421 −0.697948
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −18.4853 −2.78676
\(45\) 1.00000 0.149071
\(46\) 21.3137 3.14253
\(47\) −0.343146 −0.0500530 −0.0250265 0.999687i \(-0.507967\pi\)
−0.0250265 + 0.999687i \(0.507967\pi\)
\(48\) −6.00000 −0.866025
\(49\) −6.31371 −0.901958
\(50\) −2.41421 −0.341421
\(51\) 5.65685 0.792118
\(52\) −7.65685 −1.06181
\(53\) 7.65685 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(54\) −9.65685 −1.31413
\(55\) −4.82843 −0.651065
\(56\) −3.65685 −0.488668
\(57\) −1.65685 −0.219456
\(58\) −2.41421 −0.317002
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −7.65685 −0.988496
\(61\) 7.65685 0.980360 0.490180 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) 25.3137 3.21484
\(63\) 0.828427 0.104372
\(64\) −9.82843 −1.22855
\(65\) −2.00000 −0.248069
\(66\) −23.3137 −2.86972
\(67\) −10.4853 −1.28098 −0.640490 0.767966i \(-0.721269\pi\)
−0.640490 + 0.767966i \(0.721269\pi\)
\(68\) −10.8284 −1.31314
\(69\) 17.6569 2.12564
\(70\) −2.00000 −0.239046
\(71\) 7.31371 0.867978 0.433989 0.900918i \(-0.357106\pi\)
0.433989 + 0.900918i \(0.357106\pi\)
\(72\) −4.41421 −0.520220
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) −20.4853 −2.38137
\(75\) −2.00000 −0.230940
\(76\) 3.17157 0.363804
\(77\) −4.00000 −0.455842
\(78\) −9.65685 −1.09342
\(79\) 14.4853 1.62972 0.814861 0.579657i \(-0.196813\pi\)
0.814861 + 0.579657i \(0.196813\pi\)
\(80\) 3.00000 0.335410
\(81\) −11.0000 −1.22222
\(82\) 14.4853 1.59963
\(83\) 12.8284 1.40810 0.704051 0.710149i \(-0.251372\pi\)
0.704051 + 0.710149i \(0.251372\pi\)
\(84\) −6.34315 −0.692094
\(85\) −2.82843 −0.306786
\(86\) 14.4853 1.56199
\(87\) −2.00000 −0.214423
\(88\) 21.3137 2.27205
\(89\) 3.65685 0.387626 0.193813 0.981039i \(-0.437915\pi\)
0.193813 + 0.981039i \(0.437915\pi\)
\(90\) −2.41421 −0.254480
\(91\) −1.65685 −0.173686
\(92\) −33.7990 −3.52379
\(93\) 20.9706 2.17455
\(94\) 0.828427 0.0854457
\(95\) 0.828427 0.0849948
\(96\) −3.17157 −0.323697
\(97\) 4.48528 0.455411 0.227706 0.973730i \(-0.426878\pi\)
0.227706 + 0.973730i \(0.426878\pi\)
\(98\) 15.2426 1.53974
\(99\) −4.82843 −0.485275
\(100\) 3.82843 0.382843
\(101\) 4.34315 0.432159 0.216080 0.976376i \(-0.430673\pi\)
0.216080 + 0.976376i \(0.430673\pi\)
\(102\) −13.6569 −1.35223
\(103\) −12.1421 −1.19640 −0.598200 0.801347i \(-0.704117\pi\)
−0.598200 + 0.801347i \(0.704117\pi\)
\(104\) 8.82843 0.865699
\(105\) −1.65685 −0.161692
\(106\) −18.4853 −1.79545
\(107\) −8.14214 −0.787130 −0.393565 0.919297i \(-0.628758\pi\)
−0.393565 + 0.919297i \(0.628758\pi\)
\(108\) 15.3137 1.47356
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 11.6569 1.11144
\(111\) −16.9706 −1.61077
\(112\) 2.48528 0.234837
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) 4.00000 0.374634
\(115\) −8.82843 −0.823255
\(116\) 3.82843 0.355461
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −2.34315 −0.214796
\(120\) 8.82843 0.805921
\(121\) 12.3137 1.11943
\(122\) −18.4853 −1.67358
\(123\) 12.0000 1.08200
\(124\) −40.1421 −3.60487
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 20.5563 1.81694
\(129\) 12.0000 1.05654
\(130\) 4.82843 0.423481
\(131\) 16.1421 1.41034 0.705172 0.709036i \(-0.250870\pi\)
0.705172 + 0.709036i \(0.250870\pi\)
\(132\) 36.9706 3.21787
\(133\) 0.686292 0.0595090
\(134\) 25.3137 2.18677
\(135\) 4.00000 0.344265
\(136\) 12.4853 1.07060
\(137\) −10.8284 −0.925135 −0.462567 0.886584i \(-0.653072\pi\)
−0.462567 + 0.886584i \(0.653072\pi\)
\(138\) −42.6274 −3.62869
\(139\) 10.3431 0.877294 0.438647 0.898659i \(-0.355458\pi\)
0.438647 + 0.898659i \(0.355458\pi\)
\(140\) 3.17157 0.268047
\(141\) 0.686292 0.0577962
\(142\) −17.6569 −1.48173
\(143\) 9.65685 0.807547
\(144\) 3.00000 0.250000
\(145\) 1.00000 0.0830455
\(146\) 20.4853 1.69537
\(147\) 12.6274 1.04149
\(148\) 32.4853 2.67027
\(149\) −13.3137 −1.09070 −0.545351 0.838208i \(-0.683604\pi\)
−0.545351 + 0.838208i \(0.683604\pi\)
\(150\) 4.82843 0.394239
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −3.65685 −0.296610
\(153\) −2.82843 −0.228665
\(154\) 9.65685 0.778171
\(155\) −10.4853 −0.842198
\(156\) 15.3137 1.22608
\(157\) −16.4853 −1.31567 −0.657834 0.753163i \(-0.728527\pi\)
−0.657834 + 0.753163i \(0.728527\pi\)
\(158\) −34.9706 −2.78211
\(159\) −15.3137 −1.21446
\(160\) 1.58579 0.125367
\(161\) −7.31371 −0.576401
\(162\) 26.5563 2.08646
\(163\) −19.6569 −1.53964 −0.769822 0.638259i \(-0.779655\pi\)
−0.769822 + 0.638259i \(0.779655\pi\)
\(164\) −22.9706 −1.79370
\(165\) 9.65685 0.751785
\(166\) −30.9706 −2.40378
\(167\) 14.4853 1.12090 0.560452 0.828187i \(-0.310627\pi\)
0.560452 + 0.828187i \(0.310627\pi\)
\(168\) 7.31371 0.564265
\(169\) −9.00000 −0.692308
\(170\) 6.82843 0.523716
\(171\) 0.828427 0.0633514
\(172\) −22.9706 −1.75149
\(173\) −5.31371 −0.403994 −0.201997 0.979386i \(-0.564743\pi\)
−0.201997 + 0.979386i \(0.564743\pi\)
\(174\) 4.82843 0.366042
\(175\) 0.828427 0.0626232
\(176\) −14.4853 −1.09187
\(177\) 0 0
\(178\) −8.82843 −0.661719
\(179\) −0.686292 −0.0512958 −0.0256479 0.999671i \(-0.508165\pi\)
−0.0256479 + 0.999671i \(0.508165\pi\)
\(180\) 3.82843 0.285354
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 4.00000 0.296500
\(183\) −15.3137 −1.13202
\(184\) 38.9706 2.87295
\(185\) 8.48528 0.623850
\(186\) −50.6274 −3.71218
\(187\) 13.6569 0.998688
\(188\) −1.31371 −0.0958120
\(189\) 3.31371 0.241037
\(190\) −2.00000 −0.145095
\(191\) −15.1716 −1.09778 −0.548888 0.835896i \(-0.684949\pi\)
−0.548888 + 0.835896i \(0.684949\pi\)
\(192\) 19.6569 1.41861
\(193\) −12.4853 −0.898710 −0.449355 0.893353i \(-0.648346\pi\)
−0.449355 + 0.893353i \(0.648346\pi\)
\(194\) −10.8284 −0.777436
\(195\) 4.00000 0.286446
\(196\) −24.1716 −1.72654
\(197\) −8.34315 −0.594425 −0.297212 0.954811i \(-0.596057\pi\)
−0.297212 + 0.954811i \(0.596057\pi\)
\(198\) 11.6569 0.828417
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) −4.41421 −0.312132
\(201\) 20.9706 1.47915
\(202\) −10.4853 −0.737742
\(203\) 0.828427 0.0581442
\(204\) 21.6569 1.51628
\(205\) −6.00000 −0.419058
\(206\) 29.3137 2.04238
\(207\) −8.82843 −0.613618
\(208\) −6.00000 −0.416025
\(209\) −4.00000 −0.276686
\(210\) 4.00000 0.276026
\(211\) 4.82843 0.332403 0.166201 0.986092i \(-0.446850\pi\)
0.166201 + 0.986092i \(0.446850\pi\)
\(212\) 29.3137 2.01327
\(213\) −14.6274 −1.00225
\(214\) 19.6569 1.34371
\(215\) −6.00000 −0.409197
\(216\) −17.6569 −1.20140
\(217\) −8.68629 −0.589664
\(218\) −4.82843 −0.327022
\(219\) 16.9706 1.14676
\(220\) −18.4853 −1.24628
\(221\) 5.65685 0.380521
\(222\) 40.9706 2.74976
\(223\) 21.7990 1.45977 0.729884 0.683571i \(-0.239574\pi\)
0.729884 + 0.683571i \(0.239574\pi\)
\(224\) 1.31371 0.0877758
\(225\) 1.00000 0.0666667
\(226\) −6.82843 −0.454220
\(227\) −8.14214 −0.540413 −0.270206 0.962802i \(-0.587092\pi\)
−0.270206 + 0.962802i \(0.587092\pi\)
\(228\) −6.34315 −0.420085
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 21.3137 1.40538
\(231\) 8.00000 0.526361
\(232\) −4.41421 −0.289807
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 4.82843 0.315644
\(235\) −0.343146 −0.0223844
\(236\) 0 0
\(237\) −28.9706 −1.88184
\(238\) 5.65685 0.366679
\(239\) −23.3137 −1.50804 −0.754019 0.656852i \(-0.771887\pi\)
−0.754019 + 0.656852i \(0.771887\pi\)
\(240\) −6.00000 −0.387298
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −29.7279 −1.91098
\(243\) 10.0000 0.641500
\(244\) 29.3137 1.87662
\(245\) −6.31371 −0.403368
\(246\) −28.9706 −1.84710
\(247\) −1.65685 −0.105423
\(248\) 46.2843 2.93905
\(249\) −25.6569 −1.62594
\(250\) −2.41421 −0.152688
\(251\) 3.17157 0.200188 0.100094 0.994978i \(-0.468086\pi\)
0.100094 + 0.994978i \(0.468086\pi\)
\(252\) 3.17157 0.199790
\(253\) 42.6274 2.67996
\(254\) −14.4853 −0.908887
\(255\) 5.65685 0.354246
\(256\) −29.9706 −1.87316
\(257\) 29.3137 1.82854 0.914269 0.405107i \(-0.132766\pi\)
0.914269 + 0.405107i \(0.132766\pi\)
\(258\) −28.9706 −1.80363
\(259\) 7.02944 0.436788
\(260\) −7.65685 −0.474858
\(261\) 1.00000 0.0618984
\(262\) −38.9706 −2.40761
\(263\) −8.34315 −0.514460 −0.257230 0.966350i \(-0.582810\pi\)
−0.257230 + 0.966350i \(0.582810\pi\)
\(264\) −42.6274 −2.62354
\(265\) 7.65685 0.470357
\(266\) −1.65685 −0.101588
\(267\) −7.31371 −0.447592
\(268\) −40.1421 −2.45207
\(269\) 1.31371 0.0800982 0.0400491 0.999198i \(-0.487249\pi\)
0.0400491 + 0.999198i \(0.487249\pi\)
\(270\) −9.65685 −0.587697
\(271\) 29.7990 1.81016 0.905080 0.425242i \(-0.139811\pi\)
0.905080 + 0.425242i \(0.139811\pi\)
\(272\) −8.48528 −0.514496
\(273\) 3.31371 0.200555
\(274\) 26.1421 1.57930
\(275\) −4.82843 −0.291165
\(276\) 67.5980 4.06892
\(277\) 7.65685 0.460056 0.230028 0.973184i \(-0.426118\pi\)
0.230028 + 0.973184i \(0.426118\pi\)
\(278\) −24.9706 −1.49763
\(279\) −10.4853 −0.627737
\(280\) −3.65685 −0.218539
\(281\) −6.68629 −0.398871 −0.199435 0.979911i \(-0.563911\pi\)
−0.199435 + 0.979911i \(0.563911\pi\)
\(282\) −1.65685 −0.0986642
\(283\) −0.828427 −0.0492449 −0.0246224 0.999697i \(-0.507838\pi\)
−0.0246224 + 0.999697i \(0.507838\pi\)
\(284\) 28.0000 1.66149
\(285\) −1.65685 −0.0981436
\(286\) −23.3137 −1.37857
\(287\) −4.97056 −0.293403
\(288\) 1.58579 0.0934434
\(289\) −9.00000 −0.529412
\(290\) −2.41421 −0.141768
\(291\) −8.97056 −0.525864
\(292\) −32.4853 −1.90106
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) −30.4853 −1.77794
\(295\) 0 0
\(296\) −37.4558 −2.17708
\(297\) −19.3137 −1.12070
\(298\) 32.1421 1.86194
\(299\) 17.6569 1.02112
\(300\) −7.65685 −0.442069
\(301\) −4.97056 −0.286498
\(302\) 28.9706 1.66707
\(303\) −8.68629 −0.499014
\(304\) 2.48528 0.142541
\(305\) 7.65685 0.438430
\(306\) 6.82843 0.390355
\(307\) 10.9706 0.626123 0.313062 0.949733i \(-0.398645\pi\)
0.313062 + 0.949733i \(0.398645\pi\)
\(308\) −15.3137 −0.872580
\(309\) 24.2843 1.38148
\(310\) 25.3137 1.43772
\(311\) −2.48528 −0.140927 −0.0704637 0.997514i \(-0.522448\pi\)
−0.0704637 + 0.997514i \(0.522448\pi\)
\(312\) −17.6569 −0.999623
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 39.7990 2.24599
\(315\) 0.828427 0.0466766
\(316\) 55.4558 3.11963
\(317\) −2.82843 −0.158860 −0.0794301 0.996840i \(-0.525310\pi\)
−0.0794301 + 0.996840i \(0.525310\pi\)
\(318\) 36.9706 2.07321
\(319\) −4.82843 −0.270340
\(320\) −9.82843 −0.549426
\(321\) 16.2843 0.908899
\(322\) 17.6569 0.983978
\(323\) −2.34315 −0.130376
\(324\) −42.1127 −2.33959
\(325\) −2.00000 −0.110940
\(326\) 47.4558 2.62834
\(327\) −4.00000 −0.221201
\(328\) 26.4853 1.46241
\(329\) −0.284271 −0.0156724
\(330\) −23.3137 −1.28338
\(331\) −17.7990 −0.978321 −0.489160 0.872194i \(-0.662697\pi\)
−0.489160 + 0.872194i \(0.662697\pi\)
\(332\) 49.1127 2.69541
\(333\) 8.48528 0.464991
\(334\) −34.9706 −1.91350
\(335\) −10.4853 −0.572872
\(336\) −4.97056 −0.271166
\(337\) −6.82843 −0.371968 −0.185984 0.982553i \(-0.559547\pi\)
−0.185984 + 0.982553i \(0.559547\pi\)
\(338\) 21.7279 1.18184
\(339\) −5.65685 −0.307238
\(340\) −10.8284 −0.587254
\(341\) 50.6274 2.74163
\(342\) −2.00000 −0.108148
\(343\) −11.0294 −0.595534
\(344\) 26.4853 1.42799
\(345\) 17.6569 0.950613
\(346\) 12.8284 0.689661
\(347\) 20.1421 1.08129 0.540643 0.841252i \(-0.318181\pi\)
0.540643 + 0.841252i \(0.318181\pi\)
\(348\) −7.65685 −0.410450
\(349\) −24.6274 −1.31828 −0.659138 0.752022i \(-0.729079\pi\)
−0.659138 + 0.752022i \(0.729079\pi\)
\(350\) −2.00000 −0.106904
\(351\) −8.00000 −0.427008
\(352\) −7.65685 −0.408112
\(353\) −15.6569 −0.833330 −0.416665 0.909060i \(-0.636801\pi\)
−0.416665 + 0.909060i \(0.636801\pi\)
\(354\) 0 0
\(355\) 7.31371 0.388171
\(356\) 14.0000 0.741999
\(357\) 4.68629 0.248025
\(358\) 1.65685 0.0875675
\(359\) −32.1421 −1.69640 −0.848199 0.529678i \(-0.822313\pi\)
−0.848199 + 0.529678i \(0.822313\pi\)
\(360\) −4.41421 −0.232649
\(361\) −18.3137 −0.963879
\(362\) 14.4853 0.761329
\(363\) −24.6274 −1.29260
\(364\) −6.34315 −0.332471
\(365\) −8.48528 −0.444140
\(366\) 36.9706 1.93248
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −26.4853 −1.38064
\(369\) −6.00000 −0.312348
\(370\) −20.4853 −1.06498
\(371\) 6.34315 0.329320
\(372\) 80.2843 4.16255
\(373\) 26.9706 1.39648 0.698241 0.715862i \(-0.253966\pi\)
0.698241 + 0.715862i \(0.253966\pi\)
\(374\) −32.9706 −1.70487
\(375\) −2.00000 −0.103280
\(376\) 1.51472 0.0781156
\(377\) −2.00000 −0.103005
\(378\) −8.00000 −0.411476
\(379\) 5.51472 0.283272 0.141636 0.989919i \(-0.454764\pi\)
0.141636 + 0.989919i \(0.454764\pi\)
\(380\) 3.17157 0.162698
\(381\) −12.0000 −0.614779
\(382\) 36.6274 1.87402
\(383\) −14.4853 −0.740163 −0.370082 0.928999i \(-0.620670\pi\)
−0.370082 + 0.928999i \(0.620670\pi\)
\(384\) −41.1127 −2.09802
\(385\) −4.00000 −0.203859
\(386\) 30.1421 1.53419
\(387\) −6.00000 −0.304997
\(388\) 17.1716 0.871755
\(389\) −6.68629 −0.339008 −0.169504 0.985529i \(-0.554217\pi\)
−0.169504 + 0.985529i \(0.554217\pi\)
\(390\) −9.65685 −0.488994
\(391\) 24.9706 1.26282
\(392\) 27.8701 1.40765
\(393\) −32.2843 −1.62853
\(394\) 20.1421 1.01475
\(395\) 14.4853 0.728834
\(396\) −18.4853 −0.928920
\(397\) −8.34315 −0.418730 −0.209365 0.977838i \(-0.567140\pi\)
−0.209365 + 0.977838i \(0.567140\pi\)
\(398\) −28.9706 −1.45216
\(399\) −1.37258 −0.0687151
\(400\) 3.00000 0.150000
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) −50.6274 −2.52507
\(403\) 20.9706 1.04462
\(404\) 16.6274 0.827245
\(405\) −11.0000 −0.546594
\(406\) −2.00000 −0.0992583
\(407\) −40.9706 −2.03084
\(408\) −24.9706 −1.23623
\(409\) 30.9706 1.53140 0.765698 0.643200i \(-0.222394\pi\)
0.765698 + 0.643200i \(0.222394\pi\)
\(410\) 14.4853 0.715377
\(411\) 21.6569 1.06825
\(412\) −46.4853 −2.29017
\(413\) 0 0
\(414\) 21.3137 1.04751
\(415\) 12.8284 0.629723
\(416\) −3.17157 −0.155499
\(417\) −20.6863 −1.01301
\(418\) 9.65685 0.472332
\(419\) −4.97056 −0.242828 −0.121414 0.992602i \(-0.538743\pi\)
−0.121414 + 0.992602i \(0.538743\pi\)
\(420\) −6.34315 −0.309514
\(421\) −14.9706 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(422\) −11.6569 −0.567447
\(423\) −0.343146 −0.0166843
\(424\) −33.7990 −1.64142
\(425\) −2.82843 −0.137199
\(426\) 35.3137 1.71095
\(427\) 6.34315 0.306966
\(428\) −31.1716 −1.50673
\(429\) −19.3137 −0.932475
\(430\) 14.4853 0.698542
\(431\) −19.3137 −0.930309 −0.465154 0.885230i \(-0.654001\pi\)
−0.465154 + 0.885230i \(0.654001\pi\)
\(432\) 12.0000 0.577350
\(433\) −34.8284 −1.67375 −0.836874 0.547396i \(-0.815619\pi\)
−0.836874 + 0.547396i \(0.815619\pi\)
\(434\) 20.9706 1.00662
\(435\) −2.00000 −0.0958927
\(436\) 7.65685 0.366697
\(437\) −7.31371 −0.349862
\(438\) −40.9706 −1.95765
\(439\) −21.6569 −1.03363 −0.516813 0.856099i \(-0.672882\pi\)
−0.516813 + 0.856099i \(0.672882\pi\)
\(440\) 21.3137 1.01609
\(441\) −6.31371 −0.300653
\(442\) −13.6569 −0.649590
\(443\) 3.65685 0.173742 0.0868712 0.996220i \(-0.472313\pi\)
0.0868712 + 0.996220i \(0.472313\pi\)
\(444\) −64.9706 −3.08337
\(445\) 3.65685 0.173352
\(446\) −52.6274 −2.49198
\(447\) 26.6274 1.25943
\(448\) −8.14214 −0.384680
\(449\) 0.343146 0.0161940 0.00809702 0.999967i \(-0.497423\pi\)
0.00809702 + 0.999967i \(0.497423\pi\)
\(450\) −2.41421 −0.113807
\(451\) 28.9706 1.36417
\(452\) 10.8284 0.509326
\(453\) 24.0000 1.12762
\(454\) 19.6569 0.922542
\(455\) −1.65685 −0.0776745
\(456\) 7.31371 0.342496
\(457\) 8.34315 0.390276 0.195138 0.980776i \(-0.437485\pi\)
0.195138 + 0.980776i \(0.437485\pi\)
\(458\) 4.82843 0.225618
\(459\) −11.3137 −0.528079
\(460\) −33.7990 −1.57589
\(461\) −24.3431 −1.13377 −0.566887 0.823796i \(-0.691852\pi\)
−0.566887 + 0.823796i \(0.691852\pi\)
\(462\) −19.3137 −0.898555
\(463\) −17.7990 −0.827189 −0.413595 0.910461i \(-0.635727\pi\)
−0.413595 + 0.910461i \(0.635727\pi\)
\(464\) 3.00000 0.139272
\(465\) 20.9706 0.972487
\(466\) −43.4558 −2.01305
\(467\) −22.9706 −1.06295 −0.531475 0.847074i \(-0.678362\pi\)
−0.531475 + 0.847074i \(0.678362\pi\)
\(468\) −7.65685 −0.353938
\(469\) −8.68629 −0.401096
\(470\) 0.828427 0.0382125
\(471\) 32.9706 1.51920
\(472\) 0 0
\(473\) 28.9706 1.33207
\(474\) 69.9411 3.21250
\(475\) 0.828427 0.0380108
\(476\) −8.97056 −0.411165
\(477\) 7.65685 0.350583
\(478\) 56.2843 2.57438
\(479\) 12.8284 0.586146 0.293073 0.956090i \(-0.405322\pi\)
0.293073 + 0.956090i \(0.405322\pi\)
\(480\) −3.17157 −0.144762
\(481\) −16.9706 −0.773791
\(482\) −24.1421 −1.09964
\(483\) 14.6274 0.665571
\(484\) 47.1421 2.14282
\(485\) 4.48528 0.203666
\(486\) −24.1421 −1.09511
\(487\) −29.7990 −1.35032 −0.675161 0.737671i \(-0.735926\pi\)
−0.675161 + 0.737671i \(0.735926\pi\)
\(488\) −33.7990 −1.53001
\(489\) 39.3137 1.77783
\(490\) 15.2426 0.688592
\(491\) 43.4558 1.96113 0.980567 0.196183i \(-0.0628545\pi\)
0.980567 + 0.196183i \(0.0628545\pi\)
\(492\) 45.9411 2.07119
\(493\) −2.82843 −0.127386
\(494\) 4.00000 0.179969
\(495\) −4.82843 −0.217022
\(496\) −31.4558 −1.41241
\(497\) 6.05887 0.271778
\(498\) 61.9411 2.77565
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 3.82843 0.171212
\(501\) −28.9706 −1.29431
\(502\) −7.65685 −0.341742
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) −3.65685 −0.162889
\(505\) 4.34315 0.193267
\(506\) −102.912 −4.57498
\(507\) 18.0000 0.799408
\(508\) 22.9706 1.01915
\(509\) −44.6274 −1.97808 −0.989038 0.147663i \(-0.952825\pi\)
−0.989038 + 0.147663i \(0.952825\pi\)
\(510\) −13.6569 −0.604736
\(511\) −7.02944 −0.310964
\(512\) 31.2426 1.38074
\(513\) 3.31371 0.146304
\(514\) −70.7696 −3.12151
\(515\) −12.1421 −0.535046
\(516\) 45.9411 2.02245
\(517\) 1.65685 0.0728684
\(518\) −16.9706 −0.745644
\(519\) 10.6274 0.466492
\(520\) 8.82843 0.387152
\(521\) 1.31371 0.0575546 0.0287773 0.999586i \(-0.490839\pi\)
0.0287773 + 0.999586i \(0.490839\pi\)
\(522\) −2.41421 −0.105667
\(523\) −14.4853 −0.633397 −0.316699 0.948526i \(-0.602574\pi\)
−0.316699 + 0.948526i \(0.602574\pi\)
\(524\) 61.7990 2.69970
\(525\) −1.65685 −0.0723110
\(526\) 20.1421 0.878239
\(527\) 29.6569 1.29187
\(528\) 28.9706 1.26078
\(529\) 54.9411 2.38874
\(530\) −18.4853 −0.802949
\(531\) 0 0
\(532\) 2.62742 0.113913
\(533\) 12.0000 0.519778
\(534\) 17.6569 0.764087
\(535\) −8.14214 −0.352015
\(536\) 46.2843 1.99918
\(537\) 1.37258 0.0592313
\(538\) −3.17157 −0.136736
\(539\) 30.4853 1.31309
\(540\) 15.3137 0.658997
\(541\) −38.9706 −1.67548 −0.837738 0.546073i \(-0.816122\pi\)
−0.837738 + 0.546073i \(0.816122\pi\)
\(542\) −71.9411 −3.09014
\(543\) 12.0000 0.514969
\(544\) −4.48528 −0.192305
\(545\) 2.00000 0.0856706
\(546\) −8.00000 −0.342368
\(547\) 14.4853 0.619346 0.309673 0.950843i \(-0.399780\pi\)
0.309673 + 0.950843i \(0.399780\pi\)
\(548\) −41.4558 −1.77091
\(549\) 7.65685 0.326787
\(550\) 11.6569 0.497050
\(551\) 0.828427 0.0352922
\(552\) −77.9411 −3.31739
\(553\) 12.0000 0.510292
\(554\) −18.4853 −0.785364
\(555\) −16.9706 −0.720360
\(556\) 39.5980 1.67933
\(557\) 39.9411 1.69236 0.846180 0.532897i \(-0.178897\pi\)
0.846180 + 0.532897i \(0.178897\pi\)
\(558\) 25.3137 1.07161
\(559\) 12.0000 0.507546
\(560\) 2.48528 0.105022
\(561\) −27.3137 −1.15319
\(562\) 16.1421 0.680915
\(563\) −3.65685 −0.154118 −0.0770590 0.997027i \(-0.524553\pi\)
−0.0770590 + 0.997027i \(0.524553\pi\)
\(564\) 2.62742 0.110634
\(565\) 2.82843 0.118993
\(566\) 2.00000 0.0840663
\(567\) −9.11270 −0.382697
\(568\) −32.2843 −1.35462
\(569\) 16.3431 0.685140 0.342570 0.939492i \(-0.388703\pi\)
0.342570 + 0.939492i \(0.388703\pi\)
\(570\) 4.00000 0.167542
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 36.9706 1.54582
\(573\) 30.3431 1.26760
\(574\) 12.0000 0.500870
\(575\) −8.82843 −0.368171
\(576\) −9.82843 −0.409518
\(577\) −15.7990 −0.657721 −0.328860 0.944379i \(-0.606665\pi\)
−0.328860 + 0.944379i \(0.606665\pi\)
\(578\) 21.7279 0.903762
\(579\) 24.9706 1.03774
\(580\) 3.82843 0.158967
\(581\) 10.6274 0.440900
\(582\) 21.6569 0.897705
\(583\) −36.9706 −1.53116
\(584\) 37.4558 1.54993
\(585\) −2.00000 −0.0826898
\(586\) 20.4853 0.846239
\(587\) 9.79899 0.404448 0.202224 0.979339i \(-0.435183\pi\)
0.202224 + 0.979339i \(0.435183\pi\)
\(588\) 48.3431 1.99364
\(589\) −8.68629 −0.357912
\(590\) 0 0
\(591\) 16.6863 0.686382
\(592\) 25.4558 1.04623
\(593\) 3.65685 0.150169 0.0750845 0.997177i \(-0.476077\pi\)
0.0750845 + 0.997177i \(0.476077\pi\)
\(594\) 46.6274 1.91315
\(595\) −2.34315 −0.0960596
\(596\) −50.9706 −2.08784
\(597\) −24.0000 −0.982255
\(598\) −42.6274 −1.74316
\(599\) 1.79899 0.0735047 0.0367524 0.999324i \(-0.488299\pi\)
0.0367524 + 0.999324i \(0.488299\pi\)
\(600\) 8.82843 0.360419
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 12.0000 0.489083
\(603\) −10.4853 −0.426994
\(604\) −45.9411 −1.86932
\(605\) 12.3137 0.500623
\(606\) 20.9706 0.851871
\(607\) 42.9706 1.74412 0.872061 0.489398i \(-0.162783\pi\)
0.872061 + 0.489398i \(0.162783\pi\)
\(608\) 1.31371 0.0532779
\(609\) −1.65685 −0.0671391
\(610\) −18.4853 −0.748447
\(611\) 0.686292 0.0277644
\(612\) −10.8284 −0.437713
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −26.4853 −1.06886
\(615\) 12.0000 0.483887
\(616\) 17.6569 0.711415
\(617\) −14.8284 −0.596970 −0.298485 0.954414i \(-0.596481\pi\)
−0.298485 + 0.954414i \(0.596481\pi\)
\(618\) −58.6274 −2.35834
\(619\) 29.7990 1.19772 0.598861 0.800853i \(-0.295620\pi\)
0.598861 + 0.800853i \(0.295620\pi\)
\(620\) −40.1421 −1.61215
\(621\) −35.3137 −1.41709
\(622\) 6.00000 0.240578
\(623\) 3.02944 0.121372
\(624\) 12.0000 0.480384
\(625\) 1.00000 0.0400000
\(626\) 14.4853 0.578948
\(627\) 8.00000 0.319489
\(628\) −63.1127 −2.51847
\(629\) −24.0000 −0.956943
\(630\) −2.00000 −0.0796819
\(631\) 3.02944 0.120600 0.0603000 0.998180i \(-0.480794\pi\)
0.0603000 + 0.998180i \(0.480794\pi\)
\(632\) −63.9411 −2.54344
\(633\) −9.65685 −0.383825
\(634\) 6.82843 0.271191
\(635\) 6.00000 0.238103
\(636\) −58.6274 −2.32473
\(637\) 12.6274 0.500316
\(638\) 11.6569 0.461499
\(639\) 7.31371 0.289326
\(640\) 20.5563 0.812561
\(641\) −44.6274 −1.76268 −0.881338 0.472485i \(-0.843357\pi\)
−0.881338 + 0.472485i \(0.843357\pi\)
\(642\) −39.3137 −1.55159
\(643\) −31.4558 −1.24050 −0.620249 0.784405i \(-0.712968\pi\)
−0.620249 + 0.784405i \(0.712968\pi\)
\(644\) −28.0000 −1.10335
\(645\) 12.0000 0.472500
\(646\) 5.65685 0.222566
\(647\) 21.1127 0.830026 0.415013 0.909816i \(-0.363777\pi\)
0.415013 + 0.909816i \(0.363777\pi\)
\(648\) 48.5563 1.90747
\(649\) 0 0
\(650\) 4.82843 0.189386
\(651\) 17.3726 0.680885
\(652\) −75.2548 −2.94721
\(653\) −22.8284 −0.893345 −0.446673 0.894697i \(-0.647391\pi\)
−0.446673 + 0.894697i \(0.647391\pi\)
\(654\) 9.65685 0.377613
\(655\) 16.1421 0.630725
\(656\) −18.0000 −0.702782
\(657\) −8.48528 −0.331042
\(658\) 0.686292 0.0267544
\(659\) −37.7990 −1.47244 −0.736220 0.676743i \(-0.763391\pi\)
−0.736220 + 0.676743i \(0.763391\pi\)
\(660\) 36.9706 1.43908
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 42.9706 1.67010
\(663\) −11.3137 −0.439388
\(664\) −56.6274 −2.19757
\(665\) 0.686292 0.0266132
\(666\) −20.4853 −0.793789
\(667\) −8.82843 −0.341838
\(668\) 55.4558 2.14565
\(669\) −43.5980 −1.68560
\(670\) 25.3137 0.977954
\(671\) −36.9706 −1.42723
\(672\) −2.62742 −0.101355
\(673\) −10.9706 −0.422884 −0.211442 0.977391i \(-0.567816\pi\)
−0.211442 + 0.977391i \(0.567816\pi\)
\(674\) 16.4853 0.634989
\(675\) 4.00000 0.153960
\(676\) −34.4558 −1.32522
\(677\) 36.7696 1.41317 0.706584 0.707629i \(-0.250235\pi\)
0.706584 + 0.707629i \(0.250235\pi\)
\(678\) 13.6569 0.524488
\(679\) 3.71573 0.142597
\(680\) 12.4853 0.478789
\(681\) 16.2843 0.624015
\(682\) −122.225 −4.68025
\(683\) 40.1421 1.53600 0.767998 0.640452i \(-0.221253\pi\)
0.767998 + 0.640452i \(0.221253\pi\)
\(684\) 3.17157 0.121268
\(685\) −10.8284 −0.413733
\(686\) 26.6274 1.01664
\(687\) 4.00000 0.152610
\(688\) −18.0000 −0.686244
\(689\) −15.3137 −0.583406
\(690\) −42.6274 −1.62280
\(691\) −11.0294 −0.419580 −0.209790 0.977747i \(-0.567278\pi\)
−0.209790 + 0.977747i \(0.567278\pi\)
\(692\) −20.3431 −0.773330
\(693\) −4.00000 −0.151947
\(694\) −48.6274 −1.84587
\(695\) 10.3431 0.392338
\(696\) 8.82843 0.334641
\(697\) 16.9706 0.642806
\(698\) 59.4558 2.25044
\(699\) −36.0000 −1.36165
\(700\) 3.17157 0.119874
\(701\) 29.3137 1.10716 0.553582 0.832795i \(-0.313261\pi\)
0.553582 + 0.832795i \(0.313261\pi\)
\(702\) 19.3137 0.728949
\(703\) 7.02944 0.265120
\(704\) 47.4558 1.78856
\(705\) 0.686292 0.0258472
\(706\) 37.7990 1.42258
\(707\) 3.59798 0.135316
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) −17.6569 −0.662650
\(711\) 14.4853 0.543240
\(712\) −16.1421 −0.604952
\(713\) 92.5685 3.46672
\(714\) −11.3137 −0.423405
\(715\) 9.65685 0.361146
\(716\) −2.62742 −0.0981912
\(717\) 46.6274 1.74133
\(718\) 77.5980 2.89593
\(719\) 10.6274 0.396336 0.198168 0.980168i \(-0.436501\pi\)
0.198168 + 0.980168i \(0.436501\pi\)
\(720\) 3.00000 0.111803
\(721\) −10.0589 −0.374612
\(722\) 44.2132 1.64545
\(723\) −20.0000 −0.743808
\(724\) −22.9706 −0.853694
\(725\) 1.00000 0.0371391
\(726\) 59.4558 2.20661
\(727\) 43.9411 1.62969 0.814843 0.579682i \(-0.196823\pi\)
0.814843 + 0.579682i \(0.196823\pi\)
\(728\) 7.31371 0.271064
\(729\) 13.0000 0.481481
\(730\) 20.4853 0.758194
\(731\) 16.9706 0.627679
\(732\) −58.6274 −2.16693
\(733\) −17.1716 −0.634247 −0.317123 0.948384i \(-0.602717\pi\)
−0.317123 + 0.948384i \(0.602717\pi\)
\(734\) −43.4558 −1.60398
\(735\) 12.6274 0.465769
\(736\) −14.0000 −0.516047
\(737\) 50.6274 1.86488
\(738\) 14.4853 0.533211
\(739\) −2.48528 −0.0914226 −0.0457113 0.998955i \(-0.514555\pi\)
−0.0457113 + 0.998955i \(0.514555\pi\)
\(740\) 32.4853 1.19418
\(741\) 3.31371 0.121732
\(742\) −15.3137 −0.562184
\(743\) −7.37258 −0.270474 −0.135237 0.990813i \(-0.543180\pi\)
−0.135237 + 0.990813i \(0.543180\pi\)
\(744\) −92.5685 −3.39373
\(745\) −13.3137 −0.487777
\(746\) −65.1127 −2.38395
\(747\) 12.8284 0.469368
\(748\) 52.2843 1.91170
\(749\) −6.74517 −0.246463
\(750\) 4.82843 0.176309
\(751\) −12.1421 −0.443073 −0.221536 0.975152i \(-0.571107\pi\)
−0.221536 + 0.975152i \(0.571107\pi\)
\(752\) −1.02944 −0.0375397
\(753\) −6.34315 −0.231157
\(754\) 4.82843 0.175841
\(755\) −12.0000 −0.436725
\(756\) 12.6863 0.461396
\(757\) 36.4853 1.32608 0.663040 0.748584i \(-0.269266\pi\)
0.663040 + 0.748584i \(0.269266\pi\)
\(758\) −13.3137 −0.483576
\(759\) −85.2548 −3.09455
\(760\) −3.65685 −0.132648
\(761\) −36.6274 −1.32774 −0.663871 0.747847i \(-0.731088\pi\)
−0.663871 + 0.747847i \(0.731088\pi\)
\(762\) 28.9706 1.04949
\(763\) 1.65685 0.0599822
\(764\) −58.0833 −2.10138
\(765\) −2.82843 −0.102262
\(766\) 34.9706 1.26354
\(767\) 0 0
\(768\) 59.9411 2.16294
\(769\) −4.34315 −0.156618 −0.0783089 0.996929i \(-0.524952\pi\)
−0.0783089 + 0.996929i \(0.524952\pi\)
\(770\) 9.65685 0.348009
\(771\) −58.6274 −2.11141
\(772\) −47.7990 −1.72032
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) 14.4853 0.520663
\(775\) −10.4853 −0.376642
\(776\) −19.7990 −0.710742
\(777\) −14.0589 −0.504359
\(778\) 16.1421 0.578724
\(779\) −4.97056 −0.178089
\(780\) 15.3137 0.548319
\(781\) −35.3137 −1.26362
\(782\) −60.2843 −2.15576
\(783\) 4.00000 0.142948
\(784\) −18.9411 −0.676469
\(785\) −16.4853 −0.588385
\(786\) 77.9411 2.78007
\(787\) −21.7990 −0.777050 −0.388525 0.921438i \(-0.627015\pi\)
−0.388525 + 0.921438i \(0.627015\pi\)
\(788\) −31.9411 −1.13786
\(789\) 16.6863 0.594048
\(790\) −34.9706 −1.24420
\(791\) 2.34315 0.0833127
\(792\) 21.3137 0.757350
\(793\) −15.3137 −0.543806
\(794\) 20.1421 0.714818
\(795\) −15.3137 −0.543121
\(796\) 45.9411 1.62834
\(797\) −34.1421 −1.20938 −0.604688 0.796462i \(-0.706702\pi\)
−0.604688 + 0.796462i \(0.706702\pi\)
\(798\) 3.31371 0.117304
\(799\) 0.970563 0.0343360
\(800\) 1.58579 0.0560660
\(801\) 3.65685 0.129209
\(802\) 70.7696 2.49896
\(803\) 40.9706 1.44582
\(804\) 80.2843 2.83141
\(805\) −7.31371 −0.257774
\(806\) −50.6274 −1.78327
\(807\) −2.62742 −0.0924895
\(808\) −19.1716 −0.674454
\(809\) −14.2843 −0.502208 −0.251104 0.967960i \(-0.580794\pi\)
−0.251104 + 0.967960i \(0.580794\pi\)
\(810\) 26.5563 0.933095
\(811\) 26.3431 0.925033 0.462516 0.886611i \(-0.346947\pi\)
0.462516 + 0.886611i \(0.346947\pi\)
\(812\) 3.17157 0.111300
\(813\) −59.5980 −2.09019
\(814\) 98.9117 3.46685
\(815\) −19.6569 −0.688550
\(816\) 16.9706 0.594089
\(817\) −4.97056 −0.173898
\(818\) −74.7696 −2.61426
\(819\) −1.65685 −0.0578952
\(820\) −22.9706 −0.802167
\(821\) −45.3137 −1.58146 −0.790730 0.612165i \(-0.790299\pi\)
−0.790730 + 0.612165i \(0.790299\pi\)
\(822\) −52.2843 −1.82362
\(823\) 2.97056 0.103547 0.0517737 0.998659i \(-0.483513\pi\)
0.0517737 + 0.998659i \(0.483513\pi\)
\(824\) 53.5980 1.86717
\(825\) 9.65685 0.336209
\(826\) 0 0
\(827\) −5.31371 −0.184776 −0.0923879 0.995723i \(-0.529450\pi\)
−0.0923879 + 0.995723i \(0.529450\pi\)
\(828\) −33.7990 −1.17460
\(829\) −24.6274 −0.855346 −0.427673 0.903934i \(-0.640666\pi\)
−0.427673 + 0.903934i \(0.640666\pi\)
\(830\) −30.9706 −1.07500
\(831\) −15.3137 −0.531227
\(832\) 19.6569 0.681479
\(833\) 17.8579 0.618738
\(834\) 49.9411 1.72932
\(835\) 14.4853 0.501284
\(836\) −15.3137 −0.529636
\(837\) −41.9411 −1.44970
\(838\) 12.0000 0.414533
\(839\) −14.4853 −0.500087 −0.250044 0.968235i \(-0.580445\pi\)
−0.250044 + 0.968235i \(0.580445\pi\)
\(840\) 7.31371 0.252347
\(841\) 1.00000 0.0344828
\(842\) 36.1421 1.24554
\(843\) 13.3726 0.460576
\(844\) 18.4853 0.636290
\(845\) −9.00000 −0.309609
\(846\) 0.828427 0.0284819
\(847\) 10.2010 0.350511
\(848\) 22.9706 0.788812
\(849\) 1.65685 0.0568631
\(850\) 6.82843 0.234213
\(851\) −74.9117 −2.56794
\(852\) −56.0000 −1.91853
\(853\) −11.1127 −0.380492 −0.190246 0.981736i \(-0.560928\pi\)
−0.190246 + 0.981736i \(0.560928\pi\)
\(854\) −15.3137 −0.524024
\(855\) 0.828427 0.0283316
\(856\) 35.9411 1.22844
\(857\) 48.6274 1.66108 0.830540 0.556958i \(-0.188032\pi\)
0.830540 + 0.556958i \(0.188032\pi\)
\(858\) 46.6274 1.59183
\(859\) 28.4264 0.969896 0.484948 0.874543i \(-0.338838\pi\)
0.484948 + 0.874543i \(0.338838\pi\)
\(860\) −22.9706 −0.783290
\(861\) 9.94113 0.338793
\(862\) 46.6274 1.58814
\(863\) −7.85786 −0.267485 −0.133742 0.991016i \(-0.542699\pi\)
−0.133742 + 0.991016i \(0.542699\pi\)
\(864\) 6.34315 0.215798
\(865\) −5.31371 −0.180672
\(866\) 84.0833 2.85727
\(867\) 18.0000 0.611312
\(868\) −33.2548 −1.12874
\(869\) −69.9411 −2.37259
\(870\) 4.82843 0.163699
\(871\) 20.9706 0.710560
\(872\) −8.82843 −0.298968
\(873\) 4.48528 0.151804
\(874\) 17.6569 0.597252
\(875\) 0.828427 0.0280059
\(876\) 64.9706 2.19515
\(877\) −18.2843 −0.617416 −0.308708 0.951157i \(-0.599897\pi\)
−0.308708 + 0.951157i \(0.599897\pi\)
\(878\) 52.2843 1.76451
\(879\) 16.9706 0.572403
\(880\) −14.4853 −0.488299
\(881\) 6.68629 0.225267 0.112633 0.993637i \(-0.464071\pi\)
0.112633 + 0.993637i \(0.464071\pi\)
\(882\) 15.2426 0.513246
\(883\) 2.48528 0.0836364 0.0418182 0.999125i \(-0.486685\pi\)
0.0418182 + 0.999125i \(0.486685\pi\)
\(884\) 21.6569 0.728399
\(885\) 0 0
\(886\) −8.82843 −0.296597
\(887\) −29.3137 −0.984258 −0.492129 0.870522i \(-0.663781\pi\)
−0.492129 + 0.870522i \(0.663781\pi\)
\(888\) 74.9117 2.51387
\(889\) 4.97056 0.166707
\(890\) −8.82843 −0.295930
\(891\) 53.1127 1.77934
\(892\) 83.4558 2.79431
\(893\) −0.284271 −0.00951277
\(894\) −64.2843 −2.14999
\(895\) −0.686292 −0.0229402
\(896\) 17.0294 0.568914
\(897\) −35.3137 −1.17909
\(898\) −0.828427 −0.0276450
\(899\) −10.4853 −0.349704
\(900\) 3.82843 0.127614
\(901\) −21.6569 −0.721494
\(902\) −69.9411 −2.32878
\(903\) 9.94113 0.330820
\(904\) −12.4853 −0.415254
\(905\) −6.00000 −0.199447
\(906\) −57.9411 −1.92496
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) −31.1716 −1.03446
\(909\) 4.34315 0.144053
\(910\) 4.00000 0.132599
\(911\) 3.85786 0.127817 0.0639084 0.997956i \(-0.479643\pi\)
0.0639084 + 0.997956i \(0.479643\pi\)
\(912\) −4.97056 −0.164592
\(913\) −61.9411 −2.04995
\(914\) −20.1421 −0.666243
\(915\) −15.3137 −0.506256
\(916\) −7.65685 −0.252990
\(917\) 13.3726 0.441602
\(918\) 27.3137 0.901487
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 38.9706 1.28482
\(921\) −21.9411 −0.722985
\(922\) 58.7696 1.93547
\(923\) −14.6274 −0.481467
\(924\) 30.6274 1.00757
\(925\) 8.48528 0.278994
\(926\) 42.9706 1.41210
\(927\) −12.1421 −0.398800
\(928\) 1.58579 0.0520560
\(929\) 40.6274 1.33294 0.666471 0.745531i \(-0.267804\pi\)
0.666471 + 0.745531i \(0.267804\pi\)
\(930\) −50.6274 −1.66014
\(931\) −5.23045 −0.171421
\(932\) 68.9117 2.25728
\(933\) 4.97056 0.162729
\(934\) 55.4558 1.81457
\(935\) 13.6569 0.446627
\(936\) 8.82843 0.288566
\(937\) 8.34315 0.272559 0.136279 0.990670i \(-0.456486\pi\)
0.136279 + 0.990670i \(0.456486\pi\)
\(938\) 20.9706 0.684713
\(939\) 12.0000 0.391605
\(940\) −1.31371 −0.0428484
\(941\) 39.9411 1.30204 0.651022 0.759059i \(-0.274341\pi\)
0.651022 + 0.759059i \(0.274341\pi\)
\(942\) −79.5980 −2.59344
\(943\) 52.9706 1.72496
\(944\) 0 0
\(945\) 3.31371 0.107795
\(946\) −69.9411 −2.27398
\(947\) −56.9117 −1.84938 −0.924691 0.380719i \(-0.875676\pi\)
−0.924691 + 0.380719i \(0.875676\pi\)
\(948\) −110.912 −3.60224
\(949\) 16.9706 0.550888
\(950\) −2.00000 −0.0648886
\(951\) 5.65685 0.183436
\(952\) 10.3431 0.335223
\(953\) 6.68629 0.216590 0.108295 0.994119i \(-0.465461\pi\)
0.108295 + 0.994119i \(0.465461\pi\)
\(954\) −18.4853 −0.598483
\(955\) −15.1716 −0.490941
\(956\) −89.2548 −2.88671
\(957\) 9.65685 0.312162
\(958\) −30.9706 −1.00061
\(959\) −8.97056 −0.289675
\(960\) 19.6569 0.634422
\(961\) 78.9411 2.54649
\(962\) 40.9706 1.32094
\(963\) −8.14214 −0.262377
\(964\) 38.2843 1.23305
\(965\) −12.4853 −0.401915
\(966\) −35.3137 −1.13620
\(967\) −18.9706 −0.610052 −0.305026 0.952344i \(-0.598665\pi\)
−0.305026 + 0.952344i \(0.598665\pi\)
\(968\) −54.3553 −1.74705
\(969\) 4.68629 0.150545
\(970\) −10.8284 −0.347680
\(971\) −0.142136 −0.00456135 −0.00228067 0.999997i \(-0.500726\pi\)
−0.00228067 + 0.999997i \(0.500726\pi\)
\(972\) 38.2843 1.22797
\(973\) 8.56854 0.274695
\(974\) 71.9411 2.30514
\(975\) 4.00000 0.128103
\(976\) 22.9706 0.735270
\(977\) −25.3137 −0.809857 −0.404929 0.914348i \(-0.632704\pi\)
−0.404929 + 0.914348i \(0.632704\pi\)
\(978\) −94.9117 −3.03494
\(979\) −17.6569 −0.564316
\(980\) −24.1716 −0.772133
\(981\) 2.00000 0.0638551
\(982\) −104.912 −3.34787
\(983\) 13.3137 0.424641 0.212321 0.977200i \(-0.431898\pi\)
0.212321 + 0.977200i \(0.431898\pi\)
\(984\) −52.9706 −1.68864
\(985\) −8.34315 −0.265835
\(986\) 6.82843 0.217461
\(987\) 0.568542 0.0180969
\(988\) −6.34315 −0.201802
\(989\) 52.9706 1.68437
\(990\) 11.6569 0.370479
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) −16.6274 −0.527921
\(993\) 35.5980 1.12967
\(994\) −14.6274 −0.463953
\(995\) 12.0000 0.380426
\(996\) −98.2254 −3.11239
\(997\) 1.17157 0.0371041 0.0185520 0.999828i \(-0.494094\pi\)
0.0185520 + 0.999828i \(0.494094\pi\)
\(998\) −86.9117 −2.75114
\(999\) 33.9411 1.07385
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 145.2.a.b.1.1 2
3.2 odd 2 1305.2.a.n.1.2 2
4.3 odd 2 2320.2.a.k.1.1 2
5.2 odd 4 725.2.b.c.349.1 4
5.3 odd 4 725.2.b.c.349.4 4
5.4 even 2 725.2.a.c.1.2 2
7.6 odd 2 7105.2.a.e.1.1 2
8.3 odd 2 9280.2.a.w.1.1 2
8.5 even 2 9280.2.a.be.1.2 2
15.14 odd 2 6525.2.a.p.1.1 2
29.28 even 2 4205.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 2 1.1 even 1 trivial
725.2.a.c.1.2 2 5.4 even 2
725.2.b.c.349.1 4 5.2 odd 4
725.2.b.c.349.4 4 5.3 odd 4
1305.2.a.n.1.2 2 3.2 odd 2
2320.2.a.k.1.1 2 4.3 odd 2
4205.2.a.d.1.2 2 29.28 even 2
6525.2.a.p.1.1 2 15.14 odd 2
7105.2.a.e.1.1 2 7.6 odd 2
9280.2.a.w.1.1 2 8.3 odd 2
9280.2.a.be.1.2 2 8.5 even 2