Properties

Label 4205.2.a.d.1.2
Level $4205$
Weight $2$
Character 4205.1
Self dual yes
Analytic conductor $33.577$
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] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, 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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
Analytic rank: \(0\)
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: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} +2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} +0.828427 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} +2.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} +4.82843 q^{6} +0.828427 q^{7} +4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} +4.82843 q^{11} +7.65685 q^{12} -2.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} +3.00000 q^{16} +2.82843 q^{17} +2.41421 q^{18} -0.828427 q^{19} +3.82843 q^{20} +1.65685 q^{21} +11.6569 q^{22} -8.82843 q^{23} +8.82843 q^{24} +1.00000 q^{25} -4.82843 q^{26} -4.00000 q^{27} +3.17157 q^{28} +4.82843 q^{30} +10.4853 q^{31} -1.58579 q^{32} +9.65685 q^{33} +6.82843 q^{34} +0.828427 q^{35} +3.82843 q^{36} -8.48528 q^{37} -2.00000 q^{38} -4.00000 q^{39} +4.41421 q^{40} +6.00000 q^{41} +4.00000 q^{42} +6.00000 q^{43} +18.4853 q^{44} +1.00000 q^{45} -21.3137 q^{46} +0.343146 q^{47} +6.00000 q^{48} -6.31371 q^{49} +2.41421 q^{50} +5.65685 q^{51} -7.65685 q^{52} +7.65685 q^{53} -9.65685 q^{54} +4.82843 q^{55} +3.65685 q^{56} -1.65685 q^{57} +7.65685 q^{60} -7.65685 q^{61} +25.3137 q^{62} +0.828427 q^{63} -9.82843 q^{64} -2.00000 q^{65} +23.3137 q^{66} -10.4853 q^{67} +10.8284 q^{68} -17.6569 q^{69} +2.00000 q^{70} +7.31371 q^{71} +4.41421 q^{72} +8.48528 q^{73} -20.4853 q^{74} +2.00000 q^{75} -3.17157 q^{76} +4.00000 q^{77} -9.65685 q^{78} -14.4853 q^{79} +3.00000 q^{80} -11.0000 q^{81} +14.4853 q^{82} +12.8284 q^{83} +6.34315 q^{84} +2.82843 q^{85} +14.4853 q^{86} +21.3137 q^{88} -3.65685 q^{89} +2.41421 q^{90} -1.65685 q^{91} -33.7990 q^{92} +20.9706 q^{93} +0.828427 q^{94} -0.828427 q^{95} -3.17157 q^{96} -4.48528 q^{97} -15.2426 q^{98} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} + 2 q^{9} + 2 q^{10} + 4 q^{11} + 4 q^{12} - 4 q^{13} + 4 q^{14} + 4 q^{15} + 6 q^{16} + 2 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} + 12 q^{22} - 12 q^{23} + 12 q^{24} + 2 q^{25} - 4 q^{26} - 8 q^{27} + 12 q^{28} + 4 q^{30} + 4 q^{31} - 6 q^{32} + 8 q^{33} + 8 q^{34} - 4 q^{35} + 2 q^{36} - 4 q^{38} - 8 q^{39} + 6 q^{40} + 12 q^{41} + 8 q^{42} + 12 q^{43} + 20 q^{44} + 2 q^{45} - 20 q^{46} + 12 q^{47} + 12 q^{48} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 4 q^{53} - 8 q^{54} + 4 q^{55} - 4 q^{56} + 8 q^{57} + 4 q^{60} - 4 q^{61} + 28 q^{62} - 4 q^{63} - 14 q^{64} - 4 q^{65} + 24 q^{66} - 4 q^{67} + 16 q^{68} - 24 q^{69} + 4 q^{70} - 8 q^{71} + 6 q^{72} - 24 q^{74} + 4 q^{75} - 12 q^{76} + 8 q^{77} - 8 q^{78} - 12 q^{79} + 6 q^{80} - 22 q^{81} + 12 q^{82} + 20 q^{83} + 24 q^{84} + 12 q^{86} + 20 q^{88} + 4 q^{89} + 2 q^{90} + 8 q^{91} - 28 q^{92} + 8 q^{93} - 4 q^{94} + 4 q^{95} - 12 q^{96} + 8 q^{97} - 22 q^{98} + 4 q^{99}+O(q^{100}) \) 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 \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 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 \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 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 \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 2.41421 0.569036
\(19\) −0.828427 −0.190054 −0.0950271 0.995475i \(-0.530294\pi\)
−0.0950271 + 0.995475i \(0.530294\pi\)
\(20\) 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
\(29\) 0 0
\(30\) 4.82843 0.881546
\(31\) 10.4853 1.88321 0.941606 0.336717i \(-0.109316\pi\)
0.941606 + 0.336717i \(0.109316\pi\)
\(32\) −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 \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −2.00000 −0.324443
\(39\) −4.00000 −0.640513
\(40\) 4.41421 0.697948
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000 0.617213
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 18.4853 2.78676
\(45\) 1.00000 0.149071
\(46\) −21.3137 −3.14253
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 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
\(58\) 0 0
\(59\) 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 \(-0.663069\pi\)
−0.490180 + 0.871621i \(0.663069\pi\)
\(62\) 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 \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) −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 \(-0.803187\pi\)
−0.814861 + 0.579657i \(0.803187\pi\)
\(80\) 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
\(87\) 0 0
\(88\) 21.3137 2.27205
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 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 \(-0.573122\pi\)
−0.227706 + 0.973730i \(0.573122\pi\)
\(98\) −15.2426 −1.53974
\(99\) 4.82843 0.485275
\(100\) 3.82843 0.382843
\(101\) −4.34315 −0.432159 −0.216080 0.976376i \(-0.569327\pi\)
−0.216080 + 0.976376i \(0.569327\pi\)
\(102\) 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 \(-0.542473\pi\)
−0.133038 + 0.991111i \(0.542473\pi\)
\(114\) −4.00000 −0.374634
\(115\) −8.82843 −0.823255
\(116\) 0 0
\(117\) −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 \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) −20.5563 −1.81694
\(129\) 12.0000 1.05654
\(130\) −4.82843 −0.423481
\(131\) −16.1421 −1.41034 −0.705172 0.709036i \(-0.749130\pi\)
−0.705172 + 0.709036i \(0.749130\pi\)
\(132\) 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 \(-0.346928\pi\)
0.462567 + 0.886584i \(0.346928\pi\)
\(138\) −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
\(145\) 0 0
\(146\) 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 \(-0.271473\pi\)
0.657834 + 0.753163i \(0.271473\pi\)
\(158\) −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 \(-0.220345\pi\)
0.769822 + 0.638259i \(0.220345\pi\)
\(164\) 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 \(0.564743\pi\)
\(174\) 0 0
\(175\) 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 \(-0.315051\pi\)
0.548888 + 0.835896i \(0.315051\pi\)
\(192\) −19.6569 −1.41861
\(193\) 12.4853 0.898710 0.449355 0.893353i \(-0.351654\pi\)
0.449355 + 0.893353i \(0.351654\pi\)
\(194\) −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 0
\(204\) 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 \(-0.553150\pi\)
−0.166201 + 0.986092i \(0.553150\pi\)
\(212\) 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 \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −21.3137 −1.40538
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) 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 \(-0.531914\pi\)
−0.100094 + 0.994978i \(0.531914\pi\)
\(252\) 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
\(261\) 0 0
\(262\) −38.9706 −2.40761
\(263\) 8.34315 0.514460 0.257230 0.966350i \(-0.417190\pi\)
0.257230 + 0.966350i \(0.417190\pi\)
\(264\) 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 \(-0.512751\pi\)
−0.0400491 + 0.999198i \(0.512751\pi\)
\(270\) −9.65685 −0.587697
\(271\) −29.7990 −1.81016 −0.905080 0.425242i \(-0.860189\pi\)
−0.905080 + 0.425242i \(0.860189\pi\)
\(272\) 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
\(290\) 0 0
\(291\) −8.97056 −0.525864
\(292\) 32.4853 1.90106
\(293\) 8.48528 0.495715 0.247858 0.968796i \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) −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 \(-0.601355\pi\)
−0.313062 + 0.949733i \(0.601355\pi\)
\(308\) 15.3137 0.872580
\(309\) −24.2843 −1.38148
\(310\) 25.3137 1.43772
\(311\) 2.48528 0.140927 0.0704637 0.997514i \(-0.477552\pi\)
0.0704637 + 0.997514i \(0.477552\pi\)
\(312\) −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 \(-0.474690\pi\)
0.0794301 + 0.996840i \(0.474690\pi\)
\(318\) 36.9706 2.07321
\(319\) 0 0
\(320\) −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 \(-0.337303\pi\)
0.489160 + 0.872194i \(0.337303\pi\)
\(332\) 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 \(-0.440453\pi\)
0.185984 + 0.982553i \(0.440453\pi\)
\(338\) −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 \(0.318181\pi\)
\(348\) 0 0
\(349\) −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 \(-0.177687\pi\)
0.848199 + 0.529678i \(0.177687\pi\)
\(360\) 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 \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −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
\(377\) 0 0
\(378\) −8.00000 −0.411476
\(379\) −5.51472 −0.283272 −0.141636 0.989919i \(-0.545236\pi\)
−0.141636 + 0.989919i \(0.545236\pi\)
\(380\) −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 \(-0.445783\pi\)
0.169504 + 0.985529i \(0.445783\pi\)
\(390\) −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
\(406\) 0 0
\(407\) −40.9706 −2.03084
\(408\) 24.9706 1.23623
\(409\) −30.9706 −1.53140 −0.765698 0.643200i \(-0.777606\pi\)
−0.765698 + 0.643200i \(0.777606\pi\)
\(410\) 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 \(-0.381134\pi\)
0.364810 + 0.931082i \(0.381134\pi\)
\(422\) −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 \(-0.184381\pi\)
0.836874 + 0.547396i \(0.184381\pi\)
\(434\) 20.9706 1.00662
\(435\) 0 0
\(436\) 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 \(-0.527687\pi\)
−0.0868712 + 0.996220i \(0.527687\pi\)
\(444\) −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 \(-0.502577\pi\)
−0.00809702 + 0.999967i \(0.502577\pi\)
\(450\) 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 \(-0.308148\pi\)
0.566887 + 0.823796i \(0.308148\pi\)
\(462\) 19.3137 0.898555
\(463\) −17.7990 −0.827189 −0.413595 0.910461i \(-0.635727\pi\)
−0.413595 + 0.910461i \(0.635727\pi\)
\(464\) 0 0
\(465\) 20.9706 0.972487
\(466\) 43.4558 2.01305
\(467\) 22.9706 1.06295 0.531475 0.847074i \(-0.321638\pi\)
0.531475 + 0.847074i \(0.321638\pi\)
\(468\) −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 \(-0.594678\pi\)
−0.293073 + 0.956090i \(0.594678\pi\)
\(480\) −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 \(-0.937145\pi\)
−0.980567 + 0.196183i \(0.937145\pi\)
\(492\) 45.9411 2.07119
\(493\) 0 0
\(494\) 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 \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 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 \(0.490839\pi\)
\(522\) 0 0
\(523\) −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 \(-0.183878\pi\)
0.837738 + 0.546073i \(0.183878\pi\)
\(542\) −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 0
\(552\) −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 \(-0.475447\pi\)
0.0770590 + 0.997027i \(0.475447\pi\)
\(564\) 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 \(-0.611297\pi\)
−0.342570 + 0.939492i \(0.611297\pi\)
\(570\) −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 \(-0.393335\pi\)
0.328860 + 0.944379i \(0.393335\pi\)
\(578\) −21.7279 −0.903762
\(579\) 24.9706 1.03774
\(580\) 0 0
\(581\) 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 \(-0.511701\pi\)
−0.0367524 + 0.999324i \(0.511701\pi\)
\(600\) 8.82843 0.360419
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 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 \(-0.837217\pi\)
−0.872061 + 0.489398i \(0.837217\pi\)
\(608\) 1.31371 0.0532779
\(609\) 0 0
\(610\) −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 \(-0.403519\pi\)
0.298485 + 0.954414i \(0.403519\pi\)
\(618\) −58.6274 −2.35834
\(619\) −29.7990 −1.19772 −0.598861 0.800853i \(-0.704380\pi\)
−0.598861 + 0.800853i \(0.704380\pi\)
\(620\) 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
\(638\) 0 0
\(639\) 7.31371 0.289326
\(640\) −20.5563 −0.812561
\(641\) 44.6274 1.76268 0.881338 0.472485i \(-0.156643\pi\)
0.881338 + 0.472485i \(0.156643\pi\)
\(642\) −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 \(-0.352609\pi\)
0.446673 + 0.894697i \(0.352609\pi\)
\(654\) 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 \(-0.236609\pi\)
0.736220 + 0.676743i \(0.236609\pi\)
\(660\) 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
\(667\) 0 0
\(668\) 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 \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(678\) −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
\(696\) 0 0
\(697\) 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
\(725\) 0 0
\(726\) 59.4558 2.20661
\(727\) −43.9411 −1.62969 −0.814843 0.579682i \(-0.803177\pi\)
−0.814843 + 0.579682i \(0.803177\pi\)
\(728\) −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 \(-0.397283\pi\)
0.317123 + 0.948384i \(0.397283\pi\)
\(734\) −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 \(-0.485445\pi\)
0.0457113 + 0.998955i \(0.485445\pi\)
\(740\) −32.4853 −1.19418
\(741\) 3.31371 0.121732
\(742\) 15.3137 0.562184
\(743\) 7.37258 0.270474 0.135237 0.990813i \(-0.456820\pi\)
0.135237 + 0.990813i \(0.456820\pi\)
\(744\) 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 \(-0.428893\pi\)
0.221536 + 0.975152i \(0.428893\pi\)
\(752\) 1.02944 0.0375397
\(753\) −6.34315 −0.231157
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) −12.6863 −0.461396
\(757\) −36.4853 −1.32608 −0.663040 0.748584i \(-0.730734\pi\)
−0.663040 + 0.748584i \(0.730734\pi\)
\(758\) −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 \(-0.475048\pi\)
0.0783089 + 0.996929i \(0.475048\pi\)
\(770\) 9.65685 0.348009
\(771\) 58.6274 2.11141
\(772\) 47.7990 1.72032
\(773\) −8.48528 −0.305194 −0.152597 0.988288i \(-0.548764\pi\)
−0.152597 + 0.988288i \(0.548764\pi\)
\(774\) 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
\(783\) 0 0
\(784\) −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 \(-0.293298\pi\)
0.604688 + 0.796462i \(0.293298\pi\)
\(798\) −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 \(-0.419206\pi\)
0.251104 + 0.967960i \(0.419206\pi\)
\(810\) −26.5563 −0.933095
\(811\) 26.3431 0.925033 0.462516 0.886611i \(-0.346947\pi\)
0.462516 + 0.886611i \(0.346947\pi\)
\(812\) 0 0
\(813\) −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 \(-0.516487\pi\)
−0.0517737 + 0.998659i \(0.516487\pi\)
\(824\) −53.5980 −1.86717
\(825\) 9.65685 0.336209
\(826\) 0 0
\(827\) 5.31371 0.184776 0.0923879 0.995723i \(-0.470550\pi\)
0.0923879 + 0.995723i \(0.470550\pi\)
\(828\) −33.7990 −1.17460
\(829\) 24.6274 0.855346 0.427673 0.903934i \(-0.359334\pi\)
0.427673 + 0.903934i \(0.359334\pi\)
\(830\) 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 \(-0.419555\pi\)
0.250044 + 0.968235i \(0.419555\pi\)
\(840\) 7.31371 0.252347
\(841\) 0 0
\(842\) 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 \(-0.439072\pi\)
0.190246 + 0.981736i \(0.439072\pi\)
\(854\) −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 \(-0.661162\pi\)
−0.484948 + 0.874543i \(0.661162\pi\)
\(860\) 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
\(870\) 0 0
\(871\) 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 \(-0.535929\pi\)
−0.112633 + 0.993637i \(0.535929\pi\)
\(882\) −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 \(-0.336219\pi\)
0.492129 + 0.870522i \(0.336219\pi\)
\(888\) −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
\(899\) 0 0
\(900\) 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 \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −31.1716 −1.03446
\(909\) −4.34315 −0.144053
\(910\) −4.00000 −0.132599
\(911\) −3.85786 −0.127817 −0.0639084 0.997956i \(-0.520357\pi\)
−0.0639084 + 0.997956i \(0.520357\pi\)
\(912\) −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
\(928\) 0 0
\(929\) 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 \(-0.124324\pi\)
0.924691 + 0.380719i \(0.124324\pi\)
\(948\) −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
\(957\) 0 0
\(958\) −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 \(-0.401335\pi\)
0.305026 + 0.952344i \(0.401335\pi\)
\(968\) 54.3553 1.74705
\(969\) −4.68629 −0.150545
\(970\) −10.8284 −0.347680
\(971\) 0.142136 0.00456135 0.00228067 0.999997i \(-0.499274\pi\)
0.00228067 + 0.999997i \(0.499274\pi\)
\(972\) −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 \(-0.568102\pi\)
−0.212321 + 0.977200i \(0.568102\pi\)
\(984\) 52.9706 1.68864
\(985\) −8.34315 −0.265835
\(986\) 0 0
\(987\) 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 \(-0.505906\pi\)
−0.0185520 + 0.999828i \(0.505906\pi\)
\(998\) 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 4205.2.a.d.1.2 2
29.28 even 2 145.2.a.b.1.1 2
87.86 odd 2 1305.2.a.n.1.2 2
116.115 odd 2 2320.2.a.k.1.1 2
145.28 odd 4 725.2.b.c.349.4 4
145.57 odd 4 725.2.b.c.349.1 4
145.144 even 2 725.2.a.c.1.2 2
203.202 odd 2 7105.2.a.e.1.1 2
232.115 odd 2 9280.2.a.w.1.1 2
232.173 even 2 9280.2.a.be.1.2 2
435.434 odd 2 6525.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 2 29.28 even 2
725.2.a.c.1.2 2 145.144 even 2
725.2.b.c.349.1 4 145.57 odd 4
725.2.b.c.349.4 4 145.28 odd 4
1305.2.a.n.1.2 2 87.86 odd 2
2320.2.a.k.1.1 2 116.115 odd 2
4205.2.a.d.1.2 2 1.1 even 1 trivial
6525.2.a.p.1.1 2 435.434 odd 2
7105.2.a.e.1.1 2 203.202 odd 2
9280.2.a.w.1.1 2 232.115 odd 2
9280.2.a.be.1.2 2 232.173 even 2