Properties

Label 841.2.b.a.840.2
Level $841$
Weight $2$
Character 841.840
Analytic conductor $6.715$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [841,2,Mod(840,841)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(841, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("841.840");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(6.71541880999\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 840.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 841.840
Dual form 841.2.b.a.840.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214i q^{2} +0.414214i q^{3} +1.82843 q^{4} +1.00000 q^{5} +0.171573 q^{6} +2.82843 q^{7} -1.58579i q^{8} +2.82843 q^{9} +O(q^{10})\) \(q-0.414214i q^{2} +0.414214i q^{3} +1.82843 q^{4} +1.00000 q^{5} +0.171573 q^{6} +2.82843 q^{7} -1.58579i q^{8} +2.82843 q^{9} -0.414214i q^{10} -2.41421i q^{11} +0.757359i q^{12} -1.82843 q^{13} -1.17157i q^{14} +0.414214i q^{15} +3.00000 q^{16} +4.82843i q^{17} -1.17157i q^{18} -6.00000i q^{19} +1.82843 q^{20} +1.17157i q^{21} -1.00000 q^{22} -7.65685 q^{23} +0.656854 q^{24} -4.00000 q^{25} +0.757359i q^{26} +2.41421i q^{27} +5.17157 q^{28} +0.171573 q^{30} +4.07107i q^{31} -4.41421i q^{32} +1.00000 q^{33} +2.00000 q^{34} +2.82843 q^{35} +5.17157 q^{36} -4.00000i q^{37} -2.48528 q^{38} -0.757359i q^{39} -1.58579i q^{40} +12.4853i q^{41} +0.485281 q^{42} -6.41421i q^{43} -4.41421i q^{44} +2.82843 q^{45} +3.17157i q^{46} +5.24264i q^{47} +1.24264i q^{48} +1.00000 q^{49} +1.65685i q^{50} -2.00000 q^{51} -3.34315 q^{52} -7.48528 q^{53} +1.00000 q^{54} -2.41421i q^{55} -4.48528i q^{56} +2.48528 q^{57} +7.65685 q^{59} +0.757359i q^{60} -0.828427i q^{61} +1.68629 q^{62} +8.00000 q^{63} +4.17157 q^{64} -1.82843 q^{65} -0.414214i q^{66} +5.65685 q^{67} +8.82843i q^{68} -3.17157i q^{69} -1.17157i q^{70} +3.17157 q^{71} -4.48528i q^{72} +4.00000i q^{73} -1.65685 q^{74} -1.65685i q^{75} -10.9706i q^{76} -6.82843i q^{77} -0.313708 q^{78} -0.414214i q^{79} +3.00000 q^{80} +7.48528 q^{81} +5.17157 q^{82} -3.65685 q^{83} +2.14214i q^{84} +4.82843i q^{85} -2.65685 q^{86} -3.82843 q^{88} -4.48528i q^{89} -1.17157i q^{90} -5.17157 q^{91} -14.0000 q^{92} -1.68629 q^{93} +2.17157 q^{94} -6.00000i q^{95} +1.82843 q^{96} -12.4853i q^{97} -0.414214i q^{98} -6.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{5} + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{5} + 12 q^{6} + 4 q^{13} + 12 q^{16} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 20 q^{24} - 16 q^{25} + 32 q^{28} + 12 q^{30} + 4 q^{33} + 8 q^{34} + 32 q^{36} + 24 q^{38} - 32 q^{42} + 4 q^{49} - 8 q^{51} - 36 q^{52} + 4 q^{53} + 4 q^{54} - 24 q^{57} + 8 q^{59} + 52 q^{62} + 32 q^{63} + 28 q^{64} + 4 q^{65} + 24 q^{71} + 16 q^{74} + 44 q^{78} + 12 q^{80} - 4 q^{81} + 32 q^{82} + 8 q^{83} + 12 q^{86} - 4 q^{88} - 32 q^{91} - 56 q^{92} - 52 q^{93} + 20 q^{94} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/841\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.414214i − 0.292893i −0.989219 0.146447i \(-0.953216\pi\)
0.989219 0.146447i \(-0.0467837\pi\)
\(3\) 0.414214i 0.239146i 0.992825 + 0.119573i \(0.0381526\pi\)
−0.992825 + 0.119573i \(0.961847\pi\)
\(4\) 1.82843 0.914214
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0.171573 0.0700443
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) − 1.58579i − 0.560660i
\(9\) 2.82843 0.942809
\(10\) − 0.414214i − 0.130986i
\(11\) − 2.41421i − 0.727913i −0.931416 0.363956i \(-0.881426\pi\)
0.931416 0.363956i \(-0.118574\pi\)
\(12\) 0.757359i 0.218631i
\(13\) −1.82843 −0.507114 −0.253557 0.967320i \(-0.581601\pi\)
−0.253557 + 0.967320i \(0.581601\pi\)
\(14\) − 1.17157i − 0.313116i
\(15\) 0.414214i 0.106949i
\(16\) 3.00000 0.750000
\(17\) 4.82843i 1.17107i 0.810649 + 0.585533i \(0.199115\pi\)
−0.810649 + 0.585533i \(0.800885\pi\)
\(18\) − 1.17157i − 0.276142i
\(19\) − 6.00000i − 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 1.82843 0.408849
\(21\) 1.17157i 0.255658i
\(22\) −1.00000 −0.213201
\(23\) −7.65685 −1.59656 −0.798282 0.602284i \(-0.794258\pi\)
−0.798282 + 0.602284i \(0.794258\pi\)
\(24\) 0.656854 0.134080
\(25\) −4.00000 −0.800000
\(26\) 0.757359i 0.148530i
\(27\) 2.41421i 0.464616i
\(28\) 5.17157 0.977335
\(29\) 0 0
\(30\) 0.171573 0.0313248
\(31\) 4.07107i 0.731185i 0.930775 + 0.365593i \(0.119134\pi\)
−0.930775 + 0.365593i \(0.880866\pi\)
\(32\) − 4.41421i − 0.780330i
\(33\) 1.00000 0.174078
\(34\) 2.00000 0.342997
\(35\) 2.82843 0.478091
\(36\) 5.17157 0.861929
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) −2.48528 −0.403166
\(39\) − 0.757359i − 0.121275i
\(40\) − 1.58579i − 0.250735i
\(41\) 12.4853i 1.94987i 0.222483 + 0.974937i \(0.428584\pi\)
−0.222483 + 0.974937i \(0.571416\pi\)
\(42\) 0.485281 0.0748805
\(43\) − 6.41421i − 0.978158i −0.872239 0.489079i \(-0.837333\pi\)
0.872239 0.489079i \(-0.162667\pi\)
\(44\) − 4.41421i − 0.665468i
\(45\) 2.82843 0.421637
\(46\) 3.17157i 0.467623i
\(47\) 5.24264i 0.764718i 0.924014 + 0.382359i \(0.124888\pi\)
−0.924014 + 0.382359i \(0.875112\pi\)
\(48\) 1.24264i 0.179360i
\(49\) 1.00000 0.142857
\(50\) 1.65685i 0.234315i
\(51\) −2.00000 −0.280056
\(52\) −3.34315 −0.463611
\(53\) −7.48528 −1.02818 −0.514091 0.857736i \(-0.671871\pi\)
−0.514091 + 0.857736i \(0.671871\pi\)
\(54\) 1.00000 0.136083
\(55\) − 2.41421i − 0.325532i
\(56\) − 4.48528i − 0.599371i
\(57\) 2.48528 0.329184
\(58\) 0 0
\(59\) 7.65685 0.996838 0.498419 0.866936i \(-0.333914\pi\)
0.498419 + 0.866936i \(0.333914\pi\)
\(60\) 0.757359i 0.0977747i
\(61\) − 0.828427i − 0.106069i −0.998593 0.0530346i \(-0.983111\pi\)
0.998593 0.0530346i \(-0.0168894\pi\)
\(62\) 1.68629 0.214159
\(63\) 8.00000 1.00791
\(64\) 4.17157 0.521447
\(65\) −1.82843 −0.226788
\(66\) − 0.414214i − 0.0509862i
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 8.82843i 1.07060i
\(69\) − 3.17157i − 0.381813i
\(70\) − 1.17157i − 0.140030i
\(71\) 3.17157 0.376396 0.188198 0.982131i \(-0.439735\pi\)
0.188198 + 0.982131i \(0.439735\pi\)
\(72\) − 4.48528i − 0.528595i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −1.65685 −0.192605
\(75\) − 1.65685i − 0.191317i
\(76\) − 10.9706i − 1.25841i
\(77\) − 6.82843i − 0.778171i
\(78\) −0.313708 −0.0355205
\(79\) − 0.414214i − 0.0466027i −0.999728 0.0233013i \(-0.992582\pi\)
0.999728 0.0233013i \(-0.00741772\pi\)
\(80\) 3.00000 0.335410
\(81\) 7.48528 0.831698
\(82\) 5.17157 0.571105
\(83\) −3.65685 −0.401392 −0.200696 0.979654i \(-0.564320\pi\)
−0.200696 + 0.979654i \(0.564320\pi\)
\(84\) 2.14214i 0.233726i
\(85\) 4.82843i 0.523716i
\(86\) −2.65685 −0.286496
\(87\) 0 0
\(88\) −3.82843 −0.408112
\(89\) − 4.48528i − 0.475439i −0.971334 0.237719i \(-0.923600\pi\)
0.971334 0.237719i \(-0.0763999\pi\)
\(90\) − 1.17157i − 0.123495i
\(91\) −5.17157 −0.542128
\(92\) −14.0000 −1.45960
\(93\) −1.68629 −0.174860
\(94\) 2.17157 0.223981
\(95\) − 6.00000i − 0.615587i
\(96\) 1.82843 0.186613
\(97\) − 12.4853i − 1.26769i −0.773461 0.633844i \(-0.781476\pi\)
0.773461 0.633844i \(-0.218524\pi\)
\(98\) − 0.414214i − 0.0418419i
\(99\) − 6.82843i − 0.686283i
\(100\) −7.31371 −0.731371
\(101\) 13.6569i 1.35891i 0.733718 + 0.679454i \(0.237783\pi\)
−0.733718 + 0.679454i \(0.762217\pi\)
\(102\) 0.828427i 0.0820265i
\(103\) 0.828427 0.0816274 0.0408137 0.999167i \(-0.487005\pi\)
0.0408137 + 0.999167i \(0.487005\pi\)
\(104\) 2.89949i 0.284319i
\(105\) 1.17157i 0.114334i
\(106\) 3.10051i 0.301148i
\(107\) −9.17157 −0.886649 −0.443325 0.896361i \(-0.646201\pi\)
−0.443325 + 0.896361i \(0.646201\pi\)
\(108\) 4.41421i 0.424758i
\(109\) −1.34315 −0.128650 −0.0643250 0.997929i \(-0.520489\pi\)
−0.0643250 + 0.997929i \(0.520489\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 1.65685 0.157262
\(112\) 8.48528 0.801784
\(113\) 9.31371i 0.876160i 0.898936 + 0.438080i \(0.144341\pi\)
−0.898936 + 0.438080i \(0.855659\pi\)
\(114\) − 1.02944i − 0.0964156i
\(115\) −7.65685 −0.714005
\(116\) 0 0
\(117\) −5.17157 −0.478112
\(118\) − 3.17157i − 0.291967i
\(119\) 13.6569i 1.25192i
\(120\) 0.656854 0.0599623
\(121\) 5.17157 0.470143
\(122\) −0.343146 −0.0310670
\(123\) −5.17157 −0.466305
\(124\) 7.44365i 0.668460i
\(125\) −9.00000 −0.804984
\(126\) − 3.31371i − 0.295209i
\(127\) 15.6569i 1.38932i 0.719338 + 0.694661i \(0.244445\pi\)
−0.719338 + 0.694661i \(0.755555\pi\)
\(128\) − 10.5563i − 0.933058i
\(129\) 2.65685 0.233923
\(130\) 0.757359i 0.0664248i
\(131\) − 1.31371i − 0.114779i −0.998352 0.0573896i \(-0.981722\pi\)
0.998352 0.0573896i \(-0.0182777\pi\)
\(132\) 1.82843 0.159144
\(133\) − 16.9706i − 1.47153i
\(134\) − 2.34315i − 0.202417i
\(135\) 2.41421i 0.207782i
\(136\) 7.65685 0.656570
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) −1.31371 −0.111830
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 5.17157 0.437078
\(141\) −2.17157 −0.182879
\(142\) − 1.31371i − 0.110244i
\(143\) 4.41421i 0.369135i
\(144\) 8.48528 0.707107
\(145\) 0 0
\(146\) 1.65685 0.137122
\(147\) 0.414214i 0.0341638i
\(148\) − 7.31371i − 0.601183i
\(149\) 2.17157 0.177902 0.0889511 0.996036i \(-0.471649\pi\)
0.0889511 + 0.996036i \(0.471649\pi\)
\(150\) −0.686292 −0.0560355
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) −9.51472 −0.771746
\(153\) 13.6569i 1.10409i
\(154\) −2.82843 −0.227921
\(155\) 4.07107i 0.326996i
\(156\) − 1.38478i − 0.110871i
\(157\) − 8.48528i − 0.677199i −0.940931 0.338600i \(-0.890047\pi\)
0.940931 0.338600i \(-0.109953\pi\)
\(158\) −0.171573 −0.0136496
\(159\) − 3.10051i − 0.245886i
\(160\) − 4.41421i − 0.348974i
\(161\) −21.6569 −1.70680
\(162\) − 3.10051i − 0.243599i
\(163\) 18.0711i 1.41544i 0.706495 + 0.707718i \(0.250275\pi\)
−0.706495 + 0.707718i \(0.749725\pi\)
\(164\) 22.8284i 1.78260i
\(165\) 1.00000 0.0778499
\(166\) 1.51472i 0.117565i
\(167\) 8.82843 0.683164 0.341582 0.939852i \(-0.389037\pi\)
0.341582 + 0.939852i \(0.389037\pi\)
\(168\) 1.85786 0.143337
\(169\) −9.65685 −0.742835
\(170\) 2.00000 0.153393
\(171\) − 16.9706i − 1.29777i
\(172\) − 11.7279i − 0.894246i
\(173\) −23.6569 −1.79860 −0.899299 0.437335i \(-0.855922\pi\)
−0.899299 + 0.437335i \(0.855922\pi\)
\(174\) 0 0
\(175\) −11.3137 −0.855236
\(176\) − 7.24264i − 0.545935i
\(177\) 3.17157i 0.238390i
\(178\) −1.85786 −0.139253
\(179\) −10.4853 −0.783707 −0.391853 0.920028i \(-0.628166\pi\)
−0.391853 + 0.920028i \(0.628166\pi\)
\(180\) 5.17157 0.385466
\(181\) −14.3137 −1.06393 −0.531965 0.846766i \(-0.678546\pi\)
−0.531965 + 0.846766i \(0.678546\pi\)
\(182\) 2.14214i 0.158786i
\(183\) 0.343146 0.0253661
\(184\) 12.1421i 0.895130i
\(185\) − 4.00000i − 0.294086i
\(186\) 0.698485i 0.0512154i
\(187\) 11.6569 0.852434
\(188\) 9.58579i 0.699115i
\(189\) 6.82843i 0.496695i
\(190\) −2.48528 −0.180301
\(191\) − 2.68629i − 0.194373i −0.995266 0.0971866i \(-0.969016\pi\)
0.995266 0.0971866i \(-0.0309844\pi\)
\(192\) 1.72792i 0.124702i
\(193\) 10.8284i 0.779447i 0.920932 + 0.389724i \(0.127429\pi\)
−0.920932 + 0.389724i \(0.872571\pi\)
\(194\) −5.17157 −0.371297
\(195\) − 0.757359i − 0.0542356i
\(196\) 1.82843 0.130602
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −2.82843 −0.201008
\(199\) 16.4853 1.16861 0.584305 0.811534i \(-0.301367\pi\)
0.584305 + 0.811534i \(0.301367\pi\)
\(200\) 6.34315i 0.448528i
\(201\) 2.34315i 0.165273i
\(202\) 5.65685 0.398015
\(203\) 0 0
\(204\) −3.65685 −0.256031
\(205\) 12.4853i 0.872010i
\(206\) − 0.343146i − 0.0239081i
\(207\) −21.6569 −1.50526
\(208\) −5.48528 −0.380336
\(209\) −14.4853 −1.00197
\(210\) 0.485281 0.0334876
\(211\) 17.3848i 1.19682i 0.801191 + 0.598409i \(0.204200\pi\)
−0.801191 + 0.598409i \(0.795800\pi\)
\(212\) −13.6863 −0.939978
\(213\) 1.31371i 0.0900138i
\(214\) 3.79899i 0.259694i
\(215\) − 6.41421i − 0.437446i
\(216\) 3.82843 0.260491
\(217\) 11.5147i 0.781670i
\(218\) 0.556349i 0.0376807i
\(219\) −1.65685 −0.111960
\(220\) − 4.41421i − 0.297606i
\(221\) − 8.82843i − 0.593864i
\(222\) − 0.686292i − 0.0460609i
\(223\) −8.82843 −0.591195 −0.295598 0.955313i \(-0.595519\pi\)
−0.295598 + 0.955313i \(0.595519\pi\)
\(224\) − 12.4853i − 0.834208i
\(225\) −11.3137 −0.754247
\(226\) 3.85786 0.256621
\(227\) 20.1421 1.33688 0.668440 0.743766i \(-0.266962\pi\)
0.668440 + 0.743766i \(0.266962\pi\)
\(228\) 4.54416 0.300944
\(229\) − 20.4853i − 1.35371i −0.736118 0.676853i \(-0.763343\pi\)
0.736118 0.676853i \(-0.236657\pi\)
\(230\) 3.17157i 0.209127i
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) −4.31371 −0.282600 −0.141300 0.989967i \(-0.545128\pi\)
−0.141300 + 0.989967i \(0.545128\pi\)
\(234\) 2.14214i 0.140036i
\(235\) 5.24264i 0.341992i
\(236\) 14.0000 0.911322
\(237\) 0.171573 0.0111449
\(238\) 5.65685 0.366679
\(239\) −8.34315 −0.539673 −0.269837 0.962906i \(-0.586970\pi\)
−0.269837 + 0.962906i \(0.586970\pi\)
\(240\) 1.24264i 0.0802121i
\(241\) −4.31371 −0.277870 −0.138935 0.990301i \(-0.544368\pi\)
−0.138935 + 0.990301i \(0.544368\pi\)
\(242\) − 2.14214i − 0.137702i
\(243\) 10.3431i 0.663513i
\(244\) − 1.51472i − 0.0969699i
\(245\) 1.00000 0.0638877
\(246\) 2.14214i 0.136578i
\(247\) 10.9706i 0.698040i
\(248\) 6.45584 0.409947
\(249\) − 1.51472i − 0.0959914i
\(250\) 3.72792i 0.235774i
\(251\) − 5.92893i − 0.374231i −0.982338 0.187115i \(-0.940086\pi\)
0.982338 0.187115i \(-0.0599138\pi\)
\(252\) 14.6274 0.921441
\(253\) 18.4853i 1.16216i
\(254\) 6.48528 0.406923
\(255\) −2.00000 −0.125245
\(256\) 3.97056 0.248160
\(257\) −23.8284 −1.48638 −0.743188 0.669082i \(-0.766687\pi\)
−0.743188 + 0.669082i \(0.766687\pi\)
\(258\) − 1.10051i − 0.0685145i
\(259\) − 11.3137i − 0.703000i
\(260\) −3.34315 −0.207333
\(261\) 0 0
\(262\) −0.544156 −0.0336181
\(263\) − 11.2426i − 0.693251i −0.938004 0.346625i \(-0.887327\pi\)
0.938004 0.346625i \(-0.112673\pi\)
\(264\) − 1.58579i − 0.0975984i
\(265\) −7.48528 −0.459817
\(266\) −7.02944 −0.431002
\(267\) 1.85786 0.113699
\(268\) 10.3431 0.631808
\(269\) − 19.4558i − 1.18624i −0.805113 0.593122i \(-0.797895\pi\)
0.805113 0.593122i \(-0.202105\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 14.5563i − 0.884235i −0.896957 0.442118i \(-0.854227\pi\)
0.896957 0.442118i \(-0.145773\pi\)
\(272\) 14.4853i 0.878299i
\(273\) − 2.14214i − 0.129648i
\(274\) −4.97056 −0.300283
\(275\) 9.65685i 0.582330i
\(276\) − 5.79899i − 0.349058i
\(277\) 5.31371 0.319270 0.159635 0.987176i \(-0.448968\pi\)
0.159635 + 0.987176i \(0.448968\pi\)
\(278\) − 5.79899i − 0.347800i
\(279\) 11.5147i 0.689368i
\(280\) − 4.48528i − 0.268047i
\(281\) −1.97056 −0.117554 −0.0587770 0.998271i \(-0.518720\pi\)
−0.0587770 + 0.998271i \(0.518720\pi\)
\(282\) 0.899495i 0.0535641i
\(283\) −0.343146 −0.0203979 −0.0101989 0.999948i \(-0.503246\pi\)
−0.0101989 + 0.999948i \(0.503246\pi\)
\(284\) 5.79899 0.344107
\(285\) 2.48528 0.147215
\(286\) 1.82843 0.108117
\(287\) 35.3137i 2.08450i
\(288\) − 12.4853i − 0.735702i
\(289\) −6.31371 −0.371395
\(290\) 0 0
\(291\) 5.17157 0.303163
\(292\) 7.31371i 0.428002i
\(293\) 3.65685i 0.213636i 0.994279 + 0.106818i \(0.0340662\pi\)
−0.994279 + 0.106818i \(0.965934\pi\)
\(294\) 0.171573 0.0100063
\(295\) 7.65685 0.445799
\(296\) −6.34315 −0.368688
\(297\) 5.82843 0.338200
\(298\) − 0.899495i − 0.0521063i
\(299\) 14.0000 0.809641
\(300\) − 3.02944i − 0.174905i
\(301\) − 18.1421i − 1.04570i
\(302\) 5.85786i 0.337082i
\(303\) −5.65685 −0.324978
\(304\) − 18.0000i − 1.03237i
\(305\) − 0.828427i − 0.0474356i
\(306\) 5.65685 0.323381
\(307\) 16.8995i 0.964505i 0.876032 + 0.482253i \(0.160181\pi\)
−0.876032 + 0.482253i \(0.839819\pi\)
\(308\) − 12.4853i − 0.711415i
\(309\) 0.343146i 0.0195209i
\(310\) 1.68629 0.0957749
\(311\) − 25.3137i − 1.43541i −0.696348 0.717704i \(-0.745193\pi\)
0.696348 0.717704i \(-0.254807\pi\)
\(312\) −1.20101 −0.0679938
\(313\) 4.17157 0.235791 0.117896 0.993026i \(-0.462385\pi\)
0.117896 + 0.993026i \(0.462385\pi\)
\(314\) −3.51472 −0.198347
\(315\) 8.00000 0.450749
\(316\) − 0.757359i − 0.0426048i
\(317\) 19.4558i 1.09275i 0.837541 + 0.546375i \(0.183992\pi\)
−0.837541 + 0.546375i \(0.816008\pi\)
\(318\) −1.28427 −0.0720184
\(319\) 0 0
\(320\) 4.17157 0.233198
\(321\) − 3.79899i − 0.212039i
\(322\) 8.97056i 0.499910i
\(323\) 28.9706 1.61197
\(324\) 13.6863 0.760350
\(325\) 7.31371 0.405692
\(326\) 7.48528 0.414571
\(327\) − 0.556349i − 0.0307662i
\(328\) 19.7990 1.09322
\(329\) 14.8284i 0.817518i
\(330\) − 0.414214i − 0.0228017i
\(331\) 0.414214i 0.0227672i 0.999935 + 0.0113836i \(0.00362360\pi\)
−0.999935 + 0.0113836i \(0.996376\pi\)
\(332\) −6.68629 −0.366958
\(333\) − 11.3137i − 0.619987i
\(334\) − 3.65685i − 0.200094i
\(335\) 5.65685 0.309067
\(336\) 3.51472i 0.191744i
\(337\) − 17.7990i − 0.969573i −0.874633 0.484786i \(-0.838897\pi\)
0.874633 0.484786i \(-0.161103\pi\)
\(338\) 4.00000i 0.217571i
\(339\) −3.85786 −0.209530
\(340\) 8.82843i 0.478789i
\(341\) 9.82843 0.532239
\(342\) −7.02944 −0.380108
\(343\) −16.9706 −0.916324
\(344\) −10.1716 −0.548414
\(345\) − 3.17157i − 0.170752i
\(346\) 9.79899i 0.526797i
\(347\) 14.4853 0.777611 0.388805 0.921320i \(-0.372888\pi\)
0.388805 + 0.921320i \(0.372888\pi\)
\(348\) 0 0
\(349\) 23.1421 1.23877 0.619385 0.785087i \(-0.287382\pi\)
0.619385 + 0.785087i \(0.287382\pi\)
\(350\) 4.68629i 0.250493i
\(351\) − 4.41421i − 0.235613i
\(352\) −10.6569 −0.568012
\(353\) 6.97056 0.371006 0.185503 0.982644i \(-0.440609\pi\)
0.185503 + 0.982644i \(0.440609\pi\)
\(354\) 1.31371 0.0698228
\(355\) 3.17157 0.168330
\(356\) − 8.20101i − 0.434653i
\(357\) −5.65685 −0.299392
\(358\) 4.34315i 0.229542i
\(359\) − 18.0711i − 0.953754i −0.878970 0.476877i \(-0.841769\pi\)
0.878970 0.476877i \(-0.158231\pi\)
\(360\) − 4.48528i − 0.236395i
\(361\) −17.0000 −0.894737
\(362\) 5.92893i 0.311618i
\(363\) 2.14214i 0.112433i
\(364\) −9.45584 −0.495621
\(365\) 4.00000i 0.209370i
\(366\) − 0.142136i − 0.00742955i
\(367\) − 18.0000i − 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) −22.9706 −1.19742
\(369\) 35.3137i 1.83836i
\(370\) −1.65685 −0.0861358
\(371\) −21.1716 −1.09917
\(372\) −3.08326 −0.159860
\(373\) −3.68629 −0.190869 −0.0954345 0.995436i \(-0.530424\pi\)
−0.0954345 + 0.995436i \(0.530424\pi\)
\(374\) − 4.82843i − 0.249672i
\(375\) − 3.72792i − 0.192509i
\(376\) 8.31371 0.428747
\(377\) 0 0
\(378\) 2.82843 0.145479
\(379\) − 26.9706i − 1.38538i −0.721233 0.692692i \(-0.756424\pi\)
0.721233 0.692692i \(-0.243576\pi\)
\(380\) − 10.9706i − 0.562778i
\(381\) −6.48528 −0.332251
\(382\) −1.11270 −0.0569306
\(383\) 20.4853 1.04675 0.523374 0.852103i \(-0.324673\pi\)
0.523374 + 0.852103i \(0.324673\pi\)
\(384\) 4.37258 0.223137
\(385\) − 6.82843i − 0.348009i
\(386\) 4.48528 0.228295
\(387\) − 18.1421i − 0.922217i
\(388\) − 22.8284i − 1.15894i
\(389\) 36.9706i 1.87448i 0.348682 + 0.937241i \(0.386629\pi\)
−0.348682 + 0.937241i \(0.613371\pi\)
\(390\) −0.313708 −0.0158852
\(391\) − 36.9706i − 1.86968i
\(392\) − 1.58579i − 0.0800943i
\(393\) 0.544156 0.0274490
\(394\) − 0.828427i − 0.0417356i
\(395\) − 0.414214i − 0.0208413i
\(396\) − 12.4853i − 0.627409i
\(397\) 30.6569 1.53862 0.769312 0.638874i \(-0.220599\pi\)
0.769312 + 0.638874i \(0.220599\pi\)
\(398\) − 6.82843i − 0.342278i
\(399\) 7.02944 0.351912
\(400\) −12.0000 −0.600000
\(401\) −7.34315 −0.366699 −0.183350 0.983048i \(-0.558694\pi\)
−0.183350 + 0.983048i \(0.558694\pi\)
\(402\) 0.970563 0.0484073
\(403\) − 7.44365i − 0.370795i
\(404\) 24.9706i 1.24233i
\(405\) 7.48528 0.371947
\(406\) 0 0
\(407\) −9.65685 −0.478672
\(408\) 3.17157i 0.157016i
\(409\) − 14.9706i − 0.740247i −0.928983 0.370123i \(-0.879315\pi\)
0.928983 0.370123i \(-0.120685\pi\)
\(410\) 5.17157 0.255406
\(411\) 4.97056 0.245180
\(412\) 1.51472 0.0746248
\(413\) 21.6569 1.06566
\(414\) 8.97056i 0.440879i
\(415\) −3.65685 −0.179508
\(416\) 8.07107i 0.395717i
\(417\) 5.79899i 0.283978i
\(418\) 6.00000i 0.293470i
\(419\) 26.4853 1.29389 0.646945 0.762536i \(-0.276046\pi\)
0.646945 + 0.762536i \(0.276046\pi\)
\(420\) 2.14214i 0.104526i
\(421\) − 25.1127i − 1.22392i −0.790889 0.611959i \(-0.790382\pi\)
0.790889 0.611959i \(-0.209618\pi\)
\(422\) 7.20101 0.350540
\(423\) 14.8284i 0.720983i
\(424\) 11.8701i 0.576461i
\(425\) − 19.3137i − 0.936852i
\(426\) 0.544156 0.0263644
\(427\) − 2.34315i − 0.113393i
\(428\) −16.7696 −0.810587
\(429\) −1.82843 −0.0882773
\(430\) −2.65685 −0.128125
\(431\) 8.34315 0.401875 0.200938 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(432\) 7.24264i 0.348462i
\(433\) − 14.6274i − 0.702949i −0.936197 0.351474i \(-0.885680\pi\)
0.936197 0.351474i \(-0.114320\pi\)
\(434\) 4.76955 0.228946
\(435\) 0 0
\(436\) −2.45584 −0.117614
\(437\) 45.9411i 2.19766i
\(438\) 0.686292i 0.0327923i
\(439\) 11.6569 0.556351 0.278176 0.960530i \(-0.410270\pi\)
0.278176 + 0.960530i \(0.410270\pi\)
\(440\) −3.82843 −0.182513
\(441\) 2.82843 0.134687
\(442\) −3.65685 −0.173939
\(443\) − 35.6569i − 1.69411i −0.531507 0.847054i \(-0.678374\pi\)
0.531507 0.847054i \(-0.321626\pi\)
\(444\) 3.02944 0.143771
\(445\) − 4.48528i − 0.212623i
\(446\) 3.65685i 0.173157i
\(447\) 0.899495i 0.0425447i
\(448\) 11.7990 0.557450
\(449\) 1.02944i 0.0485821i 0.999705 + 0.0242911i \(0.00773285\pi\)
−0.999705 + 0.0242911i \(0.992267\pi\)
\(450\) 4.68629i 0.220914i
\(451\) 30.1421 1.41934
\(452\) 17.0294i 0.800997i
\(453\) − 5.85786i − 0.275226i
\(454\) − 8.34315i − 0.391563i
\(455\) −5.17157 −0.242447
\(456\) − 3.94113i − 0.184560i
\(457\) −34.9706 −1.63585 −0.817927 0.575322i \(-0.804877\pi\)
−0.817927 + 0.575322i \(0.804877\pi\)
\(458\) −8.48528 −0.396491
\(459\) −11.6569 −0.544095
\(460\) −14.0000 −0.652753
\(461\) 14.0000i 0.652045i 0.945362 + 0.326023i \(0.105709\pi\)
−0.945362 + 0.326023i \(0.894291\pi\)
\(462\) − 1.17157i − 0.0545065i
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 0 0
\(465\) −1.68629 −0.0781999
\(466\) 1.78680i 0.0827718i
\(467\) − 32.3553i − 1.49723i −0.663007 0.748613i \(-0.730720\pi\)
0.663007 0.748613i \(-0.269280\pi\)
\(468\) −9.45584 −0.437097
\(469\) 16.0000 0.738811
\(470\) 2.17157 0.100167
\(471\) 3.51472 0.161950
\(472\) − 12.1421i − 0.558887i
\(473\) −15.4853 −0.712014
\(474\) − 0.0710678i − 0.00326425i
\(475\) 24.0000i 1.10120i
\(476\) 24.9706i 1.14452i
\(477\) −21.1716 −0.969380
\(478\) 3.45584i 0.158067i
\(479\) − 12.8995i − 0.589393i −0.955591 0.294696i \(-0.904781\pi\)
0.955591 0.294696i \(-0.0952185\pi\)
\(480\) 1.82843 0.0834559
\(481\) 7.31371i 0.333476i
\(482\) 1.78680i 0.0813864i
\(483\) − 8.97056i − 0.408175i
\(484\) 9.45584 0.429811
\(485\) − 12.4853i − 0.566927i
\(486\) 4.28427 0.194338
\(487\) −28.4853 −1.29079 −0.645396 0.763848i \(-0.723307\pi\)
−0.645396 + 0.763848i \(0.723307\pi\)
\(488\) −1.31371 −0.0594688
\(489\) −7.48528 −0.338496
\(490\) − 0.414214i − 0.0187123i
\(491\) − 12.7574i − 0.575732i −0.957671 0.287866i \(-0.907054\pi\)
0.957671 0.287866i \(-0.0929457\pi\)
\(492\) −9.45584 −0.426302
\(493\) 0 0
\(494\) 4.54416 0.204451
\(495\) − 6.82843i − 0.306915i
\(496\) 12.2132i 0.548389i
\(497\) 8.97056 0.402385
\(498\) −0.627417 −0.0281152
\(499\) 14.9706 0.670174 0.335087 0.942187i \(-0.391234\pi\)
0.335087 + 0.942187i \(0.391234\pi\)
\(500\) −16.4558 −0.735928
\(501\) 3.65685i 0.163376i
\(502\) −2.45584 −0.109610
\(503\) 25.7279i 1.14715i 0.819153 + 0.573576i \(0.194444\pi\)
−0.819153 + 0.573576i \(0.805556\pi\)
\(504\) − 12.6863i − 0.565092i
\(505\) 13.6569i 0.607722i
\(506\) 7.65685 0.340389
\(507\) − 4.00000i − 0.177646i
\(508\) 28.6274i 1.27014i
\(509\) −27.4853 −1.21826 −0.609132 0.793069i \(-0.708482\pi\)
−0.609132 + 0.793069i \(0.708482\pi\)
\(510\) 0.828427i 0.0366834i
\(511\) 11.3137i 0.500489i
\(512\) − 22.7574i − 1.00574i
\(513\) 14.4853 0.639541
\(514\) 9.87006i 0.435350i
\(515\) 0.828427 0.0365049
\(516\) 4.85786 0.213856
\(517\) 12.6569 0.556648
\(518\) −4.68629 −0.205904
\(519\) − 9.79899i − 0.430128i
\(520\) 2.89949i 0.127151i
\(521\) 0.857864 0.0375837 0.0187919 0.999823i \(-0.494018\pi\)
0.0187919 + 0.999823i \(0.494018\pi\)
\(522\) 0 0
\(523\) 27.3137 1.19435 0.597173 0.802113i \(-0.296291\pi\)
0.597173 + 0.802113i \(0.296291\pi\)
\(524\) − 2.40202i − 0.104933i
\(525\) − 4.68629i − 0.204527i
\(526\) −4.65685 −0.203048
\(527\) −19.6569 −0.856266
\(528\) 3.00000 0.130558
\(529\) 35.6274 1.54902
\(530\) 3.10051i 0.134677i
\(531\) 21.6569 0.939827
\(532\) − 31.0294i − 1.34530i
\(533\) − 22.8284i − 0.988809i
\(534\) − 0.769553i − 0.0333018i
\(535\) −9.17157 −0.396522
\(536\) − 8.97056i − 0.387469i
\(537\) − 4.34315i − 0.187421i
\(538\) −8.05887 −0.347443
\(539\) − 2.41421i − 0.103988i
\(540\) 4.41421i 0.189958i
\(541\) 21.6569i 0.931101i 0.885021 + 0.465550i \(0.154144\pi\)
−0.885021 + 0.465550i \(0.845856\pi\)
\(542\) −6.02944 −0.258987
\(543\) − 5.92893i − 0.254435i
\(544\) 21.3137 0.913818
\(545\) −1.34315 −0.0575340
\(546\) −0.887302 −0.0379730
\(547\) −3.79899 −0.162433 −0.0812165 0.996696i \(-0.525881\pi\)
−0.0812165 + 0.996696i \(0.525881\pi\)
\(548\) − 21.9411i − 0.937278i
\(549\) − 2.34315i − 0.100003i
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) −5.02944 −0.214067
\(553\) − 1.17157i − 0.0498203i
\(554\) − 2.20101i − 0.0935120i
\(555\) 1.65685 0.0703295
\(556\) 25.5980 1.08560
\(557\) −5.31371 −0.225149 −0.112575 0.993643i \(-0.535910\pi\)
−0.112575 + 0.993643i \(0.535910\pi\)
\(558\) 4.76955 0.201911
\(559\) 11.7279i 0.496038i
\(560\) 8.48528 0.358569
\(561\) 4.82843i 0.203856i
\(562\) 0.816234i 0.0344307i
\(563\) − 9.24264i − 0.389531i −0.980850 0.194765i \(-0.937605\pi\)
0.980850 0.194765i \(-0.0623946\pi\)
\(564\) −3.97056 −0.167191
\(565\) 9.31371i 0.391831i
\(566\) 0.142136i 0.00597441i
\(567\) 21.1716 0.889122
\(568\) − 5.02944i − 0.211030i
\(569\) − 28.3431i − 1.18821i −0.804389 0.594103i \(-0.797507\pi\)
0.804389 0.594103i \(-0.202493\pi\)
\(570\) − 1.02944i − 0.0431184i
\(571\) −30.6274 −1.28172 −0.640859 0.767659i \(-0.721422\pi\)
−0.640859 + 0.767659i \(0.721422\pi\)
\(572\) 8.07107i 0.337468i
\(573\) 1.11270 0.0464836
\(574\) 14.6274 0.610537
\(575\) 30.6274 1.27725
\(576\) 11.7990 0.491625
\(577\) 9.79899i 0.407937i 0.978977 + 0.203969i \(0.0653841\pi\)
−0.978977 + 0.203969i \(0.934616\pi\)
\(578\) 2.61522i 0.108779i
\(579\) −4.48528 −0.186402
\(580\) 0 0
\(581\) −10.3431 −0.429106
\(582\) − 2.14214i − 0.0887944i
\(583\) 18.0711i 0.748427i
\(584\) 6.34315 0.262481
\(585\) −5.17157 −0.213818
\(586\) 1.51472 0.0625724
\(587\) −3.65685 −0.150935 −0.0754673 0.997148i \(-0.524045\pi\)
−0.0754673 + 0.997148i \(0.524045\pi\)
\(588\) 0.757359i 0.0312330i
\(589\) 24.4264 1.00647
\(590\) − 3.17157i − 0.130572i
\(591\) 0.828427i 0.0340769i
\(592\) − 12.0000i − 0.493197i
\(593\) 2.51472 0.103267 0.0516336 0.998666i \(-0.483557\pi\)
0.0516336 + 0.998666i \(0.483557\pi\)
\(594\) − 2.41421i − 0.0990564i
\(595\) 13.6569i 0.559876i
\(596\) 3.97056 0.162641
\(597\) 6.82843i 0.279469i
\(598\) − 5.79899i − 0.237138i
\(599\) 43.8701i 1.79248i 0.443567 + 0.896241i \(0.353713\pi\)
−0.443567 + 0.896241i \(0.646287\pi\)
\(600\) −2.62742 −0.107264
\(601\) 22.8284i 0.931191i 0.884998 + 0.465595i \(0.154160\pi\)
−0.884998 + 0.465595i \(0.845840\pi\)
\(602\) −7.51472 −0.306277
\(603\) 16.0000 0.651570
\(604\) −25.8579 −1.05214
\(605\) 5.17157 0.210254
\(606\) 2.34315i 0.0951838i
\(607\) 17.7279i 0.719554i 0.933038 + 0.359777i \(0.117147\pi\)
−0.933038 + 0.359777i \(0.882853\pi\)
\(608\) −26.4853 −1.07412
\(609\) 0 0
\(610\) −0.343146 −0.0138936
\(611\) − 9.58579i − 0.387799i
\(612\) 24.9706i 1.00938i
\(613\) 9.00000 0.363507 0.181753 0.983344i \(-0.441823\pi\)
0.181753 + 0.983344i \(0.441823\pi\)
\(614\) 7.00000 0.282497
\(615\) −5.17157 −0.208538
\(616\) −10.8284 −0.436290
\(617\) 23.3137i 0.938575i 0.883046 + 0.469287i \(0.155489\pi\)
−0.883046 + 0.469287i \(0.844511\pi\)
\(618\) 0.142136 0.00571753
\(619\) 36.4142i 1.46361i 0.681514 + 0.731805i \(0.261322\pi\)
−0.681514 + 0.731805i \(0.738678\pi\)
\(620\) 7.44365i 0.298944i
\(621\) − 18.4853i − 0.741789i
\(622\) −10.4853 −0.420421
\(623\) − 12.6863i − 0.508266i
\(624\) − 2.27208i − 0.0909559i
\(625\) 11.0000 0.440000
\(626\) − 1.72792i − 0.0690617i
\(627\) − 6.00000i − 0.239617i
\(628\) − 15.5147i − 0.619105i
\(629\) 19.3137 0.770088
\(630\) − 3.31371i − 0.132021i
\(631\) 31.1716 1.24092 0.620460 0.784238i \(-0.286946\pi\)
0.620460 + 0.784238i \(0.286946\pi\)
\(632\) −0.656854 −0.0261283
\(633\) −7.20101 −0.286214
\(634\) 8.05887 0.320059
\(635\) 15.6569i 0.621323i
\(636\) − 5.66905i − 0.224792i
\(637\) −1.82843 −0.0724449
\(638\) 0 0
\(639\) 8.97056 0.354870
\(640\) − 10.5563i − 0.417276i
\(641\) 21.7990i 0.861008i 0.902589 + 0.430504i \(0.141664\pi\)
−0.902589 + 0.430504i \(0.858336\pi\)
\(642\) −1.57359 −0.0621048
\(643\) −15.5147 −0.611841 −0.305920 0.952057i \(-0.598964\pi\)
−0.305920 + 0.952057i \(0.598964\pi\)
\(644\) −39.5980 −1.56038
\(645\) 2.65685 0.104614
\(646\) − 12.0000i − 0.472134i
\(647\) −28.3431 −1.11428 −0.557142 0.830417i \(-0.688102\pi\)
−0.557142 + 0.830417i \(0.688102\pi\)
\(648\) − 11.8701i − 0.466300i
\(649\) − 18.4853i − 0.725611i
\(650\) − 3.02944i − 0.118824i
\(651\) −4.76955 −0.186934
\(652\) 33.0416i 1.29401i
\(653\) − 1.85786i − 0.0727039i −0.999339 0.0363519i \(-0.988426\pi\)
0.999339 0.0363519i \(-0.0115737\pi\)
\(654\) −0.230447 −0.00901121
\(655\) − 1.31371i − 0.0513308i
\(656\) 37.4558i 1.46241i
\(657\) 11.3137i 0.441390i
\(658\) 6.14214 0.239445
\(659\) − 11.5858i − 0.451318i −0.974206 0.225659i \(-0.927546\pi\)
0.974206 0.225659i \(-0.0724535\pi\)
\(660\) 1.82843 0.0711714
\(661\) 10.6863 0.415649 0.207824 0.978166i \(-0.433362\pi\)
0.207824 + 0.978166i \(0.433362\pi\)
\(662\) 0.171573 0.00666837
\(663\) 3.65685 0.142020
\(664\) 5.79899i 0.225044i
\(665\) − 16.9706i − 0.658090i
\(666\) −4.68629 −0.181590
\(667\) 0 0
\(668\) 16.1421 0.624558
\(669\) − 3.65685i − 0.141382i
\(670\) − 2.34315i − 0.0905236i
\(671\) −2.00000 −0.0772091
\(672\) 5.17157 0.199498
\(673\) −23.6274 −0.910770 −0.455385 0.890295i \(-0.650498\pi\)
−0.455385 + 0.890295i \(0.650498\pi\)
\(674\) −7.37258 −0.283981
\(675\) − 9.65685i − 0.371692i
\(676\) −17.6569 −0.679110
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 1.59798i 0.0613700i
\(679\) − 35.3137i − 1.35522i
\(680\) 7.65685 0.293627
\(681\) 8.34315i 0.319710i
\(682\) − 4.07107i − 0.155889i
\(683\) −12.9706 −0.496305 −0.248152 0.968721i \(-0.579823\pi\)
−0.248152 + 0.968721i \(0.579823\pi\)
\(684\) − 31.0294i − 1.18644i
\(685\) − 12.0000i − 0.458496i
\(686\) 7.02944i 0.268385i
\(687\) 8.48528 0.323734
\(688\) − 19.2426i − 0.733619i
\(689\) 13.6863 0.521406
\(690\) −1.31371 −0.0500120
\(691\) 48.0000 1.82601 0.913003 0.407953i \(-0.133757\pi\)
0.913003 + 0.407953i \(0.133757\pi\)
\(692\) −43.2548 −1.64430
\(693\) − 19.3137i − 0.733667i
\(694\) − 6.00000i − 0.227757i
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) −60.2843 −2.28343
\(698\) − 9.58579i − 0.362827i
\(699\) − 1.78680i − 0.0675829i
\(700\) −20.6863 −0.781868
\(701\) −22.1127 −0.835185 −0.417593 0.908634i \(-0.637126\pi\)
−0.417593 + 0.908634i \(0.637126\pi\)
\(702\) −1.82843 −0.0690095
\(703\) −24.0000 −0.905177
\(704\) − 10.0711i − 0.379568i
\(705\) −2.17157 −0.0817862
\(706\) − 2.88730i − 0.108665i
\(707\) 38.6274i 1.45273i
\(708\) 5.79899i 0.217939i
\(709\) −0.857864 −0.0322178 −0.0161089 0.999870i \(-0.505128\pi\)
−0.0161089 + 0.999870i \(0.505128\pi\)
\(710\) − 1.31371i − 0.0493026i
\(711\) − 1.17157i − 0.0439374i
\(712\) −7.11270 −0.266560
\(713\) − 31.1716i − 1.16738i
\(714\) 2.34315i 0.0876900i
\(715\) 4.41421i 0.165082i
\(716\) −19.1716 −0.716475
\(717\) − 3.45584i − 0.129061i
\(718\) −7.48528 −0.279348
\(719\) 8.14214 0.303650 0.151825 0.988407i \(-0.451485\pi\)
0.151825 + 0.988407i \(0.451485\pi\)
\(720\) 8.48528 0.316228
\(721\) 2.34315 0.0872633
\(722\) 7.04163i 0.262062i
\(723\) − 1.78680i − 0.0664517i
\(724\) −26.1716 −0.972659
\(725\) 0 0
\(726\) 0.887302 0.0329309
\(727\) 21.3137i 0.790482i 0.918578 + 0.395241i \(0.129339\pi\)
−0.918578 + 0.395241i \(0.870661\pi\)
\(728\) 8.20101i 0.303950i
\(729\) 18.1716 0.673021
\(730\) 1.65685 0.0613229
\(731\) 30.9706 1.14549
\(732\) 0.627417 0.0231900
\(733\) 49.2548i 1.81927i 0.415410 + 0.909634i \(0.363638\pi\)
−0.415410 + 0.909634i \(0.636362\pi\)
\(734\) −7.45584 −0.275200
\(735\) 0.414214i 0.0152785i
\(736\) 33.7990i 1.24585i
\(737\) − 13.6569i − 0.503057i
\(738\) 14.6274 0.538443
\(739\) 10.0711i 0.370470i 0.982694 + 0.185235i \(0.0593047\pi\)
−0.982694 + 0.185235i \(0.940695\pi\)
\(740\) − 7.31371i − 0.268857i
\(741\) −4.54416 −0.166934
\(742\) 8.76955i 0.321940i
\(743\) 12.3431i 0.452826i 0.974031 + 0.226413i \(0.0726999\pi\)
−0.974031 + 0.226413i \(0.927300\pi\)
\(744\) 2.67410i 0.0980372i
\(745\) 2.17157 0.0795603
\(746\) 1.52691i 0.0559042i
\(747\) −10.3431 −0.378436
\(748\) 21.3137 0.779306
\(749\) −25.9411 −0.947868
\(750\) −1.54416 −0.0563846
\(751\) 2.68629i 0.0980242i 0.998798 + 0.0490121i \(0.0156073\pi\)
−0.998798 + 0.0490121i \(0.984393\pi\)
\(752\) 15.7279i 0.573538i
\(753\) 2.45584 0.0894959
\(754\) 0 0
\(755\) −14.1421 −0.514685
\(756\) 12.4853i 0.454085i
\(757\) − 42.4853i − 1.54415i −0.635529 0.772077i \(-0.719218\pi\)
0.635529 0.772077i \(-0.280782\pi\)
\(758\) −11.1716 −0.405770
\(759\) −7.65685 −0.277926
\(760\) −9.51472 −0.345135
\(761\) −33.5980 −1.21793 −0.608963 0.793199i \(-0.708414\pi\)
−0.608963 + 0.793199i \(0.708414\pi\)
\(762\) 2.68629i 0.0973141i
\(763\) −3.79899 −0.137533
\(764\) − 4.91169i − 0.177699i
\(765\) 13.6569i 0.493765i
\(766\) − 8.48528i − 0.306586i
\(767\) −14.0000 −0.505511
\(768\) 1.64466i 0.0593466i
\(769\) 13.1127i 0.472856i 0.971649 + 0.236428i \(0.0759767\pi\)
−0.971649 + 0.236428i \(0.924023\pi\)
\(770\) −2.82843 −0.101929
\(771\) − 9.87006i − 0.355461i
\(772\) 19.7990i 0.712581i
\(773\) 36.4853i 1.31228i 0.754637 + 0.656142i \(0.227813\pi\)
−0.754637 + 0.656142i \(0.772187\pi\)
\(774\) −7.51472 −0.270111
\(775\) − 16.2843i − 0.584948i
\(776\) −19.7990 −0.710742
\(777\) 4.68629 0.168120
\(778\) 15.3137 0.549023
\(779\) 74.9117 2.68399
\(780\) − 1.38478i − 0.0495829i
\(781\) − 7.65685i − 0.273984i
\(782\) −15.3137 −0.547617
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) − 8.48528i − 0.302853i
\(786\) − 0.225397i − 0.00803964i
\(787\) −42.0833 −1.50011 −0.750053 0.661378i \(-0.769972\pi\)
−0.750053 + 0.661378i \(0.769972\pi\)
\(788\) 3.65685 0.130270
\(789\) 4.65685 0.165788
\(790\) −0.171573 −0.00610429
\(791\) 26.3431i 0.936654i
\(792\) −10.8284 −0.384771
\(793\) 1.51472i 0.0537892i
\(794\) − 12.6985i − 0.450652i
\(795\) − 3.10051i − 0.109964i
\(796\) 30.1421 1.06836
\(797\) 55.7401i 1.97442i 0.159437 + 0.987208i \(0.449032\pi\)
−0.159437 + 0.987208i \(0.550968\pi\)
\(798\) − 2.91169i − 0.103073i
\(799\) −25.3137 −0.895535
\(800\) 17.6569i 0.624264i
\(801\) − 12.6863i − 0.448248i
\(802\) 3.04163i 0.107404i
\(803\) 9.65685 0.340783
\(804\) 4.28427i 0.151095i
\(805\) −21.6569 −0.763304
\(806\) −3.08326 −0.108603
\(807\) 8.05887 0.283686
\(808\) 21.6569 0.761885
\(809\) − 20.2843i − 0.713157i −0.934265 0.356578i \(-0.883943\pi\)
0.934265 0.356578i \(-0.116057\pi\)
\(810\) − 3.10051i − 0.108941i
\(811\) −5.17157 −0.181598 −0.0907992 0.995869i \(-0.528942\pi\)
−0.0907992 + 0.995869i \(0.528942\pi\)
\(812\) 0 0
\(813\) 6.02944 0.211462
\(814\) 4.00000i 0.140200i
\(815\) 18.0711i 0.633002i
\(816\) −6.00000 −0.210042
\(817\) −38.4853 −1.34643
\(818\) −6.20101 −0.216813
\(819\) −14.6274 −0.511123
\(820\) 22.8284i 0.797203i
\(821\) −15.4853 −0.540440 −0.270220 0.962799i \(-0.587096\pi\)
−0.270220 + 0.962799i \(0.587096\pi\)
\(822\) − 2.05887i − 0.0718115i
\(823\) − 2.28427i − 0.0796247i −0.999207 0.0398123i \(-0.987324\pi\)
0.999207 0.0398123i \(-0.0126760\pi\)
\(824\) − 1.31371i − 0.0457652i
\(825\) −4.00000 −0.139262
\(826\) − 8.97056i − 0.312126i
\(827\) 13.1005i 0.455549i 0.973714 + 0.227775i \(0.0731449\pi\)
−0.973714 + 0.227775i \(0.926855\pi\)
\(828\) −39.5980 −1.37612
\(829\) − 9.79899i − 0.340333i −0.985415 0.170166i \(-0.945569\pi\)
0.985415 0.170166i \(-0.0544306\pi\)
\(830\) 1.51472i 0.0525767i
\(831\) 2.20101i 0.0763522i
\(832\) −7.62742 −0.264433
\(833\) 4.82843i 0.167295i
\(834\) 2.40202 0.0831752
\(835\) 8.82843 0.305520
\(836\) −26.4853 −0.916013
\(837\) −9.82843 −0.339720
\(838\) − 10.9706i − 0.378972i
\(839\) − 22.0711i − 0.761978i −0.924580 0.380989i \(-0.875584\pi\)
0.924580 0.380989i \(-0.124416\pi\)
\(840\) 1.85786 0.0641024
\(841\) 0 0
\(842\) −10.4020 −0.358477
\(843\) − 0.816234i − 0.0281126i
\(844\) 31.7868i 1.09415i
\(845\) −9.65685 −0.332206
\(846\) 6.14214 0.211171
\(847\) 14.6274 0.502604
\(848\) −22.4558 −0.771137
\(849\) − 0.142136i − 0.00487808i
\(850\) −8.00000 −0.274398
\(851\) 30.6274i 1.04989i
\(852\) 2.40202i 0.0822919i
\(853\) 10.9706i 0.375625i 0.982205 + 0.187812i \(0.0601397\pi\)
−0.982205 + 0.187812i \(0.939860\pi\)
\(854\) −0.970563 −0.0332120
\(855\) − 16.9706i − 0.580381i
\(856\) 14.5442i 0.497109i
\(857\) −11.8284 −0.404051 −0.202026 0.979380i \(-0.564752\pi\)
−0.202026 + 0.979380i \(0.564752\pi\)
\(858\) 0.757359i 0.0258558i
\(859\) − 5.72792i − 0.195434i −0.995214 0.0977171i \(-0.968846\pi\)
0.995214 0.0977171i \(-0.0311540\pi\)
\(860\) − 11.7279i − 0.399919i
\(861\) −14.6274 −0.498501
\(862\) − 3.45584i − 0.117707i
\(863\) 45.1127 1.53565 0.767827 0.640657i \(-0.221338\pi\)
0.767827 + 0.640657i \(0.221338\pi\)
\(864\) 10.6569 0.362554
\(865\) −23.6569 −0.804357
\(866\) −6.05887 −0.205889
\(867\) − 2.61522i − 0.0888177i
\(868\) 21.0538i 0.714613i
\(869\) −1.00000 −0.0339227
\(870\) 0 0
\(871\) −10.3431 −0.350464
\(872\) 2.12994i 0.0721289i
\(873\) − 35.3137i − 1.19519i
\(874\) 19.0294 0.643680
\(875\) −25.4558 −0.860565
\(876\) −3.02944 −0.102355
\(877\) −8.85786 −0.299109 −0.149554 0.988753i \(-0.547784\pi\)
−0.149554 + 0.988753i \(0.547784\pi\)
\(878\) − 4.82843i − 0.162952i
\(879\) −1.51472 −0.0510902
\(880\) − 7.24264i − 0.244149i
\(881\) − 14.0000i − 0.471672i −0.971793 0.235836i \(-0.924217\pi\)
0.971793 0.235836i \(-0.0757828\pi\)
\(882\) − 1.17157i − 0.0394489i
\(883\) 46.4264 1.56237 0.781186 0.624298i \(-0.214615\pi\)
0.781186 + 0.624298i \(0.214615\pi\)
\(884\) − 16.1421i − 0.542919i
\(885\) 3.17157i 0.106611i
\(886\) −14.7696 −0.496193
\(887\) − 36.8995i − 1.23896i −0.785011 0.619482i \(-0.787343\pi\)
0.785011 0.619482i \(-0.212657\pi\)
\(888\) − 2.62742i − 0.0881703i
\(889\) 44.2843i 1.48525i
\(890\) −1.85786 −0.0622758
\(891\) − 18.0711i − 0.605404i
\(892\) −16.1421 −0.540479
\(893\) 31.4558 1.05263
\(894\) 0.372583 0.0124610
\(895\) −10.4853 −0.350484
\(896\) − 29.8579i − 0.997481i
\(897\) 5.79899i 0.193623i
\(898\) 0.426407 0.0142294
\(899\) 0 0
\(900\) −20.6863 −0.689543
\(901\) − 36.1421i − 1.20407i
\(902\) − 12.4853i − 0.415714i
\(903\) 7.51472 0.250074
\(904\) 14.7696 0.491228
\(905\) −14.3137 −0.475804
\(906\) −2.42641 −0.0806120
\(907\) − 34.2843i − 1.13839i −0.822202 0.569195i \(-0.807255\pi\)
0.822202 0.569195i \(-0.192745\pi\)
\(908\) 36.8284 1.22219
\(909\) 38.6274i 1.28119i
\(910\) 2.14214i 0.0710111i
\(911\) − 46.5563i − 1.54248i −0.636544 0.771240i \(-0.719637\pi\)
0.636544 0.771240i \(-0.280363\pi\)
\(912\) 7.45584 0.246888
\(913\) 8.82843i 0.292178i
\(914\) 14.4853i 0.479131i
\(915\) 0.343146 0.0113440
\(916\) − 37.4558i − 1.23758i
\(917\) − 3.71573i − 0.122704i
\(918\) 4.82843i 0.159362i
\(919\) −20.1421 −0.664428 −0.332214 0.943204i \(-0.607796\pi\)
−0.332214 + 0.943204i \(0.607796\pi\)
\(920\) 12.1421i 0.400314i
\(921\) −7.00000 −0.230658
\(922\) 5.79899 0.190980
\(923\) −5.79899 −0.190876
\(924\) 5.17157 0.170132
\(925\) 16.0000i 0.526077i
\(926\) − 10.7696i − 0.353909i
\(927\) 2.34315 0.0769590
\(928\) 0 0
\(929\) 41.3137 1.35546 0.677729 0.735311i \(-0.262964\pi\)
0.677729 + 0.735311i \(0.262964\pi\)
\(930\) 0.698485i 0.0229042i
\(931\) − 6.00000i − 0.196642i
\(932\) −7.88730 −0.258357
\(933\) 10.4853 0.343273
\(934\) −13.4020 −0.438527
\(935\) 11.6569 0.381220
\(936\) 8.20101i 0.268058i
\(937\) −28.6274 −0.935217 −0.467608 0.883936i \(-0.654884\pi\)
−0.467608 + 0.883936i \(0.654884\pi\)
\(938\) − 6.62742i − 0.216393i
\(939\) 1.72792i 0.0563886i
\(940\) 9.58579i 0.312654i
\(941\) −22.5980 −0.736673 −0.368337 0.929693i \(-0.620073\pi\)
−0.368337 + 0.929693i \(0.620073\pi\)
\(942\) − 1.45584i − 0.0474340i
\(943\) − 95.5980i − 3.11310i
\(944\) 22.9706 0.747628
\(945\) 6.82843i 0.222129i
\(946\) 6.41421i 0.208544i
\(947\) − 39.3848i − 1.27983i −0.768444 0.639917i \(-0.778969\pi\)
0.768444 0.639917i \(-0.221031\pi\)
\(948\) 0.313708 0.0101888
\(949\) − 7.31371i − 0.237413i
\(950\) 9.94113 0.322533
\(951\) −8.05887 −0.261327
\(952\) 21.6569 0.701903
\(953\) 9.62742 0.311863 0.155931 0.987768i \(-0.450162\pi\)
0.155931 + 0.987768i \(0.450162\pi\)
\(954\) 8.76955i 0.283925i
\(955\) − 2.68629i − 0.0869264i
\(956\) −15.2548 −0.493377
\(957\) 0 0
\(958\) −5.34315 −0.172629
\(959\) − 33.9411i − 1.09602i
\(960\) 1.72792i 0.0557684i
\(961\) 14.4264 0.465368
\(962\) 3.02944 0.0976730
\(963\) −25.9411 −0.835941
\(964\) −7.88730 −0.254033
\(965\) 10.8284i 0.348579i
\(966\) −3.71573 −0.119552
\(967\) − 26.7574i − 0.860459i −0.902720 0.430229i \(-0.858433\pi\)
0.902720 0.430229i \(-0.141567\pi\)
\(968\) − 8.20101i − 0.263590i
\(969\) 12.0000i 0.385496i
\(970\) −5.17157 −0.166049
\(971\) 4.34315i 0.139378i 0.997569 + 0.0696891i \(0.0222007\pi\)
−0.997569 + 0.0696891i \(0.977799\pi\)
\(972\) 18.9117i 0.606593i
\(973\) 39.5980 1.26945
\(974\) 11.7990i 0.378064i
\(975\) 3.02944i 0.0970196i
\(976\) − 2.48528i − 0.0795519i
\(977\) 41.8284 1.33821 0.669105 0.743168i \(-0.266678\pi\)
0.669105 + 0.743168i \(0.266678\pi\)
\(978\) 3.10051i 0.0991432i
\(979\) −10.8284 −0.346078
\(980\) 1.82843 0.0584070
\(981\) −3.79899 −0.121292
\(982\) −5.28427 −0.168628
\(983\) 31.8701i 1.01650i 0.861210 + 0.508248i \(0.169707\pi\)
−0.861210 + 0.508248i \(0.830293\pi\)
\(984\) 8.20101i 0.261439i
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) −6.14214 −0.195506
\(988\) 20.0589i 0.638158i
\(989\) 49.1127i 1.56169i
\(990\) −2.82843 −0.0898933
\(991\) 7.17157 0.227813 0.113906 0.993492i \(-0.463664\pi\)
0.113906 + 0.993492i \(0.463664\pi\)
\(992\) 17.9706 0.570566
\(993\) −0.171573 −0.00544470
\(994\) − 3.71573i − 0.117856i
\(995\) 16.4853 0.522619
\(996\) − 2.76955i − 0.0877566i
\(997\) 28.2843i 0.895772i 0.894091 + 0.447886i \(0.147823\pi\)
−0.894091 + 0.447886i \(0.852177\pi\)
\(998\) − 6.20101i − 0.196290i
\(999\) 9.65685 0.305529
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 841.2.b.a.840.2 4
29.2 odd 28 841.2.d.f.605.2 12
29.3 odd 28 841.2.d.f.571.2 12
29.4 even 14 841.2.e.k.651.3 24
29.5 even 14 841.2.e.k.236.2 24
29.6 even 14 841.2.e.k.196.3 24
29.7 even 7 841.2.e.k.270.3 24
29.8 odd 28 841.2.d.j.574.2 12
29.9 even 14 841.2.e.k.267.3 24
29.10 odd 28 841.2.d.j.190.1 12
29.11 odd 28 841.2.d.f.778.1 12
29.12 odd 4 29.2.a.a.1.2 2
29.13 even 14 841.2.e.k.63.2 24
29.14 odd 28 841.2.d.f.645.2 12
29.15 odd 28 841.2.d.j.645.1 12
29.16 even 7 841.2.e.k.63.3 24
29.17 odd 4 841.2.a.d.1.1 2
29.18 odd 28 841.2.d.j.778.2 12
29.19 odd 28 841.2.d.f.190.2 12
29.20 even 7 841.2.e.k.267.2 24
29.21 odd 28 841.2.d.f.574.1 12
29.22 even 14 841.2.e.k.270.2 24
29.23 even 7 841.2.e.k.196.2 24
29.24 even 7 841.2.e.k.236.3 24
29.25 even 7 841.2.e.k.651.2 24
29.26 odd 28 841.2.d.j.571.1 12
29.27 odd 28 841.2.d.j.605.1 12
29.28 even 2 inner 841.2.b.a.840.3 4
87.17 even 4 7569.2.a.c.1.2 2
87.41 even 4 261.2.a.d.1.1 2
116.99 even 4 464.2.a.h.1.2 2
145.12 even 4 725.2.b.b.349.3 4
145.99 odd 4 725.2.a.b.1.1 2
145.128 even 4 725.2.b.b.349.2 4
203.41 even 4 1421.2.a.j.1.2 2
232.99 even 4 1856.2.a.w.1.1 2
232.157 odd 4 1856.2.a.r.1.2 2
319.186 even 4 3509.2.a.j.1.1 2
348.215 odd 4 4176.2.a.bq.1.1 2
377.12 odd 4 4901.2.a.g.1.1 2
435.389 even 4 6525.2.a.o.1.2 2
493.186 odd 4 8381.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.2.a.a.1.2 2 29.12 odd 4
261.2.a.d.1.1 2 87.41 even 4
464.2.a.h.1.2 2 116.99 even 4
725.2.a.b.1.1 2 145.99 odd 4
725.2.b.b.349.2 4 145.128 even 4
725.2.b.b.349.3 4 145.12 even 4
841.2.a.d.1.1 2 29.17 odd 4
841.2.b.a.840.2 4 1.1 even 1 trivial
841.2.b.a.840.3 4 29.28 even 2 inner
841.2.d.f.190.2 12 29.19 odd 28
841.2.d.f.571.2 12 29.3 odd 28
841.2.d.f.574.1 12 29.21 odd 28
841.2.d.f.605.2 12 29.2 odd 28
841.2.d.f.645.2 12 29.14 odd 28
841.2.d.f.778.1 12 29.11 odd 28
841.2.d.j.190.1 12 29.10 odd 28
841.2.d.j.571.1 12 29.26 odd 28
841.2.d.j.574.2 12 29.8 odd 28
841.2.d.j.605.1 12 29.27 odd 28
841.2.d.j.645.1 12 29.15 odd 28
841.2.d.j.778.2 12 29.18 odd 28
841.2.e.k.63.2 24 29.13 even 14
841.2.e.k.63.3 24 29.16 even 7
841.2.e.k.196.2 24 29.23 even 7
841.2.e.k.196.3 24 29.6 even 14
841.2.e.k.236.2 24 29.5 even 14
841.2.e.k.236.3 24 29.24 even 7
841.2.e.k.267.2 24 29.20 even 7
841.2.e.k.267.3 24 29.9 even 14
841.2.e.k.270.2 24 29.22 even 14
841.2.e.k.270.3 24 29.7 even 7
841.2.e.k.651.2 24 29.25 even 7
841.2.e.k.651.3 24 29.4 even 14
1421.2.a.j.1.2 2 203.41 even 4
1856.2.a.r.1.2 2 232.157 odd 4
1856.2.a.w.1.1 2 232.99 even 4
3509.2.a.j.1.1 2 319.186 even 4
4176.2.a.bq.1.1 2 348.215 odd 4
4901.2.a.g.1.1 2 377.12 odd 4
6525.2.a.o.1.2 2 435.389 even 4
7569.2.a.c.1.2 2 87.17 even 4
8381.2.a.e.1.2 2 493.186 odd 4