Properties

Label 1110.2.d.b.889.1
Level $1110$
Weight $2$
Character 1110.889
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
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] = [1110,2,Mod(889,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.d (of order \(2\), degree \(1\), 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: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 889.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1110.889
Dual form 1110.2.d.b.889.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(2.00000 + 1.00000i) q^{10} +1.00000 q^{11} -1.00000i q^{12} +5.00000 q^{14} +(-2.00000 - 1.00000i) q^{15} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +(1.00000 - 2.00000i) q^{20} -5.00000 q^{21} -1.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{27} -5.00000i q^{28} +5.00000 q^{29} +(-1.00000 + 2.00000i) q^{30} +3.00000 q^{31} -1.00000i q^{32} +1.00000i q^{33} +3.00000 q^{34} +(-10.0000 - 5.00000i) q^{35} +1.00000 q^{36} -1.00000i q^{37} +8.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} -5.00000 q^{41} +5.00000i q^{42} -9.00000i q^{43} -1.00000 q^{44} +(1.00000 - 2.00000i) q^{45} -8.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} -18.0000 q^{49} +(-4.00000 + 3.00000i) q^{50} -3.00000 q^{51} +5.00000i q^{53} -1.00000 q^{54} +(-1.00000 + 2.00000i) q^{55} -5.00000 q^{56} -8.00000i q^{57} -5.00000i q^{58} +4.00000 q^{59} +(2.00000 + 1.00000i) q^{60} -9.00000 q^{61} -3.00000i q^{62} -5.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} +8.00000i q^{67} -3.00000i q^{68} +8.00000 q^{69} +(-5.00000 + 10.0000i) q^{70} +6.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -1.00000 q^{74} +(4.00000 - 3.00000i) q^{75} +8.00000 q^{76} +5.00000i q^{77} +8.00000 q^{79} +(-1.00000 + 2.00000i) q^{80} +1.00000 q^{81} +5.00000i q^{82} -4.00000i q^{83} +5.00000 q^{84} +(-6.00000 - 3.00000i) q^{85} -9.00000 q^{86} +5.00000i q^{87} +1.00000i q^{88} +4.00000 q^{89} +(-2.00000 - 1.00000i) q^{90} +8.00000i q^{92} +3.00000i q^{93} +12.0000 q^{94} +(8.00000 - 16.0000i) q^{95} +1.00000 q^{96} +9.00000i q^{97} +18.0000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} + 2 q^{11} + 10 q^{14} - 4 q^{15} + 2 q^{16} - 16 q^{19} + 2 q^{20} - 10 q^{21} - 2 q^{24} - 6 q^{25} + 10 q^{29} - 2 q^{30} + 6 q^{31} + 6 q^{34} - 20 q^{35} + 2 q^{36} - 4 q^{40} - 10 q^{41} - 2 q^{44} + 2 q^{45} - 16 q^{46} - 36 q^{49} - 8 q^{50} - 6 q^{51} - 2 q^{54} - 2 q^{55} - 10 q^{56} + 8 q^{59} + 4 q^{60} - 18 q^{61} - 2 q^{64} + 2 q^{66} + 16 q^{69} - 10 q^{70} + 12 q^{71} - 2 q^{74} + 8 q^{75} + 16 q^{76} + 16 q^{79} - 2 q^{80} + 2 q^{81} + 10 q^{84} - 12 q^{85} - 18 q^{86} + 8 q^{89} - 4 q^{90} + 24 q^{94} + 16 q^{95} + 2 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) 1.00000 0.408248
\(7\) 5.00000i 1.88982i 0.327327 + 0.944911i \(0.393852\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 2.00000 + 1.00000i 0.632456 + 0.316228i
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 5.00000 1.33631
\(15\) −2.00000 1.00000i −0.516398 0.258199i
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 2.00000i 0.223607 0.447214i
\(21\) −5.00000 −1.09109
\(22\) 1.00000i 0.213201i
\(23\) 8.00000i 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 5.00000i 0.944911i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) −1.00000 + 2.00000i −0.182574 + 0.365148i
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 1.00000i 0.174078i
\(34\) 3.00000 0.514496
\(35\) −10.0000 5.00000i −1.69031 0.845154i
\(36\) 1.00000 0.166667
\(37\) 1.00000i 0.164399i
\(38\) 8.00000i 1.29777i
\(39\) 0 0
\(40\) −2.00000 1.00000i −0.316228 0.158114i
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 5.00000i 0.771517i
\(43\) 9.00000i 1.37249i −0.727372 0.686244i \(-0.759258\pi\)
0.727372 0.686244i \(-0.240742\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 2.00000i 0.149071 0.298142i
\(46\) −8.00000 −1.17954
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −18.0000 −2.57143
\(50\) −4.00000 + 3.00000i −0.565685 + 0.424264i
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 5.00000i 0.686803i 0.939189 + 0.343401i \(0.111579\pi\)
−0.939189 + 0.343401i \(0.888421\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 + 2.00000i −0.134840 + 0.269680i
\(56\) −5.00000 −0.668153
\(57\) 8.00000i 1.05963i
\(58\) 5.00000i 0.656532i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 + 1.00000i 0.258199 + 0.129099i
\(61\) −9.00000 −1.15233 −0.576166 0.817333i \(-0.695452\pi\)
−0.576166 + 0.817333i \(0.695452\pi\)
\(62\) 3.00000i 0.381000i
\(63\) 5.00000i 0.629941i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 8.00000 0.963087
\(70\) −5.00000 + 10.0000i −0.597614 + 1.19523i
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −1.00000 −0.116248
\(75\) 4.00000 3.00000i 0.461880 0.346410i
\(76\) 8.00000 0.917663
\(77\) 5.00000i 0.569803i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 + 2.00000i −0.111803 + 0.223607i
\(81\) 1.00000 0.111111
\(82\) 5.00000i 0.552158i
\(83\) 4.00000i 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 5.00000 0.545545
\(85\) −6.00000 3.00000i −0.650791 0.325396i
\(86\) −9.00000 −0.970495
\(87\) 5.00000i 0.536056i
\(88\) 1.00000i 0.106600i
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) −2.00000 1.00000i −0.210819 0.105409i
\(91\) 0 0
\(92\) 8.00000i 0.834058i
\(93\) 3.00000i 0.311086i
\(94\) 12.0000 1.23771
\(95\) 8.00000 16.0000i 0.820783 1.64157i
\(96\) 1.00000 0.102062
\(97\) 9.00000i 0.913812i 0.889515 + 0.456906i \(0.151042\pi\)
−0.889515 + 0.456906i \(0.848958\pi\)
\(98\) 18.0000i 1.81827i
\(99\) −1.00000 −0.100504
\(100\) 3.00000 + 4.00000i 0.300000 + 0.400000i
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 3.00000i 0.297044i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 5.00000 10.0000i 0.487950 0.975900i
\(106\) 5.00000 0.485643
\(107\) 2.00000i 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 2.00000 + 1.00000i 0.190693 + 0.0953463i
\(111\) 1.00000 0.0949158
\(112\) 5.00000i 0.472456i
\(113\) 15.0000i 1.41108i 0.708669 + 0.705541i \(0.249296\pi\)
−0.708669 + 0.705541i \(0.750704\pi\)
\(114\) −8.00000 −0.749269
\(115\) 16.0000 + 8.00000i 1.49201 + 0.746004i
\(116\) −5.00000 −0.464238
\(117\) 0 0
\(118\) 4.00000i 0.368230i
\(119\) −15.0000 −1.37505
\(120\) 1.00000 2.00000i 0.0912871 0.182574i
\(121\) −10.0000 −0.909091
\(122\) 9.00000i 0.814822i
\(123\) 5.00000i 0.450835i
\(124\) −3.00000 −0.269408
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) −5.00000 −0.445435
\(127\) 8.00000i 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 9.00000 0.792406
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 1.00000i 0.0870388i
\(133\) 40.0000i 3.46844i
\(134\) 8.00000 0.691095
\(135\) 2.00000 + 1.00000i 0.172133 + 0.0860663i
\(136\) −3.00000 −0.257248
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 10.0000 + 5.00000i 0.845154 + 0.422577i
\(141\) −12.0000 −1.01058
\(142\) 6.00000i 0.503509i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) −5.00000 + 10.0000i −0.415227 + 0.830455i
\(146\) 2.00000 0.165521
\(147\) 18.0000i 1.48461i
\(148\) 1.00000i 0.0821995i
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) −3.00000 4.00000i −0.244949 0.326599i
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 8.00000i 0.648886i
\(153\) 3.00000i 0.242536i
\(154\) 5.00000 0.402911
\(155\) −3.00000 + 6.00000i −0.240966 + 0.481932i
\(156\) 0 0
\(157\) 11.0000i 0.877896i 0.898513 + 0.438948i \(0.144649\pi\)
−0.898513 + 0.438948i \(0.855351\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −5.00000 −0.396526
\(160\) 2.00000 + 1.00000i 0.158114 + 0.0790569i
\(161\) 40.0000 3.15244
\(162\) 1.00000i 0.0785674i
\(163\) 1.00000i 0.0783260i −0.999233 0.0391630i \(-0.987531\pi\)
0.999233 0.0391630i \(-0.0124692\pi\)
\(164\) 5.00000 0.390434
\(165\) −2.00000 1.00000i −0.155700 0.0778499i
\(166\) −4.00000 −0.310460
\(167\) 2.00000i 0.154765i 0.997001 + 0.0773823i \(0.0246562\pi\)
−0.997001 + 0.0773823i \(0.975344\pi\)
\(168\) 5.00000i 0.385758i
\(169\) 13.0000 1.00000
\(170\) −3.00000 + 6.00000i −0.230089 + 0.460179i
\(171\) 8.00000 0.611775
\(172\) 9.00000i 0.686244i
\(173\) 21.0000i 1.59660i 0.602260 + 0.798300i \(0.294267\pi\)
−0.602260 + 0.798300i \(0.705733\pi\)
\(174\) 5.00000 0.379049
\(175\) 20.0000 15.0000i 1.51186 1.13389i
\(176\) 1.00000 0.0753778
\(177\) 4.00000i 0.300658i
\(178\) 4.00000i 0.299813i
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) −1.00000 + 2.00000i −0.0745356 + 0.149071i
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) 9.00000i 0.665299i
\(184\) 8.00000 0.589768
\(185\) 2.00000 + 1.00000i 0.147043 + 0.0735215i
\(186\) 3.00000 0.219971
\(187\) 3.00000i 0.219382i
\(188\) 12.0000i 0.875190i
\(189\) 5.00000 0.363696
\(190\) −16.0000 8.00000i −1.16076 0.580381i
\(191\) −27.0000 −1.95365 −0.976826 0.214036i \(-0.931339\pi\)
−0.976826 + 0.214036i \(0.931339\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) 9.00000 0.646162
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) 6.00000i 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 1.00000i 0.0710669i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 4.00000 3.00000i 0.282843 0.212132i
\(201\) −8.00000 −0.564276
\(202\) 6.00000i 0.422159i
\(203\) 25.0000i 1.75466i
\(204\) 3.00000 0.210042
\(205\) 5.00000 10.0000i 0.349215 0.698430i
\(206\) 4.00000 0.278693
\(207\) 8.00000i 0.556038i
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) −10.0000 5.00000i −0.690066 0.345033i
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 5.00000i 0.343401i
\(213\) 6.00000i 0.411113i
\(214\) −2.00000 −0.136717
\(215\) 18.0000 + 9.00000i 1.22759 + 0.613795i
\(216\) 1.00000 0.0680414
\(217\) 15.0000i 1.01827i
\(218\) 11.0000i 0.745014i
\(219\) −2.00000 −0.135147
\(220\) 1.00000 2.00000i 0.0674200 0.134840i
\(221\) 0 0
\(222\) 1.00000i 0.0671156i
\(223\) 23.0000i 1.54019i −0.637927 0.770097i \(-0.720208\pi\)
0.637927 0.770097i \(-0.279792\pi\)
\(224\) 5.00000 0.334077
\(225\) 3.00000 + 4.00000i 0.200000 + 0.266667i
\(226\) 15.0000 0.997785
\(227\) 21.0000i 1.39382i −0.717159 0.696909i \(-0.754558\pi\)
0.717159 0.696909i \(-0.245442\pi\)
\(228\) 8.00000i 0.529813i
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 8.00000 16.0000i 0.527504 1.05501i
\(231\) −5.00000 −0.328976
\(232\) 5.00000i 0.328266i
\(233\) 6.00000i 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) −24.0000 12.0000i −1.56559 0.782794i
\(236\) −4.00000 −0.260378
\(237\) 8.00000i 0.519656i
\(238\) 15.0000i 0.972306i
\(239\) 19.0000 1.22901 0.614504 0.788914i \(-0.289356\pi\)
0.614504 + 0.788914i \(0.289356\pi\)
\(240\) −2.00000 1.00000i −0.129099 0.0645497i
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 10.0000i 0.642824i
\(243\) 1.00000i 0.0641500i
\(244\) 9.00000 0.576166
\(245\) 18.0000 36.0000i 1.14998 2.29996i
\(246\) −5.00000 −0.318788
\(247\) 0 0
\(248\) 3.00000i 0.190500i
\(249\) 4.00000 0.253490
\(250\) −2.00000 11.0000i −0.126491 0.695701i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 5.00000i 0.314970i
\(253\) 8.00000i 0.502956i
\(254\) −8.00000 −0.501965
\(255\) 3.00000 6.00000i 0.187867 0.375735i
\(256\) 1.00000 0.0625000
\(257\) 26.0000i 1.62184i 0.585160 + 0.810918i \(0.301032\pi\)
−0.585160 + 0.810918i \(0.698968\pi\)
\(258\) 9.00000i 0.560316i
\(259\) 5.00000 0.310685
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 6.00000i 0.370681i
\(263\) 5.00000i 0.308313i 0.988046 + 0.154157i \(0.0492660\pi\)
−0.988046 + 0.154157i \(0.950734\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −10.0000 5.00000i −0.614295 0.307148i
\(266\) −40.0000 −2.45256
\(267\) 4.00000i 0.244796i
\(268\) 8.00000i 0.488678i
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 1.00000 2.00000i 0.0608581 0.121716i
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −3.00000 4.00000i −0.180907 0.241209i
\(276\) −8.00000 −0.481543
\(277\) 32.0000i 1.92269i −0.275340 0.961347i \(-0.588791\pi\)
0.275340 0.961347i \(-0.411209\pi\)
\(278\) 5.00000i 0.299880i
\(279\) −3.00000 −0.179605
\(280\) 5.00000 10.0000i 0.298807 0.597614i
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 12.0000i 0.714590i
\(283\) 16.0000i 0.951101i 0.879688 + 0.475551i \(0.157751\pi\)
−0.879688 + 0.475551i \(0.842249\pi\)
\(284\) −6.00000 −0.356034
\(285\) 16.0000 + 8.00000i 0.947758 + 0.473879i
\(286\) 0 0
\(287\) 25.0000i 1.47570i
\(288\) 1.00000i 0.0589256i
\(289\) 8.00000 0.470588
\(290\) 10.0000 + 5.00000i 0.587220 + 0.293610i
\(291\) −9.00000 −0.527589
\(292\) 2.00000i 0.117041i
\(293\) 23.0000i 1.34367i 0.740699 + 0.671837i \(0.234495\pi\)
−0.740699 + 0.671837i \(0.765505\pi\)
\(294\) −18.0000 −1.04978
\(295\) −4.00000 + 8.00000i −0.232889 + 0.465778i
\(296\) 1.00000 0.0581238
\(297\) 1.00000i 0.0580259i
\(298\) 20.0000i 1.15857i
\(299\) 0 0
\(300\) −4.00000 + 3.00000i −0.230940 + 0.173205i
\(301\) 45.0000 2.59376
\(302\) 10.0000i 0.575435i
\(303\) 6.00000i 0.344691i
\(304\) −8.00000 −0.458831
\(305\) 9.00000 18.0000i 0.515339 1.03068i
\(306\) −3.00000 −0.171499
\(307\) 4.00000i 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 5.00000i 0.284901i
\(309\) −4.00000 −0.227552
\(310\) 6.00000 + 3.00000i 0.340777 + 0.170389i
\(311\) 17.0000 0.963982 0.481991 0.876176i \(-0.339914\pi\)
0.481991 + 0.876176i \(0.339914\pi\)
\(312\) 0 0
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 11.0000 0.620766
\(315\) 10.0000 + 5.00000i 0.563436 + 0.281718i
\(316\) −8.00000 −0.450035
\(317\) 11.0000i 0.617822i 0.951091 + 0.308911i \(0.0999645\pi\)
−0.951091 + 0.308911i \(0.900036\pi\)
\(318\) 5.00000i 0.280386i
\(319\) 5.00000 0.279946
\(320\) 1.00000 2.00000i 0.0559017 0.111803i
\(321\) 2.00000 0.111629
\(322\) 40.0000i 2.22911i
\(323\) 24.0000i 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −1.00000 −0.0553849
\(327\) 11.0000i 0.608301i
\(328\) 5.00000i 0.276079i
\(329\) −60.0000 −3.30791
\(330\) −1.00000 + 2.00000i −0.0550482 + 0.110096i
\(331\) 22.0000 1.20923 0.604615 0.796518i \(-0.293327\pi\)
0.604615 + 0.796518i \(0.293327\pi\)
\(332\) 4.00000i 0.219529i
\(333\) 1.00000i 0.0547997i
\(334\) 2.00000 0.109435
\(335\) −16.0000 8.00000i −0.874173 0.437087i
\(336\) −5.00000 −0.272772
\(337\) 20.0000i 1.08947i 0.838608 + 0.544735i \(0.183370\pi\)
−0.838608 + 0.544735i \(0.816630\pi\)
\(338\) 13.0000i 0.707107i
\(339\) −15.0000 −0.814688
\(340\) 6.00000 + 3.00000i 0.325396 + 0.162698i
\(341\) 3.00000 0.162459
\(342\) 8.00000i 0.432590i
\(343\) 55.0000i 2.96972i
\(344\) 9.00000 0.485247
\(345\) −8.00000 + 16.0000i −0.430706 + 0.861411i
\(346\) 21.0000 1.12897
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 5.00000i 0.268028i
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −15.0000 20.0000i −0.801784 1.06904i
\(351\) 0 0
\(352\) 1.00000i 0.0533002i
\(353\) 19.0000i 1.01127i −0.862748 0.505634i \(-0.831259\pi\)
0.862748 0.505634i \(-0.168741\pi\)
\(354\) 4.00000 0.212598
\(355\) −6.00000 + 12.0000i −0.318447 + 0.636894i
\(356\) −4.00000 −0.212000
\(357\) 15.0000i 0.793884i
\(358\) 6.00000i 0.317110i
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 2.00000 + 1.00000i 0.105409 + 0.0527046i
\(361\) 45.0000 2.36842
\(362\) 12.0000i 0.630706i
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) −4.00000 2.00000i −0.209370 0.104685i
\(366\) −9.00000 −0.470438
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) 8.00000i 0.417029i
\(369\) 5.00000 0.260290
\(370\) 1.00000 2.00000i 0.0519875 0.103975i
\(371\) −25.0000 −1.29794
\(372\) 3.00000i 0.155543i
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) 3.00000 0.155126
\(375\) 2.00000 + 11.0000i 0.103280 + 0.568038i
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 5.00000i 0.257172i
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −8.00000 + 16.0000i −0.410391 + 0.820783i
\(381\) 8.00000 0.409852
\(382\) 27.0000i 1.38144i
\(383\) 36.0000i 1.83951i −0.392488 0.919757i \(-0.628386\pi\)
0.392488 0.919757i \(-0.371614\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −10.0000 5.00000i −0.509647 0.254824i
\(386\) 10.0000 0.508987
\(387\) 9.00000i 0.457496i
\(388\) 9.00000i 0.456906i
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 18.0000i 0.909137i
\(393\) 6.00000i 0.302660i
\(394\) −6.00000 −0.302276
\(395\) −8.00000 + 16.0000i −0.402524 + 0.805047i
\(396\) 1.00000 0.0502519
\(397\) 18.0000i 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 40.0000 2.00250
\(400\) −3.00000 4.00000i −0.150000 0.200000i
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) −1.00000 + 2.00000i −0.0496904 + 0.0993808i
\(406\) 25.0000 1.24073
\(407\) 1.00000i 0.0495682i
\(408\) 3.00000i 0.148522i
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) −10.0000 5.00000i −0.493865 0.246932i
\(411\) −6.00000 −0.295958
\(412\) 4.00000i 0.197066i
\(413\) 20.0000i 0.984136i
\(414\) 8.00000 0.393179
\(415\) 8.00000 + 4.00000i 0.392705 + 0.196352i
\(416\) 0 0
\(417\) 5.00000i 0.244851i
\(418\) 8.00000i 0.391293i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −5.00000 + 10.0000i −0.243975 + 0.487950i
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 3.00000i 0.146038i
\(423\) 12.0000i 0.583460i
\(424\) −5.00000 −0.242821
\(425\) 12.0000 9.00000i 0.582086 0.436564i
\(426\) 6.00000 0.290701
\(427\) 45.0000i 2.17770i
\(428\) 2.00000i 0.0966736i
\(429\) 0 0
\(430\) 9.00000 18.0000i 0.434019 0.868037i
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 32.0000i 1.53782i 0.639356 + 0.768911i \(0.279201\pi\)
−0.639356 + 0.768911i \(0.720799\pi\)
\(434\) 15.0000 0.720023
\(435\) −10.0000 5.00000i −0.479463 0.239732i
\(436\) 11.0000 0.526804
\(437\) 64.0000i 3.06154i
\(438\) 2.00000i 0.0955637i
\(439\) 11.0000 0.525001 0.262501 0.964932i \(-0.415453\pi\)
0.262501 + 0.964932i \(0.415453\pi\)
\(440\) −2.00000 1.00000i −0.0953463 0.0476731i
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 10.0000i 0.475114i 0.971374 + 0.237557i \(0.0763467\pi\)
−0.971374 + 0.237557i \(0.923653\pi\)
\(444\) −1.00000 −0.0474579
\(445\) −4.00000 + 8.00000i −0.189618 + 0.379236i
\(446\) −23.0000 −1.08908
\(447\) 20.0000i 0.945968i
\(448\) 5.00000i 0.236228i
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 4.00000 3.00000i 0.188562 0.141421i
\(451\) −5.00000 −0.235441
\(452\) 15.0000i 0.705541i
\(453\) 10.0000i 0.469841i
\(454\) −21.0000 −0.985579
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) 13.0000i 0.608114i 0.952654 + 0.304057i \(0.0983414\pi\)
−0.952654 + 0.304057i \(0.901659\pi\)
\(458\) 4.00000i 0.186908i
\(459\) 3.00000 0.140028
\(460\) −16.0000 8.00000i −0.746004 0.373002i
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 5.00000i 0.232621i
\(463\) 20.0000i 0.929479i 0.885448 + 0.464739i \(0.153852\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) 5.00000 0.232119
\(465\) −6.00000 3.00000i −0.278243 0.139122i
\(466\) −6.00000 −0.277945
\(467\) 31.0000i 1.43451i −0.696811 0.717254i \(-0.745399\pi\)
0.696811 0.717254i \(-0.254601\pi\)
\(468\) 0 0
\(469\) −40.0000 −1.84703
\(470\) −12.0000 + 24.0000i −0.553519 + 1.10704i
\(471\) −11.0000 −0.506853
\(472\) 4.00000i 0.184115i
\(473\) 9.00000i 0.413820i
\(474\) 8.00000 0.367452
\(475\) 24.0000 + 32.0000i 1.10120 + 1.46826i
\(476\) 15.0000 0.687524
\(477\) 5.00000i 0.228934i
\(478\) 19.0000i 0.869040i
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) −1.00000 + 2.00000i −0.0456435 + 0.0912871i
\(481\) 0 0
\(482\) 10.0000i 0.455488i
\(483\) 40.0000i 1.82006i
\(484\) 10.0000 0.454545
\(485\) −18.0000 9.00000i −0.817338 0.408669i
\(486\) 1.00000 0.0453609
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) 9.00000i 0.407411i
\(489\) 1.00000 0.0452216
\(490\) −36.0000 18.0000i −1.62631 0.813157i
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 5.00000i 0.225417i
\(493\) 15.0000i 0.675566i
\(494\) 0 0
\(495\) 1.00000 2.00000i 0.0449467 0.0898933i
\(496\) 3.00000 0.134704
\(497\) 30.0000i 1.34568i
\(498\) 4.00000i 0.179244i
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) −11.0000 + 2.00000i −0.491935 + 0.0894427i
\(501\) −2.00000 −0.0893534
\(502\) 0 0
\(503\) 10.0000i 0.445878i −0.974832 0.222939i \(-0.928435\pi\)
0.974832 0.222939i \(-0.0715651\pi\)
\(504\) 5.00000 0.222718
\(505\) 6.00000 12.0000i 0.266996 0.533993i
\(506\) −8.00000 −0.355643
\(507\) 13.0000i 0.577350i
\(508\) 8.00000i 0.354943i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −6.00000 3.00000i −0.265684 0.132842i
\(511\) −10.0000 −0.442374
\(512\) 1.00000i 0.0441942i
\(513\) 8.00000i 0.353209i
\(514\) 26.0000 1.14681
\(515\) −8.00000 4.00000i −0.352522 0.176261i
\(516\) −9.00000 −0.396203
\(517\) 12.0000i 0.527759i
\(518\) 5.00000i 0.219687i
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 5.00000i 0.218844i
\(523\) 44.0000i 1.92399i 0.273075 + 0.961993i \(0.411959\pi\)
−0.273075 + 0.961993i \(0.588041\pi\)
\(524\) 6.00000 0.262111
\(525\) 15.0000 + 20.0000i 0.654654 + 0.872872i
\(526\) 5.00000 0.218010
\(527\) 9.00000i 0.392046i
\(528\) 1.00000i 0.0435194i
\(529\) −41.0000 −1.78261
\(530\) −5.00000 + 10.0000i −0.217186 + 0.434372i
\(531\) −4.00000 −0.173585
\(532\) 40.0000i 1.73422i
\(533\) 0 0
\(534\) 4.00000 0.173097
\(535\) 4.00000 + 2.00000i 0.172935 + 0.0864675i
\(536\) −8.00000 −0.345547
\(537\) 6.00000i 0.258919i
\(538\) 4.00000i 0.172452i
\(539\) −18.0000 −0.775315
\(540\) −2.00000 1.00000i −0.0860663 0.0430331i
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 14.0000i 0.601351i
\(543\) 12.0000i 0.514969i
\(544\) 3.00000 0.128624
\(545\) 11.0000 22.0000i 0.471188 0.942376i
\(546\) 0 0
\(547\) 31.0000i 1.32546i −0.748857 0.662732i \(-0.769397\pi\)
0.748857 0.662732i \(-0.230603\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 9.00000 0.384111
\(550\) −4.00000 + 3.00000i −0.170561 + 0.127920i
\(551\) −40.0000 −1.70406
\(552\) 8.00000i 0.340503i
\(553\) 40.0000i 1.70097i
\(554\) −32.0000 −1.35955
\(555\) −1.00000 + 2.00000i −0.0424476 + 0.0848953i
\(556\) 5.00000 0.212047
\(557\) 26.0000i 1.10166i −0.834619 0.550828i \(-0.814312\pi\)
0.834619 0.550828i \(-0.185688\pi\)
\(558\) 3.00000i 0.127000i
\(559\) 0 0
\(560\) −10.0000 5.00000i −0.422577 0.211289i
\(561\) −3.00000 −0.126660
\(562\) 30.0000i 1.26547i
\(563\) 23.0000i 0.969334i −0.874699 0.484667i \(-0.838941\pi\)
0.874699 0.484667i \(-0.161059\pi\)
\(564\) 12.0000 0.505291
\(565\) −30.0000 15.0000i −1.26211 0.631055i
\(566\) 16.0000 0.672530
\(567\) 5.00000i 0.209980i
\(568\) 6.00000i 0.251754i
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 8.00000 16.0000i 0.335083 0.670166i
\(571\) 35.0000 1.46470 0.732352 0.680926i \(-0.238422\pi\)
0.732352 + 0.680926i \(0.238422\pi\)
\(572\) 0 0
\(573\) 27.0000i 1.12794i
\(574\) −25.0000 −1.04348
\(575\) −32.0000 + 24.0000i −1.33449 + 1.00087i
\(576\) 1.00000 0.0416667
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 8.00000i 0.332756i
\(579\) −10.0000 −0.415586
\(580\) 5.00000 10.0000i 0.207614 0.415227i
\(581\) 20.0000 0.829740
\(582\) 9.00000i 0.373062i
\(583\) 5.00000i 0.207079i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 23.0000 0.950121
\(587\) 21.0000i 0.866763i 0.901211 + 0.433381i \(0.142680\pi\)
−0.901211 + 0.433381i \(0.857320\pi\)
\(588\) 18.0000i 0.742307i
\(589\) −24.0000 −0.988903
\(590\) 8.00000 + 4.00000i 0.329355 + 0.164677i
\(591\) 6.00000 0.246807
\(592\) 1.00000i 0.0410997i
\(593\) 36.0000i 1.47834i 0.673517 + 0.739171i \(0.264783\pi\)
−0.673517 + 0.739171i \(0.735217\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 15.0000 30.0000i 0.614940 1.22988i
\(596\) −20.0000 −0.819232
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 3.00000 + 4.00000i 0.122474 + 0.163299i
\(601\) 25.0000 1.01977 0.509886 0.860242i \(-0.329688\pi\)
0.509886 + 0.860242i \(0.329688\pi\)
\(602\) 45.0000i 1.83406i
\(603\) 8.00000i 0.325785i
\(604\) 10.0000 0.406894
\(605\) 10.0000 20.0000i 0.406558 0.813116i
\(606\) −6.00000 −0.243733
\(607\) 46.0000i 1.86708i 0.358470 + 0.933541i \(0.383298\pi\)
−0.358470 + 0.933541i \(0.616702\pi\)
\(608\) 8.00000i 0.324443i
\(609\) −25.0000 −1.01305
\(610\) −18.0000 9.00000i −0.728799 0.364399i
\(611\) 0 0
\(612\) 3.00000i 0.121268i
\(613\) 9.00000i 0.363507i −0.983344 0.181753i \(-0.941823\pi\)
0.983344 0.181753i \(-0.0581772\pi\)
\(614\) −4.00000 −0.161427
\(615\) 10.0000 + 5.00000i 0.403239 + 0.201619i
\(616\) −5.00000 −0.201456
\(617\) 2.00000i 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 4.00000i 0.160904i
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 3.00000 6.00000i 0.120483 0.240966i
\(621\) −8.00000 −0.321029
\(622\) 17.0000i 0.681638i
\(623\) 20.0000i 0.801283i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 26.0000 1.03917
\(627\) 8.00000i 0.319489i
\(628\) 11.0000i 0.438948i
\(629\) 3.00000 0.119618
\(630\) 5.00000 10.0000i 0.199205 0.398410i
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 3.00000i 0.119239i
\(634\) 11.0000 0.436866
\(635\) 16.0000 + 8.00000i 0.634941 + 0.317470i
\(636\) 5.00000 0.198263
\(637\) 0 0
\(638\) 5.00000i 0.197952i
\(639\) −6.00000 −0.237356
\(640\) −2.00000 1.00000i −0.0790569 0.0395285i
\(641\) −17.0000 −0.671460 −0.335730 0.941958i \(-0.608983\pi\)
−0.335730 + 0.941958i \(0.608983\pi\)
\(642\) 2.00000i 0.0789337i
\(643\) 19.0000i 0.749287i 0.927169 + 0.374643i \(0.122235\pi\)
−0.927169 + 0.374643i \(0.877765\pi\)
\(644\) −40.0000 −1.57622
\(645\) −9.00000 + 18.0000i −0.354375 + 0.708749i
\(646\) −24.0000 −0.944267
\(647\) 6.00000i 0.235884i −0.993020 0.117942i \(-0.962370\pi\)
0.993020 0.117942i \(-0.0376297\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) −15.0000 −0.587896
\(652\) 1.00000i 0.0391630i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) −11.0000 −0.430134
\(655\) 6.00000 12.0000i 0.234439 0.468879i
\(656\) −5.00000 −0.195217
\(657\) 2.00000i 0.0780274i
\(658\) 60.0000i 2.33904i
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 2.00000 + 1.00000i 0.0778499 + 0.0389249i
\(661\) 1.00000 0.0388955 0.0194477 0.999811i \(-0.493809\pi\)
0.0194477 + 0.999811i \(0.493809\pi\)
\(662\) 22.0000i 0.855054i
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 80.0000 + 40.0000i 3.10227 + 1.55113i
\(666\) 1.00000 0.0387492
\(667\) 40.0000i 1.54881i
\(668\) 2.00000i 0.0773823i
\(669\) 23.0000 0.889231
\(670\) −8.00000 + 16.0000i −0.309067 + 0.618134i
\(671\) −9.00000 −0.347441
\(672\) 5.00000i 0.192879i
\(673\) 32.0000i 1.23351i 0.787155 + 0.616755i \(0.211553\pi\)
−0.787155 + 0.616755i \(0.788447\pi\)
\(674\) 20.0000 0.770371
\(675\) −4.00000 + 3.00000i −0.153960 + 0.115470i
\(676\) −13.0000 −0.500000
\(677\) 18.0000i 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 15.0000i 0.576072i
\(679\) −45.0000 −1.72694
\(680\) 3.00000 6.00000i 0.115045 0.230089i
\(681\) 21.0000 0.804722
\(682\) 3.00000i 0.114876i
\(683\) 47.0000i 1.79841i −0.437533 0.899203i \(-0.644148\pi\)
0.437533 0.899203i \(-0.355852\pi\)
\(684\) −8.00000 −0.305888
\(685\) −12.0000 6.00000i −0.458496 0.229248i
\(686\) −55.0000 −2.09991
\(687\) 4.00000i 0.152610i
\(688\) 9.00000i 0.343122i
\(689\) 0 0
\(690\) 16.0000 + 8.00000i 0.609110 + 0.304555i
\(691\) −21.0000 −0.798878 −0.399439 0.916760i \(-0.630795\pi\)
−0.399439 + 0.916760i \(0.630795\pi\)
\(692\) 21.0000i 0.798300i
\(693\) 5.00000i 0.189934i
\(694\) 28.0000 1.06287
\(695\) 5.00000 10.0000i 0.189661 0.379322i
\(696\) −5.00000 −0.189525
\(697\) 15.0000i 0.568166i
\(698\) 2.00000i 0.0757011i
\(699\) 6.00000 0.226941
\(700\) −20.0000 + 15.0000i −0.755929 + 0.566947i
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 24.0000i 0.451946 0.903892i
\(706\) −19.0000 −0.715074
\(707\) 30.0000i 1.12827i
\(708\) 4.00000i 0.150329i
\(709\) −3.00000 −0.112667 −0.0563337 0.998412i \(-0.517941\pi\)
−0.0563337 + 0.998412i \(0.517941\pi\)
\(710\) 12.0000 + 6.00000i 0.450352 + 0.225176i
\(711\) −8.00000 −0.300023
\(712\) 4.00000i 0.149906i
\(713\) 24.0000i 0.898807i
\(714\) −15.0000 −0.561361
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 19.0000i 0.709568i
\(718\) 18.0000i 0.671754i
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 1.00000 2.00000i 0.0372678 0.0745356i
\(721\) −20.0000 −0.744839
\(722\) 45.0000i 1.67473i
\(723\) 10.0000i 0.371904i
\(724\) 12.0000 0.445976
\(725\) −15.0000 20.0000i −0.557086 0.742781i
\(726\) −10.0000 −0.371135
\(727\) 26.0000i 0.964287i 0.876092 + 0.482143i \(0.160142\pi\)
−0.876092 + 0.482143i \(0.839858\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) −2.00000 + 4.00000i −0.0740233 + 0.148047i
\(731\) 27.0000 0.998631
\(732\) 9.00000i 0.332650i
\(733\) 49.0000i 1.80986i −0.425564 0.904928i \(-0.639924\pi\)
0.425564 0.904928i \(-0.360076\pi\)
\(734\) 17.0000 0.627481
\(735\) 36.0000 + 18.0000i 1.32788 + 0.663940i
\(736\) −8.00000 −0.294884
\(737\) 8.00000i 0.294684i
\(738\) 5.00000i 0.184053i
\(739\) 43.0000 1.58178 0.790890 0.611958i \(-0.209618\pi\)
0.790890 + 0.611958i \(0.209618\pi\)
\(740\) −2.00000 1.00000i −0.0735215 0.0367607i
\(741\) 0 0
\(742\) 25.0000i 0.917779i
\(743\) 17.0000i 0.623670i 0.950136 + 0.311835i \(0.100944\pi\)
−0.950136 + 0.311835i \(0.899056\pi\)
\(744\) −3.00000 −0.109985
\(745\) −20.0000 + 40.0000i −0.732743 + 1.46549i
\(746\) 26.0000 0.951928
\(747\) 4.00000i 0.146352i
\(748\) 3.00000i 0.109691i
\(749\) 10.0000 0.365392
\(750\) 11.0000 2.00000i 0.401663 0.0730297i
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 0 0
\(754\) 0 0
\(755\) 10.0000 20.0000i 0.363937 0.727875i
\(756\) −5.00000 −0.181848
\(757\) 32.0000i 1.16306i 0.813525 + 0.581530i \(0.197546\pi\)
−0.813525 + 0.581530i \(0.802454\pi\)
\(758\) 28.0000i 1.01701i
\(759\) 8.00000 0.290382
\(760\) 16.0000 + 8.00000i 0.580381 + 0.290191i
\(761\) 47.0000 1.70375 0.851874 0.523746i \(-0.175466\pi\)
0.851874 + 0.523746i \(0.175466\pi\)
\(762\) 8.00000i 0.289809i
\(763\) 55.0000i 1.99113i
\(764\) 27.0000 0.976826
\(765\) 6.00000 + 3.00000i 0.216930 + 0.108465i
\(766\) −36.0000 −1.30073
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) 52.0000 1.87517 0.937584 0.347759i \(-0.113057\pi\)
0.937584 + 0.347759i \(0.113057\pi\)
\(770\) −5.00000 + 10.0000i −0.180187 + 0.360375i
\(771\) −26.0000 −0.936367
\(772\) 10.0000i 0.359908i
\(773\) 11.0000i 0.395643i 0.980238 + 0.197821i \(0.0633866\pi\)
−0.980238 + 0.197821i \(0.936613\pi\)
\(774\) 9.00000 0.323498
\(775\) −9.00000 12.0000i −0.323290 0.431053i
\(776\) −9.00000 −0.323081
\(777\) 5.00000i 0.179374i
\(778\) 9.00000i 0.322666i
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 24.0000i 0.858238i
\(783\) 5.00000i 0.178685i
\(784\) −18.0000 −0.642857
\(785\) −22.0000 11.0000i −0.785214 0.392607i
\(786\) −6.00000 −0.214013
\(787\) 26.0000i 0.926800i −0.886149 0.463400i \(-0.846629\pi\)
0.886149 0.463400i \(-0.153371\pi\)
\(788\) 6.00000i 0.213741i
\(789\) −5.00000 −0.178005
\(790\) 16.0000 + 8.00000i 0.569254 + 0.284627i
\(791\) −75.0000 −2.66669
\(792\) 1.00000i 0.0355335i
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) 5.00000 10.0000i 0.177332 0.354663i
\(796\) 16.0000 0.567105
\(797\) 24.0000i 0.850124i −0.905164 0.425062i \(-0.860252\pi\)
0.905164 0.425062i \(-0.139748\pi\)
\(798\) 40.0000i 1.41598i
\(799\) −36.0000 −1.27359
\(800\) −4.00000 + 3.00000i −0.141421 + 0.106066i
\(801\) −4.00000 −0.141333
\(802\) 14.0000i 0.494357i
\(803\) 2.00000i 0.0705785i
\(804\) 8.00000 0.282138
\(805\) −40.0000 + 80.0000i −1.40981 + 2.81963i
\(806\) 0 0
\(807\) 4.00000i 0.140807i
\(808\) 6.00000i 0.211079i
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) 2.00000 + 1.00000i 0.0702728 + 0.0351364i
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 25.0000i 0.877328i
\(813\) 14.0000i 0.491001i
\(814\) −1.00000 −0.0350500
\(815\) 2.00000 + 1.00000i 0.0700569 + 0.0350285i
\(816\) −3.00000 −0.105021
\(817\) 72.0000i 2.51896i
\(818\) 20.0000i 0.699284i
\(819\) 0 0
\(820\) −5.00000 + 10.0000i −0.174608 + 0.349215i
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 24.0000i 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) −4.00000 −0.139347
\(825\) 4.00000 3.00000i 0.139262 0.104447i
\(826\) 20.0000 0.695889
\(827\) 31.0000i 1.07798i 0.842314 + 0.538988i \(0.181193\pi\)
−0.842314 + 0.538988i \(0.818807\pi\)
\(828\) 8.00000i 0.278019i
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 4.00000 8.00000i 0.138842 0.277684i
\(831\) 32.0000 1.11007
\(832\) 0 0
\(833\) 54.0000i 1.87099i
\(834\) −5.00000 −0.173136
\(835\) −4.00000 2.00000i −0.138426 0.0692129i
\(836\) 8.00000 0.276686
\(837\) 3.00000i 0.103695i
\(838\) 12.0000i 0.414533i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 10.0000 + 5.00000i 0.345033 + 0.172516i
\(841\) −4.00000 −0.137931
\(842\) 22.0000i 0.758170i
\(843\) 30.0000i 1.03325i
\(844\) −3.00000 −0.103264
\(845\) −13.0000 + 26.0000i −0.447214 + 0.894427i
\(846\) −12.0000 −0.412568
\(847\) 50.0000i 1.71802i
\(848\) 5.00000i 0.171701i
\(849\) −16.0000 −0.549119
\(850\) −9.00000 12.0000i −0.308697 0.411597i
\(851\) −8.00000 −0.274236
\(852\) 6.00000i 0.205557i
\(853\) 30.0000i 1.02718i −0.858036 0.513590i \(-0.828315\pi\)
0.858036 0.513590i \(-0.171685\pi\)
\(854\) −45.0000 −1.53987
\(855\) −8.00000 + 16.0000i −0.273594 + 0.547188i
\(856\) 2.00000 0.0683586
\(857\) 39.0000i 1.33221i −0.745856 0.666107i \(-0.767959\pi\)
0.745856 0.666107i \(-0.232041\pi\)
\(858\) 0 0
\(859\) −24.0000 −0.818869 −0.409435 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) −18.0000 9.00000i −0.613795 0.306897i
\(861\) 25.0000 0.851998
\(862\) 3.00000i 0.102180i
\(863\) 37.0000i 1.25949i −0.776800 0.629747i \(-0.783158\pi\)
0.776800 0.629747i \(-0.216842\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −42.0000 21.0000i −1.42804 0.714021i
\(866\) 32.0000 1.08740
\(867\) 8.00000i 0.271694i
\(868\) 15.0000i 0.509133i
\(869\) 8.00000 0.271381
\(870\) −5.00000 + 10.0000i −0.169516 + 0.339032i
\(871\) 0 0
\(872\) 11.0000i 0.372507i
\(873\) 9.00000i 0.304604i
\(874\) 64.0000 2.16483
\(875\) 10.0000 + 55.0000i 0.338062 + 1.85934i
\(876\) 2.00000 0.0675737
\(877\) 17.0000i 0.574049i −0.957923 0.287025i \(-0.907334\pi\)
0.957923 0.287025i \(-0.0926662\pi\)
\(878\) 11.0000i 0.371232i
\(879\) −23.0000 −0.775771
\(880\) −1.00000 + 2.00000i −0.0337100 + 0.0674200i
\(881\) −25.0000 −0.842271 −0.421136 0.906998i \(-0.638368\pi\)
−0.421136 + 0.906998i \(0.638368\pi\)
\(882\) 18.0000i 0.606092i
\(883\) 47.0000i 1.58168i −0.612026 0.790838i \(-0.709645\pi\)
0.612026 0.790838i \(-0.290355\pi\)
\(884\) 0 0
\(885\) −8.00000 4.00000i −0.268917 0.134459i
\(886\) 10.0000 0.335957
\(887\) 3.00000i 0.100730i 0.998731 + 0.0503651i \(0.0160385\pi\)
−0.998731 + 0.0503651i \(0.983962\pi\)
\(888\) 1.00000i 0.0335578i
\(889\) 40.0000 1.34156
\(890\) 8.00000 + 4.00000i 0.268161 + 0.134080i
\(891\) 1.00000 0.0335013
\(892\) 23.0000i 0.770097i
\(893\) 96.0000i 3.21252i
\(894\) 20.0000 0.668900
\(895\) −6.00000 + 12.0000i −0.200558 + 0.401116i
\(896\) −5.00000 −0.167038
\(897\) 0 0
\(898\) 12.0000i 0.400445i
\(899\) 15.0000 0.500278
\(900\) −3.00000 4.00000i −0.100000 0.133333i
\(901\) −15.0000 −0.499722
\(902\) 5.00000i 0.166482i
\(903\) 45.0000i 1.49751i
\(904\) −15.0000 −0.498893
\(905\) 12.0000 24.0000i 0.398893 0.797787i
\(906\) −10.0000 −0.332228
\(907\) 52.0000i 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) 21.0000i 0.696909i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −60.0000 −1.98789 −0.993944 0.109885i \(-0.964952\pi\)
−0.993944 + 0.109885i \(0.964952\pi\)
\(912\) 8.00000i 0.264906i
\(913\) 4.00000i 0.132381i
\(914\) 13.0000 0.430002
\(915\) 18.0000 + 9.00000i 0.595062 + 0.297531i
\(916\) −4.00000 −0.132164
\(917\) 30.0000i 0.990687i
\(918\) 3.00000i 0.0990148i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −8.00000 + 16.0000i −0.263752 + 0.527504i
\(921\) 4.00000 0.131804
\(922\) 3.00000i 0.0987997i
\(923\) 0 0
\(924\) 5.00000 0.164488
\(925\) −4.00000 + 3.00000i −0.131519 + 0.0986394i
\(926\) 20.0000 0.657241
\(927\) 4.00000i 0.131377i
\(928\) 5.00000i 0.164133i
\(929\) −25.0000 −0.820223 −0.410112 0.912035i \(-0.634510\pi\)
−0.410112 + 0.912035i \(0.634510\pi\)
\(930\) −3.00000 + 6.00000i −0.0983739 + 0.196748i
\(931\) 144.000 4.71941
\(932\) 6.00000i 0.196537i
\(933\) 17.0000i 0.556555i
\(934\) −31.0000 −1.01435
\(935\) −6.00000 3.00000i −0.196221 0.0981105i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 40.0000i 1.30605i
\(939\) −26.0000 −0.848478
\(940\) 24.0000 + 12.0000i 0.782794 + 0.391397i
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 11.0000i 0.358399i
\(943\) 40.0000i 1.30258i
\(944\) 4.00000 0.130189
\(945\) −5.00000 + 10.0000i −0.162650 + 0.325300i
\(946\) −9.00000 −0.292615
\(947\) 15.0000i 0.487435i 0.969846 + 0.243717i \(0.0783669\pi\)
−0.969846 + 0.243717i \(0.921633\pi\)
\(948\) 8.00000i 0.259828i
\(949\) 0 0
\(950\) 32.0000 24.0000i 1.03822 0.778663i
\(951\) −11.0000 −0.356699
\(952\) 15.0000i 0.486153i
\(953\) 22.0000i 0.712650i 0.934362 + 0.356325i \(0.115970\pi\)
−0.934362 + 0.356325i \(0.884030\pi\)
\(954\) −5.00000 −0.161881
\(955\) 27.0000 54.0000i 0.873699 1.74740i
\(956\) −19.0000 −0.614504
\(957\) 5.00000i 0.161627i
\(958\) 36.0000i 1.16311i
\(959\) −30.0000 −0.968751
\(960\) 2.00000 + 1.00000i 0.0645497 + 0.0322749i
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 2.00000i 0.0644491i
\(964\) 10.0000 0.322078
\(965\) −20.0000 10.0000i −0.643823 0.321911i
\(966\) 40.0000 1.28698
\(967\) 34.0000i 1.09337i −0.837340 0.546683i \(-0.815890\pi\)
0.837340 0.546683i \(-0.184110\pi\)
\(968\) 10.0000i 0.321412i
\(969\) 24.0000 0.770991
\(970\) −9.00000 + 18.0000i −0.288973 + 0.577945i
\(971\) −5.00000 −0.160458 −0.0802288 0.996776i \(-0.525565\pi\)
−0.0802288 + 0.996776i \(0.525565\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 25.0000i 0.801463i
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −9.00000 −0.288083
\(977\) 47.0000i 1.50366i 0.659355 + 0.751832i \(0.270829\pi\)
−0.659355 + 0.751832i \(0.729171\pi\)
\(978\) 1.00000i 0.0319765i
\(979\) 4.00000 0.127841
\(980\) −18.0000 + 36.0000i −0.574989 + 1.14998i
\(981\) 11.0000 0.351203
\(982\) 20.0000i 0.638226i
\(983\) 15.0000i 0.478426i 0.970967 + 0.239213i \(0.0768894\pi\)
−0.970967 + 0.239213i \(0.923111\pi\)
\(984\) 5.00000 0.159394
\(985\) 12.0000 + 6.00000i 0.382352 + 0.191176i
\(986\) 15.0000 0.477697
\(987\) 60.0000i 1.90982i
\(988\) 0 0
\(989\) −72.0000 −2.28947
\(990\) −2.00000 1.00000i −0.0635642 0.0317821i
\(991\) 25.0000 0.794151 0.397076 0.917786i \(-0.370025\pi\)
0.397076 + 0.917786i \(0.370025\pi\)
\(992\) 3.00000i 0.0952501i
\(993\) 22.0000i 0.698149i
\(994\) 30.0000 0.951542
\(995\) 16.0000 32.0000i 0.507234 1.01447i
\(996\) −4.00000 −0.126745
\(997\) 60.0000i 1.90022i −0.311916 0.950110i \(-0.600971\pi\)
0.311916 0.950110i \(-0.399029\pi\)
\(998\) 6.00000i 0.189927i
\(999\) −1.00000 −0.0316386
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 1110.2.d.b.889.1 2
3.2 odd 2 3330.2.d.e.1999.2 2
5.2 odd 4 5550.2.a.be.1.1 1
5.3 odd 4 5550.2.a.l.1.1 1
5.4 even 2 inner 1110.2.d.b.889.2 yes 2
15.14 odd 2 3330.2.d.e.1999.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.d.b.889.1 2 1.1 even 1 trivial
1110.2.d.b.889.2 yes 2 5.4 even 2 inner
3330.2.d.e.1999.1 2 15.14 odd 2
3330.2.d.e.1999.2 2 3.2 odd 2
5550.2.a.l.1.1 1 5.3 odd 4
5550.2.a.be.1.1 1 5.2 odd 4