Properties

Label 1512.2.j.b.323.7
Level $1512$
Weight $2$
Character 1512.323
Analytic conductor $12.073$
Analytic rank $0$
Dimension $8$
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] = [1512,2,Mod(323,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("1512.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.j (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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.7
Root \(0.500000 + 1.56488i\) of defining polynomial
Character \(\chi\) \(=\) 1512.323
Dual form 1512.2.j.b.323.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} -1.12976 q^{5} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +O(q^{10})\) \(q+(1.36603 + 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} -1.12976 q^{5} -1.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +(-1.54329 - 0.413523i) q^{10} +3.86182i q^{11} -5.08658i q^{13} +(0.366025 - 1.36603i) q^{14} +(2.00000 + 3.46410i) q^{16} +2.00000i q^{17} +7.99158 q^{19} +(-1.95681 - 1.12976i) q^{20} +(-1.41352 + 5.27534i) q^{22} +7.68044 q^{23} -3.72363 q^{25} +(1.86182 - 6.94839i) q^{26} +(1.00000 - 1.73205i) q^{28} +4.73205 q^{29} +4.08658i q^{31} +(1.46410 + 5.46410i) q^{32} +(-0.732051 + 2.73205i) q^{34} +1.12976i q^{35} +8.28273i q^{37} +(10.9167 + 2.92512i) q^{38} +(-2.25953 - 2.25953i) q^{40} +8.41249i q^{41} -10.0148 q^{43} +(-3.86182 + 6.68886i) q^{44} +(10.4917 + 2.81124i) q^{46} +1.93662 q^{47} -1.00000 q^{49} +(-5.08658 - 1.36294i) q^{50} +(5.08658 - 8.81021i) q^{52} +7.72363 q^{53} -4.36294i q^{55} +(2.00000 - 2.00000i) q^{56} +(6.46410 + 1.73205i) q^{58} -10.9916i q^{59} -11.4641i q^{61} +(-1.49579 + 5.58237i) q^{62} +8.00000i q^{64} +5.74663i q^{65} +5.08658 q^{67} +(-2.00000 + 3.46410i) q^{68} +(-0.413523 + 1.54329i) q^{70} -13.6804 q^{71} -2.63706 q^{73} +(-3.03169 + 11.3144i) q^{74} +(13.8418 + 7.99158i) q^{76} +3.86182 q^{77} -7.17295i q^{79} +(-2.25953 - 3.91362i) q^{80} +(-3.07919 + 11.4917i) q^{82} -14.9052i q^{83} -2.25953i q^{85} +(-13.6804 - 3.66566i) q^{86} +(-7.72363 + 7.72363i) q^{88} +12.6889i q^{89} -5.08658 q^{91} +(13.3029 + 7.68044i) q^{92} +(2.64548 + 0.708853i) q^{94} -9.02861 q^{95} +4.17315 q^{97} +(-1.36603 - 0.366025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 8 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 8 q^{5} + 16 q^{8} + 4 q^{10} - 4 q^{14} + 16 q^{16} + 16 q^{19} - 12 q^{22} - 16 q^{23} + 32 q^{25} - 16 q^{26} + 8 q^{28} + 24 q^{29} - 16 q^{32} + 8 q^{34} + 20 q^{38} + 16 q^{40} + 8 q^{43} + 4 q^{46} + 8 q^{47} - 8 q^{49} - 8 q^{50} + 8 q^{52} + 16 q^{56} + 24 q^{58} + 12 q^{62} + 8 q^{67} - 16 q^{68} - 4 q^{70} - 32 q^{71} + 8 q^{73} - 28 q^{74} + 24 q^{76} + 16 q^{80} - 36 q^{82} - 32 q^{86} - 8 q^{91} + 24 q^{92} + 40 q^{94} - 112 q^{95} - 32 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36603 + 0.366025i 0.965926 + 0.258819i
\(3\) 0 0
\(4\) 1.73205 + 1.00000i 0.866025 + 0.500000i
\(5\) −1.12976 −0.505246 −0.252623 0.967565i \(-0.581293\pi\)
−0.252623 + 0.967565i \(0.581293\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) 0 0
\(10\) −1.54329 0.413523i −0.488030 0.130767i
\(11\) 3.86182i 1.16438i 0.813052 + 0.582191i \(0.197804\pi\)
−0.813052 + 0.582191i \(0.802196\pi\)
\(12\) 0 0
\(13\) 5.08658i 1.41076i −0.708828 0.705381i \(-0.750776\pi\)
0.708828 0.705381i \(-0.249224\pi\)
\(14\) 0.366025 1.36603i 0.0978244 0.365086i
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 7.99158 1.83339 0.916697 0.399583i \(-0.130845\pi\)
0.916697 + 0.399583i \(0.130845\pi\)
\(20\) −1.95681 1.12976i −0.437556 0.252623i
\(21\) 0 0
\(22\) −1.41352 + 5.27534i −0.301364 + 1.12471i
\(23\) 7.68044 1.60148 0.800741 0.599010i \(-0.204439\pi\)
0.800741 + 0.599010i \(0.204439\pi\)
\(24\) 0 0
\(25\) −3.72363 −0.744726
\(26\) 1.86182 6.94839i 0.365132 1.36269i
\(27\) 0 0
\(28\) 1.00000 1.73205i 0.188982 0.327327i
\(29\) 4.73205 0.878720 0.439360 0.898311i \(-0.355205\pi\)
0.439360 + 0.898311i \(0.355205\pi\)
\(30\) 0 0
\(31\) 4.08658i 0.733971i 0.930227 + 0.366985i \(0.119610\pi\)
−0.930227 + 0.366985i \(0.880390\pi\)
\(32\) 1.46410 + 5.46410i 0.258819 + 0.965926i
\(33\) 0 0
\(34\) −0.732051 + 2.73205i −0.125546 + 0.468543i
\(35\) 1.12976i 0.190965i
\(36\) 0 0
\(37\) 8.28273i 1.36167i 0.732436 + 0.680836i \(0.238383\pi\)
−0.732436 + 0.680836i \(0.761617\pi\)
\(38\) 10.9167 + 2.92512i 1.77092 + 0.474517i
\(39\) 0 0
\(40\) −2.25953 2.25953i −0.357263 0.357263i
\(41\) 8.41249i 1.31381i 0.753973 + 0.656905i \(0.228135\pi\)
−0.753973 + 0.656905i \(0.771865\pi\)
\(42\) 0 0
\(43\) −10.0148 −1.52724 −0.763620 0.645666i \(-0.776580\pi\)
−0.763620 + 0.645666i \(0.776580\pi\)
\(44\) −3.86182 + 6.68886i −0.582191 + 1.00838i
\(45\) 0 0
\(46\) 10.4917 + 2.81124i 1.54691 + 0.414494i
\(47\) 1.93662 0.282485 0.141243 0.989975i \(-0.454890\pi\)
0.141243 + 0.989975i \(0.454890\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) −5.08658 1.36294i −0.719350 0.192749i
\(51\) 0 0
\(52\) 5.08658 8.81021i 0.705381 1.22176i
\(53\) 7.72363 1.06092 0.530461 0.847709i \(-0.322019\pi\)
0.530461 + 0.847709i \(0.322019\pi\)
\(54\) 0 0
\(55\) 4.36294i 0.588299i
\(56\) 2.00000 2.00000i 0.267261 0.267261i
\(57\) 0 0
\(58\) 6.46410 + 1.73205i 0.848778 + 0.227429i
\(59\) 10.9916i 1.43098i −0.698622 0.715491i \(-0.746203\pi\)
0.698622 0.715491i \(-0.253797\pi\)
\(60\) 0 0
\(61\) 11.4641i 1.46783i −0.679243 0.733914i \(-0.737692\pi\)
0.679243 0.733914i \(-0.262308\pi\)
\(62\) −1.49579 + 5.58237i −0.189966 + 0.708961i
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 5.74663i 0.712782i
\(66\) 0 0
\(67\) 5.08658 0.621424 0.310712 0.950504i \(-0.399433\pi\)
0.310712 + 0.950504i \(0.399433\pi\)
\(68\) −2.00000 + 3.46410i −0.242536 + 0.420084i
\(69\) 0 0
\(70\) −0.413523 + 1.54329i −0.0494254 + 0.184458i
\(71\) −13.6804 −1.62357 −0.811785 0.583957i \(-0.801504\pi\)
−0.811785 + 0.583957i \(0.801504\pi\)
\(72\) 0 0
\(73\) −2.63706 −0.308644 −0.154322 0.988021i \(-0.549319\pi\)
−0.154322 + 0.988021i \(0.549319\pi\)
\(74\) −3.03169 + 11.3144i −0.352427 + 1.31527i
\(75\) 0 0
\(76\) 13.8418 + 7.99158i 1.58777 + 0.916697i
\(77\) 3.86182 0.440095
\(78\) 0 0
\(79\) 7.17295i 0.807020i −0.914975 0.403510i \(-0.867790\pi\)
0.914975 0.403510i \(-0.132210\pi\)
\(80\) −2.25953 3.91362i −0.252623 0.437556i
\(81\) 0 0
\(82\) −3.07919 + 11.4917i −0.340039 + 1.26904i
\(83\) 14.9052i 1.63606i −0.575177 0.818029i \(-0.695067\pi\)
0.575177 0.818029i \(-0.304933\pi\)
\(84\) 0 0
\(85\) 2.25953i 0.245080i
\(86\) −13.6804 3.66566i −1.47520 0.395279i
\(87\) 0 0
\(88\) −7.72363 + 7.72363i −0.823342 + 0.823342i
\(89\) 12.6889i 1.34502i 0.740090 + 0.672508i \(0.234783\pi\)
−0.740090 + 0.672508i \(0.765217\pi\)
\(90\) 0 0
\(91\) −5.08658 −0.533218
\(92\) 13.3029 + 7.68044i 1.38692 + 0.800741i
\(93\) 0 0
\(94\) 2.64548 + 0.708853i 0.272860 + 0.0731126i
\(95\) −9.02861 −0.926316
\(96\) 0 0
\(97\) 4.17315 0.423719 0.211860 0.977300i \(-0.432048\pi\)
0.211860 + 0.977300i \(0.432048\pi\)
\(98\) −1.36603 0.366025i −0.137989 0.0369741i
\(99\) 0 0
\(100\) −6.44952 3.72363i −0.644952 0.372363i
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 5.37753i 0.529863i −0.964267 0.264932i \(-0.914651\pi\)
0.964267 0.264932i \(-0.0853494\pi\)
\(104\) 10.1732 10.1732i 0.997559 0.997559i
\(105\) 0 0
\(106\) 10.5507 + 2.82705i 1.02477 + 0.274587i
\(107\) 12.8418i 1.24147i 0.784022 + 0.620733i \(0.213165\pi\)
−0.784022 + 0.620733i \(0.786835\pi\)
\(108\) 0 0
\(109\) 13.0782i 1.25266i −0.779558 0.626330i \(-0.784556\pi\)
0.779558 0.626330i \(-0.215444\pi\)
\(110\) 1.59695 5.95989i 0.152263 0.568253i
\(111\) 0 0
\(112\) 3.46410 2.00000i 0.327327 0.188982i
\(113\) 7.05496i 0.663675i −0.943337 0.331837i \(-0.892331\pi\)
0.943337 0.331837i \(-0.107669\pi\)
\(114\) 0 0
\(115\) −8.67709 −0.809143
\(116\) 8.19615 + 4.73205i 0.760994 + 0.439360i
\(117\) 0 0
\(118\) 4.02320 15.0148i 0.370365 1.38222i
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) −3.91362 −0.355784
\(122\) 4.19615 15.6603i 0.379902 1.41781i
\(123\) 0 0
\(124\) −4.08658 + 7.07816i −0.366985 + 0.635637i
\(125\) 9.85565 0.881516
\(126\) 0 0
\(127\) 4.63706i 0.411472i 0.978608 + 0.205736i \(0.0659589\pi\)
−0.978608 + 0.205736i \(0.934041\pi\)
\(128\) −2.92820 + 10.9282i −0.258819 + 0.965926i
\(129\) 0 0
\(130\) −2.10341 + 7.85005i −0.184482 + 0.688495i
\(131\) 8.36930i 0.731229i 0.930766 + 0.365615i \(0.119141\pi\)
−0.930766 + 0.365615i \(0.880859\pi\)
\(132\) 0 0
\(133\) 7.99158i 0.692958i
\(134\) 6.94839 + 1.86182i 0.600250 + 0.160836i
\(135\) 0 0
\(136\) −4.00000 + 4.00000i −0.342997 + 0.342997i
\(137\) 13.9366i 1.19069i 0.803472 + 0.595343i \(0.202984\pi\)
−0.803472 + 0.595343i \(0.797016\pi\)
\(138\) 0 0
\(139\) −13.9832 −1.18604 −0.593018 0.805189i \(-0.702064\pi\)
−0.593018 + 0.805189i \(0.702064\pi\)
\(140\) −1.12976 + 1.95681i −0.0954826 + 0.165381i
\(141\) 0 0
\(142\) −18.6878 5.00739i −1.56825 0.420211i
\(143\) 19.6434 1.64266
\(144\) 0 0
\(145\) −5.34610 −0.443970
\(146\) −3.60229 0.965230i −0.298127 0.0798830i
\(147\) 0 0
\(148\) −8.28273 + 14.3461i −0.680836 + 1.17924i
\(149\) −21.7700 −1.78347 −0.891735 0.452558i \(-0.850512\pi\)
−0.891735 + 0.452558i \(0.850512\pi\)
\(150\) 0 0
\(151\) 16.0464i 1.30584i −0.757428 0.652919i \(-0.773544\pi\)
0.757428 0.652919i \(-0.226456\pi\)
\(152\) 15.9832 + 15.9832i 1.29641 + 1.29641i
\(153\) 0 0
\(154\) 5.27534 + 1.41352i 0.425099 + 0.113905i
\(155\) 4.61687i 0.370836i
\(156\) 0 0
\(157\) 5.49572i 0.438606i −0.975657 0.219303i \(-0.929622\pi\)
0.975657 0.219303i \(-0.0703783\pi\)
\(158\) 2.62548 9.79844i 0.208872 0.779522i
\(159\) 0 0
\(160\) −1.65409 6.17315i −0.130767 0.488030i
\(161\) 7.68044i 0.605304i
\(162\) 0 0
\(163\) 1.62247 0.127082 0.0635410 0.997979i \(-0.479761\pi\)
0.0635410 + 0.997979i \(0.479761\pi\)
\(164\) −8.41249 + 14.5709i −0.656905 + 1.13779i
\(165\) 0 0
\(166\) 5.45568 20.3609i 0.423443 1.58031i
\(167\) −8.70905 −0.673926 −0.336963 0.941518i \(-0.609400\pi\)
−0.336963 + 0.941518i \(0.609400\pi\)
\(168\) 0 0
\(169\) −12.8732 −0.990250
\(170\) 0.827045 3.08658i 0.0634315 0.236729i
\(171\) 0 0
\(172\) −17.3461 10.0148i −1.32263 0.763620i
\(173\) 1.53891 0.117001 0.0585005 0.998287i \(-0.481368\pi\)
0.0585005 + 0.998287i \(0.481368\pi\)
\(174\) 0 0
\(175\) 3.72363i 0.281480i
\(176\) −13.3777 + 7.72363i −1.00838 + 0.582191i
\(177\) 0 0
\(178\) −4.64445 + 17.3333i −0.348116 + 1.29919i
\(179\) 26.7386i 1.99854i −0.0382400 0.999269i \(-0.512175\pi\)
0.0382400 0.999269i \(-0.487825\pi\)
\(180\) 0 0
\(181\) 3.10135i 0.230522i 0.993335 + 0.115261i \(0.0367704\pi\)
−0.993335 + 0.115261i \(0.963230\pi\)
\(182\) −6.94839 1.86182i −0.515049 0.138007i
\(183\) 0 0
\(184\) 15.3609 + 15.3609i 1.13242 + 1.13242i
\(185\) 9.35753i 0.687980i
\(186\) 0 0
\(187\) −7.72363 −0.564808
\(188\) 3.35433 + 1.93662i 0.244640 + 0.141243i
\(189\) 0 0
\(190\) −12.3333 3.30470i −0.894752 0.239748i
\(191\) −13.8132 −0.999489 −0.499745 0.866173i \(-0.666573\pi\)
−0.499745 + 0.866173i \(0.666573\pi\)
\(192\) 0 0
\(193\) 7.05496 0.507827 0.253913 0.967227i \(-0.418282\pi\)
0.253913 + 0.967227i \(0.418282\pi\)
\(194\) 5.70063 + 1.52748i 0.409281 + 0.109667i
\(195\) 0 0
\(196\) −1.73205 1.00000i −0.123718 0.0714286i
\(197\) 0.362748 0.0258448 0.0129224 0.999917i \(-0.495887\pi\)
0.0129224 + 0.999917i \(0.495887\pi\)
\(198\) 0 0
\(199\) 7.70029i 0.545859i 0.962034 + 0.272930i \(0.0879926\pi\)
−0.962034 + 0.272930i \(0.912007\pi\)
\(200\) −7.44726 7.44726i −0.526601 0.526601i
\(201\) 0 0
\(202\) 2.73205 + 0.732051i 0.192226 + 0.0515069i
\(203\) 4.73205i 0.332125i
\(204\) 0 0
\(205\) 9.50414i 0.663798i
\(206\) 1.96831 7.34584i 0.137139 0.511809i
\(207\) 0 0
\(208\) 17.6204 10.1732i 1.22176 0.705381i
\(209\) 30.8620i 2.13477i
\(210\) 0 0
\(211\) −14.5191 −0.999533 −0.499767 0.866160i \(-0.666581\pi\)
−0.499767 + 0.866160i \(0.666581\pi\)
\(212\) 13.3777 + 7.72363i 0.918786 + 0.530461i
\(213\) 0 0
\(214\) −4.70043 + 17.5423i −0.321315 + 1.19916i
\(215\) 11.3143 0.771632
\(216\) 0 0
\(217\) 4.08658 0.277415
\(218\) 4.78694 17.8651i 0.324212 1.20998i
\(219\) 0 0
\(220\) 4.36294 7.55684i 0.294150 0.509482i
\(221\) 10.1732 0.684320
\(222\) 0 0
\(223\) 19.2743i 1.29070i −0.763886 0.645352i \(-0.776711\pi\)
0.763886 0.645352i \(-0.223289\pi\)
\(224\) 5.46410 1.46410i 0.365086 0.0978244i
\(225\) 0 0
\(226\) 2.58229 9.63725i 0.171772 0.641060i
\(227\) 23.2341i 1.54210i −0.636773 0.771052i \(-0.719731\pi\)
0.636773 0.771052i \(-0.280269\pi\)
\(228\) 0 0
\(229\) 13.4641i 0.889733i −0.895597 0.444866i \(-0.853251\pi\)
0.895597 0.444866i \(-0.146749\pi\)
\(230\) −11.8531 3.17604i −0.781572 0.209422i
\(231\) 0 0
\(232\) 9.46410 + 9.46410i 0.621349 + 0.621349i
\(233\) 17.4641i 1.14411i 0.820215 + 0.572056i \(0.193854\pi\)
−0.820215 + 0.572056i \(0.806146\pi\)
\(234\) 0 0
\(235\) −2.18793 −0.142725
\(236\) 10.9916 19.0380i 0.715491 1.23927i
\(237\) 0 0
\(238\) 2.73205 + 0.732051i 0.177093 + 0.0474518i
\(239\) 1.82724 0.118194 0.0590972 0.998252i \(-0.481178\pi\)
0.0590972 + 0.998252i \(0.481178\pi\)
\(240\) 0 0
\(241\) −21.5193 −1.38618 −0.693089 0.720852i \(-0.743751\pi\)
−0.693089 + 0.720852i \(0.743751\pi\)
\(242\) −5.34610 1.43248i −0.343661 0.0920836i
\(243\) 0 0
\(244\) 11.4641 19.8564i 0.733914 1.27118i
\(245\) 1.12976 0.0721780
\(246\) 0 0
\(247\) 40.6498i 2.58648i
\(248\) −8.17315 + 8.17315i −0.518996 + 0.518996i
\(249\) 0 0
\(250\) 13.4631 + 3.60742i 0.851479 + 0.228153i
\(251\) 1.15818i 0.0731035i −0.999332 0.0365517i \(-0.988363\pi\)
0.999332 0.0365517i \(-0.0116374\pi\)
\(252\) 0 0
\(253\) 29.6604i 1.86474i
\(254\) −1.69728 + 6.33434i −0.106497 + 0.397452i
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) 11.0812i 0.691224i −0.938377 0.345612i \(-0.887671\pi\)
0.938377 0.345612i \(-0.112329\pi\)
\(258\) 0 0
\(259\) 8.28273 0.514664
\(260\) −5.74663 + 9.95346i −0.356391 + 0.617287i
\(261\) 0 0
\(262\) −3.06338 + 11.4327i −0.189256 + 0.706313i
\(263\) 10.7354 0.661973 0.330987 0.943635i \(-0.392619\pi\)
0.330987 + 0.943635i \(0.392619\pi\)
\(264\) 0 0
\(265\) −8.72589 −0.536027
\(266\) 2.92512 10.9167i 0.179351 0.669346i
\(267\) 0 0
\(268\) 8.81021 + 5.08658i 0.538169 + 0.310712i
\(269\) 5.75204 0.350708 0.175354 0.984505i \(-0.443893\pi\)
0.175354 + 0.984505i \(0.443893\pi\)
\(270\) 0 0
\(271\) 1.33735i 0.0812380i 0.999175 + 0.0406190i \(0.0129330\pi\)
−0.999175 + 0.0406190i \(0.987067\pi\)
\(272\) −6.92820 + 4.00000i −0.420084 + 0.242536i
\(273\) 0 0
\(274\) −5.10116 + 19.0378i −0.308172 + 1.15011i
\(275\) 14.3800i 0.867145i
\(276\) 0 0
\(277\) 3.13537i 0.188386i 0.995554 + 0.0941930i \(0.0300271\pi\)
−0.995554 + 0.0941930i \(0.969973\pi\)
\(278\) −19.1014 5.11819i −1.14562 0.306969i
\(279\) 0 0
\(280\) −2.25953 + 2.25953i −0.135033 + 0.135033i
\(281\) 13.3839i 0.798416i 0.916860 + 0.399208i \(0.130715\pi\)
−0.916860 + 0.399208i \(0.869285\pi\)
\(282\) 0 0
\(283\) 21.8104 1.29649 0.648247 0.761430i \(-0.275502\pi\)
0.648247 + 0.761430i \(0.275502\pi\)
\(284\) −23.6952 13.6804i −1.40605 0.811785i
\(285\) 0 0
\(286\) 26.8334 + 7.18999i 1.58669 + 0.425153i
\(287\) 8.41249 0.496574
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) −7.30291 1.95681i −0.428842 0.114908i
\(291\) 0 0
\(292\) −4.56752 2.63706i −0.267294 0.154322i
\(293\) −30.6982 −1.79341 −0.896705 0.442629i \(-0.854046\pi\)
−0.896705 + 0.442629i \(0.854046\pi\)
\(294\) 0 0
\(295\) 12.4179i 0.722998i
\(296\) −16.5655 + 16.5655i −0.962848 + 0.962848i
\(297\) 0 0
\(298\) −29.7384 7.96838i −1.72270 0.461596i
\(299\) 39.0671i 2.25931i
\(300\) 0 0
\(301\) 10.0148i 0.577242i
\(302\) 5.87339 21.9198i 0.337976 1.26134i
\(303\) 0 0
\(304\) 15.9832 + 27.6836i 0.916697 + 1.58777i
\(305\) 12.9517i 0.741614i
\(306\) 0 0
\(307\) 9.10977 0.519922 0.259961 0.965619i \(-0.416290\pi\)
0.259961 + 0.965619i \(0.416290\pi\)
\(308\) 6.68886 + 3.86182i 0.381133 + 0.220047i
\(309\) 0 0
\(310\) 1.68989 6.30676i 0.0959794 0.358200i
\(311\) −27.5275 −1.56094 −0.780470 0.625193i \(-0.785020\pi\)
−0.780470 + 0.625193i \(0.785020\pi\)
\(312\) 0 0
\(313\) 5.36294 0.303131 0.151566 0.988447i \(-0.451568\pi\)
0.151566 + 0.988447i \(0.451568\pi\)
\(314\) 2.01157 7.50729i 0.113520 0.423661i
\(315\) 0 0
\(316\) 7.17295 12.4239i 0.403510 0.698900i
\(317\) 7.30832 0.410476 0.205238 0.978712i \(-0.434203\pi\)
0.205238 + 0.978712i \(0.434203\pi\)
\(318\) 0 0
\(319\) 18.2743i 1.02316i
\(320\) 9.03812i 0.505246i
\(321\) 0 0
\(322\) 2.81124 10.4917i 0.156664 0.584678i
\(323\) 15.9832i 0.889327i
\(324\) 0 0
\(325\) 18.9405i 1.05063i
\(326\) 2.21634 + 0.593866i 0.122752 + 0.0328912i
\(327\) 0 0
\(328\) −16.8250 + 16.8250i −0.929004 + 0.929004i
\(329\) 1.93662i 0.106769i
\(330\) 0 0
\(331\) 10.5823 0.581655 0.290828 0.956775i \(-0.406069\pi\)
0.290828 + 0.956775i \(0.406069\pi\)
\(332\) 14.9052 25.8166i 0.818029 1.41687i
\(333\) 0 0
\(334\) −11.8968 3.18773i −0.650963 0.174425i
\(335\) −5.74663 −0.313972
\(336\) 0 0
\(337\) −4.29095 −0.233743 −0.116872 0.993147i \(-0.537287\pi\)
−0.116872 + 0.993147i \(0.537287\pi\)
\(338\) −17.5852 4.71193i −0.956508 0.256295i
\(339\) 0 0
\(340\) 2.25953 3.91362i 0.122540 0.212246i
\(341\) −15.7816 −0.854622
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −20.0296 20.0296i −1.07992 1.07992i
\(345\) 0 0
\(346\) 2.10219 + 0.563280i 0.113014 + 0.0302821i
\(347\) 10.3405i 0.555107i −0.960710 0.277554i \(-0.910476\pi\)
0.960710 0.277554i \(-0.0895236\pi\)
\(348\) 0 0
\(349\) 11.1182i 0.595143i 0.954700 + 0.297572i \(0.0961767\pi\)
−0.954700 + 0.297572i \(0.903823\pi\)
\(350\) −1.36294 + 5.08658i −0.0728524 + 0.271889i
\(351\) 0 0
\(352\) −21.1014 + 5.65409i −1.12471 + 0.301364i
\(353\) 23.3407i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(354\) 0 0
\(355\) 15.4557 0.820302
\(356\) −12.6889 + 21.9778i −0.672508 + 1.16482i
\(357\) 0 0
\(358\) 9.78701 36.5256i 0.517259 1.93044i
\(359\) −22.3923 −1.18182 −0.590910 0.806737i \(-0.701231\pi\)
−0.590910 + 0.806737i \(0.701231\pi\)
\(360\) 0 0
\(361\) 44.8654 2.36133
\(362\) −1.13517 + 4.23653i −0.0596634 + 0.222667i
\(363\) 0 0
\(364\) −8.81021 5.08658i −0.461780 0.266609i
\(365\) 2.97925 0.155941
\(366\) 0 0
\(367\) 1.93696i 0.101109i −0.998721 0.0505543i \(-0.983901\pi\)
0.998721 0.0505543i \(-0.0160988\pi\)
\(368\) 15.3609 + 26.6058i 0.800741 + 1.38692i
\(369\) 0 0
\(370\) 3.42510 12.7826i 0.178062 0.664537i
\(371\) 7.72363i 0.400991i
\(372\) 0 0
\(373\) 15.1246i 0.783120i −0.920153 0.391560i \(-0.871936\pi\)
0.920153 0.391560i \(-0.128064\pi\)
\(374\) −10.5507 2.82705i −0.545563 0.146183i
\(375\) 0 0
\(376\) 3.87325 + 3.87325i 0.199747 + 0.199747i
\(377\) 24.0699i 1.23966i
\(378\) 0 0
\(379\) 6.15837 0.316334 0.158167 0.987412i \(-0.449442\pi\)
0.158167 + 0.987412i \(0.449442\pi\)
\(380\) −15.6380 9.02861i −0.802213 0.463158i
\(381\) 0 0
\(382\) −18.8692 5.05599i −0.965433 0.258687i
\(383\) −14.8822 −0.760445 −0.380222 0.924895i \(-0.624153\pi\)
−0.380222 + 0.924895i \(0.624153\pi\)
\(384\) 0 0
\(385\) −4.36294 −0.222356
\(386\) 9.63725 + 2.58229i 0.490523 + 0.131435i
\(387\) 0 0
\(388\) 7.22811 + 4.17315i 0.366952 + 0.211860i
\(389\) 1.82685 0.0926250 0.0463125 0.998927i \(-0.485253\pi\)
0.0463125 + 0.998927i \(0.485253\pi\)
\(390\) 0 0
\(391\) 15.3609i 0.776833i
\(392\) −2.00000 2.00000i −0.101015 0.101015i
\(393\) 0 0
\(394\) 0.495523 + 0.132775i 0.0249641 + 0.00668911i
\(395\) 8.10375i 0.407744i
\(396\) 0 0
\(397\) 2.18999i 0.109912i −0.998489 0.0549562i \(-0.982498\pi\)
0.998489 0.0549562i \(-0.0175019\pi\)
\(398\) −2.81850 + 10.5188i −0.141279 + 0.527259i
\(399\) 0 0
\(400\) −7.44726 12.8990i −0.372363 0.644952i
\(401\) 3.59071i 0.179312i 0.995973 + 0.0896558i \(0.0285767\pi\)
−0.995973 + 0.0896558i \(0.971423\pi\)
\(402\) 0 0
\(403\) 20.7867 1.03546
\(404\) 3.46410 + 2.00000i 0.172345 + 0.0995037i
\(405\) 0 0
\(406\) 1.73205 6.46410i 0.0859602 0.320808i
\(407\) −31.9864 −1.58551
\(408\) 0 0
\(409\) −35.3588 −1.74838 −0.874191 0.485583i \(-0.838607\pi\)
−0.874191 + 0.485583i \(0.838607\pi\)
\(410\) 3.47876 12.9829i 0.171804 0.641179i
\(411\) 0 0
\(412\) 5.37753 9.31415i 0.264932 0.458875i
\(413\) −10.9916 −0.540860
\(414\) 0 0
\(415\) 16.8394i 0.826612i
\(416\) 27.7936 7.44726i 1.36269 0.365132i
\(417\) 0 0
\(418\) −11.2963 + 42.1583i −0.552519 + 2.06203i
\(419\) 2.44952i 0.119667i −0.998208 0.0598334i \(-0.980943\pi\)
0.998208 0.0598334i \(-0.0190569\pi\)
\(420\) 0 0
\(421\) 6.26589i 0.305381i −0.988274 0.152690i \(-0.951206\pi\)
0.988274 0.152690i \(-0.0487937\pi\)
\(422\) −19.8334 5.31434i −0.965475 0.258698i
\(423\) 0 0
\(424\) 15.4473 + 15.4473i 0.750185 + 0.750185i
\(425\) 7.44726i 0.361245i
\(426\) 0 0
\(427\) −11.4641 −0.554787
\(428\) −12.8418 + 22.2427i −0.620733 + 1.07514i
\(429\) 0 0
\(430\) 15.4557 + 4.14134i 0.745339 + 0.199713i
\(431\) −3.01177 −0.145072 −0.0725359 0.997366i \(-0.523109\pi\)
−0.0725359 + 0.997366i \(0.523109\pi\)
\(432\) 0 0
\(433\) −7.00020 −0.336408 −0.168204 0.985752i \(-0.553797\pi\)
−0.168204 + 0.985752i \(0.553797\pi\)
\(434\) 5.58237 + 1.49579i 0.267962 + 0.0718002i
\(435\) 0 0
\(436\) 13.0782 22.6520i 0.626330 1.08484i
\(437\) 61.3789 2.93615
\(438\) 0 0
\(439\) 1.33735i 0.0638281i −0.999491 0.0319140i \(-0.989840\pi\)
0.999491 0.0319140i \(-0.0101603\pi\)
\(440\) 8.72589 8.72589i 0.415990 0.415990i
\(441\) 0 0
\(442\) 13.8968 + 3.72363i 0.661002 + 0.177115i
\(443\) 1.90821i 0.0906618i 0.998972 + 0.0453309i \(0.0144342\pi\)
−0.998972 + 0.0453309i \(0.985566\pi\)
\(444\) 0 0
\(445\) 14.3354i 0.679565i
\(446\) 7.05489 26.3292i 0.334059 1.24672i
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) 10.0695i 0.475211i 0.971362 + 0.237606i \(0.0763625\pi\)
−0.971362 + 0.237606i \(0.923637\pi\)
\(450\) 0 0
\(451\) −32.4875 −1.52978
\(452\) 7.05496 12.2195i 0.331837 0.574759i
\(453\) 0 0
\(454\) 8.50428 31.7384i 0.399126 1.48956i
\(455\) 5.74663 0.269406
\(456\) 0 0
\(457\) 10.3205 0.482773 0.241387 0.970429i \(-0.422398\pi\)
0.241387 + 0.970429i \(0.422398\pi\)
\(458\) 4.92820 18.3923i 0.230280 0.859416i
\(459\) 0 0
\(460\) −15.0292 8.67709i −0.700738 0.404572i
\(461\) −17.9420 −0.835644 −0.417822 0.908529i \(-0.637206\pi\)
−0.417822 + 0.908529i \(0.637206\pi\)
\(462\) 0 0
\(463\) 21.6664i 1.00692i 0.864017 + 0.503462i \(0.167941\pi\)
−0.864017 + 0.503462i \(0.832059\pi\)
\(464\) 9.46410 + 16.3923i 0.439360 + 0.760994i
\(465\) 0 0
\(466\) −6.39230 + 23.8564i −0.296118 + 1.10513i
\(467\) 18.3525i 0.849251i −0.905369 0.424625i \(-0.860406\pi\)
0.905369 0.424625i \(-0.139594\pi\)
\(468\) 0 0
\(469\) 5.08658i 0.234876i
\(470\) −2.98877 0.800837i −0.137861 0.0369399i
\(471\) 0 0
\(472\) 21.9832 21.9832i 1.01186 1.01186i
\(473\) 38.6752i 1.77829i
\(474\) 0 0
\(475\) −29.7577 −1.36538
\(476\) 3.46410 + 2.00000i 0.158777 + 0.0916698i
\(477\) 0 0
\(478\) 2.49606 + 0.668816i 0.114167 + 0.0305910i
\(479\) −14.6518 −0.669459 −0.334730 0.942314i \(-0.608645\pi\)
−0.334730 + 0.942314i \(0.608645\pi\)
\(480\) 0 0
\(481\) 42.1307 1.92100
\(482\) −29.3958 7.87659i −1.33894 0.358769i
\(483\) 0 0
\(484\) −6.77859 3.91362i −0.308118 0.177892i
\(485\) −4.71468 −0.214083
\(486\) 0 0
\(487\) 6.60789i 0.299432i 0.988729 + 0.149716i \(0.0478360\pi\)
−0.988729 + 0.149716i \(0.952164\pi\)
\(488\) 22.9282 22.9282i 1.03791 1.03791i
\(489\) 0 0
\(490\) 1.54329 + 0.413523i 0.0697186 + 0.0186810i
\(491\) 7.08323i 0.319662i 0.987144 + 0.159831i \(0.0510949\pi\)
−0.987144 + 0.159831i \(0.948905\pi\)
\(492\) 0 0
\(493\) 9.46410i 0.426242i
\(494\) 14.8788 55.5286i 0.669431 2.49835i
\(495\) 0 0
\(496\) −14.1563 + 8.17315i −0.635637 + 0.366985i
\(497\) 13.6804i 0.613652i
\(498\) 0 0
\(499\) −7.13297 −0.319316 −0.159658 0.987172i \(-0.551039\pi\)
−0.159658 + 0.987172i \(0.551039\pi\)
\(500\) 17.0705 + 9.85565i 0.763416 + 0.440758i
\(501\) 0 0
\(502\) 0.423922 1.58210i 0.0189206 0.0706125i
\(503\) 14.7321 0.656870 0.328435 0.944527i \(-0.393479\pi\)
0.328435 + 0.944527i \(0.393479\pi\)
\(504\) 0 0
\(505\) −2.25953 −0.100548
\(506\) −10.8565 + 40.5169i −0.482629 + 1.80120i
\(507\) 0 0
\(508\) −4.63706 + 8.03162i −0.205736 + 0.356345i
\(509\) −6.21955 −0.275677 −0.137838 0.990455i \(-0.544015\pi\)
−0.137838 + 0.990455i \(0.544015\pi\)
\(510\) 0 0
\(511\) 2.63706i 0.116657i
\(512\) −16.0000 + 16.0000i −0.707107 + 0.707107i
\(513\) 0 0
\(514\) 4.05599 15.1372i 0.178902 0.667671i
\(515\) 6.07534i 0.267711i
\(516\) 0 0
\(517\) 7.47888i 0.328921i
\(518\) 11.3144 + 3.03169i 0.497127 + 0.133205i
\(519\) 0 0
\(520\) −11.4933 + 11.4933i −0.504013 + 0.504013i
\(521\) 20.0897i 0.880147i −0.897962 0.440073i \(-0.854952\pi\)
0.897962 0.440073i \(-0.145048\pi\)
\(522\) 0 0
\(523\) 25.5885 1.11891 0.559453 0.828862i \(-0.311011\pi\)
0.559453 + 0.828862i \(0.311011\pi\)
\(524\) −8.36930 + 14.4961i −0.365615 + 0.633263i
\(525\) 0 0
\(526\) 14.6648 + 3.92943i 0.639417 + 0.171331i
\(527\) −8.17315 −0.356028
\(528\) 0 0
\(529\) 35.9892 1.56475
\(530\) −11.9198 3.19390i −0.517762 0.138734i
\(531\) 0 0
\(532\) 7.99158 13.8418i 0.346479 0.600119i
\(533\) 42.7908 1.85347
\(534\) 0 0
\(535\) 14.5082i 0.627246i
\(536\) 10.1732 + 10.1732i 0.439413 + 0.439413i
\(537\) 0 0
\(538\) 7.85744 + 2.10539i 0.338758 + 0.0907700i
\(539\) 3.86182i 0.166340i
\(540\) 0 0
\(541\) 25.7932i 1.10894i 0.832205 + 0.554469i \(0.187078\pi\)
−0.832205 + 0.554469i \(0.812922\pi\)
\(542\) −0.489503 + 1.82685i −0.0210260 + 0.0784699i
\(543\) 0 0
\(544\) −10.9282 + 2.92820i −0.468543 + 0.125546i
\(545\) 14.7752i 0.632902i
\(546\) 0 0
\(547\) −1.41771 −0.0606167 −0.0303084 0.999541i \(-0.509649\pi\)
−0.0303084 + 0.999541i \(0.509649\pi\)
\(548\) −13.9366 + 24.1389i −0.595343 + 1.03116i
\(549\) 0 0
\(550\) 5.26344 19.6434i 0.224434 0.837598i
\(551\) 37.8166 1.61104
\(552\) 0 0
\(553\) −7.17295 −0.305025
\(554\) −1.14762 + 4.28299i −0.0487579 + 0.181967i
\(555\) 0 0
\(556\) −24.2195 13.9832i −1.02714 0.593018i
\(557\) 4.36275 0.184856 0.0924278 0.995719i \(-0.470537\pi\)
0.0924278 + 0.995719i \(0.470537\pi\)
\(558\) 0 0
\(559\) 50.9409i 2.15457i
\(560\) −3.91362 + 2.25953i −0.165381 + 0.0954826i
\(561\) 0 0
\(562\) −4.89884 + 18.2827i −0.206645 + 0.771210i
\(563\) 25.4007i 1.07051i 0.844690 + 0.535256i \(0.179785\pi\)
−0.844690 + 0.535256i \(0.820215\pi\)
\(564\) 0 0
\(565\) 7.97044i 0.335319i
\(566\) 29.7936 + 7.98316i 1.25232 + 0.335557i
\(567\) 0 0
\(568\) −27.3609 27.3609i −1.14804 1.14804i
\(569\) 24.9746i 1.04699i 0.852029 + 0.523495i \(0.175372\pi\)
−0.852029 + 0.523495i \(0.824628\pi\)
\(570\) 0 0
\(571\) −8.02956 −0.336026 −0.168013 0.985785i \(-0.553735\pi\)
−0.168013 + 0.985785i \(0.553735\pi\)
\(572\) 34.0234 + 19.6434i 1.42259 + 0.821332i
\(573\) 0 0
\(574\) 11.4917 + 3.07919i 0.479653 + 0.128523i
\(575\) −28.5991 −1.19267
\(576\) 0 0
\(577\) −21.7808 −0.906748 −0.453374 0.891320i \(-0.649780\pi\)
−0.453374 + 0.891320i \(0.649780\pi\)
\(578\) 17.7583 + 4.75833i 0.738649 + 0.197920i
\(579\) 0 0
\(580\) −9.25973 5.34610i −0.384489 0.221985i
\(581\) −14.9052 −0.618372
\(582\) 0 0
\(583\) 29.8272i 1.23532i
\(584\) −5.27411 5.27411i −0.218244 0.218244i
\(585\) 0 0
\(586\) −41.9346 11.2363i −1.73230 0.464169i
\(587\) 13.2913i 0.548592i 0.961645 + 0.274296i \(0.0884449\pi\)
−0.961645 + 0.274296i \(0.911555\pi\)
\(588\) 0 0
\(589\) 32.6582i 1.34566i
\(590\) −4.54527 + 16.9632i −0.187126 + 0.698363i
\(591\) 0 0
\(592\) −28.6922 + 16.5655i −1.17924 + 0.680836i
\(593\) 0.746075i 0.0306376i 0.999883 + 0.0153188i \(0.00487632\pi\)
−0.999883 + 0.0153188i \(0.995124\pi\)
\(594\) 0 0
\(595\) −2.25953 −0.0926317
\(596\) −37.7068 21.7700i −1.54453 0.891735i
\(597\) 0 0
\(598\) 14.2996 53.3667i 0.584753 2.18233i
\(599\) 42.1595 1.72259 0.861296 0.508104i \(-0.169654\pi\)
0.861296 + 0.508104i \(0.169654\pi\)
\(600\) 0 0
\(601\) −12.6371 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(602\) −3.66566 + 13.6804i −0.149401 + 0.557573i
\(603\) 0 0
\(604\) 16.0464 27.7932i 0.652919 1.13089i
\(605\) 4.42147 0.179758
\(606\) 0 0
\(607\) 44.5950i 1.81006i −0.425352 0.905028i \(-0.639850\pi\)
0.425352 0.905028i \(-0.360150\pi\)
\(608\) 11.7005 + 43.6668i 0.474517 + 1.77092i
\(609\) 0 0
\(610\) −4.74067 + 17.6924i −0.191944 + 0.716345i
\(611\) 9.85078i 0.398520i
\(612\) 0 0
\(613\) 16.6859i 0.673935i 0.941516 + 0.336968i \(0.109401\pi\)
−0.941516 + 0.336968i \(0.890599\pi\)
\(614\) 12.4442 + 3.33441i 0.502206 + 0.134566i
\(615\) 0 0
\(616\) 7.72363 + 7.72363i 0.311194 + 0.311194i
\(617\) 30.1225i 1.21269i 0.795203 + 0.606343i \(0.207364\pi\)
−0.795203 + 0.606343i \(0.792636\pi\)
\(618\) 0 0
\(619\) 32.5002 1.30629 0.653147 0.757231i \(-0.273448\pi\)
0.653147 + 0.757231i \(0.273448\pi\)
\(620\) 4.61687 7.99665i 0.185418 0.321153i
\(621\) 0 0
\(622\) −37.6032 10.0758i −1.50775 0.404001i
\(623\) 12.6889 0.508368
\(624\) 0 0
\(625\) 7.48359 0.299343
\(626\) 7.32592 + 1.96297i 0.292803 + 0.0784562i
\(627\) 0 0
\(628\) 5.49572 9.51886i 0.219303 0.379844i
\(629\) −16.5655 −0.660508
\(630\) 0 0
\(631\) 18.8734i 0.751340i −0.926754 0.375670i \(-0.877413\pi\)
0.926754 0.375670i \(-0.122587\pi\)
\(632\) 14.3459 14.3459i 0.570650 0.570650i
\(633\) 0 0
\(634\) 9.98336 + 2.67503i 0.396490 + 0.106239i
\(635\) 5.23878i 0.207895i
\(636\) 0 0
\(637\) 5.08658i 0.201537i
\(638\) −6.68886 + 24.9632i −0.264815 + 0.988301i
\(639\) 0 0
\(640\) 3.30818 12.3463i 0.130767 0.488030i
\(641\) 13.1417i 0.519067i −0.965734 0.259534i \(-0.916431\pi\)
0.965734 0.259534i \(-0.0835688\pi\)
\(642\) 0 0
\(643\) 11.3121 0.446105 0.223053 0.974806i \(-0.428398\pi\)
0.223053 + 0.974806i \(0.428398\pi\)
\(644\) 7.68044 13.3029i 0.302652 0.524208i
\(645\) 0 0
\(646\) −5.85024 + 21.8334i −0.230175 + 0.859024i
\(647\) −10.1038 −0.397219 −0.198610 0.980079i \(-0.563643\pi\)
−0.198610 + 0.980079i \(0.563643\pi\)
\(648\) 0 0
\(649\) 42.4475 1.66621
\(650\) −6.93272 + 25.8732i −0.271923 + 1.01483i
\(651\) 0 0
\(652\) 2.81021 + 1.62247i 0.110056 + 0.0635410i
\(653\) −18.2661 −0.714807 −0.357404 0.933950i \(-0.616338\pi\)
−0.357404 + 0.933950i \(0.616338\pi\)
\(654\) 0 0
\(655\) 9.45534i 0.369451i
\(656\) −29.1417 + 16.8250i −1.13779 + 0.656905i
\(657\) 0 0
\(658\) 0.708853 2.64548i 0.0276340 0.103131i
\(659\) 37.9781i 1.47942i −0.672927 0.739709i \(-0.734963\pi\)
0.672927 0.739709i \(-0.265037\pi\)
\(660\) 0 0
\(661\) 8.98899i 0.349631i −0.984601 0.174816i \(-0.944067\pi\)
0.984601 0.174816i \(-0.0559329\pi\)
\(662\) 14.4557 + 3.87339i 0.561836 + 0.150544i
\(663\) 0 0
\(664\) 29.8104 29.8104i 1.15687 1.15687i
\(665\) 9.02861i 0.350114i
\(666\) 0 0
\(667\) 36.3442 1.40725
\(668\) −15.0845 8.70905i −0.583637 0.336963i
\(669\) 0 0
\(670\) −7.85005 2.10341i −0.303274 0.0812620i
\(671\) 44.2722 1.70911
\(672\) 0 0
\(673\) 21.4937 0.828520 0.414260 0.910159i \(-0.364040\pi\)
0.414260 + 0.910159i \(0.364040\pi\)
\(674\) −5.86155 1.57060i −0.225778 0.0604971i
\(675\) 0 0
\(676\) −22.2971 12.8732i −0.857581 0.495125i
\(677\) 13.5920 0.522383 0.261192 0.965287i \(-0.415885\pi\)
0.261192 + 0.965287i \(0.415885\pi\)
\(678\) 0 0
\(679\) 4.17315i 0.160151i
\(680\) 4.51906 4.51906i 0.173298 0.173298i
\(681\) 0 0
\(682\) −21.5581 5.77647i −0.825501 0.221192i
\(683\) 13.0200i 0.498196i −0.968478 0.249098i \(-0.919866\pi\)
0.968478 0.249098i \(-0.0801342\pi\)
\(684\) 0 0
\(685\) 15.7451i 0.601590i
\(686\) −0.366025 + 1.36603i −0.0139749 + 0.0521551i
\(687\) 0 0
\(688\) −20.0296 34.6922i −0.763620 1.32263i
\(689\) 39.2868i 1.49671i
\(690\) 0 0
\(691\) −8.07556 −0.307209 −0.153604 0.988132i \(-0.549088\pi\)
−0.153604 + 0.988132i \(0.549088\pi\)
\(692\) 2.66547 + 1.53891i 0.101326 + 0.0585005i
\(693\) 0 0
\(694\) 3.78489 14.1254i 0.143672 0.536192i
\(695\) 15.7977 0.599240
\(696\) 0 0
\(697\) −16.8250 −0.637292
\(698\) −4.06954 + 15.1877i −0.154034 + 0.574864i
\(699\) 0 0
\(700\) −3.72363 + 6.44952i −0.140740 + 0.243769i
\(701\) 9.91978 0.374665 0.187333 0.982297i \(-0.440016\pi\)
0.187333 + 0.982297i \(0.440016\pi\)
\(702\) 0 0
\(703\) 66.1921i 2.49648i
\(704\) −30.8945 −1.16438
\(705\) 0 0
\(706\) −8.54329 + 31.8840i −0.321531 + 1.19997i
\(707\) 2.00000i 0.0752177i
\(708\) 0 0
\(709\) 33.4709i 1.25702i −0.777800 0.628512i \(-0.783664\pi\)
0.777800 0.628512i \(-0.216336\pi\)
\(710\) 21.1129 + 5.65717i 0.792351 + 0.212310i
\(711\) 0 0
\(712\) −25.3777 + 25.3777i −0.951070 + 0.951070i
\(713\) 31.3867i 1.17544i
\(714\) 0 0
\(715\) −22.1924 −0.829950
\(716\) 26.7386 46.3126i 0.999269 1.73078i
\(717\) 0 0
\(718\) −30.5885 8.19615i −1.14155 0.305878i
\(719\) 29.1075 1.08553 0.542764 0.839886i \(-0.317378\pi\)
0.542764 + 0.839886i \(0.317378\pi\)
\(720\) 0 0
\(721\) −5.37753 −0.200270
\(722\) 61.2872 + 16.4219i 2.28087 + 0.611158i
\(723\) 0 0
\(724\) −3.10135 + 5.37170i −0.115261 + 0.199638i
\(725\) −17.6204 −0.654406
\(726\) 0 0
\(727\) 2.39230i 0.0887257i −0.999015 0.0443628i \(-0.985874\pi\)
0.999015 0.0443628i \(-0.0141258\pi\)
\(728\) −10.1732 10.1732i −0.377042 0.377042i
\(729\) 0 0
\(730\) 4.06974 + 1.09048i 0.150628 + 0.0403606i
\(731\) 20.0296i 0.740820i
\(732\) 0 0
\(733\) 20.3315i 0.750962i 0.926830 + 0.375481i \(0.122522\pi\)
−0.926830 + 0.375481i \(0.877478\pi\)
\(734\) 0.708977 2.64594i 0.0261688 0.0976634i
\(735\) 0 0
\(736\) 11.2449 + 41.9667i 0.414494 + 1.54691i
\(737\) 19.6434i 0.723574i
\(738\) 0 0
\(739\) 36.5823 1.34570 0.672851 0.739778i \(-0.265070\pi\)
0.672851 + 0.739778i \(0.265070\pi\)
\(740\) 9.35753 16.2077i 0.343990 0.595808i
\(741\) 0 0
\(742\) 2.82705 10.5507i 0.103784 0.387328i
\(743\) 13.9864 0.513110 0.256555 0.966530i \(-0.417413\pi\)
0.256555 + 0.966530i \(0.417413\pi\)
\(744\) 0 0
\(745\) 24.5950 0.901091
\(746\) 5.53597 20.6605i 0.202686 0.756435i
\(747\) 0 0
\(748\) −13.3777 7.72363i −0.489138 0.282404i
\(749\) 12.8418 0.469230
\(750\) 0 0
\(751\) 47.5657i 1.73570i 0.496830 + 0.867848i \(0.334497\pi\)
−0.496830 + 0.867848i \(0.665503\pi\)
\(752\) 3.87325 + 6.70866i 0.141243 + 0.244640i
\(753\) 0 0
\(754\) 8.81021 32.8801i 0.320849 1.19742i
\(755\) 18.1287i 0.659769i
\(756\) 0 0
\(757\) 41.3844i 1.50414i −0.659082 0.752071i \(-0.729055\pi\)
0.659082 0.752071i \(-0.270945\pi\)
\(758\) 8.41249 + 2.25412i 0.305555 + 0.0818733i
\(759\) 0 0
\(760\) −18.0572 18.0572i −0.655004 0.655004i
\(761\) 19.8335i 0.718966i 0.933152 + 0.359483i \(0.117047\pi\)
−0.933152 + 0.359483i \(0.882953\pi\)
\(762\) 0 0
\(763\) −13.0782 −0.473461
\(764\) −23.9252 13.8132i −0.865583 0.499745i
\(765\) 0 0
\(766\) −20.3295 5.44726i −0.734533 0.196818i
\(767\) −55.9095 −2.01878
\(768\) 0 0
\(769\) −20.2739 −0.731096 −0.365548 0.930792i \(-0.619118\pi\)
−0.365548 + 0.930792i \(0.619118\pi\)
\(770\) −5.95989 1.59695i −0.214780 0.0575500i
\(771\) 0 0
\(772\) 12.2195 + 7.05496i 0.439791 + 0.253913i
\(773\) −0.271102 −0.00975088 −0.00487544 0.999988i \(-0.501552\pi\)
−0.00487544 + 0.999988i \(0.501552\pi\)
\(774\) 0 0
\(775\) 15.2169i 0.546607i
\(776\) 8.34630 + 8.34630i 0.299615 + 0.299615i
\(777\) 0 0
\(778\) 2.49552 + 0.668673i 0.0894689 + 0.0239731i
\(779\) 67.2291i 2.40873i
\(780\) 0 0
\(781\) 52.8313i 1.89045i
\(782\) −5.62247 + 20.9834i −0.201059 + 0.750363i
\(783\) 0 0
\(784\) −2.00000 3.46410i −0.0714286 0.123718i
\(785\) 6.20887i 0.221604i
\(786\) 0 0
\(787\) −50.0760 −1.78501 −0.892507 0.451033i \(-0.851056\pi\)
−0.892507 + 0.451033i \(0.851056\pi\)
\(788\) 0.628299 + 0.362748i 0.0223822 + 0.0129224i
\(789\) 0 0
\(790\) −2.96618 + 11.0699i −0.105532 + 0.393850i
\(791\) −7.05496 −0.250845
\(792\) 0 0
\(793\) −58.3130 −2.07076
\(794\) 0.801592 2.99158i 0.0284474 0.106167i
\(795\) 0 0
\(796\) −7.70029 + 13.3373i −0.272930 + 0.472728i
\(797\) −27.1762 −0.962629 −0.481314 0.876548i \(-0.659840\pi\)
−0.481314 + 0.876548i \(0.659840\pi\)
\(798\) 0 0
\(799\) 3.87325i 0.137026i
\(800\) −5.45177 20.3463i −0.192749 0.719350i
\(801\) 0 0
\(802\) −1.31429 + 4.90501i −0.0464093 + 0.173202i
\(803\) 10.1838i 0.359379i
\(804\) 0 0
\(805\) 8.67709i 0.305827i
\(806\) 28.3951 + 7.60845i 1.00018 + 0.267996i
\(807\) 0 0
\(808\) 4.00000 + 4.00000i 0.140720 + 0.140720i
\(809\) 33.5566i 1.17979i −0.807480 0.589894i \(-0.799169\pi\)
0.807480 0.589894i \(-0.200831\pi\)
\(810\) 0 0
\(811\) −3.54206 −0.124379 −0.0621893 0.998064i \(-0.519808\pi\)
−0.0621893 + 0.998064i \(0.519808\pi\)
\(812\) 4.73205 8.19615i 0.166062 0.287629i
\(813\) 0 0
\(814\) −43.6942 11.7078i −1.53148 0.410359i
\(815\) −1.83301 −0.0642077
\(816\) 0 0
\(817\) −80.0339 −2.80003
\(818\) −48.3010 12.9422i −1.68881 0.452514i
\(819\) 0 0
\(820\) 9.50414 16.4617i 0.331899 0.574866i
\(821\) −31.0847 −1.08486 −0.542431 0.840100i \(-0.682496\pi\)
−0.542431 + 0.840100i \(0.682496\pi\)
\(822\) 0 0
\(823\) 36.2323i 1.26298i −0.775385 0.631489i \(-0.782444\pi\)
0.775385 0.631489i \(-0.217556\pi\)
\(824\) 10.7551 10.7551i 0.374670 0.374670i
\(825\) 0 0
\(826\) −15.0148 4.02320i −0.522431 0.139985i
\(827\) 53.4991i 1.86034i 0.367123 + 0.930172i \(0.380343\pi\)
−0.367123 + 0.930172i \(0.619657\pi\)
\(828\) 0 0
\(829\) 46.2003i 1.60460i 0.596920 + 0.802301i \(0.296391\pi\)
−0.596920 + 0.802301i \(0.703609\pi\)
\(830\) −6.16364 + 23.0030i −0.213943 + 0.798446i
\(831\) 0 0
\(832\) 40.6926 1.41076
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) 9.83918 0.340499
\(836\) −30.8620 + 53.4546i −1.06738 + 1.84876i
\(837\) 0 0
\(838\) 0.896586 3.34610i 0.0309721 0.115589i
\(839\) −29.0211 −1.00192 −0.500960 0.865470i \(-0.667020\pi\)
−0.500960 + 0.865470i \(0.667020\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) 2.29347 8.55936i 0.0790383 0.294975i
\(843\) 0 0
\(844\) −25.1477 14.5191i −0.865621 0.499767i
\(845\) 14.5437 0.500320
\(846\) 0 0
\(847\) 3.91362i 0.134474i
\(848\) 15.4473 + 26.7554i 0.530461 + 0.918786i
\(849\) 0 0
\(850\) 2.72589 10.1732i 0.0934972 0.348936i
\(851\) 63.6150i 2.18069i
\(852\) 0 0
\(853\) 50.8777i 1.74202i 0.491266 + 0.871009i \(0.336534\pi\)
−0.491266 + 0.871009i \(0.663466\pi\)
\(854\) −15.6603 4.19615i −0.535883 0.143589i
\(855\) 0 0
\(856\) −25.6836 + 25.6836i −0.877849 + 0.877849i
\(857\) 22.3553i 0.763642i 0.924236 + 0.381821i \(0.124703\pi\)
−0.924236 + 0.381821i \(0.875297\pi\)
\(858\) 0 0
\(859\) 44.4134 1.51537 0.757684 0.652622i \(-0.226331\pi\)
0.757684 + 0.652622i \(0.226331\pi\)
\(860\) 19.5970 + 11.3143i 0.668253 + 0.385816i
\(861\) 0 0
\(862\) −4.11415 1.10238i −0.140129 0.0375473i
\(863\) 8.80145 0.299605 0.149802 0.988716i \(-0.452136\pi\)
0.149802 + 0.988716i \(0.452136\pi\)
\(864\) 0 0
\(865\) −1.73860 −0.0591143
\(866\) −9.56244 2.56225i −0.324945 0.0870688i
\(867\) 0 0
\(868\) 7.07816 + 4.08658i 0.240248 + 0.138707i
\(869\) 27.7006 0.939679
\(870\) 0 0
\(871\) 25.8732i 0.876681i
\(872\) 26.1563 26.1563i 0.885764 0.885764i
\(873\) 0 0
\(874\) 83.8451 + 22.4662i 2.83610 + 0.759931i
\(875\) 9.85565i 0.333182i
\(876\) 0 0
\(877\) 25.5280i 0.862020i 0.902347 + 0.431010i \(0.141843\pi\)
−0.902347 + 0.431010i \(0.858157\pi\)
\(878\) 0.489503 1.82685i 0.0165199 0.0616532i
\(879\) 0 0
\(880\) 15.1137 8.72589i 0.509482 0.294150i
\(881\) 41.0812i 1.38406i −0.721869 0.692030i \(-0.756717\pi\)
0.721869 0.692030i \(-0.243283\pi\)
\(882\) 0 0
\(883\) 2.80145 0.0942763 0.0471381 0.998888i \(-0.484990\pi\)
0.0471381 + 0.998888i \(0.484990\pi\)
\(884\) 17.6204 + 10.1732i 0.592639 + 0.342160i
\(885\) 0 0
\(886\) −0.698454 + 2.60666i −0.0234650 + 0.0875726i
\(887\) 45.4366 1.52561 0.762806 0.646628i \(-0.223821\pi\)
0.762806 + 0.646628i \(0.223821\pi\)
\(888\) 0 0
\(889\) 4.63706 0.155522
\(890\) 5.24713 19.5826i 0.175884 0.656409i
\(891\) 0 0
\(892\) 19.2743 33.3841i 0.645352 1.11778i
\(893\) 15.4767 0.517907
\(894\) 0 0
\(895\) 30.2083i 1.00975i
\(896\) 10.9282 + 2.92820i 0.365086 + 0.0978244i
\(897\) 0 0
\(898\) −3.68571 + 13.7552i −0.122994 + 0.459019i
\(899\) 19.3379i 0.644954i
\(900\) 0 0
\(901\) 15.4473i 0.514623i
\(902\) −44.3787 11.8912i −1.47765 0.395935i
\(903\) 0 0
\(904\) 14.1099 14.1099i 0.469289 0.469289i
\(905\) 3.50380i 0.116470i
\(906\) 0 0
\(907\) 46.5318 1.54506 0.772531 0.634977i \(-0.218990\pi\)
0.772531 + 0.634977i \(0.218990\pi\)
\(908\) 23.2341 40.2427i 0.771052 1.33550i
\(909\) 0 0
\(910\) 7.85005 + 2.10341i 0.260227 + 0.0697275i
\(911\) −24.9174 −0.825550 −0.412775 0.910833i \(-0.635440\pi\)
−0.412775 + 0.910833i \(0.635440\pi\)
\(912\) 0 0
\(913\) 57.5611 1.90500
\(914\) 14.0981 + 3.77757i 0.466323 + 0.124951i
\(915\) 0 0
\(916\) 13.4641 23.3205i 0.444866 0.770531i
\(917\) 8.36930 0.276379
\(918\) 0 0
\(919\) 8.77228i 0.289371i −0.989478 0.144685i \(-0.953783\pi\)
0.989478 0.144685i \(-0.0462170\pi\)
\(920\) −17.3542 17.3542i −0.572151 0.572151i
\(921\) 0 0
\(922\) −24.5093 6.56724i −0.807170 0.216281i
\(923\) 69.5866i 2.29047i
\(924\) 0 0
\(925\) 30.8418i 1.01407i
\(926\) −7.93046 + 29.5969i −0.260611 + 0.972614i
\(927\) 0 0
\(928\) 6.92820 + 25.8564i 0.227429 + 0.848778i
\(929\) 3.96002i 0.129924i −0.997888 0.0649620i \(-0.979307\pi\)
0.997888 0.0649620i \(-0.0206926\pi\)
\(930\) 0 0
\(931\) −7.99158 −0.261913
\(932\) −17.4641 + 30.2487i −0.572056 + 0.990829i
\(933\) 0 0
\(934\) 6.71747 25.0699i 0.219802 0.820313i
\(935\) 8.72589 0.285367
\(936\) 0 0
\(937\) −11.6077 −0.379207 −0.189603 0.981861i \(-0.560720\pi\)
−0.189603 + 0.981861i \(0.560720\pi\)
\(938\) 1.86182 6.94839i 0.0607904 0.226873i
\(939\) 0 0
\(940\) −3.78960 2.18793i −0.123603 0.0713624i
\(941\) 2.05758 0.0670751 0.0335376 0.999437i \(-0.489323\pi\)
0.0335376 + 0.999437i \(0.489323\pi\)
\(942\) 0 0
\(943\) 64.6117i 2.10404i
\(944\) 38.0760 21.9832i 1.23927 0.715491i
\(945\) 0 0
\(946\) 14.1561 52.8313i 0.460255 1.71770i
\(947\) 53.3723i 1.73437i −0.497989 0.867184i \(-0.665928\pi\)
0.497989 0.867184i \(-0.334072\pi\)
\(948\) 0 0
\(949\) 13.4136i 0.435423i
\(950\) −40.6498 10.8921i −1.31885 0.353386i
\(951\) 0 0
\(952\) 4.00000 + 4.00000i 0.129641 + 0.129641i
\(953\) 45.9433i 1.48825i −0.668040 0.744125i \(-0.732867\pi\)
0.668040 0.744125i \(-0.267133\pi\)
\(954\) 0 0
\(955\) 15.6057 0.504988
\(956\) 3.16487 + 1.82724i 0.102359 + 0.0590972i
\(957\) 0 0
\(958\) −20.0148 5.36294i −0.646648 0.173269i
\(959\) 13.9366 0.450037
\(960\) 0 0
\(961\) 14.2999 0.461287
\(962\) 57.5516 + 15.4209i 1.85554 + 0.497190i
\(963\) 0 0
\(964\) −37.2724 21.5193i −1.20046 0.693089i
\(965\) −7.97044 −0.256578
\(966\) 0 0
\(967\) 1.43455i 0.0461319i −0.999734 0.0230659i \(-0.992657\pi\)
0.999734 0.0230659i \(-0.00734276\pi\)
\(968\) −7.82724 7.82724i −0.251577 0.251577i
\(969\) 0 0
\(970\) −6.44037 1.72569i −0.206788 0.0554086i
\(971\) 13.6030i 0.436542i 0.975888 + 0.218271i \(0.0700417\pi\)
−0.975888 + 0.218271i \(0.929958\pi\)
\(972\) 0 0
\(973\) 13.9832i 0.448280i
\(974\) −2.41866 + 9.02655i −0.0774987 + 0.289229i
\(975\) 0 0
\(976\) 39.7128 22.9282i 1.27118 0.733914i
\(977\) 10.3860i 0.332278i 0.986102 + 0.166139i \(0.0531300\pi\)
−0.986102 + 0.166139i \(0.946870\pi\)
\(978\) 0 0
\(979\) −49.0020 −1.56611
\(980\) 1.95681 + 1.12976i 0.0625080 + 0.0360890i
\(981\) 0 0
\(982\) −2.59264 + 9.67587i −0.0827345 + 0.308769i
\(983\) 20.0062 0.638098 0.319049 0.947738i \(-0.396637\pi\)
0.319049 + 0.947738i \(0.396637\pi\)
\(984\) 0 0
\(985\) −0.409820 −0.0130580
\(986\) −3.46410 + 12.9282i −0.110319 + 0.411718i
\(987\) 0 0
\(988\) 40.6498 70.4075i 1.29324 2.23996i
\(989\) −76.9179 −2.44585
\(990\) 0 0
\(991\) 45.7220i 1.45241i −0.687480 0.726203i \(-0.741283\pi\)
0.687480 0.726203i \(-0.258717\pi\)
\(992\) −22.3295 + 5.98316i −0.708961 + 0.189966i
\(993\) 0 0
\(994\) −5.00739 + 18.6878i −0.158825 + 0.592742i
\(995\) 8.69952i 0.275793i
\(996\) 0 0
\(997\) 41.6964i 1.32054i 0.751030 + 0.660269i \(0.229558\pi\)
−0.751030 + 0.660269i \(0.770442\pi\)
\(998\) −9.74382 2.61085i −0.308435 0.0826450i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1512.2.j.b.323.7 yes 8
3.2 odd 2 1512.2.j.a.323.2 8
4.3 odd 2 6048.2.j.b.5615.4 8
8.3 odd 2 1512.2.j.a.323.4 yes 8
8.5 even 2 6048.2.j.a.5615.5 8
12.11 even 2 6048.2.j.a.5615.6 8
24.5 odd 2 6048.2.j.b.5615.3 8
24.11 even 2 inner 1512.2.j.b.323.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.a.323.2 8 3.2 odd 2
1512.2.j.a.323.4 yes 8 8.3 odd 2
1512.2.j.b.323.5 yes 8 24.11 even 2 inner
1512.2.j.b.323.7 yes 8 1.1 even 1 trivial
6048.2.j.a.5615.5 8 8.5 even 2
6048.2.j.a.5615.6 8 12.11 even 2
6048.2.j.b.5615.3 8 24.5 odd 2
6048.2.j.b.5615.4 8 4.3 odd 2