Properties

Label 384.2.o.a.47.4
Level $384$
Weight $2$
Character 384.47
Analytic conductor $3.066$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,2,Mod(47,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.o (of order \(8\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 47.4
Character \(\chi\) \(=\) 384.47
Dual form 384.2.o.a.335.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.18890 + 1.25957i) q^{3} +(1.06973 - 0.443098i) q^{5} +(-2.37247 - 2.37247i) q^{7} +(-0.173046 - 2.99501i) q^{9} +O(q^{10})\) \(q+(-1.18890 + 1.25957i) q^{3} +(1.06973 - 0.443098i) q^{5} +(-2.37247 - 2.37247i) q^{7} +(-0.173046 - 2.99501i) q^{9} +(5.50127 - 2.27870i) q^{11} +(0.346353 - 0.836171i) q^{13} +(-0.713689 + 1.87421i) q^{15} +0.685064 q^{17} +(3.35074 + 1.38792i) q^{19} +(5.80891 - 0.167674i) q^{21} +(2.05133 + 2.05133i) q^{23} +(-2.58754 + 2.58754i) q^{25} +(3.97816 + 3.34279i) q^{27} +(1.98869 - 4.80112i) q^{29} -6.36503i q^{31} +(-3.67025 + 9.63838i) q^{33} +(-3.58914 - 1.48667i) q^{35} +(0.112738 + 0.272172i) q^{37} +(0.641440 + 1.43038i) q^{39} +(3.26585 - 3.26585i) q^{41} +(0.993018 + 2.39736i) q^{43} +(-1.51219 - 3.12718i) q^{45} -11.7975i q^{47} +4.25719i q^{49} +(-0.814471 + 0.862888i) q^{51} +(1.56192 + 3.77080i) q^{53} +(4.87520 - 4.87520i) q^{55} +(-5.73187 + 2.57040i) q^{57} +(-2.20815 - 5.33094i) q^{59} +(-1.53549 - 0.636020i) q^{61} +(-6.69500 + 7.51609i) q^{63} -1.04795i q^{65} +(-4.69605 + 11.3373i) q^{67} +(-5.02262 + 0.144978i) q^{69} +(-7.99150 + 7.99150i) q^{71} +(-2.34760 - 2.34760i) q^{73} +(-0.182875 - 6.33551i) q^{75} +(-18.4577 - 7.64543i) q^{77} -8.91043 q^{79} +(-8.94011 + 1.03655i) q^{81} +(-3.06920 + 7.40970i) q^{83} +(0.732836 - 0.303551i) q^{85} +(3.68301 + 8.21294i) q^{87} +(1.99332 + 1.99332i) q^{89} +(-2.80550 + 1.16208i) q^{91} +(8.01722 + 7.56737i) q^{93} +4.19938 q^{95} +8.66510 q^{97} +(-7.77669 - 16.0820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{3} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 4 q^{3} + 8 q^{7} - 4 q^{9} - 8 q^{13} + 8 q^{15} + 8 q^{19} - 4 q^{21} - 8 q^{25} + 28 q^{27} - 8 q^{33} - 8 q^{37} + 28 q^{39} + 8 q^{43} - 4 q^{45} + 16 q^{51} - 24 q^{55} - 4 q^{57} - 40 q^{61} - 56 q^{67} - 4 q^{69} - 8 q^{73} - 16 q^{75} - 16 q^{79} - 48 q^{85} - 52 q^{87} - 40 q^{91} + 8 q^{93} - 16 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.18890 + 1.25957i −0.686410 + 0.727215i
\(4\) 0 0
\(5\) 1.06973 0.443098i 0.478400 0.198160i −0.130435 0.991457i \(-0.541637\pi\)
0.608834 + 0.793297i \(0.291637\pi\)
\(6\) 0 0
\(7\) −2.37247 2.37247i −0.896708 0.896708i 0.0984354 0.995143i \(-0.468616\pi\)
−0.995143 + 0.0984354i \(0.968616\pi\)
\(8\) 0 0
\(9\) −0.173046 2.99501i −0.0576820 0.998335i
\(10\) 0 0
\(11\) 5.50127 2.27870i 1.65869 0.687054i 0.660718 0.750635i \(-0.270252\pi\)
0.997977 + 0.0635809i \(0.0202521\pi\)
\(12\) 0 0
\(13\) 0.346353 0.836171i 0.0960612 0.231912i −0.868543 0.495613i \(-0.834943\pi\)
0.964604 + 0.263701i \(0.0849433\pi\)
\(14\) 0 0
\(15\) −0.713689 + 1.87421i −0.184274 + 0.483918i
\(16\) 0 0
\(17\) 0.685064 0.166152 0.0830762 0.996543i \(-0.473526\pi\)
0.0830762 + 0.996543i \(0.473526\pi\)
\(18\) 0 0
\(19\) 3.35074 + 1.38792i 0.768712 + 0.318411i 0.732351 0.680928i \(-0.238423\pi\)
0.0363614 + 0.999339i \(0.488423\pi\)
\(20\) 0 0
\(21\) 5.80891 0.167674i 1.26761 0.0365895i
\(22\) 0 0
\(23\) 2.05133 + 2.05133i 0.427732 + 0.427732i 0.887855 0.460123i \(-0.152195\pi\)
−0.460123 + 0.887855i \(0.652195\pi\)
\(24\) 0 0
\(25\) −2.58754 + 2.58754i −0.517508 + 0.517508i
\(26\) 0 0
\(27\) 3.97816 + 3.34279i 0.765597 + 0.643320i
\(28\) 0 0
\(29\) 1.98869 4.80112i 0.369291 0.891546i −0.624576 0.780964i \(-0.714728\pi\)
0.993867 0.110582i \(-0.0352716\pi\)
\(30\) 0 0
\(31\) 6.36503i 1.14319i −0.820535 0.571596i \(-0.806324\pi\)
0.820535 0.571596i \(-0.193676\pi\)
\(32\) 0 0
\(33\) −3.67025 + 9.63838i −0.638909 + 1.67783i
\(34\) 0 0
\(35\) −3.58914 1.48667i −0.606676 0.251293i
\(36\) 0 0
\(37\) 0.112738 + 0.272172i 0.0185339 + 0.0447449i 0.932876 0.360199i \(-0.117291\pi\)
−0.914342 + 0.404944i \(0.867291\pi\)
\(38\) 0 0
\(39\) 0.641440 + 1.43038i 0.102713 + 0.229044i
\(40\) 0 0
\(41\) 3.26585 3.26585i 0.510040 0.510040i −0.404499 0.914539i \(-0.632554\pi\)
0.914539 + 0.404499i \(0.132554\pi\)
\(42\) 0 0
\(43\) 0.993018 + 2.39736i 0.151434 + 0.365594i 0.981332 0.192321i \(-0.0616016\pi\)
−0.829898 + 0.557915i \(0.811602\pi\)
\(44\) 0 0
\(45\) −1.51219 3.12718i −0.225425 0.466173i
\(46\) 0 0
\(47\) 11.7975i 1.72084i −0.509589 0.860418i \(-0.670203\pi\)
0.509589 0.860418i \(-0.329797\pi\)
\(48\) 0 0
\(49\) 4.25719i 0.608171i
\(50\) 0 0
\(51\) −0.814471 + 0.862888i −0.114049 + 0.120828i
\(52\) 0 0
\(53\) 1.56192 + 3.77080i 0.214546 + 0.517959i 0.994112 0.108361i \(-0.0345602\pi\)
−0.779566 + 0.626320i \(0.784560\pi\)
\(54\) 0 0
\(55\) 4.87520 4.87520i 0.657372 0.657372i
\(56\) 0 0
\(57\) −5.73187 + 2.57040i −0.759205 + 0.340458i
\(58\) 0 0
\(59\) −2.20815 5.33094i −0.287476 0.694029i 0.712495 0.701678i \(-0.247565\pi\)
−0.999971 + 0.00764882i \(0.997565\pi\)
\(60\) 0 0
\(61\) −1.53549 0.636020i −0.196599 0.0814341i 0.282212 0.959352i \(-0.408932\pi\)
−0.478811 + 0.877918i \(0.658932\pi\)
\(62\) 0 0
\(63\) −6.69500 + 7.51609i −0.843491 + 0.946939i
\(64\) 0 0
\(65\) 1.04795i 0.129982i
\(66\) 0 0
\(67\) −4.69605 + 11.3373i −0.573714 + 1.38507i 0.324657 + 0.945832i \(0.394751\pi\)
−0.898371 + 0.439237i \(0.855249\pi\)
\(68\) 0 0
\(69\) −5.02262 + 0.144978i −0.604653 + 0.0174533i
\(70\) 0 0
\(71\) −7.99150 + 7.99150i −0.948416 + 0.948416i −0.998733 0.0503173i \(-0.983977\pi\)
0.0503173 + 0.998733i \(0.483977\pi\)
\(72\) 0 0
\(73\) −2.34760 2.34760i −0.274766 0.274766i 0.556250 0.831015i \(-0.312240\pi\)
−0.831015 + 0.556250i \(0.812240\pi\)
\(74\) 0 0
\(75\) −0.182875 6.33551i −0.0211165 0.731562i
\(76\) 0 0
\(77\) −18.4577 7.64543i −2.10345 0.871278i
\(78\) 0 0
\(79\) −8.91043 −1.00250 −0.501251 0.865302i \(-0.667127\pi\)
−0.501251 + 0.865302i \(0.667127\pi\)
\(80\) 0 0
\(81\) −8.94011 + 1.03655i −0.993346 + 0.115172i
\(82\) 0 0
\(83\) −3.06920 + 7.40970i −0.336888 + 0.813320i 0.661123 + 0.750278i \(0.270080\pi\)
−0.998011 + 0.0630420i \(0.979920\pi\)
\(84\) 0 0
\(85\) 0.732836 0.303551i 0.0794872 0.0329247i
\(86\) 0 0
\(87\) 3.68301 + 8.21294i 0.394861 + 0.880520i
\(88\) 0 0
\(89\) 1.99332 + 1.99332i 0.211291 + 0.211291i 0.804816 0.593525i \(-0.202264\pi\)
−0.593525 + 0.804816i \(0.702264\pi\)
\(90\) 0 0
\(91\) −2.80550 + 1.16208i −0.294096 + 0.121819i
\(92\) 0 0
\(93\) 8.01722 + 7.56737i 0.831347 + 0.784699i
\(94\) 0 0
\(95\) 4.19938 0.430848
\(96\) 0 0
\(97\) 8.66510 0.879807 0.439904 0.898045i \(-0.355013\pi\)
0.439904 + 0.898045i \(0.355013\pi\)
\(98\) 0 0
\(99\) −7.77669 16.0820i −0.781586 1.61630i
\(100\) 0 0
\(101\) 15.6920 6.49983i 1.56141 0.646758i 0.576077 0.817395i \(-0.304583\pi\)
0.985334 + 0.170637i \(0.0545827\pi\)
\(102\) 0 0
\(103\) 5.39788 + 5.39788i 0.531869 + 0.531869i 0.921128 0.389259i \(-0.127269\pi\)
−0.389259 + 0.921128i \(0.627269\pi\)
\(104\) 0 0
\(105\) 6.13969 2.75329i 0.599173 0.268693i
\(106\) 0 0
\(107\) 0.0169596 0.00702491i 0.00163955 0.000679124i −0.381864 0.924219i \(-0.624718\pi\)
0.383503 + 0.923540i \(0.374718\pi\)
\(108\) 0 0
\(109\) −4.94357 + 11.9348i −0.473508 + 1.14315i 0.489094 + 0.872231i \(0.337328\pi\)
−0.962602 + 0.270919i \(0.912672\pi\)
\(110\) 0 0
\(111\) −0.476854 0.181584i −0.0452610 0.0172352i
\(112\) 0 0
\(113\) −10.4984 −0.987609 −0.493804 0.869573i \(-0.664394\pi\)
−0.493804 + 0.869573i \(0.664394\pi\)
\(114\) 0 0
\(115\) 3.10332 + 1.28544i 0.289386 + 0.119868i
\(116\) 0 0
\(117\) −2.56427 0.892634i −0.237067 0.0825241i
\(118\) 0 0
\(119\) −1.62529 1.62529i −0.148990 0.148990i
\(120\) 0 0
\(121\) 17.2933 17.2933i 1.57212 1.57212i
\(122\) 0 0
\(123\) 0.230814 + 7.99633i 0.0208118 + 0.721005i
\(124\) 0 0
\(125\) −3.83694 + 9.26318i −0.343186 + 0.828524i
\(126\) 0 0
\(127\) 0.793212i 0.0703862i 0.999381 + 0.0351931i \(0.0112046\pi\)
−0.999381 + 0.0351931i \(0.988795\pi\)
\(128\) 0 0
\(129\) −4.20024 1.59943i −0.369811 0.140822i
\(130\) 0 0
\(131\) 9.39267 + 3.89057i 0.820641 + 0.339921i 0.753191 0.657802i \(-0.228514\pi\)
0.0674502 + 0.997723i \(0.478514\pi\)
\(132\) 0 0
\(133\) −4.65672 11.2423i −0.403789 0.974832i
\(134\) 0 0
\(135\) 5.73676 + 1.81318i 0.493741 + 0.156054i
\(136\) 0 0
\(137\) −2.95969 + 2.95969i −0.252863 + 0.252863i −0.822144 0.569280i \(-0.807222\pi\)
0.569280 + 0.822144i \(0.307222\pi\)
\(138\) 0 0
\(139\) 1.91155 + 4.61488i 0.162135 + 0.391429i 0.983979 0.178284i \(-0.0570545\pi\)
−0.821844 + 0.569713i \(0.807054\pi\)
\(140\) 0 0
\(141\) 14.8597 + 14.0260i 1.25142 + 1.18120i
\(142\) 0 0
\(143\) 5.38924i 0.450671i
\(144\) 0 0
\(145\) 6.01711i 0.499694i
\(146\) 0 0
\(147\) −5.36224 5.06137i −0.442270 0.417455i
\(148\) 0 0
\(149\) 5.53451 + 13.3615i 0.453405 + 1.09462i 0.971019 + 0.239002i \(0.0768202\pi\)
−0.517614 + 0.855614i \(0.673180\pi\)
\(150\) 0 0
\(151\) −2.96063 + 2.96063i −0.240932 + 0.240932i −0.817236 0.576303i \(-0.804495\pi\)
0.576303 + 0.817236i \(0.304495\pi\)
\(152\) 0 0
\(153\) −0.118548 2.05177i −0.00958400 0.165876i
\(154\) 0 0
\(155\) −2.82033 6.80889i −0.226535 0.546903i
\(156\) 0 0
\(157\) −16.1213 6.67768i −1.28662 0.532937i −0.368646 0.929570i \(-0.620178\pi\)
−0.917977 + 0.396633i \(0.870178\pi\)
\(158\) 0 0
\(159\) −6.60656 2.51575i −0.523934 0.199512i
\(160\) 0 0
\(161\) 9.73343i 0.767102i
\(162\) 0 0
\(163\) 4.62172 11.1578i 0.362001 0.873949i −0.633006 0.774147i \(-0.718179\pi\)
0.995007 0.0998018i \(-0.0318209\pi\)
\(164\) 0 0
\(165\) 0.344556 + 11.9368i 0.0268236 + 0.929278i
\(166\) 0 0
\(167\) −1.28603 + 1.28603i −0.0995157 + 0.0995157i −0.755112 0.655596i \(-0.772417\pi\)
0.655596 + 0.755112i \(0.272417\pi\)
\(168\) 0 0
\(169\) 8.61317 + 8.61317i 0.662551 + 0.662551i
\(170\) 0 0
\(171\) 3.57700 10.2756i 0.273540 0.785799i
\(172\) 0 0
\(173\) −0.0565964 0.0234430i −0.00430295 0.00178234i 0.380531 0.924768i \(-0.375741\pi\)
−0.384834 + 0.922986i \(0.625741\pi\)
\(174\) 0 0
\(175\) 12.2777 0.928107
\(176\) 0 0
\(177\) 9.33996 + 3.55662i 0.702034 + 0.267332i
\(178\) 0 0
\(179\) 1.17461 2.83576i 0.0877946 0.211955i −0.873884 0.486135i \(-0.838406\pi\)
0.961678 + 0.274180i \(0.0884064\pi\)
\(180\) 0 0
\(181\) −20.8334 + 8.62949i −1.54854 + 0.641425i −0.983051 0.183334i \(-0.941311\pi\)
−0.565485 + 0.824758i \(0.691311\pi\)
\(182\) 0 0
\(183\) 2.62665 1.17790i 0.194168 0.0870726i
\(184\) 0 0
\(185\) 0.241198 + 0.241198i 0.0177333 + 0.0177333i
\(186\) 0 0
\(187\) 3.76872 1.56105i 0.275596 0.114156i
\(188\) 0 0
\(189\) −1.50739 17.3687i −0.109647 1.26339i
\(190\) 0 0
\(191\) 23.0566 1.66832 0.834158 0.551525i \(-0.185954\pi\)
0.834158 + 0.551525i \(0.185954\pi\)
\(192\) 0 0
\(193\) 7.52832 0.541900 0.270950 0.962593i \(-0.412662\pi\)
0.270950 + 0.962593i \(0.412662\pi\)
\(194\) 0 0
\(195\) 1.31997 + 1.24590i 0.0945249 + 0.0892211i
\(196\) 0 0
\(197\) −2.14750 + 0.889524i −0.153003 + 0.0633760i −0.457871 0.889019i \(-0.651388\pi\)
0.304868 + 0.952395i \(0.401388\pi\)
\(198\) 0 0
\(199\) −7.46994 7.46994i −0.529530 0.529530i 0.390902 0.920432i \(-0.372163\pi\)
−0.920432 + 0.390902i \(0.872163\pi\)
\(200\) 0 0
\(201\) −8.69699 19.3939i −0.613439 1.36794i
\(202\) 0 0
\(203\) −16.1086 + 6.67240i −1.13060 + 0.468311i
\(204\) 0 0
\(205\) 2.04650 4.94068i 0.142934 0.345072i
\(206\) 0 0
\(207\) 5.78877 6.49872i 0.402348 0.451692i
\(208\) 0 0
\(209\) 21.5960 1.49382
\(210\) 0 0
\(211\) −15.9819 6.61994i −1.10024 0.455735i −0.242676 0.970107i \(-0.578025\pi\)
−0.857567 + 0.514372i \(0.828025\pi\)
\(212\) 0 0
\(213\) −0.564800 19.5669i −0.0386994 1.34070i
\(214\) 0 0
\(215\) 2.12453 + 2.12453i 0.144892 + 0.144892i
\(216\) 0 0
\(217\) −15.1008 + 15.1008i −1.02511 + 1.02511i
\(218\) 0 0
\(219\) 5.74803 0.165917i 0.388416 0.0112116i
\(220\) 0 0
\(221\) 0.237274 0.572831i 0.0159608 0.0385328i
\(222\) 0 0
\(223\) 18.2244i 1.22040i 0.792248 + 0.610199i \(0.208910\pi\)
−0.792248 + 0.610199i \(0.791090\pi\)
\(224\) 0 0
\(225\) 8.19746 + 7.30193i 0.546497 + 0.486795i
\(226\) 0 0
\(227\) 10.8455 + 4.49234i 0.719840 + 0.298167i 0.712369 0.701805i \(-0.247622\pi\)
0.00747022 + 0.999972i \(0.497622\pi\)
\(228\) 0 0
\(229\) 5.20986 + 12.5777i 0.344277 + 0.831159i 0.997273 + 0.0737978i \(0.0235119\pi\)
−0.652996 + 0.757362i \(0.726488\pi\)
\(230\) 0 0
\(231\) 31.5743 14.1592i 2.07744 0.931606i
\(232\) 0 0
\(233\) −11.7233 + 11.7233i −0.768016 + 0.768016i −0.977757 0.209741i \(-0.932738\pi\)
0.209741 + 0.977757i \(0.432738\pi\)
\(234\) 0 0
\(235\) −5.22743 12.6201i −0.341000 0.823247i
\(236\) 0 0
\(237\) 10.5936 11.2233i 0.688128 0.729034i
\(238\) 0 0
\(239\) 21.5949i 1.39685i 0.715681 + 0.698427i \(0.246117\pi\)
−0.715681 + 0.698427i \(0.753883\pi\)
\(240\) 0 0
\(241\) 14.6095i 0.941078i −0.882379 0.470539i \(-0.844059\pi\)
0.882379 0.470539i \(-0.155941\pi\)
\(242\) 0 0
\(243\) 9.32327 12.4931i 0.598088 0.801430i
\(244\) 0 0
\(245\) 1.88636 + 4.55406i 0.120515 + 0.290949i
\(246\) 0 0
\(247\) 2.32108 2.32108i 0.147687 0.147687i
\(248\) 0 0
\(249\) −5.68409 12.6752i −0.360215 0.803261i
\(250\) 0 0
\(251\) 1.08581 + 2.62137i 0.0685355 + 0.165459i 0.954436 0.298416i \(-0.0964584\pi\)
−0.885900 + 0.463876i \(0.846458\pi\)
\(252\) 0 0
\(253\) 15.9593 + 6.61055i 1.00335 + 0.415602i
\(254\) 0 0
\(255\) −0.488923 + 1.28395i −0.0306175 + 0.0804041i
\(256\) 0 0
\(257\) 1.35436i 0.0844827i −0.999107 0.0422414i \(-0.986550\pi\)
0.999107 0.0422414i \(-0.0134498\pi\)
\(258\) 0 0
\(259\) 0.378254 0.913186i 0.0235036 0.0567426i
\(260\) 0 0
\(261\) −14.7235 5.12532i −0.911363 0.317250i
\(262\) 0 0
\(263\) −3.52372 + 3.52372i −0.217282 + 0.217282i −0.807352 0.590070i \(-0.799100\pi\)
0.590070 + 0.807352i \(0.299100\pi\)
\(264\) 0 0
\(265\) 3.34167 + 3.34167i 0.205277 + 0.205277i
\(266\) 0 0
\(267\) −4.88057 + 0.140878i −0.298686 + 0.00862158i
\(268\) 0 0
\(269\) 16.4741 + 6.82380i 1.00444 + 0.416055i 0.823425 0.567425i \(-0.192060\pi\)
0.181020 + 0.983479i \(0.442060\pi\)
\(270\) 0 0
\(271\) 8.50293 0.516516 0.258258 0.966076i \(-0.416851\pi\)
0.258258 + 0.966076i \(0.416851\pi\)
\(272\) 0 0
\(273\) 1.87173 4.91532i 0.113282 0.297489i
\(274\) 0 0
\(275\) −8.33852 + 20.1310i −0.502832 + 1.21394i
\(276\) 0 0
\(277\) 20.2028 8.36828i 1.21387 0.502801i 0.318414 0.947952i \(-0.396850\pi\)
0.895456 + 0.445150i \(0.146850\pi\)
\(278\) 0 0
\(279\) −19.0633 + 1.10144i −1.14129 + 0.0659416i
\(280\) 0 0
\(281\) 10.3342 + 10.3342i 0.616487 + 0.616487i 0.944629 0.328142i \(-0.106422\pi\)
−0.328142 + 0.944629i \(0.606422\pi\)
\(282\) 0 0
\(283\) −8.67342 + 3.59265i −0.515581 + 0.213561i −0.625275 0.780405i \(-0.715013\pi\)
0.109694 + 0.993965i \(0.465013\pi\)
\(284\) 0 0
\(285\) −4.99264 + 5.28943i −0.295738 + 0.313319i
\(286\) 0 0
\(287\) −15.4962 −0.914714
\(288\) 0 0
\(289\) −16.5307 −0.972393
\(290\) 0 0
\(291\) −10.3019 + 10.9143i −0.603909 + 0.639809i
\(292\) 0 0
\(293\) −11.3816 + 4.71442i −0.664922 + 0.275420i −0.689508 0.724278i \(-0.742173\pi\)
0.0245862 + 0.999698i \(0.492173\pi\)
\(294\) 0 0
\(295\) −4.72426 4.72426i −0.275057 0.275057i
\(296\) 0 0
\(297\) 29.5021 + 9.32455i 1.71189 + 0.541065i
\(298\) 0 0
\(299\) 2.42575 1.00478i 0.140285 0.0581078i
\(300\) 0 0
\(301\) 3.33175 8.04355i 0.192039 0.463623i
\(302\) 0 0
\(303\) −10.4692 + 27.4928i −0.601437 + 1.57942i
\(304\) 0 0
\(305\) −1.92438 −0.110190
\(306\) 0 0
\(307\) 12.6145 + 5.22511i 0.719950 + 0.298213i 0.712415 0.701759i \(-0.247601\pi\)
0.00753514 + 0.999972i \(0.497601\pi\)
\(308\) 0 0
\(309\) −13.2166 + 0.381496i −0.751863 + 0.0217025i
\(310\) 0 0
\(311\) −18.8020 18.8020i −1.06617 1.06617i −0.997650 0.0685162i \(-0.978174\pi\)
−0.0685162 0.997650i \(-0.521826\pi\)
\(312\) 0 0
\(313\) 4.30835 4.30835i 0.243522 0.243522i −0.574783 0.818306i \(-0.694914\pi\)
0.818306 + 0.574783i \(0.194914\pi\)
\(314\) 0 0
\(315\) −3.83150 + 11.0068i −0.215881 + 0.620161i
\(316\) 0 0
\(317\) −0.543402 + 1.31189i −0.0305205 + 0.0736830i −0.938405 0.345538i \(-0.887697\pi\)
0.907884 + 0.419221i \(0.137697\pi\)
\(318\) 0 0
\(319\) 30.9439i 1.73252i
\(320\) 0 0
\(321\) −0.0113149 + 0.0297138i −0.000631535 + 0.00165846i
\(322\) 0 0
\(323\) 2.29547 + 0.950815i 0.127723 + 0.0529047i
\(324\) 0 0
\(325\) 1.26742 + 3.05983i 0.0703040 + 0.169729i
\(326\) 0 0
\(327\) −9.15540 20.4161i −0.506295 1.12901i
\(328\) 0 0
\(329\) −27.9891 + 27.9891i −1.54309 + 1.54309i
\(330\) 0 0
\(331\) 5.18815 + 12.5253i 0.285167 + 0.688453i 0.999941 0.0109024i \(-0.00347041\pi\)
−0.714774 + 0.699356i \(0.753470\pi\)
\(332\) 0 0
\(333\) 0.795649 0.384748i 0.0436013 0.0210840i
\(334\) 0 0
\(335\) 14.2087i 0.776303i
\(336\) 0 0
\(337\) 19.9878i 1.08881i 0.838824 + 0.544403i \(0.183244\pi\)
−0.838824 + 0.544403i \(0.816756\pi\)
\(338\) 0 0
\(339\) 12.4815 13.2235i 0.677905 0.718203i
\(340\) 0 0
\(341\) −14.5040 35.0157i −0.785435 1.89621i
\(342\) 0 0
\(343\) −6.50721 + 6.50721i −0.351357 + 0.351357i
\(344\) 0 0
\(345\) −5.30863 + 2.38060i −0.285807 + 0.128167i
\(346\) 0 0
\(347\) −5.84634 14.1143i −0.313848 0.757696i −0.999555 0.0298187i \(-0.990507\pi\)
0.685707 0.727877i \(-0.259493\pi\)
\(348\) 0 0
\(349\) 7.90170 + 3.27299i 0.422969 + 0.175199i 0.584007 0.811749i \(-0.301484\pi\)
−0.161038 + 0.986948i \(0.551484\pi\)
\(350\) 0 0
\(351\) 4.17299 2.16864i 0.222738 0.115753i
\(352\) 0 0
\(353\) 28.4566i 1.51459i −0.653072 0.757296i \(-0.726520\pi\)
0.653072 0.757296i \(-0.273480\pi\)
\(354\) 0 0
\(355\) −5.00776 + 12.0898i −0.265784 + 0.641659i
\(356\) 0 0
\(357\) 3.97948 0.114868i 0.210616 0.00607944i
\(358\) 0 0
\(359\) −3.48589 + 3.48589i −0.183978 + 0.183978i −0.793087 0.609109i \(-0.791527\pi\)
0.609109 + 0.793087i \(0.291527\pi\)
\(360\) 0 0
\(361\) −4.13391 4.13391i −0.217574 0.217574i
\(362\) 0 0
\(363\) 1.22221 + 42.3421i 0.0641491 + 2.22238i
\(364\) 0 0
\(365\) −3.55152 1.47109i −0.185895 0.0770003i
\(366\) 0 0
\(367\) −7.37411 −0.384925 −0.192463 0.981304i \(-0.561647\pi\)
−0.192463 + 0.981304i \(0.561647\pi\)
\(368\) 0 0
\(369\) −10.3464 9.21609i −0.538611 0.479771i
\(370\) 0 0
\(371\) 5.24050 12.6517i 0.272073 0.656843i
\(372\) 0 0
\(373\) −7.72861 + 3.20130i −0.400172 + 0.165757i −0.573687 0.819074i \(-0.694488\pi\)
0.173515 + 0.984831i \(0.444488\pi\)
\(374\) 0 0
\(375\) −7.10593 15.8459i −0.366948 0.818277i
\(376\) 0 0
\(377\) −3.32577 3.32577i −0.171286 0.171286i
\(378\) 0 0
\(379\) 22.6759 9.39268i 1.16478 0.482470i 0.285319 0.958433i \(-0.407900\pi\)
0.879465 + 0.475963i \(0.157900\pi\)
\(380\) 0 0
\(381\) −0.999108 0.943047i −0.0511858 0.0483138i
\(382\) 0 0
\(383\) −11.3729 −0.581129 −0.290565 0.956855i \(-0.593843\pi\)
−0.290565 + 0.956855i \(0.593843\pi\)
\(384\) 0 0
\(385\) −23.1325 −1.17894
\(386\) 0 0
\(387\) 7.00826 3.38895i 0.356250 0.172270i
\(388\) 0 0
\(389\) −11.2635 + 4.66550i −0.571083 + 0.236550i −0.649489 0.760371i \(-0.725017\pi\)
0.0784059 + 0.996922i \(0.475017\pi\)
\(390\) 0 0
\(391\) 1.40529 + 1.40529i 0.0710687 + 0.0710687i
\(392\) 0 0
\(393\) −16.0674 + 7.20526i −0.810492 + 0.363457i
\(394\) 0 0
\(395\) −9.53179 + 3.94820i −0.479596 + 0.198655i
\(396\) 0 0
\(397\) 3.65241 8.81769i 0.183309 0.442547i −0.805336 0.592819i \(-0.798015\pi\)
0.988645 + 0.150272i \(0.0480149\pi\)
\(398\) 0 0
\(399\) 19.6969 + 7.50048i 0.986076 + 0.375494i
\(400\) 0 0
\(401\) −5.46170 −0.272744 −0.136372 0.990658i \(-0.543544\pi\)
−0.136372 + 0.990658i \(0.543544\pi\)
\(402\) 0 0
\(403\) −5.32225 2.20455i −0.265120 0.109816i
\(404\) 0 0
\(405\) −9.10425 + 5.07018i −0.452394 + 0.251939i
\(406\) 0 0
\(407\) 1.24040 + 1.24040i 0.0614843 + 0.0614843i
\(408\) 0 0
\(409\) −21.6474 + 21.6474i −1.07039 + 1.07039i −0.0730670 + 0.997327i \(0.523279\pi\)
−0.997327 + 0.0730670i \(0.976721\pi\)
\(410\) 0 0
\(411\) −0.209176 7.24671i −0.0103179 0.357454i
\(412\) 0 0
\(413\) −7.40871 + 17.8862i −0.364559 + 0.880123i
\(414\) 0 0
\(415\) 9.28636i 0.455849i
\(416\) 0 0
\(417\) −8.08541 3.07889i −0.395944 0.150774i
\(418\) 0 0
\(419\) 16.1057 + 6.67119i 0.786814 + 0.325909i 0.739662 0.672979i \(-0.234986\pi\)
0.0471519 + 0.998888i \(0.484986\pi\)
\(420\) 0 0
\(421\) 11.0865 + 26.7651i 0.540321 + 1.30445i 0.924497 + 0.381190i \(0.124486\pi\)
−0.384176 + 0.923260i \(0.625514\pi\)
\(422\) 0 0
\(423\) −35.3334 + 2.04150i −1.71797 + 0.0992612i
\(424\) 0 0
\(425\) −1.77263 + 1.77263i −0.0859852 + 0.0859852i
\(426\) 0 0
\(427\) 2.13396 + 5.15183i 0.103270 + 0.249315i
\(428\) 0 0
\(429\) 6.78813 + 6.40725i 0.327734 + 0.309345i
\(430\) 0 0
\(431\) 20.4774i 0.986361i −0.869927 0.493180i \(-0.835834\pi\)
0.869927 0.493180i \(-0.164166\pi\)
\(432\) 0 0
\(433\) 14.6756i 0.705262i 0.935762 + 0.352631i \(0.114713\pi\)
−0.935762 + 0.352631i \(0.885287\pi\)
\(434\) 0 0
\(435\) 7.57899 + 7.15373i 0.363385 + 0.342995i
\(436\) 0 0
\(437\) 4.02639 + 9.72056i 0.192608 + 0.464997i
\(438\) 0 0
\(439\) −4.86825 + 4.86825i −0.232349 + 0.232349i −0.813672 0.581324i \(-0.802535\pi\)
0.581324 + 0.813672i \(0.302535\pi\)
\(440\) 0 0
\(441\) 12.7503 0.736690i 0.607158 0.0350805i
\(442\) 0 0
\(443\) −6.00738 14.5031i −0.285419 0.689063i 0.714525 0.699610i \(-0.246643\pi\)
−0.999944 + 0.0105468i \(0.996643\pi\)
\(444\) 0 0
\(445\) 3.01555 + 1.24908i 0.142951 + 0.0592122i
\(446\) 0 0
\(447\) −23.4097 8.91432i −1.10724 0.421633i
\(448\) 0 0
\(449\) 1.76611i 0.0833480i −0.999131 0.0416740i \(-0.986731\pi\)
0.999131 0.0416740i \(-0.0132691\pi\)
\(450\) 0 0
\(451\) 10.5244 25.4082i 0.495576 1.19643i
\(452\) 0 0
\(453\) −0.209243 7.24901i −0.00983108 0.340588i
\(454\) 0 0
\(455\) −2.48622 + 2.48622i −0.116556 + 0.116556i
\(456\) 0 0
\(457\) −22.3579 22.3579i −1.04586 1.04586i −0.998897 0.0469601i \(-0.985047\pi\)
−0.0469601 0.998897i \(-0.514953\pi\)
\(458\) 0 0
\(459\) 2.72529 + 2.29003i 0.127206 + 0.106889i
\(460\) 0 0
\(461\) −0.811938 0.336316i −0.0378157 0.0156638i 0.363695 0.931518i \(-0.381515\pi\)
−0.401511 + 0.915854i \(0.631515\pi\)
\(462\) 0 0
\(463\) −14.0009 −0.650678 −0.325339 0.945597i \(-0.605478\pi\)
−0.325339 + 0.945597i \(0.605478\pi\)
\(464\) 0 0
\(465\) 11.9294 + 4.54265i 0.553211 + 0.210661i
\(466\) 0 0
\(467\) 5.94695 14.3572i 0.275192 0.664373i −0.724498 0.689277i \(-0.757928\pi\)
0.999690 + 0.0249045i \(0.00792818\pi\)
\(468\) 0 0
\(469\) 38.0385 15.7561i 1.75646 0.727548i
\(470\) 0 0
\(471\) 27.5776 12.3669i 1.27071 0.569838i
\(472\) 0 0
\(473\) 10.9257 + 10.9257i 0.502365 + 0.502365i
\(474\) 0 0
\(475\) −12.2615 + 5.07887i −0.562595 + 0.233034i
\(476\) 0 0
\(477\) 11.0233 5.33047i 0.504722 0.244066i
\(478\) 0 0
\(479\) 34.4708 1.57501 0.787505 0.616309i \(-0.211373\pi\)
0.787505 + 0.616309i \(0.211373\pi\)
\(480\) 0 0
\(481\) 0.266630 0.0121573
\(482\) 0 0
\(483\) 12.2600 + 11.5720i 0.557847 + 0.526546i
\(484\) 0 0
\(485\) 9.26935 3.83949i 0.420899 0.174342i
\(486\) 0 0
\(487\) 13.6710 + 13.6710i 0.619490 + 0.619490i 0.945401 0.325911i \(-0.105671\pi\)
−0.325911 + 0.945401i \(0.605671\pi\)
\(488\) 0 0
\(489\) 8.55934 + 19.0869i 0.387067 + 0.863140i
\(490\) 0 0
\(491\) −16.0034 + 6.62882i −0.722223 + 0.299154i −0.713352 0.700806i \(-0.752824\pi\)
−0.00887095 + 0.999961i \(0.502824\pi\)
\(492\) 0 0
\(493\) 1.36238 3.28908i 0.0613585 0.148133i
\(494\) 0 0
\(495\) −15.4449 13.7576i −0.694196 0.618359i
\(496\) 0 0
\(497\) 37.9191 1.70090
\(498\) 0 0
\(499\) 6.36166 + 2.63509i 0.284787 + 0.117963i 0.520504 0.853859i \(-0.325744\pi\)
−0.235717 + 0.971822i \(0.575744\pi\)
\(500\) 0 0
\(501\) −0.0908900 3.14880i −0.00406067 0.140678i
\(502\) 0 0
\(503\) 16.8510 + 16.8510i 0.751350 + 0.751350i 0.974731 0.223381i \(-0.0717095\pi\)
−0.223381 + 0.974731i \(0.571709\pi\)
\(504\) 0 0
\(505\) 13.9062 13.9062i 0.618817 0.618817i
\(506\) 0 0
\(507\) −21.0891 + 0.608736i −0.936599 + 0.0270349i
\(508\) 0 0
\(509\) 6.71610 16.2141i 0.297686 0.718678i −0.702291 0.711890i \(-0.747839\pi\)
0.999977 0.00678764i \(-0.00216059\pi\)
\(510\) 0 0
\(511\) 11.1392i 0.492769i
\(512\) 0 0
\(513\) 8.69024 + 16.7222i 0.383684 + 0.738302i
\(514\) 0 0
\(515\) 8.16609 + 3.38250i 0.359841 + 0.149051i
\(516\) 0 0
\(517\) −26.8828 64.9009i −1.18231 2.85434i
\(518\) 0 0
\(519\) 0.0968155 0.0434160i 0.00424973 0.00190575i
\(520\) 0 0
\(521\) 18.2126 18.2126i 0.797908 0.797908i −0.184858 0.982765i \(-0.559182\pi\)
0.982765 + 0.184858i \(0.0591824\pi\)
\(522\) 0 0
\(523\) 0.472480 + 1.14067i 0.0206601 + 0.0498779i 0.933873 0.357605i \(-0.116407\pi\)
−0.913213 + 0.407483i \(0.866407\pi\)
\(524\) 0 0
\(525\) −14.5969 + 15.4647i −0.637062 + 0.674933i
\(526\) 0 0
\(527\) 4.36045i 0.189944i
\(528\) 0 0
\(529\) 14.5841i 0.634090i
\(530\) 0 0
\(531\) −15.5841 + 7.53590i −0.676291 + 0.327030i
\(532\) 0 0
\(533\) −1.59967 3.86195i −0.0692895 0.167280i
\(534\) 0 0
\(535\) 0.0150296 0.0150296i 0.000649785 0.000649785i
\(536\) 0 0
\(537\) 2.17536 + 4.85094i 0.0938735 + 0.209333i
\(538\) 0 0
\(539\) 9.70087 + 23.4200i 0.417846 + 1.00877i
\(540\) 0 0
\(541\) 14.2297 + 5.89412i 0.611781 + 0.253408i 0.666990 0.745067i \(-0.267582\pi\)
−0.0552087 + 0.998475i \(0.517582\pi\)
\(542\) 0 0
\(543\) 13.8993 36.5008i 0.596478 1.56640i
\(544\) 0 0
\(545\) 14.9576i 0.640713i
\(546\) 0 0
\(547\) 6.24323 15.0725i 0.266941 0.644453i −0.732395 0.680880i \(-0.761598\pi\)
0.999336 + 0.0364266i \(0.0115975\pi\)
\(548\) 0 0
\(549\) −1.63917 + 4.70886i −0.0699582 + 0.200969i
\(550\) 0 0
\(551\) 13.3272 13.3272i 0.567756 0.567756i
\(552\) 0 0
\(553\) 21.1397 + 21.1397i 0.898952 + 0.898952i
\(554\) 0 0
\(555\) −0.590567 + 0.0170467i −0.0250682 + 0.000723593i
\(556\) 0 0
\(557\) −7.09223 2.93770i −0.300507 0.124474i 0.227335 0.973817i \(-0.426999\pi\)
−0.527842 + 0.849342i \(0.676999\pi\)
\(558\) 0 0
\(559\) 2.34854 0.0993325
\(560\) 0 0
\(561\) −2.51436 + 6.60291i −0.106156 + 0.278775i
\(562\) 0 0
\(563\) 0.517007 1.24817i 0.0217892 0.0526039i −0.912610 0.408830i \(-0.865937\pi\)
0.934400 + 0.356226i \(0.115937\pi\)
\(564\) 0 0
\(565\) −11.2305 + 4.65183i −0.472471 + 0.195704i
\(566\) 0 0
\(567\) 23.6693 + 18.7509i 0.994016 + 0.787465i
\(568\) 0 0
\(569\) −4.40004 4.40004i −0.184459 0.184459i 0.608836 0.793296i \(-0.291637\pi\)
−0.793296 + 0.608836i \(0.791637\pi\)
\(570\) 0 0
\(571\) −23.9135 + 9.90528i −1.00075 + 0.414523i −0.822071 0.569385i \(-0.807181\pi\)
−0.178676 + 0.983908i \(0.557181\pi\)
\(572\) 0 0
\(573\) −27.4119 + 29.0415i −1.14515 + 1.21322i
\(574\) 0 0
\(575\) −10.6158 −0.442709
\(576\) 0 0
\(577\) −7.45244 −0.310249 −0.155124 0.987895i \(-0.549578\pi\)
−0.155124 + 0.987895i \(0.549578\pi\)
\(578\) 0 0
\(579\) −8.95040 + 9.48246i −0.371966 + 0.394078i
\(580\) 0 0
\(581\) 24.8608 10.2977i 1.03140 0.427220i
\(582\) 0 0
\(583\) 17.1850 + 17.1850i 0.711732 + 0.711732i
\(584\) 0 0
\(585\) −3.13861 + 0.181343i −0.129766 + 0.00749762i
\(586\) 0 0
\(587\) 39.1257 16.2064i 1.61489 0.668910i 0.621472 0.783436i \(-0.286535\pi\)
0.993420 + 0.114526i \(0.0365349\pi\)
\(588\) 0 0
\(589\) 8.83416 21.3275i 0.364005 0.878786i
\(590\) 0 0
\(591\) 1.43274 3.76249i 0.0589350 0.154768i
\(592\) 0 0
\(593\) −42.7694 −1.75633 −0.878164 0.478359i \(-0.841232\pi\)
−0.878164 + 0.478359i \(0.841232\pi\)
\(594\) 0 0
\(595\) −2.45879 1.01847i −0.100801 0.0417530i
\(596\) 0 0
\(597\) 18.2899 0.527938i 0.748556 0.0216071i
\(598\) 0 0
\(599\) −21.3145 21.3145i −0.870885 0.870885i 0.121684 0.992569i \(-0.461171\pi\)
−0.992569 + 0.121684i \(0.961171\pi\)
\(600\) 0 0
\(601\) −4.26621 + 4.26621i −0.174022 + 0.174022i −0.788744 0.614722i \(-0.789268\pi\)
0.614722 + 0.788744i \(0.289268\pi\)
\(602\) 0 0
\(603\) 34.7678 + 12.1028i 1.41586 + 0.492865i
\(604\) 0 0
\(605\) 10.8366 26.1618i 0.440570 1.06363i
\(606\) 0 0
\(607\) 30.5562i 1.24024i 0.784508 + 0.620118i \(0.212915\pi\)
−0.784508 + 0.620118i \(0.787085\pi\)
\(608\) 0 0
\(609\) 10.7471 28.2228i 0.435495 1.14364i
\(610\) 0 0
\(611\) −9.86469 4.08609i −0.399083 0.165305i
\(612\) 0 0
\(613\) 13.4587 + 32.4921i 0.543590 + 1.31234i 0.922174 + 0.386775i \(0.126411\pi\)
−0.378584 + 0.925567i \(0.623589\pi\)
\(614\) 0 0
\(615\) 3.79007 + 8.45168i 0.152830 + 0.340804i
\(616\) 0 0
\(617\) 32.1395 32.1395i 1.29389 1.29389i 0.361524 0.932363i \(-0.382257\pi\)
0.932363 0.361524i \(-0.117743\pi\)
\(618\) 0 0
\(619\) −14.9456 36.0820i −0.600716 1.45026i −0.872846 0.487996i \(-0.837728\pi\)
0.272129 0.962261i \(-0.412272\pi\)
\(620\) 0 0
\(621\) 1.30335 + 15.0177i 0.0523018 + 0.602639i
\(622\) 0 0
\(623\) 9.45815i 0.378933i
\(624\) 0 0
\(625\) 6.68739i 0.267496i
\(626\) 0 0
\(627\) −25.6754 + 27.2017i −1.02538 + 1.08633i
\(628\) 0 0
\(629\) 0.0772324 + 0.186456i 0.00307946 + 0.00743447i
\(630\) 0 0
\(631\) −20.5960 + 20.5960i −0.819912 + 0.819912i −0.986095 0.166183i \(-0.946856\pi\)
0.166183 + 0.986095i \(0.446856\pi\)
\(632\) 0 0
\(633\) 27.3392 12.2600i 1.08664 0.487291i
\(634\) 0 0
\(635\) 0.351471 + 0.848526i 0.0139477 + 0.0336727i
\(636\) 0 0
\(637\) 3.55974 + 1.47449i 0.141042 + 0.0584216i
\(638\) 0 0
\(639\) 25.3175 + 22.5517i 1.00154 + 0.892130i
\(640\) 0 0
\(641\) 12.6916i 0.501289i −0.968079 0.250645i \(-0.919357\pi\)
0.968079 0.250645i \(-0.0806426\pi\)
\(642\) 0 0
\(643\) −0.974363 + 2.35232i −0.0384251 + 0.0927665i −0.941927 0.335818i \(-0.890987\pi\)
0.903502 + 0.428584i \(0.140987\pi\)
\(644\) 0 0
\(645\) −5.20185 + 0.150151i −0.204823 + 0.00591221i
\(646\) 0 0
\(647\) 22.3823 22.3823i 0.879941 0.879941i −0.113587 0.993528i \(-0.536234\pi\)
0.993528 + 0.113587i \(0.0362341\pi\)
\(648\) 0 0
\(649\) −24.2952 24.2952i −0.953670 0.953670i
\(650\) 0 0
\(651\) −1.06725 36.9739i −0.0418289 1.44912i
\(652\) 0 0
\(653\) 2.02663 + 0.839459i 0.0793083 + 0.0328506i 0.421985 0.906603i \(-0.361334\pi\)
−0.342677 + 0.939453i \(0.611334\pi\)
\(654\) 0 0
\(655\) 11.7716 0.459953
\(656\) 0 0
\(657\) −6.62483 + 7.43731i −0.258459 + 0.290157i
\(658\) 0 0
\(659\) 4.65284 11.2330i 0.181249 0.437574i −0.806975 0.590585i \(-0.798897\pi\)
0.988224 + 0.153011i \(0.0488971\pi\)
\(660\) 0 0
\(661\) −22.0472 + 9.13226i −0.857538 + 0.355204i −0.767744 0.640757i \(-0.778621\pi\)
−0.0897938 + 0.995960i \(0.528621\pi\)
\(662\) 0 0
\(663\) 0.439427 + 0.979901i 0.0170659 + 0.0380562i
\(664\) 0 0
\(665\) −9.96289 9.96289i −0.386344 0.386344i
\(666\) 0 0
\(667\) 13.9282 5.76923i 0.539300 0.223386i
\(668\) 0 0
\(669\) −22.9550 21.6670i −0.887491 0.837694i
\(670\) 0 0
\(671\) −9.89643 −0.382048
\(672\) 0 0
\(673\) 1.77155 0.0682883 0.0341441 0.999417i \(-0.489129\pi\)
0.0341441 + 0.999417i \(0.489129\pi\)
\(674\) 0 0
\(675\) −18.9432 + 1.64404i −0.729126 + 0.0632793i
\(676\) 0 0
\(677\) 31.1005 12.8822i 1.19529 0.495105i 0.305815 0.952091i \(-0.401071\pi\)
0.889474 + 0.456986i \(0.151071\pi\)
\(678\) 0 0
\(679\) −20.5577 20.5577i −0.788930 0.788930i
\(680\) 0 0
\(681\) −18.5526 + 8.31973i −0.710937 + 0.318813i
\(682\) 0 0
\(683\) −2.51740 + 1.04274i −0.0963256 + 0.0398994i −0.430326 0.902674i \(-0.641601\pi\)
0.334000 + 0.942573i \(0.391601\pi\)
\(684\) 0 0
\(685\) −1.85465 + 4.47752i −0.0708624 + 0.171077i
\(686\) 0 0
\(687\) −22.0365 8.39142i −0.840747 0.320153i
\(688\) 0 0
\(689\) 3.69401 0.140731
\(690\) 0 0
\(691\) −36.0335 14.9256i −1.37078 0.567795i −0.428779 0.903409i \(-0.641056\pi\)
−0.942000 + 0.335614i \(0.891056\pi\)
\(692\) 0 0
\(693\) −19.7041 + 56.6039i −0.748496 + 2.15021i
\(694\) 0 0
\(695\) 4.08969 + 4.08969i 0.155131 + 0.155131i
\(696\) 0 0
\(697\) 2.23732 2.23732i 0.0847444 0.0847444i
\(698\) 0 0
\(699\) −0.828543 28.7040i −0.0313384 1.08569i
\(700\) 0 0
\(701\) −1.39440 + 3.36638i −0.0526658 + 0.127147i −0.948023 0.318203i \(-0.896921\pi\)
0.895357 + 0.445350i \(0.146921\pi\)
\(702\) 0 0
\(703\) 1.06845i 0.0402973i
\(704\) 0 0
\(705\) 22.1109 + 8.41972i 0.832743 + 0.317105i
\(706\) 0 0
\(707\) −52.6494 21.8081i −1.98008 0.820177i
\(708\) 0 0
\(709\) 9.10175 + 21.9736i 0.341824 + 0.825235i 0.997531 + 0.0702205i \(0.0223703\pi\)
−0.655708 + 0.755015i \(0.727630\pi\)
\(710\) 0 0
\(711\) 1.54191 + 26.6868i 0.0578263 + 1.00083i
\(712\) 0 0
\(713\) 13.0568 13.0568i 0.488980 0.488980i
\(714\) 0 0
\(715\) −2.38796 5.76505i −0.0893047 0.215601i
\(716\) 0 0
\(717\) −27.2003 25.6741i −1.01581 0.958816i
\(718\) 0 0
\(719\) 19.5450i 0.728907i 0.931222 + 0.364453i \(0.118744\pi\)
−0.931222 + 0.364453i \(0.881256\pi\)
\(720\) 0 0
\(721\) 25.6126i 0.953863i
\(722\) 0 0
\(723\) 18.4017 + 17.3692i 0.684366 + 0.645966i
\(724\) 0 0
\(725\) 7.27728 + 17.5689i 0.270271 + 0.652493i
\(726\) 0 0
\(727\) 20.8914 20.8914i 0.774821 0.774821i −0.204124 0.978945i \(-0.565435\pi\)
0.978945 + 0.204124i \(0.0654347\pi\)
\(728\) 0 0
\(729\) 4.65151 + 26.5963i 0.172278 + 0.985048i
\(730\) 0 0
\(731\) 0.680281 + 1.64234i 0.0251611 + 0.0607443i
\(732\) 0 0
\(733\) −9.09957 3.76916i −0.336100 0.139217i 0.208249 0.978076i \(-0.433223\pi\)
−0.544350 + 0.838859i \(0.683223\pi\)
\(734\) 0 0
\(735\) −7.97886 3.03831i −0.294305 0.112070i
\(736\) 0 0
\(737\) 73.0703i 2.69158i
\(738\) 0 0
\(739\) −8.03797 + 19.4054i −0.295682 + 0.713839i 0.704311 + 0.709892i \(0.251256\pi\)
−0.999992 + 0.00394665i \(0.998744\pi\)
\(740\) 0 0
\(741\) 0.164043 + 5.68309i 0.00602625 + 0.208774i
\(742\) 0 0
\(743\) −14.8827 + 14.8827i −0.545995 + 0.545995i −0.925280 0.379285i \(-0.876170\pi\)
0.379285 + 0.925280i \(0.376170\pi\)
\(744\) 0 0
\(745\) 11.8409 + 11.8409i 0.433817 + 0.433817i
\(746\) 0 0
\(747\) 22.7232 + 7.91004i 0.831398 + 0.289413i
\(748\) 0 0
\(749\) −0.0569026 0.0235698i −0.00207917 0.000861222i
\(750\) 0 0
\(751\) 47.9945 1.75135 0.875673 0.482905i \(-0.160418\pi\)
0.875673 + 0.482905i \(0.160418\pi\)
\(752\) 0 0
\(753\) −4.59272 1.74889i −0.167368 0.0637330i
\(754\) 0 0
\(755\) −1.85523 + 4.47893i −0.0675189 + 0.163005i
\(756\) 0 0
\(757\) −4.61422 + 1.91127i −0.167707 + 0.0694663i −0.464957 0.885333i \(-0.653930\pi\)
0.297251 + 0.954799i \(0.403930\pi\)
\(758\) 0 0
\(759\) −27.3004 + 12.2426i −0.990943 + 0.444379i
\(760\) 0 0
\(761\) −35.4112 35.4112i −1.28365 1.28365i −0.938573 0.345081i \(-0.887851\pi\)
−0.345081 0.938573i \(-0.612149\pi\)
\(762\) 0 0
\(763\) 40.0435 16.5865i 1.44967 0.600473i
\(764\) 0 0
\(765\) −1.03595 2.14232i −0.0374549 0.0774557i
\(766\) 0 0
\(767\) −5.22237 −0.188569
\(768\) 0 0
\(769\) 49.1306 1.77169 0.885847 0.463978i \(-0.153578\pi\)
0.885847 + 0.463978i \(0.153578\pi\)
\(770\) 0 0
\(771\) 1.70592 + 1.61020i 0.0614371 + 0.0579898i
\(772\) 0 0
\(773\) −48.9331 + 20.2688i −1.76000 + 0.729017i −0.763467 + 0.645847i \(0.776504\pi\)
−0.996535 + 0.0831696i \(0.973496\pi\)
\(774\) 0 0
\(775\) 16.4698 + 16.4698i 0.591611 + 0.591611i
\(776\) 0 0
\(777\) 0.700519 + 1.56212i 0.0251310 + 0.0560408i
\(778\) 0 0
\(779\) 15.4757 6.41026i 0.554476 0.229672i
\(780\) 0 0
\(781\) −25.7531 + 62.1736i −0.921520 + 2.22474i
\(782\) 0 0
\(783\) 23.9605 12.4519i 0.856277 0.444993i
\(784\) 0 0
\(785\) −20.2044 −0.721126
\(786\) 0 0
\(787\) 44.3943 + 18.3887i 1.58249 + 0.655487i 0.988805 0.149214i \(-0.0476743\pi\)
0.593681 + 0.804701i \(0.297674\pi\)
\(788\) 0 0
\(789\) −0.249039 8.62771i −0.00886602 0.307155i
\(790\) 0 0
\(791\) 24.9072 + 24.9072i 0.885597 + 0.885597i
\(792\) 0 0
\(793\) −1.06364 + 1.06364i −0.0377711 + 0.0377711i
\(794\) 0 0
\(795\) −8.18198 + 0.236173i −0.290185 + 0.00837619i
\(796\) 0 0
\(797\) −10.5848 + 25.5539i −0.374931 + 0.905164i 0.617968 + 0.786203i \(0.287956\pi\)
−0.992899 + 0.118961i \(0.962044\pi\)
\(798\) 0 0
\(799\) 8.08201i 0.285921i
\(800\) 0 0
\(801\) 5.62505 6.31492i 0.198752 0.223127i
\(802\) 0 0
\(803\) −18.2642 7.56530i −0.644531 0.266974i
\(804\) 0 0
\(805\) −4.31287 10.4122i −0.152009 0.366981i
\(806\) 0 0
\(807\) −28.1811 + 12.6375i −0.992022 + 0.444863i
\(808\) 0 0
\(809\) 26.4258 26.4258i 0.929083 0.929083i −0.0685638 0.997647i \(-0.521842\pi\)
0.997647 + 0.0685638i \(0.0218417\pi\)
\(810\) 0 0
\(811\) −10.2190 24.6709i −0.358838 0.866311i −0.995464 0.0951387i \(-0.969671\pi\)
0.636626 0.771173i \(-0.280329\pi\)
\(812\) 0 0
\(813\) −10.1091 + 10.7101i −0.354542 + 0.375618i
\(814\) 0 0
\(815\) 13.9838i 0.489831i
\(816\) 0 0
\(817\) 9.41115i 0.329254i
\(818\) 0 0
\(819\) 3.96590 + 8.20140i 0.138580 + 0.286580i
\(820\) 0 0
\(821\) 15.9273 + 38.4519i 0.555866 + 1.34198i 0.913013 + 0.407932i \(0.133750\pi\)
−0.357146 + 0.934048i \(0.616250\pi\)
\(822\) 0 0
\(823\) 27.1818 27.1818i 0.947499 0.947499i −0.0511904 0.998689i \(-0.516302\pi\)
0.998689 + 0.0511904i \(0.0163015\pi\)
\(824\) 0 0
\(825\) −15.4428 34.4366i −0.537648 1.19893i
\(826\) 0 0
\(827\) 12.6297 + 30.4907i 0.439176 + 1.06027i 0.976234 + 0.216720i \(0.0695358\pi\)
−0.537058 + 0.843546i \(0.680464\pi\)
\(828\) 0 0
\(829\) −34.6568 14.3553i −1.20368 0.498581i −0.311494 0.950248i \(-0.600829\pi\)
−0.892187 + 0.451667i \(0.850829\pi\)
\(830\) 0 0
\(831\) −13.4786 + 35.3960i −0.467568 + 1.22787i
\(832\) 0 0
\(833\) 2.91645i 0.101049i
\(834\) 0 0
\(835\) −0.805870 + 1.94554i −0.0278883 + 0.0673282i
\(836\) 0 0
\(837\) 21.2770 25.3211i 0.735439 0.875225i
\(838\) 0 0
\(839\) 5.13731 5.13731i 0.177360 0.177360i −0.612844 0.790204i \(-0.709975\pi\)
0.790204 + 0.612844i \(0.209975\pi\)
\(840\) 0 0
\(841\) 1.41020 + 1.41020i 0.0486277 + 0.0486277i
\(842\) 0 0
\(843\) −25.3030 + 0.730371i −0.871481 + 0.0251553i
\(844\) 0 0
\(845\) 13.0303 + 5.39732i 0.448255 + 0.185673i
\(846\) 0 0
\(847\) −82.0555 −2.81946
\(848\) 0 0
\(849\) 5.78660 15.1961i 0.198596 0.521528i
\(850\) 0 0
\(851\) −0.327054 + 0.789578i −0.0112113 + 0.0270664i
\(852\) 0 0
\(853\) 27.7734 11.5041i 0.950943 0.393893i 0.147358 0.989083i \(-0.452923\pi\)
0.803585 + 0.595190i \(0.202923\pi\)
\(854\) 0 0
\(855\) −0.726686 12.5772i −0.0248521 0.430130i
\(856\) 0 0
\(857\) −23.9037 23.9037i −0.816536 0.816536i 0.169069 0.985604i \(-0.445924\pi\)
−0.985604 + 0.169069i \(0.945924\pi\)
\(858\) 0 0
\(859\) −36.4970 + 15.1176i −1.24526 + 0.515805i −0.905356 0.424654i \(-0.860396\pi\)
−0.339907 + 0.940459i \(0.610396\pi\)
\(860\) 0 0
\(861\) 18.4234 19.5186i 0.627869 0.665193i
\(862\) 0 0
\(863\) −48.7809 −1.66052 −0.830260 0.557377i \(-0.811808\pi\)
−0.830260 + 0.557377i \(0.811808\pi\)
\(864\) 0 0
\(865\) −0.0709307 −0.00241172
\(866\) 0 0
\(867\) 19.6533 20.8216i 0.667461 0.707139i
\(868\) 0 0
\(869\) −49.0187 + 20.3042i −1.66284 + 0.688773i
\(870\) 0 0
\(871\) 7.85341 + 7.85341i 0.266103 + 0.266103i
\(872\) 0 0
\(873\) −1.49946 25.9520i −0.0507490 0.878342i
\(874\) 0 0
\(875\) 31.0796 12.8736i 1.05068 0.435207i
\(876\) 0 0
\(877\) −8.45430 + 20.4105i −0.285481 + 0.689213i −0.999945 0.0104591i \(-0.996671\pi\)
0.714464 + 0.699672i \(0.246671\pi\)
\(878\) 0 0
\(879\) 7.59343 19.9410i 0.256120 0.672592i
\(880\) 0 0
\(881\) −25.8223 −0.869976 −0.434988 0.900436i \(-0.643247\pi\)
−0.434988 + 0.900436i \(0.643247\pi\)
\(882\) 0 0
\(883\) −33.3809 13.8268i −1.12336 0.465310i −0.257839 0.966188i \(-0.583010\pi\)
−0.865518 + 0.500878i \(0.833010\pi\)
\(884\) 0 0
\(885\) 11.5672 0.333887i 0.388827 0.0112235i
\(886\) 0 0
\(887\) 34.6997 + 34.6997i 1.16510 + 1.16510i 0.983343 + 0.181760i \(0.0581796\pi\)
0.181760 + 0.983343i \(0.441820\pi\)
\(888\) 0 0
\(889\) 1.88187 1.88187i 0.0631158 0.0631158i
\(890\) 0 0
\(891\) −46.8200 + 26.0741i −1.56853 + 0.873517i
\(892\) 0 0
\(893\) 16.3739 39.5302i 0.547933 1.32283i
\(894\) 0 0
\(895\) 3.55398i 0.118796i
\(896\) 0 0
\(897\) −1.61838 + 4.24999i −0.0540360 + 0.141903i
\(898\) 0 0
\(899\) −30.5593 12.6581i −1.01921 0.422170i
\(900\) 0 0
\(901\) 1.07001 + 2.58324i 0.0356473 + 0.0860602i
\(902\) 0 0
\(903\) 6.17033 + 13.7595i 0.205336 + 0.457889i
\(904\) 0 0
\(905\) −18.4625 + 18.4625i −0.613715 + 0.613715i
\(906\) 0 0
\(907\) 7.42403 + 17.9232i 0.246511 + 0.595130i 0.997903 0.0647258i \(-0.0206173\pi\)
−0.751392 + 0.659856i \(0.770617\pi\)
\(908\) 0 0
\(909\) −22.1825 45.8728i −0.735746 1.52151i
\(910\) 0 0
\(911\) 37.5930i 1.24551i −0.782416 0.622756i \(-0.786013\pi\)
0.782416 0.622756i \(-0.213987\pi\)
\(912\) 0 0
\(913\) 47.7565i 1.58051i
\(914\) 0 0
\(915\) 2.28790 2.42390i 0.0756355 0.0801317i
\(916\) 0 0
\(917\) −13.0535 31.5140i −0.431066 1.04069i
\(918\) 0 0
\(919\) −28.3220 + 28.3220i −0.934258 + 0.934258i −0.997968 0.0637109i \(-0.979706\pi\)
0.0637109 + 0.997968i \(0.479706\pi\)
\(920\) 0 0
\(921\) −21.5788 + 9.67681i −0.711046 + 0.318862i
\(922\) 0 0
\(923\) 3.91438 + 9.45014i 0.128843 + 0.311055i
\(924\) 0 0
\(925\) −0.995970 0.412544i −0.0327473 0.0135644i
\(926\) 0 0
\(927\) 15.2326 17.1008i 0.500304 0.561663i
\(928\) 0 0
\(929\) 14.6297i 0.479986i 0.970775 + 0.239993i \(0.0771452\pi\)
−0.970775 + 0.239993i \(0.922855\pi\)
\(930\) 0 0
\(931\) −5.90865 + 14.2647i −0.193648 + 0.467508i
\(932\) 0 0
\(933\) 46.0362 1.32884i 1.50716 0.0435042i
\(934\) 0 0
\(935\) 3.33983 3.33983i 0.109224 0.109224i
\(936\) 0 0
\(937\) 5.60824 + 5.60824i 0.183213 + 0.183213i 0.792754 0.609541i \(-0.208646\pi\)
−0.609541 + 0.792754i \(0.708646\pi\)
\(938\) 0 0
\(939\) 0.304493 + 10.5489i 0.00993676 + 0.344249i
\(940\) 0 0
\(941\) −10.5273 4.36054i −0.343180 0.142150i 0.204435 0.978880i \(-0.434464\pi\)
−0.547615 + 0.836730i \(0.684464\pi\)
\(942\) 0 0
\(943\) 13.3987 0.436321
\(944\) 0 0
\(945\) −9.30855 17.9120i −0.302807 0.582676i
\(946\) 0 0
\(947\) −22.3491 + 53.9555i −0.726248 + 1.75332i −0.0715370 + 0.997438i \(0.522790\pi\)
−0.654711 + 0.755879i \(0.727210\pi\)
\(948\) 0 0
\(949\) −2.77609 + 1.14990i −0.0901158 + 0.0373272i
\(950\) 0 0
\(951\) −1.00637 2.24416i −0.0326338 0.0727718i
\(952\) 0 0
\(953\) 32.8805 + 32.8805i 1.06510 + 1.06510i 0.997728 + 0.0673750i \(0.0214624\pi\)
0.0673750 + 0.997728i \(0.478538\pi\)
\(954\) 0 0
\(955\) 24.6644 10.2163i 0.798122 0.330593i
\(956\) 0 0
\(957\) 38.9761 + 36.7891i 1.25992 + 1.18922i
\(958\) 0 0
\(959\) 14.0435 0.453489
\(960\) 0 0
\(961\) −9.51359 −0.306890
\(962\) 0 0
\(963\) −0.0239745 0.0495786i −0.000772566 0.00159765i
\(964\) 0 0
\(965\) 8.05330 3.33578i 0.259245 0.107383i
\(966\) 0 0
\(967\) −14.6204 14.6204i −0.470160 0.470160i 0.431807 0.901966i \(-0.357876\pi\)
−0.901966 + 0.431807i \(0.857876\pi\)
\(968\) 0 0
\(969\) −3.92670 + 1.76089i −0.126144 + 0.0565679i
\(970\) 0 0
\(971\) −39.9825 + 16.5613i −1.28310 + 0.531477i −0.916921 0.399068i \(-0.869334\pi\)
−0.366178 + 0.930545i \(0.619334\pi\)
\(972\) 0 0
\(973\) 6.41357 15.4837i 0.205610 0.496386i
\(974\) 0 0
\(975\) −5.36091 2.04141i −0.171687 0.0653775i
\(976\) 0 0
\(977\) 14.1567 0.452912 0.226456 0.974021i \(-0.427286\pi\)
0.226456 + 0.974021i \(0.427286\pi\)
\(978\) 0 0
\(979\) 15.5079 + 6.42359i 0.495635 + 0.205299i
\(980\) 0 0
\(981\) 36.6004 + 12.7408i 1.16856 + 0.406781i
\(982\) 0 0
\(983\) −27.5590 27.5590i −0.878995 0.878995i 0.114436 0.993431i \(-0.463494\pi\)
−0.993431 + 0.114436i \(0.963494\pi\)
\(984\) 0 0
\(985\) −1.90311 + 1.90311i −0.0606381 + 0.0606381i
\(986\) 0 0
\(987\) −1.97813 68.5304i −0.0629646 2.18135i
\(988\) 0 0
\(989\) −2.88077 + 6.95478i −0.0916030 + 0.221149i
\(990\) 0 0
\(991\) 14.9042i 0.473446i 0.971577 + 0.236723i \(0.0760734\pi\)
−0.971577 + 0.236723i \(0.923927\pi\)
\(992\) 0 0
\(993\) −21.9447 8.35645i −0.696395 0.265184i
\(994\) 0 0
\(995\) −11.3008 4.68093i −0.358258 0.148395i
\(996\) 0 0
\(997\) −4.66402 11.2599i −0.147711 0.356606i 0.832655 0.553792i \(-0.186820\pi\)
−0.980366 + 0.197186i \(0.936820\pi\)
\(998\) 0 0
\(999\) −0.461327 + 1.45960i −0.0145958 + 0.0461798i
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 384.2.o.a.47.4 56
3.2 odd 2 inner 384.2.o.a.47.7 56
4.3 odd 2 96.2.o.a.35.6 yes 56
8.3 odd 2 768.2.o.b.95.4 56
8.5 even 2 768.2.o.a.95.11 56
12.11 even 2 96.2.o.a.35.9 yes 56
24.5 odd 2 768.2.o.a.95.8 56
24.11 even 2 768.2.o.b.95.7 56
32.5 even 8 768.2.o.b.671.7 56
32.11 odd 8 inner 384.2.o.a.335.7 56
32.21 even 8 96.2.o.a.11.9 yes 56
32.27 odd 8 768.2.o.a.671.8 56
96.5 odd 8 768.2.o.b.671.4 56
96.11 even 8 inner 384.2.o.a.335.4 56
96.53 odd 8 96.2.o.a.11.6 56
96.59 even 8 768.2.o.a.671.11 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.o.a.11.6 56 96.53 odd 8
96.2.o.a.11.9 yes 56 32.21 even 8
96.2.o.a.35.6 yes 56 4.3 odd 2
96.2.o.a.35.9 yes 56 12.11 even 2
384.2.o.a.47.4 56 1.1 even 1 trivial
384.2.o.a.47.7 56 3.2 odd 2 inner
384.2.o.a.335.4 56 96.11 even 8 inner
384.2.o.a.335.7 56 32.11 odd 8 inner
768.2.o.a.95.8 56 24.5 odd 2
768.2.o.a.95.11 56 8.5 even 2
768.2.o.a.671.8 56 32.27 odd 8
768.2.o.a.671.11 56 96.59 even 8
768.2.o.b.95.4 56 8.3 odd 2
768.2.o.b.95.7 56 24.11 even 2
768.2.o.b.671.4 56 96.5 odd 8
768.2.o.b.671.7 56 32.5 even 8