LMFDB - Embedded newform 216.3.r.a.43.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [216,3,Mod(43,216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("216.43"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 4])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	216.r (of order $18$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.88557371018$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			43.2
Root			$0.483690 - 1.32893i$ of defining polynomial
Character	$\chi$	$=$	216.43
Dual form			216.3.r.a.211.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.53209 - 1.28558i) q^{2} +(1.90707 + 2.31583i) q^{3} +(0.694593 + 3.93923i) q^{4} +(0.0553729 - 5.99974i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-1.72616 + 8.83292i) q^{9} +(13.4204 - 4.88462i) q^{11} +(-7.79796 + 9.12096i) q^{12} +(-15.0351 + 5.47232i) q^{16} +(10.1424 + 17.5672i) q^{17} +(14.0000 - 11.3137i) q^{18} +(-18.8769 + 32.6958i) q^{19} +(-26.8408 - 9.76924i) q^{22} +(23.6728 - 3.94925i) q^{24} +(19.1511 + 16.0697i) q^{25} +(-23.7474 + 12.8475i) q^{27} +(30.0702 + 10.9446i) q^{32} +(36.9056 + 21.7640i) q^{33} +(7.04485 - 39.9533i) q^{34} +(-35.9939 - 0.664446i) q^{36} +(70.9540 - 25.8251i) q^{38} +(50.8983 - 42.7087i) q^{41} +(41.1752 - 14.9865i) q^{43} +(28.5633 + 49.4731i) q^{44} +(-41.3460 - 24.3826i) q^{48} +(-46.0449 - 16.7590i) q^{49} +(-8.68241 - 49.2404i) q^{50} +(-21.3403 + 56.9900i) q^{51} +(52.8997 + 10.8456i) q^{54} +(-111.718 + 18.6374i) q^{57} +(-45.1367 - 16.4284i) q^{59} +(-32.0000 - 55.4256i) q^{64} +(-28.5633 - 80.7893i) q^{66} +(-81.3715 + 68.2788i) q^{67} +(-62.1564 + 52.1554i) q^{68} +(54.2916 + 47.2908i) q^{72} +(65.2976 - 113.099i) q^{73} +(-0.692161 + 74.9968i) q^{75} +(-141.908 - 51.6503i) q^{76} +(-75.0408 - 30.4940i) q^{81} -132.886 q^{82} +(-26.7420 - 22.4392i) q^{83} +(-82.3504 - 29.9731i) q^{86} +(19.8399 - 112.518i) q^{88} +(80.5908 - 139.587i) q^{89} +(32.0000 + 90.5097i) q^{96} +(172.387 - 62.7437i) q^{97} +(49.0000 + 84.8705i) q^{98} +(19.9798 + 126.973i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 48 q^{8} - 42 q^{11} + 24 q^{12} + 168 q^{18} + 84 q^{22} - 138 q^{27} + 186 q^{33} + 12 q^{34} + 408 q^{38} + 138 q^{41} - 42 q^{43} - 588 q^{51} - 246 q^{57} - 492 q^{59} - 384 q^{64} - 186 q^{67}+ \cdots + 588 q^{98}+O(q^{100}) $ 12 * q + 48 * q^8 - 42 * q^11 + 24 * q^12 + 168 * q^18 + 84 * q^22 - 138 * q^27 + 186 * q^33 + 12 * q^34 + 408 * q^38 + 138 * q^41 - 42 * q^43 - 588 * q^51 - 246 * q^57 - 492 * q^59 - 384 * q^64 - 186 * q^67 + 48 * q^68 - 816 * q^76 + 84 * q^86 - 672 * q^88 + 438 * q^89 + 384 * q^96 + 282 * q^97 + 588 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{2}{9}\right)$

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 $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.53209	−	1.28558i	−0.766044	−	0.642788i
$2$	−1.53209	−	1.28558i	−0.766044	−	0.642788i
$3$	1.90707	+	2.31583i	0.635691	+	0.771944i
$3$	1.90707	+	2.31583i	0.635691	+	0.771944i
$4$	0.694593	+	3.93923i	0.173648	+	0.984808i
$5$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$5$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$6$	0.0553729	−	5.99974i	0.00922881	−	0.999957i
$7$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$7$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$8$	4.00000	−	6.92820i	0.500000	−	0.866025i
$9$	−1.72616	+	8.83292i	−0.191795	+	0.981435i
$10$		0			0
$11$	13.4204	−	4.88462i	1.22003	−	0.444056i	0.349861	−	0.936802i	$-0.386229\pi$
$11$	13.4204	−	4.88462i	1.22003	−	0.444056i	0.870173	+	0.492746i	$0.164007\pi$
$12$	−7.79796	+	9.12096i	−0.649830	+	0.760080i
$13$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$13$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$14$		0			0
$15$		0			0
$16$	−15.0351	+	5.47232i	−0.939693	+	0.342020i
$17$	10.1424	+	17.5672i	0.596613	+	1.03336i	0.993317	+	0.115418i	$0.0368206\pi$
$17$	10.1424	+	17.5672i	0.596613	+	1.03336i	−0.396704	+	0.917947i	$0.629846\pi$
$18$	14.0000	−	11.3137i	0.777778	−	0.628539i
$19$	−18.8769	+	32.6958i	−0.993522	+	1.72083i	−0.398343	+	0.917236i	$0.630415\pi$
$19$	−18.8769	+	32.6958i	−0.993522	+	1.72083i	−0.595178	+	0.803594i	$0.702919\pi$
$20$		0			0
$21$		0			0
$22$	−26.8408	−	9.76924i	−1.22003	−	0.444056i
$23$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$23$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$24$	23.6728	−	3.94925i	0.986368	−	0.164552i
$25$	19.1511	+	16.0697i	0.766044	+	0.642788i
$26$		0			0
$27$	−23.7474	+	12.8475i	−0.879535	+	0.475834i
$28$		0			0
$29$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$29$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$30$		0			0
$31$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$31$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$32$	30.0702	+	10.9446i	0.939693	+	0.342020i
$33$	36.9056	+	21.7640i	1.11835	+	0.659516i
$34$	7.04485	−	39.9533i	0.207202	−	1.17510i
$35$		0			0
$36$	−35.9939	−	0.664446i	−0.999830	−	0.0184568i
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	70.9540	−	25.8251i	1.86721	−	0.679609i
$39$		0			0
$40$		0			0
$41$	50.8983	−	42.7087i	1.24142	−	1.04168i	0.244009	−	0.969773i	$-0.421537\pi$
$41$	50.8983	−	42.7087i	1.24142	−	1.04168i	0.997412	−	0.0719030i	$-0.0229072\pi$
$42$		0			0
$43$	41.1752	−	14.9865i	0.957562	−	0.348524i	0.184485	−	0.982835i	$-0.440938\pi$
$43$	41.1752	−	14.9865i	0.957562	−	0.348524i	0.773078	+	0.634311i	$0.218716\pi$
$44$	28.5633	+	49.4731i	0.649167	+	1.12439i
$45$		0			0
$46$		0			0
$47$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$47$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$48$	−41.3460	−	24.3826i	−0.861374	−	0.507971i
$49$	−46.0449	−	16.7590i	−0.939693	−	0.342020i
$50$	−8.68241	−	49.2404i	−0.173648	−	0.984808i
$51$	−21.3403	+	56.9900i	−0.418438	+	1.11745i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	52.8997	+	10.8456i	0.979623	+	0.200844i
$55$		0			0
$56$		0			0
$57$	−111.718	+	18.6374i	−1.95996	+	0.326972i
$58$		0			0
$59$	−45.1367	−	16.4284i	−0.765028	−	0.278447i	−0.0701128	−	0.997539i	$-0.522336\pi$
$59$	−45.1367	−	16.4284i	−0.765028	−	0.278447i	−0.694915	+	0.719092i	$0.744558\pi$
$60$		0			0
$61$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$61$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$62$		0			0
$63$		0			0
$64$	−32.0000	−	55.4256i	−0.500000	−	0.866025i
$65$		0			0
$66$	−28.5633	−	80.7893i	−0.432778	−	1.22408i
$67$	−81.3715	+	68.2788i	−1.21450	+	1.01909i	−0.215407	+	0.976524i	$0.569108\pi$
$67$	−81.3715	+	68.2788i	−1.21450	+	1.01909i	−0.999094	+	0.0425626i	$0.986448\pi$
$68$	−62.1564	+	52.1554i	−0.914064	+	0.766991i
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$	54.2916	+	47.2908i	0.754050	+	0.656817i
$73$	65.2976	−	113.099i	0.894488	−	1.54930i	0.0600510	−	0.998195i	$-0.480874\pi$
$73$	65.2976	−	113.099i	0.894488	−	1.54930i	0.834437	−	0.551103i	$-0.185793\pi$
$74$		0			0
$75$	−0.692161	+	74.9968i	−0.00922881	+	0.999957i
$76$	−141.908	−	51.6503i	−1.86721	−	0.679609i
$77$		0			0
$78$		0			0
$79$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$79$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$80$		0			0
$81$	−75.0408	−	30.4940i	−0.926429	−	0.376469i
$82$	−132.886			−1.62056
$83$	−26.7420	−	22.4392i	−0.322193	−	0.270352i	0.467317	−	0.884090i	$-0.345221\pi$
$83$	−26.7420	−	22.4392i	−0.322193	−	0.270352i	−0.789510	+	0.613738i	$0.789665\pi$
$84$		0			0
$85$		0			0
$86$	−82.3504	−	29.9731i	−0.957562	−	0.348524i
$87$		0			0
$88$	19.8399	−	112.518i	0.225453	−	1.27861i
$89$	80.5908	−	139.587i	0.905515	−	1.56840i	0.0852901	−	0.996356i	$-0.472818\pi$
$89$	80.5908	−	139.587i	0.905515	−	1.56840i	0.820225	−	0.572041i	$-0.193848\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	32.0000	+	90.5097i	0.333333	+	0.942809i
$97$	172.387	−	62.7437i	1.77718	−	0.646842i	0.777343	−	0.629077i	$-0.216567\pi$
$97$	172.387	−	62.7437i	1.77718	−	0.646842i	0.999842	−	0.0177651i	$-0.00565510\pi$
$98$	49.0000	+	84.8705i	0.500000	+	0.866025i
$99$	19.9798	+	126.973i	0.201816	+	1.28255i
$100$	−50.0000	+	86.6025i	−0.500000	+	0.866025i
$101$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$101$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$102$	105.960	−	59.8792i	1.03883	−	0.587051i
$103$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$103$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−59.9966			−0.560716			−0.280358	−	0.959895i	$-0.590453\pi$
$107$	−59.9966			−0.560716			−0.280358	+	0.959895i	$0.590453\pi$
$108$	−67.1041	−	84.6229i	−0.621335	−	0.783545i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−119.682	−	43.5608i	−1.05914	−	0.385494i	−0.247032	−	0.969007i	$-0.579455\pi$
$113$	−119.682	−	43.5608i	−1.05914	−	0.385494i	−0.812103	+	0.583513i	$0.801678\pi$
$114$	195.121	+	115.067i	1.71159	+	1.00936i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	48.0334	+	83.1963i	0.407063	+	0.705054i
$119$		0			0
$120$		0			0
$121$	63.5557	−	53.3295i	0.525253	−	0.440740i
$122$		0			0
$123$	195.973	+	36.4232i	1.59327	+	0.296124i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$	−22.2270	+	126.055i	−0.173648	+	0.984808i
$129$	113.230	+	66.7744i	0.877755	+	0.517631i
$130$		0			0
$131$	−43.6645	−	247.634i	−0.333317	−	1.89034i	−0.443253	−	0.896397i	$-0.646176\pi$
$131$	−43.6645	−	247.634i	−0.333317	−	1.89034i	0.109936	−	0.993939i	$-0.464935\pi$
$132$	−60.0992	+	160.497i	−0.455297	+	1.21588i
$133$		0			0
$134$	212.446			1.58542
$135$		0			0
$136$	162.279			1.19323
$137$	70.7623	+	59.3766i	0.516513	+	0.433406i	0.863414	−	0.504496i	$-0.168322\pi$
$137$	70.7623	+	59.3766i	0.516513	+	0.433406i	−0.346901	+	0.937902i	$0.612766\pi$
$138$		0			0
$139$	38.1352	+	216.275i	0.274354	+	1.55594i	0.741007	+	0.671497i	$0.234348\pi$
$139$	38.1352	+	216.275i	0.274354	+	1.55594i	−0.466653	+	0.884440i	$0.654540\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−22.3837	−	142.250i	−0.155442	−	0.987845i
$145$		0			0
$146$	−245.439	+	89.3324i	−1.68109	+	0.611866i
$147$	−49.0000	−	138.593i	−0.333333	−	0.942809i
$148$		0			0
$149$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$149$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$150$	97.4745	−	114.012i	0.649830	−	0.760080i
$151$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$151$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$152$	151.015	+	261.566i	0.993522	+	1.72083i
$153$	−172.677	+	59.2635i	−1.12861	+	0.387343i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$157$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	75.7669	+	143.190i	0.467697	+	0.883889i
$163$	205.091			1.25823			0.629113	−	0.777314i	$-0.283418\pi$
$163$	205.091			1.25823			0.629113	+	0.777314i	$0.283418\pi$
$164$	203.593	+	170.835i	1.24142	+	1.04168i
$165$		0			0
$166$	12.1238	+	68.7577i	0.0730351	+	0.414203i
$167$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$167$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$168$		0			0
$169$	29.3465	−	166.433i	0.173648	−	0.984808i
$170$		0			0
$171$	−256.214	−	223.176i	−1.49833	−	1.30512i
$172$	87.6354	+	151.789i	0.509508	+	0.882494i
$173$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$173$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$174$		0			0
$175$		0			0
$176$	−175.046	+	146.881i	−0.994581	+	0.834553i
$177$	−48.0334	−	135.859i	−0.271375	−	0.767565i
$178$	−302.922	+	110.255i	−1.70181	+	0.619409i
$179$	−162.818	−	282.009i	−0.909597	−	1.57547i	−0.814625	−	0.579988i	$-0.803057\pi$
$179$	−162.818	−	282.009i	−0.909597	−	1.57547i	−0.0949721	−	0.995480i	$-0.530276\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	221.924	+	186.216i	1.18676	+	0.995810i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$191$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$192$	67.3301	−	179.807i	0.350678	−	0.936496i
$193$	6.18260	+	35.0633i	0.0320342	+	0.181675i	0.996627	−	0.0820705i	$-0.0261533\pi$
$193$	6.18260	+	35.0633i	0.0320342	+	0.181675i	−0.964592	+	0.263745i	$0.915042\pi$
$194$	−344.774	−	125.487i	−1.77718	−	0.646842i
$195$		0			0
$196$	34.0350	−	193.022i	0.173648	−	0.984808i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$	132.622	−	220.219i	0.669809	−	1.11222i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	187.939	−	68.4040i	0.939693	−	0.342020i
$201$	−313.304	−	58.2302i	−1.55872	−	0.289702i
$202$		0			0
$203$		0			0
$204$	−239.320	−	44.4796i	−1.17314	−	0.218037i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−93.6289	+	530.996i	−0.447985	+	2.54065i
$210$		0			0
$211$	−396.208	−	144.208i	−1.87776	−	0.683450i	−0.953090	−	0.302688i	$-0.902116\pi$
$211$	−396.208	−	144.208i	−1.87776	−	0.683450i	−0.924673	−	0.380762i	$-0.875662\pi$
$212$		0			0
$213$		0			0
$214$	91.9202	+	77.1302i	0.429534	+	0.360421i
$215$		0			0
$216$	−5.97959	+	215.917i	−0.0276833	+	0.999617i
$217$		0			0
$218$		0			0
$219$	386.445	−	64.4692i	1.76459	−	0.294380i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$223$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$224$		0			0
$225$	−175.000	+	141.421i	−0.777778	+	0.628539i
$226$	127.363	+	220.600i	0.563554	+	0.976105i
$227$	−269.798	+	98.1986i	−1.18854	+	0.432593i	−0.859210	−	0.511622i	$-0.829045\pi$
$227$	−269.798	+	98.1986i	−1.18854	+	0.432593i	−0.329329	+	0.944215i	$0.606822\pi$
$228$	−151.015	−	427.136i	−0.662348	−	1.87340i
$229$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$229$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	174.892	+	302.922i	0.750609	+	1.30009i	0.947528	+	0.319673i	$0.103573\pi$
$233$	174.892	+	302.922i	0.750609	+	1.30009i	−0.196919	+	0.980420i	$0.563093\pi$
$234$		0			0
$235$		0			0
$236$	33.3637	−	189.215i	0.141371	−	0.801757i
$237$		0			0
$238$		0			0
$239$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$239$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$240$		0			0
$241$	358.677	+	300.965i	1.48828	+	1.24882i	0.896748	+	0.442541i	$0.145923\pi$
$241$	358.677	+	300.965i	1.48828	+	1.24882i	0.591536	+	0.806279i	$0.298522\pi$
$242$	−165.932			−0.685670
$243$	−72.4892	−	231.936i	−0.298310	−	0.954469i
$244$		0			0
$245$		0			0
$246$	−253.423	−	307.741i	−1.03017	−	1.25098i
$247$		0			0
$248$		0			0
$249$	0.966511	−	104.723i	0.00388157	−	0.420575i
$250$		0			0
$251$	238.485	−	413.068i	0.950139	−	1.64569i	0.205020	−	0.978758i	$-0.434274\pi$
$251$	238.485	−	413.068i	0.950139	−	1.64569i	0.745119	−	0.666932i	$-0.232393\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	196.107	−	164.554i	0.766044	−	0.642788i
$257$	307.401	−	257.940i	1.19611	−	1.00366i	0.196379	−	0.980528i	$-0.437082\pi$
$257$	307.401	−	257.940i	1.19611	−	1.00366i	0.999732	−	0.0231286i	$-0.00736271\pi$
$258$	−87.6354	−	247.870i	−0.339672	−	0.960738i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−251.454	+	435.531i	−0.959748	+	1.66233i
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$	298.408	−	168.633i	1.13033	−	0.638762i
$265$		0			0
$266$		0			0
$267$	476.953	−	79.5684i	1.78634	−	0.298009i
$268$	−325.486	−	273.115i	−1.21450	−	1.01909i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−248.625	−	208.622i	−0.914064	−	0.766991i
$273$		0			0
$274$	−32.0810	−	181.941i	−0.117084	−	0.664017i
$275$	335.509	+	122.115i	1.22003	+	0.444056i
$276$		0			0
$277$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$277$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$278$	219.612	−	380.379i	0.789970	−	1.36827i
$279$		0			0
$280$		0			0
$281$	−526.142	+	191.500i	−1.87239	+	0.681494i	−0.906713	+	0.421749i	$0.861416\pi$
$281$	−526.142	+	191.500i	−1.87239	+	0.681494i	−0.965677	+	0.259746i	$0.916361\pi$
$282$		0			0
$283$	−340.123	+	285.397i	−1.20185	+	1.00847i	−0.202273	+	0.979329i	$0.564833\pi$
$283$	−340.123	+	285.397i	−1.20185	+	1.00847i	−0.999575	+	0.0291408i	$0.990723\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−148.579	+	246.715i	−0.515899	+	0.856649i
$289$	−61.2375	+	106.066i	−0.211894	+	0.367012i
$290$		0			0
$291$	474.058	+	279.562i	1.62907	+	0.960696i
$292$	490.878	+	178.665i	1.68109	+	0.611866i
$293$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$293$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$294$	−103.099	+	275.330i	−0.350678	+	0.936496i
$295$		0			0
$296$		0			0
$297$	−255.945	+	288.416i	−0.861766	+	0.971097i
$298$		0			0
$299$		0			0
$300$	−295.911	+	49.3657i	−0.986368	+	0.164552i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	104.894	−	594.884i	0.345046	−	1.95686i
$305$		0			0
$306$	340.744	+	131.192i	1.11354	+	0.428733i
$307$	303.993	+	526.531i	0.990205	+	1.71509i	0.616016	+	0.787734i	$0.288746\pi$
$307$	303.993	+	526.531i	0.990205	+	1.71509i	0.374189	+	0.927352i	$0.377921\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$311$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$312$		0			0
$313$	−355.385	+	129.350i	−1.13542	+	0.413257i	−0.840256	−	0.542191i	$-0.817595\pi$
$313$	−355.385	+	129.350i	−1.13542	+	0.413257i	−0.295160	+	0.955448i	$0.595373\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$317$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−114.418	−	138.942i	−0.356442	−	0.432842i
$322$		0			0
$323$	−765.830			−2.37099
$324$	68.0000	−	316.784i	0.209877	−	0.977728i
$325$		0			0
$326$	−314.217	−	263.660i	−0.963857	−	0.808772i
$327$		0			0
$328$	−92.3016	−	523.468i	−0.281407	−	1.59594i
$329$		0			0
$330$		0			0
$331$	98.3162	−	557.579i	0.297028	−	1.68453i	−0.361816	−	0.932249i	$-0.617843\pi$
$331$	98.3162	−	557.579i	0.297028	−	1.68453i	0.658844	−	0.752279i	$-0.271046\pi$
$332$	69.8184	−	120.929i	0.210296	−	0.364244i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−46.5235	+	39.0379i	−0.138052	+	0.115839i	−0.709199	−	0.705009i	$-0.750943\pi$
$337$	−46.5235	+	39.0379i	−0.138052	+	0.115839i	0.571147	+	0.820848i	$0.306499\pi$
$338$	−258.923	+	217.262i	−0.766044	+	0.642788i
$339$	−127.363	−	360.238i	−0.375703	−	1.06265i
$340$		0			0
$341$		0			0
$342$	105.634	+	671.309i	0.308870	+	1.96289i
$343$		0			0
$344$	60.8709	−	345.216i	0.176950	−	1.00354i
$345$		0			0
$346$		0			0
$347$	120.472	+	683.233i	0.347183	+	1.96897i	0.198761	+	0.980048i	$0.436308\pi$
$347$	120.472	+	683.233i	0.347183	+	1.96897i	0.148422	+	0.988924i	$0.452580\pi$
$348$		0			0
$349$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$349$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$350$		0			0
$351$		0			0
$352$	457.013			1.29833
$353$	448.099	+	375.999i	1.26940	+	1.06515i	0.994614	+	0.103649i	$0.0330520\pi$
$353$	448.099	+	375.999i	1.26940	+	1.06515i	0.274788	+	0.961505i	$0.411392\pi$
$354$	−101.066	+	269.899i	−0.285496	+	0.762426i
$355$		0			0
$356$	605.845	+	220.509i	1.70181	+	0.619409i
$357$		0			0
$358$	−113.092	+	641.377i	−0.315900	+	1.79156i
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−532.176	−	921.755i	−1.47417	−	2.55334i
$362$		0			0
$363$	244.707	+	45.4810i	0.674125	+	0.125292i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$367$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$368$		0			0
$369$	289.384	+	523.302i	0.784239	+	1.41816i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$373$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$374$	−100.612	−	570.600i	−0.269017	−	1.52567i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−534.348			−1.40989			−0.704944	−	0.709262i	$-0.749028\pi$
$379$	−534.348			−1.40989			−0.704944	+	0.709262i	$0.749028\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$383$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$384$	−334.312	+	188.923i	−0.870603	+	0.491986i
$385$		0			0
$386$	35.6042	−	61.6682i	0.0922388	−	0.159762i
$387$	61.3001	+	389.566i	0.158398	+	1.00663i
$388$	366.901	+	635.491i	0.945620	+	1.63786i
$389$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$389$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$390$		0			0
$391$		0			0
$392$	−300.289	+	251.973i	−0.766044	+	0.642788i
$393$	490.207	−	573.375i	1.24735	−	1.45897i
$394$		0			0
$395$		0			0
$396$	−486.297	+	166.899i	−1.22802	+	0.421463i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	−375.877	−	136.808i	−0.939693	−	0.342020i
$401$	110.882	+	628.844i	0.276514	+	1.56819i	0.734111	+	0.679029i	$0.237599\pi$
$401$	110.882	+	628.844i	0.276514	+	1.56819i	−0.457597	+	0.889160i	$0.651290\pi$
$402$	405.150	+	491.989i	1.00784	+	1.22385i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	309.477	+	375.810i	0.758523	+	0.921104i
$409$	−127.776	−	724.652i	−0.312410	−	1.77177i	−0.586388	−	0.810030i	$-0.699451\pi$
$409$	−127.776	−	724.652i	−0.312410	−	1.77177i	0.273978	−	0.961736i	$-0.411661\pi$
$410$		0			0
$411$	−2.55750	+	277.109i	−0.00622262	+	0.674232i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−428.131	+	500.767i	−1.02669	+	1.20088i
$418$	826.083	−	693.166i	1.97628	−	1.65829i
$419$	635.955	−	533.630i	1.51779	−	1.27358i	0.671291	−	0.741194i	$-0.265740\pi$
$419$	635.955	−	533.630i	1.51779	−	1.27358i	0.846502	−	0.532385i	$-0.178704\pi$
$420$		0			0
$421$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$421$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$422$	421.636	+	730.294i	0.999137	+	1.73056i
$423$		0			0
$424$		0			0
$425$	−88.0607	+	499.417i	−0.207202	+	1.17510i
$426$		0			0
$427$		0			0
$428$	−41.6732	−	236.341i	−0.0973674	−	0.552198i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	286.739	−	323.117i	0.663748	−	0.747956i
$433$	28.2519			0.0652469			0.0326235	−	0.999468i	$-0.489614\pi$
$433$	28.2519			0.0652469			0.0326235	+	0.999468i	$0.489614\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−674.948	−	398.032i	−1.54098	−	0.908748i
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	227.511	−	377.782i	0.515899	−	0.856649i
$442$		0			0
$443$	737.114	−	268.288i	1.66391	−	0.605615i	0.672943	−	0.739694i	$-0.265030\pi$
$443$	737.114	−	268.288i	1.66391	−	0.605615i	0.990971	+	0.134079i	$0.0428076\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−192.179	−	332.864i	−0.428015	−	0.741344i	0.568681	−	0.822558i	$-0.307454\pi$
$449$	−192.179	−	332.864i	−0.428015	−	0.741344i	−0.996697	+	0.0812137i	$0.974120\pi$
$450$	449.923	+	8.30558i	0.999830	+	0.0184568i
$451$	474.458	−	821.786i	1.05201	−	1.82214i
$452$	88.4656	−	501.713i	0.195720	−	1.10999i
$453$		0			0
$454$	539.597	+	196.397i	1.18854	+	0.432593i
$455$		0			0
$456$	−317.746	+	848.551i	−0.696812	+	1.86086i
$457$	402.533	+	337.765i	0.880816	+	0.739092i	0.966347	−	0.257244i	$-0.0828143\pi$
$457$	402.533	+	337.765i	0.880816	+	0.739092i	−0.0855311	+	0.996336i	$0.527259\pi$
$458$		0			0
$459$	−466.551	−	286.871i	−1.01645	−	0.624991i
$460$		0			0
$461$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$461$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$462$		0			0
$463$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$463$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$464$		0			0
$465$		0			0
$466$	121.479	−	688.940i	0.260684	−	1.47841i
$467$	−175.592	+	304.135i	−0.376001	+	0.651252i	−0.990476	−	0.137684i	$-0.956034\pi$
$467$	−175.592	+	304.135i	−0.376001	+	0.651252i	0.614476	+	0.788936i	$0.289368\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−294.366	+	247.002i	−0.623657	+	0.523310i
$473$	479.383	−	402.250i	1.01349	−	0.850423i
$474$		0			0
$475$	−886.925	+	322.814i	−1.86721	+	0.679609i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$479$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$480$		0			0
$481$		0			0
$482$	−162.611	−	922.211i	−0.337367	−	1.91330i
$483$		0			0
$484$	254.223	+	213.318i	0.525253	+	0.440740i
$485$		0			0
$486$	−187.111	+	448.537i	−0.385003	+	0.922916i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	391.123	+	474.956i	0.799842	+	0.971280i
$490$		0			0
$491$	−204.149	−	74.3040i	−0.415781	−	0.151332i	0.125655	−	0.992074i	$-0.459897\pi$
$491$	−204.149	−	74.3040i	−0.415781	−	0.151332i	−0.541436	+	0.840742i	$0.682119\pi$
$492$	−7.35827	+	797.282i	−0.0149558	+	1.62049i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−136.110	+	159.203i	−0.273314	+	0.319684i
$499$	−763.106	+	640.322i	−1.52927	+	1.28321i	−0.725663	+	0.688050i	$0.758467\pi$
$499$	−763.106	+	640.322i	−1.52927	+	1.28321i	−0.803607	+	0.595160i	$0.797089\pi$
$500$		0			0
$501$		0			0
$502$	−896.410	+	326.267i	−1.78568	+	0.649933i
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	441.396	−	249.437i	0.870603	−	0.491986i
$508$		0			0
$509$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$509$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$510$		0			0
$511$		0			0
$512$	−512.000			−1.00000
$513$	28.2190	−	1018.96i	0.0550079	−	1.98628i
$514$	−802.566			−1.56141
$515$		0			0
$516$	−184.391	+	492.421i	−0.357347	+	0.954305i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	298.053	−	516.242i	0.572078	−	0.990868i	−0.424274	−	0.905534i	$-0.639471\pi$
$521$	298.053	−	516.242i	0.572078	−	0.990868i	0.996352	−	0.0853344i	$-0.0271959\pi$
$522$		0			0
$523$	−319.363	−	553.153i	−0.610636	−	1.05765i	−0.991133	−	0.132871i	$-0.957580\pi$
$523$	−319.363	−	553.153i	−0.610636	−	1.05765i	0.380497	−	0.924782i	$-0.375753\pi$
$524$	945.158	−	344.009i	1.80374	−	0.656507i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−673.978	−	125.265i	−1.27647	−	0.237244i
$529$	−497.097	+	180.929i	−0.939693	+	0.342020i
$530$		0			0
$531$	223.024	−	370.330i	0.420007	−	0.697420i
$532$		0			0
$533$		0			0
$534$	−833.026	−	491.254i	−1.55997	−	0.919951i
$535$		0			0
$536$	147.563	+	836.874i	0.275305	+	1.56133i
$537$	342.580	−	914.870i	0.637951	−	1.70367i
$538$		0			0
$539$	−699.802			−1.29833
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	112.718	+	639.254i	0.207202	+	1.17510i
$545$		0			0
$546$		0			0
$547$	92.3813	−	523.920i	0.168887	−	0.957807i	−0.776078	−	0.630637i	$-0.782794\pi$
$547$	92.3813	−	523.920i	0.168887	−	0.957807i	0.944965	−	0.327170i	$-0.106095\pi$
$548$	−184.747	+	319.992i	−0.337130	+	0.583927i
$549$		0			0
$550$	−357.042	−	618.414i	−0.649167	−	1.12439i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−825.470	+	300.446i	−1.48466	+	0.540371i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−8.02080	+	869.067i	−0.0142973	+	1.54914i
$562$	1052.28	+	383.000i	1.87239	+	0.681494i
$563$	181.824	+	1031.18i	0.322956	+	1.83157i	0.523666	+	0.851923i	$0.324564\pi$
$563$	181.824	+	1031.18i	0.322956	+	1.83157i	−0.200710	+	0.979651i	$0.564325\pi$
$564$		0			0
$565$		0			0
$566$	887.998			1.56890
$567$		0			0
$568$		0			0
$569$	−699.618	−	587.050i	−1.22956	−	1.03172i	−0.998268	−	0.0588367i	$-0.981261\pi$
$569$	−699.618	−	587.050i	−1.22956	−	1.03172i	−0.231290	−	0.972885i	$-0.574295\pi$
$570$		0			0
$571$	16.6543	+	94.4514i	0.0291670	+	0.165414i	0.995912	−	0.0903277i	$-0.0287914\pi$
$571$	16.6543	+	94.4514i	0.0291670	+	0.165414i	−0.966745	+	0.255742i	$0.917680\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	544.807	−	186.980i	0.945845	−	0.324618i
$577$	−568.407	−	984.510i	−0.985107	−	1.70626i	−0.641459	−	0.767157i	$-0.721671\pi$
$577$	−568.407	−	984.510i	−0.985107	−	1.70626i	−0.343648	−	0.939098i	$-0.611663\pi$
$578$	230.178	−	83.7778i	0.398231	−	0.144944i
$579$	−69.4099	+	81.1860i	−0.119879	+	0.140218i
$580$		0			0
$581$		0			0
$582$	−366.901	−	1037.75i	−0.630414	−	1.78308i
$583$		0			0
$584$	−522.381	−	904.790i	−0.894488	−	1.54930i
$585$		0			0
$586$		0			0
$587$	15.0214	−	85.1908i	0.0255902	−	0.145129i	−0.969336	−	0.245741i	$-0.920969\pi$
$587$	15.0214	−	85.1908i	0.0255902	−	0.145129i	0.994926	+	0.100612i	$0.0320799\pi$
$588$	511.915	−	289.288i	0.870603	−	0.491986i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1136.45			1.91645			0.958223	−	0.286021i	$-0.0923326\pi$
$593$	1136.45			1.91645			0.958223	+	0.286021i	$0.0923326\pi$
$594$	762.910	−	112.843i	1.28436	−	0.189971i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$599$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$600$	516.824	+	304.783i	0.861374	+	0.507971i
$601$	−105.938	+	600.803i	−0.176269	+	0.999673i	0.760399	+	0.649456i	$0.225003\pi$
$601$	−105.938	+	600.803i	−0.176269	+	0.999673i	−0.936669	+	0.350217i	$0.886108\pi$
$602$		0			0
$603$	−462.641	−	836.608i	−0.767233	−	1.38741i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$607$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$608$	−925.475	+	776.566i	−1.52216	+	1.27725i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−353.393	−	639.050i	−0.577439	−	1.04420i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	211.151	−	1197.50i	0.343895	−	1.95032i
$615$		0			0
$616$		0			0
$617$	−213.221	−	1209.24i	−0.345577	−	1.95986i	−0.270665	−	0.962674i	$-0.587243\pi$
$617$	−213.221	−	1209.24i	−0.345577	−	1.95986i	−0.0749124	−	0.997190i	$-0.523868\pi$
$618$		0			0
$619$	115.542	+	96.9513i	0.186659	+	0.156626i	0.731329	−	0.682025i	$-0.238901\pi$
$619$	115.542	+	96.9513i	0.186659	+	0.156626i	−0.544670	+	0.838651i	$0.683345\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	108.530	+	615.505i	0.173648	+	0.984808i
$626$	710.770	+	258.699i	1.13542	+	0.413257i
$627$	−1408.25	+	795.819i	−2.24602	+	1.26925i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	−421.636	−	1192.57i	−0.666091	−	1.88399i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.09767	+	45.9242i	−0.0126329	+	0.0716446i	−0.990472	−	0.137711i	$-0.956025\pi$
$641$	−8.09767	+	45.9242i	−0.0126329	+	0.0716446i	0.977840	+	0.209356i	$0.0671366\pi$
$642$	−3.32219	+	359.965i	−0.00517474	+	0.560692i
$643$	194.525	+	70.8015i	0.302528	+	0.110111i	0.488824	−	0.872383i	$-0.337426\pi$
$643$	194.525	+	70.8015i	0.302528	+	0.110111i	−0.186296	+	0.982494i	$0.559648\pi$
$644$		0			0
$645$		0			0
$646$	1173.32	+	984.533i	1.81629	+	1.52404i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−511.431	+	397.922i	−0.789246	+	0.614077i
$649$	−685.997			−1.05701
$650$		0			0
$651$		0			0
$652$	142.455	+	807.900i	0.218489	+	1.23911i
$653$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$653$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$654$		0			0
$655$		0			0
$656$	−531.544	+	920.661i	−0.810280	+	1.40345i
$657$	886.278	+	771.994i	1.34898	+	1.17503i
$658$		0			0
$659$	237.314	−	86.3751i	0.360112	−	0.131070i	−0.155627	−	0.987816i	$-0.549740\pi$
$659$	237.314	−	86.3751i	0.360112	−	0.131070i	0.515739	+	0.856746i	$0.327518\pi$
$660$		0			0
$661$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$661$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$662$	−867.439	+	727.868i	−1.31033	+	1.09950i
$663$		0			0
$664$	−262.431	+	95.5172i	−0.395228	+	0.143851i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	815.001	+	683.867i	1.21100	+	1.01615i	0.999246	+	0.0388234i	$0.0123610\pi$
$673$	815.001	+	683.867i	1.21100	+	1.01615i	0.211751	+	0.977324i	$0.432083\pi$
$674$	121.464			0.180214
$675$	−661.246	−	135.570i	−0.979623	−	0.200844i
$676$	676.000			1.00000
$677$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$677$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$678$	−267.981	+	715.651i	−0.395252	+	1.05553i
$679$		0			0
$680$		0			0
$681$	−741.936	−	437.536i	−1.08948	−	0.642491i
$682$		0			0
$683$	−677.997	+	1174.32i	−0.992674	+	1.71936i	−0.391703	+	0.920092i	$0.628114\pi$
$683$	−677.997	+	1174.32i	−0.992674	+	1.71936i	−0.600971	+	0.799271i	$0.705219\pi$
$684$	701.178	−	1164.30i	1.02511	−	1.70220i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−537.061	+	450.648i	−0.780612	+	0.655011i
$689$		0			0
$690$		0			0
$691$	1297.80	−	472.360i	1.87815	−	0.683589i	0.926655	−	0.375913i	$-0.122671\pi$
$691$	1297.80	−	472.360i	1.87815	−	0.683589i	0.951491	−	0.307676i	$-0.0995512\pi$
$692$		0			0
$693$		0			0
$694$	693.773	−	1201.65i	0.999673	−	1.73149i
$695$		0			0
$696$		0			0
$697$	1266.50	+	460.970i	1.81708	+	0.661363i
$698$		0			0
$699$	−367.984	+	982.714i	−0.526444	+	1.40589i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−700.185	−	587.525i	−0.994581	−	0.834553i
$705$		0			0
$706$	−203.151	−	1152.13i	−0.287750	−	1.63191i
$707$		0			0
$708$	501.816	−	283.581i	0.708780	−	0.400539i
$709$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$709$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$710$		0			0
$711$		0			0
$712$	−644.727	−	1116.70i	−0.905515	−	1.56840i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	997.806	−	837.258i	1.39358	−	1.16936i
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	−369.645	+	2096.36i	−0.511974	+	2.90355i
$723$	−12.9633	+	1404.60i	−0.0179299	+	1.94273i
$724$		0			0
$725$		0			0
$726$	−316.444	−	384.271i	−0.435874	−	0.529299i
$727$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$727$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$728$		0			0
$729$	398.883	−	610.191i	0.547164	−	0.837025i
$730$		0			0
$731$	680.888	+	571.333i	0.931447	+	0.781577i
$732$		0			0
$733$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$733$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−758.521	+	1313.80i	−1.02920	+	1.78263i
$738$	229.382	−	1173.77i	0.310815	−	1.59047i
$739$	738.249	+	1278.68i	0.998984	+	1.73029i	0.538528	+	0.842608i	$0.318981\pi$
$739$	738.249	+	1278.68i	0.998984	+	1.73029i	0.460456	+	0.887683i	$0.347686\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$743$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	244.364	−	197.476i	0.327128	−	0.264359i
$748$	−579.403	+	1003.56i	−0.774603	+	1.34165i
$749$		0			0
$750$		0			0
$751$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$751$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$752$		0			0
$753$	1411.40	−	235.459i	1.87437	−	0.312695i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	818.669	+	686.944i	1.08004	+	0.906259i
$759$		0			0
$760$		0			0
$761$	−1309.93	−	476.776i	−1.72133	−	0.626513i	−0.723376	−	0.690455i	$-0.757410\pi$
$761$	−1309.93	−	476.776i	−1.72133	−	0.626513i	−0.997954	+	0.0639422i	$0.979633\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	755.069	+	140.336i	0.983163	+	0.182729i
$769$	1146.77	−	962.252i	1.49124	−	1.25130i	0.598190	−	0.801355i	$-0.295887\pi$
$769$	1146.77	−	962.252i	1.49124	−	1.25130i	0.893055	−	0.449948i	$-0.148558\pi$
$770$		0			0
$771$	1183.58	+	219.979i	1.53512	+	0.285316i
$772$	−133.828	+	48.7094i	−0.173352	+	0.0630950i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$	406.899	−	675.656i	0.525710	−	0.872940i
$775$		0			0
$776$	254.847	−	1445.31i	0.328410	−	1.86251i
$777$		0			0
$778$		0			0
$779$	435.592	+	2470.37i	0.559168	+	3.17120i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$		0			0
$786$	−1488.16	+	248.264i	−1.89333	+	0.315857i
$787$	111.008	+	629.558i	0.141052	+	0.799947i	0.970452	+	0.241292i	$0.0775713\pi$
$787$	111.008	+	629.558i	0.141052	+	0.799947i	−0.829400	+	0.558655i	$0.811318\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	959.611	+	369.467i	1.21163	+	0.466499i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$797$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$798$		0			0
$799$		0			0
$800$	400.000	+	692.820i	0.500000	+	0.866025i
$801$	1093.85	+	952.801i	1.36561	+	1.18951i
$802$	638.545	−	1105.99i	0.796190	−	1.37904i
$803$	323.874	−	1836.78i	0.403330	−	2.28740i
$804$	11.7637	−	1274.62i	0.0146315	−	1.58535i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1430.05			−1.76768			−0.883839	−	0.467791i	$-0.845050\pi$
$809$	−1430.05			−1.76768			−0.883839	+	0.467791i	$0.845050\pi$
$810$		0			0
$811$	1556.80			1.91961			0.959803	−	0.280673i	$-0.0905577\pi$
$811$	1556.80			1.91961			0.959803	+	0.280673i	$0.0905577\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	8.98584	−	973.631i	0.0110121	−	1.19318i
$817$	−287.264	+	1629.15i	−0.351608	+	1.99407i
$818$	−735.831	+	1274.50i	−0.899549	+	1.55807i
$819$		0			0
$820$		0			0
$821$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$821$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$822$	360.163	−	421.268i	0.438155	−	0.512492i
$823$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$823$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$824$		0			0
$825$	357.042	+	1009.87i	0.432778	+	1.22408i
$826$		0			0
$827$	−631.000	−	1092.92i	−0.762999	−	1.32155i	−0.941298	−	0.337576i	$-0.890393\pi$
$827$	−631.000	−	1092.92i	−0.762999	−	1.32155i	0.178299	−	0.983976i	$-0.442940\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−172.599	−	978.857i	−0.207202	−	1.17510i
$834$	1299.71	−	216.825i	1.55840	−	0.259983i
$835$		0			0
$836$	−2156.75			−2.57984
$837$		0			0
$838$	−1660.36			−1.98134
$839$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$839$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$840$		0			0
$841$	146.038	+	828.223i	0.173648	+	0.984808i
$842$		0			0
$843$	−1446.87	−	853.251i	−1.71634	−	1.01216i
$844$	292.865	−	1660.92i	0.346997	−	1.96792i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1309.57	−	243.395i	−1.54249	−	0.286684i
$850$	776.955	−	651.942i	0.914064	−	0.766991i
$851$		0			0
$852$		0			0
$853$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$853$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$854$		0			0
$855$		0			0
$856$	−239.987	+	415.669i	−0.280358	+	0.485595i
$857$	208.725	−	1183.74i	0.243553	−	1.38126i	−0.580276	−	0.814420i	$-0.697055\pi$
$857$	208.725	−	1183.74i	0.243553	−	1.38126i	0.823829	−	0.566839i	$-0.191834\pi$
$858$		0			0
$859$	−724.028	−	263.525i	−0.842873	−	0.306781i	−0.115742	−	0.993279i	$-0.536925\pi$
$859$	−724.028	−	263.525i	−0.842873	−	0.306781i	−0.727131	+	0.686499i	$0.759147\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−854.701	+	126.420i	−0.989237	+	0.146319i
$865$		0			0
$866$	−43.2845	−	36.3200i	−0.0499821	−	0.0419399i
$867$	−362.416	+	60.4606i	−0.418012	+	0.0697354i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	256.643	+	1630.98i	0.293979	+	1.86825i
$874$		0			0
$875$		0			0
$876$	522.381	+	1477.52i	0.596325	+	1.68666i
$877$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$877$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	81.4082	+	141.003i	0.0924043	+	0.160049i	0.908522	−	0.417836i	$-0.137211\pi$
$881$	81.4082	+	141.003i	0.0924043	+	0.160049i	−0.816118	+	0.577885i	$0.803878\pi$
$882$	−834.235	+	286.313i	−0.945845	+	0.324618i
$883$	636.706	−	1102.81i	0.721071	−	1.24893i	−0.239500	−	0.970896i	$-0.576984\pi$
$883$	636.706	−	1102.81i	0.721071	−	1.24893i	0.960571	−	0.278035i	$-0.0896831\pi$
$884$		0			0
$885$		0			0
$886$	−1474.23	−	536.575i	−1.66391	−	0.605615i
$887$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$887$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1156.03	−	42.6950i	−1.29745	−	0.0479181i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−133.486	+	757.037i	−0.148648	+	0.843025i
$899$		0			0
$900$	−678.645	−	591.135i	−0.754050	−	0.656817i
$901$		0			0
$902$	−1783.38	+	649.097i	−1.97714	+	0.719620i
$903$		0			0
$904$	−780.527	+	654.940i	−0.863415	+	0.724491i
$905$		0			0
$906$		0			0
$907$	−269.986	+	98.2668i	−0.297669	+	0.108343i	−0.486537	−	0.873660i	$-0.661740\pi$
$907$	−269.986	+	98.2668i	−0.297669	+	0.108343i	0.188868	+	0.982002i	$0.439518\pi$
$908$	−574.227	−	994.590i	−0.632409	−	1.09536i
$909$		0			0
$910$		0			0
$911$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$911$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$912$	1577.69	−	891.570i	1.72993	−	0.977598i
$913$	−468.494	−	170.518i	−0.513137	−	0.186767i
$914$	−182.494	−	1034.97i	−0.199665	−	1.13235i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	346.004	+	1039.30i	0.376911	+	1.13213i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−639.622	+	1708.13i	−0.694486	+	1.85465i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−994.195	+	361.857i	−1.07018	+	0.389513i	−0.816243	−	0.577709i	$-0.803947\pi$
$929$	−994.195	+	361.857i	−1.07018	+	0.389513i	−0.253934	+	0.967221i	$0.581725\pi$
$930$		0			0
$931$	1417.13	−	1189.12i	1.52216	−	1.27725i
$932$	−1071.80	+	899.347i	−1.15000	+	0.964964i
$933$		0			0
$934$	660.011	−	240.224i	0.706650	−	0.257200i
$935$		0			0
$936$		0			0
$937$	−929.044	+	1609.15i	−0.991509	+	1.71734i	−0.383138	+	0.923691i	$0.625157\pi$
$937$	−929.044	+	1609.15i	−0.991509	+	1.71734i	−0.608371	+	0.793653i	$0.708177\pi$
$938$		0			0
$939$	−977.297	−	576.333i	−1.04078	−	0.613773i
$940$		0			0
$941$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$941$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$942$		0			0
$943$		0			0
$944$	768.535			0.814126
$945$		0			0
$946$	−1251.58			−1.32302
$947$	−1377.16	−	1155.57i	−1.45423	−	1.22025i	−0.929418	−	0.369030i	$-0.879690\pi$
$947$	−1377.16	−	1155.57i	−1.45423	−	1.22025i	−0.524815	−	0.851216i	$-0.675866\pi$
$948$		0			0
$949$		0			0
$950$	1773.85	+	645.628i	1.86721	+	0.679609i
$951$		0			0
$952$		0			0
$953$	948.243	−	1642.40i	0.995008	−	1.72340i	0.411078	−	0.911600i	$-0.365152\pi$
$953$	948.243	−	1642.40i	0.995008	−	1.72340i	0.583930	−	0.811804i	$-0.301514\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−903.045	+	328.681i	−0.939693	+	0.342020i
$962$		0			0
$963$	103.564	−	529.945i	0.107543	−	0.550307i
$964$	−936.438	+	1621.96i	−0.971409	+	1.68253i
$965$		0			0
$966$		0			0
$967$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$967$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$968$	−115.255	−	653.645i	−0.119065	−	0.675253i
$969$	−1460.49	−	1773.53i	−1.50722	−	1.83027i
$970$		0			0
$971$	974.000			1.00309			0.501545	−	0.865132i	$-0.332765\pi$
$971$	974.000			1.00309			0.501545	+	0.865132i	$0.332765\pi$
$972$	863.299	−	446.653i	0.888168	−	0.459519i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1813.63	−	660.107i	−1.85633	−	0.675647i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$	−1813.63	−	660.107i	−1.85633	−	0.675647i	−0.874749	−	0.484576i	$-0.838974\pi$
$978$	11.3565	−	1230.49i	0.0116119	−	1.25817i
$979$	399.728	−	2266.97i	0.408302	−	2.31560i
$980$		0			0
$981$		0			0
$982$	217.250	+	376.289i	0.221233	+	0.383186i
$983$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$983$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$984$	1036.24	−	1212.05i	1.05309	−	1.23175i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	1478.76	−	835.659i	1.48918	−	0.841550i
$994$		0			0
$995$		0			0
$996$	413.200	−	68.9326i	0.414859	−	0.0692094i
$997$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$997$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$998$	1992.33			1.99632
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.a.43.2	✓	12
8.3	odd	2	CM	216.3.r.a.43.2	✓	12
27.22	even	9	inner	216.3.r.a.211.2	yes	12
216.211	odd	18	inner	216.3.r.a.211.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.a.43.2	✓	12	1.1	even	1	trivial
216.3.r.a.43.2	✓	12	8.3	odd	2	CM
216.3.r.a.211.2	yes	12	27.22	even	9	inner
216.3.r.a.211.2	yes	12	216.211	odd	18	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.r (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.53209 - 1.28558i) q^{2} +(1.90707 + 2.31583i) q^{3} +(0.694593 + 3.93923i) q^{4} +(0.0553729 - 5.99974i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-1.72616 + 8.83292i) q^{9} +(13.4204 - 4.88462i) q^{11} +(-7.79796 + 9.12096i) q^{12} +(-15.0351 + 5.47232i) q^{16} +(10.1424 + 17.5672i) q^{17} +(14.0000 - 11.3137i) q^{18} +(-18.8769 + 32.6958i) q^{19} +(-26.8408 - 9.76924i) q^{22} +(23.6728 - 3.94925i) q^{24} +(19.1511 + 16.0697i) q^{25} +(-23.7474 + 12.8475i) q^{27} +(30.0702 + 10.9446i) q^{32} +(36.9056 + 21.7640i) q^{33} +(7.04485 - 39.9533i) q^{34} +(-35.9939 - 0.664446i) q^{36} +(70.9540 - 25.8251i) q^{38} +(50.8983 - 42.7087i) q^{41} +(41.1752 - 14.9865i) q^{43} +(28.5633 + 49.4731i) q^{44} +(-41.3460 - 24.3826i) q^{48} +(-46.0449 - 16.7590i) q^{49} +(-8.68241 - 49.2404i) q^{50} +(-21.3403 + 56.9900i) q^{51} +(52.8997 + 10.8456i) q^{54} +(-111.718 + 18.6374i) q^{57} +(-45.1367 - 16.4284i) q^{59} +(-32.0000 - 55.4256i) q^{64} +(-28.5633 - 80.7893i) q^{66} +(-81.3715 + 68.2788i) q^{67} +(-62.1564 + 52.1554i) q^{68} +(54.2916 + 47.2908i) q^{72} +(65.2976 - 113.099i) q^{73} +(-0.692161 + 74.9968i) q^{75} +(-141.908 - 51.6503i) q^{76} +(-75.0408 - 30.4940i) q^{81} -132.886 q^{82} +(-26.7420 - 22.4392i) q^{83} +(-82.3504 - 29.9731i) q^{86} +(19.8399 - 112.518i) q^{88} +(80.5908 - 139.587i) q^{89} +(32.0000 + 90.5097i) q^{96} +(172.387 - 62.7437i) q^{97} +(49.0000 + 84.8705i) q^{98} +(19.9798 + 126.973i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 48 q^{8} - 42 q^{11} + 24 q^{12} + 168 q^{18} + 84 q^{22} - 138 q^{27} + 186 q^{33} + 12 q^{34} + 408 q^{38} + 138 q^{41} - 42 q^{43} - 588 q^{51} - 246 q^{57} - 492 q^{59} - 384 q^{64} - 186 q^{67}+ \cdots + 588 q^{98}+O(q^{100}) \) 12 * q + 48 * q^8 - 42 * q^11 + 24 * q^12 + 168 * q^18 + 84 * q^22 - 138 * q^27 + 186 * q^33 + 12 * q^34 + 408 * q^38 + 138 * q^41 - 42 * q^43 - 588 * q^51 - 246 * q^57 - 492 * q^59 - 384 * q^64 - 186 * q^67 + 48 * q^68 - 816 * q^76 + 84 * q^86 - 672 * q^88 + 438 * q^89 + 384 * q^96 + 282 * q^97 + 588 * q^98

Introduction

L-functions

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