LMFDB - Embedded newform 216.3.r.a.43.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(43,216)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			43.1
Root			$-0.483690 + 1.32893i$ of defining polynomial
Character	$\chi$	$=$	216.43
Dual form			216.3.r.a.211.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.53209 - 1.28558i) q^{2} +(-0.0276864 - 2.99987i) q^{3} +(0.694593 + 3.93923i) q^{4} +(-3.81414 + 4.63166i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-8.99847 + 0.166112i) q^{9} +O(q^{10})$	\(q+(-1.53209 - 1.28558i) q^{2} +(-0.0276864 - 2.99987i) q^{3} +(0.694593 + 3.93923i) q^{4} +(-3.81414 + 4.63166i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-8.99847 + 0.166112i) q^{9} +(-17.9893 + 6.54757i) q^{11} +(11.7980 - 2.19275i) q^{12} +(-15.0351 + 5.47232i) q^{16} +(-11.6745 - 20.2208i) q^{17} +(14.0000 + 11.3137i) q^{18} +(-13.0726 + 22.6425i) q^{19} +(35.9786 + 13.0951i) q^{22} +(-20.8945 - 11.8077i) q^{24} +(19.1511 + 16.0697i) q^{25} +(0.747449 + 26.9897i) q^{27} +(30.0702 + 10.9446i) q^{32} +(20.1399 + 53.7843i) q^{33} +(-8.10903 + 45.9886i) q^{34} +(-6.90462 - 35.3317i) q^{36} +(49.1370 - 17.8844i) q^{38} +(15.3276 - 12.8614i) q^{41} +(-61.3309 + 22.3226i) q^{43} +(-38.2876 - 66.3161i) q^{44} +(16.8325 + 44.9518i) q^{48} +(-46.0449 - 16.7590i) q^{49} +(-8.68241 - 49.2404i) q^{50} +(-60.3367 + 35.5819i) q^{51} +(33.5521 - 42.3114i) q^{54} +(68.2865 + 38.5894i) q^{57} +(-99.6790 - 36.2802i) q^{59} +(-32.0000 - 55.4256i) q^{64} +(38.2876 - 108.294i) q^{66} +(97.8663 - 82.1196i) q^{67} +(71.5456 - 60.0339i) q^{68} +(-34.8430 + 63.0077i) q^{72} +(43.4807 - 75.3108i) q^{73} +(47.6768 - 57.8958i) q^{75} +(-98.2741 - 35.7688i) q^{76} +(80.9448 - 2.98950i) q^{81} -40.0175 q^{82} +(-94.2930 - 79.1213i) q^{83} +(122.662 + 44.6452i) q^{86} +(-26.5943 + 150.824i) q^{88} +(-7.59082 + 13.1477i) q^{89} +(32.0000 - 90.5097i) q^{96} +(-141.710 + 51.5782i) q^{97} +(49.0000 + 84.8705i) q^{98} +(160.789 - 61.9063i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 48 q^{8}+O(q^{10}) $ 12 * q + 48 * q^8	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	−0.0276864	−	2.99987i	−0.00922881	−	0.999957i
$3$	−0.0276864	−	2.99987i	−0.00922881	−	0.999957i
$4$	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	$0.111111\pi$
$6$	−3.81414	+	4.63166i	−0.635691	+	0.771944i
$7$		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
$9$	−8.99847	+	0.166112i	−0.999830	+	0.0184568i
$10$		0			0
$11$	−17.9893	+	6.54757i	−1.63539	+	0.595234i	−0.986224	−	0.165412i	$-0.947104\pi$
$11$	−17.9893	+	6.54757i	−1.63539	+	0.595234i	−0.649167	+	0.760646i	$0.724882\pi$
$12$	11.7980	−	2.19275i	0.983163	−	0.182729i
$13$		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
$17$	−11.6745	−	20.2208i	−0.686736	−	1.18946i	−0.972888	−	0.231277i	$-0.925710\pi$
$17$	−11.6745	−	20.2208i	−0.686736	−	1.18946i	0.286152	−	0.958184i	$-0.407624\pi$
$18$	14.0000	+	11.3137i	0.777778	+	0.628539i
$19$	−13.0726	+	22.6425i	−0.688034	+	1.19171i	0.284440	+	0.958694i	$0.408192\pi$
$19$	−13.0726	+	22.6425i	−0.688034	+	1.19171i	−0.972473	+	0.233015i	$0.925141\pi$
$20$		0			0
$21$		0			0
$22$	35.9786	+	13.0951i	1.63539	+	0.595234i
$23$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$23$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$24$	−20.8945	−	11.8077i	−0.870603	−	0.491986i
$25$	19.1511	+	16.0697i	0.766044	+	0.642788i
$26$		0			0
$27$	0.747449	+	26.9897i	0.0276833	+	0.999617i
$28$		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
$33$	20.1399	+	53.7843i	0.610301	+	1.62983i
$34$	−8.10903	+	45.9886i	−0.238501	+	1.35261i
$35$		0			0
$36$	−6.90462	−	35.3317i	−0.191795	−	0.981435i
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	49.1370	−	17.8844i	1.29308	−	0.470643i
$39$		0			0
$40$		0			0
$41$	15.3276	−	12.8614i	0.373844	−	0.313692i	−0.436436	−	0.899735i	$-0.643759\pi$
$41$	15.3276	−	12.8614i	0.373844	−	0.313692i	0.810280	+	0.586043i	$0.199315\pi$
$42$		0			0
$43$	−61.3309	+	22.3226i	−1.42630	+	0.519131i	−0.935869	−	0.352349i	$-0.885383\pi$
$43$	−61.3309	+	22.3226i	−1.42630	+	0.519131i	−0.490431	+	0.871480i	$0.663161\pi$
$44$	−38.2876	−	66.3161i	−0.870173	−	1.50718i
$45$		0			0
$46$		0			0
$47$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$47$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$48$	16.8325	+	44.9518i	0.350678	+	0.936496i
$49$	−46.0449	−	16.7590i	−0.939693	−	0.342020i
$50$	−8.68241	−	49.2404i	−0.173648	−	0.984808i
$51$	−60.3367	+	35.5819i	−1.18307	+	0.697684i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	33.5521	−	42.3114i	0.621335	−	0.783545i
$55$		0			0
$56$		0			0
$57$	68.2865	+	38.5894i	1.19801	+	0.677006i
$58$		0			0
$59$	−99.6790	−	36.2802i	−1.68947	−	0.614918i	−0.694915	−	0.719092i	$-0.744558\pi$
$59$	−99.6790	−	36.2802i	−1.68947	−	0.614918i	−0.994559	+	0.104173i	$0.966780\pi$
$60$		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
$66$	38.2876	−	108.294i	0.580116	−	1.64081i
$67$	97.8663	−	82.1196i	1.46069	−	1.22567i	0.536407	−	0.843959i	$-0.319781\pi$
$67$	97.8663	−	82.1196i	1.46069	−	1.22567i	0.924284	−	0.381706i	$-0.124663\pi$
$68$	71.5456	−	60.0339i	1.05214	−	0.882851i
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$	−34.8430	+	63.0077i	−0.483931	+	0.875106i
$73$	43.4807	−	75.3108i	0.595626	−	1.03165i	−0.397832	−	0.917458i	$-0.630237\pi$
$73$	43.4807	−	75.3108i	0.595626	−	1.03165i	0.993458	−	0.114196i	$-0.0364292\pi$
$74$		0			0
$75$	47.6768	−	57.8958i	0.635691	−	0.771944i
$76$	−98.2741	−	35.7688i	−1.29308	−	0.470643i
$77$		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
$81$	80.9448	−	2.98950i	0.999319	−	0.0369074i
$82$	−40.0175			−0.488019
$83$	−94.2930	−	79.1213i	−1.13606	−	0.953268i	−0.136758	−	0.990604i	$-0.543668\pi$
$83$	−94.2930	−	79.1213i	−1.13606	−	0.953268i	−0.999303	+	0.0373364i	$0.988113\pi$
$84$		0			0
$85$		0			0
$86$	122.662	+	44.6452i	1.42630	+	0.519131i
$87$		0			0
$88$	−26.5943	+	150.824i	−0.302208	+	1.71391i
$89$	−7.59082	+	13.1477i	−0.0852901	+	0.147727i	−0.905515	−	0.424315i	$-0.860515\pi$
$89$	−7.59082	+	13.1477i	−0.0852901	+	0.147727i	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
$97$	−141.710	+	51.5782i	−1.46093	+	0.531734i	−0.945620	−	0.325272i	$-0.894544\pi$
$97$	−141.710	+	51.5782i	−1.46093	+	0.531734i	−0.515306	+	0.857006i	$0.672322\pi$
$98$	49.0000	+	84.8705i	0.500000	+	0.866025i
$99$	160.789	−	61.9063i	1.62413	−	0.625316i
$100$	−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	$0.555556\pi$
$102$	138.184	+	23.0528i	1.35475	+	0.226008i
$103$		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
$107$	−212.715			−1.98799			−0.993996	−	0.109415i	$-0.965102\pi$
$107$	−212.715			−1.98799			−0.993996	+	0.109415i	$0.965102\pi$
$108$	−105.799	+	21.6912i	−0.979623	+	0.200844i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	211.772	+	77.0788i	1.87409	+	0.682113i	0.962701	+	0.270568i	$0.0872114\pi$
$113$	211.772	+	77.0788i	1.87409	+	0.682113i	0.911389	+	0.411545i	$0.135011\pi$
$114$	−55.0114	−	146.910i	−0.482556	−	1.28868i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	106.076	+	183.729i	0.898951	+	1.55703i
$119$		0			0
$120$		0			0
$121$	188.053	−	157.795i	1.55416	−	1.30409i
$122$		0			0
$123$	−39.0069	−	45.6248i	−0.317129	−	0.370933i
$124$		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
$129$	68.6630	+	183.367i	0.532272	+	1.42145i
$130$		0			0
$131$	32.8984	+	186.576i	0.251132	+	1.42424i	0.805808	+	0.592177i	$0.201731\pi$
$131$	32.8984	+	186.576i	0.251132	+	1.42424i	−0.554676	+	0.832067i	$0.687158\pi$
$132$	−197.880	+	116.694i	−1.49909	+	0.884046i
$133$		0			0
$134$	−255.511			−1.90680
$135$		0			0
$136$	−186.792			−1.37347
$137$	−134.081	−	112.507i	−0.978693	−	0.821221i	0.00519888	−	0.999986i	$-0.498345\pi$
$137$	−134.081	−	112.507i	−0.978693	−	0.821221i	−0.983892	+	0.178766i	$0.942790\pi$
$138$		0			0
$139$	−25.7118	−	145.819i	−0.184977	−	1.04906i	−0.925985	−	0.377561i	$-0.876763\pi$
$139$	−25.7118	−	145.819i	−0.184977	−	1.04906i	0.741007	−	0.671497i	$-0.234348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	134.384	−	51.7400i	0.933220	−	0.359306i
$145$		0			0
$146$	−163.434	+	59.4851i	−1.11941	+	0.407432i
$147$	−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	$0.777778\pi$
$150$	−147.474	+	27.4094i	−0.983163	+	0.182729i
$151$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$151$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$152$	104.581	+	181.140i	0.688034	+	1.19171i
$153$	108.412	+	180.017i	0.708573	+	1.17658i
$154$		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
$162$	−127.858	−	99.4805i	−0.789246	−	0.614077i
$163$	116.909			0.717234			0.358617	−	0.933485i	$-0.383248\pi$
$163$	116.909			0.717234			0.358617	+	0.933485i	$0.383248\pi$
$164$	61.3104	+	51.4455i	0.373844	+	0.313692i
$165$		0			0
$166$	42.7490	+	242.442i	0.257524	+	1.46049i
$167$		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
$171$	113.873	−	205.919i	0.665921	−	1.20420i
$172$	−130.534	−	226.091i	−0.758918	−	1.31448i
$173$		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
$176$	234.640	−	196.887i	1.33318	−	1.11867i
$177$	−106.076	+	300.029i	−0.599300	+	1.69508i
$178$	28.5321	−	10.3848i	0.160293	−	0.0583418i
$179$	145.818	+	252.564i	0.814625	+	1.41097i	0.909597	+	0.415492i	$0.136390\pi$
$179$	145.818	+	252.564i	0.814625	+	1.41097i	−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
$187$	342.414	+	287.319i	1.83109	+	1.53647i
$188$		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	$0.222222\pi$
$192$	−165.384	+	97.5304i	−0.861374	+	0.507971i
$193$	−38.1651	−	216.445i	−0.197747	−	1.12148i	−0.908453	−	0.417987i	$-0.862736\pi$
$193$	−38.1651	−	216.445i	−0.197747	−	1.12148i	0.710706	−	0.703489i	$-0.248375\pi$
$194$	283.420	+	103.156i	1.46093	+	0.531734i
$195$		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	$0.666667\pi$
$198$	−325.928	−	111.860i	−1.64610	−	0.564948i
$199$		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
$201$	−249.058	−	291.313i	−1.23909	−	1.44932i
$202$		0			0
$203$		0			0
$204$	−182.075	−	212.965i	−0.892523	−	1.04395i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	86.9144	−	492.916i	0.415859	−	2.35845i
$210$		0			0
$211$	183.837	+	66.9114i	0.871267	+	0.317115i	0.738680	−	0.674056i	$-0.235449\pi$
$211$	183.837	+	66.9114i	0.871267	+	0.317115i	0.132587	+	0.991171i	$0.457672\pi$
$212$		0			0
$213$		0			0
$214$	325.899	+	273.461i	1.52289	+	1.27786i
$215$		0			0
$216$	189.980	+	102.780i	0.879535	+	0.475834i
$217$		0			0
$218$		0			0
$219$	−227.126	−	128.351i	−1.03711	−	0.586080i
$220$		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
$226$	−225.363	−	390.341i	−0.997183	−	1.72717i
$227$	−372.304	+	135.508i	−1.64011	+	0.596950i	−0.987058	−	0.160364i	$-0.948733\pi$
$227$	−372.304	+	135.508i	−1.64011	+	0.596950i	−0.653050	+	0.757315i	$0.726511\pi$
$228$	−104.581	+	295.800i	−0.458689	+	1.29737i
$229$		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
$233$	232.935	+	403.455i	0.999720	+	1.73157i	0.520366	+	0.853943i	$0.325795\pi$
$233$	232.935	+	403.455i	0.999720	+	1.73157i	0.479353	+	0.877622i	$0.340871\pi$
$234$		0			0
$235$		0			0
$236$	73.6797	−	417.859i	0.312202	−	1.77059i
$237$		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
$241$	−307.064	−	257.657i	−1.27412	−	1.06912i	−0.994026	−	0.109146i	$-0.965188\pi$
$241$	−307.064	−	257.657i	−1.27412	−	1.06912i	−0.280099	−	0.959971i	$-0.590367\pi$
$242$	−490.971			−2.02881
$243$	−11.2092	−	242.741i	−0.0461283	−	0.998936i
$244$		0			0
$245$		0			0
$246$	1.10794	+	120.047i	0.00450383	+	0.487998i
$247$		0			0
$248$		0			0
$249$	−234.743	+	285.058i	−0.942743	+	1.14481i
$250$		0			0
$251$	118.492	−	205.234i	0.472079	−	0.817665i	−0.527411	−	0.849610i	$-0.676837\pi$
$251$	118.492	−	205.234i	0.472079	−	0.817665i	0.999490	+	0.0319459i	$0.0101704\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	196.107	−	164.554i	0.766044	−	0.642788i
$257$	−204.707	+	171.770i	−0.796527	+	0.668366i	−0.947352	−	0.320195i	$-0.896252\pi$
$257$	−204.707	+	171.770i	−0.796527	+	0.668366i	0.150825	+	0.988561i	$0.451807\pi$
$258$	130.534	−	369.206i	0.505945	−	1.43103i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	189.454	−	328.144i	0.723107	−	1.25246i
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$	453.189	+	75.6038i	1.71662	+	0.286378i
$265$		0			0
$266$		0			0
$267$	39.6515	+	22.4075i	0.148508	+	0.0839231i
$268$	391.465	+	328.478i	1.46069	+	1.22567i
$269$		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			$\pi$
$272$	286.182	+	240.135i	1.05214	+	0.882851i
$273$		0			0
$274$	60.7874	+	344.742i	0.221852	+	1.25818i
$275$	−449.733	−	163.689i	−1.63539	−	0.595234i
$276$		0			0
$277$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$277$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$278$	−148.069	+	256.462i	−0.532621	+	0.922527i
$279$		0			0
$280$		0			0
$281$	302.495	−	110.099i	1.07649	−	0.391812i	0.257892	−	0.966174i	$-0.416972\pi$
$281$	302.495	−	110.099i	1.07649	−	0.391812i	0.818601	+	0.574362i	$0.194750\pi$
$282$		0			0
$283$	402.939	−	338.106i	1.42381	−	1.19472i	0.474551	−	0.880228i	$-0.342610\pi$
$283$	402.939	−	338.106i	1.42381	−	1.19472i	0.949260	−	0.314491i	$-0.101834\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−272.403	−	93.4900i	−0.945845	−	0.324618i
$289$	−128.088	+	221.856i	−0.443213	+	0.767667i
$290$		0			0
$291$	158.651	+	423.683i	0.545194	+	1.45596i
$292$	326.868	+	118.970i	1.11941	+	0.407432i
$293$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$293$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$294$	253.244	−	149.344i	0.861374	−	0.507971i
$295$		0			0
$296$		0			0
$297$	−190.163	−	480.631i	−0.640279	−	1.61829i
$298$		0			0
$299$		0			0
$300$	261.181	+	147.596i	0.870603	+	0.491986i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	72.6413	−	411.969i	0.238952	−	1.35516i
$305$		0			0
$306$	65.3296	−	415.174i	0.213495	−	1.35678i
$307$	205.320	+	355.625i	0.668796	+	1.15839i	0.978241	+	0.207471i	$0.0665234\pi$
$307$	205.320	+	355.625i	0.668796	+	1.15839i	−0.309445	+	0.950917i	$0.600143\pi$
$308$		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
$313$	−573.554	+	208.757i	−1.83244	+	0.666954i	−0.840256	+	0.542191i	$0.817595\pi$
$313$	−573.554	+	208.757i	−1.83244	+	0.666954i	−0.992186	+	0.124764i	$0.960183\pi$
$314$		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
$321$	5.88932	+	638.118i	0.0183468	+	1.98791i
$322$		0			0
$323$	610.467			1.88999
$324$	68.0000	+	316.784i	0.209877	+	0.977728i
$325$		0			0
$326$	−179.115	−	150.296i	−0.549433	−	0.461029i
$327$		0			0
$328$	−27.7959	−	157.638i	−0.0847435	−	0.480605i
$329$		0			0
$330$		0			0
$331$	−100.747	+	571.366i	−0.304373	+	1.72618i	0.322071	+	0.946716i	$0.395621\pi$
$331$	−100.747	+	571.366i	−0.304373	+	1.72618i	−0.626444	+	0.779467i	$0.715490\pi$
$332$	246.182	−	426.399i	0.741511	−	1.28433i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−514.480	+	431.700i	−1.52665	+	1.28101i	−0.709199	+	0.705009i	$0.750943\pi$
$337$	−514.480	+	431.700i	−1.52665	+	1.28101i	−0.817449	+	0.576001i	$0.804612\pi$
$338$	−258.923	+	217.262i	−0.766044	+	0.642788i
$339$	225.363	−	637.424i	0.664788	−	1.88031i
$340$		0			0
$341$		0			0
$342$	−439.187	+	169.095i	−1.28417	+	0.494429i
$343$		0			0
$344$	−90.6679	+	514.203i	−0.263570	+	1.49478i
$345$		0			0
$346$		0			0
$347$	94.2670	+	534.615i	0.271663	+	1.54068i	0.749366	+	0.662156i	$0.230358\pi$
$347$	94.2670	+	534.615i	0.271663	+	1.54068i	−0.477703	+	0.878521i	$0.658531\pi$
$348$		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
$352$	−612.602			−1.74035
$353$	−220.411	−	184.947i	−0.624394	−	0.523928i	0.274788	−	0.961505i	$-0.411392\pi$
$353$	−220.411	−	184.947i	−0.624394	−	0.523928i	−0.899181	+	0.437576i	$0.855837\pi$
$354$	548.228	−	323.302i	1.54867	−	0.913282i
$355$		0			0
$356$	−57.0643	−	20.7697i	−0.160293	−	0.0583418i
$357$		0			0
$358$	101.284	−	574.410i	0.282916	−	1.60450i
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−161.288	−	279.358i	−0.446780	−	0.773846i
$362$		0			0
$363$	−478.572	−	559.766i	−1.31838	−	1.54206i
$364$		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
$369$	−135.788	+	118.279i	−0.367991	+	0.320539i
$370$		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	$0.111111\pi$
$374$	−155.238	−	880.397i	−0.415074	−	2.35400i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−754.910			−1.99185			−0.995924	−	0.0901970i	$-0.971250\pi$
$379$	−754.910			−1.99185			−0.995924	+	0.0901970i	$0.971250\pi$
$380$		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	$0.111111\pi$
$384$	378.765	+	63.1880i	0.986368	+	0.164552i
$385$		0			0
$386$	−219.784	+	380.677i	−0.569388	+	0.986210i
$387$	548.176	−	211.057i	1.41648	−	0.545367i
$388$	−301.609	−	522.402i	−0.777343	−	1.34640i
$389$		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
$393$	558.793	−	103.856i	1.42186	−	0.264266i
$394$		0			0
$395$		0			0
$396$	355.546	+	590.383i	0.897843	+	1.49087i
$397$		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
$401$	139.103	+	788.894i	0.346891	+	1.96732i	0.221001	+	0.975274i	$0.429068\pi$
$401$	139.103	+	788.894i	0.346891	+	1.96732i	0.125890	+	0.992044i	$0.459821\pi$
$402$	7.07418	+	766.500i	0.0175975	+	1.90672i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	5.17161	+	560.353i	0.0126755	+	1.37341i
$409$	38.9169	+	220.709i	0.0951514	+	0.539631i	0.994701	+	0.102812i	$0.0327839\pi$
$409$	38.9169	+	220.709i	0.0951514	+	0.539631i	−0.899549	+	0.436819i	$0.856105\pi$
$410$		0			0
$411$	−333.795	+	405.341i	−0.812154	+	0.986230i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−436.727	+	81.1695i	−1.04731	+	0.194651i
$418$	−766.842	+	643.457i	−1.83455	+	1.53937i
$419$	−242.208	+	203.237i	−0.578063	+	0.485052i	−0.884310	−	0.466900i	$-0.845371\pi$
$419$	−242.208	+	203.237i	−0.578063	+	0.485052i	0.306247	+	0.951952i	$0.400927\pi$
$420$		0			0
$421$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$421$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$422$	−195.636	−	338.851i	−0.463592	−	0.802964i
$423$		0			0
$424$		0			0
$425$	101.363	−	574.857i	0.238501	−	1.35261i
$426$		0			0
$427$		0			0
$428$	−147.750	−	837.934i	−0.345211	−	1.95779i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−158.934	−	401.701i	−0.367903	−	0.929864i
$433$	857.295			1.97990			0.989949	−	0.141428i	$-0.0451693\pi$
$433$	857.295			1.97990			0.989949	+	0.141428i	$0.0451693\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	182.973	+	488.634i	0.417746	+	1.11560i
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	417.118	+	143.157i	0.945845	+	0.324618i
$442$		0			0
$443$	813.473	−	296.080i	1.83628	−	0.668352i	0.845312	−	0.534273i	$-0.179415\pi$
$443$	813.473	−	296.080i	1.83628	−	0.668352i	0.990971	−	0.134079i	$-0.0428076\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	41.7995	+	72.3989i	0.0930947	+	0.161245i	0.908812	−	0.417206i	$-0.136991\pi$
$449$	41.7995	+	72.3989i	0.0930947	+	0.161245i	−0.815717	+	0.578451i	$0.803657\pi$
$450$	86.3078	+	441.646i	0.191795	+	0.981435i
$451$	−191.522	+	331.726i	−0.424661	+	0.735534i
$452$	−156.536	+	887.758i	−0.346318	+	1.96407i
$453$		0			0
$454$	744.609	+	271.016i	1.64011	+	0.596950i
$455$		0			0
$456$	540.501	−	318.745i	1.18531	−	0.699002i
$457$	−59.8858	−	50.2502i	−0.131041	−	0.109957i	0.574911	−	0.818216i	$-0.305036\pi$
$457$	−59.8858	−	50.2502i	−0.131041	−	0.109957i	−0.705953	+	0.708259i	$0.749481\pi$
$458$		0			0
$459$	537.028	−	330.205i	1.16999	−	0.719401i
$460$		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
$466$	161.795	−	917.583i	0.347199	−	1.96906i
$467$	143.643	−	248.797i	0.307586	−	0.532755i	−0.670248	−	0.742138i	$-0.733812\pi$
$467$	143.643	−	248.797i	0.307586	−	0.532755i	0.977834	+	0.209383i	$0.0671454\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−650.072	+	545.476i	−1.37727	+	1.15567i
$473$	957.141	−	803.137i	2.02355	−	1.69796i
$474$		0			0
$475$	−614.213	+	223.555i	−1.29308	+	0.470643i
$476$		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
$482$	139.212	+	789.508i	0.288821	+	1.63798i
$483$		0			0
$484$	752.212	+	631.181i	1.55416	+	1.30409i
$485$		0			0
$486$	−294.889	+	386.312i	−0.606767	+	0.794880i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−3.23680	−	350.713i	−0.00661922	−	0.717204i
$490$		0			0
$491$	−921.691	−	335.468i	−1.87717	−	0.683234i	−0.955183	−	0.296018i	$-0.904341\pi$
$491$	−921.691	−	335.468i	−1.87717	−	0.683234i	−0.921989	−	0.387217i	$-0.873436\pi$
$492$	152.633	−	185.348i	0.310229	−	0.376723i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	726.110	−	134.954i	1.45805	−	0.270992i
$499$	−178.160	+	149.494i	−0.357034	+	0.299587i	−0.803607	−	0.595160i	$-0.797089\pi$
$499$	−178.160	+	149.494i	−0.357034	+	0.299587i	0.446573	+	0.894747i	$0.352644\pi$
$500$		0			0
$501$		0			0
$502$	−445.383	+	162.106i	−0.887218	+	0.322921i
$503$		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
$507$	−500.089	−	83.4280i	−0.986368	−	0.164552i
$508$		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
$513$	−620.884	−	335.902i	−1.21030	−	0.654779i
$514$	534.453			1.03979
$515$		0			0
$516$	−674.631	+	397.845i	−1.30742	+	0.771017i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	472.588	−	818.547i	0.907079	−	1.57111i	0.0889768	−	0.996034i	$-0.471640\pi$
$521$	472.588	−	818.547i	0.907079	−	1.57111i	0.818102	−	0.575073i	$-0.195026\pi$
$522$		0			0
$523$	518.363	+	897.831i	0.991133	+	1.71669i	0.610636	+	0.791911i	$0.290914\pi$
$523$	518.363	+	897.831i	0.991133	+	1.71669i	0.380497	+	0.924782i	$0.375753\pi$
$524$	−712.114	+	259.188i	−1.35900	+	0.494634i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−597.131	−	698.440i	−1.13093	−	1.32280i
$529$	−497.097	+	180.929i	−0.939693	+	0.342020i
$530$		0			0
$531$	902.985	+	309.908i	1.70054	+	0.583631i
$532$		0			0
$533$		0			0
$534$	−31.9432	−	85.3052i	−0.0598187	−	0.159748i
$535$		0			0
$536$	−177.476	−	1006.52i	−0.331112	−	1.87783i
$537$	753.622	−	444.428i	1.40339	−	0.827612i
$538$		0			0
$539$	938.047			1.74035
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−129.744	−	735.818i	−0.238501	−	1.35261i
$545$		0			0
$546$		0			0
$547$	179.516	−	1018.09i	0.328183	−	1.86122i	−0.158108	−	0.987422i	$-0.550539\pi$
$547$	179.516	−	1018.09i	0.328183	−	1.86122i	0.486291	−	0.873797i	$-0.338349\pi$
$548$	350.060	−	606.322i	0.638796	−	1.10643i
$549$		0			0
$550$	478.595	+	828.952i	0.870173	+	1.50718i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	556.556	−	202.570i	1.00100	−	0.364334i
$557$		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
$561$	852.441	−	1035.15i	1.51950	−	1.84519i
$562$	−604.989	−	220.198i	−1.07649	−	0.391812i
$563$	−195.454	−	1108.47i	−0.347165	−	1.96887i	−0.200710	−	0.979651i	$-0.564325\pi$
$563$	−195.454	−	1108.47i	−0.347165	−	1.96887i	−0.146454	−	0.989217i	$-0.546786\pi$
$564$		0			0
$565$		0			0
$566$	−1052.00			−1.85865
$567$		0			0
$568$		0			0
$569$	−201.629	−	169.187i	−0.354357	−	0.297341i	0.448180	−	0.893943i	$-0.352072\pi$
$569$	−201.629	−	169.187i	−0.354357	−	0.297341i	−0.802537	+	0.596603i	$0.796517\pi$
$570$		0			0
$571$	−191.712	−	1087.25i	−0.335747	−	1.90412i	−0.419730	−	0.907649i	$-0.637875\pi$
$571$	−191.712	−	1087.25i	−0.335747	−	1.90412i	0.0839829	−	0.996467i	$-0.473236\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	297.158	+	493.430i	0.515899	+	0.856649i
$577$	568.060	+	983.908i	0.984505	+	1.70521i	0.644114	+	0.764929i	$0.277226\pi$
$577$	568.060	+	983.908i	0.984505	+	1.70521i	0.340391	+	0.940284i	$0.389441\pi$
$578$	481.455	−	175.235i	0.832967	−	0.303175i
$579$	−648.250	+	120.483i	−1.11960	+	0.208088i
$580$		0			0
$581$		0			0
$582$	301.609	−	853.079i	0.518228	−	1.46577i
$583$		0			0
$584$	−347.846	−	602.486i	−0.595626	−	1.03165i
$585$		0			0
$586$		0			0
$587$	−83.6512	+	474.410i	−0.142506	+	0.808194i	0.826829	+	0.562453i	$0.190142\pi$
$587$	−83.6512	+	474.410i	−0.142506	+	0.808194i	−0.969336	+	0.245741i	$0.920969\pi$
$588$	−579.985	−	96.7567i	−0.986368	−	0.164552i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−274.453			−0.462821			−0.231411	−	0.972856i	$-0.574334\pi$
$593$	−274.453			−0.462821			−0.231411	+	0.972856i	$0.574334\pi$
$594$	−326.541	+	980.838i	−0.549733	+	1.65124i
$595$		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	$0.111111\pi$
$600$	−210.407	−	561.898i	−0.350678	−	0.936496i
$601$	161.059	−	913.410i	0.267985	−	1.51982i	−0.492415	−	0.870361i	$-0.663886\pi$
$601$	161.059	−	913.410i	0.267985	−	1.51982i	0.760399	−	0.649456i	$-0.225003\pi$
$602$		0			0
$603$	−867.006	+	755.207i	−1.43782	+	1.25242i
$604$		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	$0.777778\pi$
$608$	−640.910	+	537.787i	−1.05413	+	0.884519i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−633.828	+	552.097i	−1.03567	+	0.902120i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	142.614	−	808.804i	0.232270	−	1.31727i
$615$		0			0
$616$		0			0
$617$	193.078	+	1095.00i	0.312931	+	1.77472i	0.583595	+	0.812045i	$0.301645\pi$
$617$	193.078	+	1095.00i	0.312931	+	1.77472i	−0.270665	+	0.962674i	$0.587243\pi$
$618$		0			0
$619$	693.565	+	581.970i	1.12046	+	0.940178i	0.998628	−	0.0523724i	$-0.0166783\pi$
$619$	693.565	+	581.970i	1.12046	+	0.940178i	0.121833	+	0.992551i	$0.461123\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	108.530	+	615.505i	0.173648	+	0.984808i
$626$	1147.11	+	417.513i	1.83244	+	0.666954i
$627$	−1481.09	−	247.085i	−2.36219	−	0.394075i
$628$		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
$633$	195.636	−	553.341i	0.309061	−	0.874157i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−149.204	+	846.177i	−0.232767	+	1.32009i	0.614497	+	0.788919i	$0.289359\pi$
$641$	−149.204	+	846.177i	−0.232767	+	1.32009i	−0.847265	+	0.531171i	$0.821752\pi$
$642$	811.326	−	985.225i	1.26375	−	1.53462i
$643$	−590.717	−	215.003i	−0.918688	−	0.334375i	−0.160972	−	0.986959i	$-0.551463\pi$
$643$	−590.717	−	215.003i	−0.918688	−	0.334375i	−0.757717	+	0.652584i	$0.773685\pi$
$644$		0			0
$645$		0			0
$646$	−935.289	−	784.801i	−1.44782	−	1.21486i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	303.067	−	572.760i	0.467697	−	0.883889i
$649$	2030.70			3.12897
$650$		0			0
$651$		0			0
$652$	81.2043	+	460.532i	0.124546	+	0.706338i
$653$		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
$656$	−160.070	+	277.250i	−0.244009	+	0.422637i
$657$	−378.750	+	684.904i	−0.576483	+	1.04247i
$658$		0			0
$659$	−1171.37	+	426.343i	−1.77749	+	0.646955i	−0.777660	+	0.628685i	$0.783594\pi$
$659$	−1171.37	+	426.343i	−1.77749	+	0.646955i	−0.999833	+	0.0182698i	$0.994184\pi$
$660$		0			0
$661$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$661$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$662$	888.888	−	745.866i	1.34273	−	1.12669i
$663$		0			0
$664$	−925.340	+	336.796i	−1.39358	+	0.507223i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	139.490	+	117.046i	0.207267	+	0.173917i	0.740512	−	0.672043i	$-0.234583\pi$
$673$	139.490	+	117.046i	0.207267	+	0.173917i	−0.533245	+	0.845961i	$0.679028\pi$
$674$	1343.21			1.99290
$675$	−419.401	+	528.893i	−0.621335	+	0.783545i
$676$	676.000			1.00000
$677$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$677$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$678$	−1164.73	+	686.868i	−1.71789	+	1.01308i
$679$		0			0
$680$		0			0
$681$	416.814	+	1113.11i	0.612061	+	1.63453i
$682$		0			0
$683$	608.885	−	1054.62i	0.891485	−	1.54410i	0.0533905	−	0.998574i	$-0.482997\pi$
$683$	608.885	−	1054.62i	0.891485	−	1.54410i	0.838095	−	0.545524i	$-0.183669\pi$
$684$	890.258	+	305.540i	1.30155	+	0.446696i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	799.958	−	671.245i	1.16273	−	0.975646i
$689$		0			0
$690$		0			0
$691$	−608.065	+	221.317i	−0.879978	+	0.320286i	−0.742201	−	0.670178i	$-0.766218\pi$
$691$	−608.065	+	221.317i	−0.879978	+	0.320286i	−0.137777	+	0.990463i	$0.543996\pi$
$692$		0			0
$693$		0			0
$694$	542.862	−	940.265i	0.782222	−	1.35485i
$695$		0			0
$696$		0			0
$697$	−439.010	−	159.787i	−0.629857	−	0.229249i
$698$		0			0
$699$	1203.86	−	709.944i	1.72227	−	1.01566i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	938.561	+	787.546i	1.33318	+	1.11867i
$705$		0			0
$706$	99.9262	+	566.710i	0.141539	+	0.802705i
$707$		0			0
$708$	−1255.56	−	209.461i	−1.77339	−	0.295849i
$709$		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
$712$	60.7265	+	105.181i	0.0852901	+	0.147727i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−893.624	+	749.839i	−1.24808	+	1.04726i
$717$		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
$722$	−112.029	+	635.349i	−0.155165	+	0.879985i
$723$	−764.437	+	928.286i	−1.05731	+	1.28394i
$724$		0			0
$725$		0			0
$726$	13.5932	+	1472.85i	0.0187235	+	2.02872i
$727$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$727$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$728$		0			0
$729$	−727.883	+	40.3468i	−0.998467	+	0.0553454i
$730$		0			0
$731$	1167.39	+	979.557i	1.59698	+	1.34002i
$732$		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
$737$	−1222.86	+	2118.06i	−1.65924	+	2.87389i
$738$	360.096	−	6.64737i	0.487935	−	0.00900728i
$739$	−682.334	−	1181.84i	−0.923321	−	1.59924i	−0.794240	−	0.607605i	$-0.792130\pi$
$739$	−682.334	−	1181.84i	−0.923321	−	1.59924i	−0.129081	−	0.991634i	$-0.541203\pi$
$740$		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
$747$	861.636	+	696.307i	1.15346	+	0.932138i
$748$	−893.979	+	1548.42i	−1.19516	+	2.07008i
$749$		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
$753$	−618.956	−	349.778i	−0.821986	−	0.464513i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	1156.59	+	970.494i	1.52584	+	1.28034i
$759$		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
$768$	−499.069	−	583.741i	−0.649830	−	0.760080i
$769$	−339.356	+	284.754i	−0.441295	+	0.370291i	−0.836194	−	0.548434i	$-0.815224\pi$
$769$	−339.356	+	284.754i	−0.441295	+	0.370291i	0.394899	+	0.918725i	$0.370780\pi$
$770$		0			0
$771$	520.956	+	609.341i	0.675688	+	0.790325i
$772$	826.117	−	300.682i	1.07010	−	0.389485i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$	−1111.18	−	381.363i	−1.43564	−	0.492717i
$775$		0			0
$776$	−209.495	+	1188.11i	−0.269968	+	1.53107i
$777$		0			0
$778$		0			0
$779$	90.8414	+	515.187i	0.116613	+	0.661344i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$		0			0
$786$	−989.636	−	559.253i	−1.25908	−	0.711518i
$787$	−271.806	−	1541.49i	−0.345370	−	1.95869i	−0.276261	−	0.961083i	$-0.589095\pi$
$787$	−271.806	−	1541.49i	−0.345370	−	1.95869i	−0.0691092	−	0.997609i	$-0.522016\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	214.254	−	1361.60i	0.270523	−	1.71919i
$793$		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
$801$	66.1217	−	119.570i	0.0825490	−	0.149276i
$802$	801.064	−	1387.48i	0.998833	−	1.73003i
$803$	−289.085	+	1639.48i	−0.360006	+	2.04169i
$804$	974.555	−	1183.44i	1.21213	−	1.47194i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−993.713			−1.22832			−0.614161	−	0.789181i	$-0.710506\pi$
$809$	−993.713			−1.22832			−0.614161	+	0.789181i	$0.710506\pi$
$810$		0			0
$811$	−1618.62			−1.99583			−0.997916	−	0.0645255i	$-0.979447\pi$
$811$	−1618.62			−1.99583			−0.997916	+	0.0645255i	$0.979447\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	712.452	−	865.159i	0.873103	−	1.06024i
$817$	296.317	−	1680.50i	0.362689	−	2.05691i
$818$	224.114	−	388.176i	0.273978	−	0.474543i
$819$		0			0
$820$		0			0
$821$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$821$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$822$	1032.50	−	191.899i	1.25608	−	0.233454i
$823$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$823$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$824$		0			0
$825$	−478.595	+	1353.67i	−0.580116	+	1.64081i
$826$		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
$833$	198.671	+	1126.72i	0.238501	+	1.35261i
$834$	773.454	+	437.086i	0.927403	+	0.524084i
$835$		0			0
$836$	2002.08			2.39483
$837$		0			0
$838$	632.361			0.754608
$839$		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
$843$	−338.658	−	904.397i	−0.401730	−	1.07283i
$844$	−135.887	+	770.654i	−0.161004	+	0.913097i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1025.43	−	1199.40i	−1.20781	−	1.41272i
$850$	−894.319	+	750.423i	−1.05214	+	0.882851i
$851$		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
$856$	−850.861	+	1473.73i	−0.993996	+	1.72165i
$857$	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
$859$	186.853	+	68.0089i	0.217524	+	0.0791721i	0.448483	−	0.893791i	$-0.351964\pi$
$859$	186.853	+	68.0089i	0.217524	+	0.0791721i	−0.230960	+	0.972963i	$0.574187\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−272.916	+	819.764i	−0.315875	+	0.948801i
$865$		0			0
$866$	−1313.45	−	1102.12i	−1.51669	−	1.27265i
$867$	669.085	+	378.107i	0.771724	+	0.436109i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1266.60	−	487.664i	1.45086	−	0.558607i
$874$		0			0
$875$		0			0
$876$	347.846	−	983.856i	0.397084	−	1.12312i
$877$		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
$881$	−800.408	−	1386.35i	−0.908522	−	1.57361i	−0.816118	−	0.577885i	$-0.803878\pi$
$881$	−800.408	−	1386.35i	−0.908522	−	1.57361i	−0.0924043	−	0.995722i	$-0.529455\pi$
$882$	−455.023	−	755.565i	−0.515899	−	0.856649i
$883$	713.065	−	1235.06i	0.807548	−	1.39871i	−0.107010	−	0.994258i	$-0.534128\pi$
$883$	713.065	−	1235.06i	0.807548	−	1.39871i	0.914557	−	0.404456i	$-0.132539\pi$
$884$		0			0
$885$		0			0
$886$	−1626.95	−	592.160i	−1.83628	−	0.668352i
$887$		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
$891$	−1436.57	+	583.771i	−1.61231	+	0.655186i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	29.0336	−	164.658i	0.0323315	−	0.183361i
$899$		0			0
$900$	435.538	−	787.596i	0.483931	−	0.875106i
$901$		0			0
$902$	719.887	−	262.018i	0.798101	−	0.290485i
$903$		0			0
$904$	1381.11	−	1158.89i	1.52777	−	1.28195i
$905$		0			0
$906$		0			0
$907$	829.353	−	301.860i	0.914391	−	0.332811i	0.158386	−	0.987377i	$-0.449371\pi$
$907$	829.353	−	301.860i	0.914391	−	0.332811i	0.756005	+	0.654566i	$0.227149\pi$
$908$	−792.396	−	1372.47i	−0.872683	−	1.51153i
$909$		0			0
$910$		0			0
$911$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$911$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$912$	−1237.87	−	206.509i	−1.35731	−	0.226435i
$913$	2214.32	+	805.946i	2.42532	+	0.882745i
$914$	27.1500	+	153.976i	0.0297046	+	0.168463i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1247.28	−	184.486i	−1.35869	−	0.200965i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	1061.15	−	625.781i	1.15217	−	0.679458i
$922$		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
$931$	981.394	−	823.487i	1.05413	−	0.884519i
$932$	−1427.51	+	1197.82i	−1.53166	+	1.28521i
$933$		0			0
$934$	−539.920	+	196.515i	−0.578073	+	0.210401i
$935$		0			0
$936$		0			0
$937$	570.044	−	987.345i	0.608371	−	1.05373i	−0.383138	−	0.923691i	$-0.625157\pi$
$937$	570.044	−	987.345i	0.608371	−	1.05373i	0.991509	−	0.130039i	$-0.0415101\pi$
$938$		0			0
$939$	642.123	+	1714.81i	0.683837	+	1.82621i
$940$		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
$944$	1697.22			1.79790
$945$		0			0
$946$	−2498.92			−2.64156
$947$	210.552	+	176.674i	0.222336	+	0.186562i	0.747151	−	0.664654i	$-0.231421\pi$
$947$	210.552	+	176.674i	0.222336	+	0.186562i	−0.524815	+	0.851216i	$0.675866\pi$
$948$		0			0
$949$		0			0
$950$	1228.43	+	447.111i	1.29308	+	0.470643i
$951$		0			0
$952$		0			0
$953$	−923.585	+	1599.70i	−0.969134	+	1.67859i	−0.271061	+	0.962562i	$0.587375\pi$
$953$	−923.585	+	1599.70i	−0.969134	+	1.67859i	−0.698073	+	0.716027i	$0.745959\pi$
$954$		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
$963$	1914.11	−	35.3344i	1.98765	−	0.0366920i
$964$	801.687	−	1388.56i	0.831626	−	1.44042i
$965$		0			0
$966$		0			0
$967$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$967$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$968$	−341.025	−	1934.05i	−0.352299	−	1.99799i
$969$	−16.9016	−	1831.32i	−0.0174424	−	1.88991i
$970$		0			0
$971$	974.000			1.00309			0.501545	−	0.865132i	$-0.332765\pi$
$971$	974.000			1.00309			0.501545	+	0.865132i	$0.332765\pi$
$972$	948.428	−	212.762i	0.975749	−	0.218891i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1573.64	−	572.759i	−1.61069	−	0.586243i	−0.629113	−	0.777314i	$-0.716582\pi$
$977$	−1573.64	−	572.759i	−1.61069	−	0.586243i	−0.981576	+	0.191071i	$0.938804\pi$
$978$	−445.908	+	541.484i	−0.455939	+	0.553665i
$979$	50.4681	−	286.219i	0.0515507	−	0.292358i
$980$		0			0
$981$		0			0
$982$	980.843	+	1698.87i	0.998822	+	1.73001i
$983$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$983$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$984$	−472.125	+	87.7485i	−0.479802	+	0.0891753i
$985$		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
$993$	1716.82	+	286.410i	1.72892	+	0.288429i
$994$		0			0
$995$		0			0
$996$	−1285.96	−	726.708i	−1.29112	−	0.729626i
$997$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$997$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$998$	465.143			0.466075
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.a.43.1	✓	12
8.3	odd	2	CM	216.3.r.a.43.1	✓	12
27.22	even	9	inner	216.3.r.a.211.1	yes	12
216.211	odd	18	inner	216.3.r.a.211.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.a.43.1	✓	12	1.1	even	1	trivial
216.3.r.a.43.1	✓	12	8.3	odd	2	CM
216.3.r.a.211.1	yes	12	27.22	even	9	inner
216.3.r.a.211.1	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-1.53209 - 1.28558i) q^{2} +(-0.0276864 - 2.99987i) q^{3} +(0.694593 + 3.93923i) q^{4} +(-3.81414 + 4.63166i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-8.99847 + 0.166112i) q^{9} +O(q^{10})\)	\(q+(-1.53209 - 1.28558i) q^{2} +(-0.0276864 - 2.99987i) q^{3} +(0.694593 + 3.93923i) q^{4} +(-3.81414 + 4.63166i) q^{6} +(4.00000 - 6.92820i) q^{8} +(-8.99847 + 0.166112i) q^{9} +(-17.9893 + 6.54757i) q^{11} +(11.7980 - 2.19275i) q^{12} +(-15.0351 + 5.47232i) q^{16} +(-11.6745 - 20.2208i) q^{17} +(14.0000 + 11.3137i) q^{18} +(-13.0726 + 22.6425i) q^{19} +(35.9786 + 13.0951i) q^{22} +(-20.8945 - 11.8077i) q^{24} +(19.1511 + 16.0697i) q^{25} +(0.747449 + 26.9897i) q^{27} +(30.0702 + 10.9446i) q^{32} +(20.1399 + 53.7843i) q^{33} +(-8.10903 + 45.9886i) q^{34} +(-6.90462 - 35.3317i) q^{36} +(49.1370 - 17.8844i) q^{38} +(15.3276 - 12.8614i) q^{41} +(-61.3309 + 22.3226i) q^{43} +(-38.2876 - 66.3161i) q^{44} +(16.8325 + 44.9518i) q^{48} +(-46.0449 - 16.7590i) q^{49} +(-8.68241 - 49.2404i) q^{50} +(-60.3367 + 35.5819i) q^{51} +(33.5521 - 42.3114i) q^{54} +(68.2865 + 38.5894i) q^{57} +(-99.6790 - 36.2802i) q^{59} +(-32.0000 - 55.4256i) q^{64} +(38.2876 - 108.294i) q^{66} +(97.8663 - 82.1196i) q^{67} +(71.5456 - 60.0339i) q^{68} +(-34.8430 + 63.0077i) q^{72} +(43.4807 - 75.3108i) q^{73} +(47.6768 - 57.8958i) q^{75} +(-98.2741 - 35.7688i) q^{76} +(80.9448 - 2.98950i) q^{81} -40.0175 q^{82} +(-94.2930 - 79.1213i) q^{83} +(122.662 + 44.6452i) q^{86} +(-26.5943 + 150.824i) q^{88} +(-7.59082 + 13.1477i) q^{89} +(32.0000 - 90.5097i) q^{96} +(-141.710 + 51.5782i) q^{97} +(49.0000 + 84.8705i) q^{98} +(160.789 - 61.9063i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 48 q^{8}+O(q^{10}) \) 12 * q + 48 * q^8	\( 12 q + 48 q^{8} - 42 q^{11} + 24 q^{12} + 168 q^{18} + 84 q^{22} - 138 q^{27} + 186 q^{33} + 12 q^{34} + 408 q^{38} + 138 q^{41} - 42 q^{43} - 588 q^{51} - 246 q^{57} - 492 q^{59} - 384 q^{64} - 186 q^{67} + 48 q^{68} - 816 q^{76} + 84 q^{86} - 672 q^{88} + 438 q^{89} + 384 q^{96} + 282 q^{97} + 588 q^{98}+O(q^{100}) \) 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

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