LMFDB - Embedded newform 1503.1.f.b.1168.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,1,Mod(166,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1503.166");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1503.f (of order $6$, degree $2$, 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:	$0.750094713987$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} + \cdots)$

Embedding invariants

Embedding label			1168.6
Root			$0.723734 + 0.690079i$ of defining polynomial
Character	$\chi$	$=$	1503.1168
Dual form			1503.1.f.b.166.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0475819 + 0.0824143i) q^{2} +(-0.327068 - 0.945001i) q^{3} +(0.495472 + 0.858183i) q^{4} +(0.0934441 + 0.0180099i) q^{6} +(-0.235759 + 0.408346i) q^{7} -0.189466 q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})$	\(q+(-0.0475819 + 0.0824143i) q^{2} +(-0.327068 - 0.945001i) q^{3} +(0.495472 + 0.858183i) q^{4} +(0.0934441 + 0.0180099i) q^{6} +(-0.235759 + 0.408346i) q^{7} -0.189466 q^{8} +(-0.786053 + 0.618159i) q^{9} +(-0.580057 + 1.00469i) q^{11} +(0.648930 - 0.748905i) q^{12} +(-0.0224357 - 0.0388598i) q^{14} +(-0.486457 + 0.842568i) q^{16} +(-0.0135432 - 0.0941952i) q^{18} +1.85674 q^{19} +(0.462997 + 0.0892353i) q^{21} +(-0.0552004 - 0.0956100i) q^{22} +(0.0619682 + 0.179045i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.841254 + 0.540641i) q^{27} -0.467248 q^{28} +(-0.723734 + 1.25354i) q^{29} +(-0.0475819 - 0.0824143i) q^{31} +(-0.141026 - 0.244264i) q^{32} +(1.13915 + 0.219553i) q^{33} +(-0.919960 - 0.368297i) q^{36} +(-0.0883470 + 0.153022i) q^{38} +(-0.0293845 + 0.0339116i) q^{42} -1.14961 q^{44} +(0.786053 - 1.36148i) q^{47} +(0.955332 + 0.184125i) q^{48} +(0.388835 + 0.673483i) q^{49} +(-0.0475819 - 0.0824143i) q^{50} +(-0.0845850 + 0.0436066i) q^{54} +(0.0446683 - 0.0773677i) q^{56} +(-0.607279 - 1.75462i) q^{57} +(-0.0688733 - 0.119292i) q^{58} +(0.654861 - 1.13425i) q^{61} +0.00905615 q^{62} +(-0.0671040 - 0.466718i) q^{63} -0.946072 q^{64} +(-0.0722972 + 0.0834354i) q^{66} +(0.148930 - 0.117120i) q^{72} +(0.981929 + 0.189251i) q^{75} +(0.919960 + 1.59342i) q^{76} +(-0.273507 - 0.473728i) q^{77} +(0.235759 - 0.971812i) q^{81} +(0.152822 + 0.441549i) q^{84} +(1.42131 + 0.273935i) q^{87} +(0.109901 - 0.190354i) q^{88} -0.654136 q^{89} +(-0.0623191 + 0.0719200i) q^{93} +(0.0748038 + 0.129564i) q^{94} +(-0.184705 + 0.213161i) q^{96} +(-0.415415 + 0.719520i) q^{97} -0.0740061 q^{98} +(-0.165101 - 1.14831i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$2$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$3$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$3$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$4$	0.495472	+	0.858183i	0.495472	+	0.858183i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	0.0934441	+	0.0180099i	0.0934441	+	0.0180099i
$7$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$7$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$8$	−0.189466			−0.189466
$9$	−0.786053	+	0.618159i	−0.786053	+	0.618159i
$10$		0			0
$11$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$11$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$12$	0.648930	−	0.748905i	0.648930	−	0.748905i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	−0.0224357	−	0.0388598i	−0.0224357	−	0.0388598i
$15$		0			0
$16$	−0.486457	+	0.842568i	−0.486457	+	0.842568i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.0135432	−	0.0941952i	−0.0135432	−	0.0941952i
$19$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$19$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$20$		0			0
$21$	0.462997	+	0.0892353i	0.462997	+	0.0892353i
$22$	−0.0552004	−	0.0956100i	−0.0552004	−	0.0956100i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	0.0619682	+	0.179045i	0.0619682	+	0.179045i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	0.841254	+	0.540641i	0.841254	+	0.540641i
$28$	−0.467248			−0.467248
$29$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$29$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$30$		0			0
$31$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$31$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$32$	−0.141026	−	0.244264i	−0.141026	−	0.244264i
$33$	1.13915	+	0.219553i	1.13915	+	0.219553i
$34$		0			0
$35$		0			0
$36$	−0.919960	−	0.368297i	−0.919960	−	0.368297i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.0883470	+	0.153022i	−0.0883470	+	0.153022i
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$	−0.0293845	+	0.0339116i	−0.0293845	+	0.0339116i
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−1.14961			−1.14961
$45$		0			0
$46$		0			0
$47$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$47$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$48$	0.955332	+	0.184125i	0.955332	+	0.184125i
$49$	0.388835	+	0.673483i	0.388835	+	0.673483i
$50$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.0845850	+	0.0436066i	−0.0845850	+	0.0436066i
$55$		0			0
$56$	0.0446683	−	0.0773677i	0.0446683	−	0.0773677i
$57$	−0.607279	−	1.75462i	−0.607279	−	1.75462i
$58$	−0.0688733	−	0.119292i	−0.0688733	−	0.119292i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$61$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$62$	0.00905615			0.00905615
$63$	−0.0671040	−	0.466718i	−0.0671040	−	0.466718i
$64$	−0.946072			−0.946072
$65$		0			0
$66$	−0.0722972	+	0.0834354i	−0.0722972	+	0.0834354i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.148930	−	0.117120i	0.148930	−	0.117120i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.981929	+	0.189251i	0.981929	+	0.189251i
$76$	0.919960	+	1.59342i	0.919960	+	1.59342i
$77$	−0.273507	−	0.473728i	−0.273507	−	0.473728i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	0.235759	−	0.971812i	0.235759	−	0.971812i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$	0.152822	+	0.441549i	0.152822	+	0.441549i
$85$		0			0
$86$		0			0
$87$	1.42131	+	0.273935i	1.42131	+	0.273935i
$88$	0.109901	−	0.190354i	0.109901	−	0.190354i
$89$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$89$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.0623191	+	0.0719200i	−0.0623191	+	0.0719200i
$94$	0.0748038	+	0.129564i	0.0748038	+	0.129564i
$95$		0			0
$96$	−0.184705	+	0.213161i	−0.184705	+	0.213161i
$97$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$97$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$98$	−0.0740061			−0.0740061
$99$	−0.165101	−	1.14831i	−0.165101	−	1.14831i
$100$	−0.990944			−0.990944
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$107$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$108$	−0.0471510	+	0.989821i	−0.0471510	+	0.989821i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−0.229373	−	0.397286i	−0.229373	−	0.397286i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$	0.173501	+	0.0334396i	0.173501	+	0.0334396i
$115$		0			0
$116$	−1.43436			−1.43436
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.172932	−	0.299527i	−0.172932	−	0.299527i
$122$	0.0623191	+	0.107940i	0.0623191	+	0.107940i
$123$		0			0
$124$	0.0471510	−	0.0816679i	0.0471510	−	0.0816679i
$125$		0			0
$126$	0.0416572	+	0.0166770i	0.0416572	+	0.0166770i
$127$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$127$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$128$	0.186042	−	0.322234i	0.186042	−	0.322234i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	0.376000	+	1.08638i	0.376000	+	1.08638i
$133$	−0.437742	+	0.758192i	−0.437742	+	0.758192i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$137$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$	−1.54370	−	0.297523i	−1.54370	−	0.297523i
$142$		0			0
$143$		0			0
$144$	−0.138460	−	0.963011i	−0.138460	−	0.963011i
$145$		0			0
$146$		0			0
$147$	0.509266	−	0.587724i	0.509266	−	0.587724i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	−0.0623191	+	0.0719200i	−0.0623191	+	0.0719200i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−0.351788			−0.351788
$153$		0			0
$154$	0.0520560			0.0520560
$155$		0			0
$156$		0			0
$157$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$157$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.0688733	+	0.0656706i	0.0688733	+	0.0656706i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$168$	−0.0877221	−	0.0169070i	−0.0877221	−	0.0169070i
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	−1.45949	+	1.14776i	−1.45949	+	1.14776i
$172$		0			0
$173$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$173$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$174$	−0.0902048	+	0.104102i	−0.0902048	+	0.104102i
$175$	−0.235759	−	0.408346i	−0.235759	−	0.408346i
$176$	−0.564345	−	0.977475i	−0.564345	−	0.977475i
$177$		0			0
$178$	0.0311250	−	0.0539102i	0.0311250	−	0.0539102i
$179$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$179$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$180$		0			0
$181$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$181$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$182$		0			0
$183$	−1.28605	−	0.247866i	−1.28605	−	0.247866i
$184$		0			0
$185$		0			0
$186$	−0.00296198	−	0.00855807i	−0.00296198	−	0.00855807i
$187$		0			0
$188$	1.55787			1.55787
$189$	−0.419102	+	0.216062i	−0.419102	+	0.216062i
$190$		0			0
$191$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$191$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$192$	0.309430	+	0.894039i	0.309430	+	0.894039i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	−0.0395325	−	0.0684723i	−0.0395325	−	0.0684723i
$195$		0			0
$196$	−0.385314	+	0.667384i	−0.385314	+	0.667384i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.102493	+	0.0410319i	0.102493	+	0.0410319i
$199$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$199$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$200$	0.0947329	−	0.164082i	0.0947329	−	0.164082i
$201$		0			0
$202$		0			0
$203$	−0.341254	−	0.591068i	−0.341254	−	0.591068i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.07701	+	1.86544i	−1.07701	+	1.86544i
$210$		0			0
$211$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$211$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$212$		0			0
$213$		0			0
$214$	−0.0224357	+	0.0388598i	−0.0224357	+	0.0388598i
$215$		0			0
$216$	−0.159389	−	0.102433i	−0.159389	−	0.102433i
$217$	0.0448714			0.0448714
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$223$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$224$	0.132993			0.132993
$225$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	1.20489	−	1.39052i	1.20489	−	1.39052i
$229$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$229$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$230$		0			0
$231$	−0.358218	+	0.413406i	−0.358218	+	0.413406i
$232$	0.137123	−	0.237504i	0.137123	−	0.237504i
$233$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$233$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$239$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	0.0329138			0.0329138
$243$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$244$	1.29786			1.29786
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.00901515	+	0.0156147i	0.00901515	+	0.0156147i
$249$		0			0
$250$		0			0
$251$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$251$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$252$	0.367282	−	0.288833i	0.367282	−	0.288833i
$253$		0			0
$254$	−0.0552004	+	0.0956100i	−0.0552004	+	0.0956100i
$255$		0			0
$256$	−0.455332	−	0.788658i	−0.455332	−	0.788658i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.205996	−	1.43273i	−0.205996	−	1.43273i
$262$		0			0
$263$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$263$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$264$	−0.215830	−	0.0415978i	−0.215830	−	0.0415978i
$265$		0			0
$266$	−0.0416572	−	0.0721524i	−0.0416572	−	0.0721524i
$267$	0.213947	+	0.618159i	0.213947	+	0.618159i
$268$		0			0
$269$		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			$\pi$
$272$		0			0
$273$		0			0
$274$	0.0947329	+	0.164082i	0.0947329	+	0.164082i
$275$	−0.580057	−	1.00469i	−0.580057	−	1.00469i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	0.0883470	+	0.0353688i	0.0883470	+	0.0353688i
$280$		0			0
$281$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$281$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$282$	0.0979722	−	0.113066i	0.0979722	−	0.113066i
$283$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$283$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.261848	+	0.104828i	0.261848	+	0.104828i
$289$	1.00000			1.00000
$290$		0			0
$291$	0.815816	+	0.157236i	0.815816	+	0.157236i
$292$		0			0
$293$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$293$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$294$	0.0242050	+	0.0699359i	0.0242050	+	0.0699359i
$295$		0			0
$296$		0			0
$297$	−1.03115	+	0.531595i	−1.03115	+	0.531595i
$298$		0			0
$299$		0			0
$300$	0.324106	+	0.936443i	0.324106	+	0.936443i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.903222	+	1.56443i	−0.903222	+	1.56443i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	0.271030	−	0.469438i	0.271030	−	0.469438i
$309$		0			0
$310$		0			0
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	0.160114			0.160114
$315$		0			0
$316$		0			0
$317$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$317$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$318$		0			0
$319$	−0.839614	−	1.45425i	−0.839614	−	1.45425i
$320$		0			0
$321$	−0.154218	−	0.445585i	−0.154218	−	0.445585i
$322$		0			0
$323$		0			0
$324$	0.950804	−	0.279181i	0.950804	−	0.279181i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.370638	+	0.641964i	0.370638	+	0.641964i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	0.0951638			0.0951638
$335$		0			0
$336$	−0.300415	+	0.346697i	−0.300415	+	0.346697i
$337$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$337$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$338$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i
$339$		0			0
$340$		0			0
$341$	0.110401			0.110401
$342$	−0.0251462	−	0.174896i	−0.0251462	−	0.174896i
$343$	−0.838204			−0.838204
$344$		0			0
$345$		0			0
$346$	−0.0395325	−	0.0684723i	−0.0395325	−	0.0684723i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	0.469133	+	1.35547i	0.469133	+	1.35547i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$	0.0448714			0.0448714
$351$		0			0
$352$	0.327212			0.327212
$353$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$353$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$354$		0			0
$355$		0			0
$356$	−0.324106	−	0.561368i	−0.324106	−	0.561368i
$357$		0			0
$358$	−0.0883470	+	0.153022i	−0.0883470	+	0.153022i
$359$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$359$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$360$		0			0
$361$	2.44747			2.44747
$362$	0.0748038	−	0.129564i	0.0748038	−	0.129564i
$363$	−0.226493	+	0.261387i	−0.226493	+	0.261387i
$364$		0			0
$365$		0			0
$366$	0.0816206	−	0.0941952i	0.0816206	−	0.0941952i
$367$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$367$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.0925979	−	0.0178468i	−0.0925979	−	0.0178468i
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	−0.148930	+	0.257955i	−0.148930	+	0.257955i
$377$		0			0
$378$	0.00213507	−	0.0448206i	0.00213507	−	0.0448206i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.379436	−	1.09631i	−0.379436	−	1.09631i
$382$	0.0845850	+	0.146505i	0.0845850	+	0.146505i
$383$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$383$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$384$	−0.365360	−	0.0704173i	−0.365360	−	0.0704173i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.823306			−0.823306
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−0.0736710	−	0.127602i	−0.0736710	−	0.127602i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.903653	−	0.710640i	0.903653	−	0.710640i
$397$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$397$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$398$	−0.0395325	+	0.0684723i	−0.0395325	+	0.0684723i
$399$	0.859663	+	0.165686i	0.859663	+	0.165686i
$400$	−0.486457	−	0.842568i	−0.486457	−	0.842568i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.0649500			0.0649500
$407$		0			0
$408$		0			0
$409$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$409$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$410$		0			0
$411$	−1.95496	−	0.376789i	−1.95496	−	0.376789i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−0.102493	−	0.177522i	−0.102493	−	0.177522i
$419$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$419$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$420$		0			0
$421$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$421$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$422$	0.160114			0.160114
$423$	0.223734	+	1.55610i	0.223734	+	1.55610i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.308779	+	0.534820i	0.308779	+	0.534820i
$428$	0.233624	+	0.404648i	0.233624	+	0.404648i
$429$		0			0
$430$		0			0
$431$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$431$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$432$	−0.864760	+	0.445815i	−0.864760	+	0.445815i
$433$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$433$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$434$	−0.00213507	+	0.00369805i	−0.00213507	+	0.00369805i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.721965	−	0.289031i	−0.721965	−	0.289031i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$	0.0311250	+	0.0539102i	0.0311250	+	0.0539102i
$447$		0			0
$448$	0.223045	−	0.386325i	0.223045	−	0.386325i
$449$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$449$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$450$	0.0883470	+	0.0353688i	0.0883470	+	0.0353688i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	0.115059	+	0.332440i	0.115059	+	0.332440i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	0.137747			0.137747
$459$		0			0
$460$		0			0
$461$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$461$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$462$	−0.0170258	−	0.0491929i	−0.0170258	−	0.0491929i
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	−0.704131	−	1.21959i	−0.704131	−	1.21959i
$465$		0			0
$466$	0.0845850	−	0.146505i	0.0845850	−	0.146505i
$467$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$467$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.928368	+	1.60798i	−0.928368	+	1.60798i
$476$		0			0
$477$		0			0
$478$	−0.0622501			−0.0622501
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.171366	−	0.296815i	0.171366	−	0.296815i
$485$		0			0
$486$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−0.124074	+	0.214902i	−0.124074	+	0.214902i
$489$		0			0
$490$		0			0
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.0925862			0.0925862
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$502$	0.0913090	−	0.158152i	0.0913090	−	0.158152i
$503$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$503$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$504$	0.0127139	+	0.0884272i	0.0127139	+	0.0884272i
$505$		0			0
$506$		0			0
$507$	0.981929	+	0.189251i	0.981929	+	0.189251i
$508$	0.574804	+	0.995589i	0.574804	+	0.995589i
$509$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$509$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$		0			0
$512$	0.458746			0.458746
$513$	1.56199	+	1.00383i	1.56199	+	1.00383i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	0.911911	+	1.57948i	0.911911	+	1.57948i
$518$		0			0
$519$	0.815816	+	0.157236i	0.815816	+	0.157236i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	0.127880	+	0.0511952i	0.127880	+	0.0511952i
$523$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$523$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$524$		0			0
$525$	−0.308779	+	0.356349i	−0.308779	+	0.356349i
$526$	0.0623191	+	0.107940i	0.0623191	+	0.107940i
$527$		0			0
$528$	−0.739135	+	0.853007i	−0.739135	+	0.853007i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−0.867556			−0.867556
$533$		0			0
$534$	−0.0611251	−	0.0117809i	−0.0611251	−	0.0117809i
$535$		0			0
$536$		0			0
$537$	−0.607279	−	1.75462i	−0.607279	−	1.75462i
$538$		0			0
$539$	−0.902187			−0.902187
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	0.514186	+	1.48564i	0.514186	+	1.48564i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	1.97291			1.97291
$549$	0.186393	+	1.29639i	0.186393	+	1.29639i
$550$	0.110401			0.110401
$551$	−1.34378	+	2.32750i	−1.34378	+	2.32750i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$557$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$558$	−0.00711862	+	0.00559814i	−0.00711862	+	0.00559814i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0934441	−	0.161850i	−0.0934441	−	0.161850i
$563$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$563$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$564$	−0.509529	−	1.47219i	−0.509529	−	1.47219i
$565$		0			0
$566$	0.0790650			0.0790650
$567$	0.341254	+	0.325385i	0.341254	+	0.325385i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$	−1.74555	−	0.336426i	−1.74555	−	0.336426i
$574$		0			0
$575$		0			0
$576$	0.743663	−	0.584823i	0.743663	−	0.584823i
$577$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$577$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$578$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−0.0517765	+	0.0597533i	−0.0517765	+	0.0597533i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.169170			−0.169170
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	0.756702	+	0.145842i	0.756702	+	0.145842i
$589$	−0.0883470	−	0.153022i	−0.0883470	−	0.153022i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	0.00525309	−	0.110276i	0.00525309	−	0.110276i
$595$		0			0
$596$		0			0
$597$	−0.271738	−	0.785135i	−0.271738	−	0.785135i
$598$		0			0
$599$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$599$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$600$	−0.186042	−	0.0358566i	−0.186042	−	0.0358566i
$601$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$601$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−0.261848	−	0.453534i	−0.261848	−	0.453534i
$609$	−0.446947	+	0.515804i	−0.446947	+	0.515804i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$613$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$614$		0			0
$615$		0			0
$616$	0.0518203	+	0.0897553i	0.0518203	+	0.0897553i
$617$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$617$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$	−0.0951638			−0.0951638
$623$	0.154218	−	0.267114i	0.154218	−	0.267114i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	2.11510	+	0.407652i	2.11510	+	0.407652i
$628$	0.833635	−	1.44390i	0.833635	−	1.44390i
$629$		0			0
$630$		0			0
$631$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$631$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$632$		0			0
$633$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$634$	−0.0934441	−	0.161850i	−0.0934441	−	0.161850i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.159802			0.159802
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	0.0440606	+	0.00849198i	0.0440606	+	0.00849198i
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.0446683	+	0.184125i	−0.0446683	+	0.184125i
$649$		0			0
$650$		0			0
$651$	−0.0146760	−	0.0424036i	−0.0146760	−	0.0424036i
$652$		0			0
$653$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$653$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.0705427			−0.0705427
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.495472	−	0.858183i	0.495472	−	0.858183i
$669$	−0.642315	−	0.123796i	−0.642315	−	0.123796i
$670$		0			0
$671$	0.759713	+	1.31586i	0.759713	+	1.31586i
$672$	−0.0434976	−	0.125678i	−0.0434976	−	0.125678i
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	−0.149608			−0.149608
$675$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$676$	−0.990944			−0.990944
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$678$		0			0
$679$	−0.195876	−	0.339266i	−0.195876	−	0.339266i
$680$		0			0
$681$		0			0
$682$	−0.00525309	+	0.00909861i	−0.00525309	+	0.00909861i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−1.70812	−	0.683830i	−1.70812	−	0.683830i
$685$		0			0
$686$	0.0398833	−	0.0690800i	0.0398833	−	0.0690800i
$687$	−0.947890	+	1.09392i	−0.947890	+	1.09392i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	−0.823306			−0.823306
$693$	0.507831	+	0.203305i	0.507831	+	0.203305i
$694$		0			0
$695$		0			0
$696$	−0.269290	−	0.0519014i	−0.269290	−	0.0519014i
$697$		0			0
$698$		0			0
$699$	0.581419	+	1.67990i	0.581419	+	1.67990i
$700$	0.233624	−	0.404648i	0.233624	−	0.404648i
$701$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$701$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$702$		0			0
$703$		0			0
$704$	0.548776	−	0.950508i	0.548776	−	0.950508i
$705$		0			0
$706$	0.0913090	+	0.158152i	0.0913090	+	0.158152i
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	0.123936			0.123936
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.919960	+	1.59342i	0.919960	+	1.59342i
$717$	0.428368	−	0.494363i	0.428368	−	0.494363i
$718$	−0.0800569	+	0.138663i	−0.0800569	+	0.138663i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−0.116455	+	0.201706i	−0.116455	+	0.201706i
$723$		0			0
$724$	−0.778934	−	1.34915i	−0.778934	−	1.34915i
$725$	−0.723734	−	1.25354i	−0.723734	−	1.25354i
$726$	−0.0107650	−	0.0311035i	−0.0107650	−	0.0311035i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	0.415415	+	0.909632i	0.415415	+	0.909632i
$730$		0			0
$731$		0			0
$732$	−0.424489	−	1.22648i	−0.424489	−	1.22648i
$733$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$733$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$734$	0.0748038	+	0.129564i	0.0748038	+	0.129564i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0.0118073	−	0.0136264i	0.0118073	−	0.0136264i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.111165	+	0.192543i	−0.111165	+	0.192543i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	0.764762	+	1.32461i	0.764762	+	1.32461i
$753$	0.627639	+	1.81344i	0.627639	+	1.81344i
$754$		0			0
$755$		0			0
$756$	−0.393074	−	0.252613i	−0.393074	−	0.252613i
$757$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$757$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$761$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$762$	0.108406	+	0.0208935i	0.108406	+	0.0208935i
$763$		0			0
$764$	1.76157			1.76157
$765$		0			0
$766$	−0.182618			−0.182618
$767$		0			0
$768$	−0.596358	+	0.688234i	−0.596358	+	0.688234i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	0.0951638			0.0951638
$776$	0.0787070	−	0.136324i	0.0787070	−	0.136324i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.28656	+	0.663268i	−1.28656	+	0.663268i
$784$	−0.756607			−0.756607
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$	−1.28605	−	0.247866i	−1.28605	−	0.247866i
$790$		0			0
$791$		0			0
$792$	0.0312811	+	0.217565i	0.0312811	+	0.217565i
$793$		0			0
$794$	−0.0934441	+	0.161850i	−0.0934441	+	0.161850i
$795$		0			0
$796$	0.411653	+	0.713004i	0.411653	+	0.713004i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$	−0.0545593	+	0.0629648i	−0.0545593	+	0.0629648i
$799$		0			0
$800$	0.282052			0.282052
$801$	0.514186	−	0.404360i	0.514186	−	0.404360i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$809$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	0.338163	−	0.585716i	0.338163	−	0.585716i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.110401			0.110401
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	0.124074	−	0.143189i	0.124074	−	0.143189i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−0.759713	+	0.876756i	−0.759713	+	0.876756i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−2.13452			−2.13452
$837$	0.00452808	−	0.0950560i	0.00452808	−	0.0950560i
$838$	−0.169170			−0.169170
$839$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$839$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$840$		0			0
$841$	−0.547582	−	0.948440i	−0.547582	−	0.948440i
$842$	0.0913090	+	0.158152i	0.0913090	+	0.158152i
$843$	1.92837	+	0.371662i	1.92837	+	0.371662i
$844$	0.833635	−	1.44390i	0.833635	−	1.44390i
$845$		0			0
$846$	−0.138891	−	0.0556035i	−0.138891	−	0.0556035i
$847$	0.163081			0.163081
$848$		0			0
$849$	−0.544078	+	0.627899i	−0.544078	+	0.627899i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$853$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$854$	−0.0587691			−0.0587691
$855$		0			0
$856$	−0.0893365			−0.0893365
$857$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$857$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$858$		0			0
$859$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$859$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$860$		0			0
$861$		0			0
$862$	−0.0934441	+	0.161850i	−0.0934441	+	0.161850i
$863$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$863$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$864$	0.0134206	−	0.281733i	0.0134206	−	0.281733i
$865$		0			0
$866$	0.0135432	−	0.0234576i	0.0135432	−	0.0234576i
$867$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$868$	0.0222325	+	0.0385079i	0.0222325	+	0.0385079i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$877$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$878$		0			0
$879$	1.16413	−	1.34347i	1.16413	−	1.34347i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.0581728	−	0.0457476i	0.0581728	−	0.0457476i
$883$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$883$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−0.273507	+	0.473728i	−0.273507	+	0.473728i
$890$		0			0
$891$	0.839614	+	0.800570i	0.839614	+	0.800570i
$892$	0.648212			0.648212
$893$	1.45949	−	2.52792i	1.45949	−	2.52792i
$894$		0			0
$895$		0			0
$896$	0.0877221	+	0.151939i	0.0877221	+	0.151939i
$897$		0			0
$898$	0.0135432	−	0.0234576i	0.0135432	−	0.0234576i
$899$	0.137747			0.137747
$900$	0.778934	−	0.612561i	0.778934	−	0.612561i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$907$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$911$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$912$	1.77380	+	0.341872i	1.77380	+	0.341872i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.717180	−	1.24219i	0.717180	−	1.24219i
$917$		0			0
$918$		0			0
$919$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$919$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$920$		0			0
$921$		0			0
$922$	0.0135432	+	0.0234576i	0.0135432	+	0.0234576i
$923$		0			0
$924$	−0.532265	−	0.102586i	−0.532265	−	0.102586i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	0.408261			0.408261
$929$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$929$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$930$		0			0
$931$	0.721965	+	1.25048i	0.721965	+	1.25048i
$932$	−0.880786	−	1.52557i	−0.880786	−	1.52557i
$933$	0.654861	−	0.755750i	0.654861	−	0.755750i
$934$	−0.0688733	+	0.119292i	−0.0688733	+	0.119292i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$	−0.0523681	−	0.151308i	−0.0523681	−	0.151308i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$947$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$948$		0			0
$949$		0			0
$950$	−0.0883470	−	0.153022i	−0.0883470	−	0.153022i
$951$	1.92837	+	0.371662i	1.92837	+	0.371662i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.324106	+	0.561368i	−0.324106	+	0.561368i
$957$	−1.09966	+	1.26908i	−1.09966	+	1.26908i
$958$		0			0
$959$	0.469383	+	0.812995i	0.469383	+	0.812995i
$960$		0			0
$961$	0.495472	−	0.858183i	0.495472	−	0.858183i
$962$		0			0
$963$	−0.370638	+	0.291473i	−0.370638	+	0.291473i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$967$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$968$	0.0327647	+	0.0567502i	0.0327647	+	0.0567502i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−0.574804	−	0.807199i	−0.574804	−	0.807199i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.637123	+	1.10353i	0.637123	+	1.10353i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	0.379436	−	0.657203i	0.379436	−	0.657203i
$980$		0			0
$981$		0			0
$982$	0.190328			0.190328
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.485433	−	0.560219i	0.485433	−	0.560219i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.0134206	+	0.0232451i	−0.0134206	+	0.0232451i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$997$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.1.f.b.1168.6	yes	20
9.4	even	3	inner	1503.1.f.b.166.6	✓	20
167.166	odd	2	CM	1503.1.f.b.1168.6	yes	20
1503.166	odd	6	inner	1503.1.f.b.166.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.1.f.b.166.6	✓	20	9.4	even	3	inner
1503.1.f.b.166.6	✓	20	1503.166	odd	6	inner
1503.1.f.b.1168.6	yes	20	1.1	even	1	trivial
1503.1.f.b.1168.6	yes	20	167.166	odd	2	CM

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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1503.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0475819 + 0.0824143i) q^{2} +(-0.327068 - 0.945001i) q^{3} +(0.495472 + 0.858183i) q^{4} +(0.0934441 + 0.0180099i) q^{6} +(-0.235759 + 0.408346i) q^{7} -0.189466 q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)	\(q+(-0.0475819 + 0.0824143i) q^{2} +(-0.327068 - 0.945001i) q^{3} +(0.495472 + 0.858183i) q^{4} +(0.0934441 + 0.0180099i) q^{6} +(-0.235759 + 0.408346i) q^{7} -0.189466 q^{8} +(-0.786053 + 0.618159i) q^{9} +(-0.580057 + 1.00469i) q^{11} +(0.648930 - 0.748905i) q^{12} +(-0.0224357 - 0.0388598i) q^{14} +(-0.486457 + 0.842568i) q^{16} +(-0.0135432 - 0.0941952i) q^{18} +1.85674 q^{19} +(0.462997 + 0.0892353i) q^{21} +(-0.0552004 - 0.0956100i) q^{22} +(0.0619682 + 0.179045i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.841254 + 0.540641i) q^{27} -0.467248 q^{28} +(-0.723734 + 1.25354i) q^{29} +(-0.0475819 - 0.0824143i) q^{31} +(-0.141026 - 0.244264i) q^{32} +(1.13915 + 0.219553i) q^{33} +(-0.919960 - 0.368297i) q^{36} +(-0.0883470 + 0.153022i) q^{38} +(-0.0293845 + 0.0339116i) q^{42} -1.14961 q^{44} +(0.786053 - 1.36148i) q^{47} +(0.955332 + 0.184125i) q^{48} +(0.388835 + 0.673483i) q^{49} +(-0.0475819 - 0.0824143i) q^{50} +(-0.0845850 + 0.0436066i) q^{54} +(0.0446683 - 0.0773677i) q^{56} +(-0.607279 - 1.75462i) q^{57} +(-0.0688733 - 0.119292i) q^{58} +(0.654861 - 1.13425i) q^{61} +0.00905615 q^{62} +(-0.0671040 - 0.466718i) q^{63} -0.946072 q^{64} +(-0.0722972 + 0.0834354i) q^{66} +(0.148930 - 0.117120i) q^{72} +(0.981929 + 0.189251i) q^{75} +(0.919960 + 1.59342i) q^{76} +(-0.273507 - 0.473728i) q^{77} +(0.235759 - 0.971812i) q^{81} +(0.152822 + 0.441549i) q^{84} +(1.42131 + 0.273935i) q^{87} +(0.109901 - 0.190354i) q^{88} -0.654136 q^{89} +(-0.0623191 + 0.0719200i) q^{93} +(0.0748038 + 0.129564i) q^{94} +(-0.184705 + 0.213161i) q^{96} +(-0.415415 + 0.719520i) q^{97} -0.0740061 q^{98} +(-0.165101 - 1.14831i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

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