LMFDB - Embedded newform 1503.1.f.b.166.5

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			166.5
Root			$-0.786053 - 0.618159i$ of defining polynomial
Character	$\chi$	$=$	1503.166
Dual form			1503.1.f.b.1168.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.235759 - 0.408346i) q^{2} +(-0.995472 + 0.0950560i) q^{3} +(0.388835 - 0.673483i) q^{4} +(0.273507 + 0.384087i) q^{6} +(-0.928368 - 1.60798i) q^{7} -0.838204 q^{8} +(0.981929 - 0.189251i) q^{9} +O(q^{10})$	\(q+(-0.235759 - 0.408346i) q^{2} +(-0.995472 + 0.0950560i) q^{3} +(0.388835 - 0.673483i) q^{4} +(0.273507 + 0.384087i) q^{6} +(-0.928368 - 1.60798i) q^{7} -0.838204 q^{8} +(0.981929 - 0.189251i) q^{9} +(-0.0475819 - 0.0824143i) q^{11} +(-0.323056 + 0.707394i) q^{12} +(-0.437742 + 0.758192i) q^{14} +(-0.191221 - 0.331205i) q^{16} +(-0.308779 - 0.356349i) q^{18} -0.654136 q^{19} +(1.07701 + 1.51245i) q^{21} +(-0.0224357 + 0.0388598i) q^{22} +(0.834408 - 0.0796763i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-0.959493 + 0.281733i) q^{27} -1.44393 q^{28} +(0.786053 + 1.36148i) q^{29} +(-0.235759 + 0.408346i) q^{31} +(-0.509266 + 0.882075i) q^{32} +(0.0552004 + 0.0775182i) q^{33} +(0.254351 - 0.734900i) q^{36} +(0.154218 + 0.267114i) q^{38} +(0.363689 - 0.796368i) q^{42} -0.0740061 q^{44} +(-0.981929 - 1.70075i) q^{47} +(0.221839 + 0.311529i) q^{48} +(-1.22373 + 2.11957i) q^{49} +(-0.235759 + 0.408346i) q^{50} +(0.341254 + 0.325385i) q^{54} +(0.778161 + 1.34781i) q^{56} +(0.651174 - 0.0621796i) q^{57} +(0.370638 - 0.641964i) q^{58} +(-0.415415 - 0.719520i) q^{61} +0.222329 q^{62} +(-1.21590 - 1.40323i) q^{63} +0.0978132 q^{64} +(0.0186403 - 0.0408165i) q^{66} +(-0.823056 + 0.158631i) q^{72} +(0.580057 + 0.814576i) q^{75} +(-0.254351 + 0.440549i) q^{76} +(-0.0883470 + 0.153022i) q^{77} +(0.928368 - 0.371662i) q^{81} +(1.43739 - 0.137254i) q^{84} +(-0.911911 - 1.28060i) q^{87} +(0.0398833 + 0.0690800i) q^{88} -1.99094 q^{89} +(0.195876 - 0.428908i) q^{93} +(-0.462997 + 0.801934i) q^{94} +(0.423114 - 0.926490i) q^{96} +(-0.841254 - 1.45709i) q^{97} +1.15402 q^{98} +(-0.0623191 - 0.0719200i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$2$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$3$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$3$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$4$	0.388835	−	0.673483i	0.388835	−	0.673483i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	0.273507	+	0.384087i	0.273507	+	0.384087i
$7$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$7$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$8$	−0.838204			−0.838204
$9$	0.981929	−	0.189251i	0.981929	−	0.189251i
$10$		0			0
$11$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$11$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$12$	−0.323056	+	0.707394i	−0.323056	+	0.707394i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$	−0.437742	+	0.758192i	−0.437742	+	0.758192i
$15$		0			0
$16$	−0.191221	−	0.331205i	−0.191221	−	0.331205i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.308779	−	0.356349i	−0.308779	−	0.356349i
$19$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$19$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$20$		0			0
$21$	1.07701	+	1.51245i	1.07701	+	1.51245i
$22$	−0.0224357	+	0.0388598i	−0.0224357	+	0.0388598i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.834408	−	0.0796763i	0.834408	−	0.0796763i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$28$	−1.44393			−1.44393
$29$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$29$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$30$		0			0
$31$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$31$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$32$	−0.509266	+	0.882075i	−0.509266	+	0.882075i
$33$	0.0552004	+	0.0775182i	0.0552004	+	0.0775182i
$34$		0			0
$35$		0			0
$36$	0.254351	−	0.734900i	0.254351	−	0.734900i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	0.154218	+	0.267114i	0.154218	+	0.267114i
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$	0.363689	−	0.796368i	0.363689	−	0.796368i
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$	−0.0740061			−0.0740061
$45$		0			0
$46$		0			0
$47$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$47$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$48$	0.221839	+	0.311529i	0.221839	+	0.311529i
$49$	−1.22373	+	2.11957i	−1.22373	+	2.11957i
$50$	−0.235759	+	0.408346i	−0.235759	+	0.408346i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.341254	+	0.325385i	0.341254	+	0.325385i
$55$		0			0
$56$	0.778161	+	1.34781i	0.778161	+	1.34781i
$57$	0.651174	−	0.0621796i	0.651174	−	0.0621796i
$58$	0.370638	−	0.641964i	0.370638	−	0.641964i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$61$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$62$	0.222329			0.222329
$63$	−1.21590	−	1.40323i	−1.21590	−	1.40323i
$64$	0.0978132			0.0978132
$65$		0			0
$66$	0.0186403	−	0.0408165i	0.0186403	−	0.0408165i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.823056	+	0.158631i	−0.823056	+	0.158631i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.580057	+	0.814576i	0.580057	+	0.814576i
$76$	−0.254351	+	0.440549i	−0.254351	+	0.440549i
$77$	−0.0883470	+	0.153022i	−0.0883470	+	0.153022i
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	0.928368	−	0.371662i	0.928368	−	0.371662i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	1.43739	−	0.137254i	1.43739	−	0.137254i
$85$		0			0
$86$		0			0
$87$	−0.911911	−	1.28060i	−0.911911	−	1.28060i
$88$	0.0398833	+	0.0690800i	0.0398833	+	0.0690800i
$89$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$89$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.195876	−	0.428908i	0.195876	−	0.428908i
$94$	−0.462997	+	0.801934i	−0.462997	+	0.801934i
$95$		0			0
$96$	0.423114	−	0.926490i	0.423114	−	0.926490i
$97$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$97$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$98$	1.15402			1.15402
$99$	−0.0623191	−	0.0719200i	−0.0623191	−	0.0719200i
$100$	−0.777671			−0.777671
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$107$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$108$	−0.183343	+	0.755750i	−0.183343	+	0.755750i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−0.355048	+	0.614961i	−0.355048	+	0.614961i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$	−0.178911	−	0.251245i	−0.178911	−	0.251245i
$115$		0			0
$116$	1.22258			1.22258
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.495472	−	0.858183i	0.495472	−	0.858183i
$122$	−0.195876	+	0.339266i	−0.195876	+	0.339266i
$123$		0			0
$124$	0.183343	+	0.317559i	0.183343	+	0.317559i
$125$		0			0
$126$	−0.286343	+	0.827333i	−0.286343	+	0.827333i
$127$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$127$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$128$	0.486206	+	0.842133i	0.486206	+	0.842133i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	0.0736710	−	0.00703473i	0.0736710	−	0.00703473i
$133$	0.607279	+	1.05184i	0.607279	+	1.05184i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$137$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$	1.13915	+	1.59971i	1.13915	+	1.59971i
$142$		0			0
$143$		0			0
$144$	−0.250447	−	0.289031i	−0.250447	−	0.289031i
$145$		0			0
$146$		0			0
$147$	1.01671	−	2.22630i	1.01671	−	2.22630i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	0.195876	−	0.428908i	0.195876	−	0.428908i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	0.548299			0.548299
$153$		0			0
$154$	0.0833144			0.0833144
$155$		0			0
$156$		0			0
$157$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$157$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.370638	−	0.291473i	−0.370638	−	0.291473i
$163$		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
$168$	−0.902756	−	1.26774i	−0.902756	−	1.26774i
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	−0.642315	+	0.123796i	−0.642315	+	0.123796i
$172$		0			0
$173$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$173$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$174$	−0.307937	+	0.674289i	−0.307937	+	0.674289i
$175$	−0.928368	+	1.60798i	−0.928368	+	1.60798i
$176$	−0.0181974	+	0.0315188i	−0.0181974	+	0.0315188i
$177$		0			0
$178$	0.469383	+	0.812995i	0.469383	+	0.812995i
$179$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$179$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$180$		0			0
$181$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$181$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$182$		0			0
$183$	0.481929	+	0.676774i	0.481929	+	0.676774i
$184$		0			0
$185$		0			0
$186$	−0.221322	+	0.0211337i	−0.221322	+	0.0211337i
$187$		0			0
$188$	−1.52723			−1.52723
$189$	1.34378	+	1.28129i	1.34378	+	1.28129i
$190$		0			0
$191$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$191$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$192$	−0.0973703	+	0.00929774i	−0.0973703	+	0.00929774i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	−0.396666	+	0.687046i	−0.396666	+	0.687046i
$195$		0			0
$196$	0.951662	+	1.64833i	0.951662	+	1.64833i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.0146760	+	0.0424036i	−0.0146760	+	0.0424036i
$199$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$199$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$200$	0.419102	+	0.725906i	0.419102	+	0.725906i
$201$		0			0
$202$		0			0
$203$	1.45949	−	2.52792i	1.45949	−	2.52792i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.0311250	+	0.0539102i	0.0311250	+	0.0539102i
$210$		0			0
$211$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$211$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$212$		0			0
$213$		0			0
$214$	−0.437742	−	0.758192i	−0.437742	−	0.758192i
$215$		0			0
$216$	0.804250	−	0.236149i	0.804250	−	0.236149i
$217$	0.875484			0.875484
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$223$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$224$	1.89115			1.89115
$225$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	0.211323	−	0.462732i	0.211323	−	0.462732i
$229$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$229$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$230$		0			0
$231$	0.0734014	−	0.160727i	0.0734014	−	0.160727i
$232$	−0.658873	−	1.14120i	−0.658873	−	1.14120i
$233$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$233$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$239$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	−0.467248			−0.467248
$243$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$244$	−0.646112			−0.646112
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.197614	−	0.342277i	0.197614	−	0.342277i
$249$		0			0
$250$		0			0
$251$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$251$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$252$	−1.41784	+	0.273265i	−1.41784	+	0.273265i
$253$		0			0
$254$	−0.0224357	−	0.0388598i	−0.0224357	−	0.0388598i
$255$		0			0
$256$	0.278161	−	0.481790i	0.278161	−	0.481790i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.02951	+	1.18812i	1.02951	+	1.18812i
$262$		0			0
$263$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$263$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$264$	−0.0462692	−	0.0649760i	−0.0462692	−	0.0649760i
$265$		0			0
$266$	0.286343	−	0.495960i	0.286343	−	0.495960i
$267$	1.98193	−	0.189251i	1.98193	−	0.189251i
$268$		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
$274$	0.419102	−	0.725906i	0.419102	−	0.725906i
$275$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$	−0.154218	+	0.445585i	−0.154218	+	0.445585i
$280$		0			0
$281$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$281$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$282$	0.384672	−	0.842314i	0.384672	−	0.842314i
$283$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$283$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.333129	+	0.962514i	−0.333129	+	0.962514i
$289$	1.00000			1.00000
$290$		0			0
$291$	0.975950	+	1.37053i	0.975950	+	1.37053i
$292$		0			0
$293$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$293$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$294$	−1.14880	+	0.109697i	−1.14880	+	0.109697i
$295$		0			0
$296$		0			0
$297$	0.0688733	+	0.0656706i	0.0688733	+	0.0656706i
$298$		0			0
$299$		0			0
$300$	0.774150	−	0.0739223i	0.774150	−	0.0739223i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.125085	+	0.216653i	0.125085	+	0.216653i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	0.0687049	+	0.119000i	0.0687049	+	0.119000i
$309$		0			0
$310$		0			0
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	−0.904836			−0.904836
$315$		0			0
$316$		0			0
$317$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$317$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$318$		0			0
$319$	0.0748038	−	0.129564i	0.0748038	−	0.129564i
$320$		0			0
$321$	−1.84833	+	0.176494i	−1.84833	+	0.176494i
$322$		0			0
$323$		0			0
$324$	0.110674	−	0.769755i	0.110674	−	0.769755i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.82318	+	3.15784i	−1.82318	+	3.15784i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	0.471518			0.471518
$335$		0			0
$336$	0.294984	−	0.645926i	0.294984	−	0.645926i
$337$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$337$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$338$	−0.235759	+	0.408346i	−0.235759	+	0.408346i
$339$		0			0
$340$		0			0
$341$	0.0448714			0.0448714
$342$	0.201983	+	0.233101i	0.201983	+	0.233101i
$343$	2.68757			2.68757
$344$		0			0
$345$		0			0
$346$	−0.396666	+	0.687046i	−0.396666	+	0.687046i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−1.21705	+	0.116214i	−1.21705	+	0.116214i
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$	0.875484			0.875484
$351$		0			0
$352$	0.0969274			0.0969274
$353$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$353$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$354$		0			0
$355$		0			0
$356$	−0.774150	+	1.34087i	−0.774150	+	1.34087i
$357$		0			0
$358$	0.154218	+	0.267114i	0.154218	+	0.267114i
$359$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$359$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	−0.572106			−0.572106
$362$	−0.462997	−	0.801934i	−0.462997	−	0.801934i
$363$	−0.411653	+	0.901394i	−0.411653	+	0.901394i
$364$		0			0
$365$		0			0
$366$	0.162739	−	0.356349i	0.162739	−	0.356349i
$367$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$367$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.212699	−	0.298693i	−0.212699	−	0.298693i
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$	0.823056	+	1.42558i	0.823056	+	1.42558i
$377$		0			0
$378$	0.206403	−	0.850806i	0.206403	−	0.850806i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.0947329	+	0.00904590i	−0.0947329	+	0.00904590i
$382$	−0.341254	+	0.591068i	−0.341254	+	0.591068i
$383$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$383$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$384$	−0.564054	−	0.792103i	−0.564054	−	0.792103i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.30844			−1.30844
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	1.02574	−	1.77663i	1.02574	−	1.77663i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.0726688	+	0.0140058i	−0.0726688	+	0.0140058i
$397$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$397$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$398$	−0.396666	−	0.687046i	−0.396666	−	0.687046i
$399$	−0.704513	−	0.989349i	−0.704513	−	0.989349i
$400$	−0.191221	+	0.331205i	−0.191221	+	0.331205i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−1.37635			−1.37635
$407$		0			0
$408$		0			0
$409$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$409$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$410$		0			0
$411$	−1.03115	−	1.44805i	−1.03115	−	1.44805i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	0.0146760	−	0.0254196i	0.0146760	−	0.0254196i
$419$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$419$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$420$		0			0
$421$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$421$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$422$	−0.904836			−0.904836
$423$	−1.28605	−	1.48418i	−1.28605	−	1.48418i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.771316	+	1.33596i	−0.771316	+	1.33596i
$428$	0.721965	−	1.25048i	0.721965	−	1.25048i
$429$		0			0
$430$		0			0
$431$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$431$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$432$	0.276787	+	0.263916i	0.276787	+	0.263916i
$433$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$433$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$434$	−0.206403	−	0.357501i	−0.206403	−	0.357501i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−0.800488	+	2.31286i	−0.800488	+	2.31286i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	0.469383	−	0.812995i	0.469383	−	0.812995i
$447$		0			0
$448$	−0.0908067	−	0.157282i	−0.0908067	−	0.157282i
$449$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$449$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$450$	−0.154218	+	0.445585i	−0.154218	+	0.445585i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−0.545816	+	0.0521191i	−0.545816	+	0.0521191i
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−0.741276			−0.741276
$459$		0			0
$460$		0			0
$461$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$461$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$462$	−0.0829372	+	0.00791954i	−0.0829372	+	0.00791954i
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$	0.300620	−	0.520690i	0.300620	−	0.520690i
$465$		0			0
$466$	−0.341254	−	0.591068i	−0.341254	−	0.591068i
$467$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$467$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.327068	+	0.566498i	0.327068	+	0.566498i
$476$		0			0
$477$		0			0
$478$	−0.938766			−0.938766
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.385314	−	0.667384i	−0.385314	−	0.667384i
$485$		0			0
$486$	0.396666	+	0.254922i	0.396666	+	0.254922i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	0.348202	+	0.603104i	0.348202	+	0.603104i
$489$		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
$496$	0.180329			0.180329
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$	0.415415	−	0.909632i	0.415415	−	0.909632i
$502$	0.0671040	+	0.116228i	0.0671040	+	0.116228i
$503$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$503$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$504$	1.01917	+	1.17619i	1.01917	+	1.17619i
$505$		0			0
$506$		0			0
$507$	0.580057	+	0.814576i	0.580057	+	0.814576i
$508$	0.0370031	−	0.0640912i	0.0370031	−	0.0640912i
$509$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$509$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$510$		0			0
$511$		0			0
$512$	0.710095			0.710095
$513$	0.627639	−	0.184291i	0.627639	−	0.184291i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.0934441	+	0.161850i	−0.0934441	+	0.161850i
$518$		0			0
$519$	0.975950	+	1.37053i	0.975950	+	1.37053i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	0.242448	−	0.700507i	0.242448	−	0.700507i
$523$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$523$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$524$		0			0
$525$	0.771316	−	1.68895i	0.771316	−	1.68895i
$526$	−0.195876	+	0.339266i	−0.195876	+	0.339266i
$527$		0			0
$528$	0.0151189	−	0.0331058i	0.0151189	−	0.0331058i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	0.944526			0.944526
$533$		0			0
$534$	−0.544537	−	0.764696i	−0.544537	−	0.764696i
$535$		0			0
$536$		0			0
$537$	0.651174	−	0.0621796i	0.651174	−	0.0621796i
$538$		0			0
$539$	0.232910			0.232910
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−1.95496	+	0.186677i	−1.95496	+	0.186677i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$	1.38244			1.38244
$549$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$550$	0.0448714			0.0448714
$551$	−0.514186	−	0.890596i	−0.514186	−	0.890596i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$557$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$558$	0.218311	−	0.0420761i	0.218311	−	0.0420761i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.273507	+	0.473728i	−0.273507	+	0.473728i
$563$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$563$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$564$	1.52032	−	0.145173i	1.52032	−	0.145173i
$565$		0			0
$566$	0.793332			0.793332
$567$	−1.45949	−	1.14776i	−1.45949	−	1.14776i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	0.839614	+	1.17907i	0.839614	+	1.17907i
$574$		0			0
$575$		0			0
$576$	0.0960456	−	0.0185113i	0.0960456	−	0.0185113i
$577$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$577$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$578$	−0.235759	−	0.408346i	−0.235759	−	0.408346i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	0.329562	−	0.721640i	0.329562	−	0.721640i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.682507			0.682507
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$	−1.10404	−	1.55040i	−1.10404	−	1.55040i
$589$	0.154218	−	0.267114i	0.154218	−	0.267114i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	0.0105788	−	0.0436066i	0.0105788	−	0.0436066i
$595$		0			0
$596$		0			0
$597$	−1.67489	+	0.159932i	−1.67489	+	0.159932i
$598$		0			0
$599$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$599$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$600$	−0.486206	−	0.682780i	−0.486206	−	0.682780i
$601$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$601$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	0.333129	−	0.576997i	0.333129	−	0.576997i
$609$	−1.21259	+	2.65520i	−1.21259	+	2.65520i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$613$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$614$		0			0
$615$		0			0
$616$	0.0740528	−	0.128263i	0.0740528	−	0.128263i
$617$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$617$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$	−0.471518			−0.471518
$623$	1.84833	+	3.20140i	1.84833	+	3.20140i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$	−0.0361086	−	0.0507074i	−0.0361086	−	0.0507074i
$628$	−0.746170	−	1.29240i	−0.746170	−	1.29240i
$629$		0			0
$630$		0			0
$631$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$631$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$632$		0			0
$633$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$634$	−0.273507	+	0.473728i	−0.273507	+	0.473728i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.0705427			−0.0705427
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	0.507831	+	0.713148i	0.507831	+	0.713148i
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.778161	+	0.311529i	−0.778161	+	0.311529i
$649$		0			0
$650$		0			0
$651$	−0.871520	+	0.0832201i	−0.871520	+	0.0832201i
$652$		0			0
$653$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$653$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	1.71933			1.71933
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.388835	+	0.673483i	0.388835	+	0.673483i
$669$	−1.15486	−	1.62177i	−1.15486	−	1.62177i
$670$		0			0
$671$	−0.0395325	+	0.0684723i	−0.0395325	+	0.0684723i
$672$	−1.88258	+	0.179765i	−1.88258	+	0.179765i
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	0.925994			0.925994
$675$	0.723734	+	0.690079i	0.723734	+	0.690079i
$676$	−0.777671			−0.777671
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−1.56199	+	2.70544i	−1.56199	+	2.70544i
$680$		0			0
$681$		0			0
$682$	−0.0105788	−	0.0183231i	−0.0105788	−	0.0183231i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−0.166380	+	0.480724i	−0.166380	+	0.480724i
$685$		0			0
$686$	−0.633618	−	1.09746i	−0.633618	−	1.09746i
$687$	−0.653077	+	1.43004i	−0.653077	+	1.43004i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−1.30844			−1.30844
$693$	−0.0577910	+	0.166976i	−0.0577910	+	0.166976i
$694$		0			0
$695$		0			0
$696$	0.764367	+	1.07340i	0.764367	+	1.07340i
$697$		0			0
$698$		0			0
$699$	−1.44091	+	0.137591i	−1.44091	+	0.137591i
$700$	0.721965	+	1.25048i	0.721965	+	1.25048i
$701$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$701$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$702$		0			0
$703$		0			0
$704$	−0.00465414	−	0.00806121i	−0.00465414	−	0.00806121i
$705$		0			0
$706$	0.0671040	−	0.116228i	0.0671040	−	0.116228i
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	1.66882			1.66882
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.254351	+	0.440549i	−0.254351	+	0.440549i
$717$	−0.827068	+	1.81103i	−0.827068	+	1.81103i
$718$	0.452418	+	0.783611i	0.452418	+	0.783611i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	0.134879	+	0.233618i	0.134879	+	0.233618i
$723$		0			0
$724$	0.763617	−	1.32262i	0.763617	−	1.32262i
$725$	0.786053	−	1.36148i	0.786053	−	1.36148i
$726$	0.465132	−	0.0444147i	0.465132	−	0.0444147i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	0.841254	−	0.540641i	0.841254	−	0.540641i
$730$		0			0
$731$		0			0
$732$	0.643187	−	0.0614169i	0.643187	−	0.0614169i
$733$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$733$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$734$	−0.462997	+	0.801934i	−0.462997	+	0.801934i
$735$		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	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$744$	−0.164184	+	0.359512i	−0.164184	+	0.359512i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.72373	−	2.98559i	−1.72373	−	2.98559i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	−0.375532	+	0.650440i	−0.375532	+	0.650440i
$753$	0.283341	−	0.0270558i	0.283341	−	0.0270558i
$754$		0			0
$755$		0			0
$756$	1.38544	−	0.406802i	1.38544	−	0.406802i
$757$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$757$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$761$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$762$	0.0260280	+	0.0365512i	0.0260280	+	0.0365512i
$763$		0			0
$764$	−1.12565			−1.12565
$765$		0			0
$766$	−0.134208			−0.134208
$767$		0			0
$768$	−0.231105	+	0.506049i	−0.231105	+	0.506049i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	0.471518			0.471518
$776$	0.705142	+	1.22134i	0.705142	+	1.22134i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.13779	−	1.08488i	−1.13779	−	1.08488i
$784$	0.936017			0.936017
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$	0.481929	+	0.676774i	0.481929	+	0.676774i
$790$		0			0
$791$		0			0
$792$	0.0522361	+	0.0602836i	0.0522361	+	0.0602836i
$793$		0			0
$794$	−0.273507	−	0.473728i	−0.273507	−	0.473728i
$795$		0			0
$796$	0.654218	−	1.13314i	0.654218	−	1.13314i
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$	−0.237902	+	0.520933i	−0.237902	+	0.520933i
$799$		0			0
$800$	1.01853			1.01853
$801$	−1.95496	+	0.376789i	−1.95496	+	0.376789i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$809$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−1.13501	−	1.96589i	−1.13501	−	1.96589i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.0448714			0.0448714
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	−0.348202	+	0.762457i	−0.348202	+	0.762457i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i
$826$		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
$836$	0.0484101			0.0484101
$837$	0.111165	−	0.458227i	0.111165	−	0.458227i
$838$	0.682507			0.682507
$839$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$839$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$840$		0			0
$841$	−0.735759	+	1.27437i	−0.735759	+	1.27437i
$842$	0.0671040	−	0.116228i	0.0671040	−	0.116228i
$843$	0.672932	+	0.945001i	0.672932	+	0.945001i
$844$	−0.746170	−	1.29240i	−0.746170	−	1.29240i
$845$		0			0
$846$	−0.302863	+	0.875065i	−0.302863	+	0.875065i
$847$	−1.83992			−1.83992
$848$		0			0
$849$	0.698939	−	1.53046i	0.698939	−	1.53046i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$853$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$854$	0.727379			0.727379
$855$		0			0
$856$	−1.55632			−1.55632
$857$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$857$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$858$		0			0
$859$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$859$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$860$		0			0
$861$		0			0
$862$	−0.273507	−	0.473728i	−0.273507	−	0.473728i
$863$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$863$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$864$	0.240128	−	0.989821i	0.240128	−	0.989821i
$865$		0			0
$866$	0.308779	+	0.534820i	0.308779	+	0.534820i
$867$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$868$	0.340419	−	0.589623i	0.340419	−	0.589623i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$877$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$878$		0			0
$879$	0.601300	−	1.31666i	0.601300	−	1.31666i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.13317	−	0.218401i	1.13317	−	0.218401i
$883$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$883$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−0.0883470	−	0.153022i	−0.0883470	−	0.153022i
$890$		0			0
$891$	−0.0748038	−	0.0588264i	−0.0748038	−	0.0588264i
$892$	1.54830			1.54830
$893$	0.642315	+	1.11252i	0.642315	+	1.11252i
$894$		0			0
$895$		0			0
$896$	0.902756	−	1.56362i	0.902756	−	1.56362i
$897$		0			0
$898$	0.308779	+	0.534820i	0.308779	+	0.534820i
$899$	−0.741276			−0.741276
$900$	−0.763617	+	0.147175i	−0.763617	+	0.147175i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$907$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$911$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$912$	−0.145113	−	0.203782i	−0.145113	−	0.203782i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.611291	−	1.05879i	−0.611291	−	1.05879i
$917$		0			0
$918$		0			0
$919$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$919$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$920$		0			0
$921$		0			0
$922$	0.308779	−	0.534820i	0.308779	−	0.534820i
$923$		0			0
$924$	−0.0797055	−	0.111931i	−0.0797055	−	0.111931i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−1.60124			−1.60124
$929$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$929$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$930$		0			0
$931$	0.800488	−	1.38649i	0.800488	−	1.38649i
$932$	0.562827	−	0.974845i	0.562827	−	0.974845i
$933$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$934$	0.370638	+	0.641964i	0.370638	+	0.641964i
$935$		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	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$	0.900739	−	0.0860101i	0.900739	−	0.0860101i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$947$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$948$		0			0
$949$		0			0
$950$	0.154218	−	0.267114i	0.154218	−	0.267114i
$951$	0.672932	+	0.945001i	0.672932	+	0.945001i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.774150	−	1.34087i	−0.774150	−	1.34087i
$957$	−0.0621493	+	0.136088i	−0.0621493	+	0.136088i
$958$		0			0
$959$	1.65033	−	2.85846i	1.65033	−	2.85846i
$960$		0			0
$961$	0.388835	+	0.673483i	0.388835	+	0.673483i
$962$		0			0
$963$	1.82318	−	0.351390i	1.82318	−	0.351390i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$967$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$968$	−0.415306	+	0.719332i	−0.415306	+	0.719332i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−0.0370031	+	0.776790i	−0.0370031	+	0.776790i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.158873	+	0.275175i	−0.158873	+	0.275175i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	0.0947329	+	0.164082i	0.0947329	+	0.164082i
$980$		0			0
$981$		0			0
$982$	0.943036			0.943036
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1.51475	−	3.31685i	1.51475	−	3.31685i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.240128	−	0.415914i	−0.240128	−	0.415914i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$997$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$998$		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.166.5	✓	20
9.7	even	3	inner	1503.1.f.b.1168.5	yes	20
167.166	odd	2	CM	1503.1.f.b.166.5	✓	20
1503.1168	odd	6	inner	1503.1.f.b.1168.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.1.f.b.166.5	✓	20	1.1	even	1	trivial
1503.1.f.b.166.5	✓	20	167.166	odd	2	CM
1503.1.f.b.1168.5	yes	20	9.7	even	3	inner
1503.1.f.b.1168.5	yes	20	1503.1168	odd	6	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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1503.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.235759 - 0.408346i) q^{2} +(-0.995472 + 0.0950560i) q^{3} +(0.388835 - 0.673483i) q^{4} +(0.273507 + 0.384087i) q^{6} +(-0.928368 - 1.60798i) q^{7} -0.838204 q^{8} +(0.981929 - 0.189251i) q^{9} +O(q^{10})\)	\(q+(-0.235759 - 0.408346i) q^{2} +(-0.995472 + 0.0950560i) q^{3} +(0.388835 - 0.673483i) q^{4} +(0.273507 + 0.384087i) q^{6} +(-0.928368 - 1.60798i) q^{7} -0.838204 q^{8} +(0.981929 - 0.189251i) q^{9} +(-0.0475819 - 0.0824143i) q^{11} +(-0.323056 + 0.707394i) q^{12} +(-0.437742 + 0.758192i) q^{14} +(-0.191221 - 0.331205i) q^{16} +(-0.308779 - 0.356349i) q^{18} -0.654136 q^{19} +(1.07701 + 1.51245i) q^{21} +(-0.0224357 + 0.0388598i) q^{22} +(0.834408 - 0.0796763i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-0.959493 + 0.281733i) q^{27} -1.44393 q^{28} +(0.786053 + 1.36148i) q^{29} +(-0.235759 + 0.408346i) q^{31} +(-0.509266 + 0.882075i) q^{32} +(0.0552004 + 0.0775182i) q^{33} +(0.254351 - 0.734900i) q^{36} +(0.154218 + 0.267114i) q^{38} +(0.363689 - 0.796368i) q^{42} -0.0740061 q^{44} +(-0.981929 - 1.70075i) q^{47} +(0.221839 + 0.311529i) q^{48} +(-1.22373 + 2.11957i) q^{49} +(-0.235759 + 0.408346i) q^{50} +(0.341254 + 0.325385i) q^{54} +(0.778161 + 1.34781i) q^{56} +(0.651174 - 0.0621796i) q^{57} +(0.370638 - 0.641964i) q^{58} +(-0.415415 - 0.719520i) q^{61} +0.222329 q^{62} +(-1.21590 - 1.40323i) q^{63} +0.0978132 q^{64} +(0.0186403 - 0.0408165i) q^{66} +(-0.823056 + 0.158631i) q^{72} +(0.580057 + 0.814576i) q^{75} +(-0.254351 + 0.440549i) q^{76} +(-0.0883470 + 0.153022i) q^{77} +(0.928368 - 0.371662i) q^{81} +(1.43739 - 0.137254i) q^{84} +(-0.911911 - 1.28060i) q^{87} +(0.0398833 + 0.0690800i) q^{88} -1.99094 q^{89} +(0.195876 - 0.428908i) q^{93} +(-0.462997 + 0.801934i) q^{94} +(0.423114 - 0.926490i) q^{96} +(-0.841254 - 1.45709i) q^{97} +1.15402 q^{98} +(-0.0623191 - 0.0719200i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 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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