LMFDB - Embedded newform 1503.1.f.b.1168.9

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			1168.9
Root			$0.235759 + 0.971812i$ of defining polynomial
Character	$\chi$	$=$	1503.1168
Dual form			1503.1.f.b.166.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.888835 - 1.53951i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-1.08006 - 1.87071i) q^{4} +(0.581419 - 1.67990i) q^{6} +(-0.723734 + 1.25354i) q^{7} -2.06230 q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})$	\(q+(0.888835 - 1.53951i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-1.08006 - 1.87071i) q^{4} +(0.581419 - 1.67990i) q^{6} +(-0.723734 + 1.25354i) q^{7} -2.06230 q^{8} +(0.928368 - 0.371662i) q^{9} +(0.995472 - 1.72421i) q^{11} +(-1.41457 - 1.63251i) q^{12} +(1.28656 + 2.22839i) q^{14} +(-0.752989 + 1.30422i) q^{16} +(0.252989 - 1.75958i) q^{18} -1.57211 q^{19} +(-0.473420 + 1.36786i) q^{21} +(-1.76962 - 3.06507i) q^{22} +(-2.02503 + 0.390293i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.841254 - 0.540641i) q^{27} +3.12670 q^{28} +(-0.235759 + 0.408346i) q^{29} +(0.888835 + 1.53951i) q^{31} +(0.307416 + 0.532461i) q^{32} +(0.651174 - 1.88144i) q^{33} +(-1.69796 - 1.33529i) q^{36} +(-1.39734 + 2.42027i) q^{38} +(1.68504 + 1.94464i) q^{42} -4.30067 q^{44} +(-0.928368 + 1.60798i) q^{47} +(-0.492557 + 1.42315i) q^{48} +(-0.547582 - 0.948440i) q^{49} +(0.888835 + 1.53951i) q^{50} +(-0.0845850 - 1.77566i) q^{54} +(1.49256 - 2.58518i) q^{56} +(-1.54370 + 0.297523i) q^{57} +(0.419102 + 0.725906i) q^{58} +(0.654861 - 1.13425i) q^{61} +3.16011 q^{62} +(-0.205996 + 1.43273i) q^{63} -0.413008 q^{64} +(-2.31771 - 2.67478i) q^{66} +(-1.91457 + 0.766480i) q^{72} +(-0.327068 + 0.945001i) q^{75} +(1.69796 + 2.94096i) q^{76} +(1.44091 + 2.49574i) q^{77} +(0.723734 - 0.690079i) q^{81} +(3.07019 - 0.591731i) q^{84} +(-0.154218 + 0.445585i) q^{87} +(-2.05296 + 3.55584i) q^{88} +1.96386 q^{89} +(1.16413 + 1.34347i) q^{93} +(1.65033 + 2.85846i) q^{94} +(0.402630 + 0.464659i) q^{96} +(-0.415415 + 0.719520i) q^{97} -1.94684 q^{98} +(0.283341 - 1.97068i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$2$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$3$	0.981929	−	0.189251i	0.981929	−	0.189251i
$3$	0.981929	−	0.189251i	0.981929	−	0.189251i
$4$	−1.08006	−	1.87071i	−1.08006	−	1.87071i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	0.581419	−	1.67990i	0.581419	−	1.67990i
$7$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$7$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$8$	−2.06230			−2.06230
$9$	0.928368	−	0.371662i	0.928368	−	0.371662i
$10$		0			0
$11$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$11$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$12$	−1.41457	−	1.63251i	−1.41457	−	1.63251i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	1.28656	+	2.22839i	1.28656	+	2.22839i
$15$		0			0
$16$	−0.752989	+	1.30422i	−0.752989	+	1.30422i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.252989	−	1.75958i	0.252989	−	1.75958i
$19$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$19$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$20$		0			0
$21$	−0.473420	+	1.36786i	−0.473420	+	1.36786i
$22$	−1.76962	−	3.06507i	−1.76962	−	3.06507i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−2.02503	+	0.390293i	−2.02503	+	0.390293i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	0.841254	−	0.540641i	0.841254	−	0.540641i
$28$	3.12670			3.12670
$29$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$29$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$30$		0			0
$31$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$31$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$32$	0.307416	+	0.532461i	0.307416	+	0.532461i
$33$	0.651174	−	1.88144i	0.651174	−	1.88144i
$34$		0			0
$35$		0			0
$36$	−1.69796	−	1.33529i	−1.69796	−	1.33529i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−1.39734	+	2.42027i	−1.39734	+	2.42027i
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$	1.68504	+	1.94464i	1.68504	+	1.94464i
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−4.30067			−4.30067
$45$		0			0
$46$		0			0
$47$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$47$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$48$	−0.492557	+	1.42315i	−0.492557	+	1.42315i
$49$	−0.547582	−	0.948440i	−0.547582	−	0.948440i
$50$	0.888835	+	1.53951i	0.888835	+	1.53951i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.0845850	−	1.77566i	−0.0845850	−	1.77566i
$55$		0			0
$56$	1.49256	−	2.58518i	1.49256	−	2.58518i
$57$	−1.54370	+	0.297523i	−1.54370	+	0.297523i
$58$	0.419102	+	0.725906i	0.419102	+	0.725906i
$59$		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	$-0.0606061\pi$
$62$	3.16011			3.16011
$63$	−0.205996	+	1.43273i	−0.205996	+	1.43273i
$64$	−0.413008			−0.413008
$65$		0			0
$66$	−2.31771	−	2.67478i	−2.31771	−	2.67478i
$67$		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			$\pi$
$72$	−1.91457	+	0.766480i	−1.91457	+	0.766480i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−0.327068	+	0.945001i	−0.327068	+	0.945001i
$76$	1.69796	+	2.94096i	1.69796	+	2.94096i
$77$	1.44091	+	2.49574i	1.44091	+	2.49574i
$78$		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
$81$	0.723734	−	0.690079i	0.723734	−	0.690079i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$	3.07019	−	0.591731i	3.07019	−	0.591731i
$85$		0			0
$86$		0			0
$87$	−0.154218	+	0.445585i	−0.154218	+	0.445585i
$88$	−2.05296	+	3.55584i	−2.05296	+	3.55584i
$89$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$89$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.16413	+	1.34347i	1.16413	+	1.34347i
$94$	1.65033	+	2.85846i	1.65033	+	2.85846i
$95$		0			0
$96$	0.402630	+	0.464659i	0.402630	+	0.464659i
$97$	−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	$0.303030\pi$
$98$	−1.94684			−1.94684
$99$	0.283341	−	1.97068i	0.283341	−	1.97068i
$100$	2.16011			2.16011
$101$		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
$107$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$107$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$108$	−1.91999	−	0.989821i	−1.91999	−	0.989821i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−1.08993	−	1.88781i	−1.08993	−	1.88781i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$	−0.914053	+	2.64098i	−0.914053	+	2.64098i
$115$		0			0
$116$	1.01853			1.01853
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.48193	−	2.56678i	−1.48193	−	2.56678i
$122$	−1.16413	−	2.01633i	−1.16413	−	2.01633i
$123$		0			0
$124$	1.91999	−	3.32551i	1.91999	−	3.32551i
$125$		0			0
$126$	2.02261	+	1.59060i	2.02261	+	1.59060i
$127$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$127$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$128$	−0.674512	+	1.16829i	−0.674512	+	1.16829i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−4.22295	+	0.813906i	−4.22295	+	0.813906i
$133$	1.13779	−	1.97070i	1.13779	−	1.97070i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$137$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$138$		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
$141$	−0.607279	+	1.75462i	−0.607279	+	1.75462i
$142$		0			0
$143$		0			0
$144$	−0.214323	+	1.49065i	−0.214323	+	1.49065i
$145$		0			0
$146$		0			0
$147$	−0.717180	−	0.827670i	−0.717180	−	0.827670i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	1.16413	+	1.34347i	1.16413	+	1.34347i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	3.24216			3.24216
$153$		0			0
$154$	5.12294			5.12294
$155$		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
$162$	−0.419102	−	1.72756i	−0.419102	−	1.72756i
$163$		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
$168$	0.976335	−	2.82094i	0.976335	−	2.82094i
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	−1.45949	+	0.584293i	−1.45949	+	0.584293i
$172$		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	$0.303030\pi$
$174$	0.548907	+	0.633472i	0.548907	+	0.633472i
$175$	−0.723734	−	1.25354i	−0.723734	−	1.25354i
$176$	1.49916	+	2.59662i	1.49916	+	2.59662i
$177$		0			0
$178$	1.74555	−	3.02337i	1.74555	−	3.02337i
$179$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$179$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$180$		0			0
$181$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$181$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$182$		0			0
$183$	0.428368	−	1.23769i	0.428368	−	1.23769i
$184$		0			0
$185$		0			0
$186$	3.10301	−	0.598055i	3.10301	−	0.598055i
$187$		0			0
$188$	4.01076			4.01076
$189$	0.0688733	+	1.44583i	0.0688733	+	1.44583i
$190$		0			0
$191$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$191$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$192$	−0.405544	+	0.0781623i	−0.405544	+	0.0781623i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	0.738471	+	1.27907i	0.738471	+	1.27907i
$195$		0			0
$196$	−1.18284	+	2.04874i	−1.18284	+	2.04874i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−2.78203	−	2.18781i	−2.78203	−	2.18781i
$199$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$199$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$200$	1.03115	−	1.78600i	1.03115	−	1.78600i
$201$		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
$209$	−1.56499	+	2.71064i	−1.56499	+	2.71064i
$210$		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
$214$	1.28656	−	2.22839i	1.28656	−	2.22839i
$215$		0			0
$216$	−1.73492	+	1.11496i	−1.73492	+	1.11496i
$217$	−2.57312			−2.57312
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$223$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$224$	−0.889950			−0.889950
$225$	−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	$0.666667\pi$
$228$	2.22386	+	2.56647i	2.22386	+	2.56647i
$229$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$229$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$230$		0			0
$231$	1.88720	+	2.17794i	1.88720	+	2.17794i
$232$	0.486206	−	0.842133i	0.486206	−	0.842133i
$233$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$233$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$239$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−5.26876			−5.26876
$243$	0.580057	−	0.814576i	0.580057	−	0.814576i
$244$	−2.82915			−2.82915
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−1.83305	−	3.17493i	−1.83305	−	3.17493i
$249$		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	$0.909091\pi$
$252$	2.90272	−	1.16208i	2.90272	−	1.16208i
$253$		0			0
$254$	−1.76962	+	3.06507i	−1.76962	+	3.06507i
$255$		0			0
$256$	0.992557	+	1.71916i	0.992557	+	1.71916i
$257$		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
$261$	−0.0671040	+	0.466718i	−0.0671040	+	0.466718i
$262$		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	$-0.0606061\pi$
$264$	−1.34292	+	3.88010i	−1.34292	+	3.88010i
$265$		0			0
$266$	−2.02261	−	3.50326i	−2.02261	−	3.50326i
$267$	1.92837	−	0.371662i	1.92837	−	0.371662i
$268$		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
$274$	1.03115	+	1.78600i	1.03115	+	1.78600i
$275$	0.995472	+	1.72421i	0.995472	+	1.72421i
$276$		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
$279$	1.39734	+	1.09888i	1.39734	+	1.09888i
$280$		0			0
$281$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$281$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$282$	2.16148	+	2.49448i	2.16148	+	2.49448i
$283$	−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
$288$	0.483291	+	0.380064i	0.483291	+	0.380064i
$289$	1.00000			1.00000
$290$		0			0
$291$	−0.271738	+	0.785135i	−0.271738	+	0.785135i
$292$		0			0
$293$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$293$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$294$	−1.91166	+	0.368442i	−1.91166	+	0.368442i
$295$		0			0
$296$		0			0
$297$	−0.0947329	−	1.98869i	−0.0947329	−	1.98869i
$298$		0			0
$299$		0			0
$300$	2.12108	−	0.408804i	2.12108	−	0.408804i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	1.18378	−	2.05036i	1.18378	−	2.05036i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	3.11254	−	5.39107i	3.11254	−	5.39107i
$309$		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	$0.666667\pi$
$314$	−2.99094			−2.99094
$315$		0			0
$316$		0			0
$317$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$317$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$318$		0			0
$319$	0.469383	+	0.812995i	0.469383	+	0.812995i
$320$		0			0
$321$	1.42131	−	0.273935i	1.42131	−	0.273935i
$322$		0			0
$323$		0			0
$324$	−2.07261	−	0.608574i	−2.07261	−	0.608574i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.34378	−	2.32750i	−1.34378	−	2.32750i
$330$		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
$334$	−1.77767			−1.77767
$335$		0			0
$336$	−1.42750	−	1.64742i	−1.42750	−	1.64742i
$337$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$337$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$338$	0.888835	+	1.53951i	0.888835	+	1.53951i
$339$		0			0
$340$		0			0
$341$	3.53924			3.53924
$342$	−0.397725	+	2.76624i	−0.397725	+	2.76624i
$343$	0.137747			0.137747
$344$		0			0
$345$		0			0
$346$	0.738471	+	1.27907i	0.738471	+	1.27907i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	1.00013	−	0.192758i	1.00013	−	0.192758i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$	−2.57312			−2.57312
$351$		0			0
$352$	1.22410			1.22410
$353$	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
$356$	−2.12108	−	3.67381i	−2.12108	−	3.67381i
$357$		0			0
$358$	−1.39734	+	2.42027i	−1.39734	+	2.42027i
$359$	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
$361$	1.47152			1.47152
$362$	1.65033	−	2.85846i	1.65033	−	2.85846i
$363$	−1.94091	−	2.23993i	−1.94091	−	2.23993i
$364$		0			0
$365$		0			0
$366$	−1.52468	−	1.75958i	−1.52468	−	1.75958i
$367$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$367$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.25593	−	3.62878i	1.25593	−	3.62878i
$373$		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
$376$	1.91457	−	3.31614i	1.91457	−	3.31614i
$377$		0			0
$378$	2.28708	+	1.17907i	2.28708	+	1.17907i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.95496	+	0.376789i	−1.95496	+	0.376789i
$382$	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	$0.424242\pi$
$384$	−0.441223	+	1.27483i	−0.441223	+	1.27483i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.79469			1.79469
$389$		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
$392$	1.12928	+	1.95597i	1.12928	+	1.95597i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−3.99260	+	1.59840i	−3.99260	+	1.59840i
$397$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$397$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$398$	0.738471	−	1.27907i	0.738471	−	1.27907i
$399$	0.744267	−	2.15042i	0.744267	−	2.15042i
$400$	−0.752989	−	1.30422i	−0.752989	−	1.30422i
$401$		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
$406$	−1.21327			−1.21327
$407$		0			0
$408$		0			0
$409$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$409$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$410$		0			0
$411$	−0.379436	+	1.09631i	−0.379436	+	1.09631i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	2.78203	+	4.81862i	2.78203	+	4.81862i
$419$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$419$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$420$		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	$-0.242424\pi$
$422$	−2.99094			−2.99094
$423$	−0.264241	+	1.83784i	−0.264241	+	1.83784i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.947890	+	1.64179i	0.947890	+	1.64179i
$428$	−1.56335	−	2.70780i	−1.56335	−	2.70780i
$429$		0			0
$430$		0			0
$431$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$431$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$432$	0.0716573	+	1.50427i	0.0716573	+	1.50427i
$433$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$433$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$434$	−2.28708	+	3.96134i	−2.28708	+	3.96134i
$435$		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
$441$	−0.860857	−	0.676985i	−0.860857	−	0.676985i
$442$		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
$446$	1.74555	+	3.02337i	1.74555	+	3.02337i
$447$		0			0
$448$	0.298908	−	0.517724i	0.298908	−	0.517724i
$449$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$449$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$450$	1.39734	+	1.09888i	1.39734	+	1.09888i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	3.18357	−	0.613582i	3.18357	−	0.613582i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−0.838204			−0.838204
$459$		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	$0.121212\pi$
$462$	5.03036	−	0.969523i	5.03036	−	0.969523i
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	−0.355048	−	0.614961i	−0.355048	−	0.614961i
$465$		0			0
$466$	0.0845850	−	0.146505i	0.0845850	−	0.146505i
$467$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$467$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$468$		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
$475$	0.786053	−	1.36148i	0.786053	−	1.36148i
$476$		0			0
$477$		0			0
$478$	−3.49109			−3.49109
$479$		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
$484$	−3.20113	+	5.54453i	−3.20113	+	5.54453i
$485$		0			0
$486$	−0.738471	−	1.61703i	−0.738471	−	1.61703i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−1.35052	+	2.33917i	−1.35052	+	2.33917i
$489$		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
$496$	−2.67713			−2.67713
$497$		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
$502$	−1.70566	+	2.95429i	−1.70566	+	2.95429i
$503$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$503$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$504$	0.424826	−	2.95473i	0.424826	−	2.95473i
$505$		0			0
$506$		0			0
$507$	−0.327068	+	0.945001i	−0.327068	+	0.945001i
$508$	2.15033	+	3.72449i	2.15033	+	3.72449i
$509$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$509$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$510$		0			0
$511$		0			0
$512$	2.17985			2.17985
$513$	−1.32254	+	0.849945i	−1.32254	+	0.849945i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.84833	+	3.20140i	1.84833	+	3.20140i
$518$		0			0
$519$	−0.271738	+	0.785135i	−0.271738	+	0.785135i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	0.658873	+	0.518143i	0.658873	+	0.518143i
$523$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$523$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$524$		0			0
$525$	−0.947890	−	1.09392i	−0.947890	−	1.09392i
$526$	−1.16413	−	2.01633i	−1.16413	−	2.01633i
$527$		0			0
$528$	1.96348	+	2.26598i	1.96348	+	2.26598i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−4.91550			−4.91550
$533$		0			0
$534$	1.14182	−	3.29908i	1.14182	−	3.29908i
$535$		0			0
$536$		0			0
$537$	−1.54370	+	0.297523i	−1.54370	+	0.297523i
$538$		0			0
$539$	−2.18041			−2.18041
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	1.82318	−	0.351390i	1.82318	−	0.351390i
$544$		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	$0.666667\pi$
$548$	2.50598			2.50598
$549$	0.186393	−	1.29639i	0.186393	−	1.29639i
$550$	3.53924			3.53924
$551$	0.370638	−	0.641964i	0.370638	−	0.641964i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$557$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$558$	2.93375	−	1.17450i	2.93375	−	1.17450i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.581419	−	1.00705i	−0.581419	−	1.00705i
$563$	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	$0.787879\pi$
$564$	3.93828	−	0.759041i	3.93828	−	0.759041i
$565$		0			0
$566$	−1.47694			−1.47694
$567$	0.341254	+	1.40667i	0.341254	+	1.40667i
$568$		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
$573$	−0.0311250	+	0.0899299i	−0.0311250	+	0.0899299i
$574$		0			0
$575$		0			0
$576$	−0.383423	+	0.153500i	−0.383423	+	0.153500i
$577$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$577$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$578$	0.888835	−	1.53951i	0.888835	−	1.53951i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	0.967192	+	1.11620i	0.967192	+	1.11620i
$583$		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	$0.666667\pi$
$588$	−0.773738	+	2.23557i	−0.773738	+	2.23557i
$589$	−1.39734	−	2.42027i	−1.39734	−	2.42027i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−3.14580	−	1.62177i	−3.14580	−	1.62177i
$595$		0			0
$596$		0			0
$597$	0.815816	−	0.157236i	0.815816	−	0.157236i
$598$		0			0
$599$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$599$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$600$	0.674512	−	1.94888i	0.674512	−	1.94888i
$601$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$601$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$602$		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	$0.333333\pi$
$608$	−0.483291	−	0.837085i	−0.483291	−	0.837085i
$609$	−0.446947	−	0.515804i	−0.446947	−	0.515804i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$613$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$614$		0			0
$615$		0			0
$616$	−2.97160	−	5.14696i	−2.97160	−	5.14696i
$617$	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
$622$	1.77767			1.77767
$623$	−1.42131	+	2.46178i	−1.42131	+	2.46178i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	−1.02371	+	2.95783i	−1.02371	+	2.95783i
$628$	−1.81720	+	3.14749i	−1.81720	+	3.14749i
$629$		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
$634$	−0.581419	−	1.00705i	−0.581419	−	1.00705i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.66882			1.66882
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	0.841586	−	2.43160i	0.841586	−	2.43160i
$643$		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			$\pi$
$648$	−1.49256	+	1.42315i	−1.49256	+	1.42315i
$649$		0			0
$650$		0			0
$651$	−2.52662	+	0.486967i	−2.52662	+	0.486967i
$652$		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
$658$	−4.77761			−4.77761
$659$		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
$668$	−1.08006	+	1.87071i	−1.08006	+	1.87071i
$669$	−0.642315	+	1.85585i	−0.642315	+	1.85585i
$670$		0			0
$671$	−1.30379	−	2.25823i	−1.30379	−	2.25823i
$672$	−0.873868	+	0.168424i	−0.873868	+	0.168424i
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	−3.30067			−3.30067
$675$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$676$	2.16011			2.16011
$677$	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
$679$	−0.601300	−	1.04148i	−0.601300	−	1.04148i
$680$		0			0
$681$		0			0
$682$	3.14580	−	5.44869i	3.14580	−	5.44869i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	2.66938	+	2.09922i	2.66938	+	2.09922i
$685$		0			0
$686$	0.122434	−	0.212062i	0.122434	−	0.212062i
$687$	−0.308779	−	0.356349i	−0.308779	−	0.356349i
$688$		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	$0.666667\pi$
$692$	1.79469			1.79469
$693$	2.26527	+	1.78143i	2.26527	+	1.78143i
$694$		0			0
$695$		0			0
$696$	0.318045	−	0.918930i	0.318045	−	0.918930i
$697$		0			0
$698$		0			0
$699$	0.0934441	−	0.0180099i	0.0934441	−	0.0180099i
$700$	−1.56335	+	2.70780i	−1.56335	+	2.70780i
$701$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$701$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$702$		0			0
$703$		0			0
$704$	−0.411138	+	0.712112i	−0.411138	+	0.712112i
$705$		0			0
$706$	−1.70566	−	2.95429i	−1.70566	−	2.95429i
$707$		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
$712$	−4.05006			−4.05006
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.69796	+	2.94096i	1.69796	+	2.94096i
$717$	−1.28605	−	1.48418i	−1.28605	−	1.48418i
$718$	1.49547	−	2.59023i	1.49547	−	2.59023i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	1.30794	−	2.26541i	1.30794	−	2.26541i
$723$		0			0
$724$	−2.00538	−	3.47342i	−2.00538	−	3.47342i
$725$	−0.235759	−	0.408346i	−0.235759	−	0.408346i
$726$	−5.17355	+	0.997120i	−5.17355	+	0.997120i
$727$		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
$732$	−2.77802	+	0.535420i	−2.77802	+	0.535420i
$733$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$733$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$734$	1.65033	+	2.85846i	1.65033	+	2.85846i
$735$		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	$0.666667\pi$
$744$	−2.40078	−	2.77065i	−2.40078	−	2.77065i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.04758	+	1.81447i	−1.04758	+	1.81447i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−1.39810	−	2.42158i	−1.39810	−	2.42158i
$753$	−1.88431	+	0.363170i	−1.88431	+	0.363170i
$754$		0			0
$755$		0			0
$756$	2.63034	−	1.69042i	2.63034	−	1.69042i
$757$	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	$0.606061\pi$
$762$	−1.15757	+	3.34459i	−1.15757	+	3.34459i
$763$		0			0
$764$	0.205565			0.205565
$765$		0			0
$766$	3.41133			3.41133
$767$		0			0
$768$	1.29997	+	1.50025i	1.29997	+	1.50025i
$769$		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
$775$	−1.77767			−1.77767
$776$	0.856711	−	1.48387i	0.856711	−	1.48387i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.0224357	+	0.470984i	0.0224357	+	0.470984i
$784$	1.64929			1.64929
$785$		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
$789$	0.428368	−	1.23769i	0.428368	−	1.23769i
$790$		0			0
$791$		0			0
$792$	−0.584334	+	4.06413i	−0.584334	+	4.06413i
$793$		0			0
$794$	−0.581419	+	1.00705i	−0.581419	+	1.00705i
$795$		0			0
$796$	−0.897344	−	1.55424i	−0.897344	−	1.55424i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$	−2.64906	−	3.05717i	−2.64906	−	3.05717i
$799$		0			0
$800$	−0.614832			−0.614832
$801$	1.82318	−	0.729892i	1.82318	−	0.729892i
$802$		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			$\pi$
$812$	−0.737146	+	1.27678i	−0.737146	+	1.27678i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	3.53924			3.53924
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	1.35052	+	1.55858i	1.35052	+	1.55858i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	1.30379	+	1.50465i	1.30379	+	1.50465i
$826$		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
$836$	6.76110			6.76110
$837$	1.58006	+	0.814576i	1.58006	+	0.814576i
$838$	−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
$841$	0.388835	+	0.673483i	0.388835	+	0.673483i
$842$	−1.70566	−	2.95429i	−1.70566	−	2.95429i
$843$	0.213947	−	0.618159i	0.213947	−	0.618159i
$844$	−1.81720	+	3.14749i	−1.81720	+	3.14749i
$845$		0			0
$846$	2.59450	+	2.04034i	2.59450	+	2.04034i
$847$	4.29009			4.29009
$848$		0			0
$849$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$853$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$854$	3.37007			3.37007
$855$		0			0
$856$	−2.98511			−2.98511
$857$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$857$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$858$		0			0
$859$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$859$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$860$		0			0
$861$		0			0
$862$	−0.581419	+	1.00705i	−0.581419	+	1.00705i
$863$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$863$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$864$	0.546485	+	0.281733i	0.546485	+	0.281733i
$865$		0			0
$866$	−0.252989	+	0.438190i	−0.252989	+	0.438190i
$867$	0.981929	−	0.189251i	0.981929	−	0.189251i
$868$	2.77912	+	4.81357i	2.77912	+	4.81357i
$869$		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
$877$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$877$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$878$		0			0
$879$	−0.0623191	−	0.0719200i	−0.0623191	−	0.0719200i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.80738	+	0.723568i	−1.80738	+	0.723568i
$883$	−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
$889$	1.44091	−	2.49574i	1.44091	−	2.49574i
$890$		0			0
$891$	−0.469383	−	1.93482i	−0.469383	−	1.93482i
$892$	4.24216			4.24216
$893$	1.45949	−	2.52792i	1.45949	−	2.52792i
$894$		0			0
$895$		0			0
$896$	−0.976335	−	1.69106i	−0.976335	−	1.69106i
$897$		0			0
$898$	−0.252989	+	0.438190i	−0.252989	+	0.438190i
$899$	−0.838204			−0.838204
$900$	2.00538	−	0.802833i	2.00538	−	0.802833i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$907$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$908$		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	$0.121212\pi$
$912$	0.774352	−	2.23734i	0.774352	−	2.23734i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.509266	+	0.882075i	−0.509266	+	0.882075i
$917$		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
$922$	−0.252989	−	0.438190i	−0.252989	−	0.438190i
$923$		0			0
$924$	2.03602	−	5.88270i	2.03602	−	5.88270i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.289905			−0.289905
$929$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$929$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$930$		0			0
$931$	0.860857	+	1.49105i	0.860857	+	1.49105i
$932$	−0.102782	−	0.178024i	−0.102782	−	0.178024i
$933$	0.654861	+	0.755750i	0.654861	+	0.755750i
$934$	0.419102	−	0.725906i	0.419102	−	0.725906i
$935$		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	$0.333333\pi$
$942$	−2.93689	+	0.566040i	−2.93689	+	0.566040i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$947$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$948$		0			0
$949$		0			0
$950$	−1.39734	−	2.42027i	−1.39734	−	2.42027i
$951$	0.213947	−	0.618159i	0.213947	−	0.618159i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−2.12108	+	3.67381i	−2.12108	+	3.67381i
$957$	0.614761	+	0.709472i	0.614761	+	0.709472i
$958$		0			0
$959$	−0.839614	−	1.45425i	−0.839614	−	1.45425i
$960$		0			0
$961$	−1.08006	+	1.87071i	−1.08006	+	1.87071i
$962$		0			0
$963$	1.34378	−	0.537970i	1.34378	−	0.537970i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$967$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$968$	3.05618	+	5.29346i	3.05618	+	5.29346i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−2.15033	−	0.205332i	−2.15033	−	0.205332i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.986206	+	1.70816i	0.986206	+	1.70816i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	1.95496	−	3.38610i	1.95496	−	3.38610i
$980$		0			0
$981$		0			0
$982$	−3.55534			−3.55534
$983$		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
$987$	−1.75998	−	2.03113i	−1.75998	−	2.03113i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.546485	+	0.946540i	−0.546485	+	0.946540i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$997$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$998$		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.1168.9	yes	20
9.4	even	3	inner	1503.1.f.b.166.9	✓	20
167.166	odd	2	CM	1503.1.f.b.1168.9	yes	20
1503.166	odd	6	inner	1503.1.f.b.166.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.1.f.b.166.9	✓	20	9.4	even	3	inner
1503.1.f.b.166.9	✓	20	1503.166	odd	6	inner
1503.1.f.b.1168.9	yes	20	1.1	even	1	trivial
1503.1.f.b.1168.9	yes	20	167.166	odd	2	CM

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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)

\(f(q)\)	\(=\)	\(q+(0.888835 - 1.53951i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-1.08006 - 1.87071i) q^{4} +(0.581419 - 1.67990i) q^{6} +(-0.723734 + 1.25354i) q^{7} -2.06230 q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)	\(q+(0.888835 - 1.53951i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-1.08006 - 1.87071i) q^{4} +(0.581419 - 1.67990i) q^{6} +(-0.723734 + 1.25354i) q^{7} -2.06230 q^{8} +(0.928368 - 0.371662i) q^{9} +(0.995472 - 1.72421i) q^{11} +(-1.41457 - 1.63251i) q^{12} +(1.28656 + 2.22839i) q^{14} +(-0.752989 + 1.30422i) q^{16} +(0.252989 - 1.75958i) q^{18} -1.57211 q^{19} +(-0.473420 + 1.36786i) q^{21} +(-1.76962 - 3.06507i) q^{22} +(-2.02503 + 0.390293i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.841254 - 0.540641i) q^{27} +3.12670 q^{28} +(-0.235759 + 0.408346i) q^{29} +(0.888835 + 1.53951i) q^{31} +(0.307416 + 0.532461i) q^{32} +(0.651174 - 1.88144i) q^{33} +(-1.69796 - 1.33529i) q^{36} +(-1.39734 + 2.42027i) q^{38} +(1.68504 + 1.94464i) q^{42} -4.30067 q^{44} +(-0.928368 + 1.60798i) q^{47} +(-0.492557 + 1.42315i) q^{48} +(-0.547582 - 0.948440i) q^{49} +(0.888835 + 1.53951i) q^{50} +(-0.0845850 - 1.77566i) q^{54} +(1.49256 - 2.58518i) q^{56} +(-1.54370 + 0.297523i) q^{57} +(0.419102 + 0.725906i) q^{58} +(0.654861 - 1.13425i) q^{61} +3.16011 q^{62} +(-0.205996 + 1.43273i) q^{63} -0.413008 q^{64} +(-2.31771 - 2.67478i) q^{66} +(-1.91457 + 0.766480i) q^{72} +(-0.327068 + 0.945001i) q^{75} +(1.69796 + 2.94096i) q^{76} +(1.44091 + 2.49574i) q^{77} +(0.723734 - 0.690079i) q^{81} +(3.07019 - 0.591731i) q^{84} +(-0.154218 + 0.445585i) q^{87} +(-2.05296 + 3.55584i) q^{88} +1.96386 q^{89} +(1.16413 + 1.34347i) q^{93} +(1.65033 + 2.85846i) q^{94} +(0.402630 + 0.464659i) q^{96} +(-0.415415 + 0.719520i) q^{97} -1.94684 q^{98} +(0.283341 - 1.97068i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 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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