LMFDB - Embedded newform 1169.1.f.c.667.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1169,1,Mod(333,1169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("1169.333");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1169 = 7 \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1169.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:	$0.583406999768$
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:	$\Z[a_1, \ldots, a_{7}]$
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			667.9
Root			$-0.888835 + 0.458227i$ of defining polynomial
Character	$\chi$	$=$	1169.667
Dual form			1169.1.f.c.333.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.959493 - 1.66189i) q^{2} +(-0.580057 - 1.00469i) q^{3} +(-1.34125 - 2.32312i) q^{4} -2.22624 q^{6} +(-0.786053 - 0.618159i) q^{7} -3.22871 q^{8} +(-0.172932 + 0.299527i) q^{9} +O(q^{10})$	\(q+(0.959493 - 1.66189i) q^{2} +(-0.580057 - 1.00469i) q^{3} +(-1.34125 - 2.32312i) q^{4} -2.22624 q^{6} +(-0.786053 - 0.618159i) q^{7} -3.22871 q^{8} +(-0.172932 + 0.299527i) q^{9} +(0.888835 + 1.53951i) q^{11} +(-1.55601 + 2.69508i) q^{12} +(-1.78153 + 0.713215i) q^{14} +(-1.75667 + 3.04264i) q^{16} +(0.331854 + 0.574788i) q^{18} +(0.654861 - 1.13425i) q^{19} +(-0.165101 + 1.14831i) q^{21} +3.41133 q^{22} +(1.87283 + 3.24384i) q^{24} +(-0.500000 - 0.866025i) q^{25} -0.758872 q^{27} +(-0.381761 + 2.65520i) q^{28} +1.85674 q^{29} +(-0.723734 - 1.25354i) q^{31} +(1.75667 + 3.04264i) q^{32} +(1.03115 - 1.78600i) q^{33} +0.927783 q^{36} +(-1.25667 - 2.17661i) q^{38} +(1.74994 + 1.37617i) q^{42} +(2.38431 - 4.12974i) q^{44} +(-0.981929 + 1.70075i) q^{47} +4.07587 q^{48} +(0.235759 + 0.971812i) q^{49} -1.91899 q^{50} +(-0.728132 + 1.26116i) q^{54} +(2.53794 + 1.99585i) q^{56} -1.51943 q^{57} +(1.78153 - 3.08569i) q^{58} +(-0.415415 + 0.719520i) q^{61} -2.77767 q^{62} +(0.321089 - 0.128545i) q^{63} +3.22871 q^{64} +(-1.97876 - 3.42732i) q^{66} +(0.558347 - 0.967085i) q^{72} +(-0.580057 + 1.00469i) q^{75} -3.51334 q^{76} +(0.252989 - 1.75958i) q^{77} +(0.613121 + 1.06196i) q^{81} +(2.88909 - 1.15662i) q^{84} +(-1.07701 - 1.86544i) q^{87} +(-2.86979 - 4.97062i) q^{88} +(0.995472 - 1.72421i) q^{89} +(-0.839614 + 1.45425i) q^{93} +(1.88431 + 3.26372i) q^{94} +(2.03794 - 3.52981i) q^{96} +1.68251 q^{97} +(1.84125 + 0.540641i) q^{98} -0.614832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) $ 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9	$ 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9} - q^{11} - 3 q^{12} + 2 q^{14} - 6 q^{16} + 2 q^{19} + 2 q^{21} + 4 q^{22} - 4 q^{24} - 10 q^{25} - 2 q^{27} - 6 q^{28} + 2 q^{29} - q^{31} + 6 q^{32} + q^{33} + 22 q^{36} + 4 q^{38} + 20 q^{42} + 8 q^{44} - q^{47} - 12 q^{48} + q^{49} - 4 q^{50} - 20 q^{54} + 4 q^{56} + 4 q^{57} - 2 q^{58} + 2 q^{61} - 18 q^{62} + 8 q^{64} + 2 q^{66} + 11 q^{72} - q^{75} - 12 q^{76} + 2 q^{77} - 12 q^{81} + 19 q^{84} + q^{87} - 4 q^{88} - q^{89} + q^{93} - 2 q^{94} - 6 q^{96} - 4 q^{97} + 18 q^{98}+O(q^{100}) $ 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9 - q^11 - 3 * q^12 + 2 * q^14 - 6 * q^16 + 2 * q^19 + 2 * q^21 + 4 * q^22 - 4 * q^24 - 10 * q^25 - 2 * q^27 - 6 * q^28 + 2 * q^29 - q^31 + 6 * q^32 + q^33 + 22 * q^36 + 4 * q^38 + 20 * q^42 + 8 * q^44 - q^47 - 12 * q^48 + q^49 - 4 * q^50 - 20 * q^54 + 4 * q^56 + 4 * q^57 - 2 * q^58 + 2 * q^61 - 18 * q^62 + 8 * q^64 + 2 * q^66 + 11 * q^72 - q^75 - 12 * q^76 + 2 * q^77 - 12 * q^81 + 19 * q^84 + q^87 - 4 * q^88 - q^89 + q^93 - 2 * q^94 - 6 * q^96 - 4 * q^97 + 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1169\mathbb{Z}\right)^\times$.

$n$	$673$	$836$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$2$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$3$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$3$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$4$	−1.34125	−	2.32312i	−1.34125	−	2.32312i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	−2.22624			−2.22624
$7$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$7$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$8$	−3.22871			−3.22871
$9$	−0.172932	+	0.299527i	−0.172932	+	0.299527i
$10$		0			0
$11$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$11$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$12$	−1.55601	+	2.69508i	−1.55601	+	2.69508i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−1.78153	+	0.713215i	−1.78153	+	0.713215i
$15$		0			0
$16$	−1.75667	+	3.04264i	−1.75667	+	3.04264i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	0.331854	+	0.574788i	0.331854	+	0.574788i
$19$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$19$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$20$		0			0
$21$	−0.165101	+	1.14831i	−0.165101	+	1.14831i
$22$	3.41133			3.41133
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	1.87283	+	3.24384i	1.87283	+	3.24384i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	−0.758872			−0.758872
$28$	−0.381761	+	2.65520i	−0.381761	+	2.65520i
$29$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$29$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$30$		0			0
$31$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$31$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$32$	1.75667	+	3.04264i	1.75667	+	3.04264i
$33$	1.03115	−	1.78600i	1.03115	−	1.78600i
$34$		0			0
$35$		0			0
$36$	0.927783			0.927783
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	−1.25667	−	2.17661i	−1.25667	−	2.17661i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	1.74994	+	1.37617i	1.74994	+	1.37617i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	2.38431	−	4.12974i	2.38431	−	4.12974i
$45$		0			0
$46$		0			0
$47$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$47$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$48$	4.07587			4.07587
$49$	0.235759	+	0.971812i	0.235759	+	0.971812i
$50$	−1.91899			−1.91899
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−0.728132	+	1.26116i	−0.728132	+	1.26116i
$55$		0			0
$56$	2.53794	+	1.99585i	2.53794	+	1.99585i
$57$	−1.51943			−1.51943
$58$	1.78153	−	3.08569i	1.78153	−	3.08569i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$61$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$62$	−2.77767			−2.77767
$63$	0.321089	−	0.128545i	0.321089	−	0.128545i
$64$	3.22871			3.22871
$65$		0			0
$66$	−1.97876	−	3.42732i	−1.97876	−	3.42732i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.558347	−	0.967085i	0.558347	−	0.967085i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	−0.580057	+	1.00469i	−0.580057	+	1.00469i
$76$	−3.51334			−3.51334
$77$	0.252989	−	1.75958i	0.252989	−	1.75958i
$78$		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
$81$	0.613121	+	1.06196i	0.613121	+	1.06196i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	2.88909	−	1.15662i	2.88909	−	1.15662i
$85$		0			0
$86$		0			0
$87$	−1.07701	−	1.86544i	−1.07701	−	1.86544i
$88$	−2.86979	−	4.97062i	−2.86979	−	4.97062i
$89$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$89$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.839614	+	1.45425i	−0.839614	+	1.45425i
$94$	1.88431	+	3.26372i	1.88431	+	3.26372i
$95$		0			0
$96$	2.03794	−	3.52981i	2.03794	−	3.52981i
$97$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$97$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$98$	1.84125	+	0.540641i	1.84125	+	0.540641i
$99$	−0.614832			−0.614832
$100$	−1.34125	+	2.32312i	−1.34125	+	2.32312i
$101$		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
$107$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$107$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$108$	1.01784	+	1.76295i	1.01784	+	1.76295i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$	3.26167	−	1.30578i	3.26167	−	1.30578i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−1.45788	+	2.52512i	−1.45788	+	2.52512i
$115$		0			0
$116$	−2.49035	−	4.31342i	−2.49035	−	4.31342i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.08006	+	1.87071i	−1.08006	+	1.87071i
$122$	0.797176	+	1.38075i	0.797176	+	1.38075i
$123$		0			0
$124$	−1.94142	+	3.36264i	−1.94142	+	3.36264i
$125$		0			0
$126$	0.0944555	−	0.656953i	0.0944555	−	0.656953i
$127$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$127$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$128$	1.34125	−	2.32312i	1.34125	−	2.32312i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−5.53214			−5.53214
$133$	−1.21590	+	0.486774i	−1.21590	+	0.486774i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$137$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	2.27830			2.27830
$142$		0			0
$143$		0			0
$144$	−0.607569	−	1.05234i	−0.607569	−	1.05234i
$145$		0			0
$146$		0			0
$147$	0.839614	−	0.800570i	0.839614	−	0.800570i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	1.11312	+	1.92798i	1.11312	+	1.92798i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−2.11435	+	3.66217i	−2.11435	+	3.66217i
$153$		0			0
$154$	−2.68148	−	2.10874i	−2.68148	−	2.10874i
$155$		0			0
$156$		0			0
$157$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$157$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	2.35314			2.35314
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000			1.00000
$167$	1.00000			1.00000
$168$	0.533064	−	3.70754i	0.533064	−	3.70754i
$169$	1.00000			1.00000
$170$		0			0
$171$	0.226493	+	0.392297i	0.226493	+	0.392297i
$172$		0			0
$173$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$173$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$174$	−4.13354			−4.13354
$175$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$176$	−6.24556			−6.24556
$177$		0			0
$178$	−1.91030	−	3.30873i	−1.91030	−	3.30873i
$179$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$179$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−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
$183$	0.963857			0.963857
$184$		0			0
$185$		0			0
$186$	1.61121	+	2.79069i	1.61121	+	2.79069i
$187$		0			0
$188$	5.26806			5.26806
$189$	0.596514	+	0.469104i	0.596514	+	0.469104i
$190$		0			0
$191$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$191$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$192$	−1.87283	−	3.24384i	−1.87283	−	3.24384i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	1.61435	−	2.79614i	1.61435	−	2.79614i
$195$		0			0
$196$	1.94142	−	1.85114i	1.94142	−	1.85114i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.589927	+	1.02178i	−0.589927	+	1.02178i
$199$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$199$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$200$	1.61435	+	2.79614i	1.61435	+	2.79614i
$201$		0			0
$202$		0			0
$203$	−1.45949	−	1.14776i	−1.45949	−	1.14776i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.32825			2.32825
$210$		0			0
$211$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$211$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$212$		0			0
$213$		0			0
$214$	−1.50842	−	2.61267i	−1.50842	−	2.61267i
$215$		0			0
$216$	2.45018			2.45018
$217$	−0.205996	+	1.43273i	−0.205996	+	1.43273i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$223$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$224$	0.500000	−	3.47758i	0.500000	−	3.47758i
$225$	0.345864			0.345864
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	2.03794	+	3.52981i	2.03794	+	3.52981i
$229$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$229$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$230$		0			0
$231$	−1.91457	+	0.766480i	−1.91457	+	0.766480i
$232$	−5.99486			−5.99486
$233$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$233$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$239$	1.16011			1.16011			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	$0.333333\pi$
$242$	2.07261	+	3.58987i	2.07261	+	3.58987i
$243$	0.331854	−	0.574788i	0.331854	−	0.574788i
$244$	2.22871			2.22871
$245$		0			0
$246$		0			0
$247$		0			0
$248$	2.33673	+	4.04733i	2.33673	+	4.04733i
$249$		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	$0.545455\pi$
$252$	−0.729287	−	0.573517i	−0.729287	−	0.573517i
$253$		0			0
$254$	0.0913090	−	0.158152i	0.0913090	−	0.158152i
$255$		0			0
$256$	−0.959493	−	1.66189i	−0.959493	−	1.66189i
$257$		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
$261$	−0.321089	+	0.556143i	−0.321089	+	0.556143i
$262$		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	$0.969697\pi$
$264$	−3.32928	+	5.76649i	−3.32928	+	5.76649i
$265$		0			0
$266$	−0.357685	+	2.48775i	−0.357685	+	2.48775i
$267$	−2.30972			−2.30972
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$	−0.182618			−0.182618
$275$	0.888835	−	1.53951i	0.888835	−	1.53951i
$276$		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
$279$	0.500627			0.500627
$280$		0			0
$281$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$281$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$282$	2.18601	−	3.78628i	2.18601	−	3.78628i
$283$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$283$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.21514			−1.21514
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$	−0.975950	−	1.69039i	−0.975950	−	1.69039i
$292$		0			0
$293$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$293$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$294$	−0.524856	−	2.16349i	−0.524856	−	2.16349i
$295$		0			0
$296$		0			0
$297$	−0.674512	−	1.16829i	−0.674512	−	1.16829i
$298$		0			0
$299$		0			0
$300$	3.11201			3.11201
$301$		0			0
$302$		0			0
$303$		0			0
$304$	2.30075	+	3.98501i	2.30075	+	3.98501i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−4.42703	+	1.77232i	−4.42703	+	1.77232i
$309$		0			0
$310$		0			0
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−0.904836			−0.904836
$315$		0			0
$316$		0			0
$317$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$317$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$318$		0			0
$319$	1.65033	+	2.85846i	1.65033	+	2.85846i
$320$		0			0
$321$	−1.82382			−1.82382
$322$		0			0
$323$		0			0
$324$	1.64470	−	2.84871i	1.64470	−	2.84871i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.82318	−	0.729892i	1.82318	−	0.729892i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	0.959493	−	1.66189i	0.959493	−	1.66189i
$335$		0			0
$336$	−3.20385	−	2.51954i	−3.20385	−	2.51954i
$337$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$337$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$338$	0.959493	−	1.66189i	0.959493	−	1.66189i
$339$		0			0
$340$		0			0
$341$	1.28656	−	2.22839i	1.28656	−	2.22839i
$342$	0.869273			0.869273
$343$	0.415415	−	0.909632i	0.415415	−	0.909632i
$344$		0			0
$345$		0			0
$346$	0.0913090	+	0.158152i	0.0913090	+	0.158152i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	−2.88909	+	5.00406i	−2.88909	+	5.00406i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	1.50842	+	1.18624i	1.50842	+	1.18624i
$351$		0			0
$352$	−3.12278	+	5.40881i	−3.12278	+	5.40881i
$353$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$353$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$354$		0			0
$355$		0			0
$356$	−5.34072			−5.34072
$357$		0			0
$358$	−3.76861			−3.76861
$359$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$359$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	−0.357685	−	0.619529i	−0.357685	−	0.619529i
$362$	1.88431	−	3.26372i	1.88431	−	3.26372i
$363$	2.50598			2.50598
$364$		0			0
$365$		0			0
$366$	0.924814	−	1.60183i	0.924814	−	1.60183i
$367$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$367$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	4.50454			4.50454
$373$		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
$376$	3.17036	−	5.49123i	3.17036	−	5.49123i
$377$		0			0
$378$	1.35195	−	0.541239i	1.35195	−	0.541239i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.0552004	−	0.0956100i	−0.0552004	−	0.0956100i
$382$	0.452418	+	0.783611i	0.452418	+	0.783611i
$383$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$383$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$384$	−3.11201			−3.11201
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−2.25667	−	3.90866i	−2.25667	−	3.90866i
$389$		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
$392$	−0.761197	−	3.13770i	−0.761197	−	3.13770i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.824646	+	1.42833i	0.824646	+	1.42833i
$397$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$397$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$398$	−0.182618			−0.182618
$399$	1.19435	+	0.939247i	1.19435	+	0.939247i
$400$	3.51334			3.51334
$401$		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
$406$	−3.30782	+	1.32425i	−3.30782	+	1.32425i
$407$		0			0
$408$		0			0
$409$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$409$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$410$		0			0
$411$	−0.0552004	+	0.0956100i	−0.0552004	+	0.0956100i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	2.23394	−	3.86930i	2.23394	−	3.86930i
$419$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$419$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$420$		0			0
$421$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$421$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$422$	0.452418	−	0.783611i	0.452418	−	0.783611i
$423$	−0.339614	−	0.588228i	−0.339614	−	0.588228i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.771316	−	0.308788i	0.771316	−	0.308788i
$428$	−4.21719			−4.21719
$429$		0			0
$430$		0			0
$431$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$431$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$432$	1.33309	−	2.30897i	1.33309	−	2.30897i
$433$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$433$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$434$	2.18340	+	1.71704i	2.18340	+	1.71704i
$435$		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
$441$	−0.331854	−	0.0974412i	−0.331854	−	0.0974412i
$442$		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
$446$	−1.91030	+	3.30873i	−1.91030	+	3.30873i
$447$		0			0
$448$	−2.53794	−	1.99585i	−2.53794	−	1.99585i
$449$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$449$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$450$	0.331854	−	0.574788i	0.331854	−	0.574788i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	4.90578			4.90578
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	1.78153	+	3.08569i	1.78153	+	3.08569i
$459$		0			0
$460$		0			0
$461$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$461$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$462$	−0.563215	+	3.91724i	−0.563215	+	3.91724i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−3.26167	+	5.64938i	−3.26167	+	5.64938i
$465$		0			0
$466$	1.38884	+	2.40553i	1.38884	+	2.40553i
$467$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$467$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.273507	+	0.473728i	−0.273507	+	0.473728i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.30972			−1.30972
$476$		0			0
$477$		0			0
$478$	1.11312	−	1.92798i	1.11312	−	1.92798i
$479$		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
$484$	5.79452			5.79452
$485$		0			0
$486$	−0.636823	−	1.10301i	−0.636823	−	1.10301i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$	1.34125	−	2.32312i	1.34125	−	2.32312i
$489$		0			0
$490$		0			0
$491$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	5.08544			5.08544
$497$		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
$501$	−0.580057	−	1.00469i	−0.580057	−	1.00469i
$502$	−0.273100	+	0.473023i	−0.273100	+	0.473023i
$503$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$503$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$504$	−1.03670	+	0.415033i	−1.03670	+	0.415033i
$505$		0			0
$506$		0			0
$507$	−0.580057	−	1.00469i	−0.580057	−	1.00469i
$508$	−0.127639	−	0.221077i	−0.127639	−	0.221077i
$509$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$509$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	−0.496956	+	0.860752i	−0.496956	+	0.860752i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−3.49109			−3.49109
$518$		0			0
$519$	0.110401			0.110401
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	0.616165	+	1.06723i	0.616165	+	1.06723i
$523$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$523$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$524$		0			0
$525$	1.07701	−	0.431171i	1.07701	−	0.431171i
$526$	−1.59435			−1.59435
$527$		0			0
$528$	3.62278	+	6.27484i	3.62278	+	6.27484i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	2.76167	+	2.17180i	2.76167	+	2.17180i
$533$		0			0
$534$	−2.21616	+	3.83850i	−2.21616	+	3.83850i
$535$		0			0
$536$		0			0
$537$	−1.13915	+	1.97306i	−1.13915	+	1.97306i
$538$		0			0
$539$	−1.28656	+	1.22673i	−1.28656	+	1.22673i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	−1.13915	−	1.97306i	−1.13915	−	1.97306i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−0.127639	+	0.221077i	−0.127639	+	0.221077i
$549$	−0.143677	−	0.248856i	−0.143677	−	0.248856i
$550$	−1.70566	−	2.95429i	−1.70566	−	2.95429i
$551$	1.21590	−	2.10601i	1.21590	−	2.10601i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$557$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$558$	0.480348	−	0.831988i	0.480348	−	0.831988i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.797176	−	1.38075i	0.797176	−	1.38075i
$563$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$563$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$564$	−3.05578	−	5.29276i	−3.05578	−	5.29276i
$565$		0			0
$566$	3.41133			3.41133
$567$	0.174512	−	1.21376i	0.174512	−	1.21376i
$568$		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
$573$	0.547014			0.547014
$574$		0			0
$575$		0			0
$576$	−0.558347	+	0.967085i	−0.558347	+	0.967085i
$577$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$577$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$578$	0.959493	+	1.66189i	0.959493	+	1.66189i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−3.74567			−3.74567
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.38884	−	2.40553i	1.38884	−	2.40553i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−2.98596	−	0.876756i	−2.98596	−	0.876756i
$589$	−1.89578			−1.89578
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$	−2.58876			−2.58876
$595$		0			0
$596$		0			0
$597$	−0.0552004	+	0.0956100i	−0.0552004	+	0.0956100i
$598$		0			0
$599$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$599$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$600$	1.87283	−	3.24384i	1.87283	−	3.24384i
$601$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$601$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$602$		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	$0.666667\pi$
$608$	4.60149			4.60149
$609$	−0.306550	+	2.13210i	−0.306550	+	2.13210i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$613$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$614$		0			0
$615$		0			0
$616$	−0.816827	+	5.68116i	−0.816827	+	5.68116i
$617$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$617$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$618$		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
$622$	1.91899			1.91899
$623$	−1.84833	+	0.739959i	−1.84833	+	0.739959i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$	−1.35052	−	2.33917i	−1.35052	−	2.33917i
$628$	−0.632425	+	1.09539i	−0.632425	+	1.09539i
$629$		0			0
$630$		0			0
$631$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$631$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$632$		0			0
$633$	−0.273507	−	0.473728i	−0.273507	−	0.473728i
$634$	1.11312	+	1.92798i	1.11312	+	1.92798i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	6.33393			6.33393
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	−1.74994	+	3.03099i	−1.74994	+	3.03099i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	−1.97959	−	3.42875i	−1.97959	−	3.42875i
$649$		0			0
$650$		0			0
$651$	1.55894	−	0.624106i	1.55894	−	0.624106i
$652$		0			0
$653$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$653$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.536330	−	3.73026i	0.536330	−	3.73026i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		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
$668$	−1.34125	−	2.32312i	−1.34125	−	2.32312i
$669$	1.15486	+	2.00028i	1.15486	+	2.00028i
$670$		0			0
$671$	−1.47694			−1.47694
$672$	−3.78391	+	1.51485i	−3.78391	+	1.51485i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−1.25667	+	2.17661i	−1.25667	+	2.17661i
$675$	0.379436	+	0.657203i	0.379436	+	0.657203i
$676$	−1.34125	−	2.32312i	−1.34125	−	2.32312i
$677$	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
$679$	−1.32254	−	1.04006i	−1.32254	−	1.04006i
$680$		0			0
$681$		0			0
$682$	−2.46889	−	4.27625i	−2.46889	−	4.27625i
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	0.607569	−	1.05234i	0.607569	−	1.05234i
$685$		0			0
$686$	−1.11312	−	1.56316i	−1.11312	−	1.56316i
$687$	2.15402			2.15402
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	0.255278			0.255278
$693$	0.483291	+	0.380064i	0.483291	+	0.380064i
$694$		0			0
$695$		0			0
$696$	3.47736	+	6.02296i	3.47736	+	6.02296i
$697$		0			0
$698$		0			0
$699$	1.67923			1.67923
$700$	2.49035	−	0.996987i	2.49035	−	0.996987i
$701$	−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
$704$	2.86979	+	4.97062i	2.86979	+	4.97062i
$705$		0			0
$706$	3.01685			3.01685
$707$		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
$712$	−3.21409	+	5.56696i	−3.21409	+	5.56696i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−2.63403	+	4.56227i	−2.63403	+	4.56227i
$717$	−0.672932	−	1.16555i	−0.672932	−	1.16555i
$718$	1.38884	+	2.40553i	1.38884	+	2.40553i
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$	−1.37279			−1.37279
$723$		0			0
$724$	−2.63403	−	4.56227i	−2.63403	−	4.56227i
$725$	−0.928368	−	1.60798i	−0.928368	−	1.60798i
$726$	2.40447	−	4.16466i	2.40447	−	4.16466i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.456265			0.456265
$730$		0			0
$731$		0			0
$732$	−1.29278	−	2.23916i	−1.29278	−	2.23916i
$733$	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	$0.0606061\pi$
$734$	1.25528			1.25528
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$744$	2.71087	−	4.69536i	2.71087	−	4.69536i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.45949	+	0.584293i	−1.45949	+	0.584293i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	−3.44985	−	5.97531i	−3.44985	−	5.97531i
$753$	0.165101	+	0.285964i	0.165101	+	0.285964i
$754$		0			0
$755$		0			0
$756$	0.289707	−	2.01496i	0.289707	−	2.01496i
$757$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$757$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$758$		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	$0.303030\pi$
$762$	−0.211858			−0.211858
$763$		0			0
$764$	1.26485			1.26485
$765$		0			0
$766$	1.78153	+	3.08569i	1.78153	+	3.08569i
$767$		0			0
$768$	−1.11312	+	1.92798i	−1.11312	+	1.92798i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−0.723734	+	1.25354i	−0.723734	+	1.25354i
$776$	−5.43232			−5.43232
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.40903			−1.40903
$784$	−3.37102	−	0.989821i	−3.37102	−	0.989821i
$785$		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
$789$	−0.481929	+	0.834725i	−0.481929	+	0.834725i
$790$		0			0
$791$		0			0
$792$	1.98511			1.98511
$793$		0			0
$794$	−1.91030	−	3.30873i	−1.91030	−	3.30873i
$795$		0			0
$796$	−0.127639	+	0.221077i	−0.127639	+	0.221077i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	2.70690	−	1.08368i	2.70690	−	1.08368i
$799$		0			0
$800$	1.75667	−	3.04264i	1.75667	−	3.04264i
$801$	0.344298	+	0.596342i	0.344298	+	0.596342i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$809$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−0.708829	+	4.93001i	−0.708829	+	4.93001i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.182618			−0.182618
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	0.105929	+	0.183474i	0.105929	+	0.183474i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−2.06230			−2.06230
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−3.12278	−	5.40881i	−3.12278	−	5.40881i
$837$	0.549222	+	0.951280i	0.549222	+	0.951280i
$838$	0.452418	−	0.783611i	0.452418	−	0.783611i
$839$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$839$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$840$		0			0
$841$	2.44747			2.44747
$842$	−1.50842	+	2.61267i	−1.50842	+	2.61267i
$843$	−0.481929	−	0.834725i	−0.481929	−	0.834725i
$844$	−0.632425	−	1.09539i	−0.632425	−	1.09539i
$845$		0			0
$846$	−1.30343			−1.30343
$847$	2.00538	−	0.802833i	2.00538	−	0.802833i
$848$		0			0
$849$	1.03115	−	1.78600i	1.03115	−	1.78600i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$853$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$854$	0.226900	−	1.57812i	0.226900	−	1.57812i
$855$		0			0
$856$	−2.53794	+	4.39583i	−2.53794	+	4.39583i
$857$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$857$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$858$		0			0
$859$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$859$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$860$		0			0
$861$		0			0
$862$	3.82059			3.82059
$863$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$863$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$864$	−1.33309	−	2.30897i	−1.33309	−	2.30897i
$865$		0			0
$866$	−0.627639	+	1.08710i	−0.627639	+	1.08710i
$867$	1.16011			1.16011
$868$	3.60471	−	1.44311i	3.60471	−	1.44311i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.290959	+	0.503956i	−0.290959	+	0.503956i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$877$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$878$		0			0
$879$	−0.839614	−	1.45425i	−0.839614	−	1.45425i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.480348	+	0.458011i	−0.480348	+	0.458011i
$883$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$883$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$884$		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
$889$	−0.0748038	−	0.0588264i	−0.0748038	−	0.0588264i
$890$		0			0
$891$	−1.08993	+	1.88781i	−1.08993	+	1.88781i
$892$	2.67036	+	4.62520i	2.67036	+	4.62520i
$893$	1.28605	+	2.22751i	1.28605	+	2.22751i
$894$		0			0
$895$		0			0
$896$	−2.49035	+	0.996987i	−2.49035	+	0.996987i
$897$		0			0
$898$	1.88431	−	3.26372i	1.88431	−	3.26372i
$899$	−1.34378	−	2.32750i	−1.34378	−	2.32750i
$900$	−0.463891	−	0.803483i	−0.463891	−	0.803483i
$901$		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
$911$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$911$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$912$	2.66913	−	4.62306i	2.66913	−	4.62306i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	4.98071			4.98071
$917$		0			0
$918$		0			0
$919$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$919$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$920$		0			0
$921$		0			0
$922$	−1.25667	+	2.17661i	−1.25667	+	2.17661i
$923$		0			0
$924$	4.34855	+	3.41974i	4.34855	+	3.41974i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	3.26167	+	5.64938i	3.26167	+	5.64938i
$929$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$929$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$930$		0			0
$931$	1.25667	+	0.368991i	1.25667	+	0.368991i
$932$	3.88284			3.88284
$933$	0.580057	−	1.00469i	0.580057	−	1.00469i
$934$	−1.50842	−	2.61267i	−1.50842	−	2.61267i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$	0.524856	+	0.909078i	0.524856	+	0.909078i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$947$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$948$		0			0
$949$		0			0
$950$	−1.25667	+	2.17661i	−1.25667	+	2.17661i
$951$	1.34586			1.34586
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−1.55601	−	2.69508i	−1.55601	−	2.69508i
$957$	1.91457	−	3.31614i	1.91457	−	3.31614i
$958$		0			0
$959$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i
$960$		0			0
$961$	−0.547582	+	0.948440i	−0.547582	+	0.948440i
$962$		0			0
$963$	0.271868	+	0.470888i	0.271868	+	0.470888i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$967$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$968$	3.48719	−	6.03999i	3.48719	−	6.03999i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−1.78040			−1.78040
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.45949	−	2.52792i	−1.45949	−	2.52792i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	3.53924			3.53924
$980$		0			0
$981$		0			0
$982$	−0.959493	+	1.66189i	−0.959493	+	1.66189i
$983$		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
$987$	−1.79086	−	1.40835i	−1.79086	−	1.40835i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$	2.54272	−	4.40412i	2.54272	−	4.40412i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$997$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1169.1.f.c.667.9	yes	20
7.4	even	3	inner	1169.1.f.c.333.9	✓	20
167.166	odd	2	CM	1169.1.f.c.667.9	yes	20
1169.333	odd	6	inner	1169.1.f.c.333.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1169.1.f.c.333.9	✓	20	7.4	even	3	inner
1169.1.f.c.333.9	✓	20	1169.333	odd	6	inner
1169.1.f.c.667.9	yes	20	1.1	even	1	trivial
1169.1.f.c.667.9	yes	20	167.166	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1169 = 7 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1169.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.959493 - 1.66189i) q^{2} +(-0.580057 - 1.00469i) q^{3} +(-1.34125 - 2.32312i) q^{4} -2.22624 q^{6} +(-0.786053 - 0.618159i) q^{7} -3.22871 q^{8} +(-0.172932 + 0.299527i) q^{9} +O(q^{10})\)	\(q+(0.959493 - 1.66189i) q^{2} +(-0.580057 - 1.00469i) q^{3} +(-1.34125 - 2.32312i) q^{4} -2.22624 q^{6} +(-0.786053 - 0.618159i) q^{7} -3.22871 q^{8} +(-0.172932 + 0.299527i) q^{9} +(0.888835 + 1.53951i) q^{11} +(-1.55601 + 2.69508i) q^{12} +(-1.78153 + 0.713215i) q^{14} +(-1.75667 + 3.04264i) q^{16} +(0.331854 + 0.574788i) q^{18} +(0.654861 - 1.13425i) q^{19} +(-0.165101 + 1.14831i) q^{21} +3.41133 q^{22} +(1.87283 + 3.24384i) q^{24} +(-0.500000 - 0.866025i) q^{25} -0.758872 q^{27} +(-0.381761 + 2.65520i) q^{28} +1.85674 q^{29} +(-0.723734 - 1.25354i) q^{31} +(1.75667 + 3.04264i) q^{32} +(1.03115 - 1.78600i) q^{33} +0.927783 q^{36} +(-1.25667 - 2.17661i) q^{38} +(1.74994 + 1.37617i) q^{42} +(2.38431 - 4.12974i) q^{44} +(-0.981929 + 1.70075i) q^{47} +4.07587 q^{48} +(0.235759 + 0.971812i) q^{49} -1.91899 q^{50} +(-0.728132 + 1.26116i) q^{54} +(2.53794 + 1.99585i) q^{56} -1.51943 q^{57} +(1.78153 - 3.08569i) q^{58} +(-0.415415 + 0.719520i) q^{61} -2.77767 q^{62} +(0.321089 - 0.128545i) q^{63} +3.22871 q^{64} +(-1.97876 - 3.42732i) q^{66} +(0.558347 - 0.967085i) q^{72} +(-0.580057 + 1.00469i) q^{75} -3.51334 q^{76} +(0.252989 - 1.75958i) q^{77} +(0.613121 + 1.06196i) q^{81} +(2.88909 - 1.15662i) q^{84} +(-1.07701 - 1.86544i) q^{87} +(-2.86979 - 4.97062i) q^{88} +(0.995472 - 1.72421i) q^{89} +(-0.839614 + 1.45425i) q^{93} +(1.88431 + 3.26372i) q^{94} +(2.03794 - 3.52981i) q^{96} +1.68251 q^{97} +(1.84125 + 0.540641i) q^{98} -0.614832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9} - q^{11} - 3 q^{12} + 2 q^{14} - 6 q^{16} + 2 q^{19} + 2 q^{21} + 4 q^{22} - 4 q^{24} - 10 q^{25} - 2 q^{27} - 6 q^{28} + 2 q^{29} - q^{31} + 6 q^{32} + q^{33} + 22 q^{36} + 4 q^{38} + 20 q^{42} + 8 q^{44} - q^{47} - 12 q^{48} + q^{49} - 4 q^{50} - 20 q^{54} + 4 q^{56} + 4 q^{57} - 2 q^{58} + 2 q^{61} - 18 q^{62} + 8 q^{64} + 2 q^{66} + 11 q^{72} - q^{75} - 12 q^{76} + 2 q^{77} - 12 q^{81} + 19 q^{84} + q^{87} - 4 q^{88} - q^{89} + q^{93} - 2 q^{94} - 6 q^{96} - 4 q^{97} + 18 q^{98}+O(q^{100}) \) 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9 - q^11 - 3 * q^12 + 2 * q^14 - 6 * q^16 + 2 * q^19 + 2 * q^21 + 4 * q^22 - 4 * q^24 - 10 * q^25 - 2 * q^27 - 6 * q^28 + 2 * q^29 - q^31 + 6 * q^32 + q^33 + 22 * q^36 + 4 * q^38 + 20 * q^42 + 8 * q^44 - q^47 - 12 * q^48 + q^49 - 4 * q^50 - 20 * q^54 + 4 * q^56 + 4 * q^57 - 2 * q^58 + 2 * q^61 - 18 * q^62 + 8 * q^64 + 2 * q^66 + 11 * q^72 - q^75 - 12 * q^76 + 2 * q^77 - 12 * q^81 + 19 * q^84 + q^87 - 4 * q^88 - q^89 + q^93 - 2 * q^94 - 6 * q^96 - 4 * q^97 + 18 * q^98

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