LMFDB - Embedded newform 1169.1.f.c.667.10

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			667.10
Root			$0.0475819 + 0.998867i$ of defining polynomial
Character	$\chi$	$=$	1169.667
Dual form			1169.1.f.c.333.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.959493 - 1.66189i) q^{2} +(0.995472 + 1.72421i) q^{3} +(-1.34125 - 2.32312i) q^{4} +3.82059 q^{6} +(0.928368 + 0.371662i) q^{7} -3.22871 q^{8} +(-1.48193 + 2.56678i) q^{9} +O(q^{10})$	\(q+(0.959493 - 1.66189i) q^{2} +(0.995472 + 1.72421i) q^{3} +(-1.34125 - 2.32312i) q^{4} +3.82059 q^{6} +(0.928368 + 0.371662i) q^{7} -3.22871 q^{8} +(-1.48193 + 2.56678i) q^{9} +(-0.0475819 - 0.0824143i) q^{11} +(2.67036 - 4.62520i) q^{12} +(1.50842 - 1.18624i) q^{14} +(-1.75667 + 3.04264i) q^{16} +(2.84380 + 4.92561i) q^{18} +(0.654861 - 1.13425i) q^{19} +(0.283341 + 1.97068i) q^{21} -0.182618 q^{22} +(-3.21409 - 5.56696i) q^{24} +(-0.500000 - 0.866025i) q^{25} -3.90993 q^{27} +(-0.381761 - 2.65520i) q^{28} -1.57211 q^{29} +(-0.235759 - 0.408346i) q^{31} +(1.75667 + 3.04264i) q^{32} +(0.0947329 - 0.164082i) q^{33} +7.95057 q^{36} +(-1.25667 - 2.17661i) q^{38} +(3.54692 + 1.41997i) q^{42} +(-0.127639 + 0.221077i) q^{44} +(0.327068 - 0.566498i) q^{47} -6.99486 q^{48} +(0.723734 + 0.690079i) q^{49} -1.91899 q^{50} +(-3.75155 + 6.49788i) q^{54} +(-2.99743 - 1.19999i) q^{56} +2.60758 q^{57} +(-1.50842 + 2.61267i) q^{58} +(-0.415415 + 0.719520i) q^{61} -0.904836 q^{62} +(-2.32975 + 1.83214i) q^{63} +3.22871 q^{64} +(-0.181791 - 0.314872i) q^{66} +(4.78471 - 8.28737i) q^{72} +(0.995472 - 1.72421i) q^{75} -3.51334 q^{76} +(-0.0135432 - 0.0941952i) q^{77} +(-2.41030 - 4.17476i) q^{81} +(4.19809 - 3.30141i) q^{84} +(-1.56499 - 2.71064i) q^{87} +(0.153628 + 0.266092i) q^{88} +(-0.580057 + 1.00469i) q^{89} +(0.469383 - 0.812995i) q^{93} +(-0.627639 - 1.08710i) q^{94} +(-3.49743 + 6.05772i) q^{96} +1.68251 q^{97} +(1.84125 - 0.540641i) q^{98} +0.282052 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.242424\pi$
$3$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$3$	0.995472	+	1.72421i	0.995472	+	1.72421i	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	$0.666667\pi$
$6$	3.82059			3.82059
$7$	0.928368	+	0.371662i	0.928368	+	0.371662i
$7$	0.928368	+	0.371662i	0.928368	+	0.371662i
$8$	−3.22871			−3.22871
$9$	−1.48193	+	2.56678i	−1.48193	+	2.56678i
$10$		0			0
$11$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$11$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$12$	2.67036	−	4.62520i	2.67036	−	4.62520i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	1.50842	−	1.18624i	1.50842	−	1.18624i
$15$		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	$0.333333\pi$
$18$	2.84380	+	4.92561i	2.84380	+	4.92561i
$19$	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
$21$	0.283341	+	1.97068i	0.283341	+	1.97068i
$22$	−0.182618			−0.182618
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−3.21409	−	5.56696i	−3.21409	−	5.56696i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	−3.90993			−3.90993
$28$	−0.381761	−	2.65520i	−0.381761	−	2.65520i
$29$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$29$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$30$		0			0
$31$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$31$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$32$	1.75667	+	3.04264i	1.75667	+	3.04264i
$33$	0.0947329	−	0.164082i	0.0947329	−	0.164082i
$34$		0			0
$35$		0			0
$36$	7.95057			7.95057
$37$		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			$\pi$
$42$	3.54692	+	1.41997i	3.54692	+	1.41997i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−0.127639	+	0.221077i	−0.127639	+	0.221077i
$45$		0			0
$46$		0			0
$47$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$47$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$48$	−6.99486			−6.99486
$49$	0.723734	+	0.690079i	0.723734	+	0.690079i
$50$	−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	$0.333333\pi$
$54$	−3.75155	+	6.49788i	−3.75155	+	6.49788i
$55$		0			0
$56$	−2.99743	−	1.19999i	−2.99743	−	1.19999i
$57$	2.60758			2.60758
$58$	−1.50842	+	2.61267i	−1.50842	+	2.61267i
$59$		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	$0.303030\pi$
$62$	−0.904836			−0.904836
$63$	−2.32975	+	1.83214i	−2.32975	+	1.83214i
$64$	3.22871			3.22871
$65$		0			0
$66$	−0.181791	−	0.314872i	−0.181791	−	0.314872i
$67$		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			$\pi$
$72$	4.78471	−	8.28737i	4.78471	−	8.28737i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	0.995472	−	1.72421i	0.995472	−	1.72421i
$76$	−3.51334			−3.51334
$77$	−0.0135432	−	0.0941952i	−0.0135432	−	0.0941952i
$78$		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
$81$	−2.41030	−	4.17476i	−2.41030	−	4.17476i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	4.19809	−	3.30141i	4.19809	−	3.30141i
$85$		0			0
$86$		0			0
$87$	−1.56499	−	2.71064i	−1.56499	−	2.71064i
$88$	0.153628	+	0.266092i	0.153628	+	0.266092i
$89$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$89$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.469383	−	0.812995i	0.469383	−	0.812995i
$94$	−0.627639	−	1.08710i	−0.627639	−	1.08710i
$95$		0			0
$96$	−3.49743	+	6.05772i	−3.49743	+	6.05772i
$97$	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
$99$	0.282052			0.282052
$100$	−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
$107$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$107$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$108$	5.24421	+	9.08323i	5.24421	+	9.08323i
$109$		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
$112$	−2.76167	+	2.17180i	−2.76167	+	2.17180i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	2.50196	−	4.33352i	2.50196	−	4.33352i
$115$		0			0
$116$	2.10859	+	3.65219i	2.10859	+	3.65219i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.495472	−	0.858183i	0.495472	−	0.858183i
$122$	0.797176	+	1.38075i	0.797176	+	1.38075i
$123$		0			0
$124$	−0.632425	+	1.09539i	−0.632425	+	1.09539i
$125$		0			0
$126$	0.809430	+	5.62971i	0.809430	+	5.62971i
$127$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$127$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$128$	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	$0.666667\pi$
$132$	−0.508243			−0.508243
$133$	1.02951	−	0.809616i	1.02951	−	0.809616i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$137$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	1.30235			1.30235
$142$		0			0
$143$		0			0
$144$	−5.20652	−	9.01795i	−5.20652	−	9.01795i
$145$		0			0
$146$		0			0
$147$	−0.469383	+	1.93482i	−0.469383	+	1.93482i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−1.91030	−	3.30873i	−1.91030	−	3.30873i
$151$		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
$154$	−0.169537	−	0.0678723i	−0.169537	−	0.0678723i
$155$		0			0
$156$		0			0
$157$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$157$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−9.25065			−9.25065
$163$		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
$168$	−0.914825	−	6.36275i	−0.914825	−	6.36275i
$169$	1.00000			1.00000
$170$		0			0
$171$	1.94091	+	3.36176i	1.94091	+	3.36176i
$172$		0			0
$173$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$173$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$174$	−6.00638			−6.00638
$175$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$176$	0.334343			0.334343
$177$		0			0
$178$	1.11312	+	1.92798i	1.11312	+	1.92798i
$179$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$179$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$180$		0			0
$181$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$181$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$182$		0			0
$183$	−1.65414			−1.65414
$184$		0			0
$185$		0			0
$186$	−0.900739	−	1.56013i	−0.900739	−	1.56013i
$187$		0			0
$188$	−1.75472			−1.75472
$189$	−3.62985	−	1.45317i	−3.62985	−	1.45317i
$190$		0			0
$191$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$191$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$192$	3.21409	+	5.56696i	3.21409	+	5.56696i
$193$		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
$196$	0.632425	−	2.60689i	0.632425	−	2.60689i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.270627	−	0.468740i	0.270627	−	0.468740i
$199$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$199$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$200$	1.61435	+	2.79614i	1.61435	+	2.79614i
$201$		0			0
$202$		0			0
$203$	−1.45949	−	0.584293i	−1.45949	−	0.584293i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.124638			−0.124638
$210$		0			0
$211$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$211$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$212$		0			0
$213$		0			0
$214$	1.78153	+	3.08569i	1.78153	+	3.08569i
$215$		0			0
$216$	12.6240			12.6240
$217$	−0.0671040	−	0.466718i	−0.0671040	−	0.466718i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$223$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$224$	0.500000	+	3.47758i	0.500000	+	3.47758i
$225$	2.96386			2.96386
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−3.49743	−	6.05772i	−3.49743	−	6.05772i
$229$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$229$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$230$		0			0
$231$	0.148930	−	0.117120i	0.148930	−	0.117120i
$232$	5.07587			5.07587
$233$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$233$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$239$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	−0.950804	−	1.64684i	−0.950804	−	1.64684i
$243$	2.84380	−	4.92561i	2.84380	−	4.92561i
$244$	2.22871			2.22871
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.761197	+	1.31843i	0.761197	+	1.31843i
$249$		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	$0.545455\pi$
$252$	7.38105	+	2.95493i	7.38105	+	2.95493i
$253$		0			0
$254$	−1.70566	+	2.95429i	−1.70566	+	2.95429i
$255$		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
$261$	2.32975	−	4.03524i	2.32975	−	4.03524i
$262$		0			0
$263$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$263$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$264$	−0.305865	+	0.529774i	−0.305865	+	0.529774i
$265$		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
$274$	3.41133			3.41133
$275$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$	1.39751			1.39751
$280$		0			0
$281$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$281$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$282$	1.24959	−	2.16436i	1.24959	−	2.16436i
$283$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$283$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−10.4130			−10.4130
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$	1.67489	+	2.90099i	1.67489	+	2.90099i
$292$		0			0
$293$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$293$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$294$	2.76509	+	2.63651i	2.76509	+	2.63651i
$295$		0			0
$296$		0			0
$297$	0.186042	+	0.322234i	0.186042	+	0.322234i
$298$		0			0
$299$		0			0
$300$	−5.34072			−5.34072
$301$		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			$\pi$
$308$	−0.200662	+	0.157802i	−0.200662	+	0.157802i
$309$		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	$0.666667\pi$
$314$	−2.77767			−2.77767
$315$		0			0
$316$		0			0
$317$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$317$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$318$		0			0
$319$	0.0748038	+	0.129564i	0.0748038	+	0.129564i
$320$		0			0
$321$	−3.69666			−3.69666
$322$		0			0
$323$		0			0
$324$	−6.46564	+	11.1988i	−6.46564	+	11.1988i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.514186	−	0.404360i	0.514186	−	0.404360i
$330$		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
$336$	−6.49380	−	2.59973i	−6.49380	−	2.59973i
$337$	−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
$341$	−0.0224357	+	0.0388598i	−0.0224357	+	0.0388598i
$342$	7.44917			7.44917
$343$	0.415415	+	0.909632i	0.415415	+	0.909632i
$344$		0			0
$345$		0			0
$346$	−1.70566	−	2.95429i	−1.70566	−	2.95429i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	−4.19809	+	7.27131i	−4.19809	+	7.27131i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−1.78153	−	0.713215i	−1.78153	−	0.713215i
$351$		0			0
$352$	0.167171	−	0.289549i	0.167171	−	0.289549i
$353$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$353$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$354$		0			0
$355$		0			0
$356$	3.11201			3.11201
$357$		0			0
$358$	1.25528			1.25528
$359$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$359$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$360$		0			0
$361$	−0.357685	−	0.619529i	−0.357685	−	0.619529i
$362$	−0.627639	+	1.08710i	−0.627639	+	1.08710i
$363$	1.97291			1.97291
$364$		0			0
$365$		0			0
$366$	−1.58713	+	2.74899i	−1.58713	+	2.74899i
$367$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$367$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.51825			−2.51825
$373$		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
$376$	−1.05601	+	1.82906i	−1.05601	+	1.82906i
$377$		0			0
$378$	−5.89784	+	4.63811i	−5.89784	+	4.63811i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.76962	−	3.06507i	−1.76962	−	3.06507i
$382$	1.38884	+	2.40553i	1.38884	+	2.40553i
$383$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$383$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$384$	5.34072			5.34072
$385$		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
$392$	−2.33673	−	2.22806i	−2.33673	−	2.22806i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.378303	−	0.655240i	−0.378303	−	0.655240i
$397$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$397$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$398$	3.41133			3.41133
$399$	2.42080	+	0.969140i	2.42080	+	0.969140i
$400$	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
$406$	−2.37140	+	1.86489i	−2.37140	+	1.86489i
$407$		0			0
$408$		0			0
$409$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$409$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$410$		0			0
$411$	−1.76962	+	3.06507i	−1.76962	+	3.06507i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−0.119589	+	0.207135i	−0.119589	+	0.207135i
$419$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$419$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$420$		0			0
$421$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$421$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$422$	1.38884	−	2.40553i	1.38884	−	2.40553i
$423$	0.969383	+	1.67902i	0.969383	+	1.67902i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.653077	+	0.513585i	−0.653077	+	0.513585i
$428$	4.98071			4.98071
$429$		0			0
$430$		0			0
$431$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$431$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$432$	6.86845	−	11.8965i	6.86845	−	11.8965i
$433$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$433$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$434$	−0.840021	−	0.336294i	−0.840021	−	0.336294i
$435$		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
$441$	−2.84380	+	0.835015i	−2.84380	+	0.835015i
$442$		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
$446$	1.11312	−	1.92798i	1.11312	−	1.92798i
$447$		0			0
$448$	2.99743	+	1.19999i	2.99743	+	1.19999i
$449$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$449$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$450$	2.84380	−	4.92561i	2.84380	−	4.92561i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−8.41912			−8.41912
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−1.50842	−	2.61267i	−1.50842	−	2.61267i
$459$		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	$0.727273\pi$
$462$	−0.0517432	−	0.359882i	−0.0517432	−	0.359882i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	2.76167	−	4.78335i	2.76167	−	4.78335i
$465$		0			0
$466$	0.452418	+	0.783611i	0.452418	+	0.783611i
$467$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$467$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.44091	−	2.49574i	1.44091	−	2.49574i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.30972			−1.30972
$476$		0			0
$477$		0			0
$478$	−1.91030	+	3.30873i	−1.91030	+	3.30873i
$479$		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
$484$	−2.65821			−2.65821
$485$		0			0
$486$	−5.45721	−	9.45217i	−5.45721	−	9.45217i
$487$		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
$496$	1.65660			1.65660
$497$		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
$501$	0.995472	+	1.72421i	0.995472	+	1.72421i
$502$	−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	$0.363636\pi$
$504$	7.52208	−	5.91543i	7.52208	−	5.91543i
$505$		0			0
$506$		0			0
$507$	0.995472	+	1.72421i	0.995472	+	1.72421i
$508$	2.38431	+	4.12974i	2.38431	+	4.12974i
$509$	−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
$513$	−2.56046	+	4.43485i	−2.56046	+	4.43485i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.0622501			−0.0622501
$518$		0			0
$519$	3.53924			3.53924
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	−4.47076	−	7.74358i	−4.47076	−	7.74358i
$523$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$523$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$524$		0			0
$525$	1.56499	−	1.23072i	1.56499	−	1.23072i
$526$	−1.59435			−1.59435
$527$		0			0
$528$	0.332829	+	0.576476i	0.332829	+	0.576476i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−3.26167	−	1.30578i	−3.26167	−	1.30578i
$533$		0			0
$534$	−2.21616	+	3.83850i	−2.21616	+	3.83850i
$535$		0			0
$536$		0			0
$537$	−0.651174	+	1.12787i	−0.651174	+	1.12787i
$538$		0			0
$539$	0.0224357	−	0.0924813i	0.0224357	−	0.0924813i
$540$		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
$543$	−0.651174	−	1.12787i	−0.651174	−	1.12787i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	2.38431	−	4.12974i	2.38431	−	4.12974i
$549$	−1.23123	−	2.13255i	−1.23123	−	2.13255i
$550$	0.0913090	+	0.158152i	0.0913090	+	0.158152i
$551$	−1.02951	+	1.78316i	−1.02951	+	1.78316i
$552$		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	$-0.484848\pi$
$558$	1.34090	−	2.32251i	1.34090	−	2.32251i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.797176	−	1.38075i	0.797176	−	1.38075i
$563$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$563$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$564$	−1.74678	−	3.02551i	−1.74678	−	3.02551i
$565$		0			0
$566$	−0.182618			−0.182618
$567$	−0.686042	−	4.77153i	−0.686042	−	4.77153i
$568$		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
$573$	−2.88183			−2.88183
$574$		0			0
$575$		0			0
$576$	−4.78471	+	8.28737i	−4.78471	+	8.28737i
$577$	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
$582$	6.42818			6.42818
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.452418	−	0.783611i	0.452418	−	0.783611i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	5.12438	−	1.50465i	5.12438	−	1.50465i
$589$	−0.617557			−0.617557
$590$		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	$0.666667\pi$
$594$	0.714024			0.714024
$595$		0			0
$596$		0			0
$597$	−1.76962	+	3.06507i	−1.76962	+	3.06507i
$598$		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	$0.787879\pi$
$600$	−3.21409	+	5.56696i	−3.21409	+	5.56696i
$601$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$601$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$602$		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
$609$	−0.445442	−	3.09812i	−0.445442	−	3.09812i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$613$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$614$		0			0
$615$		0			0
$616$	0.0437271	+	0.304129i	0.0437271	+	0.304129i
$617$	−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
$623$	−0.911911	+	0.717135i	−0.911911	+	0.717135i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$	−0.124074	−	0.214902i	−0.124074	−	0.214902i
$628$	−1.94142	+	3.36264i	−1.94142	+	3.36264i
$629$		0			0
$630$		0			0
$631$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$631$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$632$		0			0
$633$	1.44091	+	2.49574i	1.44091	+	2.49574i
$634$	−1.91030	−	3.30873i	−1.91030	−	3.30873i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.287095			0.287095
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	−3.54692	+	6.14344i	−3.54692	+	6.14344i
$643$		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	$0.333333\pi$
$648$	7.78214	+	13.4791i	7.78214	+	13.4791i
$649$		0			0
$650$		0			0
$651$	0.737920	−	0.580306i	0.737920	−	0.580306i
$652$		0			0
$653$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$653$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.178645	−	1.24250i	−0.178645	−	1.24250i
$659$		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
$671$	0.0790650			0.0790650
$672$	−5.49833	+	4.32393i	−5.49833	+	4.32393i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−1.25667	+	2.17661i	−1.25667	+	2.17661i
$675$	1.95496	+	3.38610i	1.95496	+	3.38610i
$676$	−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
$679$	1.56199	+	0.625325i	1.56199	+	0.625325i
$680$		0			0
$681$		0			0
$682$	0.0430538	+	0.0745714i	0.0430538	+	0.0745714i
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	5.20652	−	9.01795i	5.20652	−	9.01795i
$685$		0			0
$686$	1.91030	+	0.182411i	1.91030	+	0.182411i
$687$	3.12998			3.12998
$688$		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	$0.666667\pi$
$692$	−4.76861			−4.76861
$693$	0.261848	+	0.104828i	0.261848	+	0.104828i
$694$		0			0
$695$		0			0
$696$	5.05289	+	8.75186i	5.05289	+	8.75186i
$697$		0			0
$698$		0			0
$699$	−0.938766			−0.938766
$700$	−2.10859	+	1.65822i	−2.10859	+	1.65822i
$701$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$701$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$702$		0			0
$703$		0			0
$704$	−0.153628	−	0.266092i	−0.153628	−	0.266092i
$705$		0			0
$706$	−3.56305			−3.56305
$707$		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
$712$	1.87283	−	3.24384i	1.87283	−	3.24384i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.877362	−	1.51964i	0.877362	−	1.51964i
$717$	−1.98193	−	3.43280i	−1.98193	−	3.43280i
$718$	0.452418	+	0.783611i	0.452418	+	0.783611i
$719$		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
$724$	0.877362	+	1.51964i	0.877362	+	1.51964i
$725$	0.786053	+	1.36148i	0.786053	+	1.36148i
$726$	1.89300	−	3.27877i	1.89300	−	3.27877i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	6.50310			6.50310
$730$		0			0
$731$		0			0
$732$	2.21862	+	3.84276i	2.21862	+	3.84276i
$733$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$733$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$734$	−3.76861			−3.76861
$735$		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	$0.666667\pi$
$744$	−1.51550	+	2.62492i	−1.51550	+	2.62492i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.45949	+	1.14776i	−1.45949	+	1.14776i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	1.14910	+	1.99030i	1.14910	+	1.99030i
$753$	−0.283341	−	0.490761i	−0.283341	−	0.490761i
$754$		0			0
$755$		0			0
$756$	1.49266	+	10.3817i	1.49266	+	10.3817i
$757$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$757$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$758$		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	$0.303030\pi$
$762$	−6.79176			−6.79176
$763$		0			0
$764$	3.88284			3.88284
$765$		0			0
$766$	−1.50842	−	2.61267i	−1.50842	−	2.61267i
$767$		0			0
$768$	1.91030	−	3.30873i	1.91030	−	3.30873i
$769$		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
$775$	−0.235759	+	0.408346i	−0.235759	+	0.408346i
$776$	−5.43232			−5.43232
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	6.14682			6.14682
$784$	−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
$789$	0.827068	−	1.43252i	0.827068	−	1.43252i
$790$		0			0
$791$		0			0
$792$	−0.910663			−0.910663
$793$		0			0
$794$	1.11312	+	1.92798i	1.11312	+	1.92798i
$795$		0			0
$796$	2.38431	−	4.12974i	2.38431	−	4.12974i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	3.93334	−	3.09321i	3.93334	−	3.09321i
$799$		0			0
$800$	1.75667	−	3.04264i	1.75667	−	3.04264i
$801$	−1.71921	−	2.97775i	−1.71921	−	2.97775i
$802$		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			$\pi$
$812$	0.600168	+	4.17426i	0.600168	+	4.17426i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	3.41133			3.41133
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	3.39588	+	5.88183i	3.39588	+	5.88183i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−0.189466			−0.189466
$826$		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
$836$	0.167171	+	0.289549i	0.167171	+	0.289549i
$837$	0.921801	+	1.59661i	0.921801	+	1.59661i
$838$	1.38884	−	2.40553i	1.38884	−	2.40553i
$839$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$839$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$840$		0			0
$841$	1.47152			1.47152
$842$	1.78153	−	3.08569i	1.78153	−	3.08569i
$843$	0.827068	+	1.43252i	0.827068	+	1.43252i
$844$	−1.94142	−	3.36264i	−1.94142	−	3.36264i
$845$		0			0
$846$	3.72046			3.72046
$847$	0.778934	−	0.612561i	0.778934	−	0.612561i
$848$		0			0
$849$	0.0947329	−	0.164082i	0.0947329	−	0.164082i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$853$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$854$	0.226900	+	1.57812i	0.226900	+	1.57812i
$855$		0			0
$856$	2.99743	−	5.19170i	2.99743	−	5.19170i
$857$	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
$862$	−2.22624			−2.22624
$863$	−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	$0.848485\pi$
$864$	−6.86845	−	11.8965i	−6.86845	−	11.8965i
$865$		0			0
$866$	1.88431	−	3.26372i	1.88431	−	3.26372i
$867$	−1.99094			−1.99094
$868$	−0.994239	+	0.781878i	−0.994239	+	0.781878i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−2.49336	+	4.31862i	−2.49336	+	4.31862i
$874$		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
$879$	0.469383	+	0.812995i	0.469383	+	0.812995i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.34090	+	5.52728i	−1.34090	+	5.52728i
$883$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$883$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$884$		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
$889$	−1.65033	−	0.660694i	−1.65033	−	0.660694i
$890$		0			0
$891$	−0.229373	+	0.397286i	−0.229373	+	0.397286i
$892$	−1.55601	−	2.69508i	−1.55601	−	2.69508i
$893$	−0.428368	−	0.741955i	−0.428368	−	0.741955i
$894$		0			0
$895$		0			0
$896$	2.10859	−	1.65822i	2.10859	−	1.65822i
$897$		0			0
$898$	−0.627639	+	1.08710i	−0.627639	+	1.08710i
$899$	0.370638	+	0.641964i	0.370638	+	0.641964i
$900$	−3.97528	−	6.88539i	−3.97528	−	6.88539i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$907$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$911$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$912$	−4.58066	+	7.93393i	−4.58066	+	7.93393i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−4.21719			−4.21719
$917$		0			0
$918$		0			0
$919$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$919$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$920$		0			0
$921$		0			0
$922$	−1.25667	+	2.17661i	−1.25667	+	2.17661i
$923$		0			0
$924$	−0.471837	−	0.188895i	−0.471837	−	0.188895i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−2.76167	−	4.78335i	−2.76167	−	4.78335i
$929$	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
$932$	1.26485			1.26485
$933$	−0.995472	+	1.72421i	−0.995472	+	1.72421i
$934$	1.78153	+	3.08569i	1.78153	+	3.08569i
$935$		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	$0.333333\pi$
$942$	−2.76509	−	4.78928i	−2.76509	−	4.78928i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$947$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$948$		0			0
$949$		0			0
$950$	−1.25667	+	2.17661i	−1.25667	+	2.17661i
$951$	3.96386			3.96386
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	2.67036	+	4.62520i	2.67036	+	4.62520i
$957$	−0.148930	+	0.257955i	−0.148930	+	0.257955i
$958$		0			0
$959$	0.252989	+	1.75958i	0.252989	+	1.75958i
$960$		0			0
$961$	0.388835	−	0.673483i	0.388835	−	0.673483i
$962$		0			0
$963$	−2.75155	−	4.76582i	−2.75155	−	4.76582i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$967$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$968$	−1.59973	+	2.77082i	−1.59973	+	2.77082i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−15.2570			−15.2570
$973$		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
$979$	0.110401			0.110401
$980$		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
$987$	1.20906	+	0.484034i	1.20906	+	0.484034i
$988$		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	$0.333333\pi$
$992$	0.828301	−	1.43466i	0.828301	−	1.43466i
$993$		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.667.10	yes	20
7.4	even	3	inner	1169.1.f.c.333.10	✓	20
167.166	odd	2	CM	1169.1.f.c.667.10	yes	20
1169.333	odd	6	inner	1169.1.f.c.333.10	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1169.1.f.c.333.10	✓	20	7.4	even	3	inner
1169.1.f.c.333.10	✓	20	1169.333	odd	6	inner
1169.1.f.c.667.10	yes	20	1.1	even	1	trivial
1169.1.f.c.667.10	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)

\(f(q)\)	\(=\)	\(q+(0.959493 - 1.66189i) q^{2} +(0.995472 + 1.72421i) q^{3} +(-1.34125 - 2.32312i) q^{4} +3.82059 q^{6} +(0.928368 + 0.371662i) q^{7} -3.22871 q^{8} +(-1.48193 + 2.56678i) q^{9} +O(q^{10})\)	\(q+(0.959493 - 1.66189i) q^{2} +(0.995472 + 1.72421i) q^{3} +(-1.34125 - 2.32312i) q^{4} +3.82059 q^{6} +(0.928368 + 0.371662i) q^{7} -3.22871 q^{8} +(-1.48193 + 2.56678i) q^{9} +(-0.0475819 - 0.0824143i) q^{11} +(2.67036 - 4.62520i) q^{12} +(1.50842 - 1.18624i) q^{14} +(-1.75667 + 3.04264i) q^{16} +(2.84380 + 4.92561i) q^{18} +(0.654861 - 1.13425i) q^{19} +(0.283341 + 1.97068i) q^{21} -0.182618 q^{22} +(-3.21409 - 5.56696i) q^{24} +(-0.500000 - 0.866025i) q^{25} -3.90993 q^{27} +(-0.381761 - 2.65520i) q^{28} -1.57211 q^{29} +(-0.235759 - 0.408346i) q^{31} +(1.75667 + 3.04264i) q^{32} +(0.0947329 - 0.164082i) q^{33} +7.95057 q^{36} +(-1.25667 - 2.17661i) q^{38} +(3.54692 + 1.41997i) q^{42} +(-0.127639 + 0.221077i) q^{44} +(0.327068 - 0.566498i) q^{47} -6.99486 q^{48} +(0.723734 + 0.690079i) q^{49} -1.91899 q^{50} +(-3.75155 + 6.49788i) q^{54} +(-2.99743 - 1.19999i) q^{56} +2.60758 q^{57} +(-1.50842 + 2.61267i) q^{58} +(-0.415415 + 0.719520i) q^{61} -0.904836 q^{62} +(-2.32975 + 1.83214i) q^{63} +3.22871 q^{64} +(-0.181791 - 0.314872i) q^{66} +(4.78471 - 8.28737i) q^{72} +(0.995472 - 1.72421i) q^{75} -3.51334 q^{76} +(-0.0135432 - 0.0941952i) q^{77} +(-2.41030 - 4.17476i) q^{81} +(4.19809 - 3.30141i) q^{84} +(-1.56499 - 2.71064i) q^{87} +(0.153628 + 0.266092i) q^{88} +(-0.580057 + 1.00469i) q^{89} +(0.469383 - 0.812995i) q^{93} +(-0.627639 - 1.08710i) q^{94} +(-3.49743 + 6.05772i) q^{96} +1.68251 q^{97} +(1.84125 - 0.540641i) q^{98} +0.282052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 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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