LMFDB - Embedded newform 1169.1.f.c.333.8

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			333.8
Root			$-0.786053 + 0.618159i$ of defining polynomial
Character	$\chi$	$=$	1169.333
Dual form			1169.1.f.c.667.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.654861 + 1.13425i) q^{2} +(-0.235759 + 0.408346i) q^{3} +(-0.357685 + 0.619529i) q^{4} -0.617557 q^{6} +(0.580057 - 0.814576i) q^{7} +0.372786 q^{8} +(0.388835 + 0.673483i) q^{9} +O(q^{10})$	\(q+(0.654861 + 1.13425i) q^{2} +(-0.235759 + 0.408346i) q^{3} +(-0.357685 + 0.619529i) q^{4} -0.617557 q^{6} +(0.580057 - 0.814576i) q^{7} +0.372786 q^{8} +(0.388835 + 0.673483i) q^{9} +(0.786053 - 1.36148i) q^{11} +(-0.168655 - 0.292119i) q^{12} +(1.30379 + 0.124497i) q^{14} +(0.601808 + 1.04236i) q^{16} +(-0.509266 + 0.882075i) q^{18} +(-0.841254 - 1.45709i) q^{19} +(0.195876 + 0.428908i) q^{21} +2.05902 q^{22} +(-0.0878875 + 0.152226i) q^{24} +(-0.500000 + 0.866025i) q^{25} -0.838204 q^{27} +(0.297176 + 0.650724i) q^{28} -1.99094 q^{29} +(-0.981929 + 1.70075i) q^{31} +(-0.601808 + 1.04236i) q^{32} +(0.370638 + 0.641964i) q^{33} -0.556323 q^{36} +(1.10181 - 1.90839i) q^{38} +(-0.358218 + 0.503047i) q^{42} +(0.562319 + 0.973965i) q^{44} +(-0.0475819 - 0.0824143i) q^{47} -0.567526 q^{48} +(-0.327068 - 0.945001i) q^{49} -1.30972 q^{50} +(-0.548907 - 0.950734i) q^{54} +(0.216237 - 0.303662i) q^{56} +0.793332 q^{57} +(-1.30379 - 2.25823i) q^{58} +(0.959493 + 1.66189i) q^{61} -2.57211 q^{62} +(0.774150 + 0.0739223i) q^{63} -0.372786 q^{64} +(-0.485433 + 0.840794i) q^{66} +(0.144952 + 0.251065i) q^{72} +(-0.235759 - 0.408346i) q^{75} +1.20362 q^{76} +(-0.653077 - 1.43004i) q^{77} +(-0.191221 + 0.331205i) q^{81} +(-0.335783 - 0.0320633i) q^{84} +(0.469383 - 0.812995i) q^{87} +(0.293029 - 0.507542i) q^{88} +(-0.723734 - 1.25354i) q^{89} +(-0.462997 - 0.801934i) q^{93} +(0.0623191 - 0.107940i) q^{94} +(-0.283763 - 0.491492i) q^{96} -0.284630 q^{97} +(0.857685 - 0.989821i) q^{98} +1.22258 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1169\mathbb{Z}\right)^\times$.

$n$	$673$	$836$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$2$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$3$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$3$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$4$	−0.357685	+	0.619529i	−0.357685	+	0.619529i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−0.617557			−0.617557
$7$	0.580057	−	0.814576i	0.580057	−	0.814576i
$7$	0.580057	−	0.814576i	0.580057	−	0.814576i
$8$	0.372786			0.372786
$9$	0.388835	+	0.673483i	0.388835	+	0.673483i
$10$		0			0
$11$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$11$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$12$	−0.168655	−	0.292119i	−0.168655	−	0.292119i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	1.30379	+	0.124497i	1.30379	+	0.124497i
$15$		0			0
$16$	0.601808	+	1.04236i	0.601808	+	1.04236i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−0.509266	+	0.882075i	−0.509266	+	0.882075i
$19$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$19$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$20$		0			0
$21$	0.195876	+	0.428908i	0.195876	+	0.428908i
$22$	2.05902			2.05902
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−0.0878875	+	0.152226i	−0.0878875	+	0.152226i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−0.838204			−0.838204
$28$	0.297176	+	0.650724i	0.297176	+	0.650724i
$29$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$29$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$30$		0			0
$31$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$31$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$32$	−0.601808	+	1.04236i	−0.601808	+	1.04236i
$33$	0.370638	+	0.641964i	0.370638	+	0.641964i
$34$		0			0
$35$		0			0
$36$	−0.556323			−0.556323
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	1.10181	−	1.90839i	1.10181	−	1.90839i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−0.358218	+	0.503047i	−0.358218	+	0.503047i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.562319	+	0.973965i	0.562319	+	0.973965i
$45$		0			0
$46$		0			0
$47$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$47$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$48$	−0.567526			−0.567526
$49$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$50$	−1.30972			−1.30972
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	−0.548907	−	0.950734i	−0.548907	−	0.950734i
$55$		0			0
$56$	0.216237	−	0.303662i	0.216237	−	0.303662i
$57$	0.793332			0.793332
$58$	−1.30379	−	2.25823i	−1.30379	−	2.25823i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$61$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$62$	−2.57211			−2.57211
$63$	0.774150	+	0.0739223i	0.774150	+	0.0739223i
$64$	−0.372786			−0.372786
$65$		0			0
$66$	−0.485433	+	0.840794i	−0.485433	+	0.840794i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.144952	+	0.251065i	0.144952	+	0.251065i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	−0.235759	−	0.408346i	−0.235759	−	0.408346i
$76$	1.20362			1.20362
$77$	−0.653077	−	1.43004i	−0.653077	−	1.43004i
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−0.191221	+	0.331205i	−0.191221	+	0.331205i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−0.335783	−	0.0320633i	−0.335783	−	0.0320633i
$85$		0			0
$86$		0			0
$87$	0.469383	−	0.812995i	0.469383	−	0.812995i
$88$	0.293029	−	0.507542i	0.293029	−	0.507542i
$89$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$89$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.462997	−	0.801934i	−0.462997	−	0.801934i
$94$	0.0623191	−	0.107940i	0.0623191	−	0.107940i
$95$		0			0
$96$	−0.283763	−	0.491492i	−0.283763	−	0.491492i
$97$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$97$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$98$	0.857685	−	0.989821i	0.857685	−	0.989821i
$99$	1.22258			1.22258
$100$	−0.357685	−	0.619529i	−0.357685	−	0.619529i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$107$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$108$	0.299813	−	0.519291i	0.299813	−	0.519291i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	1.19817	+	0.114411i	1.19817	+	0.114411i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	0.519522	+	0.899839i	0.519522	+	0.899839i
$115$		0			0
$116$	0.712131	−	1.23345i	0.712131	−	1.23345i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.735759	−	1.27437i	−0.735759	−	1.27437i
$122$	−1.25667	+	2.17661i	−1.25667	+	2.17661i
$123$		0			0
$124$	−0.702443	−	1.21667i	−0.702443	−	1.21667i
$125$		0			0
$126$	0.423114	+	0.926490i	0.423114	+	0.926490i
$127$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$127$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$128$	0.357685	+	0.619529i	0.357685	+	0.619529i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−0.530287			−0.530287
$133$	−1.67489	−	0.159932i	−1.67489	−	0.159932i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$137$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	0.0448714			0.0448714
$142$		0			0
$143$		0			0
$144$	−0.468008	+	0.810614i	−0.468008	+	0.810614i
$145$		0			0
$146$		0			0
$147$	0.462997	+	0.0892353i	0.462997	+	0.0892353i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	0.308779	−	0.534820i	0.308779	−	0.534820i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−0.313607	−	0.543184i	−0.313607	−	0.543184i
$153$		0			0
$154$	1.19435	−	1.67723i	1.19435	−	1.67723i
$155$		0			0
$156$		0			0
$157$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$157$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.500894			−0.500894
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000			1.00000
$167$	1.00000			1.00000
$168$	0.0730196	+	0.159891i	0.0730196	+	0.159891i
$169$	1.00000			1.00000
$170$		0			0
$171$	0.654218	−	1.13314i	0.654218	−	1.13314i
$172$		0			0
$173$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$173$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$174$	1.22952			1.22952
$175$	0.415415	+	0.909632i	0.415415	+	0.909632i
$176$	1.89221			1.89221
$177$		0			0
$178$	0.947890	−	1.64179i	0.947890	−	1.64179i
$179$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$179$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$180$		0			0
$181$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$181$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$182$		0			0
$183$	−0.904836			−0.904836
$184$		0			0
$185$		0			0
$186$	0.606397	−	1.05031i	0.606397	−	1.05031i
$187$		0			0
$188$	0.0680774			0.0680774
$189$	−0.486206	+	0.682780i	−0.486206	+	0.682780i
$190$		0			0
$191$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$191$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$192$	0.0878875	−	0.152226i	0.0878875	−	0.152226i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	−0.186393	−	0.322842i	−0.186393	−	0.322842i
$195$		0			0
$196$	0.702443	+	0.135385i	0.702443	+	0.135385i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.800620	+	1.38672i	0.800620	+	1.38672i
$199$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$199$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$200$	−0.186393	+	0.322842i	−0.186393	+	0.322842i
$201$		0			0
$202$		0			0
$203$	−1.15486	+	1.62177i	−1.15486	+	1.62177i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.64508			−2.64508
$210$		0			0
$211$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$211$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$212$		0			0
$213$		0			0
$214$	0.759713	−	1.31586i	0.759713	−	1.31586i
$215$		0			0
$216$	−0.312470			−0.312470
$217$	0.815816	+	1.78639i	0.815816	+	1.78639i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$223$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$224$	0.500000	+	1.09485i	0.500000	+	1.09485i
$225$	−0.777671			−0.777671
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−0.283763	+	0.491492i	−0.283763	+	0.491492i
$229$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$229$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$230$		0			0
$231$	0.737920	+	0.0704628i	0.737920	+	0.0704628i
$232$	−0.742195			−0.742195
$233$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$233$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$239$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	0.963639	−	1.66907i	0.963639	−	1.66907i
$243$	−0.509266	−	0.882075i	−0.509266	−	0.882075i
$244$	−1.37279			−1.37279
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−0.366049	+	0.634015i	−0.366049	+	0.634015i
$249$		0			0
$250$		0			0
$251$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$251$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$252$	−0.322699	+	0.453167i	−0.322699	+	0.453167i
$253$		0			0
$254$	1.21590	+	2.10601i	1.21590	+	2.10601i
$255$		0			0
$256$	−0.654861	+	1.13425i	−0.654861	+	1.13425i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.774150	−	1.34087i	−0.774150	−	1.34087i
$262$		0			0
$263$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$263$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$264$	0.138169	+	0.239315i	0.138169	+	0.239315i
$265$		0			0
$266$	−0.915415	−	2.00448i	−0.915415	−	2.00448i
$267$	0.682507			0.682507
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$	−2.43181			−2.43181
$275$	0.786053	+	1.36148i	0.786053	+	1.36148i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	−1.52723			−1.52723
$280$		0			0
$281$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$281$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$282$	0.0293845	+	0.0508955i	0.0293845	+	0.0508955i
$283$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$283$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.936017			−0.936017
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$	0.0671040	−	0.116228i	0.0671040	−	0.116228i
$292$		0			0
$293$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$293$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$294$	0.201983	+	0.583592i	0.201983	+	0.583592i
$295$		0			0
$296$		0			0
$297$	−0.658873	+	1.14120i	−0.658873	+	1.14120i
$298$		0			0
$299$		0			0
$300$	0.337310			0.337310
$301$		0			0
$302$		0			0
$303$		0			0
$304$	1.01255	−	1.75378i	1.01255	−	1.75378i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	1.11955	+	0.106904i	1.11955	+	0.106904i
$309$		0			0
$310$		0			0
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	0.856736			0.856736
$315$		0			0
$316$		0			0
$317$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$317$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$318$		0			0
$319$	−1.56499	+	2.71064i	−1.56499	+	2.71064i
$320$		0			0
$321$	0.547014			0.547014
$322$		0			0
$323$		0			0
$324$	−0.136794	−	0.236934i	−0.136794	−	0.236934i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.0947329	−	0.00904590i	−0.0947329	−	0.00904590i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	0.654861	+	1.13425i	0.654861	+	1.13425i
$335$		0			0
$336$	−0.329198	+	0.462293i	−0.329198	+	0.462293i
$337$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$337$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$338$	0.654861	+	1.13425i	0.654861	+	1.13425i
$339$		0			0
$340$		0			0
$341$	1.54370	+	2.67376i	1.54370	+	2.67376i
$342$	1.71369			1.71369
$343$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$344$		0			0
$345$		0			0
$346$	1.21590	−	2.10601i	1.21590	−	2.10601i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	0.335783	+	0.581592i	0.335783	+	0.581592i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−0.759713	+	1.06687i	−0.759713	+	1.06687i
$351$		0			0
$352$	0.946106	+	1.63870i	0.946106	+	1.63870i
$353$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$353$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$354$		0			0
$355$		0			0
$356$	1.03548			1.03548
$357$		0			0
$358$	−0.124638			−0.124638
$359$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$359$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$360$		0			0
$361$	−0.915415	+	1.58555i	−0.915415	+	1.58555i
$362$	0.0623191	+	0.107940i	0.0623191	+	0.107940i
$363$	0.693847			0.693847
$364$		0			0
$365$		0			0
$366$	−0.592542	−	1.02631i	−0.592542	−	1.02631i
$367$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$367$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.662429			0.662429
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	−0.0177379	−	0.0307229i	−0.0177379	−	0.0307229i
$377$		0			0
$378$	−1.09284	−	0.104354i	−1.09284	−	0.104354i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.437742	+	0.758192i	−0.437742	+	0.758192i
$382$	−0.428368	+	0.741955i	−0.428368	+	0.741955i
$383$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$383$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$384$	−0.337310			−0.337310
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.101808	−	0.176336i	0.101808	−	0.176336i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−0.121926	−	0.352283i	−0.121926	−	0.352283i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.437299	+	0.757424i	−0.437299	+	0.757424i
$397$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$397$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$398$	−2.43181			−2.43181
$399$	0.460178	−	0.646229i	0.460178	−	0.646229i
$400$	−1.20362			−1.20362
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−2.59577	−	0.247866i	−2.59577	−	0.247866i
$407$		0			0
$408$		0			0
$409$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$409$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$410$		0			0
$411$	−0.437742	−	0.758192i	−0.437742	−	0.758192i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−1.73216	−	3.00019i	−1.73216	−	3.00019i
$419$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$419$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$420$		0			0
$421$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$421$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$422$	−0.428368	−	0.741955i	−0.428368	−	0.741955i
$423$	0.0370031	−	0.0640912i	0.0370031	−	0.0640912i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.91030	+	0.182411i	1.91030	+	0.182411i
$428$	0.829911			0.829911
$429$		0			0
$430$		0			0
$431$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$431$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$432$	−0.504437	−	0.873711i	−0.504437	−	0.873711i
$433$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$433$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$434$	−1.49197	+	2.09518i	−1.49197	+	2.09518i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	0.509266	−	0.587724i	0.509266	−	0.587724i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	0.947890	+	1.64179i	0.947890	+	1.64179i
$447$		0			0
$448$	−0.216237	+	0.303662i	−0.216237	+	0.303662i
$449$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$449$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$450$	−0.509266	−	0.882075i	−0.509266	−	0.882075i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	0.295743			0.295743
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−1.30379	+	2.25823i	−1.30379	+	2.25823i
$459$		0			0
$460$		0			0
$461$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$461$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$462$	0.403312	+	0.883130i	0.403312	+	0.883130i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−1.19817	−	2.07528i	−1.19817	−	2.07528i
$465$		0			0
$466$	1.28605	−	2.22751i	1.28605	−	2.22751i
$467$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$467$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.154218	+	0.267114i	0.154218	+	0.267114i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.68251			1.68251
$476$		0			0
$477$		0			0
$478$	0.308779	+	0.534820i	0.308779	+	0.534820i
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.05268			1.05268
$485$		0			0
$486$	0.666997	−	1.15527i	0.666997	−	1.15527i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$	0.357685	+	0.619529i	0.357685	+	0.619529i
$489$		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
$496$	−2.36373			−2.36373
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$	−0.235759	+	0.408346i	−0.235759	+	0.408346i
$502$	0.544078	+	0.942371i	0.544078	+	0.942371i
$503$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$503$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$504$	0.288592	+	0.0275572i	0.288592	+	0.0275572i
$505$		0			0
$506$		0			0
$507$	−0.235759	+	0.408346i	−0.235759	+	0.408346i
$508$	−0.664127	+	1.15030i	−0.664127	+	1.15030i
$509$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$509$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	0.705142	+	1.22134i	0.705142	+	1.22134i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.149608			−0.149608
$518$		0			0
$519$	0.875484			0.875484
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	1.01392	−	1.75616i	1.01392	−	1.75616i
$523$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$523$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$524$		0			0
$525$	−0.469383	−	0.0448206i	−0.469383	−	0.0448206i
$526$	2.51334			2.51334
$527$		0			0
$528$	−0.446106	+	0.772678i	−0.446106	+	0.772678i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	0.698166	−	0.980436i	0.698166	−	0.980436i
$533$		0			0
$534$	0.446947	+	0.774135i	0.446947	+	0.774135i
$535$		0			0
$536$		0			0
$537$	−0.0224357	−	0.0388598i	−0.0224357	−	0.0388598i
$538$		0			0
$539$	−1.54370	−	0.297523i	−1.54370	−	0.297523i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$	−0.0224357	+	0.0388598i	−0.0224357	+	0.0388598i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−0.664127	−	1.15030i	−0.664127	−	1.15030i
$549$	−0.746170	+	1.29240i	−0.746170	+	1.29240i
$550$	−1.02951	+	1.78316i	−1.02951	+	1.78316i
$551$	1.67489	+	2.90099i	1.67489	+	2.90099i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$557$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$558$	−1.00013	−	1.73227i	−1.00013	−	1.73227i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.25667	−	2.17661i	−1.25667	−	2.17661i
$563$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$563$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$564$	−0.0160499	+	0.0277992i	−0.0160499	+	0.0277992i
$565$		0			0
$566$	2.05902			2.05902
$567$	0.158873	+	0.347882i	0.158873	+	0.347882i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	−0.308437			−0.308437
$574$		0			0
$575$		0			0
$576$	−0.144952	−	0.251065i	−0.144952	−	0.251065i
$577$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$577$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$578$	0.654861	−	1.13425i	0.654861	−	1.13425i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	0.175775			0.175775
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.28605	+	2.22751i	1.28605	+	2.22751i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−0.220891	+	0.254922i	−0.220891	+	0.254922i
$589$	3.30420			3.30420
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$	−1.72588			−1.72588
$595$		0			0
$596$		0			0
$597$	−0.437742	−	0.758192i	−0.437742	−	0.758192i
$598$		0			0
$599$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$599$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$600$	−0.0878875	−	0.152226i	−0.0878875	−	0.152226i
$601$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$601$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	2.02509			2.02509
$609$	−0.389977	−	0.853931i	−0.389977	−	0.853931i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$613$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$614$		0			0
$615$		0			0
$616$	−0.243458	−	0.533098i	−0.243458	−	0.533098i
$617$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$617$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$	1.30972			1.30972
$623$	−1.44091	−	0.137591i	−1.44091	−	0.137591i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	0.623601	−	1.08011i	0.623601	−	1.08011i
$628$	0.233975	+	0.405256i	0.233975	+	0.405256i
$629$		0			0
$630$		0			0
$631$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$631$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$632$		0			0
$633$	0.154218	−	0.267114i	0.154218	−	0.267114i
$634$	0.308779	−	0.534820i	0.308779	−	0.534820i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−4.09940			−4.09940
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	0.358218	+	0.620452i	0.358218	+	0.620452i
$643$		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	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.0712846	+	0.123469i	−0.0712846	+	0.123469i
$649$		0			0
$650$		0			0
$651$	−0.921801	−	0.0880213i	−0.921801	−	0.0880213i
$652$		0			0
$653$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$653$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.0517765	−	0.113375i	−0.0517765	−	0.113375i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.357685	+	0.619529i	−0.357685	+	0.619529i
$669$	−0.341254	+	0.591068i	−0.341254	+	0.591068i
$670$		0			0
$671$	3.01685			3.01685
$672$	−0.564956	−	0.0539468i	−0.564956	−	0.0539468i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	1.10181	+	1.90839i	1.10181	+	1.90839i
$675$	0.419102	−	0.725906i	0.419102	−	0.725906i
$676$	−0.357685	+	0.619529i	−0.357685	+	0.619529i
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−0.165101	+	0.231852i	−0.165101	+	0.231852i
$680$		0			0
$681$		0			0
$682$	−2.02181	+	3.50188i	−2.02181	+	3.50188i
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	0.468008	+	0.810614i	0.468008	+	0.810614i
$685$		0			0
$686$	−0.308779	−	1.27280i	−0.308779	−	1.27280i
$687$	−0.938766			−0.938766
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	1.32825			1.32825
$693$	0.709167	−	0.995885i	0.709167	−	0.995885i
$694$		0			0
$695$		0			0
$696$	0.174979	−	0.303073i	0.174979	−	0.303073i
$697$		0			0
$698$		0			0
$699$	0.925994			0.925994
$700$	−0.712131	−	0.0680003i	−0.712131	−	0.0680003i
$701$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$701$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$702$		0			0
$703$		0			0
$704$	−0.293029	+	0.507542i	−0.293029	+	0.507542i
$705$		0			0
$706$	−1.51943			−1.51943
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	−0.269798	−	0.467303i	−0.269798	−	0.467303i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.0340387	−	0.0589567i	−0.0340387	−	0.0589567i
$717$	−0.111165	+	0.192543i	−0.111165	+	0.192543i
$718$	1.28605	−	2.22751i	1.28605	−	2.22751i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	−2.39788			−2.39788
$723$		0			0
$724$	−0.0340387	+	0.0589567i	−0.0340387	+	0.0589567i
$725$	0.995472	−	1.72421i	0.995472	−	1.72421i
$726$	0.454373	+	0.786997i	0.454373	+	0.786997i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.0978132			0.0978132
$730$		0			0
$731$		0			0
$732$	0.323646	−	0.560572i	0.323646	−	0.560572i
$733$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$733$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$734$	2.32825			2.32825
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		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	$0.666667\pi$
$744$	−0.172599	−	0.298950i	−0.172599	−	0.298950i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.15486	−	0.110276i	−1.15486	−	0.110276i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	0.0572703	−	0.0991951i	0.0572703	−	0.0991951i
$753$	−0.195876	+	0.339266i	−0.195876	+	0.339266i
$754$		0			0
$755$		0			0
$756$	−0.249094	−	0.545439i	−0.249094	−	0.545439i
$757$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$757$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$761$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$762$	−1.14664			−1.14664
$763$		0			0
$764$	−0.467949			−0.467949
$765$		0			0
$766$	−1.30379	+	2.25823i	−1.30379	+	2.25823i
$767$		0			0
$768$	−0.308779	−	0.534820i	−0.308779	−	0.534820i
$769$		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	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	−0.981929	−	1.70075i	−0.981929	−	1.70075i
$776$	−0.106106			−0.106106
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.66882			1.66882
$784$	0.788201	−	0.909632i	0.788201	−	0.909632i
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$	0.452418	+	0.783611i	0.452418	+	0.783611i
$790$		0			0
$791$		0			0
$792$	0.455761			0.455761
$793$		0			0
$794$	0.947890	−	1.64179i	0.947890	−	1.64179i
$795$		0			0
$796$	−0.664127	−	1.15030i	−0.664127	−	1.15030i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	1.03434	+	0.0987674i	1.03434	+	0.0987674i
$799$		0			0
$800$	−0.601808	−	1.04236i	−0.601808	−	1.04236i
$801$	0.562827	−	0.974845i	0.562827	−	0.974845i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$809$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−0.591660	−	1.29555i	−0.591660	−	1.29555i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−2.43181			−2.43181
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	0.573320	−	0.993020i	0.573320	−	0.993020i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	−0.741276			−0.741276
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	0.946106	−	1.63870i	0.946106	−	1.63870i
$837$	0.823056	−	1.42558i	0.823056	−	1.42558i
$838$	−0.428368	−	0.741955i	−0.428368	−	0.741955i
$839$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$839$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$840$		0			0
$841$	2.96386			2.96386
$842$	0.759713	+	1.31586i	0.759713	+	1.31586i
$843$	0.452418	−	0.783611i	0.452418	−	0.783611i
$844$	0.233975	−	0.405256i	0.233975	−	0.405256i
$845$		0			0
$846$	0.0969274			0.0969274
$847$	−1.46485	−	0.139877i	−1.46485	−	0.139877i
$848$		0			0
$849$	0.370638	+	0.641964i	0.370638	+	0.641964i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$853$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$854$	1.04408	+	2.28621i	1.04408	+	2.28621i
$855$		0			0
$856$	−0.216237	−	0.374533i	−0.216237	−	0.374533i
$857$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$857$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$858$		0			0
$859$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$859$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$860$		0			0
$861$		0			0
$862$	−1.89578			−1.89578
$863$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$863$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$864$	0.504437	−	0.873711i	0.504437	−	0.873711i
$865$		0			0
$866$	−1.16413	−	2.01633i	−1.16413	−	2.01633i
$867$	0.471518			0.471518
$868$	−1.39852	−	0.133543i	−1.39852	−	0.133543i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.110674	−	0.191693i	−0.110674	−	0.191693i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$877$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$878$		0			0
$879$	−0.462997	+	0.801934i	−0.462997	+	0.801934i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.00013	+	0.192758i	1.00013	+	0.192758i
$883$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$883$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	1.07701	−	1.51245i	1.07701	−	1.51245i
$890$		0			0
$891$	0.300620	+	0.520690i	0.300620	+	0.520690i
$892$	−0.517738	+	0.896748i	−0.517738	+	0.896748i
$893$	−0.0800569	+	0.138663i	−0.0800569	+	0.138663i
$894$		0			0
$895$		0			0
$896$	0.712131	+	0.0680003i	0.712131	+	0.0680003i
$897$		0			0
$898$	0.0623191	+	0.107940i	0.0623191	+	0.107940i
$899$	1.95496	−	3.38610i	1.95496	−	3.38610i
$900$	0.278161	−	0.481790i	0.278161	−	0.481790i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$907$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$911$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$912$	0.477433	+	0.826939i	0.477433	+	0.826939i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.42426			−1.42426
$917$		0			0
$918$		0			0
$919$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$919$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$920$		0			0
$921$		0			0
$922$	1.10181	+	1.90839i	1.10181	+	1.90839i
$923$		0			0
$924$	−0.307597	+	0.431959i	−0.307597	+	0.431959i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	1.19817	−	2.07528i	1.19817	−	2.07528i
$929$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$929$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$930$		0			0
$931$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$932$	1.40489			1.40489
$933$	0.235759	+	0.408346i	0.235759	+	0.408346i
$934$	0.759713	−	1.31586i	0.759713	−	1.31586i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$	−0.201983	+	0.349845i	−0.201983	+	0.349845i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$947$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$948$		0			0
$949$		0			0
$950$	1.10181	+	1.90839i	1.10181	+	1.90839i
$951$	0.222329			0.222329
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.168655	+	0.292119i	−0.168655	+	0.292119i
$957$	−0.737920	−	1.27811i	−0.737920	−	1.27811i
$958$		0			0
$959$	0.771316	+	1.68895i	0.771316	+	1.68895i
$960$		0			0
$961$	−1.42837	−	2.47401i	−1.42837	−	2.47401i
$962$		0			0
$963$	0.451093	−	0.781317i	0.451093	−	0.781317i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$967$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$968$	−0.274280	−	0.475067i	−0.274280	−	0.475067i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	0.728628			0.728628
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.15486	+	2.00028i	−1.15486	+	2.00028i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	−2.27557			−2.27557
$980$		0			0
$981$		0			0
$982$	−0.654861	−	1.13425i	−0.654861	−	1.13425i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.0260280	−	0.0365512i	0.0260280	−	0.0365512i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$	−1.18186	−	2.04705i	−1.18186	−	2.04705i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$997$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$998$		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.333.8	✓	20
7.2	even	3	inner	1169.1.f.c.667.8	yes	20
167.166	odd	2	CM	1169.1.f.c.333.8	✓	20
1169.667	odd	6	inner	1169.1.f.c.667.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1169.1.f.c.333.8	✓	20	1.1	even	1	trivial
1169.1.f.c.333.8	✓	20	167.166	odd	2	CM
1169.1.f.c.667.8	yes	20	7.2	even	3	inner
1169.1.f.c.667.8	yes	20	1169.667	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.654861 + 1.13425i) q^{2} +(-0.235759 + 0.408346i) q^{3} +(-0.357685 + 0.619529i) q^{4} -0.617557 q^{6} +(0.580057 - 0.814576i) q^{7} +0.372786 q^{8} +(0.388835 + 0.673483i) q^{9} +O(q^{10})\)	\(q+(0.654861 + 1.13425i) q^{2} +(-0.235759 + 0.408346i) q^{3} +(-0.357685 + 0.619529i) q^{4} -0.617557 q^{6} +(0.580057 - 0.814576i) q^{7} +0.372786 q^{8} +(0.388835 + 0.673483i) q^{9} +(0.786053 - 1.36148i) q^{11} +(-0.168655 - 0.292119i) q^{12} +(1.30379 + 0.124497i) q^{14} +(0.601808 + 1.04236i) q^{16} +(-0.509266 + 0.882075i) q^{18} +(-0.841254 - 1.45709i) q^{19} +(0.195876 + 0.428908i) q^{21} +2.05902 q^{22} +(-0.0878875 + 0.152226i) q^{24} +(-0.500000 + 0.866025i) q^{25} -0.838204 q^{27} +(0.297176 + 0.650724i) q^{28} -1.99094 q^{29} +(-0.981929 + 1.70075i) q^{31} +(-0.601808 + 1.04236i) q^{32} +(0.370638 + 0.641964i) q^{33} -0.556323 q^{36} +(1.10181 - 1.90839i) q^{38} +(-0.358218 + 0.503047i) q^{42} +(0.562319 + 0.973965i) q^{44} +(-0.0475819 - 0.0824143i) q^{47} -0.567526 q^{48} +(-0.327068 - 0.945001i) q^{49} -1.30972 q^{50} +(-0.548907 - 0.950734i) q^{54} +(0.216237 - 0.303662i) q^{56} +0.793332 q^{57} +(-1.30379 - 2.25823i) q^{58} +(0.959493 + 1.66189i) q^{61} -2.57211 q^{62} +(0.774150 + 0.0739223i) q^{63} -0.372786 q^{64} +(-0.485433 + 0.840794i) q^{66} +(0.144952 + 0.251065i) q^{72} +(-0.235759 - 0.408346i) q^{75} +1.20362 q^{76} +(-0.653077 - 1.43004i) q^{77} +(-0.191221 + 0.331205i) q^{81} +(-0.335783 - 0.0320633i) q^{84} +(0.469383 - 0.812995i) q^{87} +(0.293029 - 0.507542i) q^{88} +(-0.723734 - 1.25354i) q^{89} +(-0.462997 - 0.801934i) q^{93} +(0.0623191 - 0.107940i) q^{94} +(-0.283763 - 0.491492i) q^{96} -0.284630 q^{97} +(0.857685 - 0.989821i) q^{98} +1.22258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 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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