LMFDB - Embedded newform 584.1.br.a.267.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [584,1,Mod(19,584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(584, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([18, 18, 31]))

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

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

chi := DirichletCharacter("584.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 584 = 2^{3} \cdot 73 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	584.br (of order $36$, degree $12$, 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:	$0.291453967378$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{36}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{36} - \cdots)$

Embedding invariants

Embedding label			267.1
Root			$0.342020 - 0.939693i$ of defining polynomial
Character	$\chi$	$=$	584.267
Dual form			584.1.br.a.35.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.766044 - 0.642788i) q^{2} +(1.62760 + 0.939693i) q^{3} +(0.173648 + 0.984808i) q^{4} +(-0.642788 - 1.76604i) q^{6} +(0.500000 - 0.866025i) q^{8} +(1.26604 + 2.19285i) q^{9} +O(q^{10})$	\(q+(-0.766044 - 0.642788i) q^{2} +(1.62760 + 0.939693i) q^{3} +(0.173648 + 0.984808i) q^{4} +(-0.642788 - 1.76604i) q^{6} +(0.500000 - 0.866025i) q^{8} +(1.26604 + 2.19285i) q^{9} +(-0.842020 - 0.0736672i) q^{11} +(-0.642788 + 1.76604i) q^{12} +(-0.939693 + 0.342020i) q^{16} +(-1.86603 - 0.500000i) q^{17} +(0.439693 - 2.49362i) q^{18} +(1.11334 - 1.32683i) q^{19} +(0.597672 + 0.597672i) q^{22} +(1.62760 - 0.939693i) q^{24} +(-0.984808 - 0.173648i) q^{25} +2.87939i q^{27} +(0.939693 + 0.342020i) q^{32} +(-1.30124 - 0.911141i) q^{33} +(1.10806 + 1.58248i) q^{34} +(-1.93969 + 1.62760i) q^{36} +(-1.70574 + 0.300767i) q^{38} +(0.642788 + 0.233956i) q^{41} +(1.10806 - 0.296905i) q^{43} +(-0.0736672 - 0.842020i) q^{44} +(-1.85083 - 0.326352i) q^{48} +(-0.866025 + 0.500000i) q^{49} +(0.642788 + 0.766044i) q^{50} +(-2.56729 - 2.56729i) q^{51} +(1.85083 - 2.20574i) q^{54} +(3.05888 - 1.11334i) q^{57} +(0.766044 - 1.64279i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(0.411141 + 1.53440i) q^{66} +(-0.118782 - 0.326352i) q^{67} +(0.168372 - 1.92450i) q^{68} +2.53209 q^{72} +(0.173648 + 0.984808i) q^{73} +(-1.43969 - 1.20805i) q^{75} +(1.50000 + 0.866025i) q^{76} +(-1.43969 + 2.49362i) q^{81} +(-0.342020 - 0.592396i) q^{82} +(-0.366025 + 0.366025i) q^{83} +(-1.03967 - 0.484808i) q^{86} +(-0.484808 + 0.692377i) q^{88} +(1.20805 + 1.43969i) q^{96} +(-1.70574 + 0.984808i) q^{97} +(0.984808 + 0.173648i) q^{98} +(-0.904494 - 1.93969i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{8} + 6 q^{9}+O(q^{10}) $ 12 * q + 6 * q^8 + 6 * q^9	$ 12 q + 6 q^{8} + 6 q^{9} - 6 q^{11} - 12 q^{17} - 6 q^{18} + 6 q^{33} - 12 q^{36} - 6 q^{64} - 6 q^{66} + 12 q^{72} - 6 q^{75} + 18 q^{76} - 6 q^{81} + 6 q^{83} + 6 q^{88} - 12 q^{99}+O(q^{100}) $ 12 * q + 6 * q^8 + 6 * q^9 - 6 * q^11 - 12 * q^17 - 6 * q^18 + 6 * q^33 - 12 * q^36 - 6 * q^64 - 6 * q^66 + 12 * q^72 - 6 * q^75 + 18 * q^76 - 6 * q^81 + 6 * q^83 + 6 * q^88 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/584\mathbb{Z}\right)^\times$.

$n$	$293$	$297$	$439$
$\chi(n)$	$-1$	$e\left(\frac{19}{36}\right)$	$-1$

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 $

$2$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$2$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$3$	1.62760	+	0.939693i	1.62760	+	0.939693i	0.984808	+	0.173648i	$0.0555556\pi$
$3$	1.62760	+	0.939693i	1.62760	+	0.939693i	0.642788	+	0.766044i	$0.277778\pi$
$4$	0.173648	+	0.984808i	0.173648	+	0.984808i
$5$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$5$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$6$	−0.642788	−	1.76604i	−0.642788	−	1.76604i
$7$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$7$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$8$	0.500000	−	0.866025i	0.500000	−	0.866025i
$9$	1.26604	+	2.19285i	1.26604	+	2.19285i
$10$		0			0
$11$	−0.842020	−	0.0736672i	−0.842020	−	0.0736672i	−0.342020	−	0.939693i	$-0.611111\pi$
$11$	−0.842020	−	0.0736672i	−0.842020	−	0.0736672i	−0.500000	+	0.866025i	$0.666667\pi$
$12$	−0.642788	+	1.76604i	−0.642788	+	1.76604i
$13$		0			0		−0.906308	−	0.422618i	$-0.861111\pi$
$13$		0			0		0.906308	+	0.422618i	$0.138889\pi$
$14$		0			0
$15$		0			0
$16$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$17$	−1.86603	−	0.500000i	−1.86603	−	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$17$	−1.86603	−	0.500000i	−1.86603	−	0.500000i	−1.00000			$\pi$
$18$	0.439693	−	2.49362i	0.439693	−	2.49362i
$19$	1.11334	−	1.32683i	1.11334	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$19$	1.11334	−	1.32683i	1.11334	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$20$		0			0
$21$		0			0
$22$	0.597672	+	0.597672i	0.597672	+	0.597672i
$23$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$23$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$24$	1.62760	−	0.939693i	1.62760	−	0.939693i
$25$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$26$		0			0
$27$			2.87939i			2.87939i
$28$		0			0
$29$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$29$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$30$		0			0
$31$		0			0		0.819152	−	0.573576i	$-0.194444\pi$
$31$		0			0		−0.819152	+	0.573576i	$0.805556\pi$
$32$	0.939693	+	0.342020i	0.939693	+	0.342020i
$33$	−1.30124	−	0.911141i	−1.30124	−	0.911141i
$34$	1.10806	+	1.58248i	1.10806	+	1.58248i
$35$		0			0
$36$	−1.93969	+	1.62760i	−1.93969	+	1.62760i
$37$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$37$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$38$	−1.70574	+	0.300767i	−1.70574	+	0.300767i
$39$		0			0
$40$		0			0
$41$	0.642788	+	0.233956i	0.642788	+	0.233956i	0.642788	−	0.766044i	$-0.277778\pi$
$41$	0.642788	+	0.233956i	0.642788	+	0.233956i			1.00000i	$0.5\pi$
$42$		0			0
$43$	1.10806	−	0.296905i	1.10806	−	0.296905i	0.342020	−	0.939693i	$-0.388889\pi$
$43$	1.10806	−	0.296905i	1.10806	−	0.296905i	0.766044	+	0.642788i	$0.222222\pi$
$44$	−0.0736672	−	0.842020i	−0.0736672	−	0.842020i
$45$		0			0
$46$		0			0
$47$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$47$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$48$	−1.85083	−	0.326352i	−1.85083	−	0.326352i
$49$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$50$	0.642788	+	0.766044i	0.642788	+	0.766044i
$51$	−2.56729	−	2.56729i	−2.56729	−	2.56729i
$52$		0			0
$53$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$53$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$54$	1.85083	−	2.20574i	1.85083	−	2.20574i
$55$		0			0
$56$		0			0
$57$	3.05888	−	1.11334i	3.05888	−	1.11334i
$58$		0			0
$59$	0.766044	−	1.64279i	0.766044	−	1.64279i		−	1.00000i	$-0.5\pi$
$59$	0.766044	−	1.64279i	0.766044	−	1.64279i	0.766044	−	0.642788i	$-0.222222\pi$
$60$		0			0
$61$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$61$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$62$		0			0
$63$		0			0
$64$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$65$		0			0
$66$	0.411141	+	1.53440i	0.411141	+	1.53440i
$67$	−0.118782	−	0.326352i	−0.118782	−	0.326352i	0.866025	−	0.500000i	$-0.166667\pi$
$67$	−0.118782	−	0.326352i	−0.118782	−	0.326352i	−0.984808	+	0.173648i	$0.944444\pi$
$68$	0.168372	−	1.92450i	0.168372	−	1.92450i
$69$		0			0
$70$		0			0
$71$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$71$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$72$	2.53209			2.53209
$73$	0.173648	+	0.984808i	0.173648	+	0.984808i
$73$	0.173648	+	0.984808i	0.173648	+	0.984808i
$74$		0			0
$75$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$76$	1.50000	+	0.866025i	1.50000	+	0.866025i
$77$		0			0
$78$		0			0
$79$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$79$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$80$		0			0
$81$	−1.43969	+	2.49362i	−1.43969	+	2.49362i
$82$	−0.342020	−	0.592396i	−0.342020	−	0.592396i
$83$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$83$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$	−1.03967	−	0.484808i	−1.03967	−	0.484808i
$87$		0			0
$88$	−0.484808	+	0.692377i	−0.484808	+	0.692377i
$89$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$89$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.20805	+	1.43969i	1.20805	+	1.43969i
$97$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$97$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$98$	0.984808	+	0.173648i	0.984808	+	0.173648i
$99$	−0.904494	−	1.93969i	−0.904494	−	1.93969i
$100$		−	1.00000i		−	1.00000i
$101$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$101$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$102$	0.316436	+	3.61688i	0.316436	+	3.61688i
$103$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$103$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.14279	−	1.63207i	−1.14279	−	1.63207i	−0.642788	−	0.766044i	$-0.722222\pi$
$107$	−1.14279	−	1.63207i	−1.14279	−	1.63207i	−0.500000	−	0.866025i	$-0.666667\pi$
$108$	−2.83564	+	0.500000i	−2.83564	+	0.500000i
$109$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$109$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.424024	+	0.296905i	0.424024	+	0.296905i	0.766044	−	0.642788i	$-0.222222\pi$
$113$	0.424024	+	0.296905i	0.424024	+	0.296905i	−0.342020	+	0.939693i	$0.611111\pi$
$114$	−3.05888	−	1.11334i	−3.05888	−	1.11334i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.64279	+	0.766044i	−1.64279	+	0.766044i
$119$		0			0
$120$		0			0
$121$	−0.281237	−	0.0495896i	−0.281237	−	0.0495896i
$122$		0			0
$123$	0.826352	+	0.984808i	0.826352	+	0.984808i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$127$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$128$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$129$	2.08248	+	0.557999i	2.08248	+	0.557999i
$130$		0			0
$131$	−0.939693	+	1.34202i	−0.939693	+	1.34202i			1.00000i	$0.5\pi$
$131$	−0.939693	+	1.34202i	−0.939693	+	1.34202i	−0.939693	+	0.342020i	$0.888889\pi$
$132$	0.671340	−	1.43969i	0.671340	−	1.43969i
$133$		0			0
$134$	−0.118782	+	0.326352i	−0.118782	+	0.326352i
$135$		0			0
$136$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$137$	0.342020	+	0.592396i	0.342020	+	0.592396i	0.984808	−	0.173648i	$-0.0555556\pi$
$137$	0.342020	+	0.592396i	0.342020	+	0.592396i	−0.642788	+	0.766044i	$0.722222\pi$
$138$		0			0
$139$	0.296905	+	1.10806i	0.296905	+	1.10806i	0.939693	+	0.342020i	$0.111111\pi$
$139$	0.296905	+	1.10806i	0.296905	+	1.10806i	−0.642788	+	0.766044i	$0.722222\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.93969	−	1.62760i	−1.93969	−	1.62760i
$145$		0			0
$146$	0.500000	−	0.866025i	0.500000	−	0.866025i
$147$	−1.87939			−1.87939
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	0.326352	+	1.85083i	0.326352	+	1.85083i
$151$		0			0		0.0871557	−	0.996195i	$-0.472222\pi$
$151$		0			0		−0.0871557	+	0.996195i	$0.527778\pi$
$152$	−0.592396	−	1.62760i	−0.592396	−	1.62760i
$153$	−1.26604	−	4.72494i	−1.26604	−	4.72494i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$157$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	2.70574	−	0.984808i	2.70574	−	0.984808i
$163$	−0.168372	−	0.0451151i	−0.168372	−	0.0451151i	0.173648	−	0.984808i	$-0.444444\pi$
$163$	−0.168372	−	0.0451151i	−0.168372	−	0.0451151i	−0.342020	+	0.939693i	$0.611111\pi$
$164$	−0.118782	+	0.673648i	−0.118782	+	0.673648i
$165$		0			0
$166$	0.515668	−	0.0451151i	0.515668	−	0.0451151i
$167$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$167$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$168$		0			0
$169$	0.642788	+	0.766044i	0.642788	+	0.766044i
$170$		0			0
$171$	4.31908	+	0.761570i	4.31908	+	0.761570i
$172$	0.484808	+	1.03967i	0.484808	+	1.03967i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	0.816436	−	0.218763i	0.816436	−	0.218763i
$177$	2.79053	−	1.95395i	2.79053	−	1.95395i
$178$		0			0
$179$	1.15846	+	0.811160i	1.15846	+	0.811160i	0.984808	−	0.173648i	$-0.0555556\pi$
$179$	1.15846	+	0.811160i	1.15846	+	0.811160i	0.173648	+	0.984808i	$0.444444\pi$
$180$		0			0
$181$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$181$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.53440	+	0.558475i	1.53440	+	0.558475i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.906308	−	0.422618i	$-0.138889\pi$
$191$		0			0		−0.906308	+	0.422618i	$0.861111\pi$
$192$		−	1.87939i		−	1.87939i
$193$	−0.766044	−	1.64279i	−0.766044	−	1.64279i	−0.766044	−	0.642788i	$-0.777778\pi$
$193$	−0.766044	−	1.64279i	−0.766044	−	1.64279i		−	1.00000i	$-0.5\pi$
$194$	1.93969	+	0.342020i	1.93969	+	0.342020i
$195$		0			0
$196$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$	−0.553928	+	2.06729i	−0.553928	+	2.06729i
$199$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$199$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$200$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$201$	0.113341	−	0.642788i	0.113341	−	0.642788i
$202$		0			0
$203$		0			0
$204$	2.08248	−	2.97409i	2.08248	−	2.97409i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.03520	+	1.03520i	−1.03520	+	1.03520i
$210$		0			0
$211$	0.984808	−	1.70574i	0.984808	−	1.70574i	0.342020	−	0.939693i	$-0.388889\pi$
$211$	0.984808	−	1.70574i	0.984808	−	1.70574i	0.642788	−	0.766044i	$-0.277778\pi$
$212$		0			0
$213$		0			0
$214$	−0.173648	+	1.98481i	−0.173648	+	1.98481i
$215$		0			0
$216$	2.49362	+	1.43969i	2.49362	+	1.43969i
$217$		0			0
$218$		0			0
$219$	−0.642788	+	1.76604i	−0.642788	+	1.76604i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$223$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$224$		0			0
$225$	−0.866025	−	2.37939i	−0.866025	−	2.37939i
$226$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$227$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$227$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$228$	1.62760	+	2.81908i	1.62760	+	2.81908i
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.842020	+	1.80572i	−0.842020	+	1.80572i	−0.342020	+	0.939693i	$0.611111\pi$
$233$	−0.842020	+	1.80572i	−0.842020	+	1.80572i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$	1.75085	+	0.469139i	1.75085	+	0.469139i
$237$		0			0
$238$		0			0
$239$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$239$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$240$		0			0
$241$	−1.40883	−	1.40883i	−1.40883	−	1.40883i	−0.766044	−	0.642788i	$-0.777778\pi$
$241$	−1.40883	−	1.40883i	−1.40883	−	1.40883i	−0.642788	−	0.766044i	$-0.722222\pi$
$242$	0.183564	+	0.218763i	0.183564	+	0.218763i
$243$	−2.19285	+	1.26604i	−2.19285	+	1.26604i
$244$		0			0
$245$		0			0
$246$		−	1.28558i		−	1.28558i
$247$		0			0
$248$		0			0
$249$	−0.939693	+	0.251790i	−0.939693	+	0.251790i
$250$		0			0
$251$	1.85083	+	0.673648i	1.85083	+	0.673648i	0.984808	+	0.173648i	$0.0555556\pi$
$251$	1.85083	+	0.673648i	1.85083	+	0.673648i	0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.766044	−	0.642788i	0.766044	−	0.642788i
$257$	0.342020	−	0.0603074i	0.342020	−	0.0603074i		−	1.00000i	$-0.5\pi$
$257$	0.342020	−	0.0603074i	0.342020	−	0.0603074i	0.342020	+	0.939693i	$0.388889\pi$
$258$	−1.23660	−	1.76604i	−1.23660	−	1.76604i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.58248	−	0.424024i	1.58248	−	0.424024i
$263$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$263$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$264$	−1.43969	+	0.671340i	−1.43969	+	0.671340i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.300767	−	0.173648i	0.300767	−	0.173648i
$269$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$269$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$270$		0			0
$271$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$271$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$272$	1.92450	−	0.168372i	1.92450	−	0.168372i
$273$		0			0
$274$	0.118782	−	0.673648i	0.118782	−	0.673648i
$275$	0.816436	+	0.218763i	0.816436	+	0.218763i
$276$		0			0
$277$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$277$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$278$	0.484808	−	1.03967i	0.484808	−	1.03967i
$279$		0			0
$280$		0			0
$281$	1.80572	+	0.157980i	1.80572	+	0.157980i	0.939693	−	0.342020i	$-0.111111\pi$
$281$	1.80572	+	0.157980i	1.80572	+	0.157980i	0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$283$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.439693	+	2.49362i	0.439693	+	2.49362i
$289$	2.36603	+	1.36603i	2.36603	+	1.36603i
$290$		0			0
$291$	−3.70167			−3.70167
$292$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	1.43969	+	1.20805i	1.43969	+	1.20805i
$295$		0			0
$296$		0			0
$297$	0.212116	−	2.42450i	0.212116	−	2.42450i
$298$		0			0
$299$		0			0
$300$	0.939693	−	1.62760i	0.939693	−	1.62760i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.592396	+	1.62760i	−0.592396	+	1.62760i
$305$		0			0
$306$	−2.06729	+	4.43331i	−2.06729	+	4.43331i
$307$	0.811160	−	1.15846i	0.811160	−	1.15846i	−0.173648	−	0.984808i	$-0.555556\pi$
$307$	0.811160	−	1.15846i	0.811160	−	1.15846i	0.984808	−	0.173648i	$-0.0555556\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$311$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$312$		0			0
$313$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$313$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$317$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.326352	−	3.73022i	−0.326352	−	3.73022i
$322$		0			0
$323$	−2.74094	+	1.91922i	−2.74094	+	1.91922i
$324$	−2.70574	−	0.984808i	−2.70574	−	0.984808i
$325$		0			0
$326$	0.0999810	+	0.142788i	0.0999810	+	0.142788i
$327$		0			0
$328$	0.524005	−	0.439693i	0.524005	−	0.439693i
$329$		0			0
$330$		0			0
$331$	0.939693	+	1.34202i	0.939693	+	1.34202i	0.939693	+	0.342020i	$0.111111\pi$
$331$	0.939693	+	1.34202i	0.939693	+	1.34202i			1.00000i	$0.5\pi$
$332$	−0.424024	−	0.296905i	−0.424024	−	0.296905i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.48481	+	0.692377i	−1.48481	+	0.692377i	−0.984808	−	0.173648i	$-0.944444\pi$
$337$	−1.48481	+	0.692377i	−1.48481	+	0.692377i	−0.500000	+	0.866025i	$0.666667\pi$
$338$		−	1.00000i		−	1.00000i
$339$	0.411141	+	0.881694i	0.411141	+	0.881694i
$340$		0			0
$341$		0			0
$342$	−2.81908	−	3.35965i	−2.81908	−	3.35965i
$343$		0			0
$344$	0.296905	−	1.10806i	0.296905	−	1.10806i
$345$		0			0
$346$		0			0
$347$	−0.223238	+	1.26604i	−0.223238	+	1.26604i	0.642788	+	0.766044i	$0.277778\pi$
$347$	−0.223238	+	1.26604i	−0.223238	+	1.26604i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$349$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$350$		0			0
$351$		0			0
$352$	−0.766044	−	0.357212i	−0.766044	−	0.357212i
$353$	0.439693	−	1.20805i	0.439693	−	1.20805i	−0.500000	−	0.866025i	$-0.666667\pi$
$353$	0.439693	−	1.20805i	0.439693	−	1.20805i	0.939693	−	0.342020i	$-0.111111\pi$
$354$	−3.39364	−	0.296905i	−3.39364	−	0.296905i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$359$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$359$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$360$		0			0
$361$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$362$		0			0
$363$	−0.411141	−	0.344988i	−0.411141	−	0.344988i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$367$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$368$		0			0
$369$	0.300767	+	1.70574i	0.300767	+	1.70574i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$	−0.816436	−	1.41411i	−0.816436	−	1.41411i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0736672	+	0.157980i	−0.0736672	+	0.157980i	−0.939693	−	0.342020i	$-0.888889\pi$
$379$	−0.0736672	+	0.157980i	−0.0736672	+	0.157980i	0.866025	+	0.500000i	$0.166667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$383$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$384$	−1.20805	+	1.43969i	−1.20805	+	1.43969i
$385$		0			0
$386$	−0.469139	+	1.75085i	−0.469139	+	1.75085i
$387$	2.05393	+	2.05393i	2.05393	+	2.05393i
$388$	−1.26604	−	1.50881i	−1.26604	−	1.50881i
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$			1.00000i			1.00000i
$393$	−2.79053	+	1.30124i	−2.79053	+	1.30124i
$394$		0			0
$395$		0			0
$396$	1.75316	−	1.22758i	1.75316	−	1.22758i
$397$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$397$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$398$		0			0
$399$		0			0
$400$	0.984808	−	0.173648i	0.984808	−	0.173648i
$401$	0.766044	−	0.642788i	0.766044	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$401$	0.766044	−	0.642788i	0.766044	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$402$	−0.500000	+	0.419550i	−0.500000	+	0.419550i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−3.50698	+	0.939693i	−3.50698	+	0.939693i
$409$	0.142788	+	1.63207i	0.142788	+	1.63207i	0.642788	+	0.766044i	$0.277778\pi$
$409$	0.142788	+	1.63207i	0.142788	+	1.63207i	−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$			1.28558i			1.28558i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.557999	+	2.08248i	−0.557999	+	2.08248i
$418$	1.45842	−	0.127595i	1.45842	−	0.127595i
$419$	0.826352	−	0.984808i	0.826352	−	0.984808i	−0.173648	−	0.984808i	$-0.555556\pi$
$419$	0.826352	−	0.984808i	0.826352	−	0.984808i	1.00000			$0$
$420$		0			0
$421$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$421$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$422$	−1.85083	+	0.673648i	−1.85083	+	0.673648i
$423$		0			0
$424$		0			0
$425$	1.75085	+	0.816436i	1.75085	+	0.816436i
$426$		0			0
$427$		0			0
$428$	1.40883	−	1.40883i	1.40883	−	1.40883i
$429$		0			0
$430$		0			0
$431$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$431$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$432$	−0.984808	−	2.70574i	−0.984808	−	2.70574i
$433$	0.0451151	−	0.515668i	0.0451151	−	0.515668i	−0.939693	−	0.342020i	$-0.888889\pi$
$433$	0.0451151	−	0.515668i	0.0451151	−	0.515668i	0.984808	−	0.173648i	$-0.0555556\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.62760	−	0.939693i	1.62760	−	0.939693i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−2.19285	−	1.26604i	−2.19285	−	1.26604i
$442$		0			0
$443$	−0.157980	+	1.80572i	−0.157980	+	1.80572i	0.342020	+	0.939693i	$0.388889\pi$
$443$	−0.157980	+	1.80572i	−0.157980	+	1.80572i	−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.14279	+	0.0999810i	1.14279	+	0.0999810i	0.642788	−	0.766044i	$-0.277778\pi$
$449$	1.14279	+	0.0999810i	1.14279	+	0.0999810i	0.500000	+	0.866025i	$0.333333\pi$
$450$	−0.866025	+	2.37939i	−0.866025	+	2.37939i
$451$	−0.524005	−	0.244348i	−0.524005	−	0.244348i
$452$	−0.218763	+	0.469139i	−0.218763	+	0.469139i
$453$		0			0
$454$	0.939693	−	0.342020i	0.939693	−	0.342020i
$455$		0			0
$456$	0.565258	−	3.20574i	0.565258	−	3.20574i
$457$	0.642788	−	0.766044i	0.642788	−	0.766044i	−0.342020	−	0.939693i	$-0.611111\pi$
$457$	0.642788	−	0.766044i	0.642788	−	0.766044i	0.984808	+	0.173648i	$0.0555556\pi$
$458$		0			0
$459$	1.43969	−	5.37301i	1.43969	−	5.37301i
$460$		0			0
$461$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$461$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$462$		0			0
$463$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$463$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$464$		0			0
$465$		0			0
$466$	1.80572	−	0.842020i	1.80572	−	0.842020i
$467$	0.0151922	+	0.173648i	0.0151922	+	0.173648i	1.00000			$0$
$467$	0.0151922	+	0.173648i	0.0151922	+	0.173648i	−0.984808	+	0.173648i	$0.944444\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.03967	−	1.48481i	−1.03967	−	1.48481i
$473$	−0.954885	+	0.168372i	−0.954885	+	0.168372i
$474$		0			0
$475$	−1.32683	+	1.11334i	−1.32683	+	1.11334i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$479$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$480$		0			0
$481$		0			0
$482$	0.173648	+	1.98481i	0.173648	+	1.98481i
$483$		0			0
$484$		−	0.285575i		−	0.285575i
$485$		0			0
$486$	2.49362	+	0.439693i	2.49362	+	0.439693i
$487$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$487$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$488$		0			0
$489$	−0.231647	−	0.231647i	−0.231647	−	0.231647i
$490$		0			0
$491$	−0.842020	+	0.0736672i	−0.842020	+	0.0736672i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−0.842020	+	0.0736672i	−0.842020	+	0.0736672i	−0.342020	+	0.939693i	$0.611111\pi$
$492$	−0.826352	+	0.984808i	−0.826352	+	0.984808i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.881694	+	0.411141i	0.881694	+	0.411141i
$499$	−0.233956	+	0.642788i	−0.233956	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$499$	−0.233956	+	0.642788i	−0.233956	+	0.642788i	−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$	−0.984808	−	1.70574i	−0.984808	−	1.70574i
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.326352	+	1.85083i	0.326352	+	1.85083i
$508$		0			0
$509$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$509$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	3.82045	+	3.20574i	3.82045	+	3.20574i
$514$	−0.300767	−	0.173648i	−0.300767	−	0.173648i
$515$		0			0
$516$	−0.187903	+	2.14774i	−0.187903	+	2.14774i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$522$		0			0
$523$	0.592396	−	1.62760i	0.592396	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$523$	0.592396	−	1.62760i	0.592396	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$524$	−1.48481	−	0.692377i	−1.48481	−	0.692377i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.53440	+	0.411141i	1.53440	+	0.411141i
$529$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$530$		0			0
$531$	4.57224	−	0.400019i	4.57224	−	0.400019i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.342020	−	0.0603074i	−0.342020	−	0.0603074i
$537$	1.12326	+	2.40883i	1.12326	+	2.40883i
$538$		0			0
$539$	0.766044	−	0.357212i	0.766044	−	0.357212i
$540$		0			0
$541$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$541$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$542$		0			0
$543$		0			0
$544$	−1.58248	−	1.10806i	−1.58248	−	1.10806i
$545$		0			0
$546$		0			0
$547$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$547$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$548$	−0.524005	+	0.439693i	−0.524005	+	0.439693i
$549$		0			0
$550$	−0.484808	−	0.692377i	−0.484808	−	0.692377i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.03967	+	0.484808i	−1.03967	+	0.484808i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.97258	+	2.35083i	1.97258	+	2.35083i
$562$	−1.28171	−	1.28171i	−1.28171	−	1.28171i
$563$	−0.0451151	+	0.168372i	−0.0451151	+	0.168372i	−0.984808	−	0.173648i	$-0.944444\pi$
$563$	−0.0451151	+	0.168372i	−0.0451151	+	0.168372i	0.939693	+	0.342020i	$0.111111\pi$
$564$		0			0
$565$		0			0
$566$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$567$		0			0
$568$		0			0
$569$	−0.811160	+	1.15846i	−0.811160	+	1.15846i	0.173648	+	0.984808i	$0.444444\pi$
$569$	−0.811160	+	1.15846i	−0.811160	+	1.15846i	−0.984808	+	0.173648i	$0.944444\pi$
$570$		0			0
$571$	0.469139	+	0.218763i	0.469139	+	0.218763i	0.642788	−	0.766044i	$-0.277778\pi$
$571$	0.469139	+	0.218763i	0.469139	+	0.218763i	−0.173648	+	0.984808i	$0.555556\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.26604	−	2.19285i	1.26604	−	2.19285i
$577$	−0.515668	−	1.92450i	−0.515668	−	1.92450i	−0.342020	−	0.939693i	$-0.611111\pi$
$577$	−0.515668	−	1.92450i	−0.515668	−	1.92450i	−0.173648	−	0.984808i	$-0.555556\pi$
$578$	−0.934416	−	2.56729i	−0.934416	−	2.56729i
$579$	0.296905	−	3.39364i	0.296905	−	3.39364i
$580$		0			0
$581$		0			0
$582$	2.83564	+	2.37939i	2.83564	+	2.37939i
$583$		0			0
$584$	0.939693	+	0.342020i	0.939693	+	0.342020i
$585$		0			0
$586$		0			0
$587$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$587$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$588$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$593$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$594$	−1.72093	+	1.72093i	−1.72093	+	1.72093i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.573576	−	0.819152i	$-0.305556\pi$
$599$		0			0		−0.573576	+	0.819152i	$0.694444\pi$
$600$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$601$	0.500000	+	0.133975i	0.500000	+	0.133975i	0.500000	−	0.866025i	$-0.333333\pi$
$601$	0.500000	+	0.133975i	0.500000	+	0.133975i			1.00000i	$0.5\pi$
$602$		0			0
$603$	0.565258	−	0.673648i	0.565258	−	0.673648i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$607$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$608$	1.50000	−	0.866025i	1.50000	−	0.866025i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	4.43331	−	2.06729i	4.43331	−	2.06729i
$613$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$613$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$614$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$615$		0			0
$616$		0			0
$617$	−0.692377	−	0.484808i	−0.692377	−	0.484808i	0.173648	−	0.984808i	$-0.444444\pi$
$617$	−0.692377	−	0.484808i	−0.692377	−	0.484808i	−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$	−1.50881	+	0.266044i	−1.50881	+	0.266044i	−0.866025	−	0.500000i	$-0.833333\pi$
$619$	−1.50881	+	0.266044i	−1.50881	+	0.266044i	−0.642788	+	0.766044i	$0.722222\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.939693	+	0.342020i	0.939693	+	0.342020i
$626$	1.15846	−	0.811160i	1.15846	−	0.811160i
$627$	−2.65765	+	0.712116i	−2.65765	+	0.712116i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$631$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$632$		0			0
$633$	3.20574	−	1.85083i	3.20574	−	1.85083i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.62760	+	0.592396i	−1.62760	+	0.592396i	−0.984808	−	0.173648i	$-0.944444\pi$
$641$	−1.62760	+	0.592396i	−1.62760	+	0.592396i	−0.642788	+	0.766044i	$0.722222\pi$
$642$	−2.14774	+	3.06729i	−2.14774	+	3.06729i
$643$	0.218763	−	0.469139i	0.218763	−	0.469139i	−0.766044	−	0.642788i	$-0.777778\pi$
$643$	0.218763	−	0.469139i	0.218763	−	0.469139i	0.984808	+	0.173648i	$0.0555556\pi$
$644$		0			0
$645$		0			0
$646$	3.33333	+	0.291629i	3.33333	+	0.291629i
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	1.43969	+	2.49362i	1.43969	+	2.49362i
$649$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$650$		0			0
$651$		0			0
$652$	0.0151922	−	0.173648i	0.0151922	−	0.173648i
$653$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$653$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$654$		0			0
$655$		0			0
$656$	−0.684040			−0.684040
$657$	−1.93969	+	1.62760i	−1.93969	+	1.62760i
$658$		0			0
$659$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	0.500000	−	0.866025i	$-0.333333\pi$
$659$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	−0.766044	+	0.642788i	$0.777778\pi$
$660$		0			0
$661$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$661$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$662$	0.142788	−	1.63207i	0.142788	−	1.63207i
$663$		0			0
$664$	0.133975	+	0.500000i	0.133975	+	0.500000i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$674$	1.58248	+	0.424024i	1.58248	+	0.424024i
$675$	0.500000	−	2.83564i	0.500000	−	2.83564i
$676$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$677$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$677$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$678$	0.251790	−	0.939693i	0.251790	−	0.939693i
$679$		0			0
$680$		0			0
$681$	−1.62760	+	0.939693i	−1.62760	+	0.939693i
$682$		0			0
$683$	0.0736672	+	0.157980i	0.0736672	+	0.157980i	0.939693	−	0.342020i	$-0.111111\pi$
$683$	0.0736672	+	0.157980i	0.0736672	+	0.157980i	−0.866025	+	0.500000i	$0.833333\pi$
$684$			4.38571i			4.38571i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.939693	+	0.657980i	−0.939693	+	0.657980i
$689$		0			0
$690$		0			0
$691$	−1.10806	−	1.58248i	−1.10806	−	1.58248i	−0.766044	−	0.642788i	$-0.777778\pi$
$691$	−1.10806	−	1.58248i	−1.10806	−	1.58248i	−0.342020	−	0.939693i	$-0.611111\pi$
$692$		0			0
$693$		0			0
$694$	0.984808	−	0.826352i	0.984808	−	0.826352i
$695$		0			0
$696$		0			0
$697$	−1.08248	−	0.757961i	−1.08248	−	0.757961i
$698$		0			0
$699$	−3.06729	+	2.14774i	−3.06729	+	2.14774i
$700$		0			0
$701$		0			0		−0.0871557	−	0.996195i	$-0.527778\pi$
$701$		0			0		0.0871557	+	0.996195i	$0.472222\pi$
$702$		0			0
$703$		0			0
$704$	0.357212	+	0.766044i	0.357212	+	0.766044i
$705$		0			0
$706$	−1.11334	+	0.642788i	−1.11334	+	0.642788i
$707$		0			0
$708$	2.40883	+	2.40883i	2.40883	+	2.40883i
$709$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$709$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.597672	+	1.28171i	−0.597672	+	1.28171i
$717$		0			0
$718$		0			0
$719$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$719$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$720$		0			0
$721$		0			0
$722$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$723$	−0.969139	−	3.61688i	−0.969139	−	3.61688i
$724$		0			0
$725$		0			0
$726$	0.0931980	+	0.528552i	0.0931980	+	0.528552i
$727$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$727$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$	−1.87939			−1.87939
$730$		0			0
$731$	−2.21613			−2.21613
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.0759757	+	0.283545i	0.0759757	+	0.283545i
$738$	0.866025	−	1.50000i	0.866025	−	1.50000i
$739$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$739$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.906308	−	0.422618i	$-0.861111\pi$
$743$		0			0		0.906308	+	0.422618i	$0.138889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.26604	−	0.339236i	−1.26604	−	0.339236i
$748$	−0.283545	+	1.60806i	−0.283545	+	1.60806i
$749$		0			0
$750$		0			0
$751$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$751$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$752$		0			0
$753$	2.37939	+	2.83564i	2.37939	+	2.83564i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	0.157980	−	0.0736672i	0.157980	−	0.0736672i
$759$		0			0
$760$		0			0
$761$	−0.692377	+	0.484808i	−0.692377	+	0.484808i	−0.866025	−	0.500000i	$-0.833333\pi$
$761$	−0.692377	+	0.484808i	−0.692377	+	0.484808i	0.173648	+	0.984808i	$0.444444\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.85083	−	0.326352i	1.85083	−	0.326352i
$769$	−1.14279	−	1.63207i	−1.14279	−	1.63207i	−0.642788	−	0.766044i	$-0.722222\pi$
$769$	−1.14279	−	1.63207i	−1.14279	−	1.63207i	−0.500000	−	0.866025i	$-0.666667\pi$
$770$		0			0
$771$	0.613341	+	0.223238i	0.613341	+	0.223238i
$772$	1.48481	−	1.03967i	1.48481	−	1.03967i
$773$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$773$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$774$	−0.253161	−	2.89364i	−0.253161	−	2.89364i
$775$		0			0
$776$			1.96962i			1.96962i
$777$		0			0
$778$		0			0
$779$	1.02606	−	0.592396i	1.02606	−	0.592396i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.642788	−	0.766044i	0.642788	−	0.766044i
$785$		0			0
$786$	2.97409	+	0.796905i	2.97409	+	0.796905i
$787$	0.642788	−	0.233956i	0.642788	−	0.233956i		−	1.00000i	$-0.5\pi$
$787$	0.642788	−	0.233956i	0.642788	−	0.233956i	0.642788	+	0.766044i	$0.277778\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−2.13207	−	0.186532i	−2.13207	−	0.186532i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$797$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$798$		0			0
$799$		0			0
$800$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$801$		0			0
$802$	−1.00000			−1.00000
$803$	−0.0736672	−	0.842020i	−0.0736672	−	0.842020i
$804$	0.652704			0.652704
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.439693	−	1.20805i	−0.439693	−	1.20805i	−0.939693	−	0.342020i	$-0.888889\pi$
$809$	−0.439693	−	1.20805i	−0.439693	−	1.20805i	0.500000	−	0.866025i	$-0.333333\pi$
$810$		0			0
$811$	0.642788	−	1.11334i	0.642788	−	1.11334i	−0.342020	−	0.939693i	$-0.611111\pi$
$811$	0.642788	−	1.11334i	0.642788	−	1.11334i	0.984808	−	0.173648i	$-0.0555556\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	3.29053	+	1.53440i	3.29053	+	1.53440i
$817$	0.839712	−	1.80077i	0.839712	−	1.80077i
$818$	0.939693	−	1.34202i	0.939693	−	1.34202i
$819$		0			0
$820$		0			0
$821$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$821$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$822$	0.826352	−	0.984808i	0.826352	−	0.984808i
$823$		0			0		0.996195	−	0.0871557i	$-0.0277778\pi$
$823$		0			0		−0.996195	+	0.0871557i	$0.972222\pi$
$824$		0			0
$825$	1.12326	+	1.12326i	1.12326	+	1.12326i
$826$		0			0
$827$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$827$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$828$		0			0
$829$		0			0		−0.422618	−	0.906308i	$-0.638889\pi$
$829$		0			0		0.422618	+	0.906308i	$0.361111\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.86603	−	0.500000i	1.86603	−	0.500000i
$834$	1.76604	−	1.23660i	1.76604	−	1.23660i
$835$		0			0
$836$	−1.19923	−	0.839712i	−1.19923	−	0.839712i
$837$		0			0
$838$	−1.26604	+	0.223238i	−1.26604	+	0.223238i
$839$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$839$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$840$		0			0
$841$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$842$		0			0
$843$	2.79053	+	1.95395i	2.79053	+	1.95395i
$844$	1.85083	+	0.673648i	1.85083	+	0.673648i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	1.87939i		−	1.87939i
$850$	−0.816436	−	1.75085i	−0.816436	−	1.75085i
$851$		0			0
$852$		0			0
$853$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$853$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$854$		0			0
$855$		0			0
$856$	−1.98481	+	0.173648i	−1.98481	+	0.173648i
$857$	−1.11334	+	1.32683i	−1.11334	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$857$	−1.11334	+	1.32683i	−1.11334	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$858$		0			0
$859$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.906308	−	0.422618i	$-0.861111\pi$
$863$		0			0		0.906308	+	0.422618i	$0.138889\pi$
$864$	−0.984808	+	2.70574i	−0.984808	+	2.70574i
$865$		0			0
$866$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$867$	2.56729	+	4.44667i	2.56729	+	4.44667i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−4.31908	−	2.49362i	−4.31908	−	2.49362i
$874$		0			0
$875$		0			0
$876$	−1.85083	−	0.326352i	−1.85083	−	0.326352i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.0999810	−	1.14279i	0.0999810	−	1.14279i	−0.766044	−	0.642788i	$-0.777778\pi$
$881$	0.0999810	−	1.14279i	0.0999810	−	1.14279i	0.866025	−	0.500000i	$-0.166667\pi$
$882$	0.866025	+	2.37939i	0.866025	+	2.37939i
$883$	0.424024	+	1.58248i	0.424024	+	1.58248i	0.766044	+	0.642788i	$0.222222\pi$
$883$	0.424024	+	1.58248i	0.424024	+	1.58248i	−0.342020	+	0.939693i	$0.611111\pi$
$884$		0			0
$885$		0			0
$886$	1.28171	−	1.28171i	1.28171	−	1.28171i
$887$		0			0		−0.996195	−	0.0871557i	$-0.972222\pi$
$887$		0			0		0.996195	+	0.0871557i	$0.0277778\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.39595	−	1.99362i	1.39595	−	1.99362i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.811160	−	0.811160i	−0.811160	−	0.811160i
$899$		0			0
$900$	2.19285	−	1.26604i	2.19285	−	1.26604i
$901$		0			0
$902$	0.244348	+	0.524005i	0.244348	+	0.524005i
$903$		0			0
$904$	0.469139	−	0.218763i	0.469139	−	0.218763i
$905$		0			0
$906$		0			0
$907$	1.58248	−	1.10806i	1.58248	−	1.10806i	0.642788	−	0.766044i	$-0.277778\pi$
$907$	1.58248	−	1.10806i	1.58248	−	1.10806i	0.939693	−	0.342020i	$-0.111111\pi$
$908$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$909$		0			0
$910$		0			0
$911$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$911$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$912$	−2.49362	+	2.09240i	−2.49362	+	2.09240i
$913$	0.335165	−	0.281237i	0.335165	−	0.281237i
$914$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−4.55657	+	3.19054i	−4.55657	+	3.19054i
$919$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$919$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$920$		0			0
$921$	2.40883	−	1.12326i	2.40883	−	1.12326i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.98481	−	0.173648i	1.98481	−	0.173648i	0.984808	−	0.173648i	$-0.0555556\pi$
$929$	1.98481	−	0.173648i	1.98481	−	0.173648i	1.00000			$0$
$930$		0			0
$931$	−0.300767	+	1.70574i	−0.300767	+	1.70574i
$932$	−1.92450	−	0.515668i	−1.92450	−	0.515668i
$933$		0			0
$934$	0.0999810	−	0.142788i	0.0999810	−	0.142788i
$935$		0			0
$936$		0			0
$937$	0.342020	−	0.939693i	0.342020	−	0.939693i	−0.642788	−	0.766044i	$-0.722222\pi$
$937$	0.342020	−	0.939693i	0.342020	−	0.939693i	0.984808	−	0.173648i	$-0.0555556\pi$
$938$		0			0
$939$	−1.87939	+	1.87939i	−1.87939	+	1.87939i
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	−0.157980	+	1.80572i	−0.157980	+	1.80572i
$945$		0			0
$946$	0.839712	+	0.484808i	0.839712	+	0.484808i
$947$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	0.500000	−	0.866025i	$-0.333333\pi$
$947$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	−0.766044	+	0.642788i	$0.777778\pi$
$948$		0			0
$949$		0			0
$950$	1.73205			1.73205
$951$		0			0
$952$		0			0
$953$	0.173648	+	0.984808i	0.173648	+	0.984808i	0.939693	+	0.342020i	$0.111111\pi$
$953$	0.173648	+	0.984808i	0.173648	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.342020	−	0.939693i	0.342020	−	0.939693i
$962$		0			0
$963$	2.13207	−	4.57224i	2.13207	−	4.57224i
$964$	1.14279	−	1.63207i	1.14279	−	1.63207i
$965$		0			0
$966$		0			0
$967$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$967$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$968$	−0.183564	+	0.218763i	−0.183564	+	0.218763i
$969$	−6.26462	+	0.548083i	−6.26462	+	0.548083i
$970$		0			0
$971$	1.40883	+	1.40883i	1.40883	+	1.40883i	0.766044	+	0.642788i	$0.222222\pi$
$971$	1.40883	+	1.40883i	1.40883	+	1.40883i	0.642788	+	0.766044i	$0.277778\pi$
$972$	−1.62760	−	1.93969i	−1.62760	−	1.93969i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.28171	−	0.597672i	1.28171	−	0.597672i	0.342020	−	0.939693i	$-0.388889\pi$
$977$	1.28171	−	0.597672i	1.28171	−	0.597672i	0.939693	+	0.342020i	$0.111111\pi$
$978$	0.0285521	+	0.326352i	0.0285521	+	0.326352i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	0.692377	+	0.484808i	0.692377	+	0.484808i
$983$		0			0		−0.573576	−	0.819152i	$-0.694444\pi$
$983$		0			0		0.573576	+	0.819152i	$0.305556\pi$
$984$	1.26604	−	0.223238i	1.26604	−	0.223238i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.819152	−	0.573576i	$-0.194444\pi$
$991$		0			0		−0.819152	+	0.573576i	$0.805556\pi$
$992$		0			0
$993$	0.268353	+	3.06729i	0.268353	+	3.06729i
$994$		0			0
$995$		0			0
$996$	−0.411141	−	0.881694i	−0.411141	−	0.881694i
$997$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$997$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$998$	0.592396	−	0.342020i	0.592396	−	0.342020i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	584.1.br.a.267.1	yes	12
4.3	odd	2		2336.1.ed.a.559.1		12
8.3	odd	2	CM	584.1.br.a.267.1	yes	12
8.5	even	2		2336.1.ed.a.559.1		12
73.35	even	36	inner	584.1.br.a.35.1	✓	12
292.35	odd	36		2336.1.ed.a.911.1		12
584.35	odd	36	inner	584.1.br.a.35.1	✓	12
584.181	even	36		2336.1.ed.a.911.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
584.1.br.a.35.1	✓	12	73.35	even	36	inner
584.1.br.a.35.1	✓	12	584.35	odd	36	inner
584.1.br.a.267.1	yes	12	1.1	even	1	trivial
584.1.br.a.267.1	yes	12	8.3	odd	2	CM
2336.1.ed.a.559.1		12	4.3	odd	2
2336.1.ed.a.559.1		12	8.5	even	2
2336.1.ed.a.911.1		12	292.35	odd	36
2336.1.ed.a.911.1		12	584.181	even	36

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:	\( N \)	\(=\)	\( 584 = 2^{3} \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	584.br (of order \(36\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.766044 - 0.642788i) q^{2} +(1.62760 + 0.939693i) q^{3} +(0.173648 + 0.984808i) q^{4} +(-0.642788 - 1.76604i) q^{6} +(0.500000 - 0.866025i) q^{8} +(1.26604 + 2.19285i) q^{9} +O(q^{10})\)	\(q+(-0.766044 - 0.642788i) q^{2} +(1.62760 + 0.939693i) q^{3} +(0.173648 + 0.984808i) q^{4} +(-0.642788 - 1.76604i) q^{6} +(0.500000 - 0.866025i) q^{8} +(1.26604 + 2.19285i) q^{9} +(-0.842020 - 0.0736672i) q^{11} +(-0.642788 + 1.76604i) q^{12} +(-0.939693 + 0.342020i) q^{16} +(-1.86603 - 0.500000i) q^{17} +(0.439693 - 2.49362i) q^{18} +(1.11334 - 1.32683i) q^{19} +(0.597672 + 0.597672i) q^{22} +(1.62760 - 0.939693i) q^{24} +(-0.984808 - 0.173648i) q^{25} +2.87939i q^{27} +(0.939693 + 0.342020i) q^{32} +(-1.30124 - 0.911141i) q^{33} +(1.10806 + 1.58248i) q^{34} +(-1.93969 + 1.62760i) q^{36} +(-1.70574 + 0.300767i) q^{38} +(0.642788 + 0.233956i) q^{41} +(1.10806 - 0.296905i) q^{43} +(-0.0736672 - 0.842020i) q^{44} +(-1.85083 - 0.326352i) q^{48} +(-0.866025 + 0.500000i) q^{49} +(0.642788 + 0.766044i) q^{50} +(-2.56729 - 2.56729i) q^{51} +(1.85083 - 2.20574i) q^{54} +(3.05888 - 1.11334i) q^{57} +(0.766044 - 1.64279i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(0.411141 + 1.53440i) q^{66} +(-0.118782 - 0.326352i) q^{67} +(0.168372 - 1.92450i) q^{68} +2.53209 q^{72} +(0.173648 + 0.984808i) q^{73} +(-1.43969 - 1.20805i) q^{75} +(1.50000 + 0.866025i) q^{76} +(-1.43969 + 2.49362i) q^{81} +(-0.342020 - 0.592396i) q^{82} +(-0.366025 + 0.366025i) q^{83} +(-1.03967 - 0.484808i) q^{86} +(-0.484808 + 0.692377i) q^{88} +(1.20805 + 1.43969i) q^{96} +(-1.70574 + 0.984808i) q^{97} +(0.984808 + 0.173648i) q^{98} +(-0.904494 - 1.93969i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{8} + 6 q^{9}+O(q^{10}) \) 12 * q + 6 * q^8 + 6 * q^9	\( 12 q + 6 q^{8} + 6 q^{9} - 6 q^{11} - 12 q^{17} - 6 q^{18} + 6 q^{33} - 12 q^{36} - 6 q^{64} - 6 q^{66} + 12 q^{72} - 6 q^{75} + 18 q^{76} - 6 q^{81} + 6 q^{83} + 6 q^{88} - 12 q^{99}+O(q^{100}) \) 12 * q + 6 * q^8 + 6 * q^9 - 6 * q^11 - 12 * q^17 - 6 * q^18 + 6 * q^33 - 12 * q^36 - 6 * q^64 - 6 * q^66 + 12 * q^72 - 6 * q^75 + 18 * q^76 - 6 * q^81 + 6 * q^83 + 6 * q^88 - 12 * q^99

Introduction

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