LMFDB - Embedded newform 1815.1.o.g.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,1,Mod(269,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 4]))

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

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

chi := DirichletCharacter("1815.269");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1815.o (of order $10$, degree $4$, not 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:	$0.905802997929$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.324000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 $ x^8 + 3x^6 + 9x^4 + 27*x^2 + 81
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.36236475.1

Embedding invariants

Embedding label			269.1
Root			$-1.40126 - 1.01807i$ of defining polynomial
Character	$\chi$	$=$	1815.269
Dual form			1815.1.o.g.614.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.535233 - 1.64728i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(-0.309017 + 0.951057i) q^{5} +(-0.535233 + 1.64728i) q^{6} +(1.40126 + 1.01807i) q^{8} +(0.309017 + 0.951057i) q^{9} +O(q^{10})$	\(q+(-0.535233 - 1.64728i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(-0.309017 + 0.951057i) q^{5} +(-0.535233 + 1.64728i) q^{6} +(1.40126 + 1.01807i) q^{8} +(0.309017 + 0.951057i) q^{9} +1.73205 q^{10} +2.00000 q^{12} +(0.809017 - 0.587785i) q^{15} +(0.309017 - 0.951057i) q^{16} +(0.535233 - 1.64728i) q^{17} +(1.40126 - 1.01807i) q^{18} +(-0.618034 - 1.90211i) q^{20} -1.00000 q^{23} +(-0.535233 - 1.64728i) q^{24} +(-0.809017 - 0.587785i) q^{25} +(0.309017 - 0.951057i) q^{27} +(-1.40126 - 1.01807i) q^{30} +(-0.309017 - 0.951057i) q^{31} -3.00000 q^{34} +(-1.61803 - 1.17557i) q^{36} +(-1.40126 + 1.01807i) q^{40} -1.00000 q^{45} +(0.535233 + 1.64728i) q^{46} +(-0.809017 - 0.587785i) q^{47} +(-0.809017 + 0.587785i) q^{48} +(0.309017 - 0.951057i) q^{49} +(-0.535233 + 1.64728i) q^{50} +(-1.40126 + 1.01807i) q^{51} +(0.309017 + 0.951057i) q^{53} -1.73205 q^{54} +(-0.618034 + 1.90211i) q^{60} +(0.535233 - 1.64728i) q^{61} +(-1.40126 + 1.01807i) q^{62} +(-0.309017 - 0.951057i) q^{64} +(1.07047 + 3.29456i) q^{68} +(0.809017 + 0.587785i) q^{69} +(-0.535233 + 1.64728i) q^{72} +(0.309017 + 0.951057i) q^{75} +(-0.535233 - 1.64728i) q^{79} +(0.809017 + 0.587785i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(1.40126 + 1.01807i) q^{85} +(0.535233 + 1.64728i) q^{90} +(1.61803 - 1.17557i) q^{92} +(-0.309017 + 0.951057i) q^{93} +(-0.535233 + 1.64728i) q^{94} -1.73205 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 - 4 * q^4 + 2 * q^5 - 2 * q^9	$ 8 q - 2 q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{9} + 16 q^{12} + 2 q^{15} - 2 q^{16} + 4 q^{20} - 8 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{31} - 24 q^{34} - 4 q^{36} - 8 q^{45} - 2 q^{47} - 2 q^{48} - 2 q^{49} - 2 q^{53} + 4 q^{60} + 2 q^{64} + 2 q^{69} - 2 q^{75} + 2 q^{80} - 2 q^{81} + 4 q^{92} + 2 q^{93}+O(q^{100}) $ 8 * q - 2 * q^3 - 4 * q^4 + 2 * q^5 - 2 * q^9 + 16 * q^12 + 2 * q^15 - 2 * q^16 + 4 * q^20 - 8 * q^23 - 2 * q^25 - 2 * q^27 + 2 * q^31 - 24 * q^34 - 4 * q^36 - 8 * q^45 - 2 * q^47 - 2 * q^48 - 2 * q^49 - 2 * q^53 + 4 * q^60 + 2 * q^64 + 2 * q^69 - 2 * q^75 + 2 * q^80 - 2 * q^81 + 4 * q^92 + 2 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$.

$n$	$727$	$1211$	$1696$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{2}{5}\right)$

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 $

$2$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$2$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$3$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$3$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$4$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$5$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$5$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$6$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$7$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$7$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$8$	1.40126	+	1.01807i	1.40126	+	1.01807i
$9$	0.309017	+	0.951057i	0.309017	+	0.951057i
$10$	1.73205			1.73205
$11$		0			0
$11$		0			0
$12$	2.00000			2.00000
$13$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$13$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$14$		0			0
$15$	0.809017	−	0.587785i	0.809017	−	0.587785i
$16$	0.309017	−	0.951057i	0.309017	−	0.951057i
$17$	0.535233	−	1.64728i	0.535233	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$17$	0.535233	−	1.64728i	0.535233	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$18$	1.40126	−	1.01807i	1.40126	−	1.01807i
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$21$		0			0
$22$		0			0
$23$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$25$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$26$		0			0
$27$	0.309017	−	0.951057i	0.309017	−	0.951057i
$28$		0			0
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$31$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$31$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$32$		0			0
$33$		0			0
$34$	−3.00000			−3.00000
$35$		0			0
$36$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$37$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$37$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$38$		0			0
$39$		0			0
$40$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$41$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$41$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−1.00000			−1.00000
$46$	0.535233	+	1.64728i	0.535233	+	1.64728i
$47$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$47$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$48$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$49$	0.309017	−	0.951057i	0.309017	−	0.951057i
$50$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$51$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$52$		0			0
$53$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$53$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$54$	−1.73205			−1.73205
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$59$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$60$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$61$	0.535233	−	1.64728i	0.535233	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$61$	0.535233	−	1.64728i	0.535233	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$62$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$63$		0			0
$64$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	1.07047	+	3.29456i	1.07047	+	3.29456i
$69$	0.809017	+	0.587785i	0.809017	+	0.587785i
$70$		0			0
$71$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$71$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$72$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$73$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$73$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$74$		0			0
$75$	0.309017	+	0.951057i	0.309017	+	0.951057i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$79$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$80$	0.809017	+	0.587785i	0.809017	+	0.587785i
$81$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$82$		0			0
$83$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$83$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$84$		0			0
$85$	1.40126	+	1.01807i	1.40126	+	1.01807i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.535233	+	1.64728i	0.535233	+	1.64728i
$91$		0			0
$92$	1.61803	−	1.17557i	1.61803	−	1.17557i
$93$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$94$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$95$		0			0
$96$		0			0
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$	−1.73205			−1.73205
$99$		0			0
$100$	2.00000			2.00000
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$	2.42705	+	1.76336i	2.42705	+	1.76336i
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$		0			0
$105$		0			0
$106$	1.40126	−	1.01807i	1.40126	−	1.01807i
$107$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$107$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$108$	0.618034	+	1.90211i	0.618034	+	1.90211i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$113$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$114$		0			0
$115$	0.309017	−	0.951057i	0.309017	−	0.951057i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	1.73205			1.73205
$121$		0			0
$122$	−3.00000			−3.00000
$123$		0			0
$124$	1.61803	+	1.17557i	1.61803	+	1.17557i
$125$	0.809017	−	0.587785i	0.809017	−	0.587785i
$126$		0			0
$127$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$127$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$128$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.809017	+	0.587785i	0.809017	+	0.587785i
$136$	2.42705	−	1.76336i	2.42705	−	1.76336i
$137$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$137$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$138$	0.535233	−	1.64728i	0.535233	−	1.64728i
$139$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$139$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$140$		0			0
$141$	0.309017	+	0.951057i	0.309017	+	0.951057i
$142$		0			0
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$148$		0			0
$149$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$149$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$150$	1.40126	−	1.01807i	1.40126	−	1.01807i
$151$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$151$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$152$		0			0
$153$	1.73205			1.73205
$154$		0			0
$155$	1.00000			1.00000
$156$		0			0
$157$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$157$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$158$	−2.42705	+	1.76336i	−2.42705	+	1.76336i
$159$	0.309017	−	0.951057i	0.309017	−	0.951057i
$160$		0			0
$161$		0			0
$162$	1.40126	+	1.01807i	1.40126	+	1.01807i
$163$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$163$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.535233	+	1.64728i	0.535233	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$167$	0.535233	+	1.64728i	0.535233	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$168$		0			0
$169$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$170$	0.927051	−	2.85317i	0.927051	−	2.85317i
$171$		0			0
$172$		0			0
$173$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$173$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$179$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$180$	1.61803	−	1.17557i	1.61803	−	1.17557i
$181$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$181$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$182$		0			0
$183$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$184$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$185$		0			0
$186$	1.73205			1.73205
$187$		0			0
$188$	2.00000			2.00000
$189$		0			0
$190$		0			0
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$193$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$193$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$194$		0			0
$195$		0			0
$196$	0.618034	+	1.90211i	0.618034	+	1.90211i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$200$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	1.07047	−	3.29456i	1.07047	−	3.29456i
$205$		0			0
$206$		0			0
$207$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$211$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$212$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$213$		0			0
$214$	−0.927051	+	2.85317i	−0.927051	+	2.85317i
$215$		0			0
$216$	1.40126	−	1.01807i	1.40126	−	1.01807i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$		0			0
$225$	0.309017	−	0.951057i	0.309017	−	0.951057i
$226$	0.535233	−	1.64728i	0.535233	−	1.64728i
$227$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$227$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$228$		0			0
$229$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$229$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$230$	−1.73205			−1.73205
$231$		0			0
$232$		0			0
$233$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$233$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$234$		0			0
$235$	0.809017	−	0.587785i	0.809017	−	0.587785i
$236$		0			0
$237$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$238$		0			0
$239$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$239$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$240$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$241$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$241$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$242$		0			0
$243$	1.00000			1.00000
$244$	1.07047	+	3.29456i	1.07047	+	3.29456i
$245$	0.809017	+	0.587785i	0.809017	+	0.587785i
$246$		0			0
$247$		0			0
$248$	0.535233	−	1.64728i	0.535233	−	1.64728i
$249$		0			0
$250$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$256$	1.61803	+	1.17557i	1.61803	+	1.17557i
$257$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$257$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$263$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	−1.00000			−1.00000
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$269$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$270$	0.535233	−	1.64728i	0.535233	−	1.64728i
$271$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.406737	+	0.913545i	$0.633333\pi$
$271$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$272$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$273$		0			0
$274$	−1.73205			−1.73205
$275$		0			0
$276$	−2.00000			−2.00000
$277$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$277$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$278$	−2.42705	−	1.76336i	−2.42705	−	1.76336i
$279$	0.809017	−	0.587785i	0.809017	−	0.587785i
$280$		0			0
$281$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$281$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$282$	1.40126	−	1.01807i	1.40126	−	1.01807i
$283$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$283$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$293$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$294$	1.40126	+	1.01807i	1.40126	+	1.01807i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$301$		0			0
$302$	−0.927051	+	2.85317i	−0.927051	+	2.85317i
$303$		0			0
$304$		0			0
$305$	1.40126	+	1.01807i	1.40126	+	1.01807i
$306$	−0.927051	−	2.85317i	−0.927051	−	2.85317i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$311$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$		0			0
$315$		0			0
$316$	2.80252	+	2.03615i	2.80252	+	2.03615i
$317$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$317$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$318$	−1.73205			−1.73205
$319$		0			0
$320$	1.00000			1.00000
$321$	0.535233	+	1.64728i	0.535233	+	1.64728i
$322$		0			0
$323$		0			0
$324$	0.618034	−	1.90211i	0.618034	−	1.90211i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$331$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	2.42705	−	1.76336i	2.42705	−	1.76336i
$335$		0			0
$336$		0			0
$337$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$337$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$338$	1.40126	+	1.01807i	1.40126	+	1.01807i
$339$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$340$	−3.46410			−3.46410
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$346$		0			0
$347$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$347$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$348$		0			0
$349$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.994522	+	0.104528i	$0.0333333\pi$
$349$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$353$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$359$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$360$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$361$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$362$	3.46410			3.46410
$363$		0			0
$364$		0			0
$365$		0			0
$366$	2.42705	+	1.76336i	2.42705	+	1.76336i
$367$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$367$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$368$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−1.00000			−1.00000
$376$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$377$		0			0
$378$		0			0
$379$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$379$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$383$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$384$	1.73205			1.73205
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$389$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$392$	1.40126	−	1.01807i	1.40126	−	1.01807i
$393$		0			0
$394$		0			0
$395$	1.73205			1.73205
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$399$		0			0
$400$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$401$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$401$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$406$		0			0
$407$		0			0
$408$	−3.00000			−3.00000
$409$	0.535233	+	1.64728i	0.535233	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$409$	0.535233	+	1.64728i	0.535233	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$410$		0			0
$411$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$412$		0			0
$413$		0			0
$414$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$415$		0			0
$416$		0			0
$417$	−1.73205			−1.73205
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$421$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$422$	−2.42705	+	1.76336i	−2.42705	+	1.76336i
$423$	0.309017	−	0.951057i	0.309017	−	0.951057i
$424$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$425$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$426$		0			0
$427$		0			0
$428$	3.46410			3.46410
$429$		0			0
$430$		0			0
$431$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$431$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$432$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$439$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	1.00000			1.00000
$442$		0			0
$443$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$443$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$449$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$450$	−1.73205			−1.73205
$451$		0			0
$452$	−2.00000			−2.00000
$453$	0.535233	+	1.64728i	0.535233	+	1.64728i
$454$	−2.42705	−	1.76336i	−2.42705	−	1.76336i
$455$		0			0
$456$		0			0
$457$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$457$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$458$	1.40126	−	1.01807i	1.40126	−	1.01807i
$459$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$460$	0.618034	+	1.90211i	0.618034	+	1.90211i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$466$	−2.42705	+	1.76336i	−2.42705	+	1.76336i
$467$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$467$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$468$		0			0
$469$		0			0
$470$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	3.00000			3.00000
$475$		0			0
$476$		0			0
$477$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$478$		0			0
$479$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$479$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$480$		0			0
$481$		0			0
$482$	0.927051	+	2.85317i	0.927051	+	2.85317i
$483$		0			0
$484$		0			0
$485$		0			0
$486$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$487$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$487$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$488$	2.42705	−	1.76336i	2.42705	−	1.76336i
$489$		0			0
$490$	0.535233	−	1.64728i	0.535233	−	1.64728i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000			−1.00000
$497$		0			0
$498$		0			0
$499$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$499$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$500$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$501$	0.535233	−	1.64728i	0.535233	−	1.64728i
$502$		0			0
$503$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.994522	+	0.104528i	$0.0333333\pi$
$503$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.00000			1.00000
$508$		0			0
$509$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$509$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$510$	−2.42705	+	1.76336i	−2.42705	+	1.76336i
$511$		0			0
$512$	0.535233	−	1.64728i	0.535233	−	1.64728i
$513$		0			0
$514$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$523$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$524$		0			0
$525$		0			0
$526$	0.927051	+	2.85317i	0.927051	+	2.85317i
$527$	−1.73205			−1.73205
$528$		0			0
$529$		0			0
$530$	0.535233	+	1.64728i	0.535233	+	1.64728i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	1.40126	−	1.01807i	1.40126	−	1.01807i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−2.00000			−2.00000
$541$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$541$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$542$	2.42705	+	1.76336i	2.42705	+	1.76336i
$543$	1.61803	−	1.17557i	1.61803	−	1.17557i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$547$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$548$	0.618034	+	1.90211i	0.618034	+	1.90211i
$549$	1.73205			1.73205
$550$		0			0
$551$		0			0
$552$	0.535233	+	1.64728i	0.535233	+	1.64728i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.07047	+	3.29456i	−1.07047	+	3.29456i
$557$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.406737	+	0.913545i	$0.633333\pi$
$557$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$558$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$563$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$564$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$565$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$571$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.809017	+	0.587785i	0.809017	+	0.587785i
$576$	0.809017	−	0.587785i	0.809017	−	0.587785i
$577$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$577$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$578$	−1.07047	+	3.29456i	−1.07047	+	3.29456i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.42705	−	1.76336i	−2.42705	−	1.76336i
$587$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$587$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$588$	0.618034	−	1.90211i	0.618034	−	1.90211i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$598$		0			0
$599$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$599$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$600$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$601$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$601$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$602$		0			0
$603$		0			0
$604$	3.46410			3.46410
$605$		0			0
$606$		0			0
$607$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$607$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$608$		0			0
$609$		0			0
$610$	0.927051	−	2.85317i	0.927051	−	2.85317i
$611$		0			0
$612$	−2.80252	+	2.03615i	−2.80252	+	2.03615i
$613$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$613$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.00000			2.00000			1.00000			$0$
$617$	2.00000			2.00000			1.00000			$0$
$618$		0			0
$619$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$619$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$620$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$621$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.309017	+	0.951057i	0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$631$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$632$	0.927051	−	2.85317i	0.927051	−	2.85317i
$633$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$634$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$635$		0			0
$636$	0.618034	+	1.90211i	0.618034	+	1.90211i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$641$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$641$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$642$	2.42705	−	1.76336i	2.42705	−	1.76336i
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$647$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$648$	−1.73205			−1.73205
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$653$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−2.00000			−2.00000			−1.00000			$\pi$
$661$	−2.00000			−2.00000			−1.00000			$\pi$
$662$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−2.80252	−	2.03615i	−2.80252	−	2.03615i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$673$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$674$		0			0
$675$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$676$	0.618034	−	1.90211i	0.618034	−	1.90211i
$677$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$677$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$678$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$679$		0			0
$680$	0.927051	+	2.85317i	0.927051	+	2.85317i
$681$	−1.73205			−1.73205
$682$		0			0
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$684$		0			0
$685$	0.809017	+	0.587785i	0.809017	+	0.587785i
$686$		0			0
$687$	0.309017	−	0.951057i	0.309017	−	0.951057i
$688$		0			0
$689$		0			0
$690$	1.40126	+	1.01807i	1.40126	+	1.01807i
$691$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$691$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$692$		0			0
$693$		0			0
$694$	3.00000			3.00000
$695$	0.535233	+	1.64728i	0.535233	+	1.64728i
$696$		0			0
$697$		0			0
$698$	0.927051	−	2.85317i	0.927051	−	2.85317i
$699$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$700$		0			0
$701$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$701$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−1.00000			−1.00000
$706$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$707$		0			0
$708$		0			0
$709$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$709$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$710$		0			0
$711$	1.40126	−	1.01807i	1.40126	−	1.01807i
$712$		0			0
$713$	0.309017	+	0.951057i	0.309017	+	0.951057i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$719$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$720$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$721$		0			0
$722$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$723$	1.40126	+	1.01807i	1.40126	+	1.01807i
$724$	−1.23607	−	3.80423i	−1.23607	−	3.80423i
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$	1.07047	−	3.29456i	1.07047	−	3.29456i
$733$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$733$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$734$		0			0
$735$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.535233	+	1.64728i	0.535233	+	1.64728i	0.743145	+	0.669131i	$0.233333\pi$
$739$	0.535233	+	1.64728i	0.535233	+	1.64728i	−0.207912	+	0.978148i	$0.566667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.535233	−	1.64728i	0.535233	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$743$	0.535233	−	1.64728i	0.535233	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$744$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$	0.535233	+	1.64728i	0.535233	+	1.64728i
$751$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$751$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$752$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$753$		0			0
$754$		0			0
$755$	1.40126	−	1.01807i	1.40126	−	1.01807i
$756$		0			0
$757$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$757$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$758$	1.73205			1.73205
$759$		0			0
$760$		0			0
$761$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$761$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$766$	−2.80252	+	2.03615i	−2.80252	+	2.03615i
$767$		0			0
$768$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$769$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$769$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	−1.00000			−1.00000
$772$		0			0
$773$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$773$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$774$		0			0
$775$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	3.00000			3.00000
$783$		0			0
$784$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$785$		0			0
$786$		0			0
$787$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$787$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$788$		0			0
$789$	1.40126	+	1.01807i	1.40126	+	1.01807i
$790$	−0.927051	−	2.85317i	−0.927051	−	2.85317i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.809017	+	0.587785i	0.809017	+	0.587785i
$796$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$797$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$797$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$798$		0			0
$799$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$809$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$810$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$811$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$811$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$812$		0			0
$813$	1.73205			1.73205
$814$		0			0
$815$		0			0
$816$	0.535233	+	1.64728i	0.535233	+	1.64728i
$817$		0			0
$818$	2.42705	−	1.76336i	2.42705	−	1.76336i
$819$		0			0
$820$		0			0
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$	1.40126	+	1.01807i	1.40126	+	1.01807i
$823$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$823$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$827$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$828$	1.61803	+	1.17557i	1.61803	+	1.17557i
$829$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$829$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$834$	0.927051	+	2.85317i	0.927051	+	2.85317i
$835$	−1.73205			−1.73205
$836$		0			0
$837$	−1.00000			−1.00000
$838$		0			0
$839$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$839$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$	0.535233	−	1.64728i	0.535233	−	1.64728i
$843$		0			0
$844$	2.80252	+	2.03615i	2.80252	+	2.03615i
$845$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$846$	−1.73205			−1.73205
$847$		0			0
$848$	1.00000			1.00000
$849$		0			0
$850$	2.42705	+	1.76336i	2.42705	+	1.76336i
$851$		0			0
$852$		0			0
$853$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$853$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$		0			0
$856$	−0.927051	−	2.85317i	−0.927051	−	2.85317i
$857$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$857$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−2.00000			−2.00000			−1.00000			$\pi$
$859$	−2.00000			−2.00000			−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$863$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.618034	+	1.90211i	0.618034	+	1.90211i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$877$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$878$	−0.927051	−	2.85317i	−0.927051	−	2.85317i
$879$	−1.73205			−1.73205
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$883$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$883$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$884$		0			0
$885$		0			0
$886$	1.07047	−	3.29456i	1.07047	−	3.29456i
$887$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$887$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.618034	+	1.90211i	0.618034	+	1.90211i
$901$	1.73205			1.73205
$902$		0			0
$903$		0			0
$904$	0.535233	+	1.64728i	0.535233	+	1.64728i
$905$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$906$	2.42705	−	1.76336i	2.42705	−	1.76336i
$907$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$907$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$908$	−1.07047	+	3.29456i	−1.07047	+	3.29456i
$909$		0			0
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−0.535233	−	1.64728i	−0.535233	−	1.64728i
$916$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$917$		0			0
$918$	−0.927051	+	2.85317i	−0.927051	+	2.85317i
$919$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$919$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$920$	1.40126	−	1.01807i	1.40126	−	1.01807i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$929$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$930$	−0.535233	+	1.64728i	−0.535233	+	1.64728i
$931$		0			0
$932$	2.80252	+	2.03615i	2.80252	+	2.03615i
$933$		0			0
$934$	1.73205			1.73205
$935$		0			0
$936$		0			0
$937$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$937$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$938$		0			0
$939$		0			0
$940$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$948$	−1.07047	−	3.29456i	−1.07047	−	3.29456i
$949$		0			0
$950$		0			0
$951$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$952$		0			0
$953$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$953$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$954$	1.40126	+	1.01807i	1.40126	+	1.01807i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$961$		0			0
$962$		0			0
$963$	0.535233	−	1.64728i	0.535233	−	1.64728i
$964$	2.80252	−	2.03615i	2.80252	−	2.03615i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$971$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$972$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.40126	−	1.01807i	−1.40126	−	1.01807i
$977$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$977$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$978$		0			0
$979$		0			0
$980$	−2.00000			−2.00000
$981$		0			0
$982$		0			0
$983$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$983$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$994$		0			0
$995$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$996$		0			0
$997$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$997$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$998$	−2.80252	−	2.03615i	−2.80252	−	2.03615i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.1.o.g.269.1		8
3.2	odd	2		1815.1.o.h.269.2		8
5.4	even	2		1815.1.o.h.269.2		8
11.2	odd	10	inner	1815.1.o.g.614.2		8
11.3	even	5		1815.1.g.h.1574.1	yes	2
11.4	even	5	inner	1815.1.o.g.1334.2		8
11.5	even	5	inner	1815.1.o.g.1049.2		8
11.6	odd	10	inner	1815.1.o.g.1049.1		8
11.7	odd	10	inner	1815.1.o.g.1334.1		8
11.8	odd	10		1815.1.g.h.1574.2	yes	2
11.9	even	5	inner	1815.1.o.g.614.1		8
11.10	odd	2	inner	1815.1.o.g.269.2		8
15.14	odd	2	CM	1815.1.o.g.269.1		8
33.2	even	10		1815.1.o.h.614.1		8
33.5	odd	10		1815.1.o.h.1049.1		8
33.8	even	10		1815.1.g.g.1574.1	✓	2
33.14	odd	10		1815.1.g.g.1574.2	yes	2
33.17	even	10		1815.1.o.h.1049.2		8
33.20	odd	10		1815.1.o.h.614.2		8
33.26	odd	10		1815.1.o.h.1334.1		8
33.29	even	10		1815.1.o.h.1334.2		8
33.32	even	2		1815.1.o.h.269.1		8
55.4	even	10		1815.1.o.h.1334.1		8
55.9	even	10		1815.1.o.h.614.2		8
55.14	even	10		1815.1.g.g.1574.2	yes	2
55.19	odd	10		1815.1.g.g.1574.1	✓	2
55.24	odd	10		1815.1.o.h.614.1		8
55.29	odd	10		1815.1.o.h.1334.2		8
55.39	odd	10		1815.1.o.h.1049.2		8
55.49	even	10		1815.1.o.h.1049.1		8
55.54	odd	2		1815.1.o.h.269.1		8
165.14	odd	10		1815.1.g.h.1574.1	yes	2
165.29	even	10	inner	1815.1.o.g.1334.1		8
165.59	odd	10	inner	1815.1.o.g.1334.2		8
165.74	even	10		1815.1.g.h.1574.2	yes	2
165.104	odd	10	inner	1815.1.o.g.1049.2		8
165.119	odd	10	inner	1815.1.o.g.614.1		8
165.134	even	10	inner	1815.1.o.g.614.2		8
165.149	even	10	inner	1815.1.o.g.1049.1		8
165.164	even	2	inner	1815.1.o.g.269.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.1.g.g.1574.1	✓	2	33.8	even	10
1815.1.g.g.1574.1	✓	2	55.19	odd	10
1815.1.g.g.1574.2	yes	2	33.14	odd	10
1815.1.g.g.1574.2	yes	2	55.14	even	10
1815.1.g.h.1574.1	yes	2	11.3	even	5
1815.1.g.h.1574.1	yes	2	165.14	odd	10
1815.1.g.h.1574.2	yes	2	11.8	odd	10
1815.1.g.h.1574.2	yes	2	165.74	even	10
1815.1.o.g.269.1		8	1.1	even	1	trivial
1815.1.o.g.269.1		8	15.14	odd	2	CM
1815.1.o.g.269.2		8	11.10	odd	2	inner
1815.1.o.g.269.2		8	165.164	even	2	inner
1815.1.o.g.614.1		8	11.9	even	5	inner
1815.1.o.g.614.1		8	165.119	odd	10	inner
1815.1.o.g.614.2		8	11.2	odd	10	inner
1815.1.o.g.614.2		8	165.134	even	10	inner
1815.1.o.g.1049.1		8	11.6	odd	10	inner
1815.1.o.g.1049.1		8	165.149	even	10	inner
1815.1.o.g.1049.2		8	11.5	even	5	inner
1815.1.o.g.1049.2		8	165.104	odd	10	inner
1815.1.o.g.1334.1		8	11.7	odd	10	inner
1815.1.o.g.1334.1		8	165.29	even	10	inner
1815.1.o.g.1334.2		8	11.4	even	5	inner
1815.1.o.g.1334.2		8	165.59	odd	10	inner
1815.1.o.h.269.1		8	33.32	even	2
1815.1.o.h.269.1		8	55.54	odd	2
1815.1.o.h.269.2		8	3.2	odd	2
1815.1.o.h.269.2		8	5.4	even	2
1815.1.o.h.614.1		8	33.2	even	10
1815.1.o.h.614.1		8	55.24	odd	10
1815.1.o.h.614.2		8	33.20	odd	10
1815.1.o.h.614.2		8	55.9	even	10
1815.1.o.h.1049.1		8	33.5	odd	10
1815.1.o.h.1049.1		8	55.49	even	10
1815.1.o.h.1049.2		8	33.17	even	10
1815.1.o.h.1049.2		8	55.39	odd	10
1815.1.o.h.1334.1		8	33.26	odd	10
1815.1.o.h.1334.1		8	55.4	even	10
1815.1.o.h.1334.2		8	33.29	even	10
1815.1.o.h.1334.2		8	55.29	odd	10

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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1815.o (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.535233 - 1.64728i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(-0.309017 + 0.951057i) q^{5} +(-0.535233 + 1.64728i) q^{6} +(1.40126 + 1.01807i) q^{8} +(0.309017 + 0.951057i) q^{9} +O(q^{10})\)	\(q+(-0.535233 - 1.64728i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(-0.309017 + 0.951057i) q^{5} +(-0.535233 + 1.64728i) q^{6} +(1.40126 + 1.01807i) q^{8} +(0.309017 + 0.951057i) q^{9} +1.73205 q^{10} +2.00000 q^{12} +(0.809017 - 0.587785i) q^{15} +(0.309017 - 0.951057i) q^{16} +(0.535233 - 1.64728i) q^{17} +(1.40126 - 1.01807i) q^{18} +(-0.618034 - 1.90211i) q^{20} -1.00000 q^{23} +(-0.535233 - 1.64728i) q^{24} +(-0.809017 - 0.587785i) q^{25} +(0.309017 - 0.951057i) q^{27} +(-1.40126 - 1.01807i) q^{30} +(-0.309017 - 0.951057i) q^{31} -3.00000 q^{34} +(-1.61803 - 1.17557i) q^{36} +(-1.40126 + 1.01807i) q^{40} -1.00000 q^{45} +(0.535233 + 1.64728i) q^{46} +(-0.809017 - 0.587785i) q^{47} +(-0.809017 + 0.587785i) q^{48} +(0.309017 - 0.951057i) q^{49} +(-0.535233 + 1.64728i) q^{50} +(-1.40126 + 1.01807i) q^{51} +(0.309017 + 0.951057i) q^{53} -1.73205 q^{54} +(-0.618034 + 1.90211i) q^{60} +(0.535233 - 1.64728i) q^{61} +(-1.40126 + 1.01807i) q^{62} +(-0.309017 - 0.951057i) q^{64} +(1.07047 + 3.29456i) q^{68} +(0.809017 + 0.587785i) q^{69} +(-0.535233 + 1.64728i) q^{72} +(0.309017 + 0.951057i) q^{75} +(-0.535233 - 1.64728i) q^{79} +(0.809017 + 0.587785i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(1.40126 + 1.01807i) q^{85} +(0.535233 + 1.64728i) q^{90} +(1.61803 - 1.17557i) q^{92} +(-0.309017 + 0.951057i) q^{93} +(-0.535233 + 1.64728i) q^{94} -1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 - 4 * q^4 + 2 * q^5 - 2 * q^9	\( 8 q - 2 q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{9} + 16 q^{12} + 2 q^{15} - 2 q^{16} + 4 q^{20} - 8 q^{23} - 2 q^{25} - 2 q^{27} + 2 q^{31} - 24 q^{34} - 4 q^{36} - 8 q^{45} - 2 q^{47} - 2 q^{48} - 2 q^{49} - 2 q^{53} + 4 q^{60} + 2 q^{64} + 2 q^{69} - 2 q^{75} + 2 q^{80} - 2 q^{81} + 4 q^{92} + 2 q^{93}+O(q^{100}) \) 8 * q - 2 * q^3 - 4 * q^4 + 2 * q^5 - 2 * q^9 + 16 * q^12 + 2 * q^15 - 2 * q^16 + 4 * q^20 - 8 * q^23 - 2 * q^25 - 2 * q^27 + 2 * q^31 - 24 * q^34 - 4 * q^36 - 8 * q^45 - 2 * q^47 - 2 * q^48 - 2 * q^49 - 2 * q^53 + 4 * q^60 + 2 * q^64 + 2 * q^69 - 2 * q^75 + 2 * q^80 - 2 * q^81 + 4 * q^92 + 2 * q^93

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