LMFDB - Embedded newform 1045.1.w.e.664.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,1,Mod(284,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 6, 5]))

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

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

chi := DirichletCharacter("1045.284");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1045.w (of order $10$, degree $4$, 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:	$0.521522938201$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.2.87298324158753125.1

Embedding invariants

Embedding label			664.1
Root			$-0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1045.664
Dual form			1045.1.w.e.949.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.53884 - 1.11803i) q^{2} +(-0.363271 + 1.11803i) q^{3} +(0.809017 + 2.48990i) q^{4} +(-0.809017 + 0.587785i) q^{5} +(1.80902 - 1.31433i) q^{6} +(0.951057 - 2.92705i) q^{8} +(-0.309017 - 0.224514i) q^{9} +O(q^{10})$	\(q+(-1.53884 - 1.11803i) q^{2} +(-0.363271 + 1.11803i) q^{3} +(0.809017 + 2.48990i) q^{4} +(-0.809017 + 0.587785i) q^{5} +(1.80902 - 1.31433i) q^{6} +(0.951057 - 2.92705i) q^{8} +(-0.309017 - 0.224514i) q^{9} +1.90211 q^{10} +(0.809017 + 0.587785i) q^{11} -3.07768 q^{12} +(1.53884 + 1.11803i) q^{13} +(-0.363271 - 1.11803i) q^{15} +(-2.61803 + 1.90211i) q^{16} +(0.224514 + 0.690983i) q^{18} +(0.309017 - 0.951057i) q^{19} +(-2.11803 - 1.53884i) q^{20} +(-0.587785 - 1.80902i) q^{22} +(2.92705 + 2.12663i) q^{24} +(0.309017 - 0.951057i) q^{25} +(-1.11803 - 3.44095i) q^{26} +(-0.587785 + 0.427051i) q^{27} +(-0.690983 + 2.12663i) q^{30} +3.07768 q^{32} +(-0.951057 + 0.690983i) q^{33} +(0.309017 - 0.951057i) q^{36} +(0.363271 + 1.11803i) q^{37} +(-1.53884 + 1.11803i) q^{38} +(-1.80902 + 1.31433i) q^{39} +(0.951057 + 2.92705i) q^{40} +(-0.809017 + 2.48990i) q^{44} +0.381966 q^{45} +(-1.17557 - 3.61803i) q^{48} +(-0.809017 + 0.587785i) q^{49} +(-1.53884 + 1.11803i) q^{50} +(-1.53884 + 4.73607i) q^{52} +1.38197 q^{54} -1.00000 q^{55} +(0.951057 + 0.690983i) q^{57} +(2.48990 - 1.80902i) q^{60} +(-1.30902 + 0.951057i) q^{61} +(-2.11803 - 1.53884i) q^{64} -1.90211 q^{65} +2.23607 q^{66} +1.17557 q^{67} +(-0.951057 + 0.690983i) q^{72} +(0.690983 - 2.12663i) q^{74} +(0.951057 + 0.690983i) q^{75} +2.61803 q^{76} +4.25325 q^{78} +(1.00000 - 3.07768i) q^{80} +(-0.381966 - 1.17557i) q^{81} +(2.48990 - 1.80902i) q^{88} +(-0.587785 - 0.427051i) q^{90} +(0.309017 + 0.951057i) q^{95} +(-1.11803 + 3.44095i) q^{96} +(-1.53884 - 1.11803i) q^{97} +1.90211 q^{98} +(-0.118034 - 0.363271i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9}+O(q^{10}) $ 8 * q + 2 * q^4 - 2 * q^5 + 10 * q^6 + 2 * q^9	$ 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9} + 2 q^{11} - 12 q^{16} - 2 q^{19} - 8 q^{20} + 10 q^{24} - 2 q^{25} - 10 q^{30} - 2 q^{36} - 10 q^{39} - 2 q^{44} + 12 q^{45} - 2 q^{49} + 20 q^{54} - 8 q^{55} - 6 q^{61} - 8 q^{64} + 10 q^{74} + 12 q^{76} + 8 q^{80} - 12 q^{81} - 2 q^{95} + 8 q^{99}+O(q^{100}) $ 8 * q + 2 * q^4 - 2 * q^5 + 10 * q^6 + 2 * q^9 + 2 * q^11 - 12 * q^16 - 2 * q^19 - 8 * q^20 + 10 * q^24 - 2 * q^25 - 10 * q^30 - 2 * q^36 - 10 * q^39 - 2 * q^44 + 12 * q^45 - 2 * q^49 + 20 * q^54 - 8 * q^55 - 6 * q^61 - 8 * q^64 + 10 * q^74 + 12 * q^76 + 8 * q^80 - 12 * q^81 - 2 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$-1$	$e\left(\frac{1}{5}\right)$	$-1$

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 $

$2$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$2$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$3$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$3$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$4$	0.809017	+	2.48990i	0.809017	+	2.48990i
$5$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$5$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$6$	1.80902	−	1.31433i	1.80902	−	1.31433i
$7$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$7$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$8$	0.951057	−	2.92705i	0.951057	−	2.92705i
$9$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$10$	1.90211			1.90211
$11$	0.809017	+	0.587785i	0.809017	+	0.587785i
$11$	0.809017	+	0.587785i	0.809017	+	0.587785i
$12$	−3.07768			−3.07768
$13$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$13$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$14$		0			0
$15$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$16$	−2.61803	+	1.90211i	−2.61803	+	1.90211i
$17$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$17$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$18$	0.224514	+	0.690983i	0.224514	+	0.690983i
$19$	0.309017	−	0.951057i	0.309017	−	0.951057i
$19$	0.309017	−	0.951057i	0.309017	−	0.951057i
$20$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$21$		0			0
$22$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	2.92705	+	2.12663i	2.92705	+	2.12663i
$25$	0.309017	−	0.951057i	0.309017	−	0.951057i
$26$	−1.11803	−	3.44095i	−1.11803	−	3.44095i
$27$	−0.587785	+	0.427051i	−0.587785	+	0.427051i
$28$		0			0
$29$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$29$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$30$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$	3.07768			3.07768
$33$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$34$		0			0
$35$		0			0
$36$	0.309017	−	0.951057i	0.309017	−	0.951057i
$37$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$37$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$38$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$39$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$40$	0.951057	+	2.92705i	0.951057	+	2.92705i
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$45$	0.381966			0.381966
$46$		0			0
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$	−1.17557	−	3.61803i	−1.17557	−	3.61803i
$49$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$50$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$51$		0			0
$52$	−1.53884	+	4.73607i	−1.53884	+	4.73607i
$53$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$53$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$54$	1.38197			1.38197
$55$	−1.00000			−1.00000
$56$		0			0
$57$	0.951057	+	0.690983i	0.951057	+	0.690983i
$58$		0			0
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$	2.48990	−	1.80902i	2.48990	−	1.80902i
$61$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$61$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$65$	−1.90211			−1.90211
$66$	2.23607			2.23607
$67$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$67$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$73$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$73$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$74$	0.690983	−	2.12663i	0.690983	−	2.12663i
$75$	0.951057	+	0.690983i	0.951057	+	0.690983i
$76$	2.61803			2.61803
$77$		0			0
$78$	4.25325			4.25325
$79$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$79$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$80$	1.00000	−	3.07768i	1.00000	−	3.07768i
$81$	−0.381966	−	1.17557i	−0.381966	−	1.17557i
$82$		0			0
$83$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$83$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	2.48990	−	1.80902i	2.48990	−	1.80902i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.587785	−	0.427051i	−0.587785	−	0.427051i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.309017	+	0.951057i	0.309017	+	0.951057i
$96$	−1.11803	+	3.44095i	−1.11803	+	3.44095i
$97$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$97$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$98$	1.90211			1.90211
$99$	−0.118034	−	0.363271i	−0.118034	−	0.363271i
$100$	2.61803			2.61803
$101$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$101$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$102$		0			0
$103$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$103$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$104$	4.73607	−	3.44095i	4.73607	−	3.44095i
$105$		0			0
$106$		0			0
$107$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$107$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$108$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	1.53884	+	1.11803i	1.53884	+	1.11803i
$111$	−1.38197			−1.38197
$112$		0			0
$113$	−0.587785	+	1.80902i	−0.587785	+	1.80902i			1.00000i	$0.5\pi$
$113$	−0.587785	+	1.80902i	−0.587785	+	1.80902i	−0.587785	+	0.809017i	$0.700000\pi$
$114$	−0.690983	−	2.12663i	−0.690983	−	2.12663i
$115$		0			0
$116$		0			0
$117$	−0.224514	−	0.690983i	−0.224514	−	0.690983i
$118$		0			0
$119$		0			0
$120$	−3.61803			−3.61803
$121$	0.309017	+	0.951057i	0.309017	+	0.951057i
$122$	3.07768			3.07768
$123$		0			0
$124$		0			0
$125$	0.309017	+	0.951057i	0.309017	+	0.951057i
$126$		0			0
$127$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$127$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$128$	0.587785	+	1.80902i	0.587785	+	1.80902i
$129$		0			0
$130$	2.92705	+	2.12663i	2.92705	+	2.12663i
$131$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$131$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$132$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$133$		0			0
$134$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$135$	0.224514	−	0.690983i	0.224514	−	0.690983i
$136$		0			0
$137$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$137$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$138$		0			0
$139$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$139$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.587785	+	1.80902i	0.587785	+	1.80902i
$144$	1.23607			1.23607
$145$		0			0
$146$		0			0
$147$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$148$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$149$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$149$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$150$	−0.690983	−	2.12663i	−0.690983	−	2.12663i
$151$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$151$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$152$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−4.73607	−	3.44095i	−4.73607	−	3.44095i
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$		0			0
$159$		0			0
$160$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$161$		0			0
$162$	−0.726543	+	2.23607i	−0.726543	+	2.23607i
$163$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$163$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$164$		0			0
$165$	0.363271	−	1.11803i	0.363271	−	1.11803i
$166$		0			0
$167$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$167$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$168$		0			0
$169$	0.809017	+	2.48990i	0.809017	+	2.48990i
$170$		0			0
$171$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$172$		0			0
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$		0			0
$176$	−3.23607			−3.23607
$177$		0			0
$178$		0			0
$179$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$179$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$180$	0.309017	+	0.951057i	0.309017	+	0.951057i
$181$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$181$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$184$		0			0
$185$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.587785	−	1.80902i	0.587785	−	1.80902i
$191$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$191$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$192$	2.48990	−	1.80902i	2.48990	−	1.80902i
$193$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$193$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$194$	1.11803	+	3.44095i	1.11803	+	3.44095i
$195$	0.690983	−	2.12663i	0.690983	−	2.12663i
$196$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.224514	+	0.690983i	−0.224514	+	0.690983i
$199$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$199$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$200$	−2.48990	−	1.80902i	−2.48990	−	1.80902i
$201$	−0.427051	+	1.31433i	−0.427051	+	1.31433i
$202$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.11803	+	3.44095i	−1.11803	+	3.44095i
$207$		0			0
$208$	−6.15537			−6.15537
$209$	0.809017	−	0.587785i	0.809017	−	0.587785i
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$		0			0
$213$		0			0
$214$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$215$		0			0
$216$	0.690983	+	2.12663i	0.690983	+	2.12663i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$221$		0			0
$222$	2.12663	+	1.54508i	2.12663	+	1.54508i
$223$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$223$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$224$		0			0
$225$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$226$	2.92705	−	2.12663i	2.92705	−	2.12663i
$227$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$227$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$228$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$229$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$229$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$233$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$234$	−0.427051	+	1.31433i	−0.427051	+	1.31433i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$239$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$240$	3.07768	+	2.23607i	3.07768	+	2.23607i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.587785	−	1.80902i	0.587785	−	1.80902i
$243$	0.726543			0.726543
$244$	−3.42705	−	2.48990i	−3.42705	−	2.48990i
$245$	0.309017	−	0.951057i	0.309017	−	0.951057i
$246$		0			0
$247$	1.53884	−	1.11803i	1.53884	−	1.11803i
$248$		0			0
$249$		0			0
$250$	0.587785	−	1.80902i	0.587785	−	1.80902i
$251$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$251$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	−	0.951057i	0.309017	−	0.951057i
$257$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$257$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$258$		0			0
$259$		0			0
$260$	−1.53884	−	4.73607i	−1.53884	−	4.73607i
$261$		0			0
$262$	2.48990	+	1.80902i	2.48990	+	1.80902i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	1.11803	+	3.44095i	1.11803	+	3.44095i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.951057	+	2.92705i	0.951057	+	2.92705i
$269$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$269$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$270$	−1.11803	+	0.812299i	−1.11803	+	0.812299i
$271$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$271$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.809017	−	0.587785i	0.809017	−	0.587785i
$276$		0			0
$277$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$277$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$278$	−1.17557	+	3.61803i	−1.17557	+	3.61803i
$279$		0			0
$280$		0			0
$281$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$281$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$282$		0			0
$283$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$283$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$284$		0			0
$285$	−1.17557			−1.17557
$286$	1.11803	−	3.44095i	1.11803	−	3.44095i
$287$		0			0
$288$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$289$	0.309017	−	0.951057i	0.309017	−	0.951057i
$290$		0			0
$291$	1.80902	−	1.31433i	1.80902	−	1.31433i
$292$		0			0
$293$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$293$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$294$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$295$		0			0
$296$	3.61803			3.61803
$297$	−0.726543			−0.726543
$298$	−3.80423			−3.80423
$299$		0			0
$300$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$301$		0			0
$302$		0			0
$303$	−0.587785	+	0.427051i	−0.587785	+	0.427051i
$304$	1.00000	+	3.07768i	1.00000	+	3.07768i
$305$	0.500000	−	1.53884i	0.500000	−	1.53884i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	2.23607			2.23607
$310$		0			0
$311$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$311$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$312$	2.12663	+	6.54508i	2.12663	+	6.54508i
$313$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$313$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$317$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$318$		0			0
$319$		0			0
$320$	2.61803			2.61803
$321$	1.11803	+	0.812299i	1.11803	+	0.812299i
$322$		0			0
$323$		0			0
$324$	2.61803	−	1.90211i	2.61803	−	1.90211i
$325$	1.53884	−	1.11803i	1.53884	−	1.11803i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	0.138757	−	0.427051i	0.138757	−	0.427051i
$334$	−0.690983	−	2.12663i	−0.690983	−	2.12663i
$335$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$336$		0			0
$337$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$337$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$338$	1.53884	−	4.73607i	1.53884	−	4.73607i
$339$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$340$		0			0
$341$		0			0
$342$	0.726543			0.726543
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$347$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$348$		0			0
$349$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$349$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$350$		0			0
$351$	−1.38197			−1.38197
$352$	2.48990	+	1.80902i	2.48990	+	1.80902i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$359$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$360$	0.363271	−	1.11803i	0.363271	−	1.11803i
$361$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$362$		0			0
$363$	−1.17557			−1.17557
$364$		0			0
$365$		0			0
$366$	−1.11803	+	3.44095i	−1.11803	+	3.44095i
$367$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$367$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$368$		0			0
$369$		0			0
$370$	0.690983	+	2.12663i	0.690983	+	2.12663i
$371$		0			0
$372$		0			0
$373$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$373$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$374$		0			0
$375$	−1.17557			−1.17557
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$379$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$380$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$381$		0			0
$382$	0.363271	−	1.11803i	0.363271	−	1.11803i
$383$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$383$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$384$	−2.23607			−2.23607
$385$		0			0
$386$	−2.23607			−2.23607
$387$		0			0
$388$	1.53884	−	4.73607i	1.53884	−	4.73607i
$389$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$389$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$390$	−3.44095	+	2.50000i	−3.44095	+	2.50000i
$391$		0			0
$392$	0.951057	+	2.92705i	0.951057	+	2.92705i
$393$	0.587785	−	1.80902i	0.587785	−	1.80902i
$394$		0			0
$395$		0			0
$396$	0.809017	−	0.587785i	0.809017	−	0.587785i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	2.48990	+	1.80902i	2.48990	+	1.80902i
$399$		0			0
$400$	1.00000	+	3.07768i	1.00000	+	3.07768i
$401$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$401$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$402$	2.12663	−	1.54508i	2.12663	−	1.54508i
$403$		0			0
$404$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$405$	1.00000	+	0.726543i	1.00000	+	0.726543i
$406$		0			0
$407$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$408$		0			0
$409$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$409$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$410$		0			0
$411$		0			0
$412$	4.02874	−	2.92705i	4.02874	−	2.92705i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	4.73607	+	3.44095i	4.73607	+	3.44095i
$417$	2.35114			2.35114
$418$	−1.90211			−1.90211
$419$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$419$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$420$		0			0
$421$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$421$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	3.07768			3.07768
$429$	−2.23607			−2.23607
$430$		0			0
$431$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$431$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$432$	0.726543	−	2.23607i	0.726543	−	2.23607i
$433$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$433$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$441$	0.381966			0.381966
$442$		0			0
$443$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$443$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$444$	−1.11803	−	3.44095i	−1.11803	−	3.44095i
$445$		0			0
$446$		0			0
$447$	0.726543	+	2.23607i	0.726543	+	2.23607i
$448$		0			0
$449$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$449$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$450$	0.726543			0.726543
$451$		0			0
$452$	−4.97980			−4.97980
$453$		0			0
$454$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$455$		0			0
$456$	2.92705	−	2.12663i	2.92705	−	2.12663i
$457$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$457$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$458$	−1.17557	−	3.61803i	−1.17557	−	3.61803i
$459$		0			0
$460$		0			0
$461$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$461$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$467$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$468$	1.53884	−	1.11803i	1.53884	−	1.11803i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$476$		0			0
$477$		0			0
$478$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$479$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$479$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$480$	−1.11803	−	3.44095i	−1.11803	−	3.44095i
$481$	−0.690983	+	2.12663i	−0.690983	+	2.12663i
$482$		0			0
$483$		0			0
$484$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$485$	1.90211			1.90211
$486$	−1.11803	−	0.812299i	−1.11803	−	0.812299i
$487$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$487$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$488$	1.53884	+	4.73607i	1.53884	+	4.73607i
$489$		0			0
$490$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$491$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$491$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$492$		0			0
$493$		0			0
$494$	−3.61803			−3.61803
$495$	0.309017	+	0.224514i	0.309017	+	0.224514i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$499$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$500$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$501$	−1.11803	+	0.812299i	−1.11803	+	0.812299i
$502$	−0.951057	−	2.92705i	−0.951057	−	2.92705i
$503$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$503$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$504$		0			0
$505$	−0.618034			−0.618034
$506$		0			0
$507$	−3.07768			−3.07768
$508$		0			0
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	0.224514	+	0.690983i	0.224514	+	0.690983i
$514$	−1.11803	+	3.44095i	−1.11803	+	3.44095i
$515$	1.53884	+	1.11803i	1.53884	+	1.11803i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−1.80902	+	5.56758i	−1.80902	+	5.56758i
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$		0			0
$523$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$523$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$524$	−1.30902	−	4.02874i	−1.30902	−	4.02874i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.17557	−	3.61803i	1.17557	−	3.61803i
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.363271	+	1.11803i	0.363271	+	1.11803i
$536$	1.11803	−	3.44095i	1.11803	−	3.44095i
$537$		0			0
$538$		0			0
$539$	−1.00000			−1.00000
$540$	1.90211			1.90211
$541$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$541$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$542$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$547$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$548$		0			0
$549$	0.618034			0.618034
$550$	−1.90211			−1.90211
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.11803	−	0.812299i	1.11803	−	0.812299i
$556$	4.23607	−	3.07768i	4.23607	−	3.07768i
$557$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$557$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$563$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$564$		0			0
$565$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$569$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$570$	1.80902	+	1.31433i	1.80902	+	1.31433i
$571$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$571$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$572$	−4.02874	+	2.92705i	−4.02874	+	2.92705i
$573$	−0.726543			−0.726543
$574$		0			0
$575$		0			0
$576$	0.309017	+	0.951057i	0.309017	+	0.951057i
$577$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$577$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$578$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$579$	0.427051	+	1.31433i	0.427051	+	1.31433i
$580$		0			0
$581$		0			0
$582$	−4.25325			−4.25325
$583$		0			0
$584$		0			0
$585$	0.587785	+	0.427051i	0.587785	+	0.427051i
$586$	1.11803	−	3.44095i	1.11803	−	3.44095i
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$	2.48990	−	1.80902i	2.48990	−	1.80902i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−3.07768	−	2.23607i	−3.07768	−	2.23607i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	1.11803	+	0.812299i	1.11803	+	0.812299i
$595$		0			0
$596$	4.23607	+	3.07768i	4.23607	+	3.07768i
$597$	0.587785	−	1.80902i	0.587785	−	1.80902i
$598$		0			0
$599$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$599$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$600$	2.92705	−	2.12663i	2.92705	−	2.12663i
$601$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$601$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$	−0.363271	−	0.263932i	−0.363271	−	0.263932i
$604$		0			0
$605$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$606$	1.38197			1.38197
$607$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$607$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$608$	0.951057	−	2.92705i	0.951057	−	2.92705i
$609$		0			0
$610$	−2.48990	+	1.80902i	−2.48990	+	1.80902i
$611$		0			0
$612$		0			0
$613$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$613$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−3.44095	−	2.50000i	−3.44095	−	2.50000i
$619$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$619$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$620$		0			0
$621$		0			0
$622$	0.951057	−	0.690983i	0.951057	−	0.690983i
$623$		0			0
$624$	2.23607	−	6.88191i	2.23607	−	6.88191i
$625$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$626$		0			0
$627$	0.363271	+	1.11803i	0.363271	+	1.11803i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$631$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.90211			−1.90211
$638$		0			0
$639$		0			0
$640$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$641$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$641$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$642$	−0.812299	−	2.50000i	−0.812299	−	2.50000i
$643$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$643$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$	−3.80423			−3.80423
$649$		0			0
$650$	−3.61803			−3.61803
$651$		0			0
$652$		0			0
$653$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$653$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$654$		0			0
$655$	1.30902	−	0.951057i	1.30902	−	0.951057i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$	3.07768			3.07768
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.690983	+	0.502029i	−0.690983	+	0.502029i
$667$		0			0
$668$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$669$		0			0
$670$	2.23607			2.23607
$671$	−1.61803			−1.61803
$672$		0			0
$673$	−0.951057	−	0.690983i	−0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$673$	−0.951057	−	0.690983i	−0.951057	−	0.690983i	−0.951057	+	0.309017i	$0.900000\pi$
$674$		0			0
$675$	0.224514	+	0.690983i	0.224514	+	0.690983i
$676$	−5.54508	+	4.02874i	−5.54508	+	4.02874i
$677$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$677$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$678$	1.31433	+	4.04508i	1.31433	+	4.04508i
$679$		0			0
$680$		0			0
$681$	1.38197			1.38197
$682$		0			0
$683$	1.90211			1.90211			0.951057	−	0.309017i	$-0.100000\pi$
$683$	1.90211			1.90211			0.951057	+	0.309017i	$0.100000\pi$
$684$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$685$		0			0
$686$		0			0
$687$	−1.90211	+	1.38197i	−1.90211	+	1.38197i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$691$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.61803	+	1.17557i	1.61803	+	1.17557i
$696$		0			0
$697$		0			0
$698$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$699$		0			0
$700$		0			0
$701$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$701$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$702$	2.12663	+	1.54508i	2.12663	+	1.54508i
$703$	1.17557			1.17557
$704$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$709$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$716$		0			0
$717$	0.587785	+	0.427051i	0.587785	+	0.427051i
$718$	0.951057	−	2.92705i	0.951057	−	2.92705i
$719$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$719$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$720$	−1.00000	+	0.726543i	−1.00000	+	0.726543i
$721$		0			0
$722$	0.587785	+	1.80902i	0.587785	+	1.80902i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	1.80902	+	1.31433i	1.80902	+	1.31433i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.118034	−	0.363271i	0.118034	−	0.363271i
$730$		0			0
$731$		0			0
$732$	4.02874	−	2.92705i	4.02874	−	2.92705i
$733$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$733$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$734$		0			0
$735$	0.951057	+	0.690983i	0.951057	+	0.690983i
$736$		0			0
$737$	0.951057	+	0.690983i	0.951057	+	0.690983i
$738$		0			0
$739$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$739$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$740$	0.951057	−	2.92705i	0.951057	−	2.92705i
$741$	0.690983	+	2.12663i	0.690983	+	2.12663i
$742$		0			0
$743$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$743$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$744$		0			0
$745$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$746$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	1.80902	+	1.31433i	1.80902	+	1.31433i
$751$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$751$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$752$		0			0
$753$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$757$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$758$		0			0
$759$		0			0
$760$	3.07768			3.07768
$761$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$761$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.951057	+	0.690983i	0.951057	+	0.690983i
$769$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$769$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$770$		0			0
$771$	2.23607			2.23607
$772$	2.48990	+	1.80902i	2.48990	+	1.80902i
$773$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$773$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$774$		0			0
$775$		0			0
$776$	−4.73607	+	3.44095i	−4.73607	+	3.44095i
$777$		0			0
$778$	0.363271	−	1.11803i	0.363271	−	1.11803i
$779$		0			0
$780$	5.85410			5.85410
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000	−	3.07768i	1.00000	−	3.07768i
$785$		0			0
$786$	−2.92705	+	2.12663i	−2.92705	+	2.12663i
$787$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$787$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1.17557			−1.17557
$793$	−3.07768			−3.07768
$794$		0			0
$795$		0			0
$796$	−1.30902	−	4.02874i	−1.30902	−	4.02874i
$797$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$797$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$798$		0			0
$799$		0			0
$800$	0.951057	−	2.92705i	0.951057	−	2.92705i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−3.61803			−3.61803
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.53884	−	1.11803i	1.53884	−	1.11803i
$809$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$809$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$810$	−0.726543	−	2.23607i	−0.726543	−	2.23607i
$811$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$811$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$812$		0			0
$813$	0.726543			0.726543
$814$	1.80902	−	1.31433i	1.80902	−	1.31433i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$821$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$822$		0			0
$823$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$823$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$824$	−5.85410			−5.85410
$825$	0.363271	+	1.11803i	0.363271	+	1.11803i
$826$		0			0
$827$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$827$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$828$		0			0
$829$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$829$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$830$		0			0
$831$		0			0
$832$	−1.53884	−	4.73607i	−1.53884	−	4.73607i
$833$		0			0
$834$	−3.61803	−	2.62866i	−3.61803	−	2.62866i
$835$	−1.17557			−1.17557
$836$	2.11803	+	1.53884i	2.11803	+	1.53884i
$837$		0			0
$838$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$839$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$839$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$840$		0			0
$841$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$	0.118034	−	0.363271i	0.118034	−	0.363271i
$856$	−2.92705	−	2.12663i	−2.92705	−	2.12663i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	3.44095	+	2.50000i	3.44095	+	2.50000i
$859$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$859$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$863$	1.53884	−	1.11803i	1.53884	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$864$	−1.80902	+	1.31433i	−1.80902	+	1.31433i
$865$		0			0
$866$	1.11803	−	3.44095i	1.11803	−	3.44095i
$867$	0.951057	+	0.690983i	0.951057	+	0.690983i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1.80902	+	1.31433i	1.80902	+	1.31433i
$872$		0			0
$873$	0.224514	+	0.690983i	0.224514	+	0.690983i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.587785	+	1.80902i	−0.587785	+	1.80902i			1.00000i	$0.5\pi$
$877$	−0.587785	+	1.80902i	−0.587785	+	1.80902i	−0.587785	+	0.809017i	$0.700000\pi$
$878$		0			0
$879$	−2.23607			−2.23607
$880$	2.61803	−	1.90211i	2.61803	−	1.90211i
$881$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$881$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$882$	−0.587785	−	0.427051i	−0.587785	−	0.427051i
$883$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$883$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$887$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$888$	−1.31433	+	4.04508i	−1.31433	+	4.04508i
$889$		0			0
$890$		0			0
$891$	0.381966	−	1.17557i	0.381966	−	1.17557i
$892$		0			0
$893$		0			0
$894$	1.38197	−	4.25325i	1.38197	−	4.25325i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	4.73607	+	3.44095i	4.73607	+	3.44095i
$905$		0			0
$906$		0			0
$907$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$907$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$908$	2.48990	−	1.80902i	2.48990	−	1.80902i
$909$	−0.0729490	−	0.224514i	−0.0729490	−	0.224514i
$910$		0			0
$911$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$911$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$912$	−3.80423			−3.80423
$913$		0			0
$914$		0			0
$915$	1.53884	+	1.11803i	1.53884	+	1.11803i
$916$	−1.61803	+	4.97980i	−1.61803	+	4.97980i
$917$		0			0
$918$		0			0
$919$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$919$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$920$		0			0
$921$		0			0
$922$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$923$		0			0
$924$		0			0
$925$	1.17557			1.17557
$926$		0			0
$927$	−0.224514	+	0.690983i	−0.224514	+	0.690983i
$928$		0			0
$929$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$929$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$	0.309017	+	0.951057i	0.309017	+	0.951057i
$932$		0			0
$933$	−0.587785	−	0.427051i	−0.587785	−	0.427051i
$934$		0			0
$935$		0			0
$936$	−2.23607			−2.23607
$937$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$937$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$941$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$	0.587785	+	1.80902i	0.587785	+	1.80902i
$951$		0			0
$952$		0			0
$953$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$953$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$954$		0			0
$955$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$956$	1.61803			1.61803
$957$		0			0
$958$	−1.17557			−1.17557
$959$		0			0
$960$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$961$	0.309017	+	0.951057i	0.309017	+	0.951057i
$962$	3.44095	−	2.50000i	3.44095	−	2.50000i
$963$	−0.363271	+	0.263932i	−0.363271	+	0.263932i
$964$		0			0
$965$	−0.363271	+	1.11803i	−0.363271	+	1.11803i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	3.07768			3.07768
$969$		0			0
$970$	−2.92705	−	2.12663i	−2.92705	−	2.12663i
$971$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$971$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$972$	0.587785	+	1.80902i	0.587785	+	1.80902i
$973$		0			0
$974$		0			0
$975$	0.690983	+	2.12663i	0.690983	+	2.12663i
$976$	1.61803	−	4.97980i	1.61803	−	4.97980i
$977$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$977$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$978$		0			0
$979$		0			0
$980$	2.61803			2.61803
$981$		0			0
$982$	−0.951057	+	2.92705i	−0.951057	+	2.92705i
$983$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$983$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	4.02874	+	2.92705i	4.02874	+	2.92705i
$989$		0			0
$990$	−0.224514	−	0.690983i	−0.224514	−	0.690983i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.30902	−	0.951057i	1.30902	−	0.951057i
$996$		0			0
$997$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$997$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$998$	−1.17557	+	3.61803i	−1.17557	+	3.61803i
$999$	−0.690983	−	0.502029i	−0.690983	−	0.502029i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.1.w.e.664.1	✓	8
5.4	even	2	inner	1045.1.w.e.664.2	yes	8
11.3	even	5	inner	1045.1.w.e.949.1	yes	8
19.18	odd	2	inner	1045.1.w.e.664.2	yes	8
55.14	even	10	inner	1045.1.w.e.949.2	yes	8
95.94	odd	2	CM	1045.1.w.e.664.1	✓	8
209.113	odd	10	inner	1045.1.w.e.949.2	yes	8
1045.949	odd	10	inner	1045.1.w.e.949.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.1.w.e.664.1	✓	8	1.1	even	1	trivial
1045.1.w.e.664.1	✓	8	95.94	odd	2	CM
1045.1.w.e.664.2	yes	8	5.4	even	2	inner
1045.1.w.e.664.2	yes	8	19.18	odd	2	inner
1045.1.w.e.949.1	yes	8	11.3	even	5	inner
1045.1.w.e.949.1	yes	8	1045.949	odd	10	inner
1045.1.w.e.949.2	yes	8	55.14	even	10	inner
1045.1.w.e.949.2	yes	8	209.113	odd	10	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1045.w (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.53884 - 1.11803i) q^{2} +(-0.363271 + 1.11803i) q^{3} +(0.809017 + 2.48990i) q^{4} +(-0.809017 + 0.587785i) q^{5} +(1.80902 - 1.31433i) q^{6} +(0.951057 - 2.92705i) q^{8} +(-0.309017 - 0.224514i) q^{9} +O(q^{10})\)	\(q+(-1.53884 - 1.11803i) q^{2} +(-0.363271 + 1.11803i) q^{3} +(0.809017 + 2.48990i) q^{4} +(-0.809017 + 0.587785i) q^{5} +(1.80902 - 1.31433i) q^{6} +(0.951057 - 2.92705i) q^{8} +(-0.309017 - 0.224514i) q^{9} +1.90211 q^{10} +(0.809017 + 0.587785i) q^{11} -3.07768 q^{12} +(1.53884 + 1.11803i) q^{13} +(-0.363271 - 1.11803i) q^{15} +(-2.61803 + 1.90211i) q^{16} +(0.224514 + 0.690983i) q^{18} +(0.309017 - 0.951057i) q^{19} +(-2.11803 - 1.53884i) q^{20} +(-0.587785 - 1.80902i) q^{22} +(2.92705 + 2.12663i) q^{24} +(0.309017 - 0.951057i) q^{25} +(-1.11803 - 3.44095i) q^{26} +(-0.587785 + 0.427051i) q^{27} +(-0.690983 + 2.12663i) q^{30} +3.07768 q^{32} +(-0.951057 + 0.690983i) q^{33} +(0.309017 - 0.951057i) q^{36} +(0.363271 + 1.11803i) q^{37} +(-1.53884 + 1.11803i) q^{38} +(-1.80902 + 1.31433i) q^{39} +(0.951057 + 2.92705i) q^{40} +(-0.809017 + 2.48990i) q^{44} +0.381966 q^{45} +(-1.17557 - 3.61803i) q^{48} +(-0.809017 + 0.587785i) q^{49} +(-1.53884 + 1.11803i) q^{50} +(-1.53884 + 4.73607i) q^{52} +1.38197 q^{54} -1.00000 q^{55} +(0.951057 + 0.690983i) q^{57} +(2.48990 - 1.80902i) q^{60} +(-1.30902 + 0.951057i) q^{61} +(-2.11803 - 1.53884i) q^{64} -1.90211 q^{65} +2.23607 q^{66} +1.17557 q^{67} +(-0.951057 + 0.690983i) q^{72} +(0.690983 - 2.12663i) q^{74} +(0.951057 + 0.690983i) q^{75} +2.61803 q^{76} +4.25325 q^{78} +(1.00000 - 3.07768i) q^{80} +(-0.381966 - 1.17557i) q^{81} +(2.48990 - 1.80902i) q^{88} +(-0.587785 - 0.427051i) q^{90} +(0.309017 + 0.951057i) q^{95} +(-1.11803 + 3.44095i) q^{96} +(-1.53884 - 1.11803i) q^{97} +1.90211 q^{98} +(-0.118034 - 0.363271i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9}+O(q^{10}) \) 8 * q + 2 * q^4 - 2 * q^5 + 10 * q^6 + 2 * q^9	\( 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9} + 2 q^{11} - 12 q^{16} - 2 q^{19} - 8 q^{20} + 10 q^{24} - 2 q^{25} - 10 q^{30} - 2 q^{36} - 10 q^{39} - 2 q^{44} + 12 q^{45} - 2 q^{49} + 20 q^{54} - 8 q^{55} - 6 q^{61} - 8 q^{64} + 10 q^{74} + 12 q^{76} + 8 q^{80} - 12 q^{81} - 2 q^{95} + 8 q^{99}+O(q^{100}) \) 8 * q + 2 * q^4 - 2 * q^5 + 10 * q^6 + 2 * q^9 + 2 * q^11 - 12 * q^16 - 2 * q^19 - 8 * q^20 + 10 * q^24 - 2 * q^25 - 10 * q^30 - 2 * q^36 - 10 * q^39 - 2 * q^44 + 12 * q^45 - 2 * q^49 + 20 * q^54 - 8 * q^55 - 6 * q^61 - 8 * q^64 + 10 * q^74 + 12 * q^76 + 8 * q^80 - 12 * q^81 - 2 * q^95 + 8 * q^99

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