LMFDB - Embedded newform 2023.1.f.b.1483.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2023,1,Mod(251,2023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2023, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("2023.251");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2023 = 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2023.f (of order $4$, degree $2$, 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:	$1.00960852056$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 119)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.14161.1
Artin image:	$C_8\times D_5$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{40} - \cdots)$

Embedding invariants

Embedding label			1483.1
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	2023.1483
Dual form			2023.1.f.b.251.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034i q^{2} +(-0.437016 - 0.437016i) q^{3} +0.618034 q^{4} +(-1.14412 - 1.14412i) q^{5} +(-0.270091 + 0.270091i) q^{6} +(0.707107 - 0.707107i) q^{7} -1.00000i q^{8} -0.618034i q^{9} +O(q^{10})$	\(q-0.618034i q^{2} +(-0.437016 - 0.437016i) q^{3} +0.618034 q^{4} +(-1.14412 - 1.14412i) q^{5} +(-0.270091 + 0.270091i) q^{6} +(0.707107 - 0.707107i) q^{7} -1.00000i q^{8} -0.618034i q^{9} +(-0.707107 + 0.707107i) q^{10} +(-0.270091 - 0.270091i) q^{12} +(-0.437016 - 0.437016i) q^{14} +1.00000i q^{15} -0.381966 q^{18} +(-0.707107 - 0.707107i) q^{20} -0.618034 q^{21} +(-0.437016 + 0.437016i) q^{24} +1.61803i q^{25} +(-0.707107 + 0.707107i) q^{27} +(0.437016 - 0.437016i) q^{28} +0.618034 q^{30} +(1.14412 + 1.14412i) q^{31} -1.00000i q^{32} -1.61803 q^{35} -0.381966i q^{36} +(-1.14412 + 1.14412i) q^{40} +(0.437016 - 0.437016i) q^{41} +0.381966i q^{42} +0.618034i q^{43} +(-0.707107 + 0.707107i) q^{45} -1.00000i q^{49} +1.00000 q^{50} +1.61803i q^{53} +(0.437016 + 0.437016i) q^{54} +(-0.707107 - 0.707107i) q^{56} +0.618034i q^{60} +(0.437016 - 0.437016i) q^{61} +(0.707107 - 0.707107i) q^{62} +(-0.437016 - 0.437016i) q^{63} -0.618034 q^{64} -1.61803 q^{67} +1.00000i q^{70} -0.618034 q^{72} +(0.437016 + 0.437016i) q^{73} +(0.707107 - 0.707107i) q^{75} +(-0.270091 - 0.270091i) q^{82} -0.381966 q^{84} +0.381966 q^{86} +(0.437016 + 0.437016i) q^{90} -1.00000i q^{93} +(-0.437016 + 0.437016i) q^{96} +(-1.14412 - 1.14412i) q^{97} -0.618034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4}+O(q^{10}) $ 8 * q - 4 * q^4	$ 8 q - 4 q^{4} - 12 q^{18} + 4 q^{21} - 4 q^{30} - 4 q^{35} + 8 q^{50} + 4 q^{64} - 4 q^{67} + 4 q^{72} - 12 q^{84} + 12 q^{86} + 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^4 - 12 * q^18 + 4 * q^21 - 4 * q^30 - 4 * q^35 + 8 * q^50 + 4 * q^64 - 4 * q^67 + 4 * q^72 - 12 * q^84 + 12 * q^86 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times$.

$n$	$290$	$1737$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$2$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$3$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$3$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$4$	0.618034			0.618034
$5$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$5$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$6$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$7$	0.707107	−	0.707107i	0.707107	−	0.707107i
$7$	0.707107	−	0.707107i	0.707107	−	0.707107i
$8$		−	1.00000i		−	1.00000i
$9$		−	0.618034i		−	0.618034i
$10$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$15$			1.00000i			1.00000i
$16$		0			0
$17$		0			0
$17$		0			0
$18$	−0.381966			−0.381966
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$21$	−0.618034			−0.618034
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$25$			1.61803i			1.61803i
$26$		0			0
$27$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$28$	0.437016	−	0.437016i	0.437016	−	0.437016i
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$	0.618034			0.618034
$31$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$31$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$32$		−	1.00000i		−	1.00000i
$33$		0			0
$34$		0			0
$35$	−1.61803			−1.61803
$36$		−	0.381966i		−	0.381966i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$41$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$41$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$42$			0.381966i			0.381966i
$43$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$43$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$44$		0			0
$45$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$		−	1.00000i		−	1.00000i
$50$	1.00000			1.00000
$51$		0			0
$52$		0			0
$53$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$53$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$54$	0.437016	+	0.437016i	0.437016	+	0.437016i
$55$		0			0
$56$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$			0.618034i			0.618034i
$61$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$61$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$62$	0.707107	−	0.707107i	0.707107	−	0.707107i
$63$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$64$	−0.618034			−0.618034
$65$		0			0
$66$		0			0
$67$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$67$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$68$		0			0
$69$		0			0
$70$			1.00000i			1.00000i
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$	−0.618034			−0.618034
$73$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$73$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$74$		0			0
$75$	0.707107	−	0.707107i	0.707107	−	0.707107i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$79$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$80$		0			0
$81$		0			0
$82$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−0.381966			−0.381966
$85$		0			0
$86$	0.381966			0.381966
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.437016	+	0.437016i	0.437016	+	0.437016i
$91$		0			0
$92$		0			0
$93$		−	1.00000i		−	1.00000i
$94$		0			0
$95$		0			0
$96$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$97$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$97$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$98$	−0.618034			−0.618034
$99$		0			0
$100$			1.00000i			1.00000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	0.707107	+	0.707107i	0.707107	+	0.707107i
$106$	1.00000			1.00000
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	1.00000			1.00000
$121$		−	1.00000i		−	1.00000i
$122$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$123$	−0.381966			−0.381966
$124$	0.707107	+	0.707107i	0.707107	+	0.707107i
$125$	0.707107	−	0.707107i	0.707107	−	0.707107i
$126$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$127$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$127$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$128$		−	0.618034i		−	0.618034i
$129$	0.270091	−	0.270091i	0.270091	−	0.270091i
$130$		0			0
$131$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$131$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$			1.00000i			1.00000i
$135$	1.61803			1.61803
$136$		0			0
$137$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$137$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$138$		0			0
$139$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$139$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$140$	−1.00000			−1.00000
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	0.270091	−	0.270091i	0.270091	−	0.270091i
$147$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$148$		0			0
$149$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$149$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$150$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$151$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$151$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.61803i		−	2.61803i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	0.707107	−	0.707107i	0.707107	−	0.707107i
$160$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$161$		0			0
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	0.270091	−	0.270091i	0.270091	−	0.270091i
$165$		0			0
$166$		0			0
$167$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$167$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$168$			0.618034i			0.618034i
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$			0.381966i			0.381966i
$173$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$173$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$174$		0			0
$175$	1.14412	+	1.14412i	1.14412	+	1.14412i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$179$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$180$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$181$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$181$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$	−0.381966			−0.381966
$184$		0			0
$185$		0			0
$186$	−0.618034			−0.618034
$187$		0			0
$188$		0			0
$189$			1.00000i			1.00000i
$190$		0			0
$191$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$191$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$192$	0.270091	+	0.270091i	0.270091	+	0.270091i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$195$		0			0
$196$		−	0.618034i		−	0.618034i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$199$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$200$	1.61803			1.61803
$201$	0.707107	+	0.707107i	0.707107	+	0.707107i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.00000			−1.00000
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$	0.437016	−	0.437016i	0.437016	−	0.437016i
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$			1.00000i			1.00000i
$213$		0			0
$214$		0			0
$215$	0.707107	−	0.707107i	0.707107	−	0.707107i
$216$	0.707107	+	0.707107i	0.707107	+	0.707107i
$217$	1.61803			1.61803
$218$		0			0
$219$		−	0.381966i		−	0.381966i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$225$	1.00000			1.00000
$226$		0			0
$227$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$227$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$239$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$241$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$242$	−0.618034			−0.618034
$243$	0.707107	+	0.707107i	0.707107	+	0.707107i
$244$	0.270091	−	0.270091i	0.270091	−	0.270091i
$245$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$246$			0.236068i			0.236068i
$247$		0			0
$248$	1.14412	−	1.14412i	1.14412	−	1.14412i
$249$		0			0
$250$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$253$		0			0
$254$	−1.00000			−1.00000
$255$		0			0
$256$	−1.00000			−1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	−0.166925	−	0.166925i	−0.166925	−	0.166925i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.874032	−	0.874032i	0.874032	−	0.874032i
$263$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$263$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$264$		0			0
$265$	1.85123	−	1.85123i	1.85123	−	1.85123i
$266$		0			0
$267$		0			0
$268$	−1.00000			−1.00000
$269$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$269$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$270$		−	1.00000i		−	1.00000i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		−	0.381966i		−	0.381966i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	0.707107	−	0.707107i	0.707107	−	0.707107i
$279$	0.707107	−	0.707107i	0.707107	−	0.707107i
$280$			1.61803i			1.61803i
$281$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$281$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$282$		0			0
$283$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$283$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	0.618034i		−	0.618034i
$288$	−0.618034			−0.618034
$289$		0			0
$290$		0			0
$291$			1.00000i			1.00000i
$292$	0.270091	+	0.270091i	0.270091	+	0.270091i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	0.270091	+	0.270091i	0.270091	+	0.270091i
$295$		0			0
$296$		0			0
$297$		0			0
$298$			0.381966i			0.381966i
$299$		0			0
$300$	0.437016	−	0.437016i	0.437016	−	0.437016i
$301$	0.437016	+	0.437016i	0.437016	+	0.437016i
$302$	−0.381966			−0.381966
$303$		0			0
$304$		0			0
$305$	−1.00000			−1.00000
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−1.61803			−1.61803
$311$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$311$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$312$		0			0
$313$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$313$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$314$		0			0
$315$			1.00000i			1.00000i
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$319$		0			0
$320$	0.707107	+	0.707107i	0.707107	+	0.707107i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$329$		0			0
$330$		0			0
$331$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$331$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$332$		0			0
$333$		0			0
$334$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$335$	1.85123	+	1.85123i	1.85123	+	1.85123i
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		−	0.618034i		−	0.618034i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$344$	0.618034			0.618034
$345$		0			0
$346$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	0.707107	−	0.707107i	0.707107	−	0.707107i
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.381966			0.381966
$359$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$359$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$360$	0.707107	+	0.707107i	0.707107	+	0.707107i
$361$	−1.00000			−1.00000
$362$	0.874032	+	0.874032i	0.874032	+	0.874032i
$363$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$364$		0			0
$365$		−	1.00000i		−	1.00000i
$366$			0.236068i			0.236068i
$367$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$367$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$368$		0			0
$369$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$370$		0			0
$371$	1.14412	+	1.14412i	1.14412	+	1.14412i
$372$		−	0.618034i		−	0.618034i
$373$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$373$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$374$		0			0
$375$	−0.618034			−0.618034
$376$		0			0
$377$		0			0
$378$	0.618034			0.618034
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$382$		−	1.00000i		−	1.00000i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$385$		0			0
$386$		0			0
$387$	0.381966			0.381966
$388$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$389$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$389$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$390$		0			0
$391$		0			0
$392$	−1.00000			−1.00000
$393$		−	1.23607i		−	1.23607i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$397$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$398$	0.270091	−	0.270091i	0.270091	−	0.270091i
$399$		0			0
$400$		0			0
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$	0.437016	−	0.437016i	0.437016	−	0.437016i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$			0.618034i			0.618034i
$411$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	1.00000i		−	1.00000i
$418$		0			0
$419$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$419$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$420$	0.437016	+	0.437016i	0.437016	+	0.437016i
$421$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$421$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$422$		0			0
$423$		0			0
$424$	1.61803			1.61803
$425$		0			0
$426$		0			0
$427$		−	0.618034i		−	0.618034i
$428$		0			0
$429$		0			0
$430$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		−	1.00000i		−	1.00000i
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.236068			−0.236068
$439$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$439$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$440$		0			0
$441$	−0.618034			−0.618034
$442$		0			0
$443$	2.00000			2.00000			1.00000			$0$
$443$	2.00000			2.00000			1.00000			$0$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	0.270091	+	0.270091i	0.270091	+	0.270091i
$448$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$449$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$449$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$450$		−	0.618034i		−	0.618034i
$451$		0			0
$452$		0			0
$453$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$454$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$455$		0			0
$456$		0			0
$457$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$457$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$463$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$464$		0			0
$465$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.00000			1.00000
$478$		−	0.381966i		−	0.381966i
$479$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$479$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$480$	1.00000			1.00000
$481$		0			0
$482$	0.707107	−	0.707107i	0.707107	−	0.707107i
$483$		0			0
$484$		−	0.618034i		−	0.618034i
$485$			2.61803i			2.61803i
$486$	0.437016	−	0.437016i	0.437016	−	0.437016i
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$489$		0			0
$490$	0.707107	+	0.707107i	0.707107	+	0.707107i
$491$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$491$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$492$	−0.236068			−0.236068
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	0.437016	−	0.437016i	0.437016	−	0.437016i
$501$			0.381966i			0.381966i
$502$		0			0
$503$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$503$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$504$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$505$		0			0
$506$		0			0
$507$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$508$		−	1.00000i		−	1.00000i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	0.618034			0.618034
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$	0.166925	−	0.166925i	0.166925	−	0.166925i
$517$		0			0
$518$		0			0
$519$			0.381966i			0.381966i
$520$		0			0
$521$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$521$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	0.874032	+	0.874032i	0.874032	+	0.874032i
$525$		−	1.00000i		−	1.00000i
$526$	1.23607			1.23607
$527$		0			0
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$			1.61803i			1.61803i
$537$	0.270091	−	0.270091i	0.270091	−	0.270091i
$538$	−0.874032	+	0.874032i	−0.874032	+	0.874032i
$539$		0			0
$540$	1.00000			1.00000
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$	1.23607			1.23607
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	0.381966			0.381966
$549$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.707107	+	0.707107i	0.707107	+	0.707107i
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$558$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.00000			−1.00000
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$567$		0			0
$568$		0			0
$569$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$569$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$570$		0			0
$571$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$571$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$574$	−0.381966			−0.381966
$575$		0			0
$576$			0.381966i			0.381966i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$	0.618034			0.618034
$583$		0			0
$584$	0.437016	−	0.437016i	0.437016	−	0.437016i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$589$		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
$596$	−0.381966			−0.381966
$597$		−	0.381966i		−	0.381966i
$598$		0			0
$599$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$599$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$600$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$601$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$601$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$602$	0.270091	−	0.270091i	0.270091	−	0.270091i
$603$			1.00000i			1.00000i
$604$		−	0.381966i		−	0.381966i
$605$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$606$		0			0
$607$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$607$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$608$		0			0
$609$		0			0
$610$			0.618034i			0.618034i
$611$		0			0
$612$		0			0
$613$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$614$		0			0
$615$	0.437016	+	0.437016i	0.437016	+	0.437016i
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$619$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$620$		−	1.61803i		−	1.61803i
$621$		0			0
$622$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$623$		0			0
$624$		0			0
$625$		0			0
$626$	0.707107	+	0.707107i	0.707107	+	0.707107i
$627$		0			0
$628$		0			0
$629$		0			0
$630$	0.618034			0.618034
$631$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$631$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.85123	+	1.85123i	−1.85123	+	1.85123i
$636$	0.437016	−	0.437016i	0.437016	−	0.437016i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$641$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$641$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$642$		0			0
$643$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$643$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$644$		0			0
$645$	−0.618034			−0.618034
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		−	3.23607i		−	3.23607i
$656$		0			0
$657$	0.270091	−	0.270091i	0.270091	−	0.270091i
$658$		0			0
$659$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$659$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−1.00000			−1.00000
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$669$		0			0
$670$	1.14412	−	1.14412i	1.14412	−	1.14412i
$671$		0			0
$672$			0.618034i			0.618034i
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$676$	0.618034			0.618034
$677$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$677$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$678$		0			0
$679$	−1.61803			−1.61803
$680$		0			0
$681$	−1.00000			−1.00000
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$686$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$691$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$692$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$693$		0			0
$694$		0			0
$695$		−	2.61803i		−	2.61803i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.707107	+	0.707107i	0.707107	+	0.707107i
$701$	−2.00000			−2.00000			−1.00000			$\pi$
$701$	−2.00000			−2.00000			−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$			0.381966i			0.381966i
$717$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$718$	1.00000			1.00000
$719$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$719$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$720$		0			0
$721$		0			0
$722$			0.618034i			0.618034i
$723$		−	1.00000i		−	1.00000i
$724$	−0.874032	+	0.874032i	−0.874032	+	0.874032i
$725$		0			0
$726$	0.270091	+	0.270091i	0.270091	+	0.270091i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	0.618034i		−	0.618034i
$730$	−0.618034			−0.618034
$731$		0			0
$732$	−0.236068			−0.236068
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	−0.270091	−	0.270091i	−0.270091	−	0.270091i
$735$	1.00000			1.00000
$736$		0			0
$737$		0			0
$738$	−0.166925	+	0.166925i	−0.166925	+	0.166925i
$739$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$739$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$740$		0			0
$741$		0			0
$742$	0.707107	−	0.707107i	0.707107	−	0.707107i
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$	−1.00000			−1.00000
$745$	0.707107	+	0.707107i	0.707107	+	0.707107i
$746$		−	0.381966i		−	0.381966i
$747$		0			0
$748$		0			0
$749$		0			0
$750$			0.381966i			0.381966i
$751$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$751$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$756$			0.618034i			0.618034i
$757$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$757$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	0.437016	+	0.437016i	0.437016	+	0.437016i
$763$		0			0
$764$	1.00000			1.00000
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.437016	+	0.437016i	0.437016	+	0.437016i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		−	0.236068i		−	0.236068i
$775$	−1.85123	+	1.85123i	−1.85123	+	1.85123i
$776$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$777$		0			0
$778$	1.00000			1.00000
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	−0.763932			−0.763932
$787$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$787$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$	0.874032	−	0.874032i	0.874032	−	0.874032i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$795$	−1.61803			−1.61803
$796$	0.270091	+	0.270091i	0.270091	+	0.270091i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	1.61803			1.61803
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.437016	+	0.437016i	0.437016	+	0.437016i
$805$		0			0
$806$		0			0
$807$			1.23607i			1.23607i
$808$		0			0
$809$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$809$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$811$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−0.618034			−0.618034
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$	−0.166925	+	0.166925i	−0.166925	+	0.166925i
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$	−0.618034			−0.618034
$835$			1.00000i			1.00000i
$836$		0			0
$837$	−1.61803			−1.61803
$838$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$839$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$839$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$840$	0.707107	−	0.707107i	0.707107	−	0.707107i
$841$			1.00000i			1.00000i
$842$			0.381966i			0.381966i
$843$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$844$		0			0
$845$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$846$		0			0
$847$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$848$		0			0
$849$	−1.00000			−1.00000
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$853$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$854$	−0.381966			−0.381966
$855$		0			0
$856$		0			0
$857$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$857$	−1.14412	+	1.14412i	−1.14412	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$	0.437016	−	0.437016i	0.437016	−	0.437016i
$861$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$862$		0			0
$863$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$863$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$864$	0.707107	+	0.707107i	0.707107	+	0.707107i
$865$			1.00000i			1.00000i
$866$		0			0
$867$		0			0
$868$	1.00000			1.00000
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$874$		0			0
$875$		−	1.00000i		−	1.00000i
$876$		−	0.236068i		−	0.236068i
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$879$		0			0
$880$		0			0
$881$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$881$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$882$			0.381966i			0.381966i
$883$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$883$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$884$		0			0
$885$		0			0
$886$		−	1.23607i		−	1.23607i
$887$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$887$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$888$		0			0
$889$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	0.166925	−	0.166925i	0.166925	−	0.166925i
$895$	0.707107	−	0.707107i	0.707107	−	0.707107i
$896$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.618034			0.618034
$901$		0			0
$902$		0			0
$903$		−	0.381966i		−	0.381966i
$904$		0			0
$905$	3.23607			3.23607
$906$	0.166925	+	0.166925i	0.166925	+	0.166925i
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	0.707107	−	0.707107i	0.707107	−	0.707107i
$909$		0			0
$910$		0			0
$911$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$911$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$	1.00000			1.00000
$915$	0.437016	+	0.437016i	0.437016	+	0.437016i
$916$		0			0
$917$	2.00000			2.00000
$918$		0			0
$919$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$919$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$			0.381966i			0.381966i
$927$		0			0
$928$		0			0
$929$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$929$	1.14412	−	1.14412i	1.14412	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$930$	0.707107	+	0.707107i	0.707107	+	0.707107i
$931$		0			0
$932$		0			0
$933$			1.00000i			1.00000i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	0.707107	+	0.707107i	0.707107	+	0.707107i
$939$	1.00000			1.00000
$940$		0			0
$941$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$941$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	1.14412	−	1.14412i	1.14412	−	1.14412i
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$953$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$954$		−	0.618034i		−	0.618034i
$955$	−1.85123	−	1.85123i	−1.85123	−	1.85123i
$956$	0.381966			0.381966
$957$		0			0
$958$	−0.270091	+	0.270091i	−0.270091	+	0.270091i
$959$	0.437016	−	0.437016i	0.437016	−	0.437016i
$960$		−	0.618034i		−	0.618034i
$961$			1.61803i			1.61803i
$962$		0			0
$963$		0			0
$964$	0.707107	+	0.707107i	0.707107	+	0.707107i
$965$		0			0
$966$		0			0
$967$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$967$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$968$	−1.00000			−1.00000
$969$		0			0
$970$	1.61803			1.61803
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	0.437016	+	0.437016i	0.437016	+	0.437016i
$973$	1.61803			1.61803
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$977$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$978$		0			0
$979$		0			0
$980$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$981$		0			0
$982$	1.00000			1.00000
$983$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$983$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$984$			0.381966i			0.381966i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$991$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$992$	1.14412	−	1.14412i	1.14412	−	1.14412i
$993$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$994$		0			0
$995$		−	1.00000i		−	1.00000i
$996$		0			0
$997$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$997$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2023.1.f.b.1483.1		8
7.6	odd	2	inner	2023.1.f.b.1483.2		8
17.2	even	8		119.1.d.b.118.2	yes	2
17.3	odd	16		2023.1.l.b.1889.1		16
17.4	even	4	inner	2023.1.f.b.251.3		8
17.5	odd	16		2023.1.l.b.1868.4		16
17.6	odd	16		2023.1.l.b.1266.3		16
17.7	odd	16		2023.1.l.b.468.1		16
17.8	even	8		2023.1.c.e.1735.2		4
17.9	even	8		2023.1.c.e.1735.1		4
17.10	odd	16		2023.1.l.b.468.2		16
17.11	odd	16		2023.1.l.b.1266.4		16
17.12	odd	16		2023.1.l.b.1868.3		16
17.13	even	4	inner	2023.1.f.b.251.4		8
17.14	odd	16		2023.1.l.b.1889.2		16
17.15	even	8		119.1.d.a.118.2	✓	2
17.16	even	2	inner	2023.1.f.b.1483.2		8
51.2	odd	8		1071.1.h.a.118.1		2
51.32	odd	8		1071.1.h.b.118.1		2
68.15	odd	8		1904.1.n.b.1665.1		2
68.19	odd	8		1904.1.n.a.1665.2		2
85.2	odd	8		2975.1.b.a.2974.3		4
85.19	even	8		2975.1.h.c.951.1		2
85.32	odd	8		2975.1.b.b.2974.3		4
85.49	even	8		2975.1.h.d.951.1		2
85.53	odd	8		2975.1.b.a.2974.2		4
85.83	odd	8		2975.1.b.b.2974.2		4
119.2	even	24		833.1.h.a.815.1		4
119.6	even	16		2023.1.l.b.1266.4		16
119.13	odd	4	inner	2023.1.f.b.251.3		8
119.19	odd	24		833.1.h.b.815.1		4
119.20	even	16		2023.1.l.b.1889.2		16
119.27	even	16		2023.1.l.b.468.1		16
119.32	even	24		833.1.h.b.509.1		4
119.41	even	16		2023.1.l.b.468.2		16
119.48	even	16		2023.1.l.b.1889.1		16
119.53	even	24		833.1.h.a.509.1		4
119.55	odd	4	inner	2023.1.f.b.251.4		8
119.62	even	16		2023.1.l.b.1266.3		16
119.66	odd	24		833.1.h.a.509.1		4
119.76	odd	8		2023.1.c.e.1735.1		4
119.83	odd	8		119.1.d.b.118.2	yes	2
119.87	odd	24		833.1.h.b.509.1		4
119.90	even	16		2023.1.l.b.1868.3		16
119.97	even	16		2023.1.l.b.1868.4		16
119.100	even	24		833.1.h.b.815.1		4
119.104	odd	8		119.1.d.a.118.2	✓	2
119.111	odd	8		2023.1.c.e.1735.2		4
119.117	odd	24		833.1.h.a.815.1		4
119.118	odd	2	CM	2023.1.f.b.1483.1		8
357.83	even	8		1071.1.h.a.118.1		2
357.104	even	8		1071.1.h.b.118.1		2
476.83	even	8		1904.1.n.a.1665.2		2
476.223	even	8		1904.1.n.b.1665.1		2
595.83	even	8		2975.1.b.a.2974.2		4
595.104	odd	8		2975.1.h.d.951.1		2
595.202	even	8		2975.1.b.a.2974.3		4
595.223	even	8		2975.1.b.b.2974.2		4
595.342	even	8		2975.1.b.b.2974.3		4
595.559	odd	8		2975.1.h.c.951.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.1.d.a.118.2	✓	2	17.15	even	8
119.1.d.a.118.2	✓	2	119.104	odd	8
119.1.d.b.118.2	yes	2	17.2	even	8
119.1.d.b.118.2	yes	2	119.83	odd	8
833.1.h.a.509.1		4	119.53	even	24
833.1.h.a.509.1		4	119.66	odd	24
833.1.h.a.815.1		4	119.2	even	24
833.1.h.a.815.1		4	119.117	odd	24
833.1.h.b.509.1		4	119.32	even	24
833.1.h.b.509.1		4	119.87	odd	24
833.1.h.b.815.1		4	119.19	odd	24
833.1.h.b.815.1		4	119.100	even	24
1071.1.h.a.118.1		2	51.2	odd	8
1071.1.h.a.118.1		2	357.83	even	8
1071.1.h.b.118.1		2	51.32	odd	8
1071.1.h.b.118.1		2	357.104	even	8
1904.1.n.a.1665.2		2	68.19	odd	8
1904.1.n.a.1665.2		2	476.83	even	8
1904.1.n.b.1665.1		2	68.15	odd	8
1904.1.n.b.1665.1		2	476.223	even	8
2023.1.c.e.1735.1		4	17.9	even	8
2023.1.c.e.1735.1		4	119.76	odd	8
2023.1.c.e.1735.2		4	17.8	even	8
2023.1.c.e.1735.2		4	119.111	odd	8
2023.1.f.b.251.3		8	17.4	even	4	inner
2023.1.f.b.251.3		8	119.13	odd	4	inner
2023.1.f.b.251.4		8	17.13	even	4	inner
2023.1.f.b.251.4		8	119.55	odd	4	inner
2023.1.f.b.1483.1		8	1.1	even	1	trivial
2023.1.f.b.1483.1		8	119.118	odd	2	CM
2023.1.f.b.1483.2		8	7.6	odd	2	inner
2023.1.f.b.1483.2		8	17.16	even	2	inner
2023.1.l.b.468.1		16	17.7	odd	16
2023.1.l.b.468.1		16	119.27	even	16
2023.1.l.b.468.2		16	17.10	odd	16
2023.1.l.b.468.2		16	119.41	even	16
2023.1.l.b.1266.3		16	17.6	odd	16
2023.1.l.b.1266.3		16	119.62	even	16
2023.1.l.b.1266.4		16	17.11	odd	16
2023.1.l.b.1266.4		16	119.6	even	16
2023.1.l.b.1868.3		16	17.12	odd	16
2023.1.l.b.1868.3		16	119.90	even	16
2023.1.l.b.1868.4		16	17.5	odd	16
2023.1.l.b.1868.4		16	119.97	even	16
2023.1.l.b.1889.1		16	17.3	odd	16
2023.1.l.b.1889.1		16	119.48	even	16
2023.1.l.b.1889.2		16	17.14	odd	16
2023.1.l.b.1889.2		16	119.20	even	16
2975.1.b.a.2974.2		4	85.53	odd	8
2975.1.b.a.2974.2		4	595.83	even	8
2975.1.b.a.2974.3		4	85.2	odd	8
2975.1.b.a.2974.3		4	595.202	even	8
2975.1.b.b.2974.2		4	85.83	odd	8
2975.1.b.b.2974.2		4	595.223	even	8
2975.1.b.b.2974.3		4	85.32	odd	8
2975.1.b.b.2974.3		4	595.342	even	8
2975.1.h.c.951.1		2	85.19	even	8
2975.1.h.c.951.1		2	595.559	odd	8
2975.1.h.d.951.1		2	85.49	even	8
2975.1.h.d.951.1		2	595.104	odd	8

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:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2023.f (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-0.618034i q^{2} +(-0.437016 - 0.437016i) q^{3} +0.618034 q^{4} +(-1.14412 - 1.14412i) q^{5} +(-0.270091 + 0.270091i) q^{6} +(0.707107 - 0.707107i) q^{7} -1.00000i q^{8} -0.618034i q^{9} +O(q^{10})\)	\(q-0.618034i q^{2} +(-0.437016 - 0.437016i) q^{3} +0.618034 q^{4} +(-1.14412 - 1.14412i) q^{5} +(-0.270091 + 0.270091i) q^{6} +(0.707107 - 0.707107i) q^{7} -1.00000i q^{8} -0.618034i q^{9} +(-0.707107 + 0.707107i) q^{10} +(-0.270091 - 0.270091i) q^{12} +(-0.437016 - 0.437016i) q^{14} +1.00000i q^{15} -0.381966 q^{18} +(-0.707107 - 0.707107i) q^{20} -0.618034 q^{21} +(-0.437016 + 0.437016i) q^{24} +1.61803i q^{25} +(-0.707107 + 0.707107i) q^{27} +(0.437016 - 0.437016i) q^{28} +0.618034 q^{30} +(1.14412 + 1.14412i) q^{31} -1.00000i q^{32} -1.61803 q^{35} -0.381966i q^{36} +(-1.14412 + 1.14412i) q^{40} +(0.437016 - 0.437016i) q^{41} +0.381966i q^{42} +0.618034i q^{43} +(-0.707107 + 0.707107i) q^{45} -1.00000i q^{49} +1.00000 q^{50} +1.61803i q^{53} +(0.437016 + 0.437016i) q^{54} +(-0.707107 - 0.707107i) q^{56} +0.618034i q^{60} +(0.437016 - 0.437016i) q^{61} +(0.707107 - 0.707107i) q^{62} +(-0.437016 - 0.437016i) q^{63} -0.618034 q^{64} -1.61803 q^{67} +1.00000i q^{70} -0.618034 q^{72} +(0.437016 + 0.437016i) q^{73} +(0.707107 - 0.707107i) q^{75} +(-0.270091 - 0.270091i) q^{82} -0.381966 q^{84} +0.381966 q^{86} +(0.437016 + 0.437016i) q^{90} -1.00000i q^{93} +(-0.437016 + 0.437016i) q^{96} +(-1.14412 - 1.14412i) q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4}+O(q^{10}) \) 8 * q - 4 * q^4	\( 8 q - 4 q^{4} - 12 q^{18} + 4 q^{21} - 4 q^{30} - 4 q^{35} + 8 q^{50} + 4 q^{64} - 4 q^{67} + 4 q^{72} - 12 q^{84} + 12 q^{86} + 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^4 - 12 * q^18 + 4 * q^21 - 4 * q^30 - 4 * q^35 + 8 * q^50 + 4 * q^64 - 4 * q^67 + 4 * q^72 - 12 * q^84 + 12 * q^86 + 4 * q^98

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