LMFDB - Embedded newform 2020.1.bt.a.1259.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2020,1,Mod(279,2020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2020, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 25, 11]))

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

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

chi := DirichletCharacter("2020.279");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2020 = 2^{2} \cdot 5 \cdot 101 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2020.bt (of order $50$, degree $20$, 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:	$1.00811132552$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{50}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{50} - \cdots)$

Embedding invariants

Embedding label			1259.1
Root			$-0.728969 + 0.684547i$ of defining polynomial
Character	$\chi$	$=$	2020.1259
Dual form			2020.1.bt.a.1659.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.968583 + 0.248690i) q^{2} +(0.876307 + 0.481754i) q^{4} +(0.425779 + 0.904827i) q^{5} +(0.728969 + 0.684547i) q^{8} +(-0.992115 - 0.125333i) q^{9} +O(q^{10})$	\(q+(0.968583 + 0.248690i) q^{2} +(0.876307 + 0.481754i) q^{4} +(0.425779 + 0.904827i) q^{5} +(0.728969 + 0.684547i) q^{8} +(-0.992115 - 0.125333i) q^{9} +(0.187381 + 0.982287i) q^{10} +(-1.53799 + 1.27233i) q^{13} +(0.535827 + 0.844328i) q^{16} +(1.11803 - 0.363271i) q^{17} +(-0.929776 - 0.368125i) q^{18} +(-0.0627905 + 0.998027i) q^{20} +(-0.637424 + 0.770513i) q^{25} +(-1.80608 + 0.849878i) q^{26} +(1.46560 - 1.21245i) q^{29} +(0.309017 + 0.951057i) q^{32} +(1.17325 - 0.0738147i) q^{34} +(-0.809017 - 0.587785i) q^{36} +(0.961606 - 0.0604991i) q^{37} +(-0.309017 + 0.951057i) q^{40} +(-1.60601 - 0.521823i) q^{41} +(-0.309017 - 0.951057i) q^{45} +(-0.187381 - 0.982287i) q^{49} +(-0.809017 + 0.587785i) q^{50} +(-1.96070 + 0.374023i) q^{52} +(1.11716 - 0.614163i) q^{53} +(1.72108 - 0.809880i) q^{58} +(0.659566 - 1.19975i) q^{61} +(0.0627905 + 0.998027i) q^{64} +(-1.80608 - 0.849878i) q^{65} +(1.15475 + 0.220280i) q^{68} +(-0.637424 - 0.770513i) q^{72} +(-0.791759 + 0.313480i) q^{73} +(0.946441 + 0.180543i) q^{74} +(-0.535827 + 0.844328i) q^{80} +(0.968583 + 0.248690i) q^{81} +(-1.42578 - 0.904827i) q^{82} +(0.804733 + 0.856954i) q^{85} +(0.211645 + 0.134314i) q^{89} +(-0.0627905 - 0.998027i) q^{90} +(0.742395 - 1.35041i) q^{97} +(0.0627905 - 0.998027i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 5 q^{13} - 5 q^{32} - 5 q^{36} + 5 q^{40} + 5 q^{45} - 5 q^{50} + 5 q^{61} - 5 q^{74} - 20 q^{82}+O(q^{100}) $ 20 * q - 5 * q^13 - 5 * q^32 - 5 * q^36 + 5 * q^40 + 5 * q^45 - 5 * q^50 + 5 * q^61 - 5 * q^74 - 20 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2020\mathbb{Z}\right)^\times$.

$n$	$1011$	$1617$	$1921$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{29}{50}\right)$

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 $

$2$	0.968583	+	0.248690i	0.968583	+	0.248690i
$2$	0.968583	+	0.248690i	0.968583	+	0.248690i
$3$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$3$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$4$	0.876307	+	0.481754i	0.876307	+	0.481754i
$5$	0.425779	+	0.904827i	0.425779	+	0.904827i
$5$	0.425779	+	0.904827i	0.425779	+	0.904827i
$6$		0			0
$7$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$7$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$8$	0.728969	+	0.684547i	0.728969	+	0.684547i
$9$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$10$	0.187381	+	0.982287i	0.187381	+	0.982287i
$11$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$11$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$12$		0			0
$13$	−1.53799	+	1.27233i	−1.53799	+	1.27233i	−0.728969	+	0.684547i	$0.760000\pi$
$13$	−1.53799	+	1.27233i	−1.53799	+	1.27233i	−0.809017	+	0.587785i	$0.800000\pi$
$14$		0			0
$15$		0			0
$16$	0.535827	+	0.844328i	0.535827	+	0.844328i
$17$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$17$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$18$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$19$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$19$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$20$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$21$		0			0
$22$		0			0
$23$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$23$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$24$		0			0
$25$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$26$	−1.80608	+	0.849878i	−1.80608	+	0.849878i
$27$		0			0
$28$		0			0
$29$	1.46560	−	1.21245i	1.46560	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$29$	1.46560	−	1.21245i	1.46560	−	1.21245i	0.929776	−	0.368125i	$-0.120000\pi$
$30$		0			0
$31$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$31$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$32$	0.309017	+	0.951057i	0.309017	+	0.951057i
$33$		0			0
$34$	1.17325	−	0.0738147i	1.17325	−	0.0738147i
$35$		0			0
$36$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$37$	0.961606	−	0.0604991i	0.961606	−	0.0604991i	0.425779	−	0.904827i	$-0.360000\pi$
$37$	0.961606	−	0.0604991i	0.961606	−	0.0604991i	0.535827	+	0.844328i	$0.320000\pi$
$38$		0			0
$39$		0			0
$40$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$41$	−1.60601	−	0.521823i	−1.60601	−	0.521823i	−0.637424	−	0.770513i	$-0.720000\pi$
$41$	−1.60601	−	0.521823i	−1.60601	−	0.521823i	−0.968583	+	0.248690i	$0.920000\pi$
$42$		0			0
$43$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$43$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$44$		0			0
$45$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$46$		0			0
$47$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$47$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$48$		0			0
$49$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$50$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$51$		0			0
$52$	−1.96070	+	0.374023i	−1.96070	+	0.374023i
$53$	1.11716	−	0.614163i	1.11716	−	0.614163i	0.187381	−	0.982287i	$-0.440000\pi$
$53$	1.11716	−	0.614163i	1.11716	−	0.614163i	0.929776	+	0.368125i	$0.120000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.72108	−	0.809880i	1.72108	−	0.809880i
$59$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$59$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$60$		0			0
$61$	0.659566	−	1.19975i	0.659566	−	1.19975i	−0.309017	−	0.951057i	$-0.600000\pi$
$61$	0.659566	−	1.19975i	0.659566	−	1.19975i	0.968583	−	0.248690i	$-0.0800000\pi$
$62$		0			0
$63$		0			0
$64$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$65$	−1.80608	−	0.849878i	−1.80608	−	0.849878i
$66$		0			0
$67$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$67$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$68$	1.15475	+	0.220280i	1.15475	+	0.220280i
$69$		0			0
$70$		0			0
$71$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$71$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$72$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$73$	−0.791759	+	0.313480i	−0.791759	+	0.313480i	−0.728969	−	0.684547i	$-0.760000\pi$
$73$	−0.791759	+	0.313480i	−0.791759	+	0.313480i	−0.0627905	+	0.998027i	$0.520000\pi$
$74$	0.946441	+	0.180543i	0.946441	+	0.180543i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$79$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$80$	−0.535827	+	0.844328i	−0.535827	+	0.844328i
$81$	0.968583	+	0.248690i	0.968583	+	0.248690i
$82$	−1.42578	−	0.904827i	−1.42578	−	0.904827i
$83$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$83$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$84$		0			0
$85$	0.804733	+	0.856954i	0.804733	+	0.856954i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.211645	+	0.134314i	0.211645	+	0.134314i	0.637424	−	0.770513i	$-0.280000\pi$
$89$	0.211645	+	0.134314i	0.211645	+	0.134314i	−0.425779	+	0.904827i	$0.640000\pi$
$90$	−0.0627905	−	0.998027i	−0.0627905	−	0.998027i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.742395	−	1.35041i	0.742395	−	1.35041i	−0.187381	−	0.982287i	$-0.560000\pi$
$97$	0.742395	−	1.35041i	0.742395	−	1.35041i	0.929776	−	0.368125i	$-0.120000\pi$
$98$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$99$		0			0
$100$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$101$	−0.876307	−	0.481754i	−0.876307	−	0.481754i
$101$	−0.876307	−	0.481754i	−0.876307	−	0.481754i
$102$		0			0
$103$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$103$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$104$	−1.99211	−	0.125333i	−1.99211	−	0.125333i
$105$		0			0
$106$	1.23480	−	0.317042i	1.23480	−	0.317042i
$107$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$107$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$108$		0			0
$109$	0.340480	−	0.362574i	0.340480	−	0.362574i	−0.535827	−	0.844328i	$-0.680000\pi$
$109$	0.340480	−	0.362574i	0.340480	−	0.362574i	0.876307	+	0.481754i	$0.160000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.03799	+	1.63560i	−1.03799	+	1.63560i	−0.309017	+	0.951057i	$0.600000\pi$
$113$	−1.03799	+	1.63560i	−1.03799	+	1.63560i	−0.728969	+	0.684547i	$0.760000\pi$
$114$		0			0
$115$		0			0
$116$	1.86842	−	0.356420i	1.86842	−	0.356420i
$117$	1.68532	−	1.06954i	1.68532	−	1.06954i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.968583	−	0.248690i	−0.968583	−	0.248690i
$122$	0.937209	−	0.998027i	0.937209	−	0.998027i
$123$		0			0
$124$		0			0
$125$	−0.968583	−	0.248690i	−0.968583	−	0.248690i
$126$		0			0
$127$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$127$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$128$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$129$		0			0
$130$	−1.53799	−	1.27233i	−1.53799	−	1.27233i
$131$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$131$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.06369	+	0.500534i	1.06369	+	0.500534i
$137$	−1.06369	+	1.46404i	−1.06369	+	1.46404i	−0.187381	+	0.982287i	$0.560000\pi$
$137$	−1.06369	+	1.46404i	−1.06369	+	1.46404i	−0.876307	+	0.481754i	$0.840000\pi$
$138$		0			0
$139$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$139$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$145$	1.72108	+	0.809880i	1.72108	+	0.809880i
$146$	−0.844844	+	0.106729i	−0.844844	+	0.106729i
$147$		0			0
$148$	0.871808	+	0.410241i	0.871808	+	0.410241i
$149$	0.871808	+	1.58581i	0.871808	+	1.58581i	0.809017	+	0.587785i	$0.200000\pi$
$149$	0.871808	+	1.58581i	0.871808	+	1.58581i	0.0627905	+	0.998027i	$0.480000\pi$
$150$		0			0
$151$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$151$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$152$		0			0
$153$	−1.15475	+	0.220280i	−1.15475	+	0.220280i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.0623382	+	0.493458i	0.0623382	+	0.493458i	0.992115	+	0.125333i	$0.0400000\pi$
$157$	0.0623382	+	0.493458i	0.0623382	+	0.493458i	−0.929776	+	0.368125i	$0.880000\pi$
$158$		0			0
$159$		0			0
$160$	−0.728969	+	0.684547i	−0.728969	+	0.684547i
$161$		0			0
$162$	0.876307	+	0.481754i	0.876307	+	0.481754i
$163$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$163$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$164$	−1.15596	−	1.23098i	−1.15596	−	1.23098i
$165$		0			0
$166$		0			0
$167$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$167$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$168$		0			0
$169$	0.559189	−	2.93137i	0.559189	−	2.93137i
$170$	0.566335	+	1.03016i	0.566335	+	1.03016i
$171$		0			0
$172$		0			0
$173$	1.11716	+	1.35041i	1.11716	+	1.35041i	0.929776	+	0.368125i	$0.120000\pi$
$173$	1.11716	+	1.35041i	1.11716	+	1.35041i	0.187381	+	0.982287i	$0.440000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.171593	+	0.182728i	0.171593	+	0.182728i
$179$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$179$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$180$	0.187381	−	0.982287i	0.187381	−	0.982287i
$181$	−1.06279	−	0.998027i	−1.06279	−	0.998027i	−0.0627905	−	0.998027i	$-0.520000\pi$
$181$	−1.06279	−	0.998027i	−1.06279	−	0.998027i	−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.464173	+	0.844328i	0.464173	+	0.844328i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$191$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$192$		0			0
$193$	0.211645	−	1.67534i	0.211645	−	1.67534i	−0.425779	−	0.904827i	$-0.640000\pi$
$193$	0.211645	−	1.67534i	0.211645	−	1.67534i	0.637424	−	0.770513i	$-0.280000\pi$
$194$	1.05491	−	1.12336i	1.05491	−	1.12336i
$195$		0			0
$196$	0.309017	−	0.951057i	0.309017	−	0.951057i
$197$	−0.473036	−	1.84235i	−0.473036	−	1.84235i	−0.535827	−	0.844328i	$-0.680000\pi$
$197$	−0.473036	−	1.84235i	−0.473036	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$198$		0			0
$199$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$199$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$200$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$201$		0			0
$202$	−0.728969	−	0.684547i	−0.728969	−	0.684547i
$203$		0			0
$204$		0			0
$205$	−0.211645	−	1.67534i	−0.211645	−	1.67534i
$206$		0			0
$207$		0			0
$208$	−1.89836	−	0.616814i	−1.89836	−	0.616814i
$209$		0			0
$210$		0			0
$211$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$211$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$212$	1.27485			1.27485
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.419952	−	0.266509i	0.419952	−	0.266509i
$219$		0			0
$220$		0			0
$221$	−1.25732	+	1.98122i	−1.25732	+	1.98122i
$222$		0			0
$223$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$223$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$224$		0			0
$225$	0.728969	−	0.684547i	0.728969	−	0.684547i
$226$	−1.41213	+	1.32608i	−1.41213	+	1.32608i
$227$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$227$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$228$		0			0
$229$	1.51373	+	0.288760i	1.51373	+	0.288760i	0.876307	−	0.481754i	$-0.160000\pi$
$229$	1.51373	+	0.288760i	1.51373	+	0.288760i	0.637424	+	0.770513i	$0.280000\pi$
$230$		0			0
$231$		0			0
$232$	1.89836	+	0.119435i	1.89836	+	0.119435i
$233$	−1.05491	−	0.872693i	−1.05491	−	0.872693i	−0.0627905	−	0.998027i	$-0.520000\pi$
$233$	−1.05491	−	0.872693i	−1.05491	−	0.872693i	−0.992115	+	0.125333i	$0.960000\pi$
$234$	1.89836	−	0.616814i	1.89836	−	0.616814i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$239$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$240$		0			0
$241$	0.432756	+	0.595638i	0.432756	+	0.595638i	0.968583	−	0.248690i	$-0.0800000\pi$
$241$	0.432756	+	0.595638i	0.432756	+	0.595638i	−0.535827	+	0.844328i	$0.680000\pi$
$242$	−0.876307	−	0.481754i	−0.876307	−	0.481754i
$243$		0			0
$244$	1.15596	−	0.733597i	1.15596	−	0.733597i
$245$	0.809017	−	0.587785i	0.809017	−	0.587785i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.876307	−	0.481754i	−0.876307	−	0.481754i
$251$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$251$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$257$	−1.44644	+	0.182728i	−1.44644	+	0.182728i	−0.809017	−	0.587785i	$-0.800000\pi$
$257$	−1.44644	+	0.182728i	−1.44644	+	0.182728i	−0.637424	+	0.770513i	$0.720000\pi$
$258$		0			0
$259$		0			0
$260$	−1.17325	−	1.61484i	−1.17325	−	1.61484i
$261$	−1.60601	+	1.01920i	−1.60601	+	1.01920i
$262$		0			0
$263$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$263$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$264$		0			0
$265$	1.03137	+	0.749337i	1.03137	+	0.749337i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.734796	+	0.0462295i	−0.734796	+	0.0462295i	−0.425779	−	0.904827i	$-0.640000\pi$
$269$	−0.734796	+	0.0462295i	−0.734796	+	0.0462295i	−0.309017	+	0.951057i	$0.600000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$	0.905793	+	0.749337i	0.905793	+	0.749337i
$273$		0			0
$274$	−1.39436	+	1.15352i	−1.39436	+	1.15352i
$275$		0			0
$276$		0			0
$277$	−0.456288	−	0.969661i	−0.456288	−	0.969661i	−0.992115	−	0.125333i	$-0.960000\pi$
$277$	−0.456288	−	0.969661i	−0.456288	−	0.969661i	0.535827	−	0.844328i	$-0.320000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.62954	+	0.645180i	1.62954	+	0.645180i	0.992115	−	0.125333i	$-0.0400000\pi$
$281$	1.62954	+	0.645180i	1.62954	+	0.645180i	0.637424	+	0.770513i	$0.280000\pi$
$282$		0			0
$283$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$283$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$289$	0.309017	−	0.224514i	0.309017	−	0.224514i
$290$	1.46560	+	1.21245i	1.46560	+	1.21245i
$291$		0			0
$292$	−0.844844	−	0.106729i	−0.844844	−	0.106729i
$293$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$293$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$294$		0			0
$295$		0			0
$296$	0.742395	+	0.614163i	0.742395	+	0.614163i
$297$		0			0
$298$	0.450043	+	1.75280i	0.450043	+	1.75280i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.36639	+	0.0859661i	1.36639	+	0.0859661i
$306$	−1.17325	−	0.0738147i	−1.17325	−	0.0738147i
$307$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$307$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$311$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$312$		0			0
$313$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$313$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$314$	−0.0623382	+	0.493458i	−0.0623382	+	0.493458i
$315$		0			0
$316$		0			0
$317$	−0.147338	−	0.202793i	−0.147338	−	0.202793i	0.728969	−	0.684547i	$-0.240000\pi$
$317$	−0.147338	−	0.202793i	−0.147338	−	0.202793i	−0.876307	+	0.481754i	$0.840000\pi$
$318$		0			0
$319$		0			0
$320$	−0.876307	+	0.481754i	−0.876307	+	0.481754i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.728969	+	0.684547i	0.728969	+	0.684547i
$325$		−	1.99605i		−	1.99605i
$326$		0			0
$327$		0			0
$328$	−0.813516	−	1.47978i	−0.813516	−	1.47978i
$329$		0			0
$330$		0			0
$331$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$331$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$332$		0			0
$333$	−0.961606	−	0.0604991i	−0.961606	−	0.0604991i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.0534698	−	0.849878i	−0.0534698	−	0.849878i	−0.929776	−	0.368125i	$-0.880000\pi$
$337$	−0.0534698	−	0.849878i	−0.0534698	−	0.849878i	0.876307	−	0.481754i	$-0.160000\pi$
$338$	1.27062	−	2.70021i	1.27062	−	2.70021i
$339$		0			0
$340$	0.292352	+	1.13864i	0.292352	+	1.13864i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.746226	+	1.58581i	0.746226	+	1.58581i
$347$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$347$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$348$		0			0
$349$	0.120759	+	0.955910i	0.120759	+	0.955910i	0.929776	+	0.368125i	$0.120000\pi$
$349$	0.120759	+	0.955910i	0.120759	+	0.955910i	−0.809017	+	0.587785i	$0.800000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.598617	+	0.153699i	−0.598617	+	0.153699i	−0.535827	−	0.844328i	$-0.680000\pi$
$353$	−0.598617	+	0.153699i	−0.598617	+	0.153699i	−0.0627905	+	0.998027i	$0.520000\pi$
$354$		0			0
$355$		0			0
$356$	0.120759	+	0.219661i	0.120759	+	0.219661i
$357$		0			0
$358$		0			0
$359$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$359$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$360$	0.425779	−	0.904827i	0.425779	−	0.904827i
$361$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$362$	−0.781202	−	1.23098i	−0.781202	−	1.23098i
$363$		0			0
$364$		0			0
$365$	−0.620759	−	0.582932i	−0.620759	−	0.582932i
$366$		0			0
$367$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$367$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$368$		0			0
$369$	1.52794	+	0.718995i	1.52794	+	0.718995i
$370$	0.239615	+	0.933237i	0.239615	+	0.933237i
$371$		0			0
$372$		0			0
$373$	−0.742395	−	0.614163i	−0.742395	−	0.614163i	0.187381	−	0.982287i	$-0.440000\pi$
$373$	−0.742395	−	0.614163i	−0.742395	−	0.614163i	−0.929776	+	0.368125i	$0.880000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.711434	+	3.72947i	−0.711434	+	3.72947i
$378$		0			0
$379$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$379$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$383$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$384$		0			0
$385$		0			0
$386$	0.621636	−	1.57007i	0.621636	−	1.57007i
$387$		0			0
$388$	1.30113	−	0.825723i	1.30113	−	0.825723i
$389$	1.15475	−	0.220280i	1.15475	−	0.220280i	0.425779	−	0.904827i	$-0.360000\pi$
$389$	1.15475	−	0.220280i	1.15475	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$390$		0			0
$391$		0			0
$392$	0.535827	−	0.844328i	0.535827	−	0.844328i
$393$		0			0
$394$		−	1.90211i		−	1.90211i
$395$		0			0
$396$		0			0
$397$	−0.0800484	+	0.0967619i	−0.0800484	+	0.0967619i	−0.809017	−	0.587785i	$-0.800000\pi$
$397$	−0.0800484	+	0.0967619i	−0.0800484	+	0.0967619i	0.728969	+	0.684547i	$0.240000\pi$
$398$		0			0
$399$		0			0
$400$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$401$	1.99211	+	0.125333i	1.99211	+	0.125333i	1.00000			$0$
$401$	1.99211	+	0.125333i	1.99211	+	0.125333i	0.992115	+	0.125333i	$0.0400000\pi$
$402$		0			0
$403$		0			0
$404$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$405$	0.187381	+	0.982287i	0.187381	+	0.982287i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.87631	+	0.481754i	−1.87631	+	0.481754i	−0.876307	+	0.481754i	$0.840000\pi$
$409$	−1.87631	+	0.481754i	−1.87631	+	0.481754i	−1.00000			$\pi$
$410$	0.211645	−	1.67534i	0.211645	−	1.67534i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.68532	−	1.06954i	−1.68532	−	1.06954i
$417$		0			0
$418$		0			0
$419$		0			0		0.982287	−	0.187381i	$-0.0600000\pi$
$419$		0			0		−0.982287	+	0.187381i	$0.940000\pi$
$420$		0			0
$421$	0.574633	+	1.76854i	0.574633	+	1.76854i	0.637424	+	0.770513i	$0.280000\pi$
$421$	0.574633	+	1.76854i	0.574633	+	1.76854i	−0.0627905	+	0.998027i	$0.520000\pi$
$422$		0			0
$423$		0			0
$424$	1.23480	+	0.317042i	1.23480	+	0.317042i
$425$	−0.432756	+	1.09302i	−0.432756	+	1.09302i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$431$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$432$		0			0
$433$	1.18532	+	1.43281i	1.18532	+	1.43281i	0.876307	+	0.481754i	$0.160000\pi$
$433$	1.18532	+	1.43281i	1.18532	+	1.43281i	0.309017	+	0.951057i	$0.400000\pi$
$434$		0			0
$435$		0			0
$436$	0.473036	−	0.153699i	0.473036	−	0.153699i
$437$		0			0
$438$		0			0
$439$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$439$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$440$		0			0
$441$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$442$	−1.71053	+	1.60629i	−1.71053	+	1.60629i
$443$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$443$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$444$		0			0
$445$	−0.0314168	+	0.248690i	−0.0314168	+	0.248690i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.84489	+	0.233064i	−1.84489	+	0.233064i	−0.968583	−	0.248690i	$-0.920000\pi$
$449$	−1.84489	+	0.233064i	−1.84489	+	0.233064i	−0.876307	+	0.481754i	$0.840000\pi$
$450$	0.876307	−	0.481754i	0.876307	−	0.481754i
$451$		0			0
$452$	−1.69755	+	0.933237i	−1.69755	+	0.933237i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.110048	+	0.0604991i	−0.110048	+	0.0604991i	−0.535827	−	0.844328i	$-0.680000\pi$
$457$	−0.110048	+	0.0604991i	−0.110048	+	0.0604991i	0.425779	+	0.904827i	$0.360000\pi$
$458$	1.39436	+	0.656137i	1.39436	+	0.656137i
$459$		0			0
$460$		0			0
$461$	0.147338	−	0.202793i	0.147338	−	0.202793i	−0.728969	−	0.684547i	$-0.760000\pi$
$461$	0.147338	−	0.202793i	0.147338	−	0.202793i	0.876307	+	0.481754i	$0.160000\pi$
$462$		0			0
$463$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$463$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$464$	1.80902	+	0.587785i	1.80902	+	0.587785i
$465$		0			0
$466$	−0.804733	−	1.10762i	−0.804733	−	1.10762i
$467$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$467$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$468$	1.99211	−	0.125333i	1.99211	−	0.125333i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.18532	+	0.469303i	−1.18532	+	0.469303i
$478$		0			0
$479$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$479$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$480$		0			0
$481$	−1.40196	+	1.31653i	−1.40196	+	1.31653i
$482$	0.271031	+	0.684547i	0.271031	+	0.684547i
$483$		0			0
$484$	−0.728969	−	0.684547i	−0.728969	−	0.684547i
$485$	1.53799	+	0.0967619i	1.53799	+	0.0967619i
$486$		0			0
$487$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$487$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$488$	1.30209	−	0.423073i	1.30209	−	0.423073i
$489$		0			0
$490$	0.929776	−	0.368125i	0.929776	−	0.368125i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$	1.19815	−	1.88797i	1.19815	−	1.88797i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$499$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$500$	−0.728969	−	0.684547i	−0.728969	−	0.684547i
$501$		0			0
$502$		0			0
$503$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$503$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$504$		0			0
$505$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.541587	+	0.297740i	0.541587	+	0.297740i	0.728969	−	0.684547i	$-0.240000\pi$
$509$	0.541587	+	0.297740i	0.541587	+	0.297740i	−0.187381	+	0.982287i	$0.560000\pi$
$510$		0			0
$511$		0			0
$512$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$513$		0			0
$514$	−1.44644	−	0.182728i	−1.44644	−	0.182728i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.734796	−	1.85588i	−0.734796	−	1.85588i
$521$	0.781202	+	1.23098i	0.781202	+	1.23098i	0.968583	+	0.248690i	$0.0800000\pi$
$521$	0.781202	+	1.23098i	0.781202	+	1.23098i	−0.187381	+	0.982287i	$0.560000\pi$
$522$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$523$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$523$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.728969	+	0.684547i	−0.728969	+	0.684547i
$530$	0.812619	+	0.982287i	0.812619	+	0.982287i
$531$		0			0
$532$		0			0
$533$	3.13395	−	1.24082i	3.13395	−	1.24082i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.723208	−	0.137959i	−0.723208	−	0.137959i
$539$		0			0
$540$		0			0
$541$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$541$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$542$		0			0
$543$		0			0
$544$	0.690983	+	0.951057i	0.690983	+	0.951057i
$545$	0.473036	+	0.153699i	0.473036	+	0.153699i
$546$		0			0
$547$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$547$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$548$	−1.63742	+	0.770513i	−1.63742	+	0.770513i
$549$	−0.804733	+	1.10762i	−0.804733	+	1.10762i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.200808	−	1.05267i	−0.200808	−	1.05267i
$555$		0			0
$556$		0			0
$557$	1.51373	−	0.288760i	1.51373	−	0.288760i	0.637424	−	0.770513i	$-0.280000\pi$
$557$	1.51373	−	0.288760i	1.51373	−	0.288760i	0.876307	+	0.481754i	$0.160000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.41789	+	1.03016i	1.41789	+	1.03016i
$563$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$563$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$564$		0			0
$565$	−1.92189	−	0.242791i	−1.92189	−	0.242791i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.0388067	+	0.616814i	0.0388067	+	0.616814i	0.968583	+	0.248690i	$0.0800000\pi$
$569$	0.0388067	+	0.616814i	0.0388067	+	0.616814i	−0.929776	+	0.368125i	$0.880000\pi$
$570$		0			0
$571$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$571$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$577$	0.683098	+	0.825723i	0.683098	+	0.825723i	0.992115	−	0.125333i	$-0.0400000\pi$
$577$	0.683098	+	0.825723i	0.683098	+	0.825723i	−0.309017	+	0.951057i	$0.600000\pi$
$578$	0.355143	−	0.140611i	0.355143	−	0.140611i
$579$		0			0
$580$	1.11803	+	1.53884i	1.11803	+	1.53884i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.791759	−	0.313480i	−0.791759	−	0.313480i
$585$	1.68532	+	1.06954i	1.68532	+	1.06954i
$586$	−1.92189	−	0.493458i	−1.92189	−	0.493458i
$587$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$587$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.566335	+	0.779494i	0.566335	+	0.779494i
$593$	−0.567290	+	0.469303i	−0.567290	+	0.469303i	−0.876307	−	0.481754i	$-0.840000\pi$
$593$	−0.567290	+	0.469303i	−0.567290	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$594$		0			0
$595$		0			0
$596$			1.80965i			1.80965i
$597$		0			0
$598$		0			0
$599$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$599$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$600$		0			0
$601$	1.56720	−	0.402389i	1.56720	−	0.402389i	0.637424	−	0.770513i	$-0.280000\pi$
$601$	1.56720	−	0.402389i	1.56720	−	0.402389i	0.929776	+	0.368125i	$0.120000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$	1.30209	+	0.423073i	1.30209	+	0.423073i
$611$		0			0
$612$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$613$	−1.18532	+	1.43281i	−1.18532	+	1.43281i	−0.309017	+	0.951057i	$0.600000\pi$
$613$	−1.18532	+	1.43281i	−1.18532	+	1.43281i	−0.876307	+	0.481754i	$0.840000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.728969	−	0.684547i	$-0.240000\pi$
$617$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.535827	+	0.844328i	$0.320000\pi$
$618$		0			0
$619$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$619$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$626$	−1.56720	−	0.402389i	−1.56720	−	0.402389i
$627$		0			0
$628$	−0.183098	+	0.462452i	−0.183098	+	0.462452i
$629$	1.05313	−	0.416964i	1.05313	−	0.416964i
$630$		0			0
$631$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$631$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$632$		0			0
$633$		0			0
$634$	−0.0922765	−	0.233064i	−0.0922765	−	0.233064i
$635$		0			0
$636$		0			0
$637$	1.53799	+	1.27233i	1.53799	+	1.27233i
$638$		0			0
$639$		0			0
$640$	−0.968583	+	0.248690i	−0.968583	+	0.248690i
$641$	−1.23879	−	0.582932i	−1.23879	−	0.582932i	−0.309017	−	0.951057i	$-0.600000\pi$
$641$	−1.23879	−	0.582932i	−1.23879	−	0.582932i	−0.929776	+	0.368125i	$0.880000\pi$
$642$		0			0
$643$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$643$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$	0.535827	+	0.844328i	0.535827	+	0.844328i
$649$		0			0
$650$	0.496398	−	1.93334i	0.496398	−	1.93334i
$651$		0			0
$652$		0			0
$653$	1.06369	+	0.500534i	1.06369	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$653$	1.06369	+	0.500534i	1.06369	+	0.500534i	0.187381	+	0.982287i	$0.440000\pi$
$654$		0			0
$655$		0			0
$656$	−0.419952	−	1.63560i	−0.419952	−	1.63560i
$657$	0.824805	−	0.211774i	0.824805	−	0.211774i
$658$		0			0
$659$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$659$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$660$		0			0
$661$	−0.211645	−	1.67534i	−0.211645	−	1.67534i	−0.637424	−	0.770513i	$-0.720000\pi$
$661$	−0.211645	−	1.67534i	−0.211645	−	1.67534i	0.425779	−	0.904827i	$-0.360000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.916350	−	0.297740i	−0.916350	−	0.297740i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.110048	+	1.74915i	0.110048	+	1.74915i	0.535827	+	0.844328i	$0.320000\pi$
$673$	0.110048	+	1.74915i	0.110048	+	1.74915i	−0.425779	+	0.904827i	$0.640000\pi$
$674$	0.159566	−	0.836475i	0.159566	−	0.836475i
$675$		0			0
$676$	1.90222	−	2.29939i	1.90222	−	2.29939i
$677$	−0.734796	−	0.0462295i	−0.734796	−	0.0462295i	−0.309017	−	0.951057i	$-0.600000\pi$
$677$	−0.734796	−	0.0462295i	−0.734796	−	0.0462295i	−0.425779	+	0.904827i	$0.640000\pi$
$678$		0			0
$679$		0			0
$680$			1.17557i			1.17557i
$681$		0			0
$682$		0			0
$683$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$683$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$684$		0			0
$685$	−1.77760	−	0.339095i	−1.77760	−	0.339095i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.936754	+	2.36597i	−0.936754	+	2.36597i
$690$		0			0
$691$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$691$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$692$	0.328407	+	1.72157i	0.328407	+	1.72157i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.98513			−1.98513
$698$	−0.120759	+	0.955910i	−0.120759	+	0.955910i
$699$		0			0
$700$		0			0
$701$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$701$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.618034			−0.618034
$707$		0			0
$708$		0			0
$709$	0.0623382	−	0.242791i	0.0623382	−	0.242791i	−0.929776	−	0.368125i	$-0.880000\pi$
$709$	0.0623382	−	0.242791i	0.0623382	−	0.242791i	0.992115	+	0.125333i	$0.0400000\pi$
$710$		0			0
$711$		0			0
$712$	0.0623382	+	0.242791i	0.0623382	+	0.242791i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$719$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$720$	0.637424	−	0.770513i	0.637424	−	0.770513i
$721$		0			0
$722$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$723$		0			0
$724$	−0.450527	−	1.38658i	−0.450527	−	1.38658i
$725$			1.90211i			1.90211i
$726$		0			0
$727$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$727$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$728$		0			0
$729$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$730$	−0.456288	−	0.718995i	−0.456288	−	0.718995i
$731$		0			0
$732$		0			0
$733$	−0.824805	−	1.75280i	−0.824805	−	1.75280i	−0.637424	−	0.770513i	$-0.720000\pi$
$733$	−0.824805	−	1.75280i	−0.824805	−	1.75280i	−0.187381	−	0.982287i	$-0.560000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.30113	+	1.07639i	1.30113	+	1.07639i
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$			0.963507i			0.963507i
$741$		0			0
$742$		0			0
$743$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$743$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$744$		0			0
$745$	−1.06369	+	1.46404i	−1.06369	+	1.46404i
$746$	−0.566335	−	0.779494i	−0.566335	−	0.779494i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$751$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$752$		0			0
$753$		0			0
$754$	−1.61656	+	3.43537i	−1.61656	+	3.43537i
$755$		0			0
$756$		0			0
$757$	1.92189	−	0.493458i	1.92189	−	0.493458i	0.929776	−	0.368125i	$-0.120000\pi$
$757$	1.92189	−	0.493458i	1.92189	−	0.493458i	0.992115	−	0.125333i	$-0.0400000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.746226	+	1.58581i	−0.746226	+	1.58581i	0.0627905	+	0.998027i	$0.480000\pi$
$761$	−0.746226	+	1.58581i	−0.746226	+	1.58581i	−0.809017	+	0.587785i	$0.800000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$769$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$770$		0			0
$771$		0			0
$772$	0.992567	−	1.36615i	0.992567	−	1.36615i
$773$	−0.362576	+	0.770513i	−0.362576	+	0.770513i	0.637424	+	0.770513i	$0.280000\pi$
$773$	−0.362576	+	0.770513i	−0.362576	+	0.770513i	−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$	1.46560	−	0.476203i	1.46560	−	0.476203i
$777$		0			0
$778$	1.17325	+	0.0738147i	1.17325	+	0.0738147i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.728969	−	0.684547i	0.728969	−	0.684547i
$785$	−0.419952	+	0.266509i	−0.419952	+	0.266509i
$786$		0			0
$787$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$787$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$788$	0.473036	−	1.84235i	0.473036	−	1.84235i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.512073	+	2.68438i	0.512073	+	2.68438i
$794$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i
$795$		0			0
$796$		0			0
$797$	−1.44644	−	0.182728i	−1.44644	−	0.182728i	−0.637424	−	0.770513i	$-0.720000\pi$
$797$	−1.44644	−	0.182728i	−1.44644	−	0.182728i	−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$801$	−0.193142	−	0.159781i	−0.193142	−	0.159781i
$802$	1.89836	+	0.616814i	1.89836	+	0.616814i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$809$	−1.45794			−1.45794			−0.728969	−	0.684547i	$-0.760000\pi$
$809$	−1.45794			−1.45794			−0.728969	+	0.684547i	$0.760000\pi$
$810$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$811$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$811$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.93717			−1.93717
$819$		0			0
$820$	0.621636	−	1.57007i	0.621636	−	1.57007i
$821$	0.683098	+	0.825723i	0.683098	+	0.825723i	0.992115	−	0.125333i	$-0.0400000\pi$
$821$	0.683098	+	0.825723i	0.683098	+	0.825723i	−0.309017	+	0.951057i	$0.600000\pi$
$822$		0			0
$823$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$823$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$827$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$828$		0			0
$829$	0.0915446	+	0.0859661i	0.0915446	+	0.0859661i	0.728969	−	0.684547i	$-0.240000\pi$
$829$	0.0915446	+	0.0859661i	0.0915446	+	0.0859661i	−0.637424	+	0.770513i	$0.720000\pi$
$830$		0			0
$831$		0			0
$832$	−1.36639	−	1.45506i	−1.36639	−	1.45506i
$833$	−0.566335	−	1.03016i	−0.566335	−	1.03016i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$839$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$840$		0			0
$841$	0.490571	−	2.57166i	0.490571	−	2.57166i
$842$	0.116762	+	1.85588i	0.116762	+	1.85588i
$843$		0			0
$844$		0			0
$845$	2.89047	−	0.742148i	2.89047	−	0.742148i
$846$		0			0
$847$		0			0
$848$	1.11716	+	0.614163i	1.11716	+	0.614163i
$849$		0			0
$850$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$851$		0			0
$852$		0			0
$853$	0.0314168	+	0.248690i	0.0314168	+	0.248690i	1.00000			$0$
$853$	0.0314168	+	0.248690i	0.0314168	+	0.248690i	−0.968583	+	0.248690i	$0.920000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.77760	+	0.339095i	−1.77760	+	0.339095i	−0.968583	−	0.248690i	$-0.920000\pi$
$857$	−1.77760	+	0.339095i	−1.77760	+	0.339095i	−0.809017	+	0.587785i	$0.800000\pi$
$858$		0			0
$859$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$859$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$863$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$864$		0			0
$865$	−0.746226	+	1.58581i	−0.746226	+	1.58581i
$866$	0.791759	+	1.68257i	0.791759	+	1.68257i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.496398	−	0.0312307i	0.496398	−	0.0312307i
$873$	−0.905793	+	1.24672i	−0.905793	+	1.24672i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.613161	+	1.88711i	0.613161	+	1.88711i	0.425779	+	0.904827i	$0.360000\pi$
$877$	0.613161	+	1.88711i	0.613161	+	1.88711i	0.187381	+	0.982287i	$0.440000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.183098	−	0.462452i	−0.183098	−	0.462452i	0.809017	−	0.587785i	$-0.200000\pi$
$881$	−0.183098	−	0.462452i	−0.183098	−	0.462452i	−0.992115	+	0.125333i	$0.960000\pi$
$882$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$883$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$883$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$884$	−2.05626	+	1.13044i	−2.05626	+	1.13044i
$885$		0			0
$886$		0			0
$887$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$887$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$888$		0			0
$889$		0			0
$890$	−0.0922765	+	0.233064i	−0.0922765	+	0.233064i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.84489	−	0.233064i	−1.84489	−	0.233064i
$899$		0			0
$900$	0.968583	−	0.248690i	0.968583	−	0.248690i
$901$	1.02591	−	1.09249i	1.02591	−	1.09249i
$902$		0			0
$903$		0			0
$904$	−1.87631	+	0.481754i	−1.87631	+	0.481754i
$905$	0.450527	−	1.38658i	0.450527	−	1.38658i
$906$		0			0
$907$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$907$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$908$		0			0
$909$	0.809017	+	0.587785i	0.809017	+	0.587785i
$910$		0			0
$911$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$911$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$912$		0			0
$913$		0			0
$914$	−0.121636	+	0.0312307i	−0.121636	+	0.0312307i
$915$		0			0
$916$	1.18738	+	0.982287i	1.18738	+	0.982287i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	0.193142	−	0.159781i	0.193142	−	0.159781i
$923$		0			0
$924$		0			0
$925$	−0.566335	+	0.779494i	−0.566335	+	0.779494i
$926$		0			0
$927$		0			0
$928$	1.60601	+	1.01920i	1.60601	+	1.01920i
$929$	−0.598617	−	0.153699i	−0.598617	−	0.153699i	−0.0627905	−	0.998027i	$-0.520000\pi$
$929$	−0.598617	−	0.153699i	−0.598617	−	0.153699i	−0.535827	+	0.844328i	$0.680000\pi$
$930$		0			0
$931$		0			0
$932$	−0.503997	−	1.27295i	−0.503997	−	1.27295i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	1.96070	+	0.374023i	1.96070	+	0.374023i
$937$	−0.116762	+	0.0462295i	−0.116762	+	0.0462295i	−0.425779	−	0.904827i	$-0.640000\pi$
$937$	−0.116762	+	0.0462295i	−0.116762	+	0.0462295i	0.309017	+	0.951057i	$0.400000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.72108	+	0.559214i	−1.72108	+	0.559214i	−0.992115	−	0.125333i	$-0.960000\pi$
$941$	−1.72108	+	0.559214i	−1.72108	+	0.559214i	−0.728969	+	0.684547i	$0.760000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$947$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$948$		0			0
$949$	0.818864	−	1.48951i	0.818864	−	1.48951i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$953$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$954$	−1.26480	+	0.159781i	−1.26480	+	0.159781i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$962$	−1.68532	+	0.926515i	−1.68532	+	0.926515i
$963$		0			0
$964$	0.0922765	+	0.730444i	0.0922765	+	0.730444i
$965$	1.60601	−	0.521823i	1.60601	−	0.521823i
$966$		0			0
$967$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$967$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$968$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$969$		0			0
$970$	1.46560	+	0.476203i	1.46560	+	0.476203i
$971$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$971$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.36639	−	0.0859661i	1.36639	−	0.0859661i
$977$	−1.96070	−	0.374023i	−1.96070	−	0.374023i	−0.992115	−	0.125333i	$-0.960000\pi$
$977$	−1.96070	−	0.374023i	−1.96070	−	0.374023i	−0.968583	−	0.248690i	$-0.920000\pi$
$978$		0			0
$979$		0			0
$980$	0.992115	−	0.125333i	0.992115	−	0.125333i
$981$	−0.383238	+	0.317042i	−0.383238	+	0.317042i
$982$		0			0
$983$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$983$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$984$		0			0
$985$	1.46560	−	1.21245i	1.46560	−	1.21245i
$986$	1.63002	−	1.53069i	1.63002	−	1.53069i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$991$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.18738	−	0.982287i	1.18738	−	0.982287i	0.187381	−	0.982287i	$-0.440000\pi$
$997$	1.18738	−	0.982287i	1.18738	−	0.982287i	1.00000			$0$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2020.1.bt.a.1259.1	✓	20
4.3	odd	2	CM	2020.1.bt.a.1259.1	✓	20
5.4	even	2		2020.1.bt.b.1259.1	yes	20
20.19	odd	2		2020.1.bt.b.1259.1	yes	20
101.43	even	50		2020.1.bt.b.1659.1	yes	20
404.43	odd	50		2020.1.bt.b.1659.1	yes	20
505.144	even	50	inner	2020.1.bt.a.1659.1	yes	20
2020.1659	odd	50	inner	2020.1.bt.a.1659.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2020.1.bt.a.1259.1	✓	20	1.1	even	1	trivial
2020.1.bt.a.1259.1	✓	20	4.3	odd	2	CM
2020.1.bt.a.1659.1	yes	20	505.144	even	50	inner
2020.1.bt.a.1659.1	yes	20	2020.1659	odd	50	inner
2020.1.bt.b.1259.1	yes	20	5.4	even	2
2020.1.bt.b.1259.1	yes	20	20.19	odd	2
2020.1.bt.b.1659.1	yes	20	101.43	even	50
2020.1.bt.b.1659.1	yes	20	404.43	odd	50

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:	\( N \)	\(=\)	\( 2020 = 2^{2} \cdot 5 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2020.bt (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.968583 + 0.248690i) q^{2} +(0.876307 + 0.481754i) q^{4} +(0.425779 + 0.904827i) q^{5} +(0.728969 + 0.684547i) q^{8} +(-0.992115 - 0.125333i) q^{9} +O(q^{10})\)	\(q+(0.968583 + 0.248690i) q^{2} +(0.876307 + 0.481754i) q^{4} +(0.425779 + 0.904827i) q^{5} +(0.728969 + 0.684547i) q^{8} +(-0.992115 - 0.125333i) q^{9} +(0.187381 + 0.982287i) q^{10} +(-1.53799 + 1.27233i) q^{13} +(0.535827 + 0.844328i) q^{16} +(1.11803 - 0.363271i) q^{17} +(-0.929776 - 0.368125i) q^{18} +(-0.0627905 + 0.998027i) q^{20} +(-0.637424 + 0.770513i) q^{25} +(-1.80608 + 0.849878i) q^{26} +(1.46560 - 1.21245i) q^{29} +(0.309017 + 0.951057i) q^{32} +(1.17325 - 0.0738147i) q^{34} +(-0.809017 - 0.587785i) q^{36} +(0.961606 - 0.0604991i) q^{37} +(-0.309017 + 0.951057i) q^{40} +(-1.60601 - 0.521823i) q^{41} +(-0.309017 - 0.951057i) q^{45} +(-0.187381 - 0.982287i) q^{49} +(-0.809017 + 0.587785i) q^{50} +(-1.96070 + 0.374023i) q^{52} +(1.11716 - 0.614163i) q^{53} +(1.72108 - 0.809880i) q^{58} +(0.659566 - 1.19975i) q^{61} +(0.0627905 + 0.998027i) q^{64} +(-1.80608 - 0.849878i) q^{65} +(1.15475 + 0.220280i) q^{68} +(-0.637424 - 0.770513i) q^{72} +(-0.791759 + 0.313480i) q^{73} +(0.946441 + 0.180543i) q^{74} +(-0.535827 + 0.844328i) q^{80} +(0.968583 + 0.248690i) q^{81} +(-1.42578 - 0.904827i) q^{82} +(0.804733 + 0.856954i) q^{85} +(0.211645 + 0.134314i) q^{89} +(-0.0627905 - 0.998027i) q^{90} +(0.742395 - 1.35041i) q^{97} +(0.0627905 - 0.998027i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 5 q^{13} - 5 q^{32} - 5 q^{36} + 5 q^{40} + 5 q^{45} - 5 q^{50} + 5 q^{61} - 5 q^{74} - 20 q^{82}+O(q^{100}) \) 20 * q - 5 * q^13 - 5 * q^32 - 5 * q^36 + 5 * q^40 + 5 * q^45 - 5 * q^50 + 5 * q^61 - 5 * q^74 - 20 * q^82

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