LMFDB - Embedded newform 3131.1.de.a.278.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3131,1,Mod(30,3131)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3131, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 47]))

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

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

chi := DirichletCharacter("3131.30");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3131 = 31 \cdot 101 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3131.de (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:	$1.56257255455$
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:	$\Z[a_1, \ldots, a_{4}]$
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			278.1
Root			$0.187381 + 0.982287i$ of defining polynomial
Character	$\chi$	$=$	3131.278
Dual form			3131.1.de.a.1115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.96070 - 0.123357i) q^{2} +(2.83700 - 0.358397i) q^{4} +(-0.110048 + 1.74915i) q^{5} +(-1.65875 + 1.05267i) q^{7} +(3.58852 - 0.684547i) q^{8} +(-0.728969 - 0.684547i) q^{9} +O(q^{10})$	\(q+(1.96070 - 0.123357i) q^{2} +(2.83700 - 0.358397i) q^{4} +(-0.110048 + 1.74915i) q^{5} +(-1.65875 + 1.05267i) q^{7} +(3.58852 - 0.684547i) q^{8} +(-0.728969 - 0.684547i) q^{9} +3.44314i q^{10} +(-3.12244 + 2.26859i) q^{14} +(4.18185 - 1.07372i) q^{16} +(-1.51373 - 1.25227i) q^{18} +(1.03799 + 0.266509i) q^{19} +(0.314687 + 5.00180i) q^{20} +(-2.05532 - 0.259647i) q^{25} +(-4.32859 + 3.58092i) q^{28} +(-0.535827 - 0.844328i) q^{31} +(4.59246 - 1.49218i) q^{32} +(-1.65875 - 3.01725i) q^{35} +(-2.31343 - 1.68080i) q^{36} +(2.06805 + 0.394502i) q^{38} +(0.802471 + 6.35221i) q^{40} +(1.72108 + 0.559214i) q^{41} +(1.27760 - 1.19975i) q^{45} +(0.541587 - 0.297740i) q^{47} +(1.21754 - 2.58740i) q^{49} +(-4.06189 - 0.255552i) q^{50} +(-5.23184 + 4.91302i) q^{56} +(-0.183098 - 0.713118i) q^{59} +(-1.15475 - 1.58937i) q^{62} +(1.92978 + 0.368125i) q^{63} +(4.80606 - 1.90285i) q^{64} +(-0.183098 - 0.462452i) q^{67} +(-3.62450 - 5.71129i) q^{70} +(-1.80113 - 0.713118i) q^{71} +(-3.08452 - 1.95750i) q^{72} +(3.04029 + 0.384077i) q^{76} +(1.41789 + 7.43286i) q^{80} +(0.0627905 + 0.998027i) q^{81} +(3.44351 + 0.884142i) q^{82} +(2.35699 - 2.50994i) q^{90} +(1.02516 - 0.650587i) q^{94} +(-0.580394 + 1.78627i) q^{95} +(0.613161 - 0.0774602i) q^{97} +(2.06805 - 5.22330i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{8}+O(q^{10}) $ 20 * q + 20 * q^8	$ 20 q + 20 q^{8} - 10 q^{14} + 5 q^{16} - 5 q^{19} + 5 q^{20} + 5 q^{28} - 5 q^{36} + 5 q^{38} - 5 q^{40} - 5 q^{45} + 5 q^{50} - 20 q^{56} + 5 q^{59} + 20 q^{63} + 20 q^{64} + 5 q^{67} - 5 q^{70} - 5 q^{71} - 20 q^{72} - 5 q^{76} + 20 q^{82} + 5 q^{98}+O(q^{100}) $ 20 * q + 20 * q^8 - 10 * q^14 + 5 * q^16 - 5 * q^19 + 5 * q^20 + 5 * q^28 - 5 * q^36 + 5 * q^38 - 5 * q^40 - 5 * q^45 + 5 * q^50 - 20 * q^56 + 5 * q^59 + 20 * q^63 + 20 * q^64 + 5 * q^67 - 5 * q^70 - 5 * q^71 - 20 * q^72 - 5 * q^76 + 20 * q^82 + 5 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3131\mathbb{Z}\right)^\times$.

$n$	$809$	$2729$
$\chi(n)$	$-1$	$e\left(\frac{49}{50}\right)$

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 $

$2$	1.96070	−	0.123357i	1.96070	−	0.123357i	0.968583	−	0.248690i	$-0.0800000\pi$
$2$	1.96070	−	0.123357i	1.96070	−	0.123357i	0.992115	+	0.125333i	$0.0400000\pi$
$3$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$3$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$4$	2.83700	−	0.358397i	2.83700	−	0.358397i
$5$	−0.110048	+	1.74915i	−0.110048	+	1.74915i	0.425779	+	0.904827i	$0.360000\pi$
$5$	−0.110048	+	1.74915i	−0.110048	+	1.74915i	−0.535827	+	0.844328i	$0.680000\pi$
$6$		0			0
$7$	−1.65875	+	1.05267i	−1.65875	+	1.05267i	−0.728969	+	0.684547i	$0.760000\pi$
$7$	−1.65875	+	1.05267i	−1.65875	+	1.05267i	−0.929776	+	0.368125i	$0.880000\pi$
$8$	3.58852	−	0.684547i	3.58852	−	0.684547i
$9$	−0.728969	−	0.684547i	−0.728969	−	0.684547i
$10$			3.44314i			3.44314i
$11$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$11$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$12$		0			0
$13$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$13$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$14$	−3.12244	+	2.26859i	−3.12244	+	2.26859i
$15$		0			0
$16$	4.18185	−	1.07372i	4.18185	−	1.07372i
$17$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$17$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$18$	−1.51373	−	1.25227i	−1.51373	−	1.25227i
$19$	1.03799	+	0.266509i	1.03799	+	0.266509i	0.728969	−	0.684547i	$-0.240000\pi$
$19$	1.03799	+	0.266509i	1.03799	+	0.266509i	0.309017	+	0.951057i	$0.400000\pi$
$20$	0.314687	+	5.00180i	0.314687	+	5.00180i
$21$		0			0
$22$		0			0
$23$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$23$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$24$		0			0
$25$	−2.05532	−	0.259647i	−2.05532	−	0.259647i
$26$		0			0
$27$		0			0
$28$	−4.32859	+	3.58092i	−4.32859	+	3.58092i
$29$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$29$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$30$		0			0
$31$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$31$	−0.535827	−	0.844328i	−0.535827	−	0.844328i
$32$	4.59246	−	1.49218i	4.59246	−	1.49218i
$33$		0			0
$34$		0			0
$35$	−1.65875	−	3.01725i	−1.65875	−	3.01725i
$36$	−2.31343	−	1.68080i	−2.31343	−	1.68080i
$37$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$37$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$38$	2.06805	+	0.394502i	2.06805	+	0.394502i
$39$		0			0
$40$	0.802471	+	6.35221i	0.802471	+	6.35221i
$41$	1.72108	+	0.559214i	1.72108	+	0.559214i	0.992115	−	0.125333i	$-0.0400000\pi$
$41$	1.72108	+	0.559214i	1.72108	+	0.559214i	0.728969	+	0.684547i	$0.240000\pi$
$42$		0			0
$43$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$43$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$44$		0			0
$45$	1.27760	−	1.19975i	1.27760	−	1.19975i
$46$		0			0
$47$	0.541587	−	0.297740i	0.541587	−	0.297740i	−0.187381	−	0.982287i	$-0.560000\pi$
$47$	0.541587	−	0.297740i	0.541587	−	0.297740i	0.728969	+	0.684547i	$0.240000\pi$
$48$		0			0
$49$	1.21754	−	2.58740i	1.21754	−	2.58740i
$50$	−4.06189	−	0.255552i	−4.06189	−	0.255552i
$51$		0			0
$52$		0			0
$53$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$53$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$54$		0			0
$55$		0			0
$56$	−5.23184	+	4.91302i	−5.23184	+	4.91302i
$57$		0			0
$58$		0			0
$59$	−0.183098	−	0.713118i	−0.183098	−	0.713118i	−0.992115	−	0.125333i	$-0.960000\pi$
$59$	−0.183098	−	0.713118i	−0.183098	−	0.713118i	0.809017	−	0.587785i	$-0.200000\pi$
$60$		0			0
$61$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$61$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$62$	−1.15475	−	1.58937i	−1.15475	−	1.58937i
$63$	1.92978	+	0.368125i	1.92978	+	0.368125i
$64$	4.80606	−	1.90285i	4.80606	−	1.90285i
$65$		0			0
$66$		0			0
$67$	−0.183098	−	0.462452i	−0.183098	−	0.462452i	0.809017	−	0.587785i	$-0.200000\pi$
$67$	−0.183098	−	0.462452i	−0.183098	−	0.462452i	−0.992115	+	0.125333i	$0.960000\pi$
$68$		0			0
$69$		0			0
$70$	−3.62450	−	5.71129i	−3.62450	−	5.71129i
$71$	−1.80113	−	0.713118i	−1.80113	−	0.713118i	−0.992115	−	0.125333i	$-0.960000\pi$
$71$	−1.80113	−	0.713118i	−1.80113	−	0.713118i	−0.809017	−	0.587785i	$-0.800000\pi$
$72$	−3.08452	−	1.95750i	−3.08452	−	1.95750i
$73$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$73$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$74$		0			0
$75$		0			0
$76$	3.04029	+	0.384077i	3.04029	+	0.384077i
$77$		0			0
$78$		0			0
$79$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$79$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$80$	1.41789	+	7.43286i	1.41789	+	7.43286i
$81$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$82$	3.44351	+	0.884142i	3.44351	+	0.884142i
$83$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$83$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$89$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$90$	2.35699	−	2.50994i	2.35699	−	2.50994i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	1.02516	−	0.650587i	1.02516	−	0.650587i
$95$	−0.580394	+	1.78627i	−0.580394	+	1.78627i
$96$		0			0
$97$	0.613161	−	0.0774602i	0.613161	−	0.0774602i	0.187381	−	0.982287i	$-0.440000\pi$
$97$	0.613161	−	0.0774602i	0.613161	−	0.0774602i	0.425779	+	0.904827i	$0.360000\pi$
$98$	2.06805	−	5.22330i	2.06805	−	5.22330i
$99$		0			0
$100$	−5.92400			−5.92400
$101$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$101$	−0.0627905	+	0.998027i	−0.0627905	+	0.998027i
$102$		0			0
$103$	0.496398	−	0.0312307i	0.496398	−	0.0312307i	0.187381	−	0.982287i	$-0.440000\pi$
$103$	0.496398	−	0.0312307i	0.496398	−	0.0312307i	0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$107$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$108$		0			0
$109$	−1.15475	+	0.220280i	−1.15475	+	0.220280i	−0.728969	−	0.684547i	$-0.760000\pi$
$109$	−1.15475	+	0.220280i	−1.15475	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$110$		0			0
$111$		0			0
$112$	−5.80635	+	6.18313i	−5.80635	+	6.18313i
$113$	0.183098	−	0.713118i	0.183098	−	0.713118i	−0.809017	−	0.587785i	$-0.800000\pi$
$113$	0.183098	−	0.713118i	0.183098	−	0.713118i	0.992115	−	0.125333i	$-0.0400000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.446967	−	1.37562i	−0.446967	−	1.37562i
$119$		0			0
$120$		0			0
$121$	−0.0627905	−	0.998027i	−0.0627905	−	0.998027i
$122$		0			0
$123$		0			0
$124$	−1.82275	−	2.20332i	−1.82275	−	2.20332i
$125$	0.351939	−	1.84493i	0.351939	−	1.84493i
$126$	3.82912	+	0.483730i	3.82912	+	0.483730i
$127$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$127$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$128$	4.81927	−	2.26778i	4.81927	−	2.26778i
$129$		0			0
$130$		0			0
$131$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.876307	−	0.481754i	$-0.160000\pi$
$131$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$132$		0			0
$133$	−2.00230	+	0.650587i	−2.00230	+	0.650587i
$134$	−0.416046	−	0.884142i	−0.416046	−	0.884142i
$135$		0			0
$136$		0			0
$137$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$137$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$138$		0			0
$139$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$139$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$140$	−5.78724	−	7.96545i	−5.78724	−	7.96545i
$141$		0			0
$142$	−3.61944	−	1.17603i	−3.61944	−	1.17603i
$143$		0			0
$144$	−3.78345	−	2.07997i	−3.78345	−	2.07997i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.226810	−	1.79538i	0.226810	−	1.79538i	−0.309017	−	0.951057i	$-0.600000\pi$
$149$	0.226810	−	1.79538i	0.226810	−	1.79538i	0.535827	−	0.844328i	$-0.320000\pi$
$150$		0			0
$151$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$151$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$152$	3.90727	+	0.245825i	3.90727	+	0.245825i
$153$		0			0
$154$		0			0
$155$	1.53583	−	0.844328i	1.53583	−	0.844328i
$156$		0			0
$157$	−1.27760	+	1.19975i	−1.27760	+	1.19975i	−0.309017	+	0.951057i	$0.600000\pi$
$157$	−1.27760	+	1.19975i	−1.27760	+	1.19975i	−0.968583	+	0.248690i	$0.920000\pi$
$158$		0			0
$159$		0			0
$160$	2.10467	+	8.19714i	2.10467	+	8.19714i
$161$		0			0
$162$	0.246226	+	1.94908i	0.246226	+	1.94908i
$163$	−1.17325	−	1.61484i	−1.17325	−	1.61484i	−0.637424	−	0.770513i	$-0.720000\pi$
$163$	−1.17325	−	1.61484i	−1.17325	−	1.61484i	−0.535827	−	0.844328i	$-0.680000\pi$
$164$	5.08314	+	0.969661i	5.08314	+	0.969661i
$165$		0			0
$166$		0			0
$167$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$167$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$168$		0			0
$169$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$170$		0			0
$171$	−0.574221	−	0.904827i	−0.574221	−	0.904827i
$172$		0			0
$173$	−0.813516	−	0.516273i	−0.813516	−	0.516273i	0.0627905	−	0.998027i	$-0.480000\pi$
$173$	−0.813516	−	0.516273i	−0.813516	−	0.516273i	−0.876307	+	0.481754i	$0.840000\pi$
$174$		0			0
$175$	3.68257	−	1.73289i	3.68257	−	1.73289i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$179$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$180$	3.19457	−	3.86157i	3.19457	−	3.86157i
$181$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$181$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	1.42978	−	1.03879i	1.42978	−	1.03879i
$189$		0			0
$190$	−0.917629	+	3.57393i	−0.917629	+	3.57393i
$191$	−1.15596	+	1.23098i	−1.15596	+	1.23098i	−0.187381	+	0.982287i	$0.560000\pi$
$191$	−1.15596	+	1.23098i	−1.15596	+	1.23098i	−0.968583	+	0.248690i	$0.920000\pi$
$192$		0			0
$193$	1.44644	+	1.35830i	1.44644	+	1.35830i	0.809017	+	0.587785i	$0.200000\pi$
$193$	1.44644	+	1.35830i	1.44644	+	1.35830i	0.637424	+	0.770513i	$0.280000\pi$
$194$	1.19267	−	0.227513i	1.19267	−	0.227513i
$195$		0			0
$196$	2.52685	−	7.77683i	2.52685	−	7.77683i
$197$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$197$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$198$		0			0
$199$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$199$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$200$	−7.55329	+	0.475213i	−7.55329	+	0.475213i
$201$		0			0
$202$			1.96457i			1.96457i
$203$		0			0
$204$		0			0
$205$	−1.16755	+	2.94890i	−1.16755	+	2.94890i
$206$	0.969435	−	0.122468i	0.969435	−	0.122468i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.450527	−	0.423073i	−0.450527	−	0.423073i	0.425779	−	0.904827i	$-0.360000\pi$
$211$	−0.450527	−	0.423073i	−0.450527	−	0.423073i	−0.876307	+	0.481754i	$0.840000\pi$
$212$		0			0
$213$		0			0
$214$	−0.416046	+	1.62039i	−0.416046	+	1.62039i
$215$		0			0
$216$		0			0
$217$	1.77760	+	0.836475i	1.77760	+	0.836475i
$218$	−2.23694	+	0.574348i	−2.23694	+	0.574348i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$223$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$224$	−6.04695	+	7.30950i	−6.04695	+	7.30950i
$225$	1.32052	+	1.59624i	1.32052	+	1.59624i
$226$	0.271031	−	1.42080i	0.271031	−	1.42080i
$227$	−1.84489	−	0.233064i	−1.84489	−	0.233064i	−0.876307	−	0.481754i	$-0.840000\pi$
$227$	−1.84489	−	0.233064i	−1.84489	−	0.233064i	−0.968583	+	0.248690i	$0.920000\pi$
$228$		0			0
$229$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$229$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.331159	+	0.521823i	0.331159	+	0.521823i	0.968583	−	0.248690i	$-0.0800000\pi$
$233$	0.331159	+	0.521823i	0.331159	+	0.521823i	−0.637424	+	0.770513i	$0.720000\pi$
$234$		0			0
$235$	0.461193	+	0.980086i	0.461193	+	0.980086i
$236$	−0.775029	−	1.95750i	−0.775029	−	1.95750i
$237$		0			0
$238$		0			0
$239$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$239$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$240$		0			0
$241$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$241$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$242$	−0.246226	−	1.94908i	−0.246226	−	1.94908i
$243$		0			0
$244$		0			0
$245$	4.39178	+	2.41440i	4.39178	+	2.41440i
$246$		0			0
$247$		0			0
$248$	−2.50081	−	2.66309i	−2.50081	−	2.66309i
$249$		0			0
$250$	0.462461	−	3.66076i	0.462461	−	3.66076i
$251$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$251$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$252$	5.60672	+	0.352745i	5.60672	+	0.352745i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.63972	−	2.55071i	4.63972	−	2.55071i
$257$	1.30209	+	1.38658i	1.30209	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$257$	1.30209	+	1.38658i	1.30209	+	1.38658i	0.425779	+	0.904827i	$0.360000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	2.38195	+	0.773944i	2.38195	+	0.773944i
$263$		0			0		−0.125333	−	0.992115i	$-0.540000\pi$
$263$		0			0		0.125333	+	0.992115i	$0.460000\pi$
$264$		0			0
$265$		0			0
$266$	−3.84565	+	1.52260i	−3.84565	+	1.52260i
$267$		0			0
$268$	−0.685190	−	1.24636i	−0.685190	−	1.24636i
$269$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$269$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$277$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$278$		0			0
$279$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$280$	−8.01789	−	9.69196i	−8.01789	−	9.69196i
$281$	1.26480	−	1.52888i	1.26480	−	1.52888i	0.535827	−	0.844328i	$-0.320000\pi$
$281$	1.26480	−	1.52888i	1.26480	−	1.52888i	0.728969	−	0.684547i	$-0.240000\pi$
$282$		0			0
$283$	0.0388067	+	0.616814i	0.0388067	+	0.616814i	0.968583	+	0.248690i	$0.0800000\pi$
$283$	0.0388067	+	0.616814i	0.0388067	+	0.616814i	−0.929776	+	0.368125i	$0.880000\pi$
$284$	−5.36540	−	1.37760i	−5.36540	−	1.37760i
$285$		0			0
$286$		0			0
$287$	−3.44351	+	0.884142i	−3.44351	+	0.884142i
$288$	−4.36923	−	2.05600i	−4.36923	−	2.05600i
$289$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$293$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$	1.26750	−	0.241789i	1.26750	−	0.241789i
$296$		0			0
$297$		0			0
$298$	0.223233	−	3.54818i	0.223233	−	3.54818i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	4.62685			4.62685
$305$		0			0
$306$		0			0
$307$	−1.60528	+	0.202793i	−1.60528	+	0.202793i	−0.876307	−	0.481754i	$-0.840000\pi$
$307$	−1.60528	+	0.202793i	−1.60528	+	0.202793i	−0.728969	+	0.684547i	$0.760000\pi$
$308$		0			0
$309$		0			0
$310$	2.90714	−	1.84493i	2.90714	−	1.84493i
$311$	−1.15475	+	0.220280i	−1.15475	+	0.220280i	−0.728969	−	0.684547i	$-0.760000\pi$
$311$	−1.15475	+	0.220280i	−1.15475	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	−2.35699	+	2.50994i	−2.35699	+	2.50994i
$315$	−0.856274	+	3.33497i	−0.856274	+	3.33497i
$316$		0			0
$317$	1.56720	−	1.13864i	1.56720	−	1.13864i	0.637424	−	0.770513i	$-0.280000\pi$
$317$	1.56720	−	1.13864i	1.56720	−	1.13864i	0.929776	−	0.368125i	$-0.120000\pi$
$318$		0			0
$319$		0			0
$320$	2.79949	+	8.61595i	2.79949	+	8.61595i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.535827	+	2.80890i	0.535827	+	2.80890i
$325$		0			0
$326$	−2.49959	−	3.02149i	−2.49959	−	3.02149i
$327$		0			0
$328$	6.55895	+	0.828588i	6.55895	+	0.828588i
$329$	−0.584933	+	1.06399i	−0.584933	+	1.06399i
$330$		0			0
$331$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$331$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.829050	−	0.269375i	0.829050	−	0.269375i
$336$		0			0
$337$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$337$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$338$	−0.946441	−	1.72157i	−0.946441	−	1.72157i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−1.23749	−	1.70326i	−1.23749	−	1.70326i
$343$	0.457871	+	3.62442i	0.457871	+	3.62442i
$344$		0			0
$345$		0			0
$346$	−1.65875	−	0.911903i	−1.65875	−	0.911903i
$347$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$347$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$348$		0			0
$349$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$349$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$350$	7.00665	−	3.85194i	7.00665	−	3.85194i
$351$		0			0
$352$		0			0
$353$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$353$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$354$		0			0
$355$	1.44556	−	3.07198i	1.44556	−	3.07198i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.929324	−	0.872693i	0.929324	−	0.872693i	−0.0627905	−	0.998027i	$-0.520000\pi$
$359$	0.929324	−	0.872693i	0.929324	−	0.872693i	0.992115	+	0.125333i	$0.0400000\pi$
$360$	3.76341	−	5.17989i	3.76341	−	5.17989i
$361$	0.130080	+	0.0715122i	0.130080	+	0.0715122i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$367$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$368$		0			0
$369$	−0.871808	−	1.58581i	−0.871808	−	1.58581i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.781202	+	1.23098i	0.781202	+	1.23098i	0.968583	+	0.248690i	$0.0800000\pi$
$373$	0.781202	+	1.23098i	0.781202	+	1.23098i	−0.187381	+	0.982287i	$0.560000\pi$
$374$		0			0
$375$		0			0
$376$	1.73968	−	1.43919i	1.73968	−	1.43919i
$377$		0			0
$378$		0			0
$379$	−1.06320	−	0.134314i	−1.06320	−	0.134314i	−0.425779	−	0.904827i	$-0.640000\pi$
$379$	−1.06320	−	0.134314i	−1.06320	−	0.134314i	−0.637424	+	0.770513i	$0.720000\pi$
$380$	−1.00639	+	5.27566i	−1.00639	+	5.27566i
$381$		0			0
$382$	−2.11465	+	2.55617i	−2.11465	+	2.55617i
$383$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$383$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$384$		0			0
$385$		0			0
$386$	3.00359	+	2.48478i	3.00359	+	2.48478i
$387$		0			0
$388$	1.71178	−	0.439510i	1.71178	−	0.439510i
$389$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$389$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$390$		0			0
$391$		0			0
$392$	2.59797	−	10.1184i	2.59797	−	10.1184i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.52794	+	0.969661i	−1.52794	+	0.969661i	−0.535827	+	0.844328i	$0.680000\pi$
$397$	−1.52794	+	0.969661i	−1.52794	+	0.969661i	−0.992115	+	0.125333i	$0.960000\pi$
$398$		0			0
$399$		0			0
$400$	−8.87382	+	1.12102i	−8.87382	+	1.12102i
$401$		0			0		0.368125	−	0.929776i	$-0.380000\pi$
$401$		0			0		−0.368125	+	0.929776i	$0.620000\pi$
$402$		0			0
$403$		0			0
$404$	0.179553	+	2.85391i	0.179553	+	2.85391i
$405$	−1.75261			−1.75261
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$409$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$410$	−1.92545	+	5.92593i	−1.92545	+	5.92593i
$411$		0			0
$412$	1.39709	−	0.266509i	1.39709	−	0.266509i
$413$	1.05439	+	0.990140i	1.05439	+	0.990140i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.72108	−	0.809880i	−1.72108	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$419$	−1.72108	−	0.809880i	−1.72108	−	0.809880i	−0.728969	−	0.684547i	$-0.760000\pi$
$420$		0			0
$421$	0.393950	+	1.21245i	0.393950	+	1.21245i	0.929776	+	0.368125i	$0.120000\pi$
$421$	0.393950	+	1.21245i	0.393950	+	1.21245i	−0.535827	+	0.844328i	$0.680000\pi$
$422$	−0.935537	−	0.773944i	−0.935537	−	0.773944i
$423$	−0.598617	−	0.153699i	−0.598617	−	0.153699i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.456288	+	2.39194i	−0.456288	+	2.39194i
$429$		0			0
$430$		0			0
$431$	−0.871808	+	0.410241i	−0.871808	+	0.410241i	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	−0.871808	+	0.410241i	−0.871808	+	0.410241i	−0.0627905	+	0.998027i	$0.520000\pi$
$432$		0			0
$433$		0			0		−0.844328	−	0.535827i	$-0.820000\pi$
$433$		0			0		0.844328	+	0.535827i	$0.180000\pi$
$434$	3.58852	+	1.42080i	3.58852	+	1.42080i
$435$		0			0
$436$	−3.19708	+	1.03879i	−3.19708	+	1.03879i
$437$		0			0
$438$		0			0
$439$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$439$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$440$		0			0
$441$	−2.65875	+	1.05267i	−2.65875	+	1.05267i
$442$		0			0
$443$	0.905793	+	1.24672i	0.905793	+	1.24672i	0.968583	+	0.248690i	$0.0800000\pi$
$443$	0.905793	+	1.24672i	0.905793	+	1.24672i	−0.0627905	+	0.998027i	$0.520000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−5.96895	+	8.21555i	−5.96895	+	8.21555i
$449$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$449$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$450$	2.78605	+	2.96684i	2.78605	+	2.96684i
$451$		0			0
$452$	0.263869	−	2.08874i	0.263869	−	2.08874i
$453$		0			0
$454$	−3.64602	−	0.229388i	−3.64602	−	0.229388i
$455$		0			0
$456$		0			0
$457$		0			0		0.125333	−	0.992115i	$-0.460000\pi$
$457$		0			0		−0.125333	+	0.992115i	$0.540000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$461$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$462$		0			0
$463$		0			0		−0.248690	−	0.968583i	$-0.580000\pi$
$463$		0			0		0.248690	+	0.968583i	$0.420000\pi$
$464$		0			0
$465$		0			0
$466$	0.713673	+	0.982287i	0.713673	+	0.982287i
$467$	−1.15475	−	0.220280i	−1.15475	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$467$	−1.15475	−	0.220280i	−1.15475	−	0.220280i	−0.728969	+	0.684547i	$0.760000\pi$
$468$		0			0
$469$	0.790523	+	0.574348i	0.790523	+	0.574348i
$470$	1.02516	+	1.86476i	1.02516	+	1.86476i
$471$		0			0
$472$	−1.14521	−	2.43370i	−1.14521	−	2.43370i
$473$		0			0
$474$		0			0
$475$	−2.06419	−	0.817271i	−2.06419	−	0.817271i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.742395	−	1.35041i	0.742395	−	1.35041i	−0.187381	−	0.982287i	$-0.560000\pi$
$479$	0.742395	−	1.35041i	0.742395	−	1.35041i	0.929776	−	0.368125i	$-0.120000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.535827	−	2.80890i	−0.535827	−	2.80890i
$485$	0.0680131	+	1.08104i	0.0680131	+	1.08104i
$486$		0			0
$487$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$487$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$488$		0			0
$489$		0			0
$490$	8.90878	+	4.19215i	8.90878	+	4.19215i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−3.14731	−	2.95553i	−3.14731	−	2.95553i
$497$	3.73830	−	0.713118i	3.73830	−	0.713118i
$498$		0			0
$499$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$499$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$500$	0.337235	−	5.36020i	0.337235	−	5.36020i
$501$		0			0
$502$		0			0
$503$	0.496398	−	0.0312307i	0.496398	−	0.0312307i	0.187381	−	0.982287i	$-0.440000\pi$
$503$	0.496398	−	0.0312307i	0.496398	−	0.0312307i	0.309017	+	0.951057i	$0.400000\pi$
$504$	7.17704			7.17704
$505$	−1.73879	−	0.219661i	−1.73879	−	0.219661i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$509$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$510$		0			0
$511$		0			0
$512$	4.28540	−	2.71960i	4.28540	−	2.71960i
$513$		0			0
$514$	2.72404	+	2.55804i	2.72404	+	2.55804i
$515$			0.871714i			0.871714i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.92189	−	0.493458i	1.92189	−	0.493458i	0.929776	−	0.368125i	$-0.120000\pi$
$521$	1.92189	−	0.493458i	1.92189	−	0.493458i	0.992115	−	0.125333i	$-0.0400000\pi$
$522$		0			0
$523$		0			0		−0.770513	−	0.637424i	$-0.780000\pi$
$523$		0			0		0.770513	+	0.637424i	$0.220000\pi$
$524$	3.53097	+	0.906598i	3.53097	+	0.906598i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$530$		0			0
$531$	−0.354691	+	0.645180i	−0.354691	+	0.645180i
$532$	−5.44737	+	2.56334i	−5.44737	+	2.56334i
$533$		0			0
$534$		0			0
$535$	−1.38765	−	0.549409i	−1.38765	−	0.549409i
$536$	−0.973620	−	1.53418i	−0.973620	−	1.53418i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$541$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.258227	−	2.04407i	−0.258227	−	2.04407i
$546$		0			0
$547$	−0.0623382	−	0.242791i	−0.0623382	−	0.242791i	0.929776	−	0.368125i	$-0.120000\pi$
$547$	−0.0623382	−	0.242791i	−0.0623382	−	0.242791i	−0.992115	+	0.125333i	$0.960000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$557$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$558$	−0.246226	+	1.94908i	−0.246226	+	1.94908i
$559$		0			0
$560$	−10.1763	−	10.8366i	−10.1763	−	10.8366i
$561$		0			0
$562$	2.29128	−	3.15368i	2.29128	−	3.15368i
$563$	−0.939097	−	0.516273i	−0.939097	−	0.516273i	−0.0627905	−	0.998027i	$-0.520000\pi$
$563$	−0.939097	−	0.516273i	−0.939097	−	0.516273i	−0.876307	+	0.481754i	$0.840000\pi$
$564$		0			0
$565$	1.22721	+	0.398743i	1.22721	+	0.398743i
$566$	0.152176	+	1.20460i	0.152176	+	1.20460i
$567$	−1.15475	−	1.58937i	−1.15475	−	1.58937i
$568$	−6.95156	−	1.32608i	−6.95156	−	1.32608i
$569$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$569$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$570$		0			0
$571$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$571$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$572$		0			0
$573$		0			0
$574$	−6.64261	+	2.15832i	−6.64261	+	2.15832i
$575$		0			0
$576$	−4.80606	−	1.90285i	−4.80606	−	1.90285i
$577$	−1.68532	−	1.06954i	−1.68532	−	1.06954i	−0.876307	−	0.481754i	$-0.840000\pi$
$577$	−1.68532	−	1.06954i	−1.68532	−	1.06954i	−0.809017	−	0.587785i	$-0.800000\pi$
$578$	−1.51373	+	1.25227i	−1.51373	+	1.25227i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.234638	+	3.72947i	0.234638	+	3.72947i
$587$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$587$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$588$		0			0
$589$	−0.331159	−	1.01920i	−0.331159	−	1.01920i
$590$	2.45537	−	0.630431i	2.45537	−	0.630431i
$591$		0			0
$592$		0			0
$593$	−0.866986	+	1.36615i	−0.866986	+	1.36615i	0.0627905	+	0.998027i	$0.480000\pi$
$593$	−0.866986	+	1.36615i	−0.866986	+	1.36615i	−0.929776	+	0.368125i	$0.880000\pi$
$594$		0			0
$595$		0			0
$596$		−	5.17480i		−	5.17480i
$597$		0			0
$598$		0			0
$599$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$599$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$600$		0			0
$601$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$601$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$602$		0			0
$603$	−0.183098	+	0.462452i	−0.183098	+	0.462452i
$604$		0			0
$605$	1.75261			1.75261
$606$		0			0
$607$	1.98423			1.98423			0.992115	−	0.125333i	$-0.0400000\pi$
$607$	1.98423			1.98423			0.992115	+	0.125333i	$0.0400000\pi$
$608$	5.16459	−	0.324928i	5.16459	−	0.324928i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.844328	−	0.535827i	$-0.180000\pi$
$613$		0			0		−0.844328	+	0.535827i	$0.820000\pi$
$614$	−3.12244	+	0.595638i	−3.12244	+	0.595638i
$615$		0			0
$616$		0			0
$617$	1.23879	−	1.31918i	1.23879	−	1.31918i	0.309017	−	0.951057i	$-0.400000\pi$
$617$	1.23879	−	1.31918i	1.23879	−	1.31918i	0.929776	−	0.368125i	$-0.120000\pi$
$618$		0			0
$619$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$619$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$620$	4.05454	−	2.94580i	4.05454	−	2.94580i
$621$		0			0
$622$	−2.23694	+	0.574348i	−2.23694	+	0.574348i
$623$		0			0
$624$		0			0
$625$	1.18176	+	0.303425i	1.18176	+	0.303425i
$626$		0			0
$627$		0			0
$628$	−3.19457	+	3.86157i	−3.19457	+	3.86157i
$629$		0			0
$630$	−1.26750	+	6.64449i	−1.26750	+	6.64449i
$631$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$631$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$632$		0			0
$633$		0			0
$634$	2.93235	−	2.42585i	2.93235	−	2.42585i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.824805	+	1.75280i	0.824805	+	1.75280i
$640$	3.43635	+	8.67922i	3.43635	+	8.67922i
$641$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$641$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$642$		0			0
$643$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$643$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$647$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$648$	0.908521	+	3.53846i	0.908521	+	3.53846i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−3.90727	−	4.16082i	−3.90727	−	4.16082i
$653$	−0.110048	+	0.0604991i	−0.110048	+	0.0604991i	−0.535827	−	0.844328i	$-0.680000\pi$
$653$	−0.110048	+	0.0604991i	−0.110048	+	0.0604991i	0.425779	+	0.904827i	$0.360000\pi$
$654$		0			0
$655$	−0.951325	+	2.02167i	−0.951325	+	2.02167i
$656$	7.79775	+	0.490593i	7.79775	+	0.490593i
$657$		0			0
$658$	−1.01563	+	2.15832i	−1.01563	+	2.15832i
$659$	0.250172	−	1.98031i	0.250172	−	1.98031i	0.0627905	−	0.998027i	$-0.480000\pi$
$659$	0.250172	−	1.98031i	0.250172	−	1.98031i	0.187381	−	0.982287i	$-0.440000\pi$
$660$		0			0
$661$	−0.340480	−	0.362574i	−0.340480	−	0.362574i	0.535827	−	0.844328i	$-0.320000\pi$
$661$	−0.340480	−	0.362574i	−0.340480	−	0.362574i	−0.876307	+	0.481754i	$0.840000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.917629	−	3.57393i	−0.917629	−	3.57393i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	1.59229	−	0.630431i	1.59229	−	0.630431i
$671$		0			0
$672$		0			0
$673$		0			0		−0.368125	−	0.929776i	$-0.620000\pi$
$673$		0			0		0.368125	+	0.929776i	$0.380000\pi$
$674$		0			0
$675$		0			0
$676$	−1.53223	−	2.41440i	−1.53223	−	2.41440i
$677$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$677$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$678$		0			0
$679$	−0.935537	+	0.773944i	−0.935537	+	0.773944i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.0235315	−	0.123357i	0.0235315	−	0.123357i	−0.968583	−	0.248690i	$-0.920000\pi$
$683$	0.0235315	−	0.123357i	0.0235315	−	0.123357i	0.992115	+	0.125333i	$0.0400000\pi$
$684$	−1.95335	−	2.36120i	−1.95335	−	2.36120i
$685$		0			0
$686$	1.34484	+	7.04992i	1.34484	+	7.04992i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.121636	+	0.0312307i	−0.121636	+	0.0312307i	−0.309017	−	0.951057i	$-0.600000\pi$
$691$	−0.121636	+	0.0312307i	−0.121636	+	0.0312307i	0.187381	+	0.982287i	$0.440000\pi$
$692$	−2.49298	−	1.17311i	−2.49298	−	1.17311i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	9.82641	−	6.23603i	9.82641	−	6.23603i
$701$	0.115808	−	0.356420i	0.115808	−	0.356420i	−0.876307	−	0.481754i	$-0.840000\pi$
$701$	0.115808	−	0.356420i	0.115808	−	0.356420i	0.992115	+	0.125333i	$0.0400000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.946441	−	1.72157i	−0.946441	−	1.72157i
$708$		0			0
$709$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$709$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$710$	2.45537	−	6.20155i	2.45537	−	6.20155i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	1.71447	−	1.82573i	1.71447	−	1.82573i
$719$		0			0		0.248690	−	0.968583i	$-0.420000\pi$
$719$		0			0		−0.248690	+	0.968583i	$0.580000\pi$
$720$	4.05454	−	6.38894i	4.05454	−	6.38894i
$721$	−0.790523	+	0.574348i	−0.790523	+	0.574348i
$722$	0.263869	+	0.124168i	0.263869	+	0.124168i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	1.00000			$0$
$727$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	−0.992115	+	0.125333i	$0.960000\pi$
$728$		0			0
$729$	0.637424	−	0.770513i	0.637424	−	0.770513i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.464173	+	0.844328i	−0.464173	+	0.844328i	0.535827	+	0.844328i	$0.320000\pi$
$733$	−0.464173	+	0.844328i	−0.464173	+	0.844328i	−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.90497	−	3.00175i	−1.90497	−	3.00175i
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$	3.11545	+	0.594303i	3.11545	+	0.594303i
$746$	1.68355	+	2.31721i	1.68355	+	2.31721i
$747$		0			0
$748$		0			0
$749$	−0.416046	−	1.62039i	−0.416046	−	1.62039i
$750$		0			0
$751$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$751$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$752$	1.94515	−	1.82662i	1.94515	−	1.82662i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$757$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$758$	−2.10119	−	0.132196i	−2.10119	−	0.132196i
$759$		0			0
$760$	−0.859971	+	6.80737i	−0.859971	+	6.80737i
$761$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$761$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$762$		0			0
$763$	1.68355	−	1.58096i	1.68355	−	1.58096i
$764$	−2.83830	+	3.90658i	−2.83830	+	3.90658i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$769$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$770$		0			0
$771$		0			0
$772$	4.59037	+	3.33510i	4.59037	+	3.33510i
$773$		0			0		−0.481754	−	0.876307i	$-0.660000\pi$
$773$		0			0		0.481754	+	0.876307i	$0.340000\pi$
$774$		0			0
$775$	0.882067	+	1.87449i	0.882067	+	1.87449i
$776$	2.14731	−	0.697705i	2.14731	−	0.697705i
$777$		0			0
$778$		0			0
$779$	1.63742	+	1.03914i	1.63742	+	1.03914i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.31343	−	12.1274i	2.31343	−	12.1274i
$785$	−1.95795	−	2.36675i	−1.95795	−	2.36675i
$786$		0			0
$787$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$787$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.446967	+	1.37562i	0.446967	+	1.37562i
$792$		0			0
$793$		0			0
$794$	−2.87622	+	2.08969i	−2.87622	+	2.08969i
$795$		0			0
$796$		0			0
$797$		0			0		0.684547	−	0.728969i	$-0.260000\pi$
$797$		0			0		−0.684547	+	0.728969i	$0.740000\pi$
$798$		0			0
$799$		0			0
$800$	−9.82641	+	1.87449i	−9.82641	+	1.87449i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.457871	+	3.62442i	0.457871	+	3.62442i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−3.43635	+	0.216197i	−3.43635	+	0.216197i
$811$	−0.723208	+	1.82662i	−0.723208	+	1.82662i	−0.187381	+	0.982287i	$0.560000\pi$
$811$	−0.723208	+	1.82662i	−0.723208	+	1.82662i	−0.535827	+	0.844328i	$0.680000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.95372	−	1.87449i	2.95372	−	1.87449i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−2.25547	+	8.78449i	−2.25547	+	8.78449i
$821$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$821$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$822$		0			0
$823$		0			0		−0.904827	−	0.425779i	$-0.860000\pi$
$823$		0			0		0.904827	+	0.425779i	$0.140000\pi$
$824$	1.75996	−	0.451880i	1.75996	−	0.451880i
$825$		0			0
$826$	2.18948	+	1.81130i	2.18948	+	1.81130i
$827$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$827$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$828$		0			0
$829$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$829$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−3.47443	−	1.37562i	−3.47443	−	1.37562i
$839$	−0.200808	−	0.316423i	−0.200808	−	0.316423i	0.728969	−	0.684547i	$-0.240000\pi$
$839$	−0.200808	−	0.316423i	−0.200808	−	0.316423i	−0.929776	+	0.368125i	$0.880000\pi$
$840$		0			0
$841$	0.425779	+	0.904827i	0.425779	+	0.904827i
$842$	0.921980	+	2.32866i	0.921980	+	2.32866i
$843$		0			0
$844$	−1.42978	−	1.03879i	−1.42978	−	1.03879i
$845$	1.62954	−	0.645180i	1.62954	−	0.645180i
$846$	−1.19267	−	0.227513i	−1.19267	−	0.227513i
$847$	1.15475	+	1.58937i	1.15475	+	1.58937i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.781202	−	0.733597i	0.781202	−	0.733597i	−0.187381	−	0.982287i	$-0.560000\pi$
$853$	0.781202	−	0.733597i	0.781202	−	0.733597i	0.968583	+	0.248690i	$0.0800000\pi$
$854$		0			0
$855$	1.64587	−	0.904827i	1.64587	−	0.904827i
$856$	−0.389904	+	3.08641i	−0.389904	+	3.08641i
$857$	0.688925	−	1.46404i	0.688925	−	1.46404i	−0.187381	−	0.982287i	$-0.560000\pi$
$857$	0.688925	−	1.46404i	0.688925	−	1.46404i	0.876307	−	0.481754i	$-0.160000\pi$
$858$		0			0
$859$		0			0		−0.998027	−	0.0627905i	$-0.980000\pi$
$859$		0			0		0.998027	+	0.0627905i	$0.0200000\pi$
$860$		0			0
$861$		0			0
$862$	−1.65875	+	0.911903i	−1.65875	+	0.911903i
$863$		0			0		−0.684547	−	0.728969i	$-0.740000\pi$
$863$		0			0		0.684547	+	0.728969i	$0.260000\pi$
$864$		0			0
$865$	0.992567	−	1.36615i	0.992567	−	1.36615i
$866$		0			0
$867$		0			0
$868$	5.34285	+	1.73600i	5.34285	+	1.73600i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−3.99305	+	1.58096i	−3.99305	+	1.58096i
$873$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$874$		0			0
$875$	1.35833	+	3.43074i	1.35833	+	3.43074i
$876$		0			0
$877$	1.30209	−	0.423073i	1.30209	−	0.423073i	0.425779	−	0.904827i	$-0.360000\pi$
$877$	1.30209	−	0.423073i	1.30209	−	0.423073i	0.876307	+	0.481754i	$0.160000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.770513	−	0.637424i	$-0.220000\pi$
$881$		0			0		−0.770513	+	0.637424i	$0.780000\pi$
$882$	−5.08314	+	2.39194i	−5.08314	+	2.39194i
$883$		0			0		0.481754	−	0.876307i	$-0.340000\pi$
$883$		0			0		−0.481754	+	0.876307i	$0.660000\pi$
$884$		0			0
$885$		0			0
$886$	1.92978	+	2.33270i	1.92978	+	2.33270i
$887$	0.542804	−	0.656137i	0.542804	−	0.656137i	−0.425779	−	0.904827i	$-0.640000\pi$
$887$	0.542804	−	0.656137i	0.542804	−	0.656137i	0.968583	+	0.248690i	$0.0800000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.641510	−	0.164712i	0.641510	−	0.164712i
$894$		0			0
$895$		0			0
$896$	−5.60672	+	8.83478i	−5.60672	+	8.83478i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	4.31841	+	4.05526i	4.31841	+	4.05526i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.168887	−	2.68438i	0.168887	−	2.68438i
$905$		0			0
$906$		0			0
$907$	−1.80608	+	0.113629i	−1.80608	+	0.113629i	−0.929776	−	0.368125i	$-0.880000\pi$
$907$	−1.80608	+	0.113629i	−1.80608	+	0.113629i	−0.876307	+	0.481754i	$0.840000\pi$
$908$	−5.31749			−5.31749
$909$	0.728969	−	0.684547i	0.728969	−	0.684547i
$910$		0			0
$911$		0			0		0.998027	−	0.0627905i	$-0.0200000\pi$
$911$		0			0		−0.998027	+	0.0627905i	$0.980000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.46017	+	0.469303i	−2.46017	+	0.469303i
$918$		0			0
$919$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$919$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.383238	−	0.317042i	−0.383238	−	0.317042i
$928$		0			0
$929$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$929$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$930$		0			0
$931$	1.95335	−	2.36120i	1.95335	−	2.36120i
$932$	1.12652	+	1.36173i	1.12652	+	1.36173i
$933$		0			0
$934$	−2.29128	−	0.289457i	−2.29128	−	0.289457i
$935$		0			0
$936$		0			0
$937$	−0.742395	+	0.614163i	−0.742395	+	0.614163i	−0.929776	−	0.368125i	$-0.880000\pi$
$937$	−0.742395	+	0.614163i	−0.742395	+	0.614163i	0.187381	+	0.982287i	$0.440000\pi$
$938$	1.62083	+	1.02861i	1.62083	+	1.02861i
$939$		0			0
$940$	1.65967	+	2.61522i	1.65967	+	2.61522i
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$	−1.53137	−	2.78556i	−1.53137	−	2.78556i
$945$		0			0
$946$		0			0
$947$		0			0		−0.982287	−	0.187381i	$-0.940000\pi$
$947$		0			0		0.982287	+	0.187381i	$0.0600000\pi$
$948$		0			0
$949$		0			0
$950$	−4.14807	−	1.34779i	−4.14807	−	1.34779i
$951$		0			0
$952$		0			0
$953$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$953$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$954$		0			0
$955$	−2.02596	−	2.15743i	−2.02596	−	2.15743i
$956$		0			0
$957$		0			0
$958$	1.28903	−	2.73933i	1.28903	−	2.73933i
$959$		0			0
$960$		0			0
$961$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$962$		0			0
$963$	0.746226	−	0.410241i	0.746226	−	0.410241i
$964$		0			0
$965$	−2.53505	+	2.38057i	−2.53505	+	2.38057i
$966$		0			0
$967$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$967$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$968$	−0.908521	−	3.53846i	−0.908521	−	3.53846i
$969$		0			0
$970$	0.266706	+	2.11120i	0.266706	+	2.11120i
$971$	1.17325	+	1.61484i	1.17325	+	1.61484i	0.637424	+	0.770513i	$0.280000\pi$
$971$	1.17325	+	1.61484i	1.17325	+	1.61484i	0.535827	+	0.844328i	$0.320000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.824805	+	1.75280i	0.824805	+	1.75280i	0.637424	+	0.770513i	$0.280000\pi$
$977$	0.824805	+	1.75280i	0.824805	+	1.75280i	0.187381	+	0.982287i	$0.440000\pi$
$978$		0			0
$979$		0			0
$980$	13.3248	+	5.27566i	13.3248	+	5.27566i
$981$	0.992567	+	0.629902i	0.992567	+	0.629902i
$982$		0			0
$983$		0			0		0.904827	−	0.425779i	$-0.140000\pi$
$983$		0			0		−0.904827	+	0.425779i	$0.860000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$991$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$992$	−3.72066	−	3.07799i	−3.72066	−	3.07799i
$993$		0			0
$994$	7.24171	−	1.85935i	7.24171	−	1.85935i
$995$		0			0
$996$		0			0
$997$	−0.331159	+	0.521823i	−0.331159	+	0.521823i	−0.968583	−	0.248690i	$-0.920000\pi$
$997$	−0.331159	+	0.521823i	−0.331159	+	0.521823i	0.637424	+	0.770513i	$0.280000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3131.1.de.a.278.1	✓	20
31.30	odd	2	CM	3131.1.de.a.278.1	✓	20
101.4	even	50	inner	3131.1.de.a.1115.1	yes	20
3131.1115	odd	50	inner	3131.1.de.a.1115.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3131.1.de.a.278.1	✓	20	1.1	even	1	trivial
3131.1.de.a.278.1	✓	20	31.30	odd	2	CM
3131.1.de.a.1115.1	yes	20	101.4	even	50	inner
3131.1.de.a.1115.1	yes	20	3131.1115	odd	50	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3131 = 31 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3131.de (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(1.96070 - 0.123357i) q^{2} +(2.83700 - 0.358397i) q^{4} +(-0.110048 + 1.74915i) q^{5} +(-1.65875 + 1.05267i) q^{7} +(3.58852 - 0.684547i) q^{8} +(-0.728969 - 0.684547i) q^{9} +O(q^{10})\)	\(q+(1.96070 - 0.123357i) q^{2} +(2.83700 - 0.358397i) q^{4} +(-0.110048 + 1.74915i) q^{5} +(-1.65875 + 1.05267i) q^{7} +(3.58852 - 0.684547i) q^{8} +(-0.728969 - 0.684547i) q^{9} +3.44314i q^{10} +(-3.12244 + 2.26859i) q^{14} +(4.18185 - 1.07372i) q^{16} +(-1.51373 - 1.25227i) q^{18} +(1.03799 + 0.266509i) q^{19} +(0.314687 + 5.00180i) q^{20} +(-2.05532 - 0.259647i) q^{25} +(-4.32859 + 3.58092i) q^{28} +(-0.535827 - 0.844328i) q^{31} +(4.59246 - 1.49218i) q^{32} +(-1.65875 - 3.01725i) q^{35} +(-2.31343 - 1.68080i) q^{36} +(2.06805 + 0.394502i) q^{38} +(0.802471 + 6.35221i) q^{40} +(1.72108 + 0.559214i) q^{41} +(1.27760 - 1.19975i) q^{45} +(0.541587 - 0.297740i) q^{47} +(1.21754 - 2.58740i) q^{49} +(-4.06189 - 0.255552i) q^{50} +(-5.23184 + 4.91302i) q^{56} +(-0.183098 - 0.713118i) q^{59} +(-1.15475 - 1.58937i) q^{62} +(1.92978 + 0.368125i) q^{63} +(4.80606 - 1.90285i) q^{64} +(-0.183098 - 0.462452i) q^{67} +(-3.62450 - 5.71129i) q^{70} +(-1.80113 - 0.713118i) q^{71} +(-3.08452 - 1.95750i) q^{72} +(3.04029 + 0.384077i) q^{76} +(1.41789 + 7.43286i) q^{80} +(0.0627905 + 0.998027i) q^{81} +(3.44351 + 0.884142i) q^{82} +(2.35699 - 2.50994i) q^{90} +(1.02516 - 0.650587i) q^{94} +(-0.580394 + 1.78627i) q^{95} +(0.613161 - 0.0774602i) q^{97} +(2.06805 - 5.22330i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{8}+O(q^{10}) \) 20 * q + 20 * q^8	\( 20 q + 20 q^{8} - 10 q^{14} + 5 q^{16} - 5 q^{19} + 5 q^{20} + 5 q^{28} - 5 q^{36} + 5 q^{38} - 5 q^{40} - 5 q^{45} + 5 q^{50} - 20 q^{56} + 5 q^{59} + 20 q^{63} + 20 q^{64} + 5 q^{67} - 5 q^{70} - 5 q^{71} - 20 q^{72} - 5 q^{76} + 20 q^{82} + 5 q^{98}+O(q^{100}) \) 20 * q + 20 * q^8 - 10 * q^14 + 5 * q^16 - 5 * q^19 + 5 * q^20 + 5 * q^28 - 5 * q^36 + 5 * q^38 - 5 * q^40 - 5 * q^45 + 5 * q^50 - 20 * q^56 + 5 * q^59 + 20 * q^63 + 20 * q^64 + 5 * q^67 - 5 * q^70 - 5 * q^71 - 20 * q^72 - 5 * q^76 + 20 * q^82 + 5 * q^98

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