LMFDB - Embedded newform 869.1.j.a.631.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [869,1,Mod(157,869)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(869, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8, 5]))

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

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

chi := DirichletCharacter("869.157");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 869 = 11 \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	869.j (of order $10$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.433687495978$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{10})$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

Embedding invariants

Embedding label			631.2
Root			$0.637424 + 0.770513i$ of defining polynomial
Character	$\chi$	$=$	869.631
Dual form			869.1.j.a.157.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866986 - 0.629902i) q^{2} +(0.0458709 + 0.141176i) q^{4} +(-1.41789 + 1.03016i) q^{5} +(-0.282001 + 0.867911i) q^{8} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})$	\(q+(-0.866986 - 0.629902i) q^{2} +(0.0458709 + 0.141176i) q^{4} +(-1.41789 + 1.03016i) q^{5} +(-0.282001 + 0.867911i) q^{8} +(-0.809017 - 0.587785i) q^{9} +1.87819 q^{10} +(0.535827 - 0.844328i) q^{11} +(1.03137 + 0.749337i) q^{13} +(0.911282 - 0.662085i) q^{16} +(0.331159 + 1.01920i) q^{18} +(0.450527 - 1.38658i) q^{19} +(-0.210474 - 0.152918i) q^{20} +(-0.996398 + 0.394502i) q^{22} +1.45794 q^{23} +(0.640176 - 1.97026i) q^{25} +(-0.422178 - 1.29933i) q^{26} +(1.50441 + 1.09302i) q^{31} -0.294542 q^{32} +(0.0458709 - 0.141176i) q^{36} +(-1.26401 + 0.918358i) q^{38} +(-0.494239 - 1.52111i) q^{40} +(0.143778 + 0.0369159i) q^{44} +1.75261 q^{45} +(-1.26401 - 0.918358i) q^{46} +(-0.809017 + 0.587785i) q^{49} +(-1.79609 + 1.30494i) q^{50} +(-0.0584785 + 0.179978i) q^{52} +(0.110048 + 1.74915i) q^{55} +(-0.615808 - 1.89526i) q^{62} +(-0.655918 - 0.476553i) q^{64} -2.23432 q^{65} -0.374763 q^{67} +(0.738289 - 0.536399i) q^{72} +(-0.263146 - 0.809880i) q^{73} +0.216418 q^{76} +(-0.809017 - 0.587785i) q^{79} +(-0.610048 + 1.87753i) q^{80} +(0.309017 + 0.951057i) q^{81} +(1.30902 - 0.951057i) q^{83} +(0.581698 + 0.703152i) q^{88} +1.93717 q^{89} +(-1.51949 - 1.10397i) q^{90} +(0.0668769 + 0.205826i) q^{92} +(0.789600 + 2.43014i) q^{95} +(1.60528 + 1.16630i) q^{97} +1.07165 q^{98} +(-0.929776 + 0.368125i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) $ 20 * q - 5 * q^4 - 5 * q^9	$ 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} + 15 q^{20} - 5 q^{22} - 5 q^{25} + 15 q^{26} - 10 q^{32} - 5 q^{36} - 10 q^{40} - 5 q^{49} - 10 q^{50} - 10 q^{62} - 5 q^{64} - 10 q^{76} - 5 q^{79} - 10 q^{80} - 5 q^{81} + 15 q^{83} - 5 q^{88} + 15 q^{92} + 15 q^{95}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 + 15 * q^20 - 5 * q^22 - 5 * q^25 + 15 * q^26 - 10 * q^32 - 5 * q^36 - 10 * q^40 - 5 * q^49 - 10 * q^50 - 10 * q^62 - 5 * q^64 - 10 * q^76 - 5 * q^79 - 10 * q^80 - 5 * q^81 + 15 * q^83 - 5 * q^88 + 15 * q^92 + 15 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/869\mathbb{Z}\right)^\times$.

$n$	$475$	$793$
$\chi(n)$	$e\left(\frac{1}{5}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$2$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$3$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$3$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$4$	0.0458709	+	0.141176i	0.0458709	+	0.141176i
$5$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$5$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$6$		0			0
$7$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$7$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$8$	−0.282001	+	0.867911i	−0.282001	+	0.867911i
$9$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$10$	1.87819			1.87819
$11$	0.535827	−	0.844328i	0.535827	−	0.844328i
$11$	0.535827	−	0.844328i	0.535827	−	0.844328i
$12$		0			0
$13$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$13$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$14$		0			0
$15$		0			0
$16$	0.911282	−	0.662085i	0.911282	−	0.662085i
$17$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$17$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$18$	0.331159	+	1.01920i	0.331159	+	1.01920i
$19$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$19$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$20$	−0.210474	−	0.152918i	−0.210474	−	0.152918i
$21$		0			0
$22$	−0.996398	+	0.394502i	−0.996398	+	0.394502i
$23$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$23$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$24$		0			0
$25$	0.640176	−	1.97026i	0.640176	−	1.97026i
$26$	−0.422178	−	1.29933i	−0.422178	−	1.29933i
$27$		0			0
$28$		0			0
$29$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$29$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$30$		0			0
$31$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$31$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$32$	−0.294542			−0.294542
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.0458709	−	0.141176i	0.0458709	−	0.141176i
$37$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$37$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$38$	−1.26401	+	0.918358i	−1.26401	+	0.918358i
$39$		0			0
$40$	−0.494239	−	1.52111i	−0.494239	−	1.52111i
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.143778	+	0.0369159i	0.143778	+	0.0369159i
$45$	1.75261			1.75261
$46$	−1.26401	−	0.918358i	−1.26401	−	0.918358i
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$		0			0
$49$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$50$	−1.79609	+	1.30494i	−1.79609	+	1.30494i
$51$		0			0
$52$	−0.0584785	+	0.179978i	−0.0584785	+	0.179978i
$53$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$53$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$54$		0			0
$55$	0.110048	+	1.74915i	0.110048	+	1.74915i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$61$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$62$	−0.615808	−	1.89526i	−0.615808	−	1.89526i
$63$		0			0
$64$	−0.655918	−	0.476553i	−0.655918	−	0.476553i
$65$	−2.23432			−2.23432
$66$		0			0
$67$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$67$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$	0.738289	−	0.536399i	0.738289	−	0.536399i
$73$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$73$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$74$		0			0
$75$		0			0
$76$	0.216418			0.216418
$77$		0			0
$78$		0			0
$79$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$79$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$80$	−0.610048	+	1.87753i	−0.610048	+	1.87753i
$81$	0.309017	+	0.951057i	0.309017	+	0.951057i
$82$		0			0
$83$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$83$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.581698	+	0.703152i	0.581698	+	0.703152i
$89$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$89$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$90$	−1.51949	−	1.10397i	−1.51949	−	1.10397i
$91$		0			0
$92$	0.0668769	+	0.205826i	0.0668769	+	0.205826i
$93$		0			0
$94$		0			0
$95$	0.789600	+	2.43014i	0.789600	+	2.43014i
$96$		0			0
$97$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$97$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$98$	1.07165			1.07165
$99$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$100$	0.307519			0.307519
$101$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$101$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$102$		0			0
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$	−0.941207	+	0.683827i	−0.941207	+	0.683827i
$105$		0			0
$106$		0			0
$107$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$107$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	1.00639	−	1.58581i	1.00639	−	1.58581i
$111$		0			0
$112$		0			0
$113$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$113$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$114$		0			0
$115$	−2.06720	+	1.50191i	−2.06720	+	1.50191i
$116$		0			0
$117$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$122$		0			0
$123$		0			0
$124$	−0.0852994	+	0.262525i	−0.0852994	+	0.262525i
$125$	0.580394	+	1.78627i	0.580394	+	1.78627i
$126$		0			0
$127$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$127$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$128$	0.359509	+	1.10645i	0.359509	+	1.10645i
$129$		0			0
$130$	1.93712	+	1.40740i	1.93712	+	1.40740i
$131$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$131$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$132$		0			0
$133$		0			0
$134$	0.324914	+	0.236064i	0.324914	+	0.236064i
$135$		0			0
$136$		0			0
$137$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$137$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$138$		0			0
$139$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$139$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.18532	−	0.469303i	1.18532	−	0.469303i
$144$	−1.12641			−1.12641
$145$		0			0
$146$	−0.282001	+	0.867911i	−0.282001	+	0.867911i
$147$		0			0
$148$		0			0
$149$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$149$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$151$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$152$	1.07638	+	0.782036i	1.07638	+	0.782036i
$153$		0			0
$154$		0			0
$155$	−3.25908			−3.25908
$156$		0			0
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$	0.331159	+	1.01920i	0.331159	+	1.01920i
$159$		0			0
$160$	0.417629	−	0.303425i	0.417629	−	0.303425i
$161$		0			0
$162$	0.331159	−	1.01920i	0.331159	−	1.01920i
$163$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$163$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$164$		0			0
$165$		0			0
$166$	−1.73397			−1.73397
$167$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$167$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$168$		0			0
$169$	0.193209	+	0.594636i	0.193209	+	0.594636i
$170$		0			0
$171$	−1.17950	+	0.856954i	−1.17950	+	0.856954i
$172$		0			0
$173$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$173$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$174$		0			0
$175$		0			0
$176$	−0.0707277	−	1.12418i	−0.0707277	−	1.12418i
$177$		0			0
$178$	−1.67950	−	1.22023i	−1.67950	−	1.22023i
$179$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$179$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$180$	0.0803940	+	0.247427i	0.0803940	+	0.247427i
$181$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$181$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$182$		0			0
$183$		0			0
$184$	−0.411140	+	1.26536i	−0.411140	+	1.26536i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.846178	−	2.60427i	0.846178	−	2.60427i
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$		0			0
$193$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$193$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$194$	−0.657096	−	2.02233i	−0.657096	−	2.02233i
$195$		0			0
$196$	−0.120092	−	0.0872517i	−0.120092	−	0.0872517i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	1.03799	+	0.266509i	1.03799	+	0.266509i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	1.52948	+	1.11123i	1.52948	+	1.11123i
$201$		0			0
$202$	0.641510	+	1.97437i	0.641510	+	1.97437i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.17950	−	0.856954i	−1.17950	−	0.856954i
$208$	1.43600			1.43600
$209$	−0.929324	−	1.12336i	−0.929324	−	1.12336i
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.241891	+	0.0957714i	−0.241891	+	0.0957714i
$221$		0			0
$222$		0			0
$223$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$223$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	−1.67600	+	1.21769i	−1.67600	+	1.21769i
$226$		0			0
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$		0			0
$229$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$229$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$230$	2.73829			2.73829
$231$		0			0
$232$		0			0
$233$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$233$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$234$	−0.422178	+	1.29933i	−0.422178	+	1.29933i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$239$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$240$		0			0
$241$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$241$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$242$	−0.200808	+	1.05267i	−0.200808	+	1.05267i
$243$		0			0
$244$		0			0
$245$	0.541587	−	1.66683i	0.541587	−	1.66683i
$246$		0			0
$247$	1.50368	−	1.09249i	1.50368	−	1.09249i
$248$	−1.37289	+	0.997462i	−1.37289	+	0.997462i
$249$		0			0
$250$	0.621981	−	1.91426i	0.621981	−	1.91426i
$251$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$251$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$252$		0			0
$253$	0.781202	−	1.23098i	0.781202	−	1.23098i
$254$		0			0
$255$		0			0
$256$	0.134731	−	0.414658i	0.134731	−	0.414658i
$257$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$257$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$258$		0			0
$259$		0			0
$260$	−0.102490	−	0.315432i	−0.102490	−	0.315432i
$261$		0			0
$262$	0.738289	+	0.536399i	0.738289	+	0.536399i
$263$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$263$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.0171907	−	0.0529076i	−0.0171907	−	0.0529076i
$269$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$269$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$270$		0			0
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.32052	−	1.59624i	−1.32052	−	1.59624i
$276$		0			0
$277$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$277$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$278$		0			0
$279$	−0.574633	−	1.76854i	−0.574633	−	1.76854i
$280$		0			0
$281$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$281$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$282$		0			0
$283$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$283$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$284$		0			0
$285$		0			0
$286$	−1.32327	−	0.339759i	−1.32327	−	0.339759i
$287$		0			0
$288$	0.238289	+	0.173127i	0.238289	+	0.173127i
$289$	0.309017	−	0.951057i	0.309017	−	0.951057i
$290$		0			0
$291$		0			0
$292$	0.102265	−	0.0742999i	0.102265	−	0.0742999i
$293$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$293$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.50368	+	1.09249i	1.50368	+	1.09249i
$300$		0			0
$301$		0			0
$302$	0.324914	−	0.236064i	0.324914	−	0.236064i
$303$		0			0
$304$	−0.507477	−	1.56185i	−0.507477	−	1.56185i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	2.82557	+	2.05290i	2.82557	+	2.05290i
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$313$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$314$		0			0
$315$		0			0
$316$	0.0458709	−	0.141176i	0.0458709	−	0.141176i
$317$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$317$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$318$		0			0
$319$		0			0
$320$	1.42095			1.42095
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.120092	+	0.0872517i	−0.120092	+	0.0872517i
$325$	2.13665	−	1.55237i	2.13665	−	1.55237i
$326$	0.0415873	+	0.127993i	0.0415873	+	0.127993i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	0.194312	+	0.141176i	0.194312	+	0.141176i
$333$		0			0
$334$	−0.615808	−	1.89526i	−0.615808	−	1.89526i
$335$	0.531374	−	0.386066i	0.531374	−	0.386066i
$336$		0			0
$337$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$337$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$338$	0.207053	−	0.637244i	0.207053	−	0.637244i
$339$		0			0
$340$		0			0
$341$	1.72897	−	0.684547i	1.72897	−	0.684547i
$342$	1.56240			1.56240
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$347$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$348$		0			0
$349$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$349$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$350$		0			0
$351$		0			0
$352$	−0.157823	+	0.248690i	−0.157823	+	0.248690i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	0.0888596	+	0.273482i	0.0888596	+	0.273482i
$357$		0			0
$358$	1.40281	−	1.01920i	1.40281	−	1.01920i
$359$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$359$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$360$	−0.494239	+	1.52111i	−0.494239	+	1.52111i
$361$	−0.910614	−	0.661600i	−0.910614	−	0.661600i
$362$	−1.36620			−1.36620
$363$		0			0
$364$		0			0
$365$	1.20742	+	0.877242i	1.20742	+	0.877242i
$366$		0			0
$367$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$367$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$368$	1.32859	−	0.965279i	1.32859	−	0.965279i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$379$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$380$	−0.306858	+	0.222946i	−0.306858	+	0.222946i
$381$		0			0
$382$		0			0
$383$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$383$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.0910184	+	0.280126i	−0.0910184	+	0.280126i
$389$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$389$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$390$		0			0
$391$		0			0
$392$	−0.282001	−	0.867911i	−0.282001	−	0.867911i
$393$		0			0
$394$		0			0
$395$	1.75261			1.75261
$396$	−0.0946201	−	0.114376i	−0.0946201	−	0.114376i
$397$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$397$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$398$		0			0
$399$		0			0
$400$	−0.721099	−	2.21931i	−0.721099	−	2.21931i
$401$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$401$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$	0.732570	+	2.25462i	0.732570	+	2.25462i
$404$	0.0888596	−	0.273482i	0.0888596	−	0.273482i
$405$	−1.41789	−	1.03016i	−1.41789	−	1.03016i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$409$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.482809	+	1.48593i	0.482809	+	1.48593i
$415$	−0.876307	+	2.69699i	−0.876307	+	2.69699i
$416$	−0.303783	−	0.220711i	−0.303783	−	0.220711i
$417$		0			0
$418$	0.0981041	+	1.55932i	0.0981041	+	1.55932i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$421$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$431$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$432$		0			0
$433$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$433$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.656841	−	2.02155i	0.656841	−	2.02155i
$438$		0			0
$439$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$439$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$440$	−1.54914	−	0.397753i	−1.54914	−	0.397753i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$443$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$444$		0			0
$445$	−2.74670	+	1.99559i	−2.74670	+	1.99559i
$446$	1.40281	−	1.01920i	1.40281	−	1.01920i
$447$		0			0
$448$		0			0
$449$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$449$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$450$	2.22010			2.22010
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$457$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$458$		0			0
$459$		0			0
$460$	−0.306858	−	0.222946i	−0.306858	−	0.222946i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$467$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$468$	0.153099	−	0.111233i	0.153099	−	0.111233i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.44351	−	1.77531i	−2.44351	−	1.77531i
$476$		0			0
$477$		0			0
$478$	−1.51949	+	1.10397i	−1.51949	+	1.10397i
$479$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$479$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$480$		0			0
$481$		0			0
$482$	−0.108877	−	0.0791038i	−0.108877	−	0.0791038i
$483$		0			0
$484$	0.108209	−	0.101615i	0.108209	−	0.101615i
$485$	−3.47759			−3.47759
$486$		0			0
$487$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$487$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$488$		0			0
$489$		0			0
$490$	−1.51949	+	1.10397i	−1.51949	+	1.10397i
$491$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$491$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$		0			0
$494$	−1.99183			−1.99183
$495$	0.939097	−	1.47978i	0.939097	−	1.47978i
$496$	2.09461			2.09461
$497$		0			0
$498$		0			0
$499$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$499$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$500$	−0.225555	+	0.163876i	−0.225555	+	0.163876i
$501$		0			0
$502$		0			0
$503$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$503$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$504$		0			0
$505$	3.39510			3.39510
$506$	−1.45269	+	0.575159i	−1.45269	+	0.575159i
$507$		0			0
$508$		0			0
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$		0			0
$512$	0.563203	−	0.409191i	0.563203	−	0.409191i
$513$		0			0
$514$	−0.422178	+	1.29933i	−0.422178	+	1.29933i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.630080	−	1.93919i	0.630080	−	1.93919i
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$		0			0
$523$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$523$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$524$	−0.0390618	−	0.120220i	−0.0390618	−	0.120220i
$525$		0			0
$526$	−0.108877	−	0.0791038i	−0.108877	−	0.0791038i
$527$		0			0
$528$		0			0
$529$	1.12558			1.12558
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.105684	−	0.325261i	0.105684	−	0.325261i
$537$		0			0
$538$	−0.401616			−0.401616
$539$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$540$		0			0
$541$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$541$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$547$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$548$		0			0
$549$		0			0
$550$	0.139401	+	2.21571i	0.139401	+	2.21571i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.580394	+	1.78627i	0.580394	+	1.78627i
$555$		0			0
$556$		0			0
$557$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$557$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$558$	−0.615808	+	1.89526i	−0.615808	+	1.89526i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.87819			1.87819
$563$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$563$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$564$		0			0
$565$		0			0
$566$	1.61221	−	1.17134i	1.61221	−	1.17134i
$567$		0			0
$568$		0			0
$569$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$569$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$570$		0			0
$571$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$571$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$572$	0.120626	+	0.145812i	0.120626	+	0.145812i
$573$		0			0
$574$		0			0
$575$	0.933337	−	2.87251i	0.933337	−	2.87251i
$576$	0.250539	+	0.771078i	0.250539	+	0.771078i
$577$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$577$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$578$	−0.866986	+	0.629902i	−0.866986	+	0.629902i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.777112			0.777112
$585$	1.80760	+	1.31330i	1.80760	+	1.31330i
$586$		0			0
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$		0			0
$589$	2.19334	−	1.59355i	2.19334	−	1.59355i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$593$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−0.615508	−	1.89434i	−0.615508	−	1.89434i
$599$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$599$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$600$		0			0
$601$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$601$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$	0.303189	+	0.220280i	0.303189	+	0.220280i
$604$	−0.0556303			−0.0556303
$605$	1.53583	+	0.844328i	1.53583	+	0.844328i
$606$		0			0
$607$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$607$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$608$	−0.132699	+	0.408406i	−0.132699	+	0.408406i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$613$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$617$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$618$		0			0
$619$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$619$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$620$	−0.149497	−	0.460104i	−0.149497	−	0.460104i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.987078	−	0.717154i	−0.987078	−	0.717154i
$626$	−1.99280			−1.99280
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$631$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$632$	0.738289	−	0.536399i	0.738289	−	0.536399i
$633$		0			0
$634$	0.204668	+	0.629902i	0.204668	+	0.629902i
$635$		0			0
$636$		0			0
$637$	−1.27485			−1.27485
$638$		0			0
$639$		0			0
$640$	−1.64957	−	1.19848i	−1.64957	−	1.19848i
$641$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$641$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$642$		0			0
$643$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$643$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$	−0.912576			−0.912576
$649$		0			0
$650$	−2.83028			−2.83028
$651$		0			0
$652$	0.00576052	−	0.0177291i	0.00576052	−	0.0177291i
$653$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$653$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$654$		0			0
$655$	1.20742	−	0.877242i	1.20742	−	0.877242i
$656$		0			0
$657$	−0.263146	+	0.809880i	−0.263146	+	0.809880i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	0.456288	+	1.40431i	0.456288	+	1.40431i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.0852994	+	0.262525i	−0.0852994	+	0.262525i
$669$		0			0
$670$	−0.703877			−0.703877
$671$		0			0
$672$		0			0
$673$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$673$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$674$	−0.657096	+	2.02233i	−0.657096	+	2.02233i
$675$		0			0
$676$	−0.0750858	+	0.0545530i	−0.0750858	+	0.0545530i
$677$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$677$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−1.93019	−	0.495588i	−1.93019	−	0.495588i
$683$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$683$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$684$	−0.175086	−	0.127207i	−0.175086	−	0.127207i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$691$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$692$		0			0
$693$		0			0
$694$	−0.401616			−0.401616
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$701$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$702$		0			0
$703$		0			0
$704$	−0.753825	+	0.298461i	−0.753825	+	0.298461i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$709$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$710$		0			0
$711$	0.309017	+	0.951057i	0.309017	+	0.951057i
$712$	−0.546284	+	1.68129i	−0.546284	+	1.68129i
$713$	2.19334	+	1.59355i	2.19334	+	1.59355i
$714$		0			0
$715$	−1.19721	+	1.88650i	−1.19721	+	1.88650i
$716$	−0.240183			−0.240183
$717$		0			0
$718$		0			0
$719$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$719$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$720$	1.59713	−	1.16038i	1.59713	−	1.16038i
$721$		0			0
$722$	0.372746	+	1.14720i	0.372746	+	1.14720i
$723$		0			0
$724$	0.153099	+	0.111233i	0.153099	+	0.111233i
$725$		0			0
$726$		0			0
$727$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$727$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$728$		0			0
$729$	0.309017	−	0.951057i	0.309017	−	0.951057i
$730$	−0.494239	−	1.52111i	−0.494239	−	1.52111i
$731$		0			0
$732$		0			0
$733$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$733$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$734$	0.641510	−	1.97437i	0.641510	−	1.97437i
$735$		0			0
$736$	−0.429423			−0.429423
$737$	−0.200808	+	0.316423i	−0.200808	+	0.316423i
$738$		0			0
$739$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$739$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$743$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.61803			−1.61803
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$751$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.202967	−	0.624667i	−0.202967	−	0.624667i
$756$		0			0
$757$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$757$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$		0			0
$760$	−2.33181			−2.33181
$761$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$761$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.354888	+	1.09223i	0.354888	+	1.09223i
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$773$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$774$		0			0
$775$	3.11662	−	2.26435i	3.11662	−	2.26435i
$776$	−1.46494	+	1.06434i	−1.46494	+	1.06434i
$777$		0			0
$778$	−0.615808	+	1.89526i	−0.615808	+	1.89526i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.348079	+	1.07128i	−0.348079	+	1.07128i
$785$		0			0
$786$		0			0
$787$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$787$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$788$		0			0
$789$		0			0
$790$	−1.51949	−	1.10397i	−1.51949	−	1.10397i
$791$		0			0
$792$	−0.0573011	−	0.910775i	−0.0573011	−	0.910775i
$793$		0			0
$794$	1.40281	+	1.01920i	1.40281	+	1.01920i
$795$		0			0
$796$		0			0
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	−0.188559	+	0.580324i	−0.188559	+	0.580324i
$801$	−1.56720	−	1.13864i	−1.56720	−	1.13864i
$802$		0			0
$803$	−0.824805	−	0.211774i	−0.824805	−	0.211774i
$804$		0			0
$805$		0			0
$806$	0.785062	−	2.41617i	0.785062	−	2.41617i
$807$		0			0
$808$	1.43019	−	1.03909i	1.43019	−	1.03909i
$809$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$809$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$810$	0.580394	+	1.78627i	0.580394	+	1.78627i
$811$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$811$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.220095			0.220095
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$821$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$822$		0			0
$823$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$823$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$827$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$828$	0.0668769	−	0.205826i	0.0668769	−	0.205826i
$829$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$829$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$830$	2.45859	−	1.78627i	2.45859	−	1.78627i
$831$		0			0
$832$	−0.319399	−	0.983008i	−0.319399	−	0.983008i
$833$		0			0
$834$		0			0
$835$	−3.25908			−3.25908
$836$	0.115963	−	0.182728i	0.115963	−	0.182728i
$837$		0			0
$838$		0			0
$839$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$839$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$840$		0			0
$841$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$842$	1.72030	−	1.24987i	1.72030	−	1.24987i
$843$		0			0
$844$		0			0
$845$	−0.886520	−	0.644095i	−0.886520	−	0.644095i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$	0.789600	−	2.43014i	0.789600	−	2.43014i
$856$		0			0
$857$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$857$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−0.282001	−	0.867911i	−0.282001	−	0.867911i
$863$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$863$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$864$		0			0
$865$		0			0
$866$	0.482809	−	1.48593i	0.482809	−	1.48593i
$867$		0			0
$868$		0			0
$869$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$870$		0			0
$871$	−0.386520	−	0.280823i	−0.386520	−	0.280823i
$872$		0			0
$873$	−0.613161	−	1.88711i	−0.613161	−	1.88711i
$874$	−1.84285	+	1.33891i	−1.84285	+	1.33891i
$875$		0			0
$876$		0			0
$877$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$877$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$878$	1.72030	+	1.24987i	1.72030	+	1.24987i
$879$		0			0
$880$	1.25837	+	1.52111i	1.25837	+	1.52111i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.866986	−	0.629902i	−0.866986	−	0.629902i
$883$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$883$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$887$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$888$		0			0
$889$		0			0
$890$	3.63837			3.63837
$891$	0.968583	+	0.248690i	0.968583	+	0.248690i
$892$	−0.240183			−0.240183
$893$		0			0
$894$		0			0
$895$	−0.876307	−	2.69699i	−0.876307	−	2.69699i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.248788	−	0.180755i	−0.248788	−	0.180755i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.690441	+	2.12496i	−0.690441	+	2.12496i
$906$		0			0
$907$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$907$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$908$		0			0
$909$	0.598617	+	1.84235i	0.598617	+	1.84235i
$910$		0			0
$911$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$911$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$912$		0			0
$913$	−0.101597	−	1.61484i	−0.101597	−	1.61484i
$914$	2.07597			2.07597
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$919$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$920$	−0.720570	−	2.21769i	−0.720570	−	2.21769i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$	0.450527	+	1.38658i	0.450527	+	1.38658i
$932$		0			0
$933$		0			0
$934$	1.14844			1.14844
$935$		0			0
$936$	1.16340			1.16340
$937$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$937$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$941$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	0.335471	−	1.03247i	0.335471	−	1.03247i
$950$	1.00021	+	3.07834i	1.00021	+	3.07834i
$951$		0			0
$952$		0			0
$953$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$953$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$954$		0			0
$955$		0			0
$956$	0.260160			0.260160
$957$		0			0
$958$	−1.73397			−1.73397
$959$		0			0
$960$		0			0
$961$	0.759544	+	2.33764i	0.759544	+	2.33764i
$962$		0			0
$963$		0			0
$964$	0.00576052	+	0.0177291i	0.00576052	+	0.0177291i
$965$		0			0
$966$		0			0
$967$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$967$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$968$	0.905380	−	0.114376i	0.905380	−	0.114376i
$969$		0			0
$970$	3.01502	+	2.19054i	3.01502	+	2.19054i
$971$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$971$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$972$		0			0
$973$		0			0
$974$	1.10528	−	0.803030i	1.10528	−	0.803030i
$975$		0			0
$976$		0			0
$977$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$977$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$978$		0			0
$979$	1.03799	−	1.63560i	1.03799	−	1.63560i
$980$	0.260160			0.260160
$981$		0			0
$982$		0			0
$983$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$983$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.223208	+	0.162170i	0.223208	+	0.162170i
$989$		0			0
$990$	−1.74630	+	0.691409i	−1.74630	+	0.691409i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.443112	−	0.321939i	−0.443112	−	0.321939i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$997$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$998$	0.354888	−	1.09223i	0.354888	−	1.09223i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.1.j.a.631.2	yes	20
11.3	even	5	inner	869.1.j.a.157.2	✓	20
79.78	odd	2	CM	869.1.j.a.631.2	yes	20
869.157	odd	10	inner	869.1.j.a.157.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.1.j.a.157.2	✓	20	11.3	even	5	inner
869.1.j.a.157.2	✓	20	869.157	odd	10	inner
869.1.j.a.631.2	yes	20	1.1	even	1	trivial
869.1.j.a.631.2	yes	20	79.78	odd	2	CM

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:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	869.j (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866986 - 0.629902i) q^{2} +(0.0458709 + 0.141176i) q^{4} +(-1.41789 + 1.03016i) q^{5} +(-0.282001 + 0.867911i) q^{8} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})\)	\(q+(-0.866986 - 0.629902i) q^{2} +(0.0458709 + 0.141176i) q^{4} +(-1.41789 + 1.03016i) q^{5} +(-0.282001 + 0.867911i) q^{8} +(-0.809017 - 0.587785i) q^{9} +1.87819 q^{10} +(0.535827 - 0.844328i) q^{11} +(1.03137 + 0.749337i) q^{13} +(0.911282 - 0.662085i) q^{16} +(0.331159 + 1.01920i) q^{18} +(0.450527 - 1.38658i) q^{19} +(-0.210474 - 0.152918i) q^{20} +(-0.996398 + 0.394502i) q^{22} +1.45794 q^{23} +(0.640176 - 1.97026i) q^{25} +(-0.422178 - 1.29933i) q^{26} +(1.50441 + 1.09302i) q^{31} -0.294542 q^{32} +(0.0458709 - 0.141176i) q^{36} +(-1.26401 + 0.918358i) q^{38} +(-0.494239 - 1.52111i) q^{40} +(0.143778 + 0.0369159i) q^{44} +1.75261 q^{45} +(-1.26401 - 0.918358i) q^{46} +(-0.809017 + 0.587785i) q^{49} +(-1.79609 + 1.30494i) q^{50} +(-0.0584785 + 0.179978i) q^{52} +(0.110048 + 1.74915i) q^{55} +(-0.615808 - 1.89526i) q^{62} +(-0.655918 - 0.476553i) q^{64} -2.23432 q^{65} -0.374763 q^{67} +(0.738289 - 0.536399i) q^{72} +(-0.263146 - 0.809880i) q^{73} +0.216418 q^{76} +(-0.809017 - 0.587785i) q^{79} +(-0.610048 + 1.87753i) q^{80} +(0.309017 + 0.951057i) q^{81} +(1.30902 - 0.951057i) q^{83} +(0.581698 + 0.703152i) q^{88} +1.93717 q^{89} +(-1.51949 - 1.10397i) q^{90} +(0.0668769 + 0.205826i) q^{92} +(0.789600 + 2.43014i) q^{95} +(1.60528 + 1.16630i) q^{97} +1.07165 q^{98} +(-0.929776 + 0.368125i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 20 * q - 5 * q^4 - 5 * q^9	\( 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} + 15 q^{20} - 5 q^{22} - 5 q^{25} + 15 q^{26} - 10 q^{32} - 5 q^{36} - 10 q^{40} - 5 q^{49} - 10 q^{50} - 10 q^{62} - 5 q^{64} - 10 q^{76} - 5 q^{79} - 10 q^{80} - 5 q^{81} + 15 q^{83} - 5 q^{88} + 15 q^{92} + 15 q^{95}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 + 15 * q^20 - 5 * q^22 - 5 * q^25 + 15 * q^26 - 10 * q^32 - 5 * q^36 - 10 * q^40 - 5 * q^49 - 10 * q^50 - 10 * q^62 - 5 * q^64 - 10 * q^76 - 5 * q^79 - 10 * q^80 - 5 * q^81 + 15 * q^83 - 5 * q^88 + 15 * q^92 + 15 * q^95

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