LMFDB - Embedded newform 869.1.j.a.157.4

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			157.4
Root			$0.929776 + 0.368125i$ of defining polynomial
Character	$\chi$	$=$	869.157
Dual form			869.1.j.a.631.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.03137 - 0.749337i) q^{2} +(0.193209 - 0.594636i) q^{4} +(0.688925 + 0.500534i) q^{5} +(0.147638 + 0.454382i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})$	\(q+(1.03137 - 0.749337i) q^{2} +(0.193209 - 0.594636i) q^{4} +(0.688925 + 0.500534i) q^{5} +(0.147638 + 0.454382i) q^{8} +(-0.809017 + 0.587785i) q^{9} +1.08561 q^{10} +(-0.637424 + 0.770513i) q^{11} +(1.50441 - 1.09302i) q^{13} +(0.998582 + 0.725513i) q^{16} +(-0.393950 + 1.21245i) q^{18} +(-0.613161 - 1.88711i) q^{19} +(0.430742 - 0.312952i) q^{20} +(-0.0800484 + 1.27233i) q^{22} -1.98423 q^{23} +(-0.0849327 - 0.261396i) q^{25} +(0.732570 - 2.25462i) q^{26} +(-0.101597 + 0.0738147i) q^{31} +1.09580 q^{32} +(0.193209 + 0.594636i) q^{36} +(-2.04648 - 1.48686i) q^{38} +(-0.125722 + 0.386933i) q^{40} +(0.335019 + 0.527905i) q^{44} -0.851559 q^{45} +(-2.04648 + 1.48686i) q^{46} +(-0.809017 - 0.587785i) q^{49} +(-0.283471 - 0.205954i) q^{50} +(-0.359282 - 1.10576i) q^{52} +(-0.824805 + 0.211774i) q^{55} +(-0.0494726 + 0.152261i) q^{62} +(0.131596 - 0.0956103i) q^{64} +1.58352 q^{65} +1.45794 q^{67} +(-0.386520 - 0.280823i) q^{72} +(-0.115808 + 0.356420i) q^{73} -1.24061 q^{76} +(-0.809017 + 0.587785i) q^{79} +(0.324805 + 0.999648i) q^{80} +(0.309017 - 0.951057i) q^{81} +(1.30902 + 0.951057i) q^{83} +(-0.444215 - 0.175877i) q^{88} +1.07165 q^{89} +(-0.878275 + 0.638104i) q^{90} +(-0.383371 + 1.17989i) q^{92} +(0.522142 - 1.60699i) q^{95} +(-1.41789 + 1.03016i) q^{97} -1.27485 q^{98} +(0.0627905 - 0.998027i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/869\mathbb{Z}\right)^\times$.

$n$	$475$	$793$
$\chi(n)$	$e\left(\frac{4}{5}\right)$	$-1$

Coefficient data

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

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

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

$2$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$2$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$3$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$3$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$4$	0.193209	−	0.594636i	0.193209	−	0.594636i
$5$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$5$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$6$		0			0
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$	0.147638	+	0.454382i	0.147638	+	0.454382i
$9$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$10$	1.08561			1.08561
$11$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$11$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$12$		0			0
$13$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$13$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$14$		0			0
$15$		0			0
$16$	0.998582	+	0.725513i	0.998582	+	0.725513i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$	−0.393950	+	1.21245i	−0.393950	+	1.21245i
$19$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$19$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$20$	0.430742	−	0.312952i	0.430742	−	0.312952i
$21$		0			0
$22$	−0.0800484	+	1.27233i	−0.0800484	+	1.27233i
$23$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$23$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$24$		0			0
$25$	−0.0849327	−	0.261396i	−0.0849327	−	0.261396i
$26$	0.732570	−	2.25462i	0.732570	−	2.25462i
$27$		0			0
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$31$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$32$	1.09580			1.09580
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.193209	+	0.594636i	0.193209	+	0.594636i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$39$		0			0
$40$	−0.125722	+	0.386933i	−0.125722	+	0.386933i
$41$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$41$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.335019	+	0.527905i	0.335019	+	0.527905i
$45$	−0.851559			−0.851559
$46$	−2.04648	+	1.48686i	−2.04648	+	1.48686i
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$		0			0
$49$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$50$	−0.283471	−	0.205954i	−0.283471	−	0.205954i
$51$		0			0
$52$	−0.359282	−	1.10576i	−0.359282	−	1.10576i
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$		0			0
$55$	−0.824805	+	0.211774i	−0.824805	+	0.211774i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$59$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$60$		0			0
$61$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$61$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$62$	−0.0494726	+	0.152261i	−0.0494726	+	0.152261i
$63$		0			0
$64$	0.131596	−	0.0956103i	0.131596	−	0.0956103i
$65$	1.58352			1.58352
$66$		0			0
$67$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$67$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$	−0.386520	−	0.280823i	−0.386520	−	0.280823i
$73$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$73$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$74$		0			0
$75$		0			0
$76$	−1.24061			−1.24061
$77$		0			0
$78$		0			0
$79$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$79$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$80$	0.324805	+	0.999648i	0.324805	+	0.999648i
$81$	0.309017	−	0.951057i	0.309017	−	0.951057i
$82$		0			0
$83$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$83$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.444215	−	0.175877i	−0.444215	−	0.175877i
$89$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$89$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$90$	−0.878275	+	0.638104i	−0.878275	+	0.638104i
$91$		0			0
$92$	−0.383371	+	1.17989i	−0.383371	+	1.17989i
$93$		0			0
$94$		0			0
$95$	0.522142	−	1.60699i	0.522142	−	1.60699i
$96$		0			0
$97$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$97$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$98$	−1.27485			−1.27485
$99$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$100$	−0.171845			−0.171845
$101$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$101$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$102$		0			0
$103$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$103$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$104$	0.718755	+	0.522206i	0.718755	+	0.522206i
$105$		0			0
$106$		0			0
$107$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$107$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−0.691992	+	0.836475i	−0.691992	+	0.836475i
$111$		0			0
$112$		0			0
$113$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$113$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$114$		0			0
$115$	−1.36699	−	0.993173i	−1.36699	−	0.993173i
$116$		0			0
$117$	−0.574633	+	1.76854i	−0.574633	+	1.76854i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$122$		0			0
$123$		0			0
$124$	0.0242634	+	0.0746750i	0.0242634	+	0.0746750i
$125$	0.335471	−	1.03247i	0.335471	−	1.03247i
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$	−0.274540	+	0.844947i	−0.274540	+	0.844947i
$129$		0			0
$130$	1.63320	−	1.18659i	1.63320	−	1.18659i
$131$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$131$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$132$		0			0
$133$		0			0
$134$	1.50368	−	1.09249i	1.50368	−	1.09249i
$135$		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		0.309017	−	0.951057i	$-0.400000\pi$
$139$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.116762	+	1.85588i	−0.116762	+	1.85588i
$144$	−1.23432			−1.23432
$145$		0			0
$146$	0.147638	+	0.454382i	0.147638	+	0.454382i
$147$		0			0
$148$		0			0
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$		0			0
$151$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$151$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$152$	0.766945	−	0.557218i	0.766945	−	0.557218i
$153$		0			0
$154$		0			0
$155$	−0.106940			−0.106940
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$	−0.393950	+	1.21245i	−0.393950	+	1.21245i
$159$		0			0
$160$	0.754924	+	0.548484i	0.754924	+	0.548484i
$161$		0			0
$162$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$163$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$163$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$164$		0			0
$165$		0			0
$166$	2.06275			2.06275
$167$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$167$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$168$		0			0
$169$	0.759544	−	2.33764i	0.759544	−	2.33764i
$170$		0			0
$171$	1.60528	+	1.16630i	1.60528	+	1.16630i
$172$		0			0
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$	−1.19554	+	0.306962i	−1.19554	+	0.306962i
$177$		0			0
$178$	1.10528	−	0.803030i	1.10528	−	0.803030i
$179$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$179$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$180$	−0.164529	+	0.506367i	−0.164529	+	0.506367i
$181$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$181$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$182$		0			0
$183$		0			0
$184$	−0.292947	−	0.901598i	−0.292947	−	0.901598i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.665652	−	2.04867i	−0.665652	−	2.04867i
$191$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$191$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$192$		0			0
$193$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$193$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$194$	−0.690441	+	2.12496i	−0.690441	+	2.12496i
$195$		0			0
$196$	−0.505828	+	0.367505i	−0.505828	+	0.367505i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.683098	−	1.07639i	−0.683098	−	1.07639i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.106234	−	0.0771838i	0.106234	−	0.0771838i
$201$		0			0
$202$	−0.422178	+	1.29933i	−0.422178	+	1.29933i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.60528	−	1.16630i	1.60528	−	1.16630i
$208$	2.29528			2.29528
$209$	1.84489	+	0.730444i	1.84489	+	0.730444i
$210$		0			0
$211$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$211$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.0334313	+	0.531376i	−0.0334313	+	0.531376i
$221$		0			0
$222$		0			0
$223$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$223$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$224$		0			0
$225$	0.222357	+	0.161552i	0.222357	+	0.161552i
$226$		0			0
$227$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$227$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$228$		0			0
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$	−2.15409			−2.15409
$231$		0			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$	0.732570	+	2.25462i	0.732570	+	2.25462i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$239$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$240$		0			0
$241$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$241$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$242$	−0.929324	−	0.872693i	−0.929324	−	0.872693i
$243$		0			0
$244$		0			0
$245$	−0.263146	−	0.809880i	−0.263146	−	0.809880i
$246$		0			0
$247$	−2.98509	−	2.16880i	−2.98509	−	2.16880i
$248$	−0.0485396	−	0.0352661i	−0.0485396	−	0.0352661i
$249$		0			0
$250$	−0.427675	−	1.31625i	−0.427675	−	1.31625i
$251$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$251$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$252$		0			0
$253$	1.26480	−	1.52888i	1.26480	−	1.52888i
$254$		0			0
$255$		0			0
$256$	0.400262	+	1.23188i	0.400262	+	1.23188i
$257$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$257$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$258$		0			0
$259$		0			0
$260$	0.305950	−	0.941617i	0.305950	−	0.941617i
$261$		0			0
$262$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$263$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$263$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.281687	−	0.866942i	0.281687	−	0.866942i
$269$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$269$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$270$		0			0
$271$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$271$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.255547	+	0.101178i	0.255547	+	0.101178i
$276$		0			0
$277$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$277$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$278$		0			0
$279$	0.0388067	−	0.119435i	0.0388067	−	0.119435i
$280$		0			0
$281$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$281$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$282$		0			0
$283$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$283$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$284$		0			0
$285$		0			0
$286$	1.27026	+	2.00160i	1.27026	+	2.00160i
$287$		0			0
$288$	−0.886520	+	0.644095i	−0.886520	+	0.644095i
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$		0			0
$292$	0.189565	+	0.137727i	0.189565	+	0.137727i
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.98509	+	2.16880i	−2.98509	+	2.16880i
$300$		0			0
$301$		0			0
$302$	1.50368	+	1.09249i	1.50368	+	1.09249i
$303$		0			0
$304$	0.756834	−	2.32929i	0.756834	−	2.32929i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−0.110295	+	0.0801338i	−0.110295	+	0.0801338i
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$313$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$314$		0			0
$315$		0			0
$316$	0.193209	+	0.594636i	0.193209	+	0.594636i
$317$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$317$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$318$		0			0
$319$		0			0
$320$	0.138516			0.138516
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.505828	−	0.367505i	−0.505828	−	0.367505i
$325$	−0.413484	−	0.300414i	−0.413484	−	0.300414i
$326$	−0.763146	+	2.34872i	−0.763146	+	2.34872i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	0.818446	−	0.594636i	0.818446	−	0.594636i
$333$		0			0
$334$	−0.0494726	+	0.152261i	−0.0494726	+	0.152261i
$335$	1.00441	+	0.729747i	1.00441	+	0.729747i
$336$		0			0
$337$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$337$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$338$	−0.968304	−	2.98013i	−0.968304	−	2.98013i
$339$		0			0
$340$		0			0
$341$	0.00788530	−	0.125333i	0.00788530	−	0.125333i
$342$	2.52959			2.52959
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$347$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$348$		0			0
$349$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$349$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$		0			0
$352$	−0.698489	+	0.844328i	−0.698489	+	0.844328i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	0.207053	−	0.637244i	0.207053	−	0.637244i
$357$		0			0
$358$	−1.66880	−	1.21245i	−1.66880	−	1.21245i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$	−0.125722	−	0.386933i	−0.125722	−	0.386933i
$361$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$362$	2.37065			2.37065
$363$		0			0
$364$		0			0
$365$	−0.258183	+	0.187581i	−0.258183	+	0.187581i
$366$		0			0
$367$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$367$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$368$	−1.98142	−	1.43958i	−1.98142	−	1.43958i
$369$		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	$-0.800000\pi$
$379$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$380$	−0.854691	−	0.620969i	−0.854691	−	0.620969i
$381$		0			0
$382$		0			0
$383$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$383$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.338621	+	1.04217i	0.338621	+	1.04217i
$389$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$389$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$390$		0			0
$391$		0			0
$392$	0.147638	−	0.454382i	0.147638	−	0.454382i
$393$		0			0
$394$		0			0
$395$	−0.851559			−0.851559
$396$	−0.581331	−	0.230165i	−0.581331	−	0.230165i
$397$	−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
$400$	0.104834	−	0.322645i	0.104834	−	0.322645i
$401$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$401$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$402$		0			0
$403$	−0.0721631	+	0.222095i	−0.0721631	+	0.222095i
$404$	0.207053	+	0.637244i	0.207053	+	0.637244i
$405$	0.688925	−	0.500534i	0.688925	−	0.500534i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$409$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.781687	−	2.40578i	0.781687	−	2.40578i
$415$	0.425779	+	1.31041i	0.425779	+	1.31041i
$416$	1.64853	−	1.19773i	1.64853	−	1.19773i
$417$		0			0
$418$	2.45012	−	0.629084i	2.45012	−	0.629084i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$421$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$422$		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
$431$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$431$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$432$		0			0
$433$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$433$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.21665	+	3.74447i	1.21665	+	3.74447i
$438$		0			0
$439$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$439$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$440$	−0.217999	−	0.343511i	−0.217999	−	0.343511i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$443$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$444$		0			0
$445$	0.738289	+	0.536399i	0.738289	+	0.536399i
$446$	−1.66880	−	1.21245i	−1.66880	−	1.21245i
$447$		0			0
$448$		0			0
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$	0.350389			0.350389
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$457$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$458$		0			0
$459$		0			0
$460$	−0.854691	+	0.620969i	−0.854691	+	0.620969i
$461$		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
$467$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$467$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$468$	0.940613	+	0.683396i	0.940613	+	0.683396i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.441207	+	0.320555i	−0.441207	+	0.320555i
$476$		0			0
$477$		0			0
$478$	−0.878275	−	0.638104i	−0.878275	−	0.638104i
$479$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$479$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$480$		0			0
$481$		0			0
$482$	1.99794	−	1.45159i	1.99794	−	1.45159i
$483$		0			0
$484$	−0.620307	−	0.0783630i	−0.620307	−	0.0783630i
$485$	−1.49245			−1.49245
$486$		0			0
$487$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$487$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$488$		0			0
$489$		0			0
$490$	−0.878275	−	0.638104i	−0.878275	−	0.638104i
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$	−4.70391			−4.70391
$495$	0.542804	−	0.656137i	0.542804	−	0.656137i
$496$	−0.155007			−0.155007
$497$		0			0
$498$		0			0
$499$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$499$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$500$	−0.549130	−	0.398967i	−0.549130	−	0.398967i
$501$		0			0
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$	−0.912576			−0.912576
$506$	0.158834	−	2.52460i	0.158834	−	2.52460i
$507$		0			0
$508$		0			0
$509$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$509$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$		0			0
$512$	0.617158	+	0.448391i	0.617158	+	0.448391i
$513$		0			0
$514$	0.732570	+	2.25462i	0.732570	+	2.25462i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.233787	+	0.719522i	0.233787	+	0.719522i
$521$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$521$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$522$		0			0
$523$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$523$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$524$	−0.0724075	+	0.222847i	−0.0724075	+	0.222847i
$525$		0			0
$526$	1.99794	−	1.45159i	1.99794	−	1.45159i
$527$		0			0
$528$		0			0
$529$	2.93717			2.93717
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.215246	+	0.662460i	0.215246	+	0.662460i
$537$		0			0
$538$	−1.85865			−1.85865
$539$	0.968583	−	0.248690i	0.968583	−	0.248690i
$540$		0			0
$541$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$541$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$547$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$548$		0			0
$549$		0			0
$550$	0.339381	−	0.0871383i	0.339381	−	0.0871383i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.335471	−	1.03247i	0.335471	−	1.03247i
$555$		0			0
$556$		0			0
$557$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$557$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$558$	−0.0494726	−	0.152261i	−0.0494726	−	0.152261i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.08561			1.08561
$563$	−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
$566$	0.129521	+	0.0941025i	0.129521	+	0.0941025i
$567$		0			0
$568$		0			0
$569$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$569$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$570$		0			0
$571$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$571$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$572$	1.08102	+	0.428004i	1.08102	+	0.428004i
$573$		0			0
$574$		0			0
$575$	0.168526	+	0.518670i	0.168526	+	0.518670i
$576$	−0.0502653	+	0.154701i	−0.0502653	+	0.154701i
$577$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$577$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$578$	1.03137	+	0.749337i	1.03137	+	0.749337i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.179049			−0.179049
$585$	−1.28109	+	0.930769i	−1.28109	+	0.930769i
$586$		0			0
$587$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$587$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$588$		0			0
$589$	0.201592	+	0.146465i	0.201592	+	0.146465i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$593$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−1.45359	+	4.47368i	−1.45359	+	4.47368i
$599$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$599$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$600$		0			0
$601$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$601$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$602$		0			0
$603$	−1.17950	+	0.856954i	−1.17950	+	0.856954i
$604$	0.911557			0.911557
$605$	0.362576	−	0.770513i	0.362576	−	0.770513i
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$	−0.671901	−	2.06790i	−0.671901	−	2.06790i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$613$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$617$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$	−0.0206617	+	0.0635902i	−0.0206617	+	0.0635902i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.525546	−	0.381832i	0.525546	−	0.381832i
$626$	−0.160097			−0.160097
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$631$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$632$	−0.386520	−	0.280823i	−0.386520	−	0.280823i
$633$		0			0
$634$	−0.243474	+	0.749337i	−0.243474	+	0.749337i
$635$		0			0
$636$		0			0
$637$	−1.85955			−1.85955
$638$		0			0
$639$		0			0
$640$	−0.612062	+	0.444689i	−0.612062	+	0.444689i
$641$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$641$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$642$		0			0
$643$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$643$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$647$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$648$	0.477765			0.477765
$649$		0			0
$650$	−0.651568			−0.651568
$651$		0			0
$652$	0.374278	+	1.15191i	0.374278	+	1.15191i
$653$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$653$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$654$		0			0
$655$	−0.258183	−	0.187581i	−0.258183	−	0.187581i
$656$		0			0
$657$	−0.115808	−	0.356420i	−0.115808	−	0.356420i
$658$		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
$664$	−0.238883	+	0.735205i	−0.238883	+	0.735205i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.0242634	+	0.0746750i	0.0242634	+	0.0746750i
$669$		0			0
$670$	1.58275			1.58275
$671$		0			0
$672$		0			0
$673$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$673$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$674$	−0.690441	−	2.12496i	−0.690441	−	2.12496i
$675$		0			0
$676$	−1.24329	−	0.903305i	−1.24329	−	0.903305i
$677$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$677$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−0.0857841	−	0.135174i	−0.0857841	−	0.135174i
$683$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$683$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$684$	1.00368	−	0.729215i	1.00368	−	0.729215i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$691$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$		0			0
$694$	−1.85865			−1.85865
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$	−0.0102136	+	0.162341i	−0.0102136	+	0.162341i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$709$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$	0.309017	−	0.951057i	0.309017	−	0.951057i
$712$	0.158216	+	0.486940i	0.158216	+	0.486940i
$713$	0.201592	−	0.146465i	0.201592	−	0.146465i
$714$		0			0
$715$	−1.00937	+	1.22012i	−1.00937	+	1.22012i
$716$	−1.01166			−1.01166
$717$		0			0
$718$		0			0
$719$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$719$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$720$	−0.850351	−	0.617816i	−0.850351	−	0.617816i
$721$		0			0
$722$	−1.15710	+	3.56117i	−1.15710	+	3.56117i
$723$		0			0
$724$	0.940613	−	0.683396i	0.940613	−	0.683396i
$725$		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
$730$	−0.125722	+	0.386933i	−0.125722	+	0.386933i
$731$		0			0
$732$		0			0
$733$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$733$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$734$	−0.422178	−	1.29933i	−0.422178	−	1.29933i
$735$		0			0
$736$	−2.17432			−2.17432
$737$	−0.929324	+	1.12336i	−0.929324	+	1.12336i
$738$		0			0
$739$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$739$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$743$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.61803			−1.61803
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$751$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.383650	+	1.18075i	−0.383650	+	1.18075i
$756$		0			0
$757$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$757$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$758$		0			0
$759$		0			0
$760$	0.807274			0.807274
$761$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$761$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.502226	−	1.54569i	0.502226	−	1.54569i
$767$		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
$773$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$773$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$774$		0			0
$775$	0.0279238	+	0.0202878i	0.0279238	+	0.0202878i
$776$	−0.677421	−	0.492175i	−0.677421	−	0.492175i
$777$		0			0
$778$	−0.0494726	−	0.152261i	−0.0494726	−	0.152261i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.381424	−	1.17390i	−0.381424	−	1.17390i
$785$		0			0
$786$		0			0
$787$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$787$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$788$		0			0
$789$		0			0
$790$	−0.878275	+	0.638104i	−0.878275	+	0.638104i
$791$		0			0
$792$	0.462756	−	0.118815i	0.462756	−	0.118815i
$793$		0			0
$794$	−1.66880	+	1.21245i	−1.66880	+	1.21245i
$795$		0			0
$796$		0			0
$797$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$797$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$798$		0			0
$799$		0			0
$800$	−0.0930692	−	0.286438i	−0.0930692	−	0.286438i
$801$	−0.866986	+	0.629902i	−0.866986	+	0.629902i
$802$		0			0
$803$	−0.200808	−	0.316423i	−0.200808	−	0.316423i
$804$		0			0
$805$		0			0
$806$	0.0919969	+	0.283137i	0.0919969	+	0.283137i
$807$		0			0
$808$	−0.414216	−	0.300945i	−0.414216	−	0.300945i
$809$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$809$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$810$	0.335471	−	1.03247i	0.335471	−	1.03247i
$811$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$811$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.64961			−1.64961
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$821$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$822$		0			0
$823$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$823$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$	−0.383371	−	1.17989i	−0.383371	−	1.17989i
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$	1.42108	+	1.03247i	1.42108	+	1.03247i
$831$		0			0
$832$	0.0934710	−	0.287674i	0.0934710	−	0.287674i
$833$		0			0
$834$		0			0
$835$	−0.106940			−0.106940
$836$	0.790797	−	0.955910i	0.790797	−	0.955910i
$837$		0			0
$838$		0			0
$839$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$839$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$	1.80760	+	1.31330i	1.80760	+	1.31330i
$843$		0			0
$844$		0			0
$845$	1.69334	−	1.23028i	1.69334	−	1.23028i
$846$		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	$-0.800000\pi$
$853$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$	0.522142	+	1.60699i	0.522142	+	1.60699i
$856$		0			0
$857$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$857$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	0.147638	−	0.454382i	0.147638	−	0.454382i
$863$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$863$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$864$		0			0
$865$		0			0
$866$	0.781687	+	2.40578i	0.781687	+	2.40578i
$867$		0			0
$868$		0			0
$869$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$870$		0			0
$871$	2.19334	−	1.59355i	2.19334	−	1.59355i
$872$		0			0
$873$	0.541587	−	1.66683i	0.541587	−	1.66683i
$874$	4.06069	+	2.95026i	4.06069	+	2.95026i
$875$		0			0
$876$		0			0
$877$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$877$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$878$	1.80760	−	1.31330i	1.80760	−	1.31330i
$879$		0			0
$880$	−0.977281	−	0.386933i	−0.977281	−	0.386933i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.03137	−	0.749337i	1.03137	−	0.749337i
$883$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$883$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$887$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$888$		0			0
$889$		0			0
$890$	1.16340			1.16340
$891$	0.535827	+	0.844328i	0.535827	+	0.844328i
$892$	−1.01166			−1.01166
$893$		0			0
$894$		0			0
$895$	0.425779	−	1.31041i	0.425779	−	1.31041i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.139026	−	0.101008i	0.139026	−	0.101008i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.489334	+	1.50602i	0.489334	+	1.50602i
$906$		0			0
$907$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$907$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$908$		0			0
$909$	0.331159	−	1.01920i	0.331159	−	1.01920i
$910$		0			0
$911$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$911$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$912$		0			0
$913$	−1.56720	+	0.402389i	−1.56720	+	0.402389i
$914$	−1.36620			−1.36620
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$919$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$920$	0.249461	−	0.767763i	0.249461	−	0.767763i
$921$		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	$-0.800000\pi$
$929$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$	−0.613161	+	1.88711i	−0.613161	+	1.88711i
$932$		0			0
$933$		0			0
$934$	1.62524			1.62524
$935$		0			0
$936$	−0.888430			−0.888430
$937$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$937$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$941$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	0.215351	+	0.662783i	0.215351	+	0.662783i
$950$	−0.214845	+	0.661225i	−0.214845	+	0.661225i
$951$		0			0
$952$		0			0
$953$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$953$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$954$		0			0
$955$		0			0
$956$	−0.532426			−0.532426
$957$		0			0
$958$	2.06275			2.06275
$959$		0			0
$960$		0			0
$961$	−0.304144	+	0.936058i	−0.304144	+	0.936058i
$962$		0			0
$963$		0			0
$964$	0.374278	−	1.15191i	0.374278	−	1.15191i
$965$		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	$0.800000\pi$
$968$	0.418669	−	0.230165i	0.418669	−	0.230165i
$969$		0			0
$970$	−1.53928	+	1.11835i	−1.53928	+	1.11835i
$971$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$971$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$972$		0			0
$973$		0			0
$974$	−1.91789	−	1.39343i	−1.91789	−	1.39343i
$975$		0			0
$976$		0			0
$977$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$977$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$978$		0			0
$979$	−0.683098	+	0.825723i	−0.683098	+	0.825723i
$980$	−0.532426			−0.532426
$981$		0			0
$982$		0			0
$983$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$983$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.86639	+	1.35601i	−1.86639	+	1.35601i
$989$		0			0
$990$	0.0681659	−	1.08347i	0.0681659	−	1.08347i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.111330	+	0.0808861i	−0.111330	+	0.0808861i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$997$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$998$	0.502226	+	1.54569i	0.502226	+	1.54569i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.1.j.a.157.4	✓	20
11.4	even	5	inner	869.1.j.a.631.4	yes	20
79.78	odd	2	CM	869.1.j.a.157.4	✓	20
869.631	odd	10	inner	869.1.j.a.631.4	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.1.j.a.157.4	✓	20	1.1	even	1	trivial
869.1.j.a.157.4	✓	20	79.78	odd	2	CM
869.1.j.a.631.4	yes	20	11.4	even	5	inner
869.1.j.a.631.4	yes	20	869.631	odd	10	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.03137 - 0.749337i) q^{2} +(0.193209 - 0.594636i) q^{4} +(0.688925 + 0.500534i) q^{5} +(0.147638 + 0.454382i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\)	\(q+(1.03137 - 0.749337i) q^{2} +(0.193209 - 0.594636i) q^{4} +(0.688925 + 0.500534i) q^{5} +(0.147638 + 0.454382i) q^{8} +(-0.809017 + 0.587785i) q^{9} +1.08561 q^{10} +(-0.637424 + 0.770513i) q^{11} +(1.50441 - 1.09302i) q^{13} +(0.998582 + 0.725513i) q^{16} +(-0.393950 + 1.21245i) q^{18} +(-0.613161 - 1.88711i) q^{19} +(0.430742 - 0.312952i) q^{20} +(-0.0800484 + 1.27233i) q^{22} -1.98423 q^{23} +(-0.0849327 - 0.261396i) q^{25} +(0.732570 - 2.25462i) q^{26} +(-0.101597 + 0.0738147i) q^{31} +1.09580 q^{32} +(0.193209 + 0.594636i) q^{36} +(-2.04648 - 1.48686i) q^{38} +(-0.125722 + 0.386933i) q^{40} +(0.335019 + 0.527905i) q^{44} -0.851559 q^{45} +(-2.04648 + 1.48686i) q^{46} +(-0.809017 - 0.587785i) q^{49} +(-0.283471 - 0.205954i) q^{50} +(-0.359282 - 1.10576i) q^{52} +(-0.824805 + 0.211774i) q^{55} +(-0.0494726 + 0.152261i) q^{62} +(0.131596 - 0.0956103i) q^{64} +1.58352 q^{65} +1.45794 q^{67} +(-0.386520 - 0.280823i) q^{72} +(-0.115808 + 0.356420i) q^{73} -1.24061 q^{76} +(-0.809017 + 0.587785i) q^{79} +(0.324805 + 0.999648i) q^{80} +(0.309017 - 0.951057i) q^{81} +(1.30902 + 0.951057i) q^{83} +(-0.444215 - 0.175877i) q^{88} +1.07165 q^{89} +(-0.878275 + 0.638104i) q^{90} +(-0.383371 + 1.17989i) q^{92} +(0.522142 - 1.60699i) q^{95} +(-1.41789 + 1.03016i) q^{97} -1.27485 q^{98} +(0.0627905 - 0.998027i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 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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