LMFDB - Embedded newform 869.1.j.a.157.1

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			157.1
Root			$-0.535827 - 0.844328i$ of defining polynomial
Character	$\chi$	$=$	869.157
Dual form			869.1.j.a.631.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.56720 + 1.13864i) q^{2} +(0.850604 - 2.61789i) q^{4} +(1.60528 + 1.16630i) q^{5} +(1.04914 + 3.22894i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})$	\(q+(-1.56720 + 1.13864i) q^{2} +(0.850604 - 2.61789i) q^{4} +(1.60528 + 1.16630i) q^{5} +(1.04914 + 3.22894i) q^{8} +(-0.809017 + 0.587785i) q^{9} -3.84378 q^{10} +(0.968583 - 0.248690i) q^{11} +(-0.866986 + 0.629902i) q^{13} +(-3.09390 - 2.24785i) q^{16} +(0.598617 - 1.84235i) q^{18} +(-0.115808 - 0.356420i) q^{19} +(4.41870 - 3.21038i) q^{20} +(-1.23480 + 1.49261i) q^{22} -0.374763 q^{23} +(0.907634 + 2.79341i) q^{25} +(0.641510 - 1.97437i) q^{26} +(1.03137 - 0.749337i) q^{31} +4.01314 q^{32} +(0.850604 + 2.61789i) q^{36} +(0.587328 + 0.426719i) q^{38} +(-2.08174 + 6.40695i) q^{40} +(0.172838 - 2.74718i) q^{44} -1.98423 q^{45} +(0.587328 - 0.426719i) q^{46} +(-0.809017 - 0.587785i) q^{49} +(-4.60313 - 3.34437i) q^{50} +(0.911553 + 2.80547i) q^{52} +(1.84489 + 0.730444i) q^{55} +(-0.763146 + 2.34872i) q^{62} +(-3.19549 + 2.32166i) q^{64} -2.12641 q^{65} -0.851559 q^{67} +(-2.74670 - 1.99559i) q^{72} +(0.541587 - 1.66683i) q^{73} -1.03158 q^{76} +(-0.809017 + 0.587785i) q^{79} +(-2.34489 - 7.21683i) q^{80} +(0.309017 - 0.951057i) q^{81} +(1.30902 + 0.951057i) q^{83} +(1.81919 + 2.86658i) q^{88} +0.125581 q^{89} +(3.10969 - 2.25932i) q^{90} +(-0.318775 + 0.981088i) q^{92} +(0.229790 - 0.707220i) q^{95} +(-1.17950 + 0.856954i) q^{97} +1.93717 q^{98} +(-0.637424 + 0.770513i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$2$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$3$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$3$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$4$	0.850604	−	2.61789i	0.850604	−	2.61789i
$5$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$5$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$6$		0			0
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$	1.04914	+	3.22894i	1.04914	+	3.22894i
$9$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$10$	−3.84378			−3.84378
$11$	0.968583	−	0.248690i	0.968583	−	0.248690i
$11$	0.968583	−	0.248690i	0.968583	−	0.248690i
$12$		0			0
$13$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$13$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$14$		0			0
$15$		0			0
$16$	−3.09390	−	2.24785i	−3.09390	−	2.24785i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$	0.598617	−	1.84235i	0.598617	−	1.84235i
$19$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$19$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$20$	4.41870	−	3.21038i	4.41870	−	3.21038i
$21$		0			0
$22$	−1.23480	+	1.49261i	−1.23480	+	1.49261i
$23$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$23$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$24$		0			0
$25$	0.907634	+	2.79341i	0.907634	+	2.79341i
$26$	0.641510	−	1.97437i	0.641510	−	1.97437i
$27$		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
$31$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$31$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$32$	4.01314			4.01314
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.850604	+	2.61789i	0.850604	+	2.61789i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$	0.587328	+	0.426719i	0.587328	+	0.426719i
$39$		0			0
$40$	−2.08174	+	6.40695i	−2.08174	+	6.40695i
$41$		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			$\pi$
$44$	0.172838	−	2.74718i	0.172838	−	2.74718i
$45$	−1.98423			−1.98423
$46$	0.587328	−	0.426719i	0.587328	−	0.426719i
$47$		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
$50$	−4.60313	−	3.34437i	−4.60313	−	3.34437i
$51$		0			0
$52$	0.911553	+	2.80547i	0.911553	+	2.80547i
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$		0			0
$55$	1.84489	+	0.730444i	1.84489	+	0.730444i
$56$		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	$0.200000\pi$
$62$	−0.763146	+	2.34872i	−0.763146	+	2.34872i
$63$		0			0
$64$	−3.19549	+	2.32166i	−3.19549	+	2.32166i
$65$	−2.12641			−2.12641
$66$		0			0
$67$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$67$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$68$		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	$0.200000\pi$
$72$	−2.74670	−	1.99559i	−2.74670	−	1.99559i
$73$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$73$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$74$		0			0
$75$		0			0
$76$	−1.03158			−1.03158
$77$		0			0
$78$		0			0
$79$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$79$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$80$	−2.34489	−	7.21683i	−2.34489	−	7.21683i
$81$	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
$88$	1.81919	+	2.86658i	1.81919	+	2.86658i
$89$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$89$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$90$	3.10969	−	2.25932i	3.10969	−	2.25932i
$91$		0			0
$92$	−0.318775	+	0.981088i	−0.318775	+	0.981088i
$93$		0			0
$94$		0			0
$95$	0.229790	−	0.707220i	0.229790	−	0.707220i
$96$		0			0
$97$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$97$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$98$	1.93717			1.93717
$99$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$100$	8.08488			8.08488
$101$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$101$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$102$		0			0
$103$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$103$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$104$	−2.94351	−	2.13858i	−2.94351	−	2.13858i
$105$		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			$\pi$
$110$	−3.72302	+	0.955910i	−3.72302	+	0.955910i
$111$		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
$115$	−0.601597	−	0.437086i	−0.601597	−	0.437086i
$116$		0			0
$117$	0.331159	−	1.01920i	0.331159	−	1.01920i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.876307	−	0.481754i	0.876307	−	0.481754i
$122$		0			0
$123$		0			0
$124$	−1.08439	−	3.33741i	−1.08439	−	3.33741i
$125$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$	1.12432	−	3.46030i	1.12432	−	3.46030i
$129$		0			0
$130$	3.33251	−	2.42121i	3.33251	−	2.42121i
$131$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$131$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$132$		0			0
$133$		0			0
$134$	1.33456	−	0.969617i	1.33456	−	0.969617i
$135$		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
$143$	−0.683098	+	0.825723i	−0.683098	+	0.825723i
$144$	3.82427			3.82427
$145$		0			0
$146$	1.04914	+	3.22894i	1.04914	+	3.22894i
$147$		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
$151$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$151$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$152$	1.02936	−	0.747873i	1.02936	−	0.747873i
$153$		0			0
$154$		0			0
$155$	2.52959			2.52959
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$	0.598617	−	1.84235i	0.598617	−	1.84235i
$159$		0			0
$160$	6.44219	+	4.68053i	6.44219	+	4.68053i
$161$		0			0
$162$	0.598617	+	1.84235i	0.598617	+	1.84235i
$163$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$163$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$164$		0			0
$165$		0			0
$166$	−3.13440			−3.13440
$167$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$167$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$168$		0			0
$169$	0.0458709	−	0.141176i	0.0458709	−	0.141176i
$170$		0			0
$171$	0.303189	+	0.220280i	0.303189	+	0.220280i
$172$		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
$176$	−3.55571	−	1.40781i	−3.55571	−	1.40781i
$177$		0			0
$178$	−0.196811	+	0.142991i	−0.196811	+	0.142991i
$179$	−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	$-0.400000\pi$
$180$	−1.68779	+	5.19450i	−1.68779	+	5.19450i
$181$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$181$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$182$		0			0
$183$		0			0
$184$	−0.393180	−	1.21008i	−0.393180	−	1.21008i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.445141	+	1.37000i	0.445141	+	1.37000i
$191$		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	$0.200000\pi$
$194$	0.872746	−	2.68604i	0.872746	−	2.68604i
$195$		0			0
$196$	−2.22691	+	1.61795i	−2.22691	+	1.61795i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.121636	−	1.93334i	0.121636	−	1.93334i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−8.06750	+	5.86139i	−8.06750	+	5.86139i
$201$		0			0
$202$	0.0751750	−	0.231365i	0.0751750	−	0.231365i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.303189	−	0.220280i	0.303189	−	0.220280i
$208$	4.09829			4.09829
$209$	−0.200808	−	0.316423i	−0.200808	−	0.316423i
$210$		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
$220$	3.48149	−	4.20840i	3.48149	−	4.20840i
$221$		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
$225$	−2.37622	−	1.72642i	−2.37622	−	1.72642i
$226$		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	$0.800000\pi$
$230$	1.44051			1.44051
$231$		0			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$	0.641510	+	1.97437i	0.641510	+	1.97437i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$239$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$240$		0			0
$241$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$241$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$242$	−0.824805	+	1.75280i	−0.824805	+	1.75280i
$243$		0			0
$244$		0			0
$245$	−0.613161	−	1.88711i	−0.613161	−	1.88711i
$246$		0			0
$247$	0.324914	+	0.236064i	0.324914	+	0.236064i
$248$	3.50162	+	2.54408i	3.50162	+	2.54408i
$249$		0			0
$250$	−2.30095	−	7.08161i	−2.30095	−	7.08161i
$251$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$251$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$252$		0			0
$253$	−0.362989	+	0.0931997i	−0.362989	+	0.0931997i
$254$		0			0
$255$		0			0
$256$	0.957424	+	2.94665i	0.957424	+	2.94665i
$257$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$257$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$258$		0			0
$259$		0			0
$260$	−1.80873	+	5.56670i	−1.80873	+	5.56670i
$261$		0			0
$262$	−2.74670	+	1.99559i	−2.74670	+	1.99559i
$263$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$263$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.724339	+	2.22929i	−0.724339	+	2.22929i
$269$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$269$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$270$		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
$275$	1.57381	+	2.47993i	1.57381	+	2.47993i
$276$		0			0
$277$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$277$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$278$		0			0
$279$	−0.393950	+	1.21245i	−0.393950	+	1.21245i
$280$		0			0
$281$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$281$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$282$		0			0
$283$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$283$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$284$		0			0
$285$		0			0
$286$	0.130351	−	2.07187i	0.130351	−	2.07187i
$287$		0			0
$288$	−3.24670	+	2.35886i	−3.24670	+	2.35886i
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$		0			0
$292$	−3.90291	−	2.83563i	−3.90291	−	2.83563i
$293$		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
$299$	0.324914	−	0.236064i	0.324914	−	0.236064i
$300$		0			0
$301$		0			0
$302$	1.33456	+	0.969617i	1.33456	+	0.969617i
$303$		0			0
$304$	−0.442881	+	1.36305i	−0.442881	+	1.36305i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−3.96438	+	2.88029i	−3.96438	+	2.88029i
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$313$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$314$		0			0
$315$		0			0
$316$	0.850604	+	2.61789i	0.850604	+	2.61789i
$317$	−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
$320$	−7.83740			−7.83740
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.22691	−	1.61795i	−2.22691	−	1.61795i
$325$	−2.54648	−	1.85013i	−2.54648	−	1.85013i
$326$	−1.11316	+	3.42596i	−1.11316	+	3.42596i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	3.60322	−	2.61789i	3.60322	−	2.61789i
$333$		0			0
$334$	−0.763146	+	2.34872i	−0.763146	+	2.34872i
$335$	−1.36699	−	0.993173i	−1.36699	−	0.993173i
$336$		0			0
$337$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$337$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$338$	0.0888596	+	0.273482i	0.0888596	+	0.273482i
$339$		0			0
$340$		0			0
$341$	0.812619	−	0.982287i	0.812619	−	0.982287i
$342$	−0.725978			−0.725978
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$347$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$348$		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
$352$	3.88706	−	0.998027i	3.88706	−	0.998027i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	0.106820	−	0.328757i	0.106820	−	0.328757i
$357$		0			0
$358$	2.53578	+	1.84235i	2.53578	+	1.84235i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$	−2.08174	−	6.40695i	−2.08174	−	6.40695i
$361$	0.695393	−	0.505233i	0.695393	−	0.505233i
$362$	2.07597			2.07597
$363$		0			0
$364$		0			0
$365$	2.81343	−	2.04407i	2.81343	−	2.04407i
$366$		0			0
$367$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$367$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$368$	1.15948	+	0.842409i	1.15948	+	0.842409i
$369$		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	$0.200000\pi$
$380$	−1.65596	−	1.20313i	−1.65596	−	1.20313i
$381$		0			0
$382$		0			0
$383$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$383$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.24013	+	3.81672i	1.24013	+	3.81672i
$389$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$389$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$390$		0			0
$391$		0			0
$392$	1.04914	−	3.22894i	1.04914	−	3.22894i
$393$		0			0
$394$		0			0
$395$	−1.98423			−1.98423
$396$	1.47492	+	2.32411i	1.47492	+	2.32411i
$397$	−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
$400$	3.47104	−	10.6828i	3.47104	−	10.6828i
$401$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$401$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$402$		0			0
$403$	−0.422178	+	1.29933i	−0.422178	+	1.29933i
$404$	0.106820	+	0.328757i	0.106820	+	0.328757i
$405$	1.60528	−	1.16630i	1.60528	−	1.16630i
$406$		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
$414$	−0.224339	+	0.690446i	−0.224339	+	0.690446i
$415$	0.992115	+	3.05342i	0.992115	+	3.05342i
$416$	−3.47933	+	2.52788i	−3.47933	+	2.52788i
$417$		0			0
$418$	0.674997	+	0.267250i	0.674997	+	0.267250i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$421$	0.450527	+	1.38658i	0.450527	+	1.38658i	−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
$431$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$431$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$432$		0			0
$433$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$433$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.0434005	+	0.133573i	0.0434005	+	0.133573i
$438$		0			0
$439$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$439$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$440$	−0.422999	+	6.72337i	−0.422999	+	6.72337i
$441$	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
$445$	0.201592	+	0.146465i	0.201592	+	0.146465i
$446$	2.53578	+	1.84235i	2.53578	+	1.84235i
$447$		0			0
$448$		0			0
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$	5.68978			5.68978
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$457$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$458$		0			0
$459$		0			0
$460$	−1.65596	+	1.20313i	−1.65596	+	1.20313i
$461$		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
$467$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$467$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$468$	−2.38648	−	1.73388i	−2.38648	−	1.73388i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.890518	−	0.646999i	0.890518	−	0.646999i
$476$		0			0
$477$		0			0
$478$	3.10969	+	2.25932i	3.10969	+	2.25932i
$479$	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
$482$	2.91429	−	2.11736i	2.91429	−	2.11736i
$483$		0			0
$484$	−0.515788	−	2.70386i	−0.515788	−	2.70386i
$485$	−2.89288			−2.89288
$486$		0			0
$487$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$487$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$488$		0			0
$489$		0			0
$490$	3.10969	+	2.25932i	3.10969	+	2.25932i
$491$		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
$494$	−0.777996			−0.777996
$495$	−1.92189	+	0.493458i	−1.92189	+	0.493458i
$496$	−4.87536			−4.87536
$497$		0			0
$498$		0			0
$499$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$499$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$500$	8.55976	+	6.21903i	8.55976	+	6.21903i
$501$		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
$505$	−0.249182			−0.249182
$506$	0.462756	−	0.559375i	0.462756	−	0.559375i
$507$		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
$512$	−1.91213	−	1.38925i	−1.91213	−	1.38925i
$513$		0			0
$514$	0.641510	+	1.97437i	0.641510	+	1.97437i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−2.23091	−	6.86603i	−2.23091	−	6.86603i
$521$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$521$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$522$		0			0
$523$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$523$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$524$	1.49078	−	4.58815i	1.49078	−	4.58815i
$525$		0			0
$526$	2.91429	−	2.11736i	2.91429	−	2.11736i
$527$		0			0
$528$		0			0
$529$	−0.859553			−0.859553
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.893408	−	2.74963i	−0.893408	−	2.74963i
$537$		0			0
$538$	−1.64961			−1.64961
$539$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$540$		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
$547$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$547$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$548$		0			0
$549$		0			0
$550$	−5.29022	−	2.09455i	−5.29022	−	2.09455i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$555$		0			0
$556$		0			0
$557$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$557$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$558$	−0.763146	−	2.34872i	−0.763146	−	2.34872i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−3.84378			−3.84378
$563$	−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
$566$	1.99794	+	1.45159i	1.99794	+	1.45159i
$567$		0			0
$568$		0			0
$569$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$569$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$570$		0			0
$571$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$571$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$572$	1.58061	+	2.49064i	1.58061	+	2.49064i
$573$		0			0
$574$		0			0
$575$	−0.340147	−	1.04687i	−0.340147	−	1.04687i
$576$	1.22057	−	3.75653i	1.22057	−	3.75653i
$577$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$577$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$578$	−1.56720	−	1.13864i	−1.56720	−	1.13864i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	5.95030			5.95030
$585$	1.72030	−	1.24987i	1.72030	−	1.24987i
$586$		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
$589$	−0.386520	−	0.280823i	−0.386520	−	0.280823i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$593$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−0.240414	+	0.739919i	−0.240414	+	0.739919i
$599$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$599$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−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
$603$	0.688925	−	0.500534i	0.688925	−	0.500534i
$604$	−2.34401			−2.34401
$605$	1.96858	+	0.248690i	1.96858	+	0.248690i
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$	−0.464754	−	1.43036i	−0.464754	−	1.43036i
$609$		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
$617$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$617$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$	2.15168	−	6.62219i	2.15168	−	6.62219i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.79411	+	2.75658i	−3.79411	+	2.75658i
$626$	−2.46959			−2.46959
$627$		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	$0.600000\pi$
$632$	−2.74670	−	1.99559i	−2.74670	−	1.99559i
$633$		0			0
$634$	0.369966	−	1.13864i	0.369966	−	1.13864i
$635$		0			0
$636$		0			0
$637$	1.07165			1.07165
$638$		0			0
$639$		0			0
$640$	5.84059	−	4.24344i	5.84059	−	4.24344i
$641$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$641$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$642$		0			0
$643$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$643$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−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	$0.800000\pi$
$648$	3.39510			3.39510
$649$		0			0
$650$	6.09747			6.09747
$651$		0			0
$652$	−1.58174	−	4.86811i	−1.58174	−	4.86811i
$653$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$653$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$654$		0			0
$655$	2.81343	+	2.04407i	2.81343	+	2.04407i
$656$		0			0
$657$	0.541587	+	1.66683i	0.541587	+	1.66683i
$658$		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
$664$	−1.69755	+	5.22453i	−1.69755	+	5.22453i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.08439	−	3.33741i	−1.08439	−	3.33741i
$669$		0			0
$670$	3.27321			3.27321
$671$		0			0
$672$		0			0
$673$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$673$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$674$	0.872746	+	2.68604i	0.872746	+	2.68604i
$675$		0			0
$676$	−0.330566	−	0.240170i	−0.330566	−	0.240170i
$677$	−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
$682$	−0.155067	+	2.46472i	−0.155067	+	2.46472i
$683$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$683$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$684$	0.834563	−	0.606346i	0.834563	−	0.606346i
$685$		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
$694$	−1.64961			−1.64961
$695$		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
$704$	−2.51773	+	3.04341i	−2.51773	+	3.04341i
$705$		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
$712$	0.131753	+	0.405493i	0.131753	+	0.405493i
$713$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$714$		0			0
$715$	−2.05960	+	0.528816i	−2.05960	+	0.528816i
$716$	−4.45382			−4.45382
$717$		0			0
$718$		0			0
$719$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$719$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$720$	6.13900	+	4.46025i	6.13900	+	4.46025i
$721$		0			0
$722$	−0.514543	+	1.58360i	−0.514543	+	1.58360i
$723$		0			0
$724$	−2.38648	+	1.73388i	−2.38648	+	1.73388i
$725$		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
$730$	−2.08174	+	6.40695i	−2.08174	+	6.40695i
$731$		0			0
$732$		0			0
$733$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$733$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$734$	0.0751750	+	0.231365i	0.0751750	+	0.231365i
$735$		0			0
$736$	−1.50397			−1.50397
$737$	−0.824805	+	0.211774i	−0.824805	+	0.211774i
$738$		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
$743$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$743$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.61803			−1.61803
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$751$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.522142	−	1.60699i	0.522142	−	1.60699i
$756$		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
$760$	2.52465			2.52465
$761$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$761$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1.15962	−	3.56895i	1.15962	−	3.56895i
$767$		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
$773$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$773$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$774$		0			0
$775$	3.02932	+	2.20093i	3.02932	+	2.20093i
$776$	−4.00451	−	2.90945i	−4.00451	−	2.90945i
$777$		0			0
$778$	−0.763146	−	2.34872i	−0.763146	−	2.34872i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.18176	+	3.63709i	1.18176	+	3.63709i
$785$		0			0
$786$		0			0
$787$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$787$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$788$		0			0
$789$		0			0
$790$	3.10969	−	2.25932i	3.10969	−	2.25932i
$791$		0			0
$792$	−3.15669	−	1.24982i	−3.15669	−	1.24982i
$793$		0			0
$794$	2.53578	−	1.84235i	2.53578	−	1.84235i
$795$		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
$800$	3.64246	+	11.2103i	3.64246	+	11.2103i
$801$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i
$802$		0			0
$803$	0.110048	−	1.74915i	0.110048	−	1.74915i
$804$		0			0
$805$		0			0
$806$	−0.817828	−	2.51702i	−0.817828	−	2.51702i
$807$		0			0
$808$	−0.344933	−	0.250608i	−0.344933	−	0.250608i
$809$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$809$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$810$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$811$	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
$815$	3.68978			3.68978
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$821$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$822$		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	$0.800000\pi$
$828$	−0.318775	−	0.981088i	−0.318775	−	0.981088i
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$	−5.03158	−	3.65565i	−5.03158	−	3.65565i
$831$		0			0
$832$	1.30803	−	4.02570i	1.30803	−	4.02570i
$833$		0			0
$834$		0			0
$835$	2.52959			2.52959
$836$	−0.999168	+	0.256543i	−0.999168	+	0.256543i
$837$		0			0
$838$		0			0
$839$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$839$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$	−2.28488	−	1.66006i	−2.28488	−	1.66006i
$843$		0			0
$844$		0			0
$845$	0.238289	−	0.173127i	0.238289	−	0.173127i
$846$		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
$855$	0.229790	+	0.707220i	0.229790	+	0.707220i
$856$		0			0
$857$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$857$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	1.04914	−	3.22894i	1.04914	−	3.22894i
$863$	−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
$866$	−0.224339	−	0.690446i	−0.224339	−	0.690446i
$867$		0			0
$868$		0			0
$869$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$870$		0			0
$871$	0.738289	−	0.536399i	0.738289	−	0.536399i
$872$		0			0
$873$	0.450527	−	1.38658i	0.450527	−	1.38658i
$874$	−0.220109	−	0.159918i	−0.220109	−	0.159918i
$875$		0			0
$876$		0			0
$877$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$877$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$878$	−2.28488	+	1.66006i	−2.28488	+	1.66006i
$879$		0			0
$880$	−4.06597	−	6.40695i	−4.06597	−	6.40695i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.56720	+	1.13864i	−1.56720	+	1.13864i
$883$		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
$887$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$887$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$888$		0			0
$889$		0			0
$890$	−0.482706			−0.482706
$891$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$892$	−4.45382			−4.45382
$893$		0			0
$894$		0			0
$895$	0.992115	−	3.05342i	0.992115	−	3.05342i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−6.54081	+	4.75218i	−6.54081	+	4.75218i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.657096	−	2.02233i	−0.657096	−	2.02233i
$906$		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
$909$	0.0388067	−	0.119435i	0.0388067	−	0.119435i
$910$		0			0
$911$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$911$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$912$		0			0
$913$	1.50441	+	0.595638i	1.50441	+	0.595638i
$914$	0.243271			0.243271
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$919$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$920$	0.780160	−	2.40108i	0.780160	−	2.40108i
$921$		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
$931$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$932$		0			0
$933$		0			0
$934$	3.75261			3.75261
$935$		0			0
$936$	3.63837			3.63837
$937$		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
$949$	0.580394	+	1.78627i	0.580394	+	1.78627i
$950$	−0.658922	+	2.02795i	−0.658922	+	2.02795i
$951$		0			0
$952$		0			0
$953$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$953$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$954$		0			0
$955$		0			0
$956$	−5.46182			−5.46182
$957$		0			0
$958$	−3.13440			−3.13440
$959$		0			0
$960$		0			0
$961$	0.193209	−	0.594636i	0.193209	−	0.594636i
$962$		0			0
$963$		0			0
$964$	−1.58174	+	4.86811i	−1.58174	+	4.86811i
$965$		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	$0.800000\pi$
$968$	2.47492	+	2.32411i	2.47492	+	2.32411i
$969$		0			0
$970$	4.53373	−	3.29394i	4.53373	−	3.29394i
$971$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$971$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$972$		0			0
$973$		0			0
$974$	−1.67950	−	1.22023i	−1.67950	−	1.22023i
$975$		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
$979$	0.121636	−	0.0312307i	0.121636	−	0.0312307i
$980$	−5.46182			−5.46182
$981$		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
$988$	0.894362	−	0.649792i	0.894362	−	0.649792i
$989$		0			0
$990$	2.45012	−	2.96169i	2.45012	−	2.96169i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	4.13904	−	3.00719i	4.13904	−	3.00719i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$997$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$998$	1.15962	+	3.56895i	1.15962	+	3.56895i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.1.j.a.157.1	✓	20
11.4	even	5	inner	869.1.j.a.631.1	yes	20
79.78	odd	2	CM	869.1.j.a.157.1	✓	20
869.631	odd	10	inner	869.1.j.a.631.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.1.j.a.157.1	✓	20	1.1	even	1	trivial
869.1.j.a.157.1	✓	20	79.78	odd	2	CM
869.1.j.a.631.1	yes	20	11.4	even	5	inner
869.1.j.a.631.1	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)

\(f(q)\)	\(=\)	\(q+(-1.56720 + 1.13864i) q^{2} +(0.850604 - 2.61789i) q^{4} +(1.60528 + 1.16630i) q^{5} +(1.04914 + 3.22894i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\)	\(q+(-1.56720 + 1.13864i) q^{2} +(0.850604 - 2.61789i) q^{4} +(1.60528 + 1.16630i) q^{5} +(1.04914 + 3.22894i) q^{8} +(-0.809017 + 0.587785i) q^{9} -3.84378 q^{10} +(0.968583 - 0.248690i) q^{11} +(-0.866986 + 0.629902i) q^{13} +(-3.09390 - 2.24785i) q^{16} +(0.598617 - 1.84235i) q^{18} +(-0.115808 - 0.356420i) q^{19} +(4.41870 - 3.21038i) q^{20} +(-1.23480 + 1.49261i) q^{22} -0.374763 q^{23} +(0.907634 + 2.79341i) q^{25} +(0.641510 - 1.97437i) q^{26} +(1.03137 - 0.749337i) q^{31} +4.01314 q^{32} +(0.850604 + 2.61789i) q^{36} +(0.587328 + 0.426719i) q^{38} +(-2.08174 + 6.40695i) q^{40} +(0.172838 - 2.74718i) q^{44} -1.98423 q^{45} +(0.587328 - 0.426719i) q^{46} +(-0.809017 - 0.587785i) q^{49} +(-4.60313 - 3.34437i) q^{50} +(0.911553 + 2.80547i) q^{52} +(1.84489 + 0.730444i) q^{55} +(-0.763146 + 2.34872i) q^{62} +(-3.19549 + 2.32166i) q^{64} -2.12641 q^{65} -0.851559 q^{67} +(-2.74670 - 1.99559i) q^{72} +(0.541587 - 1.66683i) q^{73} -1.03158 q^{76} +(-0.809017 + 0.587785i) q^{79} +(-2.34489 - 7.21683i) q^{80} +(0.309017 - 0.951057i) q^{81} +(1.30902 + 0.951057i) q^{83} +(1.81919 + 2.86658i) q^{88} +0.125581 q^{89} +(3.10969 - 2.25932i) q^{90} +(-0.318775 + 0.981088i) q^{92} +(0.229790 - 0.707220i) q^{95} +(-1.17950 + 0.856954i) q^{97} +1.93717 q^{98} +(-0.637424 + 0.770513i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more