LMFDB - Embedded newform 869.1.j.a.631.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("869.157");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.433687495978$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

Embedding invariants

Embedding label			631.5
Root			$-0.0627905 - 0.998027i$ of defining polynomial
Character	$\chi$	$=$	869.631
Dual form			869.1.j.a.157.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.303189 - 0.220280i) q^{5} +(-0.837780 + 2.57842i) q^{8} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})$	\(q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.303189 - 0.220280i) q^{5} +(-0.837780 + 2.57842i) q^{8} +(-0.809017 - 0.587785i) q^{9} +0.696891 q^{10} +(-0.929776 + 0.368125i) q^{11} +(-0.101597 - 0.0738147i) q^{13} +(-2.09011 + 1.51855i) q^{16} +(-0.574633 - 1.76854i) q^{18} +(0.541587 - 1.66683i) q^{19} +(0.745220 + 0.541434i) q^{20} +(-1.80113 - 0.462452i) q^{22} +1.75261 q^{23} +(-0.265616 + 0.817483i) q^{25} +(-0.0721631 - 0.222095i) q^{26} +(-1.56720 - 1.13864i) q^{31} -2.09308 q^{32} +(0.759544 - 2.33764i) q^{36} +(2.63665 - 1.91564i) q^{38} +(0.313968 + 0.966296i) q^{40} +(-1.56675 - 1.89387i) q^{44} -0.374763 q^{45} +(2.63665 + 1.91564i) q^{46} +(-0.809017 + 0.587785i) q^{49} +(-1.29312 + 0.939507i) q^{50} +(0.0953844 - 0.293563i) q^{52} +(-0.200808 + 0.316423i) q^{55} +(-1.11316 - 3.42596i) q^{62} +(-1.05874 - 0.769217i) q^{64} -0.0470631 q^{65} -1.98423 q^{67} +(2.19334 - 1.59355i) q^{72} +(0.450527 + 1.38658i) q^{73} +4.30781 q^{76} +(-0.809017 - 0.587785i) q^{79} +(-0.299192 + 0.920819i) q^{80} +(0.309017 + 0.951057i) q^{81} +(1.30902 - 0.951057i) q^{83} +(-0.170232 - 2.70576i) q^{88} -1.27485 q^{89} +(-0.563797 - 0.409622i) q^{90} +(1.33119 + 4.09697i) q^{92} +(-0.202967 - 0.624667i) q^{95} +(0.688925 + 0.500534i) q^{97} -1.85955 q^{98} +(0.968583 + 0.248690i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) $ 20 * q - 5 * q^4 - 5 * q^9	$ 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} + 15 q^{20} - 5 q^{22} - 5 q^{25} + 15 q^{26} - 10 q^{32} - 5 q^{36} - 10 q^{40} - 5 q^{49} - 10 q^{50} - 10 q^{62} - 5 q^{64} - 10 q^{76} - 5 q^{79} - 10 q^{80} - 5 q^{81} + 15 q^{83} - 5 q^{88} + 15 q^{92} + 15 q^{95}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 + 15 * q^20 - 5 * q^22 - 5 * q^25 + 15 * q^26 - 10 * q^32 - 5 * q^36 - 10 * q^40 - 5 * q^49 - 10 * q^50 - 10 * q^62 - 5 * q^64 - 10 * q^76 - 5 * q^79 - 10 * q^80 - 5 * q^81 + 15 * q^83 - 5 * q^88 + 15 * q^92 + 15 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$2$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$3$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$3$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$4$	0.759544	+	2.33764i	0.759544	+	2.33764i
$5$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$5$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$6$		0			0
$7$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$7$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$8$	−0.837780	+	2.57842i	−0.837780	+	2.57842i
$9$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$10$	0.696891			0.696891
$11$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$11$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$12$		0			0
$13$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$13$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$14$		0			0
$15$		0			0
$16$	−2.09011	+	1.51855i	−2.09011	+	1.51855i
$17$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$17$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$18$	−0.574633	−	1.76854i	−0.574633	−	1.76854i
$19$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$19$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$20$	0.745220	+	0.541434i	0.745220	+	0.541434i
$21$		0			0
$22$	−1.80113	−	0.462452i	−1.80113	−	0.462452i
$23$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$23$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$24$		0			0
$25$	−0.265616	+	0.817483i	−0.265616	+	0.817483i
$26$	−0.0721631	−	0.222095i	−0.0721631	−	0.222095i
$27$		0			0
$28$		0			0
$29$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$29$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$30$		0			0
$31$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$31$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$32$	−2.09308			−2.09308
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.759544	−	2.33764i	0.759544	−	2.33764i
$37$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$37$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$38$	2.63665	−	1.91564i	2.63665	−	1.91564i
$39$		0			0
$40$	0.313968	+	0.966296i	0.313968	+	0.966296i
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−1.56675	−	1.89387i	−1.56675	−	1.89387i
$45$	−0.374763			−0.374763
$46$	2.63665	+	1.91564i	2.63665	+	1.91564i
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$		0			0
$49$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$50$	−1.29312	+	0.939507i	−1.29312	+	0.939507i
$51$		0			0
$52$	0.0953844	−	0.293563i	0.0953844	−	0.293563i
$53$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$53$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$54$		0			0
$55$	−0.200808	+	0.316423i	−0.200808	+	0.316423i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$61$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$62$	−1.11316	−	3.42596i	−1.11316	−	3.42596i
$63$		0			0
$64$	−1.05874	−	0.769217i	−1.05874	−	0.769217i
$65$	−0.0470631			−0.0470631
$66$		0			0
$67$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$67$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$	2.19334	−	1.59355i	2.19334	−	1.59355i
$73$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$73$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$74$		0			0
$75$		0			0
$76$	4.30781			4.30781
$77$		0			0
$78$		0			0
$79$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$79$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$80$	−0.299192	+	0.920819i	−0.299192	+	0.920819i
$81$	0.309017	+	0.951057i	0.309017	+	0.951057i
$82$		0			0
$83$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$83$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.170232	−	2.70576i	−0.170232	−	2.70576i
$89$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$89$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$90$	−0.563797	−	0.409622i	−0.563797	−	0.409622i
$91$		0			0
$92$	1.33119	+	4.09697i	1.33119	+	4.09697i
$93$		0			0
$94$		0			0
$95$	−0.202967	−	0.624667i	−0.202967	−	0.624667i
$96$		0			0
$97$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$97$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$98$	−1.85955			−1.85955
$99$	0.968583	+	0.248690i	0.968583	+	0.248690i
$100$	−2.11273			−2.11273
$101$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$101$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$102$		0			0
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$	0.275441	−	0.200120i	0.275441	−	0.200120i
$105$		0			0
$106$		0			0
$107$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$107$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−0.647953	+	0.256543i	−0.647953	+	0.256543i
$111$		0			0
$112$		0			0
$113$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$113$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$114$		0			0
$115$	0.531374	−	0.386066i	0.531374	−	0.386066i
$116$		0			0
$117$	0.0388067	+	0.119435i	0.0388067	+	0.119435i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.728969	−	0.684547i	0.728969	−	0.684547i
$122$		0			0
$123$		0			0
$124$	1.47136	−	4.52839i	1.47136	−	4.52839i
$125$	0.215351	+	0.662783i	0.215351	+	0.662783i
$126$		0			0
$127$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$127$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$128$	−0.105209	−	0.323801i	−0.105209	−	0.323801i
$129$		0			0
$130$	−0.0708022	−	0.0514408i	−0.0708022	−	0.0514408i
$131$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$131$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$132$		0			0
$133$		0			0
$134$	−2.98509	−	2.16880i	−2.98509	−	2.16880i
$135$		0			0
$136$		0			0
$137$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$137$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$138$		0			0
$139$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$139$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.121636	+	0.0312307i	0.121636	+	0.0312307i
$144$	2.58352			2.58352
$145$		0			0
$146$	−0.837780	+	2.57842i	−0.837780	+	2.57842i
$147$		0			0
$148$		0			0
$149$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$149$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$151$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$152$	3.84407	+	2.79288i	3.84407	+	2.79288i
$153$		0			0
$154$		0			0
$155$	−0.725978			−0.725978
$156$		0			0
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$	−0.574633	−	1.76854i	−0.574633	−	1.76854i
$159$		0			0
$160$	−0.634599	+	0.461063i	−0.634599	+	0.461063i
$161$		0			0
$162$	−0.574633	+	1.76854i	−0.574633	+	1.76854i
$163$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$163$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$164$		0			0
$165$		0			0
$166$	3.00882			3.00882
$167$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$167$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$168$		0			0
$169$	−0.304144	−	0.936058i	−0.304144	−	0.936058i
$170$		0			0
$171$	−1.41789	+	1.03016i	−1.41789	+	1.03016i
$172$		0			0
$173$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$173$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$174$		0			0
$175$		0			0
$176$	1.38432	−	2.18134i	1.38432	−	2.18134i
$177$		0			0
$178$	−1.91789	−	1.39343i	−1.91789	−	1.39343i
$179$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$179$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$180$	−0.284649	−	0.876059i	−0.284649	−	0.876059i
$181$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$181$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$182$		0			0
$183$		0			0
$184$	−1.46830	+	4.51897i	−1.46830	+	4.51897i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.377427	−	1.16160i	0.377427	−	1.16160i
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$		0			0
$193$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$193$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$194$	0.489334	+	1.50602i	0.489334	+	1.50602i
$195$		0			0
$196$	−1.98851	−	1.44474i	−1.98851	−	1.44474i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	1.18532	+	1.43281i	1.18532	+	1.43281i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−1.88529	−	1.36974i	−1.88529	−	1.36974i
$201$		0			0
$202$	0.732570	+	2.25462i	0.732570	+	2.25462i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.41789	−	1.03016i	−1.41789	−	1.03016i
$208$	0.324441			0.324441
$209$	0.110048	+	1.74915i	0.110048	+	1.74915i
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.892204	−	0.229079i	−0.892204	−	0.229079i
$221$		0			0
$222$		0			0
$223$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$223$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	0.695393	−	0.505233i	0.695393	−	0.505233i
$226$		0			0
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$		0			0
$229$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$229$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$230$	1.22138			1.22138
$231$		0			0
$232$		0			0
$233$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$233$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$234$	−0.0721631	+	0.222095i	−0.0721631	+	0.222095i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$239$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$240$		0			0
$241$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$241$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$242$	1.84489	−	0.233064i	1.84489	−	0.233064i
$243$		0			0
$244$		0			0
$245$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$246$		0			0
$247$	−0.178061	+	0.129369i	−0.178061	+	0.129369i
$248$	4.24886	−	3.08697i	4.24886	−	3.08697i
$249$		0			0
$250$	−0.400457	+	1.23248i	−0.400457	+	1.23248i
$251$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$251$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$252$		0			0
$253$	−1.62954	+	0.645180i	−1.62954	+	0.645180i
$254$		0			0
$255$		0			0
$256$	−0.208759	+	0.642495i	−0.208759	+	0.642495i
$257$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$257$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$258$		0			0
$259$		0			0
$260$	−0.0357465	−	0.110016i	−0.0357465	−	0.110016i
$261$		0			0
$262$	2.19334	+	1.59355i	2.19334	+	1.59355i
$263$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$263$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.50711	−	4.63841i	−1.50711	−	4.63841i
$269$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$269$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$270$		0			0
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.0539718	−	0.857857i	−0.0539718	−	0.857857i
$276$		0			0
$277$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$277$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$278$		0			0
$279$	0.598617	+	1.84235i	0.598617	+	1.84235i
$280$		0			0
$281$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$281$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$282$		0			0
$283$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$283$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$284$		0			0
$285$		0			0
$286$	0.148854	+	0.179934i	0.148854	+	0.179934i
$287$		0			0
$288$	1.69334	+	1.23028i	1.69334	+	1.23028i
$289$	0.309017	−	0.951057i	0.309017	−	0.951057i
$290$		0			0
$291$		0			0
$292$	−2.89913	+	2.10634i	−2.89913	+	2.10634i
$293$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$293$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.178061	−	0.129369i	−0.178061	−	0.129369i
$300$		0			0
$301$		0			0
$302$	−2.98509	+	2.16880i	−2.98509	+	2.16880i
$303$		0			0
$304$	1.39920	+	4.30630i	1.39920	+	4.30630i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−1.09217	−	0.793506i	−1.09217	−	0.793506i
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$313$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$314$		0			0
$315$		0			0
$316$	0.759544	−	2.33764i	0.759544	−	2.33764i
$317$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$317$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$318$		0			0
$319$		0			0
$320$	−0.490441			−0.490441
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.98851	+	1.44474i	−1.98851	+	1.44474i
$325$	0.0873282	−	0.0634476i	0.0873282	−	0.0634476i
$326$	−0.615808	−	1.89526i	−0.615808	−	1.89526i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	3.21748	+	2.33764i	3.21748	+	2.33764i
$333$		0			0
$334$	−1.11316	−	3.42596i	−1.11316	−	3.42596i
$335$	−0.601597	+	0.437086i	−0.601597	+	0.437086i
$336$		0			0
$337$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$337$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$338$	0.565571	−	1.74065i	0.565571	−	1.74065i
$339$		0			0
$340$		0			0
$341$	1.87631	+	0.481754i	1.87631	+	0.481754i
$342$	−3.25908			−3.25908
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$347$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$348$		0			0
$349$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$349$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$350$		0			0
$351$		0			0
$352$	1.94609	−	0.770513i	1.94609	−	0.770513i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	−0.968304	−	2.98013i	−0.968304	−	2.98013i
$357$		0			0
$358$	−2.43419	+	1.76854i	−2.43419	+	1.76854i
$359$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$359$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$360$	0.313968	−	0.966296i	0.313968	−	0.966296i
$361$	−1.67600	−	1.21769i	−1.67600	−	1.21769i
$362$	−0.233525			−0.233525
$363$		0			0
$364$		0			0
$365$	0.442031	+	0.321154i	0.442031	+	0.321154i
$366$		0			0
$367$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$367$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$368$	−3.66316	+	2.66144i	−3.66316	+	2.66144i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$379$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$380$	1.30608	−	0.948925i	1.30608	−	0.948925i
$381$		0			0
$382$		0			0
$383$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$383$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.646797	+	1.99064i	−0.646797	+	1.99064i
$389$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$389$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$390$		0			0
$391$		0			0
$392$	−0.837780	−	2.57842i	−0.837780	−	2.57842i
$393$		0			0
$394$		0			0
$395$	−0.374763			−0.374763
$396$	0.154335	+	2.45309i	0.154335	+	2.45309i
$397$	−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
$400$	−0.686225	−	2.11198i	−0.686225	−	2.11198i
$401$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$401$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$	0.0751750	+	0.231365i	0.0751750	+	0.231365i
$404$	−0.968304	+	2.98013i	−0.968304	+	2.98013i
$405$	0.303189	+	0.220280i	0.303189	+	0.220280i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$409$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−1.00711	−	3.09957i	−1.00711	−	3.09957i
$415$	0.187381	−	0.576700i	0.187381	−	0.576700i
$416$	0.212651	+	0.154500i	0.212651	+	0.154500i
$417$		0			0
$418$	−1.74630	+	2.75173i	−1.74630	+	2.75173i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$421$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$422$		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
$431$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$431$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$432$		0			0
$433$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$433$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.949193	−	2.92132i	0.949193	−	2.92132i
$438$		0			0
$439$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$439$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$440$	−0.647638	−	0.782859i	−0.647638	−	0.782859i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$443$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$444$		0			0
$445$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$446$	−2.43419	+	1.76854i	−2.43419	+	1.76854i
$447$		0			0
$448$		0			0
$449$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$449$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$450$	1.59838			1.59838
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$457$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$458$		0			0
$459$		0			0
$460$	1.30608	+	0.948925i	1.30608	+	0.948925i
$461$		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
$467$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$467$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$468$	−0.249720	+	0.181432i	−0.249720	+	0.181432i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.21875	+	0.885477i	1.21875	+	0.885477i
$476$		0			0
$477$		0			0
$478$	−0.563797	+	0.409622i	−0.563797	+	0.409622i
$479$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$479$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$480$		0			0
$481$		0			0
$482$	1.61221	+	1.17134i	1.61221	+	1.17134i
$483$		0			0
$484$	2.15391	+	1.18412i	2.15391	+	1.18412i
$485$	0.319132			0.319132
$486$		0			0
$487$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$487$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$488$		0			0
$489$		0			0
$490$	−0.563797	+	0.409622i	−0.563797	+	0.409622i
$491$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$491$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$		0			0
$494$	−0.409278			−0.409278
$495$	0.348445	−	0.137959i	0.348445	−	0.137959i
$496$	5.00470			5.00470
$497$		0			0
$498$		0			0
$499$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$499$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$500$	−1.38578	+	1.00683i	−1.38578	+	1.00683i
$501$		0			0
$502$		0			0
$503$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$503$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$504$		0			0
$505$	0.477765			0.477765
$506$	−3.15669	−	0.810500i	−3.15669	−	0.810500i
$507$		0			0
$508$		0			0
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$		0			0
$512$	−1.29176	+	0.938518i	−1.29176	+	0.938518i
$513$		0			0
$514$	−0.0721631	+	0.222095i	−0.0721631	+	0.222095i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.0394285	−	0.121348i	0.0394285	−	0.121348i
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$		0			0
$523$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$523$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$524$	1.10737	+	3.40813i	1.10737	+	3.40813i
$525$		0			0
$526$	1.61221	+	1.17134i	1.61221	+	1.17134i
$527$		0			0
$528$		0			0
$529$	2.07165			2.07165
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	1.66235	−	5.11618i	1.66235	−	5.11618i
$537$		0			0
$538$	3.68978			3.68978
$539$	0.535827	−	0.844328i	0.535827	−	0.844328i
$540$		0			0
$541$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$541$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$547$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$548$		0			0
$549$		0			0
$550$	0.856457	−	1.34956i	0.856457	−	1.34956i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.215351	+	0.662783i	0.215351	+	0.662783i
$555$		0			0
$556$		0			0
$557$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$557$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$558$	−1.11316	+	3.42596i	−1.11316	+	3.42596i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.696891			0.696891
$563$	−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
$566$	2.91429	−	2.11736i	2.91429	−	2.11736i
$567$		0			0
$568$		0			0
$569$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$569$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$570$		0			0
$571$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$571$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$572$	0.0193816	+	0.308061i	0.0193816	+	0.308061i
$573$		0			0
$574$		0			0
$575$	−0.465523	+	1.43273i	−0.465523	+	1.43273i
$576$	0.404401	+	1.24462i	0.404401	+	1.24462i
$577$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$577$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$578$	1.50441	−	1.09302i	1.50441	−	1.09302i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−3.95263			−3.95263
$585$	0.0380748	+	0.0276630i	0.0380748	+	0.0276630i
$586$		0			0
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$		0			0
$589$	−2.74670	+	1.99559i	−2.74670	+	1.99559i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$593$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−0.126474	−	0.389247i	−0.126474	−	0.389247i
$599$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$599$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$600$		0			0
$601$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$601$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$	1.60528	+	1.16630i	1.60528	+	1.16630i
$604$	−4.87711			−4.87711
$605$	0.0702235	−	0.368125i	0.0702235	−	0.368125i
$606$		0			0
$607$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$607$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$608$	−1.13358	+	3.48881i	−1.13358	+	3.48881i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$613$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$617$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$618$		0			0
$619$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$619$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$620$	−0.551412	−	1.69707i	−0.551412	−	1.69707i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.484103	−	0.351721i	−0.484103	−	0.351721i
$626$	−3.60226			−3.60226
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$631$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$632$	2.19334	−	1.59355i	2.19334	−	1.59355i
$633$		0			0
$634$	−0.355143	−	1.09302i	−0.355143	−	1.09302i
$635$		0			0
$636$		0			0
$637$	0.125581			0.125581
$638$		0			0
$639$		0			0
$640$	−0.103225	−	0.0749974i	−0.103225	−	0.0749974i
$641$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$641$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$642$		0			0
$643$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$643$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$	−2.71111			−2.71111
$649$		0			0
$650$	0.200727			0.200727
$651$		0			0
$652$	0.813968	−	2.50514i	0.813968	−	2.50514i
$653$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$653$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$654$		0			0
$655$	0.442031	−	0.321154i	0.442031	−	0.321154i
$656$		0			0
$657$	0.450527	−	1.38658i	0.450527	−	1.38658i
$658$		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
$664$	1.35556	+	4.17197i	1.35556	+	4.17197i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	1.47136	−	4.52839i	1.47136	−	4.52839i
$669$		0			0
$670$	−1.38279			−1.38279
$671$		0			0
$672$		0			0
$673$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$673$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$674$	0.489334	−	1.50602i	0.489334	−	1.50602i
$675$		0			0
$676$	1.95715	−	1.42195i	1.95715	−	1.42195i
$677$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$677$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	2.29617	+	2.77559i	2.29617	+	2.77559i
$683$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$683$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$684$	−3.48509	−	2.53207i	−3.48509	−	2.53207i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$691$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$692$		0			0
$693$		0			0
$694$	3.68978			3.68978
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$701$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$702$		0			0
$703$		0			0
$704$	1.26756	+	0.325453i	1.26756	+	0.325453i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$709$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$710$		0			0
$711$	0.309017	+	0.951057i	0.309017	+	0.951057i
$712$	1.06804	−	3.28709i	1.06804	−	3.28709i
$713$	−2.74670	−	1.99559i	−2.74670	−	1.99559i
$714$		0			0
$715$	0.0437581	−	0.0173251i	0.0437581	−	0.0173251i
$716$	−3.97703			−3.97703
$717$		0			0
$718$		0			0
$719$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$719$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$720$	0.783295	−	0.569097i	0.783295	−	0.569097i
$721$		0			0
$722$	−1.19044	−	3.66380i	−1.19044	−	3.66380i
$723$		0			0
$724$	−0.249720	−	0.181432i	−0.249720	−	0.181432i
$725$		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
$730$	0.313968	+	0.966296i	0.313968	+	0.966296i
$731$		0			0
$732$		0			0
$733$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$733$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$734$	0.732570	−	2.25462i	0.732570	−	2.25462i
$735$		0			0
$736$	−3.66836			−3.66836
$737$	1.84489	−	0.730444i	1.84489	−	0.730444i
$738$		0			0
$739$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$739$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$743$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	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
$751$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$751$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.229790	+	0.707220i	0.229790	+	0.707220i
$756$		0			0
$757$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$757$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$		0			0
$760$	1.78070			1.78070
$761$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$761$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1.06856	+	3.28869i	1.06856	+	3.28869i
$767$		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
$773$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$773$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$774$		0			0
$775$	1.34709	−	0.978720i	1.34709	−	0.978720i
$776$	−1.86775	+	1.35700i	−1.86775	+	1.35700i
$777$		0			0
$778$	−1.11316	+	3.42596i	−1.11316	+	3.42596i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.798351	−	2.45707i	0.798351	−	2.45707i
$785$		0			0
$786$		0			0
$787$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$787$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$788$		0			0
$789$		0			0
$790$	−0.563797	−	0.409622i	−0.563797	−	0.409622i
$791$		0			0
$792$	−1.45269	+	2.28907i	−1.45269	+	2.28907i
$793$		0			0
$794$	−2.43419	−	1.76854i	−2.43419	−	1.76854i
$795$		0			0
$796$		0			0
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	0.555956	−	1.71106i	0.555956	−	1.71106i
$801$	1.03137	+	0.749337i	1.03137	+	0.749337i
$802$		0			0
$803$	−0.929324	−	1.12336i	−0.929324	−	1.12336i
$804$		0			0
$805$		0			0
$806$	−0.139792	+	0.430235i	−0.139792	+	0.430235i
$807$		0			0
$808$	−2.79617	+	2.03154i	−2.79617	+	2.03154i
$809$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$809$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$810$	0.215351	+	0.662783i	0.215351	+	0.662783i
$811$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$811$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.401616			−0.401616
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$821$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$822$		0			0
$823$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$823$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$827$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$828$	1.33119	−	4.09697i	1.33119	−	4.09697i
$829$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$829$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$830$	0.912242	−	0.662783i	0.912242	−	0.662783i
$831$		0			0
$832$	0.0507851	+	0.156301i	0.0507851	+	0.156301i
$833$		0			0
$834$		0			0
$835$	−0.725978			−0.725978
$836$	−4.00530	+	1.58581i	−4.00530	+	1.58581i
$837$		0			0
$838$		0			0
$839$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$839$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$840$		0			0
$841$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$842$	−1.28109	+	0.930769i	−1.28109	+	0.930769i
$843$		0			0
$844$		0			0
$845$	−0.298408	−	0.216806i	−0.298408	−	0.216806i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$	−0.202967	+	0.624667i	−0.202967	+	0.624667i
$856$		0			0
$857$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$857$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−0.837780	−	2.57842i	−0.837780	−	2.57842i
$863$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$863$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$864$		0			0
$865$		0			0
$866$	−1.00711	+	3.09957i	−1.00711	+	3.09957i
$867$		0			0
$868$		0			0
$869$	0.968583	+	0.248690i	0.968583	+	0.248690i
$870$		0			0
$871$	0.201592	+	0.146465i	0.201592	+	0.146465i
$872$		0			0
$873$	−0.263146	−	0.809880i	−0.263146	−	0.809880i
$874$	4.62103	−	3.35737i	4.62103	−	3.35737i
$875$		0			0
$876$		0			0
$877$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$877$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$878$	−1.28109	−	0.930769i	−1.28109	−	0.930769i
$879$		0			0
$880$	−0.0607942	−	0.966296i	−0.0607942	−	0.966296i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.50441	+	1.09302i	1.50441	+	1.09302i
$883$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$883$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$887$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$888$		0			0
$889$		0			0
$890$	−0.888430			−0.888430
$891$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$892$	−3.97703			−3.97703
$893$		0			0
$894$		0			0
$895$	0.187381	+	0.576700i	0.187381	+	0.576700i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.70923	+	1.24183i	1.70923	+	1.24183i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.0145433	+	0.0447596i	−0.0145433	+	0.0447596i
$906$		0			0
$907$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$907$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$908$		0			0
$909$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$910$		0			0
$911$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$911$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$912$		0			0
$913$	−0.866986	+	1.36615i	−0.866986	+	1.36615i
$914$	2.37065			2.37065
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$919$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$920$	0.550265	+	1.69354i	0.550265	+	1.69354i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$	0.541587	+	1.66683i	0.541587	+	1.66683i
$932$		0			0
$933$		0			0
$934$	3.45794			3.45794
$935$		0			0
$936$	−0.340464			−0.340464
$937$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$937$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$941$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	0.0565777	−	0.174128i	0.0565777	−	0.174128i
$950$	0.865665	+	2.66424i	0.865665	+	2.66424i
$951$		0			0
$952$		0			0
$953$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$953$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$954$		0			0
$955$		0			0
$956$	−0.921143			−0.921143
$957$		0			0
$958$	3.00882			3.00882
$959$		0			0
$960$		0			0
$961$	0.850604	+	2.61789i	0.850604	+	2.61789i
$962$		0			0
$963$		0			0
$964$	0.813968	+	2.50514i	0.813968	+	2.50514i
$965$		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	$0.800000\pi$
$968$	1.15434	+	2.45309i	1.15434	+	2.45309i
$969$		0			0
$970$	0.480106	+	0.348817i	0.480106	+	0.348817i
$971$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$971$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$972$		0			0
$973$		0			0
$974$	0.188925	−	0.137262i	0.188925	−	0.137262i
$975$		0			0
$976$		0			0
$977$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$977$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$978$		0			0
$979$	1.18532	−	0.469303i	1.18532	−	0.469303i
$980$	−0.921143			−0.921143
$981$		0			0
$982$		0			0
$983$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$983$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.437662	−	0.317980i	−0.437662	−	0.317980i
$989$		0			0
$990$	0.674997	+	0.173310i	0.674997	+	0.173310i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	3.28027	+	2.38326i	3.28027	+	2.38326i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$997$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$998$	1.06856	−	3.28869i	1.06856	−	3.28869i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.1.j.a.631.5	yes	20
11.3	even	5	inner	869.1.j.a.157.5	✓	20
79.78	odd	2	CM	869.1.j.a.631.5	yes	20
869.157	odd	10	inner	869.1.j.a.157.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.1.j.a.157.5	✓	20	11.3	even	5	inner
869.1.j.a.157.5	✓	20	869.157	odd	10	inner
869.1.j.a.631.5	yes	20	1.1	even	1	trivial
869.1.j.a.631.5	yes	20	79.78	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.303189 - 0.220280i) q^{5} +(-0.837780 + 2.57842i) q^{8} +(-0.809017 - 0.587785i) q^{9} +O(q^{10})\)	\(q+(1.50441 + 1.09302i) q^{2} +(0.759544 + 2.33764i) q^{4} +(0.303189 - 0.220280i) q^{5} +(-0.837780 + 2.57842i) q^{8} +(-0.809017 - 0.587785i) q^{9} +0.696891 q^{10} +(-0.929776 + 0.368125i) q^{11} +(-0.101597 - 0.0738147i) q^{13} +(-2.09011 + 1.51855i) q^{16} +(-0.574633 - 1.76854i) q^{18} +(0.541587 - 1.66683i) q^{19} +(0.745220 + 0.541434i) q^{20} +(-1.80113 - 0.462452i) q^{22} +1.75261 q^{23} +(-0.265616 + 0.817483i) q^{25} +(-0.0721631 - 0.222095i) q^{26} +(-1.56720 - 1.13864i) q^{31} -2.09308 q^{32} +(0.759544 - 2.33764i) q^{36} +(2.63665 - 1.91564i) q^{38} +(0.313968 + 0.966296i) q^{40} +(-1.56675 - 1.89387i) q^{44} -0.374763 q^{45} +(2.63665 + 1.91564i) q^{46} +(-0.809017 + 0.587785i) q^{49} +(-1.29312 + 0.939507i) q^{50} +(0.0953844 - 0.293563i) q^{52} +(-0.200808 + 0.316423i) q^{55} +(-1.11316 - 3.42596i) q^{62} +(-1.05874 - 0.769217i) q^{64} -0.0470631 q^{65} -1.98423 q^{67} +(2.19334 - 1.59355i) q^{72} +(0.450527 + 1.38658i) q^{73} +4.30781 q^{76} +(-0.809017 - 0.587785i) q^{79} +(-0.299192 + 0.920819i) q^{80} +(0.309017 + 0.951057i) q^{81} +(1.30902 - 0.951057i) q^{83} +(-0.170232 - 2.70576i) q^{88} -1.27485 q^{89} +(-0.563797 - 0.409622i) q^{90} +(1.33119 + 4.09697i) q^{92} +(-0.202967 - 0.624667i) q^{95} +(0.688925 + 0.500534i) q^{97} -1.85955 q^{98} +(0.968583 + 0.248690i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 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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