LMFDB - Embedded newform 820.1.bz.a.263.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [820,1,Mod(7,820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(820, base_ring=CyclotomicField(40))

chi = DirichletCharacter(H, H._module([20, 10, 39]))

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

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

chi := DirichletCharacter("820.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 820 = 2^{2} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	820.bz (of order $40$, degree $16$, 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:	$0.409233310359$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} - \cdots)$

Embedding invariants

Embedding label			263.1
Root			$0.891007 + 0.453990i$ of defining polynomial
Character	$\chi$	$=$	820.263
Dual form			820.1.bz.a.767.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.809017 + 0.587785i) q^{2} +(0.309017 + 0.951057i) q^{4} +(0.453990 + 0.891007i) q^{5} +(-0.309017 + 0.951057i) q^{8} +(-0.707107 - 0.707107i) q^{9} +(-0.156434 + 0.987688i) q^{10} +(1.10749 - 0.678671i) q^{13} +(-0.809017 + 0.587785i) q^{16} +(-1.04178 + 0.0819895i) q^{17} +(-0.156434 - 0.987688i) q^{18} +(-0.707107 + 0.707107i) q^{20} +(-0.587785 + 0.809017i) q^{25} +(1.29489 + 0.101910i) q^{26} +(-1.20002 - 1.40505i) q^{29} -1.00000 q^{32} +(-0.891007 - 0.546010i) q^{34} +(0.453990 - 0.891007i) q^{36} +(1.69480 + 0.863541i) q^{37} +(-0.987688 + 0.156434i) q^{40} +(0.587785 - 0.809017i) q^{41} +(0.309017 - 0.951057i) q^{45} +(0.891007 - 0.453990i) q^{49} +(-0.951057 + 0.309017i) q^{50} +(0.987688 + 0.843566i) q^{52} +(0.0366318 - 0.465451i) q^{53} +(-0.144974 - 1.84206i) q^{58} +(-1.76007 - 0.278768i) q^{61} +(-0.809017 - 0.587785i) q^{64} +(1.10749 + 0.678671i) q^{65} +(-0.399903 - 0.965451i) q^{68} +(0.891007 - 0.453990i) q^{72} +0.907981i q^{73} +(0.863541 + 1.69480i) q^{74} +(-0.891007 - 0.453990i) q^{80} +1.00000i q^{81} +(0.951057 - 0.309017i) q^{82} +(-0.546010 - 0.891007i) q^{85} +(-0.399903 + 0.652583i) q^{89} +(0.809017 - 0.587785i) q^{90} +(0.101910 + 0.119322i) q^{97} +(0.987688 + 0.156434i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} - 4 q^{4} + 4 q^{8} - 4 q^{16} + 4 q^{29} - 16 q^{32} - 4 q^{45} - 4 q^{58} - 4 q^{61} - 4 q^{64} - 16 q^{85} + 4 q^{90} + 4 q^{97}+O(q^{100}) $ 16 * q + 4 * q^2 - 4 * q^4 + 4 * q^8 - 4 * q^16 + 4 * q^29 - 16 * q^32 - 4 * q^45 - 4 * q^58 - 4 * q^61 - 4 * q^64 - 16 * q^85 + 4 * q^90 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$.

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$e\left(\frac{33}{40}\right)$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$	0.809017	+	0.587785i	0.809017	+	0.587785i
$2$	0.809017	+	0.587785i	0.809017	+	0.587785i
$3$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$3$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$4$	0.309017	+	0.951057i	0.309017	+	0.951057i
$5$	0.453990	+	0.891007i	0.453990	+	0.891007i
$5$	0.453990	+	0.891007i	0.453990	+	0.891007i
$6$		0			0
$7$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$7$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$8$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$9$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$10$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$11$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$11$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$12$		0			0
$13$	1.10749	−	0.678671i	1.10749	−	0.678671i	0.156434	−	0.987688i	$-0.450000\pi$
$13$	1.10749	−	0.678671i	1.10749	−	0.678671i	0.951057	+	0.309017i	$0.100000\pi$
$14$		0			0
$15$		0			0
$16$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$17$	−1.04178	+	0.0819895i	−1.04178	+	0.0819895i	−0.587785	−	0.809017i	$-0.700000\pi$
$17$	−1.04178	+	0.0819895i	−1.04178	+	0.0819895i	−0.453990	+	0.891007i	$0.650000\pi$
$18$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$19$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$19$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$20$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$21$		0			0
$22$		0			0
$23$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$23$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$24$		0			0
$25$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$26$	1.29489	+	0.101910i	1.29489	+	0.101910i
$27$		0			0
$28$		0			0
$29$	−1.20002	−	1.40505i	−1.20002	−	1.40505i	−0.891007	−	0.453990i	$-0.850000\pi$
$29$	−1.20002	−	1.40505i	−1.20002	−	1.40505i	−0.309017	−	0.951057i	$-0.600000\pi$
$30$		0			0
$31$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$31$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$32$	−1.00000			−1.00000
$33$		0			0
$34$	−0.891007	−	0.546010i	−0.891007	−	0.546010i
$35$		0			0
$36$	0.453990	−	0.891007i	0.453990	−	0.891007i
$37$	1.69480	+	0.863541i	1.69480	+	0.863541i	0.987688	+	0.156434i	$0.0500000\pi$
$37$	1.69480	+	0.863541i	1.69480	+	0.863541i	0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$41$	0.587785	−	0.809017i	0.587785	−	0.809017i
$41$	0.587785	−	0.809017i	0.587785	−	0.809017i
$42$		0			0
$43$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$43$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$44$		0			0
$45$	0.309017	−	0.951057i	0.309017	−	0.951057i
$46$		0			0
$47$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$47$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$48$		0			0
$49$	0.891007	−	0.453990i	0.891007	−	0.453990i
$50$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$51$		0			0
$52$	0.987688	+	0.843566i	0.987688	+	0.843566i
$53$	0.0366318	−	0.465451i	0.0366318	−	0.465451i	−0.951057	−	0.309017i	$-0.900000\pi$
$53$	0.0366318	−	0.465451i	0.0366318	−	0.465451i	0.987688	−	0.156434i	$-0.0500000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.144974	−	1.84206i	−0.144974	−	1.84206i
$59$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$59$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$60$		0			0
$61$	−1.76007	−	0.278768i	−1.76007	−	0.278768i	−0.809017	−	0.587785i	$-0.800000\pi$
$61$	−1.76007	−	0.278768i	−1.76007	−	0.278768i	−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$		0			0
$64$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$65$	1.10749	+	0.678671i	1.10749	+	0.678671i
$66$		0			0
$67$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$67$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$68$	−0.399903	−	0.965451i	−0.399903	−	0.965451i
$69$		0			0
$70$		0			0
$71$		0			0		0.996917	−	0.0784591i	$-0.0250000\pi$
$71$		0			0		−0.996917	+	0.0784591i	$0.975000\pi$
$72$	0.891007	−	0.453990i	0.891007	−	0.453990i
$73$			0.907981i			0.907981i	0.891007	+	0.453990i	$0.150000\pi$
$73$			0.907981i			0.907981i	−0.891007	+	0.453990i	$0.850000\pi$
$74$	0.863541	+	1.69480i	0.863541	+	1.69480i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$79$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$80$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$81$			1.00000i			1.00000i
$82$	0.951057	−	0.309017i	0.951057	−	0.309017i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−0.546010	−	0.891007i	−0.546010	−	0.891007i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.399903	+	0.652583i	−0.399903	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$89$	−0.399903	+	0.652583i	−0.399903	+	0.652583i	0.587785	+	0.809017i	$0.300000\pi$
$90$	0.809017	−	0.587785i	0.809017	−	0.587785i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.101910	+	0.119322i	0.101910	+	0.119322i	0.809017	−	0.587785i	$-0.200000\pi$
$97$	0.101910	+	0.119322i	0.101910	+	0.119322i	−0.707107	+	0.707107i	$0.750000\pi$
$98$	0.987688	+	0.156434i	0.987688	+	0.156434i
$99$		0			0
$100$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$101$	−0.453990	−	1.89101i	−0.453990	−	1.89101i	−0.453990	−	0.891007i	$-0.650000\pi$
$101$	−0.453990	−	1.89101i	−0.453990	−	1.89101i		−	1.00000i	$-0.5\pi$
$102$		0			0
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$	0.303221	+	1.26301i	0.303221	+	1.26301i
$105$		0			0
$106$	0.303221	−	0.355026i	0.303221	−	0.355026i
$107$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$107$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$108$		0			0
$109$	0.652583	−	1.57547i	0.652583	−	1.57547i	−0.156434	−	0.987688i	$-0.550000\pi$
$109$	0.652583	−	1.57547i	0.652583	−	1.57547i	0.809017	−	0.587785i	$-0.200000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$113$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	−0.987688	+	0.156434i	$0.950000\pi$
$114$		0			0
$115$		0			0
$116$	0.965451	−	1.57547i	0.965451	−	1.57547i
$117$	−1.26301	−	0.303221i	−1.26301	−	0.303221i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$122$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$123$		0			0
$124$		0			0
$125$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$126$		0			0
$127$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$127$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$128$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$129$		0			0
$130$	0.497066	+	1.20002i	0.497066	+	1.20002i
$131$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$131$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.243950	−	1.01612i	0.243950	−	1.01612i
$137$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$137$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$138$		0			0
$139$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$139$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.987688	+	0.156434i	0.987688	+	0.156434i
$145$	0.707107	−	1.70711i	0.707107	−	1.70711i
$146$	−0.533698	+	0.734572i	−0.533698	+	0.734572i
$147$		0			0
$148$	−0.297556	+	1.87869i	−0.297556	+	1.87869i
$149$	−0.156434	+	1.98769i	−0.156434	+	1.98769i			1.00000i	$0.5\pi$
$149$	−0.156434	+	1.98769i	−0.156434	+	1.98769i	−0.156434	+	0.987688i	$0.550000\pi$
$150$		0			0
$151$		0			0		−0.972370	−	0.233445i	$-0.925000\pi$
$151$		0			0		0.972370	+	0.233445i	$0.0750000\pi$
$152$		0			0
$153$	0.794622	+	0.678671i	0.794622	+	0.678671i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.465451	+	1.93874i	0.465451	+	1.93874i	0.309017	+	0.951057i	$0.400000\pi$
$157$	0.465451	+	1.93874i	0.465451	+	1.93874i	0.156434	+	0.987688i	$0.450000\pi$
$158$		0			0
$159$		0			0
$160$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$161$		0			0
$162$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	0.951057	+	0.309017i	0.951057	+	0.309017i
$165$		0			0
$166$		0			0
$167$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$167$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$168$		0			0
$169$	0.311951	−	0.612238i	0.311951	−	0.612238i
$170$	0.0819895	−	1.04178i	0.0819895	−	1.04178i
$171$		0			0
$172$		0			0
$173$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$173$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$179$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$179$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$180$	1.00000			1.00000
$181$	−1.15643	−	0.987688i	−1.15643	−	0.987688i	−0.156434	−	0.987688i	$-0.550000\pi$
$181$	−1.15643	−	0.987688i	−1.15643	−	0.987688i	−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			1.90211i			1.90211i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$191$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$192$		0			0
$193$	1.70002	+	0.133795i	1.70002	+	0.133795i	0.891007	−	0.453990i	$-0.150000\pi$
$193$	1.70002	+	0.133795i	1.70002	+	0.133795i	0.809017	+	0.587785i	$0.200000\pi$
$194$	0.0123117	+	0.156434i	0.0123117	+	0.156434i
$195$		0			0
$196$	0.707107	+	0.707107i	0.707107	+	0.707107i
$197$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$197$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$198$		0			0
$199$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$199$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$200$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$201$		0			0
$202$	0.744220	−	1.79671i	0.744220	−	1.79671i
$203$		0			0
$204$		0			0
$205$	0.987688	+	0.156434i	0.987688	+	0.156434i
$206$		0			0
$207$		0			0
$208$	−0.497066	+	1.20002i	−0.497066	+	1.20002i
$209$		0			0
$210$		0			0
$211$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$211$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$212$	0.453990	−	0.108993i	0.453990	−	0.108993i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.45399	−	0.891007i	1.45399	−	0.891007i
$219$		0			0
$220$		0			0
$221$	−1.09811	+	0.797826i	−1.09811	+	0.797826i
$222$		0			0
$223$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$223$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$224$		0			0
$225$	0.987688	−	0.156434i	0.987688	−	0.156434i
$226$	0.0966818	−	0.610425i	0.0966818	−	0.610425i
$227$		0			0		0.522499	−	0.852640i	$-0.325000\pi$
$227$		0			0		−0.522499	+	0.852640i	$0.675000\pi$
$228$		0			0
$229$	−0.119322	−	0.101910i	−0.119322	−	0.101910i	0.587785	−	0.809017i	$-0.300000\pi$
$229$	−0.119322	−	0.101910i	−0.119322	−	0.101910i	−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$	1.70711	−	0.707107i	1.70711	−	0.707107i
$233$	1.01612	+	1.65816i	1.01612	+	1.65816i	0.707107	+	0.707107i	$0.250000\pi$
$233$	1.01612	+	1.65816i	1.01612	+	1.65816i	0.309017	+	0.951057i	$0.400000\pi$
$234$	−0.843566	−	0.987688i	−0.843566	−	0.987688i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$239$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$240$		0			0
$241$	−0.896802	+	1.76007i	−0.896802	+	1.76007i	−0.309017	+	0.951057i	$0.600000\pi$
$241$	−0.896802	+	1.76007i	−0.896802	+	1.76007i	−0.587785	+	0.809017i	$0.700000\pi$
$242$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$243$		0			0
$244$	−0.278768	−	1.76007i	−0.278768	−	1.76007i
$245$	0.809017	+	0.587785i	0.809017	+	0.587785i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$251$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$251$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	−	0.951057i	0.309017	−	0.951057i
$257$	−0.581990	−	0.497066i	−0.581990	−	0.497066i	0.309017	−	0.951057i	$-0.400000\pi$
$257$	−0.581990	−	0.497066i	−0.581990	−	0.497066i	−0.891007	+	0.453990i	$0.850000\pi$
$258$		0			0
$259$		0			0
$260$	−0.303221	+	1.26301i	−0.303221	+	1.26301i
$261$	−0.144974	+	1.84206i	−0.144974	+	1.84206i
$262$		0			0
$263$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$263$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$264$		0			0
$265$	0.431351	−	0.178671i	0.431351	−	0.178671i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$269$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$271$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$272$	0.794622	−	0.678671i	0.794622	−	0.678671i
$273$		0			0
$274$	−0.355026	+	1.47879i	−0.355026	+	1.47879i
$275$		0			0
$276$		0			0
$277$	−1.04744	+	0.533698i	−1.04744	+	0.533698i	−0.891007	−	0.453990i	$-0.850000\pi$
$277$	−1.04744	+	0.533698i	−1.04744	+	0.533698i	−0.156434	+	0.987688i	$0.550000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.794622	+	1.29671i	0.794622	+	1.29671i	0.951057	+	0.309017i	$0.100000\pi$
$281$	0.794622	+	1.29671i	0.794622	+	1.29671i	−0.156434	+	0.987688i	$0.550000\pi$
$282$		0			0
$283$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$283$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.707107	+	0.707107i	0.707107	+	0.707107i
$289$	0.0908861	−	0.0143949i	0.0908861	−	0.0143949i
$290$	1.57547	−	0.965451i	1.57547	−	0.965451i
$291$		0			0
$292$	−0.863541	+	0.280582i	−0.863541	+	0.280582i
$293$	−1.89101	−	0.453990i	−1.89101	−	0.453990i	−0.891007	−	0.453990i	$-0.850000\pi$
$293$	−1.89101	−	0.453990i	−1.89101	−	0.453990i	−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	−1.34500	+	1.34500i	−1.34500	+	1.34500i
$297$		0			0
$298$	−1.29489	+	1.51612i	−1.29489	+	1.51612i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.550672	−	1.69480i	−0.550672	−	1.69480i
$306$	0.243950	+	1.01612i	0.243950	+	1.01612i
$307$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$307$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$311$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$312$		0			0
$313$	−0.497066	−	0.581990i	−0.497066	−	0.581990i	0.453990	−	0.891007i	$-0.350000\pi$
$313$	−0.497066	−	0.581990i	−0.497066	−	0.581990i	−0.951057	+	0.309017i	$0.900000\pi$
$314$	−0.763007	+	1.84206i	−0.763007	+	1.84206i
$315$		0			0
$316$		0			0
$317$	1.29489	−	1.51612i	1.29489	−	1.51612i	0.587785	−	0.809017i	$-0.300000\pi$
$317$	1.29489	−	1.51612i	1.29489	−	1.51612i	0.707107	−	0.707107i	$-0.250000\pi$
$318$		0			0
$319$		0			0
$320$	0.156434	−	0.987688i	0.156434	−	0.987688i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$325$	−0.101910	+	1.29489i	−0.101910	+	1.29489i
$326$		0			0
$327$		0			0
$328$	0.587785	+	0.809017i	0.587785	+	0.809017i
$329$		0			0
$330$		0			0
$331$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$331$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$332$		0			0
$333$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	1.97538i		−	1.97538i	−0.156434	−	0.987688i	$-0.550000\pi$
$337$		−	1.97538i		−	1.97538i	0.156434	−	0.987688i	$-0.450000\pi$
$338$	0.612238	−	0.311951i	0.612238	−	0.311951i
$339$		0			0
$340$	0.678671	−	0.794622i	0.678671	−	0.794622i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$347$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$347$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$348$		0			0
$349$	1.16110	+	0.183900i	1.16110	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$349$	1.16110	+	0.183900i	1.16110	+	0.183900i	0.453990	+	0.891007i	$0.350000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.253116	+	1.59811i	−0.253116	+	1.59811i	0.453990	+	0.891007i	$0.350000\pi$
$353$	−0.253116	+	1.59811i	−0.253116	+	1.59811i	−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$	−0.744220	−	0.178671i	−0.744220	−	0.178671i
$357$		0			0
$358$		0			0
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$	0.809017	+	0.587785i	0.809017	+	0.587785i
$361$	0.891007	−	0.453990i	0.891007	−	0.453990i
$362$	−0.355026	−	1.47879i	−0.355026	−	1.47879i
$363$		0			0
$364$		0			0
$365$	−0.809017	+	0.412215i	−0.809017	+	0.412215i
$366$		0			0
$367$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$367$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$368$		0			0
$369$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$370$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$371$		0			0
$372$		0			0
$373$	1.58779	+	0.809017i	1.58779	+	0.809017i	1.00000			$0$
$373$	1.58779	+	0.809017i	1.58779	+	0.809017i	0.587785	+	0.809017i	$0.300000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.28258	−	0.741655i	−2.28258	−	0.741655i
$378$		0			0
$379$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$379$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$383$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$384$		0			0
$385$		0			0
$386$	1.29671	+	1.10749i	1.29671	+	1.10749i
$387$		0			0
$388$	−0.0819895	+	0.133795i	−0.0819895	+	0.133795i
$389$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i	0.951057	+	0.309017i	$0.100000\pi$
$389$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i	−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.156434	+	0.987688i	0.156434	+	0.987688i
$393$		0			0
$394$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$395$		0			0
$396$		0			0
$397$	1.45399	−	0.891007i	1.45399	−	0.891007i	0.453990	−	0.891007i	$-0.350000\pi$
$397$	1.45399	−	0.891007i	1.45399	−	0.891007i	1.00000			$0$
$398$		0			0
$399$		0			0
$400$		−	1.00000i		−	1.00000i
$401$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$401$	1.14412	+	1.14412i	1.14412	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$402$		0			0
$403$		0			0
$404$	1.65816	−	1.01612i	1.65816	−	1.01612i
$405$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$409$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$410$	0.707107	+	0.707107i	0.707107	+	0.707107i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.10749	+	0.678671i	−1.10749	+	0.678671i
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	−0.0366318	−	0.465451i	−0.0366318	−	0.465451i	−0.987688	−	0.156434i	$-0.950000\pi$
$421$	−0.0366318	−	0.465451i	−0.0366318	−	0.465451i	0.951057	−	0.309017i	$-0.100000\pi$
$422$		0			0
$423$		0			0
$424$	0.431351	+	0.178671i	0.431351	+	0.178671i
$425$	0.546010	−	0.891007i	0.546010	−	0.891007i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$431$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$432$		0			0
$433$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i	−0.309017	−	0.951057i	$-0.600000\pi$
$433$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i			1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	1.70002	+	0.133795i	1.70002	+	0.133795i
$437$		0			0
$438$		0			0
$439$		0			0		−0.649448	−	0.760406i	$-0.725000\pi$
$439$		0			0		0.649448	+	0.760406i	$0.275000\pi$
$440$		0			0
$441$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$442$	−1.35734			−1.35734
$443$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$443$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$444$		0			0
$445$	−0.763007	−	0.0600500i	−0.763007	−	0.0600500i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.0489435	+	0.309017i	0.0489435	+	0.309017i	1.00000			$0$
$449$	0.0489435	+	0.309017i	0.0489435	+	0.309017i	−0.951057	+	0.309017i	$0.900000\pi$
$450$	0.891007	+	0.453990i	0.891007	+	0.453990i
$451$		0			0
$452$	0.437016	−	0.437016i	0.437016	−	0.437016i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.0366318	+	0.152583i	−0.0366318	+	0.152583i	−0.987688	−	0.156434i	$-0.950000\pi$
$457$	−0.0366318	+	0.152583i	−0.0366318	+	0.152583i	0.951057	+	0.309017i	$0.100000\pi$
$458$	−0.0366318	−	0.152583i	−0.0366318	−	0.152583i
$459$		0			0
$460$		0			0
$461$	0.280582	−	0.863541i	0.280582	−	0.863541i	−0.707107	−	0.707107i	$-0.750000\pi$
$461$	0.280582	−	0.863541i	0.280582	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$462$		0			0
$463$		0			0		0.0784591	−	0.996917i	$-0.475000\pi$
$463$		0			0		−0.0784591	+	0.996917i	$0.525000\pi$
$464$	1.79671	+	0.431351i	1.79671	+	0.431351i
$465$		0			0
$466$	−0.152583	+	1.93874i	−0.152583	+	1.93874i
$467$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$467$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$468$	−0.101910	−	1.29489i	−0.101910	−	1.29489i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.355026	+	0.303221i	−0.355026	+	0.303221i
$478$		0			0
$479$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$479$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$480$		0			0
$481$	2.46303	−	0.193845i	2.46303	−	0.193845i
$482$	−1.76007	+	0.896802i	−1.76007	+	0.896802i
$483$		0			0
$484$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$485$	−0.0600500	+	0.144974i	−0.0600500	+	0.144974i
$486$		0			0
$487$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$487$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$488$	0.809017	−	1.58779i	0.809017	−	1.58779i
$489$		0			0
$490$	0.309017	+	0.951057i	0.309017	+	0.951057i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	1.36535	+	1.36535i	1.36535	+	1.36535i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.522499	−	0.852640i	$-0.325000\pi$
$499$		0			0		−0.522499	+	0.852640i	$0.675000\pi$
$500$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$501$		0			0
$502$		0			0
$503$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$503$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$504$		0			0
$505$	1.47879	−	1.26301i	1.47879	−	1.26301i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.10749	−	1.29671i	1.10749	−	1.29671i	0.156434	−	0.987688i	$-0.450000\pi$
$509$	1.10749	−	1.29671i	1.10749	−	1.29671i	0.951057	−	0.309017i	$-0.100000\pi$
$510$		0			0
$511$		0			0
$512$	0.809017	−	0.587785i	0.809017	−	0.587785i
$513$		0			0
$514$	−0.178671	−	0.744220i	−0.178671	−	0.744220i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.987688	+	0.843566i	−0.987688	+	0.843566i
$521$	0.119322	−	0.101910i	0.119322	−	0.101910i	−0.587785	−	0.809017i	$-0.700000\pi$
$521$	0.119322	−	0.101910i	0.119322	−	0.101910i	0.707107	+	0.707107i	$0.250000\pi$
$522$	−1.20002	+	1.40505i	−1.20002	+	1.40505i
$523$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$523$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.951057	−	0.309017i	0.951057	−	0.309017i
$530$	0.453990	+	0.108993i	0.453990	+	0.108993i
$531$		0			0
$532$		0			0
$533$	0.101910	−	1.29489i	0.101910	−	1.29489i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$539$		0			0
$540$		0			0
$541$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	−0.891007	−	0.453990i	$-0.850000\pi$
$541$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	0.156434	−	0.987688i	$-0.450000\pi$
$542$		0			0
$543$		0			0
$544$	1.04178	−	0.0819895i	1.04178	−	0.0819895i
$545$	1.70002	−	0.133795i	1.70002	−	0.133795i
$546$		0			0
$547$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$547$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$548$	−1.15643	+	0.987688i	−1.15643	+	0.987688i
$549$	1.04744	+	1.44168i	1.04744	+	1.44168i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.16110	−	0.183900i	−1.16110	−	0.183900i
$555$		0			0
$556$		0			0
$557$	−0.0600500	−	0.763007i	−0.0600500	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$557$	−0.0600500	−	0.763007i	−0.0600500	−	0.763007i	0.891007	−	0.453990i	$-0.150000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.119322	+	1.51612i	−0.119322	+	1.51612i
$563$		0			0		−0.760406	−	0.649448i	$-0.775000\pi$
$563$		0			0		0.760406	+	0.649448i	$0.225000\pi$
$564$		0			0
$565$	0.363271	−	0.500000i	0.363271	−	0.500000i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.69480	+	0.863541i	1.69480	+	0.863541i	0.987688	+	0.156434i	$0.0500000\pi$
$569$	1.69480	+	0.863541i	1.69480	+	0.863541i	0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$571$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.156434	+	0.987688i	0.156434	+	0.987688i
$577$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.987688	−	0.156434i	$-0.950000\pi$
$577$	−1.79671	−	0.744220i	−1.79671	−	0.744220i	−0.809017	−	0.587785i	$-0.800000\pi$
$578$	0.0819895	+	0.0417758i	0.0819895	+	0.0417758i
$579$		0			0
$580$	1.84206	+	0.144974i	1.84206	+	0.144974i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.863541	−	0.280582i	−0.863541	−	0.280582i
$585$	−0.303221	−	1.26301i	−0.303221	−	1.26301i
$586$	−1.26301	−	1.47879i	−1.26301	−	1.47879i
$587$		0			0		−0.522499	−	0.852640i	$-0.675000\pi$
$587$		0			0		0.522499	+	0.852640i	$0.325000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.87869	+	0.297556i	−1.87869	+	0.297556i
$593$	−1.04178	+	1.70002i	−1.04178	+	1.70002i	−0.453990	+	0.891007i	$0.650000\pi$
$593$	−1.04178	+	1.70002i	−1.04178	+	1.70002i	−0.587785	+	0.809017i	$0.700000\pi$
$594$		0			0
$595$		0			0
$596$	−1.93874	+	0.465451i	−1.93874	+	0.465451i
$597$		0			0
$598$		0			0
$599$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$599$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$600$		0			0
$601$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.951057	−	0.309017i	$-0.900000\pi$
$601$	−1.84206	−	0.763007i	−1.84206	−	0.763007i	−0.891007	−	0.453990i	$-0.850000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$606$		0			0
$607$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$607$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$608$		0			0
$609$		0			0
$610$	0.550672	−	1.69480i	0.550672	−	1.69480i
$611$		0			0
$612$	−0.399903	+	0.965451i	−0.399903	+	0.965451i
$613$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.987688	−	0.156434i	$-0.950000\pi$
$613$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.453990	−	0.891007i	$-0.650000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$617$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$626$	−0.0600500	−	0.763007i	−0.0600500	−	0.763007i
$627$		0			0
$628$	−1.70002	+	1.04178i	−1.70002	+	1.04178i
$629$	−1.83640	−	0.760661i	−1.83640	−	0.760661i
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	1.93874	−	0.465451i	1.93874	−	0.465451i
$635$		0			0
$636$		0			0
$637$	0.678671	−	1.10749i	0.678671	−	1.10749i
$638$		0			0
$639$		0			0
$640$	0.707107	−	0.707107i	0.707107	−	0.707107i
$641$	0.465451	+	0.0366318i	0.465451	+	0.0366318i	0.309017	−	0.951057i	$-0.400000\pi$
$641$	0.465451	+	0.0366318i	0.465451	+	0.0366318i	0.156434	+	0.987688i	$0.450000\pi$
$642$		0			0
$643$		0			0		−0.522499	−	0.852640i	$-0.675000\pi$
$643$		0			0		0.522499	+	0.852640i	$0.325000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$649$		0			0
$650$	−0.843566	+	0.987688i	−0.843566	+	0.987688i
$651$		0			0
$652$		0			0
$653$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	0.309017	−	0.951057i	$-0.400000\pi$
$653$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	−0.453990	+	0.891007i	$0.650000\pi$
$654$		0			0
$655$		0			0
$656$			1.00000i			1.00000i
$657$	0.642040	−	0.642040i	0.642040	−	0.642040i
$658$		0			0
$659$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$659$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$660$		0			0
$661$	1.69480	+	0.863541i	1.69480	+	0.863541i	0.987688	+	0.156434i	$0.0500000\pi$
$661$	1.69480	+	0.863541i	1.69480	+	0.863541i	0.707107	+	0.707107i	$0.250000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.587785	−	1.80902i	0.587785	−	1.80902i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.119322	−	1.51612i	−0.119322	−	1.51612i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	−0.119322	−	1.51612i	−0.119322	−	1.51612i	0.587785	−	0.809017i	$-0.300000\pi$
$674$	1.16110	−	1.59811i	1.16110	−	1.59811i
$675$		0			0
$676$	0.678671	+	0.107491i	0.678671	+	0.107491i
$677$	−0.533698	−	0.734572i	−0.533698	−	0.734572i	0.453990	−	0.891007i	$-0.350000\pi$
$677$	−0.533698	−	0.734572i	−0.533698	−	0.734572i	−0.987688	+	0.156434i	$0.950000\pi$
$678$		0			0
$679$		0			0
$680$	1.01612	−	0.243950i	1.01612	−	0.243950i
$681$		0			0
$682$		0			0
$683$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$683$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$684$		0			0
$685$	−0.987688	+	1.15643i	−0.987688	+	1.15643i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.275319	−	0.540344i	−0.275319	−	0.540344i
$690$		0			0
$691$		0			0		−0.522499	−	0.852640i	$-0.675000\pi$
$691$		0			0		0.522499	+	0.852640i	$0.325000\pi$
$692$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.546010	+	0.891007i	−0.546010	+	0.891007i
$698$	0.831254	+	0.831254i	0.831254	+	0.831254i
$699$		0			0
$700$		0			0
$701$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.891007	−	0.453990i	$-0.150000\pi$
$701$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.987688	−	0.156434i	$-0.0500000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.14412	+	1.14412i	−1.14412	+	1.14412i
$707$		0			0
$708$		0			0
$709$	−1.47879	+	1.26301i	−1.47879	+	1.26301i	−0.587785	+	0.809017i	$0.700000\pi$
$709$	−1.47879	+	1.26301i	−1.47879	+	1.26301i	−0.891007	+	0.453990i	$0.850000\pi$
$710$		0			0
$711$		0			0
$712$	−0.497066	−	0.581990i	−0.497066	−	0.581990i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$719$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$720$	0.309017	+	0.951057i	0.309017	+	0.951057i
$721$		0			0
$722$	0.987688	+	0.156434i	0.987688	+	0.156434i
$723$		0			0
$724$	0.581990	−	1.40505i	0.581990	−	1.40505i
$725$	1.84206	−	0.144974i	1.84206	−	0.144974i
$726$		0			0
$727$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$727$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$728$		0			0
$729$	0.707107	−	0.707107i	0.707107	−	0.707107i
$730$	−0.896802	−	0.142040i	−0.896802	−	0.142040i
$731$		0			0
$732$		0			0
$733$	0.587785	−	0.190983i	0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$733$	0.587785	−	0.190983i	0.587785	−	0.190983i	0.587785	+	0.809017i	$0.300000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$741$		0			0
$742$		0			0
$743$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$743$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$	−1.84206	+	0.763007i	−1.84206	+	0.763007i
$746$	0.809017	+	1.58779i	0.809017	+	1.58779i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$751$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$752$		0			0
$753$		0			0
$754$	−1.41071	−	1.94168i	−1.41071	−	1.94168i
$755$		0			0
$756$		0			0
$757$	−1.65816	−	1.01612i	−1.65816	−	1.01612i	−0.951057	−	0.309017i	$-0.900000\pi$
$757$	−1.65816	−	1.01612i	−1.65816	−	1.01612i	−0.707107	−	0.707107i	$-0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.16110	−	1.59811i	1.16110	−	1.59811i	0.453990	−	0.891007i	$-0.350000\pi$
$761$	1.16110	−	1.59811i	1.16110	−	1.59811i	0.707107	−	0.707107i	$-0.250000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.243950	+	1.01612i	−0.243950	+	1.01612i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$769$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$770$		0			0
$771$		0			0
$772$	0.398090	+	1.65816i	0.398090	+	1.65816i
$773$	0.431351	−	1.79671i	0.431351	−	1.79671i	−0.156434	−	0.987688i	$-0.550000\pi$
$773$	0.431351	−	1.79671i	0.431351	−	1.79671i	0.587785	−	0.809017i	$-0.300000\pi$
$774$		0			0
$775$		0			0
$776$	−0.144974	+	0.0600500i	−0.144974	+	0.0600500i
$777$		0			0
$778$	−0.221232	+	0.221232i	−0.221232	+	0.221232i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$785$	−1.51612	+	1.29489i	−1.51612	+	1.29489i
$786$		0			0
$787$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$787$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$788$	−1.17557			−1.17557
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.13846	+	0.885778i	−2.13846	+	0.885778i
$794$	1.70002	+	0.133795i	1.70002	+	0.133795i
$795$		0			0
$796$		0			0
$797$	1.95106	−	0.309017i	1.95106	−	0.309017i	0.951057	−	0.309017i	$-0.100000\pi$
$797$	1.95106	−	0.309017i	1.95106	−	0.309017i	1.00000			$0$
$798$		0			0
$799$		0			0
$800$	0.587785	−	0.809017i	0.587785	−	0.809017i
$801$	0.744220	−	0.178671i	0.744220	−	0.178671i
$802$	0.253116	+	1.59811i	0.253116	+	1.59811i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.93874	+	0.152583i	1.93874	+	0.152583i
$809$	−0.0123117	−	0.156434i	−0.0123117	−	0.156434i	0.987688	−	0.156434i	$-0.0500000\pi$
$809$	−0.0123117	−	0.156434i	−0.0123117	−	0.156434i	−1.00000			$\pi$
$810$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$819$		0			0
$820$	0.156434	+	0.987688i	0.156434	+	0.987688i
$821$	0.907981			0.907981			0.453990	−	0.891007i	$-0.350000\pi$
$821$	0.907981			0.907981			0.453990	+	0.891007i	$0.350000\pi$
$822$		0			0
$823$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$823$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$827$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$828$		0			0
$829$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.891007	+	0.453990i	$0.150000\pi$
$829$	1.34500	+	1.34500i	1.34500	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$830$		0			0
$831$		0			0
$832$	−1.29489	−	0.101910i	−1.29489	−	0.101910i
$833$	−0.891007	+	0.546010i	−0.891007	+	0.546010i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$839$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$840$		0			0
$841$	−0.377666	+	2.38449i	−0.377666	+	2.38449i
$842$	0.243950	−	0.398090i	0.243950	−	0.398090i
$843$		0			0
$844$		0			0
$845$	0.687131			0.687131
$846$		0			0
$847$		0			0
$848$	0.243950	+	0.398090i	0.243950	+	0.398090i
$849$		0			0
$850$	0.965451	−	0.399903i	0.965451	−	0.399903i
$851$		0			0
$852$		0			0
$853$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$853$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$857$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.891007	+	0.453990i	$0.850000\pi$
$858$		0			0
$859$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$859$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$863$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$864$		0			0
$865$	−0.734572	−	1.44168i	−0.734572	−	1.44168i
$866$	−0.278768	−	0.142040i	−0.278768	−	0.142040i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.29671	+	1.10749i	1.29671	+	1.10749i
$873$	0.0123117	−	0.156434i	0.0123117	−	0.156434i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.309017	−	1.95106i	0.309017	−	1.95106i		−	1.00000i	$-0.5\pi$
$877$	0.309017	−	1.95106i	0.309017	−	1.95106i	0.309017	−	0.951057i	$-0.400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.39680	−	0.221232i	−1.39680	−	0.221232i	−0.587785	−	0.809017i	$-0.700000\pi$
$881$	−1.39680	−	0.221232i	−1.39680	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$882$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$883$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$883$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$884$	−1.09811	−	0.797826i	−1.09811	−	0.797826i
$885$		0			0
$886$		0			0
$887$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$887$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$888$		0			0
$889$		0			0
$890$	−0.581990	−	0.497066i	−0.581990	−	0.497066i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.142040	+	0.278768i	−0.142040	+	0.278768i
$899$		0			0
$900$	0.453990	+	0.891007i	0.453990	+	0.891007i
$901$			0.487899i			0.487899i
$902$		0			0
$903$		0			0
$904$	0.610425	−	0.0966818i	0.610425	−	0.0966818i
$905$	0.355026	−	1.47879i	0.355026	−	1.47879i
$906$		0			0
$907$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$907$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$908$		0			0
$909$	−1.01612	+	1.65816i	−1.01612	+	1.65816i
$910$		0			0
$911$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$911$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$	−0.119322	+	0.101910i	−0.119322	+	0.101910i
$915$		0			0
$916$	0.0600500	−	0.144974i	0.0600500	−	0.144974i
$917$		0			0
$918$		0			0
$919$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$919$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$920$		0			0
$921$		0			0
$922$	0.734572	−	0.533698i	0.734572	−	0.533698i
$923$		0			0
$924$		0			0
$925$	−1.69480	+	0.863541i	−1.69480	+	0.863541i
$926$		0			0
$927$		0			0
$928$	1.20002	+	1.40505i	1.20002	+	1.40505i
$929$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$929$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$930$		0			0
$931$		0			0
$932$	−1.26301	+	1.47879i	−1.26301	+	1.47879i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.678671	−	1.10749i	0.678671	−	1.10749i
$937$	1.79671	+	0.431351i	1.79671	+	0.431351i	0.987688	−	0.156434i	$-0.0500000\pi$
$937$	1.79671	+	0.431351i	1.79671	+	0.431351i	0.809017	+	0.587785i	$0.200000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.76007	−	0.278768i	1.76007	−	0.278768i	0.809017	−	0.587785i	$-0.200000\pi$
$941$	1.76007	−	0.278768i	1.76007	−	0.278768i	0.951057	+	0.309017i	$0.100000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$947$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$948$		0			0
$949$	0.616221	+	1.00558i	0.616221	+	1.00558i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$953$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$954$	−0.465451	+	0.0366318i	−0.465451	+	0.0366318i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.809017	+	0.587785i	0.809017	+	0.587785i
$962$	2.10657	+	1.29091i	2.10657	+	1.29091i
$963$		0			0
$964$	−1.95106	−	0.309017i	−1.95106	−	0.309017i
$965$	0.652583	+	1.57547i	0.652583	+	1.57547i
$966$		0			0
$967$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$967$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$968$	0.156434	−	0.987688i	0.156434	−	0.987688i
$969$		0			0
$970$	−0.133795	+	0.0819895i	−0.133795	+	0.0819895i
$971$		0			0		−0.972370	−	0.233445i	$-0.925000\pi$
$971$		0			0		0.972370	+	0.233445i	$0.0750000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.58779	−	0.809017i	1.58779	−	0.809017i
$977$	0.431351	+	1.79671i	0.431351	+	1.79671i	0.587785	+	0.809017i	$0.300000\pi$
$977$	0.431351	+	1.79671i	0.431351	+	1.79671i	−0.156434	+	0.987688i	$0.550000\pi$
$978$		0			0
$979$		0			0
$980$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$981$	−1.57547	+	0.652583i	−1.57547	+	0.652583i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	−1.16110	+	0.183900i	−1.16110	+	0.183900i
$986$	0.302060	+	1.90713i	0.302060	+	1.90713i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$991$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.243950	−	0.398090i	−0.243950	−	0.398090i	0.707107	−	0.707107i	$-0.250000\pi$
$997$	−0.243950	−	0.398090i	−0.243950	−	0.398090i	−0.951057	+	0.309017i	$0.900000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	820.1.bz.a.263.1	yes	16
4.3	odd	2	CM	820.1.bz.a.263.1	yes	16
5.2	odd	4		820.1.by.a.427.1	✓	16
20.7	even	4		820.1.by.a.427.1	✓	16
41.29	odd	40		820.1.by.a.603.1	yes	16
164.111	even	40		820.1.by.a.603.1	yes	16
205.152	even	40	inner	820.1.bz.a.767.1	yes	16
820.767	odd	40	inner	820.1.bz.a.767.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
820.1.by.a.427.1	✓	16	5.2	odd	4
820.1.by.a.427.1	✓	16	20.7	even	4
820.1.by.a.603.1	yes	16	41.29	odd	40
820.1.by.a.603.1	yes	16	164.111	even	40
820.1.bz.a.263.1	yes	16	1.1	even	1	trivial
820.1.bz.a.263.1	yes	16	4.3	odd	2	CM
820.1.bz.a.767.1	yes	16	205.152	even	40	inner
820.1.bz.a.767.1	yes	16	820.767	odd	40	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:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	820.bz (of order \(40\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.809017 + 0.587785i) q^{2} +(0.309017 + 0.951057i) q^{4} +(0.453990 + 0.891007i) q^{5} +(-0.309017 + 0.951057i) q^{8} +(-0.707107 - 0.707107i) q^{9} +(-0.156434 + 0.987688i) q^{10} +(1.10749 - 0.678671i) q^{13} +(-0.809017 + 0.587785i) q^{16} +(-1.04178 + 0.0819895i) q^{17} +(-0.156434 - 0.987688i) q^{18} +(-0.707107 + 0.707107i) q^{20} +(-0.587785 + 0.809017i) q^{25} +(1.29489 + 0.101910i) q^{26} +(-1.20002 - 1.40505i) q^{29} -1.00000 q^{32} +(-0.891007 - 0.546010i) q^{34} +(0.453990 - 0.891007i) q^{36} +(1.69480 + 0.863541i) q^{37} +(-0.987688 + 0.156434i) q^{40} +(0.587785 - 0.809017i) q^{41} +(0.309017 - 0.951057i) q^{45} +(0.891007 - 0.453990i) q^{49} +(-0.951057 + 0.309017i) q^{50} +(0.987688 + 0.843566i) q^{52} +(0.0366318 - 0.465451i) q^{53} +(-0.144974 - 1.84206i) q^{58} +(-1.76007 - 0.278768i) q^{61} +(-0.809017 - 0.587785i) q^{64} +(1.10749 + 0.678671i) q^{65} +(-0.399903 - 0.965451i) q^{68} +(0.891007 - 0.453990i) q^{72} +0.907981i q^{73} +(0.863541 + 1.69480i) q^{74} +(-0.891007 - 0.453990i) q^{80} +1.00000i q^{81} +(0.951057 - 0.309017i) q^{82} +(-0.546010 - 0.891007i) q^{85} +(-0.399903 + 0.652583i) q^{89} +(0.809017 - 0.587785i) q^{90} +(0.101910 + 0.119322i) q^{97} +(0.987688 + 0.156434i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 4 q^{4} + 4 q^{8} - 4 q^{16} + 4 q^{29} - 16 q^{32} - 4 q^{45} - 4 q^{58} - 4 q^{61} - 4 q^{64} - 16 q^{85} + 4 q^{90} + 4 q^{97}+O(q^{100}) \) 16 * q + 4 * q^2 - 4 * q^4 + 4 * q^8 - 4 * q^16 + 4 * q^29 - 16 * q^32 - 4 * q^45 - 4 * q^58 - 4 * q^61 - 4 * q^64 - 16 * q^85 + 4 * q^90 + 4 * q^97

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