LMFDB - Embedded newform 563.1.b.b.562.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [563,1,Mod(562,563)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(563, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("563.562");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 563 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	563.b (of order $2$, degree $1$, 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:	$0.280973602112$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.563.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.178453547.2

Embedding invariants

Embedding label			562.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	563.562
Dual form			563.1.b.b.562.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000 q^{7} +O(q^{10})$	$q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000 q^{7} -2.00000 q^{10} +1.00000 q^{12} +1.41421i q^{14} -1.41421i q^{15} -1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{19} -1.41421i q^{20} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{30} -1.41421i q^{31} -1.41421i q^{32} -1.41421i q^{34} +1.41421i q^{35} +1.41421i q^{38} -1.41421i q^{42} +1.41421i q^{43} -1.41421i q^{46} +1.00000 q^{47} +1.00000 q^{48} -1.41421i q^{50} +1.00000 q^{51} +1.41421i q^{53} +1.41421i q^{54} -1.00000 q^{57} +1.00000 q^{59} +1.41421i q^{60} +1.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{67} +1.00000 q^{68} +1.00000 q^{69} -2.00000 q^{70} +1.00000 q^{71} +1.00000 q^{75} -1.00000 q^{76} +1.41421i q^{79} -1.41421i q^{80} -1.00000 q^{81} -1.41421i q^{83} +1.00000 q^{84} -1.41421i q^{85} -2.00000 q^{86} -1.41421i q^{89} +1.00000 q^{92} +1.41421i q^{93} +1.41421i q^{94} +1.41421i q^{95} +1.41421i q^{96} +1.41421i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^7	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{7} - 4 q^{10} + 2 q^{12} - 2 q^{16} - 2 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{23} - 2 q^{25} + 2 q^{27} - 2 q^{28} + 4 q^{30} + 2 q^{47} + 2 q^{48} + 2 q^{51} - 2 q^{57} + 2 q^{59} + 2 q^{61} + 4 q^{62} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 2 q^{69} - 4 q^{70} + 2 q^{71} + 2 q^{75} - 2 q^{76} - 2 q^{81} + 2 q^{84} - 4 q^{86} + 2 q^{92}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^7 - 4 * q^10 + 2 * q^12 - 2 * q^16 - 2 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^23 - 2 * q^25 + 2 * q^27 - 2 * q^28 + 4 * q^30 + 2 * q^47 + 2 * q^48 + 2 * q^51 - 2 * q^57 + 2 * q^59 + 2 * q^61 + 4 * q^62 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 2 * q^69 - 4 * q^70 + 2 * q^71 + 2 * q^75 - 2 * q^76 - 2 * q^81 + 2 * q^84 - 4 * q^86 + 2 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/563\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.00000	−1.00000
$5$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$5$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$6$	− 1.41421i	− 1.41421i
$7$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$7$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$8$	0	0
$9$	0	0
$10$	−2.00000	−2.00000
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	1.00000
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.41421i	1.41421i
$15$	− 1.41421i	− 1.41421i
$16$	−1.00000	−1.00000
$17$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$18$	0	0
$19$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$20$	− 1.41421i	− 1.41421i
$21$	−1.00000	−1.00000
$22$	0	0
$23$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$24$	0	0
$25$	−1.00000	−1.00000
$26$	0	0
$27$	1.00000	1.00000
$28$	−1.00000	−1.00000
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	2.00000	2.00000
$31$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$31$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$32$	− 1.41421i	− 1.41421i
$33$	0	0
$34$	− 1.41421i	− 1.41421i
$35$	1.41421i	1.41421i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.41421i	1.41421i
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	− 1.41421i	− 1.41421i
$43$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$43$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$44$	0	0
$45$	0	0
$46$	− 1.41421i	− 1.41421i
$47$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$47$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$48$	1.00000	1.00000
$49$	0	0
$50$	− 1.41421i	− 1.41421i
$51$	1.00000	1.00000
$52$	0	0
$53$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$53$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$54$	1.41421i	1.41421i
$55$	0	0
$56$	0	0
$57$	−1.00000	−1.00000
$58$	0	0
$59$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$59$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$60$	1.41421i	1.41421i
$61$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$61$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$62$	2.00000	2.00000
$63$	0	0
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$67$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$68$	1.00000	1.00000
$69$	1.00000	1.00000
$70$	−2.00000	−2.00000
$71$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.00000	1.00000
$76$	−1.00000	−1.00000
$77$	0	0
$78$	0	0
$79$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$79$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$80$	− 1.41421i	− 1.41421i
$81$	−1.00000	−1.00000
$82$	0	0
$83$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$83$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$84$	1.00000	1.00000
$85$	− 1.41421i	− 1.41421i
$86$	−2.00000	−2.00000
$87$	0	0
$88$	0	0
$89$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$89$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	1.00000
$93$	1.41421i	1.41421i
$94$	1.41421i	1.41421i
$95$	1.41421i	1.41421i
$96$	1.41421i	1.41421i
$97$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$97$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	1.00000
$101$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$102$	1.41421i	1.41421i
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	− 1.41421i	− 1.41421i
$106$	−2.00000	−2.00000
$107$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$107$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$108$	−1.00000	−1.00000
$109$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$109$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−1.00000
$113$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$114$	− 1.41421i	− 1.41421i
$115$	− 1.41421i	− 1.41421i
$116$	0	0
$117$	0	0
$118$	1.41421i	1.41421i
$119$	−1.00000	−1.00000
$120$	0	0
$121$	−1.00000	−1.00000
$122$	1.41421i	1.41421i
$123$	0	0
$124$	1.41421i	1.41421i
$125$	0	0
$126$	0	0
$127$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$127$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$128$	0	0
$129$	− 1.41421i	− 1.41421i
$130$	0	0
$131$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$131$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$132$	0	0
$133$	1.00000	1.00000
$134$	1.41421i	1.41421i
$135$	1.41421i	1.41421i
$136$	0	0
$137$	2.00000	2.00000	1.00000			$0$
$137$	2.00000	2.00000	1.00000			$0$
$138$	1.41421i	1.41421i
$139$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$139$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$140$	− 1.41421i	− 1.41421i
$141$	−1.00000	−1.00000
$142$	1.41421i	1.41421i
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$149$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$150$	1.41421i	1.41421i
$151$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$151$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	2.00000
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−2.00000	−2.00000
$159$	− 1.41421i	− 1.41421i
$160$	2.00000	2.00000
$161$	−1.00000	−1.00000
$162$	− 1.41421i	− 1.41421i
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	2.00000	2.00000
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−1.00000	−1.00000
$170$	2.00000	2.00000
$171$	0	0
$172$	− 1.41421i	− 1.41421i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−1.00000	−1.00000
$176$	0	0
$177$	−1.00000	−1.00000
$178$	2.00000	2.00000
$179$	−2.00000	−2.00000	−1.00000			$\pi$
$179$	−2.00000	−2.00000	−1.00000			$\pi$
$180$	0	0
$181$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$182$	0	0
$183$	−1.00000	−1.00000
$184$	0	0
$185$	0	0
$186$	−2.00000	−2.00000
$187$	0	0
$188$	−1.00000	−1.00000
$189$	1.00000	1.00000
$190$	−2.00000	−2.00000
$191$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$191$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$192$	−1.00000	−1.00000
$193$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$193$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$194$	−2.00000	−2.00000
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$199$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$200$	0	0
$201$	−1.00000	−1.00000
$202$	− 1.41421i	− 1.41421i
$203$	0	0
$204$	−1.00000	−1.00000
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	2.00000	2.00000
$211$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$211$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$212$	− 1.41421i	− 1.41421i
$213$	−1.00000	−1.00000
$214$	1.41421i	1.41421i
$215$	−2.00000	−2.00000
$216$	0	0
$217$	− 1.41421i	− 1.41421i
$218$	2.00000	2.00000
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$224$	− 1.41421i	− 1.41421i
$225$	0	0
$226$	− 1.41421i	− 1.41421i
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	1.00000	1.00000
$229$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$229$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$230$	2.00000	2.00000
$231$	0	0
$232$	0	0
$233$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$233$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$234$	0	0
$235$	1.41421i	1.41421i
$236$	−1.00000	−1.00000
$237$	− 1.41421i	− 1.41421i
$238$	− 1.41421i	− 1.41421i
$239$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$239$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$240$	1.41421i	1.41421i
$241$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$241$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$242$	− 1.41421i	− 1.41421i
$243$	0	0
$244$	−1.00000	−1.00000
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.41421i	1.41421i
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	− 1.41421i	− 1.41421i
$255$	1.41421i	1.41421i
$256$	1.00000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	2.00000	2.00000
$259$	0	0
$260$	0	0
$261$	0	0
$262$	2.00000	2.00000
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−2.00000	−2.00000
$266$	1.41421i	1.41421i
$267$	1.41421i	1.41421i
$268$	−1.00000	−1.00000
$269$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$269$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$270$	−2.00000	−2.00000
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	1.00000	1.00000
$273$	0	0
$274$	2.82843i	2.82843i
$275$	0	0
$276$	−1.00000	−1.00000
$277$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$277$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$278$	2.00000	2.00000
$279$	0	0
$280$	0	0
$281$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$282$	− 1.41421i	− 1.41421i
$283$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$283$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$284$	−1.00000	−1.00000
$285$	− 1.41421i	− 1.41421i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0	0
$290$	0	0
$291$	− 1.41421i	− 1.41421i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	1.41421i	1.41421i
$296$	0	0
$297$	0	0
$298$	1.41421i	1.41421i
$299$	0	0
$300$	−1.00000	−1.00000
$301$	1.41421i	1.41421i
$302$	−2.00000	−2.00000
$303$	1.00000	1.00000
$304$	−1.00000	−1.00000
$305$	1.41421i	1.41421i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	2.82843i	2.82843i
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$313$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$314$	0	0
$315$	0	0
$316$	− 1.41421i	− 1.41421i
$317$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$317$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$318$	2.00000	2.00000
$319$	0	0
$320$	1.41421i	1.41421i
$321$	−1.00000	−1.00000
$322$	− 1.41421i	− 1.41421i
$323$	−1.00000	−1.00000
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	1.41421i	1.41421i
$328$	0	0
$329$	1.00000	1.00000
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1.41421i	1.41421i
$333$	0	0
$334$	0	0
$335$	1.41421i	1.41421i
$336$	1.00000	1.00000
$337$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$338$	− 1.41421i	− 1.41421i
$339$	1.00000	1.00000
$340$	1.41421i	1.41421i
$341$	0	0
$342$	0	0
$343$	−1.00000	−1.00000
$344$	0	0
$345$	1.41421i	1.41421i
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$349$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$350$	− 1.41421i	− 1.41421i
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	− 1.41421i	− 1.41421i
$355$	1.41421i	1.41421i
$356$	1.41421i	1.41421i
$357$	1.00000	1.00000
$358$	− 2.82843i	− 2.82843i
$359$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$359$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$360$	0	0
$361$	0	0
$362$	− 1.41421i	− 1.41421i
$363$	1.00000	1.00000
$364$	0	0
$365$	0	0
$366$	− 1.41421i	− 1.41421i
$367$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$367$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$368$	1.00000	1.00000
$369$	0	0
$370$	0	0
$371$	1.41421i	1.41421i
$372$	− 1.41421i	− 1.41421i
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	1.41421i	1.41421i
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	− 1.41421i	− 1.41421i
$381$	1.00000	1.00000
$382$	1.41421i	1.41421i
$383$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$384$	0	0
$385$	0	0
$386$	1.41421i	1.41421i
$387$	0	0
$388$	− 1.41421i	− 1.41421i
$389$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$389$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$390$	0	0
$391$	1.00000	1.00000
$392$	0	0
$393$	1.41421i	1.41421i
$394$	0	0
$395$	−2.00000	−2.00000
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	2.00000	2.00000
$399$	−1.00000	−1.00000
$400$	1.00000	1.00000
$401$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$401$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$402$	− 1.41421i	− 1.41421i
$403$	0	0
$404$	1.00000	1.00000
$405$	− 1.41421i	− 1.41421i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$410$	0	0
$411$	−2.00000	−2.00000
$412$	0	0
$413$	1.00000	1.00000
$414$	0	0
$415$	2.00000	2.00000
$416$	0	0
$417$	1.41421i	1.41421i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	1.41421i	1.41421i
$421$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$421$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$422$	1.41421i	1.41421i
$423$	0	0
$424$	0	0
$425$	1.00000	1.00000
$426$	− 1.41421i	− 1.41421i
$427$	1.00000	1.00000
$428$	−1.00000	−1.00000
$429$	0	0
$430$	− 2.82843i	− 2.82843i
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.00000	−1.00000
$433$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$433$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$434$	2.00000	2.00000
$435$	0	0
$436$	1.41421i	1.41421i
$437$	−1.00000	−1.00000
$438$	0	0
$439$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$439$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$443$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$444$	0	0
$445$	2.00000	2.00000
$446$	− 1.41421i	− 1.41421i
$447$	−1.00000	−1.00000
$448$	1.00000	1.00000
$449$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$450$	0	0
$451$	0	0
$452$	1.00000	1.00000
$453$	− 1.41421i	− 1.41421i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−2.00000	−2.00000
$459$	−1.00000	−1.00000
$460$	1.41421i	1.41421i
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	−2.00000	−2.00000
$466$	2.00000	2.00000
$467$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$467$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$468$	0	0
$469$	1.00000	1.00000
$470$	−2.00000	−2.00000
$471$	0	0
$472$	0	0
$473$	0	0
$474$	2.00000	2.00000
$475$	−1.00000	−1.00000
$476$	1.00000	1.00000
$477$	0	0
$478$	2.00000	2.00000
$479$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$479$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$480$	−2.00000	−2.00000
$481$	0	0
$482$	1.41421i	1.41421i
$483$	1.00000	1.00000
$484$	1.00000	1.00000
$485$	−2.00000	−2.00000
$486$	0	0
$487$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.41421i	1.41421i
$497$	1.00000	1.00000
$498$	−2.00000	−2.00000
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$503$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$504$	0	0
$505$	− 1.41421i	− 1.41421i
$506$	0	0
$507$	1.00000	1.00000
$508$	1.00000	1.00000
$509$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$509$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$510$	−2.00000	−2.00000
$511$	0	0
$512$	1.41421i	1.41421i
$513$	1.00000	1.00000
$514$	0	0
$515$	0	0
$516$	1.41421i	1.41421i
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1.41421i	1.41421i
$525$	1.00000	1.00000
$526$	0	0
$527$	1.41421i	1.41421i
$528$	0	0
$529$	0	0
$530$	− 2.82843i	− 2.82843i
$531$	0	0
$532$	−1.00000	−1.00000
$533$	0	0
$534$	−2.00000	−2.00000
$535$	1.41421i	1.41421i
$536$	0	0
$537$	2.00000	2.00000
$538$	1.41421i	1.41421i
$539$	0	0
$540$	− 1.41421i	− 1.41421i
$541$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$541$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$542$	0	0
$543$	1.00000	1.00000
$544$	1.41421i	1.41421i
$545$	2.00000	2.00000
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−2.00000	−2.00000
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.41421i	1.41421i
$554$	− 1.41421i	− 1.41421i
$555$	0	0
$556$	1.41421i	1.41421i
$557$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$557$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$558$	0	0
$559$	0	0
$560$	− 1.41421i	− 1.41421i
$561$	0	0
$562$	− 1.41421i	− 1.41421i
$563$	−1.00000	−1.00000
$563$	−1.00000	−1.00000
$564$	1.00000	1.00000
$565$	− 1.41421i	− 1.41421i
$566$	−2.00000	−2.00000
$567$	−1.00000	−1.00000
$568$	0	0
$569$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$569$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$570$	2.00000	2.00000
$571$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$571$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$572$	0	0
$573$	−1.00000	−1.00000
$574$	0	0
$575$	1.00000	1.00000
$576$	0	0
$577$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$577$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$578$	0	0
$579$	−1.00000	−1.00000
$580$	0	0
$581$	− 1.41421i	− 1.41421i
$582$	2.00000	2.00000
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	− 1.41421i	− 1.41421i
$590$	−2.00000	−2.00000
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	− 1.41421i	− 1.41421i
$596$	−1.00000	−1.00000
$597$	1.41421i	1.41421i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$601$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$602$	−2.00000	−2.00000
$603$	0	0
$604$	− 1.41421i	− 1.41421i
$605$	− 1.41421i	− 1.41421i
$606$	1.41421i	1.41421i
$607$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$607$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$608$	− 1.41421i	− 1.41421i
$609$	0	0
$610$	−2.00000	−2.00000
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$617$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−2.00000	−2.00000
$621$	−1.00000	−1.00000
$622$	0	0
$623$	− 1.41421i	− 1.41421i
$624$	0	0
$625$	−1.00000	−1.00000
$626$	2.00000	2.00000
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−1.00000	−1.00000
$634$	−2.00000	−2.00000
$635$	− 1.41421i	− 1.41421i
$636$	1.41421i	1.41421i
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$642$	− 1.41421i	− 1.41421i
$643$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$643$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$644$	1.00000	1.00000
$645$	2.00000	2.00000
$646$	− 1.41421i	− 1.41421i
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.41421i	1.41421i
$652$	0	0
$653$	−2.00000	−2.00000	−1.00000			$\pi$
$653$	−2.00000	−2.00000	−1.00000			$\pi$
$654$	−2.00000	−2.00000
$655$	2.00000	2.00000
$656$	0	0
$657$	0	0
$658$	1.41421i	1.41421i
$659$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$659$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.41421i	1.41421i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1.00000	1.00000
$670$	−2.00000	−2.00000
$671$	0	0
$672$	1.41421i	1.41421i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	− 1.41421i	− 1.41421i
$675$	−1.00000	−1.00000
$676$	1.00000	1.00000
$677$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$677$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$678$	1.41421i	1.41421i
$679$	1.41421i	1.41421i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$684$	0	0
$685$	2.82843i	2.82843i
$686$	− 1.41421i	− 1.41421i
$687$	− 1.41421i	− 1.41421i
$688$	− 1.41421i	− 1.41421i
$689$	0	0
$690$	−2.00000	−2.00000
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.00000	2.00000
$696$	0	0
$697$	0	0
$698$	1.41421i	1.41421i
$699$	1.41421i	1.41421i
$700$	1.00000	1.00000
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 1.41421i	− 1.41421i
$706$	0	0
$707$	−1.00000	−1.00000
$708$	1.00000	1.00000
$709$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$709$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$710$	−2.00000	−2.00000
$711$	0	0
$712$	0	0
$713$	1.41421i	1.41421i
$714$	1.41421i	1.41421i
$715$	0	0
$716$	2.00000	2.00000
$717$	1.41421i	1.41421i
$718$	2.00000	2.00000
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.00000	−1.00000
$724$	1.00000	1.00000
$725$	0	0
$726$	1.41421i	1.41421i
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	− 1.41421i	− 1.41421i
$732$	1.00000	1.00000
$733$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$734$	2.00000	2.00000
$735$	0	0
$736$	1.41421i	1.41421i
$737$	0	0
$738$	0	0
$739$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$740$	0	0
$741$	0	0
$742$	−2.00000	−2.00000
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1.41421i	1.41421i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.00000	1.00000
$750$	0	0
$751$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$751$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$752$	−1.00000	−1.00000
$753$	0	0
$754$	0	0
$755$	−2.00000	−2.00000
$756$	−1.00000	−1.00000
$757$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$757$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	1.41421i	1.41421i
$763$	− 1.41421i	− 1.41421i
$764$	−1.00000	−1.00000
$765$	0	0
$766$	− 1.41421i	− 1.41421i
$767$	0	0
$768$	−1.00000	−1.00000
$769$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$769$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$770$	0	0
$771$	0	0
$772$	−1.00000	−1.00000
$773$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$773$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$774$	0	0
$775$	1.41421i	1.41421i
$776$	0	0
$777$	0	0
$778$	−2.00000	−2.00000
$779$	0	0
$780$	0	0
$781$	0	0
$782$	1.41421i	1.41421i
$783$	0	0
$784$	0	0
$785$	0	0
$786$	−2.00000	−2.00000
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	− 2.82843i	− 2.82843i
$791$	−1.00000	−1.00000
$792$	0	0
$793$	0	0
$794$	0	0
$795$	2.00000	2.00000
$796$	1.41421i	1.41421i
$797$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$797$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$798$	− 1.41421i	− 1.41421i
$799$	−1.00000	−1.00000
$800$	1.41421i	1.41421i
$801$	0	0
$802$	1.41421i	1.41421i
$803$	0	0
$804$	1.00000	1.00000
$805$	− 1.41421i	− 1.41421i
$806$	0	0
$807$	−1.00000	−1.00000
$808$	0	0
$809$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$809$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$810$	2.00000	2.00000
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−1.00000	−1.00000
$817$	1.41421i	1.41421i
$818$	− 1.41421i	− 1.41421i
$819$	0	0
$820$	0	0
$821$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$822$	− 2.82843i	− 2.82843i
$823$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$823$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$824$	0	0
$825$	0	0
$826$	1.41421i	1.41421i
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	2.82843i	2.82843i
$831$	1.00000	1.00000
$832$	0	0
$833$	0	0
$834$	−2.00000	−2.00000
$835$	0	0
$836$	0	0
$837$	− 1.41421i	− 1.41421i
$838$	0	0
$839$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$839$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	1.41421i	1.41421i
$843$	1.00000	1.00000
$844$	−1.00000	−1.00000
$845$	− 1.41421i	− 1.41421i
$846$	0	0
$847$	−1.00000	−1.00000
$848$	− 1.41421i	− 1.41421i
$849$	− 1.41421i	− 1.41421i
$850$	1.41421i	1.41421i
$851$	0	0
$852$	1.00000	1.00000
$853$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$853$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$854$	1.41421i	1.41421i
$855$	0	0
$856$	0	0
$857$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$857$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$858$	0	0
$859$	2.00000	2.00000	1.00000			$0$
$859$	2.00000	2.00000	1.00000			$0$
$860$	2.00000	2.00000
$861$	0	0
$862$	0	0
$863$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$863$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$864$	− 1.41421i	− 1.41421i
$865$	0	0
$866$	2.00000	2.00000
$867$	0	0
$868$	1.41421i	1.41421i
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	− 1.41421i	− 1.41421i
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	1.41421i	1.41421i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	− 1.41421i	− 1.41421i
$886$	2.00000	2.00000
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−1.00000	−1.00000
$890$	2.82843i	2.82843i
$891$	0	0
$892$	1.00000	1.00000
$893$	1.00000	1.00000
$894$	− 1.41421i	− 1.41421i
$895$	− 2.82843i	− 2.82843i
$896$	0	0
$897$	0	0
$898$	− 1.41421i	− 1.41421i
$899$	0	0
$900$	0	0
$901$	− 1.41421i	− 1.41421i
$902$	0	0
$903$	− 1.41421i	− 1.41421i
$904$	0	0
$905$	− 1.41421i	− 1.41421i
$906$	2.00000	2.00000
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.00000	1.00000
$913$	0	0
$914$	0	0
$915$	− 1.41421i	− 1.41421i
$916$	− 1.41421i	− 1.41421i
$917$	− 1.41421i	− 1.41421i
$918$	− 1.41421i	− 1.41421i
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$929$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$930$	− 2.82843i	− 2.82843i
$931$	0	0
$932$	1.41421i	1.41421i
$933$	0	0
$934$	− 1.41421i	− 1.41421i
$935$	0	0
$936$	0	0
$937$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$937$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$938$	1.41421i	1.41421i
$939$	1.41421i	1.41421i
$940$	− 1.41421i	− 1.41421i
$941$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$941$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$942$	0	0
$943$	0	0
$944$	−1.00000	−1.00000
$945$	1.41421i	1.41421i
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	1.41421i	1.41421i
$949$	0	0
$950$	− 1.41421i	− 1.41421i
$951$	− 1.41421i	− 1.41421i
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	1.41421i	1.41421i
$956$	1.41421i	1.41421i
$957$	0	0
$958$	2.00000	2.00000
$959$	2.00000	2.00000
$960$	− 1.41421i	− 1.41421i
$961$	−1.00000	−1.00000
$962$	0	0
$963$	0	0
$964$	−1.00000	−1.00000
$965$	1.41421i	1.41421i
$966$	1.41421i	1.41421i
$967$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$967$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$968$	0	0
$969$	1.00000	1.00000
$970$	− 2.82843i	− 2.82843i
$971$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$971$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$972$	0	0
$973$	− 1.41421i	− 1.41421i
$974$	2.00000	2.00000
$975$	0	0
$976$	−1.00000	−1.00000
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 1.41421i	− 1.41421i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.00000	−1.00000
$988$	0	0
$989$	− 1.41421i	− 1.41421i
$990$	0	0
$991$	2.00000	2.00000	1.00000			$0$
$991$	2.00000	2.00000	1.00000			$0$
$992$	−2.00000	−2.00000
$993$	0	0
$994$	1.41421i	1.41421i
$995$	2.00000	2.00000
$996$	− 1.41421i	− 1.41421i
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	563.1.b.b.562.2	yes	2
563.562	odd	2	inner	563.1.b.b.562.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
563.1.b.b.562.1	✓	2	563.562	odd	2	inner
563.1.b.b.562.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 563 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	563.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000 q^{7} +O(q^{10})\)	\(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000 q^{7} -2.00000 q^{10} +1.00000 q^{12} +1.41421i q^{14} -1.41421i q^{15} -1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{19} -1.41421i q^{20} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{30} -1.41421i q^{31} -1.41421i q^{32} -1.41421i q^{34} +1.41421i q^{35} +1.41421i q^{38} -1.41421i q^{42} +1.41421i q^{43} -1.41421i q^{46} +1.00000 q^{47} +1.00000 q^{48} -1.41421i q^{50} +1.00000 q^{51} +1.41421i q^{53} +1.41421i q^{54} -1.00000 q^{57} +1.00000 q^{59} +1.41421i q^{60} +1.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{67} +1.00000 q^{68} +1.00000 q^{69} -2.00000 q^{70} +1.00000 q^{71} +1.00000 q^{75} -1.00000 q^{76} +1.41421i q^{79} -1.41421i q^{80} -1.00000 q^{81} -1.41421i q^{83} +1.00000 q^{84} -1.41421i q^{85} -2.00000 q^{86} -1.41421i q^{89} +1.00000 q^{92} +1.41421i q^{93} +1.41421i q^{94} +1.41421i q^{95} +1.41421i q^{96} +1.41421i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^7	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{7} - 4 q^{10} + 2 q^{12} - 2 q^{16} - 2 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{23} - 2 q^{25} + 2 q^{27} - 2 q^{28} + 4 q^{30} + 2 q^{47} + 2 q^{48} + 2 q^{51} - 2 q^{57} + 2 q^{59} + 2 q^{61} + 4 q^{62} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 2 q^{69} - 4 q^{70} + 2 q^{71} + 2 q^{75} - 2 q^{76} - 2 q^{81} + 2 q^{84} - 4 q^{86} + 2 q^{92}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^7 - 4 * q^10 + 2 * q^12 - 2 * q^16 - 2 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^23 - 2 * q^25 + 2 * q^27 - 2 * q^28 + 4 * q^30 + 2 * q^47 + 2 * q^48 + 2 * q^51 - 2 * q^57 + 2 * q^59 + 2 * q^61 + 4 * q^62 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 2 * q^69 - 4 * q^70 + 2 * q^71 + 2 * q^75 - 2 * q^76 - 2 * q^81 + 2 * q^84 - 4 * q^86 + 2 * q^92

Introduction

L-functions

Modular forms

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