LMFDB - Embedded newform 643.1.b.b.642.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [643,1,Mod(642,643)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(643, 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("643.642");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 643 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	643.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:	$0.320898803123$
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 4.2.643.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.265847707.1

Embedding invariants

Embedding label			642.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	643.642
Dual form			643.1.b.b.642.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{9} -1.41421i q^{11} +1.41421i q^{12} +1.41421i q^{13} -1.41421i q^{14} -1.00000 q^{16} +1.41421i q^{17} +1.41421i q^{18} +1.41421i q^{19} -1.41421i q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +1.00000 q^{29} -1.00000 q^{31} +1.41421i q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{38} +2.00000 q^{39} +1.41421i q^{41} -2.00000 q^{42} +1.41421i q^{44} +1.41421i q^{46} +1.41421i q^{48} -1.41421i q^{50} +2.00000 q^{51} -1.41421i q^{52} -1.00000 q^{53} +2.00000 q^{57} -1.41421i q^{58} -1.41421i q^{59} +1.41421i q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.82843i q^{66} -1.41421i q^{67} -1.41421i q^{68} +1.41421i q^{69} -1.41421i q^{73} -1.41421i q^{75} -1.41421i q^{76} -1.41421i q^{77} -2.82843i q^{78} +1.41421i q^{79} -1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{83} +1.41421i q^{84} -1.41421i q^{87} +1.00000 q^{89} +1.41421i q^{91} +1.00000 q^{92} +1.41421i q^{93} +2.00000 q^{96} -1.00000 q^{97} +1.41421i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{6} + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^6 + 2 * q^7 - 2 * q^9	$ 2 q - 2 q^{4} - 4 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{16} - 4 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 2 q^{29} - 2 q^{31} - 4 q^{33} + 4 q^{34} + 2 q^{36} + 4 q^{38} + 4 q^{39} - 4 q^{42} + 4 q^{51} - 2 q^{53} + 4 q^{57} - 2 q^{63} + 2 q^{64} - 2 q^{81} + 4 q^{82} - 2 q^{83} + 2 q^{89} + 2 q^{92} + 4 q^{96} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 + 2 * q^7 - 2 * q^9 - 2 * q^16 - 4 * q^22 - 2 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 2 * q^29 - 2 * q^31 - 4 * q^33 + 4 * q^34 + 2 * q^36 + 4 * q^38 + 4 * q^39 - 4 * q^42 + 4 * q^51 - 2 * q^53 + 4 * q^57 - 2 * q^63 + 2 * q^64 - 2 * q^81 + 4 * q^82 - 2 * q^83 + 2 * q^89 + 2 * q^92 + 4 * q^96 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/643\mathbb{Z}\right)^\times$.

$n$	$11$
$\chi(n)$	$-1$

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	$-0.750000\pi$
$2$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$	−1.00000	−1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−2.00000	−2.00000
$7$	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
$9$	−1.00000	−1.00000
$10$	0	0
$11$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$11$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$12$	1.41421i	1.41421i
$13$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$13$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$14$	− 1.41421i	− 1.41421i
$15$	0	0
$16$	−1.00000	−1.00000
$17$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$17$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$18$	1.41421i	1.41421i
$19$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$19$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$20$	0	0
$21$	− 1.41421i	− 1.41421i
$22$	−2.00000	−2.00000
$23$	−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
$26$	2.00000	2.00000
$27$	0	0
$28$	−1.00000	−1.00000
$29$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$29$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$30$	0	0
$31$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$31$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$32$	1.41421i	1.41421i
$33$	−2.00000	−2.00000
$34$	2.00000	2.00000
$35$	0	0
$36$	1.00000	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	2.00000	2.00000
$39$	2.00000	2.00000
$40$	0	0
$41$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$41$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$42$	−2.00000	−2.00000
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	1.41421i	1.41421i
$45$	0	0
$46$	1.41421i	1.41421i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.41421i	1.41421i
$49$	0	0
$50$	− 1.41421i	− 1.41421i
$51$	2.00000	2.00000
$52$	− 1.41421i	− 1.41421i
$53$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	2.00000
$58$	− 1.41421i	− 1.41421i
$59$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.41421i	1.41421i
$63$	−1.00000	−1.00000
$64$	1.00000	1.00000
$65$	0	0
$66$	2.82843i	2.82843i
$67$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$67$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$68$	− 1.41421i	− 1.41421i
$69$	1.41421i	1.41421i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$73$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$74$	0	0
$75$	− 1.41421i	− 1.41421i
$76$	− 1.41421i	− 1.41421i
$77$	− 1.41421i	− 1.41421i
$78$	− 2.82843i	− 2.82843i
$79$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$79$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$80$	0	0
$81$	−1.00000	−1.00000
$82$	2.00000	2.00000
$83$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$84$	1.41421i	1.41421i
$85$	0	0
$86$	0	0
$87$	− 1.41421i	− 1.41421i
$88$	0	0
$89$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$89$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$90$	0	0
$91$	1.41421i	1.41421i
$92$	1.00000	1.00000
$93$	1.41421i	1.41421i
$94$	0	0
$95$	0	0
$96$	2.00000	2.00000
$97$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$97$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$98$	0	0
$99$	1.41421i	1.41421i
$100$	−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	$0.666667\pi$
$102$	− 2.82843i	− 2.82843i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	1.41421i	1.41421i
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 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
$113$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$114$	− 2.82843i	− 2.82843i
$115$	0	0
$116$	−1.00000	−1.00000
$117$	− 1.41421i	− 1.41421i
$118$	−2.00000	−2.00000
$119$	1.41421i	1.41421i
$120$	0	0
$121$	−1.00000	−1.00000
$122$	0	0
$123$	2.00000	2.00000
$124$	1.00000	1.00000
$125$	0	0
$126$	1.41421i	1.41421i
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$132$	2.00000	2.00000
$133$	1.41421i	1.41421i
$134$	−2.00000	−2.00000
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	2.00000	2.00000
$139$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.00000	2.00000
$144$	1.00000	1.00000
$145$	0	0
$146$	−2.00000	−2.00000
$147$	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	$0.333333\pi$
$150$	−2.00000	−2.00000
$151$	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
$153$	− 1.41421i	− 1.41421i
$154$	−2.00000	−2.00000
$155$	0	0
$156$	−2.00000	−2.00000
$157$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$157$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$158$	2.00000	2.00000
$159$	1.41421i	1.41421i
$160$	0	0
$161$	−1.00000	−1.00000
$162$	1.41421i	1.41421i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	− 1.41421i	− 1.41421i
$165$	0	0
$166$	1.41421i	1.41421i
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−1.00000	−1.00000
$170$	0	0
$171$	− 1.41421i	− 1.41421i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−2.00000	−2.00000
$175$	1.00000	1.00000
$176$	1.41421i	1.41421i
$177$	−2.00000	−2.00000
$178$	− 1.41421i	− 1.41421i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	2.00000	2.00000
$183$	0	0
$184$	0	0
$185$	0	0
$186$	2.00000	2.00000
$187$	2.00000	2.00000
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 1.41421i	− 1.41421i
$193$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$194$	1.41421i	1.41421i
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	2.00000	2.00000
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−2.00000	−2.00000
$202$	1.41421i	1.41421i
$203$	1.00000	1.00000
$204$	−2.00000	−2.00000
$205$	0	0
$206$	0	0
$207$	1.00000	1.00000
$208$	− 1.41421i	− 1.41421i
$209$	2.00000	2.00000
$210$	0	0
$211$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$212$	1.00000	1.00000
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.00000	−1.00000
$218$	−2.00000	−2.00000
$219$	−2.00000	−2.00000
$220$	0	0
$221$	−2.00000	−2.00000
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.41421i	1.41421i
$225$	−1.00000	−1.00000
$226$	−2.00000	−2.00000
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−2.00000	−2.00000
$229$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$229$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$230$	0	0
$231$	−2.00000	−2.00000
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−2.00000	−2.00000
$235$	0	0
$236$	1.41421i	1.41421i
$237$	2.00000	2.00000
$238$	2.00000	2.00000
$239$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$239$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$240$	0	0
$241$	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
$243$	1.41421i	1.41421i
$244$	0	0
$245$	0	0
$246$	− 2.82843i	− 2.82843i
$247$	−2.00000	−2.00000
$248$	0	0
$249$	1.41421i	1.41421i
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	1.00000	1.00000
$253$	1.41421i	1.41421i
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$257$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.00000	−1.00000
$262$	− 1.41421i	− 1.41421i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	2.00000
$267$	− 1.41421i	− 1.41421i
$268$	1.41421i	1.41421i
$269$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$269$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 1.41421i	− 1.41421i
$273$	2.00000	2.00000
$274$	0	0
$275$	− 1.41421i	− 1.41421i
$276$	− 1.41421i	− 1.41421i
$277$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$277$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$278$	− 1.41421i	− 1.41421i
$279$	1.00000	1.00000
$280$	0	0
$281$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$282$	0	0
$283$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$283$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$284$	0	0
$285$	0	0
$286$	− 2.82843i	− 2.82843i
$287$	1.41421i	1.41421i
$288$	− 1.41421i	− 1.41421i
$289$	−1.00000	−1.00000
$290$	0	0
$291$	1.41421i	1.41421i
$292$	1.41421i	1.41421i
$293$	−2.00000	−2.00000	−1.00000			$\pi$
$293$	−2.00000	−2.00000	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	− 1.41421i	− 1.41421i
$299$	− 1.41421i	− 1.41421i
$300$	1.41421i	1.41421i
$301$	0	0
$302$	2.00000	2.00000
$303$	1.41421i	1.41421i
$304$	− 1.41421i	− 1.41421i
$305$	0	0
$306$	−2.00000	−2.00000
$307$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$307$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$308$	1.41421i	1.41421i
$309$	0	0
$310$	0	0
$311$	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	$-0.250000\pi$
$314$	2.00000	2.00000
$315$	0	0
$316$	− 1.41421i	− 1.41421i
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	2.00000	2.00000
$319$	− 1.41421i	− 1.41421i
$320$	0	0
$321$	0	0
$322$	1.41421i	1.41421i
$323$	−2.00000	−2.00000
$324$	1.00000	1.00000
$325$	1.41421i	1.41421i
$326$	0	0
$327$	−2.00000	−2.00000
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$331$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$332$	1.00000	1.00000
$333$	0	0
$334$	0	0
$335$	0	0
$336$	1.41421i	1.41421i
$337$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$337$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$338$	1.41421i	1.41421i
$339$	−2.00000	−2.00000
$340$	0	0
$341$	1.41421i	1.41421i
$342$	−2.00000	−2.00000
$343$	−1.00000	−1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	1.41421i	1.41421i
$349$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$349$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$350$	− 1.41421i	− 1.41421i
$351$	0	0
$352$	2.00000	2.00000
$353$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$353$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$354$	2.82843i	2.82843i
$355$	0	0
$356$	−1.00000	−1.00000
$357$	2.00000	2.00000
$358$	0	0
$359$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$359$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$360$	0	0
$361$	−1.00000	−1.00000
$362$	0	0
$363$	1.41421i	1.41421i
$364$	− 1.41421i	− 1.41421i
$365$	0	0
$366$	0	0
$367$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$368$	1.00000	1.00000
$369$	− 1.41421i	− 1.41421i
$370$	0	0
$371$	−1.00000	−1.00000
$372$	− 1.41421i	− 1.41421i
$373$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$374$	− 2.82843i	− 2.82843i
$375$	0	0
$376$	0	0
$377$	1.41421i	1.41421i
$378$	0	0
$379$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$379$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	1.41421i	1.41421i
$387$	0	0
$388$	1.00000	1.00000
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	− 1.41421i	− 1.41421i
$392$	0	0
$393$	− 1.41421i	− 1.41421i
$394$	0	0
$395$	0	0
$396$	− 1.41421i	− 1.41421i
$397$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$398$	0	0
$399$	2.00000	2.00000
$400$	−1.00000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	2.82843i	2.82843i
$403$	− 1.41421i	− 1.41421i
$404$	1.00000	1.00000
$405$	0	0
$406$	− 1.41421i	− 1.41421i
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.41421i	− 1.41421i
$414$	− 1.41421i	− 1.41421i
$415$	0	0
$416$	−2.00000	−2.00000
$417$	− 1.41421i	− 1.41421i
$418$	− 2.82843i	− 2.82843i
$419$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$420$	0	0
$421$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$422$	1.41421i	1.41421i
$423$	0	0
$424$	0	0
$425$	1.41421i	1.41421i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 2.82843i	− 2.82843i
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	1.41421i	1.41421i
$435$	0	0
$436$	1.41421i	1.41421i
$437$	− 1.41421i	− 1.41421i
$438$	2.82843i	2.82843i
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	2.82843i	2.82843i
$443$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.41421i	− 1.41421i
$448$	1.00000	1.00000
$449$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$450$	1.41421i	1.41421i
$451$	2.00000	2.00000
$452$	1.41421i	1.41421i
$453$	2.00000	2.00000
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	− 1.41421i	− 1.41421i
$459$	0	0
$460$	0	0
$461$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$461$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$462$	2.82843i	2.82843i
$463$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$463$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$464$	−1.00000	−1.00000
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	1.41421i	1.41421i
$469$	− 1.41421i	− 1.41421i
$470$	0	0
$471$	2.00000	2.00000
$472$	0	0
$473$	0	0
$474$	− 2.82843i	− 2.82843i
$475$	1.41421i	1.41421i
$476$	− 1.41421i	− 1.41421i
$477$	1.00000	1.00000
$478$	2.00000	2.00000
$479$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$479$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$480$	0	0
$481$	0	0
$482$	− 1.41421i	− 1.41421i
$483$	1.41421i	1.41421i
$484$	1.00000	1.00000
$485$	0	0
$486$	2.00000	2.00000
$487$	− 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
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−2.00000	−2.00000
$493$	1.41421i	1.41421i
$494$	2.82843i	2.82843i
$495$	0	0
$496$	1.00000	1.00000
$497$	0	0
$498$	2.00000	2.00000
$499$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$499$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.00000	−2.00000	−1.00000			$\pi$
$503$	−2.00000	−2.00000	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	2.00000	2.00000
$507$	1.41421i	1.41421i
$508$	0	0
$509$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$509$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$510$	0	0
$511$	− 1.41421i	− 1.41421i
$512$	− 1.41421i	− 1.41421i
$513$	0	0
$514$	1.41421i	1.41421i
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	1.41421i	1.41421i
$523$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$524$	−1.00000	−1.00000
$525$	− 1.41421i	− 1.41421i
$526$	0	0
$527$	− 1.41421i	− 1.41421i
$528$	2.00000	2.00000
$529$	0	0
$530$	0	0
$531$	1.41421i	1.41421i
$532$	− 1.41421i	− 1.41421i
$533$	−2.00000	−2.00000
$534$	−2.00000	−2.00000
$535$	0	0
$536$	0	0
$537$	0	0
$538$	− 1.41421i	− 1.41421i
$539$	0	0
$540$	0	0
$541$	2.00000	2.00000	1.00000			$0$
$541$	2.00000	2.00000	1.00000			$0$
$542$	0	0
$543$	0	0
$544$	−2.00000	−2.00000
$545$	0	0
$546$	− 2.82843i	− 2.82843i
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−2.00000	−2.00000
$551$	1.41421i	1.41421i
$552$	0	0
$553$	1.41421i	1.41421i
$554$	− 1.41421i	− 1.41421i
$555$	0	0
$556$	−1.00000	−1.00000
$557$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$557$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$558$	− 1.41421i	− 1.41421i
$559$	0	0
$560$	0	0
$561$	− 2.82843i	− 2.82843i
$562$	1.41421i	1.41421i
$563$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$563$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$564$	0	0
$565$	0	0
$566$	−2.00000	−2.00000
$567$	−1.00000	−1.00000
$568$	0	0
$569$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$569$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$570$	0	0
$571$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$572$	−2.00000	−2.00000
$573$	0	0
$574$	2.00000	2.00000
$575$	−1.00000	−1.00000
$576$	−1.00000	−1.00000
$577$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$577$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$578$	1.41421i	1.41421i
$579$	1.41421i	1.41421i
$580$	0	0
$581$	−1.00000	−1.00000
$582$	2.00000	2.00000
$583$	1.41421i	1.41421i
$584$	0	0
$585$	0	0
$586$	2.82843i	2.82843i
$587$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$587$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$588$	0	0
$589$	− 1.41421i	− 1.41421i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$593$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$594$	0	0
$595$	0	0
$596$	−1.00000	−1.00000
$597$	0	0
$598$	−2.00000	−2.00000
$599$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$599$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	1.41421i	1.41421i
$604$	− 1.41421i	− 1.41421i
$605$	0	0
$606$	2.00000	2.00000
$607$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$607$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$608$	−2.00000	−2.00000
$609$	− 1.41421i	− 1.41421i
$610$	0	0
$611$	0	0
$612$	1.41421i	1.41421i
$613$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$613$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$614$	1.41421i	1.41421i
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.00000	1.00000
$624$	−2.00000	−2.00000
$625$	1.00000	1.00000
$626$	−2.00000	−2.00000
$627$	− 2.82843i	− 2.82843i
$628$	− 1.41421i	− 1.41421i
$629$	0	0
$630$	0	0
$631$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$632$	0	0
$633$	1.41421i	1.41421i
$634$	0	0
$635$	0	0
$636$	− 1.41421i	− 1.41421i
$637$	0	0
$638$	−2.00000	−2.00000
$639$	0	0
$640$	0	0
$641$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$642$	0	0
$643$	−1.00000	−1.00000
$643$	−1.00000	−1.00000
$644$	1.00000	1.00000
$645$	0	0
$646$	2.82843i	2.82843i
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−2.00000	−2.00000
$650$	2.00000	2.00000
$651$	1.41421i	1.41421i
$652$	0	0
$653$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$653$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$654$	2.82843i	2.82843i
$655$	0	0
$656$	− 1.41421i	− 1.41421i
$657$	1.41421i	1.41421i
$658$	0	0
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$660$	0	0
$661$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$661$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$662$	− 1.41421i	− 1.41421i
$663$	2.82843i	2.82843i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.00000	−1.00000
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	2.00000	2.00000
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	2.00000	2.00000
$675$	0	0
$676$	1.00000	1.00000
$677$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$677$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$678$	2.82843i	2.82843i
$679$	−1.00000	−1.00000
$680$	0	0
$681$	0	0
$682$	2.00000	2.00000
$683$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$684$	1.41421i	1.41421i
$685$	0	0
$686$	1.41421i	1.41421i
$687$	− 1.41421i	− 1.41421i
$688$	0	0
$689$	− 1.41421i	− 1.41421i
$690$	0	0
$691$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$691$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$692$	0	0
$693$	1.41421i	1.41421i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.00000	−2.00000
$698$	−2.00000	−2.00000
$699$	0	0
$700$	−1.00000	−1.00000
$701$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$701$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$702$	0	0
$703$	0	0
$704$	− 1.41421i	− 1.41421i
$705$	0	0
$706$	2.00000	2.00000
$707$	−1.00000	−1.00000
$708$	2.00000	2.00000
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	− 1.41421i	− 1.41421i
$712$	0	0
$713$	1.00000	1.00000
$714$	− 2.82843i	− 2.82843i
$715$	0	0
$716$	0	0
$717$	2.00000	2.00000
$718$	1.41421i	1.41421i
$719$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$719$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$720$	0	0
$721$	0	0
$722$	1.41421i	1.41421i
$723$	− 1.41421i	− 1.41421i
$724$	0	0
$725$	1.00000	1.00000
$726$	2.00000	2.00000
$727$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$727$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$733$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$734$	− 1.41421i	− 1.41421i
$735$	0	0
$736$	− 1.41421i	− 1.41421i
$737$	−2.00000	−2.00000
$738$	−2.00000	−2.00000
$739$	−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
$741$	2.82843i	2.82843i
$742$	1.41421i	1.41421i
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	− 1.41421i	− 1.41421i
$747$	1.00000	1.00000
$748$	−2.00000	−2.00000
$749$	0	0
$750$	0	0
$751$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$752$	0	0
$753$	0	0
$754$	2.00000	2.00000
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	− 1.41421i	− 1.41421i
$759$	2.00000	2.00000
$760$	0	0
$761$	2.00000	2.00000	1.00000			$0$
$761$	2.00000	2.00000	1.00000			$0$
$762$	0	0
$763$	− 1.41421i	− 1.41421i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.00000	2.00000
$768$	− 1.41421i	− 1.41421i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	1.41421i	1.41421i
$772$	1.00000	1.00000
$773$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$773$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$774$	0	0
$775$	−1.00000	−1.00000
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−2.00000
$780$	0	0
$781$	0	0
$782$	−2.00000	−2.00000
$783$	0	0
$784$	0	0
$785$	0	0
$786$	−2.00000	−2.00000
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.41421i	− 1.41421i
$792$	0	0
$793$	0	0
$794$	1.41421i	1.41421i
$795$	0	0
$796$	0	0
$797$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$797$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$798$	− 2.82843i	− 2.82843i
$799$	0	0
$800$	1.41421i	1.41421i
$801$	−1.00000	−1.00000
$802$	0	0
$803$	−2.00000	−2.00000
$804$	2.00000	2.00000
$805$	0	0
$806$	−2.00000	−2.00000
$807$	− 1.41421i	− 1.41421i
$808$	0	0
$809$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$809$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	−1.00000	−1.00000
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−2.00000	−2.00000
$817$	0	0
$818$	0	0
$819$	− 1.41421i	− 1.41421i
$820$	0	0
$821$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$821$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$822$	0	0
$823$	− 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
$825$	−2.00000	−2.00000
$826$	−2.00000	−2.00000
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−1.00000	−1.00000
$829$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$830$	0	0
$831$	− 1.41421i	− 1.41421i
$832$	1.41421i	1.41421i
$833$	0	0
$834$	−2.00000	−2.00000
$835$	0	0
$836$	−2.00000	−2.00000
$837$	0	0
$838$	− 1.41421i	− 1.41421i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	0	0
$842$	1.41421i	1.41421i
$843$	1.41421i	1.41421i
$844$	1.00000	1.00000
$845$	0	0
$846$	0	0
$847$	−1.00000	−1.00000
$848$	1.00000	1.00000
$849$	−2.00000	−2.00000
$850$	2.00000	2.00000
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$857$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$858$	−4.00000	−4.00000
$859$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$859$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$860$	0	0
$861$	2.00000	2.00000
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.41421i	1.41421i
$868$	1.00000	1.00000
$869$	2.00000	2.00000
$870$	0	0
$871$	2.00000	2.00000
$872$	0	0
$873$	1.00000	1.00000
$874$	−2.00000	−2.00000
$875$	0	0
$876$	2.00000	2.00000
$877$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$878$	0	0
$879$	2.82843i	2.82843i
$880$	0	0
$881$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$882$	0	0
$883$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$883$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$884$	2.00000	2.00000
$885$	0	0
$886$	1.41421i	1.41421i
$887$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$887$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.41421i	1.41421i
$892$	0	0
$893$	0	0
$894$	−2.00000	−2.00000
$895$	0	0
$896$	0	0
$897$	−2.00000	−2.00000
$898$	− 1.41421i	− 1.41421i
$899$	−1.00000	−1.00000
$900$	1.00000	1.00000
$901$	− 1.41421i	− 1.41421i
$902$	− 2.82843i	− 2.82843i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	− 2.82843i	− 2.82843i
$907$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$907$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$908$	0	0
$909$	1.00000	1.00000
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−2.00000	−2.00000
$913$	1.41421i	1.41421i
$914$	0	0
$915$	0	0
$916$	−1.00000	−1.00000
$917$	1.00000	1.00000
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1.41421i	1.41421i
$922$	−2.00000	−2.00000
$923$	0	0
$924$	2.00000	2.00000
$925$	0	0
$926$	− 1.41421i	− 1.41421i
$927$	0	0
$928$	1.41421i	1.41421i
$929$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$929$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$937$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$938$	−2.00000	−2.00000
$939$	−2.00000	−2.00000
$940$	0	0
$941$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$941$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$942$	− 2.82843i	− 2.82843i
$943$	− 1.41421i	− 1.41421i
$944$	1.41421i	1.41421i
$945$	0	0
$946$	0	0
$947$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$948$	−2.00000	−2.00000
$949$	2.00000	2.00000
$950$	2.00000	2.00000
$951$	0	0
$952$	0	0
$953$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$954$	− 1.41421i	− 1.41421i
$955$	0	0
$956$	− 1.41421i	− 1.41421i
$957$	−2.00000	−2.00000
$958$	− 1.41421i	− 1.41421i
$959$	0	0
$960$	0	0
$961$	0	0
$962$	0	0
$963$	0	0
$964$	−1.00000	−1.00000
$965$	0	0
$966$	2.00000	2.00000
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	2.82843i	2.82843i
$970$	0	0
$971$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$972$	− 1.41421i	− 1.41421i
$973$	1.00000	1.00000
$974$	−2.00000	−2.00000
$975$	2.00000	2.00000
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	− 1.41421i	− 1.41421i
$980$	0	0
$981$	1.41421i	1.41421i
$982$	0	0
$983$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$983$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$984$	0	0
$985$	0	0
$986$	2.00000	2.00000
$987$	0	0
$988$	2.00000	2.00000
$989$	0	0
$990$	0	0
$991$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$991$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$992$	− 1.41421i	− 1.41421i
$993$	− 1.41421i	− 1.41421i
$994$	0	0
$995$	0	0
$996$	− 1.41421i	− 1.41421i
$997$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$997$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$998$	2.00000	2.00000
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	643.1.b.b.642.1	✓	2
643.642	odd	2	inner	643.1.b.b.642.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
643.1.b.b.642.1	✓	2	1.1	even	1	trivial
643.1.b.b.642.2	yes	2	643.642	odd	2	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:	\( N \)	\(=\)	\( 643 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	643.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{9} -1.41421i q^{11} +1.41421i q^{12} +1.41421i q^{13} -1.41421i q^{14} -1.00000 q^{16} +1.41421i q^{17} +1.41421i q^{18} +1.41421i q^{19} -1.41421i q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +1.00000 q^{29} -1.00000 q^{31} +1.41421i q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{38} +2.00000 q^{39} +1.41421i q^{41} -2.00000 q^{42} +1.41421i q^{44} +1.41421i q^{46} +1.41421i q^{48} -1.41421i q^{50} +2.00000 q^{51} -1.41421i q^{52} -1.00000 q^{53} +2.00000 q^{57} -1.41421i q^{58} -1.41421i q^{59} +1.41421i q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.82843i q^{66} -1.41421i q^{67} -1.41421i q^{68} +1.41421i q^{69} -1.41421i q^{73} -1.41421i q^{75} -1.41421i q^{76} -1.41421i q^{77} -2.82843i q^{78} +1.41421i q^{79} -1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{83} +1.41421i q^{84} -1.41421i q^{87} +1.00000 q^{89} +1.41421i q^{91} +1.00000 q^{92} +1.41421i q^{93} +2.00000 q^{96} -1.00000 q^{97} +1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 + 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{16} - 4 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 2 q^{29} - 2 q^{31} - 4 q^{33} + 4 q^{34} + 2 q^{36} + 4 q^{38} + 4 q^{39} - 4 q^{42} + 4 q^{51} - 2 q^{53} + 4 q^{57} - 2 q^{63} + 2 q^{64} - 2 q^{81} + 4 q^{82} - 2 q^{83} + 2 q^{89} + 2 q^{92} + 4 q^{96} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 + 2 * q^7 - 2 * q^9 - 2 * q^16 - 4 * q^22 - 2 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 2 * q^29 - 2 * q^31 - 4 * q^33 + 4 * q^34 + 2 * q^36 + 4 * q^38 + 4 * q^39 - 4 * q^42 + 4 * q^51 - 2 * q^53 + 4 * q^57 - 2 * q^63 + 2 * q^64 - 2 * q^81 + 4 * q^82 - 2 * q^83 + 2 * q^89 + 2 * q^92 + 4 * q^96 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more