LMFDB - Embedded newform 2243.1.b.b.2242.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2243,1,Mod(2242,2243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2243, 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("2243.2242");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2243 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2243.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:	$1.11940282334$
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 4.2.2243.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.11284642907.1

Embedding invariants

Embedding label			2242.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	2243.2242
Dual form			2243.1.b.b.2242.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.00000 q^{7} +O(q^{10})$	$q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.00000 q^{7} +1.00000 q^{12} -1.41421i q^{13} +1.41421i q^{14} -1.00000 q^{16} -1.00000 q^{17} +1.41421i q^{19} -1.00000 q^{21} +1.41421i q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +1.41421i q^{29} -1.00000 q^{31} -1.41421i q^{32} -1.41421i q^{34} +1.41421i q^{37} -2.00000 q^{38} +1.41421i q^{39} -1.41421i q^{42} -2.00000 q^{46} +1.41421i q^{47} +1.00000 q^{48} +1.41421i q^{50} +1.00000 q^{51} +1.41421i q^{52} +1.41421i q^{54} -1.41421i q^{57} -2.00000 q^{58} -1.41421i q^{62} +1.00000 q^{64} +1.00000 q^{67} +1.00000 q^{68} -1.41421i q^{69} +1.00000 q^{71} -2.00000 q^{74} -1.00000 q^{75} -1.41421i q^{76} -2.00000 q^{78} -1.41421i q^{79} -1.00000 q^{81} -1.00000 q^{83} +1.00000 q^{84} -1.41421i q^{87} -1.00000 q^{89} -1.41421i q^{91} -1.41421i q^{92} +1.00000 q^{93} -2.00000 q^{94} +1.41421i q^{96} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{12} - 2 q^{16} - 2 q^{17} - 2 q^{21} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{31} - 4 q^{38} - 4 q^{46} + 2 q^{48} + 2 q^{51} - 4 q^{58} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 2 q^{71} - 4 q^{74} - 2 q^{75} - 4 q^{78} - 2 q^{81} - 2 q^{83} + 2 q^{84} - 2 q^{89} + 2 q^{93} - 4 q^{94} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^7 + 2 * q^12 - 2 * q^16 - 2 * q^17 - 2 * q^21 + 2 * q^25 + 4 * q^26 + 2 * q^27 - 2 * q^28 - 2 * q^31 - 4 * q^38 - 4 * q^46 + 2 * q^48 + 2 * q^51 - 4 * q^58 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 2 * q^71 - 4 * q^74 - 2 * q^75 - 4 * q^78 - 2 * q^81 - 2 * q^83 + 2 * q^84 - 2 * q^89 + 2 * q^93 - 4 * q^94 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2243\mathbb{Z}\right)^\times$.

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

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2243.1.b.b.2242.2	yes	2
2243.2242	odd	2	inner	2243.1.b.b.2242.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2243.1.b.b.2242.1	✓	2	2243.2242	odd	2	inner
2243.1.b.b.2242.2	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:	\( N \)	\(=\)	\( 2243 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2243.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.00000 q^{7} +O(q^{10})\)	\(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.00000 q^{7} +1.00000 q^{12} -1.41421i q^{13} +1.41421i q^{14} -1.00000 q^{16} -1.00000 q^{17} +1.41421i q^{19} -1.00000 q^{21} +1.41421i q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +1.41421i q^{29} -1.00000 q^{31} -1.41421i q^{32} -1.41421i q^{34} +1.41421i q^{37} -2.00000 q^{38} +1.41421i q^{39} -1.41421i q^{42} -2.00000 q^{46} +1.41421i q^{47} +1.00000 q^{48} +1.41421i q^{50} +1.00000 q^{51} +1.41421i q^{52} +1.41421i q^{54} -1.41421i q^{57} -2.00000 q^{58} -1.41421i q^{62} +1.00000 q^{64} +1.00000 q^{67} +1.00000 q^{68} -1.41421i q^{69} +1.00000 q^{71} -2.00000 q^{74} -1.00000 q^{75} -1.41421i q^{76} -2.00000 q^{78} -1.41421i q^{79} -1.00000 q^{81} -1.00000 q^{83} +1.00000 q^{84} -1.41421i q^{87} -1.00000 q^{89} -1.41421i q^{91} -1.41421i q^{92} +1.00000 q^{93} -2.00000 q^{94} +1.41421i q^{96} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^7	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{7} + 2 q^{12} - 2 q^{16} - 2 q^{17} - 2 q^{21} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{31} - 4 q^{38} - 4 q^{46} + 2 q^{48} + 2 q^{51} - 4 q^{58} + 2 q^{64} + 2 q^{67} + 2 q^{68} + 2 q^{71} - 4 q^{74} - 2 q^{75} - 4 q^{78} - 2 q^{81} - 2 q^{83} + 2 q^{84} - 2 q^{89} + 2 q^{93} - 4 q^{94} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^7 + 2 * q^12 - 2 * q^16 - 2 * q^17 - 2 * q^21 + 2 * q^25 + 4 * q^26 + 2 * q^27 - 2 * q^28 - 2 * q^31 - 4 * q^38 - 4 * q^46 + 2 * q^48 + 2 * q^51 - 4 * q^58 + 2 * q^64 + 2 * q^67 + 2 * q^68 + 2 * q^71 - 4 * q^74 - 2 * q^75 - 4 * q^78 - 2 * q^81 - 2 * q^83 + 2 * q^84 - 2 * q^89 + 2 * q^93 - 4 * q^94 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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