LMFDB - Embedded newform 1107.1.b.d.1106.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1107,1,Mod(1106,1107)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1107, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1107.1106");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1107 = 3^{3} \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1107.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:	$0.552464968985$
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 4.2.1107.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.150730227.4

Embedding invariants

Embedding label			1106.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	1107.1106
Dual form			1107.1.b.d.1106.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.00000 q^{4} +1.41421i q^{5} +1.41421i q^{7} +O(q^{10})$	$q+1.41421i q^{2} -1.00000 q^{4} +1.41421i q^{5} +1.41421i q^{7} -2.00000 q^{10} +1.00000 q^{11} -1.41421i q^{13} -2.00000 q^{14} -1.00000 q^{16} -1.41421i q^{19} -1.41421i q^{20} +1.41421i q^{22} -1.00000 q^{25} +2.00000 q^{26} -1.41421i q^{28} +1.00000 q^{29} +1.00000 q^{31} -1.41421i q^{32} -2.00000 q^{35} -1.00000 q^{37} +2.00000 q^{38} -1.00000 q^{41} +1.00000 q^{43} -1.00000 q^{44} -1.00000 q^{47} -1.00000 q^{49} -1.41421i q^{50} +1.41421i q^{52} -1.00000 q^{53} +1.41421i q^{55} +1.41421i q^{58} -1.41421i q^{59} +1.00000 q^{61} +1.41421i q^{62} +1.00000 q^{64} +2.00000 q^{65} -2.82843i q^{70} -1.00000 q^{73} -1.41421i q^{74} +1.41421i q^{76} +1.41421i q^{77} +1.41421i q^{79} -1.41421i q^{80} -1.41421i q^{82} +1.41421i q^{86} +1.00000 q^{89} +2.00000 q^{91} -1.41421i q^{94} +2.00000 q^{95} -1.41421i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 4 q^{10} + 2 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{25} + 4 q^{26} + 2 q^{29} + 2 q^{31} - 4 q^{35} - 2 q^{37} + 4 q^{38} - 2 q^{41} + 2 q^{43} - 2 q^{44} - 2 q^{47} - 2 q^{49} - 2 q^{53} + 2 q^{61} + 2 q^{64} + 4 q^{65} - 2 q^{73} + 2 q^{89} + 4 q^{91} + 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^10 + 2 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^25 + 4 * q^26 + 2 * q^29 + 2 * q^31 - 4 * q^35 - 2 * q^37 + 4 * q^38 - 2 * q^41 + 2 * q^43 - 2 * q^44 - 2 * q^47 - 2 * q^49 - 2 * q^53 + 2 * q^61 + 2 * q^64 + 4 * q^65 - 2 * q^73 + 2 * q^89 + 4 * q^91 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1107\mathbb{Z}\right)^\times$.

$n$	$83$	$703$
$\chi(n)$	$-1$	$-1$

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	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	−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	$0.750000\pi$
$6$	0	0
$7$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$7$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$8$	0	0
$9$	0	0
$10$	−2.00000	−2.00000
$11$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$11$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$12$	0	0
$13$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$13$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$14$	−2.00000	−2.00000
$15$	0	0
$16$	−1.00000	−1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$19$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$20$	− 1.41421i	− 1.41421i
$21$	0	0
$22$	1.41421i	1.41421i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−1.00000	−1.00000
$26$	2.00000	2.00000
$27$	0	0
$28$	− 1.41421i	− 1.41421i
$29$	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	$-0.333333\pi$
$31$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$32$	− 1.41421i	− 1.41421i
$33$	0	0
$34$	0	0
$35$	−2.00000	−2.00000
$36$	0	0
$37$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$38$	2.00000	2.00000
$39$	0	0
$40$	0	0
$41$	−1.00000	−1.00000
$41$	−1.00000	−1.00000
$42$	0	0
$43$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$43$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$44$	−1.00000	−1.00000
$45$	0	0
$46$	0	0
$47$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$47$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$48$	0	0
$49$	−1.00000	−1.00000
$50$	− 1.41421i	− 1.41421i
$51$	0	0
$52$	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
$55$	1.41421i	1.41421i
$56$	0	0
$57$	0	0
$58$	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
$61$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$61$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$62$	1.41421i	1.41421i
$63$	0	0
$64$	1.00000	1.00000
$65$	2.00000	2.00000
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	− 2.82843i	− 2.82843i
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$74$	− 1.41421i	− 1.41421i
$75$	0	0
$76$	1.41421i	1.41421i
$77$	1.41421i	1.41421i
$78$	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
$81$	0	0
$82$	− 1.41421i	− 1.41421i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	1.41421i	1.41421i
$87$	0	0
$88$	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
$91$	2.00000	2.00000
$92$	0	0
$93$	0	0
$94$	− 1.41421i	− 1.41421i
$95$	2.00000	2.00000
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 1.41421i	− 1.41421i
$99$	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	$0.666667\pi$
$102$	0	0
$103$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$103$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$104$	0	0
$105$	0	0
$106$	− 1.41421i	− 1.41421i
$107$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$107$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−2.00000	−2.00000
$111$	0	0
$112$	− 1.41421i	− 1.41421i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−1.00000
$117$	0	0
$118$	2.00000	2.00000
$119$	0	0
$120$	0	0
$121$	0	0
$122$	1.41421i	1.41421i
$123$	0	0
$124$	−1.00000	−1.00000
$125$	0	0
$126$	0	0
$127$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$127$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$128$	0	0
$129$	0	0
$130$	2.82843i	2.82843i
$131$	− 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
$133$	2.00000	2.00000
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$138$	0	0
$139$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$140$	2.00000	2.00000
$141$	0	0
$142$	0	0
$143$	− 1.41421i	− 1.41421i
$144$	0	0
$145$	1.41421i	1.41421i
$146$	− 1.41421i	− 1.41421i
$147$	0	0
$148$	1.00000	1.00000
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$151$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$152$	0	0
$153$	0	0
$154$	−2.00000	−2.00000
$155$	1.41421i	1.41421i
$156$	0	0
$157$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$157$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$158$	−2.00000	−2.00000
$159$	0	0
$160$	2.00000	2.00000
$161$	0	0
$162$	0	0
$163$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$164$	1.00000	1.00000
$165$	0	0
$166$	0	0
$167$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$167$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$168$	0	0
$169$	−1.00000	−1.00000
$170$	0	0
$171$	0	0
$172$	−1.00000	−1.00000
$173$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$173$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$174$	0	0
$175$	− 1.41421i	− 1.41421i
$176$	−1.00000	−1.00000
$177$	0	0
$178$	1.41421i	1.41421i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$181$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$182$	2.82843i	2.82843i
$183$	0	0
$184$	0	0
$185$	− 1.41421i	− 1.41421i
$186$	0	0
$187$	0	0
$188$	1.00000	1.00000
$189$	0	0
$190$	2.82843i	2.82843i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	1.00000	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	− 1.41421i	− 1.41421i
$203$	1.41421i	1.41421i
$204$	0	0
$205$	− 1.41421i	− 1.41421i
$206$	1.41421i	1.41421i
$207$	0	0
$208$	1.41421i	1.41421i
$209$	− 1.41421i	− 1.41421i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.00000	1.00000
$213$	0	0
$214$	−2.00000	−2.00000
$215$	1.41421i	1.41421i
$216$	0	0
$217$	1.41421i	1.41421i
$218$	0	0
$219$	0	0
$220$	− 1.41421i	− 1.41421i
$221$	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	$0.666667\pi$
$224$	2.00000	2.00000
$225$	0	0
$226$	0	0
$227$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$227$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$228$	0	0
$229$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$229$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$234$	0	0
$235$	− 1.41421i	− 1.41421i
$236$	1.41421i	1.41421i
$237$	0	0
$238$	0	0
$239$	−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
$241$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$242$	0	0
$243$	0	0
$244$	−1.00000	−1.00000
$245$	− 1.41421i	− 1.41421i
$246$	0	0
$247$	−2.00000	−2.00000
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$251$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$252$	0	0
$253$	0	0
$254$	1.41421i	1.41421i
$255$	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
$259$	− 1.41421i	− 1.41421i
$260$	−2.00000	−2.00000
$261$	0	0
$262$	2.00000	2.00000
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	− 1.41421i	− 1.41421i
$266$	2.82843i	2.82843i
$267$	0	0
$268$	0	0
$269$	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	$0.333333\pi$
$272$	0	0
$273$	0	0
$274$	− 1.41421i	− 1.41421i
$275$	−1.00000	−1.00000
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	1.41421i	1.41421i
$279$	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	$0.666667\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	2.00000	2.00000
$287$	− 1.41421i	− 1.41421i
$288$	0	0
$289$	−1.00000	−1.00000
$290$	−2.00000	−2.00000
$291$	0	0
$292$	1.00000	1.00000
$293$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$294$	0	0
$295$	2.00000	2.00000
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.41421i	1.41421i
$302$	2.00000	2.00000
$303$	0	0
$304$	1.41421i	1.41421i
$305$	1.41421i	1.41421i
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	− 1.41421i	− 1.41421i
$309$	0	0
$310$	−2.00000	−2.00000
$311$	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
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	2.00000	2.00000
$315$	0	0
$316$	− 1.41421i	− 1.41421i
$317$	2.00000	2.00000	1.00000			$0$
$317$	2.00000	2.00000	1.00000			$0$
$318$	0	0
$319$	1.00000	1.00000
$320$	1.41421i	1.41421i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.41421i	1.41421i
$326$	− 1.41421i	− 1.41421i
$327$	0	0
$328$	0	0
$329$	− 1.41421i	− 1.41421i
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	1.41421i	1.41421i
$335$	0	0
$336$	0	0
$337$	−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
$339$	0	0
$340$	0	0
$341$	1.00000	1.00000
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−2.00000	−2.00000
$347$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$347$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$348$	0	0
$349$	−2.00000	−2.00000	−1.00000			$\pi$
$349$	−2.00000	−2.00000	−1.00000			$\pi$
$350$	2.00000	2.00000
$351$	0	0
$352$	− 1.41421i	− 1.41421i
$353$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$353$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$354$	0	0
$355$	0	0
$356$	−1.00000	−1.00000
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−1.00000	−1.00000
$362$	−2.00000	−2.00000
$363$	0	0
$364$	−2.00000	−2.00000
$365$	− 1.41421i	− 1.41421i
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	2.00000	2.00000
$371$	− 1.41421i	− 1.41421i
$372$	0	0
$373$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.41421i	− 1.41421i
$378$	0	0
$379$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$380$	−2.00000	−2.00000
$381$	0	0
$382$	0	0
$383$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$383$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$384$	0	0
$385$	−2.00000	−2.00000
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.00000	−2.00000
$396$	0	0
$397$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$397$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	1.00000
$401$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$401$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$402$	0	0
$403$	− 1.41421i	− 1.41421i
$404$	1.00000	1.00000
$405$	0	0
$406$	−2.00000	−2.00000
$407$	−1.00000	−1.00000
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	2.00000	2.00000
$411$	0	0
$412$	−1.00000	−1.00000
$413$	2.00000	2.00000
$414$	0	0
$415$	0	0
$416$	−2.00000	−2.00000
$417$	0	0
$418$	2.00000	2.00000
$419$	− 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
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.41421i	1.41421i
$428$	− 1.41421i	− 1.41421i
$429$	0	0
$430$	−2.00000	−2.00000
$431$	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	$0.5\pi$
$434$	−2.00000	−2.00000
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$439$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	1.41421i	1.41421i
$446$	− 1.41421i	− 1.41421i
$447$	0	0
$448$	1.41421i	1.41421i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	−1.00000	−1.00000
$452$	0	0
$453$	0	0
$454$	1.41421i	1.41421i
$455$	2.82843i	2.82843i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−2.00000	−2.00000
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−1.00000	−1.00000
$465$	0	0
$466$	− 1.41421i	− 1.41421i
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	2.00000	2.00000
$471$	0	0
$472$	0	0
$473$	1.00000	1.00000
$474$	0	0
$475$	1.41421i	1.41421i
$476$	0	0
$477$	0	0
$478$	− 1.41421i	− 1.41421i
$479$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$480$	0	0
$481$	1.41421i	1.41421i
$482$	− 1.41421i	− 1.41421i
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	2.00000	2.00000
$491$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$491$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$492$	0	0
$493$	0	0
$494$	− 2.82843i	− 2.82843i
$495$	0	0
$496$	−1.00000	−1.00000
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	2.00000	2.00000
$503$	−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
$507$	0	0
$508$	−1.00000	−1.00000
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	− 1.41421i	− 1.41421i
$512$	1.41421i	1.41421i
$513$	0	0
$514$	− 1.41421i	− 1.41421i
$515$	1.41421i	1.41421i
$516$	0	0
$517$	−1.00000	−1.00000
$518$	2.00000	2.00000
$519$	0	0
$520$	0	0
$521$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$522$	0	0
$523$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$523$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$524$	1.41421i	1.41421i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	2.00000	2.00000
$531$	0	0
$532$	−2.00000	−2.00000
$533$	1.41421i	1.41421i
$534$	0	0
$535$	−2.00000	−2.00000
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.00000	−1.00000
$540$	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
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$547$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$548$	1.00000	1.00000
$549$	0	0
$550$	− 1.41421i	− 1.41421i
$551$	− 1.41421i	− 1.41421i
$552$	0	0
$553$	−2.00000	−2.00000
$554$	0	0
$555$	0	0
$556$	−1.00000	−1.00000
$557$	−2.00000	−2.00000	−1.00000			$\pi$
$557$	−2.00000	−2.00000	−1.00000			$\pi$
$558$	0	0
$559$	− 1.41421i	− 1.41421i
$560$	2.00000	2.00000
$561$	0	0
$562$	− 1.41421i	− 1.41421i
$563$	2.00000	2.00000	1.00000			$0$
$563$	2.00000	2.00000	1.00000			$0$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$571$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$572$	1.41421i	1.41421i
$573$	0	0
$574$	2.00000	2.00000
$575$	0	0
$576$	0	0
$577$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$577$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$578$	− 1.41421i	− 1.41421i
$579$	0	0
$580$	− 1.41421i	− 1.41421i
$581$	0	0
$582$	0	0
$583$	−1.00000	−1.00000
$584$	0	0
$585$	0	0
$586$	− 1.41421i	− 1.41421i
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	− 1.41421i	− 1.41421i
$590$	2.82843i	2.82843i
$591$	0	0
$592$	1.00000	1.00000
$593$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$593$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−2.00000	−2.00000
$603$	0	0
$604$	1.41421i	1.41421i
$605$	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
$609$	0	0
$610$	−2.00000	−2.00000
$611$	1.41421i	1.41421i
$612$	0	0
$613$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$613$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$614$	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		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	− 1.41421i	− 1.41421i
$621$	0	0
$622$	1.41421i	1.41421i
$623$	1.41421i	1.41421i
$624$	0	0
$625$	−1.00000	−1.00000
$626$	0	0
$627$	0	0
$628$	1.41421i	1.41421i
$629$	0	0
$630$	0	0
$631$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$632$	0	0
$633$	0	0
$634$	2.82843i	2.82843i
$635$	1.41421i	1.41421i
$636$	0	0
$637$	1.41421i	1.41421i
$638$	1.41421i	1.41421i
$639$	0	0
$640$	0	0
$641$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$641$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$642$	0	0
$643$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$643$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	− 1.41421i	− 1.41421i
$650$	−2.00000	−2.00000
$651$	0	0
$652$	1.00000	1.00000
$653$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$653$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$654$	0	0
$655$	2.00000	2.00000
$656$	1.00000	1.00000
$657$	0	0
$658$	2.00000	2.00000
$659$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$659$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$660$	0	0
$661$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$661$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.82843i	2.82843i
$666$	0	0
$667$	0	0
$668$	−1.00000	−1.00000
$669$	0	0
$670$	0	0
$671$	1.00000	1.00000
$672$	0	0
$673$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$674$	− 1.41421i	− 1.41421i
$675$	0	0
$676$	1.00000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	1.41421i	1.41421i
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	− 1.41421i	− 1.41421i
$686$	0	0
$687$	0	0
$688$	−1.00000	−1.00000
$689$	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	$-0.250000\pi$
$692$	− 1.41421i	− 1.41421i
$693$	0	0
$694$	1.41421i	1.41421i
$695$	1.41421i	1.41421i
$696$	0	0
$697$	0	0
$698$	− 2.82843i	− 2.82843i
$699$	0	0
$700$	1.41421i	1.41421i
$701$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$701$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$702$	0	0
$703$	1.41421i	1.41421i
$704$	1.00000	1.00000
$705$	0	0
$706$	2.00000	2.00000
$707$	− 1.41421i	− 1.41421i
$708$	0	0
$709$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$709$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	2.00000	2.00000
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$720$	0	0
$721$	1.41421i	1.41421i
$722$	− 1.41421i	− 1.41421i
$723$	0	0
$724$	− 1.41421i	− 1.41421i
$725$	−1.00000	−1.00000
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	2.00000	2.00000
$731$	0	0
$732$	0	0
$733$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	1.41421i	1.41421i
$741$	0	0
$742$	2.00000	2.00000
$743$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$743$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$744$	0	0
$745$	0	0
$746$	1.41421i	1.41421i
$747$	0	0
$748$	0	0
$749$	−2.00000	−2.00000
$750$	0	0
$751$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$752$	1.00000	1.00000
$753$	0	0
$754$	2.00000	2.00000
$755$	2.00000	2.00000
$756$	0	0
$757$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$757$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$758$	− 1.41421i	− 1.41421i
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	1.41421i	1.41421i
$767$	−2.00000	−2.00000
$768$	0	0
$769$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$770$	− 2.82843i	− 2.82843i
$771$	0	0
$772$	0	0
$773$	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
$775$	−1.00000	−1.00000
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.41421i	1.41421i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	1.00000
$785$	2.00000	2.00000
$786$	0	0
$787$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$787$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$788$	0	0
$789$	0	0
$790$	− 2.82843i	− 2.82843i
$791$	0	0
$792$	0	0
$793$	− 1.41421i	− 1.41421i
$794$	2.00000	2.00000
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	1.41421i	1.41421i
$801$	0	0
$802$	2.00000	2.00000
$803$	−1.00000	−1.00000
$804$	0	0
$805$	0	0
$806$	2.00000	2.00000
$807$	0	0
$808$	0	0
$809$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$810$	0	0
$811$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$811$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$812$	− 1.41421i	− 1.41421i
$813$	0	0
$814$	− 1.41421i	− 1.41421i
$815$	− 1.41421i	− 1.41421i
$816$	0	0
$817$	− 1.41421i	− 1.41421i
$818$	0	0
$819$	0	0
$820$	1.41421i	1.41421i
$821$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$821$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	2.82843i	2.82843i
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	− 1.41421i	− 1.41421i
$833$	0	0
$834$	0	0
$835$	1.41421i	1.41421i
$836$	1.41421i	1.41421i
$837$	0	0
$838$	2.00000	2.00000
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.41421i	− 1.41421i
$846$	0	0
$847$	0	0
$848$	1.00000	1.00000
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$853$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$854$	−2.00000	−2.00000
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$860$	− 1.41421i	− 1.41421i
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−2.00000	−2.00000
$866$	0	0
$867$	0	0
$868$	− 1.41421i	− 1.41421i
$869$	1.41421i	1.41421i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000	2.00000	1.00000			$0$
$877$	2.00000	2.00000	1.00000			$0$
$878$	−2.00000	−2.00000
$879$	0	0
$880$	− 1.41421i	− 1.41421i
$881$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$881$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−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
$889$	1.41421i	1.41421i
$890$	−2.00000	−2.00000
$891$	0	0
$892$	1.00000	1.00000
$893$	1.41421i	1.41421i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.00000	1.00000
$900$	0	0
$901$	0	0
$902$	− 1.41421i	− 1.41421i
$903$	0	0
$904$	0	0
$905$	−2.00000	−2.00000
$906$	0	0
$907$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$908$	−1.00000	−1.00000
$909$	0	0
$910$	−4.00000	−4.00000
$911$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$911$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	− 1.41421i	− 1.41421i
$917$	2.00000	2.00000
$918$	0	0
$919$	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
$925$	1.00000	1.00000
$926$	0	0
$927$	0	0
$928$	− 1.41421i	− 1.41421i
$929$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$930$	0	0
$931$	1.41421i	1.41421i
$932$	1.00000	1.00000
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	1.41421i	1.41421i
$941$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$941$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$942$	0	0
$943$	0	0
$944$	1.41421i	1.41421i
$945$	0	0
$946$	1.41421i	1.41421i
$947$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$948$	0	0
$949$	1.41421i	1.41421i
$950$	−2.00000	−2.00000
$951$	0	0
$952$	0	0
$953$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$953$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$954$	0	0
$955$	0	0
$956$	1.00000	1.00000
$957$	0	0
$958$	− 1.41421i	− 1.41421i
$959$	− 1.41421i	− 1.41421i
$960$	0	0
$961$	0	0
$962$	−2.00000	−2.00000
$963$	0	0
$964$	1.00000	1.00000
$965$	0	0
$966$	0	0
$967$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$967$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$968$	0	0
$969$	0	0
$970$	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
$973$	1.41421i	1.41421i
$974$	0	0
$975$	0	0
$976$	−1.00000	−1.00000
$977$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$977$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$978$	0	0
$979$	1.00000	1.00000
$980$	1.41421i	1.41421i
$981$	0	0
$982$	−2.00000	−2.00000
$983$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$983$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	2.00000	2.00000
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	− 1.41421i	− 1.41421i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$997$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1107.1.b.d.1106.2	yes	2
3.2	odd	2		1107.1.b.c.1106.1	✓	2
9.2	odd	6		3321.1.k.d.1106.1		4
9.4	even	3		3321.1.k.c.2213.1		4
9.5	odd	6		3321.1.k.d.2213.2		4
9.7	even	3		3321.1.k.c.1106.2		4
41.40	even	2		1107.1.b.c.1106.2	yes	2
123.122	odd	2	inner	1107.1.b.d.1106.1	yes	2
369.40	even	6		3321.1.k.d.2213.1		4
369.122	odd	6		3321.1.k.c.2213.2		4
369.245	odd	6		3321.1.k.c.1106.1		4
369.286	even	6		3321.1.k.d.1106.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1107.1.b.c.1106.1	✓	2	3.2	odd	2
1107.1.b.c.1106.2	yes	2	41.40	even	2
1107.1.b.d.1106.1	yes	2	123.122	odd	2	inner
1107.1.b.d.1106.2	yes	2	1.1	even	1	trivial
3321.1.k.c.1106.1		4	369.245	odd	6
3321.1.k.c.1106.2		4	9.7	even	3
3321.1.k.c.2213.1		4	9.4	even	3
3321.1.k.c.2213.2		4	369.122	odd	6
3321.1.k.d.1106.1		4	9.2	odd	6
3321.1.k.d.1106.2		4	369.286	even	6
3321.1.k.d.2213.1		4	369.40	even	6
3321.1.k.d.2213.2		4	9.5	odd	6

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:	\( N \)	\(=\)	\( 1107 = 3^{3} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1107.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.00000 q^{4} +1.41421i q^{5} +1.41421i q^{7} +O(q^{10})\)	\(q+1.41421i q^{2} -1.00000 q^{4} +1.41421i q^{5} +1.41421i q^{7} -2.00000 q^{10} +1.00000 q^{11} -1.41421i q^{13} -2.00000 q^{14} -1.00000 q^{16} -1.41421i q^{19} -1.41421i q^{20} +1.41421i q^{22} -1.00000 q^{25} +2.00000 q^{26} -1.41421i q^{28} +1.00000 q^{29} +1.00000 q^{31} -1.41421i q^{32} -2.00000 q^{35} -1.00000 q^{37} +2.00000 q^{38} -1.00000 q^{41} +1.00000 q^{43} -1.00000 q^{44} -1.00000 q^{47} -1.00000 q^{49} -1.41421i q^{50} +1.41421i q^{52} -1.00000 q^{53} +1.41421i q^{55} +1.41421i q^{58} -1.41421i q^{59} +1.00000 q^{61} +1.41421i q^{62} +1.00000 q^{64} +2.00000 q^{65} -2.82843i q^{70} -1.00000 q^{73} -1.41421i q^{74} +1.41421i q^{76} +1.41421i q^{77} +1.41421i q^{79} -1.41421i q^{80} -1.41421i q^{82} +1.41421i q^{86} +1.00000 q^{89} +2.00000 q^{91} -1.41421i q^{94} +2.00000 q^{95} -1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 4 q^{10} + 2 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{25} + 4 q^{26} + 2 q^{29} + 2 q^{31} - 4 q^{35} - 2 q^{37} + 4 q^{38} - 2 q^{41} + 2 q^{43} - 2 q^{44} - 2 q^{47} - 2 q^{49} - 2 q^{53} + 2 q^{61} + 2 q^{64} + 4 q^{65} - 2 q^{73} + 2 q^{89} + 4 q^{91} + 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^10 + 2 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^25 + 4 * q^26 + 2 * q^29 + 2 * q^31 - 4 * q^35 - 2 * q^37 + 4 * q^38 - 2 * q^41 + 2 * q^43 - 2 * q^44 - 2 * q^47 - 2 * q^49 - 2 * q^53 + 2 * q^61 + 2 * q^64 + 4 * q^65 - 2 * q^73 + 2 * q^89 + 4 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more