LMFDB - Embedded newform 2200.1.o.b.549.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,1,Mod(549,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2200.549");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2200.o (of order $2$, degree $1$, not 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:	$1.09794302779$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.2200.1
Artin image:	$C_4\times D_6$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

Embedding invariants

Embedding label			549.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2200.549
Dual form			2200.1.o.b.549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{13} +1.00000 q^{16} -1.00000i q^{18} +1.00000 q^{19} +1.00000i q^{22} +1.00000i q^{23} -1.00000 q^{26} +1.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +1.00000 q^{36} +1.00000i q^{38} +1.00000i q^{43} -1.00000 q^{44} -1.00000 q^{46} +2.00000i q^{47} -1.00000 q^{49} -1.00000i q^{52} +1.00000i q^{58} +2.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -1.00000 q^{71} +1.00000i q^{72} -1.00000 q^{76} +1.00000 q^{81} +1.00000i q^{83} -1.00000 q^{86} -1.00000i q^{88} +1.00000 q^{89} -1.00000i q^{92} -2.00000 q^{94} -1.00000i q^{97} -1.00000i q^{98} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{9} + 2 q^{11} + 2 q^{16} + 2 q^{19} - 2 q^{26} + 2 q^{29} - 2 q^{31} + 2 q^{36} - 2 q^{44} - 2 q^{46} - 2 q^{49} + 4 q^{61} - 2 q^{64} - 2 q^{71} - 2 q^{76} + 2 q^{81} - 2 q^{86} + 2 q^{89}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^9 + 2 * q^11 + 2 * q^16 + 2 * q^19 - 2 * q^26 + 2 * q^29 - 2 * q^31 + 2 * q^36 - 2 * q^44 - 2 * q^46 - 2 * q^49 + 4 * q^61 - 2 * q^64 - 2 * q^71 - 2 * q^76 + 2 * q^81 - 2 * q^86 + 2 * q^89 - 4 * q^94 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$.

$n$	$177$	$551$	$1101$	$1201$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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 $

$2$	1.00000i	1.00000i
$2$	1.00000i	1.00000i
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	− 1.00000i	− 1.00000i
$9$	−1.00000	−1.00000
$10$	0	0
$11$	1.00000	1.00000
$11$	1.00000	1.00000
$12$	0	0
$13$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$13$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 1.00000i	− 1.00000i
$19$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$20$	0	0
$21$	0	0
$22$	1.00000i	1.00000i
$23$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$23$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−1.00000
$27$	0	0
$28$	0	0
$29$	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	$0.666667\pi$
$32$	1.00000i	1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.00000	1.00000
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.00000i	1.00000i
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$43$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$44$	−1.00000	−1.00000
$45$	0	0
$46$	−1.00000	−1.00000
$47$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$47$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$48$	0	0
$49$	−1.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 1.00000i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	1.00000i	1.00000i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	2.00000	2.00000	1.00000			$0$
$61$	2.00000	2.00000	1.00000			$0$
$62$	− 1.00000i	− 1.00000i
$63$	0	0
$64$	−1.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$72$	1.00000i	1.00000i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	−1.00000	−1.00000
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	0	0
$83$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$83$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$84$	0	0
$85$	0	0
$86$	−1.00000	−1.00000
$87$	0	0
$88$	− 1.00000i	− 1.00000i
$89$	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
$91$	0	0
$92$	− 1.00000i	− 1.00000i
$93$	0	0
$94$	−2.00000	−2.00000
$95$	0	0
$96$	0	0
$97$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$97$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$98$	− 1.00000i	− 1.00000i
$99$	−1.00000	−1.00000
$100$	0	0
$101$	−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
$103$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$103$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$104$	1.00000	1.00000
$105$	0	0
$106$	0	0
$107$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$107$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	0	0
$113$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$113$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−1.00000
$117$	− 1.00000i	− 1.00000i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	2.00000i	2.00000i
$123$	0	0
$124$	1.00000	1.00000
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	− 1.00000i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$137$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$138$	0	0
$139$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$140$	0	0
$141$	0	0
$142$	− 1.00000i	− 1.00000i
$143$	1.00000i	1.00000i
$144$	−1.00000	−1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−2.00000	−1.00000			$\pi$
$149$	−2.00000	−2.00000	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	− 1.00000i	− 1.00000i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	1.00000i	1.00000i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	−1.00000	−1.00000
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1.00000	−1.00000
$172$	− 1.00000i	− 1.00000i
$173$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$173$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	1.00000
$177$	0	0
$178$	1.00000i	1.00000i
$179$	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			$\pi$
$182$	0	0
$183$	0	0
$184$	1.00000	1.00000
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 2.00000i	− 2.00000i
$189$	0	0
$190$	0	0
$191$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	1.00000	1.00000
$195$	0	0
$196$	1.00000	1.00000
$197$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$197$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$198$	− 1.00000i	− 1.00000i
$199$	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
$201$	0	0
$202$	− 1.00000i	− 1.00000i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.00000	−1.00000
$207$	− 1.00000i	− 1.00000i
$208$	1.00000i	1.00000i
$209$	1.00000	1.00000
$210$	0	0
$211$	2.00000	2.00000	1.00000			$0$
$211$	2.00000	2.00000	1.00000			$0$
$212$	0	0
$213$	0	0
$214$	1.00000	1.00000
$215$	0	0
$216$	0	0
$217$	0	0
$218$	1.00000i	1.00000i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$223$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	2.00000
$227$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$227$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	− 1.00000i	− 1.00000i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	1.00000	1.00000
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.00000i	1.00000i
$243$	0	0
$244$	−2.00000	−2.00000
$245$	0	0
$246$	0	0
$247$	1.00000i	1.00000i
$248$	1.00000i	1.00000i
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	1.00000i	1.00000i
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$257$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.00000	−1.00000
$262$	− 1.00000i	− 1.00000i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	1.00000	1.00000
$275$	0	0
$276$	0	0
$277$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$277$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$278$	1.00000i	1.00000i
$279$	1.00000	1.00000
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$283$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$284$	1.00000	1.00000
$285$	0	0
$286$	−1.00000	−1.00000
$287$	0	0
$288$	− 1.00000i	− 1.00000i
$289$	−1.00000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$293$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	− 2.00000i	− 2.00000i
$299$	−1.00000	−1.00000
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	1.00000	1.00000
$305$	0	0
$306$	0	0
$307$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$307$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$312$	0	0
$313$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$313$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	1.00000	1.00000
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−1.00000	−1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 1.00000i	− 1.00000i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.00000	−1.00000
$342$	− 1.00000i	− 1.00000i
$343$	0	0
$344$	1.00000	1.00000
$345$	0	0
$346$	−1.00000	−1.00000
$347$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$347$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$348$	0	0
$349$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$349$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$350$	0	0
$351$	0	0
$352$	1.00000i	1.00000i
$353$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$353$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$354$	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
$361$	0	0
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$367$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$368$	1.00000i	1.00000i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$373$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$374$	0	0
$375$	0	0
$376$	2.00000	2.00000
$377$	1.00000i	1.00000i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	− 1.00000i	− 1.00000i
$383$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$383$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.00000i	− 1.00000i
$388$	1.00000i	1.00000i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.00000i	1.00000i
$393$	0	0
$394$	1.00000	1.00000
$395$	0	0
$396$	1.00000	1.00000
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	1.00000i	1.00000i
$399$	0	0
$400$	0	0
$401$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$402$	0	0
$403$	− 1.00000i	− 1.00000i
$404$	1.00000	1.00000
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	− 1.00000i	− 1.00000i
$413$	0	0
$414$	1.00000	1.00000
$415$	0	0
$416$	−1.00000	−1.00000
$417$	0	0
$418$	1.00000i	1.00000i
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	2.00000i	2.00000i
$423$	− 2.00000i	− 2.00000i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.00000i	1.00000i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$433$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$434$	0	0
$435$	0	0
$436$	−1.00000	−1.00000
$437$	1.00000i	1.00000i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	2.00000	2.00000
$447$	0	0
$448$	0	0
$449$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$450$	0	0
$451$	0	0
$452$	2.00000i	2.00000i
$453$	0	0
$454$	1.00000	1.00000
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.00000	2.00000	1.00000			$0$
$461$	2.00000	2.00000	1.00000			$0$
$462$	0	0
$463$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$463$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$464$	1.00000	1.00000
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	1.00000i	1.00000i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.00000i	1.00000i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−1.00000	−1.00000
$485$	0	0
$486$	0	0
$487$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$487$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$488$	− 2.00000i	− 2.00000i
$489$	0	0
$490$	0	0
$491$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$492$	0	0
$493$	0	0
$494$	−1.00000	−1.00000
$495$	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
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	−1.00000	−1.00000
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	1.00000i
$513$	0	0
$514$	1.00000	1.00000
$515$	0	0
$516$	0	0
$517$	2.00000i	2.00000i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$522$	− 1.00000i	− 1.00000i
$523$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$523$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$	1.00000	1.00000
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	0	0
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	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	$-0.666667\pi$
$541$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$547$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$548$	1.00000i	1.00000i
$549$	−2.00000	−2.00000
$550$	0	0
$551$	1.00000	1.00000
$552$	0	0
$553$	0	0
$554$	−2.00000	−2.00000
$555$	0	0
$556$	−1.00000	−1.00000
$557$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$557$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$558$	1.00000i	1.00000i
$559$	−1.00000	−1.00000
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$563$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$564$	0	0
$565$	0	0
$566$	2.00000	2.00000
$567$	0	0
$568$	1.00000i	1.00000i
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$572$	− 1.00000i	− 1.00000i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	1.00000
$577$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$577$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$578$	− 1.00000i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	2.00000	2.00000
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−1.00000	−1.00000
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	2.00000	2.00000
$597$	0	0
$598$	− 1.00000i	− 1.00000i
$599$	−2.00000	−2.00000	−1.00000			$\pi$
$599$	−2.00000	−2.00000	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	1.00000i	1.00000i
$609$	0	0
$610$	0	0
$611$	−2.00000	−2.00000
$612$	0	0
$613$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$613$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$614$	−2.00000	−2.00000
$615$	0	0
$616$	0	0
$617$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$617$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	− 1.00000i	− 1.00000i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	2.00000	2.00000
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	2.00000	2.00000	1.00000			$0$
$631$	2.00000	2.00000	1.00000			$0$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.00000i	− 1.00000i
$638$	1.00000i	1.00000i
$639$	1.00000	1.00000
$640$	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
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$647$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$648$	− 1.00000i	− 1.00000i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	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
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	1.00000	1.00000
$665$	0	0
$666$	0	0
$667$	1.00000i	1.00000i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.00000	2.00000
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$677$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	− 1.00000i	− 1.00000i
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	1.00000	1.00000
$685$	0	0
$686$	0	0
$687$	0	0
$688$	1.00000i	1.00000i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	− 1.00000i	− 1.00000i
$693$	0	0
$694$	−2.00000	−2.00000
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.00000i	1.00000i
$699$	0	0
$700$	0	0
$701$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−1.00000
$705$	0	0
$706$	−1.00000	−1.00000
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	− 1.00000i	− 1.00000i
$713$	− 1.00000i	− 1.00000i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.00000	−2.00000	−1.00000			$\pi$
$719$	−2.00000	−2.00000	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$727$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$728$	0	0
$729$	−1.00000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$733$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$734$	1.00000	1.00000
$735$	0	0
$736$	−1.00000	−1.00000
$737$	0	0
$738$	0	0
$739$	−2.00000	−2.00000	−1.00000			$\pi$
$739$	−2.00000	−2.00000	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	2.00000	2.00000
$747$	− 1.00000i	− 1.00000i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$751$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$752$	2.00000i	2.00000i
$753$	0	0
$754$	−1.00000	−1.00000
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	1.00000	1.00000
$765$	0	0
$766$	−1.00000	−1.00000
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	1.00000	1.00000
$775$	0	0
$776$	−1.00000	−1.00000
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−1.00000	−1.00000
$782$	0	0
$783$	0	0
$784$	−1.00000	−1.00000
$785$	0	0
$786$	0	0
$787$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$787$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$788$	1.00000i	1.00000i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.00000i	1.00000i
$793$	2.00000i	2.00000i
$794$	0	0
$795$	0	0
$796$	−1.00000	−1.00000
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−1.00000	−1.00000
$802$	− 1.00000i	− 1.00000i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	1.00000	1.00000
$807$	0	0
$808$	1.00000i	1.00000i
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	2.00000	2.00000	1.00000			$0$
$811$	2.00000	2.00000	1.00000			$0$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.00000i	1.00000i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$822$	0	0
$823$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$823$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$824$	1.00000	1.00000
$825$	0	0
$826$	0	0
$827$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$827$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$828$	1.00000i	1.00000i
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 1.00000i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−1.00000	−1.00000
$837$	0	0
$838$	0	0
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	0	0
$844$	−2.00000	−2.00000
$845$	0	0
$846$	2.00000	2.00000
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$853$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$854$	0	0
$855$	0	0
$856$	−1.00000	−1.00000
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$863$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$864$	0	0
$865$	0	0
$866$	−1.00000	−1.00000
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 1.00000i	− 1.00000i
$873$	1.00000i	1.00000i
$874$	−1.00000	−1.00000
$875$	0	0
$876$	0	0
$877$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$877$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$882$	1.00000i	1.00000i
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	1.00000
$892$	2.00000i	2.00000i
$893$	2.00000i	2.00000i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	1.00000i	1.00000i
$899$	−1.00000	−1.00000
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−2.00000	−2.00000
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	1.00000i	1.00000i
$909$	1.00000	1.00000
$910$	0	0
$911$	2.00000	2.00000	1.00000			$0$
$911$	2.00000	2.00000	1.00000			$0$
$912$	0	0
$913$	1.00000i	1.00000i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	2.00000i	2.00000i
$923$	− 1.00000i	− 1.00000i
$924$	0	0
$925$	0	0
$926$	−1.00000	−1.00000
$927$	− 1.00000i	− 1.00000i
$928$	1.00000i	1.00000i
$929$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$929$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$930$	0	0
$931$	−1.00000	−1.00000
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−1.00000	−1.00000
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.00000	2.00000	1.00000			$0$
$941$	2.00000	2.00000	1.00000			$0$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	−1.00000	−1.00000
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0	0
$962$	0	0
$963$	1.00000i	1.00000i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	− 1.00000i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	1.00000	1.00000
$975$	0	0
$976$	2.00000	2.00000
$977$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$977$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$978$	0	0
$979$	1.00000	1.00000
$980$	0	0
$981$	−1.00000	−1.00000
$982$	− 1.00000i	− 1.00000i
$983$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$983$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 1.00000i	− 1.00000i
$989$	−1.00000	−1.00000
$990$	0	0
$991$	2.00000	2.00000	1.00000			$0$
$991$	2.00000	2.00000	1.00000			$0$
$992$	− 1.00000i	− 1.00000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$997$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.1.o.b.549.2		2
5.2	odd	4		2200.1.d.b.901.1	yes	1
5.3	odd	4		2200.1.d.d.901.1	yes	1
5.4	even	2	inner	2200.1.o.b.549.1		2
8.5	even	2		2200.1.o.a.549.1		2
11.10	odd	2		2200.1.o.a.549.1		2
40.13	odd	4		2200.1.d.a.901.1	✓	1
40.29	even	2		2200.1.o.a.549.2		2
40.37	odd	4		2200.1.d.c.901.1	yes	1
55.32	even	4		2200.1.d.c.901.1	yes	1
55.43	even	4		2200.1.d.a.901.1	✓	1
55.54	odd	2		2200.1.o.a.549.2		2
88.21	odd	2	CM	2200.1.o.b.549.2		2
440.109	odd	2	inner	2200.1.o.b.549.1		2
440.197	even	4		2200.1.d.b.901.1	yes	1
440.373	even	4		2200.1.d.d.901.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.1.d.a.901.1	✓	1	40.13	odd	4
2200.1.d.a.901.1	✓	1	55.43	even	4
2200.1.d.b.901.1	yes	1	5.2	odd	4
2200.1.d.b.901.1	yes	1	440.197	even	4
2200.1.d.c.901.1	yes	1	40.37	odd	4
2200.1.d.c.901.1	yes	1	55.32	even	4
2200.1.d.d.901.1	yes	1	5.3	odd	4
2200.1.d.d.901.1	yes	1	440.373	even	4
2200.1.o.a.549.1		2	8.5	even	2
2200.1.o.a.549.1		2	11.10	odd	2
2200.1.o.a.549.2		2	40.29	even	2
2200.1.o.a.549.2		2	55.54	odd	2
2200.1.o.b.549.1		2	5.4	even	2	inner
2200.1.o.b.549.1		2	440.109	odd	2	inner
2200.1.o.b.549.2		2	1.1	even	1	trivial
2200.1.o.b.549.2		2	88.21	odd	2	CM

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:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2200.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{13} +1.00000 q^{16} -1.00000i q^{18} +1.00000 q^{19} +1.00000i q^{22} +1.00000i q^{23} -1.00000 q^{26} +1.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +1.00000 q^{36} +1.00000i q^{38} +1.00000i q^{43} -1.00000 q^{44} -1.00000 q^{46} +2.00000i q^{47} -1.00000 q^{49} -1.00000i q^{52} +1.00000i q^{58} +2.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -1.00000 q^{71} +1.00000i q^{72} -1.00000 q^{76} +1.00000 q^{81} +1.00000i q^{83} -1.00000 q^{86} -1.00000i q^{88} +1.00000 q^{89} -1.00000i q^{92} -2.00000 q^{94} -1.00000i q^{97} -1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{9} + 2 q^{11} + 2 q^{16} + 2 q^{19} - 2 q^{26} + 2 q^{29} - 2 q^{31} + 2 q^{36} - 2 q^{44} - 2 q^{46} - 2 q^{49} + 4 q^{61} - 2 q^{64} - 2 q^{71} - 2 q^{76} + 2 q^{81} - 2 q^{86} + 2 q^{89}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^9 + 2 * q^11 + 2 * q^16 + 2 * q^19 - 2 * q^26 + 2 * q^29 - 2 * q^31 + 2 * q^36 - 2 * q^44 - 2 * q^46 - 2 * q^49 + 4 * q^61 - 2 * q^64 - 2 * q^71 - 2 * q^76 + 2 * q^81 - 2 * q^86 + 2 * q^89 - 4 * q^94 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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