LMFDB - Embedded newform 984.1.m.e.245.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [984,1,Mod(245,984)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("984.245"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 984 = 2^{3} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	984.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.491079972431$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.23616.1

Embedding invariants

Embedding label			245.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	984.245
Dual form			984.1.m.e.245.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{3} -1.00000 q^{4} +(0.707107 + 0.707107i) q^{6} +1.00000i q^{8} -1.00000i q^{9} +1.41421i q^{11} +(0.707107 - 0.707107i) q^{12} -1.41421 q^{13} +1.00000 q^{16} -1.00000 q^{18} +1.41421 q^{19} +1.41421 q^{22} +2.00000i q^{23} +(-0.707107 - 0.707107i) q^{24} -1.00000 q^{25} +1.41421i q^{26} +(0.707107 + 0.707107i) q^{27} +1.41421i q^{29} -1.00000i q^{32} +(-1.00000 - 1.00000i) q^{33} +1.00000i q^{36} -1.41421i q^{38} +(1.00000 - 1.00000i) q^{39} +1.00000i q^{41} -1.41421i q^{44} +2.00000 q^{46} +(-0.707107 + 0.707107i) q^{48} +1.00000 q^{49} +1.00000i q^{50} +1.41421 q^{52} -1.41421i q^{53} +(0.707107 - 0.707107i) q^{54} +(-1.00000 + 1.00000i) q^{57} +1.41421 q^{58} -1.00000 q^{64} +(-1.00000 + 1.00000i) q^{66} -1.41421 q^{67} +(-1.41421 - 1.41421i) q^{69} +1.00000 q^{72} +2.00000 q^{73} +(0.707107 - 0.707107i) q^{75} -1.41421 q^{76} +(-1.00000 - 1.00000i) q^{78} -1.00000 q^{81} +1.00000 q^{82} +(-1.00000 - 1.00000i) q^{87} -1.41421 q^{88} -2.00000i q^{92} +(0.707107 + 0.707107i) q^{96} -1.00000i q^{98} +1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{16} - 4 q^{18} - 4 q^{25} - 4 q^{33} + 4 q^{39} + 8 q^{46} + 4 q^{49} - 4 q^{57} - 4 q^{64} - 4 q^{66} + 4 q^{72} + 8 q^{73} - 4 q^{78} - 4 q^{81} + 4 q^{82} - 4 q^{87}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^16 - 4 * q^18 - 4 * q^25 - 4 * q^33 + 4 * q^39 + 8 * q^46 + 4 * q^49 - 4 * q^57 - 4 * q^64 - 4 * q^66 + 4 * q^72 + 8 * q^73 - 4 * q^78 - 4 * q^81 + 4 * q^82 - 4 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/984\mathbb{Z}\right)^\times$.

$n$	$247$	$329$	$457$	$493$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$4$	−1.00000			−1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	0.707107	+	0.707107i	0.707107	+	0.707107i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			1.00000i			1.00000i
$9$		−	1.00000i		−	1.00000i
$10$		0			0
$11$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$11$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$12$	0.707107	−	0.707107i	0.707107	−	0.707107i
$13$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$13$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−1.00000			−1.00000
$19$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$19$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$	1.41421			1.41421
$23$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$23$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$24$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$25$	−1.00000			−1.00000
$26$			1.41421i			1.41421i
$27$	0.707107	+	0.707107i	0.707107	+	0.707107i
$28$		0			0
$29$			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
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		−	1.00000i		−	1.00000i
$33$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$34$		0			0
$35$		0			0
$36$			1.00000i			1.00000i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	1.41421i		−	1.41421i
$39$	1.00000	−	1.00000i	1.00000	−	1.00000i
$40$		0			0
$41$			1.00000i			1.00000i
$41$			1.00000i			1.00000i
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		−	1.41421i		−	1.41421i
$45$		0			0
$46$	2.00000			2.00000
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$49$	1.00000			1.00000
$50$			1.00000i			1.00000i
$51$		0			0
$52$	1.41421			1.41421
$53$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$53$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$54$	0.707107	−	0.707107i	0.707107	−	0.707107i
$55$		0			0
$56$		0			0
$57$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$58$	1.41421			1.41421
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$67$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$67$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	1.00000			1.00000
$73$	2.00000			2.00000			1.00000			$0$
$73$	2.00000			2.00000			1.00000			$0$
$74$		0			0
$75$	0.707107	−	0.707107i	0.707107	−	0.707107i
$76$	−1.41421			−1.41421
$77$		0			0
$78$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−1.00000			−1.00000
$82$	1.00000			1.00000
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$88$	−1.41421			−1.41421
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		−	2.00000i		−	2.00000i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.707107	+	0.707107i	0.707107	+	0.707107i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	1.00000i		−	1.00000i
$99$	1.41421			1.41421
$100$	1.00000			1.00000
$101$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$101$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$102$		0			0
$103$	−2.00000			−2.00000			−1.00000			$\pi$
$103$	−2.00000			−2.00000			−1.00000			$\pi$
$104$		−	1.41421i		−	1.41421i
$105$		0			0
$106$	−1.41421			−1.41421
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$109$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$109$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	1.00000	+	1.00000i	1.00000	+	1.00000i
$115$		0			0
$116$		−	1.41421i		−	1.41421i
$117$			1.41421i			1.41421i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−1.00000
$122$		0			0
$123$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$124$		0			0
$125$		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
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	1.00000	+	1.00000i	1.00000	+	1.00000i
$133$		0			0
$134$			1.41421i			1.41421i
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.00000i		−	2.00000i
$144$		−	1.00000i		−	1.00000i
$145$		0			0
$146$		−	2.00000i		−	2.00000i
$147$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$148$		0			0
$149$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$149$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$150$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$			1.41421i			1.41421i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$157$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$157$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$	1.00000	+	1.00000i	1.00000	+	1.00000i
$160$		0			0
$161$		0			0
$162$			1.00000i			1.00000i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		−	1.00000i		−	1.00000i
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	1.00000			1.00000
$170$		0			0
$171$		−	1.41421i		−	1.41421i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$175$		0			0
$176$			1.41421i			1.41421i
$177$		0			0
$178$		0			0
$179$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$179$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$181$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$	−2.00000			−2.00000
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	0.707107	−	0.707107i	0.707107	−	0.707107i
$193$		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			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		−	1.41421i		−	1.41421i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		−	1.00000i		−	1.00000i
$201$	1.00000	−	1.00000i	1.00000	−	1.00000i
$202$	−1.41421			−1.41421
$203$		0			0
$204$		0			0
$205$		0			0
$206$			2.00000i			2.00000i
$207$	2.00000			2.00000
$208$	−1.41421			−1.41421
$209$			2.00000i			2.00000i
$210$		0			0
$211$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$211$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$212$			1.41421i			1.41421i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$217$		0			0
$218$			1.41421i			1.41421i
$219$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	2.00000			2.00000			1.00000			$0$
$223$	2.00000			2.00000			1.00000			$0$
$224$		0			0
$225$			1.00000i			1.00000i
$226$		0			0
$227$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$227$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$228$	1.00000	−	1.00000i	1.00000	−	1.00000i
$229$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$229$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$	−1.41421			−1.41421
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	1.41421			1.41421
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$			1.00000i			1.00000i
$243$	0.707107	−	0.707107i	0.707107	−	0.707107i
$244$		0			0
$245$		0			0
$246$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$247$	−2.00000			−2.00000
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−2.82843			−2.82843
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.41421			1.41421
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	1.00000	−	1.00000i	1.00000	−	1.00000i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.41421			1.41421
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1.41421i		−	1.41421i
$276$	1.41421	+	1.41421i	1.41421	+	1.41421i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	−2.00000			−2.00000
$287$		0			0
$288$	−1.00000			−1.00000
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$	−2.00000			−2.00000
$293$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$293$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$294$	0.707107	+	0.707107i	0.707107	+	0.707107i
$295$		0			0
$296$		0			0
$297$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$298$	−1.41421			−1.41421
$299$		−	2.82843i		−	2.82843i
$300$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$301$		0			0
$302$		0			0
$303$	1.00000	+	1.00000i	1.00000	+	1.00000i
$304$	1.41421			1.41421
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	1.41421	−	1.41421i	1.41421	−	1.41421i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	1.00000	+	1.00000i	1.00000	+	1.00000i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		−	1.41421i		−	1.41421i
$315$		0			0
$316$		0			0
$317$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$317$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$318$	1.00000	−	1.00000i	1.00000	−	1.00000i
$319$	−2.00000			−2.00000
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.00000			1.00000
$325$	1.41421			1.41421
$326$		0			0
$327$	1.00000	−	1.00000i	1.00000	−	1.00000i
$328$	−1.00000			−1.00000
$329$		0			0
$330$		0			0
$331$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$331$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		−	1.00000i		−	1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−1.41421			−1.41421
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$347$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$348$	1.00000	+	1.00000i	1.00000	+	1.00000i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$352$	1.41421			1.41421
$353$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$353$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.41421			1.41421
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		−	1.41421i		−	1.41421i
$363$	0.707107	−	0.707107i	0.707107	−	0.707107i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$			2.00000i			2.00000i
$369$	1.00000			1.00000
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.00000i		−	2.00000i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$			1.00000i			1.00000i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−1.41421			−1.41421
$397$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$397$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$403$		0			0
$404$			1.41421i			1.41421i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$	2.00000			2.00000
$413$		0			0
$414$		−	2.00000i		−	2.00000i
$415$		0			0
$416$			1.41421i			1.41421i
$417$		0			0
$418$	2.00000			2.00000
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$421$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$422$		−	1.41421i		−	1.41421i
$423$		0			0
$424$	1.41421			1.41421
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	1.41421	+	1.41421i	1.41421	+	1.41421i
$430$		0			0
$431$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$431$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$432$	0.707107	+	0.707107i	0.707107	+	0.707107i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	1.41421			1.41421
$437$			2.82843i			2.82843i
$438$	1.41421	+	1.41421i	1.41421	+	1.41421i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		−	1.00000i		−	1.00000i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		−	2.00000i		−	2.00000i
$447$	1.00000	+	1.00000i	1.00000	+	1.00000i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	1.00000			1.00000
$451$	−1.41421			−1.41421
$452$		0			0
$453$		0			0
$454$	−1.41421			−1.41421
$455$		0			0
$456$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		−	1.41421i		−	1.41421i
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$			1.41421i			1.41421i
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		−	1.41421i		−	1.41421i
$469$		0			0
$470$		0			0
$471$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.41421			−1.41421
$476$		0			0
$477$	−1.41421			−1.41421
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$487$	−2.00000			−2.00000			−1.00000			$\pi$
$487$	−2.00000			−2.00000			−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	0.707107	+	0.707107i	0.707107	+	0.707107i
$493$		0			0
$494$			2.00000i			2.00000i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$499$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$500$		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
$506$			2.82843i			2.82843i
$507$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$508$		0			0
$509$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$509$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$	1.00000	+	1.00000i	1.00000	+	1.00000i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		−	1.41421i		−	1.41421i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$529$	−3.00000			−3.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.41421i		−	1.41421i
$534$		0			0
$535$		0			0
$536$		−	1.41421i		−	1.41421i
$537$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$538$		0			0
$539$			1.41421i			1.41421i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$547$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$	−1.41421			−1.41421
$551$			2.00000i			2.00000i
$552$	1.41421	−	1.41421i	1.41421	−	1.41421i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$557$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$564$		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
$571$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$571$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$572$			2.00000i			2.00000i
$573$		0			0
$574$		0			0
$575$		−	2.00000i		−	2.00000i
$576$			1.00000i			1.00000i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			1.00000i			1.00000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.00000			2.00000
$584$			2.00000i			2.00000i
$585$		0			0
$586$	1.41421			1.41421
$587$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$587$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$588$	0.707107	−	0.707107i	0.707107	−	0.707107i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	1.00000	+	1.00000i	1.00000	+	1.00000i
$595$		0			0
$596$			1.41421i			1.41421i
$597$		0			0
$598$	−2.82843			−2.82843
$599$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$599$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$600$	0.707107	+	0.707107i	0.707107	+	0.707107i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			1.41421i			1.41421i
$604$		0			0
$605$		0			0
$606$	1.00000	−	1.00000i	1.00000	−	1.00000i
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		−	1.41421i		−	1.41421i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$622$		0			0
$623$		0			0
$624$	1.00000	−	1.00000i	1.00000	−	1.00000i
$625$	1.00000			1.00000
$626$		0			0
$627$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$628$	−1.41421			−1.41421
$629$		0			0
$630$		0			0
$631$	2.00000			2.00000			1.00000			$0$
$631$	2.00000			2.00000			1.00000			$0$
$632$		0			0
$633$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$634$	1.41421			1.41421
$635$		0			0
$636$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$637$	−1.41421			−1.41421
$638$			2.00000i			2.00000i
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$643$	−1.41421			−1.41421			−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			$\pi$
$648$		−	1.00000i		−	1.00000i
$649$		0			0
$650$		−	1.41421i		−	1.41421i
$651$		0			0
$652$		0			0
$653$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$653$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$654$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$655$		0			0
$656$			1.00000i			1.00000i
$657$		−	2.00000i		−	2.00000i
$658$		0			0
$659$			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			$\pi$
$662$		−	1.41421i		−	1.41421i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.82843			−2.82843
$668$		0			0
$669$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$676$	−1.00000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.00000	+	1.00000i	1.00000	+	1.00000i
$682$		0			0
$683$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$683$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$684$			1.41421i			1.41421i
$685$		0			0
$686$		0			0
$687$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$688$		0			0
$689$			2.00000i			2.00000i
$690$		0			0
$691$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$691$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$	1.41421			1.41421
$695$		0			0
$696$	1.00000	−	1.00000i	1.00000	−	1.00000i
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$703$		0			0
$704$		−	1.41421i		−	1.41421i
$705$		0			0
$706$	−2.00000			−2.00000
$707$		0			0
$708$		0			0
$709$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$709$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	1.41421i		−	1.41421i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		−	1.00000i		−	1.00000i
$723$		0			0
$724$	−1.41421			−1.41421
$725$		−	1.41421i		−	1.41421i
$726$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	2.00000			2.00000
$737$		−	2.00000i		−	2.00000i
$738$		−	1.00000i		−	1.00000i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	1.41421	−	1.41421i	1.41421	−	1.41421i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$	−2.00000			−2.00000
$755$		0			0
$756$		0			0
$757$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$757$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$	2.00000	−	2.00000i	2.00000	−	2.00000i
$760$		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
$766$		0			0
$767$		0			0
$768$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$769$	2.00000			2.00000			1.00000			$0$
$769$	2.00000			2.00000			1.00000			$0$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$773$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1.41421i			1.41421i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$			1.41421i			1.41421i
$793$		0			0
$794$		−	1.41421i		−	1.41421i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$			1.00000i			1.00000i
$801$		0			0
$802$		0			0
$803$			2.82843i			2.82843i
$804$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.41421			1.41421
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		−	2.00000i		−	2.00000i
$825$	1.00000	+	1.00000i	1.00000	+	1.00000i
$826$		0			0
$827$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$827$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$828$	−2.00000			−2.00000
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	1.41421			1.41421
$833$		0			0
$834$		0			0
$835$		0			0
$836$		−	2.00000i		−	2.00000i
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		−	1.41421i		−	1.41421i
$843$		0			0
$844$	−1.41421			−1.41421
$845$		0			0
$846$		0			0
$847$		0			0
$848$		−	1.41421i		−	1.41421i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$857$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$858$	1.41421	−	1.41421i	1.41421	−	1.41421i
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	2.00000			2.00000
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$864$	0.707107	−	0.707107i	0.707107	−	0.707107i
$865$		0			0
$866$		0			0
$867$	0.707107	−	0.707107i	0.707107	−	0.707107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	2.00000			2.00000
$872$		−	1.41421i		−	1.41421i
$873$		0			0
$874$	2.82843			2.82843
$875$		0			0
$876$	1.41421	−	1.41421i	1.41421	−	1.41421i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$880$		0			0
$881$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$881$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$882$	−1.00000			−1.00000
$883$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$883$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$884$		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
$891$		−	1.41421i		−	1.41421i
$892$	−2.00000			−2.00000
$893$		0			0
$894$	1.00000	−	1.00000i	1.00000	−	1.00000i
$895$		0			0
$896$		0			0
$897$	2.00000	+	2.00000i	2.00000	+	2.00000i
$898$		0			0
$899$		0			0
$900$		−	1.00000i		−	1.00000i
$901$		0			0
$902$			1.41421i			1.41421i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$			1.41421i			1.41421i
$909$	−1.41421			−1.41421
$910$		0			0
$911$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$911$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$912$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.41421			−1.41421
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$			2.00000i			2.00000i
$928$	1.41421			1.41421
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	1.41421			1.41421
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.41421			−1.41421
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$	1.00000	+	1.00000i	1.00000	+	1.00000i
$943$	−2.00000			−2.00000
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	−2.82843			−2.82843
$950$			1.41421i			1.41421i
$951$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$952$		0			0
$953$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$953$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$954$			1.41421i			1.41421i
$955$		0			0
$956$		0			0
$957$	1.41421	−	1.41421i	1.41421	−	1.41421i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.00000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		−	1.00000i		−	1.00000i
$969$		0			0
$970$		0			0
$971$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$971$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$972$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$973$		0			0
$974$			2.00000i			2.00000i
$975$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			1.41421i			1.41421i
$982$		0			0
$983$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$983$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$984$	0.707107	−	0.707107i	0.707107	−	0.707107i
$985$		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			$\pi$
$992$		0			0
$993$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$997$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$998$			1.41421i			1.41421i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	984.1.m.e.245.1	✓	4
3.2	odd	2	inner	984.1.m.e.245.3	yes	4
4.3	odd	2		3936.1.m.e.2705.3		4
8.3	odd	2		3936.1.m.e.2705.2		4
8.5	even	2	inner	984.1.m.e.245.2	yes	4
12.11	even	2		3936.1.m.e.2705.4		4
24.5	odd	2	inner	984.1.m.e.245.4	yes	4
24.11	even	2		3936.1.m.e.2705.1		4
41.40	even	2	inner	984.1.m.e.245.2	yes	4
123.122	odd	2	inner	984.1.m.e.245.4	yes	4
164.163	odd	2		3936.1.m.e.2705.2		4
328.163	odd	2		3936.1.m.e.2705.3		4
328.245	even	2	RM	984.1.m.e.245.1	✓	4
492.491	even	2		3936.1.m.e.2705.1		4
984.245	odd	2	inner	984.1.m.e.245.3	yes	4
984.491	even	2		3936.1.m.e.2705.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.1.m.e.245.1	✓	4	1.1	even	1	trivial
984.1.m.e.245.1	✓	4	328.245	even	2	RM
984.1.m.e.245.2	yes	4	8.5	even	2	inner
984.1.m.e.245.2	yes	4	41.40	even	2	inner
984.1.m.e.245.3	yes	4	3.2	odd	2	inner
984.1.m.e.245.3	yes	4	984.245	odd	2	inner
984.1.m.e.245.4	yes	4	24.5	odd	2	inner
984.1.m.e.245.4	yes	4	123.122	odd	2	inner
3936.1.m.e.2705.1		4	24.11	even	2
3936.1.m.e.2705.1		4	492.491	even	2
3936.1.m.e.2705.2		4	8.3	odd	2
3936.1.m.e.2705.2		4	164.163	odd	2
3936.1.m.e.2705.3		4	4.3	odd	2
3936.1.m.e.2705.3		4	328.163	odd	2
3936.1.m.e.2705.4		4	12.11	even	2
3936.1.m.e.2705.4		4	984.491	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 984 = 2^{3} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	984.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{3} -1.00000 q^{4} +(0.707107 + 0.707107i) q^{6} +1.00000i q^{8} -1.00000i q^{9} +1.41421i q^{11} +(0.707107 - 0.707107i) q^{12} -1.41421 q^{13} +1.00000 q^{16} -1.00000 q^{18} +1.41421 q^{19} +1.41421 q^{22} +2.00000i q^{23} +(-0.707107 - 0.707107i) q^{24} -1.00000 q^{25} +1.41421i q^{26} +(0.707107 + 0.707107i) q^{27} +1.41421i q^{29} -1.00000i q^{32} +(-1.00000 - 1.00000i) q^{33} +1.00000i q^{36} -1.41421i q^{38} +(1.00000 - 1.00000i) q^{39} +1.00000i q^{41} -1.41421i q^{44} +2.00000 q^{46} +(-0.707107 + 0.707107i) q^{48} +1.00000 q^{49} +1.00000i q^{50} +1.41421 q^{52} -1.41421i q^{53} +(0.707107 - 0.707107i) q^{54} +(-1.00000 + 1.00000i) q^{57} +1.41421 q^{58} -1.00000 q^{64} +(-1.00000 + 1.00000i) q^{66} -1.41421 q^{67} +(-1.41421 - 1.41421i) q^{69} +1.00000 q^{72} +2.00000 q^{73} +(0.707107 - 0.707107i) q^{75} -1.41421 q^{76} +(-1.00000 - 1.00000i) q^{78} -1.00000 q^{81} +1.00000 q^{82} +(-1.00000 - 1.00000i) q^{87} -1.41421 q^{88} -2.00000i q^{92} +(0.707107 + 0.707107i) q^{96} -1.00000i q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{16} - 4 q^{18} - 4 q^{25} - 4 q^{33} + 4 q^{39} + 8 q^{46} + 4 q^{49} - 4 q^{57} - 4 q^{64} - 4 q^{66} + 4 q^{72} + 8 q^{73} - 4 q^{78} - 4 q^{81} + 4 q^{82} - 4 q^{87}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^16 - 4 * q^18 - 4 * q^25 - 4 * q^33 + 4 * q^39 + 8 * q^46 + 4 * q^49 - 4 * q^57 - 4 * q^64 - 4 * q^66 + 4 * q^72 + 8 * q^73 - 4 * q^78 - 4 * q^81 + 4 * q^82 - 4 * q^87

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