LMFDB - Embedded newform 2664.1.m.c.739.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2664,1,Mod(739,2664)] 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("2664.739"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2664 = 2^{3} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2664.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$1.32950919365$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.2.207267213312.2

Embedding invariants

Embedding label			739.2
Root			$0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	2664.739
Dual form			2664.1.m.c.739.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.923880 + 0.382683i) q^{2} +(0.707107 - 0.707107i) q^{4} -0.765367 q^{5} +1.41421i q^{7} +(-0.382683 + 0.923880i) q^{8} +(0.707107 - 0.292893i) q^{10} +(-0.541196 - 1.30656i) q^{14} -1.00000i q^{16} +0.765367i q^{17} +(-0.541196 + 0.541196i) q^{20} +0.765367 q^{23} -0.414214 q^{25} +(1.00000 + 1.00000i) q^{28} -1.84776 q^{29} +(0.382683 + 0.923880i) q^{32} +(-0.292893 - 0.707107i) q^{34} -1.08239i q^{35} +1.00000i q^{37} +(0.292893 - 0.707107i) q^{40} +(-0.707107 + 0.292893i) q^{46} -1.00000 q^{49} +(0.382683 - 0.158513i) q^{50} +(-1.30656 - 0.541196i) q^{56} +(1.70711 - 0.707107i) q^{58} -0.765367i q^{59} +(-0.707107 - 0.707107i) q^{64} -1.41421 q^{67} +(0.541196 + 0.541196i) q^{68} +(0.414214 + 1.00000i) q^{70} -1.41421 q^{73} +(-0.382683 - 0.923880i) q^{74} +0.765367i q^{80} -0.585786i q^{85} +1.84776i q^{89} +(0.541196 - 0.541196i) q^{92} +(0.923880 - 0.382683i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{25} + 8 q^{28} - 8 q^{34} + 8 q^{40} - 8 q^{49} + 8 q^{58} - 8 q^{70}+O(q^{100}) $ 8 * q + 8 * q^25 + 8 * q^28 - 8 * q^34 + 8 * q^40 - 8 * q^49 + 8 * q^58 - 8 * q^70

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$2$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$3$		0			0
$3$		0			0
$4$	0.707107	−	0.707107i	0.707107	−	0.707107i
$5$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$5$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$6$		0			0
$7$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$7$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$8$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$9$		0			0
$10$	0.707107	−	0.292893i	0.707107	−	0.292893i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$15$		0			0
$16$		−	1.00000i		−	1.00000i
$17$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$17$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$21$		0			0
$22$		0			0
$23$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$23$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$24$		0			0
$25$	−0.414214			−0.414214
$26$		0			0
$27$		0			0
$28$	1.00000	+	1.00000i	1.00000	+	1.00000i
$29$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$29$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	0.382683	+	0.923880i	0.382683	+	0.923880i
$33$		0			0
$34$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$35$		−	1.08239i		−	1.08239i
$36$		0			0
$37$			1.00000i			1.00000i
$37$			1.00000i			1.00000i
$38$		0			0
$39$		0			0
$40$	0.292893	−	0.707107i	0.292893	−	0.707107i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$	0.382683	−	0.158513i	0.382683	−	0.158513i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$57$		0			0
$58$	1.70711	−	0.707107i	1.70711	−	0.707107i
$59$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$59$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$67$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$68$	0.541196	+	0.541196i	0.541196	+	0.541196i
$69$		0			0
$70$	0.414214	+	1.00000i	0.414214	+	1.00000i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$73$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$74$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$			0.765367i			0.765367i
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		−	0.585786i		−	0.585786i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$89$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$90$		0			0
$91$		0			0
$92$	0.541196	−	0.541196i	0.541196	−	0.541196i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.923880	−	0.382683i	0.923880	−	0.382683i
$99$		0			0
$100$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	1.41421			1.41421
$113$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$113$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$114$		0			0
$115$	−0.585786			−0.585786
$116$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$117$		0			0
$118$	0.292893	+	0.707107i	0.292893	+	0.707107i
$119$	−1.08239			−1.08239
$120$		0			0
$121$	−1.00000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.08239			1.08239
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.923880	+	0.382683i	0.923880	+	0.382683i
$129$		0			0
$130$		0			0
$131$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$131$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$132$		0			0
$133$		0			0
$134$	1.30656	−	0.541196i	1.30656	−	0.541196i
$135$		0			0
$136$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$139$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$140$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.41421			1.41421
$146$	1.30656	−	0.541196i	1.30656	−	0.541196i
$147$		0			0
$148$	0.707107	+	0.707107i	0.707107	+	0.707107i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$151$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$157$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$161$			1.08239i			1.08239i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$167$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	0.224171	+	0.541196i	0.224171	+	0.541196i
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	0.585786i		−	0.585786i
$176$		0			0
$177$		0			0
$178$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$179$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$179$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$180$		0			0
$181$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$181$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$182$		0			0
$183$		0			0
$184$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$185$		−	0.765367i		−	0.765367i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$191$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	0.158513	−	0.382683i	0.158513	−	0.382683i
$201$		0			0
$202$		0			0
$203$		−	2.61313i		−	2.61313i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	2.00000			2.00000			1.00000			$0$
$211$	2.00000			2.00000			1.00000			$0$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		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	$-0.5\pi$
$224$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$225$		0			0
$226$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$227$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$227$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$228$		0			0
$229$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$229$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$230$	0.541196	−	0.224171i	0.541196	−	0.224171i
$231$		0			0
$232$	0.707107	−	1.70711i	0.707107	−	1.70711i
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$237$		0			0
$238$	1.00000	−	0.414214i	1.00000	−	0.414214i
$239$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$239$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.923880	−	0.382683i	0.923880	−	0.382683i
$243$		0			0
$244$		0			0
$245$	0.765367			0.765367
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−1.00000	+	0.414214i	−1.00000	+	0.414214i
$251$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$251$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.00000			−1.00000
$257$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$257$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$258$		0			0
$259$	−1.41421			−1.41421
$260$		0			0
$261$		0			0
$262$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$271$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$272$	0.765367			0.765367
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	1.30656	−	0.541196i	1.30656	−	0.541196i
$279$		0			0
$280$	1.00000	+	0.414214i	1.00000	+	0.414214i
$281$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$281$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.414214			0.414214
$290$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$291$		0			0
$292$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$			0.585786i			0.585786i
$296$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.765367	−	1.84776i	−0.765367	−	1.84776i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	2.00000			2.00000			1.00000			$0$
$307$	2.00000			2.00000			1.00000			$0$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$311$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	0.541196	+	0.541196i	0.541196	+	0.541196i
$321$		0			0
$322$	−0.414214	−	1.00000i	−0.414214	−	1.00000i
$323$		0			0
$324$		0			0
$325$		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			$\pi$
$332$		0			0
$333$		0			0
$334$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$335$	1.08239			1.08239
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	0.923880	−	0.382683i	0.923880	−	0.382683i
$339$		0			0
$340$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$347$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$348$		0			0
$349$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$349$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$350$	0.224171	+	0.541196i	0.224171	+	0.541196i
$351$		0			0
$352$		0			0
$353$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$353$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$354$		0			0
$355$		0			0
$356$	1.30656	+	1.30656i	1.30656	+	1.30656i
$357$		0			0
$358$	0.707107	+	1.70711i	0.707107	+	1.70711i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$	0.541196	+	1.30656i	0.541196	+	1.30656i
$363$		0			0
$364$		0			0
$365$	1.08239			1.08239
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	0.765367i		−	0.765367i
$369$		0			0
$370$	0.292893	+	0.707107i	0.292893	+	0.707107i
$371$		0			0
$372$		0			0
$373$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$373$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$379$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$		0			0
$382$	1.70711	−	0.707107i	1.70711	−	0.707107i
$383$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$383$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$389$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$390$		0			0
$391$			0.585786i			0.585786i
$392$	0.382683	−	0.923880i	0.382683	−	0.923880i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		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
$400$			0.414214i			0.414214i
$401$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$401$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	1.00000	+	2.41421i	1.00000	+	2.41421i
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.08239			1.08239
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$423$		0			0
$424$		0			0
$425$		−	0.317025i		−	0.317025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$431$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$432$		0			0
$433$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$433$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		−	1.41421i		−	1.41421i
$446$	0.765367	+	1.84776i	0.765367	+	1.84776i
$447$		0			0
$448$	1.00000	−	1.00000i	1.00000	−	1.00000i
$449$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$449$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$450$		0			0
$451$		0			0
$452$	1.30656	+	1.30656i	1.30656	+	1.30656i
$453$		0			0
$454$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−0.765367	−	1.84776i	−0.765367	−	1.84776i
$459$		0			0
$460$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$461$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$461$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$			1.84776i			1.84776i
$465$		0			0
$466$		0			0
$467$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$467$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$468$		0			0
$469$		−	2.00000i		−	2.00000i
$470$		0			0
$471$		0			0
$472$	0.707107	+	0.292893i	0.707107	+	0.292893i
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$477$		0			0
$478$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$479$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$479$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$485$		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
$490$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		−	1.41421i		−	1.41421i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	0.765367	−	0.765367i	0.765367	−	0.765367i
$501$		0			0
$502$	0.707107	+	1.70711i	0.707107	+	1.70711i
$503$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$503$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		−	2.00000i		−	2.00000i
$512$	0.923880	−	0.382683i	0.923880	−	0.382683i
$513$		0			0
$514$	0.707107	+	1.70711i	0.707107	+	1.70711i
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.30656	−	0.541196i	1.30656	−	0.541196i
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	0.541196	+	0.541196i	0.541196	+	0.541196i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.414214			−0.414214
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.541196	−	1.30656i	0.541196	−	1.30656i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$	0.541196	+	1.30656i	0.541196	+	1.30656i
$543$		0			0
$544$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$557$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$557$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$558$		0			0
$559$		0			0
$560$	−1.08239			−1.08239
$561$		0			0
$562$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$563$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$563$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$564$		0			0
$565$		−	1.41421i		−	1.41421i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$569$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$570$		0			0
$571$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$571$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.317025			−0.317025
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.382683	+	0.158513i	−0.382683	+	0.158513i
$579$		0			0
$580$	1.00000	−	1.00000i	1.00000	−	1.00000i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.541196	−	1.30656i	0.541196	−	1.30656i
$585$		0			0
$586$		0			0
$587$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$587$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$	−0.224171	−	0.541196i	−0.224171	−	0.541196i
$591$		0			0
$592$	1.00000			1.00000
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$	0.828427			0.828427
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$601$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$602$		0			0
$603$		0			0
$604$	1.41421	+	1.41421i	1.41421	+	1.41421i
$605$	0.765367			0.765367
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	0.707107	−	0.292893i	0.707107	−	0.292893i
$623$	−2.61313			−2.61313
$624$		0			0
$625$	−0.414214			−0.414214
$626$		0			0
$627$		0			0
$628$	1.00000	+	1.00000i	1.00000	+	1.00000i
$629$	−0.765367			−0.765367
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	0.765367	+	0.765367i	0.765367	+	0.765367i
$645$		0			0
$646$		0			0
$647$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$647$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$653$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$654$		0			0
$655$		−	0.585786i		−	0.585786i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.41421			−1.41421
$668$	1.30656	−	1.30656i	1.30656	−	1.30656i
$669$		0			0
$670$	−1.00000	+	0.414214i	−1.00000	+	0.414214i
$671$		0			0
$672$		0			0
$673$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$673$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$	0.541196	+	0.224171i	0.541196	+	0.224171i
$681$		0			0
$682$		0			0
$683$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$683$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		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
$694$	0.292893	+	0.707107i	0.292893	+	0.707107i
$695$	1.08239			1.08239
$696$		0			0
$697$		0			0
$698$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$699$		0			0
$700$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$701$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$701$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.292893	+	0.707107i	0.292893	+	0.707107i
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$723$		0			0
$724$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$725$	0.765367			0.765367
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$	−1.00000	+	0.414214i	−1.00000	+	0.414214i
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	0.292893	+	0.707107i	0.292893	+	0.707107i
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1.53073i		−	1.53073i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$765$		0			0
$766$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$767$		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		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.707107	−	0.292893i	0.707107	−	0.292893i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.224171	−	0.541196i	−0.224171	−	0.541196i
$783$		0			0
$784$			1.00000i			1.00000i
$785$		−	1.08239i		−	1.08239i
$786$		0			0
$787$	−2.00000			−2.00000			−1.00000			$\pi$
$787$	−2.00000			−2.00000			−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.61313			−2.61313
$792$		0			0
$793$		0			0
$794$	0.541196	+	1.30656i	0.541196	+	1.30656i
$795$		0			0
$796$		0			0
$797$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$797$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$798$		0			0
$799$		0			0
$800$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$801$		0			0
$802$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$803$		0			0
$804$		0			0
$805$		−	0.828427i		−	0.828427i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$809$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$810$		0			0
$811$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$811$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$812$	−1.84776	−	1.84776i	−1.84776	−	1.84776i
$813$		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		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$823$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$	−1.00000	+	0.414214i	−1.00000	+	0.414214i
$827$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$827$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	0.765367i		−	0.765367i
$834$		0			0
$835$	−1.41421			−1.41421
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	2.41421			2.41421
$842$		0			0
$843$		0			0
$844$	1.41421	−	1.41421i	1.41421	−	1.41421i
$845$	0.765367			0.765367
$846$		0			0
$847$		−	1.41421i		−	1.41421i
$848$		0			0
$849$		0			0
$850$	0.121320	+	0.292893i	0.121320	+	0.292893i
$851$			0.765367i			0.765367i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$857$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			1.53073i			1.53073i
$876$		0			0
$877$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$877$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	0.541196	+	1.30656i	0.541196	+	1.30656i
$891$		0			0
$892$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$893$		0			0
$894$		0			0
$895$			1.41421i			1.41421i
$896$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$897$		0			0
$898$	0.292893	+	0.707107i	0.292893	+	0.707107i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$905$			1.08239i			1.08239i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	1.30656	+	1.30656i	1.30656	+	1.30656i
$909$		0			0
$910$		0			0
$911$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$911$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.41421	+	1.41421i	1.41421	+	1.41421i
$917$	−1.08239			−1.08239
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	0.224171	−	0.541196i	0.224171	−	0.541196i
$921$		0			0
$922$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$923$		0			0
$924$		0			0
$925$		−	0.414214i		−	0.414214i
$926$		0			0
$927$		0			0
$928$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	0.765367	+	1.84776i	0.765367	+	1.84776i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	−0.765367			−0.765367
$945$		0			0
$946$		0			0
$947$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$947$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.414214	−	1.00000i	0.414214	−	1.00000i
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	1.41421			1.41421
$956$	0.541196	−	0.541196i	0.541196	−	0.541196i
$957$		0			0
$958$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$959$		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			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	0.382683	−	0.923880i	0.382683	−	0.923880i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		−	2.00000i		−	2.00000i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$977$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$978$		0			0
$979$		0			0
$980$	0.541196	−	0.541196i	0.541196	−	0.541196i
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$	0.541196	+	1.30656i	0.541196	+	1.30656i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				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	2664.1.m.c.739.2	yes	8
3.2	odd	2	inner	2664.1.m.c.739.7	yes	8
8.3	odd	2	inner	2664.1.m.c.739.8	yes	8
24.11	even	2	inner	2664.1.m.c.739.1	✓	8
37.36	even	2	inner	2664.1.m.c.739.7	yes	8
111.110	odd	2	CM	2664.1.m.c.739.2	yes	8
296.147	odd	2	inner	2664.1.m.c.739.1	✓	8
888.443	even	2	inner	2664.1.m.c.739.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2664.1.m.c.739.1	✓	8	24.11	even	2	inner
2664.1.m.c.739.1	✓	8	296.147	odd	2	inner
2664.1.m.c.739.2	yes	8	1.1	even	1	trivial
2664.1.m.c.739.2	yes	8	111.110	odd	2	CM
2664.1.m.c.739.7	yes	8	3.2	odd	2	inner
2664.1.m.c.739.7	yes	8	37.36	even	2	inner
2664.1.m.c.739.8	yes	8	8.3	odd	2	inner
2664.1.m.c.739.8	yes	8	888.443	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2664.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.923880 + 0.382683i) q^{2} +(0.707107 - 0.707107i) q^{4} -0.765367 q^{5} +1.41421i q^{7} +(-0.382683 + 0.923880i) q^{8} +(0.707107 - 0.292893i) q^{10} +(-0.541196 - 1.30656i) q^{14} -1.00000i q^{16} +0.765367i q^{17} +(-0.541196 + 0.541196i) q^{20} +0.765367 q^{23} -0.414214 q^{25} +(1.00000 + 1.00000i) q^{28} -1.84776 q^{29} +(0.382683 + 0.923880i) q^{32} +(-0.292893 - 0.707107i) q^{34} -1.08239i q^{35} +1.00000i q^{37} +(0.292893 - 0.707107i) q^{40} +(-0.707107 + 0.292893i) q^{46} -1.00000 q^{49} +(0.382683 - 0.158513i) q^{50} +(-1.30656 - 0.541196i) q^{56} +(1.70711 - 0.707107i) q^{58} -0.765367i q^{59} +(-0.707107 - 0.707107i) q^{64} -1.41421 q^{67} +(0.541196 + 0.541196i) q^{68} +(0.414214 + 1.00000i) q^{70} -1.41421 q^{73} +(-0.382683 - 0.923880i) q^{74} +0.765367i q^{80} -0.585786i q^{85} +1.84776i q^{89} +(0.541196 - 0.541196i) q^{92} +(0.923880 - 0.382683i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{25} + 8 q^{28} - 8 q^{34} + 8 q^{40} - 8 q^{49} + 8 q^{58} - 8 q^{70}+O(q^{100}) \) 8 * q + 8 * q^25 + 8 * q^28 - 8 * q^34 + 8 * q^40 - 8 * q^49 + 8 * q^58 - 8 * q^70

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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