LMFDB - Embedded newform 880.1.x.a.549.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,1,Mod(109,880)] 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("880.109"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	880.x (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.439177211117$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
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 8.0.7676100608000.1

Embedding invariants

Embedding label			549.4
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	880.549
Dual form			880.1.x.a.109.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.923880 - 0.382683i) q^{2} +(0.707107 - 0.707107i) q^{4} +(0.707107 + 0.707107i) q^{5} +0.765367i q^{7} +(0.382683 - 0.923880i) q^{8} -1.00000i q^{9} +(0.923880 + 0.382683i) q^{10} +(-0.707107 - 0.707107i) q^{11} +(-1.30656 + 1.30656i) q^{13} +(0.292893 + 0.707107i) q^{14} -1.00000i q^{16} +0.765367 q^{17} +(-0.382683 - 0.923880i) q^{18} +1.00000 q^{20} +(-0.923880 - 0.382683i) q^{22} +1.00000i q^{25} +(-0.707107 + 1.70711i) q^{26} +(0.541196 + 0.541196i) q^{28} -1.41421 q^{31} +(-0.382683 - 0.923880i) q^{32} +(0.707107 - 0.292893i) q^{34} +(-0.541196 + 0.541196i) q^{35} +(-0.707107 - 0.707107i) q^{36} +(0.923880 - 0.382683i) q^{40} +(-1.30656 - 1.30656i) q^{43} -1.00000 q^{44} +(0.707107 - 0.707107i) q^{45} +0.414214 q^{49} +(0.382683 + 0.923880i) q^{50} +1.84776i q^{52} -1.00000i q^{55} +(0.707107 + 0.292893i) q^{56} +(1.00000 + 1.00000i) q^{59} +(-1.30656 + 0.541196i) q^{62} +0.765367 q^{63} +(-0.707107 - 0.707107i) q^{64} -1.84776 q^{65} +(0.541196 - 0.541196i) q^{68} +(-0.292893 + 0.707107i) q^{70} +(-0.923880 - 0.382683i) q^{72} -1.84776i q^{73} +(0.541196 - 0.541196i) q^{77} +(0.707107 - 0.707107i) q^{80} -1.00000 q^{81} +(-1.30656 + 1.30656i) q^{83} +(0.541196 + 0.541196i) q^{85} +(-1.70711 - 0.707107i) q^{86} +(-0.923880 + 0.382683i) q^{88} +1.41421i q^{89} +(0.382683 - 0.923880i) q^{90} +(-1.00000 - 1.00000i) q^{91} +(0.382683 - 0.158513i) q^{98} +(-0.707107 + 0.707107i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{14} + 8 q^{20} - 8 q^{44} - 8 q^{49} + 8 q^{59} - 8 q^{70} - 8 q^{81} - 8 q^{86} - 8 q^{91}+O(q^{100}) $ 8 * q + 8 * q^14 + 8 * q^20 - 8 * q^44 - 8 * q^49 + 8 * q^59 - 8 * q^70 - 8 * q^81 - 8 * q^86 - 8 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	0.707107	−	0.707107i	0.707107	−	0.707107i
$5$	0.707107	+	0.707107i	0.707107	+	0.707107i
$5$	0.707107	+	0.707107i	0.707107	+	0.707107i
$6$		0			0
$7$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$7$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$8$	0.382683	−	0.923880i	0.382683	−	0.923880i
$9$		−	1.00000i		−	1.00000i
$10$	0.923880	+	0.382683i	0.923880	+	0.382683i
$11$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$11$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$12$		0			0
$13$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$13$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$14$	0.292893	+	0.707107i	0.292893	+	0.707107i
$15$		0			0
$16$		−	1.00000i		−	1.00000i
$17$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$17$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$18$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	1.00000			1.00000
$21$		0			0
$22$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$			1.00000i			1.00000i
$26$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$27$		0			0
$28$	0.541196	+	0.541196i	0.541196	+	0.541196i
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$31$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$32$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$33$		0			0
$34$	0.707107	−	0.292893i	0.707107	−	0.292893i
$35$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$36$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$	0.923880	−	0.382683i	0.923880	−	0.382683i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$43$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$44$	−1.00000			−1.00000
$45$	0.707107	−	0.707107i	0.707107	−	0.707107i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	0.414214			0.414214
$50$	0.382683	+	0.923880i	0.382683	+	0.923880i
$51$		0			0
$52$			1.84776i			1.84776i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		−	1.00000i		−	1.00000i
$56$	0.707107	+	0.292893i	0.707107	+	0.292893i
$57$		0			0
$58$		0			0
$59$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$59$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$63$	0.765367			0.765367
$64$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$65$	−1.84776			−1.84776
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	0.541196	−	0.541196i	0.541196	−	0.541196i
$69$		0			0
$70$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$73$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$73$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.541196	−	0.541196i	0.541196	−	0.541196i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	0.707107	−	0.707107i	0.707107	−	0.707107i
$81$	−1.00000			−1.00000
$82$		0			0
$83$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$83$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$84$		0			0
$85$	0.541196	+	0.541196i	0.541196	+	0.541196i
$86$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$87$		0			0
$88$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$89$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$89$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$90$	0.382683	−	0.923880i	0.382683	−	0.923880i
$91$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.382683	−	0.158513i	0.382683	−	0.158513i
$99$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$100$	0.707107	+	0.707107i	0.707107	+	0.707107i
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	0.707107	+	1.70711i	0.707107	+	1.70711i
$105$		0			0
$106$		0			0
$107$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$107$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$108$		0			0
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$111$		0			0
$112$	0.765367			0.765367
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.30656	+	1.30656i	1.30656	+	1.30656i
$118$	1.30656	+	0.541196i	1.30656	+	0.541196i
$119$			0.585786i			0.585786i
$120$		0			0
$121$			1.00000i			1.00000i
$122$		0			0
$123$		0			0
$124$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$125$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$126$	0.707107	−	0.292893i	0.707107	−	0.292893i
$127$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$127$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$128$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$129$		0			0
$130$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$131$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.292893	−	0.707107i	0.292893	−	0.707107i
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$139$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$140$			0.765367i			0.765367i
$141$		0			0
$142$		0			0
$143$	1.84776			1.84776
$144$	−1.00000			−1.00000
$145$		0			0
$146$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$147$		0			0
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	0.765367i		−	0.765367i
$154$	0.292893	−	0.707107i	0.292893	−	0.707107i
$155$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	0.382683	−	0.923880i	0.382683	−	0.923880i
$161$		0			0
$162$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$167$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$167$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$168$		0			0
$169$		−	2.41421i		−	2.41421i
$170$	0.707107	+	0.292893i	0.707107	+	0.292893i
$171$		0			0
$172$	−1.84776			−1.84776
$173$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$173$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$174$		0			0
$175$	−0.765367			−0.765367
$176$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$177$		0			0
$178$	0.541196	+	1.30656i	0.541196	+	1.30656i
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		−	1.00000i		−	1.00000i
$181$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$181$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$182$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$191$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$192$		0			0
$193$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$193$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$194$		0			0
$195$		0			0
$196$	0.292893	−	0.292893i	0.292893	−	0.292893i
$197$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$197$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$198$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.923880	+	0.382683i	0.923880	+	0.382683i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.30656	+	1.30656i	1.30656	+	1.30656i
$209$		0			0
$210$		0			0
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$		0			0
$213$		0			0
$214$	0.707107	+	0.292893i	0.707107	+	0.292893i
$215$		−	1.84776i		−	1.84776i
$216$		0			0
$217$		−	1.08239i		−	1.08239i
$218$		0			0
$219$		0			0
$220$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$221$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	0.707107	−	0.292893i	0.707107	−	0.292893i
$225$	1.00000			1.00000
$226$		0			0
$227$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$227$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$228$		0			0
$229$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$229$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$233$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$234$	1.70711	+	0.707107i	1.70711	+	0.707107i
$235$		0			0
$236$	1.41421			1.41421
$237$		0			0
$238$	0.224171	+	0.541196i	0.224171	+	0.541196i
$239$		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			$\pi$
$242$	0.382683	+	0.923880i	0.382683	+	0.923880i
$243$		0			0
$244$		0			0
$245$	0.292893	+	0.292893i	0.292893	+	0.292893i
$246$		0			0
$247$		0			0
$248$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$249$		0			0
$250$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$251$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$251$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$252$	0.541196	−	0.541196i	0.541196	−	0.541196i
$253$		0			0
$254$	1.70711	−	0.707107i	1.70711	−	0.707107i
$255$		0			0
$256$	−1.00000			−1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$261$		0			0
$262$		0			0
$263$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$263$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$269$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		−	0.765367i		−	0.765367i
$273$		0			0
$274$		0			0
$275$	0.707107	−	0.707107i	0.707107	−	0.707107i
$276$		0			0
$277$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$277$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$278$		0			0
$279$			1.41421i			1.41421i
$280$	0.292893	+	0.707107i	0.292893	+	0.707107i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$283$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$284$		0			0
$285$		0			0
$286$	1.70711	−	0.707107i	1.70711	−	0.707107i
$287$		0			0
$288$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$289$	−0.414214			−0.414214
$290$		0			0
$291$		0			0
$292$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$293$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$293$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$294$		0			0
$295$			1.41421i			1.41421i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.00000	−	1.00000i	1.00000	−	1.00000i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$307$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$307$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$308$		−	0.765367i		−	0.765367i
$309$		0			0
$310$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$311$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$311$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$	0.541196	+	0.541196i	0.541196	+	0.541196i
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		−	1.00000i		−	1.00000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$325$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$331$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$332$			1.84776i			1.84776i
$333$		0			0
$334$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$335$		0			0
$336$		0			0
$337$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$337$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$338$	−0.923880	−	2.23044i	−0.923880	−	2.23044i
$339$		0			0
$340$	0.765367			0.765367
$341$	1.00000	+	1.00000i	1.00000	+	1.00000i
$342$		0			0
$343$			1.08239i			1.08239i
$344$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$345$		0			0
$346$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$347$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$347$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$351$		0			0
$352$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	1.00000	+	1.00000i	1.00000	+	1.00000i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$361$		−	1.00000i		−	1.00000i
$362$		0			0
$363$		0			0
$364$	−1.41421			−1.41421
$365$	1.30656	−	1.30656i	1.30656	−	1.30656i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$373$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$374$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$379$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	1.30656	−	0.541196i	1.30656	−	0.541196i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	0.765367			0.765367
$386$	1.70711	−	0.707107i	1.70711	−	0.707107i
$387$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$388$		0			0
$389$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$389$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.158513	−	0.382683i	0.158513	−	0.382683i
$393$		0			0
$394$	0.707107	+	0.292893i	0.707107	+	0.292893i
$395$		0			0
$396$			1.00000i			1.00000i
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$401$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$	1.84776	−	1.84776i	1.84776	−	1.84776i
$404$		0			0
$405$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$406$		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
$412$		0			0
$413$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$414$		0			0
$415$	−1.84776			−1.84776
$416$	1.70711	+	0.707107i	1.70711	+	0.707107i
$417$		0			0
$418$		0			0
$419$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$419$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			0.765367i			0.765367i
$426$		0			0
$427$		0			0
$428$	0.765367			0.765367
$429$		0			0
$430$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	−0.414214	−	1.00000i	−0.414214	−	1.00000i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$441$		−	0.414214i		−	0.414214i
$442$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$446$		0			0
$447$		0			0
$448$	0.541196	−	0.541196i	0.541196	−	0.541196i
$449$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$449$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$450$	0.923880	−	0.382683i	0.923880	−	0.382683i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	0.707107	−	1.70711i	0.707107	−	1.70711i
$455$		−	1.41421i		−	1.41421i
$456$		0			0
$457$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$457$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$458$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	1.84776			1.84776
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.30656	−	0.541196i	1.30656	−	0.541196i
$473$			1.84776i			1.84776i
$474$		0			0
$475$		0			0
$476$	0.414214	+	0.414214i	0.414214	+	0.414214i
$477$		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
$484$	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
$490$	0.382683	+	0.158513i	0.382683	+	0.158513i
$491$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$491$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−1.00000			−1.00000
$496$			1.41421i			1.41421i
$497$		0			0
$498$		0			0
$499$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$499$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$500$			1.00000i			1.00000i
$501$		0			0
$502$	1.30656	+	0.541196i	1.30656	+	0.541196i
$503$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$503$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$504$	0.292893	−	0.707107i	0.292893	−	0.707107i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.30656	−	1.30656i	1.30656	−	1.30656i
$509$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$509$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$510$		0			0
$511$	1.41421			1.41421
$512$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$521$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$521$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$522$		0			0
$523$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$523$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$524$		0			0
$525$		0			0
$526$	0.707107	+	1.70711i	0.707107	+	1.70711i
$527$	−1.08239			−1.08239
$528$		0			0
$529$	−1.00000			−1.00000
$530$		0			0
$531$	1.00000	−	1.00000i	1.00000	−	1.00000i
$532$		0			0
$533$		0			0
$534$		0			0
$535$			0.765367i			0.765367i
$536$		0			0
$537$		0			0
$538$	0.541196	−	1.30656i	0.541196	−	1.30656i
$539$	−0.292893	−	0.292893i	−0.292893	−	0.292893i
$540$		0			0
$541$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$541$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$		0			0
$544$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$545$		0			0
$546$		0			0
$547$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$547$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$548$		0			0
$549$		0			0
$550$	0.382683	−	0.923880i	0.382683	−	0.923880i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$555$		0			0
$556$		0			0
$557$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$557$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$558$	0.541196	+	1.30656i	0.541196	+	1.30656i
$559$	3.41421			3.41421
$560$	0.541196	+	0.541196i	0.541196	+	0.541196i
$561$		0			0
$562$		0			0
$563$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$563$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$564$		0			0
$565$		0			0
$566$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$567$		−	0.765367i		−	0.765367i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$	1.30656	−	1.30656i	1.30656	−	1.30656i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.382683	+	0.158513i	−0.382683	+	0.158513i
$579$		0			0
$580$		0			0
$581$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$582$		0			0
$583$		0			0
$584$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$585$			1.84776i			1.84776i
$586$	1.70711	+	0.707107i	1.70711	+	0.707107i
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$	0.541196	+	1.30656i	0.541196	+	1.30656i
$591$		0			0
$592$		0			0
$593$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$593$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$594$		0			0
$595$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$599$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$	0.541196	−	1.30656i	0.541196	−	1.30656i
$603$		0			0
$604$		0			0
$605$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$606$		0			0
$607$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$607$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$613$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$613$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$614$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$615$		0			0
$616$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$	−1.41421			−1.41421
$621$		0			0
$622$	−0.765367	−	1.84776i	−0.765367	−	1.84776i
$623$	−1.08239			−1.08239
$624$		0			0
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	0.707107	+	0.292893i	0.707107	+	0.292893i
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.30656	+	1.30656i	1.30656	+	1.30656i
$636$		0			0
$637$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$638$		0			0
$639$		0			0
$640$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$641$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$641$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$649$		−	1.41421i		−	1.41421i
$650$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.84776			−1.84776
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$661$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$662$	1.84776	+	0.765367i	1.84776	+	0.765367i
$663$		0			0
$664$	0.707107	+	1.70711i	0.707107	+	1.70711i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$673$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$674$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$675$		0			0
$676$	−1.70711	−	1.70711i	−1.70711	−	1.70711i
$677$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$677$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$678$		0			0
$679$		0			0
$680$	0.707107	−	0.292893i	0.707107	−	0.292893i
$681$		0			0
$682$	1.30656	+	0.541196i	1.30656	+	0.541196i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$	0.414214	+	1.00000i	0.414214	+	1.00000i
$687$		0			0
$688$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$689$		0			0
$690$		0			0
$691$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$691$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$692$			0.765367i			0.765367i
$693$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$694$	1.70711	+	0.707107i	1.70711	+	0.707107i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$			1.00000i			1.00000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$709$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$710$		0			0
$711$		0			0
$712$	1.30656	+	0.541196i	1.30656	+	0.541196i
$713$		0			0
$714$		0			0
$715$	1.30656	+	1.30656i	1.30656	+	1.30656i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$719$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$720$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$721$		0			0
$722$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$729$			1.00000i			1.00000i
$730$	0.707107	−	1.70711i	0.707107	−	1.70711i
$731$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$732$		0			0
$733$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$733$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$743$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$744$		0			0
$745$		0			0
$746$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$747$	1.30656	+	1.30656i	1.30656	+	1.30656i
$748$	−0.765367			−0.765367
$749$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$750$		0			0
$751$	−2.00000			−2.00000			−1.00000			$\pi$
$751$	−2.00000			−2.00000			−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	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
$764$	1.00000	−	1.00000i	1.00000	−	1.00000i
$765$	0.541196	−	0.541196i	0.541196	−	0.541196i
$766$		0			0
$767$	−2.61313			−2.61313
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	0.707107	−	0.292893i	0.707107	−	0.292893i
$771$		0			0
$772$	1.30656	−	1.30656i	1.30656	−	1.30656i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$775$		−	1.41421i		−	1.41421i
$776$		0			0
$777$		0			0
$778$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	0.414214i		−	0.414214i
$785$		0			0
$786$		0			0
$787$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$787$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$788$	0.765367			0.765367
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.382683	+	0.923880i	0.382683	+	0.923880i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	0.923880	−	0.382683i	0.923880	−	0.382683i
$801$	1.41421			1.41421
$802$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$803$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$804$		0			0
$805$		0			0
$806$	1.00000	−	2.41421i	1.00000	−	2.41421i
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$	−0.414214	+	1.00000i	−0.414214	+	1.00000i
$827$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$827$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$828$		0			0
$829$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$829$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$830$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$831$		0			0
$832$	1.84776			1.84776
$833$	0.317025			0.317025
$834$		0			0
$835$	1.30656	−	1.30656i	1.30656	−	1.30656i
$836$		0			0
$837$		0			0
$838$	0.765367	−	1.84776i	0.765367	−	1.84776i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$		−	1.00000i		−	1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.70711	−	1.70711i	1.70711	−	1.70711i
$846$		0			0
$847$	−0.765367			−0.765367
$848$		0			0
$849$		0			0
$850$	0.292893	+	0.707107i	0.292893	+	0.707107i
$851$		0			0
$852$		0			0
$853$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$853$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$854$		0			0
$855$		0			0
$856$	0.707107	−	0.292893i	0.707107	−	0.292893i
$857$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$857$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$858$		0			0
$859$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$859$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$860$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−0.765367			−0.765367
$866$		0			0
$867$		0			0
$868$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$876$		0			0
$877$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$877$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$878$		0			0
$879$		0			0
$880$	−1.00000			−1.00000
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$			1.41421i			1.41421i
$885$		0			0
$886$		0			0
$887$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$887$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$888$		0			0
$889$			1.41421i			1.41421i
$890$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$891$	0.707107	+	0.707107i	0.707107	+	0.707107i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.292893	−	0.707107i	0.292893	−	0.707107i
$897$		0			0
$898$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$899$		0			0
$900$	0.707107	−	0.707107i	0.707107	−	0.707107i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		−	1.84776i		−	1.84776i
$909$		0			0
$910$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$911$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$911$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$	1.84776			1.84776
$914$	0.707107	+	1.70711i	0.707107	+	1.70711i
$915$		0			0
$916$	−1.41421			−1.41421
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$933$		0			0
$934$		0			0
$935$		−	0.765367i		−	0.765367i
$936$	1.70711	−	0.707107i	1.70711	−	0.707107i
$937$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$937$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$	1.00000	−	1.00000i	1.00000	−	1.00000i
$945$		0			0
$946$	0.707107	+	1.70711i	0.707107	+	1.70711i
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$	2.41421	+	2.41421i	2.41421	+	2.41421i
$950$		0			0
$951$		0			0
$952$	0.541196	+	0.224171i	0.541196	+	0.224171i
$953$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$953$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$954$		0			0
$955$	1.00000	+	1.00000i	1.00000	+	1.00000i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$	0.541196	−	0.541196i	0.541196	−	0.541196i
$964$		0			0
$965$	1.30656	+	1.30656i	1.30656	+	1.30656i
$966$		0			0
$967$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$967$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$968$	0.923880	+	0.382683i	0.923880	+	0.382683i
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$	1.00000	−	1.00000i	1.00000	−	1.00000i
$980$	0.414214			0.414214
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$			0.765367i			0.765367i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$991$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$991$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$992$	0.541196	+	1.30656i	0.541196	+	1.30656i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$997$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$998$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.1.x.a.549.4	yes	8
4.3	odd	2		3520.1.x.a.1649.3		8
5.4	even	2	inner	880.1.x.a.549.1	yes	8
11.10	odd	2	inner	880.1.x.a.549.1	yes	8
16.3	odd	4		3520.1.x.a.3409.4		8
16.13	even	4	inner	880.1.x.a.109.4	yes	8
20.19	odd	2		3520.1.x.a.1649.4		8
44.43	even	2		3520.1.x.a.1649.4		8
55.54	odd	2	CM	880.1.x.a.549.4	yes	8
80.19	odd	4		3520.1.x.a.3409.3		8
80.29	even	4	inner	880.1.x.a.109.1	✓	8
176.109	odd	4	inner	880.1.x.a.109.1	✓	8
176.131	even	4		3520.1.x.a.3409.3		8
220.219	even	2		3520.1.x.a.1649.3		8
880.109	odd	4	inner	880.1.x.a.109.4	yes	8
880.659	even	4		3520.1.x.a.3409.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
880.1.x.a.109.1	✓	8	80.29	even	4	inner
880.1.x.a.109.1	✓	8	176.109	odd	4	inner
880.1.x.a.109.4	yes	8	16.13	even	4	inner
880.1.x.a.109.4	yes	8	880.109	odd	4	inner
880.1.x.a.549.1	yes	8	5.4	even	2	inner
880.1.x.a.549.1	yes	8	11.10	odd	2	inner
880.1.x.a.549.4	yes	8	1.1	even	1	trivial
880.1.x.a.549.4	yes	8	55.54	odd	2	CM
3520.1.x.a.1649.3		8	4.3	odd	2
3520.1.x.a.1649.3		8	220.219	even	2
3520.1.x.a.1649.4		8	20.19	odd	2
3520.1.x.a.1649.4		8	44.43	even	2
3520.1.x.a.3409.3		8	80.19	odd	4
3520.1.x.a.3409.3		8	176.131	even	4
3520.1.x.a.3409.4		8	16.3	odd	4
3520.1.x.a.3409.4		8	880.659	even	4

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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	880.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.923880 - 0.382683i) q^{2} +(0.707107 - 0.707107i) q^{4} +(0.707107 + 0.707107i) q^{5} +0.765367i q^{7} +(0.382683 - 0.923880i) q^{8} -1.00000i q^{9} +(0.923880 + 0.382683i) q^{10} +(-0.707107 - 0.707107i) q^{11} +(-1.30656 + 1.30656i) q^{13} +(0.292893 + 0.707107i) q^{14} -1.00000i q^{16} +0.765367 q^{17} +(-0.382683 - 0.923880i) q^{18} +1.00000 q^{20} +(-0.923880 - 0.382683i) q^{22} +1.00000i q^{25} +(-0.707107 + 1.70711i) q^{26} +(0.541196 + 0.541196i) q^{28} -1.41421 q^{31} +(-0.382683 - 0.923880i) q^{32} +(0.707107 - 0.292893i) q^{34} +(-0.541196 + 0.541196i) q^{35} +(-0.707107 - 0.707107i) q^{36} +(0.923880 - 0.382683i) q^{40} +(-1.30656 - 1.30656i) q^{43} -1.00000 q^{44} +(0.707107 - 0.707107i) q^{45} +0.414214 q^{49} +(0.382683 + 0.923880i) q^{50} +1.84776i q^{52} -1.00000i q^{55} +(0.707107 + 0.292893i) q^{56} +(1.00000 + 1.00000i) q^{59} +(-1.30656 + 0.541196i) q^{62} +0.765367 q^{63} +(-0.707107 - 0.707107i) q^{64} -1.84776 q^{65} +(0.541196 - 0.541196i) q^{68} +(-0.292893 + 0.707107i) q^{70} +(-0.923880 - 0.382683i) q^{72} -1.84776i q^{73} +(0.541196 - 0.541196i) q^{77} +(0.707107 - 0.707107i) q^{80} -1.00000 q^{81} +(-1.30656 + 1.30656i) q^{83} +(0.541196 + 0.541196i) q^{85} +(-1.70711 - 0.707107i) q^{86} +(-0.923880 + 0.382683i) q^{88} +1.41421i q^{89} +(0.382683 - 0.923880i) q^{90} +(-1.00000 - 1.00000i) q^{91} +(0.382683 - 0.158513i) q^{98} +(-0.707107 + 0.707107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{14} + 8 q^{20} - 8 q^{44} - 8 q^{49} + 8 q^{59} - 8 q^{70} - 8 q^{81} - 8 q^{86} - 8 q^{91}+O(q^{100}) \) 8 * q + 8 * q^14 + 8 * q^20 - 8 * q^44 - 8 * q^49 + 8 * q^59 - 8 * q^70 - 8 * q^81 - 8 * q^86 - 8 * q^91

Introduction

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