LMFDB - Embedded newform 1085.1.bh.a.774.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1085,1,Mod(464,1085)] 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("1085.464"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1085, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1085 = 5 \cdot 7 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1085.bh (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,3,0,0,0,0,3,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$0.541485538707$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{18})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

Embedding invariants

Embedding label			774.3
Root			$0.939693 - 0.342020i$ of defining polynomial
Character	$\chi$	$=$	1085.774
Dual form			1085.1.bh.a.464.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.70574 - 0.984808i) q^{2} +(1.43969 - 2.49362i) q^{4} +(-0.766044 - 0.642788i) q^{5} +(-0.939693 + 0.342020i) q^{7} -3.70167i q^{8} +(0.500000 + 0.866025i) q^{9} +(-1.93969 - 0.342020i) q^{10} +(-1.26604 + 1.50881i) q^{14} +(-2.20574 - 3.82045i) q^{16} +(1.70574 + 0.984808i) q^{18} +(0.766044 + 1.32683i) q^{19} +(-2.70574 + 0.984808i) q^{20} +(0.173648 + 0.984808i) q^{25} +(-0.500000 + 2.83564i) q^{28} +(0.500000 - 0.866025i) q^{31} +(-4.31908 - 2.49362i) q^{32} +(0.939693 + 0.342020i) q^{35} +2.87939 q^{36} +(2.61334 + 1.50881i) q^{38} +(-2.37939 + 2.83564i) q^{40} +0.347296 q^{41} +(0.173648 - 0.984808i) q^{45} +(-1.50000 + 0.866025i) q^{47} +(0.766044 - 0.642788i) q^{49} +(1.26604 + 1.50881i) q^{50} +(1.26604 + 3.47843i) q^{56} +(-0.173648 + 0.300767i) q^{59} -1.96962i q^{62} +(-0.766044 - 0.642788i) q^{63} -5.41147 q^{64} +(1.50000 + 0.866025i) q^{67} +(1.93969 - 0.342020i) q^{70} -1.53209 q^{71} +(3.20574 - 1.85083i) q^{72} +4.41147 q^{76} +(-0.766044 + 4.34445i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.592396 - 0.342020i) q^{82} +(-0.673648 - 1.85083i) q^{90} +(-1.70574 + 2.95442i) q^{94} +(0.266044 - 1.50881i) q^{95} -1.28558i q^{97} +(0.673648 - 1.85083i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{4} + 3 q^{9} - 6 q^{10} - 3 q^{14} - 3 q^{16} - 6 q^{20} - 3 q^{28} + 3 q^{31} - 9 q^{32} + 6 q^{36} + 9 q^{38} - 3 q^{40} - 9 q^{47} + 3 q^{50} + 3 q^{56} - 12 q^{64} + 9 q^{67} + 6 q^{70} + 9 q^{72}+ \cdots + 3 q^{98}+O(q^{100}) $ 6 * q + 3 * q^4 + 3 * q^9 - 6 * q^10 - 3 * q^14 - 3 * q^16 - 6 * q^20 - 3 * q^28 + 3 * q^31 - 9 * q^32 + 6 * q^36 + 9 * q^38 - 3 * q^40 - 9 * q^47 + 3 * q^50 + 3 * q^56 - 12 * q^64 + 9 * q^67 + 6 * q^70 + 9 * q^72 + 6 * q^76 - 3 * q^81 - 3 * q^90 - 3 * q^95 + 3 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1085\mathbb{Z}\right)^\times$.

$n$	$311$	$561$	$652$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$

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 $

$2$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$2$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$	1.43969	−	2.49362i	1.43969	−	2.49362i
$5$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$5$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$6$		0			0
$7$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$7$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$8$		−	3.70167i		−	3.70167i
$9$	0.500000	+	0.866025i	0.500000	+	0.866025i
$10$	−1.93969	−	0.342020i	−1.93969	−	0.342020i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−1.26604	+	1.50881i	−1.26604	+	1.50881i
$15$		0			0
$16$	−2.20574	−	3.82045i	−2.20574	−	3.82045i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	1.70574	+	0.984808i	1.70574	+	0.984808i
$19$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$19$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$20$	−2.70574	+	0.984808i	−2.70574	+	0.984808i
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	0.173648	+	0.984808i	0.173648	+	0.984808i
$26$		0			0
$27$		0			0
$28$	−0.500000	+	2.83564i	−0.500000	+	2.83564i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.500000	−	0.866025i
$31$	0.500000	−	0.866025i	0.500000	−	0.866025i
$32$	−4.31908	−	2.49362i	−4.31908	−	2.49362i
$33$		0			0
$34$		0			0
$35$	0.939693	+	0.342020i	0.939693	+	0.342020i
$36$	2.87939			2.87939
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$	2.61334	+	1.50881i	2.61334	+	1.50881i
$39$		0			0
$40$	−2.37939	+	2.83564i	−2.37939	+	2.83564i
$41$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$41$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	0.173648	−	0.984808i	0.173648	−	0.984808i
$46$		0			0
$47$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$47$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$48$		0			0
$49$	0.766044	−	0.642788i	0.766044	−	0.642788i
$50$	1.26604	+	1.50881i	1.26604	+	1.50881i
$51$		0			0
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$	1.26604	+	3.47843i	1.26604	+	3.47843i
$57$		0			0
$58$		0			0
$59$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$59$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		−	1.96962i		−	1.96962i
$63$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$64$	−5.41147			−5.41147
$65$		0			0
$66$		0			0
$67$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$67$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	1.93969	−	0.342020i	1.93969	−	0.342020i
$71$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$71$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$72$	3.20574	−	1.85083i	3.20574	−	1.85083i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$		0			0
$76$	4.41147			4.41147
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	−0.766044	+	4.34445i	−0.766044	+	4.34445i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$	0.592396	−	0.342020i	0.592396	−	0.342020i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$	−0.673648	−	1.85083i	−0.673648	−	1.85083i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−1.70574	+	2.95442i	−1.70574	+	2.95442i
$95$	0.266044	−	1.50881i	0.266044	−	1.50881i
$96$		0			0
$97$		−	1.28558i		−	1.28558i	−0.766044	−	0.642788i	$-0.777778\pi$
$97$		−	1.28558i		−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$98$	0.673648	−	1.85083i	0.673648	−	1.85083i
$99$		0			0
$100$	2.70574	+	0.984808i	2.70574	+	0.984808i
$101$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$101$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$102$		0			0
$103$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$103$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$107$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$108$		0			0
$109$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$109$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$110$		0			0
$111$		0			0
$112$	3.37939	+	2.83564i	3.37939	+	2.83564i
$113$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$113$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$			0.684040i			0.684040i
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.43969	−	2.49362i	−1.43969	−	2.49362i
$125$	0.500000	−	0.866025i	0.500000	−	0.866025i
$126$	−1.93969	−	0.342020i	−1.93969	−	0.342020i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−4.91147	+	2.83564i	−4.91147	+	2.83564i
$129$		0			0
$130$		0			0
$131$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$131$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$132$		0			0
$133$	−1.17365	−	0.984808i	−1.17365	−	0.984808i
$134$	3.41147			3.41147
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	2.20574	−	1.85083i	2.20574	−	1.85083i
$141$		0			0
$142$	−2.61334	+	1.50881i	−2.61334	+	1.50881i
$143$		0			0
$144$	2.20574	−	3.82045i	2.20574	−	3.82045i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	4.91147	−	2.83564i	4.91147	−	2.83564i
$153$		0			0
$154$		0			0
$155$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$156$		0			0
$157$	1.70574	+	0.984808i	1.70574	+	0.984808i	0.939693	+	0.342020i	$0.111111\pi$
$157$	1.70574	+	0.984808i	1.70574	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$158$		0			0
$159$		0			0
$160$	1.70574	+	4.68647i	1.70574	+	4.68647i
$161$		0			0
$162$			1.96962i			1.96962i
$163$	0.592396	−	0.342020i	0.592396	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$163$	0.592396	−	0.342020i	0.592396	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$164$	0.500000	−	0.866025i	0.500000	−	0.866025i
$165$		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
$171$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	−2.20574	−	1.85083i	−2.20574	−	1.85083i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$			4.98724i			4.98724i
$189$		0			0
$190$	−1.03209	−	2.83564i	−1.03209	−	2.83564i
$191$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$191$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$192$		0			0
$193$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$193$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$194$	−1.26604	−	2.19285i	−1.26604	−	2.19285i
$195$		0			0
$196$	−0.500000	−	2.83564i	−0.500000	−	2.83564i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	3.64543	−	0.642788i	3.64543	−	0.642788i
$201$		0			0
$202$		−	3.01763i		−	3.01763i
$203$		0			0
$204$		0			0
$205$	−0.266044	−	0.223238i	−0.266044	−	0.223238i
$206$	−1.93969	+	3.35965i	−1.93969	+	3.35965i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$211$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$212$		0			0
$213$		0			0
$214$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$215$		0			0
$216$		0			0
$217$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$218$			3.70167i			3.70167i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	4.91147	+	0.866025i	4.91147	+	0.866025i
$225$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$226$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.592396	−	0.342020i	0.592396	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$233$	0.592396	−	0.342020i	0.592396	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$234$		0			0
$235$	1.70574	+	0.300767i	1.70574	+	0.300767i
$236$	0.500000	+	0.866025i	0.500000	+	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−1.70574	−	0.984808i	−1.70574	−	0.984808i
$243$		0			0
$244$		0			0
$245$	−1.00000			−1.00000
$246$		0			0
$247$		0			0
$248$	−3.20574	−	1.85083i	−3.20574	−	1.85083i
$249$		0			0
$250$		−	1.96962i		−	1.96962i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−2.70574	+	0.984808i	−2.70574	+	0.984808i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2.87939	+	4.98724i	−2.87939	+	4.98724i
$257$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$257$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.70574	−	0.984808i	−1.70574	−	0.984808i
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$	−2.97178	−	0.524005i	−2.97178	−	0.524005i
$267$		0			0
$268$	4.31908	−	2.49362i	4.31908	−	2.49362i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$277$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$278$		0			0
$279$	1.00000			1.00000
$280$	1.26604	−	3.47843i	1.26604	−	3.47843i
$281$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$281$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$282$		0			0
$283$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$283$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$284$	−2.20574	+	3.82045i	−2.20574	+	3.82045i
$285$		0			0
$286$		0			0
$287$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$288$		−	4.98724i		−	4.98724i
$289$	0.500000	+	0.866025i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$293$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$294$		0			0
$295$	0.326352	−	0.118782i	0.326352	−	0.118782i
$296$		0			0
$297$		0			0
$298$	1.70574	+	0.984808i	1.70574	+	0.984808i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	3.37939	−	5.85327i	3.37939	−	5.85327i
$305$		0			0
$306$		0			0
$307$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$307$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$308$		0			0
$309$		0			0
$310$	−1.26604	+	1.50881i	−1.26604	+	1.50881i
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$312$		0			0
$313$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$	3.87939			3.87939
$315$	0.173648	+	0.984808i	0.173648	+	0.984808i
$316$		0			0
$317$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$317$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$318$		0			0
$319$		0			0
$320$	4.14543	+	3.47843i	4.14543	+	3.47843i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.43969	+	2.49362i	1.43969	+	2.49362i
$325$		0			0
$326$	0.673648	−	1.16679i	0.673648	−	1.16679i
$327$		0			0
$328$		−	1.28558i		−	1.28558i
$329$	1.11334	−	1.32683i	1.11334	−	1.32683i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.592396	−	1.62760i	−0.592396	−	1.62760i
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−1.70574	+	0.984808i	−1.70574	+	0.984808i
$339$		0			0
$340$		0			0
$341$		0			0
$342$			3.01763i			3.01763i
$343$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$350$	−1.70574	−	0.984808i	−1.70574	−	0.984808i
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$	1.17365	+	0.984808i	1.17365	+	0.984808i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$359$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$360$	−3.64543	−	0.642788i	−3.64543	−	0.642788i
$361$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	0.173648	+	0.300767i	0.173648	+	0.300767i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$373$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$374$		0			0
$375$		0			0
$376$	3.20574	+	5.55250i	3.20574	+	5.55250i
$377$		0			0
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$	−3.37939	−	2.83564i	−3.37939	−	2.83564i
$381$		0			0
$382$	−0.592396	−	0.342020i	−0.592396	−	0.342020i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$	−3.87939			−3.87939
$387$		0			0
$388$	−3.20574	−	1.85083i	−3.20574	−	1.85083i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−2.37939	−	2.83564i	−2.37939	−	2.83564i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$397$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$398$		0			0
$399$		0			0
$400$	3.37939	−	2.83564i	3.37939	−	2.83564i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$	−2.20574	−	3.82045i	−2.20574	−	3.82045i
$405$	0.939693	−	0.342020i	0.939693	−	0.342020i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$	−0.673648	−	0.118782i	−0.673648	−	0.118782i
$411$		0			0
$412$			5.67128i			5.67128i
$413$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$419$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$420$		0			0
$421$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$421$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$422$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$423$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$			3.70167i			3.70167i
$429$		0			0
$430$		0			0
$431$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$431$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$	0.673648	+	1.85083i	0.673648	+	1.85083i
$435$		0			0
$436$	2.70574	+	4.68647i	2.70574	+	4.68647i
$437$		0			0
$438$		0			0
$439$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$439$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$440$		0			0
$441$	0.939693	+	0.342020i	0.939693	+	0.342020i
$442$		0			0
$443$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$443$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	5.08512	−	1.85083i	5.08512	−	1.85083i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.673648	+	1.85083i	−0.673648	+	1.85083i
$451$		0			0
$452$	−1.70574	−	0.984808i	−1.70574	−	0.984808i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$457$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	0.673648	−	1.16679i	0.673648	−	1.16679i
$467$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$467$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$468$		0			0
$469$	−1.70574	−	0.300767i	−1.70574	−	0.300767i
$470$	3.20574	−	1.16679i	3.20574	−	1.16679i
$471$		0			0
$472$	1.11334	+	0.642788i	1.11334	+	0.642788i
$473$		0			0
$474$		0			0
$475$	−1.17365	+	0.984808i	−1.17365	+	0.984808i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$479$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.87939			−2.87939
$485$	−0.826352	+	0.984808i	−0.826352	+	0.984808i
$486$		0			0
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$		0			0
$490$	−1.70574	+	0.984808i	−1.70574	+	0.984808i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−4.41147			−4.41147
$497$	1.43969	−	0.524005i	1.43969	−	0.524005i
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	−1.43969	−	2.49362i	−1.43969	−	2.49362i
$501$		0			0
$502$		0			0
$503$			1.28558i			1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$503$			1.28558i			1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$504$	−2.37939	+	2.83564i	−2.37939	+	2.83564i
$505$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$			5.67128i			5.67128i
$513$		0			0
$514$	1.26604	−	2.19285i	1.26604	−	2.19285i
$515$	1.93969	+	0.342020i	1.93969	+	0.342020i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$	−2.87939			−2.87939
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$		0			0
$531$	−0.347296			−0.347296
$532$	−4.14543	+	1.50881i	−4.14543	+	1.50881i
$533$		0			0
$534$		0			0
$535$	1.26604	+	0.223238i	1.26604	+	0.223238i
$536$	3.20574	−	5.55250i	3.20574	−	5.55250i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$541$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.76604	−	0.642788i	1.76604	−	0.642788i
$546$		0			0
$547$			1.28558i			1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$547$			1.28558i			1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$	1.70574	−	0.984808i	1.70574	−	0.984808i
$559$		0			0
$560$	−0.766044	−	4.34445i	−0.766044	−	4.34445i
$561$		0			0
$562$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$563$	0.592396	+	0.342020i	0.592396	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$563$	0.592396	+	0.342020i	0.592396	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$564$		0			0
$565$	−0.439693	+	0.524005i	−0.439693	+	0.524005i
$566$	3.41147			3.41147
$567$	0.173648	−	0.984808i	0.173648	−	0.984808i
$568$			5.67128i			5.67128i
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$	−0.439693	+	0.524005i	−0.439693	+	0.524005i
$575$		0			0
$576$	−2.70574	−	4.68647i	−2.70574	−	4.68647i
$577$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$578$	1.70574	+	0.984808i	1.70574	+	0.984808i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−1.70574	−	2.95442i	−1.70574	−	2.95442i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	1.53209			1.53209
$590$	0.439693	−	0.524005i	0.439693	−	0.524005i
$591$		0			0
$592$		0			0
$593$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$593$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$594$		0			0
$595$		0			0
$596$	2.87939			2.87939
$597$		0			0
$598$		0			0
$599$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$599$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			1.73205i			1.73205i
$604$		0			0
$605$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$606$		0			0
$607$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$607$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$608$		−	7.64090i		−	7.64090i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$613$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$614$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$615$		0			0
$616$		0			0
$617$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$	−0.500000	+	2.83564i	−0.500000	+	2.83564i
$621$		0			0
$622$		−	3.70167i		−	3.70167i
$623$		0			0
$624$		0			0
$625$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$626$		0			0
$627$		0			0
$628$	4.91147	−	2.83564i	4.91147	−	2.83564i
$629$		0			0
$630$	1.26604	+	1.50881i	1.26604	+	1.50881i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	1.26604	−	2.19285i	1.26604	−	2.19285i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$640$	5.58512	+	0.984808i	5.58512	+	0.984808i
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	3.20574	+	1.85083i	3.20574	+	1.85083i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		−	1.96962i		−	1.96962i
$653$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$653$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$654$		0			0
$655$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$656$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$657$		0			0
$658$	0.592396	−	3.35965i	0.592396	−	3.35965i
$659$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$659$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$660$		0			0
$661$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$661$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.266044	+	1.50881i	0.266044	+	1.50881i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−2.61334	−	2.19285i	−2.61334	−	2.19285i
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	−1.43969	+	2.49362i	−1.43969	+	2.49362i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	0.439693	+	1.20805i	0.439693	+	1.20805i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.592396	+	0.342020i	0.592396	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$683$	0.592396	+	0.342020i	0.592396	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$684$	2.20574	+	3.82045i	2.20574	+	3.82045i
$685$		0			0
$686$			1.96962i			1.96962i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$691$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−1.70574	+	0.984808i	−1.70574	+	0.984808i
$699$		0			0
$700$	−2.87939			−2.87939
$701$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$701$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.266044	+	1.50881i	−0.266044	+	1.50881i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$	2.97178	+	0.524005i	2.97178	+	0.524005i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−3.20574	−	1.85083i	−3.20574	−	1.85083i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	−4.14543	+	1.50881i	−4.14543	+	1.50881i
$721$	1.26604	−	1.50881i	1.26604	−	1.50881i
$722$			2.65366i			2.65366i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$727$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$733$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.592396	+	0.342020i	0.592396	+	0.342020i
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	0.173648	−	0.984808i	0.173648	−	0.984808i
$746$	1.26604	−	2.19285i	1.26604	−	2.19285i
$747$		0			0
$748$		0			0
$749$	0.826352	−	0.984808i	0.826352	−	0.984808i
$750$		0			0
$751$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$751$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$752$	6.61721	+	3.82045i	6.61721	+	3.82045i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−1.70574	+	0.984808i	−1.70574	+	0.984808i
$759$		0			0
$760$	−5.58512	−	0.984808i	−5.58512	−	0.984808i
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	0.326352	−	1.85083i	0.326352	−	1.85083i
$764$	−1.00000			−1.00000
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$769$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$770$		0			0
$771$		0			0
$772$	−4.91147	+	2.83564i	−4.91147	+	2.83564i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	0.939693	+	0.342020i	0.939693	+	0.342020i
$776$	−4.75877			−4.75877
$777$		0			0
$778$		0			0
$779$	0.266044	+	0.460802i	0.266044	+	0.460802i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−4.14543	−	1.50881i	−4.14543	−	1.50881i
$785$	−0.673648	−	1.85083i	−0.673648	−	1.85083i
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.233956	+	0.642788i	0.233956	+	0.642788i
$792$		0			0
$793$		0			0
$794$	1.93969	−	3.35965i	1.93969	−	3.35965i
$795$		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
$800$	1.70574	−	4.68647i	1.70574	−	4.68647i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−4.91147	−	2.83564i	−4.91147	−	2.83564i
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$	1.26604	−	1.50881i	1.26604	−	1.50881i
$811$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$811$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.673648	−	0.118782i	−0.673648	−	0.118782i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$823$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$824$	3.64543	+	6.31407i	3.64543	+	6.31407i
$825$		0			0
$826$	−0.233956	−	0.642788i	−0.233956	−	0.642788i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−3.20574	+	1.85083i	−3.20574	+	1.85083i
$839$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$839$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	0.592396	−	0.342020i	0.592396	−	0.342020i
$843$		0			0
$844$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$845$	0.766044	+	0.642788i	0.766044	+	0.642788i
$846$	−3.41147			−3.41147
$847$	0.766044	+	0.642788i	0.766044	+	0.642788i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$853$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$	1.43969	−	0.524005i	1.43969	−	0.524005i
$856$	2.37939	+	4.12122i	2.37939	+	4.12122i
$857$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$857$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$			1.96962i			1.96962i
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	2.20574	+	1.85083i	2.20574	+	1.85083i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	6.02481	+	3.47843i	6.02481	+	3.47843i
$873$	1.11334	−	0.642788i	1.11334	−	0.642788i
$874$		0			0
$875$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$876$		0			0
$877$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$877$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$878$	3.20574	+	1.85083i	3.20574	+	1.85083i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.93969	−	0.342020i	1.93969	−	0.342020i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	1.93969	−	3.35965i	1.93969	−	3.35965i
$887$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$887$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.29813	−	1.32683i	−2.29813	−	1.32683i
$894$		0			0
$895$		0			0
$896$	3.64543	−	4.34445i	3.64543	−	4.34445i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.500000	+	2.83564i	0.500000	+	2.83564i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−2.53209			−2.53209
$905$		0			0
$906$		0			0
$907$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$907$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	−0.766044	+	0.642788i	$0.777778\pi$
$908$		0			0
$909$	1.53209			1.53209
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.766044	+	0.642788i	0.766044	+	0.642788i
$918$		0			0
$919$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$919$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.70574	−	0.984808i	−1.70574	−	0.984808i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	1.43969	+	0.524005i	1.43969	+	0.524005i
$932$		−	1.96962i		−	1.96962i
$933$		0			0
$934$	−1.93969	+	3.35965i	−1.93969	+	3.35965i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−3.20574	+	1.16679i	−3.20574	+	1.16679i
$939$		0			0
$940$	3.20574	−	3.82045i	3.20574	−	3.82045i
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	1.53209			1.53209
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$		0			0
$950$	−1.03209	+	2.83564i	−1.03209	+	2.83564i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i
$956$		0			0
$957$		0			0
$958$			3.01763i			3.01763i
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$964$		0			0
$965$	0.673648	+	1.85083i	0.673648	+	1.85083i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−3.20574	+	1.85083i	−3.20574	+	1.85083i
$969$		0			0
$970$	−0.439693	+	2.49362i	−0.439693	+	2.49362i
$971$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.11334	+	0.642788i	1.11334	+	0.642788i	0.939693	−	0.342020i	$-0.111111\pi$
$977$	1.11334	+	0.642788i	1.11334	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$978$		0			0
$979$		0			0
$980$	−1.43969	+	2.49362i	−1.43969	+	2.49362i
$981$	−1.87939			−1.87939
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$	−4.31908	+	2.49362i	−4.31908	+	2.49362i
$993$		0			0
$994$	1.93969	−	2.31164i	1.93969	−	2.31164i
$995$		0			0
$996$		0			0
$997$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$997$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1085.1.bh.a.774.3	yes	6
5.4	even	2		1085.1.bh.b.774.1	yes	6
7.2	even	3		1085.1.bh.b.464.1	yes	6
31.30	odd	2	CM	1085.1.bh.a.774.3	yes	6
35.9	even	6	inner	1085.1.bh.a.464.3	✓	6
155.154	odd	2		1085.1.bh.b.774.1	yes	6
217.30	odd	6		1085.1.bh.b.464.1	yes	6
1085.464	odd	6	inner	1085.1.bh.a.464.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1085.1.bh.a.464.3	✓	6	35.9	even	6	inner
1085.1.bh.a.464.3	✓	6	1085.464	odd	6	inner
1085.1.bh.a.774.3	yes	6	1.1	even	1	trivial
1085.1.bh.a.774.3	yes	6	31.30	odd	2	CM
1085.1.bh.b.464.1	yes	6	7.2	even	3
1085.1.bh.b.464.1	yes	6	217.30	odd	6
1085.1.bh.b.774.1	yes	6	5.4	even	2
1085.1.bh.b.774.1	yes	6	155.154	odd	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:	\( N \)	\(=\)	\( 1085 = 5 \cdot 7 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1085.bh (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.70574 - 0.984808i) q^{2} +(1.43969 - 2.49362i) q^{4} +(-0.766044 - 0.642788i) q^{5} +(-0.939693 + 0.342020i) q^{7} -3.70167i q^{8} +(0.500000 + 0.866025i) q^{9} +(-1.93969 - 0.342020i) q^{10} +(-1.26604 + 1.50881i) q^{14} +(-2.20574 - 3.82045i) q^{16} +(1.70574 + 0.984808i) q^{18} +(0.766044 + 1.32683i) q^{19} +(-2.70574 + 0.984808i) q^{20} +(0.173648 + 0.984808i) q^{25} +(-0.500000 + 2.83564i) q^{28} +(0.500000 - 0.866025i) q^{31} +(-4.31908 - 2.49362i) q^{32} +(0.939693 + 0.342020i) q^{35} +2.87939 q^{36} +(2.61334 + 1.50881i) q^{38} +(-2.37939 + 2.83564i) q^{40} +0.347296 q^{41} +(0.173648 - 0.984808i) q^{45} +(-1.50000 + 0.866025i) q^{47} +(0.766044 - 0.642788i) q^{49} +(1.26604 + 1.50881i) q^{50} +(1.26604 + 3.47843i) q^{56} +(-0.173648 + 0.300767i) q^{59} -1.96962i q^{62} +(-0.766044 - 0.642788i) q^{63} -5.41147 q^{64} +(1.50000 + 0.866025i) q^{67} +(1.93969 - 0.342020i) q^{70} -1.53209 q^{71} +(3.20574 - 1.85083i) q^{72} +4.41147 q^{76} +(-0.766044 + 4.34445i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.592396 - 0.342020i) q^{82} +(-0.673648 - 1.85083i) q^{90} +(-1.70574 + 2.95442i) q^{94} +(0.266044 - 1.50881i) q^{95} -1.28558i q^{97} +(0.673648 - 1.85083i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{4} + 3 q^{9} - 6 q^{10} - 3 q^{14} - 3 q^{16} - 6 q^{20} - 3 q^{28} + 3 q^{31} - 9 q^{32} + 6 q^{36} + 9 q^{38} - 3 q^{40} - 9 q^{47} + 3 q^{50} + 3 q^{56} - 12 q^{64} + 9 q^{67} + 6 q^{70} + 9 q^{72}+ \cdots + 3 q^{98}+O(q^{100}) \) 6 * q + 3 * q^4 + 3 * q^9 - 6 * q^10 - 3 * q^14 - 3 * q^16 - 6 * q^20 - 3 * q^28 + 3 * q^31 - 9 * q^32 + 6 * q^36 + 9 * q^38 - 3 * q^40 - 9 * q^47 + 3 * q^50 + 3 * q^56 - 12 * q^64 + 9 * q^67 + 6 * q^70 + 9 * q^72 + 6 * q^76 - 3 * q^81 - 3 * q^90 - 3 * q^95 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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