LMFDB - Embedded newform 2160.1.r.a.1459.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,1,Mod(379,2160)] 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("2160.379"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2160.r (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:	$1.07798042729$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			1459.3
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2160.1459
Dual form			2160.1.r.a.379.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.707107 - 0.707107i) q^{8} +(0.500000 - 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{16} -0.517638i q^{17} +(0.366025 - 0.366025i) q^{19} +(0.965926 + 0.258819i) q^{20} -1.93185i q^{23} +1.00000i q^{25} -1.00000i q^{31} +(0.965926 + 0.258819i) q^{32} +(0.500000 - 0.133975i) q^{34} +(0.448288 + 0.258819i) q^{38} +1.00000i q^{40} +(1.86603 - 0.500000i) q^{46} +1.41421 q^{47} +1.00000 q^{49} +(-0.965926 + 0.258819i) q^{50} +(-1.22474 - 1.22474i) q^{53} +(1.36603 + 1.36603i) q^{61} +(0.965926 - 0.258819i) q^{62} +1.00000i q^{64} +(0.258819 + 0.448288i) q^{68} +(-0.133975 + 0.500000i) q^{76} -1.73205i q^{79} +(-0.965926 + 0.258819i) q^{80} +(-0.707107 - 0.707107i) q^{83} +(-0.366025 + 0.366025i) q^{85} +(0.965926 + 1.67303i) q^{92} +(0.366025 + 1.36603i) q^{94} -0.517638 q^{95} +(0.258819 + 0.965926i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{10} + 4 q^{16} - 4 q^{19} + 4 q^{34} + 8 q^{46} + 8 q^{49} + 4 q^{61} - 8 q^{76} + 4 q^{85} - 4 q^{94}+O(q^{100}) $ 8 * q + 4 * q^10 + 4 * q^16 - 4 * q^19 + 4 * q^34 + 8 * q^46 + 8 * q^49 + 4 * q^61 - 8 * q^76 + 4 * q^85 - 4 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

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 $

$2$	0.258819	+	0.965926i	0.258819	+	0.965926i
$2$	0.258819	+	0.965926i	0.258819	+	0.965926i
$3$		0			0
$3$		0			0
$4$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$5$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$5$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$9$		0			0
$10$	0.500000	−	0.866025i	0.500000	−	0.866025i
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	0.500000	−	0.866025i	0.500000	−	0.866025i
$17$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$17$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$18$		0			0
$19$	0.366025	−	0.366025i	0.366025	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$19$	0.366025	−	0.366025i	0.366025	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$20$	0.965926	+	0.258819i	0.965926	+	0.258819i
$21$		0			0
$22$		0			0
$23$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$23$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$24$		0			0
$25$			1.00000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$31$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$32$	0.965926	+	0.258819i	0.965926	+	0.258819i
$33$		0			0
$34$	0.500000	−	0.133975i	0.500000	−	0.133975i
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	0.448288	+	0.258819i	0.448288	+	0.258819i
$39$		0			0
$40$			1.00000i			1.00000i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	1.86603	−	0.500000i	1.86603	−	0.500000i
$47$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$47$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$	1.00000			1.00000
$50$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$51$		0			0
$52$		0			0
$53$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$53$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$61$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$62$	0.965926	−	0.258819i	0.965926	−	0.258819i
$63$		0			0
$64$			1.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	0.258819	+	0.448288i	0.258819	+	0.448288i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	−0.133975	+	0.500000i	−0.133975	+	0.500000i
$77$		0			0
$78$		0			0
$79$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$79$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$80$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$81$		0			0
$82$		0			0
$83$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$83$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$84$		0			0
$85$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	0.965926	+	1.67303i	0.965926	+	1.67303i
$93$		0			0
$94$	0.366025	+	1.36603i	0.366025	+	1.36603i
$95$	−0.517638			−0.517638
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.258819	+	0.965926i	0.258819	+	0.965926i
$99$		0			0
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$		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		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	0.866025	−	1.50000i	0.866025	−	1.50000i
$107$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$107$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$109$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$113$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	1.00000i		−	1.00000i
$122$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$123$		0			0
$124$	0.500000	+	0.866025i	0.500000	+	0.866025i
$125$	0.707107	−	0.707107i	0.707107	−	0.707107i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$129$		0			0
$130$		0			0
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$137$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$137$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$138$		0			0
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		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
$151$	−2.00000			−2.00000			−1.00000			$\pi$
$151$	−2.00000			−2.00000			−1.00000			$\pi$
$152$	−0.517638			−0.517638
$153$		0			0
$154$		0			0
$155$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	1.67303	−	0.448288i	1.67303	−	0.448288i
$159$		0			0
$160$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$161$		0			0
$162$		0			0
$163$		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
$166$	0.500000	−	0.866025i	0.500000	−	0.866025i
$167$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$167$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$168$		0			0
$169$			1.00000i			1.00000i
$170$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$171$		0			0
$172$		0			0
$173$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$173$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$181$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$182$		0			0
$183$		0			0
$184$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$189$		0			0
$190$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$197$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$197$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	0.707107	−	0.707107i	0.707107	−	0.707107i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$211$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$212$	1.67303	+	0.448288i	1.67303	+	0.448288i
$213$		0			0
$214$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.258819	−	0.448288i	0.258819	−	0.448288i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$227$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$227$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$229$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$230$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$231$		0			0
$232$		0			0
$233$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$233$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$241$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$242$	0.965926	−	0.258819i	0.965926	−	0.258819i
$243$		0			0
$244$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$245$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$246$		0			0
$247$		0			0
$248$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$249$		0			0
$250$	0.866025	+	0.500000i	0.866025	+	0.500000i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$257$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$263$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$264$		0			0
$265$			1.73205i			1.73205i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$269$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$271$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$272$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$273$		0			0
$274$	0.500000	+	1.86603i	0.500000	+	1.86603i
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	0.707107	−	1.22474i	0.707107	−	1.22474i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.732051			0.732051
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$293$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.517638	−	1.93185i	−0.517638	−	1.93185i
$303$		0			0
$304$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$305$		−	1.93185i		−	1.93185i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				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			0
$316$	0.866025	+	1.50000i	0.866025	+	1.50000i
$317$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$317$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$318$		0			0
$319$		0			0
$320$	0.707107	−	0.707107i	0.707107	−	0.707107i
$321$		0			0
$322$		0			0
$323$	−0.189469	−	0.189469i	−0.189469	−	0.189469i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$331$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$332$	0.965926	+	0.258819i	0.965926	+	0.258819i
$333$		0			0
$334$	0.500000	−	0.133975i	0.500000	−	0.133975i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$339$		0			0
$340$	0.133975	−	0.500000i	0.133975	−	0.500000i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	0.366025	+	0.366025i	0.366025	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$349$	0.366025	+	0.366025i	0.366025	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$353$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$			0.732051i			0.732051i
$362$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$377$		0			0
$378$		0			0
$379$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$379$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$380$	0.448288	−	0.258819i	0.448288	−	0.258819i
$381$		0			0
$382$		0			0
$383$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$383$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$	−1.00000			−1.00000
$392$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$393$		0			0
$394$	0.500000	−	0.866025i	0.500000	−	0.866025i
$395$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$	0.866025	+	0.500000i	0.866025	+	0.500000i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$			1.00000i			1.00000i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	0.366025	−	0.366025i	0.366025	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.366025	−	0.366025i	0.366025	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$422$	1.67303	+	0.965926i	1.67303	+	0.965926i
$423$		0			0
$424$			1.73205i			1.73205i
$425$	0.517638			0.517638
$426$		0			0
$427$		0			0
$428$	0.517638	−	1.93185i	0.517638	−	1.93185i
$429$		0			0
$430$		0			0
$431$		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			0
$435$		0			0
$436$	0.500000	+	0.133975i	0.500000	+	0.133975i
$437$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$438$		0			0
$439$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$439$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$443$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$453$		0			0
$454$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	1.67303	+	0.965926i	1.67303	+	0.965926i
$459$		0			0
$460$	0.500000	−	1.86603i	0.500000	−	1.86603i
$461$		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			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	0.366025	+	1.36603i	0.366025	+	1.36603i
$467$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$467$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$468$		0			0
$469$		0			0
$470$	0.707107	−	1.22474i	0.707107	−	1.22474i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.366025	+	0.366025i	0.366025	+	0.366025i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$483$		0			0
$484$	0.500000	+	0.866025i	0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		−	1.93185i		−	1.93185i
$489$		0			0
$490$	0.500000	−	0.866025i	0.500000	−	0.866025i
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$497$		0			0
$498$		0			0
$499$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$499$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$500$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$501$		0			0
$502$		0			0
$503$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$503$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$	0.707107	−	0.707107i	0.707107	−	0.707107i
$513$		0			0
$514$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$	1.36603	−	0.366025i	1.36603	−	0.366025i
$527$	−0.517638			−0.517638
$528$		0			0
$529$	−2.73205			−2.73205
$530$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	2.00000			2.00000
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$541$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$542$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$543$		0			0
$544$	0.133975	−	0.500000i	0.133975	−	0.500000i
$545$			0.517638i			0.517638i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.36603	+	0.366025i	1.36603	+	0.366025i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	1.00000	−	1.00000i	1.00000	−	1.00000i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$571$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.93185			1.93185
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.189469	+	0.707107i	0.189469	+	0.707107i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$587$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$587$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$588$		0			0
$589$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$593$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$602$		0			0
$603$		0			0
$604$	1.73205	−	1.00000i	1.73205	−	1.00000i
$605$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	0.448288	−	0.258819i	0.448288	−	0.258819i
$609$		0			0
$610$	1.86603	−	0.500000i	1.86603	−	0.500000i
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$617$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$618$		0			0
$619$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$619$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$620$	0.258819	−	0.965926i	0.258819	−	0.965926i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$631$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$632$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$633$		0			0
$634$	0.866025	+	0.500000i	0.866025	+	0.500000i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.866025	+	0.500000i	0.866025	+	0.500000i
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		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
$646$	0.133975	−	0.232051i	0.133975	−	0.232051i
$647$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$647$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$653$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$661$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$662$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$663$		0			0
$664$			1.00000i			1.00000i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.258819	+	0.448288i	0.258819	+	0.448288i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$	0.517638			0.517638
$681$		0			0
$682$		0			0
$683$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$683$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$684$		0			0
$685$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$691$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$692$	0.448288	−	1.67303i	0.448288	−	1.67303i
$693$		0			0
$694$		0			0
$695$			1.41421i			1.41421i
$696$		0			0
$697$		0			0
$698$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$699$		0			0
$700$		0			0
$701$		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
$704$		0			0
$705$		0			0
$706$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$707$		0			0
$708$		0			0
$709$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$709$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.93185			−1.93185
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−0.707107	+	0.189469i	−0.707107	+	0.189469i
$723$		0			0
$724$	0.500000	−	1.86603i	0.500000	−	1.86603i
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$	0.500000	−	1.86603i	0.500000	−	1.86603i
$737$		0			0
$738$		0			0
$739$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$739$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$743$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$751$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$752$	0.707107	−	1.22474i	0.707107	−	1.22474i
$753$		0			0
$754$		0			0
$755$	1.41421	+	1.41421i	1.41421	+	1.41421i
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	0.965926	−	1.67303i	0.965926	−	1.67303i
$759$		0			0
$760$	0.366025	+	0.366025i	0.366025	+	0.366025i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.500000	+	1.86603i	0.500000	+	1.86603i
$767$		0			0
$768$		0			0
$769$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$769$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$773$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$774$		0			0
$775$	1.00000			1.00000
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$783$		0			0
$784$	0.500000	−	0.866025i	0.500000	−	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	0.965926	+	0.258819i	0.965926	+	0.258819i
$789$		0			0
$790$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$797$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$798$		0			0
$799$		−	0.732051i		−	0.732051i
$800$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$811$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.67303	−	0.448288i	1.67303	−	0.448288i
$819$		0			0
$820$		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		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$827$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$828$		0			0
$829$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$829$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$830$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$831$		0			0
$832$		0			0
$833$		−	0.517638i		−	0.517638i
$834$		0			0
$835$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$		−	1.00000i		−	1.00000i
$842$	0.448288	+	0.258819i	0.448288	+	0.258819i
$843$		0			0
$844$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$845$	0.707107	−	0.707107i	0.707107	−	0.707107i
$846$		0			0
$847$		0			0
$848$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$849$		0			0
$850$	0.133975	+	0.500000i	0.133975	+	0.500000i
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	2.00000			2.00000
$857$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$857$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$858$		0			0
$859$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$859$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$863$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$864$		0			0
$865$	1.73205			1.73205
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			0.517638i			0.517638i
$873$		0			0
$874$	0.500000	−	0.866025i	0.500000	−	0.866025i
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	0.448288	+	1.67303i	0.448288	+	1.67303i
$879$		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		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$887$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$887$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.517638	−	0.517638i	0.517638	−	0.517638i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.633975	+	0.633975i	−0.633975	+	0.633975i
$902$		0			0
$903$		0			0
$904$	1.00000	−	1.00000i	1.00000	−	1.00000i
$905$	1.93185			1.93185
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$909$		0			0
$910$		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
$916$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$917$		0			0
$918$		0			0
$919$	2.00000			2.00000			1.00000			$0$
$919$	2.00000			2.00000			1.00000			$0$
$920$	1.93185			1.93185
$921$		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
$931$	0.366025	−	0.366025i	0.366025	−	0.366025i
$932$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$933$		0			0
$934$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	1.36603	+	0.366025i	1.36603	+	0.366025i
$941$		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
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$947$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$948$		0			0
$949$		0			0
$950$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$951$		0			0
$952$		0			0
$953$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$953$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$	1.50000	−	0.866025i	1.50000	−	0.866025i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$969$		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
$976$	1.86603	−	0.500000i	1.86603	−	0.500000i
$977$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$977$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$978$		0			0
$979$		0			0
$980$	0.965926	+	0.258819i	0.965926	+	0.258819i
$981$		0			0
$982$		0			0
$983$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$983$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$984$		0			0
$985$			1.00000i			1.00000i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$991$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$992$	0.258819	−	0.965926i	0.258819	−	0.965926i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.1.r.a.1459.3	yes	8
3.2	odd	2	inner	2160.1.r.a.1459.2	yes	8
5.4	even	2	inner	2160.1.r.a.1459.2	yes	8
15.14	odd	2	CM	2160.1.r.a.1459.3	yes	8
16.11	odd	4	inner	2160.1.r.a.379.2	✓	8
48.11	even	4	inner	2160.1.r.a.379.3	yes	8
80.59	odd	4	inner	2160.1.r.a.379.3	yes	8
240.59	even	4	inner	2160.1.r.a.379.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.1.r.a.379.2	✓	8	16.11	odd	4	inner
2160.1.r.a.379.2	✓	8	240.59	even	4	inner
2160.1.r.a.379.3	yes	8	48.11	even	4	inner
2160.1.r.a.379.3	yes	8	80.59	odd	4	inner
2160.1.r.a.1459.2	yes	8	3.2	odd	2	inner
2160.1.r.a.1459.2	yes	8	5.4	even	2	inner
2160.1.r.a.1459.3	yes	8	1.1	even	1	trivial
2160.1.r.a.1459.3	yes	8	15.14	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2160.r (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.707107 - 0.707107i) q^{8} +(0.500000 - 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{16} -0.517638i q^{17} +(0.366025 - 0.366025i) q^{19} +(0.965926 + 0.258819i) q^{20} -1.93185i q^{23} +1.00000i q^{25} -1.00000i q^{31} +(0.965926 + 0.258819i) q^{32} +(0.500000 - 0.133975i) q^{34} +(0.448288 + 0.258819i) q^{38} +1.00000i q^{40} +(1.86603 - 0.500000i) q^{46} +1.41421 q^{47} +1.00000 q^{49} +(-0.965926 + 0.258819i) q^{50} +(-1.22474 - 1.22474i) q^{53} +(1.36603 + 1.36603i) q^{61} +(0.965926 - 0.258819i) q^{62} +1.00000i q^{64} +(0.258819 + 0.448288i) q^{68} +(-0.133975 + 0.500000i) q^{76} -1.73205i q^{79} +(-0.965926 + 0.258819i) q^{80} +(-0.707107 - 0.707107i) q^{83} +(-0.366025 + 0.366025i) q^{85} +(0.965926 + 1.67303i) q^{92} +(0.366025 + 1.36603i) q^{94} -0.517638 q^{95} +(0.258819 + 0.965926i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{10} + 4 q^{16} - 4 q^{19} + 4 q^{34} + 8 q^{46} + 8 q^{49} + 4 q^{61} - 8 q^{76} + 4 q^{85} - 4 q^{94}+O(q^{100}) \) 8 * q + 4 * q^10 + 4 * q^16 - 4 * q^19 + 4 * q^34 + 8 * q^46 + 8 * q^49 + 4 * q^61 - 8 * q^76 + 4 * q^85 - 4 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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