LMFDB - Embedded newform 3240.1.bh.f.1349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,1,Mod(269,3240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3240.269");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3240.bh (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.61697064093$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.46656000.2

Embedding invariants

Embedding label			1349.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3240.1349
Dual form			3240.1.bh.f.269.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.00000i q^{5} +(-1.50000 + 0.866025i) q^{7} -1.00000i q^{8} +O(q^{10})$	$q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.00000i q^{5} +(-1.50000 + 0.866025i) q^{7} -1.00000i q^{8} +(-0.500000 - 0.866025i) q^{10} +(-0.866025 - 1.50000i) q^{11} +(-0.866025 + 1.50000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 - 0.500000i) q^{20} +(-1.50000 - 0.866025i) q^{22} -1.00000 q^{25} +1.73205i q^{28} +(-0.500000 + 0.866025i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(0.866025 + 1.50000i) q^{35} -1.00000 q^{40} -1.73205 q^{44} +(1.00000 - 1.73205i) q^{49} +(-0.866025 + 0.500000i) q^{50} -1.00000i q^{53} +(-1.50000 + 0.866025i) q^{55} +(0.866025 + 1.50000i) q^{56} +1.00000i q^{62} -1.00000 q^{64} +(1.50000 + 0.866025i) q^{70} -1.73205i q^{73} +(2.59808 + 1.50000i) q^{77} +(-1.00000 - 1.73205i) q^{79} +(-0.866025 + 0.500000i) q^{80} +(0.866025 - 0.500000i) q^{83} +(-1.50000 + 0.866025i) q^{88} +(1.50000 - 0.866025i) q^{97} -2.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 6 q^{7}+O(q^{10}) $ 4 * q + 2 * q^4 - 6 * q^7	$ 4 q + 2 q^{4} - 6 q^{7} - 2 q^{10} - 2 q^{16} - 6 q^{22} - 4 q^{25} - 2 q^{31} - 4 q^{40} + 4 q^{49} - 6 q^{55} - 4 q^{64} + 6 q^{70} - 4 q^{79} - 6 q^{88} + 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 - 6 * q^7 - 2 * q^10 - 2 * q^16 - 6 * q^22 - 4 * q^25 - 2 * q^31 - 4 * q^40 + 4 * q^49 - 6 * q^55 - 4 * q^64 + 6 * q^70 - 4 * q^79 - 6 * q^88 + 6 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	0.866025	−	0.500000i	0.866025	−	0.500000i
$2$	0.866025	−	0.500000i	0.866025	−	0.500000i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$		−	1.00000i		−	1.00000i
$5$		−	1.00000i		−	1.00000i
$6$		0			0
$7$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$7$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$8$		−	1.00000i		−	1.00000i
$9$		0			0
$10$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$11$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$11$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$12$		0			0
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$21$		0			0
$22$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−1.00000			−1.00000
$26$		0			0
$27$		0			0
$28$			1.73205i			1.73205i
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$32$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$33$		0			0
$34$		0			0
$35$	0.866025	+	1.50000i	0.866025	+	1.50000i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$	−1.00000			−1.00000
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$	−1.73205			−1.73205
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	1.00000	−	1.73205i	1.00000	−	1.73205i
$50$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$51$		0			0
$52$		0			0
$53$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$53$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$54$		0			0
$55$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$56$	0.866025	+	1.50000i	0.866025	+	1.50000i
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$			1.00000i			1.00000i
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$		0			0
$70$	1.50000	+	0.866025i	1.50000	+	0.866025i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$73$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.59808	+	1.50000i	2.59808	+	1.50000i
$78$		0			0
$79$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$80$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$81$		0			0
$82$		0			0
$83$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$83$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$97$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$98$		−	2.00000i		−	2.00000i
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$101$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$107$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$107$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$111$		0			0
$112$	1.50000	+	0.866025i	1.50000	+	0.866025i
$113$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$113$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$122$		0			0
$123$		0			0
$124$	0.500000	+	0.866025i	0.500000	+	0.866025i
$125$			1.00000i			1.00000i
$126$		0			0
$127$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$127$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$128$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$131$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$	1.73205			1.73205
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$147$		0			0
$148$		0			0
$149$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$149$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$150$		0			0
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$	3.00000			3.00000
$155$	0.866025	+	0.500000i	0.866025	+	0.500000i
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$159$		0			0
$160$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	0.500000	−	0.866025i	0.500000	−	0.866025i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	0.500000	+	0.866025i	0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$173$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	1.50000	−	0.866025i	1.50000	−	0.866025i
$176$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$177$		0			0
$178$		0			0
$179$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$179$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$		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
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$194$	0.866025	−	1.50000i	0.866025	−	1.50000i
$195$		0			0
$196$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$197$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$197$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$200$			1.00000i			1.00000i
$201$		0			0
$202$	1.50000	+	0.866025i	1.50000	+	0.866025i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$213$		0			0
$214$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$215$		0			0
$216$		0			0
$217$		−	1.73205i		−	1.73205i
$218$		0			0
$219$		0			0
$220$			1.73205i			1.73205i
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$	1.73205			1.73205
$225$		0			0
$226$		0			0
$227$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$227$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$241$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$242$			2.00000i			2.00000i
$243$		0			0
$244$		0			0
$245$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$246$		0			0
$247$		0			0
$248$	0.866025	+	0.500000i	0.866025	+	0.500000i
$249$		0			0
$250$	0.500000	+	0.866025i	0.500000	+	0.866025i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	0.866025	+	1.50000i	0.866025	+	1.50000i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		−	1.73205i		−	1.73205i
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	−1.00000			−1.00000
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$271$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.866025	+	1.50000i	0.866025	+	1.50000i
$276$		0			0
$277$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$277$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$278$		0			0
$279$		0			0
$280$	1.50000	−	0.866025i	1.50000	−	0.866025i
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$293$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$293$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		−	1.73205i		−	1.73205i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.866025	+	0.500000i	0.866025	+	0.500000i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	2.59808	−	1.50000i	2.59808	−	1.50000i
$309$		0			0
$310$	1.00000			1.00000
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$313$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$314$		0			0
$315$		0			0
$316$	−2.00000			−2.00000
$317$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$317$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$			1.00000i			1.00000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		−	1.00000i		−	1.00000i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	0.866025	+	0.500000i	0.866025	+	0.500000i
$339$		0			0
$340$		0			0
$341$	1.73205			1.73205
$342$		0			0
$343$			1.73205i			1.73205i
$344$		0			0
$345$		0			0
$346$	0.500000	−	0.866025i	0.500000	−	0.866025i
$347$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$347$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$	0.866025	−	1.50000i	0.866025	−	1.50000i
$351$		0			0
$352$			1.73205i			1.73205i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.50000	−	0.866025i	1.50000	−	0.866025i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.73205			−1.73205
$366$		0			0
$367$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.866025	+	1.50000i	0.866025	+	1.50000i
$372$		0			0
$373$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$373$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$	1.50000	−	2.59808i	1.50000	−	2.59808i
$386$	−1.73205			−1.73205
$387$		0			0
$388$		−	1.73205i		−	1.73205i
$389$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$389$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$390$		0			0
$391$		0			0
$392$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$393$		0			0
$394$	0.500000	+	0.866025i	0.500000	+	0.866025i
$395$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	0.866025	−	0.500000i	0.866025	−	0.500000i
$399$		0			0
$400$	0.500000	+	0.866025i	0.500000	+	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$	1.73205			1.73205
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$	−1.00000			−1.00000
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$433$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$434$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$439$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$440$	0.866025	+	1.50000i	0.866025	+	1.50000i
$441$		0			0
$442$		0			0
$443$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$443$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.50000	−	0.866025i	1.50000	−	0.866025i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$455$		0			0
$456$		0			0
$457$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$457$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$461$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$462$		0			0
$463$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$463$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$467$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$	1.73205	+	1.00000i	1.73205	+	1.00000i
$483$		0			0
$484$	1.00000	+	1.73205i	1.00000	+	1.73205i
$485$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	−2.00000			−2.00000
$491$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$491$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.00000			1.00000
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	0.866025	+	0.500000i	0.866025	+	0.500000i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	1.50000	−	0.866025i	1.50000	−	0.866025i
$506$		0			0
$507$		0			0
$508$	1.50000	+	0.866025i	1.50000	+	0.866025i
$509$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$509$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	1.50000	+	2.59808i	1.50000	+	2.59808i
$512$			1.00000i			1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	+	0.866025i	0.500000	+	0.866025i
$530$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.00000			−1.00000
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.46410			−3.46410
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$547$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$548$		0			0
$549$		0			0
$550$	1.50000	+	0.866025i	1.50000	+	0.866025i
$551$		0			0
$552$		0			0
$553$	3.00000	+	1.73205i	3.00000	+	1.73205i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$557$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$	0.866025	−	1.50000i	0.866025	−	1.50000i
$561$		0			0
$562$		0			0
$563$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$563$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		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			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$579$		0			0
$580$		0			0
$581$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$582$		0			0
$583$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$584$	−1.73205			−1.73205
$585$		0			0
$586$	−2.00000			−2.00000
$587$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$587$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	1.00000			1.00000
$605$	1.73205	+	1.00000i	1.73205	+	1.00000i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$	1.50000	−	2.59808i	1.50000	−	2.59808i
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$	0.866025	−	0.500000i	0.866025	−	0.500000i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$	0.866025	−	1.50000i	0.866025	−	1.50000i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$632$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$633$		0			0
$634$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$635$	1.73205			1.73205
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	+	0.866025i	0.500000	+	0.866025i
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$643$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$644$		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			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$653$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$659$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$673$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	1.00000			1.00000
$677$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$677$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$678$		0			0
$679$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$680$		0			0
$681$		0			0
$682$	1.50000	−	0.866025i	1.50000	−	0.866025i
$683$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$683$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$684$		0			0
$685$		0			0
$686$	0.866025	+	1.50000i	0.866025	+	1.50000i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		−	1.00000i		−	1.00000i
$693$		0			0
$694$	−1.00000			−1.00000
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		−	1.73205i		−	1.73205i
$701$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$701$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$702$		0			0
$703$		0			0
$704$	0.866025	+	1.50000i	0.866025	+	1.50000i
$705$		0			0
$706$		0			0
$707$	−2.59808	−	1.50000i	−2.59808	−	1.50000i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.866025	−	1.50000i	0.866025	−	1.50000i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	0.866025	−	0.500000i	0.866025	−	0.500000i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$727$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$728$		0			0
$729$		0			0
$730$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$	0.866025	−	1.50000i	0.866025	−	1.50000i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$	1.50000	+	0.866025i	1.50000	+	0.866025i
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.866025	+	1.50000i	0.866025	+	1.50000i
$750$		0			0
$751$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$751$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.866025	−	0.500000i	0.866025	−	0.500000i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$769$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$770$		−	3.00000i		−	3.00000i
$771$		0			0
$772$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$773$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$773$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$774$		0			0
$775$	0.500000	−	0.866025i	0.500000	−	0.866025i
$776$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$777$		0			0
$778$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−2.00000			−2.00000
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$	0.866025	+	0.500000i	0.866025	+	0.500000i
$789$		0			0
$790$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.500000	−	0.866025i	0.500000	−	0.866025i
$797$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$797$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	0.866025	+	0.500000i	0.866025	+	0.500000i
$801$		0			0
$802$		0			0
$803$	−2.59808	+	1.50000i	−2.59808	+	1.50000i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.50000	−	0.866025i	1.50000	−	0.866025i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	1.00000i		−	1.00000i
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$827$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	0.500000	−	0.866025i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.866025	−	0.500000i	0.866025	−	0.500000i
$846$		0			0
$847$		−	3.46410i		−	3.46410i
$848$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$853$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$854$		0			0
$855$		0			0
$856$	−1.00000			−1.00000
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$866$	0.866025	+	1.50000i	0.866025	+	1.50000i
$867$		0			0
$868$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$869$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$879$		0			0
$880$	1.50000	+	0.866025i	1.50000	+	0.866025i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	1.00000	−	1.73205i	1.00000	−	1.73205i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	1.73205i		−	1.73205i
$896$	0.866025	−	1.50000i	0.866025	−	1.50000i
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$907$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$908$			2.00000i			2.00000i
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$914$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$915$		0			0
$916$		0			0
$917$			3.00000i			3.00000i
$918$		0			0
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	1.73205			1.73205
$927$		0			0
$928$		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
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$935$		0			0
$936$		0			0
$937$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$937$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$941$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$947$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		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			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
$964$	2.00000			2.00000
$965$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$966$		0			0
$967$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$967$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$968$	1.73205	+	1.00000i	1.73205	+	1.00000i
$969$		0			0
$970$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$971$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$971$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$981$		0			0
$982$			1.73205i			1.73205i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	1.00000			1.00000
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$	0.866025	−	0.500000i	0.866025	−	0.500000i
$993$		0			0
$994$		0			0
$995$		−	1.00000i		−	1.00000i
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.1.bh.f.1349.2		4
3.2	odd	2	inner	3240.1.bh.f.1349.1		4
5.4	even	2		3240.1.bh.i.1349.1		4
8.5	even	2	inner	3240.1.bh.f.1349.1		4
9.2	odd	6		3240.1.bh.i.269.2		4
9.4	even	3		1080.1.i.e.269.3	yes	4
9.5	odd	6		1080.1.i.e.269.2	yes	4
9.7	even	3		3240.1.bh.i.269.1		4
15.14	odd	2		3240.1.bh.i.1349.2		4
24.5	odd	2	CM	3240.1.bh.f.1349.2		4
40.29	even	2		3240.1.bh.i.1349.2		4
45.4	even	6		1080.1.i.e.269.1	✓	4
45.14	odd	6		1080.1.i.e.269.4	yes	4
45.29	odd	6	inner	3240.1.bh.f.269.1		4
45.34	even	6	inner	3240.1.bh.f.269.2		4
72.5	odd	6		1080.1.i.e.269.3	yes	4
72.13	even	6		1080.1.i.e.269.2	yes	4
72.29	odd	6		3240.1.bh.i.269.1		4
72.61	even	6		3240.1.bh.i.269.2		4
120.29	odd	2		3240.1.bh.i.1349.1		4
360.29	odd	6	inner	3240.1.bh.f.269.2		4
360.149	odd	6		1080.1.i.e.269.1	✓	4
360.229	even	6		1080.1.i.e.269.4	yes	4
360.349	even	6	inner	3240.1.bh.f.269.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.1.i.e.269.1	✓	4	45.4	even	6
1080.1.i.e.269.1	✓	4	360.149	odd	6
1080.1.i.e.269.2	yes	4	9.5	odd	6
1080.1.i.e.269.2	yes	4	72.13	even	6
1080.1.i.e.269.3	yes	4	9.4	even	3
1080.1.i.e.269.3	yes	4	72.5	odd	6
1080.1.i.e.269.4	yes	4	45.14	odd	6
1080.1.i.e.269.4	yes	4	360.229	even	6
3240.1.bh.f.269.1		4	45.29	odd	6	inner
3240.1.bh.f.269.1		4	360.349	even	6	inner
3240.1.bh.f.269.2		4	45.34	even	6	inner
3240.1.bh.f.269.2		4	360.29	odd	6	inner
3240.1.bh.f.1349.1		4	3.2	odd	2	inner
3240.1.bh.f.1349.1		4	8.5	even	2	inner
3240.1.bh.f.1349.2		4	1.1	even	1	trivial
3240.1.bh.f.1349.2		4	24.5	odd	2	CM
3240.1.bh.i.269.1		4	9.7	even	3
3240.1.bh.i.269.1		4	72.29	odd	6
3240.1.bh.i.269.2		4	9.2	odd	6
3240.1.bh.i.269.2		4	72.61	even	6
3240.1.bh.i.1349.1		4	5.4	even	2
3240.1.bh.i.1349.1		4	120.29	odd	2
3240.1.bh.i.1349.2		4	15.14	odd	2
3240.1.bh.i.1349.2		4	40.29	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3240.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.00000i q^{5} +(-1.50000 + 0.866025i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.00000i q^{5} +(-1.50000 + 0.866025i) q^{7} -1.00000i q^{8} +(-0.500000 - 0.866025i) q^{10} +(-0.866025 - 1.50000i) q^{11} +(-0.866025 + 1.50000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 - 0.500000i) q^{20} +(-1.50000 - 0.866025i) q^{22} -1.00000 q^{25} +1.73205i q^{28} +(-0.500000 + 0.866025i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(0.866025 + 1.50000i) q^{35} -1.00000 q^{40} -1.73205 q^{44} +(1.00000 - 1.73205i) q^{49} +(-0.866025 + 0.500000i) q^{50} -1.00000i q^{53} +(-1.50000 + 0.866025i) q^{55} +(0.866025 + 1.50000i) q^{56} +1.00000i q^{62} -1.00000 q^{64} +(1.50000 + 0.866025i) q^{70} -1.73205i q^{73} +(2.59808 + 1.50000i) q^{77} +(-1.00000 - 1.73205i) q^{79} +(-0.866025 + 0.500000i) q^{80} +(0.866025 - 0.500000i) q^{83} +(-1.50000 + 0.866025i) q^{88} +(1.50000 - 0.866025i) q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 6 q^{7}+O(q^{10}) \) 4 * q + 2 * q^4 - 6 * q^7	\( 4 q + 2 q^{4} - 6 q^{7} - 2 q^{10} - 2 q^{16} - 6 q^{22} - 4 q^{25} - 2 q^{31} - 4 q^{40} + 4 q^{49} - 6 q^{55} - 4 q^{64} + 6 q^{70} - 4 q^{79} - 6 q^{88} + 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 - 6 * q^7 - 2 * q^10 - 2 * q^16 - 6 * q^22 - 4 * q^25 - 2 * q^31 - 4 * q^40 + 4 * q^49 - 6 * q^55 - 4 * q^64 + 6 * q^70 - 4 * q^79 - 6 * q^88 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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