LMFDB - Embedded newform 2520.1.ek.c.1909.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,1,Mod(829,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2520.829");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2520.ek (of order $6$, degree $2$, 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:	$1.25764383184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.3630312000.4

Embedding invariants

Embedding label			1909.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2520.1909
Dual form			2520.1.ek.c.829.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{10} +(-1.50000 + 0.866025i) q^{11} +1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{20} +1.73205i q^{22} +1.00000 q^{25} +(0.500000 - 0.866025i) q^{28} +1.73205i q^{29} +(-1.50000 + 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-0.500000 - 0.866025i) q^{35} +1.00000 q^{40} +(1.50000 + 0.866025i) q^{44} +(-0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{50} +(0.500000 + 0.866025i) q^{53} +(1.50000 - 0.866025i) q^{55} +(-0.500000 - 0.866025i) q^{56} +(1.50000 + 0.866025i) q^{58} +(0.500000 + 0.866025i) q^{59} +1.73205i q^{62} +1.00000 q^{64} -1.00000 q^{70} +(-1.00000 - 1.73205i) q^{73} +(-1.50000 - 0.866025i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(0.500000 - 0.866025i) q^{80} -1.73205i q^{83} +(1.50000 - 0.866025i) q^{88} +1.00000 q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^5 + q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8} - q^{10} - 3 q^{11} + 2 q^{14} - q^{16} + q^{20} + 2 q^{25} + q^{28} - 3 q^{31} + q^{32} - q^{35} + 2 q^{40} + 3 q^{44} - q^{49} + q^{50} + q^{53} + 3 q^{55} - q^{56} + 3 q^{58} + q^{59} + 2 q^{64} - 2 q^{70} - 2 q^{73} - 3 q^{77} - q^{79} + q^{80} + 3 q^{88} + 2 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^5 + q^7 - 2 * q^8 - q^10 - 3 * q^11 + 2 * q^14 - q^16 + q^20 + 2 * q^25 + q^28 - 3 * q^31 + q^32 - q^35 + 2 * q^40 + 3 * q^44 - q^49 + q^50 + q^53 + 3 * q^55 - q^56 + 3 * q^58 + q^59 + 2 * q^64 - 2 * q^70 - 2 * q^73 - 3 * q^77 - q^79 + q^80 + 3 * q^88 + 2 * q^97 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$

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 $

$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	−1.00000			−1.00000
$5$	−1.00000			−1.00000
$6$		0			0
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i
$8$	−1.00000			−1.00000
$9$		0			0
$10$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$11$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$11$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	1.00000			1.00000
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	0.500000	+	0.866025i	0.500000	+	0.866025i
$21$		0			0
$22$			1.73205i			1.73205i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	1.00000			1.00000
$26$		0			0
$27$		0			0
$28$	0.500000	−	0.866025i	0.500000	−	0.866025i
$29$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$29$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$31$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$		0			0
$34$		0			0
$35$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$36$		0			0
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$		0			0
$39$		0			0
$40$	1.00000			1.00000
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	1.50000	+	0.866025i	1.50000	+	0.866025i
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	0.500000	−	0.866025i	0.500000	−	0.866025i
$51$		0			0
$52$		0			0
$53$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$53$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$	1.50000	−	0.866025i	1.50000	−	0.866025i
$56$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$57$		0			0
$58$	1.50000	+	0.866025i	1.50000	+	0.866025i
$59$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$59$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$			1.73205i			1.73205i
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$		0			0
$69$		0			0
$70$	−1.00000			−1.00000
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$79$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$80$	0.500000	−	0.866025i	0.500000	−	0.866025i
$81$		0			0
$82$		0			0
$83$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$83$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.50000	−	0.866025i	1.50000	−	0.866025i
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$97$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$98$	0.500000	+	0.866025i	0.500000	+	0.866025i
$99$		0			0
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$102$		0			0
$103$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$103$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	1.00000			1.00000
$107$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$107$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		−	1.73205i		−	1.73205i
$111$		0			0
$112$	−1.00000			−1.00000
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	1.50000	−	0.866025i	1.50000	−	0.866025i
$117$		0			0
$118$	1.00000			1.00000
$119$		0			0
$120$		0			0
$121$	1.00000	−	1.73205i	1.00000	−	1.73205i
$122$		0			0
$123$		0			0
$124$	1.50000	+	0.866025i	1.50000	+	0.866025i
$125$	−1.00000			−1.00000
$126$		0			0
$127$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$127$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$		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
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	1.73205i		−	1.73205i
$146$	−2.00000			−2.00000
$147$		0			0
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$151$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$155$	1.50000	−	0.866025i	1.50000	−	0.866025i
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	0.500000	+	0.866025i	0.500000	+	0.866025i
$159$		0			0
$160$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$161$		0			0
$162$		0			0
$163$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$163$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$164$		0			0
$165$		0			0
$166$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$167$		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			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.500000	+	0.866025i
$176$		−	1.73205i		−	1.73205i
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		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			$\pi$
$194$	0.500000	−	0.866025i	0.500000	−	0.866025i
$195$		0			0
$196$	1.00000			1.00000
$197$	2.00000			2.00000			1.00000			$0$
$197$	2.00000			2.00000			1.00000			$0$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−1.00000			−1.00000
$201$		0			0
$202$	−2.00000			−2.00000
$203$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$204$		0			0
$205$		0			0
$206$	1.00000	+	1.73205i	1.00000	+	1.73205i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	0.500000	−	0.866025i	0.500000	−	0.866025i
$213$		0			0
$214$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$215$		0			0
$216$		0			0
$217$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$218$		0			0
$219$		0			0
$220$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$221$		0			0
$222$		0			0
$223$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$224$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$225$		0			0
$226$		0			0
$227$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$227$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$		−	1.73205i		−	1.73205i
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$	0.500000	−	0.866025i	0.500000	−	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$241$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$242$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$243$		0			0
$244$		0			0
$245$	0.500000	−	0.866025i	0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$	1.50000	−	0.866025i	1.50000	−	0.866025i
$249$		0			0
$250$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$251$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$251$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$	1.50000	+	0.866025i	1.50000	+	0.866025i
$255$		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
$262$	0.500000	+	0.866025i	0.500000	+	0.866025i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$271$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$276$		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
$280$	0.500000	+	0.866025i	0.500000	+	0.866025i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.500000	−	0.866025i	0.500000	−	0.866025i
$290$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$291$		0			0
$292$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$293$		−	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
$295$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−1.00000			−1.00000
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$			1.73205i			1.73205i
$309$		0			0
$310$		−	1.73205i		−	1.73205i
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$314$		0			0
$315$		0			0
$316$	1.00000			1.00000
$317$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$317$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$318$		0			0
$319$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$320$	−1.00000			−1.00000
$321$		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	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$	1.50000	−	2.59808i	1.50000	−	2.59808i
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$347$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	1.00000			1.00000
$351$		0			0
$352$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$353$		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
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.00000	+	1.73205i	1.00000	+	1.73205i
$366$		0			0
$367$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$372$		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	+	0.866025i	1.50000	+	0.866025i
$386$			1.73205i			1.73205i
$387$		0			0
$388$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	0.500000	−	0.866025i	0.500000	−	0.866025i
$393$		0			0
$394$	1.00000	−	1.73205i	1.00000	−	1.73205i
$395$	0.500000	−	0.866025i	0.500000	−	0.866025i
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$		0			0
$404$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$405$		0			0
$406$			1.73205i			1.73205i
$407$		0			0
$408$		0			0
$409$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$409$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$410$		0			0
$411$		0			0
$412$	2.00000			2.00000
$413$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$414$		0			0
$415$			1.73205i			1.73205i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.00000			2.00000			1.00000			$0$
$419$	2.00000			2.00000			1.00000			$0$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.00000			−1.00000
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	−2.00000			−2.00000			−1.00000			$\pi$
$433$	−2.00000			−2.00000			−1.00000			$\pi$
$434$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$439$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$440$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$441$		0			0
$442$		0			0
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$447$		0			0
$448$	0.500000	+	0.866025i	0.500000	+	0.866025i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$			1.73205i			1.73205i
$455$		0			0
$456$		0			0
$457$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$457$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.00000			2.00000			1.00000			$0$
$461$	2.00000			2.00000			1.00000			$0$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$473$		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
$482$			1.73205i			1.73205i
$483$		0			0
$484$	−2.00000			−2.00000
$485$	−1.00000			−1.00000
$486$		0			0
$487$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$487$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$488$		0			0
$489$		0			0
$490$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$491$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$491$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	1.73205i		−	1.73205i
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	0.500000	+	0.866025i	0.500000	+	0.866025i
$501$		0			0
$502$	0.500000	−	0.866025i	0.500000	−	0.866025i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	1.00000	+	1.73205i	1.00000	+	1.73205i
$506$		0			0
$507$		0			0
$508$	1.50000	−	0.866025i	1.50000	−	0.866025i
$509$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$509$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$510$		0			0
$511$	1.00000	−	1.73205i	1.00000	−	1.73205i
$512$	−1.00000			−1.00000
$513$		0			0
$514$		0			0
$515$	1.00000	−	1.73205i	1.00000	−	1.73205i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$	1.00000			1.00000
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$	−1.00000			−1.00000
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$536$		0			0
$537$		0			0
$538$	1.00000			1.00000
$539$		−	1.73205i		−	1.73205i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$	1.50000	−	0.866025i	1.50000	−	0.866025i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$			1.73205i			1.73205i
$551$		0			0
$552$		0			0
$553$	−1.00000			−1.00000
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$557$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	1.00000			1.00000
$561$		0			0
$562$		0			0
$563$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$578$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$579$		0			0
$580$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$581$	1.50000	−	0.866025i	1.50000	−	0.866025i
$582$		0			0
$583$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$584$	1.00000	+	1.73205i	1.00000	+	1.73205i
$585$		0			0
$586$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$587$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$588$		0			0
$589$		0			0
$590$	−1.00000			−1.00000
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$601$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$605$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$606$		0			0
$607$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$607$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$613$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$	1.50000	+	0.866025i	1.50000	+	0.866025i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$	0.500000	−	0.866025i	0.500000	−	0.866025i
$633$		0			0
$634$	0.500000	+	0.866025i	0.500000	+	0.866025i
$635$		−	1.73205i		−	1.73205i
$636$		0			0
$637$		0			0
$638$	−3.00000			−3.00000
$639$		0			0
$640$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$653$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$654$		0			0
$655$	0.500000	−	0.866025i	0.500000	−	0.866025i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$		0			0
$663$		0			0
$664$			1.73205i			1.73205i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$673$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$674$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$675$		0			0
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$677$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	0.500000	+	0.866025i	0.500000	+	0.866025i
$680$		0			0
$681$		0			0
$682$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$	2.00000			2.00000
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.500000	−	0.866025i	0.500000	−	0.866025i
$701$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$701$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$702$		0			0
$703$		0			0
$704$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$705$		0			0
$706$		0			0
$707$	1.00000	−	1.73205i	1.00000	−	1.73205i
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	−2.00000			−2.00000
$722$	1.00000			1.00000
$723$		0			0
$724$		0			0
$725$			1.73205i			1.73205i
$726$		0			0
$727$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$	2.00000			2.00000
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	−1.00000			−1.00000
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$	0.500000	+	0.866025i	0.500000	+	0.866025i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.00000			1.00000
$750$		0			0
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.500000	+	0.866025i	0.500000	+	0.866025i
$756$		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
$769$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$769$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$770$	1.50000	−	0.866025i	1.50000	−	0.866025i
$771$		0			0
$772$	1.50000	+	0.866025i	1.50000	+	0.866025i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$776$	−1.00000			−1.00000
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$789$		0			0
$790$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$797$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	0.500000	+	0.866025i	0.500000	+	0.866025i
$801$		0			0
$802$		0			0
$803$	3.00000	+	1.73205i	3.00000	+	1.73205i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.00000	+	1.73205i	1.00000	+	1.73205i
$809$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	1.50000	+	0.866025i	1.50000	+	0.866025i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	1.73205i		−	1.73205i
$819$		0			0
$820$		0			0
$821$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$	1.00000	−	1.73205i	1.00000	−	1.73205i
$825$		0			0
$826$	0.500000	+	0.866025i	0.500000	+	0.866025i
$827$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$827$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$	1.50000	+	0.866025i	1.50000	+	0.866025i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	1.00000	−	1.73205i	1.00000	−	1.73205i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−2.00000			−2.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.00000			−1.00000
$846$		0			0
$847$	2.00000			2.00000
$848$	−1.00000			−1.00000
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$857$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$867$		0			0
$868$			1.73205i			1.73205i
$869$		−	1.73205i		−	1.73205i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$	1.50000	−	0.866025i	1.50000	−	0.866025i
$879$		0			0
$880$			1.73205i			1.73205i
$881$		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
$886$	0.500000	+	0.866025i	0.500000	+	0.866025i
$887$		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	+	0.866025i	−1.50000	+	0.866025i
$890$		0			0
$891$		0			0
$892$	0.500000	+	0.866025i	0.500000	+	0.866025i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	1.00000			1.00000
$897$		0			0
$898$		0			0
$899$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$900$		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	$0.833333\pi$
$908$	1.50000	+	0.866025i	1.50000	+	0.866025i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	1.50000	+	2.59808i	1.50000	+	2.59808i
$914$	1.50000	−	0.866025i	1.50000	−	0.866025i
$915$		0			0
$916$		0			0
$917$	−1.00000			−1.00000
$918$		0			0
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$	1.00000	−	1.73205i	1.00000	−	1.73205i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$937$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	−1.00000			−1.00000
$945$		0			0
$946$		0			0
$947$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$947$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.00000	−	1.73205i	1.00000	−	1.73205i
$962$		0			0
$963$		0			0
$964$	1.50000	+	0.866025i	1.50000	+	0.866025i
$965$	1.50000	−	0.866025i	1.50000	−	0.866025i
$966$		0			0
$967$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$967$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$968$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$969$		0			0
$970$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$971$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$971$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		−	1.73205i		−	1.73205i
$975$		0			0
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$	−1.00000			−1.00000
$981$		0			0
$982$	1.50000	+	0.866025i	1.50000	+	0.866025i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	−2.00000			−2.00000
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$992$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.1.ek.c.1909.1	yes	2
3.2	odd	2		2520.1.ek.b.1909.1	yes	2
5.4	even	2		2520.1.ek.a.1909.1	yes	2
7.3	odd	6		2520.1.ek.d.829.1	yes	2
8.5	even	2		2520.1.ek.b.1909.1	yes	2
15.14	odd	2		2520.1.ek.d.1909.1	yes	2
21.17	even	6		2520.1.ek.a.829.1	✓	2
24.5	odd	2	CM	2520.1.ek.c.1909.1	yes	2
35.24	odd	6		2520.1.ek.b.829.1	yes	2
40.29	even	2		2520.1.ek.d.1909.1	yes	2
56.45	odd	6		2520.1.ek.a.829.1	✓	2
105.59	even	6	inner	2520.1.ek.c.829.1	yes	2
120.29	odd	2		2520.1.ek.a.1909.1	yes	2
168.101	even	6		2520.1.ek.d.829.1	yes	2
280.269	odd	6	inner	2520.1.ek.c.829.1	yes	2
840.269	even	6		2520.1.ek.b.829.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.1.ek.a.829.1	✓	2	21.17	even	6
2520.1.ek.a.829.1	✓	2	56.45	odd	6
2520.1.ek.a.1909.1	yes	2	5.4	even	2
2520.1.ek.a.1909.1	yes	2	120.29	odd	2
2520.1.ek.b.829.1	yes	2	35.24	odd	6
2520.1.ek.b.829.1	yes	2	840.269	even	6
2520.1.ek.b.1909.1	yes	2	3.2	odd	2
2520.1.ek.b.1909.1	yes	2	8.5	even	2
2520.1.ek.c.829.1	yes	2	105.59	even	6	inner
2520.1.ek.c.829.1	yes	2	280.269	odd	6	inner
2520.1.ek.c.1909.1	yes	2	1.1	even	1	trivial
2520.1.ek.c.1909.1	yes	2	24.5	odd	2	CM
2520.1.ek.d.829.1	yes	2	7.3	odd	6
2520.1.ek.d.829.1	yes	2	168.101	even	6
2520.1.ek.d.1909.1	yes	2	15.14	odd	2
2520.1.ek.d.1909.1	yes	2	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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2520.ek (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{10} +(-1.50000 + 0.866025i) q^{11} +1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{20} +1.73205i q^{22} +1.00000 q^{25} +(0.500000 - 0.866025i) q^{28} +1.73205i q^{29} +(-1.50000 + 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-0.500000 - 0.866025i) q^{35} +1.00000 q^{40} +(1.50000 + 0.866025i) q^{44} +(-0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{50} +(0.500000 + 0.866025i) q^{53} +(1.50000 - 0.866025i) q^{55} +(-0.500000 - 0.866025i) q^{56} +(1.50000 + 0.866025i) q^{58} +(0.500000 + 0.866025i) q^{59} +1.73205i q^{62} +1.00000 q^{64} -1.00000 q^{70} +(-1.00000 - 1.73205i) q^{73} +(-1.50000 - 0.866025i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(0.500000 - 0.866025i) q^{80} -1.73205i q^{83} +(1.50000 - 0.866025i) q^{88} +1.00000 q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^5 + q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} - 2 q^{5} + q^{7} - 2 q^{8} - q^{10} - 3 q^{11} + 2 q^{14} - q^{16} + q^{20} + 2 q^{25} + q^{28} - 3 q^{31} + q^{32} - q^{35} + 2 q^{40} + 3 q^{44} - q^{49} + q^{50} + q^{53} + 3 q^{55} - q^{56} + 3 q^{58} + q^{59} + 2 q^{64} - 2 q^{70} - 2 q^{73} - 3 q^{77} - q^{79} + q^{80} + 3 q^{88} + 2 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^5 + q^7 - 2 * q^8 - q^10 - 3 * q^11 + 2 * q^14 - q^16 + q^20 + 2 * q^25 + q^28 - 3 * q^31 + q^32 - q^35 + 2 * q^40 + 3 * q^44 - q^49 + q^50 + q^53 + 3 * q^55 - q^56 + 3 * q^58 + q^59 + 2 * q^64 - 2 * q^70 - 2 * q^73 - 3 * q^77 - q^79 + q^80 + 3 * q^88 + 2 * q^97 + q^98

Introduction

L-functions

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Database

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