LMFDB - Embedded newform 2268.1.s.d.1511.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,1,Mod(755,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("2268.755");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2268.s (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:	$1.13187944865$
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:	no (minimal twist has level 756)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.756.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			1511.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2268.1511
Dual form			2268.1.s.d.755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +1.00000 q^{10} +(-0.500000 - 0.866025i) q^{11} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +2.00000 q^{17} +1.00000 q^{19} +(0.500000 + 0.866025i) q^{20} +(0.500000 - 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{23} -1.00000 q^{28} +(-0.500000 + 0.866025i) q^{31} +(0.500000 - 0.866025i) q^{32} +(1.00000 + 1.73205i) q^{34} +1.00000 q^{35} -1.00000 q^{37} +(0.500000 + 0.866025i) q^{38} +(-0.500000 + 0.866025i) q^{40} +(0.500000 - 0.866025i) q^{41} +1.00000 q^{44} -1.00000 q^{46} +(-0.500000 + 0.866025i) q^{49} -1.00000 q^{55} +(-0.500000 - 0.866025i) q^{56} -1.00000 q^{62} +1.00000 q^{64} +(-1.00000 + 1.73205i) q^{68} +(0.500000 + 0.866025i) q^{70} +1.00000 q^{71} +(-0.500000 - 0.866025i) q^{74} +(-0.500000 + 0.866025i) q^{76} +(0.500000 - 0.866025i) q^{77} -1.00000 q^{80} +1.00000 q^{82} +(1.00000 - 1.73205i) q^{85} +(0.500000 + 0.866025i) q^{88} -1.00000 q^{89} +(-0.500000 - 0.866025i) q^{92} +(0.500000 - 0.866025i) q^{95} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + q^{5} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 + q^5 + q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} + q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - q^{14} - q^{16} + 4 q^{17} + 2 q^{19} + q^{20} + q^{22} - q^{23} - 2 q^{28} - q^{31} + q^{32} + 2 q^{34} + 2 q^{35} - 2 q^{37} + q^{38} - q^{40} + q^{41} + 2 q^{44} - 2 q^{46} - q^{49} - 2 q^{55} - q^{56} - 2 q^{62} + 2 q^{64} - 2 q^{68} + q^{70} + 2 q^{71} - q^{74} - q^{76} + q^{77} - 2 q^{80} + 2 q^{82} + 2 q^{85} + q^{88} - 2 q^{89} - q^{92} + q^{95} - 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 - q^14 - q^16 + 4 * q^17 + 2 * q^19 + q^20 + q^22 - q^23 - 2 * q^28 - q^31 + q^32 + 2 * q^34 + 2 * q^35 - 2 * q^37 + q^38 - q^40 + q^41 + 2 * q^44 - 2 * q^46 - q^49 - 2 * q^55 - q^56 - 2 * q^62 + 2 * q^64 - 2 * q^68 + q^70 + 2 * q^71 - q^74 - q^76 + q^77 - 2 * q^80 + 2 * q^82 + 2 * q^85 + q^88 - 2 * q^89 - q^92 + q^95 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	0.500000	+	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
$5$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$6$		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
$10$	1.00000			1.00000
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$	2.00000			2.00000			1.00000			$0$
$17$	2.00000			2.00000			1.00000			$0$
$18$		0			0
$19$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$20$	0.500000	+	0.866025i	0.500000	+	0.866025i
$21$		0			0
$22$	0.500000	−	0.866025i	0.500000	−	0.866025i
$23$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$23$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	−1.00000			−1.00000
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		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			$\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$	1.00000	+	1.73205i	1.00000	+	1.73205i
$35$	1.00000			1.00000
$36$		0			0
$37$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$38$	0.500000	+	0.866025i	0.500000	+	0.866025i
$39$		0			0
$40$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$41$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$41$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$	1.00000			1.00000
$45$		0			0
$46$	−1.00000			−1.00000
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	−1.00000			−1.00000
$56$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	−1.00000			−1.00000
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$69$		0			0
$70$	0.500000	+	0.866025i	0.500000	+	0.866025i
$71$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$75$		0			0
$76$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$77$	0.500000	−	0.866025i	0.500000	−	0.866025i
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	−1.00000			−1.00000
$81$		0			0
$82$	1.00000			1.00000
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$	1.00000	−	1.73205i	1.00000	−	1.73205i
$86$		0			0
$87$		0			0
$88$	0.500000	+	0.866025i	0.500000	+	0.866025i
$89$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$93$		0			0
$94$		0			0
$95$	0.500000	−	0.866025i	0.500000	−	0.866025i
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	−1.00000			−1.00000
$99$		0			0
$100$		0			0
$101$	−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
$103$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$103$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.00000			−2.00000			−1.00000			$\pi$
$107$	−2.00000			−2.00000			−1.00000			$\pi$
$108$		0			0
$109$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$110$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$111$		0			0
$112$	0.500000	−	0.866025i	0.500000	−	0.866025i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$	0.500000	+	0.866025i	0.500000	+	0.866025i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.00000	+	1.73205i	1.00000	+	1.73205i
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	0.500000	+	0.866025i	0.500000	+	0.866025i
$134$		0			0
$135$		0			0
$136$	−2.00000			−2.00000
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$140$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$141$		0			0
$142$	0.500000	+	0.866025i	0.500000	+	0.866025i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.500000	−	0.866025i	0.500000	−	0.866025i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−1.00000			−1.00000
$153$		0			0
$154$	1.00000			1.00000
$155$	0.500000	+	0.866025i	0.500000	+	0.866025i
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$161$	−1.00000			−1.00000
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	0.500000	+	0.866025i	0.500000	+	0.866025i
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$	2.00000			2.00000
$171$		0			0
$172$		0			0
$173$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$173$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$177$		0			0
$178$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$179$	−2.00000			−2.00000			−1.00000			$\pi$
$179$	−2.00000			−2.00000			−1.00000			$\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	0.500000	−	0.866025i	0.500000	−	0.866025i
$185$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$186$		0			0
$187$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$188$		0			0
$189$		0			0
$190$	1.00000			1.00000
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$192$		0			0
$193$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$193$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$		0			0
$202$	1.00000	−	1.73205i	1.00000	−	1.73205i
$203$		0			0
$204$		0			0
$205$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$206$	−1.00000			−1.00000
$207$		0			0
$208$		0			0
$209$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$		0			0
$214$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$215$		0			0
$216$		0			0
$217$	−1.00000			−1.00000
$218$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$219$		0			0
$220$	0.500000	−	0.866025i	0.500000	−	0.866025i
$221$		0			0
$222$		0			0
$223$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$223$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$224$	1.00000			1.00000
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$239$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$239$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.500000	+	0.866025i	0.500000	+	0.866025i
$246$		0			0
$247$		0			0
$248$	0.500000	−	0.866025i	0.500000	−	0.866025i
$249$		0			0
$250$	0.500000	+	0.866025i	0.500000	+	0.866025i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	1.00000			1.00000
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$258$		0			0
$259$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$263$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$267$		0			0
$268$		0			0
$269$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−2.00000			−2.00000			−1.00000			$\pi$
$271$	−2.00000			−2.00000			−1.00000			$\pi$
$272$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$	2.00000			2.00000
$279$		0			0
$280$	−1.00000			−1.00000
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$283$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$284$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$285$		0			0
$286$		0			0
$287$	1.00000			1.00000
$288$		0			0
$289$	3.00000			3.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$293$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$	1.00000			1.00000
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$305$		0			0
$306$		0			0
$307$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$307$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$308$	0.500000	+	0.866025i	0.500000	+	0.866025i
$309$		0			0
$310$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$	0.500000	−	0.866025i	0.500000	−	0.866025i
$321$		0			0
$322$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$323$	2.00000			2.00000
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$		0			0
$340$	1.00000	+	1.73205i	1.00000	+	1.73205i
$341$	1.00000			1.00000
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$		0			0
$346$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$347$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$347$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	−1.00000			−1.00000
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	0.500000	−	0.866025i	0.500000	−	0.866025i
$356$	0.500000	−	0.866025i	0.500000	−	0.866025i
$357$		0			0
$358$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$359$	−2.00000			−2.00000			−1.00000			$\pi$
$359$	−2.00000			−2.00000			−1.00000			$\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		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			$\pi$
$368$	1.00000			1.00000
$369$		0			0
$370$	−1.00000			−1.00000
$371$		0			0
$372$		0			0
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$374$	1.00000	−	1.73205i	1.00000	−	1.73205i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	0.500000	+	0.866025i	0.500000	+	0.866025i
$381$		0			0
$382$	0.500000	−	0.866025i	0.500000	−	0.866025i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$386$	−2.00000			−2.00000
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$392$	0.500000	−	0.866025i	0.500000	−	0.866025i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	0.500000	+	0.866025i	0.500000	+	0.866025i
$399$		0			0
$400$		0			0
$401$		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
$404$	2.00000			2.00000
$405$		0			0
$406$		0			0
$407$	0.500000	+	0.866025i	0.500000	+	0.866025i
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$	0.500000	−	0.866025i	0.500000	−	0.866025i
$411$		0			0
$412$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	0.500000	−	0.866025i	0.500000	−	0.866025i
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	1.00000	−	1.73205i	1.00000	−	1.73205i
$429$		0			0
$430$		0			0
$431$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$431$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$435$		0			0
$436$	0.500000	−	0.866025i	0.500000	−	0.866025i
$437$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$438$		0			0
$439$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$439$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$440$	1.00000			1.00000
$441$		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
$445$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$446$	0.500000	−	0.866025i	0.500000	−	0.866025i
$447$		0			0
$448$	0.500000	+	0.866025i	0.500000	+	0.866025i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$	−1.00000			−1.00000
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$457$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$	−1.00000			−1.00000
$461$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$461$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−2.00000			−2.00000
$477$		0			0
$478$	2.00000			2.00000
$479$		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
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$491$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$491$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.00000			1.00000
$497$	0.500000	+	0.866025i	0.500000	+	0.866025i
$498$		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			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−2.00000			−2.00000
$506$	0.500000	+	0.866025i	0.500000	+	0.866025i
$507$		0			0
$508$		0			0
$509$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$509$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	1.00000			1.00000
$515$	0.500000	+	0.866025i	0.500000	+	0.866025i
$516$		0			0
$517$		0			0
$518$	0.500000	−	0.866025i	0.500000	−	0.866025i
$519$		0			0
$520$		0			0
$521$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$523$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$	0.500000	−	0.866025i	0.500000	−	0.866025i
$527$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$	−1.00000			−1.00000
$533$		0			0
$534$		0			0
$535$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$536$		0			0
$537$		0			0
$538$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$539$	1.00000			1.00000
$540$		0			0
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$543$		0			0
$544$	1.00000	−	1.73205i	1.00000	−	1.73205i
$545$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$555$		0			0
$556$	1.00000	+	1.73205i	1.00000	+	1.73205i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$	2.00000			2.00000
$567$		0			0
$568$	−1.00000			−1.00000
$569$		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
$574$	0.500000	+	0.866025i	0.500000	+	0.866025i
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.50000	+	2.59808i	1.50000	+	2.59808i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.00000			−2.00000
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$590$		0			0
$591$		0			0
$592$	0.500000	+	0.866025i	0.500000	+	0.866025i
$593$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$593$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$	2.00000			2.00000
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$607$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$608$	0.500000	−	0.866025i	0.500000	−	0.866025i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$613$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$614$	0.500000	+	0.866025i	0.500000	+	0.866025i
$615$		0			0
$616$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$619$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$620$	−1.00000			−1.00000
$621$		0			0
$622$		0			0
$623$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$624$		0			0
$625$	0.500000	−	0.866025i	0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.00000			−2.00000
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.00000			1.00000
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$643$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$644$	0.500000	−	0.866025i	0.500000	−	0.866025i
$645$		0			0
$646$	1.00000	+	1.73205i	1.00000	+	1.73205i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$	−1.00000			−1.00000
$657$		0			0
$658$		0			0
$659$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$659$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$660$		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			0
$665$	1.00000			1.00000
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$674$	1.00000			1.00000
$675$		0			0
$676$	1.00000			1.00000
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$681$		0			0
$682$	0.500000	+	0.866025i	0.500000	+	0.866025i
$683$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$684$		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
$691$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$691$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$692$	−1.00000			−1.00000
$693$		0			0
$694$	−1.00000			−1.00000
$695$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$696$		0			0
$697$	1.00000	−	1.73205i	1.00000	−	1.73205i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−1.00000			−1.00000
$704$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$705$		0			0
$706$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$707$	1.00000	−	1.73205i	1.00000	−	1.73205i
$708$		0			0
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$710$	1.00000			1.00000
$711$		0			0
$712$	1.00000			1.00000
$713$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$714$		0			0
$715$		0			0
$716$	1.00000	−	1.73205i	1.00000	−	1.73205i
$717$		0			0
$718$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−1.00000			−1.00000
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$727$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	0.500000	−	0.866025i	0.500000	−	0.866025i
$735$		0			0
$736$	0.500000	+	0.866025i	0.500000	+	0.866025i
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$741$		0			0
$742$		0			0
$743$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$743$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	1.00000			1.00000
$747$		0			0
$748$	2.00000			2.00000
$749$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.00000			2.00000			1.00000			$0$
$757$	2.00000			2.00000			1.00000			$0$
$758$		0			0
$759$		0			0
$760$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$761$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$761$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$764$	1.00000			1.00000
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$	0.500000	−	0.866025i	0.500000	−	0.866025i
$771$		0			0
$772$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$773$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.500000	−	0.866025i	0.500000	−	0.866025i
$780$		0			0
$781$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$782$	−2.00000			−2.00000
$783$		0			0
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$787$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$797$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$797$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$806$		0			0
$807$		0			0
$808$	1.00000	+	1.73205i	1.00000	+	1.73205i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$811$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$812$		0			0
$813$		0			0
$814$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	1.00000			1.00000
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$	0.500000	−	0.866025i	0.500000	−	0.866025i
$825$		0			0
$826$		0			0
$827$	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		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$834$		0			0
$835$		0			0
$836$	1.00000			1.00000
$837$		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
$842$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$843$		0			0
$844$		0			0
$845$	−1.00000			−1.00000
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.500000	−	0.866025i	0.500000	−	0.866025i
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	2.00000			2.00000
$857$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$857$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$859$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	0.500000	+	0.866025i	0.500000	+	0.866025i
$863$	−2.00000			−2.00000			−1.00000			$\pi$
$863$	−2.00000			−2.00000			−1.00000			$\pi$
$864$		0			0
$865$	1.00000			1.00000
$866$		0			0
$867$		0			0
$868$	0.500000	−	0.866025i	0.500000	−	0.866025i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.00000			1.00000
$873$		0			0
$874$	−1.00000			−1.00000
$875$	0.500000	+	0.866025i	0.500000	+	0.866025i
$876$		0			0
$877$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$877$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$878$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$879$		0			0
$880$	0.500000	+	0.866025i	0.500000	+	0.866025i
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	0.500000	−	0.866025i	0.500000	−	0.866025i
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	−1.00000			−1.00000
$891$		0			0
$892$	1.00000			1.00000
$893$		0			0
$894$		0			0
$895$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$896$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$911$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$		0			0
$914$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$921$		0			0
$922$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$930$		0			0
$931$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.00000			−2.00000
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$941$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$942$		0			0
$943$	0.500000	+	0.866025i	0.500000	+	0.866025i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$947$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−1.00000			−1.00000
$956$	1.00000	+	1.73205i	1.00000	+	1.73205i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.00000	+	1.73205i	1.00000	+	1.73205i
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	2.00000			2.00000
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	0.500000	+	0.866025i	0.500000	+	0.866025i
$980$	−1.00000			−1.00000
$981$		0			0
$982$	−1.00000			−1.00000
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.500000	+	0.866025i	0.500000	+	0.866025i
$993$		0			0
$994$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$995$	0.500000	−	0.866025i	0.500000	−	0.866025i
$996$		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	2268.1.s.d.1511.1		2
3.2	odd	2		2268.1.s.a.1511.1		2
4.3	odd	2		2268.1.s.b.1511.1		2
7.6	odd	2		2268.1.s.c.1511.1		2
9.2	odd	6		2268.1.s.a.755.1		2
9.4	even	3		756.1.h.a.755.1	✓	1
9.5	odd	6		756.1.h.d.755.1	yes	1
9.7	even	3	inner	2268.1.s.d.755.1		2
12.11	even	2		2268.1.s.c.1511.1		2
21.20	even	2		2268.1.s.b.1511.1		2
28.27	even	2		2268.1.s.a.1511.1		2
36.7	odd	6		2268.1.s.b.755.1		2
36.11	even	6		2268.1.s.c.755.1		2
36.23	even	6		756.1.h.b.755.1	yes	1
36.31	odd	6		756.1.h.c.755.1	yes	1
63.13	odd	6		756.1.h.b.755.1	yes	1
63.20	even	6		2268.1.s.b.755.1		2
63.34	odd	6		2268.1.s.c.755.1		2
63.41	even	6		756.1.h.c.755.1	yes	1
84.83	odd	2	CM	2268.1.s.d.1511.1		2
252.83	odd	6	inner	2268.1.s.d.755.1		2
252.139	even	6		756.1.h.d.755.1	yes	1
252.167	odd	6		756.1.h.a.755.1	✓	1
252.223	even	6		2268.1.s.a.755.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.1.h.a.755.1	✓	1	9.4	even	3
756.1.h.a.755.1	✓	1	252.167	odd	6
756.1.h.b.755.1	yes	1	36.23	even	6
756.1.h.b.755.1	yes	1	63.13	odd	6
756.1.h.c.755.1	yes	1	36.31	odd	6
756.1.h.c.755.1	yes	1	63.41	even	6
756.1.h.d.755.1	yes	1	9.5	odd	6
756.1.h.d.755.1	yes	1	252.139	even	6
2268.1.s.a.755.1		2	9.2	odd	6
2268.1.s.a.755.1		2	252.223	even	6
2268.1.s.a.1511.1		2	3.2	odd	2
2268.1.s.a.1511.1		2	28.27	even	2
2268.1.s.b.755.1		2	36.7	odd	6
2268.1.s.b.755.1		2	63.20	even	6
2268.1.s.b.1511.1		2	4.3	odd	2
2268.1.s.b.1511.1		2	21.20	even	2
2268.1.s.c.755.1		2	36.11	even	6
2268.1.s.c.755.1		2	63.34	odd	6
2268.1.s.c.1511.1		2	7.6	odd	2
2268.1.s.c.1511.1		2	12.11	even	2
2268.1.s.d.755.1		2	9.7	even	3	inner
2268.1.s.d.755.1		2	252.83	odd	6	inner
2268.1.s.d.1511.1		2	1.1	even	1	trivial
2268.1.s.d.1511.1		2	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2268.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +1.00000 q^{10} +(-0.500000 - 0.866025i) q^{11} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +2.00000 q^{17} +1.00000 q^{19} +(0.500000 + 0.866025i) q^{20} +(0.500000 - 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{23} -1.00000 q^{28} +(-0.500000 + 0.866025i) q^{31} +(0.500000 - 0.866025i) q^{32} +(1.00000 + 1.73205i) q^{34} +1.00000 q^{35} -1.00000 q^{37} +(0.500000 + 0.866025i) q^{38} +(-0.500000 + 0.866025i) q^{40} +(0.500000 - 0.866025i) q^{41} +1.00000 q^{44} -1.00000 q^{46} +(-0.500000 + 0.866025i) q^{49} -1.00000 q^{55} +(-0.500000 - 0.866025i) q^{56} -1.00000 q^{62} +1.00000 q^{64} +(-1.00000 + 1.73205i) q^{68} +(0.500000 + 0.866025i) q^{70} +1.00000 q^{71} +(-0.500000 - 0.866025i) q^{74} +(-0.500000 + 0.866025i) q^{76} +(0.500000 - 0.866025i) q^{77} -1.00000 q^{80} +1.00000 q^{82} +(1.00000 - 1.73205i) q^{85} +(0.500000 + 0.866025i) q^{88} -1.00000 q^{89} +(-0.500000 - 0.866025i) q^{92} +(0.500000 - 0.866025i) q^{95} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 + q^5 + q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} + q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - q^{14} - q^{16} + 4 q^{17} + 2 q^{19} + q^{20} + q^{22} - q^{23} - 2 q^{28} - q^{31} + q^{32} + 2 q^{34} + 2 q^{35} - 2 q^{37} + q^{38} - q^{40} + q^{41} + 2 q^{44} - 2 q^{46} - q^{49} - 2 q^{55} - q^{56} - 2 q^{62} + 2 q^{64} - 2 q^{68} + q^{70} + 2 q^{71} - q^{74} - q^{76} + q^{77} - 2 q^{80} + 2 q^{82} + 2 q^{85} + q^{88} - 2 q^{89} - q^{92} + q^{95} - 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 - q^14 - q^16 + 4 * q^17 + 2 * q^19 + q^20 + q^22 - q^23 - 2 * q^28 - q^31 + q^32 + 2 * q^34 + 2 * q^35 - 2 * q^37 + q^38 - q^40 + q^41 + 2 * q^44 - 2 * q^46 - q^49 - 2 * q^55 - q^56 - 2 * q^62 + 2 * q^64 - 2 * q^68 + q^70 + 2 * q^71 - q^74 - q^76 + q^77 - 2 * q^80 + 2 * q^82 + 2 * q^85 + q^88 - 2 * q^89 - q^92 + q^95 - 2 * q^98

Introduction

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