LMFDB - Embedded newform 1368.1.bz.a.235.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,1,Mod(163,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 0, 4]))

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

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

chi := DirichletCharacter("1368.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1368.bz (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:	$0.682720937282$
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 152)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.2888.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			235.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1368.235
Dual form			1368.1.bz.a.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{8} +O(q^{10})$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{8} +1.00000 q^{11} +(-0.500000 - 0.866025i) q^{16} +(1.00000 + 1.73205i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(0.500000 + 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{25} +(0.500000 - 0.866025i) q^{32} +(-1.00000 + 1.73205i) q^{34} -1.00000 q^{38} +(-0.500000 - 0.866025i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(-0.500000 + 0.866025i) q^{44} +1.00000 q^{49} -1.00000 q^{50} +(-0.500000 - 0.866025i) q^{59} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{67} -2.00000 q^{68} +(0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{76} +(0.500000 - 0.866025i) q^{82} +1.00000 q^{83} +(1.00000 - 1.73205i) q^{86} -1.00000 q^{88} +(1.00000 - 1.73205i) q^{89} +(0.500000 + 0.866025i) q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^8	$ 2 q + q^{2} - q^{4} - 2 q^{8} + 2 q^{11} - q^{16} + 2 q^{17} - q^{19} + q^{22} - q^{25} + q^{32} - 2 q^{34} - 2 q^{38} - q^{41} - 2 q^{43} - q^{44} + 2 q^{49} - 2 q^{50} - q^{59} + 2 q^{64} + q^{67} - 4 q^{68} + q^{73} - q^{76} + q^{82} + 2 q^{83} + 2 q^{86} - 2 q^{88} + 2 q^{89} + q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^8 + 2 * q^11 - q^16 + 2 * q^17 - q^19 + q^22 - q^25 + q^32 - 2 * q^34 - 2 * q^38 - q^41 - 2 * q^43 - q^44 + 2 * q^49 - 2 * q^50 - q^59 + 2 * q^64 + q^67 - 4 * q^68 + q^73 - q^76 + q^82 + 2 * q^83 + 2 * q^86 - 2 * q^88 + 2 * q^89 + q^97 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.500000	+	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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−1.00000			−1.00000
$9$		0			0
$10$		0			0
$11$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$11$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$17$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$20$		0			0
$21$		0			0
$22$	0.500000	+	0.866025i	0.500000	+	0.866025i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−1.00000			−1.00000
$39$		0			0
$40$		0			0
$41$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$41$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$42$		0			0
$43$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$43$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$44$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	1.00000			1.00000
$50$	−1.00000			−1.00000
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$59$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$67$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$68$	−2.00000			−2.00000
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$73$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$		0			0
$76$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$		0			0
$82$	0.500000	−	0.866025i	0.500000	−	0.866025i
$83$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$83$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$	1.00000	−	1.73205i	1.00000	−	1.73205i
$87$		0			0
$88$	−1.00000			−1.00000
$89$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$89$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$97$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$98$	0.500000	+	0.866025i	0.500000	+	0.866025i
$99$		0			0
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−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
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$113$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.500000	−	0.866025i	0.500000	−	0.866025i
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−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
$134$	1.00000			1.00000
$135$		0			0
$136$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$137$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$137$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$138$		0			0
$139$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	0.500000	−	0.866025i	0.500000	−	0.866025i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$163$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$164$	1.00000			1.00000
$165$		0			0
$166$	0.500000	+	0.866025i	0.500000	+	0.866025i
$167$		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
$170$		0			0
$171$		0			0
$172$	2.00000			2.00000
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$177$		0			0
$178$	2.00000			2.00000
$179$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$179$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.00000	+	1.73205i	1.00000	+	1.73205i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−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	$-0.666667\pi$
$194$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$195$		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
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	0.500000	−	0.866025i	0.500000	−	0.866025i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$210$		0			0
$211$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−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
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$		0			0
$226$	0.500000	+	0.866025i	0.500000	+	0.866025i
$227$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$227$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$233$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$	1.00000			1.00000
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$251$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		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	$0.333333\pi$
$257$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$258$		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			0
$266$		0			0
$267$		0			0
$268$	0.500000	+	0.866025i	0.500000	+	0.866025i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$	1.00000	−	1.73205i	1.00000	−	1.73205i
$273$		0			0
$274$	−1.00000			−1.00000
$275$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	1.00000			1.00000
$279$		0			0
$280$		0			0
$281$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$281$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$282$		0			0
$283$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$283$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$290$		0			0
$291$		0			0
$292$	−1.00000			−1.00000
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	1.00000			1.00000
$305$		0			0
$306$		0			0
$307$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$307$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		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
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.00000			−2.00000
$324$		0			0
$325$		0			0
$326$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$327$		0			0
$328$	0.500000	+	0.866025i	0.500000	+	0.866025i
$329$		0			0
$330$		0			0
$331$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$331$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$332$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$337$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	1.00000	+	1.73205i	1.00000	+	1.73205i
$345$		0			0
$346$		0			0
$347$	−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		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	0.500000	−	0.866025i	0.500000	−	0.866025i
$353$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$353$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$	1.00000	+	1.73205i	1.00000	+	1.73205i
$357$		0			0
$358$	0.500000	+	0.866025i	0.500000	+	0.866025i
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	2.00000			2.00000			1.00000			$0$
$379$	2.00000			2.00000			1.00000			$0$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$	1.00000	−	1.73205i	1.00000	−	1.73205i
$387$		0			0
$388$	−1.00000			−1.00000
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−1.00000			−1.00000
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		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			0
$416$		0			0
$417$		0			0
$418$	−1.00000			−1.00000
$419$	−2.00000			−2.00000			−1.00000			$\pi$
$419$	−2.00000			−2.00000			−1.00000			$\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$	1.00000	−	1.73205i	1.00000	−	1.73205i
$423$		0			0
$424$		0			0
$425$	−2.00000			−2.00000
$426$		0			0
$427$		0			0
$428$	1.00000	−	1.73205i	1.00000	−	1.73205i
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$433$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		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			0
$447$		0			0
$448$		0			0
$449$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$452$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$453$		0			0
$454$	0.500000	+	0.866025i	0.500000	+	0.866025i
$455$		0			0
$456$		0			0
$457$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	0.500000	−	0.866025i	0.500000	−	0.866025i
$467$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$467$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.500000	+	0.866025i	0.500000	+	0.866025i
$473$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$474$		0			0
$475$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	1.00000			1.00000
$483$		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			0
$491$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$491$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$	−1.00000			−1.00000
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−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			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$523$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−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
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$537$		0			0
$538$		0			0
$539$	1.00000			1.00000
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$	2.00000			2.00000
$545$		0			0
$546$		0			0
$547$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$547$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$548$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$549$		0			0
$550$	−1.00000			−1.00000
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.500000	+	0.866025i	0.500000	+	0.866025i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.00000			−1.00000
$563$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$567$		0			0
$568$		0			0
$569$	−2.00000			−2.00000			−1.00000			$\pi$
$569$	−2.00000			−2.00000			−1.00000			$\pi$
$570$		0			0
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$577$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$578$	−3.00000			−3.00000
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$585$		0			0
$586$		0			0
$587$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$587$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$593$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		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
$601$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	0.500000	+	0.866025i	0.500000	+	0.866025i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$615$		0			0
$616$		0			0
$617$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$617$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$	2.00000			2.00000			1.00000			$0$
$619$	2.00000			2.00000			1.00000			$0$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$	1.00000			1.00000
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$641$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$642$		0			0
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		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
$649$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$650$		0			0
$651$		0			0
$652$	0.500000	−	0.866025i	0.500000	−	0.866025i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$657$		0			0
$658$		0			0
$659$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$659$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$663$		0			0
$664$	−1.00000			−1.00000
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	2.00000			2.00000			1.00000			$0$
$673$	2.00000			2.00000			1.00000			$0$
$674$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$675$		0			0
$676$	1.00000			1.00000
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$689$		0			0
$690$		0			0
$691$	2.00000			2.00000			1.00000			$0$
$691$	2.00000			2.00000			1.00000			$0$
$692$		0			0
$693$		0			0
$694$	0.500000	−	0.866025i	0.500000	−	0.866025i
$695$		0			0
$696$		0			0
$697$	1.00000	−	1.73205i	1.00000	−	1.73205i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		0			0
$704$	1.00000			1.00000
$705$		0			0
$706$	0.500000	+	0.866025i	0.500000	+	0.866025i
$707$		0			0
$708$		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
$712$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	0.500000	−	0.866025i	0.500000	−	0.866025i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.00000	−	3.46410i	2.00000	−	3.46410i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.500000	−	0.866025i	0.500000	−	0.866025i
$738$		0			0
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	−2.00000			−2.00000
$749$		0			0
$750$		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
$757$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$757$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$758$	1.00000	+	1.73205i	1.00000	+	1.73205i
$759$		0			0
$760$		0			0
$761$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$761$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$769$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$	2.00000			2.00000
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$777$		0			0
$778$		0			0
$779$	1.00000			1.00000
$780$		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
$787$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$787$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$788$		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			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	0.500000	+	0.866025i	0.500000	+	0.866025i
$801$		0			0
$802$	0.500000	−	0.866025i	0.500000	−	0.866025i
$803$	0.500000	+	0.866025i	0.500000	+	0.866025i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$811$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.00000			2.00000
$818$	1.00000			1.00000
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$827$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$828$		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
$836$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$837$		0			0
$838$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$839$		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
$844$	2.00000			2.00000
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$851$		0			0
$852$		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	0.500000	−	0.866025i	$-0.333333\pi$
$857$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$858$		0			0
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$	−2.00000			−2.00000
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$881$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$883$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$884$		0			0
$885$		0			0
$886$	−1.00000			−1.00000
$887$		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
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.500000	+	0.866025i	0.500000	+	0.866025i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	0.500000	−	0.866025i	0.500000	−	0.866025i
$903$		0			0
$904$	−1.00000			−1.00000
$905$		0			0
$906$		0			0
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$908$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	1.00000			1.00000
$914$	−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			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$929$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$930$		0			0
$931$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$932$	1.00000			1.00000
$933$		0			0
$934$	0.500000	+	0.866025i	0.500000	+	0.866025i
$935$		0			0
$936$		0			0
$937$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$937$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$945$		0			0
$946$	1.00000	−	1.73205i	1.00000	−	1.73205i
$947$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$947$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$	0.500000	−	0.866025i	0.500000	−	0.866025i
$951$		0			0
$952$		0			0
$953$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$953$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−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			1.00000
$962$		0			0
$963$		0			0
$964$	0.500000	+	0.866025i	0.500000	+	0.866025i
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−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
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$977$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	1.00000	−	1.73205i	1.00000	−	1.73205i
$980$		0			0
$981$		0			0
$982$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$983$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.1.bz.a.235.1		2
3.2	odd	2		152.1.k.a.83.1	yes	2
8.3	odd	2	CM	1368.1.bz.a.235.1		2
12.11	even	2		608.1.o.a.463.1		2
15.2	even	4		3800.1.bn.b.1299.2		4
15.8	even	4		3800.1.bn.b.1299.1		4
15.14	odd	2		3800.1.bd.c.1451.1		2
19.11	even	3	inner	1368.1.bz.a.163.1		2
24.5	odd	2		608.1.o.a.463.1		2
24.11	even	2		152.1.k.a.83.1	yes	2
57.2	even	18		2888.1.u.d.2555.1		6
57.5	odd	18		2888.1.u.c.99.1		6
57.8	even	6		2888.1.k.a.2595.1		2
57.11	odd	6		152.1.k.a.11.1	✓	2
57.14	even	18		2888.1.u.d.99.1		6
57.17	odd	18		2888.1.u.c.2555.1		6
57.23	odd	18		2888.1.u.c.2411.1		6
57.26	odd	6		2888.1.f.b.723.1		1
57.29	even	18		2888.1.u.d.1859.1		6
57.32	even	18		2888.1.u.d.1867.1		6
57.35	odd	18		2888.1.u.c.595.1		6
57.41	even	18		2888.1.u.d.595.1		6
57.44	odd	18		2888.1.u.c.1867.1		6
57.47	odd	18		2888.1.u.c.1859.1		6
57.50	even	6		2888.1.f.a.723.1		1
57.53	even	18		2888.1.u.d.2411.1		6
57.56	even	2		2888.1.k.a.2819.1		2
120.59	even	2		3800.1.bd.c.1451.1		2
120.83	odd	4		3800.1.bn.b.1299.1		4
120.107	odd	4		3800.1.bn.b.1299.2		4
152.11	odd	6	inner	1368.1.bz.a.163.1		2
228.11	even	6		608.1.o.a.239.1		2
285.68	even	12		3800.1.bn.b.2899.2		4
285.182	even	12		3800.1.bn.b.2899.1		4
285.239	odd	6		3800.1.bd.c.3051.1		2
456.11	even	6		152.1.k.a.11.1	✓	2
456.35	even	18		2888.1.u.c.595.1		6
456.59	odd	18		2888.1.u.d.2555.1		6
456.83	even	6		2888.1.f.b.723.1		1
456.107	odd	6		2888.1.f.a.723.1		1
456.125	odd	6		608.1.o.a.239.1		2
456.131	even	18		2888.1.u.c.2555.1		6
456.155	odd	18		2888.1.u.d.595.1		6
456.179	odd	6		2888.1.k.a.2595.1		2
456.203	odd	18		2888.1.u.d.1867.1		6
456.227	odd	2		2888.1.k.a.2819.1		2
456.251	even	18		2888.1.u.c.2411.1		6
456.275	even	18		2888.1.u.c.1859.1		6
456.299	odd	18		2888.1.u.d.99.1		6
456.347	even	18		2888.1.u.c.99.1		6
456.371	odd	18		2888.1.u.d.1859.1		6
456.395	odd	18		2888.1.u.d.2411.1		6
456.443	even	18		2888.1.u.c.1867.1		6
2280.467	odd	12		3800.1.bn.b.2899.1		4
2280.923	odd	12		3800.1.bn.b.2899.2		4
2280.1379	even	6		3800.1.bd.c.3051.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.1.k.a.11.1	✓	2	57.11	odd	6
152.1.k.a.11.1	✓	2	456.11	even	6
152.1.k.a.83.1	yes	2	3.2	odd	2
152.1.k.a.83.1	yes	2	24.11	even	2
608.1.o.a.239.1		2	228.11	even	6
608.1.o.a.239.1		2	456.125	odd	6
608.1.o.a.463.1		2	12.11	even	2
608.1.o.a.463.1		2	24.5	odd	2
1368.1.bz.a.163.1		2	19.11	even	3	inner
1368.1.bz.a.163.1		2	152.11	odd	6	inner
1368.1.bz.a.235.1		2	1.1	even	1	trivial
1368.1.bz.a.235.1		2	8.3	odd	2	CM
2888.1.f.a.723.1		1	57.50	even	6
2888.1.f.a.723.1		1	456.107	odd	6
2888.1.f.b.723.1		1	57.26	odd	6
2888.1.f.b.723.1		1	456.83	even	6
2888.1.k.a.2595.1		2	57.8	even	6
2888.1.k.a.2595.1		2	456.179	odd	6
2888.1.k.a.2819.1		2	57.56	even	2
2888.1.k.a.2819.1		2	456.227	odd	2
2888.1.u.c.99.1		6	57.5	odd	18
2888.1.u.c.99.1		6	456.347	even	18
2888.1.u.c.595.1		6	57.35	odd	18
2888.1.u.c.595.1		6	456.35	even	18
2888.1.u.c.1859.1		6	57.47	odd	18
2888.1.u.c.1859.1		6	456.275	even	18
2888.1.u.c.1867.1		6	57.44	odd	18
2888.1.u.c.1867.1		6	456.443	even	18
2888.1.u.c.2411.1		6	57.23	odd	18
2888.1.u.c.2411.1		6	456.251	even	18
2888.1.u.c.2555.1		6	57.17	odd	18
2888.1.u.c.2555.1		6	456.131	even	18
2888.1.u.d.99.1		6	57.14	even	18
2888.1.u.d.99.1		6	456.299	odd	18
2888.1.u.d.595.1		6	57.41	even	18
2888.1.u.d.595.1		6	456.155	odd	18
2888.1.u.d.1859.1		6	57.29	even	18
2888.1.u.d.1859.1		6	456.371	odd	18
2888.1.u.d.1867.1		6	57.32	even	18
2888.1.u.d.1867.1		6	456.203	odd	18
2888.1.u.d.2411.1		6	57.53	even	18
2888.1.u.d.2411.1		6	456.395	odd	18
2888.1.u.d.2555.1		6	57.2	even	18
2888.1.u.d.2555.1		6	456.59	odd	18
3800.1.bd.c.1451.1		2	15.14	odd	2
3800.1.bd.c.1451.1		2	120.59	even	2
3800.1.bd.c.3051.1		2	285.239	odd	6
3800.1.bd.c.3051.1		2	2280.1379	even	6
3800.1.bn.b.1299.1		4	15.8	even	4
3800.1.bn.b.1299.1		4	120.83	odd	4
3800.1.bn.b.1299.2		4	15.2	even	4
3800.1.bn.b.1299.2		4	120.107	odd	4
3800.1.bn.b.2899.1		4	285.182	even	12
3800.1.bn.b.2899.1		4	2280.467	odd	12
3800.1.bn.b.2899.2		4	285.68	even	12
3800.1.bn.b.2899.2		4	2280.923	odd	12

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1368.bz (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{8} +1.00000 q^{11} +(-0.500000 - 0.866025i) q^{16} +(1.00000 + 1.73205i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(0.500000 + 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{25} +(0.500000 - 0.866025i) q^{32} +(-1.00000 + 1.73205i) q^{34} -1.00000 q^{38} +(-0.500000 - 0.866025i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(-0.500000 + 0.866025i) q^{44} +1.00000 q^{49} -1.00000 q^{50} +(-0.500000 - 0.866025i) q^{59} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{67} -2.00000 q^{68} +(0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{76} +(0.500000 - 0.866025i) q^{82} +1.00000 q^{83} +(1.00000 - 1.73205i) q^{86} -1.00000 q^{88} +(1.00000 - 1.73205i) q^{89} +(0.500000 + 0.866025i) q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^8	\( 2 q + q^{2} - q^{4} - 2 q^{8} + 2 q^{11} - q^{16} + 2 q^{17} - q^{19} + q^{22} - q^{25} + q^{32} - 2 q^{34} - 2 q^{38} - q^{41} - 2 q^{43} - q^{44} + 2 q^{49} - 2 q^{50} - q^{59} + 2 q^{64} + q^{67} - 4 q^{68} + q^{73} - q^{76} + q^{82} + 2 q^{83} + 2 q^{86} - 2 q^{88} + 2 q^{89} + q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^8 + 2 * q^11 - q^16 + 2 * q^17 - q^19 + q^22 - q^25 + q^32 - 2 * q^34 - 2 * q^38 - q^41 - 2 * q^43 - q^44 + 2 * q^49 - 2 * q^50 - q^59 + 2 * q^64 + q^67 - 4 * q^68 + q^73 - q^76 + q^82 + 2 * q^83 + 2 * q^86 - 2 * q^88 + 2 * q^89 + q^97 + q^98

Introduction

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