LMFDB - Embedded newform 1352.1.g.a.339.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,1,Mod(339,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1352.339");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1352.g (of order $2$, degree $1$, 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:	$0.674735897080$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.104.1
Artin image:	$C_4\times S_3$
Artin field:	Galois closure of 12.0.43436029431808.1

Embedding invariants

Embedding label			339.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1352.339
Dual form			1352.1.g.a.339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{10} +1.00000 q^{12} -1.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000i q^{20} -1.00000i q^{21} +1.00000i q^{24} +1.00000 q^{27} -1.00000i q^{28} -1.00000 q^{30} +2.00000i q^{31} +1.00000i q^{32} +1.00000i q^{34} +1.00000 q^{35} +1.00000i q^{37} -1.00000 q^{40} +1.00000 q^{42} +1.00000 q^{43} +1.00000i q^{47} -1.00000 q^{48} -1.00000 q^{51} +1.00000i q^{54} +1.00000 q^{56} -1.00000i q^{60} -2.00000 q^{62} -1.00000 q^{64} -1.00000 q^{68} +1.00000i q^{70} -1.00000i q^{71} -1.00000 q^{74} -1.00000i q^{80} -1.00000 q^{81} +1.00000i q^{84} -1.00000i q^{85} +1.00000i q^{86} -2.00000i q^{93} -1.00000 q^{94} -1.00000i q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{10} + 2 q^{12} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{27} - 2 q^{30} + 2 q^{35} - 2 q^{40} + 2 q^{42} + 2 q^{43} - 2 q^{48} - 2 q^{51} + 2 q^{56} - 4 q^{62} - 2 q^{64} - 2 q^{68} - 2 q^{74} - 2 q^{81} - 2 q^{94}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^10 + 2 * q^12 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^27 - 2 * q^30 + 2 * q^35 - 2 * q^40 + 2 * q^42 + 2 * q^43 - 2 * q^48 - 2 * q^51 + 2 * q^56 - 4 * q^62 - 2 * q^64 - 2 * q^68 - 2 * q^74 - 2 * q^81 - 2 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1185$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	1.00000i	1.00000i
$2$	1.00000i	1.00000i
$3$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.00000	−1.00000
$5$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$5$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$	− 1.00000i	− 1.00000i
$7$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$	− 1.00000i	− 1.00000i
$9$	0	0
$10$	1.00000	1.00000
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	1.00000
$13$	0	0
$13$	0	0
$14$	−1.00000	−1.00000
$15$	1.00000i	1.00000i
$16$	1.00000	1.00000
$17$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$17$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000i	1.00000i
$21$	− 1.00000i	− 1.00000i
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000i	1.00000i
$25$	0	0
$26$	0	0
$27$	1.00000	1.00000
$28$	− 1.00000i	− 1.00000i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−1.00000	−1.00000
$31$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$31$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$32$	1.00000i	1.00000i
$33$	0	0
$34$	1.00000i	1.00000i
$35$	1.00000	1.00000
$36$	0	0
$37$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$37$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−1.00000
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	1.00000	1.00000
$43$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$43$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$47$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$48$	−1.00000	−1.00000
$49$	0	0
$50$	0	0
$51$	−1.00000	−1.00000
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	1.00000i	1.00000i
$55$	0	0
$56$	1.00000	1.00000
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	− 1.00000i	− 1.00000i
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−2.00000	−2.00000
$63$	0	0
$64$	−1.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−1.00000	−1.00000
$69$	0	0
$70$	1.00000i	1.00000i
$71$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$71$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−1.00000	−1.00000
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 1.00000i	− 1.00000i
$81$	−1.00000	−1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	1.00000i	1.00000i
$85$	− 1.00000i	− 1.00000i
$86$	1.00000i	1.00000i
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.00000i	− 2.00000i
$94$	−1.00000	−1.00000
$95$	0	0
$96$	− 1.00000i	− 1.00000i
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	− 1.00000i	− 1.00000i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	−1.00000	−1.00000
$106$	0	0
$107$	2.00000	2.00000	1.00000			$0$
$107$	2.00000	2.00000	1.00000			$0$
$108$	−1.00000	−1.00000
$109$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$109$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$110$	0	0
$111$	− 1.00000i	− 1.00000i
$112$	1.00000i	1.00000i
$113$	2.00000	2.00000	1.00000			$0$
$113$	2.00000	2.00000	1.00000			$0$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000i	1.00000i
$120$	1.00000	1.00000
$121$	−1.00000	−1.00000
$122$	0	0
$123$	0	0
$124$	− 2.00000i	− 2.00000i
$125$	− 1.00000i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1.00000i	− 1.00000i
$129$	−1.00000	−1.00000
$130$	0	0
$131$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1.00000i	− 1.00000i
$136$	− 1.00000i	− 1.00000i
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$140$	−1.00000	−1.00000
$141$	− 1.00000i	− 1.00000i
$142$	1.00000	1.00000
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 1.00000i	− 1.00000i
$149$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$149$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$150$	0	0
$151$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$151$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	2.00000
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	1.00000	1.00000
$161$	0	0
$162$	− 1.00000i	− 1.00000i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$167$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$168$	−1.00000	−1.00000
$169$	0	0
$170$	1.00000	1.00000
$171$	0	0
$172$	−1.00000	−1.00000
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	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	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.00000	1.00000
$186$	2.00000	2.00000
$187$	0	0
$188$	− 1.00000i	− 1.00000i
$189$	1.00000i	1.00000i
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.00000	1.00000
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$197$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	1.00000	1.00000
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	− 1.00000i	− 1.00000i
$211$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$212$	0	0
$213$	1.00000i	1.00000i
$214$	2.00000i	2.00000i
$215$	− 1.00000i	− 1.00000i
$216$	− 1.00000i	− 1.00000i
$217$	−2.00000	−2.00000
$218$	1.00000	1.00000
$219$	0	0
$220$	0	0
$221$	0	0
$222$	1.00000	1.00000
$223$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$223$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$224$	−1.00000	−1.00000
$225$	0	0
$226$	2.00000i	2.00000i
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$229$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$234$	0	0
$235$	1.00000	1.00000
$236$	0	0
$237$	0	0
$238$	−1.00000	−1.00000
$239$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$239$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$240$	1.00000i	1.00000i
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	− 1.00000i	− 1.00000i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	2.00000	2.00000
$249$	0	0
$250$	1.00000	1.00000
$251$	−2.00000	−2.00000	−1.00000			$\pi$
$251$	−2.00000	−2.00000	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.00000i	1.00000i
$256$	1.00000	1.00000
$257$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$257$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$258$	− 1.00000i	− 1.00000i
$259$	−1.00000	−1.00000
$260$	0	0
$261$	0	0
$262$	− 1.00000i	− 1.00000i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	1.00000	1.00000
$271$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$271$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$272$	1.00000	1.00000
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	− 1.00000i	− 1.00000i
$279$	0	0
$280$	− 1.00000i	− 1.00000i
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	1.00000	1.00000
$283$	−2.00000	−2.00000	−1.00000			$\pi$
$283$	−2.00000	−2.00000	−1.00000			$\pi$
$284$	1.00000i	1.00000i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0	0
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$293$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$294$	0	0
$295$	0	0
$296$	1.00000	1.00000
$297$	0	0
$298$	−2.00000	−2.00000
$299$	0	0
$300$	0	0
$301$	1.00000i	1.00000i
$302$	−1.00000	−1.00000
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	2.00000i	2.00000i
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$317$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	1.00000i	1.00000i
$321$	−2.00000	−2.00000
$322$	0	0
$323$	0	0
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	1.00000i	1.00000i
$328$	0	0
$329$	−1.00000	−1.00000
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	2.00000	2.00000
$335$	0	0
$336$	− 1.00000i	− 1.00000i
$337$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$337$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$338$	0	0
$339$	−2.00000	−2.00000
$340$	1.00000i	1.00000i
$341$	0	0
$342$	0	0
$343$	1.00000i	1.00000i
$344$	− 1.00000i	− 1.00000i
$345$	0	0
$346$	0	0
$347$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$347$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$348$	0	0
$349$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$349$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−1.00000	−1.00000
$356$	0	0
$357$	− 1.00000i	− 1.00000i
$358$	1.00000i	1.00000i
$359$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$359$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$360$	0	0
$361$	−1.00000	−1.00000
$362$	0	0
$363$	1.00000	1.00000
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	1.00000i	1.00000i
$371$	0	0
$372$	2.00000i	2.00000i
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	1.00000i	1.00000i
$376$	1.00000	1.00000
$377$	0	0
$378$	−1.00000	−1.00000
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$383$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$384$	1.00000i	1.00000i
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.00000	1.00000
$394$	1.00000	1.00000
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$397$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000i	1.00000i
$406$	0	0
$407$	0	0
$408$	1.00000i	1.00000i
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.00000	1.00000
$418$	0	0
$419$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$420$	1.00000	1.00000
$421$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$421$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$422$	− 1.00000i	− 1.00000i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	−1.00000	−1.00000
$427$	0	0
$428$	−2.00000	−2.00000
$429$	0	0
$430$	1.00000	1.00000
$431$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$431$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$432$	1.00000	1.00000
$433$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$433$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$434$	− 2.00000i	− 2.00000i
$435$	0	0
$436$	1.00000i	1.00000i
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$444$	1.00000i	1.00000i
$445$	0	0
$446$	1.00000	1.00000
$447$	− 2.00000i	− 2.00000i
$448$	− 1.00000i	− 1.00000i
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−2.00000	−2.00000
$453$	− 1.00000i	− 1.00000i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	−1.00000	−1.00000
$459$	1.00000	1.00000
$460$	0	0
$461$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$461$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$462$	0	0
$463$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$463$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$464$	0	0
$465$	−2.00000	−2.00000
$466$	1.00000i	1.00000i
$467$	−2.00000	−2.00000	−1.00000			$\pi$
$467$	−2.00000	−2.00000	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	1.00000i	1.00000i
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	− 1.00000i	− 1.00000i
$477$	0	0
$478$	1.00000	1.00000
$479$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$479$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$480$	−1.00000	−1.00000
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1.00000	1.00000
$485$	0	0
$486$	0	0
$487$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$487$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$491$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	2.00000i	2.00000i
$497$	1.00000	1.00000
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	1.00000i	1.00000i
$501$	2.00000i	2.00000i
$502$	− 2.00000i	− 2.00000i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$509$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$510$	−1.00000	−1.00000
$511$	0	0
$512$	1.00000i	1.00000i
$513$	0	0
$514$	1.00000i	1.00000i
$515$	0	0
$516$	1.00000	1.00000
$517$	0	0
$518$	− 1.00000i	− 1.00000i
$519$	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
$523$	2.00000	2.00000	1.00000			$0$
$523$	2.00000	2.00000	1.00000			$0$
$524$	1.00000	1.00000
$525$	0	0
$526$	0	0
$527$	2.00000i	2.00000i
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 2.00000i	− 2.00000i
$536$	0	0
$537$	−1.00000	−1.00000
$538$	0	0
$539$	0	0
$540$	1.00000i	1.00000i
$541$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$541$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$542$	−1.00000	−1.00000
$543$	0	0
$544$	1.00000i	1.00000i
$545$	−1.00000	−1.00000
$546$	0	0
$547$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$547$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.00000	−1.00000
$556$	1.00000	1.00000
$557$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$557$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$558$	0	0
$559$	0	0
$560$	1.00000	1.00000
$561$	0	0
$562$	0	0
$563$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$564$	1.00000i	1.00000i
$565$	− 2.00000i	− 2.00000i
$566$	− 2.00000i	− 2.00000i
$567$	− 1.00000i	− 1.00000i
$568$	−1.00000	−1.00000
$569$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$569$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$570$	0	0
$571$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−1.00000	−1.00000
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1.00000i	1.00000i
$592$	1.00000i	1.00000i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	1.00000	1.00000
$596$	− 2.00000i	− 2.00000i
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$602$	−1.00000	−1.00000
$603$	0	0
$604$	− 1.00000i	− 1.00000i
$605$	1.00000i	1.00000i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$613$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	−2.00000	−2.00000
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−1.00000
$626$	− 1.00000i	− 1.00000i
$627$	0	0
$628$	0	0
$629$	1.00000i	1.00000i
$630$	0	0
$631$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$631$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$632$	0	0
$633$	1.00000	1.00000
$634$	−2.00000	−2.00000
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−1.00000	−1.00000
$641$	−2.00000	−2.00000	−1.00000			$\pi$
$641$	−2.00000	−2.00000	−1.00000			$\pi$
$642$	− 2.00000i	− 2.00000i
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	1.00000i	1.00000i
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.00000i	1.00000i
$649$	0	0
$650$	0	0
$651$	2.00000	2.00000
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−1.00000	−1.00000
$655$	1.00000i	1.00000i
$656$	0	0
$657$	0	0
$658$	− 1.00000i	− 1.00000i
$659$	2.00000	2.00000	1.00000			$0$
$659$	2.00000	2.00000	1.00000			$0$
$660$	0	0
$661$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$661$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	2.00000i	2.00000i
$669$	1.00000i	1.00000i
$670$	0	0
$671$	0	0
$672$	1.00000	1.00000
$673$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$673$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$674$	1.00000i	1.00000i
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	− 2.00000i	− 2.00000i
$679$	0	0
$680$	−1.00000	−1.00000
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−1.00000
$687$	− 1.00000i	− 1.00000i
$688$	1.00000	1.00000
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	− 1.00000i	− 1.00000i
$695$	1.00000i	1.00000i
$696$	0	0
$697$	0	0
$698$	−1.00000	−1.00000
$699$	−1.00000	−1.00000
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−1.00000	−1.00000
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$709$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$710$	− 1.00000i	− 1.00000i
$711$	0	0
$712$	0	0
$713$	0	0
$714$	1.00000	1.00000
$715$	0	0
$716$	−1.00000	−1.00000
$717$	1.00000i	1.00000i
$718$	2.00000	2.00000
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	− 1.00000i	− 1.00000i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	1.00000i	1.00000i
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	1.00000	1.00000
$732$	0	0
$733$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$733$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−1.00000	−1.00000
$741$	0	0
$742$	0	0
$743$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$744$	−2.00000	−2.00000
$745$	2.00000	2.00000
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.00000i	2.00000i
$750$	−1.00000	−1.00000
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.00000i	1.00000i
$753$	2.00000	2.00000
$754$	0	0
$755$	1.00000	1.00000
$756$	− 1.00000i	− 1.00000i
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	1.00000	1.00000
$764$	0	0
$765$	0	0
$766$	1.00000	1.00000
$767$	0	0
$768$	−1.00000	−1.00000
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	−1.00000	−1.00000
$772$	0	0
$773$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$773$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.00000	1.00000
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	1.00000i	1.00000i
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	1.00000i	1.00000i
$789$	0	0
$790$	0	0
$791$	2.00000i	2.00000i
$792$	0	0
$793$	0	0
$794$	2.00000	2.00000
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	1.00000i	1.00000i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$809$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$810$	−1.00000	−1.00000
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	− 1.00000i	− 1.00000i
$814$	0	0
$815$	0	0
$816$	−1.00000	−1.00000
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$821$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	1.00000i	1.00000i
$835$	−2.00000	−2.00000
$836$	0	0
$837$	2.00000i	2.00000i
$838$	− 1.00000i	− 1.00000i
$839$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$839$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$840$	1.00000i	1.00000i
$841$	1.00000	1.00000
$842$	1.00000	1.00000
$843$	0	0
$844$	1.00000	1.00000
$845$	0	0
$846$	0	0
$847$	− 1.00000i	− 1.00000i
$848$	0	0
$849$	2.00000	2.00000
$850$	0	0
$851$	0	0
$852$	− 1.00000i	− 1.00000i
$853$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$853$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$854$	0	0
$855$	0	0
$856$	− 2.00000i	− 2.00000i
$857$	−2.00000	−2.00000	−1.00000			$\pi$
$857$	−2.00000	−2.00000	−1.00000			$\pi$
$858$	0	0
$859$	2.00000	2.00000	1.00000			$0$
$859$	2.00000	2.00000	1.00000			$0$
$860$	1.00000i	1.00000i
$861$	0	0
$862$	1.00000	1.00000
$863$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$863$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$864$	1.00000i	1.00000i
$865$	0	0
$866$	1.00000i	1.00000i
$867$	0	0
$868$	2.00000	2.00000
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1.00000	−1.00000
$873$	0	0
$874$	0	0
$875$	1.00000	1.00000
$876$	0	0
$877$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$877$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$878$	0	0
$879$	− 1.00000i	− 1.00000i
$880$	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
$883$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$883$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$884$	0	0
$885$	0	0
$886$	− 1.00000i	− 1.00000i
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−1.00000	−1.00000
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1.00000i	1.00000i
$893$	0	0
$894$	2.00000	2.00000
$895$	− 1.00000i	− 1.00000i
$896$	1.00000	1.00000
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 1.00000i	− 1.00000i
$904$	− 2.00000i	− 2.00000i
$905$	0	0
$906$	1.00000	1.00000
$907$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$907$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	− 1.00000i	− 1.00000i
$917$	− 1.00000i	− 1.00000i
$918$	1.00000i	1.00000i
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1.00000	1.00000
$923$	0	0
$924$	0	0
$925$	0	0
$926$	2.00000	2.00000
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	− 2.00000i	− 2.00000i
$931$	0	0
$932$	−1.00000	−1.00000
$933$	0	0
$934$	− 2.00000i	− 2.00000i
$935$	0	0
$936$	0	0
$937$	2.00000	2.00000	1.00000			$0$
$937$	2.00000	2.00000	1.00000			$0$
$938$	0	0
$939$	1.00000	1.00000
$940$	−1.00000	−1.00000
$941$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$941$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.00000	1.00000
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 2.00000i	− 2.00000i
$952$	1.00000	1.00000
$953$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$953$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$954$	0	0
$955$	0	0
$956$	1.00000i	1.00000i
$957$	0	0
$958$	−1.00000	−1.00000
$959$	0	0
$960$	− 1.00000i	− 1.00000i
$961$	−3.00000	−3.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$967$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$968$	1.00000i	1.00000i
$969$	0	0
$970$	0	0
$971$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$972$	0	0
$973$	− 1.00000i	− 1.00000i
$974$	−2.00000	−2.00000
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	1.00000i	1.00000i
$983$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$983$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$984$	0	0
$985$	−1.00000	−1.00000
$986$	0	0
$987$	1.00000	1.00000
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−2.00000	−2.00000
$993$	0	0
$994$	1.00000i	1.00000i
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	1.00000i	1.00000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.1.g.a.339.2		2
8.3	odd	2	inner	1352.1.g.a.339.1		2
13.2	odd	12		1352.1.p.a.147.1		2
13.3	even	3		1352.1.n.a.867.1		4
13.4	even	6		1352.1.n.a.315.1		4
13.5	odd	4		104.1.h.b.51.1	yes	1
13.6	odd	12		1352.1.p.a.699.1		2
13.7	odd	12		1352.1.p.b.699.1		2
13.8	odd	4		104.1.h.a.51.1	✓	1
13.9	even	3		1352.1.n.a.315.2		4
13.10	even	6		1352.1.n.a.867.2		4
13.11	odd	12		1352.1.p.b.147.1		2
13.12	even	2	inner	1352.1.g.a.339.1		2
39.5	even	4		936.1.o.a.883.1		1
39.8	even	4		936.1.o.b.883.1		1
52.31	even	4		416.1.h.a.207.1		1
52.47	even	4		416.1.h.b.207.1		1
65.8	even	4		2600.1.b.b.1299.2		2
65.18	even	4		2600.1.b.a.1299.1		2
65.34	odd	4		2600.1.o.d.51.1		1
65.44	odd	4		2600.1.o.b.51.1		1
65.47	even	4		2600.1.b.b.1299.1		2
65.57	even	4		2600.1.b.a.1299.2		2
104.3	odd	6		1352.1.n.a.867.2		4
104.5	odd	4		416.1.h.b.207.1		1
104.11	even	12		1352.1.p.a.147.1		2
104.19	even	12		1352.1.p.b.699.1		2
104.21	odd	4		416.1.h.a.207.1		1
104.35	odd	6		1352.1.n.a.315.1		4
104.43	odd	6		1352.1.n.a.315.2		4
104.51	odd	2	CM	1352.1.g.a.339.2		2
104.59	even	12		1352.1.p.a.699.1		2
104.67	even	12		1352.1.p.b.147.1		2
104.75	odd	6		1352.1.n.a.867.1		4
104.83	even	4		104.1.h.a.51.1	✓	1
104.99	even	4		104.1.h.b.51.1	yes	1
156.47	odd	4		3744.1.o.a.2287.1		1
156.83	odd	4		3744.1.o.b.2287.1		1
208.5	odd	4		3328.1.c.e.3327.2		2
208.21	odd	4		3328.1.c.a.3327.2		2
208.83	even	4		3328.1.c.a.3327.2		2
208.99	even	4		3328.1.c.e.3327.2		2
208.109	odd	4		3328.1.c.e.3327.1		2
208.125	odd	4		3328.1.c.a.3327.1		2
208.187	even	4		3328.1.c.a.3327.1		2
208.203	even	4		3328.1.c.e.3327.1		2
312.5	even	4		3744.1.o.a.2287.1		1
312.83	odd	4		936.1.o.b.883.1		1
312.125	even	4		3744.1.o.b.2287.1		1
312.203	odd	4		936.1.o.a.883.1		1
520.83	odd	4		2600.1.b.b.1299.2		2
520.99	even	4		2600.1.o.b.51.1		1
520.187	odd	4		2600.1.b.b.1299.1		2
520.203	odd	4		2600.1.b.a.1299.1		2
520.307	odd	4		2600.1.b.a.1299.2		2
520.499	even	4		2600.1.o.d.51.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.1.h.a.51.1	✓	1	13.8	odd	4
104.1.h.a.51.1	✓	1	104.83	even	4
104.1.h.b.51.1	yes	1	13.5	odd	4
104.1.h.b.51.1	yes	1	104.99	even	4
416.1.h.a.207.1		1	52.31	even	4
416.1.h.a.207.1		1	104.21	odd	4
416.1.h.b.207.1		1	52.47	even	4
416.1.h.b.207.1		1	104.5	odd	4
936.1.o.a.883.1		1	39.5	even	4
936.1.o.a.883.1		1	312.203	odd	4
936.1.o.b.883.1		1	39.8	even	4
936.1.o.b.883.1		1	312.83	odd	4
1352.1.g.a.339.1		2	8.3	odd	2	inner
1352.1.g.a.339.1		2	13.12	even	2	inner
1352.1.g.a.339.2		2	1.1	even	1	trivial
1352.1.g.a.339.2		2	104.51	odd	2	CM
1352.1.n.a.315.1		4	13.4	even	6
1352.1.n.a.315.1		4	104.35	odd	6
1352.1.n.a.315.2		4	13.9	even	3
1352.1.n.a.315.2		4	104.43	odd	6
1352.1.n.a.867.1		4	13.3	even	3
1352.1.n.a.867.1		4	104.75	odd	6
1352.1.n.a.867.2		4	13.10	even	6
1352.1.n.a.867.2		4	104.3	odd	6
1352.1.p.a.147.1		2	13.2	odd	12
1352.1.p.a.147.1		2	104.11	even	12
1352.1.p.a.699.1		2	13.6	odd	12
1352.1.p.a.699.1		2	104.59	even	12
1352.1.p.b.147.1		2	13.11	odd	12
1352.1.p.b.147.1		2	104.67	even	12
1352.1.p.b.699.1		2	13.7	odd	12
1352.1.p.b.699.1		2	104.19	even	12
2600.1.b.a.1299.1		2	65.18	even	4
2600.1.b.a.1299.1		2	520.203	odd	4
2600.1.b.a.1299.2		2	65.57	even	4
2600.1.b.a.1299.2		2	520.307	odd	4
2600.1.b.b.1299.1		2	65.47	even	4
2600.1.b.b.1299.1		2	520.187	odd	4
2600.1.b.b.1299.2		2	65.8	even	4
2600.1.b.b.1299.2		2	520.83	odd	4
2600.1.o.b.51.1		1	65.44	odd	4
2600.1.o.b.51.1		1	520.99	even	4
2600.1.o.d.51.1		1	65.34	odd	4
2600.1.o.d.51.1		1	520.499	even	4
3328.1.c.a.3327.1		2	208.125	odd	4
3328.1.c.a.3327.1		2	208.187	even	4
3328.1.c.a.3327.2		2	208.21	odd	4
3328.1.c.a.3327.2		2	208.83	even	4
3328.1.c.e.3327.1		2	208.109	odd	4
3328.1.c.e.3327.1		2	208.203	even	4
3328.1.c.e.3327.2		2	208.5	odd	4
3328.1.c.e.3327.2		2	208.99	even	4
3744.1.o.a.2287.1		1	156.47	odd	4
3744.1.o.a.2287.1		1	312.5	even	4
3744.1.o.b.2287.1		1	156.83	odd	4
3744.1.o.b.2287.1		1	312.125	even	4

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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1352.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{10} +1.00000 q^{12} -1.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000i q^{20} -1.00000i q^{21} +1.00000i q^{24} +1.00000 q^{27} -1.00000i q^{28} -1.00000 q^{30} +2.00000i q^{31} +1.00000i q^{32} +1.00000i q^{34} +1.00000 q^{35} +1.00000i q^{37} -1.00000 q^{40} +1.00000 q^{42} +1.00000 q^{43} +1.00000i q^{47} -1.00000 q^{48} -1.00000 q^{51} +1.00000i q^{54} +1.00000 q^{56} -1.00000i q^{60} -2.00000 q^{62} -1.00000 q^{64} -1.00000 q^{68} +1.00000i q^{70} -1.00000i q^{71} -1.00000 q^{74} -1.00000i q^{80} -1.00000 q^{81} +1.00000i q^{84} -1.00000i q^{85} +1.00000i q^{86} -2.00000i q^{93} -1.00000 q^{94} -1.00000i q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{10} + 2 q^{12} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{27} - 2 q^{30} + 2 q^{35} - 2 q^{40} + 2 q^{42} + 2 q^{43} - 2 q^{48} - 2 q^{51} + 2 q^{56} - 4 q^{62} - 2 q^{64} - 2 q^{68} - 2 q^{74} - 2 q^{81} - 2 q^{94}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^10 + 2 * q^12 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^27 - 2 * q^30 + 2 * q^35 - 2 * q^40 + 2 * q^42 + 2 * q^43 - 2 * q^48 - 2 * q^51 + 2 * q^56 - 4 * q^62 - 2 * q^64 - 2 * q^68 - 2 * q^74 - 2 * q^81 - 2 * q^94

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