LMFDB - Embedded newform 1316.1.h.h.1315.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1316,1,Mod(1315,1316)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1316, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1316.1315");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1316 = 2^{2} \cdot 7 \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1316.h (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:	yes
Analytic conductor:	$0.656769556625$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.81397232.1

Embedding invariants

Embedding label			1315.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1316.1315

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.73205 q^{10} -1.73205 q^{11} +1.00000 q^{12} -1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} +1.73205 q^{20} +1.00000 q^{21} +1.73205 q^{22} -1.00000 q^{24} +2.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -1.73205 q^{30} -1.00000 q^{32} -1.73205 q^{33} +1.73205 q^{35} -1.00000 q^{37} -1.73205 q^{40} -1.73205 q^{41} -1.00000 q^{42} -1.73205 q^{43} -1.73205 q^{44} +1.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.00000 q^{50} +1.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -1.00000 q^{56} -1.00000 q^{59} +1.73205 q^{60} +1.00000 q^{64} +1.73205 q^{66} -1.73205 q^{70} +1.73205 q^{73} +1.00000 q^{74} +2.00000 q^{75} -1.73205 q^{77} +1.73205 q^{80} -1.00000 q^{81} +1.73205 q^{82} -1.00000 q^{83} +1.00000 q^{84} +1.73205 q^{86} +1.73205 q^{88} -1.00000 q^{94} -1.00000 q^{96} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{12} - 2 q^{14} + 2 q^{16} + 2 q^{21} - 2 q^{24} + 4 q^{25} - 2 q^{27} + 2 q^{28} - 2 q^{32} - 2 q^{37} - 2 q^{42} + 2 q^{47} + 2 q^{48} + 2 q^{49} - 4 q^{50} + 2 q^{53} + 2 q^{54} - 6 q^{55} - 2 q^{56} - 2 q^{59} + 2 q^{64} + 2 q^{74} + 4 q^{75} - 2 q^{81} - 2 q^{83} + 2 q^{84} - 2 q^{94} - 2 q^{96} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^12 - 2 * q^14 + 2 * q^16 + 2 * q^21 - 2 * q^24 + 4 * q^25 - 2 * q^27 + 2 * q^28 - 2 * q^32 - 2 * q^37 - 2 * q^42 + 2 * q^47 + 2 * q^48 + 2 * q^49 - 4 * q^50 + 2 * q^53 + 2 * q^54 - 6 * q^55 - 2 * q^56 - 2 * q^59 + 2 * q^64 + 2 * q^74 + 4 * q^75 - 2 * q^81 - 2 * q^83 + 2 * q^84 - 2 * q^94 - 2 * q^96 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1316\mathbb{Z}\right)^\times$.

$n$	$659$	$757$	$941$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.00000	−1.00000
$2$	−1.00000	−1.00000
$3$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	1.00000	1.00000
$5$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$5$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$6$	−1.00000	−1.00000
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	−1.00000	−1.00000
$9$	0	0
$10$	−1.73205	−1.73205
$11$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$11$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$12$	1.00000	1.00000
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−1.00000	−1.00000
$15$	1.73205	1.73205
$16$	1.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.73205	1.73205
$21$	1.00000	1.00000
$22$	1.73205	1.73205
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−1.00000
$25$	2.00000	2.00000
$26$	0	0
$27$	−1.00000	−1.00000
$28$	1.00000	1.00000
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−1.73205	−1.73205
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.00000	−1.00000
$33$	−1.73205	−1.73205
$34$	0	0
$35$	1.73205	1.73205
$36$	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	0
$39$	0	0
$40$	−1.73205	−1.73205
$41$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$41$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$42$	−1.00000	−1.00000
$43$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$43$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$44$	−1.73205	−1.73205
$45$	0	0
$46$	0	0
$47$	1.00000	1.00000
$47$	1.00000	1.00000
$48$	1.00000	1.00000
$49$	1.00000	1.00000
$50$	−2.00000	−2.00000
$51$	0	0
$52$	0	0
$53$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$53$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$54$	1.00000	1.00000
$55$	−3.00000	−3.00000
$56$	−1.00000	−1.00000
$57$	0	0
$58$	0	0
$59$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$60$	1.73205	1.73205
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	1.00000
$65$	0	0
$66$	1.73205	1.73205
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	−1.73205	−1.73205
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$73$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$74$	1.00000	1.00000
$75$	2.00000	2.00000
$76$	0	0
$77$	−1.73205	−1.73205
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.73205	1.73205
$81$	−1.00000	−1.00000
$82$	1.73205	1.73205
$83$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$84$	1.00000	1.00000
$85$	0	0
$86$	1.73205	1.73205
$87$	0	0
$88$	1.73205	1.73205
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−1.00000	−1.00000
$95$	0	0
$96$	−1.00000	−1.00000
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.00000	−1.00000
$99$	0	0
$100$	2.00000	2.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$103$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$104$	0	0
$105$	1.73205	1.73205
$106$	−1.00000	−1.00000
$107$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$107$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$108$	−1.00000	−1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	3.00000	3.00000
$111$	−1.00000	−1.00000
$112$	1.00000	1.00000
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	1.00000	1.00000
$119$	0	0
$120$	−1.73205	−1.73205
$121$	2.00000	2.00000
$122$	0	0
$123$	−1.73205	−1.73205
$124$	0	0
$125$	1.73205	1.73205
$126$	0	0
$127$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$127$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$128$	−1.00000	−1.00000
$129$	−1.73205	−1.73205
$130$	0	0
$131$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$132$	−1.73205	−1.73205
$133$	0	0
$134$	0	0
$135$	−1.73205	−1.73205
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	1.73205	1.73205
$141$	1.00000	1.00000
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−1.73205	−1.73205
$147$	1.00000	1.00000
$148$	−1.00000	−1.00000
$149$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$150$	−2.00000	−2.00000
$151$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$151$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$152$	0	0
$153$	0	0
$154$	1.73205	1.73205
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	1.00000	1.00000
$160$	−1.73205	−1.73205
$161$	0	0
$162$	1.00000	1.00000
$163$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$163$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$164$	−1.73205	−1.73205
$165$	−3.00000	−3.00000
$166$	1.00000	1.00000
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−1.00000	−1.00000
$169$	−1.00000	−1.00000
$170$	0	0
$171$	0	0
$172$	−1.73205	−1.73205
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	2.00000	2.00000
$176$	−1.73205	−1.73205
$177$	−1.00000	−1.00000
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$181$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.73205	−1.73205
$186$	0	0
$187$	0	0
$188$	1.00000	1.00000
$189$	−1.00000	−1.00000
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.00000	1.00000
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	1.00000	1.00000
$197$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−2.00000	−2.00000
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.00000	−3.00000
$206$	1.00000	1.00000
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−1.73205	−1.73205
$211$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$211$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$212$	1.00000	1.00000
$213$	0	0
$214$	−1.73205	−1.73205
$215$	−3.00000	−3.00000
$216$	1.00000	1.00000
$217$	0	0
$218$	0	0
$219$	1.73205	1.73205
$220$	−3.00000	−3.00000
$221$	0	0
$222$	1.00000	1.00000
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−1.00000	−1.00000
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−1.73205	−1.73205
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	1.73205	1.73205
$236$	−1.00000	−1.00000
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.73205	1.73205
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−2.00000	−2.00000
$243$	0	0
$244$	0	0
$245$	1.73205	1.73205
$246$	1.73205	1.73205
$247$	0	0
$248$	0	0
$249$	−1.00000	−1.00000
$250$	−1.73205	−1.73205
$251$	2.00000	2.00000	1.00000			$0$
$251$	2.00000	2.00000	1.00000			$0$
$252$	0	0
$253$	0	0
$254$	1.73205	1.73205
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	1.73205	1.73205
$259$	−1.00000	−1.00000
$260$	0	0
$261$	0	0
$262$	−1.00000	−1.00000
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1.73205	1.73205
$265$	1.73205	1.73205
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	1.73205	1.73205
$271$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$271$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.46410	−3.46410
$276$	0	0
$277$	−2.00000	−2.00000	−1.00000			$\pi$
$277$	−2.00000	−2.00000	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	−1.73205	−1.73205
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−1.00000	−1.00000
$283$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$283$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.73205	−1.73205
$288$	0	0
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	1.73205	1.73205
$293$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$293$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$294$	−1.00000	−1.00000
$295$	−1.73205	−1.73205
$296$	1.00000	1.00000
$297$	1.73205	1.73205
$298$	1.00000	1.00000
$299$	0	0
$300$	2.00000	2.00000
$301$	−1.73205	−1.73205
$302$	−1.73205	−1.73205
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.00000	−2.00000	−1.00000			$\pi$
$307$	−2.00000	−2.00000	−1.00000			$\pi$
$308$	−1.73205	−1.73205
$309$	−1.00000	−1.00000
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$313$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−1.00000	−1.00000
$319$	0	0
$320$	1.73205	1.73205
$321$	1.73205	1.73205
$322$	0	0
$323$	0	0
$324$	−1.00000	−1.00000
$325$	0	0
$326$	−1.73205	−1.73205
$327$	0	0
$328$	1.73205	1.73205
$329$	1.00000	1.00000
$330$	3.00000	3.00000
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−1.00000	−1.00000
$333$	0	0
$334$	0	0
$335$	0	0
$336$	1.00000	1.00000
$337$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$338$	1.00000	1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	1.00000
$344$	1.73205	1.73205
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$349$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$350$	−2.00000	−2.00000
$351$	0	0
$352$	1.73205	1.73205
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	1.00000	1.00000
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$359$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	−1.73205	−1.73205
$363$	2.00000	2.00000
$364$	0	0
$365$	3.00000	3.00000
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	1.73205	1.73205
$371$	1.00000	1.00000
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	1.73205	1.73205
$376$	−1.00000	−1.00000
$377$	0	0
$378$	1.00000	1.00000
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−1.73205	−1.73205
$382$	0	0
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$384$	−1.00000	−1.00000
$385$	−3.00000	−3.00000
$386$	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
$392$	−1.00000	−1.00000
$393$	1.00000	1.00000
$394$	1.00000	1.00000
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	2.00000	2.00000
$401$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$401$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.73205	−1.73205
$406$	0	0
$407$	1.73205	1.73205
$408$	0	0
$409$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$409$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$410$	3.00000	3.00000
$411$	0	0
$412$	−1.00000	−1.00000
$413$	−1.00000	−1.00000
$414$	0	0
$415$	−1.73205	−1.73205
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	1.73205	1.73205
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−1.73205	−1.73205
$423$	0	0
$424$	−1.00000	−1.00000
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.73205	1.73205
$429$	0	0
$430$	3.00000	3.00000
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.00000	−1.00000
$433$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$433$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−1.73205	−1.73205
$439$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$439$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$440$	3.00000	3.00000
$441$	0	0
$442$	0	0
$443$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$443$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$444$	−1.00000	−1.00000
$445$	0	0
$446$	0	0
$447$	−1.00000	−1.00000
$448$	1.00000	1.00000
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	3.00000	3.00000
$452$	0	0
$453$	1.73205	1.73205
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$457$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	1.73205	1.73205
$463$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$463$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$464$	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
$470$	−1.73205	−1.73205
$471$	0	0
$472$	1.00000	1.00000
$473$	3.00000	3.00000
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$479$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$480$	−1.73205	−1.73205
$481$	0	0
$482$	0	0
$483$	0	0
$484$	2.00000	2.00000
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.73205	1.73205
$490$	−1.73205	−1.73205
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−1.73205	−1.73205
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	1.00000	1.00000
$499$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$499$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$500$	1.73205	1.73205
$501$	0	0
$502$	−2.00000	−2.00000
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−1.00000
$508$	−1.73205	−1.73205
$509$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$509$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$510$	0	0
$511$	1.73205	1.73205
$512$	−1.00000	−1.00000
$513$	0	0
$514$	0	0
$515$	−1.73205	−1.73205
$516$	−1.73205	−1.73205
$517$	−1.73205	−1.73205
$518$	1.00000	1.00000
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$524$	1.00000	1.00000
$525$	2.00000	2.00000
$526$	0	0
$527$	0	0
$528$	−1.73205	−1.73205
$529$	−1.00000	−1.00000
$530$	−1.73205	−1.73205
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3.00000	3.00000
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.73205	−1.73205
$540$	−1.73205	−1.73205
$541$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$541$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$542$	−1.00000	−1.00000
$543$	1.73205	1.73205
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	3.46410	3.46410
$551$	0	0
$552$	0	0
$553$	0	0
$554$	2.00000	2.00000
$555$	−1.73205	−1.73205
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	1.73205	1.73205
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1.00000	1.00000
$565$	0	0
$566$	1.00000	1.00000
$567$	−1.00000	−1.00000
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	1.73205	1.73205
$575$	0	0
$576$	0	0
$577$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$577$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$578$	−1.00000	−1.00000
$579$	0	0
$580$	0	0
$581$	−1.00000	−1.00000
$582$	0	0
$583$	−1.73205	−1.73205
$584$	−1.73205	−1.73205
$585$	0	0
$586$	1.73205	1.73205
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	1.00000	1.00000
$589$	0	0
$590$	1.73205	1.73205
$591$	−1.00000	−1.00000
$592$	−1.00000	−1.00000
$593$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$593$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$594$	−1.73205	−1.73205
$595$	0	0
$596$	−1.00000	−1.00000
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−2.00000	−2.00000
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	1.73205	1.73205
$603$	0	0
$604$	1.73205	1.73205
$605$	3.46410	3.46410
$606$	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
$613$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$613$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$614$	2.00000	2.00000
$615$	−3.00000	−3.00000
$616$	1.73205	1.73205
$617$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$618$	1.00000	1.00000
$619$	−2.00000	−2.00000	−1.00000			$\pi$
$619$	−2.00000	−2.00000	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	1.73205	1.73205
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$631$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$632$	0	0
$633$	1.73205	1.73205
$634$	0	0
$635$	−3.00000	−3.00000
$636$	1.00000	1.00000
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−1.73205	−1.73205
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−1.73205	−1.73205
$643$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$643$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$644$	0	0
$645$	−3.00000	−3.00000
$646$	0	0
$647$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$647$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$648$	1.00000	1.00000
$649$	1.73205	1.73205
$650$	0	0
$651$	0	0
$652$	1.73205	1.73205
$653$	2.00000	2.00000	1.00000			$0$
$653$	2.00000	2.00000	1.00000			$0$
$654$	0	0
$655$	1.73205	1.73205
$656$	−1.73205	−1.73205
$657$	0	0
$658$	−1.00000	−1.00000
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	−3.00000	−3.00000
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	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
$672$	−1.00000	−1.00000
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	1.00000	1.00000
$675$	−2.00000	−2.00000
$676$	−1.00000	−1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−1.00000
$687$	0	0
$688$	−1.73205	−1.73205
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.73205	1.73205
$699$	0	0
$700$	2.00000	2.00000
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−1.73205	−1.73205
$705$	1.73205	1.73205
$706$	0	0
$707$	0	0
$708$	−1.00000	−1.00000
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$709$	−2.00000	−2.00000	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−1.73205	−1.73205
$719$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$719$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$720$	0	0
$721$	−1.00000	−1.00000
$722$	−1.00000	−1.00000
$723$	0	0
$724$	1.73205	1.73205
$725$	0	0
$726$	−2.00000	−2.00000
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	−3.00000	−3.00000
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	1.73205	1.73205
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−1.73205	−1.73205
$741$	0	0
$742$	−1.00000	−1.00000
$743$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$743$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$744$	0	0
$745$	−1.73205	−1.73205
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.73205	1.73205
$750$	−1.73205	−1.73205
$751$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$751$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$752$	1.00000	1.00000
$753$	2.00000	2.00000
$754$	0	0
$755$	3.00000	3.00000
$756$	−1.00000	−1.00000
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	1.73205	1.73205
$763$	0	0
$764$	0	0
$765$	0	0
$766$	2.00000	2.00000
$767$	0	0
$768$	1.00000	1.00000
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	3.00000	3.00000
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	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
$784$	1.00000	1.00000
$785$	0	0
$786$	−1.00000	−1.00000
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.00000	−1.00000
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	1.73205	1.73205
$796$	0	0
$797$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$797$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$798$	0	0
$799$	0	0
$800$	−2.00000	−2.00000
$801$	0	0
$802$	−1.00000	−1.00000
$803$	−3.00000	−3.00000
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.73205	1.73205
$811$	−2.00000	−2.00000	−1.00000			$\pi$
$811$	−2.00000	−2.00000	−1.00000			$\pi$
$812$	0	0
$813$	1.00000	1.00000
$814$	−1.73205	−1.73205
$815$	3.00000	3.00000
$816$	0	0
$817$	0	0
$818$	−1.73205	−1.73205
$819$	0	0
$820$	−3.00000	−3.00000
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	1.00000	1.00000
$825$	−3.46410	−3.46410
$826$	1.00000	1.00000
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$829$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$830$	1.73205	1.73205
$831$	−2.00000	−2.00000
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−1.73205	−1.73205
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	1.73205	1.73205
$845$	−1.73205	−1.73205
$846$	0	0
$847$	2.00000	2.00000
$848$	1.00000	1.00000
$849$	−1.00000	−1.00000
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−1.73205	−1.73205
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−3.00000	−3.00000
$861$	−1.73205	−1.73205
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.00000	1.00000
$865$	0	0
$866$	1.73205	1.73205
$867$	1.00000	1.00000
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.73205	1.73205
$876$	1.73205	1.73205
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−1.00000	−1.00000
$879$	−1.73205	−1.73205
$880$	−3.00000	−3.00000
$881$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$881$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−1.73205	−1.73205
$886$	−1.73205	−1.73205
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	1.00000	1.00000
$889$	−1.73205	−1.73205
$890$	0	0
$891$	1.73205	1.73205
$892$	0	0
$893$	0	0
$894$	1.00000	1.00000
$895$	0	0
$896$	−1.00000	−1.00000
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	−3.00000	−3.00000
$903$	−1.73205	−1.73205
$904$	0	0
$905$	3.00000	3.00000
$906$	−1.73205	−1.73205
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	1.73205	1.73205
$914$	−1.00000	−1.00000
$915$	0	0
$916$	0	0
$917$	1.00000	1.00000
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−2.00000	−2.00000
$922$	0	0
$923$	0	0
$924$	−1.73205	−1.73205
$925$	−2.00000	−2.00000
$926$	−1.73205	−1.73205
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$937$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$938$	0	0
$939$	−1.73205	−1.73205
$940$	1.73205	1.73205
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−1.00000	−1.00000
$945$	−1.73205	−1.73205
$946$	−3.00000	−3.00000
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−1.00000	−1.00000
$959$	0	0
$960$	1.73205	1.73205
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.00000	−2.00000
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−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	$-0.666667\pi$
$977$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$978$	−1.73205	−1.73205
$979$	0	0
$980$	1.73205	1.73205
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.73205	1.73205
$985$	−1.73205	−1.73205
$986$	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			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−1.00000	−1.00000
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	1.73205	1.73205
$999$	1.00000	1.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1316.1.h.h.1315.2	yes	2
4.3	odd	2		1316.1.h.g.1315.2	yes	2
7.6	odd	2		1316.1.h.g.1315.1	✓	2
28.27	even	2	inner	1316.1.h.h.1315.1	yes	2
47.46	odd	2	inner	1316.1.h.h.1315.1	yes	2
188.187	even	2		1316.1.h.g.1315.1	✓	2
329.328	even	2		1316.1.h.g.1315.2	yes	2
1316.1315	odd	2	CM	1316.1.h.h.1315.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1316.1.h.g.1315.1	✓	2	7.6	odd	2
1316.1.h.g.1315.1	✓	2	188.187	even	2
1316.1.h.g.1315.2	yes	2	4.3	odd	2
1316.1.h.g.1315.2	yes	2	329.328	even	2
1316.1.h.h.1315.1	yes	2	28.27	even	2	inner
1316.1.h.h.1315.1	yes	2	47.46	odd	2	inner
1316.1.h.h.1315.2	yes	2	1.1	even	1	trivial
1316.1.h.h.1315.2	yes	2	1316.1315	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:	\( N \)	\(=\)	\( 1316 = 2^{2} \cdot 7 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1316.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.73205 q^{10} -1.73205 q^{11} +1.00000 q^{12} -1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} +1.73205 q^{20} +1.00000 q^{21} +1.73205 q^{22} -1.00000 q^{24} +2.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -1.73205 q^{30} -1.00000 q^{32} -1.73205 q^{33} +1.73205 q^{35} -1.00000 q^{37} -1.73205 q^{40} -1.73205 q^{41} -1.00000 q^{42} -1.73205 q^{43} -1.73205 q^{44} +1.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.00000 q^{50} +1.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -1.00000 q^{56} -1.00000 q^{59} +1.73205 q^{60} +1.00000 q^{64} +1.73205 q^{66} -1.73205 q^{70} +1.73205 q^{73} +1.00000 q^{74} +2.00000 q^{75} -1.73205 q^{77} +1.73205 q^{80} -1.00000 q^{81} +1.73205 q^{82} -1.00000 q^{83} +1.00000 q^{84} +1.73205 q^{86} +1.73205 q^{88} -1.00000 q^{94} -1.00000 q^{96} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{12} - 2 q^{14} + 2 q^{16} + 2 q^{21} - 2 q^{24} + 4 q^{25} - 2 q^{27} + 2 q^{28} - 2 q^{32} - 2 q^{37} - 2 q^{42} + 2 q^{47} + 2 q^{48} + 2 q^{49} - 4 q^{50} + 2 q^{53} + 2 q^{54} - 6 q^{55} - 2 q^{56} - 2 q^{59} + 2 q^{64} + 2 q^{74} + 4 q^{75} - 2 q^{81} - 2 q^{83} + 2 q^{84} - 2 q^{94} - 2 q^{96} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^12 - 2 * q^14 + 2 * q^16 + 2 * q^21 - 2 * q^24 + 4 * q^25 - 2 * q^27 + 2 * q^28 - 2 * q^32 - 2 * q^37 - 2 * q^42 + 2 * q^47 + 2 * q^48 + 2 * q^49 - 4 * q^50 + 2 * q^53 + 2 * q^54 - 6 * q^55 - 2 * q^56 - 2 * q^59 + 2 * q^64 + 2 * q^74 + 4 * q^75 - 2 * q^81 - 2 * q^83 + 2 * q^84 - 2 * q^94 - 2 * q^96 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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