LMFDB - Embedded newform 3064.1.e.f.765.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3064,1,Mod(765,3064)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3064, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3064.765");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3064 = 2^{3} \cdot 383 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3064.e (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:	$1.52913519871$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			765.3
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	3064.765

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.41421 q^{10} +0.517638 q^{11} +1.93185 q^{13} +1.73205 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +1.41421 q^{20} -0.517638 q^{22} +1.00000 q^{23} +1.00000 q^{25} -1.93185 q^{26} -1.73205 q^{28} -1.00000 q^{32} +1.00000 q^{34} -2.44949 q^{35} +1.00000 q^{36} -1.93185 q^{37} -1.41421 q^{40} +0.517638 q^{44} +1.41421 q^{45} -1.00000 q^{46} +2.00000 q^{49} -1.00000 q^{50} +1.93185 q^{52} -0.517638 q^{53} +0.732051 q^{55} +1.73205 q^{56} +1.93185 q^{59} -0.517638 q^{61} -1.73205 q^{63} +1.00000 q^{64} +2.73205 q^{65} -1.00000 q^{68} +2.44949 q^{70} -1.00000 q^{71} -1.00000 q^{72} +1.73205 q^{73} +1.93185 q^{74} -0.896575 q^{77} +1.41421 q^{80} +1.00000 q^{81} -1.41421 q^{83} -1.41421 q^{85} -0.517638 q^{88} -1.41421 q^{90} -3.34607 q^{91} +1.00000 q^{92} -2.00000 q^{98} +0.517638 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9} + 4 q^{16} - 4 q^{17} - 4 q^{18} + 4 q^{23} + 4 q^{25} - 4 q^{32} + 4 q^{34} + 4 q^{36} - 4 q^{46} + 8 q^{49} - 4 q^{50} - 4 q^{55} + 4 q^{64} + 4 q^{65} - 4 q^{68} - 4 q^{71} - 4 q^{72} + 4 q^{81} + 4 q^{92} - 8 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9 + 4 * q^16 - 4 * q^17 - 4 * q^18 + 4 * q^23 + 4 * q^25 - 4 * q^32 + 4 * q^34 + 4 * q^36 - 4 * q^46 + 8 * q^49 - 4 * q^50 - 4 * q^55 + 4 * q^64 + 4 * q^65 - 4 * q^68 - 4 * q^71 - 4 * q^72 + 4 * q^81 + 4 * q^92 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3064\mathbb{Z}\right)^\times$.

$n$	$767$	$1533$	$1537$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.00000	1.00000
$5$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$5$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$6$	0	0
$7$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$7$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$8$	−1.00000	−1.00000
$9$	1.00000	1.00000
$10$	−1.41421	−1.41421
$11$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$11$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$12$	0	0
$13$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$13$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$14$	1.73205	1.73205
$15$	0	0
$16$	1.00000	1.00000
$17$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$18$	−1.00000	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.41421	1.41421
$21$	0	0
$22$	−0.517638	−0.517638
$23$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$23$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$24$	0	0
$25$	1.00000	1.00000
$26$	−1.93185	−1.93185
$27$	0	0
$28$	−1.73205	−1.73205
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	1.00000	1.00000
$35$	−2.44949	−2.44949
$36$	1.00000	1.00000
$37$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$37$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$38$	0	0
$39$	0	0
$40$	−1.41421	−1.41421
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0.517638	0.517638
$45$	1.41421	1.41421
$46$	−1.00000	−1.00000
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	2.00000	2.00000
$50$	−1.00000	−1.00000
$51$	0	0
$52$	1.93185	1.93185
$53$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$53$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$54$	0	0
$55$	0.732051	0.732051
$56$	1.73205	1.73205
$57$	0	0
$58$	0	0
$59$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$59$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$60$	0	0
$61$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$61$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$62$	0	0
$63$	−1.73205	−1.73205
$64$	1.00000	1.00000
$65$	2.73205	2.73205
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.00000	−1.00000
$69$	0	0
$70$	2.44949	2.44949
$71$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$72$	−1.00000	−1.00000
$73$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$73$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$74$	1.93185	1.93185
$75$	0	0
$76$	0	0
$77$	−0.896575	−0.896575
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.41421	1.41421
$81$	1.00000	1.00000
$82$	0	0
$83$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$83$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$84$	0	0
$85$	−1.41421	−1.41421
$86$	0	0
$87$	0	0
$88$	−0.517638	−0.517638
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−1.41421	−1.41421
$91$	−3.34607	−3.34607
$92$	1.00000	1.00000
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−2.00000	−2.00000
$99$	0.517638	0.517638
$100$	1.00000	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$103$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$104$	−1.93185	−1.93185
$105$	0	0
$106$	0.517638	0.517638
$107$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$107$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$108$	0	0
$109$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$109$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$110$	−0.732051	−0.732051
$111$	0	0
$112$	−1.73205	−1.73205
$113$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$113$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$114$	0	0
$115$	1.41421	1.41421
$116$	0	0
$117$	1.93185	1.93185
$118$	−1.93185	−1.93185
$119$	1.73205	1.73205
$120$	0	0
$121$	−0.732051	−0.732051
$122$	0.517638	0.517638
$123$	0	0
$124$	0	0
$125$	0	0
$126$	1.73205	1.73205
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	−2.73205	−2.73205
$131$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$131$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	1.00000	1.00000
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−2.44949	−2.44949
$141$	0	0
$142$	1.00000	1.00000
$143$	1.00000	1.00000
$144$	1.00000	1.00000
$145$	0	0
$146$	−1.73205	−1.73205
$147$	0	0
$148$	−1.93185	−1.93185
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−1.00000	−1.00000
$154$	0.896575	0.896575
$155$	0	0
$156$	0	0
$157$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$157$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$158$	0	0
$159$	0	0
$160$	−1.41421	−1.41421
$161$	−1.73205	−1.73205
$162$	−1.00000	−1.00000
$163$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$163$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$164$	0	0
$165$	0	0
$166$	1.41421	1.41421
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	2.73205	2.73205
$170$	1.41421	1.41421
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−1.73205	−1.73205
$176$	0.517638	0.517638
$177$	0	0
$178$	0	0
$179$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$179$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$180$	1.41421	1.41421
$181$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$181$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$182$	3.34607	3.34607
$183$	0	0
$184$	−1.00000	−1.00000
$185$	−2.73205	−2.73205
$186$	0	0
$187$	−0.517638	−0.517638
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$193$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$194$	0	0
$195$	0	0
$196$	2.00000	2.00000
$197$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$197$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$198$	−0.517638	−0.517638
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−1.00000	−1.00000
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.73205	−1.73205
$207$	1.00000	1.00000
$208$	1.93185	1.93185
$209$	0	0
$210$	0	0
$211$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$211$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$212$	−0.517638	−0.517638
$213$	0	0
$214$	−1.93185	−1.93185
$215$	0	0
$216$	0	0
$217$	0	0
$218$	1.41421	1.41421
$219$	0	0
$220$	0.732051	0.732051
$221$	−1.93185	−1.93185
$222$	0	0
$223$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$224$	1.73205	1.73205
$225$	1.00000	1.00000
$226$	1.73205	1.73205
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	−1.41421	−1.41421
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−1.93185	−1.93185
$235$	0	0
$236$	1.93185	1.93185
$237$	0	0
$238$	−1.73205	−1.73205
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.732051	0.732051
$243$	0	0
$244$	−0.517638	−0.517638
$245$	2.82843	2.82843
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.73205	−1.73205
$253$	0.517638	0.517638
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	3.34607	3.34607
$260$	2.73205	2.73205
$261$	0	0
$262$	0.517638	0.517638
$263$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$263$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$264$	0	0
$265$	−0.732051	−0.732051
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$269$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−1.00000	−1.00000
$273$	0	0
$274$	0	0
$275$	0.517638	0.517638
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	2.44949	2.44949
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$283$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$284$	−1.00000	−1.00000
$285$	0	0
$286$	−1.00000	−1.00000
$287$	0	0
$288$	−1.00000	−1.00000
$289$	0	0
$290$	0	0
$291$	0	0
$292$	1.73205	1.73205
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	2.73205	2.73205
$296$	1.93185	1.93185
$297$	0	0
$298$	0	0
$299$	1.93185	1.93185
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.732051	−0.732051
$306$	1.00000	1.00000
$307$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$307$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$308$	−0.896575	−0.896575
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	−0.517638	−0.517638
$315$	−2.44949	−2.44949
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	1.41421	1.41421
$321$	0	0
$322$	1.73205	1.73205
$323$	0	0
$324$	1.00000	1.00000
$325$	1.93185	1.93185
$326$	−1.41421	−1.41421
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−1.41421	−1.41421
$333$	−1.93185	−1.93185
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−2.73205	−2.73205
$339$	0	0
$340$	−1.41421	−1.41421
$341$	0	0
$342$	0	0
$343$	−1.73205	−1.73205
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$347$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$348$	0	0
$349$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$349$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$350$	1.73205	1.73205
$351$	0	0
$352$	−0.517638	−0.517638
$353$	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
$355$	−1.41421	−1.41421
$356$	0	0
$357$	0	0
$358$	1.41421	1.41421
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.41421	−1.41421
$361$	1.00000	1.00000
$362$	−0.517638	−0.517638
$363$	0	0
$364$	−3.34607	−3.34607
$365$	2.44949	2.44949
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.00000	1.00000
$369$	0	0
$370$	2.73205	2.73205
$371$	0.896575	0.896575
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0.517638	0.517638
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$379$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.00000	−1.00000
$383$	−1.00000	−1.00000
$384$	0	0
$385$	−1.26795	−1.26795
$386$	−1.73205	−1.73205
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	−1.00000	−1.00000
$392$	−2.00000	−2.00000
$393$	0	0
$394$	1.93185	1.93185
$395$	0	0
$396$	0.517638	0.517638
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	1.00000
$401$	−2.00000	−2.00000	−1.00000			$\pi$
$401$	−2.00000	−2.00000	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.41421	1.41421
$406$	0	0
$407$	−1.00000	−1.00000
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	1.73205	1.73205
$413$	−3.34607	−3.34607
$414$	−1.00000	−1.00000
$415$	−2.00000	−2.00000
$416$	−1.93185	−1.93185
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−1.41421	−1.41421
$423$	0	0
$424$	0.517638	0.517638
$425$	−1.00000	−1.00000
$426$	0	0
$427$	0.896575	0.896575
$428$	1.93185	1.93185
$429$	0	0
$430$	0	0
$431$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$431$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$432$	0	0
$433$	−2.00000	−2.00000	−1.00000			$\pi$
$433$	−2.00000	−2.00000	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	−1.41421	−1.41421
$437$	0	0
$438$	0	0
$439$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$439$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$440$	−0.732051	−0.732051
$441$	2.00000	2.00000
$442$	1.93185	1.93185
$443$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$443$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$444$	0	0
$445$	0	0
$446$	1.00000	1.00000
$447$	0	0
$448$	−1.73205	−1.73205
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.00000	−1.00000
$451$	0	0
$452$	−1.73205	−1.73205
$453$	0	0
$454$	0	0
$455$	−4.73205	−4.73205
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	1.41421	1.41421
$461$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$461$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.93185	1.93185
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.93185	−1.93185
$473$	0	0
$474$	0	0
$475$	0	0
$476$	1.73205	1.73205
$477$	−0.517638	−0.517638
$478$	0	0
$479$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$479$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$480$	0	0
$481$	−3.73205	−3.73205
$482$	0	0
$483$	0	0
$484$	−0.732051	−0.732051
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0.517638	0.517638
$489$	0	0
$490$	−2.82843	−2.82843
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.732051	0.732051
$496$	0	0
$497$	1.73205	1.73205
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.73205	1.73205
$505$	0	0
$506$	−0.517638	−0.517638
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−3.00000	−3.00000
$512$	−1.00000	−1.00000
$513$	0	0
$514$	0	0
$515$	2.44949	2.44949
$516$	0	0
$517$	0	0
$518$	−3.34607	−3.34607
$519$	0	0
$520$	−2.73205	−2.73205
$521$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$521$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$522$	0	0
$523$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$523$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$524$	−0.517638	−0.517638
$525$	0	0
$526$	−1.73205	−1.73205
$527$	0	0
$528$	0	0
$529$	0	0
$530$	0.732051	0.732051
$531$	1.93185	1.93185
$532$	0	0
$533$	0	0
$534$	0	0
$535$	2.73205	2.73205
$536$	0	0
$537$	0	0
$538$	1.93185	1.93185
$539$	1.03528	1.03528
$540$	0	0
$541$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$541$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$542$	0	0
$543$	0	0
$544$	1.00000	1.00000
$545$	−2.00000	−2.00000
$546$	0	0
$547$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$547$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$548$	0	0
$549$	−0.517638	−0.517638
$550$	−0.517638	−0.517638
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−2.44949	−2.44949
$561$	0	0
$562$	0	0
$563$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$563$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$564$	0	0
$565$	−2.44949	−2.44949
$566$	−0.517638	−0.517638
$567$	−1.73205	−1.73205
$568$	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
$571$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$571$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$572$	1.00000	1.00000
$573$	0	0
$574$	0	0
$575$	1.00000	1.00000
$576$	1.00000	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.44949	2.44949
$582$	0	0
$583$	−0.267949	−0.267949
$584$	−1.73205	−1.73205
$585$	2.73205	2.73205
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	−2.73205	−2.73205
$591$	0	0
$592$	−1.93185	−1.93185
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	2.44949	2.44949
$596$	0	0
$597$	0	0
$598$	−1.93185	−1.93185
$599$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.03528	−1.03528
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0.732051	0.732051
$611$	0	0
$612$	−1.00000	−1.00000
$613$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$613$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$614$	1.41421	1.41421
$615$	0	0
$616$	0.896575	0.896575
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$619$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−1.00000
$626$	0	0
$627$	0	0
$628$	0.517638	0.517638
$629$	1.93185	1.93185
$630$	2.44949	2.44949
$631$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.86370	3.86370
$638$	0	0
$639$	−1.00000	−1.00000
$640$	−1.41421	−1.41421
$641$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$641$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−1.73205	−1.73205
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.00000	−1.00000
$649$	1.00000	1.00000
$650$	−1.93185	−1.93185
$651$	0	0
$652$	1.41421	1.41421
$653$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$653$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$654$	0	0
$655$	−0.732051	−0.732051
$656$	0	0
$657$	1.73205	1.73205
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	1.41421	1.41421
$665$	0	0
$666$	1.93185	1.93185
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.267949	−0.267949
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	2.73205	2.73205
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	1.41421	1.41421
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	1.73205	1.73205
$687$	0	0
$688$	0	0
$689$	−1.00000	−1.00000
$690$	0	0
$691$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$691$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$692$	0	0
$693$	−0.896575	−0.896575
$694$	1.41421	1.41421
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0.517638	0.517638
$699$	0	0
$700$	−1.73205	−1.73205
$701$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$701$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$702$	0	0
$703$	0	0
$704$	0.517638	0.517638
$705$	0	0
$706$	−1.00000	−1.00000
$707$	0	0
$708$	0	0
$709$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$709$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$710$	1.41421	1.41421
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.41421	1.41421
$716$	−1.41421	−1.41421
$717$	0	0
$718$	0	0
$719$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$720$	1.41421	1.41421
$721$	−3.00000	−3.00000
$722$	−1.00000	−1.00000
$723$	0	0
$724$	0.517638	0.517638
$725$	0	0
$726$	0	0
$727$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$727$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$728$	3.34607	3.34607
$729$	1.00000	1.00000
$730$	−2.44949	−2.44949
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	−1.00000	−1.00000
$737$	0	0
$738$	0	0
$739$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$739$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$740$	−2.73205	−2.73205
$741$	0	0
$742$	−0.896575	−0.896575
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.41421	−1.41421
$748$	−0.517638	−0.517638
$749$	−3.34607	−3.34607
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$757$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$758$	0.517638	0.517638
$759$	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
$763$	2.44949	2.44949
$764$	0	0
$765$	−1.41421	−1.41421
$766$	1.00000	1.00000
$767$	3.73205	3.73205
$768$	0	0
$769$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$769$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$770$	1.26795	1.26795
$771$	0	0
$772$	1.73205	1.73205
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−0.517638	−0.517638
$782$	1.00000	1.00000
$783$	0	0
$784$	2.00000	2.00000
$785$	0.732051	0.732051
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.93185	−1.93185
$789$	0	0
$790$	0	0
$791$	3.00000	3.00000
$792$	−0.517638	−0.517638
$793$	−1.00000	−1.00000
$794$	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
$800$	−1.00000	−1.00000
$801$	0	0
$802$	2.00000	2.00000
$803$	0.896575	0.896575
$804$	0	0
$805$	−2.44949	−2.44949
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	−1.41421	−1.41421
$811$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$811$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$812$	0	0
$813$	0	0
$814$	1.00000	1.00000
$815$	2.00000	2.00000
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−3.34607	−3.34607
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$824$	−1.73205	−1.73205
$825$	0	0
$826$	3.34607	3.34607
$827$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$827$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$828$	1.00000	1.00000
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	2.00000	2.00000
$831$	0	0
$832$	1.93185	1.93185
$833$	−2.00000	−2.00000
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$839$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	1.41421	1.41421
$845$	3.86370	3.86370
$846$	0	0
$847$	1.26795	1.26795
$848$	−0.517638	−0.517638
$849$	0	0
$850$	1.00000	1.00000
$851$	−1.93185	−1.93185
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	−0.896575	−0.896575
$855$	0	0
$856$	−1.93185	−1.93185
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	1.00000	1.00000
$863$	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
$872$	1.41421	1.41421
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$877$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$878$	1.73205	1.73205
$879$	0	0
$880$	0.732051	0.732051
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−2.00000	−2.00000
$883$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$884$	−1.93185	−1.93185
$885$	0	0
$886$	1.93185	1.93185
$887$	2.00000	2.00000	1.00000			$0$
$887$	2.00000	2.00000	1.00000			$0$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0.517638	0.517638
$892$	−1.00000	−1.00000
$893$	0	0
$894$	0	0
$895$	−2.00000	−2.00000
$896$	1.73205	1.73205
$897$	0	0
$898$	0	0
$899$	0	0
$900$	1.00000	1.00000
$901$	0.517638	0.517638
$902$	0	0
$903$	0	0
$904$	1.73205	1.73205
$905$	0.732051	0.732051
$906$	0	0
$907$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$907$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$908$	0	0
$909$	0	0
$910$	4.73205	4.73205
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−0.732051	−0.732051
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.896575	0.896575
$918$	0	0
$919$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$919$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$920$	−1.41421	−1.41421
$921$	0	0
$922$	1.41421	1.41421
$923$	−1.93185	−1.93185
$924$	0	0
$925$	−1.93185	−1.93185
$926$	0	0
$927$	1.73205	1.73205
$928$	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
$935$	−0.732051	−0.732051
$936$	−1.93185	−1.93185
$937$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$937$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1.93185	1.93185
$945$	0	0
$946$	0	0
$947$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$947$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$948$	0	0
$949$	3.34607	3.34607
$950$	0	0
$951$	0	0
$952$	−1.73205	−1.73205
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0.517638	0.517638
$955$	0	0
$956$	0	0
$957$	0	0
$958$	1.73205	1.73205
$959$	0	0
$960$	0	0
$961$	−1.00000	−1.00000
$962$	3.73205	3.73205
$963$	1.93185	1.93185
$964$	0	0
$965$	2.44949	2.44949
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0.732051	0.732051
$969$	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
$976$	−0.517638	−0.517638
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	2.82843	2.82843
$981$	−1.41421	−1.41421
$982$	0	0
$983$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$983$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$984$	0	0
$985$	−2.73205	−2.73205
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−0.732051	−0.732051
$991$	2.00000	2.00000	1.00000			$0$
$991$	2.00000	2.00000	1.00000			$0$
$992$	0	0
$993$	0	0
$994$	−1.73205	−1.73205
$995$	0	0
$996$	0	0
$997$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$997$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3064.1.e.f.765.3	yes	4
8.5	even	2	inner	3064.1.e.f.765.1	✓	4
383.382	odd	2	inner	3064.1.e.f.765.1	✓	4
3064.765	odd	2	CM	3064.1.e.f.765.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3064.1.e.f.765.1	✓	4	8.5	even	2	inner
3064.1.e.f.765.1	✓	4	383.382	odd	2	inner
3064.1.e.f.765.3	yes	4	1.1	even	1	trivial
3064.1.e.f.765.3	yes	4	3064.765	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:	\( N \)	\(=\)	\( 3064 = 2^{3} \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3064.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.41421 q^{10} +0.517638 q^{11} +1.93185 q^{13} +1.73205 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +1.41421 q^{20} -0.517638 q^{22} +1.00000 q^{23} +1.00000 q^{25} -1.93185 q^{26} -1.73205 q^{28} -1.00000 q^{32} +1.00000 q^{34} -2.44949 q^{35} +1.00000 q^{36} -1.93185 q^{37} -1.41421 q^{40} +0.517638 q^{44} +1.41421 q^{45} -1.00000 q^{46} +2.00000 q^{49} -1.00000 q^{50} +1.93185 q^{52} -0.517638 q^{53} +0.732051 q^{55} +1.73205 q^{56} +1.93185 q^{59} -0.517638 q^{61} -1.73205 q^{63} +1.00000 q^{64} +2.73205 q^{65} -1.00000 q^{68} +2.44949 q^{70} -1.00000 q^{71} -1.00000 q^{72} +1.73205 q^{73} +1.93185 q^{74} -0.896575 q^{77} +1.41421 q^{80} +1.00000 q^{81} -1.41421 q^{83} -1.41421 q^{85} -0.517638 q^{88} -1.41421 q^{90} -3.34607 q^{91} +1.00000 q^{92} -2.00000 q^{98} +0.517638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9} + 4 q^{16} - 4 q^{17} - 4 q^{18} + 4 q^{23} + 4 q^{25} - 4 q^{32} + 4 q^{34} + 4 q^{36} - 4 q^{46} + 8 q^{49} - 4 q^{50} - 4 q^{55} + 4 q^{64} + 4 q^{65} - 4 q^{68} - 4 q^{71} - 4 q^{72} + 4 q^{81} + 4 q^{92} - 8 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9 + 4 * q^16 - 4 * q^17 - 4 * q^18 + 4 * q^23 + 4 * q^25 - 4 * q^32 + 4 * q^34 + 4 * q^36 - 4 * q^46 + 8 * q^49 - 4 * q^50 - 4 * q^55 + 4 * q^64 + 4 * q^65 - 4 * q^68 - 4 * q^71 - 4 * q^72 + 4 * q^81 + 4 * q^92 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more