LMFDB - Embedded newform 519.1.d.c.518.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [519,1,Mod(518,519)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(519, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("519.518");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 519 = 3 \cdot 173 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	519.d (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:	$0.259014741557$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.72555348321.1

Embedding invariants

Embedding label			518.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	519.518

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.347296 q^{2} -1.00000 q^{3} -0.879385 q^{4} -1.53209 q^{5} +0.347296 q^{6} +0.652704 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.347296 q^{2} -1.00000 q^{3} -0.879385 q^{4} -1.53209 q^{5} +0.347296 q^{6} +0.652704 q^{8} +1.00000 q^{9} +0.532089 q^{10} +1.87939 q^{11} +0.879385 q^{12} -1.00000 q^{13} +1.53209 q^{15} +0.652704 q^{16} +1.00000 q^{17} -0.347296 q^{18} +1.34730 q^{20} -0.652704 q^{22} -0.652704 q^{24} +1.34730 q^{25} +0.347296 q^{26} -1.00000 q^{27} -0.532089 q^{30} +0.347296 q^{31} -0.879385 q^{32} -1.87939 q^{33} -0.347296 q^{34} -0.879385 q^{36} +1.53209 q^{37} +1.00000 q^{39} -1.00000 q^{40} -1.87939 q^{43} -1.65270 q^{44} -1.53209 q^{45} -0.652704 q^{48} +1.00000 q^{49} -0.467911 q^{50} -1.00000 q^{51} +0.879385 q^{52} +1.87939 q^{53} +0.347296 q^{54} -2.87939 q^{55} +1.00000 q^{59} -1.34730 q^{60} -0.120615 q^{62} -0.347296 q^{64} +1.53209 q^{65} +0.652704 q^{66} +0.347296 q^{67} -0.879385 q^{68} -0.347296 q^{71} +0.652704 q^{72} +1.53209 q^{73} -0.532089 q^{74} -1.34730 q^{75} -0.347296 q^{78} -1.00000 q^{80} +1.00000 q^{81} -1.53209 q^{85} +0.652704 q^{86} +1.22668 q^{88} +0.532089 q^{90} -0.347296 q^{93} +0.879385 q^{96} -0.347296 q^{98} +1.87939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{12} - 3 q^{13} + 3 q^{16} + 3 q^{17} + 3 q^{20} - 3 q^{22} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{30} + 3 q^{32} + 3 q^{36} + 3 q^{39} - 3 q^{40} - 6 q^{44} - 3 q^{48} + 3 q^{49} - 6 q^{50} - 3 q^{51} - 3 q^{52} - 3 q^{55} + 3 q^{59} - 3 q^{60} - 6 q^{62} + 3 q^{66} + 3 q^{68} + 3 q^{72} + 3 q^{74} - 3 q^{75} - 3 q^{80} + 3 q^{81} + 3 q^{86} - 3 q^{88} - 3 q^{90} - 3 q^{96}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^12 - 3 * q^13 + 3 * q^16 + 3 * q^17 + 3 * q^20 - 3 * q^22 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^30 + 3 * q^32 + 3 * q^36 + 3 * q^39 - 3 * q^40 - 6 * q^44 - 3 * q^48 + 3 * q^49 - 6 * q^50 - 3 * q^51 - 3 * q^52 - 3 * q^55 + 3 * q^59 - 3 * q^60 - 6 * q^62 + 3 * q^66 + 3 * q^68 + 3 * q^72 + 3 * q^74 - 3 * q^75 - 3 * q^80 + 3 * q^81 + 3 * q^86 - 3 * q^88 - 3 * q^90 - 3 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/519\mathbb{Z}\right)^\times$.

$n$	$175$	$347$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$2$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$3$	−1.00000	−1.00000
$3$	−1.00000	−1.00000
$4$	−0.879385	−0.879385
$5$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$5$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$6$	0.347296	0.347296
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0.652704	0.652704
$9$	1.00000	1.00000
$10$	0.532089	0.532089
$11$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$11$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$12$	0.879385	0.879385
$13$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$14$	0	0
$15$	1.53209	1.53209
$16$	0.652704	0.652704
$17$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$17$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$18$	−0.347296	−0.347296
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.34730	1.34730
$21$	0	0
$22$	−0.652704	−0.652704
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−0.652704	−0.652704
$25$	1.34730	1.34730
$26$	0.347296	0.347296
$27$	−1.00000	−1.00000
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−0.532089	−0.532089
$31$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$31$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$32$	−0.879385	−0.879385
$33$	−1.87939	−1.87939
$34$	−0.347296	−0.347296
$35$	0	0
$36$	−0.879385	−0.879385
$37$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$37$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$38$	0	0
$39$	1.00000	1.00000
$40$	−1.00000	−1.00000
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$43$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$44$	−1.65270	−1.65270
$45$	−1.53209	−1.53209
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.652704	−0.652704
$49$	1.00000	1.00000
$50$	−0.467911	−0.467911
$51$	−1.00000	−1.00000
$52$	0.879385	0.879385
$53$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$53$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$54$	0.347296	0.347296
$55$	−2.87939	−2.87939
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$59$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$60$	−1.34730	−1.34730
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−0.120615	−0.120615
$63$	0	0
$64$	−0.347296	−0.347296
$65$	1.53209	1.53209
$66$	0.652704	0.652704
$67$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$67$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$68$	−0.879385	−0.879385
$69$	0	0
$70$	0	0
$71$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$71$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$72$	0.652704	0.652704
$73$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$73$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$74$	−0.532089	−0.532089
$75$	−1.34730	−1.34730
$76$	0	0
$77$	0	0
$78$	−0.347296	−0.347296
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.00000	−1.00000
$81$	1.00000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−1.53209	−1.53209
$86$	0.652704	0.652704
$87$	0	0
$88$	1.22668	1.22668
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.532089	0.532089
$91$	0	0
$92$	0	0
$93$	−0.347296	−0.347296
$94$	0	0
$95$	0	0
$96$	0.879385	0.879385
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.347296	−0.347296
$99$	1.87939	1.87939
$100$	−1.18479	−1.18479
$101$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$101$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$102$	0.347296	0.347296
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−0.652704	−0.652704
$105$	0	0
$106$	−0.652704	−0.652704
$107$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$107$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$108$	0.879385	0.879385
$109$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$109$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$110$	1.00000	1.00000
$111$	−1.53209	−1.53209
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−1.00000
$118$	−0.347296	−0.347296
$119$	0	0
$120$	1.00000	1.00000
$121$	2.53209	2.53209
$122$	0	0
$123$	0	0
$124$	−0.305407	−0.305407
$125$	−0.532089	−0.532089
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	1.87939	1.87939
$130$	−0.532089	−0.532089
$131$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$131$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$132$	1.65270	1.65270
$133$	0	0
$134$	−0.120615	−0.120615
$135$	1.53209	1.53209
$136$	0.652704	0.652704
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$139$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$140$	0	0
$141$	0	0
$142$	0.120615	0.120615
$143$	−1.87939	−1.87939
$144$	0.652704	0.652704
$145$	0	0
$146$	−0.532089	−0.532089
$147$	−1.00000	−1.00000
$148$	−1.34730	−1.34730
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.467911	0.467911
$151$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$152$	0	0
$153$	1.00000	1.00000
$154$	0	0
$155$	−0.532089	−0.532089
$156$	−0.879385	−0.879385
$157$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$157$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$158$	0	0
$159$	−1.87939	−1.87939
$160$	1.34730	1.34730
$161$	0	0
$162$	−0.347296	−0.347296
$163$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$164$	0	0
$165$	2.87939	2.87939
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	0	0
$170$	0.532089	0.532089
$171$	0	0
$172$	1.65270	1.65270
$173$	−1.00000	−1.00000
$173$	−1.00000	−1.00000
$174$	0	0
$175$	0	0
$176$	1.22668	1.22668
$177$	−1.00000	−1.00000
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	1.34730	1.34730
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.34730	−2.34730
$186$	0.120615	0.120615
$187$	1.87939	1.87939
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$191$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$192$	0.347296	0.347296
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	−1.53209	−1.53209
$196$	−0.879385	−0.879385
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−0.652704	−0.652704
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0.879385	0.879385
$201$	−0.347296	−0.347296
$202$	0.120615	0.120615
$203$	0	0
$204$	0.879385	0.879385
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−0.652704	−0.652704
$209$	0	0
$210$	0	0
$211$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$211$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$212$	−1.65270	−1.65270
$213$	0.347296	0.347296
$214$	0.532089	0.532089
$215$	2.87939	2.87939
$216$	−0.652704	−0.652704
$217$	0	0
$218$	0.652704	0.652704
$219$	−1.53209	−1.53209
$220$	2.53209	2.53209
$221$	−1.00000	−1.00000
$222$	0.532089	0.532089
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	1.34730	1.34730
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$229$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0.347296	0.347296
$235$	0	0
$236$	−0.879385	−0.879385
$237$	0	0
$238$	0	0
$239$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$239$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$240$	1.00000	1.00000
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−0.879385	−0.879385
$243$	−1.00000	−1.00000
$244$	0	0
$245$	−1.53209	−1.53209
$246$	0	0
$247$	0	0
$248$	0.226682	0.226682
$249$	0	0
$250$	0.184793	0.184793
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.53209	1.53209
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−0.652704	−0.652704
$259$	0	0
$260$	−1.34730	−1.34730
$261$	0	0
$262$	0.120615	0.120615
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−1.22668	−1.22668
$265$	−2.87939	−2.87939
$266$	0	0
$267$	0	0
$268$	−0.305407	−0.305407
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.532089	−0.532089
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0.652704	0.652704
$273$	0	0
$274$	0	0
$275$	2.53209	2.53209
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−0.532089	−0.532089
$279$	0.347296	0.347296
$280$	0	0
$281$	−2.00000	−2.00000	−1.00000			$\pi$
$281$	−2.00000	−2.00000	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0.305407	0.305407
$285$	0	0
$286$	0.652704	0.652704
$287$	0	0
$288$	−0.879385	−0.879385
$289$	0	0
$290$	0	0
$291$	0	0
$292$	−1.34730	−1.34730
$293$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$293$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$294$	0.347296	0.347296
$295$	−1.53209	−1.53209
$296$	1.00000	1.00000
$297$	−1.87939	−1.87939
$298$	0	0
$299$	0	0
$300$	1.18479	1.18479
$301$	0	0
$302$	0.347296	0.347296
$303$	0.347296	0.347296
$304$	0	0
$305$	0	0
$306$	−0.347296	−0.347296
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0.184793	0.184793
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0.652704	0.652704
$313$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$313$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$314$	−0.120615	−0.120615
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.652704	0.652704
$319$	0	0
$320$	0.532089	0.532089
$321$	1.53209	1.53209
$322$	0	0
$323$	0	0
$324$	−0.879385	−0.879385
$325$	−1.34730	−1.34730
$326$	0.347296	0.347296
$327$	1.87939	1.87939
$328$	0	0
$329$	0	0
$330$	−1.00000	−1.00000
$331$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$331$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$332$	0	0
$333$	1.53209	1.53209
$334$	0	0
$335$	−0.532089	−0.532089
$336$	0	0
$337$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$337$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$338$	0	0
$339$	0	0
$340$	1.34730	1.34730
$341$	0.652704	0.652704
$342$	0	0
$343$	0	0
$344$	−1.22668	−1.22668
$345$	0	0
$346$	0.347296	0.347296
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	1.00000	1.00000
$352$	−1.65270	−1.65270
$353$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$353$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$354$	0.347296	0.347296
$355$	0.532089	0.532089
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.00000	−1.00000
$361$	1.00000	1.00000
$362$	0	0
$363$	−2.53209	−2.53209
$364$	0	0
$365$	−2.34730	−2.34730
$366$	0	0
$367$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$368$	0	0
$369$	0	0
$370$	0.815207	0.815207
$371$	0	0
$372$	0.305407	0.305407
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−0.652704	−0.652704
$375$	0.532089	0.532089
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$380$	0	0
$381$	0	0
$382$	−0.652704	−0.652704
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.00000	−1.00000
$385$	0	0
$386$	0	0
$387$	−1.87939	−1.87939
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0.532089	0.532089
$391$	0	0
$392$	0.652704	0.652704
$393$	0.347296	0.347296
$394$	0	0
$395$	0	0
$396$	−1.65270	−1.65270
$397$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$397$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$398$	0	0
$399$	0	0
$400$	0.879385	0.879385
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0.120615	0.120615
$403$	−0.347296	−0.347296
$404$	0.305407	0.305407
$405$	−1.53209	−1.53209
$406$	0	0
$407$	2.87939	2.87939
$408$	−0.652704	−0.652704
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0.879385	0.879385
$417$	−1.53209	−1.53209
$418$	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			$\pi$
$422$	−0.532089	−0.532089
$423$	0	0
$424$	1.22668	1.22668
$425$	1.34730	1.34730
$426$	−0.120615	−0.120615
$427$	0	0
$428$	1.34730	1.34730
$429$	1.87939	1.87939
$430$	−1.00000	−1.00000
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.652704	−0.652704
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	1.65270	1.65270
$437$	0	0
$438$	0.532089	0.532089
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−1.87939	−1.87939
$441$	1.00000	1.00000
$442$	0.347296	0.347296
$443$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$443$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$444$	1.34730	1.34730
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$449$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$450$	−0.467911	−0.467911
$451$	0	0
$452$	0	0
$453$	1.00000	1.00000
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−0.532089	−0.532089
$459$	−1.00000	−1.00000
$460$	0	0
$461$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$461$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$462$	0	0
$463$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$463$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$464$	0	0
$465$	0.532089	0.532089
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0.879385	0.879385
$469$	0	0
$470$	0	0
$471$	−0.347296	−0.347296
$472$	0.652704	0.652704
$473$	−3.53209	−3.53209
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.87939	1.87939
$478$	0.532089	0.532089
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−1.34730	−1.34730
$481$	−1.53209	−1.53209
$482$	0	0
$483$	0	0
$484$	−2.22668	−2.22668
$485$	0	0
$486$	0.347296	0.347296
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.00000	1.00000
$490$	0.532089	0.532089
$491$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$491$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.87939	−2.87939
$496$	0.226682	0.226682
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.467911	0.467911
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0.532089	0.532089
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−0.532089	−0.532089
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−1.65270	−1.65270
$517$	0	0
$518$	0	0
$519$	1.00000	1.00000
$520$	1.00000	1.00000
$521$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$521$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$522$	0	0
$523$	2.00000	2.00000	1.00000			$0$
$523$	2.00000	2.00000	1.00000			$0$
$524$	0.305407	0.305407
$525$	0	0
$526$	0	0
$527$	0.347296	0.347296
$528$	−1.22668	−1.22668
$529$	1.00000	1.00000
$530$	1.00000	1.00000
$531$	1.00000	1.00000
$532$	0	0
$533$	0	0
$534$	0	0
$535$	2.34730	2.34730
$536$	0.226682	0.226682
$537$	0	0
$538$	0	0
$539$	1.87939	1.87939
$540$	−1.34730	−1.34730
$541$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$541$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$542$	0	0
$543$	0	0
$544$	−0.879385	−0.879385
$545$	2.87939	2.87939
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−0.879385	−0.879385
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.34730	2.34730
$556$	−1.34730	−1.34730
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−0.120615	−0.120615
$559$	1.87939	1.87939
$560$	0	0
$561$	−1.87939	−1.87939
$562$	0.694593	0.694593
$563$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−0.226682	−0.226682
$569$	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
$571$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$571$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$572$	1.65270	1.65270
$573$	−1.87939	−1.87939
$574$	0	0
$575$	0	0
$576$	−0.347296	−0.347296
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.53209	3.53209
$584$	1.00000	1.00000
$585$	1.53209	1.53209
$586$	−0.652704	−0.652704
$587$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$587$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$588$	0.879385	0.879385
$589$	0	0
$590$	0.532089	0.532089
$591$	0	0
$592$	1.00000	1.00000
$593$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$593$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$594$	0.652704	0.652704
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$599$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$600$	−0.879385	−0.879385
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0.347296	0.347296
$604$	0.879385	0.879385
$605$	−3.87939	−3.87939
$606$	−0.120615	−0.120615
$607$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$607$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−0.879385	−0.879385
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$617$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$618$	0	0
$619$	2.00000	2.00000	1.00000			$0$
$619$	2.00000	2.00000	1.00000			$0$
$620$	0.467911	0.467911
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0.652704	0.652704
$625$	−0.532089	−0.532089
$626$	0.652704	0.652704
$627$	0	0
$628$	−0.305407	−0.305407
$629$	1.53209	1.53209
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−1.53209	−1.53209
$634$	0	0
$635$	0	0
$636$	1.65270	1.65270
$637$	−1.00000	−1.00000
$638$	0	0
$639$	−0.347296	−0.347296
$640$	−1.53209	−1.53209
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−0.532089	−0.532089
$643$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$643$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$644$	0	0
$645$	−2.87939	−2.87939
$646$	0	0
$647$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$647$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$648$	0.652704	0.652704
$649$	1.87939	1.87939
$650$	0.467911	0.467911
$651$	0	0
$652$	0.879385	0.879385
$653$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$653$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$654$	−0.652704	−0.652704
$655$	0.532089	0.532089
$656$	0	0
$657$	1.53209	1.53209
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	−2.53209	−2.53209
$661$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$662$	0.652704	0.652704
$663$	1.00000	1.00000
$664$	0	0
$665$	0	0
$666$	−0.532089	−0.532089
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0.184793	0.184793
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−0.120615	−0.120615
$675$	−1.34730	−1.34730
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	−1.00000	−1.00000
$681$	0	0
$682$	−0.226682	−0.226682
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.53209	−1.53209
$688$	−1.22668	−1.22668
$689$	−1.87939	−1.87939
$690$	0	0
$691$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$692$	0.879385	0.879385
$693$	0	0
$694$	0	0
$695$	−2.34730	−2.34730
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−0.347296	−0.347296
$703$	0	0
$704$	−0.652704	−0.652704
$705$	0	0
$706$	−0.652704	−0.652704
$707$	0	0
$708$	0.879385	0.879385
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−0.184793	−0.184793
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	2.87939	2.87939
$716$	0	0
$717$	1.53209	1.53209
$718$	0	0
$719$	−2.00000	−2.00000	−1.00000			$\pi$
$719$	−2.00000	−2.00000	−1.00000			$\pi$
$720$	−1.00000	−1.00000
$721$	0	0
$722$	−0.347296	−0.347296
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0.879385	0.879385
$727$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$727$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0.815207	0.815207
$731$	−1.87939	−1.87939
$732$	0	0
$733$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$733$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$734$	0.347296	0.347296
$735$	1.53209	1.53209
$736$	0	0
$737$	0.652704	0.652704
$738$	0	0
$739$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$739$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$740$	2.06418	2.06418
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−0.226682	−0.226682
$745$	0	0
$746$	0	0
$747$	0	0
$748$	−1.65270	−1.65270
$749$	0	0
$750$	−0.184793	−0.184793
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.53209	1.53209
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0.347296	0.347296
$759$	0	0
$760$	0	0
$761$	−2.00000	−2.00000	−1.00000			$\pi$
$761$	−2.00000	−2.00000	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−1.65270	−1.65270
$765$	−1.53209	−1.53209
$766$	0	0
$767$	−1.00000	−1.00000
$768$	0	0
$769$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0.652704	0.652704
$775$	0.467911	0.467911
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	1.34730	1.34730
$781$	−0.652704	−0.652704
$782$	0	0
$783$	0	0
$784$	0.652704	0.652704
$785$	−0.532089	−0.532089
$786$	−0.120615	−0.120615
$787$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$787$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.22668	1.22668
$793$	0	0
$794$	0.652704	0.652704
$795$	2.87939	2.87939
$796$	0	0
$797$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$797$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$798$	0	0
$799$	0	0
$800$	−1.18479	−1.18479
$801$	0	0
$802$	0	0
$803$	2.87939	2.87939
$804$	0.305407	0.305407
$805$	0	0
$806$	0.120615	0.120615
$807$	0	0
$808$	−0.226682	−0.226682
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0.532089	0.532089
$811$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$811$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$812$	0	0
$813$	0	0
$814$	−1.00000	−1.00000
$815$	1.53209	1.53209
$816$	−0.652704	−0.652704
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$821$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−2.53209	−2.53209
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$829$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$830$	0	0
$831$	0	0
$832$	0.347296	0.347296
$833$	1.00000	1.00000
$834$	0.532089	0.532089
$835$	0	0
$836$	0	0
$837$	−0.347296	−0.347296
$838$	0	0
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	2.00000	2.00000
$844$	−1.34730	−1.34730
$845$	0	0
$846$	0	0
$847$	0	0
$848$	1.22668	1.22668
$849$	0	0
$850$	−0.467911	−0.467911
$851$	0	0
$852$	−0.305407	−0.305407
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−1.00000	−1.00000
$857$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$857$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$858$	−0.652704	−0.652704
$859$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$859$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$860$	−2.53209	−2.53209
$861$	0	0
$862$	0	0
$863$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$863$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$864$	0.879385	0.879385
$865$	1.53209	1.53209
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.347296	−0.347296
$872$	−1.22668	−1.22668
$873$	0	0
$874$	0	0
$875$	0	0
$876$	1.34730	1.34730
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−1.87939	−1.87939
$880$	−1.87939	−1.87939
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.347296	−0.347296
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0.879385	0.879385
$885$	1.53209	1.53209
$886$	0.120615	0.120615
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−1.00000	−1.00000
$889$	0	0
$890$	0	0
$891$	1.87939	1.87939
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0.532089	0.532089
$899$	0	0
$900$	−1.18479	−1.18479
$901$	1.87939	1.87939
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−0.347296	−0.347296
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−0.347296	−0.347296
$910$	0	0
$911$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$911$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−1.34730	−1.34730
$917$	0	0
$918$	0.347296	0.347296
$919$	2.00000	2.00000	1.00000			$0$
$919$	2.00000	2.00000	1.00000			$0$
$920$	0	0
$921$	0	0
$922$	−0.652704	−0.652704
$923$	0.347296	0.347296
$924$	0	0
$925$	2.06418	2.06418
$926$	0.652704	0.652704
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	−0.184793	−0.184793
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.87939	−2.87939
$936$	−0.652704	−0.652704
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	1.87939	1.87939
$940$	0	0
$941$	−2.00000	−2.00000	−1.00000			$\pi$
$941$	−2.00000	−2.00000	−1.00000			$\pi$
$942$	0.120615	0.120615
$943$	0	0
$944$	0.652704	0.652704
$945$	0	0
$946$	1.22668	1.22668
$947$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$947$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$948$	0	0
$949$	−1.53209	−1.53209
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−0.652704	−0.652704
$955$	−2.87939	−2.87939
$956$	1.34730	1.34730
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−0.532089	−0.532089
$961$	−0.879385	−0.879385
$962$	0.532089	0.532089
$963$	−1.53209	−1.53209
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.65270	1.65270
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.879385	0.879385
$973$	0	0
$974$	0	0
$975$	1.34730	1.34730
$976$	0	0
$977$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$977$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$978$	−0.347296	−0.347296
$979$	0	0
$980$	1.34730	1.34730
$981$	−1.87939	−1.87939
$982$	0.532089	0.532089
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	1.00000	1.00000
$991$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$991$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$992$	−0.305407	−0.305407
$993$	1.87939	1.87939
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$997$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$998$	0	0
$999$	−1.53209	−1.53209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	519.1.d.c.518.2	✓	3
3.2	odd	2		519.1.d.d.518.2	yes	3
173.172	even	2		519.1.d.d.518.2	yes	3
519.518	odd	2	CM	519.1.d.c.518.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
519.1.d.c.518.2	✓	3	1.1	even	1	trivial
519.1.d.c.518.2	✓	3	519.518	odd	2	CM
519.1.d.d.518.2	yes	3	3.2	odd	2
519.1.d.d.518.2	yes	3	173.172	even	2

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:	\( N \)	\(=\)	\( 519 = 3 \cdot 173 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	519.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.347296 q^{2} -1.00000 q^{3} -0.879385 q^{4} -1.53209 q^{5} +0.347296 q^{6} +0.652704 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.347296 q^{2} -1.00000 q^{3} -0.879385 q^{4} -1.53209 q^{5} +0.347296 q^{6} +0.652704 q^{8} +1.00000 q^{9} +0.532089 q^{10} +1.87939 q^{11} +0.879385 q^{12} -1.00000 q^{13} +1.53209 q^{15} +0.652704 q^{16} +1.00000 q^{17} -0.347296 q^{18} +1.34730 q^{20} -0.652704 q^{22} -0.652704 q^{24} +1.34730 q^{25} +0.347296 q^{26} -1.00000 q^{27} -0.532089 q^{30} +0.347296 q^{31} -0.879385 q^{32} -1.87939 q^{33} -0.347296 q^{34} -0.879385 q^{36} +1.53209 q^{37} +1.00000 q^{39} -1.00000 q^{40} -1.87939 q^{43} -1.65270 q^{44} -1.53209 q^{45} -0.652704 q^{48} +1.00000 q^{49} -0.467911 q^{50} -1.00000 q^{51} +0.879385 q^{52} +1.87939 q^{53} +0.347296 q^{54} -2.87939 q^{55} +1.00000 q^{59} -1.34730 q^{60} -0.120615 q^{62} -0.347296 q^{64} +1.53209 q^{65} +0.652704 q^{66} +0.347296 q^{67} -0.879385 q^{68} -0.347296 q^{71} +0.652704 q^{72} +1.53209 q^{73} -0.532089 q^{74} -1.34730 q^{75} -0.347296 q^{78} -1.00000 q^{80} +1.00000 q^{81} -1.53209 q^{85} +0.652704 q^{86} +1.22668 q^{88} +0.532089 q^{90} -0.347296 q^{93} +0.879385 q^{96} -0.347296 q^{98} +1.87939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{12} - 3 q^{13} + 3 q^{16} + 3 q^{17} + 3 q^{20} - 3 q^{22} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{30} + 3 q^{32} + 3 q^{36} + 3 q^{39} - 3 q^{40} - 6 q^{44} - 3 q^{48} + 3 q^{49} - 6 q^{50} - 3 q^{51} - 3 q^{52} - 3 q^{55} + 3 q^{59} - 3 q^{60} - 6 q^{62} + 3 q^{66} + 3 q^{68} + 3 q^{72} + 3 q^{74} - 3 q^{75} - 3 q^{80} + 3 q^{81} + 3 q^{86} - 3 q^{88} - 3 q^{90} - 3 q^{96}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^12 - 3 * q^13 + 3 * q^16 + 3 * q^17 + 3 * q^20 - 3 * q^22 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^30 + 3 * q^32 + 3 * q^36 + 3 * q^39 - 3 * q^40 - 6 * q^44 - 3 * q^48 + 3 * q^49 - 6 * q^50 - 3 * q^51 - 3 * q^52 - 3 * q^55 + 3 * q^59 - 3 * q^60 - 6 * q^62 + 3 * q^66 + 3 * q^68 + 3 * q^72 + 3 * q^74 - 3 * q^75 - 3 * q^80 + 3 * q^81 + 3 * q^86 - 3 * q^88 - 3 * q^90 - 3 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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