LMFDB - Embedded newform 335.1.d.d.334.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [335,1,Mod(334,335)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(335, 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("335.334");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 335 = 5 \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	335.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:	$0.167186779232$
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 9.1.12594450625.1
Artin image:	$D_9$
Artin field:	Galois closure of 9.1.12594450625.1

Embedding invariants

Embedding label			334.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	335.334

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53209 q^{2} -1.87939 q^{3} +1.34730 q^{4} +1.00000 q^{5} -2.87939 q^{6} +0.347296 q^{7} +0.532089 q^{8} +2.53209 q^{9} +O(q^{10})$	\(q+1.53209 q^{2} -1.87939 q^{3} +1.34730 q^{4} +1.00000 q^{5} -2.87939 q^{6} +0.347296 q^{7} +0.532089 q^{8} +2.53209 q^{9} +1.53209 q^{10} -2.53209 q^{12} -1.00000 q^{13} +0.532089 q^{14} -1.87939 q^{15} -0.532089 q^{16} +3.87939 q^{18} -1.00000 q^{19} +1.34730 q^{20} -0.652704 q^{21} -1.00000 q^{24} +1.00000 q^{25} -1.53209 q^{26} -2.87939 q^{27} +0.467911 q^{28} -1.87939 q^{29} -2.87939 q^{30} -1.34730 q^{32} +0.347296 q^{35} +3.41147 q^{36} -1.53209 q^{38} +1.87939 q^{39} +0.532089 q^{40} -1.00000 q^{42} +1.53209 q^{43} +2.53209 q^{45} +1.00000 q^{48} -0.879385 q^{49} +1.53209 q^{50} -1.34730 q^{52} +0.347296 q^{53} -4.41147 q^{54} +0.184793 q^{56} +1.87939 q^{57} -2.87939 q^{58} +1.53209 q^{59} -2.53209 q^{60} +0.879385 q^{63} -1.53209 q^{64} -1.00000 q^{65} +1.00000 q^{67} +0.532089 q^{70} -1.00000 q^{71} +1.34730 q^{72} -1.87939 q^{75} -1.34730 q^{76} +2.87939 q^{78} -0.532089 q^{80} +2.87939 q^{81} -0.879385 q^{84} +2.34730 q^{86} +3.53209 q^{87} +0.347296 q^{89} +3.87939 q^{90} -0.347296 q^{91} -1.00000 q^{95} +2.53209 q^{96} +1.53209 q^{97} -1.34730 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{12} - 3 q^{13} - 3 q^{14} + 3 q^{16} + 6 q^{18} - 3 q^{19} + 3 q^{20} - 3 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} - 3 q^{30} - 3 q^{32} - 3 q^{40} - 3 q^{42} + 3 q^{45} + 3 q^{48} + 3 q^{49} - 3 q^{52} - 3 q^{54} - 3 q^{56} - 3 q^{58} - 3 q^{60} - 3 q^{63} - 3 q^{65} + 3 q^{67} - 3 q^{70} - 3 q^{71} + 3 q^{72} - 3 q^{76} + 3 q^{78} + 3 q^{80} + 3 q^{81} + 3 q^{84} + 6 q^{86} + 6 q^{87} + 6 q^{90} - 3 q^{95} + 3 q^{96} - 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 - 3 * q^12 - 3 * q^13 - 3 * q^14 + 3 * q^16 + 6 * q^18 - 3 * q^19 + 3 * q^20 - 3 * q^21 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 6 * q^28 - 3 * q^30 - 3 * q^32 - 3 * q^40 - 3 * q^42 + 3 * q^45 + 3 * q^48 + 3 * q^49 - 3 * q^52 - 3 * q^54 - 3 * q^56 - 3 * q^58 - 3 * q^60 - 3 * q^63 - 3 * q^65 + 3 * q^67 - 3 * q^70 - 3 * q^71 + 3 * q^72 - 3 * q^76 + 3 * q^78 + 3 * q^80 + 3 * q^81 + 3 * q^84 + 6 * q^86 + 6 * q^87 + 6 * q^90 - 3 * q^95 + 3 * q^96 - 3 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/335\mathbb{Z}\right)^\times$.

$n$	$136$	$202$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$2$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$3$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$3$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$4$	1.34730	1.34730
$5$	1.00000	1.00000
$5$	1.00000	1.00000
$6$	−2.87939	−2.87939
$7$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$7$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$8$	0.532089	0.532089
$9$	2.53209	2.53209
$10$	1.53209	1.53209
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−2.53209	−2.53209
$13$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$14$	0.532089	0.532089
$15$	−1.87939	−1.87939
$16$	−0.532089	−0.532089
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	3.87939	3.87939
$19$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$20$	1.34730	1.34730
$21$	−0.652704	−0.652704
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−1.00000
$25$	1.00000	1.00000
$26$	−1.53209	−1.53209
$27$	−2.87939	−2.87939
$28$	0.467911	0.467911
$29$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$29$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$30$	−2.87939	−2.87939
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.34730	−1.34730
$33$	0	0
$34$	0	0
$35$	0.347296	0.347296
$36$	3.41147	3.41147
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.53209	−1.53209
$39$	1.87939	1.87939
$40$	0.532089	0.532089
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.00000	−1.00000
$43$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$43$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$44$	0	0
$45$	2.53209	2.53209
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000	1.00000
$49$	−0.879385	−0.879385
$50$	1.53209	1.53209
$51$	0	0
$52$	−1.34730	−1.34730
$53$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$53$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$54$	−4.41147	−4.41147
$55$	0	0
$56$	0.184793	0.184793
$57$	1.87939	1.87939
$58$	−2.87939	−2.87939
$59$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$59$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$60$	−2.53209	−2.53209
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0.879385	0.879385
$64$	−1.53209	−1.53209
$65$	−1.00000	−1.00000
$66$	0	0
$67$	1.00000	1.00000
$67$	1.00000	1.00000
$68$	0	0
$69$	0	0
$70$	0.532089	0.532089
$71$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$72$	1.34730	1.34730
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.87939	−1.87939
$76$	−1.34730	−1.34730
$77$	0	0
$78$	2.87939	2.87939
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.532089	−0.532089
$81$	2.87939	2.87939
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−0.879385	−0.879385
$85$	0	0
$86$	2.34730	2.34730
$87$	3.53209	3.53209
$88$	0	0
$89$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$89$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$90$	3.87939	3.87939
$91$	−0.347296	−0.347296
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−1.00000
$96$	2.53209	2.53209
$97$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$97$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$98$	−1.34730	−1.34730
$99$	0	0
$100$	1.34730	1.34730
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−0.532089	−0.532089
$105$	−0.652704	−0.652704
$106$	0.532089	0.532089
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−3.87939	−3.87939
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−0.184793	−0.184793
$113$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$113$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$114$	2.87939	2.87939
$115$	0	0
$116$	−2.53209	−2.53209
$117$	−2.53209	−2.53209
$118$	2.34730	2.34730
$119$	0	0
$120$	−1.00000	−1.00000
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	1.00000
$126$	1.34730	1.34730
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	−2.87939	−2.87939
$130$	−1.53209	−1.53209
$131$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$131$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$132$	0	0
$133$	−0.347296	−0.347296
$134$	1.53209	1.53209
$135$	−2.87939	−2.87939
$136$	0	0
$137$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0.467911	0.467911
$141$	0	0
$142$	−1.53209	−1.53209
$143$	0	0
$144$	−1.34730	−1.34730
$145$	−1.87939	−1.87939
$146$	0	0
$147$	1.65270	1.65270
$148$	0	0
$149$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$149$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$150$	−2.87939	−2.87939
$151$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$151$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$152$	−0.532089	−0.532089
$153$	0	0
$154$	0	0
$155$	0	0
$156$	2.53209	2.53209
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−0.652704	−0.652704
$160$	−1.34730	−1.34730
$161$	0	0
$162$	4.41147	4.41147
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−0.347296	−0.347296
$169$	0	0
$170$	0	0
$171$	−2.53209	−2.53209
$172$	2.06418	2.06418
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	5.41147	5.41147
$175$	0.347296	0.347296
$176$	0	0
$177$	−2.87939	−2.87939
$178$	0.532089	0.532089
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	3.41147	3.41147
$181$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$181$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$182$	−0.532089	−0.532089
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−1.00000
$190$	−1.53209	−1.53209
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	2.87939	2.87939
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	2.34730	2.34730
$195$	1.87939	1.87939
$196$	−1.18479	−1.18479
$197$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$197$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$198$	0	0
$199$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$199$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$200$	0.532089	0.532089
$201$	−1.87939	−1.87939
$202$	0	0
$203$	−0.652704	−0.652704
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0.532089	0.532089
$209$	0	0
$210$	−1.00000	−1.00000
$211$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$211$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$212$	0.467911	0.467911
$213$	1.87939	1.87939
$214$	0	0
$215$	1.53209	1.53209
$216$	−1.53209	−1.53209
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−0.467911	−0.467911
$225$	2.53209	2.53209
$226$	−2.87939	−2.87939
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	2.53209	2.53209
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−1.00000	−1.00000
$233$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$233$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$234$	−3.87939	−3.87939
$235$	0	0
$236$	2.06418	2.06418
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.00000	1.00000
$241$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$241$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$242$	1.53209	1.53209
$243$	−2.53209	−2.53209
$244$	0	0
$245$	−0.879385	−0.879385
$246$	0	0
$247$	1.00000	1.00000
$248$	0	0
$249$	0	0
$250$	1.53209	1.53209
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.18479	1.18479
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−4.41147	−4.41147
$259$	0	0
$260$	−1.34730	−1.34730
$261$	−4.75877	−4.75877
$262$	−2.87939	−2.87939
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0.347296	0.347296
$266$	−0.532089	−0.532089
$267$	−0.652704	−0.652704
$268$	1.34730	1.34730
$269$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$269$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$270$	−4.41147	−4.41147
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0.652704	0.652704
$274$	−1.53209	−1.53209
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0.184793	0.184793
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−1.34730	−1.34730
$285$	1.87939	1.87939
$286$	0	0
$287$	0	0
$288$	−3.41147	−3.41147
$289$	1.00000	1.00000
$290$	−2.87939	−2.87939
$291$	−2.87939	−2.87939
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	2.53209	2.53209
$295$	1.53209	1.53209
$296$	0	0
$297$	0	0
$298$	2.34730	2.34730
$299$	0	0
$300$	−2.53209	−2.53209
$301$	0.532089	0.532089
$302$	2.34730	2.34730
$303$	0	0
$304$	0.532089	0.532089
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	1.00000	1.00000
$313$	2.00000	2.00000	1.00000			$0$
$313$	2.00000	2.00000	1.00000			$0$
$314$	0	0
$315$	0.879385	0.879385
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−1.00000	−1.00000
$319$	0	0
$320$	−1.53209	−1.53209
$321$	0	0
$322$	0	0
$323$	0	0
$324$	3.87939	3.87939
$325$	−1.00000	−1.00000
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.00000	1.00000
$336$	0.347296	0.347296
$337$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$337$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$338$	0	0
$339$	3.53209	3.53209
$340$	0	0
$341$	0	0
$342$	−3.87939	−3.87939
$343$	−0.652704	−0.652704
$344$	0.815207	0.815207
$345$	0	0
$346$	0	0
$347$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$347$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$348$	4.75877	4.75877
$349$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$349$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$350$	0.532089	0.532089
$351$	2.87939	2.87939
$352$	0	0
$353$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$353$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$354$	−4.41147	−4.41147
$355$	−1.00000	−1.00000
$356$	0.467911	0.467911
$357$	0	0
$358$	0	0
$359$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$359$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$360$	1.34730	1.34730
$361$	0	0
$362$	2.34730	2.34730
$363$	−1.87939	−1.87939
$364$	−0.467911	−0.467911
$365$	0	0
$366$	0	0
$367$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$367$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.120615	0.120615
$372$	0	0
$373$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$373$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$374$	0	0
$375$	−1.87939	−1.87939
$376$	0	0
$377$	1.87939	1.87939
$378$	−1.53209	−1.53209
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−1.34730	−1.34730
$381$	0	0
$382$	0	0
$383$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$383$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$384$	1.87939	1.87939
$385$	0	0
$386$	0	0
$387$	3.87939	3.87939
$388$	2.06418	2.06418
$389$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$390$	2.87939	2.87939
$391$	0	0
$392$	−0.467911	−0.467911
$393$	3.53209	3.53209
$394$	−2.87939	−2.87939
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0.532089	0.532089
$399$	0.652704	0.652704
$400$	−0.532089	−0.532089
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−2.87939	−2.87939
$403$	0	0
$404$	0	0
$405$	2.87939	2.87939
$406$	−1.00000	−1.00000
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	1.87939	1.87939
$412$	0	0
$413$	0.532089	0.532089
$414$	0	0
$415$	0	0
$416$	1.34730	1.34730
$417$	0	0
$418$	0	0
$419$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$420$	−0.879385	−0.879385
$421$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$421$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$422$	0.532089	0.532089
$423$	0	0
$424$	0.184793	0.184793
$425$	0	0
$426$	2.87939	2.87939
$427$	0	0
$428$	0	0
$429$	0	0
$430$	2.34730	2.34730
$431$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$431$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$432$	1.53209	1.53209
$433$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$433$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$434$	0	0
$435$	3.53209	3.53209
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$439$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$440$	0	0
$441$	−2.22668	−2.22668
$442$	0	0
$443$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$443$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$444$	0	0
$445$	0.347296	0.347296
$446$	0	0
$447$	−2.87939	−2.87939
$448$	−0.532089	−0.532089
$449$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$449$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$450$	3.87939	3.87939
$451$	0	0
$452$	−2.53209	−2.53209
$453$	−2.87939	−2.87939
$454$	0	0
$455$	−0.347296	−0.347296
$456$	1.00000	1.00000
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$461$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$462$	0	0
$463$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$463$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$464$	1.00000	1.00000
$465$	0	0
$466$	0.532089	0.532089
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−3.41147	−3.41147
$469$	0.347296	0.347296
$470$	0	0
$471$	0	0
$472$	0.815207	0.815207
$473$	0	0
$474$	0	0
$475$	−1.00000	−1.00000
$476$	0	0
$477$	0.879385	0.879385
$478$	0	0
$479$	2.00000	2.00000	1.00000			$0$
$479$	2.00000	2.00000	1.00000			$0$
$480$	2.53209	2.53209
$481$	0	0
$482$	0.532089	0.532089
$483$	0	0
$484$	1.34730	1.34730
$485$	1.53209	1.53209
$486$	−3.87939	−3.87939
$487$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$487$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$488$	0	0
$489$	0	0
$490$	−1.34730	−1.34730
$491$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$492$	0	0
$493$	0	0
$494$	1.53209	1.53209
$495$	0	0
$496$	0	0
$497$	−0.347296	−0.347296
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1.34730	1.34730
$501$	0	0
$502$	0	0
$503$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$503$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$504$	0.467911	0.467911
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	2.87939	2.87939
$514$	0	0
$515$	0	0
$516$	−3.87939	−3.87939
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−0.532089	−0.532089
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−7.29086	−7.29086
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−2.53209	−2.53209
$525$	−0.652704	−0.652704
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0.532089	0.532089
$531$	3.87939	3.87939
$532$	−0.467911	−0.467911
$533$	0	0
$534$	−1.00000	−1.00000
$535$	0	0
$536$	0.532089	0.532089
$537$	0	0
$538$	0.532089	0.532089
$539$	0	0
$540$	−3.87939	−3.87939
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	−2.87939	−2.87939
$544$	0	0
$545$	0	0
$546$	1.00000	1.00000
$547$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$547$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$548$	−1.34730	−1.34730
$549$	0	0
$550$	0	0
$551$	1.87939	1.87939
$552$	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
$559$	−1.53209	−1.53209
$560$	−0.184793	−0.184793
$561$	0	0
$562$	0	0
$563$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$563$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$564$	0	0
$565$	−1.87939	−1.87939
$566$	0	0
$567$	1.00000	1.00000
$568$	−0.532089	−0.532089
$569$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$569$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$570$	2.87939	2.87939
$571$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$571$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−3.87939	−3.87939
$577$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$577$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$578$	1.53209	1.53209
$579$	0	0
$580$	−2.53209	−2.53209
$581$	0	0
$582$	−4.41147	−4.41147
$583$	0	0
$584$	0	0
$585$	−2.53209	−2.53209
$586$	0	0
$587$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$587$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$588$	2.22668	2.22668
$589$	0	0
$590$	2.34730	2.34730
$591$	3.53209	3.53209
$592$	0	0
$593$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$593$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$594$	0	0
$595$	0	0
$596$	2.06418	2.06418
$597$	−0.652704	−0.652704
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1.00000	−1.00000
$601$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$601$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$602$	0.815207	0.815207
$603$	2.53209	2.53209
$604$	2.06418	2.06418
$605$	1.00000	1.00000
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.34730	1.34730
$609$	1.22668	1.22668
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$619$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.120615	0.120615
$624$	−1.00000	−1.00000
$625$	1.00000	1.00000
$626$	3.06418	3.06418
$627$	0	0
$628$	0	0
$629$	0	0
$630$	1.34730	1.34730
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−0.652704	−0.652704
$634$	0	0
$635$	0	0
$636$	−0.879385	−0.879385
$637$	0.879385	0.879385
$638$	0	0
$639$	−2.53209	−2.53209
$640$	−1.00000	−1.00000
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	−2.87939	−2.87939
$646$	0	0
$647$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$648$	1.53209	1.53209
$649$	0	0
$650$	−1.53209	−1.53209
$651$	0	0
$652$	0	0
$653$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$653$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$654$	0	0
$655$	−1.87939	−1.87939
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.00000	2.00000	1.00000			$0$
$659$	2.00000	2.00000	1.00000			$0$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.347296	−0.347296
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	1.53209	1.53209
$671$	0	0
$672$	0.879385	0.879385
$673$	2.00000	2.00000	1.00000			$0$
$673$	2.00000	2.00000	1.00000			$0$
$674$	−2.87939	−2.87939
$675$	−2.87939	−2.87939
$676$	0	0
$677$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$677$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$678$	5.41147	5.41147
$679$	0.532089	0.532089
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.00000	2.00000	1.00000			$0$
$683$	2.00000	2.00000	1.00000			$0$
$684$	−3.41147	−3.41147
$685$	−1.00000	−1.00000
$686$	−1.00000	−1.00000
$687$	0	0
$688$	−0.815207	−0.815207
$689$	−0.347296	−0.347296
$690$	0	0
$691$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$691$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$692$	0	0
$693$	0	0
$694$	0.532089	0.532089
$695$	0	0
$696$	1.87939	1.87939
$697$	0	0
$698$	−2.87939	−2.87939
$699$	−0.652704	−0.652704
$700$	0.467911	0.467911
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	4.41147	4.41147
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1.53209	−1.53209
$707$	0	0
$708$	−3.87939	−3.87939
$709$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$709$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$710$	−1.53209	−1.53209
$711$	0	0
$712$	0.184793	0.184793
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−2.87939	−2.87939
$719$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$720$	−1.34730	−1.34730
$721$	0	0
$722$	0	0
$723$	−0.652704	−0.652704
$724$	2.06418	2.06418
$725$	−1.87939	−1.87939
$726$	−2.87939	−2.87939
$727$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$727$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$728$	−0.184793	−0.184793
$729$	1.87939	1.87939
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$733$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$734$	2.34730	2.34730
$735$	1.65270	1.65270
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	−1.87939	−1.87939
$742$	0.184793	0.184793
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1.53209	1.53209
$746$	−2.87939	−2.87939
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−2.87939	−2.87939
$751$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$751$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$752$	0	0
$753$	0	0
$754$	2.87939	2.87939
$755$	1.53209	1.53209
$756$	−1.34730	−1.34730
$757$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$757$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$758$	0	0
$759$	0	0
$760$	−0.532089	−0.532089
$761$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	2.34730	2.34730
$767$	−1.53209	−1.53209
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	5.94356	5.94356
$775$	0	0
$776$	0.815207	0.815207
$777$	0	0
$778$	−1.53209	−1.53209
$779$	0	0
$780$	2.53209	2.53209
$781$	0	0
$782$	0	0
$783$	5.41147	5.41147
$784$	0.467911	0.467911
$785$	0	0
$786$	5.41147	5.41147
$787$	2.00000	2.00000	1.00000			$0$
$787$	2.00000	2.00000	1.00000			$0$
$788$	−2.53209	−2.53209
$789$	0	0
$790$	0	0
$791$	−0.652704	−0.652704
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−0.652704	−0.652704
$796$	0.467911	0.467911
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	1.00000	1.00000
$799$	0	0
$800$	−1.34730	−1.34730
$801$	0.879385	0.879385
$802$	0	0
$803$	0	0
$804$	−2.53209	−2.53209
$805$	0	0
$806$	0	0
$807$	−0.652704	−0.652704
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	4.41147	4.41147
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	−0.879385	−0.879385
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.53209	−1.53209
$818$	0	0
$819$	−0.879385	−0.879385
$820$	0	0
$821$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$821$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$822$	2.87939	2.87939
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0.815207	0.815207
$827$	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
$832$	1.53209	1.53209
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−1.53209	−1.53209
$839$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$839$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$840$	−0.347296	−0.347296
$841$	2.53209	2.53209
$842$	0.532089	0.532089
$843$	0	0
$844$	0.467911	0.467911
$845$	0	0
$846$	0	0
$847$	0.347296	0.347296
$848$	−0.184793	−0.184793
$849$	0	0
$850$	0	0
$851$	0	0
$852$	2.53209	2.53209
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−2.53209	−2.53209
$856$	0	0
$857$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$857$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$858$	0	0
$859$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$859$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$860$	2.06418	2.06418
$861$	0	0
$862$	0.532089	0.532089
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	3.87939	3.87939
$865$	0	0
$866$	0.532089	0.532089
$867$	−1.87939	−1.87939
$868$	0	0
$869$	0	0
$870$	5.41147	5.41147
$871$	−1.00000	−1.00000
$872$	0	0
$873$	3.87939	3.87939
$874$	0	0
$875$	0.347296	0.347296
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−2.87939	−2.87939
$879$	0	0
$880$	0	0
$881$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$882$	−3.41147	−3.41147
$883$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$884$	0	0
$885$	−2.87939	−2.87939
$886$	2.34730	2.34730
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0.532089	0.532089
$891$	0	0
$892$	0	0
$893$	0	0
$894$	−4.41147	−4.41147
$895$	0	0
$896$	−0.347296	−0.347296
$897$	0	0
$898$	−2.87939	−2.87939
$899$	0	0
$900$	3.41147	3.41147
$901$	0	0
$902$	0	0
$903$	−1.00000	−1.00000
$904$	−1.00000	−1.00000
$905$	1.53209	1.53209
$906$	−4.41147	−4.41147
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	−0.532089	−0.532089
$911$	2.00000	2.00000	1.00000			$0$
$911$	2.00000	2.00000	1.00000			$0$
$912$	−1.00000	−1.00000
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.652704	−0.652704
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	2.34730	2.34730
$923$	1.00000	1.00000
$924$	0	0
$925$	0	0
$926$	−2.87939	−2.87939
$927$	0	0
$928$	2.53209	2.53209
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0.879385	0.879385
$932$	0.467911	0.467911
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−1.34730	−1.34730
$937$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$937$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$938$	0.532089	0.532089
$939$	−3.75877	−3.75877
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−0.815207	−0.815207
$945$	−1.00000	−1.00000
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−1.53209	−1.53209
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	1.34730	1.34730
$955$	0	0
$956$	0	0
$957$	0	0
$958$	3.06418	3.06418
$959$	−0.347296	−0.347296
$960$	2.87939	2.87939
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0.467911	0.467911
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.532089	0.532089
$969$	0	0
$970$	2.34730	2.34730
$971$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$971$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$972$	−3.41147	−3.41147
$973$	0	0
$974$	−2.87939	−2.87939
$975$	1.87939	1.87939
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	−1.18479	−1.18479
$981$	0	0
$982$	−1.53209	−1.53209
$983$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$983$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$984$	0	0
$985$	−1.87939	−1.87939
$986$	0	0
$987$	0	0
$988$	1.34730	1.34730
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	−0.532089	−0.532089
$995$	0.347296	0.347296
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.1.d.d.334.3	yes	3
3.2	odd	2		3015.1.h.c.334.1		3
5.2	odd	4		1675.1.b.d.401.5		6
5.3	odd	4		1675.1.b.d.401.2		6
5.4	even	2		335.1.d.c.334.1	✓	3
15.14	odd	2		3015.1.h.d.334.3		3
67.66	odd	2		335.1.d.c.334.1	✓	3
201.200	even	2		3015.1.h.d.334.3		3
335.133	even	4		1675.1.b.d.401.5		6
335.267	even	4		1675.1.b.d.401.2		6
335.334	odd	2	CM	335.1.d.d.334.3	yes	3
1005.1004	even	2		3015.1.h.c.334.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.1.d.c.334.1	✓	3	5.4	even	2
335.1.d.c.334.1	✓	3	67.66	odd	2
335.1.d.d.334.3	yes	3	1.1	even	1	trivial
335.1.d.d.334.3	yes	3	335.334	odd	2	CM
1675.1.b.d.401.2		6	5.3	odd	4
1675.1.b.d.401.2		6	335.267	even	4
1675.1.b.d.401.5		6	5.2	odd	4
1675.1.b.d.401.5		6	335.133	even	4
3015.1.h.c.334.1		3	3.2	odd	2
3015.1.h.c.334.1		3	1005.1004	even	2
3015.1.h.d.334.3		3	15.14	odd	2
3015.1.h.d.334.3		3	201.200	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:	\( N \)	\(=\)	\( 335 = 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	335.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.53209 q^{2} -1.87939 q^{3} +1.34730 q^{4} +1.00000 q^{5} -2.87939 q^{6} +0.347296 q^{7} +0.532089 q^{8} +2.53209 q^{9} +O(q^{10})\)	\(q+1.53209 q^{2} -1.87939 q^{3} +1.34730 q^{4} +1.00000 q^{5} -2.87939 q^{6} +0.347296 q^{7} +0.532089 q^{8} +2.53209 q^{9} +1.53209 q^{10} -2.53209 q^{12} -1.00000 q^{13} +0.532089 q^{14} -1.87939 q^{15} -0.532089 q^{16} +3.87939 q^{18} -1.00000 q^{19} +1.34730 q^{20} -0.652704 q^{21} -1.00000 q^{24} +1.00000 q^{25} -1.53209 q^{26} -2.87939 q^{27} +0.467911 q^{28} -1.87939 q^{29} -2.87939 q^{30} -1.34730 q^{32} +0.347296 q^{35} +3.41147 q^{36} -1.53209 q^{38} +1.87939 q^{39} +0.532089 q^{40} -1.00000 q^{42} +1.53209 q^{43} +2.53209 q^{45} +1.00000 q^{48} -0.879385 q^{49} +1.53209 q^{50} -1.34730 q^{52} +0.347296 q^{53} -4.41147 q^{54} +0.184793 q^{56} +1.87939 q^{57} -2.87939 q^{58} +1.53209 q^{59} -2.53209 q^{60} +0.879385 q^{63} -1.53209 q^{64} -1.00000 q^{65} +1.00000 q^{67} +0.532089 q^{70} -1.00000 q^{71} +1.34730 q^{72} -1.87939 q^{75} -1.34730 q^{76} +2.87939 q^{78} -0.532089 q^{80} +2.87939 q^{81} -0.879385 q^{84} +2.34730 q^{86} +3.53209 q^{87} +0.347296 q^{89} +3.87939 q^{90} -0.347296 q^{91} -1.00000 q^{95} +2.53209 q^{96} +1.53209 q^{97} -1.34730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{12} - 3 q^{13} - 3 q^{14} + 3 q^{16} + 6 q^{18} - 3 q^{19} + 3 q^{20} - 3 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} - 3 q^{30} - 3 q^{32} - 3 q^{40} - 3 q^{42} + 3 q^{45} + 3 q^{48} + 3 q^{49} - 3 q^{52} - 3 q^{54} - 3 q^{56} - 3 q^{58} - 3 q^{60} - 3 q^{63} - 3 q^{65} + 3 q^{67} - 3 q^{70} - 3 q^{71} + 3 q^{72} - 3 q^{76} + 3 q^{78} + 3 q^{80} + 3 q^{81} + 3 q^{84} + 6 q^{86} + 6 q^{87} + 6 q^{90} - 3 q^{95} + 3 q^{96} - 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 - 3 * q^12 - 3 * q^13 - 3 * q^14 + 3 * q^16 + 6 * q^18 - 3 * q^19 + 3 * q^20 - 3 * q^21 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 6 * q^28 - 3 * q^30 - 3 * q^32 - 3 * q^40 - 3 * q^42 + 3 * q^45 + 3 * q^48 + 3 * q^49 - 3 * q^52 - 3 * q^54 - 3 * q^56 - 3 * q^58 - 3 * q^60 - 3 * q^63 - 3 * q^65 + 3 * q^67 - 3 * q^70 - 3 * q^71 + 3 * q^72 - 3 * q^76 + 3 * q^78 + 3 * q^80 + 3 * q^81 + 3 * q^84 + 6 * q^86 + 6 * q^87 + 6 * q^90 - 3 * q^95 + 3 * q^96 - 3 * q^98

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