LMFDB - Embedded newform 3311.1.h.l.3310.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3311,1,Mod(3310,3311)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3311, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3311.3310");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3311 = 7 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3311.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.65240425683$
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 9.1.120181251723841.1
Artin image:	$D_9$
Artin field:	Galois closure of 9.1.120181251723841.1

Embedding invariants

Embedding label			3310.2
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	3311.3310

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} +0.347296 q^{5} -0.652704 q^{6} +1.00000 q^{7} -0.652704 q^{8} +2.53209 q^{9} +O(q^{10})$	\(q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} +0.347296 q^{5} -0.652704 q^{6} +1.00000 q^{7} -0.652704 q^{8} +2.53209 q^{9} +0.120615 q^{10} +1.00000 q^{11} +1.65270 q^{12} -1.00000 q^{13} +0.347296 q^{14} -0.652704 q^{15} +0.652704 q^{16} +1.53209 q^{17} +0.879385 q^{18} -0.305407 q^{20} -1.87939 q^{21} +0.347296 q^{22} -1.00000 q^{23} +1.22668 q^{24} -0.879385 q^{25} -0.347296 q^{26} -2.87939 q^{27} -0.879385 q^{28} -1.87939 q^{29} -0.226682 q^{30} +0.879385 q^{32} -1.87939 q^{33} +0.532089 q^{34} +0.347296 q^{35} -2.22668 q^{36} +1.87939 q^{39} -0.226682 q^{40} +1.53209 q^{41} -0.652704 q^{42} +1.00000 q^{43} -0.879385 q^{44} +0.879385 q^{45} -0.347296 q^{46} -1.22668 q^{48} +1.00000 q^{49} -0.305407 q^{50} -2.87939 q^{51} +0.879385 q^{52} +1.53209 q^{53} -1.00000 q^{54} +0.347296 q^{55} -0.652704 q^{56} -0.652704 q^{58} +0.573978 q^{60} +2.53209 q^{63} -0.347296 q^{64} -0.347296 q^{65} -0.652704 q^{66} +0.347296 q^{67} -1.34730 q^{68} +1.87939 q^{69} +0.120615 q^{70} -1.65270 q^{72} +1.65270 q^{75} +1.00000 q^{77} +0.652704 q^{78} +0.226682 q^{80} +2.87939 q^{81} +0.532089 q^{82} -1.87939 q^{83} +1.65270 q^{84} +0.532089 q^{85} +0.347296 q^{86} +3.53209 q^{87} -0.652704 q^{88} +2.00000 q^{89} +0.305407 q^{90} -1.00000 q^{91} +0.879385 q^{92} -1.65270 q^{96} +0.347296 q^{98} +2.53209 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{4} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^4 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{4} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 6 q^{10} + 3 q^{11} + 6 q^{12} - 3 q^{13} - 3 q^{15} + 3 q^{16} - 3 q^{18} - 3 q^{20} - 3 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{28} + 6 q^{30} - 3 q^{32} - 3 q^{34} + 6 q^{40} - 3 q^{42} + 3 q^{43} + 3 q^{44} - 3 q^{45} + 3 q^{48} + 3 q^{49} - 3 q^{50} - 3 q^{51} - 3 q^{52} - 3 q^{54} - 3 q^{56} - 3 q^{58} - 6 q^{60} + 3 q^{63} - 3 q^{66} - 3 q^{68} + 6 q^{70} - 6 q^{72} + 6 q^{75} + 3 q^{77} + 3 q^{78} - 6 q^{80} + 3 q^{81} - 3 q^{82} + 6 q^{84} - 3 q^{85} + 6 q^{87} - 3 q^{88} + 6 q^{89} + 3 q^{90} - 3 q^{91} - 3 q^{92} - 6 q^{96} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^4 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 6 * q^10 + 3 * q^11 + 6 * q^12 - 3 * q^13 - 3 * q^15 + 3 * q^16 - 3 * q^18 - 3 * q^20 - 3 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^28 + 6 * q^30 - 3 * q^32 - 3 * q^34 + 6 * q^40 - 3 * q^42 + 3 * q^43 + 3 * q^44 - 3 * q^45 + 3 * q^48 + 3 * q^49 - 3 * q^50 - 3 * q^51 - 3 * q^52 - 3 * q^54 - 3 * q^56 - 3 * q^58 - 6 * q^60 + 3 * q^63 - 3 * q^66 - 3 * q^68 + 6 * q^70 - 6 * q^72 + 6 * q^75 + 3 * q^77 + 3 * q^78 - 6 * q^80 + 3 * q^81 - 3 * q^82 + 6 * q^84 - 3 * q^85 + 6 * q^87 - 3 * q^88 + 6 * q^89 + 3 * q^90 - 3 * q^91 - 3 * q^92 - 6 * q^96 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times$.

$n$	$904$	$1893$	$2927$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$2$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$3$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$3$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$4$	−0.879385	−0.879385
$5$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$5$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$6$	−0.652704	−0.652704
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	−0.652704	−0.652704
$9$	2.53209	2.53209
$10$	0.120615	0.120615
$11$	1.00000	1.00000
$11$	1.00000	1.00000
$12$	1.65270	1.65270
$13$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$14$	0.347296	0.347296
$15$	−0.652704	−0.652704
$16$	0.652704	0.652704
$17$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$17$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$18$	0.879385	0.879385
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−0.305407	−0.305407
$21$	−1.87939	−1.87939
$22$	0.347296	0.347296
$23$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$24$	1.22668	1.22668
$25$	−0.879385	−0.879385
$26$	−0.347296	−0.347296
$27$	−2.87939	−2.87939
$28$	−0.879385	−0.879385
$29$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$29$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$30$	−0.226682	−0.226682
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.879385	0.879385
$33$	−1.87939	−1.87939
$34$	0.532089	0.532089
$35$	0.347296	0.347296
$36$	−2.22668	−2.22668
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	1.87939	1.87939
$40$	−0.226682	−0.226682
$41$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$41$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$42$	−0.652704	−0.652704
$43$	1.00000	1.00000
$43$	1.00000	1.00000
$44$	−0.879385	−0.879385
$45$	0.879385	0.879385
$46$	−0.347296	−0.347296
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.22668	−1.22668
$49$	1.00000	1.00000
$50$	−0.305407	−0.305407
$51$	−2.87939	−2.87939
$52$	0.879385	0.879385
$53$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$53$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$54$	−1.00000	−1.00000
$55$	0.347296	0.347296
$56$	−0.652704	−0.652704
$57$	0	0
$58$	−0.652704	−0.652704
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0.573978	0.573978
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	2.53209	2.53209
$64$	−0.347296	−0.347296
$65$	−0.347296	−0.347296
$66$	−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	$0.444444\pi$
$68$	−1.34730	−1.34730
$69$	1.87939	1.87939
$70$	0.120615	0.120615
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.65270	−1.65270
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.65270	1.65270
$76$	0	0
$77$	1.00000	1.00000
$78$	0.652704	0.652704
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.226682	0.226682
$81$	2.87939	2.87939
$82$	0.532089	0.532089
$83$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$83$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$84$	1.65270	1.65270
$85$	0.532089	0.532089
$86$	0.347296	0.347296
$87$	3.53209	3.53209
$88$	−0.652704	−0.652704
$89$	2.00000	2.00000	1.00000			$0$
$89$	2.00000	2.00000	1.00000			$0$
$90$	0.305407	0.305407
$91$	−1.00000	−1.00000
$92$	0.879385	0.879385
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.65270	−1.65270
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.347296	0.347296
$99$	2.53209	2.53209
$100$	0.773318	0.773318
$101$	2.00000	2.00000	1.00000			$0$
$101$	2.00000	2.00000	1.00000			$0$
$102$	−1.00000	−1.00000
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0.652704	0.652704
$105$	−0.652704	−0.652704
$106$	0.532089	0.532089
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	2.53209	2.53209
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0.120615	0.120615
$111$	0	0
$112$	0.652704	0.652704
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−0.347296	−0.347296
$116$	1.65270	1.65270
$117$	−2.53209	−2.53209
$118$	0	0
$119$	1.53209	1.53209
$120$	0.426022	0.426022
$121$	1.00000	1.00000
$122$	0	0
$123$	−2.87939	−2.87939
$124$	0	0
$125$	−0.652704	−0.652704
$126$	0.879385	0.879385
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	−1.87939	−1.87939
$130$	−0.120615	−0.120615
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	1.65270	1.65270
$133$	0	0
$134$	0.120615	0.120615
$135$	−1.00000	−1.00000
$136$	−1.00000	−1.00000
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0.652704	0.652704
$139$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$139$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$140$	−0.305407	−0.305407
$141$	0	0
$142$	0	0
$143$	−1.00000	−1.00000
$144$	1.65270	1.65270
$145$	−0.652704	−0.652704
$146$	0	0
$147$	−1.87939	−1.87939
$148$	0	0
$149$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$149$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$150$	0.573978	0.573978
$151$	−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
$153$	3.87939	3.87939
$154$	0.347296	0.347296
$155$	0	0
$156$	−1.65270	−1.65270
$157$	2.00000	2.00000	1.00000			$0$
$157$	2.00000	2.00000	1.00000			$0$
$158$	0	0
$159$	−2.87939	−2.87939
$160$	0.305407	0.305407
$161$	−1.00000	−1.00000
$162$	1.00000	1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−1.34730	−1.34730
$165$	−0.652704	−0.652704
$166$	−0.652704	−0.652704
$167$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$167$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$168$	1.22668	1.22668
$169$	0	0
$170$	0.184793	0.184793
$171$	0	0
$172$	−0.879385	−0.879385
$173$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$173$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$174$	1.22668	1.22668
$175$	−0.879385	−0.879385
$176$	0.652704	0.652704
$177$	0	0
$178$	0.694593	0.694593
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.773318	−0.773318
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−0.347296	−0.347296
$183$	0	0
$184$	0.652704	0.652704
$185$	0	0
$186$	0	0
$187$	1.53209	1.53209
$188$	0	0
$189$	−2.87939	−2.87939
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0.652704	0.652704
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0.652704	0.652704
$196$	−0.879385	−0.879385
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0.879385	0.879385
$199$	2.00000	2.00000	1.00000			$0$
$199$	2.00000	2.00000	1.00000			$0$
$200$	0.573978	0.573978
$201$	−0.652704	−0.652704
$202$	0.694593	0.694593
$203$	−1.87939	−1.87939
$204$	2.53209	2.53209
$205$	0.532089	0.532089
$206$	0	0
$207$	−2.53209	−2.53209
$208$	−0.652704	−0.652704
$209$	0	0
$210$	−0.226682	−0.226682
$211$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$211$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$212$	−1.34730	−1.34730
$213$	0	0
$214$	0	0
$215$	0.347296	0.347296
$216$	1.87939	1.87939
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−0.305407	−0.305407
$221$	−1.53209	−1.53209
$222$	0	0
$223$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$223$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$224$	0.879385	0.879385
$225$	−2.22668	−2.22668
$226$	0	0
$227$	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			$\pi$
$230$	−0.120615	−0.120615
$231$	−1.87939	−1.87939
$232$	1.22668	1.22668
$233$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$233$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$234$	−0.879385	−0.879385
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0.532089	0.532089
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−0.426022	−0.426022
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.347296	0.347296
$243$	−2.53209	−2.53209
$244$	0	0
$245$	0.347296	0.347296
$246$	−1.00000	−1.00000
$247$	0	0
$248$	0	0
$249$	3.53209	3.53209
$250$	−0.226682	−0.226682
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−2.22668	−2.22668
$253$	−1.00000	−1.00000
$254$	0	0
$255$	−1.00000	−1.00000
$256$	0	0
$257$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$257$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$258$	−0.652704	−0.652704
$259$	0	0
$260$	0.305407	0.305407
$261$	−4.75877	−4.75877
$262$	0	0
$263$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$263$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$264$	1.22668	1.22668
$265$	0.532089	0.532089
$266$	0	0
$267$	−3.75877	−3.75877
$268$	−0.305407	−0.305407
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.347296	−0.347296
$271$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$271$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$272$	1.00000	1.00000
$273$	1.87939	1.87939
$274$	0	0
$275$	−0.879385	−0.879385
$276$	−1.65270	−1.65270
$277$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$277$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$278$	0.120615	0.120615
$279$	0	0
$280$	−0.226682	−0.226682
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$283$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$284$	0	0
$285$	0	0
$286$	−0.347296	−0.347296
$287$	1.53209	1.53209
$288$	2.22668	2.22668
$289$	1.34730	1.34730
$290$	−0.226682	−0.226682
$291$	0	0
$292$	0	0
$293$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$294$	−0.652704	−0.652704
$295$	0	0
$296$	0	0
$297$	−2.87939	−2.87939
$298$	0.120615	0.120615
$299$	1.00000	1.00000
$300$	−1.45336	−1.45336
$301$	1.00000	1.00000
$302$	−0.347296	−0.347296
$303$	−3.75877	−3.75877
$304$	0	0
$305$	0	0
$306$	1.34730	1.34730
$307$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$307$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$308$	−0.879385	−0.879385
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−1.22668	−1.22668
$313$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$313$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$314$	0.694593	0.694593
$315$	0.879385	0.879385
$316$	0	0
$317$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$318$	−1.00000	−1.00000
$319$	−1.87939	−1.87939
$320$	−0.120615	−0.120615
$321$	0	0
$322$	−0.347296	−0.347296
$323$	0	0
$324$	−2.53209	−2.53209
$325$	0.879385	0.879385
$326$	0	0
$327$	0	0
$328$	−1.00000	−1.00000
$329$	0	0
$330$	−0.226682	−0.226682
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1.65270	1.65270
$333$	0	0
$334$	−0.652704	−0.652704
$335$	0.120615	0.120615
$336$	−1.22668	−1.22668
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	−0.467911	−0.467911
$341$	0	0
$342$	0	0
$343$	1.00000	1.00000
$344$	−0.652704	−0.652704
$345$	0.652704	0.652704
$346$	0.120615	0.120615
$347$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$347$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$348$	−3.10607	−3.10607
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−0.305407	−0.305407
$351$	2.87939	2.87939
$352$	0.879385	0.879385
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−1.75877	−1.75877
$357$	−2.87939	−2.87939
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.573978	−0.573978
$361$	1.00000	1.00000
$362$	0	0
$363$	−1.87939	−1.87939
$364$	0.879385	0.879385
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.652704	−0.652704
$369$	3.87939	3.87939
$370$	0	0
$371$	1.53209	1.53209
$372$	0	0
$373$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$373$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$374$	0.532089	0.532089
$375$	1.22668	1.22668
$376$	0	0
$377$	1.87939	1.87939
$378$	−1.00000	−1.00000
$379$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$379$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$384$	1.87939	1.87939
$385$	0.347296	0.347296
$386$	0	0
$387$	2.53209	2.53209
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0.226682	0.226682
$391$	−1.53209	−1.53209
$392$	−0.652704	−0.652704
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−2.22668	−2.22668
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0.694593	0.694593
$399$	0	0
$400$	−0.573978	−0.573978
$401$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$402$	−0.226682	−0.226682
$403$	0	0
$404$	−1.75877	−1.75877
$405$	1.00000	1.00000
$406$	−0.652704	−0.652704
$407$	0	0
$408$	1.87939	1.87939
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0.184793	0.184793
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−0.879385	−0.879385
$415$	−0.652704	−0.652704
$416$	−0.879385	−0.879385
$417$	−0.652704	−0.652704
$418$	0	0
$419$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$419$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$420$	0.573978	0.573978
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0.532089	0.532089
$423$	0	0
$424$	−1.00000	−1.00000
$425$	−1.34730	−1.34730
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.87939	1.87939
$430$	0.120615	0.120615
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.87939	−1.87939
$433$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$434$	0	0
$435$	1.22668	1.22668
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$440$	−0.226682	−0.226682
$441$	2.53209	2.53209
$442$	−0.532089	−0.532089
$443$	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
$445$	0.694593	0.694593
$446$	0.532089	0.532089
$447$	−0.652704	−0.652704
$448$	−0.347296	−0.347296
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−0.773318	−0.773318
$451$	1.53209	1.53209
$452$	0	0
$453$	1.87939	1.87939
$454$	0	0
$455$	−0.347296	−0.347296
$456$	0	0
$457$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$457$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$458$	0	0
$459$	−4.41147	−4.41147
$460$	0.305407	0.305407
$461$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$461$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$462$	−0.652704	−0.652704
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−1.22668	−1.22668
$465$	0	0
$466$	0.532089	0.532089
$467$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$467$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$468$	2.22668	2.22668
$469$	0.347296	0.347296
$470$	0	0
$471$	−3.75877	−3.75877
$472$	0	0
$473$	1.00000	1.00000
$474$	0	0
$475$	0	0
$476$	−1.34730	−1.34730
$477$	3.87939	3.87939
$478$	0	0
$479$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$480$	−0.573978	−0.573978
$481$	0	0
$482$	0	0
$483$	1.87939	1.87939
$484$	−0.879385	−0.879385
$485$	0	0
$486$	−0.879385	−0.879385
$487$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$487$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$488$	0	0
$489$	0	0
$490$	0.120615	0.120615
$491$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$491$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$492$	2.53209	2.53209
$493$	−2.87939	−2.87939
$494$	0	0
$495$	0.879385	0.879385
$496$	0	0
$497$	0	0
$498$	1.22668	1.22668
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.573978	0.573978
$501$	3.53209	3.53209
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−1.65270	−1.65270
$505$	0.694593	0.694593
$506$	−0.347296	−0.347296
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−0.347296	−0.347296
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	0.532089	0.532089
$515$	0	0
$516$	1.65270	1.65270
$517$	0	0
$518$	0	0
$519$	−0.652704	−0.652704
$520$	0.226682	0.226682
$521$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$522$	−1.65270	−1.65270
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	1.65270	1.65270
$526$	−0.347296	−0.347296
$527$	0	0
$528$	−1.22668	−1.22668
$529$	0	0
$530$	0.184793	0.184793
$531$	0	0
$532$	0	0
$533$	−1.53209	−1.53209
$534$	−1.30541	−1.30541
$535$	0	0
$536$	−0.226682	−0.226682
$537$	0	0
$538$	0	0
$539$	1.00000	1.00000
$540$	0.879385	0.879385
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0.532089	0.532089
$543$	0	0
$544$	1.34730	1.34730
$545$	0	0
$546$	0.652704	0.652704
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−0.305407	−0.305407
$551$	0	0
$552$	−1.22668	−1.22668
$553$	0	0
$554$	−0.652704	−0.652704
$555$	0	0
$556$	−0.305407	−0.305407
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−1.00000	−1.00000
$560$	0.226682	0.226682
$561$	−2.87939	−2.87939
$562$	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
$565$	0	0
$566$	−0.347296	−0.347296
$567$	2.87939	2.87939
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$571$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$572$	0.879385	0.879385
$573$	0	0
$574$	0.532089	0.532089
$575$	0.879385	0.879385
$576$	−0.879385	−0.879385
$577$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$577$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$578$	0.467911	0.467911
$579$	0	0
$580$	0.573978	0.573978
$581$	−1.87939	−1.87939
$582$	0	0
$583$	1.53209	1.53209
$584$	0	0
$585$	−0.879385	−0.879385
$586$	−0.347296	−0.347296
$587$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$588$	1.65270	1.65270
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−1.00000	−1.00000
$595$	0.532089	0.532089
$596$	−0.305407	−0.305407
$597$	−3.75877	−3.75877
$598$	0.347296	0.347296
$599$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$599$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$600$	−1.07873	−1.07873
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0.347296	0.347296
$603$	0.879385	0.879385
$604$	0.879385	0.879385
$605$	0.347296	0.347296
$606$	−1.30541	−1.30541
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	3.53209	3.53209
$610$	0	0
$611$	0	0
$612$	−3.41147	−3.41147
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−0.652704	−0.652704
$615$	−1.00000	−1.00000
$616$	−0.652704	−0.652704
$617$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$617$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	2.87939	2.87939
$622$	0	0
$623$	2.00000	2.00000
$624$	1.22668	1.22668
$625$	0.652704	0.652704
$626$	0.532089	0.532089
$627$	0	0
$628$	−1.75877	−1.75877
$629$	0	0
$630$	0.305407	0.305407
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−2.87939	−2.87939
$634$	−0.347296	−0.347296
$635$	0	0
$636$	2.53209	2.53209
$637$	−1.00000	−1.00000
$638$	−0.652704	−0.652704
$639$	0	0
$640$	−0.347296	−0.347296
$641$	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			$\pi$
$644$	0.879385	0.879385
$645$	−0.652704	−0.652704
$646$	0	0
$647$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$648$	−1.87939	−1.87939
$649$	0	0
$650$	0.305407	0.305407
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	1.00000	1.00000
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0.573978	0.573978
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	2.87939	2.87939
$664$	1.22668	1.22668
$665$	0	0
$666$	0	0
$667$	1.87939	1.87939
$668$	1.65270	1.65270
$669$	−2.87939	−2.87939
$670$	0.0418891	0.0418891
$671$	0	0
$672$	−1.65270	−1.65270
$673$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$674$	0	0
$675$	2.53209	2.53209
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	−0.347296	−0.347296
$681$	0	0
$682$	0	0
$683$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$683$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$684$	0	0
$685$	0	0
$686$	0.347296	0.347296
$687$	0	0
$688$	0.652704	0.652704
$689$	−1.53209	−1.53209
$690$	0.226682	0.226682
$691$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$691$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$692$	−0.305407	−0.305407
$693$	2.53209	2.53209
$694$	0.120615	0.120615
$695$	0.120615	0.120615
$696$	−2.30541	−2.30541
$697$	2.34730	2.34730
$698$	0	0
$699$	−2.87939	−2.87939
$700$	0.773318	0.773318
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	1.00000	1.00000
$703$	0	0
$704$	−0.347296	−0.347296
$705$	0	0
$706$	0	0
$707$	2.00000	2.00000
$708$	0	0
$709$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$709$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$710$	0	0
$711$	0	0
$712$	−1.30541	−1.30541
$713$	0	0
$714$	−1.00000	−1.00000
$715$	−0.347296	−0.347296
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.573978	0.573978
$721$	0	0
$722$	0.347296	0.347296
$723$	0	0
$724$	0	0
$725$	1.65270	1.65270
$726$	−0.652704	−0.652704
$727$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$727$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$728$	0.652704	0.652704
$729$	1.87939	1.87939
$730$	0	0
$731$	1.53209	1.53209
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−0.652704	−0.652704
$736$	−0.879385	−0.879385
$737$	0.347296	0.347296
$738$	1.34730	1.34730
$739$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$739$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$740$	0	0
$741$	0	0
$742$	0.532089	0.532089
$743$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$743$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$744$	0	0
$745$	0.120615	0.120615
$746$	0.120615	0.120615
$747$	−4.75877	−4.75877
$748$	−1.34730	−1.34730
$749$	0	0
$750$	0.426022	0.426022
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0.652704	0.652704
$755$	−0.347296	−0.347296
$756$	2.53209	2.53209
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0.532089	0.532089
$759$	1.87939	1.87939
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.34730	1.34730
$766$	−0.347296	−0.347296
$767$	0	0
$768$	0	0
$769$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$769$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$770$	0.120615	0.120615
$771$	−2.87939	−2.87939
$772$	0	0
$773$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$773$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$774$	0.879385	0.879385
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	−0.573978	−0.573978
$781$	0	0
$782$	−0.532089	−0.532089
$783$	5.41147	5.41147
$784$	0.652704	0.652704
$785$	0.694593	0.694593
$786$	0	0
$787$	2.00000	2.00000	1.00000			$0$
$787$	2.00000	2.00000	1.00000			$0$
$788$	0	0
$789$	1.87939	1.87939
$790$	0	0
$791$	0	0
$792$	−1.65270	−1.65270
$793$	0	0
$794$	0	0
$795$	−1.00000	−1.00000
$796$	−1.75877	−1.75877
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−0.773318	−0.773318
$801$	5.06418	5.06418
$802$	−0.347296	−0.347296
$803$	0	0
$804$	0.573978	0.573978
$805$	−0.347296	−0.347296
$806$	0	0
$807$	0	0
$808$	−1.30541	−1.30541
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0.347296	0.347296
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1.65270	1.65270
$813$	−2.87939	−2.87939
$814$	0	0
$815$	0	0
$816$	−1.87939	−1.87939
$817$	0	0
$818$	0	0
$819$	−2.53209	−2.53209
$820$	−0.467911	−0.467911
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$823$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$824$	0	0
$825$	1.65270	1.65270
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	2.22668	2.22668
$829$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$829$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$830$	−0.226682	−0.226682
$831$	3.53209	3.53209
$832$	0.347296	0.347296
$833$	1.53209	1.53209
$834$	−0.226682	−0.226682
$835$	−0.652704	−0.652704
$836$	0	0
$837$	0	0
$838$	0.120615	0.120615
$839$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$839$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$840$	0.426022	0.426022
$841$	2.53209	2.53209
$842$	0	0
$843$	0	0
$844$	−1.34730	−1.34730
$845$	0	0
$846$	0	0
$847$	1.00000	1.00000
$848$	1.00000	1.00000
$849$	1.87939	1.87939
$850$	−0.467911	−0.467911
$851$	0	0
$852$	0	0
$853$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$853$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$857$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$858$	0.652704	0.652704
$859$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$859$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$860$	−0.305407	−0.305407
$861$	−2.87939	−2.87939
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−2.53209	−2.53209
$865$	0.120615	0.120615
$866$	−0.347296	−0.347296
$867$	−2.53209	−2.53209
$868$	0	0
$869$	0	0
$870$	0.426022	0.426022
$871$	−0.347296	−0.347296
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.652704	−0.652704
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.347296	−0.347296
$879$	1.87939	1.87939
$880$	0.226682	0.226682
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.879385	0.879385
$883$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$884$	1.34730	1.34730
$885$	0	0
$886$	0.532089	0.532089
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0.241230	0.241230
$891$	2.87939	2.87939
$892$	−1.34730	−1.34730
$893$	0	0
$894$	−0.226682	−0.226682
$895$	0	0
$896$	−1.00000	−1.00000
$897$	−1.87939	−1.87939
$898$	0	0
$899$	0	0
$900$	1.95811	1.95811
$901$	2.34730	2.34730
$902$	0.532089	0.532089
$903$	−1.87939	−1.87939
$904$	0	0
$905$	0	0
$906$	0.652704	0.652704
$907$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$907$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$908$	0	0
$909$	5.06418	5.06418
$910$	−0.120615	−0.120615
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1.87939	−1.87939
$914$	−0.652704	−0.652704
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.53209	−1.53209
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0.226682	0.226682
$921$	3.53209	3.53209
$922$	−0.652704	−0.652704
$923$	0	0
$924$	1.65270	1.65270
$925$	0	0
$926$	0	0
$927$	0	0
$928$	−1.65270	−1.65270
$929$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$930$	0	0
$931$	0	0
$932$	−1.34730	−1.34730
$933$	0	0
$934$	−0.652704	−0.652704
$935$	0.532089	0.532089
$936$	1.65270	1.65270
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0.120615	0.120615
$939$	−2.87939	−2.87939
$940$	0	0
$941$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$941$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$942$	−1.30541	−1.30541
$943$	−1.53209	−1.53209
$944$	0	0
$945$	−1.00000	−1.00000
$946$	0.347296	0.347296
$947$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$947$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.87939	1.87939
$952$	−1.00000	−1.00000
$953$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$953$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$954$	1.34730	1.34730
$955$	0	0
$956$	0	0
$957$	3.53209	3.53209
$958$	−0.347296	−0.347296
$959$	0	0
$960$	0.226682	0.226682
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0.652704	0.652704
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.652704	−0.652704
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	2.22668	2.22668
$973$	0.347296	0.347296
$974$	0.532089	0.532089
$975$	−1.65270	−1.65270
$976$	0	0
$977$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$977$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$978$	0	0
$979$	2.00000	2.00000
$980$	−0.305407	−0.305407
$981$	0	0
$982$	0.532089	0.532089
$983$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$983$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$984$	1.87939	1.87939
$985$	0	0
$986$	−1.00000	−1.00000
$987$	0	0
$988$	0	0
$989$	−1.00000	−1.00000
$990$	0.305407	0.305407
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.694593	0.694593
$996$	−3.10607	−3.10607
$997$	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	3311.1.h.l.3310.2	yes	3
7.6	odd	2		3311.1.h.n.3310.2	yes	3
11.10	odd	2		3311.1.h.m.3310.2	yes	3
43.42	odd	2		3311.1.h.k.3310.2	✓	3
77.76	even	2		3311.1.h.k.3310.2	✓	3
301.300	even	2		3311.1.h.m.3310.2	yes	3
473.472	even	2		3311.1.h.n.3310.2	yes	3
3311.3310	odd	2	CM	3311.1.h.l.3310.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3311.1.h.k.3310.2	✓	3	43.42	odd	2
3311.1.h.k.3310.2	✓	3	77.76	even	2
3311.1.h.l.3310.2	yes	3	1.1	even	1	trivial
3311.1.h.l.3310.2	yes	3	3311.3310	odd	2	CM
3311.1.h.m.3310.2	yes	3	11.10	odd	2
3311.1.h.m.3310.2	yes	3	301.300	even	2
3311.1.h.n.3310.2	yes	3	7.6	odd	2
3311.1.h.n.3310.2	yes	3	473.472	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:	\( N \)	\(=\)	\( 3311 = 7 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3311.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} +0.347296 q^{5} -0.652704 q^{6} +1.00000 q^{7} -0.652704 q^{8} +2.53209 q^{9} +O(q^{10})\)	\(q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} +0.347296 q^{5} -0.652704 q^{6} +1.00000 q^{7} -0.652704 q^{8} +2.53209 q^{9} +0.120615 q^{10} +1.00000 q^{11} +1.65270 q^{12} -1.00000 q^{13} +0.347296 q^{14} -0.652704 q^{15} +0.652704 q^{16} +1.53209 q^{17} +0.879385 q^{18} -0.305407 q^{20} -1.87939 q^{21} +0.347296 q^{22} -1.00000 q^{23} +1.22668 q^{24} -0.879385 q^{25} -0.347296 q^{26} -2.87939 q^{27} -0.879385 q^{28} -1.87939 q^{29} -0.226682 q^{30} +0.879385 q^{32} -1.87939 q^{33} +0.532089 q^{34} +0.347296 q^{35} -2.22668 q^{36} +1.87939 q^{39} -0.226682 q^{40} +1.53209 q^{41} -0.652704 q^{42} +1.00000 q^{43} -0.879385 q^{44} +0.879385 q^{45} -0.347296 q^{46} -1.22668 q^{48} +1.00000 q^{49} -0.305407 q^{50} -2.87939 q^{51} +0.879385 q^{52} +1.53209 q^{53} -1.00000 q^{54} +0.347296 q^{55} -0.652704 q^{56} -0.652704 q^{58} +0.573978 q^{60} +2.53209 q^{63} -0.347296 q^{64} -0.347296 q^{65} -0.652704 q^{66} +0.347296 q^{67} -1.34730 q^{68} +1.87939 q^{69} +0.120615 q^{70} -1.65270 q^{72} +1.65270 q^{75} +1.00000 q^{77} +0.652704 q^{78} +0.226682 q^{80} +2.87939 q^{81} +0.532089 q^{82} -1.87939 q^{83} +1.65270 q^{84} +0.532089 q^{85} +0.347296 q^{86} +3.53209 q^{87} -0.652704 q^{88} +2.00000 q^{89} +0.305407 q^{90} -1.00000 q^{91} +0.879385 q^{92} -1.65270 q^{96} +0.347296 q^{98} +2.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{4} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^4 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{4} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 6 q^{10} + 3 q^{11} + 6 q^{12} - 3 q^{13} - 3 q^{15} + 3 q^{16} - 3 q^{18} - 3 q^{20} - 3 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{28} + 6 q^{30} - 3 q^{32} - 3 q^{34} + 6 q^{40} - 3 q^{42} + 3 q^{43} + 3 q^{44} - 3 q^{45} + 3 q^{48} + 3 q^{49} - 3 q^{50} - 3 q^{51} - 3 q^{52} - 3 q^{54} - 3 q^{56} - 3 q^{58} - 6 q^{60} + 3 q^{63} - 3 q^{66} - 3 q^{68} + 6 q^{70} - 6 q^{72} + 6 q^{75} + 3 q^{77} + 3 q^{78} - 6 q^{80} + 3 q^{81} - 3 q^{82} + 6 q^{84} - 3 q^{85} + 6 q^{87} - 3 q^{88} + 6 q^{89} + 3 q^{90} - 3 q^{91} - 3 q^{92} - 6 q^{96} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^4 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 6 * q^10 + 3 * q^11 + 6 * q^12 - 3 * q^13 - 3 * q^15 + 3 * q^16 - 3 * q^18 - 3 * q^20 - 3 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^28 + 6 * q^30 - 3 * q^32 - 3 * q^34 + 6 * q^40 - 3 * q^42 + 3 * q^43 + 3 * q^44 - 3 * q^45 + 3 * q^48 + 3 * q^49 - 3 * q^50 - 3 * q^51 - 3 * q^52 - 3 * q^54 - 3 * q^56 - 3 * q^58 - 6 * q^60 + 3 * q^63 - 3 * q^66 - 3 * q^68 + 6 * q^70 - 6 * q^72 + 6 * q^75 + 3 * q^77 + 3 * q^78 - 6 * q^80 + 3 * q^81 - 3 * q^82 + 6 * q^84 - 3 * q^85 + 6 * q^87 - 3 * q^88 + 6 * q^89 + 3 * q^90 - 3 * q^91 - 3 * q^92 - 6 * q^96 + 3 * q^99

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