LMFDB - Embedded newform 3311.1.h.l.3310.3

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			3310.3
Root			$-0.347296$ 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)

$f(q)$	$=$	$q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +1.53209 q^{5} +0.532089 q^{6} +1.00000 q^{7} +0.532089 q^{8} -0.879385 q^{9} +O(q^{10})$	\(q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +1.53209 q^{5} +0.532089 q^{6} +1.00000 q^{7} +0.532089 q^{8} -0.879385 q^{9} +2.34730 q^{10} +1.00000 q^{11} +0.467911 q^{12} -1.00000 q^{13} +1.53209 q^{14} +0.532089 q^{15} -0.532089 q^{16} -1.87939 q^{17} -1.34730 q^{18} +2.06418 q^{20} +0.347296 q^{21} +1.53209 q^{22} -1.00000 q^{23} +0.184793 q^{24} +1.34730 q^{25} -1.53209 q^{26} -0.652704 q^{27} +1.34730 q^{28} +0.347296 q^{29} +0.815207 q^{30} -1.34730 q^{32} +0.347296 q^{33} -2.87939 q^{34} +1.53209 q^{35} -1.18479 q^{36} -0.347296 q^{39} +0.815207 q^{40} -1.87939 q^{41} +0.532089 q^{42} +1.00000 q^{43} +1.34730 q^{44} -1.34730 q^{45} -1.53209 q^{46} -0.184793 q^{48} +1.00000 q^{49} +2.06418 q^{50} -0.652704 q^{51} -1.34730 q^{52} -1.87939 q^{53} -1.00000 q^{54} +1.53209 q^{55} +0.532089 q^{56} +0.532089 q^{58} +0.716881 q^{60} -0.879385 q^{63} -1.53209 q^{64} -1.53209 q^{65} +0.532089 q^{66} +1.53209 q^{67} -2.53209 q^{68} -0.347296 q^{69} +2.34730 q^{70} -0.467911 q^{72} +0.467911 q^{75} +1.00000 q^{77} -0.532089 q^{78} -0.815207 q^{80} +0.652704 q^{81} -2.87939 q^{82} +0.347296 q^{83} +0.467911 q^{84} -2.87939 q^{85} +1.53209 q^{86} +0.120615 q^{87} +0.532089 q^{88} +2.00000 q^{89} -2.06418 q^{90} -1.00000 q^{91} -1.34730 q^{92} -0.467911 q^{96} +1.53209 q^{98} -0.879385 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$2$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$3$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$3$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$4$	1.34730	1.34730
$5$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$5$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$6$	0.532089	0.532089
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	0.532089	0.532089
$9$	−0.879385	−0.879385
$10$	2.34730	2.34730
$11$	1.00000	1.00000
$11$	1.00000	1.00000
$12$	0.467911	0.467911
$13$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$14$	1.53209	1.53209
$15$	0.532089	0.532089
$16$	−0.532089	−0.532089
$17$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$17$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$18$	−1.34730	−1.34730
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	2.06418	2.06418
$21$	0.347296	0.347296
$22$	1.53209	1.53209
$23$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.184793	0.184793
$25$	1.34730	1.34730
$26$	−1.53209	−1.53209
$27$	−0.652704	−0.652704
$28$	1.34730	1.34730
$29$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$29$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$30$	0.815207	0.815207
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.34730	−1.34730
$33$	0.347296	0.347296
$34$	−2.87939	−2.87939
$35$	1.53209	1.53209
$36$	−1.18479	−1.18479
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−0.347296	−0.347296
$40$	0.815207	0.815207
$41$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$41$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$42$	0.532089	0.532089
$43$	1.00000	1.00000
$43$	1.00000	1.00000
$44$	1.34730	1.34730
$45$	−1.34730	−1.34730
$46$	−1.53209	−1.53209
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.184793	−0.184793
$49$	1.00000	1.00000
$50$	2.06418	2.06418
$51$	−0.652704	−0.652704
$52$	−1.34730	−1.34730
$53$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$53$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$54$	−1.00000	−1.00000
$55$	1.53209	1.53209
$56$	0.532089	0.532089
$57$	0	0
$58$	0.532089	0.532089
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0.716881	0.716881
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−0.879385	−0.879385
$64$	−1.53209	−1.53209
$65$	−1.53209	−1.53209
$66$	0.532089	0.532089
$67$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$67$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$68$	−2.53209	−2.53209
$69$	−0.347296	−0.347296
$70$	2.34730	2.34730
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.467911	−0.467911
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0.467911	0.467911
$76$	0	0
$77$	1.00000	1.00000
$78$	−0.532089	−0.532089
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.815207	−0.815207
$81$	0.652704	0.652704
$82$	−2.87939	−2.87939
$83$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$83$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$84$	0.467911	0.467911
$85$	−2.87939	−2.87939
$86$	1.53209	1.53209
$87$	0.120615	0.120615
$88$	0.532089	0.532089
$89$	2.00000	2.00000	1.00000			$0$
$89$	2.00000	2.00000	1.00000			$0$
$90$	−2.06418	−2.06418
$91$	−1.00000	−1.00000
$92$	−1.34730	−1.34730
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−0.467911	−0.467911
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.53209	1.53209
$99$	−0.879385	−0.879385
$100$	1.81521	1.81521
$101$	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			$\pi$
$104$	−0.532089	−0.532089
$105$	0.532089	0.532089
$106$	−2.87939	−2.87939
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.879385	−0.879385
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	2.34730	2.34730
$111$	0	0
$112$	−0.532089	−0.532089
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−1.53209	−1.53209
$116$	0.467911	0.467911
$117$	0.879385	0.879385
$118$	0	0
$119$	−1.87939	−1.87939
$120$	0.283119	0.283119
$121$	1.00000	1.00000
$122$	0	0
$123$	−0.652704	−0.652704
$124$	0	0
$125$	0.532089	0.532089
$126$	−1.34730	−1.34730
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	0.347296	0.347296
$130$	−2.34730	−2.34730
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0.467911	0.467911
$133$	0	0
$134$	2.34730	2.34730
$135$	−1.00000	−1.00000
$136$	−1.00000	−1.00000
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−0.532089	−0.532089
$139$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$139$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$140$	2.06418	2.06418
$141$	0	0
$142$	0	0
$143$	−1.00000	−1.00000
$144$	0.467911	0.467911
$145$	0.532089	0.532089
$146$	0	0
$147$	0.347296	0.347296
$148$	0	0
$149$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$149$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$150$	0.716881	0.716881
$151$	−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
$153$	1.65270	1.65270
$154$	1.53209	1.53209
$155$	0	0
$156$	−0.467911	−0.467911
$157$	2.00000	2.00000	1.00000			$0$
$157$	2.00000	2.00000	1.00000			$0$
$158$	0	0
$159$	−0.652704	−0.652704
$160$	−2.06418	−2.06418
$161$	−1.00000	−1.00000
$162$	1.00000	1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−2.53209	−2.53209
$165$	0.532089	0.532089
$166$	0.532089	0.532089
$167$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$167$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$168$	0.184793	0.184793
$169$	0	0
$170$	−4.41147	−4.41147
$171$	0	0
$172$	1.34730	1.34730
$173$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$173$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$174$	0.184793	0.184793
$175$	1.34730	1.34730
$176$	−0.532089	−0.532089
$177$	0	0
$178$	3.06418	3.06418
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−1.81521	−1.81521
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.53209	−1.53209
$183$	0	0
$184$	−0.532089	−0.532089
$185$	0	0
$186$	0	0
$187$	−1.87939	−1.87939
$188$	0	0
$189$	−0.652704	−0.652704
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.532089	−0.532089
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	−0.532089	−0.532089
$196$	1.34730	1.34730
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.34730	−1.34730
$199$	2.00000	2.00000	1.00000			$0$
$199$	2.00000	2.00000	1.00000			$0$
$200$	0.716881	0.716881
$201$	0.532089	0.532089
$202$	3.06418	3.06418
$203$	0.347296	0.347296
$204$	−0.879385	−0.879385
$205$	−2.87939	−2.87939
$206$	0	0
$207$	0.879385	0.879385
$208$	0.532089	0.532089
$209$	0	0
$210$	0.815207	0.815207
$211$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$211$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$212$	−2.53209	−2.53209
$213$	0	0
$214$	0	0
$215$	1.53209	1.53209
$216$	−0.347296	−0.347296
$217$	0	0
$218$	0	0
$219$	0	0
$220$	2.06418	2.06418
$221$	1.87939	1.87939
$222$	0	0
$223$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$223$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$224$	−1.34730	−1.34730
$225$	−1.18479	−1.18479
$226$	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			$\pi$
$230$	−2.34730	−2.34730
$231$	0.347296	0.347296
$232$	0.184793	0.184793
$233$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$233$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$234$	1.34730	1.34730
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−2.87939	−2.87939
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−0.283119	−0.283119
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.53209	1.53209
$243$	0.879385	0.879385
$244$	0	0
$245$	1.53209	1.53209
$246$	−1.00000	−1.00000
$247$	0	0
$248$	0	0
$249$	0.120615	0.120615
$250$	0.815207	0.815207
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.18479	−1.18479
$253$	−1.00000	−1.00000
$254$	0	0
$255$	−1.00000	−1.00000
$256$	0	0
$257$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$257$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$258$	0.532089	0.532089
$259$	0	0
$260$	−2.06418	−2.06418
$261$	−0.305407	−0.305407
$262$	0	0
$263$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$263$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$264$	0.184793	0.184793
$265$	−2.87939	−2.87939
$266$	0	0
$267$	0.694593	0.694593
$268$	2.06418	2.06418
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−1.53209	−1.53209
$271$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$271$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$272$	1.00000	1.00000
$273$	−0.347296	−0.347296
$274$	0	0
$275$	1.34730	1.34730
$276$	−0.467911	−0.467911
$277$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$277$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$278$	2.34730	2.34730
$279$	0	0
$280$	0.815207	0.815207
$281$	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
$286$	−1.53209	−1.53209
$287$	−1.87939	−1.87939
$288$	1.18479	1.18479
$289$	2.53209	2.53209
$290$	0.815207	0.815207
$291$	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	$0.666667\pi$
$294$	0.532089	0.532089
$295$	0	0
$296$	0	0
$297$	−0.652704	−0.652704
$298$	2.34730	2.34730
$299$	1.00000	1.00000
$300$	0.630415	0.630415
$301$	1.00000	1.00000
$302$	−1.53209	−1.53209
$303$	0.694593	0.694593
$304$	0	0
$305$	0	0
$306$	2.53209	2.53209
$307$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$307$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$308$	1.34730	1.34730
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−0.184793	−0.184793
$313$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$313$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$314$	3.06418	3.06418
$315$	−1.34730	−1.34730
$316$	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
$319$	0.347296	0.347296
$320$	−2.34730	−2.34730
$321$	0	0
$322$	−1.53209	−1.53209
$323$	0	0
$324$	0.879385	0.879385
$325$	−1.34730	−1.34730
$326$	0	0
$327$	0	0
$328$	−1.00000	−1.00000
$329$	0	0
$330$	0.815207	0.815207
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0.467911	0.467911
$333$	0	0
$334$	0.532089	0.532089
$335$	2.34730	2.34730
$336$	−0.184793	−0.184793
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	−3.87939	−3.87939
$341$	0	0
$342$	0	0
$343$	1.00000	1.00000
$344$	0.532089	0.532089
$345$	−0.532089	−0.532089
$346$	2.34730	2.34730
$347$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$347$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$348$	0.162504	0.162504
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	2.06418	2.06418
$351$	0.652704	0.652704
$352$	−1.34730	−1.34730
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	2.69459	2.69459
$357$	−0.652704	−0.652704
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.716881	−0.716881
$361$	1.00000	1.00000
$362$	0	0
$363$	0.347296	0.347296
$364$	−1.34730	−1.34730
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0.532089	0.532089
$369$	1.65270	1.65270
$370$	0	0
$371$	−1.87939	−1.87939
$372$	0	0
$373$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$373$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$374$	−2.87939	−2.87939
$375$	0.184793	0.184793
$376$	0	0
$377$	−0.347296	−0.347296
$378$	−1.00000	−1.00000
$379$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$379$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$380$	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	$0.666667\pi$
$384$	−0.347296	−0.347296
$385$	1.53209	1.53209
$386$	0	0
$387$	−0.879385	−0.879385
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	−0.815207	−0.815207
$391$	1.87939	1.87939
$392$	0.532089	0.532089
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−1.18479	−1.18479
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	3.06418	3.06418
$399$	0	0
$400$	−0.716881	−0.716881
$401$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$402$	0.815207	0.815207
$403$	0	0
$404$	2.69459	2.69459
$405$	1.00000	1.00000
$406$	0.532089	0.532089
$407$	0	0
$408$	−0.347296	−0.347296
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−4.41147	−4.41147
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.34730	1.34730
$415$	0.532089	0.532089
$416$	1.34730	1.34730
$417$	0.532089	0.532089
$418$	0	0
$419$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$419$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$420$	0.716881	0.716881
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−2.87939	−2.87939
$423$	0	0
$424$	−1.00000	−1.00000
$425$	−2.53209	−2.53209
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.347296	−0.347296
$430$	2.34730	2.34730
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.347296	0.347296
$433$	−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
$435$	0.184793	0.184793
$436$	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	$0.666667\pi$
$440$	0.815207	0.815207
$441$	−0.879385	−0.879385
$442$	2.87939	2.87939
$443$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$443$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$444$	0	0
$445$	3.06418	3.06418
$446$	−2.87939	−2.87939
$447$	0.532089	0.532089
$448$	−1.53209	−1.53209
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.81521	−1.81521
$451$	−1.87939	−1.87939
$452$	0	0
$453$	−0.347296	−0.347296
$454$	0	0
$455$	−1.53209	−1.53209
$456$	0	0
$457$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$457$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$458$	0	0
$459$	1.22668	1.22668
$460$	−2.06418	−2.06418
$461$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$461$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$462$	0.532089	0.532089
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−0.184793	−0.184793
$465$	0	0
$466$	−2.87939	−2.87939
$467$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$467$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$468$	1.18479	1.18479
$469$	1.53209	1.53209
$470$	0	0
$471$	0.694593	0.694593
$472$	0	0
$473$	1.00000	1.00000
$474$	0	0
$475$	0	0
$476$	−2.53209	−2.53209
$477$	1.65270	1.65270
$478$	0	0
$479$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$480$	−0.716881	−0.716881
$481$	0	0
$482$	0	0
$483$	−0.347296	−0.347296
$484$	1.34730	1.34730
$485$	0	0
$486$	1.34730	1.34730
$487$	−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
$490$	2.34730	2.34730
$491$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$491$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$492$	−0.879385	−0.879385
$493$	−0.652704	−0.652704
$494$	0	0
$495$	−1.34730	−1.34730
$496$	0	0
$497$	0	0
$498$	0.184793	0.184793
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.716881	0.716881
$501$	0.120615	0.120615
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−0.467911	−0.467911
$505$	3.06418	3.06418
$506$	−1.53209	−1.53209
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−1.53209	−1.53209
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	−2.87939	−2.87939
$515$	0	0
$516$	0.467911	0.467911
$517$	0	0
$518$	0	0
$519$	0.532089	0.532089
$520$	−0.815207	−0.815207
$521$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$522$	−0.467911	−0.467911
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0.467911	0.467911
$526$	−1.53209	−1.53209
$527$	0	0
$528$	−0.184793	−0.184793
$529$	0	0
$530$	−4.41147	−4.41147
$531$	0	0
$532$	0	0
$533$	1.87939	1.87939
$534$	1.06418	1.06418
$535$	0	0
$536$	0.815207	0.815207
$537$	0	0
$538$	0	0
$539$	1.00000	1.00000
$540$	−1.34730	−1.34730
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−2.87939	−2.87939
$543$	0	0
$544$	2.53209	2.53209
$545$	0	0
$546$	−0.532089	−0.532089
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	2.06418	2.06418
$551$	0	0
$552$	−0.184793	−0.184793
$553$	0	0
$554$	0.532089	0.532089
$555$	0	0
$556$	2.06418	2.06418
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−1.00000	−1.00000
$560$	−0.815207	−0.815207
$561$	−0.652704	−0.652704
$562$	0	0
$563$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$563$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$564$	0	0
$565$	0	0
$566$	−1.53209	−1.53209
$567$	0.652704	0.652704
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$571$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$572$	−1.34730	−1.34730
$573$	0	0
$574$	−2.87939	−2.87939
$575$	−1.34730	−1.34730
$576$	1.34730	1.34730
$577$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$577$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$578$	3.87939	3.87939
$579$	0	0
$580$	0.716881	0.716881
$581$	0.347296	0.347296
$582$	0	0
$583$	−1.87939	−1.87939
$584$	0	0
$585$	1.34730	1.34730
$586$	−1.53209	−1.53209
$587$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$588$	0.467911	0.467911
$589$	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
$595$	−2.87939	−2.87939
$596$	2.06418	2.06418
$597$	0.694593	0.694593
$598$	1.53209	1.53209
$599$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$599$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$600$	0.248970	0.248970
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	1.53209	1.53209
$603$	−1.34730	−1.34730
$604$	−1.34730	−1.34730
$605$	1.53209	1.53209
$606$	1.06418	1.06418
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0.120615	0.120615
$610$	0	0
$611$	0	0
$612$	2.22668	2.22668
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0.532089	0.532089
$615$	−1.00000	−1.00000
$616$	0.532089	0.532089
$617$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$617$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0.652704	0.652704
$622$	0	0
$623$	2.00000	2.00000
$624$	0.184793	0.184793
$625$	−0.532089	−0.532089
$626$	−2.87939	−2.87939
$627$	0	0
$628$	2.69459	2.69459
$629$	0	0
$630$	−2.06418	−2.06418
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−0.652704	−0.652704
$634$	−1.53209	−1.53209
$635$	0	0
$636$	−0.879385	−0.879385
$637$	−1.00000	−1.00000
$638$	0.532089	0.532089
$639$	0	0
$640$	−1.53209	−1.53209
$641$	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			$\pi$
$644$	−1.34730	−1.34730
$645$	0.532089	0.532089
$646$	0	0
$647$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.347296	0.347296
$649$	0	0
$650$	−2.06418	−2.06418
$651$	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			$\pi$
$660$	0.716881	0.716881
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0.652704	0.652704
$664$	0.184793	0.184793
$665$	0	0
$666$	0	0
$667$	−0.347296	−0.347296
$668$	0.467911	0.467911
$669$	−0.652704	−0.652704
$670$	3.59627	3.59627
$671$	0	0
$672$	−0.467911	−0.467911
$673$	−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
$675$	−0.879385	−0.879385
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	−1.53209	−1.53209
$681$	0	0
$682$	0	0
$683$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$683$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$684$	0	0
$685$	0	0
$686$	1.53209	1.53209
$687$	0	0
$688$	−0.532089	−0.532089
$689$	1.87939	1.87939
$690$	−0.815207	−0.815207
$691$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$691$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$692$	2.06418	2.06418
$693$	−0.879385	−0.879385
$694$	2.34730	2.34730
$695$	2.34730	2.34730
$696$	0.0641778	0.0641778
$697$	3.53209	3.53209
$698$	0	0
$699$	−0.652704	−0.652704
$700$	1.81521	1.81521
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	1.00000	1.00000
$703$	0	0
$704$	−1.53209	−1.53209
$705$	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
$712$	1.06418	1.06418
$713$	0	0
$714$	−1.00000	−1.00000
$715$	−1.53209	−1.53209
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.716881	0.716881
$721$	0	0
$722$	1.53209	1.53209
$723$	0	0
$724$	0	0
$725$	0.467911	0.467911
$726$	0.532089	0.532089
$727$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$727$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$728$	−0.532089	−0.532089
$729$	−0.347296	−0.347296
$730$	0	0
$731$	−1.87939	−1.87939
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0.532089	0.532089
$736$	1.34730	1.34730
$737$	1.53209	1.53209
$738$	2.53209	2.53209
$739$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$739$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$740$	0	0
$741$	0	0
$742$	−2.87939	−2.87939
$743$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$743$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$744$	0	0
$745$	2.34730	2.34730
$746$	2.34730	2.34730
$747$	−0.305407	−0.305407
$748$	−2.53209	−2.53209
$749$	0	0
$750$	0.283119	0.283119
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	−0.532089	−0.532089
$755$	−1.53209	−1.53209
$756$	−0.879385	−0.879385
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−2.87939	−2.87939
$759$	−0.347296	−0.347296
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.53209	2.53209
$766$	−1.53209	−1.53209
$767$	0	0
$768$	0	0
$769$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$769$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$770$	2.34730	2.34730
$771$	−0.652704	−0.652704
$772$	0	0
$773$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$773$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$774$	−1.34730	−1.34730
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	−0.716881	−0.716881
$781$	0	0
$782$	2.87939	2.87939
$783$	−0.226682	−0.226682
$784$	−0.532089	−0.532089
$785$	3.06418	3.06418
$786$	0	0
$787$	2.00000	2.00000	1.00000			$0$
$787$	2.00000	2.00000	1.00000			$0$
$788$	0	0
$789$	−0.347296	−0.347296
$790$	0	0
$791$	0	0
$792$	−0.467911	−0.467911
$793$	0	0
$794$	0	0
$795$	−1.00000	−1.00000
$796$	2.69459	2.69459
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1.81521	−1.81521
$801$	−1.75877	−1.75877
$802$	−1.53209	−1.53209
$803$	0	0
$804$	0.716881	0.716881
$805$	−1.53209	−1.53209
$806$	0	0
$807$	0	0
$808$	1.06418	1.06418
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.53209	1.53209
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0.467911	0.467911
$813$	−0.652704	−0.652704
$814$	0	0
$815$	0	0
$816$	0.347296	0.347296
$817$	0	0
$818$	0	0
$819$	0.879385	0.879385
$820$	−3.87939	−3.87939
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$823$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$824$	0	0
$825$	0.467911	0.467911
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	1.18479	1.18479
$829$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$829$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$830$	0.815207	0.815207
$831$	0.120615	0.120615
$832$	1.53209	1.53209
$833$	−1.87939	−1.87939
$834$	0.815207	0.815207
$835$	0.532089	0.532089
$836$	0	0
$837$	0	0
$838$	2.34730	2.34730
$839$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$839$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$840$	0.283119	0.283119
$841$	−0.879385	−0.879385
$842$	0	0
$843$	0	0
$844$	−2.53209	−2.53209
$845$	0	0
$846$	0	0
$847$	1.00000	1.00000
$848$	1.00000	1.00000
$849$	−0.347296	−0.347296
$850$	−3.87939	−3.87939
$851$	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
$857$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$857$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$858$	−0.532089	−0.532089
$859$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$859$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$860$	2.06418	2.06418
$861$	−0.652704	−0.652704
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.879385	0.879385
$865$	2.34730	2.34730
$866$	−1.53209	−1.53209
$867$	0.879385	0.879385
$868$	0	0
$869$	0	0
$870$	0.283119	0.283119
$871$	−1.53209	−1.53209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.532089	0.532089
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−1.53209	−1.53209
$879$	−0.347296	−0.347296
$880$	−0.815207	−0.815207
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−1.34730	−1.34730
$883$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$884$	2.53209	2.53209
$885$	0	0
$886$	−2.87939	−2.87939
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	4.69459	4.69459
$891$	0.652704	0.652704
$892$	−2.53209	−2.53209
$893$	0	0
$894$	0.815207	0.815207
$895$	0	0
$896$	−1.00000	−1.00000
$897$	0.347296	0.347296
$898$	0	0
$899$	0	0
$900$	−1.59627	−1.59627
$901$	3.53209	3.53209
$902$	−2.87939	−2.87939
$903$	0.347296	0.347296
$904$	0	0
$905$	0	0
$906$	−0.532089	−0.532089
$907$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$907$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$908$	0	0
$909$	−1.75877	−1.75877
$910$	−2.34730	−2.34730
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0.347296	0.347296
$914$	0.532089	0.532089
$915$	0	0
$916$	0	0
$917$	0	0
$918$	1.87939	1.87939
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	−0.815207	−0.815207
$921$	0.120615	0.120615
$922$	0.532089	0.532089
$923$	0	0
$924$	0.467911	0.467911
$925$	0	0
$926$	0	0
$927$	0	0
$928$	−0.467911	−0.467911
$929$	−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
$932$	−2.53209	−2.53209
$933$	0	0
$934$	0.532089	0.532089
$935$	−2.87939	−2.87939
$936$	0.467911	0.467911
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	2.34730	2.34730
$939$	−0.652704	−0.652704
$940$	0	0
$941$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$941$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$942$	1.06418	1.06418
$943$	1.87939	1.87939
$944$	0	0
$945$	−1.00000	−1.00000
$946$	1.53209	1.53209
$947$	−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
$951$	−0.347296	−0.347296
$952$	−1.00000	−1.00000
$953$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$953$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$954$	2.53209	2.53209
$955$	0	0
$956$	0	0
$957$	0.120615	0.120615
$958$	−1.53209	−1.53209
$959$	0	0
$960$	−0.815207	−0.815207
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	−0.532089	−0.532089
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.532089	0.532089
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	1.18479	1.18479
$973$	1.53209	1.53209
$974$	−2.87939	−2.87939
$975$	−0.467911	−0.467911
$976$	0	0
$977$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$977$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$978$	0	0
$979$	2.00000	2.00000
$980$	2.06418	2.06418
$981$	0	0
$982$	−2.87939	−2.87939
$983$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$983$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$984$	−0.347296	−0.347296
$985$	0	0
$986$	−1.00000	−1.00000
$987$	0	0
$988$	0	0
$989$	−1.00000	−1.00000
$990$	−2.06418	−2.06418
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.06418	3.06418
$996$	0.162504	0.162504
$997$	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.3310.3	yes	3
7.6	odd	2		3311.1.h.n.3310.3	yes	3
11.10	odd	2		3311.1.h.m.3310.1	yes	3
43.42	odd	2		3311.1.h.k.3310.1	✓	3
77.76	even	2		3311.1.h.k.3310.1	✓	3
301.300	even	2		3311.1.h.m.3310.1	yes	3
473.472	even	2		3311.1.h.n.3310.3	yes	3
3311.3310	odd	2	CM	3311.1.h.l.3310.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3311.1.h.k.3310.1	✓	3	43.42	odd	2
3311.1.h.k.3310.1	✓	3	77.76	even	2
3311.1.h.l.3310.3	yes	3	1.1	even	1	trivial
3311.1.h.l.3310.3	yes	3	3311.3310	odd	2	CM
3311.1.h.m.3310.1	yes	3	11.10	odd	2
3311.1.h.m.3310.1	yes	3	301.300	even	2
3311.1.h.n.3310.3	yes	3	7.6	odd	2
3311.1.h.n.3310.3	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)

\(f(q)\)	\(=\)	\(q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +1.53209 q^{5} +0.532089 q^{6} +1.00000 q^{7} +0.532089 q^{8} -0.879385 q^{9} +O(q^{10})\)	\(q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +1.53209 q^{5} +0.532089 q^{6} +1.00000 q^{7} +0.532089 q^{8} -0.879385 q^{9} +2.34730 q^{10} +1.00000 q^{11} +0.467911 q^{12} -1.00000 q^{13} +1.53209 q^{14} +0.532089 q^{15} -0.532089 q^{16} -1.87939 q^{17} -1.34730 q^{18} +2.06418 q^{20} +0.347296 q^{21} +1.53209 q^{22} -1.00000 q^{23} +0.184793 q^{24} +1.34730 q^{25} -1.53209 q^{26} -0.652704 q^{27} +1.34730 q^{28} +0.347296 q^{29} +0.815207 q^{30} -1.34730 q^{32} +0.347296 q^{33} -2.87939 q^{34} +1.53209 q^{35} -1.18479 q^{36} -0.347296 q^{39} +0.815207 q^{40} -1.87939 q^{41} +0.532089 q^{42} +1.00000 q^{43} +1.34730 q^{44} -1.34730 q^{45} -1.53209 q^{46} -0.184793 q^{48} +1.00000 q^{49} +2.06418 q^{50} -0.652704 q^{51} -1.34730 q^{52} -1.87939 q^{53} -1.00000 q^{54} +1.53209 q^{55} +0.532089 q^{56} +0.532089 q^{58} +0.716881 q^{60} -0.879385 q^{63} -1.53209 q^{64} -1.53209 q^{65} +0.532089 q^{66} +1.53209 q^{67} -2.53209 q^{68} -0.347296 q^{69} +2.34730 q^{70} -0.467911 q^{72} +0.467911 q^{75} +1.00000 q^{77} -0.532089 q^{78} -0.815207 q^{80} +0.652704 q^{81} -2.87939 q^{82} +0.347296 q^{83} +0.467911 q^{84} -2.87939 q^{85} +1.53209 q^{86} +0.120615 q^{87} +0.532089 q^{88} +2.00000 q^{89} -2.06418 q^{90} -1.00000 q^{91} -1.34730 q^{92} -0.467911 q^{96} +1.53209 q^{98} -0.879385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{4} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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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