LMFDB - Embedded newform 3311.1.h.p.3310.6

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:	$6$
Coefficient field:	$\Q(\zeta_{36})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 9x^{2} - 3 $ x^6 - 6x^4 + 9x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

Embedding invariants

Embedding label			3310.6
Root			$-1.96962$ 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)

$f(q)$	$=$	$q+1.53209 q^{2} +1.96962 q^{3} +1.34730 q^{4} -1.28558 q^{5} +3.01763 q^{6} +1.00000 q^{7} +0.532089 q^{8} +2.87939 q^{9} +O(q^{10})$	\(q+1.53209 q^{2} +1.96962 q^{3} +1.34730 q^{4} -1.28558 q^{5} +3.01763 q^{6} +1.00000 q^{7} +0.532089 q^{8} +2.87939 q^{9} -1.96962 q^{10} -1.00000 q^{11} +2.65366 q^{12} -1.73205 q^{13} +1.53209 q^{14} -2.53209 q^{15} -0.532089 q^{16} +0.684040 q^{17} +4.41147 q^{18} -1.73205 q^{20} +1.96962 q^{21} -1.53209 q^{22} +1.00000 q^{23} +1.04801 q^{24} +0.652704 q^{25} -2.65366 q^{26} +3.70167 q^{27} +1.34730 q^{28} -0.347296 q^{29} -3.87939 q^{30} -1.34730 q^{32} -1.96962 q^{33} +1.04801 q^{34} -1.28558 q^{35} +3.87939 q^{36} -3.41147 q^{39} -0.684040 q^{40} -0.684040 q^{41} +3.01763 q^{42} -1.00000 q^{43} -1.34730 q^{44} -3.70167 q^{45} +1.53209 q^{46} -1.04801 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.34730 q^{51} -2.33359 q^{52} -1.87939 q^{53} +5.67128 q^{54} +1.28558 q^{55} +0.532089 q^{56} -0.532089 q^{58} -3.41147 q^{60} +2.87939 q^{63} -1.53209 q^{64} +2.22668 q^{65} -3.01763 q^{66} +1.53209 q^{67} +0.921605 q^{68} +1.96962 q^{69} -1.96962 q^{70} +1.53209 q^{72} +1.28558 q^{75} -1.00000 q^{77} -5.22668 q^{78} +0.684040 q^{80} +4.41147 q^{81} -1.04801 q^{82} -1.96962 q^{83} +2.65366 q^{84} -0.879385 q^{85} -1.53209 q^{86} -0.684040 q^{87} -0.532089 q^{88} -5.67128 q^{90} -1.73205 q^{91} +1.34730 q^{92} -2.65366 q^{96} +1.53209 q^{98} -2.87939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} + 6 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^4 + 6 * q^7 - 6 * q^8 + 6 * q^9	$ 6 q + 6 q^{4} + 6 q^{7} - 6 q^{8} + 6 q^{9} - 6 q^{11} - 6 q^{15} + 6 q^{16} + 6 q^{18} + 6 q^{23} + 6 q^{25} + 6 q^{28} - 12 q^{30} - 6 q^{32} + 12 q^{36} - 6 q^{43} - 6 q^{44} + 6 q^{49} + 6 q^{50} + 6 q^{51} - 6 q^{56} + 6 q^{58} + 6 q^{63} - 6 q^{77} - 18 q^{78} + 6 q^{81} + 6 q^{85} + 6 q^{88} + 6 q^{92} - 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^4 + 6 * q^7 - 6 * q^8 + 6 * q^9 - 6 * q^11 - 6 * q^15 + 6 * q^16 + 6 * q^18 + 6 * q^23 + 6 * q^25 + 6 * q^28 - 12 * q^30 - 6 * q^32 + 12 * q^36 - 6 * q^43 - 6 * q^44 + 6 * q^49 + 6 * q^50 + 6 * q^51 - 6 * q^56 + 6 * q^58 + 6 * q^63 - 6 * q^77 - 18 * q^78 + 6 * q^81 + 6 * q^85 + 6 * q^88 + 6 * q^92 - 6 * 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	$0.222222\pi$
$3$	1.96962	1.96962	0.984808	−	0.173648i	$-0.0555556\pi$
$3$	1.96962	1.96962	0.984808	+	0.173648i	$0.0555556\pi$
$4$	1.34730	1.34730
$5$	−1.28558	−1.28558	−0.642788	−	0.766044i	$-0.722222\pi$
$5$	−1.28558	−1.28558	−0.642788	+	0.766044i	$0.722222\pi$
$6$	3.01763	3.01763
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	0.532089	0.532089
$9$	2.87939	2.87939
$10$	−1.96962	−1.96962
$11$	−1.00000	−1.00000
$11$	−1.00000	−1.00000
$12$	2.65366	2.65366
$13$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$14$	1.53209	1.53209
$15$	−2.53209	−2.53209
$16$	−0.532089	−0.532089
$17$	0.684040	0.684040	0.342020	−	0.939693i	$-0.388889\pi$
$17$	0.684040	0.684040	0.342020	+	0.939693i	$0.388889\pi$
$18$	4.41147	4.41147
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.73205	−1.73205
$21$	1.96962	1.96962
$22$	−1.53209	−1.53209
$23$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$23$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$24$	1.04801	1.04801
$25$	0.652704	0.652704
$26$	−2.65366	−2.65366
$27$	3.70167	3.70167
$28$	1.34730	1.34730
$29$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$29$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$30$	−3.87939	−3.87939
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.34730	−1.34730
$33$	−1.96962	−1.96962
$34$	1.04801	1.04801
$35$	−1.28558	−1.28558
$36$	3.87939	3.87939
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−3.41147	−3.41147
$40$	−0.684040	−0.684040
$41$	−0.684040	−0.684040	−0.342020	−	0.939693i	$-0.611111\pi$
$41$	−0.684040	−0.684040	−0.342020	+	0.939693i	$0.611111\pi$
$42$	3.01763	3.01763
$43$	−1.00000	−1.00000
$43$	−1.00000	−1.00000
$44$	−1.34730	−1.34730
$45$	−3.70167	−3.70167
$46$	1.53209	1.53209
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.04801	−1.04801
$49$	1.00000	1.00000
$50$	1.00000	1.00000
$51$	1.34730	1.34730
$52$	−2.33359	−2.33359
$53$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$53$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$54$	5.67128	5.67128
$55$	1.28558	1.28558
$56$	0.532089	0.532089
$57$	0	0
$58$	−0.532089	−0.532089
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−3.41147	−3.41147
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	2.87939	2.87939
$64$	−1.53209	−1.53209
$65$	2.22668	2.22668
$66$	−3.01763	−3.01763
$67$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$67$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$68$	0.921605	0.921605
$69$	1.96962	1.96962
$70$	−1.96962	−1.96962
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.53209	1.53209
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.28558	1.28558
$76$	0	0
$77$	−1.00000	−1.00000
$78$	−5.22668	−5.22668
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.684040	0.684040
$81$	4.41147	4.41147
$82$	−1.04801	−1.04801
$83$	−1.96962	−1.96962	−0.984808	−	0.173648i	$-0.944444\pi$
$83$	−1.96962	−1.96962	−0.984808	+	0.173648i	$0.944444\pi$
$84$	2.65366	2.65366
$85$	−0.879385	−0.879385
$86$	−1.53209	−1.53209
$87$	−0.684040	−0.684040
$88$	−0.532089	−0.532089
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−5.67128	−5.67128
$91$	−1.73205	−1.73205
$92$	1.34730	1.34730
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−2.65366	−2.65366
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.53209	1.53209
$99$	−2.87939	−2.87939
$100$	0.879385	0.879385
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	2.06418	2.06418
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−0.921605	−0.921605
$105$	−2.53209	−2.53209
$106$	−2.87939	−2.87939
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	4.98724	4.98724
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	1.96962	1.96962
$111$	0	0
$112$	−0.532089	−0.532089
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−1.28558	−1.28558
$116$	−0.467911	−0.467911
$117$	−4.98724	−4.98724
$118$	0	0
$119$	0.684040	0.684040
$120$	−1.34730	−1.34730
$121$	1.00000	1.00000
$122$	0	0
$123$	−1.34730	−1.34730
$124$	0	0
$125$	0.446476	0.446476
$126$	4.41147	4.41147
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	−1.96962	−1.96962
$130$	3.41147	3.41147
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−2.65366	−2.65366
$133$	0	0
$134$	2.34730	2.34730
$135$	−4.75877	−4.75877
$136$	0.363970	0.363970
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	3.01763	3.01763
$139$	1.28558	1.28558	0.642788	−	0.766044i	$-0.277778\pi$
$139$	1.28558	1.28558	0.642788	+	0.766044i	$0.277778\pi$
$140$	−1.73205	−1.73205
$141$	0	0
$142$	0	0
$143$	1.73205	1.73205
$144$	−1.53209	−1.53209
$145$	0.446476	0.446476
$146$	0	0
$147$	1.96962	1.96962
$148$	0	0
$149$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$149$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$150$	1.96962	1.96962
$151$	−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
$153$	1.96962	1.96962
$154$	−1.53209	−1.53209
$155$	0	0
$156$	−4.59627	−4.59627
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−3.70167	−3.70167
$160$	1.73205	1.73205
$161$	1.00000	1.00000
$162$	6.75877	6.75877
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−0.921605	−0.921605
$165$	2.53209	2.53209
$166$	−3.01763	−3.01763
$167$	1.96962	1.96962	0.984808	−	0.173648i	$-0.0555556\pi$
$167$	1.96962	1.96962	0.984808	+	0.173648i	$0.0555556\pi$
$168$	1.04801	1.04801
$169$	2.00000	2.00000
$170$	−1.34730	−1.34730
$171$	0	0
$172$	−1.34730	−1.34730
$173$	1.28558	1.28558	0.642788	−	0.766044i	$-0.277778\pi$
$173$	1.28558	1.28558	0.642788	+	0.766044i	$0.277778\pi$
$174$	−1.04801	−1.04801
$175$	0.652704	0.652704
$176$	0.532089	0.532089
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−4.98724	−4.98724
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−2.65366	−2.65366
$183$	0	0
$184$	0.532089	0.532089
$185$	0	0
$186$	0	0
$187$	−0.684040	−0.684040
$188$	0	0
$189$	3.70167	3.70167
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−3.01763	−3.01763
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	4.38571	4.38571
$196$	1.34730	1.34730
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−4.41147	−4.41147
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0.347296	0.347296
$201$	3.01763	3.01763
$202$	0	0
$203$	−0.347296	−0.347296
$204$	1.81521	1.81521
$205$	0.879385	0.879385
$206$	0	0
$207$	2.87939	2.87939
$208$	0.921605	0.921605
$209$	0	0
$210$	−3.87939	−3.87939
$211$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$211$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$212$	−2.53209	−2.53209
$213$	0	0
$214$	0	0
$215$	1.28558	1.28558
$216$	1.96962	1.96962
$217$	0	0
$218$	0	0
$219$	0	0
$220$	1.73205	1.73205
$221$	−1.18479	−1.18479
$222$	0	0
$223$	−0.684040	−0.684040	−0.342020	−	0.939693i	$-0.611111\pi$
$223$	−0.684040	−0.684040	−0.342020	+	0.939693i	$0.611111\pi$
$224$	−1.34730	−1.34730
$225$	1.87939	1.87939
$226$	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			$\pi$
$230$	−1.96962	−1.96962
$231$	−1.96962	−1.96962
$232$	−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	$0.888889\pi$
$234$	−7.64090	−7.64090
$235$	0	0
$236$	0	0
$237$	0	0
$238$	1.04801	1.04801
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.34730	1.34730
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.53209	1.53209
$243$	4.98724	4.98724
$244$	0	0
$245$	−1.28558	−1.28558
$246$	−2.06418	−2.06418
$247$	0	0
$248$	0	0
$249$	−3.87939	−3.87939
$250$	0.684040	0.684040
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	3.87939	3.87939
$253$	−1.00000	−1.00000
$254$	0	0
$255$	−1.73205	−1.73205
$256$	0	0
$257$	0.684040	0.684040	0.342020	−	0.939693i	$-0.388889\pi$
$257$	0.684040	0.684040	0.342020	+	0.939693i	$0.388889\pi$
$258$	−3.01763	−3.01763
$259$	0	0
$260$	3.00000	3.00000
$261$	−1.00000	−1.00000
$262$	0	0
$263$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$263$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$264$	−1.04801	−1.04801
$265$	2.41609	2.41609
$266$	0	0
$267$	0	0
$268$	2.06418	2.06418
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−7.29086	−7.29086
$271$	0.684040	0.684040	0.342020	−	0.939693i	$-0.388889\pi$
$271$	0.684040	0.684040	0.342020	+	0.939693i	$0.388889\pi$
$272$	−0.363970	−0.363970
$273$	−3.41147	−3.41147
$274$	0	0
$275$	−0.652704	−0.652704
$276$	2.65366	2.65366
$277$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$277$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$278$	1.96962	1.96962
$279$	0	0
$280$	−0.684040	−0.684040
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$283$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$284$	0	0
$285$	0	0
$286$	2.65366	2.65366
$287$	−0.684040	−0.684040
$288$	−3.87939	−3.87939
$289$	−0.532089	−0.532089
$290$	0.684040	0.684040
$291$	0	0
$292$	0	0
$293$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$293$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$294$	3.01763	3.01763
$295$	0	0
$296$	0	0
$297$	−3.70167	−3.70167
$298$	−2.34730	−2.34730
$299$	−1.73205	−1.73205
$300$	1.73205	1.73205
$301$	−1.00000	−1.00000
$302$	−1.53209	−1.53209
$303$	0	0
$304$	0	0
$305$	0	0
$306$	3.01763	3.01763
$307$	1.96962	1.96962	0.984808	−	0.173648i	$-0.0555556\pi$
$307$	1.96962	1.96962	0.984808	+	0.173648i	$0.0555556\pi$
$308$	−1.34730	−1.34730
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−1.81521	−1.81521
$313$	−0.684040	−0.684040	−0.342020	−	0.939693i	$-0.611111\pi$
$313$	−0.684040	−0.684040	−0.342020	+	0.939693i	$0.611111\pi$
$314$	0	0
$315$	−3.70167	−3.70167
$316$	0	0
$317$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$318$	−5.67128	−5.67128
$319$	0.347296	0.347296
$320$	1.96962	1.96962
$321$	0	0
$322$	1.53209	1.53209
$323$	0	0
$324$	5.94356	5.94356
$325$	−1.13052	−1.13052
$326$	0	0
$327$	0	0
$328$	−0.363970	−0.363970
$329$	0	0
$330$	3.87939	3.87939
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−2.65366	−2.65366
$333$	0	0
$334$	3.01763	3.01763
$335$	−1.96962	−1.96962
$336$	−1.04801	−1.04801
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	3.06418	3.06418
$339$	0	0
$340$	−1.18479	−1.18479
$341$	0	0
$342$	0	0
$343$	1.00000	1.00000
$344$	−0.532089	−0.532089
$345$	−2.53209	−2.53209
$346$	1.96962	1.96962
$347$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$347$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$348$	−0.921605	−0.921605
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	1.00000	1.00000
$351$	−6.41147	−6.41147
$352$	1.34730	1.34730
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.34730	1.34730
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.96962	−1.96962
$361$	1.00000	1.00000
$362$	0	0
$363$	1.96962	1.96962
$364$	−2.33359	−2.33359
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.532089	−0.532089
$369$	−1.96962	−1.96962
$370$	0	0
$371$	−1.87939	−1.87939
$372$	0	0
$373$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$373$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$374$	−1.04801	−1.04801
$375$	0.879385	0.879385
$376$	0	0
$377$	0.601535	0.601535
$378$	5.67128	5.67128
$379$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$379$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$383$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$384$	−1.96962	−1.96962
$385$	1.28558	1.28558
$386$	0	0
$387$	−2.87939	−2.87939
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	6.71929	6.71929
$391$	0.684040	0.684040
$392$	0.532089	0.532089
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−3.87939	−3.87939
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−0.347296	−0.347296
$401$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$402$	4.62327	4.62327
$403$	0	0
$404$	0	0
$405$	−5.67128	−5.67128
$406$	−0.532089	−0.532089
$407$	0	0
$408$	0.716881	0.716881
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	1.34730	1.34730
$411$	0	0
$412$	0	0
$413$	0	0
$414$	4.41147	4.41147
$415$	2.53209	2.53209
$416$	2.33359	2.33359
$417$	2.53209	2.53209
$418$	0	0
$419$	1.28558	1.28558	0.642788	−	0.766044i	$-0.277778\pi$
$419$	1.28558	1.28558	0.642788	+	0.766044i	$0.277778\pi$
$420$	−3.41147	−3.41147
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	2.87939	2.87939
$423$	0	0
$424$	−1.00000	−1.00000
$425$	0.446476	0.446476
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.41147	3.41147
$430$	1.96962	1.96962
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.96962	−1.96962
$433$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$433$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$434$	0	0
$435$	0.879385	0.879385
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$439$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$440$	0.684040	0.684040
$441$	2.87939	2.87939
$442$	−1.81521	−1.81521
$443$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$443$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$444$	0	0
$445$	0	0
$446$	−1.04801	−1.04801
$447$	−3.01763	−3.01763
$448$	−1.53209	−1.53209
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	2.87939	2.87939
$451$	0.684040	0.684040
$452$	0	0
$453$	−1.96962	−1.96962
$454$	0	0
$455$	2.22668	2.22668
$456$	0	0
$457$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$457$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$458$	0	0
$459$	2.53209	2.53209
$460$	−1.73205	−1.73205
$461$	−1.96962	−1.96962	−0.984808	−	0.173648i	$-0.944444\pi$
$461$	−1.96962	−1.96962	−0.984808	+	0.173648i	$0.944444\pi$
$462$	−3.01763	−3.01763
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0.184793	0.184793
$465$	0	0
$466$	−2.87939	−2.87939
$467$	−1.96962	−1.96962	−0.984808	−	0.173648i	$-0.944444\pi$
$467$	−1.96962	−1.96962	−0.984808	+	0.173648i	$0.944444\pi$
$468$	−6.71929	−6.71929
$469$	1.53209	1.53209
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.00000	1.00000
$474$	0	0
$475$	0	0
$476$	0.921605	0.921605
$477$	−5.41147	−5.41147
$478$	0	0
$479$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$479$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$480$	3.41147	3.41147
$481$	0	0
$482$	0	0
$483$	1.96962	1.96962
$484$	1.34730	1.34730
$485$	0	0
$486$	7.64090	7.64090
$487$	−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
$490$	−1.96962	−1.96962
$491$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$491$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$492$	−1.81521	−1.81521
$493$	−0.237565	−0.237565
$494$	0	0
$495$	3.70167	3.70167
$496$	0	0
$497$	0	0
$498$	−5.94356	−5.94356
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.601535	0.601535
$501$	3.87939	3.87939
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.53209	1.53209
$505$	0	0
$506$	−1.53209	−1.53209
$507$	3.93923	3.93923
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−2.65366	−2.65366
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	1.04801	1.04801
$515$	0	0
$516$	−2.65366	−2.65366
$517$	0	0
$518$	0	0
$519$	2.53209	2.53209
$520$	1.18479	1.18479
$521$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$521$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$522$	−1.53209	−1.53209
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	1.28558	1.28558
$526$	1.53209	1.53209
$527$	0	0
$528$	1.04801	1.04801
$529$	0	0
$530$	3.70167	3.70167
$531$	0	0
$532$	0	0
$533$	1.18479	1.18479
$534$	0	0
$535$	0	0
$536$	0.815207	0.815207
$537$	0	0
$538$	0	0
$539$	−1.00000	−1.00000
$540$	−6.41147	−6.41147
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	1.04801	1.04801
$543$	0	0
$544$	−0.921605	−0.921605
$545$	0	0
$546$	−5.22668	−5.22668
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−1.00000	−1.00000
$551$	0	0
$552$	1.04801	1.04801
$553$	0	0
$554$	0.532089	0.532089
$555$	0	0
$556$	1.73205	1.73205
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	1.73205	1.73205
$560$	0.684040	0.684040
$561$	−1.34730	−1.34730
$562$	0	0
$563$	1.28558	1.28558	0.642788	−	0.766044i	$-0.277778\pi$
$563$	1.28558	1.28558	0.642788	+	0.766044i	$0.277778\pi$
$564$	0	0
$565$	0	0
$566$	2.65366	2.65366
$567$	4.41147	4.41147
$568$	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	$-0.777778\pi$
$571$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$572$	2.33359	2.33359
$573$	0	0
$574$	−1.04801	−1.04801
$575$	0.652704	0.652704
$576$	−4.41147	−4.41147
$577$	1.28558	1.28558	0.642788	−	0.766044i	$-0.277778\pi$
$577$	1.28558	1.28558	0.642788	+	0.766044i	$0.277778\pi$
$578$	−0.815207	−0.815207
$579$	0	0
$580$	0.601535	0.601535
$581$	−1.96962	−1.96962
$582$	0	0
$583$	1.87939	1.87939
$584$	0	0
$585$	6.41147	6.41147
$586$	2.65366	2.65366
$587$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$587$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$588$	2.65366	2.65366
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−5.67128	−5.67128
$595$	−0.879385	−0.879385
$596$	−2.06418	−2.06418
$597$	0	0
$598$	−2.65366	−2.65366
$599$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$599$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$600$	0.684040	0.684040
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−1.53209	−1.53209
$603$	4.41147	4.41147
$604$	−1.34730	−1.34730
$605$	−1.28558	−1.28558
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	−0.684040	−0.684040
$610$	0	0
$611$	0	0
$612$	2.65366	2.65366
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	3.01763	3.01763
$615$	1.73205	1.73205
$616$	−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
$621$	3.70167	3.70167
$622$	0	0
$623$	0	0
$624$	1.81521	1.81521
$625$	−1.22668	−1.22668
$626$	−1.04801	−1.04801
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−5.67128	−5.67128
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	3.70167	3.70167
$634$	−1.53209	−1.53209
$635$	0	0
$636$	−4.98724	−4.98724
$637$	−1.73205	−1.73205
$638$	0.532089	0.532089
$639$	0	0
$640$	1.28558	1.28558
$641$	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
$645$	2.53209	2.53209
$646$	0	0
$647$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$647$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$648$	2.34730	2.34730
$649$	0	0
$650$	−1.73205	−1.73205
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0.363970	0.363970
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	3.41147	3.41147
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	−2.33359	−2.33359
$664$	−1.04801	−1.04801
$665$	0	0
$666$	0	0
$667$	−0.347296	−0.347296
$668$	2.65366	2.65366
$669$	−1.34730	−1.34730
$670$	−3.01763	−3.01763
$671$	0	0
$672$	−2.65366	−2.65366
$673$	−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
$675$	2.41609	2.41609
$676$	2.69459	2.69459
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	−0.467911	−0.467911
$681$	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
$689$	3.25519	3.25519
$690$	−3.87939	−3.87939
$691$	−1.28558	−1.28558	−0.642788	−	0.766044i	$-0.722222\pi$
$691$	−1.28558	−1.28558	−0.642788	+	0.766044i	$0.722222\pi$
$692$	1.73205	1.73205
$693$	−2.87939	−2.87939
$694$	−2.34730	−2.34730
$695$	−1.65270	−1.65270
$696$	−0.363970	−0.363970
$697$	−0.467911	−0.467911
$698$	0	0
$699$	−3.70167	−3.70167
$700$	0.879385	0.879385
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−9.82295	−9.82295
$703$	0	0
$704$	1.53209	1.53209
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	2.06418	2.06418
$715$	−2.22668	−2.22668
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	1.96962	1.96962
$721$	0	0
$722$	1.53209	1.53209
$723$	0	0
$724$	0	0
$725$	−0.226682	−0.226682
$726$	3.01763	3.01763
$727$	−1.28558	−1.28558	−0.642788	−	0.766044i	$-0.722222\pi$
$727$	−1.28558	−1.28558	−0.642788	+	0.766044i	$0.722222\pi$
$728$	−0.921605	−0.921605
$729$	5.41147	5.41147
$730$	0	0
$731$	−0.684040	−0.684040
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−2.53209	−2.53209
$736$	−1.34730	−1.34730
$737$	−1.53209	−1.53209
$738$	−3.01763	−3.01763
$739$	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
$745$	1.96962	1.96962
$746$	−2.34730	−2.34730
$747$	−5.67128	−5.67128
$748$	−0.921605	−0.921605
$749$	0	0
$750$	1.34730	1.34730
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0.921605	0.921605
$755$	1.28558	1.28558
$756$	4.98724	4.98724
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	2.87939	2.87939
$759$	−1.96962	−1.96962
$760$	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
$766$	−2.65366	−2.65366
$767$	0	0
$768$	0	0
$769$	−1.96962	−1.96962	−0.984808	−	0.173648i	$-0.944444\pi$
$769$	−1.96962	−1.96962	−0.984808	+	0.173648i	$0.944444\pi$
$770$	1.96962	1.96962
$771$	1.34730	1.34730
$772$	0	0
$773$	1.28558	1.28558	0.642788	−	0.766044i	$-0.277778\pi$
$773$	1.28558	1.28558	0.642788	+	0.766044i	$0.277778\pi$
$774$	−4.41147	−4.41147
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	5.90885	5.90885
$781$	0	0
$782$	1.04801	1.04801
$783$	−1.28558	−1.28558
$784$	−0.532089	−0.532089
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	1.96962	1.96962
$790$	0	0
$791$	0	0
$792$	−1.53209	−1.53209
$793$	0	0
$794$	0	0
$795$	4.75877	4.75877
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−0.879385	−0.879385
$801$	0	0
$802$	−1.53209	−1.53209
$803$	0	0
$804$	4.06564	4.06564
$805$	−1.28558	−1.28558
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−8.68891	−8.68891
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	−0.467911	−0.467911
$813$	1.34730	1.34730
$814$	0	0
$815$	0	0
$816$	−0.716881	−0.716881
$817$	0	0
$818$	0	0
$819$	−4.98724	−4.98724
$820$	1.18479	1.18479
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$823$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$824$	0	0
$825$	−1.28558	−1.28558
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	3.87939	3.87939
$829$	0.684040	0.684040	0.342020	−	0.939693i	$-0.388889\pi$
$829$	0.684040	0.684040	0.342020	+	0.939693i	$0.388889\pi$
$830$	3.87939	3.87939
$831$	0.684040	0.684040
$832$	2.65366	2.65366
$833$	0.684040	0.684040
$834$	3.87939	3.87939
$835$	−2.53209	−2.53209
$836$	0	0
$837$	0	0
$838$	1.96962	1.96962
$839$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$839$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$840$	−1.34730	−1.34730
$841$	−0.879385	−0.879385
$842$	0	0
$843$	0	0
$844$	2.53209	2.53209
$845$	−2.57115	−2.57115
$846$	0	0
$847$	1.00000	1.00000
$848$	1.00000	1.00000
$849$	3.41147	3.41147
$850$	0.684040	0.684040
$851$	0	0
$852$	0	0
$853$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$853$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.28558	−1.28558	−0.642788	−	0.766044i	$-0.722222\pi$
$857$	−1.28558	−1.28558	−0.642788	+	0.766044i	$0.722222\pi$
$858$	5.22668	5.22668
$859$	−0.684040	−0.684040	−0.342020	−	0.939693i	$-0.611111\pi$
$859$	−0.684040	−0.684040	−0.342020	+	0.939693i	$0.611111\pi$
$860$	1.73205	1.73205
$861$	−1.34730	−1.34730
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−4.98724	−4.98724
$865$	−1.65270	−1.65270
$866$	−2.65366	−2.65366
$867$	−1.04801	−1.04801
$868$	0	0
$869$	0	0
$870$	1.34730	1.34730
$871$	−2.65366	−2.65366
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.446476	0.446476
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	2.65366	2.65366
$879$	3.41147	3.41147
$880$	−0.684040	−0.684040
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	4.41147	4.41147
$883$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$884$	−1.59627	−1.59627
$885$	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
$890$	0	0
$891$	−4.41147	−4.41147
$892$	−0.921605	−0.921605
$893$	0	0
$894$	−4.62327	−4.62327
$895$	0	0
$896$	−1.00000	−1.00000
$897$	−3.41147	−3.41147
$898$	0	0
$899$	0	0
$900$	2.53209	2.53209
$901$	−1.28558	−1.28558
$902$	1.04801	1.04801
$903$	−1.96962	−1.96962
$904$	0	0
$905$	0	0
$906$	−3.01763	−3.01763
$907$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$907$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$908$	0	0
$909$	0	0
$910$	3.41147	3.41147
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	1.96962	1.96962
$914$	−0.532089	−0.532089
$915$	0	0
$916$	0	0
$917$	0	0
$918$	3.87939	3.87939
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	−0.684040	−0.684040
$921$	3.87939	3.87939
$922$	−3.01763	−3.01763
$923$	0	0
$924$	−2.65366	−2.65366
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0.467911	0.467911
$929$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$929$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$930$	0	0
$931$	0	0
$932$	−2.53209	−2.53209
$933$	0	0
$934$	−3.01763	−3.01763
$935$	0.879385	0.879385
$936$	−2.65366	−2.65366
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	2.34730	2.34730
$939$	−1.34730	−1.34730
$940$	0	0
$941$	−1.28558	−1.28558	−0.642788	−	0.766044i	$-0.722222\pi$
$941$	−1.28558	−1.28558	−0.642788	+	0.766044i	$0.722222\pi$
$942$	0	0
$943$	−0.684040	−0.684040
$944$	0	0
$945$	−4.75877	−4.75877
$946$	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
$951$	−1.96962	−1.96962
$952$	0.363970	0.363970
$953$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$953$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$954$	−8.29086	−8.29086
$955$	0	0
$956$	0	0
$957$	0.684040	0.684040
$958$	−2.65366	−2.65366
$959$	0	0
$960$	3.87939	3.87939
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	3.01763	3.01763
$967$	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			$\pi$
$972$	6.71929	6.71929
$973$	1.28558	1.28558
$974$	−2.87939	−2.87939
$975$	−2.22668	−2.22668
$976$	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
$979$	0	0
$980$	−1.73205	−1.73205
$981$	0	0
$982$	2.87939	2.87939
$983$	1.28558	1.28558	0.642788	−	0.766044i	$-0.277778\pi$
$983$	1.28558	1.28558	0.642788	+	0.766044i	$0.277778\pi$
$984$	−0.716881	−0.716881
$985$	0	0
$986$	−0.363970	−0.363970
$987$	0	0
$988$	0	0
$989$	−1.00000	−1.00000
$990$	5.67128	5.67128
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−5.22668	−5.22668
$997$	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.p.3310.6	yes	6
7.6	odd	2	inner	3311.1.h.p.3310.5	yes	6
11.10	odd	2		3311.1.h.o.3310.2	yes	6
43.42	odd	2		3311.1.h.o.3310.1	✓	6
77.76	even	2		3311.1.h.o.3310.1	✓	6
301.300	even	2		3311.1.h.o.3310.2	yes	6
473.472	even	2	inner	3311.1.h.p.3310.5	yes	6
3311.3310	odd	2	CM	3311.1.h.p.3310.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3311.1.h.o.3310.1	✓	6	43.42	odd	2
3311.1.h.o.3310.1	✓	6	77.76	even	2
3311.1.h.o.3310.2	yes	6	11.10	odd	2
3311.1.h.o.3310.2	yes	6	301.300	even	2
3311.1.h.p.3310.5	yes	6	7.6	odd	2	inner
3311.1.h.p.3310.5	yes	6	473.472	even	2	inner
3311.1.h.p.3310.6	yes	6	1.1	even	1	trivial
3311.1.h.p.3310.6	yes	6	3311.3310	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.53209 q^{2} +1.96962 q^{3} +1.34730 q^{4} -1.28558 q^{5} +3.01763 q^{6} +1.00000 q^{7} +0.532089 q^{8} +2.87939 q^{9} +O(q^{10})\)	\(q+1.53209 q^{2} +1.96962 q^{3} +1.34730 q^{4} -1.28558 q^{5} +3.01763 q^{6} +1.00000 q^{7} +0.532089 q^{8} +2.87939 q^{9} -1.96962 q^{10} -1.00000 q^{11} +2.65366 q^{12} -1.73205 q^{13} +1.53209 q^{14} -2.53209 q^{15} -0.532089 q^{16} +0.684040 q^{17} +4.41147 q^{18} -1.73205 q^{20} +1.96962 q^{21} -1.53209 q^{22} +1.00000 q^{23} +1.04801 q^{24} +0.652704 q^{25} -2.65366 q^{26} +3.70167 q^{27} +1.34730 q^{28} -0.347296 q^{29} -3.87939 q^{30} -1.34730 q^{32} -1.96962 q^{33} +1.04801 q^{34} -1.28558 q^{35} +3.87939 q^{36} -3.41147 q^{39} -0.684040 q^{40} -0.684040 q^{41} +3.01763 q^{42} -1.00000 q^{43} -1.34730 q^{44} -3.70167 q^{45} +1.53209 q^{46} -1.04801 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.34730 q^{51} -2.33359 q^{52} -1.87939 q^{53} +5.67128 q^{54} +1.28558 q^{55} +0.532089 q^{56} -0.532089 q^{58} -3.41147 q^{60} +2.87939 q^{63} -1.53209 q^{64} +2.22668 q^{65} -3.01763 q^{66} +1.53209 q^{67} +0.921605 q^{68} +1.96962 q^{69} -1.96962 q^{70} +1.53209 q^{72} +1.28558 q^{75} -1.00000 q^{77} -5.22668 q^{78} +0.684040 q^{80} +4.41147 q^{81} -1.04801 q^{82} -1.96962 q^{83} +2.65366 q^{84} -0.879385 q^{85} -1.53209 q^{86} -0.684040 q^{87} -0.532089 q^{88} -5.67128 q^{90} -1.73205 q^{91} +1.34730 q^{92} -2.65366 q^{96} +1.53209 q^{98} -2.87939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} + 6 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^4 + 6 * q^7 - 6 * q^8 + 6 * q^9	\( 6 q + 6 q^{4} + 6 q^{7} - 6 q^{8} + 6 q^{9} - 6 q^{11} - 6 q^{15} + 6 q^{16} + 6 q^{18} + 6 q^{23} + 6 q^{25} + 6 q^{28} - 12 q^{30} - 6 q^{32} + 12 q^{36} - 6 q^{43} - 6 q^{44} + 6 q^{49} + 6 q^{50} + 6 q^{51} - 6 q^{56} + 6 q^{58} + 6 q^{63} - 6 q^{77} - 18 q^{78} + 6 q^{81} + 6 q^{85} + 6 q^{88} + 6 q^{92} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^4 + 6 * q^7 - 6 * q^8 + 6 * q^9 - 6 * q^11 - 6 * q^15 + 6 * q^16 + 6 * q^18 + 6 * q^23 + 6 * q^25 + 6 * q^28 - 12 * q^30 - 6 * q^32 + 12 * q^36 - 6 * q^43 - 6 * q^44 + 6 * q^49 + 6 * q^50 + 6 * q^51 - 6 * q^56 + 6 * q^58 + 6 * q^63 - 6 * q^77 - 18 * q^78 + 6 * q^81 + 6 * q^85 + 6 * q^88 + 6 * q^92 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more