LMFDB - Embedded newform 331.1.b.b.330.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [331,1,Mod(330,331)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(331, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("331.330");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 331 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	331.b (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:	no
Analytic conductor:	$0.165190519182$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.331.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.36264691.2

Embedding invariants

Embedding label			330.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	331.330
Dual form			331.1.b.b.330.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.41421i q^{7} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.41421i q^{7} -1.00000 q^{9} +1.41421i q^{10} -1.41421i q^{11} +1.41421i q^{12} +2.00000 q^{14} +1.41421i q^{15} -1.00000 q^{16} +1.00000 q^{17} +1.41421i q^{18} +1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -1.41421i q^{28} +1.41421i q^{29} +2.00000 q^{30} +1.00000 q^{31} +1.41421i q^{32} -2.00000 q^{33} -1.41421i q^{34} -1.41421i q^{35} +1.00000 q^{36} -1.41421i q^{37} -1.41421i q^{38} +1.41421i q^{41} -2.82843i q^{42} -1.00000 q^{43} +1.41421i q^{44} +1.00000 q^{45} +1.41421i q^{48} -1.00000 q^{49} -1.41421i q^{51} -1.00000 q^{53} +1.41421i q^{55} -1.41421i q^{57} +2.00000 q^{58} -1.41421i q^{60} +1.41421i q^{61} -1.41421i q^{62} -1.41421i q^{63} +1.00000 q^{64} +2.82843i q^{66} +1.00000 q^{67} -1.00000 q^{68} -2.00000 q^{70} +1.00000 q^{71} -2.00000 q^{74} -1.00000 q^{76} +2.00000 q^{77} +1.00000 q^{79} +1.00000 q^{80} -1.00000 q^{81} +2.00000 q^{82} -2.00000 q^{84} -1.00000 q^{85} +1.41421i q^{86} +2.00000 q^{87} -1.41421i q^{90} -1.41421i q^{93} -1.00000 q^{95} +2.00000 q^{96} +1.41421i q^{97} +1.41421i q^{98} +1.41421i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9} + 4 q^{14} - 2 q^{16} + 2 q^{17} + 2 q^{19} + 2 q^{20} + 4 q^{21} - 4 q^{22} + 4 q^{30} + 2 q^{31} - 4 q^{33} + 2 q^{36} - 2 q^{43} + 2 q^{45} - 2 q^{49} - 2 q^{53} + 4 q^{58} + 2 q^{64} + 2 q^{67} - 2 q^{68} - 4 q^{70} + 2 q^{71} - 4 q^{74} - 2 q^{76} + 4 q^{77} + 2 q^{79} + 2 q^{80} - 2 q^{81} + 4 q^{82} - 4 q^{84} - 2 q^{85} + 4 q^{87} - 2 q^{95} + 4 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^9 + 4 * q^14 - 2 * q^16 + 2 * q^17 + 2 * q^19 + 2 * q^20 + 4 * q^21 - 4 * q^22 + 4 * q^30 + 2 * q^31 - 4 * q^33 + 2 * q^36 - 2 * q^43 + 2 * q^45 - 2 * q^49 - 2 * q^53 + 4 * q^58 + 2 * q^64 + 2 * q^67 - 2 * q^68 - 4 * q^70 + 2 * q^71 - 4 * q^74 - 2 * q^76 + 4 * q^77 + 2 * q^79 + 2 * q^80 - 2 * q^81 + 4 * q^82 - 4 * q^84 - 2 * q^85 + 4 * q^87 - 2 * q^95 + 4 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/331\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$	−1.00000	−1.00000
$5$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	−2.00000	−2.00000
$7$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$7$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$8$	0	0
$9$	−1.00000	−1.00000
$10$	1.41421i	1.41421i
$11$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$11$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$12$	1.41421i	1.41421i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	2.00000	2.00000
$15$	1.41421i	1.41421i
$16$	−1.00000	−1.00000
$17$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$17$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$18$	1.41421i	1.41421i
$19$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$20$	1.00000	1.00000
$21$	2.00000	2.00000
$22$	−2.00000	−2.00000
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	− 1.41421i	− 1.41421i
$29$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$29$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$30$	2.00000	2.00000
$31$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$31$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$32$	1.41421i	1.41421i
$33$	−2.00000	−2.00000
$34$	− 1.41421i	− 1.41421i
$35$	− 1.41421i	− 1.41421i
$36$	1.00000	1.00000
$37$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$37$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$38$	− 1.41421i	− 1.41421i
$39$	0	0
$40$	0	0
$41$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$41$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$42$	− 2.82843i	− 2.82843i
$43$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$43$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$44$	1.41421i	1.41421i
$45$	1.00000	1.00000
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.41421i	1.41421i
$49$	−1.00000	−1.00000
$50$	0	0
$51$	− 1.41421i	− 1.41421i
$52$	0	0
$53$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$54$	0	0
$55$	1.41421i	1.41421i
$56$	0	0
$57$	− 1.41421i	− 1.41421i
$58$	2.00000	2.00000
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	− 1.41421i	− 1.41421i
$61$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$61$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$62$	− 1.41421i	− 1.41421i
$63$	− 1.41421i	− 1.41421i
$64$	1.00000	1.00000
$65$	0	0
$66$	2.82843i	2.82843i
$67$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$67$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$68$	−1.00000	−1.00000
$69$	0	0
$70$	−2.00000	−2.00000
$71$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−2.00000	−2.00000
$75$	0	0
$76$	−1.00000	−1.00000
$77$	2.00000	2.00000
$78$	0	0
$79$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$79$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$80$	1.00000	1.00000
$81$	−1.00000	−1.00000
$82$	2.00000	2.00000
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−2.00000	−2.00000
$85$	−1.00000	−1.00000
$86$	1.41421i	1.41421i
$87$	2.00000	2.00000
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	− 1.41421i	− 1.41421i
$91$	0	0
$92$	0	0
$93$	− 1.41421i	− 1.41421i
$94$	0	0
$95$	−1.00000	−1.00000
$96$	2.00000	2.00000
$97$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$97$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$98$	1.41421i	1.41421i
$99$	1.41421i	1.41421i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−2.00000	−2.00000
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−2.00000	−2.00000
$106$	1.41421i	1.41421i
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$110$	2.00000	2.00000
$111$	−2.00000	−2.00000
$112$	− 1.41421i	− 1.41421i
$113$	−2.00000	−2.00000	−1.00000			$\pi$
$113$	−2.00000	−2.00000	−1.00000			$\pi$
$114$	−2.00000	−2.00000
$115$	0	0
$116$	− 1.41421i	− 1.41421i
$117$	0	0
$118$	0	0
$119$	1.41421i	1.41421i
$120$	0	0
$121$	−1.00000	−1.00000
$122$	2.00000	2.00000
$123$	2.00000	2.00000
$124$	−1.00000	−1.00000
$125$	1.00000	1.00000
$126$	−2.00000	−2.00000
$127$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$127$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$128$	0	0
$129$	1.41421i	1.41421i
$130$	0	0
$131$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$132$	2.00000	2.00000
$133$	1.41421i	1.41421i
$134$	− 1.41421i	− 1.41421i
$135$	0	0
$136$	0	0
$137$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$137$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	1.41421i	1.41421i
$141$	0	0
$142$	− 1.41421i	− 1.41421i
$143$	0	0
$144$	1.00000	1.00000
$145$	− 1.41421i	− 1.41421i
$146$	0	0
$147$	1.41421i	1.41421i
$148$	1.41421i	1.41421i
$149$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−1.00000	−1.00000
$154$	− 2.82843i	− 2.82843i
$155$	−1.00000	−1.00000
$156$	0	0
$157$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$158$	− 1.41421i	− 1.41421i
$159$	1.41421i	1.41421i
$160$	− 1.41421i	− 1.41421i
$161$	0	0
$162$	1.41421i	1.41421i
$163$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$163$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$164$	− 1.41421i	− 1.41421i
$165$	2.00000	2.00000
$166$	0	0
$167$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$167$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	1.41421i	1.41421i
$171$	−1.00000	−1.00000
$172$	1.00000	1.00000
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	− 2.82843i	− 2.82843i
$175$	0	0
$176$	1.41421i	1.41421i
$177$	0	0
$178$	0	0
$179$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$180$	−1.00000	−1.00000
$181$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$181$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$182$	0	0
$183$	2.00000	2.00000
$184$	0	0
$185$	1.41421i	1.41421i
$186$	−2.00000	−2.00000
$187$	− 1.41421i	− 1.41421i
$188$	0	0
$189$	0	0
$190$	1.41421i	1.41421i
$191$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$192$	− 1.41421i	− 1.41421i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	2.00000	2.00000
$195$	0	0
$196$	1.00000	1.00000
$197$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$197$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$198$	2.00000	2.00000
$199$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$199$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$200$	0	0
$201$	− 1.41421i	− 1.41421i
$202$	0	0
$203$	−2.00000	−2.00000
$204$	1.41421i	1.41421i
$205$	− 1.41421i	− 1.41421i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.41421i	− 1.41421i
$210$	2.82843i	2.82843i
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.00000	1.00000
$213$	− 1.41421i	− 1.41421i
$214$	0	0
$215$	1.00000	1.00000
$216$	0	0
$217$	1.41421i	1.41421i
$218$	1.41421i	1.41421i
$219$	0	0
$220$	− 1.41421i	− 1.41421i
$221$	0	0
$222$	2.82843i	2.82843i
$223$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$224$	−2.00000	−2.00000
$225$	0	0
$226$	2.82843i	2.82843i
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	1.41421i	1.41421i
$229$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$229$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$230$	0	0
$231$	− 2.82843i	− 2.82843i
$232$	0	0
$233$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.41421i	− 1.41421i
$238$	2.00000	2.00000
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	− 1.41421i	− 1.41421i
$241$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$241$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$242$	1.41421i	1.41421i
$243$	1.41421i	1.41421i
$244$	− 1.41421i	− 1.41421i
$245$	1.00000	1.00000
$246$	− 2.82843i	− 2.82843i
$247$	0	0
$248$	0	0
$249$	0	0
$250$	− 1.41421i	− 1.41421i
$251$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$251$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$252$	1.41421i	1.41421i
$253$	0	0
$254$	1.41421i	1.41421i
$255$	1.41421i	1.41421i
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	2.00000	2.00000
$259$	2.00000	2.00000
$260$	0	0
$261$	− 1.41421i	− 1.41421i
$262$	− 1.41421i	− 1.41421i
$263$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$263$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$264$	0	0
$265$	1.00000	1.00000
$266$	2.00000	2.00000
$267$	0	0
$268$	−1.00000	−1.00000
$269$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$270$	0	0
$271$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$271$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$272$	−1.00000	−1.00000
$273$	0	0
$274$	−2.00000	−2.00000
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.00000	−1.00000
$280$	0	0
$281$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$282$	0	0
$283$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$283$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$284$	−1.00000	−1.00000
$285$	1.41421i	1.41421i
$286$	0	0
$287$	−2.00000	−2.00000
$288$	− 1.41421i	− 1.41421i
$289$	0	0
$290$	−2.00000	−2.00000
$291$	2.00000	2.00000
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	2.00000	2.00000
$295$	0	0
$296$	0	0
$297$	0	0
$298$	1.41421i	1.41421i
$299$	0	0
$300$	0	0
$301$	− 1.41421i	− 1.41421i
$302$	0	0
$303$	0	0
$304$	−1.00000	−1.00000
$305$	− 1.41421i	− 1.41421i
$306$	1.41421i	1.41421i
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−2.00000	−2.00000
$309$	0	0
$310$	1.41421i	1.41421i
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$314$	1.41421i	1.41421i
$315$	1.41421i	1.41421i
$316$	−1.00000	−1.00000
$317$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$317$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$318$	2.00000	2.00000
$319$	2.00000	2.00000
$320$	−1.00000	−1.00000
$321$	0	0
$322$	0	0
$323$	1.00000	1.00000
$324$	1.00000	1.00000
$325$	0	0
$326$	− 1.41421i	− 1.41421i
$327$	1.41421i	1.41421i
$328$	0	0
$329$	0	0
$330$	− 2.82843i	− 2.82843i
$331$	−1.00000	−1.00000
$331$	−1.00000	−1.00000
$332$	0	0
$333$	1.41421i	1.41421i
$334$	1.41421i	1.41421i
$335$	−1.00000	−1.00000
$336$	−2.00000	−2.00000
$337$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$338$	− 1.41421i	− 1.41421i
$339$	2.82843i	2.82843i
$340$	1.00000	1.00000
$341$	− 1.41421i	− 1.41421i
$342$	1.41421i	1.41421i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−2.00000	−2.00000
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	2.00000
$353$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$353$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$354$	0	0
$355$	−1.00000	−1.00000
$356$	0	0
$357$	2.00000	2.00000
$358$	1.41421i	1.41421i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	0	0
$362$	2.00000	2.00000
$363$	1.41421i	1.41421i
$364$	0	0
$365$	0	0
$366$	− 2.82843i	− 2.82843i
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	− 1.41421i	− 1.41421i
$370$	2.00000	2.00000
$371$	− 1.41421i	− 1.41421i
$372$	1.41421i	1.41421i
$373$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$373$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$374$	−2.00000	−2.00000
$375$	− 1.41421i	− 1.41421i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	1.00000	1.00000
$381$	1.41421i	1.41421i
$382$	1.41421i	1.41421i
$383$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$383$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$384$	0	0
$385$	−2.00000	−2.00000
$386$	0	0
$387$	1.00000	1.00000
$388$	− 1.41421i	− 1.41421i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 1.41421i	− 1.41421i
$394$	− 1.41421i	− 1.41421i
$395$	−1.00000	−1.00000
$396$	− 1.41421i	− 1.41421i
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−2.00000	−2.00000
$399$	2.00000	2.00000
$400$	0	0
$401$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$401$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$402$	−2.00000	−2.00000
$403$	0	0
$404$	0	0
$405$	1.00000	1.00000
$406$	2.82843i	2.82843i
$407$	−2.00000	−2.00000
$408$	0	0
$409$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$409$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$410$	−2.00000	−2.00000
$411$	−2.00000	−2.00000
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−2.00000	−2.00000
$419$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$420$	2.00000	2.00000
$421$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$421$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	−2.00000	−2.00000
$427$	−2.00000	−2.00000
$428$	0	0
$429$	0	0
$430$	− 1.41421i	− 1.41421i
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$433$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$434$	2.00000	2.00000
$435$	−2.00000	−2.00000
$436$	1.00000	1.00000
$437$	0	0
$438$	0	0
$439$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$439$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	2.00000	2.00000
$445$	0	0
$446$	1.41421i	1.41421i
$447$	1.41421i	1.41421i
$448$	1.41421i	1.41421i
$449$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$450$	0	0
$451$	2.00000	2.00000
$452$	2.00000	2.00000
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$458$	−2.00000	−2.00000
$459$	0	0
$460$	0	0
$461$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$461$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$462$	−4.00000	−4.00000
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	− 1.41421i	− 1.41421i
$465$	1.41421i	1.41421i
$466$	− 1.41421i	− 1.41421i
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	1.41421i	1.41421i
$470$	0	0
$471$	1.41421i	1.41421i
$472$	0	0
$473$	1.41421i	1.41421i
$474$	−2.00000	−2.00000
$475$	0	0
$476$	− 1.41421i	− 1.41421i
$477$	1.00000	1.00000
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−2.00000	−2.00000
$481$	0	0
$482$	− 1.41421i	− 1.41421i
$483$	0	0
$484$	1.00000	1.00000
$485$	− 1.41421i	− 1.41421i
$486$	2.00000	2.00000
$487$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$487$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$488$	0	0
$489$	− 1.41421i	− 1.41421i
$490$	− 1.41421i	− 1.41421i
$491$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$491$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$492$	−2.00000	−2.00000
$493$	1.41421i	1.41421i
$494$	0	0
$495$	− 1.41421i	− 1.41421i
$496$	−1.00000	−1.00000
$497$	1.41421i	1.41421i
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−1.00000	−1.00000
$501$	1.41421i	1.41421i
$502$	2.00000	2.00000
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.41421i	− 1.41421i
$508$	1.00000	1.00000
$509$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$509$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$510$	2.00000	2.00000
$511$	0	0
$512$	− 1.41421i	− 1.41421i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	− 1.41421i	− 1.41421i
$517$	0	0
$518$	− 2.82843i	− 2.82843i
$519$	0	0
$520$	0	0
$521$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$521$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$522$	−2.00000	−2.00000
$523$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$523$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$524$	−1.00000	−1.00000
$525$	0	0
$526$	−2.00000	−2.00000
$527$	1.00000	1.00000
$528$	2.00000	2.00000
$529$	1.00000	1.00000
$530$	− 1.41421i	− 1.41421i
$531$	0	0
$532$	− 1.41421i	− 1.41421i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.41421i	1.41421i
$538$	1.41421i	1.41421i
$539$	1.41421i	1.41421i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	− 1.41421i	− 1.41421i
$543$	2.00000	2.00000
$544$	1.41421i	1.41421i
$545$	1.00000	1.00000
$546$	0	0
$547$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$547$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$548$	1.41421i	1.41421i
$549$	− 1.41421i	− 1.41421i
$550$	0	0
$551$	1.41421i	1.41421i
$552$	0	0
$553$	1.41421i	1.41421i
$554$	0	0
$555$	2.00000	2.00000
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.41421i	1.41421i
$559$	0	0
$560$	1.41421i	1.41421i
$561$	−2.00000	−2.00000
$562$	1.41421i	1.41421i
$563$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$564$	0	0
$565$	2.00000	2.00000
$566$	1.41421i	1.41421i
$567$	− 1.41421i	− 1.41421i
$568$	0	0
$569$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$569$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$570$	2.00000	2.00000
$571$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$571$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$572$	0	0
$573$	1.41421i	1.41421i
$574$	2.82843i	2.82843i
$575$	0	0
$576$	−1.00000	−1.00000
$577$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$577$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$578$	0	0
$579$	0	0
$580$	1.41421i	1.41421i
$581$	0	0
$582$	− 2.82843i	− 2.82843i
$583$	1.41421i	1.41421i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	− 1.41421i	− 1.41421i
$589$	1.00000	1.00000
$590$	0	0
$591$	− 1.41421i	− 1.41421i
$592$	1.41421i	1.41421i
$593$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$593$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$594$	0	0
$595$	− 1.41421i	− 1.41421i
$596$	1.00000	1.00000
$597$	−2.00000	−2.00000
$598$	0	0
$599$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$599$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$600$	0	0
$601$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$601$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$602$	−2.00000	−2.00000
$603$	−1.00000	−1.00000
$604$	0	0
$605$	1.00000	1.00000
$606$	0	0
$607$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$607$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$608$	1.41421i	1.41421i
$609$	2.82843i	2.82843i
$610$	−2.00000	−2.00000
$611$	0	0
$612$	1.00000	1.00000
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	−2.00000	−2.00000
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	1.00000	1.00000
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−1.00000
$626$	1.41421i	1.41421i
$627$	−2.00000	−2.00000
$628$	1.00000	1.00000
$629$	− 1.41421i	− 1.41421i
$630$	2.00000	2.00000
$631$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$631$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$632$	0	0
$633$	0	0
$634$	2.00000	2.00000
$635$	1.00000	1.00000
$636$	− 1.41421i	− 1.41421i
$637$	0	0
$638$	− 2.82843i	− 2.82843i
$639$	−1.00000	−1.00000
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	− 1.41421i	− 1.41421i
$646$	− 1.41421i	− 1.41421i
$647$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$647$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.00000	2.00000
$652$	−1.00000	−1.00000
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	2.00000	2.00000
$655$	−1.00000	−1.00000
$656$	− 1.41421i	− 1.41421i
$657$	0	0
$658$	0	0
$659$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$659$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$660$	−2.00000	−2.00000
$661$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$661$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$662$	1.41421i	1.41421i
$663$	0	0
$664$	0	0
$665$	− 1.41421i	− 1.41421i
$666$	2.00000	2.00000
$667$	0	0
$668$	1.00000	1.00000
$669$	1.41421i	1.41421i
$670$	1.41421i	1.41421i
$671$	2.00000	2.00000
$672$	2.82843i	2.82843i
$673$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$674$	1.41421i	1.41421i
$675$	0	0
$676$	−1.00000	−1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	4.00000	4.00000
$679$	−2.00000	−2.00000
$680$	0	0
$681$	0	0
$682$	−2.00000	−2.00000
$683$	−2.00000	−2.00000	−1.00000			$\pi$
$683$	−2.00000	−2.00000	−1.00000			$\pi$
$684$	1.00000	1.00000
$685$	1.41421i	1.41421i
$686$	0	0
$687$	−2.00000	−2.00000
$688$	1.00000	1.00000
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	−2.00000	−2.00000
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.41421i	1.41421i
$698$	0	0
$699$	− 1.41421i	− 1.41421i
$700$	0	0
$701$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$702$	0	0
$703$	− 1.41421i	− 1.41421i
$704$	− 1.41421i	− 1.41421i
$705$	0	0
$706$	− 1.41421i	− 1.41421i
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	1.41421i	1.41421i
$711$	−1.00000	−1.00000
$712$	0	0
$713$	0	0
$714$	− 2.82843i	− 2.82843i
$715$	0	0
$716$	1.00000	1.00000
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−1.00000	−1.00000
$721$	0	0
$722$	0	0
$723$	− 1.41421i	− 1.41421i
$724$	− 1.41421i	− 1.41421i
$725$	0	0
$726$	2.00000	2.00000
$727$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$727$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	−1.00000	−1.00000
$732$	−2.00000	−2.00000
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	− 1.41421i	− 1.41421i
$736$	0	0
$737$	− 1.41421i	− 1.41421i
$738$	−2.00000	−2.00000
$739$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$740$	− 1.41421i	− 1.41421i
$741$	0	0
$742$	−2.00000	−2.00000
$743$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$743$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$744$	0	0
$745$	1.00000	1.00000
$746$	2.00000	2.00000
$747$	0	0
$748$	1.41421i	1.41421i
$749$	0	0
$750$	−2.00000	−2.00000
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	2.00000	2.00000
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000	2.00000	1.00000			$0$
$757$	2.00000	2.00000	1.00000			$0$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	2.00000	2.00000
$763$	− 1.41421i	− 1.41421i
$764$	1.00000	1.00000
$765$	1.00000	1.00000
$766$	−2.00000	−2.00000
$767$	0	0
$768$	− 1.41421i	− 1.41421i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	2.82843i	2.82843i
$771$	0	0
$772$	0	0
$773$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$774$	− 1.41421i	− 1.41421i
$775$	0	0
$776$	0	0
$777$	− 2.82843i	− 2.82843i
$778$	0	0
$779$	1.41421i	1.41421i
$780$	0	0
$781$	− 1.41421i	− 1.41421i
$782$	0	0
$783$	0	0
$784$	1.00000	1.00000
$785$	1.00000	1.00000
$786$	−2.00000	−2.00000
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−1.00000	−1.00000
$789$	−2.00000	−2.00000
$790$	1.41421i	1.41421i
$791$	− 2.82843i	− 2.82843i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 1.41421i	− 1.41421i
$796$	1.41421i	1.41421i
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	− 2.82843i	− 2.82843i
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 1.41421i	− 1.41421i
$803$	0	0
$804$	1.41421i	1.41421i
$805$	0	0
$806$	0	0
$807$	1.41421i	1.41421i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	− 1.41421i	− 1.41421i
$811$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$811$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$812$	2.00000	2.00000
$813$	− 1.41421i	− 1.41421i
$814$	2.82843i	2.82843i
$815$	−1.00000	−1.00000
$816$	1.41421i	1.41421i
$817$	−1.00000	−1.00000
$818$	−2.00000	−2.00000
$819$	0	0
$820$	1.41421i	1.41421i
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.82843i	2.82843i
$823$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$823$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$827$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$828$	0	0
$829$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.00000	−1.00000
$834$	0	0
$835$	1.00000	1.00000
$836$	1.41421i	1.41421i
$837$	0	0
$838$	1.41421i	1.41421i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−1.00000	−1.00000
$842$	−2.00000	−2.00000
$843$	1.41421i	1.41421i
$844$	0	0
$845$	−1.00000	−1.00000
$846$	0	0
$847$	− 1.41421i	− 1.41421i
$848$	1.00000	1.00000
$849$	1.41421i	1.41421i
$850$	0	0
$851$	0	0
$852$	1.41421i	1.41421i
$853$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$853$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$854$	2.82843i	2.82843i
$855$	1.00000	1.00000
$856$	0	0
$857$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$857$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$858$	0	0
$859$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$860$	−1.00000	−1.00000
$861$	2.82843i	2.82843i
$862$	0	0
$863$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$863$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$864$	0	0
$865$	0	0
$866$	− 1.41421i	− 1.41421i
$867$	0	0
$868$	− 1.41421i	− 1.41421i
$869$	− 1.41421i	− 1.41421i
$870$	2.82843i	2.82843i
$871$	0	0
$872$	0	0
$873$	− 1.41421i	− 1.41421i
$874$	0	0
$875$	1.41421i	1.41421i
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−2.00000	−2.00000
$879$	0	0
$880$	− 1.41421i	− 1.41421i
$881$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$882$	− 1.41421i	− 1.41421i
$883$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$887$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$888$	0	0
$889$	− 1.41421i	− 1.41421i
$890$	0	0
$891$	1.41421i	1.41421i
$892$	1.00000	1.00000
$893$	0	0
$894$	2.00000	2.00000
$895$	1.00000	1.00000
$896$	0	0
$897$	0	0
$898$	− 1.41421i	− 1.41421i
$899$	1.41421i	1.41421i
$900$	0	0
$901$	−1.00000	−1.00000
$902$	− 2.82843i	− 2.82843i
$903$	−2.00000	−2.00000
$904$	0	0
$905$	− 1.41421i	− 1.41421i
$906$	0	0
$907$	2.00000	2.00000	1.00000			$0$
$907$	2.00000	2.00000	1.00000			$0$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$911$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$912$	1.41421i	1.41421i
$913$	0	0
$914$	1.41421i	1.41421i
$915$	−2.00000	−2.00000
$916$	1.41421i	1.41421i
$917$	1.41421i	1.41421i
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	− 1.41421i	− 1.41421i
$923$	0	0
$924$	2.82843i	2.82843i
$925$	0	0
$926$	0	0
$927$	0	0
$928$	−2.00000	−2.00000
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	2.00000	2.00000
$931$	−1.00000	−1.00000
$932$	−1.00000	−1.00000
$933$	0	0
$934$	0	0
$935$	1.41421i	1.41421i
$936$	0	0
$937$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$937$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$938$	2.00000	2.00000
$939$	1.41421i	1.41421i
$940$	0	0
$941$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$941$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$942$	2.00000	2.00000
$943$	0	0
$944$	0	0
$945$	0	0
$946$	2.00000	2.00000
$947$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$947$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$948$	1.41421i	1.41421i
$949$	0	0
$950$	0	0
$951$	2.00000	2.00000
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	− 1.41421i	− 1.41421i
$955$	1.00000	1.00000
$956$	0	0
$957$	− 2.82843i	− 2.82843i
$958$	0	0
$959$	2.00000	2.00000
$960$	1.41421i	1.41421i
$961$	0	0
$962$	0	0
$963$	0	0
$964$	−1.00000	−1.00000
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	− 1.41421i	− 1.41421i
$970$	−2.00000	−2.00000
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 1.41421i	− 1.41421i
$973$	0	0
$974$	− 1.41421i	− 1.41421i
$975$	0	0
$976$	− 1.41421i	− 1.41421i
$977$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$977$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$978$	−2.00000	−2.00000
$979$	0	0
$980$	−1.00000	−1.00000
$981$	1.00000	1.00000
$982$	−2.00000	−2.00000
$983$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$983$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$984$	0	0
$985$	−1.00000	−1.00000
$986$	2.00000	2.00000
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−2.00000	−2.00000
$991$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$992$	1.41421i	1.41421i
$993$	1.41421i	1.41421i
$994$	2.00000	2.00000
$995$	1.41421i	1.41421i
$996$	0	0
$997$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$997$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	331.1.b.b.330.1	✓	2
3.2	odd	2		2979.1.c.d.2647.2		2
331.330	odd	2	inner	331.1.b.b.330.2	yes	2
993.992	even	2		2979.1.c.d.2647.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
331.1.b.b.330.1	✓	2	1.1	even	1	trivial
331.1.b.b.330.2	yes	2	331.330	odd	2	inner
2979.1.c.d.2647.1		2	993.992	even	2
2979.1.c.d.2647.2		2	3.2	odd	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:	\( N \)	\(=\)	\( 331 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	331.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.41421i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.41421i q^{7} -1.00000 q^{9} +1.41421i q^{10} -1.41421i q^{11} +1.41421i q^{12} +2.00000 q^{14} +1.41421i q^{15} -1.00000 q^{16} +1.00000 q^{17} +1.41421i q^{18} +1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -1.41421i q^{28} +1.41421i q^{29} +2.00000 q^{30} +1.00000 q^{31} +1.41421i q^{32} -2.00000 q^{33} -1.41421i q^{34} -1.41421i q^{35} +1.00000 q^{36} -1.41421i q^{37} -1.41421i q^{38} +1.41421i q^{41} -2.82843i q^{42} -1.00000 q^{43} +1.41421i q^{44} +1.00000 q^{45} +1.41421i q^{48} -1.00000 q^{49} -1.41421i q^{51} -1.00000 q^{53} +1.41421i q^{55} -1.41421i q^{57} +2.00000 q^{58} -1.41421i q^{60} +1.41421i q^{61} -1.41421i q^{62} -1.41421i q^{63} +1.00000 q^{64} +2.82843i q^{66} +1.00000 q^{67} -1.00000 q^{68} -2.00000 q^{70} +1.00000 q^{71} -2.00000 q^{74} -1.00000 q^{76} +2.00000 q^{77} +1.00000 q^{79} +1.00000 q^{80} -1.00000 q^{81} +2.00000 q^{82} -2.00000 q^{84} -1.00000 q^{85} +1.41421i q^{86} +2.00000 q^{87} -1.41421i q^{90} -1.41421i q^{93} -1.00000 q^{95} +2.00000 q^{96} +1.41421i q^{97} +1.41421i q^{98} +1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{9} + 4 q^{14} - 2 q^{16} + 2 q^{17} + 2 q^{19} + 2 q^{20} + 4 q^{21} - 4 q^{22} + 4 q^{30} + 2 q^{31} - 4 q^{33} + 2 q^{36} - 2 q^{43} + 2 q^{45} - 2 q^{49} - 2 q^{53} + 4 q^{58} + 2 q^{64} + 2 q^{67} - 2 q^{68} - 4 q^{70} + 2 q^{71} - 4 q^{74} - 2 q^{76} + 4 q^{77} + 2 q^{79} + 2 q^{80} - 2 q^{81} + 4 q^{82} - 4 q^{84} - 2 q^{85} + 4 q^{87} - 2 q^{95} + 4 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 - 4 * q^6 - 2 * q^9 + 4 * q^14 - 2 * q^16 + 2 * q^17 + 2 * q^19 + 2 * q^20 + 4 * q^21 - 4 * q^22 + 4 * q^30 + 2 * q^31 - 4 * q^33 + 2 * q^36 - 2 * q^43 + 2 * q^45 - 2 * q^49 - 2 * q^53 + 4 * q^58 + 2 * q^64 + 2 * q^67 - 2 * q^68 - 4 * q^70 + 2 * q^71 - 4 * q^74 - 2 * q^76 + 4 * q^77 + 2 * q^79 + 2 * q^80 - 2 * q^81 + 4 * q^82 - 4 * q^84 - 2 * q^85 + 4 * q^87 - 2 * q^95 + 4 * q^96

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