LMFDB - Embedded newform 491.1.b.b.490.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [491,1,Mod(490,491)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(491, 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("491.490");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 491 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	491.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:	$0.245040921203$
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 4.2.491.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.118370771.2

Embedding invariants

Embedding label			490.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	491.490
Dual form			491.1.b.b.490.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.41421i q^{7} +O(q^{10})$	\(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.41421i q^{7} +1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} -1.41421i q^{19} -1.41421i q^{21} +1.41421i q^{22} +1.41421i q^{23} -1.00000 q^{25} -1.41421i q^{26} +1.00000 q^{27} -1.41421i q^{28} +1.41421i q^{29} +1.00000 q^{31} -1.41421i q^{32} -1.00000 q^{33} -1.41421i q^{34} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{39} +1.00000 q^{41} +2.00000 q^{42} +1.00000 q^{43} -1.00000 q^{44} -2.00000 q^{46} +1.00000 q^{48} -1.00000 q^{49} -1.41421i q^{50} +1.00000 q^{51} +1.00000 q^{52} +1.41421i q^{54} +1.41421i q^{57} -2.00000 q^{58} +1.41421i q^{59} +1.41421i q^{62} +1.00000 q^{64} -1.41421i q^{66} -1.41421i q^{67} +1.00000 q^{68} -1.41421i q^{69} +1.00000 q^{71} -1.41421i q^{73} +1.41421i q^{74} +1.00000 q^{75} +1.41421i q^{76} +1.41421i q^{77} +1.41421i q^{78} -1.00000 q^{81} +1.41421i q^{82} +1.00000 q^{83} +1.41421i q^{84} +1.41421i q^{86} -1.41421i q^{87} -1.41421i q^{91} -1.41421i q^{92} -1.00000 q^{93} +1.41421i q^{96} -1.00000 q^{97} -1.41421i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{11} + 2 q^{12} - 2 q^{13} - 4 q^{14} - 2 q^{16} - 2 q^{17} - 2 q^{25} + 2 q^{27} + 2 q^{31} - 2 q^{33} + 2 q^{37} + 4 q^{38} + 2 q^{39} + 2 q^{41} + 4 q^{42} + 2 q^{43} - 2 q^{44} - 4 q^{46} + 2 q^{48} - 2 q^{49} + 2 q^{51} + 2 q^{52} - 4 q^{58} + 2 q^{64} + 2 q^{68} + 2 q^{71} + 2 q^{75} - 2 q^{81} + 2 q^{83} - 2 q^{93} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^11 + 2 * q^12 - 2 * q^13 - 4 * q^14 - 2 * q^16 - 2 * q^17 - 2 * q^25 + 2 * q^27 + 2 * q^31 - 2 * q^33 + 2 * q^37 + 4 * q^38 + 2 * q^39 + 2 * q^41 + 4 * q^42 + 2 * q^43 - 2 * q^44 - 4 * q^46 + 2 * q^48 - 2 * q^49 + 2 * q^51 + 2 * q^52 - 4 * q^58 + 2 * q^64 + 2 * q^68 + 2 * q^71 + 2 * q^75 - 2 * q^81 + 2 * q^83 - 2 * q^93 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/491\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	491.1.b.b.490.2	yes	2
491.490	odd	2	inner	491.1.b.b.490.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
491.1.b.b.490.1	✓	2	491.490	odd	2	inner
491.1.b.b.490.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 491 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	491.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.41421i q^{7} +O(q^{10})\)	\(q+1.41421i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.41421i q^{6} +1.41421i q^{7} +1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} -1.41421i q^{19} -1.41421i q^{21} +1.41421i q^{22} +1.41421i q^{23} -1.00000 q^{25} -1.41421i q^{26} +1.00000 q^{27} -1.41421i q^{28} +1.41421i q^{29} +1.00000 q^{31} -1.41421i q^{32} -1.00000 q^{33} -1.41421i q^{34} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{39} +1.00000 q^{41} +2.00000 q^{42} +1.00000 q^{43} -1.00000 q^{44} -2.00000 q^{46} +1.00000 q^{48} -1.00000 q^{49} -1.41421i q^{50} +1.00000 q^{51} +1.00000 q^{52} +1.41421i q^{54} +1.41421i q^{57} -2.00000 q^{58} +1.41421i q^{59} +1.41421i q^{62} +1.00000 q^{64} -1.41421i q^{66} -1.41421i q^{67} +1.00000 q^{68} -1.41421i q^{69} +1.00000 q^{71} -1.41421i q^{73} +1.41421i q^{74} +1.00000 q^{75} +1.41421i q^{76} +1.41421i q^{77} +1.41421i q^{78} -1.00000 q^{81} +1.41421i q^{82} +1.00000 q^{83} +1.41421i q^{84} +1.41421i q^{86} -1.41421i q^{87} -1.41421i q^{91} -1.41421i q^{92} -1.00000 q^{93} +1.41421i q^{96} -1.00000 q^{97} -1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{11} + 2 q^{12} - 2 q^{13} - 4 q^{14} - 2 q^{16} - 2 q^{17} - 2 q^{25} + 2 q^{27} + 2 q^{31} - 2 q^{33} + 2 q^{37} + 4 q^{38} + 2 q^{39} + 2 q^{41} + 4 q^{42} + 2 q^{43} - 2 q^{44} - 4 q^{46} + 2 q^{48} - 2 q^{49} + 2 q^{51} + 2 q^{52} - 4 q^{58} + 2 q^{64} + 2 q^{68} + 2 q^{71} + 2 q^{75} - 2 q^{81} + 2 q^{83} - 2 q^{93} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^11 + 2 * q^12 - 2 * q^13 - 4 * q^14 - 2 * q^16 - 2 * q^17 - 2 * q^25 + 2 * q^27 + 2 * q^31 - 2 * q^33 + 2 * q^37 + 4 * q^38 + 2 * q^39 + 2 * q^41 + 4 * q^42 + 2 * q^43 - 2 * q^44 - 4 * q^46 + 2 * q^48 - 2 * q^49 + 2 * q^51 + 2 * q^52 - 4 * q^58 + 2 * q^64 + 2 * q^68 + 2 * q^71 + 2 * q^75 - 2 * q^81 + 2 * q^83 - 2 * q^93 - 2 * q^97

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