LMFDB - Embedded newform 1099.1.b.c.1098.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1099,1,Mod(1098,1099)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1099, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1099.1098");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1099 = 7 \cdot 157 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1099.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:	$0.548472448884$
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 4.2.1099.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.1327373299.4

Embedding invariants

Embedding label			1098.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1099.1098
Dual form			1099.1.b.c.1098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{9} -1.41421i q^{10} -1.00000 q^{11} +1.41421i q^{12} +1.41421i q^{13} -1.41421i q^{14} +1.41421i q^{15} -1.00000 q^{16} +1.41421i q^{17} -1.41421i q^{18} -1.41421i q^{19} +1.00000 q^{20} +1.41421i q^{21} -1.41421i q^{22} +1.41421i q^{23} -2.00000 q^{26} +1.00000 q^{28} -2.00000 q^{30} +1.41421i q^{31} -1.41421i q^{32} +1.41421i q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -2.00000 q^{42} -1.41421i q^{43} +1.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +1.41421i q^{48} +1.00000 q^{49} +2.00000 q^{51} -1.41421i q^{52} +1.00000 q^{55} -2.00000 q^{57} -1.00000 q^{59} -1.41421i q^{60} -1.00000 q^{61} -2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.41421i q^{65} -2.00000 q^{66} +1.00000 q^{67} -1.41421i q^{68} +2.00000 q^{69} +1.41421i q^{70} -1.00000 q^{71} -1.00000 q^{73} -1.41421i q^{74} +1.41421i q^{76} +1.00000 q^{77} +2.82843i q^{78} -1.41421i q^{79} +1.00000 q^{80} -1.00000 q^{81} -1.41421i q^{84} -1.41421i q^{85} +2.00000 q^{86} +1.41421i q^{90} -1.41421i q^{91} -1.41421i q^{92} +2.00000 q^{93} +1.41421i q^{95} -2.00000 q^{96} +1.00000 q^{97} +1.41421i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5} + 4 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^6 - 2 * q^7 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{5} + 4 q^{6} - 2 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{16} + 2 q^{20} - 4 q^{26} + 2 q^{28} - 4 q^{30} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{37} + 4 q^{38} + 4 q^{39} - 4 q^{42} + 2 q^{44} + 2 q^{45} - 4 q^{46} + 2 q^{49} + 4 q^{51} + 2 q^{55} - 4 q^{57} - 2 q^{59} - 2 q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{66} + 2 q^{67} + 4 q^{69} - 2 q^{71} - 2 q^{73} + 2 q^{77} + 2 q^{80} - 2 q^{81} + 4 q^{86} + 4 q^{93} - 4 q^{96} + 2 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^6 - 2 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^16 + 2 * q^20 - 4 * q^26 + 2 * q^28 - 4 * q^30 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 2 * q^37 + 4 * q^38 + 4 * q^39 - 4 * q^42 + 2 * q^44 + 2 * q^45 - 4 * q^46 + 2 * q^49 + 4 * q^51 + 2 * q^55 - 4 * q^57 - 2 * q^59 - 2 * q^61 - 4 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^66 + 2 * q^67 + 4 * q^69 - 2 * q^71 - 2 * q^73 + 2 * q^77 + 2 * q^80 - 2 * q^81 + 4 * q^86 + 4 * q^93 - 4 * q^96 + 2 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1099\mathbb{Z}\right)^\times$.

$n$	$162$	$472$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	− 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
$7$	−1.00000	−1.00000
$7$	−1.00000	−1.00000
$8$	0	0
$9$	−1.00000	−1.00000
$10$	− 1.41421i	− 1.41421i
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	1.41421i	1.41421i
$13$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$13$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$14$	− 1.41421i	− 1.41421i
$15$	1.41421i	1.41421i
$16$	−1.00000	−1.00000
$17$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$17$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$18$	− 1.41421i	− 1.41421i
$19$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$19$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$20$	1.00000	1.00000
$21$	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
$25$	0	0
$26$	−2.00000	−2.00000
$27$	0	0
$28$	1.00000	1.00000
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−2.00000	−2.00000
$31$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$31$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$32$	− 1.41421i	− 1.41421i
$33$	1.41421i	1.41421i
$34$	−2.00000	−2.00000
$35$	1.00000	1.00000
$36$	1.00000	1.00000
$37$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$38$	2.00000	2.00000
$39$	2.00000	2.00000
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−2.00000	−2.00000
$43$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$43$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$44$	1.00000	1.00000
$45$	1.00000	1.00000
$46$	−2.00000	−2.00000
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.41421i	1.41421i
$49$	1.00000	1.00000
$50$	0	0
$51$	2.00000	2.00000
$52$	− 1.41421i	− 1.41421i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	1.00000	1.00000
$56$	0	0
$57$	−2.00000	−2.00000
$58$	0	0
$59$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$60$	− 1.41421i	− 1.41421i
$61$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$62$	−2.00000	−2.00000
$63$	1.00000	1.00000
$64$	1.00000	1.00000
$65$	− 1.41421i	− 1.41421i
$66$	−2.00000	−2.00000
$67$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$67$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$68$	− 1.41421i	− 1.41421i
$69$	2.00000	2.00000
$70$	1.41421i	1.41421i
$71$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$72$	0	0
$73$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$74$	− 1.41421i	− 1.41421i
$75$	0	0
$76$	1.41421i	1.41421i
$77$	1.00000	1.00000
$78$	2.82843i	2.82843i
$79$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$79$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$80$	1.00000	1.00000
$81$	−1.00000	−1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	− 1.41421i	− 1.41421i
$85$	− 1.41421i	− 1.41421i
$86$	2.00000	2.00000
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	1.41421i	1.41421i
$91$	− 1.41421i	− 1.41421i
$92$	− 1.41421i	− 1.41421i
$93$	2.00000	2.00000
$94$	0	0
$95$	1.41421i	1.41421i
$96$	−2.00000	−2.00000
$97$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$97$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$98$	1.41421i	1.41421i
$99$	1.00000	1.00000
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	2.82843i	2.82843i
$103$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$103$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$104$	0	0
$105$	− 1.41421i	− 1.41421i
$106$	0	0
$107$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$107$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$108$	0	0
$109$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$109$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$110$	1.41421i	1.41421i
$111$	1.41421i	1.41421i
$112$	1.00000	1.00000
$113$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$113$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$114$	− 2.82843i	− 2.82843i
$115$	− 1.41421i	− 1.41421i
$116$	0	0
$117$	− 1.41421i	− 1.41421i
$118$	− 1.41421i	− 1.41421i
$119$	− 1.41421i	− 1.41421i
$120$	0	0
$121$	0	0
$122$	− 1.41421i	− 1.41421i
$123$	0	0
$124$	− 1.41421i	− 1.41421i
$125$	1.00000	1.00000
$126$	1.41421i	1.41421i
$127$	−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
$129$	−2.00000	−2.00000
$130$	2.00000	2.00000
$131$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$132$	− 1.41421i	− 1.41421i
$133$	1.41421i	1.41421i
$134$	1.41421i	1.41421i
$135$	0	0
$136$	0	0
$137$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$137$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$138$	2.82843i	2.82843i
$139$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$140$	−1.00000	−1.00000
$141$	0	0
$142$	− 1.41421i	− 1.41421i
$143$	− 1.41421i	− 1.41421i
$144$	1.00000	1.00000
$145$	0	0
$146$	− 1.41421i	− 1.41421i
$147$	− 1.41421i	− 1.41421i
$148$	1.00000	1.00000
$149$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$149$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	− 1.41421i	− 1.41421i
$154$	1.41421i	1.41421i
$155$	− 1.41421i	− 1.41421i
$156$	−2.00000	−2.00000
$157$	1.00000	1.00000
$157$	1.00000	1.00000
$158$	2.00000	2.00000
$159$	0	0
$160$	1.41421i	1.41421i
$161$	− 1.41421i	− 1.41421i
$162$	− 1.41421i	− 1.41421i
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	− 1.41421i	− 1.41421i
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−1.00000	−1.00000
$170$	2.00000	2.00000
$171$	1.41421i	1.41421i
$172$	1.41421i	1.41421i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	1.00000
$177$	1.41421i	1.41421i
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−1.00000	−1.00000
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	2.00000	2.00000
$183$	1.41421i	1.41421i
$184$	0	0
$185$	1.00000	1.00000
$186$	2.82843i	2.82843i
$187$	− 1.41421i	− 1.41421i
$188$	0	0
$189$	0	0
$190$	−2.00000	−2.00000
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 1.41421i	− 1.41421i
$193$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$194$	1.41421i	1.41421i
$195$	−2.00000	−2.00000
$196$	−1.00000	−1.00000
$197$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$198$	1.41421i	1.41421i
$199$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$199$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$200$	0	0
$201$	− 1.41421i	− 1.41421i
$202$	0	0
$203$	0	0
$204$	−2.00000	−2.00000
$205$	0	0
$206$	1.41421i	1.41421i
$207$	− 1.41421i	− 1.41421i
$208$	− 1.41421i	− 1.41421i
$209$	1.41421i	1.41421i
$210$	2.00000	2.00000
$211$	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
$213$	1.41421i	1.41421i
$214$	−2.00000	−2.00000
$215$	1.41421i	1.41421i
$216$	0	0
$217$	− 1.41421i	− 1.41421i
$218$	1.41421i	1.41421i
$219$	1.41421i	1.41421i
$220$	−1.00000	−1.00000
$221$	−2.00000	−2.00000
$222$	−2.00000	−2.00000
$223$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$223$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$224$	1.41421i	1.41421i
$225$	0	0
$226$	1.41421i	1.41421i
$227$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$228$	2.00000	2.00000
$229$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$230$	2.00000	2.00000
$231$	− 1.41421i	− 1.41421i
$232$	0	0
$233$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$234$	2.00000	2.00000
$235$	0	0
$236$	1.00000	1.00000
$237$	−2.00000	−2.00000
$238$	2.00000	2.00000
$239$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$239$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$240$	− 1.41421i	− 1.41421i
$241$	−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
$243$	1.41421i	1.41421i
$244$	1.00000	1.00000
$245$	−1.00000	−1.00000
$246$	0	0
$247$	2.00000	2.00000
$248$	0	0
$249$	0	0
$250$	1.41421i	1.41421i
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−1.00000	−1.00000
$253$	− 1.41421i	− 1.41421i
$254$	− 1.41421i	− 1.41421i
$255$	−2.00000	−2.00000
$256$	1.00000	1.00000
$257$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$257$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$258$	− 2.82843i	− 2.82843i
$259$	1.00000	1.00000
$260$	1.41421i	1.41421i
$261$	0	0
$262$	1.41421i	1.41421i
$263$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$263$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−2.00000
$267$	0	0
$268$	−1.00000	−1.00000
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$271$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$272$	− 1.41421i	− 1.41421i
$273$	−2.00000	−2.00000
$274$	−2.00000	−2.00000
$275$	0	0
$276$	−2.00000	−2.00000
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	− 1.41421i	− 1.41421i
$279$	− 1.41421i	− 1.41421i
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.00000	1.00000
$285$	2.00000	2.00000
$286$	2.00000	2.00000
$287$	0	0
$288$	1.41421i	1.41421i
$289$	−1.00000	−1.00000
$290$	0	0
$291$	− 1.41421i	− 1.41421i
$292$	1.00000	1.00000
$293$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$293$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$294$	2.00000	2.00000
$295$	1.00000	1.00000
$296$	0	0
$297$	0	0
$298$	−2.00000	−2.00000
$299$	−2.00000	−2.00000
$300$	0	0
$301$	1.41421i	1.41421i
$302$	0	0
$303$	0	0
$304$	1.41421i	1.41421i
$305$	1.00000	1.00000
$306$	2.00000	2.00000
$307$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$307$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$308$	−1.00000	−1.00000
$309$	− 1.41421i	− 1.41421i
$310$	2.00000	2.00000
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	1.41421i	1.41421i
$315$	−1.00000	−1.00000
$316$	1.41421i	1.41421i
$317$	−2.00000	−2.00000	−1.00000			$\pi$
$317$	−2.00000	−2.00000	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−1.00000
$321$	2.00000	2.00000
$322$	2.00000	2.00000
$323$	2.00000	2.00000
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	− 1.41421i	− 1.41421i
$328$	0	0
$329$	0	0
$330$	2.00000	2.00000
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	1.00000	1.00000
$334$	0	0
$335$	−1.00000	−1.00000
$336$	− 1.41421i	− 1.41421i
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 1.41421i	− 1.41421i
$339$	− 1.41421i	− 1.41421i
$340$	1.41421i	1.41421i
$341$	− 1.41421i	− 1.41421i
$342$	−2.00000	−2.00000
$343$	−1.00000	−1.00000
$344$	0	0
$345$	−2.00000	−2.00000
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	1.41421i	1.41421i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−2.00000	−2.00000
$355$	1.00000	1.00000
$356$	0	0
$357$	−2.00000	−2.00000
$358$	0	0
$359$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$359$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$360$	0	0
$361$	−1.00000	−1.00000
$362$	0	0
$363$	0	0
$364$	1.41421i	1.41421i
$365$	1.00000	1.00000
$366$	−2.00000	−2.00000
$367$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$368$	− 1.41421i	− 1.41421i
$369$	0	0
$370$	1.41421i	1.41421i
$371$	0	0
$372$	−2.00000	−2.00000
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	2.00000	2.00000
$375$	− 1.41421i	− 1.41421i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$379$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$380$	− 1.41421i	− 1.41421i
$381$	1.41421i	1.41421i
$382$	0	0
$383$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$384$	0	0
$385$	−1.00000	−1.00000
$386$	− 1.41421i	− 1.41421i
$387$	1.41421i	1.41421i
$388$	−1.00000	−1.00000
$389$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$390$	− 2.82843i	− 2.82843i
$391$	−2.00000	−2.00000
$392$	0	0
$393$	− 1.41421i	− 1.41421i
$394$	− 1.41421i	− 1.41421i
$395$	1.41421i	1.41421i
$396$	−1.00000	−1.00000
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−2.00000	−2.00000
$399$	2.00000	2.00000
$400$	0	0
$401$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$401$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$402$	2.00000	2.00000
$403$	−2.00000	−2.00000
$404$	0	0
$405$	1.00000	1.00000
$406$	0	0
$407$	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
$411$	2.00000	2.00000
$412$	−1.00000	−1.00000
$413$	1.00000	1.00000
$414$	2.00000	2.00000
$415$	0	0
$416$	2.00000	2.00000
$417$	1.41421i	1.41421i
$418$	−2.00000	−2.00000
$419$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$419$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$420$	1.41421i	1.41421i
$421$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$421$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$422$	−2.00000	−2.00000
$423$	0	0
$424$	0	0
$425$	0	0
$426$	−2.00000	−2.00000
$427$	1.00000	1.00000
$428$	− 1.41421i	− 1.41421i
$429$	−2.00000	−2.00000
$430$	−2.00000	−2.00000
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	2.00000	2.00000
$435$	0	0
$436$	−1.00000	−1.00000
$437$	2.00000	2.00000
$438$	−2.00000	−2.00000
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−1.00000	−1.00000
$442$	− 2.82843i	− 2.82843i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	− 1.41421i	− 1.41421i
$445$	0	0
$446$	1.41421i	1.41421i
$447$	2.00000	2.00000
$448$	−1.00000	−1.00000
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	−1.00000	−1.00000
$453$	0	0
$454$	− 1.41421i	− 1.41421i
$455$	1.41421i	1.41421i
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	− 1.41421i	− 1.41421i
$459$	0	0
$460$	1.41421i	1.41421i
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	2.00000	2.00000
$463$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$463$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$464$	0	0
$465$	−2.00000	−2.00000
$466$	1.41421i	1.41421i
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.41421i	1.41421i
$469$	−1.00000	−1.00000
$470$	0	0
$471$	− 1.41421i	− 1.41421i
$472$	0	0
$473$	1.41421i	1.41421i
$474$	− 2.82843i	− 2.82843i
$475$	0	0
$476$	1.41421i	1.41421i
$477$	0	0
$478$	− 1.41421i	− 1.41421i
$479$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$480$	2.00000	2.00000
$481$	− 1.41421i	− 1.41421i
$482$	− 1.41421i	− 1.41421i
$483$	−2.00000	−2.00000
$484$	0	0
$485$	−1.00000	−1.00000
$486$	−2.00000	−2.00000
$487$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$487$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$488$	0	0
$489$	0	0
$490$	− 1.41421i	− 1.41421i
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	2.82843i	2.82843i
$495$	−1.00000	−1.00000
$496$	− 1.41421i	− 1.41421i
$497$	1.00000	1.00000
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−1.00000	−1.00000
$501$	0	0
$502$	0	0
$503$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$503$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$504$	0	0
$505$	0	0
$506$	2.00000	2.00000
$507$	1.41421i	1.41421i
$508$	1.00000	1.00000
$509$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$510$	− 2.82843i	− 2.82843i
$511$	1.00000	1.00000
$512$	1.41421i	1.41421i
$513$	0	0
$514$	−2.00000	−2.00000
$515$	−1.00000	−1.00000
$516$	2.00000	2.00000
$517$	0	0
$518$	1.41421i	1.41421i
$519$	0	0
$520$	0	0
$521$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$522$	0	0
$523$	− 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
$526$	1.41421i	1.41421i
$527$	−2.00000	−2.00000
$528$	− 1.41421i	− 1.41421i
$529$	−1.00000	−1.00000
$530$	0	0
$531$	1.00000	1.00000
$532$	− 1.41421i	− 1.41421i
$533$	0	0
$534$	0	0
$535$	− 1.41421i	− 1.41421i
$536$	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	$0.250000\pi$
$541$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$542$	1.41421i	1.41421i
$543$	0	0
$544$	2.00000	2.00000
$545$	−1.00000	−1.00000
$546$	− 2.82843i	− 2.82843i
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	− 1.41421i	− 1.41421i
$549$	1.00000	1.00000
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.41421i	1.41421i
$554$	0	0
$555$	− 1.41421i	− 1.41421i
$556$	1.00000	1.00000
$557$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$558$	2.00000	2.00000
$559$	2.00000	2.00000
$560$	−1.00000	−1.00000
$561$	−2.00000	−2.00000
$562$	0	0
$563$	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
$565$	−1.00000	−1.00000
$566$	0	0
$567$	1.00000	1.00000
$568$	0	0
$569$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$569$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$570$	2.82843i	2.82843i
$571$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$572$	1.41421i	1.41421i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−1.00000	−1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 1.41421i	− 1.41421i
$579$	1.41421i	1.41421i
$580$	0	0
$581$	0	0
$582$	2.00000	2.00000
$583$	0	0
$584$	0	0
$585$	1.41421i	1.41421i
$586$	1.41421i	1.41421i
$587$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$587$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$588$	1.41421i	1.41421i
$589$	2.00000	2.00000
$590$	1.41421i	1.41421i
$591$	1.41421i	1.41421i
$592$	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	$0.750000\pi$
$594$	0	0
$595$	1.41421i	1.41421i
$596$	− 1.41421i	− 1.41421i
$597$	2.00000	2.00000
$598$	− 2.82843i	− 2.82843i
$599$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$599$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−2.00000	−2.00000
$603$	−1.00000	−1.00000
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$607$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$608$	−2.00000	−2.00000
$609$	0	0
$610$	1.41421i	1.41421i
$611$	0	0
$612$	1.41421i	1.41421i
$613$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$613$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$614$	1.41421i	1.41421i
$615$	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	$0.250000\pi$
$619$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$620$	1.41421i	1.41421i
$621$	0	0
$622$	0	0
$623$	0	0
$624$	−2.00000	−2.00000
$625$	−1.00000	−1.00000
$626$	0	0
$627$	2.00000	2.00000
$628$	−1.00000	−1.00000
$629$	− 1.41421i	− 1.41421i
$630$	− 1.41421i	− 1.41421i
$631$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$632$	0	0
$633$	2.00000	2.00000
$634$	− 2.82843i	− 2.82843i
$635$	1.00000	1.00000
$636$	0	0
$637$	1.41421i	1.41421i
$638$	0	0
$639$	1.00000	1.00000
$640$	0	0
$641$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$642$	2.82843i	2.82843i
$643$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$644$	1.41421i	1.41421i
$645$	2.00000	2.00000
$646$	2.82843i	2.82843i
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	1.00000	1.00000
$650$	0	0
$651$	−2.00000	−2.00000
$652$	0	0
$653$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$653$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$654$	2.00000	2.00000
$655$	−1.00000	−1.00000
$656$	0	0
$657$	1.00000	1.00000
$658$	0	0
$659$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$659$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$660$	1.41421i	1.41421i
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	2.82843i	2.82843i
$664$	0	0
$665$	− 1.41421i	− 1.41421i
$666$	1.41421i	1.41421i
$667$	0	0
$668$	0	0
$669$	− 1.41421i	− 1.41421i
$670$	− 1.41421i	− 1.41421i
$671$	1.00000	1.00000
$672$	2.00000	2.00000
$673$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$673$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$674$	0	0
$675$	0	0
$676$	1.00000	1.00000
$677$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$677$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$678$	2.00000	2.00000
$679$	−1.00000	−1.00000
$680$	0	0
$681$	1.41421i	1.41421i
$682$	2.00000	2.00000
$683$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$683$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$684$	− 1.41421i	− 1.41421i
$685$	− 1.41421i	− 1.41421i
$686$	− 1.41421i	− 1.41421i
$687$	1.41421i	1.41421i
$688$	1.41421i	1.41421i
$689$	0	0
$690$	− 2.82843i	− 2.82843i
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−1.00000	−1.00000
$694$	0	0
$695$	1.00000	1.00000
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 1.41421i	− 1.41421i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	1.41421i	1.41421i
$704$	−1.00000	−1.00000
$705$	0	0
$706$	0	0
$707$	0	0
$708$	− 1.41421i	− 1.41421i
$709$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$710$	1.41421i	1.41421i
$711$	1.41421i	1.41421i
$712$	0	0
$713$	−2.00000	−2.00000
$714$	− 2.82843i	− 2.82843i
$715$	1.41421i	1.41421i
$716$	0	0
$717$	1.41421i	1.41421i
$718$	2.00000	2.00000
$719$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$720$	−1.00000	−1.00000
$721$	−1.00000	−1.00000
$722$	− 1.41421i	− 1.41421i
$723$	1.41421i	1.41421i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	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
$730$	1.41421i	1.41421i
$731$	2.00000	2.00000
$732$	− 1.41421i	− 1.41421i
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	1.41421i	1.41421i
$735$	1.41421i	1.41421i
$736$	2.00000	2.00000
$737$	−1.00000	−1.00000
$738$	0	0
$739$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$739$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$740$	−1.00000	−1.00000
$741$	− 2.82843i	− 2.82843i
$742$	0	0
$743$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0	0
$745$	− 1.41421i	− 1.41421i
$746$	0	0
$747$	0	0
$748$	1.41421i	1.41421i
$749$	− 1.41421i	− 1.41421i
$750$	2.00000	2.00000
$751$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−2.00000	−2.00000
$759$	−2.00000	−2.00000
$760$	0	0
$761$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$762$	−2.00000	−2.00000
$763$	−1.00000	−1.00000
$764$	0	0
$765$	1.41421i	1.41421i
$766$	− 1.41421i	− 1.41421i
$767$	− 1.41421i	− 1.41421i
$768$	− 1.41421i	− 1.41421i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	− 1.41421i	− 1.41421i
$771$	2.00000	2.00000
$772$	1.00000	1.00000
$773$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$773$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$774$	−2.00000	−2.00000
$775$	0	0
$776$	0	0
$777$	− 1.41421i	− 1.41421i
$778$	1.41421i	1.41421i
$779$	0	0
$780$	2.00000	2.00000
$781$	1.00000	1.00000
$782$	− 2.82843i	− 2.82843i
$783$	0	0
$784$	−1.00000	−1.00000
$785$	−1.00000	−1.00000
$786$	2.00000	2.00000
$787$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$787$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$788$	1.00000	1.00000
$789$	− 1.41421i	− 1.41421i
$790$	−2.00000	−2.00000
$791$	−1.00000	−1.00000
$792$	0	0
$793$	− 1.41421i	− 1.41421i
$794$	0	0
$795$	0	0
$796$	− 1.41421i	− 1.41421i
$797$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$797$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$798$	2.82843i	2.82843i
$799$	0	0
$800$	0	0
$801$	0	0
$802$	2.00000	2.00000
$803$	1.00000	1.00000
$804$	1.41421i	1.41421i
$805$	1.41421i	1.41421i
$806$	− 2.82843i	− 2.82843i
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.41421i	1.41421i
$811$	−2.00000	−2.00000	−1.00000			$\pi$
$811$	−2.00000	−2.00000	−1.00000			$\pi$
$812$	0	0
$813$	− 1.41421i	− 1.41421i
$814$	1.41421i	1.41421i
$815$	0	0
$816$	−2.00000	−2.00000
$817$	−2.00000	−2.00000
$818$	0	0
$819$	1.41421i	1.41421i
$820$	0	0
$821$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$821$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$822$	2.82843i	2.82843i
$823$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$823$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$824$	0	0
$825$	0	0
$826$	1.41421i	1.41421i
$827$	−2.00000	−2.00000	−1.00000			$\pi$
$827$	−2.00000	−2.00000	−1.00000			$\pi$
$828$	1.41421i	1.41421i
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	1.41421i	1.41421i
$833$	1.41421i	1.41421i
$834$	−2.00000	−2.00000
$835$	0	0
$836$	− 1.41421i	− 1.41421i
$837$	0	0
$838$	2.00000	2.00000
$839$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$839$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	2.00000	2.00000
$843$	0	0
$844$	− 1.41421i	− 1.41421i
$845$	1.00000	1.00000
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.41421i	− 1.41421i
$852$	− 1.41421i	− 1.41421i
$853$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$853$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$854$	1.41421i	1.41421i
$855$	− 1.41421i	− 1.41421i
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	− 2.82843i	− 2.82843i
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	− 1.41421i	− 1.41421i
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.41421i	1.41421i
$868$	1.41421i	1.41421i
$869$	1.41421i	1.41421i
$870$	0	0
$871$	1.41421i	1.41421i
$872$	0	0
$873$	−1.00000	−1.00000
$874$	2.82843i	2.82843i
$875$	−1.00000	−1.00000
$876$	− 1.41421i	− 1.41421i
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	− 1.41421i	− 1.41421i
$880$	−1.00000	−1.00000
$881$	−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
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	2.00000	2.00000
$885$	− 1.41421i	− 1.41421i
$886$	0	0
$887$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$887$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$888$	0	0
$889$	1.00000	1.00000
$890$	0	0
$891$	1.00000	1.00000
$892$	−1.00000	−1.00000
$893$	0	0
$894$	2.82843i	2.82843i
$895$	0	0
$896$	0	0
$897$	2.82843i	2.82843i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	2.00000	2.00000
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$908$	1.00000	1.00000
$909$	0	0
$910$	−2.00000	−2.00000
$911$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$911$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$912$	2.00000	2.00000
$913$	0	0
$914$	0	0
$915$	− 1.41421i	− 1.41421i
$916$	1.00000	1.00000
$917$	−1.00000	−1.00000
$918$	0	0
$919$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$919$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$920$	0	0
$921$	− 1.41421i	− 1.41421i
$922$	0	0
$923$	− 1.41421i	− 1.41421i
$924$	1.41421i	1.41421i
$925$	0	0
$926$	−2.00000	−2.00000
$927$	−1.00000	−1.00000
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	− 2.82843i	− 2.82843i
$931$	− 1.41421i	− 1.41421i
$932$	−1.00000	−1.00000
$933$	0	0
$934$	0	0
$935$	1.41421i	1.41421i
$936$	0	0
$937$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$937$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$938$	− 1.41421i	− 1.41421i
$939$	0	0
$940$	0	0
$941$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$941$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$942$	2.00000	2.00000
$943$	0	0
$944$	1.00000	1.00000
$945$	0	0
$946$	−2.00000	−2.00000
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	2.00000	2.00000
$949$	− 1.41421i	− 1.41421i
$950$	0	0
$951$	2.82843i	2.82843i
$952$	0	0
$953$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$954$	0	0
$955$	0	0
$956$	1.00000	1.00000
$957$	0	0
$958$	− 1.41421i	− 1.41421i
$959$	− 1.41421i	− 1.41421i
$960$	1.41421i	1.41421i
$961$	−1.00000	−1.00000
$962$	2.00000	2.00000
$963$	− 1.41421i	− 1.41421i
$964$	1.00000	1.00000
$965$	1.00000	1.00000
$966$	− 2.82843i	− 2.82843i
$967$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$968$	0	0
$969$	− 2.82843i	− 2.82843i
$970$	− 1.41421i	− 1.41421i
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	− 1.41421i	− 1.41421i
$973$	1.00000	1.00000
$974$	− 1.41421i	− 1.41421i
$975$	0	0
$976$	1.00000	1.00000
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	1.00000	1.00000
$981$	−1.00000	−1.00000
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	1.00000	1.00000
$986$	0	0
$987$	0	0
$988$	−2.00000	−2.00000
$989$	2.00000	2.00000
$990$	− 1.41421i	− 1.41421i
$991$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$991$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$992$	2.00000	2.00000
$993$	0	0
$994$	1.41421i	1.41421i
$995$	− 1.41421i	− 1.41421i
$996$	0	0
$997$	−2.00000	−2.00000	−1.00000			$\pi$
$997$	−2.00000	−2.00000	−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	1099.1.b.c.1098.2	yes	2
7.6	odd	2		1099.1.b.d.1098.2	yes	2
157.156	even	2		1099.1.b.d.1098.1	yes	2
1099.1098	odd	2	inner	1099.1.b.c.1098.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1099.1.b.c.1098.1	✓	2	1099.1098	odd	2	inner
1099.1.b.c.1098.2	yes	2	1.1	even	1	trivial
1099.1.b.d.1098.1	yes	2	157.156	even	2
1099.1.b.d.1098.2	yes	2	7.6	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:	\( N \)	\(=\)	\( 1099 = 7 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1099.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -1.41421i q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{9} -1.41421i q^{10} -1.00000 q^{11} +1.41421i q^{12} +1.41421i q^{13} -1.41421i q^{14} +1.41421i q^{15} -1.00000 q^{16} +1.41421i q^{17} -1.41421i q^{18} -1.41421i q^{19} +1.00000 q^{20} +1.41421i q^{21} -1.41421i q^{22} +1.41421i q^{23} -2.00000 q^{26} +1.00000 q^{28} -2.00000 q^{30} +1.41421i q^{31} -1.41421i q^{32} +1.41421i q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -2.00000 q^{42} -1.41421i q^{43} +1.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +1.41421i q^{48} +1.00000 q^{49} +2.00000 q^{51} -1.41421i q^{52} +1.00000 q^{55} -2.00000 q^{57} -1.00000 q^{59} -1.41421i q^{60} -1.00000 q^{61} -2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.41421i q^{65} -2.00000 q^{66} +1.00000 q^{67} -1.41421i q^{68} +2.00000 q^{69} +1.41421i q^{70} -1.00000 q^{71} -1.00000 q^{73} -1.41421i q^{74} +1.41421i q^{76} +1.00000 q^{77} +2.82843i q^{78} -1.41421i q^{79} +1.00000 q^{80} -1.00000 q^{81} -1.41421i q^{84} -1.41421i q^{85} +2.00000 q^{86} +1.41421i q^{90} -1.41421i q^{91} -1.41421i q^{92} +2.00000 q^{93} +1.41421i q^{95} -2.00000 q^{96} +1.00000 q^{97} +1.41421i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5} + 4 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^6 - 2 * q^7 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{5} + 4 q^{6} - 2 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{16} + 2 q^{20} - 4 q^{26} + 2 q^{28} - 4 q^{30} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{37} + 4 q^{38} + 4 q^{39} - 4 q^{42} + 2 q^{44} + 2 q^{45} - 4 q^{46} + 2 q^{49} + 4 q^{51} + 2 q^{55} - 4 q^{57} - 2 q^{59} - 2 q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{66} + 2 q^{67} + 4 q^{69} - 2 q^{71} - 2 q^{73} + 2 q^{77} + 2 q^{80} - 2 q^{81} + 4 q^{86} + 4 q^{93} - 4 q^{96} + 2 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^6 - 2 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^16 + 2 * q^20 - 4 * q^26 + 2 * q^28 - 4 * q^30 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 2 * q^37 + 4 * q^38 + 4 * q^39 - 4 * q^42 + 2 * q^44 + 2 * q^45 - 4 * q^46 + 2 * q^49 + 4 * q^51 + 2 * q^55 - 4 * q^57 - 2 * q^59 - 2 * q^61 - 4 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^66 + 2 * q^67 + 4 * q^69 - 2 * q^71 - 2 * q^73 + 2 * q^77 + 2 * q^80 - 2 * q^81 + 4 * q^86 + 4 * q^93 - 4 * q^96 + 2 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more