LMFDB - Embedded newform 1089.1.r.a.245.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,1,Mod(245,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([5, 24]))

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

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

chi := DirichletCharacter("1089.245");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1089 = 3^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1089.r (of order $30$ , degree $8$ , not 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:	$0.543481798757$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.26198073.1

Embedding invariants

Embedding label			245.1
Root			$-0.104528 + 0.994522i$ of defining polynomial
Character	$\chi$	$=$	1089.245
Dual form			1089.1.r.a.1049.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.669131 + 0.743145i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-1.72256 - 0.181049i) q^{5} +(-0.104528 - 0.994522i) q^{9} +(0.500000 - 0.866025i) q^{12} +(1.28716 - 1.15897i) q^{15} +(0.913545 - 0.406737i) q^{16} +(1.72256 - 0.181049i) q^{20} +(1.95630 + 0.415823i) q^{25} +(0.809017 + 0.587785i) q^{27} +(0.913545 + 0.406737i) q^{31} +(0.309017 + 0.951057i) q^{36} +(0.309017 - 0.951057i) q^{37} +1.73205i q^{45} +(0.360114 - 1.69420i) q^{47} +(-0.309017 + 0.951057i) q^{48} +(0.104528 - 0.994522i) q^{49} +(-1.01807 - 1.40126i) q^{53} +(0.360114 + 1.69420i) q^{59} +(-1.01807 + 1.40126i) q^{60} +(-0.809017 + 0.587785i) q^{64} +(-0.500000 - 0.866025i) q^{67} +(1.01807 - 1.40126i) q^{71} +(-1.61803 + 1.17557i) q^{75} +(-1.64728 + 0.535233i) q^{80} +(-0.978148 + 0.207912i) q^{81} +(-0.913545 + 0.406737i) q^{93} +(0.104528 + 0.994522i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - q^{3} + q^{4} - 3 q^{5} + q^{9} + 4 q^{12} - 3 q^{15} + q^{16} + 3 q^{20} - 2 q^{25} + 2 q^{27} + q^{31} - 2 q^{36} - 2 q^{37} + 3 q^{47} + 2 q^{48} - q^{49} + 3 q^{59} - 2 q^{64} - 4 q^{67}+ \cdots - q^{97}+O(q^{100})$ 8 * q - q^3 + q^4 - 3 * q^5 + q^9 + 4 * q^12 - 3 * q^15 + q^16 + 3 * q^20 - 2 * q^25 + 2 * q^27 + q^31 - 2 * q^36 - 2 * q^37 + 3 * q^47 + 2 * q^48 - q^49 + 3 * q^59 - 2 * q^64 - 4 * q^67 - 4 * q^75 + q^81 - q^93 - q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$ .

$n$	$244$	$848$
$\chi(n)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{6}\right)$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$2$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$3$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$3$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$4$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$5$	−1.72256	−	0.181049i	−1.72256	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−1.72256	−	0.181049i	−1.72256	−	0.181049i	−0.913545	+	0.406737i	$0.866667\pi$
$6$		0			0
$7$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$7$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$8$		0			0
$9$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	0.500000	−	0.866025i	0.500000	−	0.866025i
$13$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$13$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$14$		0			0
$15$	1.28716	−	1.15897i	1.28716	−	1.15897i
$16$	0.913545	−	0.406737i	0.913545	−	0.406737i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$		0			0
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$	1.72256	−	0.181049i	1.72256	−	0.181049i
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	1.95630	+	0.415823i	1.95630	+	0.415823i
$26$		0			0
$27$	0.809017	+	0.587785i	0.809017	+	0.587785i
$28$		0			0
$29$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$29$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$30$		0			0
$31$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.809017	−	0.587785i	$-0.200000\pi$
$31$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.104528	+	0.994522i	$0.466667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.309017	+	0.951057i	0.309017	+	0.951057i
$37$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$37$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$41$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$42$		0			0
$43$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$43$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$44$		0			0
$45$			1.73205i			1.73205i
$46$		0			0
$47$	0.360114	−	1.69420i	0.360114	−	1.69420i	−0.309017	−	0.951057i	$-0.600000\pi$
$47$	0.360114	−	1.69420i	0.360114	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$48$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$49$	0.104528	−	0.994522i	0.104528	−	0.994522i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.01807	−	1.40126i	−1.01807	−	1.40126i	−0.913545	−	0.406737i	$-0.866667\pi$
$53$	−1.01807	−	1.40126i	−1.01807	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.360114	+	1.69420i	0.360114	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$59$	0.360114	+	1.69420i	0.360114	+	1.69420i	−0.309017	+	0.951057i	$0.600000\pi$
$60$	−1.01807	+	1.40126i	−1.01807	+	1.40126i
$61$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$61$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$62$		0			0
$63$		0			0
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.01807	−	1.40126i	1.01807	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$71$	1.01807	−	1.40126i	1.01807	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$72$		0			0
$73$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$73$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$74$		0			0
$75$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$79$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$80$	−1.64728	+	0.535233i	−1.64728	+	0.535233i
$81$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$82$		0			0
$83$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$83$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.104528	+	0.994522i	0.104528	+	0.994522i	0.913545	+	0.406737i	$0.133333\pi$
$97$	0.104528	+	0.994522i	0.104528	+	0.994522i	−0.809017	+	0.587785i	$0.800000\pi$
$98$		0			0
$99$		0			0
$100$	−2.00000			−2.00000
$101$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$101$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$102$		0			0
$103$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.309017	−	0.951057i	$-0.400000\pi$
$103$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.669131	+	0.743145i	$0.266667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$107$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$108$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	0.500000	+	0.866025i	0.500000	+	0.866025i
$112$		0			0
$113$	−1.28716	−	1.15897i	−1.28716	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$113$	−1.28716	−	1.15897i	−1.28716	−	1.15897i	−0.309017	−	0.951057i	$-0.600000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$125$	−1.64728	−	0.535233i	−1.64728	−	0.535233i
$126$		0			0
$127$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$127$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.28716	−	1.15897i	−1.28716	−	1.15897i
$136$		0			0
$137$	0.704489	+	1.58231i	0.704489	+	1.58231i	0.809017	+	0.587785i	$0.200000\pi$
$137$	0.704489	+	1.58231i	0.704489	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$138$		0			0
$139$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$139$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$140$		0			0
$141$	1.01807	+	1.40126i	1.01807	+	1.40126i
$142$		0			0
$143$		0			0
$144$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	0.669131	+	0.743145i	0.669131	+	0.743145i
$148$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$149$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$149$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$150$		0			0
$151$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$151$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$156$		0			0
$157$	0.669131	−	0.743145i	0.669131	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$157$	0.669131	−	0.743145i	0.669131	−	0.743145i	0.978148	+	0.207912i	$0.0666667\pi$
$158$		0			0
$159$	1.72256	+	0.181049i	1.72256	+	0.181049i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$163$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$167$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$168$		0			0
$169$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$173$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$178$		0			0
$179$	1.64728	−	0.535233i	1.64728	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$179$	1.64728	−	0.535233i	1.64728	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$180$	−0.360114	−	1.69420i	−0.360114	−	1.69420i
$181$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$181$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.704489	+	1.58231i	−0.704489	+	1.58231i
$186$		0			0
$187$		0			0
$188$			1.73205i			1.73205i
$189$		0			0
$190$		0			0
$191$	1.28716	−	1.15897i	1.28716	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$191$	1.28716	−	1.15897i	1.28716	−	1.15897i	0.978148	−	0.207912i	$-0.0666667\pi$
$192$	0.104528	−	0.994522i	0.104528	−	0.994522i
$193$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$193$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$194$		0			0
$195$		0			0
$196$	0.104528	+	0.994522i	0.104528	+	0.994522i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$	0.978148	+	0.207912i	0.978148	+	0.207912i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$211$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$212$	1.28716	+	1.15897i	1.28716	+	1.15897i
$213$	0.360114	+	1.69420i	0.360114	+	1.69420i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.95630	−	0.415823i	−1.95630	−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$223$	−1.95630	−	0.415823i	−1.95630	−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$224$		0			0
$225$	0.209057	−	1.98904i	0.209057	−	1.98904i
$226$		0			0
$227$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$227$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$228$		0			0
$229$	−1.82709	−	0.813473i	−1.82709	−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$229$	−1.82709	−	0.813473i	−1.82709	−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$		0			0
$235$	−0.927051	+	2.85317i	−0.927051	+	2.85317i
$236$	−0.704489	−	1.58231i	−0.704489	−	1.58231i
$237$		0			0
$238$		0			0
$239$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$239$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$240$	0.704489	−	1.58231i	0.704489	−	1.58231i
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$		0			0
$245$	−0.360114	+	1.69420i	−0.360114	+	1.69420i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$251$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.669131	−	0.743145i	0.669131	−	0.743145i
$257$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$257$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	1.50000	+	2.59808i	1.50000	+	2.59808i
$266$		0			0
$267$		0			0
$268$	0.669131	+	0.743145i	0.669131	+	0.743145i
$269$	−1.01807	+	1.40126i	−1.01807	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$269$	−1.01807	+	1.40126i	−1.01807	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$270$		0			0
$271$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$271$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$277$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$278$		0			0
$279$	0.309017	−	0.951057i	0.309017	−	0.951057i
$280$		0			0
$281$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$281$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$282$		0			0
$283$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$283$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$284$	−0.704489	+	1.58231i	−0.704489	+	1.58231i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$292$		0			0
$293$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$293$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$294$		0			0
$295$	−0.313585	−	2.98357i	−0.313585	−	2.98357i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	1.33826	−	1.48629i	1.33826	−	1.48629i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$310$		0			0
$311$	1.28716	+	1.15897i	1.28716	+	1.15897i	0.978148	+	0.207912i	$0.0666667\pi$
$311$	1.28716	+	1.15897i	1.28716	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$	−1.82709	+	0.813473i	−1.82709	+	0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$313$	−1.82709	+	0.813473i	−1.82709	+	0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$317$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$318$		0			0
$319$		0			0
$320$	1.50000	−	0.866025i	1.50000	−	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.913545	−	0.406737i	0.913545	−	0.406737i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$331$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$332$		0			0
$333$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$334$		0			0
$335$	0.704489	+	1.58231i	0.704489	+	1.58231i
$336$		0			0
$337$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$337$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$338$		0			0
$339$	1.72256	−	0.181049i	1.72256	−	0.181049i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$347$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$348$		0			0
$349$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$349$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	−2.00739	+	2.22943i	−2.00739	+	2.22943i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	0.809017	−	0.587785i	0.809017	−	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	0.309017	−	0.951057i	$-0.400000\pi$
$367$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	−0.978148	+	0.207912i	$0.933333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.809017	−	0.587785i	0.809017	−	0.587785i
$373$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$373$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$374$		0			0
$375$	1.50000	−	0.866025i	1.50000	−	0.866025i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$379$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.704489	−	1.58231i	0.704489	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$383$	0.704489	−	1.58231i	0.704489	−	1.58231i	0.809017	−	0.587785i	$-0.200000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$389$	1.28716	−	1.15897i	1.28716	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$389$	1.28716	−	1.15897i	1.28716	−	1.15897i	0.978148	−	0.207912i	$-0.0666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	1.95630	−	0.415823i	1.95630	−	0.415823i
$401$	1.72256	+	0.181049i	1.72256	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$401$	1.72256	+	0.181049i	1.72256	+	0.181049i	0.809017	+	0.587785i	$0.200000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.72256	−	0.181049i	1.72256	−	0.181049i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$409$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$410$		0			0
$411$	−1.64728	−	0.535233i	−1.64728	−	0.535233i
$412$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$420$		0			0
$421$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.309017	−	0.951057i	$-0.600000\pi$
$421$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.669131	+	0.743145i	$0.733333\pi$
$422$		0			0
$423$	−1.72256	−	0.181049i	−1.72256	−	0.181049i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$	0.978148	+	0.207912i	0.978148	+	0.207912i
$433$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$433$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$	−0.360114	+	1.69420i	−0.360114	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$443$	−0.360114	+	1.69420i	−0.360114	+	1.69420i	−0.669131	+	0.743145i	$0.733333\pi$
$444$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.01807	+	1.40126i	1.01807	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$449$	1.01807	+	1.40126i	1.01807	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$450$		0			0
$451$		0			0
$452$	1.50000	+	0.866025i	1.50000	+	0.866025i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$457$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$464$		0			0
$465$	1.64728	−	0.535233i	1.64728	−	0.535233i
$466$		0			0
$467$	1.01807	−	1.40126i	1.01807	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$467$	1.01807	−	1.40126i	1.01807	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.104528	+	0.994522i	0.104528	+	0.994522i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.28716	+	1.15897i	−1.28716	+	1.15897i
$478$		0			0
$479$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$479$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	1.73205i		−	1.73205i
$486$		0			0
$487$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$487$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$488$		0			0
$489$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$490$		0			0
$491$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$491$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.00000			1.00000
$497$		0			0
$498$		0			0
$499$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.669131	−	0.743145i	$-0.733333\pi$
$499$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.309017	+	0.951057i	$0.600000\pi$
$500$	1.72256	+	0.181049i	1.72256	+	0.181049i
$501$		0			0
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.00000			1.00000
$508$		0			0
$509$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$509$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.72256	+	0.181049i	−1.72256	+	0.181049i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.64728	−	0.535233i	−1.64728	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$521$	−1.64728	−	0.535233i	−1.64728	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$522$		0			0
$523$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$523$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$	1.64728	−	0.535233i	1.64728	−	0.535233i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.704489	+	1.58231i	−0.704489	+	1.58231i
$538$		0			0
$539$		0			0
$540$	1.50000	+	0.866025i	1.50000	+	0.866025i
$541$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$541$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$542$		0			0
$543$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$547$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$548$	−1.01807	−	1.40126i	−1.01807	−	1.40126i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.704489	−	1.58231i	−0.704489	−	1.58231i
$556$		0			0
$557$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$557$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$563$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$564$	−1.28716	−	1.15897i	−1.28716	−	1.15897i
$565$	2.00739	+	2.22943i	2.00739	+	2.22943i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$569$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$570$		0			0
$571$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$571$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$572$		0			0
$573$			1.73205i			1.73205i
$574$		0			0
$575$		0			0
$576$	0.669131	+	0.743145i	0.669131	+	0.743145i
$577$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$577$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.28716	+	1.15897i	−1.28716	+	1.15897i	−0.309017	+	0.951057i	$0.600000\pi$
$587$	−1.28716	+	1.15897i	−1.28716	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$588$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$598$		0			0
$599$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$599$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$600$		0			0
$601$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$601$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$602$		0			0
$603$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$607$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$613$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$617$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.669131	−	0.743145i	$-0.266667\pi$
$619$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.309017	+	0.951057i	$0.400000\pi$
$620$	1.64728	+	0.535233i	1.64728	+	0.535233i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.913545	+	0.406737i	0.913545	+	0.406737i
$626$		0			0
$627$		0			0
$628$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$629$		0			0
$630$		0			0
$631$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$631$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.72256	+	0.181049i	−1.72256	+	0.181049i
$637$		0			0
$638$		0			0
$639$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$640$		0			0
$641$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$641$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$642$		0			0
$643$	−0.209057	+	1.98904i	−0.209057	+	1.98904i	−0.104528	+	0.994522i	$0.533333\pi$
$643$	−0.209057	+	1.98904i	−0.209057	+	1.98904i	−0.104528	+	0.994522i	$0.533333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$647$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$653$	−0.360114	−	1.69420i	−0.360114	−	1.69420i	−0.669131	−	0.743145i	$-0.733333\pi$
$653$	−0.360114	−	1.69420i	−0.360114	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$661$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.61803	−	1.17557i	1.61803	−	1.17557i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$673$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$674$		0			0
$675$	1.33826	+	1.48629i	1.33826	+	1.48629i
$676$	0.809017	+	0.587785i	0.809017	+	0.587785i
$677$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$677$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$683$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	−0.927051	−	2.85317i	−0.927051	−	2.85317i
$686$		0			0
$687$	1.82709	−	0.813473i	1.82709	−	0.813473i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.104528	+	0.994522i	0.104528	+	0.994522i	0.913545	+	0.406737i	$0.133333\pi$
$691$	0.104528	+	0.994522i	0.104528	+	0.994522i	−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$706$		0			0
$707$		0			0
$708$	1.64728	+	0.535233i	1.64728	+	0.535233i
$709$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$709$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$717$		0			0
$718$		0			0
$719$	1.64728	+	0.535233i	1.64728	+	0.535233i	0.978148	−	0.207912i	$-0.0666667\pi$
$719$	1.64728	+	0.535233i	1.64728	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$720$	0.704489	+	1.58231i	0.704489	+	1.58231i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$725$		0			0
$726$		0			0
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$728$		0			0
$729$	0.309017	+	0.951057i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$733$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$734$		0			0
$735$	−1.01807	−	1.40126i	−1.01807	−	1.40126i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$739$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$740$	0.360114	−	1.69420i	0.360114	−	1.69420i
$741$		0			0
$742$		0			0
$743$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$743$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.669131	−	0.743145i	0.669131	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$751$	0.669131	−	0.743145i	0.669131	−	0.743145i	0.978148	+	0.207912i	$0.0666667\pi$
$752$	−0.360114	−	1.69420i	−0.360114	−	1.69420i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$757$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$761$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$762$		0			0
$763$		0			0
$764$	−1.01807	+	1.40126i	−1.01807	+	1.40126i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.104528	+	0.994522i	0.104528	+	0.994522i
$769$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$769$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$		0			0
$775$	1.61803	+	1.17557i	1.61803	+	1.17557i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$785$	−1.28716	+	1.15897i	−1.28716	+	1.15897i
$786$		0			0
$787$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$787$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−2.93444	−	0.623735i	−2.93444	−	0.623735i
$796$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$797$	−1.72256	−	0.181049i	−1.72256	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$797$	−1.72256	−	0.181049i	−1.72256	−	0.181049i	−0.913545	+	0.406737i	$0.866667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.00000			−1.00000
$805$		0			0
$806$		0			0
$807$	−0.360114	−	1.69420i	−0.360114	−	1.69420i
$808$		0			0
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$821$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$822$		0			0
$823$	1.82709	+	0.813473i	1.82709	+	0.813473i	0.913545	+	0.406737i	$0.133333\pi$
$823$	1.82709	+	0.813473i	1.82709	+	0.813473i	0.913545	+	0.406737i	$0.133333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$		0			0
$829$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$829$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.500000	+	0.866025i	0.500000	+	0.866025i
$838$		0			0
$839$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$839$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$840$		0			0
$841$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.01807	+	1.40126i	1.01807	+	1.40126i
$846$		0			0
$847$		0			0
$848$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−0.704489	−	1.58231i	−0.704489	−	1.58231i
$853$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$853$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$863$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.978148	−	0.207912i	0.978148	−	0.207912i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$877$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$881$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$882$		0			0
$883$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$883$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$884$		0			0
$885$	2.42705	+	1.76336i	2.42705	+	1.76336i
$886$		0			0
$887$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$887$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	2.00000			2.00000
$893$		0			0
$894$		0			0
$895$	−2.93444	+	0.623735i	−2.93444	+	0.623735i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.209057	+	1.98904i	0.209057	+	1.98904i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.28716	−	1.15897i	−1.28716	−	1.15897i
$906$		0			0
$907$	1.82709	−	0.813473i	1.82709	−	0.813473i	0.913545	−	0.406737i	$-0.133333\pi$
$907$	1.82709	−	0.813473i	1.82709	−	0.813473i	0.913545	−	0.406737i	$-0.133333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.72256	+	0.181049i	−1.72256	+	0.181049i	−0.913545	−	0.406737i	$-0.866667\pi$
$911$	−1.72256	+	0.181049i	−1.72256	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.95630	+	0.415823i	1.95630	+	0.415823i
$917$		0			0
$918$		0			0
$919$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$919$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.00000	−	1.73205i	1.00000	−	1.73205i
$926$		0			0
$927$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$928$		0			0
$929$	−0.704489	−	1.58231i	−0.704489	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$929$	−0.704489	−	1.58231i	−0.704489	−	1.58231i	0.104528	−	0.994522i	$-0.466667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1.72256	+	0.181049i	−1.72256	+	0.181049i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$937$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$938$		0			0
$939$	0.618034	−	1.90211i	0.618034	−	1.90211i
$940$	0.313585	−	2.98357i	0.313585	−	2.98357i
$941$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$941$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$942$		0			0
$943$		0			0
$944$	1.01807	+	1.40126i	1.01807	+	1.40126i
$945$		0			0
$946$		0			0
$947$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$947$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$953$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$954$		0			0
$955$	−2.42705	+	1.76336i	−2.42705	+	1.76336i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.360114	+	1.69420i	−0.360114	+	1.69420i
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$967$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$971$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$972$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$977$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$978$		0			0
$979$		0			0
$980$		−	1.73205i		−	1.73205i
$981$		0			0
$982$		0			0
$983$	−1.28716	+	1.15897i	−1.28716	+	1.15897i	−0.309017	+	0.951057i	$0.600000\pi$
$983$	−1.28716	+	1.15897i	−1.28716	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−2.00000			−2.00000			−1.00000			$\pi$
$991$	−2.00000			−2.00000			−1.00000			$\pi$
$992$		0			0
$993$	0.309017	+	0.951057i	0.309017	+	0.951057i
$994$		0			0
$995$	−1.72256	−	0.181049i	−1.72256	−	0.181049i
$996$		0			0
$997$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$997$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$998$		0			0
$999$	0.809017	−	0.587785i	0.809017	−	0.587785i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.1.r.a.245.1		8
3.2	odd	2		3267.1.v.a.2060.1		8
9.4	even	3		3267.1.v.a.3149.1		8
9.5	odd	6	inner	1089.1.r.a.608.1		8
11.2	odd	10		1089.1.i.a.848.1	yes	2
11.3	even	5	inner	1089.1.r.a.632.1		8
11.4	even	5	inner	1089.1.r.a.686.1		8
11.5	even	5	inner	1089.1.r.a.614.1		8
11.6	odd	10	inner	1089.1.r.a.614.1		8
11.7	odd	10	inner	1089.1.r.a.686.1		8
11.8	odd	10	inner	1089.1.r.a.632.1		8
11.9	even	5		1089.1.i.a.848.1	yes	2
11.10	odd	2	CM	1089.1.r.a.245.1		8
33.2	even	10		3267.1.i.a.1574.1		2
33.5	odd	10		3267.1.v.a.251.1		8
33.8	even	10		3267.1.v.a.1358.1		8
33.14	odd	10		3267.1.v.a.1358.1		8
33.17	even	10		3267.1.v.a.251.1		8
33.20	odd	10		3267.1.i.a.1574.1		2
33.26	odd	10		3267.1.v.a.1412.1		8
33.29	even	10		3267.1.v.a.1412.1		8
33.32	even	2		3267.1.v.a.2060.1		8
99.4	even	15		3267.1.v.a.2501.1		8
99.5	odd	30	inner	1089.1.r.a.977.1		8
99.13	odd	30		3267.1.i.a.2663.1		2
99.14	odd	30	inner	1089.1.r.a.995.1		8
99.31	even	15		3267.1.i.a.2663.1		2
99.32	even	6	inner	1089.1.r.a.608.1		8
99.40	odd	30		3267.1.v.a.2501.1		8
99.41	even	30	inner	1089.1.r.a.995.1		8
99.49	even	15		3267.1.v.a.1340.1		8
99.50	even	30	inner	1089.1.r.a.977.1		8
99.58	even	15		3267.1.v.a.2447.1		8
99.59	odd	30	inner	1089.1.r.a.1049.1		8
99.68	even	30		1089.1.i.a.122.1	✓	2
99.76	odd	6		3267.1.v.a.3149.1		8
99.85	odd	30		3267.1.v.a.2447.1		8
99.86	odd	30		1089.1.i.a.122.1	✓	2
99.94	odd	30		3267.1.v.a.1340.1		8
99.95	even	30	inner	1089.1.r.a.1049.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.1.i.a.122.1	✓	2	99.68	even	30
1089.1.i.a.122.1	✓	2	99.86	odd	30
1089.1.i.a.848.1	yes	2	11.2	odd	10
1089.1.i.a.848.1	yes	2	11.9	even	5
1089.1.r.a.245.1		8	1.1	even	1	trivial
1089.1.r.a.245.1		8	11.10	odd	2	CM
1089.1.r.a.608.1		8	9.5	odd	6	inner
1089.1.r.a.608.1		8	99.32	even	6	inner
1089.1.r.a.614.1		8	11.5	even	5	inner
1089.1.r.a.614.1		8	11.6	odd	10	inner
1089.1.r.a.632.1		8	11.3	even	5	inner
1089.1.r.a.632.1		8	11.8	odd	10	inner
1089.1.r.a.686.1		8	11.4	even	5	inner
1089.1.r.a.686.1		8	11.7	odd	10	inner
1089.1.r.a.977.1		8	99.5	odd	30	inner
1089.1.r.a.977.1		8	99.50	even	30	inner
1089.1.r.a.995.1		8	99.14	odd	30	inner
1089.1.r.a.995.1		8	99.41	even	30	inner
1089.1.r.a.1049.1		8	99.59	odd	30	inner
1089.1.r.a.1049.1		8	99.95	even	30	inner
3267.1.i.a.1574.1		2	33.2	even	10
3267.1.i.a.1574.1		2	33.20	odd	10
3267.1.i.a.2663.1		2	99.13	odd	30
3267.1.i.a.2663.1		2	99.31	even	15
3267.1.v.a.251.1		8	33.5	odd	10
3267.1.v.a.251.1		8	33.17	even	10
3267.1.v.a.1340.1		8	99.49	even	15
3267.1.v.a.1340.1		8	99.94	odd	30
3267.1.v.a.1358.1		8	33.8	even	10
3267.1.v.a.1358.1		8	33.14	odd	10
3267.1.v.a.1412.1		8	33.26	odd	10
3267.1.v.a.1412.1		8	33.29	even	10
3267.1.v.a.2060.1		8	3.2	odd	2
3267.1.v.a.2060.1		8	33.32	even	2
3267.1.v.a.2447.1		8	99.58	even	15
3267.1.v.a.2447.1		8	99.85	odd	30
3267.1.v.a.2501.1		8	99.4	even	15
3267.1.v.a.2501.1		8	99.40	odd	30
3267.1.v.a.3149.1		8	9.4	even	3
3267.1.v.a.3149.1		8	99.76	odd	6

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1089.1.r.a.245.1

Level	$1089$
Weight	$1$
Character	1089.245
Analytic conductor	$0.543$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{6}$
CM discriminant	-11
Inner twists	$16$