LMFDB - Embedded newform 1027.1.o.b.315.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1027,1,Mod(315,1027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1027, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("1027.315");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1027 = 13 \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1027.o (of order $6$, degree $2$, 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:	$0.512539767974$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{15} - \cdots)$

Embedding invariants

Embedding label			315.3
Root			$-0.104528 - 0.994522i$ of defining polynomial
Character	$\chi$	$=$	1027.315
Dual form			1027.1.o.b.789.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.104528 + 0.181049i) q^{2} +(0.478148 - 0.828176i) q^{4} -1.95630 q^{5} +0.408977 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(0.104528 + 0.181049i) q^{2} +(0.478148 - 0.828176i) q^{4} -1.95630 q^{5} +0.408977 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.204489 - 0.354185i) q^{10} +(-0.913545 - 1.58231i) q^{11} +(-0.809017 - 0.587785i) q^{13} +(-0.435398 - 0.754131i) q^{16} -0.209057 q^{18} +(-0.669131 + 1.15897i) q^{19} +(-0.935398 + 1.62016i) q^{20} +(0.190983 - 0.330792i) q^{22} +(-0.669131 - 1.15897i) q^{23} +2.82709 q^{25} +(0.0218524 - 0.207912i) q^{26} -0.209057 q^{31} +(0.295511 - 0.511841i) q^{32} +(0.478148 + 0.828176i) q^{36} -0.279773 q^{38} -0.800080 q^{40} -1.74724 q^{44} +(0.978148 - 1.69420i) q^{45} +(0.139886 - 0.242290i) q^{46} +(-0.500000 - 0.866025i) q^{49} +(0.295511 + 0.511841i) q^{50} +(-0.873619 + 0.388960i) q^{52} +(1.78716 + 3.09546i) q^{55} +(-0.0218524 - 0.0378495i) q^{62} -0.747238 q^{64} +(1.58268 + 1.14988i) q^{65} +(-0.309017 - 0.535233i) q^{67} +(-0.204489 + 0.354185i) q^{72} +1.33826 q^{73} +(0.639886 + 1.10832i) q^{76} +1.00000 q^{79} +(0.851767 + 1.47530i) q^{80} +(-0.500000 - 0.866025i) q^{81} -1.00000 q^{83} +(-0.373619 - 0.647127i) q^{88} +(0.809017 + 1.40126i) q^{89} +0.408977 q^{90} -1.27977 q^{92} +(1.30902 - 2.26728i) q^{95} +(0.978148 - 1.69420i) q^{97} +(0.104528 - 0.181049i) q^{98} +1.82709 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{8} - 4 q^{9}+O(q^{10}) $ 8 * q - q^2 - 5 * q^4 + 2 * q^5 - 2 * q^8 - 4 * q^9	$ 8 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{8} - 4 q^{9} + q^{10} - q^{11} - 2 q^{13} - 6 q^{16} + 2 q^{18} - q^{19} - 10 q^{20} + 6 q^{22} - q^{23} + 10 q^{25} + 9 q^{26} + 2 q^{31} + 5 q^{32} - 5 q^{36} - 2 q^{38} - 8 q^{40} - q^{45} + q^{46} - 4 q^{49} + 5 q^{50} + q^{55} - 9 q^{62} + 8 q^{64} + 2 q^{65} + 2 q^{67} + q^{72} + 2 q^{73} + 5 q^{76} + 8 q^{79} - 9 q^{80} - 4 q^{81} - 8 q^{83} + 4 q^{88} + 2 q^{89} - 2 q^{90} - 10 q^{92} + 6 q^{95} - q^{97} - q^{98} + 2 q^{99}+O(q^{100}) $ 8 * q - q^2 - 5 * q^4 + 2 * q^5 - 2 * q^8 - 4 * q^9 + q^10 - q^11 - 2 * q^13 - 6 * q^16 + 2 * q^18 - q^19 - 10 * q^20 + 6 * q^22 - q^23 + 10 * q^25 + 9 * q^26 + 2 * q^31 + 5 * q^32 - 5 * q^36 - 2 * q^38 - 8 * q^40 - q^45 + q^46 - 4 * q^49 + 5 * q^50 + q^55 - 9 * q^62 + 8 * q^64 + 2 * q^65 + 2 * q^67 + q^72 + 2 * q^73 + 5 * q^76 + 8 * q^79 - 9 * q^80 - 4 * q^81 - 8 * q^83 + 4 * q^88 + 2 * q^89 - 2 * q^90 - 10 * q^92 + 6 * q^95 - q^97 - q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1027\mathbb{Z}\right)^\times$.

$n$	$80$	$872$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$2$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$3$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$	0.478148	−	0.828176i	0.478148	−	0.828176i
$5$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$5$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	0.408977			0.408977
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$11$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$11$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$12$		0			0
$13$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$13$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$14$		0			0
$15$		0			0
$16$	−0.435398	−	0.754131i	−0.435398	−	0.754131i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−0.209057			−0.209057
$19$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$19$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$20$	−0.935398	+	1.62016i	−0.935398	+	1.62016i
$21$		0			0
$22$	0.190983	−	0.330792i	0.190983	−	0.330792i
$23$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$23$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$24$		0			0
$25$	2.82709			2.82709
$26$	0.0218524	−	0.207912i	0.0218524	−	0.207912i
$27$		0			0
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$31$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$32$	0.295511	−	0.511841i	0.295511	−	0.511841i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.478148	+	0.828176i	0.478148	+	0.828176i
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	−0.279773			−0.279773
$39$		0			0
$40$	−0.800080			−0.800080
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−1.74724			−1.74724
$45$	0.978148	−	1.69420i	0.978148	−	1.69420i
$46$	0.139886	−	0.242290i	0.139886	−	0.242290i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	0.295511	+	0.511841i	0.295511	+	0.511841i
$51$		0			0
$52$	−0.873619	+	0.388960i	−0.873619	+	0.388960i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	1.78716	+	3.09546i	1.78716	+	3.09546i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$	−0.0218524	−	0.0378495i	−0.0218524	−	0.0378495i
$63$		0			0
$64$	−0.747238			−0.747238
$65$	1.58268	+	1.14988i	1.58268	+	1.14988i
$66$		0			0
$67$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$67$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$73$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$73$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$74$		0			0
$75$		0			0
$76$	0.639886	+	1.10832i	0.639886	+	1.10832i
$77$		0			0
$78$		0			0
$79$	1.00000			1.00000
$79$	1.00000			1.00000
$80$	0.851767	+	1.47530i	0.851767	+	1.47530i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.373619	−	0.647127i	−0.373619	−	0.647127i
$89$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$89$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$90$	0.408977			0.408977
$91$		0			0
$92$	−1.27977			−1.27977
$93$		0			0
$94$		0			0
$95$	1.30902	−	2.26728i	1.30902	−	2.26728i
$96$		0			0
$97$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$97$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$98$	0.104528	−	0.181049i	0.104528	−	0.181049i
$99$	1.82709			1.82709
$100$	1.35177	−	2.34133i	1.35177	−	2.34133i
$101$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$101$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−0.330869	−	0.240391i	−0.330869	−	0.240391i
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−0.373619	+	0.647127i	−0.373619	+	0.647127i
$111$		0			0
$112$		0			0
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$	1.30902	+	2.26728i	1.30902	+	2.26728i
$116$		0			0
$117$	0.913545	−	0.406737i	0.913545	−	0.406737i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.16913	+	2.02499i	−1.16913	+	2.02499i
$122$		0			0
$123$		0			0
$124$	−0.0999601	+	0.173136i	−0.0999601	+	0.173136i
$125$	−3.57433			−3.57433
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$	−0.373619	−	0.647127i	−0.373619	−	0.647127i
$129$		0			0
$130$	−0.0427497	+	0.406737i	−0.0427497	+	0.406737i
$131$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$131$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$132$		0			0
$133$		0			0
$134$	0.0646021	−	0.111894i	0.0646021	−	0.111894i
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.190983	+	1.81708i	−0.190983	+	1.81708i
$144$	0.870796			0.870796
$145$		0			0
$146$	0.139886	+	0.242290i	0.139886	+	0.242290i
$147$		0			0
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$151$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$152$	−0.273659	+	0.473991i	−0.273659	+	0.473991i
$153$		0			0
$154$		0			0
$155$	0.408977			0.408977
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	0.104528	+	0.181049i	0.104528	+	0.181049i
$159$		0			0
$160$	−0.578108	+	1.00131i	−0.578108	+	1.00131i
$161$		0			0
$162$	0.104528	−	0.181049i	0.104528	−	0.181049i
$163$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$163$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$164$		0			0
$165$		0			0
$166$	−0.104528	−	0.181049i	−0.104528	−	0.181049i
$167$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$167$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$168$		0			0
$169$	0.309017	+	0.951057i	0.309017	+	0.951057i
$170$		0			0
$171$	−0.669131	−	1.15897i	−0.669131	−	1.15897i
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	−0.795511	+	1.37787i	−0.795511	+	1.37787i
$177$		0			0
$178$	−0.169131	+	0.292943i	−0.169131	+	0.292943i
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$180$	−0.935398	−	1.62016i	−0.935398	−	1.62016i
$181$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$181$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$		0			0
$184$	−0.273659	−	0.473991i	−0.273659	−	0.473991i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.547318			0.547318
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	0.408977			0.408977
$195$		0			0
$196$	−0.956295			−0.956295
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$	0.190983	+	0.330792i	0.190983	+	0.330792i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	1.15622			1.15622
$201$		0			0
$202$	−0.0218524	+	0.0378495i	−0.0218524	+	0.0378495i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.33826			1.33826
$208$	−0.0910229	+	0.866025i	−0.0910229	+	0.866025i
$209$	2.44512			2.44512
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	3.41811			3.41811
$221$		0			0
$222$		0			0
$223$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$224$		0			0
$225$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	−0.273659	+	0.473991i	−0.273659	+	0.473991i
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	0.169131	+	0.122881i	0.169131	+	0.122881i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$239$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$240$		0			0
$241$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$241$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$242$	−0.488830			−0.488830
$243$		0			0
$244$		0			0
$245$	0.978148	+	1.69420i	0.978148	+	1.69420i
$246$		0			0
$247$	1.22256	−	0.544320i	1.22256	−	0.544320i
$248$	−0.0854995			−0.0854995
$249$		0			0
$250$	−0.373619	−	0.647127i	−0.373619	−	0.647127i
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$254$		0			0
$255$		0			0
$256$	−0.295511	+	0.511841i	−0.295511	+	0.511841i
$257$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$257$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$258$		0			0
$259$		0			0
$260$	1.70906	−	0.760921i	1.70906	−	0.760921i
$261$		0			0
$262$	0.139886	+	0.242290i	0.139886	+	0.242290i
$263$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$263$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.591023			−0.591023
$269$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$269$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.58268	−	4.47333i	−2.58268	−	4.47333i
$276$		0			0
$277$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$277$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$278$		0			0
$279$	0.104528	−	0.181049i	0.104528	−	0.181049i
$280$		0			0
$281$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$281$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$283$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$284$		0			0
$285$		0			0
$286$	−0.348943	+	0.155360i	−0.348943	+	0.155360i
$287$		0			0
$288$	0.295511	+	0.511841i	0.295511	+	0.511841i
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	0.639886	−	1.10832i	0.639886	−	1.10832i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.139886	+	1.33093i	−0.139886	+	1.33093i
$300$		0			0
$301$		0			0
$302$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$303$		0			0
$304$	1.16535			1.16535
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	0.0427497	+	0.0740447i	0.0427497	+	0.0740447i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$313$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$314$		0			0
$315$		0			0
$316$	0.478148	−	0.828176i	0.478148	−	0.828176i
$317$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	1.46182			1.46182
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.956295			−0.956295
$325$	−2.28716	−	1.66172i	−2.28716	−	1.66172i
$326$	0.338261			0.338261
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−0.478148	+	0.828176i	−0.478148	+	0.828176i
$333$		0			0
$334$	0.190983	−	0.330792i	0.190983	−	0.330792i
$335$	0.604528	+	1.04707i	0.604528	+	1.04707i
$336$		0			0
$337$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$337$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$338$	−0.139886	+	0.155360i	−0.139886	+	0.155360i
$339$		0			0
$340$		0			0
$341$	0.190983	+	0.330792i	0.190983	+	0.330792i
$342$	0.139886	−	0.242290i	0.139886	−	0.242290i
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$347$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	−1.07985			−1.07985
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$	1.54732			1.54732
$357$		0			0
$358$	−0.104528	+	0.181049i	−0.104528	+	0.181049i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	0.400040	−	0.692889i	0.400040	−	0.692889i
$361$	−0.395472	−	0.684977i	−0.395472	−	0.684977i
$362$	−0.169131	−	0.292943i	−0.169131	−	0.292943i
$363$		0			0
$364$		0			0
$365$	−2.61803			−2.61803
$366$		0			0
$367$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$367$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$368$	−0.582676	+	1.00922i	−0.582676	+	1.00922i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$379$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$380$	−1.25181	−	2.16819i	−1.25181	−	2.16819i
$381$		0			0
$382$		0			0
$383$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$383$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.935398	−	1.62016i	−0.935398	−	1.62016i
$389$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$389$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$390$		0			0
$391$		0			0
$392$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$393$		0			0
$394$		0			0
$395$	−1.95630			−1.95630
$396$	0.873619	−	1.51315i	0.873619	−	1.51315i
$397$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$398$		0			0
$399$		0			0
$400$	−1.23091	−	2.13200i	−1.23091	−	2.13200i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	0.169131	+	0.122881i	0.169131	+	0.122881i
$404$	0.199920			0.199920
$405$	0.978148	+	1.69420i	0.978148	+	1.69420i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.139886	+	0.242290i	0.139886	+	0.242290i
$415$	1.95630			1.95630
$416$	−0.539926	+	0.240391i	−0.539926	+	0.240391i
$417$		0			0
$418$	0.255585	+	0.442686i	0.255585	+	0.442686i
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$421$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$431$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$432$		0			0
$433$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$433$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.79094			1.79094
$438$		0			0
$439$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$439$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$440$	0.730909	+	1.26597i	0.730909	+	1.26597i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$446$	0.209057	−	0.362097i	0.209057	−	0.362097i
$447$		0			0
$448$		0			0
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$	−0.591023			−0.591023
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$457$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$458$		0			0
$459$		0			0
$460$	2.50361			2.50361
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$467$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$468$	0.0999601	−	0.951057i	0.0999601	−	0.951057i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.89169	+	3.27651i	−1.89169	+	3.27651i
$476$		0			0
$477$		0			0
$478$	0.139886	+	0.242290i	0.139886	+	0.242290i
$479$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$479$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	−0.381966			−0.381966
$483$		0			0
$484$	1.11803	+	1.93649i	1.11803	+	1.93649i
$485$	−1.91355	+	3.31436i	−1.91355	+	3.31436i
$486$		0			0
$487$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$487$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$488$		0			0
$489$		0			0
$490$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$	0.226341	+	0.164446i	0.226341	+	0.164446i
$495$	−3.57433			−3.57433
$496$	0.0910229	+	0.157656i	0.0910229	+	0.157656i
$497$		0			0
$498$		0			0
$499$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$499$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$500$	−1.70906	+	2.96017i	−1.70906	+	2.96017i
$501$		0			0
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$506$	−0.511170			−0.511170
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−0.870796			−0.870796
$513$		0			0
$514$	−0.169131	+	0.292943i	−0.169131	+	0.292943i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.647278	+	0.470275i	0.647278	+	0.470275i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$523$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$524$	0.639886	−	1.10832i	0.639886	−	1.10832i
$525$		0			0
$526$	0.190983	−	0.330792i	0.190983	−	0.330792i
$527$		0			0
$528$		0			0
$529$	−0.395472	+	0.684977i	−0.395472	+	0.684977i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.126381	−	0.218898i	−0.126381	−	0.218898i
$537$		0			0
$538$	0.408977			0.408977
$539$	−0.913545	+	1.58231i	−0.913545	+	1.58231i
$540$		0			0
$541$	2.00000			2.00000			1.00000			$0$
$541$	2.00000			2.00000			1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$547$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$548$		0			0
$549$		0			0
$550$	0.539926	−	0.935180i	0.539926	−	0.935180i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.129204			−0.129204
$555$		0			0
$556$		0			0
$557$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$557$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$558$	0.0437048			0.0437048
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.0646021	+	0.111894i	0.0646021	+	0.111894i
$563$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$563$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$564$		0			0
$565$		0			0
$566$	0.190983	−	0.330792i	0.190983	−	0.330792i
$567$		0			0
$568$		0			0
$569$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$569$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$570$		0			0
$571$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$571$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$572$	1.41355	+	1.02700i	1.41355	+	1.02700i
$573$		0			0
$574$		0			0
$575$	−1.89169	−	3.27651i	−1.89169	−	3.27651i
$576$	0.373619	−	0.647127i	0.373619	−	0.647127i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.104528	−	0.181049i	0.104528	−	0.181049i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.547318			0.547318
$585$	−1.78716	+	0.795697i	−1.78716	+	0.795697i
$586$		0			0
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	0.139886	−	0.242290i	0.139886	−	0.242290i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$593$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−0.255585	+	0.113794i	−0.255585	+	0.113794i
$599$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$599$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	0.618034			0.618034
$604$	−0.935398	+	1.62016i	−0.935398	+	1.62016i
$605$	2.28716	−	3.96149i	2.28716	−	3.96149i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	0.395472	+	0.684977i	0.395472	+	0.684977i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$617$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$	0.195551	−	0.338705i	0.195551	−	0.338705i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.16535			4.16535
$626$	−0.169131	−	0.292943i	−0.169131	−	0.292943i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$	0.408977			0.408977
$633$		0			0
$634$	−0.104528	−	0.181049i	−0.104528	−	0.181049i
$635$		0			0
$636$		0			0
$637$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$638$		0			0
$639$		0			0
$640$	0.730909	+	1.26597i	0.730909	+	1.26597i
$641$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$641$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$642$		0			0
$643$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$643$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$649$		0			0
$650$	0.0617787	−	0.587785i	0.0617787	−	0.587785i
$651$		0			0
$652$	−0.773659	−	1.34002i	−0.773659	−	1.34002i
$653$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$653$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$	−2.61803			−2.61803
$656$		0			0
$657$	−0.669131	+	1.15897i	−0.669131	+	1.15897i
$658$		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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$	−0.408977			−0.408977
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.74724			−1.74724
$669$		0			0
$670$	−0.126381	+	0.218898i	−0.126381	+	0.218898i
$671$		0			0
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$675$		0			0
$676$	0.935398	+	0.198825i	0.935398	+	0.198825i
$677$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$677$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−0.0399263	+	0.0691544i	−0.0399263	+	0.0691544i
$683$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$683$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$684$	−1.27977			−1.27977
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$		0			0
$694$	−0.129204			−0.129204
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	0.682636	+	1.18236i	0.682636	+	1.18236i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$712$	0.330869	+	0.573083i	0.330869	+	0.573083i
$713$	0.139886	+	0.242290i	0.139886	+	0.242290i
$714$		0			0
$715$	0.373619	−	3.55475i	0.373619	−	3.55475i
$716$	0.956295			0.956295
$717$		0			0
$718$		0			0
$719$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$719$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$720$	−1.70353			−1.70353
$721$		0			0
$722$	0.0826761	−	0.143199i	0.0826761	−	0.143199i
$723$		0			0
$724$	−0.773659	+	1.34002i	−0.773659	+	1.34002i
$725$		0			0
$726$		0			0
$727$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$	−0.273659	−	0.473991i	−0.273659	−	0.473991i
$731$		0			0
$732$		0			0
$733$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$733$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$734$	−0.0218524	+	0.0378495i	−0.0218524	+	0.0378495i
$735$		0			0
$736$	−0.790943			−0.790943
$737$	−0.564602	+	0.977920i	−0.564602	+	0.977920i
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$743$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.500000	−	0.866025i	0.500000	−	0.866025i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$751$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.82709			3.82709
$756$		0			0
$757$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$757$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$	0.535358	−	0.927267i	0.535358	−	0.927267i
$761$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$761$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.338261			0.338261
$767$		0			0
$768$		0			0
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$773$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$774$		0			0
$775$	−0.591023			−0.591023
$776$	0.400040	−	0.692889i	0.400040	−	0.692889i
$777$		0			0
$778$	−0.0218524	−	0.0378495i	−0.0218524	−	0.0378495i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.435398	+	0.754131i	−0.435398	+	0.754131i
$785$		0			0
$786$		0			0
$787$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$787$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$788$		0			0
$789$		0			0
$790$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$791$		0			0
$792$	0.747238			0.747238
$793$		0			0
$794$	0.209057			0.209057
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	0.835438	−	1.44702i	0.835438	−	1.44702i
$801$	−1.61803			−1.61803
$802$		0			0
$803$	−1.22256	−	2.11754i	−1.22256	−	2.11754i
$804$		0			0
$805$		0			0
$806$	−0.00456840	+	0.0434654i	−0.00456840	+	0.0434654i
$807$		0			0
$808$	0.0427497	+	0.0740447i	0.0427497	+	0.0740447i
$809$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$809$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$810$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$811$	2.00000			2.00000			1.00000			$0$
$811$	2.00000			2.00000			1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$821$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	0.639886	−	1.10832i	0.639886	−	1.10832i
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$	0.204489	+	0.354185i	0.204489	+	0.354185i
$831$		0			0
$832$	0.604528	+	0.439216i	0.604528	+	0.439216i
$833$		0			0
$834$		0			0
$835$	1.78716	+	3.09546i	1.78716	+	3.09546i
$836$	1.16913	−	2.02499i	1.16913	−	2.02499i
$837$		0			0
$838$		0			0
$839$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$839$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$840$		0			0
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$	0.0646021	+	0.111894i	0.0646021	+	0.111894i
$843$		0			0
$844$		0			0
$845$	−0.604528	−	1.86055i	−0.604528	−	1.86055i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$	1.30902	+	2.26728i	1.30902	+	2.26728i
$856$		0			0
$857$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$857$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	0.0646021	−	0.111894i	0.0646021	−	0.111894i
$863$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$	−0.129204			−0.129204
$867$		0			0
$868$		0			0
$869$	−0.913545	−	1.58231i	−0.913545	−	1.58231i
$870$		0			0
$871$	−0.0646021	+	0.614648i	−0.0646021	+	0.614648i
$872$		0			0
$873$	0.978148	+	1.69420i	0.978148	+	1.69420i
$874$	0.187205	+	0.324248i	0.187205	+	0.324248i
$875$		0			0
$876$		0			0
$877$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$877$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$878$	0.0646021	−	0.111894i	0.0646021	−	0.111894i
$879$		0			0
$880$	1.55626	−	2.69551i	1.55626	−	2.69551i
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$	0.104528	+	0.181049i	0.104528	+	0.181049i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$887$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$888$		0			0
$889$		0			0
$890$	0.330869	−	0.573083i	0.330869	−	0.573083i
$891$	−0.913545	+	1.58231i	−0.913545	+	1.58231i
$892$	−1.91259			−1.91259
$893$		0			0
$894$		0			0
$895$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.35177	+	2.34133i	1.35177	+	2.34133i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.16535			3.16535
$906$		0			0
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$	−0.209057			−0.209057
$910$		0			0
$911$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$911$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$912$		0			0
$913$	0.913545	+	1.58231i	0.913545	+	1.58231i
$914$	−0.169131	+	0.292943i	−0.169131	+	0.292943i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$919$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$920$	0.535358	+	0.927267i	0.535358	+	0.927267i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	1.33826			1.33826
$932$		0			0
$933$		0			0
$934$	−0.0218524	−	0.0378495i	−0.0218524	−	0.0378495i
$935$		0			0
$936$	0.373619	−	0.166346i	0.373619	−	0.166346i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$941$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−1.08268	−	0.786610i	−1.08268	−	0.786610i
$950$	−0.790943			−0.790943
$951$		0			0
$952$		0			0
$953$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$953$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$954$		0			0
$955$		0			0
$956$	0.639886	−	1.10832i	0.639886	−	1.10832i
$957$		0			0
$958$	−0.104528	+	0.181049i	−0.104528	+	0.181049i
$959$		0			0
$960$		0			0
$961$	−0.956295			−0.956295
$962$		0			0
$963$		0			0
$964$	0.873619	+	1.51315i	0.873619	+	1.51315i
$965$		0			0
$966$		0			0
$967$	2.00000			2.00000			1.00000			$0$
$967$	2.00000			2.00000			1.00000			$0$
$968$	−0.478148	+	0.828176i	−0.478148	+	0.828176i
$969$		0			0
$970$	−0.800080			−0.800080
$971$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$971$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$972$		0			0
$973$		0			0
$974$	0.338261			0.338261
$975$		0			0
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	1.47815	−	2.56023i	1.47815	−	2.56023i
$980$	1.87080			1.87080
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.133773	−	1.27276i	0.133773	−	1.27276i
$989$		0			0
$990$	−0.373619	−	0.647127i	−0.373619	−	0.647127i
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$	−0.0617787	+	0.107004i	−0.0617787	+	0.107004i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$997$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$998$	0.190983	+	0.330792i	0.190983	+	0.330792i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1027.1.o.b.315.3	✓	8
13.9	even	3	inner	1027.1.o.b.789.3	yes	8
79.78	odd	2	CM	1027.1.o.b.315.3	✓	8
1027.789	odd	6	inner	1027.1.o.b.789.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1027.1.o.b.315.3	✓	8	1.1	even	1	trivial
1027.1.o.b.315.3	✓	8	79.78	odd	2	CM
1027.1.o.b.789.3	yes	8	13.9	even	3	inner
1027.1.o.b.789.3	yes	8	1027.789	odd	6	inner

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:	\( N \)	\(=\)	\( 1027 = 13 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1027.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.181049i) q^{2} +(0.478148 - 0.828176i) q^{4} -1.95630 q^{5} +0.408977 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.104528 + 0.181049i) q^{2} +(0.478148 - 0.828176i) q^{4} -1.95630 q^{5} +0.408977 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.204489 - 0.354185i) q^{10} +(-0.913545 - 1.58231i) q^{11} +(-0.809017 - 0.587785i) q^{13} +(-0.435398 - 0.754131i) q^{16} -0.209057 q^{18} +(-0.669131 + 1.15897i) q^{19} +(-0.935398 + 1.62016i) q^{20} +(0.190983 - 0.330792i) q^{22} +(-0.669131 - 1.15897i) q^{23} +2.82709 q^{25} +(0.0218524 - 0.207912i) q^{26} -0.209057 q^{31} +(0.295511 - 0.511841i) q^{32} +(0.478148 + 0.828176i) q^{36} -0.279773 q^{38} -0.800080 q^{40} -1.74724 q^{44} +(0.978148 - 1.69420i) q^{45} +(0.139886 - 0.242290i) q^{46} +(-0.500000 - 0.866025i) q^{49} +(0.295511 + 0.511841i) q^{50} +(-0.873619 + 0.388960i) q^{52} +(1.78716 + 3.09546i) q^{55} +(-0.0218524 - 0.0378495i) q^{62} -0.747238 q^{64} +(1.58268 + 1.14988i) q^{65} +(-0.309017 - 0.535233i) q^{67} +(-0.204489 + 0.354185i) q^{72} +1.33826 q^{73} +(0.639886 + 1.10832i) q^{76} +1.00000 q^{79} +(0.851767 + 1.47530i) q^{80} +(-0.500000 - 0.866025i) q^{81} -1.00000 q^{83} +(-0.373619 - 0.647127i) q^{88} +(0.809017 + 1.40126i) q^{89} +0.408977 q^{90} -1.27977 q^{92} +(1.30902 - 2.26728i) q^{95} +(0.978148 - 1.69420i) q^{97} +(0.104528 - 0.181049i) q^{98} +1.82709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{8} - 4 q^{9}+O(q^{10}) \) 8 * q - q^2 - 5 * q^4 + 2 * q^5 - 2 * q^8 - 4 * q^9	\( 8 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{8} - 4 q^{9} + q^{10} - q^{11} - 2 q^{13} - 6 q^{16} + 2 q^{18} - q^{19} - 10 q^{20} + 6 q^{22} - q^{23} + 10 q^{25} + 9 q^{26} + 2 q^{31} + 5 q^{32} - 5 q^{36} - 2 q^{38} - 8 q^{40} - q^{45} + q^{46} - 4 q^{49} + 5 q^{50} + q^{55} - 9 q^{62} + 8 q^{64} + 2 q^{65} + 2 q^{67} + q^{72} + 2 q^{73} + 5 q^{76} + 8 q^{79} - 9 q^{80} - 4 q^{81} - 8 q^{83} + 4 q^{88} + 2 q^{89} - 2 q^{90} - 10 q^{92} + 6 q^{95} - q^{97} - q^{98} + 2 q^{99}+O(q^{100}) \) 8 * q - q^2 - 5 * q^4 + 2 * q^5 - 2 * q^8 - 4 * q^9 + q^10 - q^11 - 2 * q^13 - 6 * q^16 + 2 * q^18 - q^19 - 10 * q^20 + 6 * q^22 - q^23 + 10 * q^25 + 9 * q^26 + 2 * q^31 + 5 * q^32 - 5 * q^36 - 2 * q^38 - 8 * q^40 - q^45 + q^46 - 4 * q^49 + 5 * q^50 + q^55 - 9 * q^62 + 8 * q^64 + 2 * q^65 + 2 * q^67 + q^72 + 2 * q^73 + 5 * q^76 + 8 * q^79 - 9 * q^80 - 4 * q^81 - 8 * q^83 + 4 * q^88 + 2 * q^89 - 2 * q^90 - 10 * q^92 + 6 * q^95 - q^97 - q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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