LMFDB - Embedded newform 961.1.h.a.414.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [961,1,Mod(115,961)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(961, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([17]))

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

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

chi := DirichletCharacter("961.115");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	961.h (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:	$0.479601477140$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.31.1
Artin image:	$S_3\times C_{15}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{45} - \cdots)$

Embedding invariants

Embedding label			414.1
Root			$0.913545 + 0.406737i$ of defining polynomial
Character	$\chi$	$=$	961.414
Dual form			961.1.h.a.513.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.309017 - 0.951057i) q^{2} +(0.500000 + 0.866025i) q^{5} +(0.104528 + 0.994522i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(-0.104528 + 0.994522i) q^{9} +(0.669131 - 0.743145i) q^{10} +(0.913545 - 0.406737i) q^{14} +(-0.309017 + 0.951057i) q^{16} +(0.978148 - 0.207912i) q^{18} +(0.978148 + 0.207912i) q^{19} +(-0.809017 + 0.587785i) q^{35} +(-0.104528 - 0.994522i) q^{38} +(0.104528 - 0.994522i) q^{40} +(-0.669131 + 0.743145i) q^{41} +(-0.913545 + 0.406737i) q^{45} +(0.618034 - 1.90211i) q^{47} +(0.500000 - 0.866025i) q^{56} +(-0.669131 - 0.743145i) q^{59} -1.00000 q^{63} +(0.309017 + 0.951057i) q^{64} +(-1.00000 - 1.73205i) q^{67} +(0.809017 + 0.587785i) q^{70} +(0.104528 - 0.994522i) q^{71} +(0.669131 - 0.743145i) q^{72} +(-0.978148 + 0.207912i) q^{80} +(-0.978148 - 0.207912i) q^{81} +(0.913545 + 0.406737i) q^{82} +(0.669131 + 0.743145i) q^{90} -2.00000 q^{94} +(0.309017 + 0.951057i) q^{95} +(0.809017 - 0.587785i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 4 q^{5} - q^{7} - 2 q^{8} + q^{9} + q^{10} + q^{14} + 2 q^{16} - q^{18} - q^{19} - 2 q^{35} + q^{38} - q^{40} - q^{41} - q^{45} - 4 q^{47} + 4 q^{56} - q^{59} - 8 q^{63} - 2 q^{64}+ \cdots + 2 q^{97}+O(q^{100}) $ 8 * q + 2 * q^2 + 4 * q^5 - q^7 - 2 * q^8 + q^9 + q^10 + q^14 + 2 * q^16 - q^18 - q^19 - 2 * q^35 + q^38 - q^40 - q^41 - q^45 - 4 * q^47 + 4 * q^56 - q^59 - 8 * q^63 - 2 * q^64 - 8 * q^67 + 2 * q^70 - q^71 + q^72 + q^80 + q^81 + q^82 + q^90 - 16 * q^94 - 2 * q^95 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/961\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$e\left(\frac{23}{30}\right)$

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 $

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.1.h.a.414.1		8
31.2	even	5	inner	961.1.h.a.726.1		8
31.3	odd	30		961.1.e.a.440.1		2
31.4	even	5	inner	961.1.h.a.145.1		8
31.5	even	3		961.1.f.a.430.1		4
31.6	odd	6	inner	961.1.h.a.117.1		8
31.7	even	15	inner	961.1.h.a.229.1		8
31.8	even	5	inner	961.1.h.a.115.1		8
31.9	even	15		961.1.f.a.333.1		4
31.10	even	15		961.1.f.a.573.1		4
31.11	odd	30		961.1.f.a.587.1		4
31.12	odd	30	inner	961.1.h.a.623.1		8
31.13	odd	30		31.1.b.a.30.1	✓	1
31.14	even	15	inner	961.1.h.a.513.1		8
31.15	odd	10		961.1.e.a.522.1		2
31.16	even	5		961.1.e.a.522.1		2
31.17	odd	30	inner	961.1.h.a.513.1		8
31.18	even	15		31.1.b.a.30.1	✓	1
31.19	even	15	inner	961.1.h.a.623.1		8
31.20	even	15		961.1.f.a.587.1		4
31.21	odd	30		961.1.f.a.573.1		4
31.22	odd	30		961.1.f.a.333.1		4
31.23	odd	10	inner	961.1.h.a.115.1		8
31.24	odd	30	inner	961.1.h.a.229.1		8
31.25	even	3	inner	961.1.h.a.117.1		8
31.26	odd	6		961.1.f.a.430.1		4
31.27	odd	10	inner	961.1.h.a.145.1		8
31.28	even	15		961.1.e.a.440.1		2
31.29	odd	10	inner	961.1.h.a.726.1		8
31.30	odd	2	CM	961.1.h.a.414.1		8
93.44	even	30		279.1.d.b.154.1		1
93.80	odd	30		279.1.d.b.154.1		1
124.75	even	30		496.1.e.a.433.1		1
124.111	odd	30		496.1.e.a.433.1		1
155.13	even	60		775.1.c.a.774.2		2
155.18	odd	60		775.1.c.a.774.2		2
155.44	odd	30		775.1.d.b.526.1		1
155.49	even	30		775.1.d.b.526.1		1
155.137	even	60		775.1.c.a.774.1		2
155.142	odd	60		775.1.c.a.774.1		2
217.13	even	30		1519.1.c.a.1177.1		1
217.18	even	15		1519.1.n.b.30.1		2
217.44	odd	30		1519.1.n.b.557.1		2
217.75	even	30		1519.1.n.a.557.1		2
217.80	odd	30		1519.1.n.a.30.1		2
217.111	odd	30		1519.1.c.a.1177.1		1
217.137	odd	30		1519.1.n.b.30.1		2
217.142	even	15		1519.1.n.b.557.1		2
217.173	odd	30		1519.1.n.a.557.1		2
217.199	even	30		1519.1.n.a.30.1		2
248.13	odd	30		1984.1.e.a.1921.1		1
248.75	even	30		1984.1.e.b.1921.1		1
248.173	even	30		1984.1.e.a.1921.1		1
248.235	odd	30		1984.1.e.b.1921.1		1
279.13	odd	30		2511.1.m.e.1270.1		2
279.49	even	15		2511.1.m.e.1270.1		2
279.106	odd	30		2511.1.m.e.433.1		2
279.137	even	30		2511.1.m.a.433.1		2
279.142	even	15		2511.1.m.e.433.1		2
279.173	odd	30		2511.1.m.a.433.1		2
279.230	even	30		2511.1.m.a.1270.1		2
279.266	odd	30		2511.1.m.a.1270.1		2
341.13	even	30		3751.1.t.a.2138.1		4
341.18	odd	30		3751.1.t.a.3657.1		4
341.49	even	15		3751.1.t.c.2665.1		4
341.75	odd	30		3751.1.t.c.2138.1		4
341.80	even	15		3751.1.t.c.2913.1		4
341.106	even	30		3751.1.t.a.3657.1		4
341.137	odd	30		3751.1.t.c.2665.1		4
341.142	odd	30		3751.1.d.b.1332.1		1
341.168	odd	30		3751.1.t.c.2913.1		4
341.173	odd	30		3751.1.t.a.2913.1		4
341.204	odd	30		3751.1.t.a.2665.1		4
341.230	even	30		3751.1.d.b.1332.1		1
341.235	even	15		3751.1.t.c.3657.1		4
341.261	even	30		3751.1.t.a.2913.1		4
341.266	odd	30		3751.1.t.a.2138.1		4
341.292	even	30		3751.1.t.a.2665.1		4
341.323	odd	30		3751.1.t.c.3657.1		4
341.328	even	15		3751.1.t.c.2138.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.1.b.a.30.1	✓	1	31.13	odd	30
31.1.b.a.30.1	✓	1	31.18	even	15
279.1.d.b.154.1		1	93.44	even	30
279.1.d.b.154.1		1	93.80	odd	30
496.1.e.a.433.1		1	124.75	even	30
496.1.e.a.433.1		1	124.111	odd	30
775.1.c.a.774.1		2	155.137	even	60
775.1.c.a.774.1		2	155.142	odd	60
775.1.c.a.774.2		2	155.13	even	60
775.1.c.a.774.2		2	155.18	odd	60
775.1.d.b.526.1		1	155.44	odd	30
775.1.d.b.526.1		1	155.49	even	30
961.1.e.a.440.1		2	31.3	odd	30
961.1.e.a.440.1		2	31.28	even	15
961.1.e.a.522.1		2	31.15	odd	10
961.1.e.a.522.1		2	31.16	even	5
961.1.f.a.333.1		4	31.9	even	15
961.1.f.a.333.1		4	31.22	odd	30
961.1.f.a.430.1		4	31.5	even	3
961.1.f.a.430.1		4	31.26	odd	6
961.1.f.a.573.1		4	31.10	even	15
961.1.f.a.573.1		4	31.21	odd	30
961.1.f.a.587.1		4	31.11	odd	30
961.1.f.a.587.1		4	31.20	even	15
961.1.h.a.115.1		8	31.8	even	5	inner
961.1.h.a.115.1		8	31.23	odd	10	inner
961.1.h.a.117.1		8	31.6	odd	6	inner
961.1.h.a.117.1		8	31.25	even	3	inner
961.1.h.a.145.1		8	31.4	even	5	inner
961.1.h.a.145.1		8	31.27	odd	10	inner
961.1.h.a.229.1		8	31.7	even	15	inner
961.1.h.a.229.1		8	31.24	odd	30	inner
961.1.h.a.414.1		8	1.1	even	1	trivial
961.1.h.a.414.1		8	31.30	odd	2	CM
961.1.h.a.513.1		8	31.14	even	15	inner
961.1.h.a.513.1		8	31.17	odd	30	inner
961.1.h.a.623.1		8	31.12	odd	30	inner
961.1.h.a.623.1		8	31.19	even	15	inner
961.1.h.a.726.1		8	31.2	even	5	inner
961.1.h.a.726.1		8	31.29	odd	10	inner
1519.1.c.a.1177.1		1	217.13	even	30
1519.1.c.a.1177.1		1	217.111	odd	30
1519.1.n.a.30.1		2	217.80	odd	30
1519.1.n.a.30.1		2	217.199	even	30
1519.1.n.a.557.1		2	217.75	even	30
1519.1.n.a.557.1		2	217.173	odd	30
1519.1.n.b.30.1		2	217.18	even	15
1519.1.n.b.30.1		2	217.137	odd	30
1519.1.n.b.557.1		2	217.44	odd	30
1519.1.n.b.557.1		2	217.142	even	15
1984.1.e.a.1921.1		1	248.13	odd	30
1984.1.e.a.1921.1		1	248.173	even	30
1984.1.e.b.1921.1		1	248.75	even	30
1984.1.e.b.1921.1		1	248.235	odd	30
2511.1.m.a.433.1		2	279.137	even	30
2511.1.m.a.433.1		2	279.173	odd	30
2511.1.m.a.1270.1		2	279.230	even	30
2511.1.m.a.1270.1		2	279.266	odd	30
2511.1.m.e.433.1		2	279.106	odd	30
2511.1.m.e.433.1		2	279.142	even	15
2511.1.m.e.1270.1		2	279.13	odd	30
2511.1.m.e.1270.1		2	279.49	even	15
3751.1.d.b.1332.1		1	341.142	odd	30
3751.1.d.b.1332.1		1	341.230	even	30
3751.1.t.a.2138.1		4	341.13	even	30
3751.1.t.a.2138.1		4	341.266	odd	30
3751.1.t.a.2665.1		4	341.204	odd	30
3751.1.t.a.2665.1		4	341.292	even	30
3751.1.t.a.2913.1		4	341.173	odd	30
3751.1.t.a.2913.1		4	341.261	even	30
3751.1.t.a.3657.1		4	341.18	odd	30
3751.1.t.a.3657.1		4	341.106	even	30
3751.1.t.c.2138.1		4	341.75	odd	30
3751.1.t.c.2138.1		4	341.328	even	15
3751.1.t.c.2665.1		4	341.49	even	15
3751.1.t.c.2665.1		4	341.137	odd	30
3751.1.t.c.2913.1		4	341.80	even	15
3751.1.t.c.2913.1		4	341.168	odd	30
3751.1.t.c.3657.1		4	341.235	even	15
3751.1.t.c.3657.1		4	341.323	odd	30

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:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	961.h (of order \(30\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.309017 - 0.951057i) q^{2} +(0.500000 + 0.866025i) q^{5} +(0.104528 + 0.994522i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(-0.104528 + 0.994522i) q^{9} +(0.669131 - 0.743145i) q^{10} +(0.913545 - 0.406737i) q^{14} +(-0.309017 + 0.951057i) q^{16} +(0.978148 - 0.207912i) q^{18} +(0.978148 + 0.207912i) q^{19} +(-0.809017 + 0.587785i) q^{35} +(-0.104528 - 0.994522i) q^{38} +(0.104528 - 0.994522i) q^{40} +(-0.669131 + 0.743145i) q^{41} +(-0.913545 + 0.406737i) q^{45} +(0.618034 - 1.90211i) q^{47} +(0.500000 - 0.866025i) q^{56} +(-0.669131 - 0.743145i) q^{59} -1.00000 q^{63} +(0.309017 + 0.951057i) q^{64} +(-1.00000 - 1.73205i) q^{67} +(0.809017 + 0.587785i) q^{70} +(0.104528 - 0.994522i) q^{71} +(0.669131 - 0.743145i) q^{72} +(-0.978148 + 0.207912i) q^{80} +(-0.978148 - 0.207912i) q^{81} +(0.913545 + 0.406737i) q^{82} +(0.669131 + 0.743145i) q^{90} -2.00000 q^{94} +(0.309017 + 0.951057i) q^{95} +(0.809017 - 0.587785i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 4 q^{5} - q^{7} - 2 q^{8} + q^{9} + q^{10} + q^{14} + 2 q^{16} - q^{18} - q^{19} - 2 q^{35} + q^{38} - q^{40} - q^{41} - q^{45} - 4 q^{47} + 4 q^{56} - q^{59} - 8 q^{63} - 2 q^{64}+ \cdots + 2 q^{97}+O(q^{100}) \) 8 * q + 2 * q^2 + 4 * q^5 - q^7 - 2 * q^8 + q^9 + q^10 + q^14 + 2 * q^16 - q^18 - q^19 - 2 * q^35 + q^38 - q^40 - q^41 - q^45 - 4 * q^47 + 4 * q^56 - q^59 - 8 * q^63 - 2 * q^64 - 8 * q^67 + 2 * q^70 - q^71 + q^72 + q^80 + q^81 + q^82 + q^90 - 16 * q^94 - 2 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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