LMFDB - Embedded newform 3871.1.m.c.3791.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3871,1,Mod(1500,3871)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3871, 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("3871.1500");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3871 = 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3871.m (of order $6$, degree $2$, 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:	$1.93188066390$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 79)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.6241.1
Artin image:	$C_3\times D_5$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{15} - \cdots)$

Embedding invariants

Embedding label			3791.1
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	3871.3791
Dual form			3871.1.m.c.1500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.309017 - 0.535233i) q^{2} +(0.309017 - 0.535233i) q^{4} +(0.809017 + 1.40126i) q^{5} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.309017 - 0.535233i) q^{2} +(0.309017 - 0.535233i) q^{4} +(0.809017 + 1.40126i) q^{5} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.309017 + 0.535233i) q^{11} +0.618034 q^{13} +(-0.309017 + 0.535233i) q^{18} +(0.809017 + 1.40126i) q^{19} +1.00000 q^{20} +0.381966 q^{22} +(0.809017 + 1.40126i) q^{23} +(-0.809017 + 1.40126i) q^{25} +(-0.190983 - 0.330792i) q^{26} +(-0.309017 + 0.535233i) q^{31} +(-0.500000 + 0.866025i) q^{32} -0.618034 q^{36} +(0.500000 - 0.866025i) q^{38} +(-0.809017 - 1.40126i) q^{40} +(0.190983 + 0.330792i) q^{44} +(0.809017 - 1.40126i) q^{45} +(0.500000 - 0.866025i) q^{46} +1.00000 q^{50} +(0.190983 - 0.330792i) q^{52} -1.00000 q^{55} +0.381966 q^{62} +0.618034 q^{64} +(0.500000 + 0.866025i) q^{65} +(0.809017 - 1.40126i) q^{67} +(0.500000 + 0.866025i) q^{72} +(0.809017 - 1.40126i) q^{73} +1.00000 q^{76} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +2.00000 q^{83} +(0.309017 - 0.535233i) q^{88} +(-0.309017 - 0.535233i) q^{89} -1.00000 q^{90} +1.00000 q^{92} +(-1.30902 + 2.26728i) q^{95} -1.61803 q^{97} +0.618034 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - q^{4} + q^{5} - 4 q^{8} - 2 q^{9}+O(q^{10}) $ 4 * q + q^2 - q^4 + q^5 - 4 * q^8 - 2 * q^9	$ 4 q + q^{2} - q^{4} + q^{5} - 4 q^{8} - 2 q^{9} + 2 q^{10} + q^{11} - 2 q^{13} + q^{18} + q^{19} + 4 q^{20} + 6 q^{22} + q^{23} - q^{25} - 3 q^{26} + q^{31} - 2 q^{32} + 2 q^{36} + 2 q^{38} - q^{40} + 3 q^{44} + q^{45} + 2 q^{46} + 4 q^{50} + 3 q^{52} - 4 q^{55} + 6 q^{62} - 2 q^{64} + 2 q^{65} + q^{67} + 2 q^{72} + q^{73} + 4 q^{76} - 2 q^{79} - 2 q^{81} + 8 q^{83} - q^{88} + q^{89} - 4 q^{90} + 4 q^{92} - 3 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q + q^2 - q^4 + q^5 - 4 * q^8 - 2 * q^9 + 2 * q^10 + q^11 - 2 * q^13 + q^18 + q^19 + 4 * q^20 + 6 * q^22 + q^23 - q^25 - 3 * q^26 + q^31 - 2 * q^32 + 2 * q^36 + 2 * q^38 - q^40 + 3 * q^44 + q^45 + 2 * q^46 + 4 * q^50 + 3 * q^52 - 4 * q^55 + 6 * q^62 - 2 * q^64 + 2 * q^65 + q^67 + 2 * q^72 + q^73 + 4 * q^76 - 2 * q^79 - 2 * q^81 + 8 * q^83 - q^88 + q^89 - 4 * q^90 + 4 * q^92 - 3 * q^95 - 2 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times$.

$n$	$1030$	$2845$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$2$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	0.309017	−	0.535233i	0.309017	−	0.535233i
$5$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$5$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.00000			−1.00000
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	0.500000	−	0.866025i	0.500000	−	0.866025i
$11$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$11$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$12$		0			0
$13$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$13$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$19$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$19$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$20$	1.00000			1.00000
$21$		0			0
$22$	0.381966			0.381966
$23$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$23$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$24$		0			0
$25$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$26$	−0.190983	−	0.330792i	−0.190983	−	0.330792i
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$31$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$32$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.618034			−0.618034
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	0.500000	−	0.866025i	0.500000	−	0.866025i
$39$		0			0
$40$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.190983	+	0.330792i	0.190983	+	0.330792i
$45$	0.809017	−	1.40126i	0.809017	−	1.40126i
$46$	0.500000	−	0.866025i	0.500000	−	0.866025i
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$		0			0
$50$	1.00000			1.00000
$51$		0			0
$52$	0.190983	−	0.330792i	0.190983	−	0.330792i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$	−1.00000			−1.00000
$56$		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	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	0.381966			0.381966
$63$		0			0
$64$	0.618034			0.618034
$65$	0.500000	+	0.866025i	0.500000	+	0.866025i
$66$		0			0
$67$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$67$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.500000	+	0.866025i	0.500000	+	0.866025i
$73$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$73$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$74$		0			0
$75$		0			0
$76$	1.00000			1.00000
$77$		0			0
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$79$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$	2.00000			2.00000			1.00000			$0$
$83$	2.00000			2.00000			1.00000			$0$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.309017	−	0.535233i	0.309017	−	0.535233i
$89$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$89$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$90$	−1.00000			−1.00000
$91$		0			0
$92$	1.00000			1.00000
$93$		0			0
$94$		0			0
$95$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$96$		0			0
$97$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$97$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$98$		0			0
$99$	0.618034			0.618034
$100$	0.500000	+	0.866025i	0.500000	+	0.866025i
$101$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$101$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	−0.618034			−0.618034
$105$		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		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$	0.309017	+	0.535233i	0.309017	+	0.535233i
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$116$		0			0
$117$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.309017	+	0.535233i	0.309017	+	0.535233i
$122$		0			0
$123$		0			0
$124$	0.190983	+	0.330792i	0.190983	+	0.330792i
$125$	−1.00000			−1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.309017	+	0.535233i	0.309017	+	0.535233i
$129$		0			0
$130$	0.309017	−	0.535233i	0.309017	−	0.535233i
$131$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$131$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$132$		0			0
$133$		0			0
$134$	−1.00000			−1.00000
$135$		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		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$144$		0			0
$145$		0			0
$146$	−1.00000			−1.00000
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$151$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$152$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$153$		0			0
$154$		0			0
$155$	−1.00000			−1.00000
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$159$		0			0
$160$	−1.61803			−1.61803
$161$		0			0
$162$	0.618034			0.618034
$163$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$163$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$164$		0			0
$165$		0			0
$166$	−0.618034	−	1.07047i	−0.618034	−	1.07047i
$167$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$167$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$168$		0			0
$169$	−0.618034			−0.618034
$170$		0			0
$171$	0.809017	−	1.40126i	0.809017	−	1.40126i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$179$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$179$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$180$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$181$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$181$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	1.61803			1.61803
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	0.500000	+	0.866025i	0.500000	+	0.866025i
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−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	$0.666667\pi$
$200$	0.809017	−	1.40126i	0.809017	−	1.40126i
$201$		0			0
$202$	0.381966			0.381966
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.809017	−	1.40126i	0.809017	−	1.40126i
$208$		0			0
$209$	−1.00000			−1.00000
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$221$		0			0
$222$		0			0
$223$	2.00000			2.00000			1.00000			$0$
$223$	2.00000			2.00000			1.00000			$0$
$224$		0			0
$225$	1.61803			1.61803
$226$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$	1.61803			1.61803
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$239$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$240$		0			0
$241$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$241$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$242$	0.190983	−	0.330792i	0.190983	−	0.330792i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.500000	+	0.866025i	0.500000	+	0.866025i
$248$	0.309017	−	0.535233i	0.309017	−	0.535233i
$249$		0			0
$250$	0.309017	+	0.535233i	0.309017	+	0.535233i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−1.00000			−1.00000
$254$		0			0
$255$		0			0
$256$	0.500000	−	0.866025i	0.500000	−	0.866025i
$257$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$257$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$258$		0			0
$259$		0			0
$260$	0.618034			0.618034
$261$		0			0
$262$	0.500000	−	0.866025i	0.500000	−	0.866025i
$263$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$263$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$269$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$269$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$270$		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
$275$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$276$		0			0
$277$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$277$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$278$		0			0
$279$	0.618034			0.618034
$280$		0			0
$281$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$281$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$282$		0			0
$283$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$283$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$284$		0			0
$285$		0			0
$286$	0.236068			0.236068
$287$		0			0
$288$	1.00000			1.00000
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.500000	+	0.866025i	0.500000	+	0.866025i
$300$		0			0
$301$		0			0
$302$	−1.00000			−1.00000
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	0.309017	+	0.535233i	0.309017	+	0.535233i
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$313$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$314$		0			0
$315$		0			0
$316$	−0.618034			−0.618034
$317$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$318$		0			0
$319$		0			0
$320$	0.500000	+	0.866025i	0.500000	+	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.309017	+	0.535233i	0.309017	+	0.535233i
$325$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$326$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	0.618034	−	1.07047i	0.618034	−	1.07047i
$333$		0			0
$334$	−0.190983	−	0.330792i	−0.190983	−	0.330792i
$335$	2.61803			2.61803
$336$		0			0
$337$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$337$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$338$	0.190983	+	0.330792i	0.190983	+	0.330792i
$339$		0			0
$340$		0			0
$341$	−0.190983	−	0.330792i	−0.190983	−	0.330792i
$342$	−1.00000			−1.00000
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$347$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$	−0.381966			−0.381966
$357$		0			0
$358$	1.23607			1.23607
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$361$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$362$	−0.190983	−	0.330792i	−0.190983	−	0.330792i
$363$		0			0
$364$		0			0
$365$	2.61803			2.61803
$366$		0			0
$367$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$367$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	0.809017	+	1.40126i	0.809017	+	1.40126i
$381$		0			0
$382$		0			0
$383$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$383$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$389$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$389$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.809017	−	1.40126i	0.809017	−	1.40126i
$396$	0.190983	−	0.330792i	0.190983	−	0.330792i
$397$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$404$	0.190983	+	0.330792i	0.190983	+	0.330792i
$405$	−1.61803			−1.61803
$406$		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
$414$	−1.00000			−1.00000
$415$	1.61803	+	2.80252i	1.61803	+	2.80252i
$416$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$417$		0			0
$418$	0.309017	+	0.535233i	0.309017	+	0.535233i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$421$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$422$		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
$431$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$431$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$432$		0			0
$433$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$433$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$438$		0			0
$439$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$439$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$440$	1.00000			1.00000
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$	0.500000	−	0.866025i	0.500000	−	0.866025i
$446$	−0.618034	−	1.07047i	−0.618034	−	1.07047i
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$457$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$458$		0			0
$459$		0			0
$460$	0.809017	+	1.40126i	0.809017	+	1.40126i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$467$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$468$	−0.381966			−0.381966
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.61803			−2.61803
$476$		0			0
$477$		0			0
$478$	0.500000	+	0.866025i	0.500000	+	0.866025i
$479$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$479$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	0.381966			0.381966
$483$		0			0
$484$	0.381966			0.381966
$485$	−1.30902	−	2.26728i	−1.30902	−	2.26728i
$486$		0			0
$487$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$487$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	0.309017	−	0.535233i	0.309017	−	0.535233i
$495$	0.500000	+	0.866025i	0.500000	+	0.866025i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$499$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$500$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−1.00000			−1.00000
$506$	0.309017	+	0.535233i	0.309017	+	0.535233i
$507$		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			0
$513$		0			0
$514$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$523$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$524$	1.00000			1.00000
$525$		0			0
$526$	0.381966			0.381966
$527$		0			0
$528$		0			0
$529$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$537$		0			0
$538$	−1.00000			−1.00000
$539$		0			0
$540$		0			0
$541$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$547$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$548$		0			0
$549$		0			0
$550$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.00000			−1.00000
$555$		0			0
$556$		0			0
$557$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$557$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$558$	−0.190983	−	0.330792i	−0.190983	−	0.330792i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.500000	+	0.866025i	0.500000	+	0.866025i
$563$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$563$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$	0.381966			0.381966
$567$		0			0
$568$		0			0
$569$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$569$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$570$		0			0
$571$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$571$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$572$	0.118034	+	0.204441i	0.118034	+	0.204441i
$573$		0			0
$574$		0			0
$575$	−2.61803			−2.61803
$576$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$585$	0.500000	−	0.866025i	0.500000	−	0.866025i
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−1.00000			−1.00000
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$593$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.309017	−	0.535233i	0.309017	−	0.535233i
$599$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$599$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−1.61803			−1.61803
$604$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$605$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−1.61803			−1.61803
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$617$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	0.500000	+	0.866025i	0.500000	+	0.866025i
$633$		0			0
$634$	−0.618034	+	1.07047i	−0.618034	+	1.07047i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$641$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$641$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$642$		0			0
$643$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$643$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$		0			0
$650$	0.618034			0.618034
$651$		0			0
$652$	−0.381966			−0.381966
$653$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$653$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$654$		0			0
$655$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$656$		0			0
$657$	−1.61803			−1.61803
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$	−2.00000			−2.00000
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.190983	−	0.330792i	0.190983	−	0.330792i
$669$		0			0
$670$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.500000	+	0.866025i	0.500000	+	0.866025i
$675$		0			0
$676$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$677$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−0.118034	+	0.204441i	−0.118034	+	0.204441i
$683$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$683$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$684$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$685$		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
$694$	−1.00000			−1.00000
$695$		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
$704$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$712$	0.309017	+	0.535233i	0.309017	+	0.535233i
$713$	−1.00000			−1.00000
$714$		0			0
$715$	−0.618034			−0.618034
$716$	0.618034	+	1.07047i	0.618034	+	1.07047i
$717$		0			0
$718$		0			0
$719$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$719$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$720$		0			0
$721$		0			0
$722$	1.00000			1.00000
$723$		0			0
$724$	0.190983	−	0.330792i	0.190983	−	0.330792i
$725$		0			0
$726$		0			0
$727$	2.00000			2.00000			1.00000			$0$
$727$	2.00000			2.00000			1.00000			$0$
$728$		0			0
$729$	1.00000			1.00000
$730$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$731$		0			0
$732$		0			0
$733$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$733$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$734$	0.381966			0.381966
$735$		0			0
$736$	−1.61803			−1.61803
$737$	0.500000	+	0.866025i	0.500000	+	0.866025i
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$743$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$751$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.61803			2.61803
$756$		0			0
$757$	2.00000			2.00000			1.00000			$0$
$757$	2.00000			2.00000			1.00000			$0$
$758$		0			0
$759$		0			0
$760$	1.30902	−	2.26728i	1.30902	−	2.26728i
$761$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$761$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$773$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$774$		0			0
$775$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$776$	1.61803			1.61803
$777$		0			0
$778$	0.381966			0.381966
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$787$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$788$		0			0
$789$		0			0
$790$	−1.00000			−1.00000
$791$		0			0
$792$	−0.618034			−0.618034
$793$		0			0
$794$	−0.618034	+	1.07047i	−0.618034	+	1.07047i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−0.809017	−	1.40126i	−0.809017	−	1.40126i
$801$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$802$		0			0
$803$	0.500000	+	0.866025i	0.500000	+	0.866025i
$804$		0			0
$805$		0			0
$806$	0.236068			0.236068
$807$		0			0
$808$	0.309017	−	0.535233i	0.309017	−	0.535233i
$809$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$809$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$810$	0.500000	+	0.866025i	0.500000	+	0.866025i
$811$	2.00000			2.00000			1.00000			$0$
$811$	2.00000			2.00000			1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.500000	−	0.866025i	0.500000	−	0.866025i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$821$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$822$		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			$\pi$
$828$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$	1.00000	−	1.73205i	1.00000	−	1.73205i
$831$		0			0
$832$	0.381966			0.381966
$833$		0			0
$834$		0			0
$835$	0.500000	+	0.866025i	0.500000	+	0.866025i
$836$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$837$		0			0
$838$		0			0
$839$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$839$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	0.500000	+	0.866025i	0.500000	+	0.866025i
$843$		0			0
$844$		0			0
$845$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$846$		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
$855$	2.61803			2.61803
$856$		0			0
$857$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$857$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−1.00000			−1.00000
$863$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$864$		0			0
$865$		0			0
$866$	0.500000	+	0.866025i	0.500000	+	0.866025i
$867$		0			0
$868$		0			0
$869$	0.618034			0.618034
$870$		0			0
$871$	0.500000	−	0.866025i	0.500000	−	0.866025i
$872$		0			0
$873$	0.809017	+	1.40126i	0.809017	+	1.40126i
$874$	1.61803			1.61803
$875$		0			0
$876$		0			0
$877$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$877$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$878$	0.500000	−	0.866025i	0.500000	−	0.866025i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$887$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$888$		0			0
$889$		0			0
$890$	−0.618034			−0.618034
$891$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$892$	0.618034	−	1.07047i	0.618034	−	1.07047i
$893$		0			0
$894$		0			0
$895$	−3.23607			−3.23607
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.500000	−	0.866025i	0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.500000	+	0.866025i	0.500000	+	0.866025i
$906$		0			0
$907$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$907$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$	0.618034			0.618034
$910$		0			0
$911$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$911$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$912$		0			0
$913$	−0.618034	+	1.07047i	−0.618034	+	1.07047i
$914$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$919$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$920$	1.30902	−	2.26728i	1.30902	−	2.26728i
$921$		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	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$935$		0			0
$936$	0.309017	+	0.535233i	0.309017	+	0.535233i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$941$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−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
$949$	0.500000	−	0.866025i	0.500000	−	0.866025i
$950$	0.809017	+	1.40126i	0.809017	+	1.40126i
$951$		0			0
$952$		0			0
$953$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$953$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$954$		0			0
$955$		0			0
$956$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$957$		0			0
$958$	1.23607			1.23607
$959$		0			0
$960$		0			0
$961$	0.309017	+	0.535233i	0.309017	+	0.535233i
$962$		0			0
$963$		0			0
$964$	0.190983	+	0.330792i	0.190983	+	0.330792i
$965$		0			0
$966$		0			0
$967$	2.00000			2.00000			1.00000			$0$
$967$	2.00000			2.00000			1.00000			$0$
$968$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$969$		0			0
$970$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$971$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$971$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$972$		0			0
$973$		0			0
$974$	0.381966			0.381966
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	0.381966			0.381966
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.618034			0.618034
$989$		0			0
$990$	0.309017	−	0.535233i	0.309017	−	0.535233i
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$997$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$998$	−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	3871.1.m.c.3791.1		4
7.2	even	3	inner	3871.1.m.c.1500.1		4
7.3	odd	6		3871.1.c.c.2843.2		2
7.4	even	3		79.1.b.a.78.2	✓	2
7.5	odd	6		3871.1.m.b.1500.1		4
7.6	odd	2		3871.1.m.b.3791.1		4
21.11	odd	6		711.1.d.b.631.1		2
28.11	odd	6		1264.1.e.a.1105.1		2
35.4	even	6		1975.1.d.c.1026.1		2
35.18	odd	12		1975.1.c.a.1974.2		4
35.32	odd	12		1975.1.c.a.1974.3		4
79.78	odd	2	CM	3871.1.m.c.3791.1		4
553.157	even	6		3871.1.c.c.2843.2		2
553.236	even	6		3871.1.m.b.1500.1		4
553.394	odd	6	inner	3871.1.m.c.1500.1		4
553.473	odd	6		79.1.b.a.78.2	✓	2
553.552	even	2		3871.1.m.b.3791.1		4
1659.473	even	6		711.1.d.b.631.1		2
2212.1579	even	6		1264.1.e.a.1105.1		2
2765.473	even	12		1975.1.c.a.1974.2		4
2765.1579	odd	6		1975.1.d.c.1026.1		2
2765.2132	even	12		1975.1.c.a.1974.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
79.1.b.a.78.2	✓	2	7.4	even	3
79.1.b.a.78.2	✓	2	553.473	odd	6
711.1.d.b.631.1		2	21.11	odd	6
711.1.d.b.631.1		2	1659.473	even	6
1264.1.e.a.1105.1		2	28.11	odd	6
1264.1.e.a.1105.1		2	2212.1579	even	6
1975.1.c.a.1974.2		4	35.18	odd	12
1975.1.c.a.1974.2		4	2765.473	even	12
1975.1.c.a.1974.3		4	35.32	odd	12
1975.1.c.a.1974.3		4	2765.2132	even	12
1975.1.d.c.1026.1		2	35.4	even	6
1975.1.d.c.1026.1		2	2765.1579	odd	6
3871.1.c.c.2843.2		2	7.3	odd	6
3871.1.c.c.2843.2		2	553.157	even	6
3871.1.m.b.1500.1		4	7.5	odd	6
3871.1.m.b.1500.1		4	553.236	even	6
3871.1.m.b.3791.1		4	7.6	odd	2
3871.1.m.b.3791.1		4	553.552	even	2
3871.1.m.c.1500.1		4	7.2	even	3	inner
3871.1.m.c.1500.1		4	553.394	odd	6	inner
3871.1.m.c.3791.1		4	1.1	even	1	trivial
3871.1.m.c.3791.1		4	79.78	odd	2	CM

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:	\( N \)	\(=\)	\( 3871 = 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3871.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.309017 - 0.535233i) q^{2} +(0.309017 - 0.535233i) q^{4} +(0.809017 + 1.40126i) q^{5} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.309017 - 0.535233i) q^{2} +(0.309017 - 0.535233i) q^{4} +(0.809017 + 1.40126i) q^{5} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.309017 + 0.535233i) q^{11} +0.618034 q^{13} +(-0.309017 + 0.535233i) q^{18} +(0.809017 + 1.40126i) q^{19} +1.00000 q^{20} +0.381966 q^{22} +(0.809017 + 1.40126i) q^{23} +(-0.809017 + 1.40126i) q^{25} +(-0.190983 - 0.330792i) q^{26} +(-0.309017 + 0.535233i) q^{31} +(-0.500000 + 0.866025i) q^{32} -0.618034 q^{36} +(0.500000 - 0.866025i) q^{38} +(-0.809017 - 1.40126i) q^{40} +(0.190983 + 0.330792i) q^{44} +(0.809017 - 1.40126i) q^{45} +(0.500000 - 0.866025i) q^{46} +1.00000 q^{50} +(0.190983 - 0.330792i) q^{52} -1.00000 q^{55} +0.381966 q^{62} +0.618034 q^{64} +(0.500000 + 0.866025i) q^{65} +(0.809017 - 1.40126i) q^{67} +(0.500000 + 0.866025i) q^{72} +(0.809017 - 1.40126i) q^{73} +1.00000 q^{76} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +2.00000 q^{83} +(0.309017 - 0.535233i) q^{88} +(-0.309017 - 0.535233i) q^{89} -1.00000 q^{90} +1.00000 q^{92} +(-1.30902 + 2.26728i) q^{95} -1.61803 q^{97} +0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - q^{4} + q^{5} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) 4 * q + q^2 - q^4 + q^5 - 4 * q^8 - 2 * q^9	\( 4 q + q^{2} - q^{4} + q^{5} - 4 q^{8} - 2 q^{9} + 2 q^{10} + q^{11} - 2 q^{13} + q^{18} + q^{19} + 4 q^{20} + 6 q^{22} + q^{23} - q^{25} - 3 q^{26} + q^{31} - 2 q^{32} + 2 q^{36} + 2 q^{38} - q^{40} + 3 q^{44} + q^{45} + 2 q^{46} + 4 q^{50} + 3 q^{52} - 4 q^{55} + 6 q^{62} - 2 q^{64} + 2 q^{65} + q^{67} + 2 q^{72} + q^{73} + 4 q^{76} - 2 q^{79} - 2 q^{81} + 8 q^{83} - q^{88} + q^{89} - 4 q^{90} + 4 q^{92} - 3 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q + q^2 - q^4 + q^5 - 4 * q^8 - 2 * q^9 + 2 * q^10 + q^11 - 2 * q^13 + q^18 + q^19 + 4 * q^20 + 6 * q^22 + q^23 - q^25 - 3 * q^26 + q^31 - 2 * q^32 + 2 * q^36 + 2 * q^38 - q^40 + 3 * q^44 + q^45 + 2 * q^46 + 4 * q^50 + 3 * q^52 - 4 * q^55 + 6 * q^62 - 2 * q^64 + 2 * q^65 + q^67 + 2 * q^72 + q^73 + 4 * q^76 - 2 * q^79 - 2 * q^81 + 8 * q^83 - q^88 + q^89 - 4 * q^90 + 4 * q^92 - 3 * q^95 - 2 * q^97 - 2 * q^99

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