LMFDB - Embedded newform 3871.1.m.b.1500.1

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:	$C_6\times D_5$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{30} - \cdots)$

Embedding invariants

Embedding label			1500.1
Root			$-0.309017 - 0.535233i$ of defining polynomial
Character	$\chi$	$=$	3871.1500
Dual form			3871.1.m.b.3791.1

$q$-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 $\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times$.

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

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	−0.978148	−	0.207912i	$-0.933333\pi$
$2$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$3$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$	0.309017	+	0.535233i	0.309017	+	0.535233i
$5$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$5$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$6$		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	0.669131	−	0.743145i	$-0.266667\pi$
$11$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$12$		0			0
$13$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$13$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$19$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	0.104528	+	0.994522i	$0.466667\pi$
$19$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$20$	−1.00000			−1.00000
$21$		0			0
$22$	0.381966			0.381966
$23$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$23$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$24$		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	$-0.0666667\pi$
$31$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$32$	−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	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	−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	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		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	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$	1.00000			1.00000
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$	−0.381966			−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	0.913545	+	0.406737i	$0.133333\pi$
$67$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$68$		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	−0.913545	−	0.406737i	$-0.866667\pi$
$73$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$74$		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			$\pi$
$83$	−2.00000			−2.00000			−1.00000			$\pi$
$84$		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	$-0.733333\pi$
$89$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$90$	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	$-0.200000\pi$
$97$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$98$		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	$-0.0666667\pi$
$101$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	0.618034			0.618034
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	−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	0.104528	+	0.994522i	$0.466667\pi$
$131$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$132$		0			0
$133$		0			0
$134$	−1.00000			−1.00000
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		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	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$151$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$152$	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	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−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	−0.978148	−	0.207912i	$-0.933333\pi$
$163$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$164$		0			0
$165$		0			0
$166$	0.618034	−	1.07047i	0.618034	−	1.07047i
$167$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$167$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$168$		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	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		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	$-0.600000\pi$
$181$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$182$		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	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	−0.500000	+	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	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	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			$\pi$
$223$	−2.00000			−2.00000			−1.00000			$\pi$
$224$		0			0
$225$	1.61803			1.61803
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$	−1.61803			−1.61803
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$	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	$-0.0666667\pi$
$241$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$242$	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	$-0.733333\pi$
$257$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$258$		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	0.669131	−	0.743145i	$-0.266667\pi$
$263$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$264$		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	−0.913545	−	0.406737i	$-0.866667\pi$
$269$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		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	0.913545	+	0.406737i	$0.133333\pi$
$277$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$278$		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	$-0.0666667\pi$
$283$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$284$		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	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$313$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$314$		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	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−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	0.913545	+	0.406737i	$0.133333\pi$
$347$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$348$		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	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		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	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	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	$-0.0666667\pi$
$367$	0.309017	+	0.535233i	0.309017	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		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	$-0.733333\pi$
$383$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$384$		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	0.669131	−	0.743145i	$-0.266667\pi$
$389$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$390$		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	$-0.333333\pi$
$397$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		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	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		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	0.913545	+	0.406737i	$0.133333\pi$
$431$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$432$		0			0
$433$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$433$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$434$		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	0.104528	+	0.994522i	$0.466667\pi$
$439$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$440$	−1.00000			−1.00000
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		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	−0.978148	−	0.207912i	$-0.933333\pi$
$457$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$458$		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	$-0.733333\pi$
$467$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$468$	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	$0.333333\pi$
$479$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$480$		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	0.669131	−	0.743145i	$-0.266667\pi$
$487$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$488$		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	−0.978148	−	0.207912i	$-0.933333\pi$
$499$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$500$	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	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		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	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	0.309017	−	0.535233i	0.309017	−	0.535233i	−0.669131	−	0.743145i	$-0.733333\pi$
$523$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$524$	−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	0.913545	+	0.406737i	$0.133333\pi$
$557$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$558$	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	$0.333333\pi$
$563$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$564$		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	−0.104528	−	0.994522i	$-0.533333\pi$
$569$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$570$		0			0
$571$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$571$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$572$	−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	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	−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	0.104528	+	0.994522i	$0.466667\pi$
$593$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$594$		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	0.913545	+	0.406737i	$0.133333\pi$
$599$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$600$		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	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	1.61803			1.61803
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		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	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$	−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	0.913545	+	0.406737i	$0.133333\pi$
$641$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$642$		0			0
$643$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$643$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	0.500000	+	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	−0.978148	−	0.207912i	$-0.933333\pi$
$653$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$654$		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	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		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	$-0.333333\pi$
$677$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$678$		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	0.913545	+	0.406737i	$0.133333\pi$
$683$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$684$	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	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		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	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$712$	−0.309017	+	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	$-0.733333\pi$
$719$	0.309017	−	0.535233i	0.309017	−	0.535233i	0.978148	+	0.207912i	$0.0666667\pi$
$720$		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			$\pi$
$727$	−2.00000			−2.00000			−1.00000			$\pi$
$728$		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	0.104528	+	0.994522i	$0.466667\pi$
$733$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$734$	−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	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		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	−0.978148	−	0.207912i	$-0.933333\pi$
$751$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$752$		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	0.104528	+	0.994522i	$0.466667\pi$
$761$	−0.809017	+	1.40126i	−0.809017	+	1.40126i	−0.913545	+	0.406737i	$0.866667\pi$
$762$		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	−0.913545	−	0.406737i	$-0.866667\pi$
$773$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$774$		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	−0.913545	−	0.406737i	$-0.866667\pi$
$787$	−0.809017	−	1.40126i	−0.809017	−	1.40126i	0.104528	−	0.994522i	$-0.466667\pi$
$788$		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

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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