LMFDB - Embedded newform 4000.1.bf.a.1599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,1,Mod(799,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 0, 7]))

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

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

chi := DirichletCharacter("4000.799");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	4000.bf (of order $10$, degree $4$, 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:	$1.99626005053$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 800)
Projective image:	$A_{5}$
Projective field:	Galois closure of 5.1.25000000.2

Embedding invariants

Embedding label			1599.2
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	4000.1599
Dual form			4000.1.bf.a.2399.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.809017 + 0.587785i) q^{3} -0.618034 q^{7} +O(q^{10})$	$q+(-0.809017 + 0.587785i) q^{3} -0.618034 q^{7} +(0.587785 + 0.190983i) q^{13} +(0.951057 - 1.30902i) q^{19} +(0.500000 - 0.363271i) q^{21} +(0.309017 + 0.951057i) q^{23} +(-0.309017 - 0.951057i) q^{27} +(1.30902 - 0.951057i) q^{29} +(-0.587785 + 0.809017i) q^{31} +(0.951057 + 0.309017i) q^{37} +(-0.587785 + 0.190983i) q^{39} -1.61803 q^{43} +(-0.500000 + 0.363271i) q^{47} -0.618034 q^{49} +(0.587785 + 0.809017i) q^{53} +1.61803i q^{57} +(1.53884 + 0.500000i) q^{59} +(0.309017 + 0.951057i) q^{61} +(-0.809017 - 0.587785i) q^{69} +(0.363271 + 0.500000i) q^{71} +(-0.951057 + 0.309017i) q^{73} +(0.951057 + 1.30902i) q^{79} +(0.809017 + 0.587785i) q^{81} +(0.809017 + 0.587785i) q^{83} +(-0.500000 + 1.53884i) q^{87} +(-0.363271 - 0.118034i) q^{91} -1.00000i q^{93} +(-0.363271 - 0.500000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 4 q^{7}+O(q^{10}) $ 8 * q - 2 * q^3 + 4 * q^7	$ 8 q - 2 q^{3} + 4 q^{7} + 4 q^{21} - 2 q^{23} + 2 q^{27} + 6 q^{29} - 4 q^{43} - 4 q^{47} + 4 q^{49} - 2 q^{61} - 2 q^{69} + 2 q^{81} + 2 q^{83} - 4 q^{87}+O(q^{100}) $ 8 * q - 2 * q^3 + 4 * q^7 + 4 * q^21 - 2 * q^23 + 2 * q^27 + 6 * q^29 - 4 * q^43 - 4 * q^47 + 4 * q^49 - 2 * q^61 - 2 * q^69 + 2 * q^81 + 2 * q^83 - 4 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$e\left(\frac{9}{10}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$3$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$7$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$11$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$12$		0			0
$13$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$13$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$		0			0
$19$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$19$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$20$		0			0
$21$	0.500000	−	0.363271i	0.500000	−	0.363271i
$22$		0			0
$23$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$23$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$28$		0			0
$29$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$29$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$30$		0			0
$31$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.994522	−	0.104528i	$-0.966667\pi$
$31$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	0.406737	+	0.913545i	$0.366667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$37$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$38$		0			0
$39$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$43$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$47$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$48$		0			0
$49$	−0.618034			−0.618034
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.587785	+	0.809017i	0.587785	+	0.809017i	0.994522	−	0.104528i	$-0.0333333\pi$
$53$	0.587785	+	0.809017i	0.587785	+	0.809017i	−0.406737	+	0.913545i	$0.633333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			1.61803i			1.61803i
$58$		0			0
$59$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$59$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$60$		0			0
$61$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$61$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$67$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$68$		0			0
$69$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$70$		0			0
$71$	0.363271	+	0.500000i	0.363271	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$71$	0.363271	+	0.500000i	0.363271	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$72$		0			0
$73$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.743145	−	0.669131i	$-0.766667\pi$
$73$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.207912	+	0.978148i	$0.566667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$79$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$80$		0			0
$81$	0.809017	+	0.587785i	0.809017	+	0.587785i
$82$		0			0
$83$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$83$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$88$		0			0
$89$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$89$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$90$		0			0
$91$	−0.363271	−	0.118034i	−0.363271	−	0.118034i
$92$		0			0
$93$		−	1.00000i		−	1.00000i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$97$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$103$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$109$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$110$		0			0
$111$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$112$		0			0
$113$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$113$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$127$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$128$		0			0
$129$	1.30902	−	0.951057i	1.30902	−	0.951057i
$130$		0			0
$131$	0.363271	−	0.500000i	0.363271	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$131$	0.363271	−	0.500000i	0.363271	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$132$		0			0
$133$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$137$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$138$		0			0
$139$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.743145	−	0.669131i	$-0.766667\pi$
$139$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.207912	+	0.978148i	$0.566667\pi$
$140$		0			0
$141$	0.190983	−	0.587785i	0.190983	−	0.587785i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.500000	−	0.363271i	0.500000	−	0.363271i
$148$		0			0
$149$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$151$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$157$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$158$		0			0
$159$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$160$		0			0
$161$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$162$		0			0
$163$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$163$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$167$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$168$		0			0
$169$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.207912	−	0.978148i	$-0.433333\pi$
$173$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.743145	+	0.669131i	$0.233333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$178$		0			0
$179$	0.587785	+	0.809017i	0.587785	+	0.809017i	0.994522	−	0.104528i	$-0.0333333\pi$
$179$	0.587785	+	0.809017i	0.587785	+	0.809017i	−0.406737	+	0.913545i	$0.633333\pi$
$180$		0			0
$181$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$181$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$182$		0			0
$183$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.190983	+	0.587785i	0.190983	+	0.587785i
$190$		0			0
$191$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$191$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$197$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$198$		0			0
$199$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$199$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.207912	−	0.978148i	$-0.433333\pi$
$211$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.743145	+	0.669131i	$0.233333\pi$
$212$		0			0
$213$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.363271	−	0.500000i	0.363271	−	0.500000i
$218$		0			0
$219$	0.587785	−	0.809017i	0.587785	−	0.809017i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$223$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$		0			0
$229$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$229$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$233$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$238$		0			0
$239$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.743145	−	0.669131i	$-0.766667\pi$
$239$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.207912	+	0.978148i	$0.566667\pi$
$240$		0			0
$241$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$241$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.809017	−	0.587785i	0.809017	−	0.587785i
$248$		0			0
$249$	−1.00000			−1.00000
$250$		0			0
$251$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$251$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$257$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$258$		0			0
$259$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$263$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$269$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$270$		0			0
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$	0.363271	−	0.118034i	0.363271	−	0.118034i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.743145	−	0.669131i	$-0.766667\pi$
$277$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.207912	+	0.978148i	$0.566667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$281$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$282$		0			0
$283$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$283$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$	0.587785	+	0.190983i	0.587785	+	0.190983i
$292$		0			0
$293$		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
$299$			0.618034i			0.618034i
$300$		0			0
$301$	1.00000			1.00000
$302$		0			0
$303$	0.809017	−	0.587785i	0.809017	−	0.587785i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$310$		0			0
$311$	0.587785	−	0.190983i	0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$311$	0.587785	−	0.190983i	0.587785	−	0.190983i	0.587785	+	0.809017i	$0.300000\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$317$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.809017	−	0.587785i	0.809017	−	0.587785i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.309017	−	0.224514i	0.309017	−	0.224514i
$330$		0			0
$331$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$331$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$337$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$338$		0			0
$339$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.00000			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$347$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		−	0.618034i		−	0.618034i
$352$		0			0
$353$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$353$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.587785	−	0.190983i	−0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$359$	−0.587785	−	0.190983i	−0.587785	−	0.190983i	−0.587785	+	0.809017i	$0.700000\pi$
$360$		0			0
$361$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$362$		0			0
$363$	0.309017	−	0.951057i	0.309017	−	0.951057i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$367$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$372$		0			0
$373$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$373$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.951057	−	0.309017i	0.951057	−	0.309017i
$378$		0			0
$379$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$379$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$383$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$389$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			0.618034i			0.618034i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.951057	−	1.30902i	−0.951057	−	1.30902i	−0.951057	−	0.309017i	$-0.900000\pi$
$397$	−0.951057	−	1.30902i	−0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$398$		0			0
$399$		−	1.00000i		−	1.00000i
$400$		0			0
$401$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$409$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$410$		0			0
$411$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$412$		0			0
$413$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.587785	−	0.809017i	0.587785	−	0.809017i
$418$		0			0
$419$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$419$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$420$		0			0
$421$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$421$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$431$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$432$		0			0
$433$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$433$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.53884	+	0.500000i	1.53884	+	0.500000i
$438$		0			0
$439$	0.587785	−	0.190983i	0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$439$	0.587785	−	0.190983i	0.587785	−	0.190983i	0.587785	+	0.809017i	$0.300000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$443$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	0.809017	−	0.587785i	0.809017	−	0.587785i
$448$		0			0
$449$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$449$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$461$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$462$		0			0
$463$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$463$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$467$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$479$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$480$		0			0
$481$	0.500000	+	0.363271i	0.500000	+	0.363271i
$482$		0			0
$483$	0.500000	+	0.363271i	0.500000	+	0.363271i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$487$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$488$		0			0
$489$	0.309017	+	0.951057i	0.309017	+	0.951057i
$490$		0			0
$491$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$491$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.224514	−	0.309017i	−0.224514	−	0.309017i
$498$		0			0
$499$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$499$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$500$		0			0
$501$	−1.61803			−1.61803
$502$		0			0
$503$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$503$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.618034			0.618034
$508$		0			0
$509$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$509$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$510$		0			0
$511$	0.587785	−	0.190983i	0.587785	−	0.190983i
$512$		0			0
$513$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$520$		0			0
$521$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$521$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$522$		0			0
$523$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$523$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$541$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$547$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2.61803i		−	2.61803i
$552$		0			0
$553$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$563$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$568$		0			0
$569$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$569$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$570$		0			0
$571$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$571$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$577$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$587$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$588$		0			0
$589$	0.500000	+	1.53884i	0.500000	+	1.53884i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$593$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.363271	+	0.500000i	0.363271	+	0.500000i
$598$		0			0
$599$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$599$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$600$		0			0
$601$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$601$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$607$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$608$		0			0
$609$	0.309017	−	0.951057i	0.309017	−	0.951057i
$610$		0			0
$611$	−0.363271	+	0.118034i	−0.363271	+	0.118034i
$612$		0			0
$613$	−0.587785	−	0.190983i	−0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$613$	−0.587785	−	0.190983i	−0.587785	−	0.190983i	−0.587785	+	0.809017i	$0.700000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$617$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$618$		0			0
$619$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.994522	−	0.104528i	$-0.966667\pi$
$619$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	0.406737	+	0.913545i	$0.366667\pi$
$620$		0			0
$621$	0.809017	−	0.587785i	0.809017	−	0.587785i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$631$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$632$		0			0
$633$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.363271	−	0.118034i	−0.363271	−	0.118034i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$641$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$647$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$			0.618034i			0.618034i
$652$		0			0
$653$	0.363271	+	0.500000i	0.363271	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$653$	0.363271	+	0.500000i	0.363271	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$659$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$660$		0			0
$661$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$661$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.30902	+	0.951057i	1.30902	+	0.951057i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$673$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.587785	+	0.190983i	−0.587785	+	0.190983i	−0.587785	−	0.809017i	$-0.700000\pi$
$677$	−0.587785	+	0.190983i	−0.587785	+	0.190983i			1.00000i	$0.5\pi$
$678$		0			0
$679$	0.224514	+	0.309017i	0.224514	+	0.309017i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$683$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.190983	+	0.587785i	0.190983	+	0.587785i
$690$		0			0
$691$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$691$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	1.30902	−	0.951057i	1.30902	−	0.951057i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.618034			0.618034
$708$		0			0
$709$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$709$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.587785	−	0.809017i	0.587785	−	0.809017i
$718$		0			0
$719$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$719$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$720$		0			0
$721$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$722$		0			0
$723$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$727$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$733$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$739$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$740$		0			0
$741$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.618034			0.618034
$750$		0			0
$751$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$751$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$	0.587785	+	0.809017i	0.587785	+	0.809017i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$757$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$761$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.809017	+	0.587785i	0.809017	+	0.587785i
$768$		0			0
$769$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$769$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$770$		0			0
$771$	0.951057	+	1.30902i	0.951057	+	1.30902i
$772$		0			0
$773$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$773$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	0.587785	−	0.190983i	0.587785	−	0.190983i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$787$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$788$		0			0
$789$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$790$		0			0
$791$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$792$		0			0
$793$			0.618034i			0.618034i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$797$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$798$		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			0
$809$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$809$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$810$		0			0
$811$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.207912	−	0.978148i	$-0.433333\pi$
$811$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.743145	+	0.669131i	$0.233333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.53884	+	2.11803i	−1.53884	+	2.11803i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$821$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$822$		0			0
$823$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$823$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$827$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$828$		0			0
$829$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$829$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$830$		0			0
$831$	0.587785	−	0.809017i	0.587785	−	0.809017i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.951057	+	0.309017i	0.951057	+	0.309017i
$838$		0			0
$839$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$839$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	0.500000	−	1.53884i	0.500000	−	1.53884i
$842$		0			0
$843$	1.61803			1.61803
$844$		0			0
$845$		0			0
$846$		0			0
$847$	0.500000	−	0.363271i	0.500000	−	0.363271i
$848$		0			0
$849$	−0.618034			−0.618034
$850$		0			0
$851$			1.00000i			1.00000i
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$857$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$858$		0			0
$859$	−0.951057	−	0.309017i	−0.951057	−	0.309017i	−0.207912	−	0.978148i	$-0.566667\pi$
$859$	−0.951057	−	0.309017i	−0.951057	−	0.309017i	−0.743145	+	0.669131i	$0.766667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$863$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$877$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$881$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$882$		0			0
$883$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$883$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$887$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1.00000i			1.00000i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$898$		0			0
$899$			1.61803i			1.61803i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.207912	−	0.978148i	$-0.433333\pi$
$911$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.743145	+	0.669131i	$0.233333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.224514	+	0.309017i	−0.224514	+	0.309017i
$918$		0			0
$919$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$919$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.118034	+	0.363271i	0.118034	+	0.363271i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$929$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$932$		0			0
$933$	−0.363271	+	0.500000i	−0.363271	+	0.500000i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$937$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$941$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$947$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$948$		0			0
$949$	−0.618034			−0.618034
$950$		0			0
$951$			1.00000i			1.00000i
$952$		0			0
$953$	0.363271	+	0.500000i	0.363271	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$953$	0.363271	+	0.500000i	0.363271	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$967$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$971$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$972$		0			0
$973$	0.587785	−	0.190983i	0.587785	−	0.190983i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.743145	−	0.669131i	$-0.766667\pi$
$977$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.207912	+	0.978148i	$0.566667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$983$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$988$		0			0
$989$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$990$		0			0
$991$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$991$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.587785	+	0.809017i	0.587785	+	0.809017i	0.994522	−	0.104528i	$-0.0333333\pi$
$997$	0.587785	+	0.809017i	0.587785	+	0.809017i	−0.406737	+	0.913545i	$0.633333\pi$
$998$		0			0
$999$		−	1.00000i		−	1.00000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.1.bf.a.1599.2		8
4.3	odd	2		4000.1.bf.b.1599.2		8
5.2	odd	4		800.1.bh.a.31.2	yes	8
5.3	odd	4		4000.1.bh.a.1151.1		8
5.4	even	2		4000.1.bf.b.1599.1		8
20.3	even	4		4000.1.bh.a.1151.2		8
20.7	even	4		800.1.bh.a.31.1	✓	8
20.19	odd	2	inner	4000.1.bf.a.1599.1		8
25.3	odd	20		4000.1.bh.a.351.2		8
25.4	even	10		4000.1.bf.b.2399.2		8
25.21	even	5	inner	4000.1.bf.a.2399.1		8
25.22	odd	20		800.1.bh.a.671.1	yes	8
40.27	even	4		1600.1.bh.b.831.2		8
40.37	odd	4		1600.1.bh.b.831.1		8
100.3	even	20		4000.1.bh.a.351.1		8
100.47	even	20		800.1.bh.a.671.2	yes	8
100.71	odd	10		4000.1.bf.b.2399.1		8
100.79	odd	10	inner	4000.1.bf.a.2399.2		8
200.147	even	20		1600.1.bh.b.1471.1		8
200.197	odd	20		1600.1.bh.b.1471.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.1.bh.a.31.1	✓	8	20.7	even	4
800.1.bh.a.31.2	yes	8	5.2	odd	4
800.1.bh.a.671.1	yes	8	25.22	odd	20
800.1.bh.a.671.2	yes	8	100.47	even	20
1600.1.bh.b.831.1		8	40.37	odd	4
1600.1.bh.b.831.2		8	40.27	even	4
1600.1.bh.b.1471.1		8	200.147	even	20
1600.1.bh.b.1471.2		8	200.197	odd	20
4000.1.bf.a.1599.1		8	20.19	odd	2	inner
4000.1.bf.a.1599.2		8	1.1	even	1	trivial
4000.1.bf.a.2399.1		8	25.21	even	5	inner
4000.1.bf.a.2399.2		8	100.79	odd	10	inner
4000.1.bf.b.1599.1		8	5.4	even	2
4000.1.bf.b.1599.2		8	4.3	odd	2
4000.1.bf.b.2399.1		8	100.71	odd	10
4000.1.bf.b.2399.2		8	25.4	even	10
4000.1.bh.a.351.1		8	100.3	even	20
4000.1.bh.a.351.2		8	25.3	odd	20
4000.1.bh.a.1151.1		8	5.3	odd	4
4000.1.bh.a.1151.2		8	20.3	even	4

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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	4000.bf (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.809017 + 0.587785i) q^{3} -0.618034 q^{7} +O(q^{10})\)	\(q+(-0.809017 + 0.587785i) q^{3} -0.618034 q^{7} +(0.587785 + 0.190983i) q^{13} +(0.951057 - 1.30902i) q^{19} +(0.500000 - 0.363271i) q^{21} +(0.309017 + 0.951057i) q^{23} +(-0.309017 - 0.951057i) q^{27} +(1.30902 - 0.951057i) q^{29} +(-0.587785 + 0.809017i) q^{31} +(0.951057 + 0.309017i) q^{37} +(-0.587785 + 0.190983i) q^{39} -1.61803 q^{43} +(-0.500000 + 0.363271i) q^{47} -0.618034 q^{49} +(0.587785 + 0.809017i) q^{53} +1.61803i q^{57} +(1.53884 + 0.500000i) q^{59} +(0.309017 + 0.951057i) q^{61} +(-0.809017 - 0.587785i) q^{69} +(0.363271 + 0.500000i) q^{71} +(-0.951057 + 0.309017i) q^{73} +(0.951057 + 1.30902i) q^{79} +(0.809017 + 0.587785i) q^{81} +(0.809017 + 0.587785i) q^{83} +(-0.500000 + 1.53884i) q^{87} +(-0.363271 - 0.118034i) q^{91} -1.00000i q^{93} +(-0.363271 - 0.500000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \) 8 * q - 2 * q^3 + 4 * q^7	\( 8 q - 2 q^{3} + 4 q^{7} + 4 q^{21} - 2 q^{23} + 2 q^{27} + 6 q^{29} - 4 q^{43} - 4 q^{47} + 4 q^{49} - 2 q^{61} - 2 q^{69} + 2 q^{81} + 2 q^{83} - 4 q^{87}+O(q^{100}) \) 8 * q - 2 * q^3 + 4 * q^7 + 4 * q^21 - 2 * q^23 + 2 * q^27 + 6 * q^29 - 4 * q^43 - 4 * q^47 + 4 * q^49 - 2 * q^61 - 2 * q^69 + 2 * q^81 + 2 * q^83 - 4 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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