LMFDB - Embedded newform 3388.1.s.a.475.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3388,1,Mod(475,3388)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3388, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 1]))

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

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

chi := DirichletCharacter("3388.475");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3388 = 2^{2} \cdot 7 \cdot 11^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3388.s (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:	$1.69083226280$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + x^{2} - x + 1$ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 308)
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.2.5797306783837184.1

Embedding invariants

Embedding label			475.1
Root			$-0.309017 - 0.951057i$ of defining polynomial
Character	$\chi$	$=$	3388.475
Dual form			3388.1.s.a.699.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(-0.809017 - 0.587785i) q^{7} -1.00000 q^{8} +(-0.309017 + 0.951057i) q^{9} +(0.809017 + 0.587785i) q^{14} +1.00000 q^{16} +(0.309017 - 0.951057i) q^{18} -1.90211i q^{23} +(-0.809017 + 0.587785i) q^{25} +(-0.809017 - 0.587785i) q^{28} +(0.690983 - 0.951057i) q^{29} -1.00000 q^{32} +(-0.309017 + 0.951057i) q^{36} +(0.500000 + 0.363271i) q^{37} +1.61803 q^{43} +1.90211i q^{46} +(0.309017 + 0.951057i) q^{49} +(0.809017 - 0.587785i) q^{50} +(0.500000 - 1.53884i) q^{53} +(0.809017 + 0.587785i) q^{56} +(-0.690983 + 0.951057i) q^{58} +(0.809017 - 0.587785i) q^{63} +1.00000 q^{64} -1.17557i q^{67} +(1.11803 - 0.363271i) q^{71} +(0.309017 - 0.951057i) q^{72} +(-0.500000 - 0.363271i) q^{74} +(0.500000 - 1.53884i) q^{79} +(-0.809017 - 0.587785i) q^{81} -1.61803 q^{86} -1.90211i q^{92} +(-0.309017 - 0.951057i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + 4 q^{4} - q^{7} - 4 q^{8} + q^{9} + q^{14} + 4 q^{16} - q^{18} - q^{25} - q^{28} + 5 q^{29} - 4 q^{32} + q^{36} + 2 q^{37} + 2 q^{43} - q^{49} + q^{50} + 2 q^{53} + q^{56} - 5 q^{58}+ \cdots + q^{98}+O(q^{100})$ 4 * q - 4 * q^2 + 4 * q^4 - q^7 - 4 * q^8 + q^9 + q^14 + 4 * q^16 - q^18 - q^25 - q^28 + 5 * q^29 - 4 * q^32 + q^36 + 2 * q^37 + 2 * q^43 - q^49 + q^50 + 2 * q^53 + q^56 - 5 * q^58 + q^63 + 4 * q^64 - q^72 - 2 * q^74 + 2 * q^79 - q^81 - 2 * q^86 + q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3388\mathbb{Z}\right)^\times$ .

$n$	$365$	$969$	$1695$
$\chi(n)$	$e\left(\frac{1}{10}\right)$	$-1$	$-1$

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

)

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

$2$	−1.00000			−1.00000
$2$	−1.00000			−1.00000
$3$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$3$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$4$	1.00000			1.00000
$5$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$5$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$6$		0			0
$7$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$7$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$8$	−1.00000			−1.00000
$9$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$13$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$14$	0.809017	+	0.587785i	0.809017	+	0.587785i
$15$		0			0
$16$	1.00000			1.00000
$17$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$17$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$18$	0.309017	−	0.951057i	0.309017	−	0.951057i
$19$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$19$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$23$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$24$		0			0
$25$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$26$		0			0
$27$		0			0
$28$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$29$	0.690983	−	0.951057i	0.690983	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$29$	0.690983	−	0.951057i	0.690983	−	0.951057i	1.00000			$0$
$30$		0			0
$31$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$31$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$32$	−1.00000			−1.00000
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$37$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$37$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$41$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$42$		0			0
$43$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$43$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$44$		0			0
$45$		0			0
$46$			1.90211i			1.90211i
$47$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$47$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$48$		0			0
$49$	0.309017	+	0.951057i	0.309017	+	0.951057i
$50$	0.809017	−	0.587785i	0.809017	−	0.587785i
$51$		0			0
$52$		0			0
$53$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$53$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$54$		0			0
$55$		0			0
$56$	0.809017	+	0.587785i	0.809017	+	0.587785i
$57$		0			0
$58$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$59$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$59$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$	0.809017	−	0.587785i	0.809017	−	0.587785i
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$67$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$71$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$72$	0.309017	−	0.951057i	0.309017	−	0.951057i
$73$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$73$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$74$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$79$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$80$		0			0
$81$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$82$		0			0
$83$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$83$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$84$		0			0
$85$		0			0
$86$	−1.61803			−1.61803
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		−	1.90211i		−	1.90211i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$97$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$98$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$99$		0			0
$100$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$		0			0
$103$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$103$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$104$		0			0
$105$		0			0
$106$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$107$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$107$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$108$		0			0
$109$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$109$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$110$		0			0
$111$		0			0
$112$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$113$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$113$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$114$		0			0
$115$		0			0
$116$	0.690983	−	0.951057i	0.690983	−	0.951057i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$127$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$127$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$128$	−1.00000			−1.00000
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$			1.17557i			1.17557i
$135$		0			0
$136$		0			0
$137$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$137$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$138$		0			0
$139$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$139$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$140$		0			0
$141$		0			0
$142$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$143$		0			0
$144$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.500000	+	0.363271i	0.500000	+	0.363271i
$149$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$149$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$150$		0			0
$151$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$151$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$157$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$158$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$159$		0			0
$160$		0			0
$161$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$162$	0.809017	+	0.587785i	0.809017	+	0.587785i
$163$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$163$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$167$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$168$		0			0
$169$	0.809017	+	0.587785i	0.809017	+	0.587785i
$170$		0			0
$171$		0			0
$172$	1.61803			1.61803
$173$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$173$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$174$		0			0
$175$	1.00000			1.00000
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$179$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$180$		0			0
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$			1.90211i			1.90211i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$191$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	−1.00000			$1.00000\pi$
$192$		0			0
$193$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$193$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$194$		0			0
$195$		0			0
$196$	0.309017	+	0.951057i	0.309017	+	0.951057i
$197$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$197$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	0.809017	−	0.587785i	0.809017	−	0.587785i
$201$		0			0
$202$		0			0
$203$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.80902	+	0.587785i	1.80902	+	0.587785i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$211$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$212$	0.500000	−	1.53884i	0.500000	−	1.53884i
$213$		0			0
$214$	0.500000	−	0.363271i	0.500000	−	0.363271i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		−	1.90211i		−	1.90211i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$223$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$224$	0.809017	+	0.587785i	0.809017	+	0.587785i
$225$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$226$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$227$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$227$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$228$		0			0
$229$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$229$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$230$		0			0
$231$		0			0
$232$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$233$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$233$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$239$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$251$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$252$	0.809017	−	0.587785i	0.809017	−	0.587785i
$253$		0			0
$254$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$257$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$258$		0			0
$259$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$260$		0			0
$261$	0.690983	+	0.951057i	0.690983	+	0.951057i
$262$		0			0
$263$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$263$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		−	1.17557i		−	1.17557i
$269$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$269$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$		0			0
$273$		0			0
$274$	0.190983	+	0.587785i	0.190983	+	0.587785i
$275$		0			0
$276$		0			0
$277$	1.80902	+	0.587785i	1.80902	+	0.587785i	1.00000			$0$
$277$	1.80902	+	0.587785i	1.80902	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$281$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$282$		0			0
$283$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$283$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$284$	1.11803	−	0.363271i	1.11803	−	0.363271i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.309017	−	0.951057i	0.309017	−	0.951057i
$289$	0.809017	−	0.587785i	0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$293$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$294$		0			0
$295$		0			0
$296$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$302$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$303$		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			0
$311$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$311$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$312$		0			0
$313$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$313$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$314$		0			0
$315$		0			0
$316$	0.500000	−	1.53884i	0.500000	−	1.53884i
$317$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$317$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	1.11803	−	1.53884i	1.11803	−	1.53884i
$323$		0			0
$324$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$325$		0			0
$326$	1.80902	+	0.587785i	1.80902	+	0.587785i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$331$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$332$		0			0
$333$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$337$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$338$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.309017	−	0.951057i	0.309017	−	0.951057i
$344$	−1.61803			−1.61803
$345$		0			0
$346$		0			0
$347$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$347$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$	−1.00000			−1.00000
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.11803	+	1.53884i	1.11803	+	1.53884i
$359$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$359$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	0.309017	−	0.951057i	0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$367$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$368$		−	1.90211i		−	1.90211i
$369$		0			0
$370$		0			0
$371$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$379$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	0.690983	−	0.951057i	0.690983	−	0.951057i
$383$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$383$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$384$		0			0
$385$		0			0
$386$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$387$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$388$		0			0
$389$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$389$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$		0			0
$392$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$393$		0			0
$394$			1.90211i			1.90211i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$401$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$401$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	1.11803	−	0.363271i	1.11803	−	0.363271i
$407$		0			0
$408$		0			0
$409$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$421$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$422$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$423$		0			0
$424$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$429$		0			0
$430$		0			0
$431$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$431$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$432$		0			0
$433$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$433$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$434$		0			0
$435$		0			0
$436$			1.90211i			1.90211i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$443$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$449$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$449$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$450$	0.309017	+	0.951057i	0.309017	+	0.951057i
$451$		0			0
$452$	1.30902	−	0.951057i	1.30902	−	0.951057i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$457$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$			1.17557i			1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$463$			1.17557i			1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$464$	0.690983	−	0.951057i	0.690983	−	0.951057i
$465$		0			0
$466$		0			0
$467$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$467$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$468$		0			0
$469$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.30902	+	0.951057i	1.30902	+	0.951057i
$478$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$479$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$479$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.690983	+	0.951057i	0.690983	+	0.951057i	1.00000			$0$
$487$	0.690983	+	0.951057i	0.690983	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$491$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$498$		0			0
$499$	0.690983	−	0.951057i	0.690983	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$499$	0.690983	−	0.951057i	0.690983	−	0.951057i	1.00000			$0$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$503$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$504$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.190983	+	0.587785i	0.190983	+	0.587785i
$509$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$509$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	0.190983	+	0.587785i	0.190983	+	0.587785i
$519$		0			0
$520$		0			0
$521$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$521$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$522$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$		0			0
$525$		0			0
$526$	1.61803			1.61803
$527$		0			0
$528$		0			0
$529$	−2.61803			−2.61803
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$			1.17557i			1.17557i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$547$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$548$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$554$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$555$		0			0
$556$		0			0
$557$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$557$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	−1.00000			$1.00000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.11803	−	0.363271i	1.11803	−	0.363271i
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.309017	+	0.951057i	0.309017	+	0.951057i
$568$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$569$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$569$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$570$		0			0
$571$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$571$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.11803	+	1.53884i	1.11803	+	1.53884i
$576$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$577$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$577$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$578$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$587$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.500000	+	0.363271i	0.500000	+	0.363271i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.80902	−	0.587785i	1.80902	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$599$	1.80902	−	0.587785i	1.80902	−	0.587785i	1.00000			$0$
$600$		0			0
$601$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$601$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$602$	1.30902	+	0.951057i	1.30902	+	0.951057i
$603$	1.11803	+	0.363271i	1.11803	+	0.363271i
$604$	0.500000	−	0.363271i	0.500000	−	0.363271i
$605$		0			0
$606$		0			0
$607$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$607$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$613$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$617$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$618$		0			0
$619$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$619$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.309017	−	0.951057i	0.309017	−	0.951057i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.11803	+	1.53884i	−1.11803	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$631$	−1.11803	+	1.53884i	−1.11803	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$632$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$633$		0			0
$634$	0.500000	−	1.53884i	0.500000	−	1.53884i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$			1.17557i			1.17557i
$640$		0			0
$641$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$641$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$642$		0			0
$643$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$643$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$644$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$645$		0			0
$646$		0			0
$647$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$647$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$648$	0.809017	+	0.587785i	0.809017	+	0.587785i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$653$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$653$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.00000			−2.00000			−1.00000			$\pi$
$659$	−2.00000			−2.00000			−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	1.90211i		−	1.90211i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.500000	−	0.363271i	0.500000	−	0.363271i
$667$	−1.80902	−	1.31433i	−1.80902	−	1.31433i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$673$	1.11803	+	0.363271i	1.11803	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$674$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$675$		0			0
$676$	0.809017	+	0.587785i	0.809017	+	0.587785i
$677$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$677$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$683$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$		0			0
$686$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$687$		0			0
$688$	1.61803			1.61803
$689$		0			0
$690$		0			0
$691$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$691$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$692$		0			0
$693$		0			0
$694$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.00000			1.00000
$701$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$701$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$709$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$710$		0			0
$711$	1.30902	+	0.951057i	1.30902	+	0.951057i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$717$		0			0
$718$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$719$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$719$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$720$		0			0
$721$		0			0
$722$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$723$		0			0
$724$		0			0
$725$			1.17557i			1.17557i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	0.809017	−	0.587785i	0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$733$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$734$		0			0
$735$		0			0
$736$			1.90211i			1.90211i
$737$		0			0
$738$		0			0
$739$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$739$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$	1.30902	−	0.951057i	1.30902	−	0.951057i
$743$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$743$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.618034			0.618034
$750$		0			0
$751$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$751$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$757$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$758$	1.80902	−	0.587785i	1.80902	−	0.587785i
$759$		0			0
$760$		0			0
$761$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$761$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$762$		0			0
$763$	1.11803	−	1.53884i	1.11803	−	1.53884i
$764$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	1.11803	−	0.363271i	1.11803	−	0.363271i
$773$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$773$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$774$	0.500000	−	1.53884i	0.500000	−	1.53884i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.500000	+	0.363271i	0.500000	+	0.363271i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.309017	+	0.951057i	0.309017	+	0.951057i
$785$		0			0
$786$		0			0
$787$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$787$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$788$		−	1.90211i		−	1.90211i
$789$		0			0
$790$		0			0
$791$	−1.61803			−1.61803
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$797$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$798$		0			0
$799$		0			0
$800$	0.809017	−	0.587785i	0.809017	−	0.587785i
$801$		0			0
$802$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.80902	−	0.587785i	1.80902	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$809$	1.80902	−	0.587785i	1.80902	−	0.587785i	1.00000			$0$
$810$		0			0
$811$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$811$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$812$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$821$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$822$		0			0
$823$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$823$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$827$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$828$	1.80902	+	0.587785i	1.80902	+	0.587785i
$829$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$829$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$839$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$840$		0			0
$841$	−0.118034	−	0.363271i	−0.118034	−	0.363271i
$842$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$843$		0			0
$844$	0.190983	−	0.587785i	0.190983	−	0.587785i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	0.500000	−	1.53884i	0.500000	−	1.53884i
$849$		0			0
$850$		0			0
$851$	0.690983	−	0.951057i	0.690983	−	0.951057i
$852$		0			0
$853$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$853$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$854$		0			0
$855$		0			0
$856$	0.500000	−	0.363271i	0.500000	−	0.363271i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	0.190983	−	0.587785i	0.190983	−	0.587785i
$863$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$863$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	1.90211i		−	1.90211i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$877$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.00000			1.00000
$883$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$883$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$884$		0			0
$885$		0			0
$886$	0.690983	+	0.951057i	0.690983	+	0.951057i
$887$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$887$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$888$		0			0
$889$	0.190983	−	0.587785i	0.190983	−	0.587785i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.809017	+	0.587785i	0.809017	+	0.587785i
$897$		0			0
$898$	0.190983	−	0.587785i	0.190983	−	0.587785i
$899$		0			0
$900$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$905$		0			0
$906$		0			0
$907$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$907$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$		0			0
$913$		0			0
$914$	1.80902	−	0.587785i	1.80902	−	0.587785i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$919$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.618034			−0.618034
$926$		−	1.17557i		−	1.17557i
$927$		0			0
$928$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$929$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$929$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$937$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$938$	0.690983	−	0.951057i	0.690983	−	0.951057i
$939$		0			0
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$953$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$954$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$955$		0			0
$956$	1.30902	−	0.951057i	1.30902	−	0.951057i
$957$		0			0
$958$		0			0
$959$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$960$		0			0
$961$	0.809017	+	0.587785i	0.809017	+	0.587785i
$962$		0			0
$963$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$967$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$971$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$972$		0			0
$973$		0			0
$974$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$975$		0			0
$976$		0			0
$977$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$977$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$982$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$983$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$983$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	3.07768i		−	3.07768i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$	1.11803	+	0.363271i	1.11803	+	0.363271i
$995$		0			0
$996$		0			0
$997$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$997$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$998$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3388.1.s.a.475.1		4
4.3	odd	2		3388.1.s.d.475.1		4
7.6	odd	2	CM	3388.1.s.a.475.1		4
11.2	odd	10		308.1.s.a.195.1	yes	4
11.3	even	5		308.1.s.b.139.1	yes	4
11.4	even	5		3388.1.g.b.3387.1		4
11.5	even	5		3388.1.s.e.699.1		4
11.6	odd	10		3388.1.s.d.699.1		4
11.7	odd	10		3388.1.g.a.3387.4		4
11.8	odd	10		3388.1.s.b.1371.1		4
11.9	even	5		3388.1.s.g.2659.1		4
11.10	odd	2		3388.1.s.h.475.1		4
28.27	even	2		3388.1.s.d.475.1		4
33.2	even	10		2772.1.dl.b.811.1		4
33.14	odd	10		2772.1.dl.a.1063.1		4
44.3	odd	10		308.1.s.a.139.1	✓	4
44.7	even	10		3388.1.g.b.3387.2		4
44.15	odd	10		3388.1.g.a.3387.3		4
44.19	even	10		3388.1.s.g.1371.1		4
44.27	odd	10		3388.1.s.h.699.1		4
44.31	odd	10		3388.1.s.b.2659.1		4
44.35	even	10		308.1.s.b.195.1	yes	4
44.39	even	10	inner	3388.1.s.a.699.1		4
44.43	even	2		3388.1.s.e.475.1		4
77.2	odd	30		2156.1.bm.b.1207.1		8
77.3	odd	30		2156.1.bm.a.1195.1		8
77.6	even	10		3388.1.s.d.699.1		4
77.13	even	10		308.1.s.a.195.1	yes	4
77.20	odd	10		3388.1.s.g.2659.1		4
77.24	even	30		2156.1.bm.b.19.1		8
77.25	even	15		2156.1.bm.a.1195.1		8
77.27	odd	10		3388.1.s.e.699.1		4
77.41	even	10		3388.1.s.b.1371.1		4
77.46	odd	30		2156.1.bm.b.19.1		8
77.47	odd	30		2156.1.bm.a.227.1		8
77.48	odd	10		3388.1.g.b.3387.1		4
77.58	even	15		2156.1.bm.a.227.1		8
77.62	even	10		3388.1.g.a.3387.4		4
77.68	even	30		2156.1.bm.b.1207.1		8
77.69	odd	10		308.1.s.b.139.1	yes	4
77.76	even	2		3388.1.s.h.475.1		4
132.35	odd	10		2772.1.dl.a.811.1		4
132.47	even	10		2772.1.dl.b.1063.1		4
231.146	even	10		2772.1.dl.a.1063.1		4
231.167	odd	10		2772.1.dl.b.811.1		4
308.3	even	30		2156.1.bm.b.1195.1		8
308.27	even	10		3388.1.s.h.699.1		4
308.47	even	30		2156.1.bm.b.227.1		8
308.79	even	30		2156.1.bm.a.1207.1		8
308.83	odd	10	inner	3388.1.s.a.699.1		4
308.123	even	30		2156.1.bm.a.19.1		8
308.135	odd	30		2156.1.bm.b.227.1		8
308.139	odd	10		3388.1.g.b.3387.2		4
308.167	odd	10		308.1.s.b.195.1	yes	4
308.179	odd	30		2156.1.bm.b.1195.1		8
308.195	odd	10		3388.1.s.g.1371.1		4
308.223	even	10		308.1.s.a.139.1	✓	4
308.251	even	10		3388.1.s.b.2659.1		4
308.255	odd	30		2156.1.bm.a.19.1		8
308.279	even	10		3388.1.g.a.3387.3		4
308.299	odd	30		2156.1.bm.a.1207.1		8
308.307	odd	2		3388.1.s.e.475.1		4
924.167	even	10		2772.1.dl.a.811.1		4
924.839	odd	10		2772.1.dl.b.1063.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.1.s.a.139.1	✓	4	44.3	odd	10
308.1.s.a.139.1	✓	4	308.223	even	10
308.1.s.a.195.1	yes	4	11.2	odd	10
308.1.s.a.195.1	yes	4	77.13	even	10
308.1.s.b.139.1	yes	4	11.3	even	5
308.1.s.b.139.1	yes	4	77.69	odd	10
308.1.s.b.195.1	yes	4	44.35	even	10
308.1.s.b.195.1	yes	4	308.167	odd	10
2156.1.bm.a.19.1		8	308.123	even	30
2156.1.bm.a.19.1		8	308.255	odd	30
2156.1.bm.a.227.1		8	77.47	odd	30
2156.1.bm.a.227.1		8	77.58	even	15
2156.1.bm.a.1195.1		8	77.3	odd	30
2156.1.bm.a.1195.1		8	77.25	even	15
2156.1.bm.a.1207.1		8	308.79	even	30
2156.1.bm.a.1207.1		8	308.299	odd	30
2156.1.bm.b.19.1		8	77.24	even	30
2156.1.bm.b.19.1		8	77.46	odd	30
2156.1.bm.b.227.1		8	308.47	even	30
2156.1.bm.b.227.1		8	308.135	odd	30
2156.1.bm.b.1195.1		8	308.3	even	30
2156.1.bm.b.1195.1		8	308.179	odd	30
2156.1.bm.b.1207.1		8	77.2	odd	30
2156.1.bm.b.1207.1		8	77.68	even	30
2772.1.dl.a.811.1		4	132.35	odd	10
2772.1.dl.a.811.1		4	924.167	even	10
2772.1.dl.a.1063.1		4	33.14	odd	10
2772.1.dl.a.1063.1		4	231.146	even	10
2772.1.dl.b.811.1		4	33.2	even	10
2772.1.dl.b.811.1		4	231.167	odd	10
2772.1.dl.b.1063.1		4	132.47	even	10
2772.1.dl.b.1063.1		4	924.839	odd	10
3388.1.g.a.3387.3		4	44.15	odd	10
3388.1.g.a.3387.3		4	308.279	even	10
3388.1.g.a.3387.4		4	11.7	odd	10
3388.1.g.a.3387.4		4	77.62	even	10
3388.1.g.b.3387.1		4	11.4	even	5
3388.1.g.b.3387.1		4	77.48	odd	10
3388.1.g.b.3387.2		4	44.7	even	10
3388.1.g.b.3387.2		4	308.139	odd	10
3388.1.s.a.475.1		4	1.1	even	1	trivial
3388.1.s.a.475.1		4	7.6	odd	2	CM
3388.1.s.a.699.1		4	44.39	even	10	inner
3388.1.s.a.699.1		4	308.83	odd	10	inner
3388.1.s.b.1371.1		4	11.8	odd	10
3388.1.s.b.1371.1		4	77.41	even	10
3388.1.s.b.2659.1		4	44.31	odd	10
3388.1.s.b.2659.1		4	308.251	even	10
3388.1.s.d.475.1		4	4.3	odd	2
3388.1.s.d.475.1		4	28.27	even	2
3388.1.s.d.699.1		4	11.6	odd	10
3388.1.s.d.699.1		4	77.6	even	10
3388.1.s.e.475.1		4	44.43	even	2
3388.1.s.e.475.1		4	308.307	odd	2
3388.1.s.e.699.1		4	11.5	even	5
3388.1.s.e.699.1		4	77.27	odd	10
3388.1.s.g.1371.1		4	44.19	even	10
3388.1.s.g.1371.1		4	308.195	odd	10
3388.1.s.g.2659.1		4	11.9	even	5
3388.1.s.g.2659.1		4	77.20	odd	10
3388.1.s.h.475.1		4	11.10	odd	2
3388.1.s.h.475.1		4	77.76	even	2
3388.1.s.h.699.1		4	44.27	odd	10
3388.1.s.h.699.1		4	308.27	even	10

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3388.1.s.a.475.1

Level	$3388$
Weight	$1$
Character	3388.475
Analytic conductor	$1.691$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{10}$
CM discriminant	-7
Inner twists	$4$