LMFDB - Embedded newform 3960.1.eg.a.3779.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,1,Mod(899,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 5, 5, 3]))

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

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

chi := DirichletCharacter("3960.899");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3960.eg (of order $10$ , degree $4$ , 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:	$1.97629745003$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{12} + x^{8} - x^{4} + 1$ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			3779.2
Root			$-0.453990 + 0.891007i$ of defining polynomial
Character	$\chi$	$=$	3960.3779
Dual form			3960.1.eg.a.1619.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.309017 + 0.951057i) q^{2} +(-0.809017 + 0.587785i) q^{4} +(-0.951057 - 0.309017i) q^{5} +(0.183900 + 0.253116i) q^{7} +(-0.809017 - 0.587785i) q^{8} -1.00000i q^{10} +(-0.987688 - 0.156434i) q^{11} +(1.69480 - 0.550672i) q^{13} +(-0.183900 + 0.253116i) q^{14} +(0.309017 - 0.951057i) q^{16} +(-0.363271 + 0.500000i) q^{19} +(0.951057 - 0.309017i) q^{20} +(-0.156434 - 0.987688i) q^{22} -1.61803i q^{23} +(0.809017 + 0.587785i) q^{25} +(1.04744 + 1.44168i) q^{26} +(-0.297556 - 0.0966818i) q^{28} +1.00000 q^{32} +(-0.0966818 - 0.297556i) q^{35} +(-0.734572 + 0.533698i) q^{37} +(-0.587785 - 0.190983i) q^{38} +(0.587785 + 0.809017i) q^{40} +(1.59811 + 1.16110i) q^{41} +(0.891007 - 0.453990i) q^{44} +(1.53884 - 0.500000i) q^{46} +(1.11803 - 1.53884i) q^{47} +(0.278768 - 0.857960i) q^{49} +(-0.309017 + 0.951057i) q^{50} +(-1.04744 + 1.44168i) q^{52} +(1.11803 - 0.363271i) q^{53} +(0.891007 + 0.453990i) q^{55} -0.312869i q^{56} +(0.533698 + 0.734572i) q^{59} +(0.309017 + 0.951057i) q^{64} -1.78201 q^{65} +(0.253116 - 0.183900i) q^{70} +(-0.734572 - 0.533698i) q^{74} -0.618034i q^{76} +(-0.142040 - 0.278768i) q^{77} +(-0.587785 + 0.809017i) q^{80} +(-0.610425 + 1.87869i) q^{82} +(0.707107 + 0.707107i) q^{88} +1.97538i q^{89} +(0.451057 + 0.327712i) q^{91} +(0.951057 + 1.30902i) q^{92} +(1.80902 + 0.587785i) q^{94} +(0.500000 - 0.363271i) q^{95} +0.902113 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 4 q^{2} - 4 q^{4} - 4 q^{8} - 4 q^{16} + 4 q^{25} + 16 q^{32} + 4 q^{49} + 4 q^{50} - 4 q^{64} + 4 q^{77} - 8 q^{91} + 20 q^{94} + 8 q^{95} - 16 q^{98}+O(q^{100})$ 16 * q - 4 * q^2 - 4 * q^4 - 4 * q^8 - 4 * q^16 + 4 * q^25 + 16 * q^32 + 4 * q^49 + 4 * q^50 - 4 * q^64 + 4 * q^77 - 8 * q^91 + 20 * q^94 + 8 * q^95 - 16 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$ .

$n$	$991$	$1981$	$2377$	$2521$	$3521$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{9}{10}\right)$	$-1$

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

)

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

$2$	0.309017	+	0.951057i	0.309017	+	0.951057i
$2$	0.309017	+	0.951057i	0.309017	+	0.951057i
$3$		0			0
$3$		0			0
$4$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$5$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$5$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$6$		0			0
$7$	0.183900	+	0.253116i	0.183900	+	0.253116i	0.891007	−	0.453990i	$-0.150000\pi$
$7$	0.183900	+	0.253116i	0.183900	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$8$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$9$		0			0
$10$		−	1.00000i		−	1.00000i
$11$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$11$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$12$		0			0
$13$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$13$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.987688	+	0.156434i	$0.0500000\pi$
$14$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$15$		0			0
$16$	0.309017	−	0.951057i	0.309017	−	0.951057i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$		0			0
$19$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$19$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$20$	0.951057	−	0.309017i	0.951057	−	0.309017i
$21$		0			0
$22$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$23$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$23$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$24$		0			0
$25$	0.809017	+	0.587785i	0.809017	+	0.587785i
$26$	1.04744	+	1.44168i	1.04744	+	1.44168i
$27$		0			0
$28$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$		0			0
$31$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$31$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$32$	1.00000			1.00000
$33$		0			0
$34$		0			0
$35$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$36$		0			0
$37$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	−0.891007	−	0.453990i	$-0.850000\pi$
$37$	−0.734572	+	0.533698i	−0.734572	+	0.533698i	0.156434	+	0.987688i	$0.450000\pi$
$38$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$39$		0			0
$40$	0.587785	+	0.809017i	0.587785	+	0.809017i
$41$	1.59811	+	1.16110i	1.59811	+	1.16110i	0.891007	+	0.453990i	$0.150000\pi$
$41$	1.59811	+	1.16110i	1.59811	+	1.16110i	0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	0.891007	−	0.453990i	0.891007	−	0.453990i
$45$		0			0
$46$	1.53884	−	0.500000i	1.53884	−	0.500000i
$47$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$47$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$48$		0			0
$49$	0.278768	−	0.857960i	0.278768	−	0.857960i
$50$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$51$		0			0
$52$	−1.04744	+	1.44168i	−1.04744	+	1.44168i
$53$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$53$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$54$		0			0
$55$	0.891007	+	0.453990i	0.891007	+	0.453990i
$56$		−	0.312869i		−	0.312869i
$57$		0			0
$58$		0			0
$59$	0.533698	+	0.734572i	0.533698	+	0.734572i	0.987688	−	0.156434i	$-0.0500000\pi$
$59$	0.533698	+	0.734572i	0.533698	+	0.734572i	−0.453990	+	0.891007i	$0.650000\pi$
$60$		0			0
$61$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$61$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$62$		0			0
$63$		0			0
$64$	0.309017	+	0.951057i	0.309017	+	0.951057i
$65$	−1.78201			−1.78201
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$	0.253116	−	0.183900i	0.253116	−	0.183900i
$71$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$71$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$72$		0			0
$73$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$73$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$74$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$75$		0			0
$76$		−	0.618034i		−	0.618034i
$77$	−0.142040	−	0.278768i	−0.142040	−	0.278768i
$78$		0			0
$79$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$79$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$80$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$81$		0			0
$82$	−0.610425	+	1.87869i	−0.610425	+	1.87869i
$83$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$83$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.707107	+	0.707107i	0.707107	+	0.707107i
$89$			1.97538i			1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$89$			1.97538i			1.97538i	−0.156434	+	0.987688i	$0.550000\pi$
$90$		0			0
$91$	0.451057	+	0.327712i	0.451057	+	0.327712i
$92$	0.951057	+	1.30902i	0.951057	+	1.30902i
$93$		0			0
$94$	1.80902	+	0.587785i	1.80902	+	0.587785i
$95$	0.500000	−	0.363271i	0.500000	−	0.363271i
$96$		0			0
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$	0.902113			0.902113
$99$		0			0
$100$	−1.00000			−1.00000
$101$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$101$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$102$		0			0
$103$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.453990	−	0.891007i	$-0.350000\pi$
$103$	1.44168	−	1.04744i	1.44168	−	1.04744i	0.987688	−	0.156434i	$-0.0500000\pi$
$104$	−1.69480	−	0.550672i	−1.69480	−	0.550672i
$105$		0			0
$106$	0.690983	+	0.951057i	0.690983	+	0.951057i
$107$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$107$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$111$		0			0
$112$	0.297556	−	0.0966818i	0.297556	−	0.0966818i
$113$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$113$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$114$		0			0
$115$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$116$		0			0
$117$		0			0
$118$	−0.533698	+	0.734572i	−0.533698	+	0.734572i
$119$		0			0
$120$		0			0
$121$	0.951057	+	0.309017i	0.951057	+	0.309017i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$126$		0			0
$127$	0.863541	+	0.280582i	0.863541	+	0.280582i	0.707107	−	0.707107i	$-0.250000\pi$
$127$	0.863541	+	0.280582i	0.863541	+	0.280582i	0.156434	+	0.987688i	$0.450000\pi$
$128$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$129$		0			0
$130$	−0.550672	−	1.69480i	−0.550672	−	1.69480i
$131$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$131$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$	−0.193364			−0.193364
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$137$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$138$		0			0
$139$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$139$	−0.690983	−	0.951057i	−0.690983	−	0.951057i	−1.00000			$\pi$
$140$	0.253116	+	0.183900i	0.253116	+	0.183900i
$141$		0			0
$142$		0			0
$143$	−1.76007	+	0.278768i	−1.76007	+	0.278768i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.280582	−	0.863541i	0.280582	−	0.863541i
$149$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$149$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$150$		0			0
$151$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$151$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$152$	0.587785	−	0.190983i	0.587785	−	0.190983i
$153$		0			0
$154$	0.221232	−	0.221232i	0.221232	−	0.221232i
$155$		0			0
$156$		0			0
$157$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	0.453990	−	0.891007i	$-0.350000\pi$
$157$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$161$	0.409551	−	0.297556i	0.409551	−	0.297556i
$162$		0			0
$163$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$163$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$164$	−1.97538			−1.97538
$165$		0			0
$166$		0			0
$167$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$167$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$168$		0			0
$169$	1.76007	−	1.27877i	1.76007	−	1.27877i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$173$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$174$		0			0
$175$			0.312869i			0.312869i
$176$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$177$		0			0
$178$	−1.87869	+	0.610425i	−1.87869	+	0.610425i
$179$	1.16110	−	1.59811i	1.16110	−	1.59811i	0.453990	−	0.891007i	$-0.350000\pi$
$179$	1.16110	−	1.59811i	1.16110	−	1.59811i	0.707107	−	0.707107i	$-0.250000\pi$
$180$		0			0
$181$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$181$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$182$	−0.172288	+	0.530249i	−0.172288	+	0.530249i
$183$		0			0
$184$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$185$	0.863541	−	0.280582i	0.863541	−	0.280582i
$186$		0			0
$187$		0			0
$188$			1.90211i			1.90211i
$189$		0			0
$190$	0.500000	+	0.363271i	0.500000	+	0.363271i
$191$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$191$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$192$		0			0
$193$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$193$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$194$		0			0
$195$		0			0
$196$	0.278768	+	0.857960i	0.278768	+	0.857960i
$197$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$197$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.16110	−	1.59811i	−1.16110	−	1.59811i
$206$	1.44168	+	1.04744i	1.44168	+	1.04744i
$207$		0			0
$208$		−	1.78201i		−	1.78201i
$209$	0.437016	−	0.437016i	0.437016	−	0.437016i
$210$		0			0
$211$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$211$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$212$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$221$		0			0
$222$		0			0
$223$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.891007	−	0.453990i	$-0.850000\pi$
$223$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$224$	0.183900	+	0.253116i	0.183900	+	0.253116i
$225$		0			0
$226$		0			0
$227$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$227$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$228$		0			0
$229$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$229$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$230$	−1.61803			−1.61803
$231$		0			0
$232$		0			0
$233$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$233$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$234$		0			0
$235$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$236$	−0.863541	−	0.280582i	−0.863541	−	0.280582i
$237$		0			0
$238$		0			0
$239$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$239$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$240$		0			0
$241$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$241$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$242$			1.00000i			1.00000i
$243$		0			0
$244$		0			0
$245$	−0.530249	+	0.729825i	−0.530249	+	0.729825i
$246$		0			0
$247$	−0.340334	+	1.04744i	−0.340334	+	1.04744i
$248$		0			0
$249$		0			0
$250$	0.587785	−	0.809017i	0.587785	−	0.809017i
$251$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.891007	−	0.453990i	$-0.150000\pi$
$251$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.987688	−	0.156434i	$-0.0500000\pi$
$252$		0			0
$253$	−0.253116	+	1.59811i	−0.253116	+	1.59811i
$254$			0.907981i			0.907981i
$255$		0			0
$256$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$257$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$257$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$258$		0			0
$259$	−0.270175	−	0.0877853i	−0.270175	−	0.0877853i
$260$	1.44168	−	1.04744i	1.44168	−	1.04744i
$261$		0			0
$262$	−0.437016	−	1.34500i	−0.437016	−	1.34500i
$263$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$263$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$264$		0			0
$265$	−1.17557			−1.17557
$266$	−0.0597526	−	0.183900i	−0.0597526	−	0.183900i
$267$		0			0
$268$		0			0
$269$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$269$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$270$		0			0
$271$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$271$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$276$		0			0
$277$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.891007	−	0.453990i	$-0.150000\pi$
$277$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.987688	−	0.156434i	$-0.0500000\pi$
$278$	0.690983	−	0.951057i	0.690983	−	0.951057i
$279$		0			0
$280$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i
$281$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$281$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$282$		0			0
$283$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$283$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$284$		0			0
$285$		0			0
$286$	−0.809017	−	1.58779i	−0.809017	−	1.58779i
$287$			0.618034i			0.618034i
$288$		0			0
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$293$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$294$		0			0
$295$	−0.280582	−	0.863541i	−0.280582	−	0.863541i
$296$	0.907981			0.907981
$297$		0			0
$298$		0			0
$299$	−0.891007	−	2.74224i	−0.891007	−	2.74224i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.363271	+	0.500000i	0.363271	+	0.500000i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	0.278768	+	0.142040i	0.278768	+	0.142040i
$309$		0			0
$310$		0			0
$311$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$311$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$	0.0966818	−	0.297556i	0.0966818	−	0.297556i
$315$		0			0
$316$		0			0
$317$	−0.587785	+	0.190983i	−0.587785	+	0.190983i	−0.587785	−	0.809017i	$-0.700000\pi$
$317$	−0.587785	+	0.190983i	−0.587785	+	0.190983i			1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		−	1.00000i		−	1.00000i
$321$		0			0
$322$	0.409551	+	0.297556i	0.409551	+	0.297556i
$323$		0			0
$324$		0			0
$325$	1.69480	+	0.550672i	1.69480	+	0.550672i
$326$		0			0
$327$		0			0
$328$	−0.610425	−	1.87869i	−0.610425	−	1.87869i
$329$	0.595112			0.595112
$330$		0			0
$331$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$331$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$332$		0			0
$333$		0			0
$334$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$335$		0			0
$336$		0			0
$337$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$337$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$338$	1.76007	+	1.27877i	1.76007	+	1.27877i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.565985	−	0.183900i	0.565985	−	0.183900i
$344$		0			0
$345$		0			0
$346$	0.500000	−	1.53884i	0.500000	−	1.53884i
$347$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$347$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$348$		0			0
$349$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$349$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$350$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i
$351$		0			0
$352$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$	−1.16110	−	1.59811i	−1.16110	−	1.59811i
$357$		0			0
$358$	1.87869	+	0.610425i	1.87869	+	0.610425i
$359$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$359$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$360$		0			0
$361$	0.190983	+	0.587785i	0.190983	+	0.587785i
$362$		0			0
$363$		0			0
$364$	−0.557537			−0.557537
$365$		0			0
$366$		0			0
$367$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$367$	−1.14412	+	0.831254i	−1.14412	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$368$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$369$		0			0
$370$	0.533698	+	0.734572i	0.533698	+	0.734572i
$371$	0.297556	+	0.216187i	0.297556	+	0.216187i
$372$		0			0
$373$			0.907981i			0.907981i	0.891007	+	0.453990i	$0.150000\pi$
$373$			0.907981i			0.907981i	−0.891007	+	0.453990i	$0.850000\pi$
$374$		0			0
$375$		0			0
$376$	−1.80902	+	0.587785i	−1.80902	+	0.587785i
$377$		0			0
$378$		0			0
$379$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$379$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$380$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$381$		0			0
$382$		0			0
$383$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$383$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$384$		0			0
$385$	0.0489435	+	0.309017i	0.0489435	+	0.309017i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$389$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$390$		0			0
$391$		0			0
$392$	−0.729825	+	0.530249i	−0.729825	+	0.530249i
$393$		0			0
$394$	0.363271	+	1.11803i	0.363271	+	1.11803i
$395$		0			0
$396$		0			0
$397$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$397$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$398$		0			0
$399$		0			0
$400$	0.809017	−	0.587785i	0.809017	−	0.587785i
$401$	−0.863541	−	0.280582i	−0.863541	−	0.280582i	−0.156434	−	0.987688i	$-0.550000\pi$
$401$	−0.863541	−	0.280582i	−0.863541	−	0.280582i	−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.809017	−	0.412215i	0.809017	−	0.412215i
$408$		0			0
$409$	1.80902	−	0.587785i	1.80902	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$409$	1.80902	−	0.587785i	1.80902	−	0.587785i	1.00000			$0$
$410$	1.16110	−	1.59811i	1.16110	−	1.59811i
$411$		0			0
$412$	−0.550672	+	1.69480i	−0.550672	+	1.69480i
$413$	−0.0877853	+	0.270175i	−0.0877853	+	0.270175i
$414$		0			0
$415$		0			0
$416$	1.69480	−	0.550672i	1.69480	−	0.550672i
$417$		0			0
$418$	0.550672	+	0.280582i	0.550672	+	0.280582i
$419$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$419$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$420$		0			0
$421$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$421$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$422$		0			0
$423$		0			0
$424$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$431$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$432$		0			0
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.809017	+	0.587785i	0.809017	+	0.587785i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$441$		0			0
$442$		0			0
$443$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$443$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$444$		0			0
$445$	0.610425	−	1.87869i	0.610425	−	1.87869i
$446$	0.610425	−	1.87869i	0.610425	−	1.87869i
$447$		0			0
$448$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$449$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$449$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.987688	+	0.156434i	$0.0500000\pi$
$450$		0			0
$451$	−1.39680	−	1.39680i	−1.39680	−	1.39680i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.327712	−	0.451057i	−0.327712	−	0.451057i
$456$		0			0
$457$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$457$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$458$		0			0
$459$		0			0
$460$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$463$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$467$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$468$		0			0
$469$		0			0
$470$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$471$		0			0
$472$		−	0.907981i		−	0.907981i
$473$		0			0
$474$		0			0
$475$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$479$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$480$		0			0
$481$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$482$	1.53884	−	0.500000i	1.53884	−	0.500000i
$483$		0			0
$484$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$485$		0			0
$486$		0			0
$487$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$487$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$488$		0			0
$489$		0			0
$490$	−0.857960	−	0.278768i	−0.857960	−	0.278768i
$491$	−1.44168	+	1.04744i	−1.44168	+	1.04744i	−0.453990	+	0.891007i	$0.650000\pi$
$491$	−1.44168	+	1.04744i	−1.44168	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$492$		0			0
$493$		0			0
$494$	−1.10134			−1.10134
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$499$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$500$	0.951057	+	0.309017i	0.951057	+	0.309017i
$501$		0			0
$502$	1.16110	+	1.59811i	1.16110	+	1.59811i
$503$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$503$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$504$		0			0
$505$		0			0
$506$	−1.59811	+	0.253116i	−1.59811	+	0.253116i
$507$		0			0
$508$	−0.863541	+	0.280582i	−0.863541	+	0.280582i
$509$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$509$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$510$		0			0
$511$		0			0
$512$	0.309017	−	0.951057i	0.309017	−	0.951057i
$513$		0			0
$514$		0			0
$515$	−1.69480	+	0.550672i	−1.69480	+	0.550672i
$516$		0			0
$517$	−1.34500	+	1.34500i	−1.34500	+	1.34500i
$518$		−	0.284079i		−	0.284079i
$519$		0			0
$520$	1.44168	+	1.04744i	1.44168	+	1.04744i
$521$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.707107	−	0.707107i	$-0.750000\pi$
$521$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.453990	−	0.891007i	$-0.650000\pi$
$522$		0			0
$523$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$523$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$524$	1.14412	−	0.831254i	1.14412	−	0.831254i
$525$		0			0
$526$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$527$		0			0
$528$		0			0
$529$	−1.61803			−1.61803
$530$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$531$		0			0
$532$	0.156434	−	0.113656i	0.156434	−	0.113656i
$533$	3.34786	+	1.08779i	3.34786	+	1.08779i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.409551	+	0.803789i	−0.409551	+	0.803789i
$540$		0			0
$541$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$541$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$547$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$548$		0			0
$549$		0			0
$550$	0.453990	−	0.891007i	0.453990	−	0.891007i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.16110	+	1.59811i	1.16110	+	1.59811i
$555$		0			0
$556$	1.11803	+	0.363271i	1.11803	+	0.363271i
$557$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$557$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$558$		0			0
$559$		0			0
$560$	−0.312869			−0.312869
$561$		0			0
$562$	−1.41421			−1.41421
$563$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$563$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.253116	+	0.183900i	0.253116	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$569$	0.253116	+	0.183900i	0.253116	+	0.183900i	−0.453990	+	0.891007i	$0.650000\pi$
$570$		0			0
$571$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$571$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$572$	1.26007	−	1.26007i	1.26007	−	1.26007i
$573$		0			0
$574$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$575$	0.951057	−	1.30902i	0.951057	−	1.30902i
$576$		0			0
$577$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$577$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$578$	0.309017	−	0.951057i	0.309017	−	0.951057i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.16110	+	0.183900i	−1.16110	+	0.183900i
$584$		0			0
$585$		0			0
$586$	0.951057	+	0.690983i	0.951057	+	0.690983i
$587$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$587$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$588$		0			0
$589$		0			0
$590$	0.734572	−	0.533698i	0.734572	−	0.533698i
$591$		0			0
$592$	0.280582	+	0.863541i	0.280582	+	0.863541i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	2.33269	−	1.69480i	2.33269	−	1.69480i
$599$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$599$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$600$		0			0
$601$	0.363271	+	0.500000i	0.363271	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$601$	0.363271	+	0.500000i	0.363271	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$606$		0			0
$607$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$607$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$608$	−0.363271	+	0.500000i	−0.363271	+	0.500000i
$609$		0			0
$610$		0			0
$611$	1.04744	−	3.22369i	1.04744	−	3.22369i
$612$		0			0
$613$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$613$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$614$		0			0
$615$		0			0
$616$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$619$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$624$		0			0
$625$	0.309017	+	0.951057i	0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$	0.312869			0.312869
$629$		0			0
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$635$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$636$		0			0
$637$		−	1.60758i		−	1.60758i
$638$		0			0
$639$		0			0
$640$	0.951057	−	0.309017i	0.951057	−	0.309017i
$641$	−1.04744	+	1.44168i	−1.04744	+	1.44168i	−0.156434	+	0.987688i	$0.550000\pi$
$641$	−1.04744	+	1.44168i	−1.04744	+	1.44168i	−0.891007	+	0.453990i	$0.850000\pi$
$642$		0			0
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$	−0.156434	+	0.481456i	−0.156434	+	0.481456i
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$		0			0
$649$	−0.412215	−	0.809017i	−0.412215	−	0.809017i
$650$			1.78201i			1.78201i
$651$		0			0
$652$		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
$655$	1.34500	+	0.437016i	1.34500	+	0.437016i
$656$	1.59811	−	1.16110i	1.59811	−	1.16110i
$657$		0			0
$658$	0.183900	+	0.565985i	0.183900	+	0.565985i
$659$	1.97538			1.97538			0.987688	−	0.156434i	$-0.0500000\pi$
$659$	1.97538			1.97538			0.987688	+	0.156434i	$0.0500000\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$663$		0			0
$664$		0			0
$665$	0.183900	+	0.0597526i	0.183900	+	0.0597526i
$666$		0			0
$667$		0			0
$668$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$673$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$674$		0			0
$675$		0			0
$676$	−0.672288	+	2.06909i	−0.672288	+	2.06909i
$677$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$677$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	0.349798	+	0.481456i	0.349798	+	0.481456i
$687$		0			0
$688$		0			0
$689$	1.69480	−	1.23134i	1.69480	−	1.23134i
$690$		0			0
$691$	0.363271	+	1.11803i	0.363271	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$691$	0.363271	+	1.11803i	0.363271	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$692$	1.61803			1.61803
$693$		0			0
$694$		0			0
$695$	0.363271	+	1.11803i	0.363271	+	1.11803i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.183900	−	0.253116i	−0.183900	−	0.253116i
$701$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$701$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$702$		0			0
$703$		−	0.561163i		−	0.561163i
$704$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$709$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$710$		0			0
$711$		0			0
$712$	1.16110	−	1.59811i	1.16110	−	1.59811i
$713$		0			0
$714$		0			0
$715$	1.76007	+	0.278768i	1.76007	+	0.278768i
$716$			1.97538i			1.97538i
$717$		0			0
$718$		0			0
$719$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$719$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$720$		0			0
$721$	0.530249	+	0.172288i	0.530249	+	0.172288i
$722$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$727$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$728$	−0.172288	−	0.530249i	−0.172288	−	0.530249i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$733$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$734$	−1.14412	−	0.831254i	−1.14412	−	0.831254i
$735$		0			0
$736$		−	1.61803i		−	1.61803i
$737$		0			0
$738$		0			0
$739$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$739$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$740$	−0.533698	+	0.734572i	−0.533698	+	0.734572i
$741$		0			0
$742$	−0.113656	+	0.349798i	−0.113656	+	0.349798i
$743$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$743$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$744$		0			0
$745$		0			0
$746$	−0.863541	+	0.280582i	−0.863541	+	0.280582i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$751$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$752$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.550672	+	1.69480i	0.550672	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$757$	0.550672	+	1.69480i	0.550672	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$758$	1.61803			1.61803
$759$		0			0
$760$	−0.618034			−0.618034
$761$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$761$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.690983	−	0.951057i	−0.690983	−	0.951057i
$767$	1.30902	+	0.951057i	1.30902	+	0.951057i
$768$		0			0
$769$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$769$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$770$	−0.278768	+	0.142040i	−0.278768	+	0.142040i
$771$		0			0
$772$		0			0
$773$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$773$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.16110	+	0.377263i	−1.16110	+	0.377263i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.729825	−	0.530249i	−0.729825	−	0.530249i
$785$	0.183900	+	0.253116i	0.183900	+	0.253116i
$786$		0			0
$787$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$787$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$788$	−0.951057	+	0.690983i	−0.951057	+	0.690983i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$795$		0			0
$796$		0			0
$797$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$797$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	0.809017	+	0.587785i	0.809017	+	0.587785i
$801$		0			0
$802$		−	0.907981i		−	0.907981i
$803$		0			0
$804$		0			0
$805$	−0.481456	+	0.156434i	−0.481456	+	0.156434i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.550672	+	1.69480i	−0.550672	+	1.69480i	0.156434	+	0.987688i	$0.450000\pi$
$809$	−0.550672	+	1.69480i	−0.550672	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$811$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	−1.00000			$1.00000\pi$
$812$		0			0
$813$		0			0
$814$	0.642040	+	0.642040i	0.642040	+	0.642040i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.11803	+	1.53884i	1.11803	+	1.53884i
$819$		0			0
$820$	1.87869	+	0.610425i	1.87869	+	0.610425i
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$	0.550672	+	1.69480i	0.550672	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$823$	0.550672	+	1.69480i	0.550672	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$824$	−1.78201			−1.78201
$825$		0			0
$826$	−0.284079			−0.284079
$827$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$827$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$828$		0			0
$829$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$829$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$830$		0			0
$831$		0			0
$832$	1.04744	+	1.44168i	1.04744	+	1.44168i
$833$		0			0
$834$		0			0
$835$		−	1.90211i		−	1.90211i
$836$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i
$837$		0			0
$838$	−1.69480	+	0.550672i	−1.69480	+	0.550672i
$839$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$839$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.06909	+	0.672288i	−2.06909	+	0.672288i
$846$		0			0
$847$	0.0966818	+	0.297556i	0.0966818	+	0.297556i
$848$		−	1.17557i		−	1.17557i
$849$		0			0
$850$		0			0
$851$	0.863541	+	1.18856i	0.863541	+	1.18856i
$852$		0			0
$853$	−1.87869	−	0.610425i	−1.87869	−	0.610425i	−0.987688	−	0.156434i	$-0.950000\pi$
$853$	−1.87869	−	0.610425i	−1.87869	−	0.610425i	−0.891007	−	0.453990i	$-0.850000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$859$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$863$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$864$		0			0
$865$	0.951057	+	1.30902i	0.951057	+	1.30902i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$875$	0.0966818	−	0.297556i	0.0966818	−	0.297556i
$876$		0			0
$877$	0.533698	−	0.734572i	0.533698	−	0.734572i	−0.453990	−	0.891007i	$-0.650000\pi$
$877$	0.533698	−	0.734572i	0.533698	−	0.734572i	0.987688	+	0.156434i	$0.0500000\pi$
$878$		0			0
$879$		0			0
$880$	0.707107	−	0.707107i	0.707107	−	0.707107i
$881$		−	0.312869i		−	0.312869i	−0.987688	−	0.156434i	$-0.950000\pi$
$881$		−	0.312869i		−	0.312869i	0.987688	−	0.156434i	$-0.0500000\pi$
$882$		0			0
$883$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$883$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$887$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$888$		0			0
$889$	0.0877853	+	0.270175i	0.0877853	+	0.270175i
$890$	1.97538			1.97538
$891$		0			0
$892$	1.97538			1.97538
$893$	0.363271	+	1.11803i	0.363271	+	1.11803i
$894$		0			0
$895$	−1.59811	+	1.16110i	−1.59811	+	1.16110i
$896$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i
$897$		0			0
$898$	1.04744	+	1.44168i	1.04744	+	1.44168i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	0.896802	−	1.76007i	0.896802	−	1.76007i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$907$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$908$		0			0
$909$		0			0
$910$	0.327712	−	0.451057i	0.327712	−	0.451057i
$911$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$911$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.260074	−	0.357960i	−0.260074	−	0.357960i
$918$		0			0
$919$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$919$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$920$	1.30902	−	0.951057i	1.30902	−	0.951057i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.907981			−0.907981
$926$	−0.280582	−	0.863541i	−0.280582	−	0.863541i
$927$		0			0
$928$		0			0
$929$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$929$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i	−0.453990	+	0.891007i	$0.650000\pi$
$930$		0			0
$931$	0.327712	+	0.451057i	0.327712	+	0.451057i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$937$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$938$		0			0
$939$		0			0
$940$	0.587785	−	1.80902i	0.587785	−	1.80902i
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		0			0
$943$	1.87869	−	2.58580i	1.87869	−	2.58580i
$944$	0.863541	−	0.280582i	0.863541	−	0.280582i
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$	−0.363271	−	0.500000i	−0.363271	−	0.500000i
$951$		0			0
$952$		0			0
$953$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$953$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$962$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$963$		0			0
$964$	0.951057	+	1.30902i	0.951057	+	1.30902i
$965$		0			0
$966$		0			0
$967$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$967$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$968$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$969$		0			0
$970$		0			0
$971$	−0.183900	+	0.253116i	−0.183900	+	0.253116i	−0.891007	−	0.453990i	$-0.850000\pi$
$971$	−0.183900	+	0.253116i	−0.183900	+	0.253116i	0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$	0.113656	−	0.349798i	0.113656	−	0.349798i
$974$	0.437016	−	1.34500i	0.437016	−	1.34500i
$975$		0			0
$976$		0			0
$977$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$977$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$978$		0			0
$979$	0.309017	−	1.95106i	0.309017	−	1.95106i
$980$		−	0.902113i		−	0.902113i
$981$		0			0
$982$	−1.44168	−	1.04744i	−1.44168	−	1.04744i
$983$	0.690983	+	0.951057i	0.690983	+	0.951057i	1.00000			$0$
$983$	0.690983	+	0.951057i	0.690983	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$984$		0			0
$985$	−1.11803	−	0.363271i	−1.11803	−	0.363271i
$986$		0			0
$987$		0			0
$988$	−0.340334	−	1.04744i	−0.340334	−	1.04744i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$997$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$998$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$999$		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	3960.1.eg.a.3779.2	yes	16
3.2	odd	2		3960.1.eg.b.3779.4	yes	16
5.4	even	2		3960.1.eg.b.3779.3	yes	16
8.3	odd	2		3960.1.eg.b.3779.3	yes	16
11.2	odd	10		3960.1.eg.b.1619.4	yes	16
15.14	odd	2	inner	3960.1.eg.a.3779.1	yes	16
24.11	even	2	inner	3960.1.eg.a.3779.1	yes	16
33.2	even	10	inner	3960.1.eg.a.1619.2	yes	16
40.19	odd	2	CM	3960.1.eg.a.3779.2	yes	16
55.24	odd	10	inner	3960.1.eg.a.1619.1	✓	16
88.35	even	10	inner	3960.1.eg.a.1619.1	✓	16
120.59	even	2		3960.1.eg.b.3779.4	yes	16
165.134	even	10		3960.1.eg.b.1619.3	yes	16
264.35	odd	10		3960.1.eg.b.1619.3	yes	16
440.299	even	10		3960.1.eg.b.1619.4	yes	16
1320.299	odd	10	inner	3960.1.eg.a.1619.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.1.eg.a.1619.1	✓	16	55.24	odd	10	inner
3960.1.eg.a.1619.1	✓	16	88.35	even	10	inner
3960.1.eg.a.1619.2	yes	16	33.2	even	10	inner
3960.1.eg.a.1619.2	yes	16	1320.299	odd	10	inner
3960.1.eg.a.3779.1	yes	16	15.14	odd	2	inner
3960.1.eg.a.3779.1	yes	16	24.11	even	2	inner
3960.1.eg.a.3779.2	yes	16	1.1	even	1	trivial
3960.1.eg.a.3779.2	yes	16	40.19	odd	2	CM
3960.1.eg.b.1619.3	yes	16	165.134	even	10
3960.1.eg.b.1619.3	yes	16	264.35	odd	10
3960.1.eg.b.1619.4	yes	16	11.2	odd	10
3960.1.eg.b.1619.4	yes	16	440.299	even	10
3960.1.eg.b.3779.3	yes	16	5.4	even	2
3960.1.eg.b.3779.3	yes	16	8.3	odd	2
3960.1.eg.b.3779.4	yes	16	3.2	odd	2
3960.1.eg.b.3779.4	yes	16	120.59	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Varieties

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Groups

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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	3960.1.eg.a.3779.2

Level	$3960$
Weight	$1$
Character	3960.3779
Analytic conductor	$1.976$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{20}$
CM discriminant	-40
Inner twists	$8$