LMFDB - Embedded newform 3872.1.r.b.481.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3872,1,Mod(161,3872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3872, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("3872.161");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3872 = 2^{5} \cdot 11^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3872.r (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:	$1.93237972891$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	16.0.6879707136000000000000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1$ x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of 12.0.18698185645162496.1

Embedding invariants

Embedding label			481.2
Root			$1.83730 - 0.596975i$ of defining polynomial
Character	$\chi$	$=$	3872.481
Dual form			3872.1.r.b.161.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.40126 - 1.01807i) q^{5} +(0.809017 - 0.587785i) q^{9} +(1.13551 + 1.56290i) q^{13} +(0.304260 - 0.418778i) q^{17} +(0.618034 + 1.90211i) q^{25} +(-0.492303 - 0.159959i) q^{29} +(0.309017 - 0.951057i) q^{37} +(1.83730 - 0.596975i) q^{41} -1.73205 q^{45} +(-0.809017 - 0.587785i) q^{49} +(-0.809017 + 0.587785i) q^{53} +(0.831254 - 1.14412i) q^{61} -3.34607i q^{65} +(1.34500 + 0.437016i) q^{73} +(0.309017 - 0.951057i) q^{81} +(-0.852694 + 0.277057i) q^{85} -1.00000 q^{89} +(1.40126 - 1.01807i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 4 q^{9} - 8 q^{25} - 4 q^{37} - 4 q^{49} - 4 q^{53} - 4 q^{81} - 16 q^{89}+O(q^{100})$ 16 * q + 4 * q^9 - 8 * q^25 - 4 * q^37 - 4 * q^49 - 4 * q^53 - 4 * q^81 - 16 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times$ .

$n$	$485$	$1695$	$2785$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{10}\right)$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$3$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$4$		0			0
$5$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$5$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$6$		0			0
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$		0			0
$9$	0.809017	−	0.587785i	0.809017	−	0.587785i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$	1.13551	+	1.56290i	1.13551	+	1.56290i	0.777146	+	0.629320i	$0.216667\pi$
$13$	1.13551	+	1.56290i	1.13551	+	1.56290i	0.358368	+	0.933580i	$0.383333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.304260	−	0.418778i	0.304260	−	0.418778i	−0.629320	−	0.777146i	$-0.716667\pi$
$17$	0.304260	−	0.418778i	0.304260	−	0.418778i	0.933580	+	0.358368i	$0.116667\pi$
$18$		0			0
$19$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$19$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	0.618034	+	1.90211i	0.618034	+	1.90211i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.492303	−	0.159959i	−0.492303	−	0.159959i	0.0523360	−	0.998630i	$-0.483333\pi$
$29$	−0.492303	−	0.159959i	−0.492303	−	0.159959i	−0.544639	+	0.838671i	$0.683333\pi$
$30$		0			0
$31$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$31$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$37$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.83730	−	0.596975i	1.83730	−	0.596975i	0.838671	−	0.544639i	$-0.183333\pi$
$41$	1.83730	−	0.596975i	1.83730	−	0.596975i	0.998630	−	0.0523360i	$-0.0166667\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−1.73205			−1.73205
$46$		0			0
$47$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$47$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$48$		0			0
$49$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$53$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$59$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$60$		0			0
$61$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$61$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.34607i		−	3.34607i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$71$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$72$		0			0
$73$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$73$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$		0			0
$81$	0.309017	−	0.951057i	0.309017	−	0.951057i
$82$		0			0
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$		0			0
$85$	−0.852694	+	0.277057i	−0.852694	+	0.277057i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$97$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$101$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$102$		0			0
$103$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$103$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$107$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$108$		0			0
$109$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$109$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$113$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.83730	+	0.596975i	1.83730	+	0.596975i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.535233	−	1.64728i	0.535233	−	1.64728i
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$137$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$138$		0			0
$139$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$139$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.526994	+	0.725345i	0.526994	+	0.725345i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.13551	−	1.56290i	1.13551	−	1.56290i	0.358368	−	0.933580i	$-0.383333\pi$
$149$	1.13551	−	1.56290i	1.13551	−	1.56290i	0.777146	−	0.629320i	$-0.216667\pi$
$150$		0			0
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$		0			0
$153$		−	0.517638i		−	0.517638i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$157$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$163$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		0			0
$169$	−0.844250	+	2.59833i	−0.844250	+	2.59833i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$173$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$179$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$180$		0			0
$181$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$181$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$191$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$192$		0			0
$193$	−1.13551	+	1.56290i	−1.13551	+	1.56290i	−0.358368	+	0.933580i	$0.616667\pi$
$193$	−1.13551	+	1.56290i	−1.13551	+	1.56290i	−0.777146	+	0.629320i	$0.783333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$197$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−3.18230	−	1.03399i	−3.18230	−	1.03399i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$211$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.00000			1.00000
$222$		0			0
$223$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$223$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$224$		0			0
$225$	1.61803	+	1.17557i	1.61803	+	1.17557i
$226$		0			0
$227$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$227$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$228$		0			0
$229$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$229$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.304260	+	0.418778i	0.304260	+	0.418778i	0.933580	−	0.358368i	$-0.116667\pi$
$233$	0.304260	+	0.418778i	0.304260	+	0.418778i	−0.629320	+	0.777146i	$0.716667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$241$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.535233	+	1.64728i	0.535233	+	1.64728i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$251$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$257$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.492303	+	0.159959i	−0.492303	+	0.159959i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	1.73205			1.73205
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$269$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$270$		0			0
$271$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$271$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.304260	+	0.418778i	0.304260	+	0.418778i	0.933580	−	0.358368i	$-0.116667\pi$
$277$	0.304260	+	0.418778i	0.304260	+	0.418778i	−0.629320	+	0.777146i	$0.716667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$281$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$282$		0			0
$283$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.226216	+	0.696222i	0.226216	+	0.696222i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.492303	+	0.159959i	0.492303	+	0.159959i	0.544639	−	0.838671i	$-0.316667\pi$
$293$	0.492303	+	0.159959i	0.492303	+	0.159959i	−0.0523360	+	0.998630i	$0.516667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.32960	+	0.756934i	−2.32960	+	0.756934i
$306$		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		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$313$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$317$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−2.27103	+	3.12580i	−2.27103	+	3.12580i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.83730	−	0.596975i	−1.83730	−	0.596975i	−0.998630	−	0.0523360i	$-0.983333\pi$
$337$	−1.83730	−	0.596975i	−1.83730	−	0.596975i	−0.838671	−	0.544639i	$-0.816667\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$347$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$348$		0			0
$349$	−1.83730	+	0.596975i	−1.83730	+	0.596975i	−0.838671	+	0.544639i	$0.816667\pi$
$349$	−1.83730	+	0.596975i	−1.83730	+	0.596975i	−0.998630	+	0.0523360i	$0.983333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$353$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.43977	−	1.98168i	−1.43977	−	1.98168i
$366$		0			0
$367$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$367$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$368$		0			0
$369$	1.13551	−	1.56290i	1.13551	−	1.56290i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$373$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$378$		0			0
$379$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$379$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$383$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$389$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$397$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.994522	+	0.104528i	$0.0333333\pi$
$401$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.40126	+	1.01807i	−1.40126	+	1.01807i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.13551	−	1.56290i	−1.13551	−	1.56290i	−0.777146	−	0.629320i	$-0.783333\pi$
$409$	−1.13551	−	1.56290i	−1.13551	−	1.56290i	−0.358368	−	0.933580i	$-0.616667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		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
$421$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	−0.743145	−	0.669131i	$-0.766667\pi$
$421$	−0.535233	−	1.64728i	−0.535233	−	1.64728i	0.207912	−	0.978148i	$-0.433333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.984606	+	0.319918i	0.984606	+	0.319918i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$		0			0
$433$	0.535233	−	1.64728i	0.535233	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$433$	0.535233	−	1.64728i	0.535233	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		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			0		0.951057	−	0.309017i	$-0.100000\pi$
$443$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$444$		0			0
$445$	1.40126	+	1.01807i	1.40126	+	1.01807i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$449$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.304260	+	0.418778i	−0.304260	+	0.418778i	−0.933580	−	0.358368i	$-0.883333\pi$
$457$	−0.304260	+	0.418778i	−0.304260	+	0.418778i	0.629320	+	0.777146i	$0.283333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$461$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$467$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$	1.83730	−	0.596975i	1.83730	−	0.596975i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.00000			−3.00000
$486$		0			0
$487$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$487$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$	−0.216775	+	0.157497i	−0.216775	+	0.157497i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$499$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$			2.44949i			2.44949i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$509$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$521$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$522$		0			0
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.00000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.01929	+	2.19364i	3.01929	+	2.19364i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$541$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.526994	−	0.725345i	0.526994	−	0.725345i
$546$		0			0
$547$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$547$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$548$		0			0
$549$		−	1.41421i		−	1.41421i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$557$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$563$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$564$		0			0
$565$	−0.927051	+	2.85317i	−0.927051	+	2.85317i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$569$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.994522	+	0.104528i	$0.0333333\pi$
$577$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.96677	−	2.70702i	−1.96677	−	2.70702i
$586$		0			0
$587$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$587$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$593$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$599$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$600$		0			0
$601$	0.492303	+	0.159959i	0.492303	+	0.159959i	0.544639	−	0.838671i	$-0.316667\pi$
$601$	0.492303	+	0.159959i	0.492303	+	0.159959i	−0.0523360	+	0.998630i	$0.516667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.83730	−	0.596975i	1.83730	−	0.596975i	0.838671	−	0.544639i	$-0.183333\pi$
$613$	1.83730	−	0.596975i	1.83730	−	0.596975i	0.998630	−	0.0523360i	$-0.0166667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$617$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$619$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.304260	−	0.418778i	−0.304260	−	0.418778i
$630$		0			0
$631$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$631$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	1.93185i		−	1.93185i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$641$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$642$		0			0
$643$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$643$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$647$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$653$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.34500	−	0.437016i	1.34500	−	0.437016i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$661$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$673$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.304260	+	0.418778i	−0.304260	+	0.418778i	−0.933580	−	0.358368i	$-0.883333\pi$
$677$	−0.304260	+	0.418778i	−0.304260	+	0.418778i	0.629320	+	0.777146i	$0.283333\pi$
$678$		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			0
$687$		0			0
$688$		0			0
$689$	−1.83730	−	0.596975i	−1.83730	−	0.596975i
$690$		0			0
$691$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$691$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.309017	−	0.951057i	0.309017	−	0.951057i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.83730	+	0.596975i	−1.83730	+	0.596975i	−0.838671	+	0.544639i	$0.816667\pi$
$701$	−1.83730	+	0.596975i	−1.83730	+	0.596975i	−0.998630	+	0.0523360i	$0.983333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$709$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$719$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1.03528i		−	1.03528i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.492303	+	0.159959i	0.492303	+	0.159959i	0.544639	−	0.838671i	$-0.316667\pi$
$733$	0.492303	+	0.159959i	0.492303	+	0.159959i	−0.0523360	+	0.998630i	$0.516667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$739$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$	−3.18230	+	1.03399i	−3.18230	+	1.03399i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$751$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.406737	+	0.913545i	$0.633333\pi$
$757$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.13551	+	1.56290i	1.13551	+	1.56290i	0.777146	+	0.629320i	$0.216667\pi$
$761$	1.13551	+	1.56290i	1.13551	+	1.56290i	0.358368	+	0.933580i	$0.383333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.526994	+	0.725345i	−0.526994	+	0.725345i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$769$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$773$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$787$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.73205			2.73205
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$797$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$809$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$810$		0			0
$811$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$811$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$821$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$822$		0			0
$823$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$823$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$		0			0
$829$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$829$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.492303	+	0.159959i	−0.492303	+	0.159959i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$839$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$840$		0			0
$841$	−0.592242	−	0.430289i	−0.592242	−	0.430289i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.82831	−	2.78143i	3.82831	−	2.78143i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.304260	−	0.418778i	0.304260	−	0.418778i	−0.629320	−	0.777146i	$-0.716667\pi$
$853$	0.304260	−	0.418778i	0.304260	−	0.418778i	0.933580	+	0.358368i	$0.116667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$857$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$858$		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			0
$863$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$863$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$864$		0			0
$865$	−2.32960	−	0.756934i	−2.32960	−	0.756934i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.535233	−	1.64728i	0.535233	−	1.64728i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.492303	+	0.159959i	−0.492303	+	0.159959i	−0.544639	−	0.838671i	$-0.683333\pi$
$877$	−0.492303	+	0.159959i	−0.492303	+	0.159959i	0.0523360	+	0.998630i	$0.483333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$881$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$882$		0			0
$883$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$883$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			0.517638i			0.517638i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.535233	+	1.64728i	0.535233	+	1.64728i
$906$		0			0
$907$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$907$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$908$		0			0
$909$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$910$		0			0
$911$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$911$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$919$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.00000			2.00000
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$929$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.13551	+	1.56290i	1.13551	+	1.56290i	0.777146	+	0.629320i	$0.216667\pi$
$937$	1.13551	+	1.56290i	1.13551	+	1.56290i	0.358368	+	0.933580i	$0.383333\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.13551	+	1.56290i	−1.13551	+	1.56290i	−0.358368	+	0.933580i	$0.616667\pi$
$941$	−1.13551	+	1.56290i	−1.13551	+	1.56290i	−0.777146	+	0.629320i	$0.783333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	0.844250	+	2.59833i	0.844250	+	2.59833i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.83730	+	0.596975i	1.83730	+	0.596975i	0.998630	+	0.0523360i	$0.0166667\pi$
$953$	1.83730	+	0.596975i	1.83730	+	0.596975i	0.838671	+	0.544639i	$0.183333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.18230	−	1.03399i	3.18230	−	1.03399i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$971$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.406737	+	0.913545i	$0.633333\pi$
$977$	−1.40126	+	1.01807i	−1.40126	+	1.01807i	−0.994522	+	0.104528i	$0.966667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.304260	+	0.418778i	0.304260	+	0.418778i
$982$		0			0
$983$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$983$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$984$		0			0
$985$	1.96677	−	2.70702i	1.96677	−	2.70702i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.83730	−	0.596975i	−1.83730	−	0.596975i	−0.998630	−	0.0523360i	$-0.983333\pi$
$997$	−1.83730	−	0.596975i	−1.83730	−	0.596975i	−0.838671	−	0.544639i	$-0.816667\pi$
$998$		0			0
$999$		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	3872.1.r.b.481.2		16
4.3	odd	2	CM	3872.1.r.b.481.2		16
11.2	odd	10		3872.1.h.b.2177.4	yes	4
11.3	even	5	inner	3872.1.r.b.3361.4		16
11.4	even	5	inner	3872.1.r.b.161.1		16
11.5	even	5	inner	3872.1.r.b.3137.3		16
11.6	odd	10	inner	3872.1.r.b.3137.4		16
11.7	odd	10	inner	3872.1.r.b.161.2		16
11.8	odd	10	inner	3872.1.r.b.3361.3		16
11.9	even	5		3872.1.h.b.2177.3	✓	4
11.10	odd	2	inner	3872.1.r.b.481.1		16
44.3	odd	10	inner	3872.1.r.b.3361.4		16
44.7	even	10	inner	3872.1.r.b.161.2		16
44.15	odd	10	inner	3872.1.r.b.161.1		16
44.19	even	10	inner	3872.1.r.b.3361.3		16
44.27	odd	10	inner	3872.1.r.b.3137.3		16
44.31	odd	10		3872.1.h.b.2177.3	✓	4
44.35	even	10		3872.1.h.b.2177.4	yes	4
44.39	even	10	inner	3872.1.r.b.3137.4		16
44.43	even	2	inner	3872.1.r.b.481.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3872.1.h.b.2177.3	✓	4	11.9	even	5
3872.1.h.b.2177.3	✓	4	44.31	odd	10
3872.1.h.b.2177.4	yes	4	11.2	odd	10
3872.1.h.b.2177.4	yes	4	44.35	even	10
3872.1.r.b.161.1		16	11.4	even	5	inner
3872.1.r.b.161.1		16	44.15	odd	10	inner
3872.1.r.b.161.2		16	11.7	odd	10	inner
3872.1.r.b.161.2		16	44.7	even	10	inner
3872.1.r.b.481.1		16	11.10	odd	2	inner
3872.1.r.b.481.1		16	44.43	even	2	inner
3872.1.r.b.481.2		16	1.1	even	1	trivial
3872.1.r.b.481.2		16	4.3	odd	2	CM
3872.1.r.b.3137.3		16	11.5	even	5	inner
3872.1.r.b.3137.3		16	44.27	odd	10	inner
3872.1.r.b.3137.4		16	11.6	odd	10	inner
3872.1.r.b.3137.4		16	44.39	even	10	inner
3872.1.r.b.3361.3		16	11.8	odd	10	inner
3872.1.r.b.3361.3		16	44.19	even	10	inner
3872.1.r.b.3361.4		16	11.3	even	5	inner
3872.1.r.b.3361.4		16	44.3	odd	10	inner

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

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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	3872.1.r.b.481.2

Level	$3872$
Weight	$1$
Character	3872.481
Analytic conductor	$1.932$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{12}$
CM discriminant	-4
Inner twists	$16$