LMFDB - Embedded newform 1287.1.bu.a.415.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,1,Mod(415,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

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

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

chi := DirichletCharacter("1287.415");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1287.bu (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:	$0.642296671259$
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:	$\Z[a_1, \ldots, a_{5}]$
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			415.2
Root			$-0.891007 - 0.453990i$ of defining polynomial
Character	$\chi$	$=$	1287.415
Dual form			1287.1.bu.a.766.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.253116 + 0.183900i) q^{2} +(-0.278768 + 0.857960i) q^{4} +(1.04744 - 1.44168i) q^{5} +(-0.183900 - 0.565985i) q^{8} +O(q^{10})$	\(q+(-0.253116 + 0.183900i) q^{2} +(-0.278768 + 0.857960i) q^{4} +(1.04744 - 1.44168i) q^{5} +(-0.183900 - 0.565985i) q^{8} +0.557537i q^{10} +(0.987688 - 0.156434i) q^{11} +(-0.587785 - 0.809017i) q^{13} +(-0.579192 - 0.420808i) q^{16} +(0.944910 + 1.30056i) q^{20} +(-0.221232 + 0.221232i) q^{22} +(-0.672288 - 2.06909i) q^{25} +(0.297556 + 0.0966818i) q^{26} +0.819101 q^{32} +(-1.00859 - 0.327712i) q^{40} +(-0.437016 - 1.34500i) q^{41} +1.90211i q^{43} +(-0.141122 + 0.891007i) q^{44} +(-0.297556 + 0.0966818i) q^{47} +(0.809017 + 0.587785i) q^{49} +(0.550672 + 0.400087i) q^{50} +(0.857960 - 0.278768i) q^{52} +(0.809017 - 1.58779i) q^{55} +(1.87869 + 0.610425i) q^{59} +(-0.951057 + 1.30902i) q^{61} +(0.371864 - 0.270175i) q^{64} -1.78201 q^{65} +(0.831254 - 1.14412i) q^{71} +(0.951057 + 1.30902i) q^{79} +(-1.21334 + 0.394238i) q^{80} +(0.357960 + 0.260074i) q^{82} +(-1.44168 - 1.04744i) q^{83} +(-0.349798 - 0.481456i) q^{86} +(-0.270175 - 0.530249i) q^{88} +1.41421i q^{89} +(0.0575365 - 0.0791922i) q^{94} -0.312869 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{4}+O(q^{10}) $ 16 * q - 4 * q^4	$ 16 q - 4 q^{4} - 16 q^{16} - 4 q^{22} + 4 q^{25} + 4 q^{49} + 20 q^{52} + 4 q^{55} - 16 q^{64} + 12 q^{82} - 20 q^{88}+O(q^{100}) $ 16 * q - 4 * q^4 - 16 * q^16 - 4 * q^22 + 4 * q^25 + 4 * q^49 + 20 * q^52 + 4 * q^55 - 16 * q^64 + 12 * q^82 - 20 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times$.

$n$	$496$	$937$	$1145$
$\chi(n)$	$-1$	$e\left(\frac{3}{10}\right)$	$1$

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 $

$2$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	0.453990	+	0.891007i	$0.350000\pi$
$3$		0			0
$3$		0			0
$4$	−0.278768	+	0.857960i	−0.278768	+	0.857960i
$5$	1.04744	−	1.44168i	1.04744	−	1.44168i	0.156434	−	0.987688i	$-0.450000\pi$
$5$	1.04744	−	1.44168i	1.04744	−	1.44168i	0.891007	−	0.453990i	$-0.150000\pi$
$6$		0			0
$7$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$7$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$8$	−0.183900	−	0.565985i	−0.183900	−	0.565985i
$9$		0			0
$10$			0.557537i			0.557537i
$11$	0.987688	−	0.156434i	0.987688	−	0.156434i
$11$	0.987688	−	0.156434i	0.987688	−	0.156434i
$12$		0			0
$13$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$13$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$14$		0			0
$15$		0			0
$16$	−0.579192	−	0.420808i	−0.579192	−	0.420808i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$		0			0
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$	0.944910	+	1.30056i	0.944910	+	1.30056i
$21$		0			0
$22$	−0.221232	+	0.221232i	−0.221232	+	0.221232i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−0.672288	−	2.06909i	−0.672288	−	2.06909i
$26$	0.297556	+	0.0966818i	0.297556	+	0.0966818i
$27$		0			0
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$	0.819101			0.819101
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$		0			0
$39$		0			0
$40$	−1.00859	−	0.327712i	−1.00859	−	0.327712i
$41$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$41$	−0.437016	−	1.34500i	−0.437016	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$42$		0			0
$43$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$43$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$44$	−0.141122	+	0.891007i	−0.141122	+	0.891007i
$45$		0			0
$46$		0			0
$47$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i	−0.453990	−	0.891007i	$-0.650000\pi$
$47$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i	0.156434	+	0.987688i	$0.450000\pi$
$48$		0			0
$49$	0.809017	+	0.587785i	0.809017	+	0.587785i
$50$	0.550672	+	0.400087i	0.550672	+	0.400087i
$51$		0			0
$52$	0.857960	−	0.278768i	0.857960	−	0.278768i
$53$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$53$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$54$		0			0
$55$	0.809017	−	1.58779i	0.809017	−	1.58779i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.87869	+	0.610425i	1.87869	+	0.610425i	0.987688	+	0.156434i	$0.0500000\pi$
$59$	1.87869	+	0.610425i	1.87869	+	0.610425i	0.891007	+	0.453990i	$0.150000\pi$
$60$		0			0
$61$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$61$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$		0			0
$64$	0.371864	−	0.270175i	0.371864	−	0.270175i
$65$	−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			0
$71$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$71$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$72$		0			0
$73$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$73$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$79$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$80$	−1.21334	+	0.394238i	−1.21334	+	0.394238i
$81$		0			0
$82$	0.357960	+	0.260074i	0.357960	+	0.260074i
$83$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.987688	−	0.156434i	$-0.950000\pi$
$83$	−1.44168	−	1.04744i	−1.44168	−	1.04744i	−0.453990	−	0.891007i	$-0.650000\pi$
$84$		0			0
$85$		0			0
$86$	−0.349798	−	0.481456i	−0.349798	−	0.481456i
$87$		0			0
$88$	−0.270175	−	0.530249i	−0.270175	−	0.530249i
$89$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$89$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.0575365	−	0.0791922i	0.0575365	−	0.0791922i
$95$		0			0
$96$		0			0
$97$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$97$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$98$	−0.312869			−0.312869
$99$		0			0
$100$	1.96261			1.96261
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$103$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$104$	−0.349798	+	0.481456i	−0.349798	+	0.481456i
$105$		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			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	0.0872179	+	0.550672i	0.0872179	+	0.550672i
$111$		0			0
$112$		0			0
$113$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$113$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$119$		0			0
$120$		0			0
$121$	0.951057	−	0.309017i	0.951057	−	0.309017i
$122$		−	0.506233i		−	0.506233i
$123$		0			0
$124$		0			0
$125$	−1.99235	−	0.647354i	−1.99235	−	0.647354i
$126$		0			0
$127$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$127$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$128$	−0.297556	+	0.915783i	−0.297556	+	0.915783i
$129$		0			0
$130$	0.451057	−	0.327712i	0.451057	−	0.327712i
$131$		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
$137$	−1.16110	+	1.59811i	−1.16110	+	1.59811i	−0.453990	+	0.891007i	$0.650000\pi$
$137$	−1.16110	+	1.59811i	−1.16110	+	1.59811i	−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$139$	−1.11803	−	0.363271i	−1.11803	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$140$		0			0
$141$		0			0
$142$			0.442463i			0.442463i
$143$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.734572	−	0.533698i	−0.734572	−	0.533698i	0.156434	−	0.987688i	$-0.450000\pi$
$149$	−0.734572	−	0.533698i	−0.734572	−	0.533698i	−0.891007	+	0.453990i	$0.850000\pi$
$150$		0			0
$151$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$151$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$157$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$158$	−0.481456	−	0.156434i	−0.481456	−	0.156434i
$159$		0			0
$160$	0.857960	−	1.18088i	0.857960	−	1.18088i
$161$		0			0
$162$		0			0
$163$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$163$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$164$	1.27578			1.27578
$165$		0			0
$166$	0.557537			0.557537
$167$	0.734572	−	0.533698i	0.734572	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$167$	0.734572	−	0.533698i	0.734572	−	0.533698i	0.891007	+	0.453990i	$0.150000\pi$
$168$		0			0
$169$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$170$		0			0
$171$		0			0
$172$	−1.63194	−	0.530249i	−1.63194	−	0.530249i
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$	−0.637890	−	0.325021i	−0.637890	−	0.325021i
$177$		0			0
$178$	−0.260074	−	0.357960i	−0.260074	−	0.357960i
$179$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$179$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$180$		0			0
$181$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$181$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		−	0.282243i		−	0.282243i
$189$		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
$193$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$193$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$194$		0			0
$195$		0			0
$196$	−0.729825	+	0.530249i	−0.729825	+	0.530249i
$197$	−1.97538			−1.97538			−0.987688	−	0.156434i	$-0.950000\pi$
$197$	−1.97538			−1.97538			−0.987688	+	0.156434i	$0.950000\pi$
$198$		0			0
$199$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$199$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$200$	−1.04744	+	0.761010i	−1.04744	+	0.761010i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.39680	−	0.778768i	−2.39680	−	0.778768i
$206$	−0.0597526	−	0.183900i	−0.0597526	−	0.183900i
$207$		0			0
$208$			0.715921i			0.715921i
$209$		0			0
$210$		0			0
$211$	0.363271	+	0.500000i	0.363271	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$211$	0.363271	+	0.500000i	0.363271	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.74224	+	1.99235i	2.74224	+	1.99235i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.13673	+	1.13673i	1.13673	+	1.13673i
$221$		0			0
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.610425	+	1.87869i	−0.610425	+	1.87869i	−0.156434	+	0.987688i	$0.550000\pi$
$227$	−0.610425	+	1.87869i	−0.610425	+	1.87869i	−0.453990	+	0.891007i	$0.650000\pi$
$228$		0			0
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$		0			0
$235$	−0.172288	+	0.530249i	−0.172288	+	0.530249i
$236$	−1.04744	+	1.44168i	−1.04744	+	1.44168i
$237$		0			0
$238$		0			0
$239$	0.550672	+	1.69480i	0.550672	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$239$	0.550672	+	1.69480i	0.550672	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$243$		0			0
$244$	−0.857960	−	1.18088i	−0.857960	−	1.18088i
$245$	1.69480	−	0.550672i	1.69480	−	0.550672i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.623345	−	0.202537i	0.623345	−	0.202537i
$251$		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
$254$		−	0.193364i		−	0.193364i
$255$		0			0
$256$	0.0489435	+	0.150633i	0.0489435	+	0.150633i
$257$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$257$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$258$		0			0
$259$		0			0
$260$	0.496769	−	1.52890i	0.496769	−	1.52890i
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$269$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$270$		0			0
$271$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$271$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$272$		0			0
$273$		0			0
$274$		−	0.618034i		−	0.618034i
$275$	−0.987688	−	1.93845i	−0.987688	−	1.93845i
$276$		0			0
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$	0.349798	−	0.113656i	0.349798	−	0.113656i
$279$		0			0
$280$		0			0
$281$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.891007	−	0.453990i	$-0.850000\pi$
$281$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$282$		0			0
$283$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$283$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$284$	0.749885	+	1.03213i	0.749885	+	1.03213i
$285$		0			0
$286$	0.309017	+	0.0489435i	0.309017	+	0.0489435i
$287$		0			0
$288$		0			0
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.550672	−	1.69480i	0.550672	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$293$	0.550672	−	1.69480i	0.550672	−	1.69480i	0.707107	−	0.707107i	$-0.250000\pi$
$294$		0			0
$295$	2.84786	−	2.06909i	2.84786	−	2.06909i
$296$		0			0
$297$		0			0
$298$	0.284079			0.284079
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.891007	+	2.74224i	0.891007	+	2.74224i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		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
$313$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.951057	+	0.309017i	$0.100000\pi$
$313$	1.53884	+	1.11803i	1.53884	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$314$	0.409551	+	0.297556i	0.409551	+	0.297556i
$315$		0			0
$316$	−1.38821	+	0.451057i	−1.38821	+	0.451057i
$317$	−0.533698	−	0.734572i	−0.533698	−	0.734572i	0.453990	−	0.891007i	$-0.350000\pi$
$317$	−0.533698	−	0.734572i	−0.533698	−	0.734572i	−0.987688	+	0.156434i	$0.950000\pi$
$318$		0			0
$319$		0			0
$320$		−	0.819101i		−	0.819101i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.27877	+	1.76007i	−1.27877	+	1.76007i
$326$		0			0
$327$		0			0
$328$	−0.680881	+	0.494689i	−0.680881	+	0.494689i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	1.30056	−	0.944910i	1.30056	−	0.944910i
$333$		0			0
$334$	−0.0877853	+	0.270175i	−0.0877853	+	0.270175i
$335$		0			0
$336$		0			0
$337$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$337$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$338$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	1.07657	−	0.349798i	1.07657	−	0.349798i
$345$		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
$349$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$349$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$350$		0			0
$351$		0			0
$352$	0.809017	−	0.128136i	0.809017	−	0.128136i
$353$		−	0.907981i		−	0.907981i	−0.891007	−	0.453990i	$-0.850000\pi$
$353$		−	0.907981i		−	0.907981i	0.891007	−	0.453990i	$-0.150000\pi$
$354$		0			0
$355$	−0.778768	−	2.39680i	−0.778768	−	2.39680i
$356$	−1.21334	−	0.394238i	−1.21334	−	0.394238i
$357$		0			0
$358$		0			0
$359$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	0.453990	+	0.891007i	$0.350000\pi$
$359$	−0.437016	+	1.34500i	−0.437016	+	1.34500i	−0.891007	+	0.453990i	$0.850000\pi$
$360$		0			0
$361$	0.809017	−	0.587785i	0.809017	−	0.587785i
$362$	0.193364			0.193364
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$367$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$373$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$374$		0			0
$375$		0			0
$376$	0.109441	+	0.150633i	0.109441	+	0.150633i
$377$		0			0
$378$		0			0
$379$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$379$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.707107	−	0.707107i	$-0.750000\pi$
$383$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.453990	−	0.891007i	$-0.650000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$389$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$390$		0			0
$391$		0			0
$392$	0.183900	−	0.565985i	0.183900	−	0.565985i
$393$		0			0
$394$	0.500000	−	0.363271i	0.500000	−	0.363271i
$395$	2.88336			2.88336
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	0.297556	−	0.216187i	0.297556	−	0.216187i
$399$		0			0
$400$	−0.481305	+	1.48131i	−0.481305	+	1.48131i
$401$	−0.533698	+	0.734572i	−0.533698	+	0.734572i	−0.987688	−	0.156434i	$-0.950000\pi$
$401$	−0.533698	+	0.734572i	−0.533698	+	0.734572i	0.453990	+	0.891007i	$0.350000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$	0.749885	−	0.243652i	0.749885	−	0.243652i
$411$		0			0
$412$	−0.451057	−	0.327712i	−0.451057	−	0.327712i
$413$		0			0
$414$		0			0
$415$	−3.02015	+	0.981305i	−3.02015	+	0.981305i
$416$	−0.481456	−	0.662667i	−0.481456	−	0.662667i
$417$		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			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$	−0.183900	−	0.0597526i	−0.183900	−	0.0597526i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−1.06050			−1.06050
$431$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.707107	−	0.707107i	$-0.250000\pi$
$431$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.891007	−	0.453990i	$-0.150000\pi$
$432$		0			0
$433$	0.587785	−	1.80902i	0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$433$	0.587785	−	1.80902i	0.587785	−	1.80902i	0.587785	−	0.809017i	$-0.300000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$439$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$440$	−1.04744	−	0.165898i	−1.04744	−	0.165898i
$441$		0			0
$442$		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
$445$	2.03884	+	1.48131i	2.03884	+	1.48131i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.183900	+	0.253116i	0.183900	+	0.253116i	0.891007	−	0.453990i	$-0.150000\pi$
$449$	0.183900	+	0.253116i	0.183900	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$450$		0			0
$451$	−0.642040	−	1.26007i	−0.642040	−	1.26007i
$452$		0			0
$453$		0			0
$454$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$455$		0			0
$456$		0			0
$457$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$457$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$461$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$467$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$468$		0			0
$469$		0			0
$470$	−0.0539036	−	0.165898i	−0.0539036	−	0.165898i
$471$		0			0
$472$		−	1.17557i		−	1.17557i
$473$	0.297556	+	1.87869i	0.297556	+	1.87869i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.451057	−	0.327712i	−0.451057	−	0.327712i
$479$	0.734572	+	0.533698i	0.734572	+	0.533698i	0.891007	−	0.453990i	$-0.150000\pi$
$479$	0.734572	+	0.533698i	0.734572	+	0.533698i	−0.156434	+	0.987688i	$0.550000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$			0.902113i			0.902113i
$485$		0			0
$486$		0			0
$487$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$487$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$488$	0.915783	+	0.297556i	0.915783	+	0.297556i
$489$		0			0
$490$	−0.327712	+	0.451057i	−0.327712	+	0.451057i
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$499$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$500$	1.11081	−	1.52890i	1.11081	−	1.52890i
$501$		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
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−0.327712	−	0.451057i	−0.327712	−	0.451057i
$509$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.156434	−	0.987688i	$-0.450000\pi$
$509$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$		0			0
$512$	−0.819101	−	0.595112i	−0.819101	−	0.595112i
$513$		0			0
$514$		0			0
$515$	0.647354	+	0.891007i	0.647354	+	0.891007i
$516$		0			0
$517$	−0.278768	+	0.142040i	−0.278768	+	0.142040i
$518$		0			0
$519$		0			0
$520$	0.327712	+	1.00859i	0.327712	+	1.00859i
$521$		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
$533$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.891007	+	0.453990i	0.891007	+	0.453990i
$540$		0			0
$541$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$541$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$547$	1.11803	−	0.363271i	1.11803	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$548$	−1.04744	−	1.44168i	−1.04744	−	1.44168i
$549$		0			0
$550$	0.606480	+	0.309017i	0.606480	+	0.309017i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.623345	−	0.857960i	0.623345	−	0.857960i
$557$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.156434	−	0.987688i	$-0.450000\pi$
$557$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.453990	−	0.891007i	$-0.350000\pi$
$558$		0			0
$559$	1.53884	−	1.11803i	1.53884	−	1.11803i
$560$		0			0
$561$		0			0
$562$	0.618034			0.618034
$563$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$563$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$564$		0			0
$565$		0			0
$566$	−0.297556	+	0.409551i	−0.297556	+	0.409551i
$567$		0			0
$568$	−0.800424	−	0.260074i	−0.800424	−	0.260074i
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$571$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$572$	0.803789	−	0.409551i	0.803789	−	0.409551i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$577$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$578$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.172288	+	0.530249i	0.172288	+	0.530249i
$587$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$587$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$588$		0			0
$589$		0			0
$590$	−0.340334	+	1.04744i	−0.340334	+	1.04744i
$591$		0			0
$592$		0			0
$593$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$593$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$594$		0			0
$595$		0			0
$596$	0.662667	−	0.481456i	0.662667	−	0.481456i
$597$		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
$601$	−0.587785	−	0.190983i	−0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$601$	−0.587785	−	0.190983i	−0.587785	−	0.190983i	−0.587785	+	0.809017i	$0.700000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.550672	−	1.69480i	0.550672	−	1.69480i
$606$		0			0
$607$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$607$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$608$		0			0
$609$		0			0
$610$	−0.729825	−	0.530249i	−0.729825	−	0.530249i
$611$	0.253116	+	0.183900i	0.253116	+	0.183900i
$612$		0			0
$613$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$613$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.97538i			1.97538i	0.156434	+	0.987688i	$0.450000\pi$
$617$			1.97538i			1.97538i	−0.156434	+	0.987688i	$0.550000\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.26007	+	0.915497i	−1.26007	+	0.915497i
$626$	−0.595112			−0.595112
$627$		0			0
$628$	1.45965			1.45965
$629$		0			0
$630$		0			0
$631$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$631$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$632$	0.565985	−	0.779012i	0.565985	−	0.779012i
$633$		0			0
$634$	0.270175	+	0.0877853i	0.270175	+	0.0877853i
$635$	0.340334	+	1.04744i	0.340334	+	1.04744i
$636$		0			0
$637$		−	1.00000i		−	1.00000i
$638$		0			0
$639$		0			0
$640$	1.00859	+	1.38821i	1.00859	+	1.38821i
$641$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$641$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$642$		0			0
$643$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$643$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$644$		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
$649$	1.95106	+	0.309017i	1.95106	+	0.309017i
$650$		−	0.680668i		−	0.680668i
$651$		0			0
$652$		0			0
$653$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$653$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$654$		0			0
$655$		0			0
$656$	−0.312869	+	0.962912i	−0.312869	+	0.962912i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	−0.327712	+	1.00859i	−0.327712	+	1.00859i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.253116	+	0.779012i	0.253116	+	0.779012i
$669$		0			0
$670$		0			0
$671$	−0.734572	+	1.44168i	−0.734572	+	1.44168i
$672$		0			0
$673$	0.690983	+	0.951057i	0.690983	+	0.951057i	1.00000			$0$
$673$	0.690983	+	0.951057i	0.690983	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$674$	−0.481456	+	0.156434i	−0.481456	+	0.156434i
$675$		0			0
$676$	−0.729825	−	0.530249i	−0.729825	−	0.530249i
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	0.312869i		−	0.312869i	−0.987688	−	0.156434i	$-0.950000\pi$
$683$		−	0.312869i		−	0.312869i	0.987688	−	0.156434i	$-0.0500000\pi$
$684$		0			0
$685$	1.08779	+	3.34786i	1.08779	+	3.34786i
$686$		0			0
$687$		0			0
$688$	0.800424	−	1.10169i	0.800424	−	1.10169i
$689$		0			0
$690$		0			0
$691$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$691$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.69480	+	1.23134i	−1.69480	+	1.23134i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$	0.325021	−	0.325021i	0.325021	−	0.325021i
$705$		0			0
$706$	0.166977	+	0.229825i	0.166977	+	0.229825i
$707$		0			0
$708$		0			0
$709$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$709$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$710$	0.637890	+	0.463454i	0.637890	+	0.463454i
$711$		0			0
$712$	0.800424	−	0.260074i	0.800424	−	0.260074i
$713$		0			0
$714$		0			0
$715$	−1.76007	+	0.278768i	−1.76007	+	0.278768i
$716$		0			0
$717$		0			0
$718$	−0.136729	−	0.420808i	−0.136729	−	0.420808i
$719$		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
$722$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i
$723$		0			0
$724$	0.451057	−	0.327712i	0.451057	−	0.327712i
$725$		0			0
$726$		0			0
$727$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$727$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$733$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$734$	−0.156434	−	0.481456i	−0.156434	−	0.481456i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$739$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	0.453990	−	0.891007i	$-0.350000\pi$
$743$	−0.253116	−	0.183900i	−0.253116	−	0.183900i	−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$746$	−0.113656	−	0.156434i	−0.113656	−	0.156434i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$751$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$752$	0.213027	+	0.0692165i	0.213027	+	0.0692165i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.951057	+	0.690983i	−0.951057	+	0.690983i	−0.951057	−	0.309017i	$-0.900000\pi$
$757$	−0.951057	+	0.690983i	−0.951057	+	0.690983i			1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	−0.707107	−	0.707107i	$-0.750000\pi$
$761$	−0.253116	+	0.183900i	−0.253116	+	0.183900i	0.453990	+	0.891007i	$0.350000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.587785	+	0.190983i	0.587785	+	0.190983i
$767$	−0.610425	−	1.87869i	−0.610425	−	1.87869i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.87869	+	0.610425i	−1.87869	+	0.610425i	−0.891007	+	0.453990i	$0.850000\pi$
$773$	−1.87869	+	0.610425i	−1.87869	+	0.610425i	−0.987688	+	0.156434i	$0.950000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.642040	−	1.26007i	0.642040	−	1.26007i
$782$		0			0
$783$		0			0
$784$	−0.221232	−	0.680881i	−0.221232	−	0.680881i
$785$	−2.74224	−	0.891007i	−2.74224	−	0.891007i
$786$		0			0
$787$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$787$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$788$	0.550672	−	1.69480i	0.550672	−	1.69480i
$789$		0			0
$790$	−0.729825	+	0.530249i	−0.729825	+	0.530249i
$791$		0			0
$792$		0			0
$793$	1.61803			1.61803
$794$		0			0
$795$		0			0
$796$	0.327712	−	1.00859i	0.327712	−	1.00859i
$797$		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
$800$	−0.550672	−	1.69480i	−0.550672	−	1.69480i
$801$		0			0
$802$		−	0.284079i		−	0.284079i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	1.33630	−	1.83927i	1.33630	−	1.83927i
$821$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	−0.987688	−	0.156434i	$-0.950000\pi$
$821$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	0.891007	+	0.453990i	$0.150000\pi$
$822$		0			0
$823$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$823$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$824$	0.367799			0.367799
$825$		0			0
$826$		0			0
$827$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.707107	+	0.707107i	$0.750000\pi$
$827$	−1.59811	+	1.16110i	−1.59811	+	1.16110i	−0.891007	+	0.453990i	$0.850000\pi$
$828$		0			0
$829$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$829$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$830$	0.583987	−	0.803789i	0.583987	−	0.803789i
$831$		0			0
$832$	−0.437153	−	0.142040i	−0.437153	−	0.142040i
$833$		0			0
$834$		0			0
$835$		−	1.61803i		−	1.61803i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.87869	+	0.610425i	−1.87869	+	0.610425i	−0.891007	+	0.453990i	$0.850000\pi$
$839$	−1.87869	+	0.610425i	−1.87869	+	0.610425i	−0.987688	+	0.156434i	$0.950000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$	−0.530249	+	0.172288i	−0.530249	+	0.172288i
$845$	1.04744	+	1.44168i	1.04744	+	1.44168i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$853$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$859$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$860$	−2.47381	+	1.79733i	−2.47381	+	1.79733i
$861$		0			0
$862$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$863$	1.16110	−	1.59811i	1.16110	−	1.59811i	0.453990	−	0.891007i	$-0.350000\pi$
$863$	1.16110	−	1.59811i	1.16110	−	1.59811i	0.707107	−	0.707107i	$-0.250000\pi$
$864$		0			0
$865$		0			0
$866$	0.183900	+	0.565985i	0.183900	+	0.565985i
$867$		0			0
$868$		0			0
$869$	1.14412	+	1.14412i	1.14412	+	1.14412i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$877$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$878$	0.349798	+	0.481456i	0.349798	+	0.481456i
$879$		0			0
$880$	−1.13673	+	0.579192i	−1.13673	+	0.579192i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−0.587785	−	1.80902i	−0.587785	−	1.80902i	−0.587785	−	0.809017i	$-0.700000\pi$
$883$	−0.587785	−	1.80902i	−0.587785	−	1.80902i		−	1.00000i	$-0.5\pi$
$884$		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
$890$	−0.788476			−0.788476
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.0930960	−	0.0302487i	−0.0930960	−	0.0302487i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	0.394238	+	0.200874i	0.394238	+	0.200874i
$903$		0			0
$904$		0			0
$905$	−1.04744	+	0.340334i	−1.04744	+	0.340334i
$906$		0			0
$907$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$907$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$908$	−1.44168	−	1.04744i	−1.44168	−	1.04744i
$909$		0			0
$910$		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
$913$	−1.58779	−	0.809017i	−1.58779	−	0.809017i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$919$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$	−0.357960	+	0.260074i	−0.357960	+	0.260074i
$923$	−1.41421			−1.41421
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.533698	−	0.734572i	0.533698	−	0.734572i	−0.453990	−	0.891007i	$-0.650000\pi$
$929$	0.533698	−	0.734572i	0.533698	−	0.734572i	0.987688	+	0.156434i	$0.0500000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$937$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$938$		0			0
$939$		0			0
$940$	−0.406904	−	0.295633i	−0.406904	−	0.295633i
$941$	0.253116	+	0.183900i	0.253116	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$941$	0.253116	+	0.183900i	0.253116	+	0.183900i	−0.453990	+	0.891007i	$0.650000\pi$
$942$		0			0
$943$		0			0
$944$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$945$		0			0
$946$	−0.420808	−	0.420808i	−0.420808	−	0.420808i
$947$		−	1.78201i		−	1.78201i	−0.453990	−	0.891007i	$-0.650000\pi$
$947$		−	1.78201i		−	1.78201i	0.453990	−	0.891007i	$-0.350000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$953$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$954$		0			0
$955$		0			0
$956$	−1.60758			−1.60758
$957$		0			0
$958$	−0.284079			−0.284079
$959$		0			0
$960$		0			0
$961$	0.309017	−	0.951057i	0.309017	−	0.951057i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−0.349798	−	0.481456i	−0.349798	−	0.481456i
$969$		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
$976$	1.10169	−	0.357960i	1.10169	−	0.357960i
$977$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$977$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$978$		0			0
$979$	0.221232	+	1.39680i	0.221232	+	1.39680i
$980$			1.60758i			1.60758i
$981$		0			0
$982$		0			0
$983$	1.69480	+	0.550672i	1.69480	+	0.550672i	0.987688	−	0.156434i	$-0.0500000\pi$
$983$	1.69480	+	0.550672i	1.69480	+	0.550672i	0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	−2.06909	+	2.84786i	−2.06909	+	2.84786i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$991$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.23134	+	1.69480i	−1.23134	+	1.69480i
$996$		0			0
$997$	1.80902	+	0.587785i	1.80902	+	0.587785i	1.00000			$0$
$997$	1.80902	+	0.587785i	1.80902	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.1.bu.a.415.2	✓	16
3.2	odd	2	inner	1287.1.bu.a.415.3	yes	16
11.7	odd	10	inner	1287.1.bu.a.766.3	yes	16
13.12	even	2	inner	1287.1.bu.a.415.3	yes	16
33.29	even	10	inner	1287.1.bu.a.766.2	yes	16
39.38	odd	2	CM	1287.1.bu.a.415.2	✓	16
143.51	odd	10	inner	1287.1.bu.a.766.2	yes	16
429.194	even	10	inner	1287.1.bu.a.766.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1287.1.bu.a.415.2	✓	16	1.1	even	1	trivial
1287.1.bu.a.415.2	✓	16	39.38	odd	2	CM
1287.1.bu.a.415.3	yes	16	3.2	odd	2	inner
1287.1.bu.a.415.3	yes	16	13.12	even	2	inner
1287.1.bu.a.766.2	yes	16	33.29	even	10	inner
1287.1.bu.a.766.2	yes	16	143.51	odd	10	inner
1287.1.bu.a.766.3	yes	16	11.7	odd	10	inner
1287.1.bu.a.766.3	yes	16	429.194	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1287.bu (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.253116 + 0.183900i) q^{2} +(-0.278768 + 0.857960i) q^{4} +(1.04744 - 1.44168i) q^{5} +(-0.183900 - 0.565985i) q^{8} +O(q^{10})\)	\(q+(-0.253116 + 0.183900i) q^{2} +(-0.278768 + 0.857960i) q^{4} +(1.04744 - 1.44168i) q^{5} +(-0.183900 - 0.565985i) q^{8} +0.557537i q^{10} +(0.987688 - 0.156434i) q^{11} +(-0.587785 - 0.809017i) q^{13} +(-0.579192 - 0.420808i) q^{16} +(0.944910 + 1.30056i) q^{20} +(-0.221232 + 0.221232i) q^{22} +(-0.672288 - 2.06909i) q^{25} +(0.297556 + 0.0966818i) q^{26} +0.819101 q^{32} +(-1.00859 - 0.327712i) q^{40} +(-0.437016 - 1.34500i) q^{41} +1.90211i q^{43} +(-0.141122 + 0.891007i) q^{44} +(-0.297556 + 0.0966818i) q^{47} +(0.809017 + 0.587785i) q^{49} +(0.550672 + 0.400087i) q^{50} +(0.857960 - 0.278768i) q^{52} +(0.809017 - 1.58779i) q^{55} +(1.87869 + 0.610425i) q^{59} +(-0.951057 + 1.30902i) q^{61} +(0.371864 - 0.270175i) q^{64} -1.78201 q^{65} +(0.831254 - 1.14412i) q^{71} +(0.951057 + 1.30902i) q^{79} +(-1.21334 + 0.394238i) q^{80} +(0.357960 + 0.260074i) q^{82} +(-1.44168 - 1.04744i) q^{83} +(-0.349798 - 0.481456i) q^{86} +(-0.270175 - 0.530249i) q^{88} +1.41421i q^{89} +(0.0575365 - 0.0791922i) q^{94} -0.312869 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{4}+O(q^{10}) \) 16 * q - 4 * q^4	\( 16 q - 4 q^{4} - 16 q^{16} - 4 q^{22} + 4 q^{25} + 4 q^{49} + 20 q^{52} + 4 q^{55} - 16 q^{64} + 12 q^{82} - 20 q^{88}+O(q^{100}) \) 16 * q - 4 * q^4 - 16 * q^16 - 4 * q^22 + 4 * q^25 + 4 * q^49 + 20 * q^52 + 4 * q^55 - 16 * q^64 + 12 * q^82 - 20 * q^88

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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