LMFDB - Embedded newform 723.1.bc.a.455.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [723,1,Mod(5,723)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(723, base_ring=CyclotomicField(40))

chi = DirichletCharacter(H, H._module([20, 23]))

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

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

chi := DirichletCharacter("723.5");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 723 = 3 \cdot 241 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	723.bc (of order $40$, degree $16$, 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:	$0.360824004134$
Analytic rank:	$0$
Dimension:	$16$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} - \cdots)$

Embedding invariants

Embedding label			455.1
Root			$0.891007 - 0.453990i$ of defining polynomial
Character	$\chi$	$=$	723.455
Dual form			723.1.bc.a.116.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.987688 + 0.156434i) q^{3} +1.00000i q^{4} +(-1.47879 + 1.26301i) q^{7} +(0.951057 + 0.309017i) q^{9} +O(q^{10})$	$q+(0.987688 + 0.156434i) q^{3} +1.00000i q^{4} +(-1.47879 + 1.26301i) q^{7} +(0.951057 + 0.309017i) q^{9} +(-0.156434 + 0.987688i) q^{12} +(-0.243950 - 1.01612i) q^{13} -1.00000 q^{16} +(0.431351 - 0.178671i) q^{19} +(-1.65816 + 1.01612i) q^{21} +(0.951057 - 0.309017i) q^{25} +(0.891007 + 0.453990i) q^{27} +(-1.26301 - 1.47879i) q^{28} +(0.763007 + 0.0600500i) q^{31} +(-0.309017 + 0.951057i) q^{36} +(0.303221 - 1.26301i) q^{37} +(-0.0819895 - 1.04178i) q^{39} +(0.101910 - 0.119322i) q^{43} +(-0.987688 - 0.156434i) q^{48} +(0.435203 - 2.74776i) q^{49} +(1.01612 - 0.243950i) q^{52} +(0.453990 - 0.108993i) q^{57} +(-0.297556 + 1.87869i) q^{61} +(-1.79671 + 0.744220i) q^{63} -1.00000i q^{64} +(-0.533698 + 1.04744i) q^{67} +(0.465451 - 1.93874i) q^{73} +(0.987688 - 0.156434i) q^{75} +(0.178671 + 0.431351i) q^{76} +(0.253116 + 1.59811i) q^{79} +(0.809017 + 0.587785i) q^{81} +(-1.01612 - 1.65816i) q^{84} +(1.64412 + 1.19453i) q^{91} +(0.744220 + 0.178671i) q^{93} +(-1.87869 - 0.610425i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{16} - 4 q^{28} - 4 q^{31} + 4 q^{36} + 4 q^{39} + 4 q^{43} + 4 q^{49} - 4 q^{52} - 4 q^{63} - 4 q^{73} - 4 q^{76} + 4 q^{81} + 4 q^{84} + 8 q^{91}+O(q^{100}) $ 16 * q - 16 * q^16 - 4 * q^28 - 4 * q^31 + 4 * q^36 + 4 * q^39 + 4 * q^43 + 4 * q^49 - 4 * q^52 - 4 * q^63 - 4 * q^73 - 4 * q^76 + 4 * q^81 + 4 * q^84 + 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/723\mathbb{Z}\right)^\times$.

$n$	$7$	$242$
$\chi(n)$	$e\left(\frac{31}{40}\right)$	$-1$

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			0		−0.707107	−	0.707107i	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$	0.987688	+	0.156434i	0.987688	+	0.156434i
$3$	0.987688	+	0.156434i	0.987688	+	0.156434i
$4$			1.00000i			1.00000i
$5$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$5$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$6$		0			0
$7$	−1.47879	+	1.26301i	−1.47879	+	1.26301i	−0.587785	+	0.809017i	$0.700000\pi$
$7$	−1.47879	+	1.26301i	−1.47879	+	1.26301i	−0.891007	+	0.453990i	$0.850000\pi$
$8$		0			0
$9$	0.951057	+	0.309017i	0.951057	+	0.309017i
$10$		0			0
$11$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$11$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$12$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$13$	−0.243950	−	1.01612i	−0.243950	−	1.01612i	−0.951057	−	0.309017i	$-0.900000\pi$
$13$	−0.243950	−	1.01612i	−0.243950	−	1.01612i	0.707107	−	0.707107i	$-0.250000\pi$
$14$		0			0
$15$		0			0
$16$	−1.00000			−1.00000
$17$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$17$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$18$		0			0
$19$	0.431351	−	0.178671i	0.431351	−	0.178671i	−0.156434	−	0.987688i	$-0.550000\pi$
$19$	0.431351	−	0.178671i	0.431351	−	0.178671i	0.587785	+	0.809017i	$0.300000\pi$
$20$		0			0
$21$	−1.65816	+	1.01612i	−1.65816	+	1.01612i
$22$		0			0
$23$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$23$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$24$		0			0
$25$	0.951057	−	0.309017i	0.951057	−	0.309017i
$26$		0			0
$27$	0.891007	+	0.453990i	0.891007	+	0.453990i
$28$	−1.26301	−	1.47879i	−1.26301	−	1.47879i
$29$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$29$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$30$		0			0
$31$	0.763007	+	0.0600500i	0.763007	+	0.0600500i	0.453990	−	0.891007i	$-0.350000\pi$
$31$	0.763007	+	0.0600500i	0.763007	+	0.0600500i	0.309017	+	0.951057i	$0.400000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$37$	0.303221	−	1.26301i	0.303221	−	1.26301i	−0.587785	−	0.809017i	$-0.700000\pi$
$37$	0.303221	−	1.26301i	0.303221	−	1.26301i	0.891007	−	0.453990i	$-0.150000\pi$
$38$		0			0
$39$	−0.0819895	−	1.04178i	−0.0819895	−	1.04178i
$40$		0			0
$41$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$41$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$42$		0			0
$43$	0.101910	−	0.119322i	0.101910	−	0.119322i	−0.707107	−	0.707107i	$-0.750000\pi$
$43$	0.101910	−	0.119322i	0.101910	−	0.119322i	0.809017	+	0.587785i	$0.200000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$47$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$48$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$49$	0.435203	−	2.74776i	0.435203	−	2.74776i
$50$		0			0
$51$		0			0
$52$	1.01612	−	0.243950i	1.01612	−	0.243950i
$53$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$53$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.453990	−	0.108993i	0.453990	−	0.108993i
$58$		0			0
$59$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$59$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$60$		0			0
$61$	−0.297556	+	1.87869i	−0.297556	+	1.87869i	0.156434	+	0.987688i	$0.450000\pi$
$61$	−0.297556	+	1.87869i	−0.297556	+	1.87869i	−0.453990	+	0.891007i	$0.650000\pi$
$62$		0			0
$63$	−1.79671	+	0.744220i	−1.79671	+	0.744220i
$64$		−	1.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	0.453990	+	0.891007i	$0.350000\pi$
$67$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.760406	−	0.649448i	$-0.775000\pi$
$71$		0			0		0.760406	+	0.649448i	$0.225000\pi$
$72$		0			0
$73$	0.465451	−	1.93874i	0.465451	−	1.93874i	0.156434	−	0.987688i	$-0.450000\pi$
$73$	0.465451	−	1.93874i	0.465451	−	1.93874i	0.309017	−	0.951057i	$-0.400000\pi$
$74$		0			0
$75$	0.987688	−	0.156434i	0.987688	−	0.156434i
$76$	0.178671	+	0.431351i	0.178671	+	0.431351i
$77$		0			0
$78$		0			0
$79$	0.253116	+	1.59811i	0.253116	+	1.59811i	0.707107	+	0.707107i	$0.250000\pi$
$79$	0.253116	+	1.59811i	0.253116	+	1.59811i	−0.453990	+	0.891007i	$0.650000\pi$
$80$		0			0
$81$	0.809017	+	0.587785i	0.809017	+	0.587785i
$82$		0			0
$83$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$83$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$84$	−1.01612	−	1.65816i	−1.01612	−	1.65816i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$89$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$90$		0			0
$91$	1.64412	+	1.19453i	1.64412	+	1.19453i
$92$		0			0
$93$	0.744220	+	0.178671i	0.744220	+	0.178671i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.87869	−	0.610425i	−1.87869	−	0.610425i	−0.987688	−	0.156434i	$-0.950000\pi$
$97$	−1.87869	−	0.610425i	−1.87869	−	0.610425i	−0.891007	−	0.453990i	$-0.850000\pi$
$98$		0			0
$99$		0			0
$100$	0.309017	+	0.951057i	0.309017	+	0.951057i
$101$		0			0		−0.760406	−	0.649448i	$-0.775000\pi$
$101$		0			0		0.760406	+	0.649448i	$0.225000\pi$
$102$		0			0
$103$	−0.133795	−	1.70002i	−0.133795	−	1.70002i	−0.587785	−	0.809017i	$-0.700000\pi$
$103$	−0.133795	−	1.70002i	−0.133795	−	1.70002i	0.453990	−	0.891007i	$-0.350000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$107$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$108$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$109$	−1.79671	+	0.431351i	−1.79671	+	0.431351i	−0.987688	−	0.156434i	$-0.950000\pi$
$109$	−1.79671	+	0.431351i	−1.79671	+	0.431351i	−0.809017	+	0.587785i	$0.800000\pi$
$110$		0			0
$111$	0.497066	−	1.20002i	0.497066	−	1.20002i
$112$	1.47879	−	1.26301i	1.47879	−	1.26301i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.0819895	−	1.04178i	0.0819895	−	1.04178i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$122$		0			0
$123$		0			0
$124$	−0.0600500	+	0.763007i	−0.0600500	+	0.763007i
$125$		0			0
$126$		0			0
$127$	−1.29671	−	0.794622i	−1.29671	−	0.794622i	−0.309017	−	0.951057i	$-0.600000\pi$
$127$	−1.29671	−	0.794622i	−1.29671	−	0.794622i	−0.987688	+	0.156434i	$0.950000\pi$
$128$		0			0
$129$	0.119322	−	0.101910i	0.119322	−	0.101910i
$130$		0			0
$131$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$131$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$132$		0			0
$133$	−0.412215	+	0.809017i	−0.412215	+	0.809017i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$137$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$138$		0			0
$139$	−0.965451	−	1.57547i	−0.965451	−	1.57547i	−0.809017	−	0.587785i	$-0.800000\pi$
$139$	−0.965451	−	1.57547i	−0.965451	−	1.57547i	−0.156434	−	0.987688i	$-0.550000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$145$		0			0
$146$		0			0
$147$	0.859689	−	2.64585i	0.859689	−	2.64585i
$148$	1.26301	+	0.303221i	1.26301	+	0.303221i
$149$		0			0		0.522499	−	0.852640i	$-0.325000\pi$
$149$		0			0		−0.522499	+	0.852640i	$0.675000\pi$
$150$		0			0
$151$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.707107	−	0.707107i	$-0.750000\pi$
$151$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.453990	−	0.891007i	$-0.650000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.04178	−	0.0819895i	1.04178	−	0.0819895i
$157$	0.794622	+	1.29671i	0.794622	+	1.29671i	0.951057	+	0.309017i	$0.100000\pi$
$157$	0.794622	+	1.29671i	0.794622	+	1.29671i	−0.156434	+	0.987688i	$0.550000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.10749	+	1.29671i	−1.10749	+	1.29671i	−0.156434	+	0.987688i	$0.550000\pi$
$163$	−1.10749	+	1.29671i	−1.10749	+	1.29671i	−0.951057	+	0.309017i	$0.900000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$167$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$168$		0			0
$169$	−0.0819895	+	0.0417758i	−0.0819895	+	0.0417758i
$170$		0			0
$171$	0.465451	−	0.0366318i	0.465451	−	0.0366318i
$172$	0.119322	+	0.101910i	0.119322	+	0.101910i
$173$		0			0		0.0784591	−	0.996917i	$-0.475000\pi$
$173$		0			0		−0.0784591	+	0.996917i	$0.525000\pi$
$174$		0			0
$175$	−1.01612	+	1.65816i	−1.01612	+	1.65816i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.649448	−	0.760406i	$-0.725000\pi$
$179$		0			0		0.649448	+	0.760406i	$0.275000\pi$
$180$		0			0
$181$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$181$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$182$		0			0
$183$	−0.587785	+	1.80902i	−0.587785	+	1.80902i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.89101	+	0.453990i	−1.89101	+	0.453990i
$190$		0			0
$191$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$191$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$192$	0.156434	−	0.987688i	0.156434	−	0.987688i
$193$	−0.309017	−	0.0489435i	−0.309017	−	0.0489435i		−	1.00000i	$-0.5\pi$
$193$	−0.309017	−	0.0489435i	−0.309017	−	0.0489435i	−0.309017	+	0.951057i	$0.600000\pi$
$194$		0			0
$195$		0			0
$196$	2.74776	+	0.435203i	2.74776	+	0.435203i
$197$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$197$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$198$		0			0
$199$	1.93874	+	0.465451i	1.93874	+	0.465451i	0.987688	+	0.156434i	$0.0500000\pi$
$199$	1.93874	+	0.465451i	1.93874	+	0.465451i	0.951057	+	0.309017i	$0.100000\pi$
$200$		0			0
$201$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.243950	+	1.01612i	0.243950	+	1.01612i
$209$		0			0
$210$		0			0
$211$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$211$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.20417	+	0.874883i	−1.20417	+	0.874883i
$218$		0			0
$219$	0.763007	−	1.84206i	0.763007	−	1.84206i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.142040	−	0.278768i	−0.142040	−	0.278768i	0.809017	−	0.587785i	$-0.200000\pi$
$223$	−0.142040	−	0.278768i	−0.142040	−	0.278768i	−0.951057	+	0.309017i	$0.900000\pi$
$224$		0			0
$225$	1.00000			1.00000
$226$		0			0
$227$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$227$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$228$	0.108993	+	0.453990i	0.108993	+	0.453990i
$229$	0.253116	−	1.59811i	0.253116	−	1.59811i	−0.453990	−	0.891007i	$-0.650000\pi$
$229$	0.253116	−	1.59811i	0.253116	−	1.59811i	0.707107	−	0.707107i	$-0.250000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			1.61803i			1.61803i
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	0.987688	+	0.156434i	0.987688	+	0.156434i
$241$	0.987688	+	0.156434i	0.987688	+	0.156434i
$242$		0			0
$243$	0.707107	+	0.707107i	0.707107	+	0.707107i
$244$	−1.87869	−	0.297556i	−1.87869	−	0.297556i
$245$		0			0
$246$		0			0
$247$	−0.286780	−	0.394719i	−0.286780	−	0.394719i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$251$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$252$	−0.744220	−	1.79671i	−0.744220	−	1.79671i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	1.14679	+	2.25070i	1.14679	+	2.25070i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$263$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.04744	−	0.533698i	−1.04744	−	0.533698i
$269$		0			0		−0.649448	−	0.760406i	$-0.725000\pi$
$269$		0			0		0.649448	+	0.760406i	$0.275000\pi$
$270$		0			0
$271$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$271$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$	1.43702	+	1.43702i	1.43702	+	1.43702i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	−0.987688	−	0.156434i	$-0.950000\pi$
$277$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i	0.891007	+	0.453990i	$0.150000\pi$
$278$		0			0
$279$	0.707107	+	0.292893i	0.707107	+	0.292893i
$280$		0			0
$281$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$281$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$282$		0			0
$283$	1.47879	+	0.355026i	1.47879	+	0.355026i	0.891007	−	0.453990i	$-0.150000\pi$
$283$	1.47879	+	0.355026i	1.47879	+	0.355026i	0.587785	+	0.809017i	$0.300000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.987688	+	0.156434i	0.987688	+	0.156434i
$290$		0			0
$291$	−1.76007	−	0.896802i	−1.76007	−	0.896802i
$292$	1.93874	+	0.465451i	1.93874	+	0.465451i
$293$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$293$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.156434	+	0.987688i	0.156434	+	0.987688i
$301$			0.305165i			0.305165i
$302$		0			0
$303$		0			0
$304$	−0.431351	+	0.178671i	−0.431351	+	0.178671i
$305$		0			0
$306$		0			0
$307$	−0.891007	+	1.45399i	−0.891007	+	1.45399i			1.00000i	$0.5\pi$
$307$	−0.891007	+	1.45399i	−0.891007	+	1.45399i	−0.891007	+	0.453990i	$0.850000\pi$
$308$		0			0
$309$	0.133795	−	1.70002i	0.133795	−	1.70002i
$310$		0			0
$311$		0			0		0.996917	−	0.0784591i	$-0.0250000\pi$
$311$		0			0		−0.996917	+	0.0784591i	$0.975000\pi$
$312$		0			0
$313$	−1.58779	+	0.809017i	−1.58779	+	0.809017i	−0.587785	+	0.809017i	$0.700000\pi$
$313$	−1.58779	+	0.809017i	−1.58779	+	0.809017i	−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	−1.59811	+	0.253116i	−1.59811	+	0.253116i
$317$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$317$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$325$	−0.546010	−	0.891007i	−0.546010	−	0.891007i
$326$		0			0
$327$	−1.84206	+	0.144974i	−1.84206	+	0.144974i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.533698	+	0.734572i	0.533698	+	0.734572i	0.987688	−	0.156434i	$-0.0500000\pi$
$331$	0.533698	+	0.734572i	0.533698	+	0.734572i	−0.453990	+	0.891007i	$0.650000\pi$
$332$		0			0
$333$	0.678671	−	1.10749i	0.678671	−	1.10749i
$334$		0			0
$335$		0			0
$336$	1.65816	−	1.01612i	1.65816	−	1.01612i
$337$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$337$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.81075	+	2.95487i	1.81075	+	2.95487i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$347$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$348$		0			0
$349$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$349$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$350$		0			0
$351$	0.243950	−	1.01612i	0.243950	−	1.01612i
$352$		0			0
$353$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$353$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$359$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$360$		0			0
$361$	−0.552967	+	0.552967i	−0.552967	+	0.552967i
$362$		0			0
$363$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$364$	−1.19453	+	1.64412i	−1.19453	+	1.64412i
$365$		0			0
$366$		0			0
$367$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	0.309017	−	0.951057i	$-0.400000\pi$
$367$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i	−0.453990	+	0.891007i	$0.650000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.178671	+	0.744220i	−0.178671	+	0.744220i
$373$	0.152583	−	0.0366318i	0.152583	−	0.0366318i	−0.156434	−	0.987688i	$-0.550000\pi$
$373$	0.152583	−	0.0366318i	0.152583	−	0.0366318i	0.309017	+	0.951057i	$0.400000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.144974	+	1.84206i	0.144974	+	1.84206i	0.453990	+	0.891007i	$0.350000\pi$
$379$	0.144974	+	1.84206i	0.144974	+	1.84206i	−0.309017	+	0.951057i	$0.600000\pi$
$380$		0			0
$381$	−1.15643	−	0.987688i	−1.15643	−	0.987688i
$382$		0			0
$383$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$383$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.133795	−	0.0819895i	0.133795	−	0.0819895i
$388$	0.610425	−	1.87869i	0.610425	−	1.87869i
$389$		0			0		−0.972370	−	0.233445i	$-0.925000\pi$
$389$		0			0		0.972370	+	0.233445i	$0.0750000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.29489	−	0.101910i	1.29489	−	0.101910i	0.587785	−	0.809017i	$-0.300000\pi$
$397$	1.29489	−	0.101910i	1.29489	−	0.101910i	0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$	−0.533698	+	0.734572i	−0.533698	+	0.734572i
$400$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$401$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$401$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$402$		0			0
$403$	−0.125117	−	0.789959i	−0.125117	−	0.789959i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.465451	−	1.93874i	0.465451	−	1.93874i	0.156434	−	0.987688i	$-0.450000\pi$
$409$	0.465451	−	1.93874i	0.465451	−	1.93874i	0.309017	−	0.951057i	$-0.400000\pi$
$410$		0			0
$411$		0			0
$412$	1.70002	−	0.133795i	1.70002	−	0.133795i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$418$		0			0
$419$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$419$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$420$		0			0
$421$	0.142040	−	0.896802i	0.142040	−	0.896802i	−0.809017	−	0.587785i	$-0.800000\pi$
$421$	0.142040	−	0.896802i	0.142040	−	0.896802i	0.951057	−	0.309017i	$-0.100000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.93278	−	3.15401i	−1.93278	−	3.15401i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.972370	−	0.233445i	$-0.925000\pi$
$431$		0			0		0.972370	+	0.233445i	$0.0750000\pi$
$432$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$433$	0.183900	−	1.16110i	0.183900	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$433$	0.183900	−	1.16110i	0.183900	−	1.16110i	0.891007	−	0.453990i	$-0.150000\pi$
$434$		0			0
$435$		0			0
$436$	−0.431351	−	1.79671i	−0.431351	−	1.79671i
$437$		0			0
$438$		0			0
$439$	1.20002	−	1.40505i	1.20002	−	1.40505i	0.309017	−	0.951057i	$-0.400000\pi$
$439$	1.20002	−	1.40505i	1.20002	−	1.40505i	0.891007	−	0.453990i	$-0.150000\pi$
$440$		0			0
$441$	1.26301	−	2.47879i	1.26301	−	2.47879i
$442$		0			0
$443$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$443$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$444$	1.20002	+	0.497066i	1.20002	+	0.497066i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.26301	+	1.47879i	1.26301	+	1.47879i
$449$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$449$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.896802	−	1.76007i	−0.896802	−	1.76007i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.156434	−	0.987688i	$-0.450000\pi$
$457$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$461$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$462$		0			0
$463$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.891007	−	0.453990i	$-0.850000\pi$
$463$	−1.20002	+	0.497066i	−1.20002	+	0.497066i	−0.309017	+	0.951057i	$0.600000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	1.04178	+	0.0819895i	1.04178	+	0.0819895i
$469$	−0.533698	−	2.22301i	−0.533698	−	2.22301i
$470$		0			0
$471$	0.581990	+	1.40505i	0.581990	+	1.40505i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.355026	−	0.303221i	0.355026	−	0.303221i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$479$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$480$		0			0
$481$	−1.35734			−1.35734
$482$		0			0
$483$		0			0
$484$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$485$		0			0
$486$		0			0
$487$	1.39680	−	0.221232i	1.39680	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$487$	1.39680	−	0.221232i	1.39680	−	0.221232i	0.809017	+	0.587785i	$0.200000\pi$
$488$		0			0
$489$	−1.29671	+	1.10749i	−1.29671	+	1.10749i
$490$		0			0
$491$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$491$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.763007	−	0.0600500i	−0.763007	−	0.0600500i
$497$		0			0
$498$		0			0
$499$	−1.51612	−	0.119322i	−1.51612	−	0.119322i	−0.707107	−	0.707107i	$-0.750000\pi$
$499$	−1.51612	−	0.119322i	−1.51612	−	0.119322i	−0.809017	+	0.587785i	$0.800000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$503$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.0875153	+	0.0284354i	−0.0875153	+	0.0284354i
$508$	0.794622	−	1.29671i	0.794622	−	1.29671i
$509$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$509$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$510$		0			0
$511$	1.76034	+	3.45487i	1.76034	+	3.45487i
$512$		0			0
$513$	0.465451	+	0.0366318i	0.465451	+	0.0366318i
$514$		0			0
$515$		0			0
$516$	0.101910	+	0.119322i	0.101910	+	0.119322i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$521$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$522$		0			0
$523$	0.642040	−	1.26007i	0.642040	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$523$	0.642040	−	1.26007i	0.642040	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$524$		0			0
$525$	−1.26301	+	1.47879i	−1.26301	+	1.47879i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.453990	+	0.891007i	0.453990	+	0.891007i
$530$		0			0
$531$		0			0
$532$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$541$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$542$		0			0
$543$	0.0966818	−	0.610425i	0.0966818	−	0.610425i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$547$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$548$		0			0
$549$	−0.863541	+	1.69480i	−0.863541	+	1.69480i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.39274	−	2.04359i	−2.39274	−	2.04359i
$554$		0			0
$555$		0			0
$556$	1.57547	−	0.965451i	1.57547	−	0.965451i
$557$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$557$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$558$		0			0
$559$	−0.146106	−	0.0744449i	−0.146106	−	0.0744449i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$563$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.93874	+	0.152583i	−1.93874	+	0.152583i
$568$		0			0
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	0.156434	+	0.987688i	$0.450000\pi$
$571$	−0.652583	+	1.57547i	−0.652583	+	1.57547i	−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.309017	−	0.951057i	0.309017	−	0.951057i
$577$	−0.652583	+	0.399903i	−0.652583	+	0.399903i	−0.809017	−	0.587785i	$-0.800000\pi$
$577$	−0.652583	+	0.399903i	−0.652583	+	0.399903i	0.156434	+	0.987688i	$0.450000\pi$
$578$		0			0
$579$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$587$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$588$	2.64585	+	0.859689i	2.64585	+	0.859689i
$589$	0.339853	−	0.110425i	0.339853	−	0.110425i
$590$		0			0
$591$		0			0
$592$	−0.303221	+	1.26301i	−0.303221	+	1.26301i
$593$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$593$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.84206	+	0.763007i	1.84206	+	0.763007i
$598$		0			0
$599$		0			0		0.0784591	−	0.996917i	$-0.475000\pi$
$599$		0			0		−0.0784591	+	0.996917i	$0.525000\pi$
$600$		0			0
$601$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$601$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$602$		0			0
$603$	−0.831254	+	0.831254i	−0.831254	+	0.831254i
$604$	1.59811	−	1.16110i	1.59811	−	1.16110i
$605$		0			0
$606$		0			0
$607$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	−0.707107	−	0.707107i	$-0.750000\pi$
$607$	−0.550672	+	0.280582i	−0.550672	+	0.280582i	0.156434	+	0.987688i	$0.450000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.431351	−	1.79671i	0.431351	−	1.79671i	−0.156434	−	0.987688i	$-0.550000\pi$
$613$	0.431351	−	1.79671i	0.431351	−	1.79671i	0.587785	−	0.809017i	$-0.300000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$617$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$618$		0			0
$619$	0.133795	+	0.0819895i	0.133795	+	0.0819895i	0.587785	−	0.809017i	$-0.300000\pi$
$619$	0.133795	+	0.0819895i	0.133795	+	0.0819895i	−0.453990	+	0.891007i	$0.650000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.0819895	+	1.04178i	0.0819895	+	1.04178i
$625$	0.809017	−	0.587785i	0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$	−1.29671	+	0.794622i	−1.29671	+	0.794622i
$629$		0			0
$630$		0			0
$631$	−0.891007	+	1.45399i	−0.891007	+	1.45399i			1.00000i	$0.5\pi$
$631$	−0.891007	+	1.45399i	−0.891007	+	1.45399i	−0.891007	+	0.453990i	$0.850000\pi$
$632$		0			0
$633$	0.831254	+	1.14412i	0.831254	+	1.14412i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.89823	+	0.228096i	−2.89823	+	0.228096i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$641$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$642$		0			0
$643$	−1.39680	+	0.221232i	−1.39680	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$643$	−1.39680	+	0.221232i	−1.39680	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$647$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.32621	+	0.675737i	−1.32621	+	0.675737i
$652$	−1.29671	−	1.10749i	−1.29671	−	1.10749i
$653$		0			0		0.996917	−	0.0784591i	$-0.0250000\pi$
$653$		0			0		−0.996917	+	0.0784591i	$0.975000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.04178	−	1.70002i	1.04178	−	1.70002i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	1.29489	+	1.51612i	1.29489	+	1.51612i	0.707107	+	0.707107i	$0.250000\pi$
$661$	1.29489	+	1.51612i	1.29489	+	1.51612i	0.587785	+	0.809017i	$0.300000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.0966818	−	0.297556i	−0.0966818	−	0.297556i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.44168	−	0.734572i	−1.44168	−	0.734572i	−0.453990	−	0.891007i	$-0.650000\pi$
$673$	−1.44168	−	0.734572i	−1.44168	−	0.734572i	−0.987688	+	0.156434i	$0.950000\pi$
$674$		0			0
$675$	0.987688	+	0.156434i	0.987688	+	0.156434i
$676$	−0.0417758	−	0.0819895i	−0.0417758	−	0.0819895i
$677$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$677$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$678$		0			0
$679$	3.54917	−	1.47011i	3.54917	−	1.47011i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$683$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$684$	0.0366318	+	0.465451i	0.0366318	+	0.465451i
$685$		0			0
$686$		0			0
$687$	0.500000	−	1.53884i	0.500000	−	1.53884i
$688$	−0.101910	+	0.119322i	−0.101910	+	0.119322i
$689$		0			0
$690$		0			0
$691$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$691$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.65816	−	1.01612i	−1.65816	−	1.01612i
$701$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$701$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$702$		0			0
$703$	−0.0948685	−	0.598976i	−0.0948685	−	0.598976i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.84206	−	0.144974i	−1.84206	−	0.144974i	−0.891007	−	0.453990i	$-0.850000\pi$
$709$	−1.84206	−	0.144974i	−1.84206	−	0.144974i	−0.951057	+	0.309017i	$0.900000\pi$
$710$		0			0
$711$	−0.253116	+	1.59811i	−0.253116	+	1.59811i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	2.34500	+	2.34500i	2.34500	+	2.34500i
$722$		0			0
$723$	0.951057	+	0.309017i	0.951057	+	0.309017i
$724$	0.618034			0.618034
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.587785	+	0.809017i	0.587785	+	0.809017i
$730$		0			0
$731$		0			0
$732$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$733$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$733$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$739$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$740$		0			0
$741$	−0.221502	−	0.434722i	−0.221502	−	0.434722i
$742$		0			0
$743$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$743$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.843566	−	0.987688i	−0.843566	−	0.987688i	0.156434	−	0.987688i	$-0.450000\pi$
$751$	−0.843566	−	0.987688i	−0.843566	−	0.987688i	−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.453990	−	1.89101i	−0.453990	−	1.89101i
$757$	−1.10749	−	1.29671i	−1.10749	−	1.29671i	−0.951057	−	0.309017i	$-0.900000\pi$
$757$	−1.10749	−	1.29671i	−1.10749	−	1.29671i	−0.156434	−	0.987688i	$-0.550000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$761$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$762$		0			0
$763$	2.11215	−	2.90713i	2.11215	−	2.90713i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.987688	+	0.156434i	0.987688	+	0.156434i
$769$	0.108993	+	0.453990i	0.108993	+	0.453990i	1.00000			$0$
$769$	0.108993	+	0.453990i	0.108993	+	0.453990i	−0.891007	+	0.453990i	$0.850000\pi$
$770$		0			0
$771$		0			0
$772$	0.0489435	−	0.309017i	0.0489435	−	0.309017i
$773$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$773$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$774$		0			0
$775$	0.744220	−	0.178671i	0.744220	−	0.178671i
$776$		0			0
$777$	0.780582	+	2.40238i	0.780582	+	2.40238i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.435203	+	2.74776i	−0.435203	+	2.74776i
$785$		0			0
$786$		0			0
$787$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$787$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.98158	−	0.155953i	1.98158	−	0.155953i
$794$		0			0
$795$		0			0
$796$	−0.465451	+	1.93874i	−0.465451	+	1.93874i
$797$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$797$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.996917	−	0.0784591i	$-0.0250000\pi$
$809$		0			0		−0.996917	+	0.0784591i	$0.975000\pi$
$810$		0			0
$811$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$811$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$	−0.183900	−	0.253116i	−0.183900	−	0.253116i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.0226397	−	0.0696779i	0.0226397	−	0.0696779i
$818$		0			0
$819$	1.19453	+	1.64412i	1.19453	+	1.64412i
$820$		0			0
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$823$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$827$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$828$		0			0
$829$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$829$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$830$		0			0
$831$	−0.142040	+	0.278768i	−0.142040	+	0.278768i
$832$	−1.01612	+	0.243950i	−1.01612	+	0.243950i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.652583	+	0.399903i	0.652583	+	0.399903i
$838$		0			0
$839$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$839$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$840$		0			0
$841$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$842$		0			0
$843$		0			0
$844$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$845$		0			0
$846$		0			0
$847$	0.152583	−	1.93874i	0.152583	−	1.93874i
$848$		0			0
$849$	1.40505	+	0.581990i	1.40505	+	0.581990i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	0.309017	+	0.951057i	$0.400000\pi$
$853$	−0.581990	+	1.40505i	−0.581990	+	1.40505i	−0.891007	+	0.453990i	$0.850000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$857$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$858$		0			0
$859$	−0.355026	+	0.303221i	−0.355026	+	0.303221i	−0.809017	−	0.587785i	$-0.800000\pi$
$859$	−0.355026	+	0.303221i	−0.355026	+	0.303221i	0.453990	+	0.891007i	$0.350000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.760406	−	0.649448i	$-0.775000\pi$
$863$		0			0		0.760406	+	0.649448i	$0.225000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.951057	+	0.309017i	0.951057	+	0.309017i
$868$	−0.874883	−	1.20417i	−0.874883	−	1.20417i
$869$		0			0
$870$		0			0
$871$	1.19453	+	0.286780i	1.19453	+	0.286780i
$872$		0			0
$873$	−1.59811	−	1.16110i	−1.59811	−	1.16110i
$874$		0			0
$875$		0			0
$876$	1.84206	+	0.763007i	1.84206	+	0.763007i
$877$	0.280582	+	0.863541i	0.280582	+	0.863541i	0.987688	+	0.156434i	$0.0500000\pi$
$877$	0.280582	+	0.863541i	0.280582	+	0.863541i	−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$881$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$882$		0			0
$883$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$883$	−1.53884	−	1.11803i	−1.53884	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$887$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$888$		0			0
$889$	2.92117	−	0.462668i	2.92117	−	0.462668i
$890$		0			0
$891$		0			0
$892$	0.278768	−	0.142040i	0.278768	−	0.142040i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$			1.00000i			1.00000i
$901$		0			0
$902$		0			0
$903$	−0.0477383	+	0.301408i	−0.0477383	+	0.301408i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.93874	+	0.465451i	−1.93874	+	0.465451i	−0.951057	+	0.309017i	$0.900000\pi$
$907$	−1.93874	+	0.465451i	−1.93874	+	0.465451i	−0.987688	+	0.156434i	$0.950000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$911$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$912$	−0.453990	+	0.108993i	−0.453990	+	0.108993i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.59811	+	0.253116i	1.59811	+	0.253116i
$917$		0			0
$918$		0			0
$919$	−1.95106	−	0.309017i	−1.95106	−	0.309017i	−0.951057	−	0.309017i	$-0.900000\pi$
$919$	−1.95106	−	0.309017i	−1.95106	−	0.309017i	−1.00000			$\pi$
$920$		0			0
$921$	−1.10749	+	1.29671i	−1.10749	+	1.29671i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.101910	−	1.29489i	−0.101910	−	1.29489i
$926$		0			0
$927$	0.398090	−	1.65816i	0.398090	−	1.65816i
$928$		0			0
$929$		0			0		0.649448	−	0.760406i	$-0.275000\pi$
$929$		0			0		−0.649448	+	0.760406i	$0.725000\pi$
$930$		0			0
$931$	−0.303221	−	1.26301i	−0.303221	−	1.26301i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.58779	+	0.809017i	1.58779	+	0.809017i	1.00000			$0$
$937$	1.58779	+	0.809017i	1.58779	+	0.809017i	0.587785	+	0.809017i	$0.300000\pi$
$938$		0			0
$939$	−1.69480	+	0.550672i	−1.69480	+	0.550672i
$940$		0			0
$941$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$941$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$947$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$948$	−1.61803			−1.61803
$949$	−2.08355			−2.08355
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$953$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.409114	−	0.0647973i	−0.409114	−	0.0647973i
$962$		0			0
$963$		0			0
$964$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$965$		0			0
$966$		0			0
$967$	−0.610425	−	0.0966818i	−0.610425	−	0.0966818i	−0.156434	−	0.987688i	$-0.550000\pi$
$967$	−0.610425	−	0.0966818i	−0.610425	−	0.0966818i	−0.453990	+	0.891007i	$0.650000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$971$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$972$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$973$	3.41754	+	1.11042i	3.41754	+	1.11042i
$974$		0			0
$975$	−0.399903	−	0.965451i	−0.399903	−	0.965451i
$976$	0.297556	−	1.87869i	0.297556	−	1.87869i
$977$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$977$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.84206	−	0.144974i	−1.84206	−	0.144974i
$982$		0			0
$983$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$983$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.394719	−	0.286780i	0.394719	−	0.286780i
$989$		0			0
$990$		0			0
$991$	1.04744	+	0.533698i	1.04744	+	0.533698i	0.891007	−	0.453990i	$-0.150000\pi$
$991$	1.04744	+	0.533698i	1.04744	+	0.533698i	0.156434	+	0.987688i	$0.450000\pi$
$992$		0			0
$993$	0.412215	+	0.809017i	0.412215	+	0.809017i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.465451	+	1.93874i	0.465451	+	1.93874i	0.309017	+	0.951057i	$0.400000\pi$
$997$	0.465451	+	1.93874i	0.465451	+	1.93874i	0.156434	+	0.987688i	$0.450000\pi$
$998$		0			0
$999$	0.843566	−	0.987688i	0.843566	−	0.987688i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	723.1.bc.a.455.1	yes	16
3.2	odd	2	CM	723.1.bc.a.455.1	yes	16
241.116	even	40	inner	723.1.bc.a.116.1	✓	16
723.116	odd	40	inner	723.1.bc.a.116.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
723.1.bc.a.116.1	✓	16	241.116	even	40	inner
723.1.bc.a.116.1	✓	16	723.116	odd	40	inner
723.1.bc.a.455.1	yes	16	1.1	even	1	trivial
723.1.bc.a.455.1	yes	16	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 723 = 3 \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	723.bc (of order \(40\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.987688 + 0.156434i) q^{3} +1.00000i q^{4} +(-1.47879 + 1.26301i) q^{7} +(0.951057 + 0.309017i) q^{9} +O(q^{10})\)	\(q+(0.987688 + 0.156434i) q^{3} +1.00000i q^{4} +(-1.47879 + 1.26301i) q^{7} +(0.951057 + 0.309017i) q^{9} +(-0.156434 + 0.987688i) q^{12} +(-0.243950 - 1.01612i) q^{13} -1.00000 q^{16} +(0.431351 - 0.178671i) q^{19} +(-1.65816 + 1.01612i) q^{21} +(0.951057 - 0.309017i) q^{25} +(0.891007 + 0.453990i) q^{27} +(-1.26301 - 1.47879i) q^{28} +(0.763007 + 0.0600500i) q^{31} +(-0.309017 + 0.951057i) q^{36} +(0.303221 - 1.26301i) q^{37} +(-0.0819895 - 1.04178i) q^{39} +(0.101910 - 0.119322i) q^{43} +(-0.987688 - 0.156434i) q^{48} +(0.435203 - 2.74776i) q^{49} +(1.01612 - 0.243950i) q^{52} +(0.453990 - 0.108993i) q^{57} +(-0.297556 + 1.87869i) q^{61} +(-1.79671 + 0.744220i) q^{63} -1.00000i q^{64} +(-0.533698 + 1.04744i) q^{67} +(0.465451 - 1.93874i) q^{73} +(0.987688 - 0.156434i) q^{75} +(0.178671 + 0.431351i) q^{76} +(0.253116 + 1.59811i) q^{79} +(0.809017 + 0.587785i) q^{81} +(-1.01612 - 1.65816i) q^{84} +(1.64412 + 1.19453i) q^{91} +(0.744220 + 0.178671i) q^{93} +(-1.87869 - 0.610425i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{16} - 4 q^{28} - 4 q^{31} + 4 q^{36} + 4 q^{39} + 4 q^{43} + 4 q^{49} - 4 q^{52} - 4 q^{63} - 4 q^{73} - 4 q^{76} + 4 q^{81} + 4 q^{84} + 8 q^{91}+O(q^{100}) \) 16 * q - 16 * q^16 - 4 * q^28 - 4 * q^31 + 4 * q^36 + 4 * q^39 + 4 * q^43 + 4 * q^49 - 4 * q^52 - 4 * q^63 - 4 * q^73 - 4 * q^76 + 4 * q^81 + 4 * q^84 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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