LMFDB - Embedded newform 1472.1.s.a.597.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,1,Mod(45,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([0, 7, 8]))

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

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

chi := DirichletCharacter("1472.45");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1472.s (of order $16$, degree $8$, 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:	$0.734623698596$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} + \cdots)$

Embedding invariants

Embedding label			597.1
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1472.597
Dual form			1472.1.s.a.1149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.382683 + 0.923880i) q^{2} +(0.382683 + 1.92388i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(-1.63099 + 1.08979i) q^{6} +(-0.923880 - 0.382683i) q^{8} +(-2.63099 + 1.08979i) q^{9} +(-1.63099 - 1.08979i) q^{12} +(1.63099 + 1.08979i) q^{13} -1.00000i q^{16} +(-2.01367 - 2.01367i) q^{18} +(-0.382683 - 0.923880i) q^{23} +(0.382683 - 1.92388i) q^{24} +(0.382683 - 0.923880i) q^{25} +(-0.382683 + 1.92388i) q^{26} +(-2.01367 - 3.01367i) q^{27} +(1.08979 - 0.216773i) q^{29} +0.765367i q^{31} +(0.923880 - 0.382683i) q^{32} +(1.08979 - 2.63099i) q^{36} +(-1.47247 + 3.55487i) q^{39} +(-0.765367 - 1.84776i) q^{41} +(0.707107 - 0.707107i) q^{46} +(-1.00000 + 1.00000i) q^{47} +(1.92388 - 0.382683i) q^{48} +(0.707107 + 0.707107i) q^{49} +1.00000 q^{50} +(-1.92388 + 0.382683i) q^{52} +(2.01367 - 3.01367i) q^{54} +(0.617317 + 0.923880i) q^{58} +(-0.324423 + 0.216773i) q^{59} +(-0.707107 + 0.292893i) q^{62} +(0.707107 + 0.707107i) q^{64} +(1.63099 - 1.08979i) q^{69} +(1.70711 + 0.707107i) q^{71} +2.84776 q^{72} +(-1.30656 + 0.541196i) q^{73} +(1.92388 + 0.382683i) q^{75} -3.84776 q^{78} +(3.01367 - 3.01367i) q^{81} +(1.41421 - 1.41421i) q^{82} +(0.834089 + 2.01367i) q^{87} +(0.923880 + 0.382683i) q^{92} +(-1.47247 + 0.292893i) q^{93} +(-1.30656 - 0.541196i) q^{94} +(1.08979 + 1.63099i) q^{96} +(-0.382683 + 0.923880i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 8 q^{47} + 8 q^{48} + 8 q^{50} - 8 q^{52} + 8 q^{58} + 8 q^{71} + 8 q^{72} + 8 q^{75} - 16 q^{78} + 8 q^{81} + 8 q^{87}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^47 + 8 * q^48 + 8 * q^50 - 8 * q^52 + 8 * q^58 + 8 * q^71 + 8 * q^72 + 8 * q^75 - 16 * q^78 + 8 * q^81 + 8 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$.

$n$	$645$	$833$	$1151$
$\chi(n)$	$e\left(\frac{13}{16}\right)$	$-1$	$1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0.382683	+	0.923880i	0.382683	+	0.923880i
$2$	0.382683	+	0.923880i	0.382683	+	0.923880i
$3$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$3$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$4$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$5$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$5$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$6$	−1.63099	+	1.08979i	−1.63099	+	1.08979i
$7$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$7$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$8$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$9$	−2.63099	+	1.08979i	−2.63099	+	1.08979i
$10$		0			0
$11$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$11$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$12$	−1.63099	−	1.08979i	−1.63099	−	1.08979i
$13$	1.63099	+	1.08979i	1.63099	+	1.08979i	0.923880	+	0.382683i	$0.125000\pi$
$13$	1.63099	+	1.08979i	1.63099	+	1.08979i	0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		−	1.00000i		−	1.00000i
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	−2.01367	−	2.01367i	−2.01367	−	2.01367i
$19$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$19$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$23$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$24$	0.382683	−	1.92388i	0.382683	−	1.92388i
$25$	0.382683	−	0.923880i	0.382683	−	0.923880i
$26$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$27$	−2.01367	−	3.01367i	−2.01367	−	3.01367i
$28$		0			0
$29$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$29$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$31$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$32$	0.923880	−	0.382683i	0.923880	−	0.382683i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.08979	−	2.63099i	1.08979	−	2.63099i
$37$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$37$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$38$		0			0
$39$	−1.47247	+	3.55487i	−1.47247	+	3.55487i
$40$		0			0
$41$	−0.765367	−	1.84776i	−0.765367	−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$41$	−0.765367	−	1.84776i	−0.765367	−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$42$		0			0
$43$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$43$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$44$		0			0
$45$		0			0
$46$	0.707107	−	0.707107i	0.707107	−	0.707107i
$47$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$47$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$48$	1.92388	−	0.382683i	1.92388	−	0.382683i
$49$	0.707107	+	0.707107i	0.707107	+	0.707107i
$50$	1.00000			1.00000
$51$		0			0
$52$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$53$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$53$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$54$	2.01367	−	3.01367i	2.01367	−	3.01367i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.617317	+	0.923880i	0.617317	+	0.923880i
$59$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	0.382683	+	0.923880i	$0.375000\pi$
$60$		0			0
$61$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$61$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$62$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$63$		0			0
$64$	0.707107	+	0.707107i	0.707107	+	0.707107i
$65$		0			0
$66$		0			0
$67$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$67$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$68$		0			0
$69$	1.63099	−	1.08979i	1.63099	−	1.08979i
$70$		0			0
$71$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$71$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$72$	2.84776			2.84776
$73$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$73$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$74$		0			0
$75$	1.92388	+	0.382683i	1.92388	+	0.382683i
$76$		0			0
$77$		0			0
$78$	−3.84776			−3.84776
$79$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$79$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$80$		0			0
$81$	3.01367	−	3.01367i	3.01367	−	3.01367i
$82$	1.41421	−	1.41421i	1.41421	−	1.41421i
$83$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$83$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.834089	+	2.01367i	0.834089	+	2.01367i
$88$		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
$91$		0			0
$92$	0.923880	+	0.382683i	0.923880	+	0.382683i
$93$	−1.47247	+	0.292893i	−1.47247	+	0.292893i
$94$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$95$		0			0
$96$	1.08979	+	1.63099i	1.08979	+	1.63099i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$99$		0			0
$100$	0.382683	+	0.923880i	0.382683	+	0.923880i
$101$	0.216773	+	0.324423i	0.216773	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$101$	0.216773	+	0.324423i	0.216773	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$103$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$104$	−1.08979	−	1.63099i	−1.08979	−	1.63099i
$105$		0			0
$106$		0			0
$107$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$107$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$108$	3.55487	+	0.707107i	3.55487	+	0.707107i
$109$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$109$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$117$	−5.47875	−	1.08979i	−5.47875	−	1.08979i
$118$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$119$		0			0
$120$		0			0
$121$	0.923880	+	0.382683i	0.923880	+	0.382683i
$122$		0			0
$123$	3.26197	−	2.17958i	3.26197	−	2.17958i
$124$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$125$		0			0
$126$		0			0
$127$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$127$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$128$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$129$		0			0
$130$		0			0
$131$	−0.216773	−	1.08979i	−0.216773	−	1.08979i	−0.923880	−	0.382683i	$-0.875000\pi$
$131$	−0.216773	−	1.08979i	−0.216773	−	1.08979i	0.707107	−	0.707107i	$-0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$137$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$138$	1.63099	+	1.08979i	1.63099	+	1.08979i
$139$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$139$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.923880	+	0.382683i	$0.875000\pi$
$140$		0			0
$141$	−2.30656	−	1.54120i	−2.30656	−	1.54120i
$142$			1.84776i			1.84776i
$143$		0			0
$144$	1.08979	+	2.63099i	1.08979	+	2.63099i
$145$		0			0
$146$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$147$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$148$		0			0
$149$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$149$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$150$	0.382683	+	1.92388i	0.382683	+	1.92388i
$151$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$151$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.47247	−	3.55487i	−1.47247	−	3.55487i
$157$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$157$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	3.93755	+	1.63099i	3.93755	+	1.63099i
$163$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$163$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$164$	1.84776	+	0.765367i	1.84776	+	0.765367i
$165$		0			0
$166$		0			0
$167$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$167$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$168$		0			0
$169$	1.08979	+	2.63099i	1.08979	+	2.63099i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.923880	−	1.38268i	0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$173$	0.923880	−	1.38268i	0.923880	−	1.38268i	0.923880	−	0.382683i	$-0.125000\pi$
$174$	−1.54120	+	1.54120i	−1.54120	+	1.54120i
$175$		0			0
$176$		0			0
$177$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$178$		0			0
$179$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$179$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$181$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$182$		0			0
$183$		0			0
$184$			1.00000i			1.00000i
$185$		0			0
$186$	−0.834089	−	1.24830i	−0.834089	−	1.24830i
$187$		0			0
$188$		−	1.41421i		−	1.41421i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−1.00000			−1.00000
$197$	0.923880	−	0.617317i	0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$197$	0.923880	−	0.617317i	0.923880	−	0.617317i	0.923880	+	0.382683i	$0.125000\pi$
$198$		0			0
$199$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$199$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$200$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$201$		0			0
$202$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.01367	+	2.01367i	2.01367	+	2.01367i
$208$	1.08979	−	1.63099i	1.08979	−	1.63099i
$209$		0			0
$210$		0			0
$211$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.382683	−	0.923880i	$-0.375000\pi$
$211$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$212$		0			0
$213$	−0.707107	+	3.55487i	−0.707107	+	3.55487i
$214$		0			0
$215$		0			0
$216$	0.707107	+	3.55487i	0.707107	+	3.55487i
$217$		0			0
$218$		0			0
$219$	−1.54120	−	2.30656i	−1.54120	−	2.30656i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$223$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$224$		0			0
$225$			2.84776i			2.84776i
$226$		0			0
$227$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$227$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$228$		0			0
$229$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$229$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$230$		0			0
$231$		0			0
$232$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$233$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$233$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$234$	−1.08979	−	5.47875i	−1.08979	−	5.47875i
$235$		0			0
$236$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$237$		0			0
$238$		0			0
$239$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$239$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$240$		0			0
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$			1.00000i			1.00000i
$243$	3.93755	+	2.63099i	3.93755	+	2.63099i
$244$		0			0
$245$		0			0
$246$	3.26197	+	2.17958i	3.26197	+	2.17958i
$247$		0			0
$248$	0.292893	−	0.707107i	0.292893	−	0.707107i
$249$		0			0
$250$		0			0
$251$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$251$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$252$		0			0
$253$		0			0
$254$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$255$		0			0
$256$	−1.00000			−1.00000
$257$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$257$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.63099	+	1.75797i	−2.63099	+	1.75797i
$262$	0.923880	−	0.617317i	0.923880	−	0.617317i
$263$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$263$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−0.382683	−	0.923880i	$-0.625000\pi$
$269$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−1.00000			$\pi$
$270$		0			0
$271$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$271$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$277$	0.0761205	−	0.382683i	0.0761205	−	0.382683i	−0.923880	−	0.382683i	$-0.875000\pi$
$277$	0.0761205	−	0.382683i	0.0761205	−	0.382683i	1.00000			$0$
$278$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$279$	−0.834089	−	2.01367i	−0.834089	−	2.01367i
$280$		0			0
$281$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$281$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$282$	0.541196	−	2.72078i	0.541196	−	2.72078i
$283$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$283$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$284$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.01367	+	2.01367i	−2.01367	+	2.01367i
$289$		−	1.00000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$	0.541196	−	1.30656i	0.541196	−	1.30656i
$293$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$293$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$294$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.382683	−	1.92388i	0.382683	−	1.92388i
$300$	−1.63099	+	1.08979i	−1.63099	+	1.08979i
$301$		0			0
$302$		0			0
$303$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.923880	−	0.617317i	−0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$307$	−0.923880	−	0.617317i	−0.923880	−	0.617317i	−0.923880	+	0.382683i	$0.875000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$311$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$312$	2.72078	−	2.72078i	2.72078	−	2.72078i
$313$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$313$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.216773	+	1.08979i	0.216773	+	1.08979i	0.923880	+	0.382683i	$0.125000\pi$
$317$	0.216773	+	1.08979i	0.216773	+	1.08979i	−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$			4.26197i			4.26197i
$325$	1.63099	−	1.08979i	1.63099	−	1.08979i
$326$	0.216773	+	0.324423i	0.216773	+	0.324423i
$327$		0			0
$328$			2.00000i			2.00000i
$329$		0			0
$330$		0			0
$331$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$331$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$332$		0			0
$333$		0			0
$334$	−0.765367			−0.765367
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−2.01367	+	2.01367i	−2.01367	+	2.01367i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.63099	+	0.324423i	1.63099	+	0.324423i
$347$	0.617317	+	0.923880i	0.617317	+	0.923880i	1.00000			$0$
$347$	0.617317	+	0.923880i	0.617317	+	0.923880i	−0.382683	+	0.923880i	$0.625000\pi$
$348$	−2.01367	−	0.834089i	−2.01367	−	0.834089i
$349$	1.92388	−	0.382683i	1.92388	−	0.382683i	0.923880	−	0.382683i	$-0.125000\pi$
$349$	1.92388	−	0.382683i	1.92388	−	0.382683i	1.00000			$0$
$350$		0			0
$351$		−	7.10973i		−	7.10973i
$352$		0			0
$353$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$353$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$354$	0.292893	−	0.707107i	0.292893	−	0.707107i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$359$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$359$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$360$		0			0
$361$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$362$		0			0
$363$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$369$	4.02734	+	4.02734i	4.02734	+	4.02734i
$370$		0			0
$371$		0			0
$372$	0.834089	−	1.24830i	0.834089	−	1.24830i
$373$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$373$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$374$		0			0
$375$		0			0
$376$	1.30656	−	0.541196i	1.30656	−	0.541196i
$377$	2.01367	+	0.834089i	2.01367	+	0.834089i
$378$		0			0
$379$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$379$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$380$		0			0
$381$	−0.541196	−	2.72078i	−0.541196	−	2.72078i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$389$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$390$		0			0
$391$		0			0
$392$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$393$	2.01367	−	0.834089i	2.01367	−	0.834089i
$394$	0.923880	+	0.617317i	0.923880	+	0.617317i
$395$		0			0
$396$		0			0
$397$	1.38268	+	0.923880i	1.38268	+	0.923880i	1.00000			$0$
$397$	1.38268	+	0.923880i	1.38268	+	0.923880i	0.382683	+	0.923880i	$0.375000\pi$
$398$		0			0
$399$		0			0
$400$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$	−0.834089	+	1.24830i	−0.834089	+	1.24830i
$404$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$409$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−1.08979	+	2.63099i	−1.08979	+	2.63099i
$415$		0			0
$416$	1.92388	+	0.382683i	1.92388	+	0.382683i
$417$		−	3.26197i		−	3.26197i
$418$		0			0
$419$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$419$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$420$		0			0
$421$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$421$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$422$	1.92388	+	0.382683i	1.92388	+	0.382683i
$423$	1.54120	−	3.72078i	1.54120	−	3.72078i
$424$		0			0
$425$		0			0
$426$	−3.55487	+	0.707107i	−3.55487	+	0.707107i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$	−3.01367	+	2.01367i	−3.01367	+	2.01367i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.54120	−	2.30656i	1.54120	−	2.30656i
$439$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$439$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$440$		0			0
$441$	−2.63099	−	1.08979i	−2.63099	−	1.08979i
$442$		0			0
$443$	−1.38268	+	0.923880i	−1.38268	+	0.923880i	−0.382683	+	0.923880i	$0.625000\pi$
$443$	−1.38268	+	0.923880i	−1.38268	+	0.923880i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$447$		0			0
$448$		0			0
$449$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$449$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$450$	−2.63099	+	1.08979i	−2.63099	+	1.08979i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$457$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.324423	+	0.216773i	0.324423	+	0.216773i	0.707107	−	0.707107i	$-0.250000\pi$
$461$	0.324423	+	0.216773i	0.324423	+	0.216773i	−0.382683	+	0.923880i	$0.625000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$465$		0			0
$466$	0.541196	−	0.541196i	0.541196	−	0.541196i
$467$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$467$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$468$	4.64466	−	3.10346i	4.64466	−	3.10346i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.70711	+	0.707107i	1.70711	+	0.707107i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$485$		0			0
$486$	−0.923880	+	4.64466i	−0.923880	+	4.64466i
$487$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$487$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$488$		0			0
$489$	0.292893	+	0.707107i	0.292893	+	0.707107i
$490$		0			0
$491$	−0.216773	+	1.08979i	−0.216773	+	1.08979i	0.707107	+	0.707107i	$0.250000\pi$
$491$	−0.216773	+	1.08979i	−0.216773	+	1.08979i	−0.923880	+	0.382683i	$0.875000\pi$
$492$	−0.765367	+	3.84776i	−0.765367	+	3.84776i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.765367			0.765367
$497$		0			0
$498$		0			0
$499$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−0.382683	−	0.923880i	$-0.625000\pi$
$499$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−1.00000			$\pi$
$500$		0			0
$501$	−1.47247	−	0.292893i	−1.47247	−	0.292893i
$502$		0			0
$503$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$503$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−4.64466	+	3.10346i	−4.64466	+	3.10346i
$508$	1.00000	−	1.00000i	1.00000	−	1.00000i
$509$	0.0761205	+	0.382683i	0.0761205	+	0.382683i	1.00000			$0$
$509$	0.0761205	+	0.382683i	0.0761205	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$510$		0			0
$511$		0			0
$512$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$513$		0			0
$514$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	3.01367	+	1.24830i	3.01367	+	1.24830i
$520$		0			0
$521$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$521$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$522$	−2.63099	−	1.75797i	−2.63099	−	1.75797i
$523$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$523$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$524$	0.923880	+	0.617317i	0.923880	+	0.617317i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$530$		0			0
$531$	0.617317	−	0.923880i	0.617317	−	0.923880i
$532$		0			0
$533$	0.765367	−	3.84776i	0.765367	−	3.84776i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.292893	−	0.707107i	0.292893	−	0.707107i
$538$	0.324423	−	1.63099i	0.324423	−	1.63099i
$539$		0			0
$540$		0			0
$541$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$541$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$542$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$547$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$553$		0			0
$554$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$555$		0			0
$556$	1.38268	−	0.923880i	1.38268	−	0.923880i
$557$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$557$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$558$	1.54120	−	1.54120i	1.54120	−	1.54120i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$563$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$564$	2.72078	−	0.541196i	2.72078	−	0.541196i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$569$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$569$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$570$		0			0
$571$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$571$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−1.00000
$576$	−2.63099	−	1.08979i	−2.63099	−	1.08979i
$577$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$577$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$578$	0.923880	−	0.382683i	0.923880	−	0.382683i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.41421			1.41421
$585$		0			0
$586$		0			0
$587$	1.63099	+	0.324423i	1.63099	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$587$	1.63099	+	0.324423i	1.63099	+	0.324423i	0.707107	+	0.707107i	$0.250000\pi$
$588$	−0.382683	−	1.92388i	−0.382683	−	1.92388i
$589$		0			0
$590$		0			0
$591$	1.54120	+	1.54120i	1.54120	+	1.54120i
$592$		0			0
$593$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$593$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.92388	−	0.382683i	1.92388	−	0.382683i
$599$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$599$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$600$	−1.63099	−	1.08979i	−1.63099	−	1.08979i
$601$	−0.765367	+	1.84776i	−0.765367	+	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$601$	−0.765367	+	1.84776i	−0.765367	+	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$607$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$607$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.72078	+	0.541196i	−2.72078	+	0.541196i
$612$		0			0
$613$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$613$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$614$	0.216773	−	1.08979i	0.216773	−	1.08979i
$615$		0			0
$616$		0			0
$617$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$617$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$618$		0			0
$619$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$619$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$620$		0			0
$621$	−2.01367	+	3.01367i	−2.01367	+	3.01367i
$622$		0			0
$623$		0			0
$624$	3.55487	+	1.47247i	3.55487	+	1.47247i
$625$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$631$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$	3.55487	+	1.47247i	3.55487	+	1.47247i
$634$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$635$		0			0
$636$		0			0
$637$	0.382683	+	1.92388i	0.382683	+	1.92388i
$638$		0			0
$639$	−5.26197			−5.26197
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$643$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.84776	−	0.765367i	−1.84776	−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$647$	−1.84776	−	0.765367i	−1.84776	−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$648$	−3.93755	+	1.63099i	−3.93755	+	1.63099i
$649$		0			0
$650$	1.63099	+	1.08979i	1.63099	+	1.08979i
$651$		0			0
$652$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$653$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$653$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$657$	2.84776	−	2.84776i	2.84776	−	2.84776i
$658$		0			0
$659$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$659$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$660$		0			0
$661$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$661$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$662$	0.382683	+	1.92388i	0.382683	+	1.92388i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.617317	−	0.923880i	−0.617317	−	0.923880i
$668$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$669$	−1.47247	+	0.292893i	−1.47247	+	0.292893i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$673$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$674$		0			0
$675$	−3.55487	+	0.707107i	−3.55487	+	0.707107i
$676$	−2.63099	−	1.08979i	−2.63099	−	1.08979i
$677$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$677$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.324423	−	1.63099i	0.324423	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$683$	0.324423	−	1.63099i	0.324423	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.923880	+	0.617317i	0.923880	+	0.617317i	0.923880	−	0.382683i	$-0.125000\pi$
$691$	0.923880	+	0.617317i	0.923880	+	0.617317i			1.00000i	$0.5\pi$
$692$	0.324423	+	1.63099i	0.324423	+	1.63099i
$693$		0			0
$694$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$695$		0			0
$696$		−	2.17958i		−	2.17958i
$697$		0			0
$698$	1.08979	+	1.63099i	1.08979	+	1.63099i
$699$	1.24830	−	0.834089i	1.24830	−	0.834089i
$700$		0			0
$701$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$701$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$702$	6.56854	−	2.72078i	6.56854	−	2.72078i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.30656	−	0.541196i	1.30656	−	0.541196i
$707$		0			0
$708$	0.765367			0.765367
$709$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$709$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.707107	−	0.292893i	0.707107	−	0.292893i
$714$		0			0
$715$		0			0
$716$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$717$	3.01367	+	2.01367i	3.01367	+	2.01367i
$718$		0			0
$719$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$719$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$720$		0			0
$721$		0			0
$722$	0.707107	−	0.707107i	0.707107	−	0.707107i
$723$		0			0
$724$		0			0
$725$	0.216773	−	1.08979i	0.216773	−	1.08979i
$726$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$727$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$727$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$728$		0			0
$729$	−1.92388	+	4.64466i	−1.92388	+	4.64466i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$733$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$734$		0			0
$735$		0			0
$736$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$737$		0			0
$738$	−2.17958	+	5.26197i	−2.17958	+	5.26197i
$739$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$739$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.382683	+	0.923880i	$0.625000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$743$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$744$	1.47247	+	0.292893i	1.47247	+	0.292893i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$751$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$752$	1.00000	+	1.00000i	1.00000	+	1.00000i
$753$		0			0
$754$			2.17958i			2.17958i
$755$		0			0
$756$		0			0
$757$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$757$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$761$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$762$	2.30656	−	1.54120i	2.30656	−	1.54120i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.765367			−0.765367
$768$	−0.382683	−	1.92388i	−0.382683	−	1.92388i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−0.707107	−	3.55487i	−0.707107	−	3.55487i
$772$		0			0
$773$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$773$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$774$		0			0
$775$	0.707107	+	0.292893i	0.707107	+	0.292893i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−2.84776	−	2.84776i	−2.84776	−	2.84776i
$784$	0.707107	−	0.707107i	0.707107	−	0.707107i
$785$		0			0
$786$	1.54120	+	1.54120i	1.54120	+	1.54120i
$787$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$787$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$788$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.324423	+	1.63099i	−0.324423	+	1.63099i
$795$		0			0
$796$		0			0
$797$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$797$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$798$		0			0
$799$		0			0
$800$		−	1.00000i		−	1.00000i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−1.47247	−	0.292893i	−1.47247	−	0.292893i
$807$	1.24830	−	3.01367i	1.24830	−	3.01367i
$808$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$809$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$809$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$810$		0			0
$811$	0.324423	−	1.63099i	0.324423	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$811$	0.324423	−	1.63099i	0.324423	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$812$		0			0
$813$	−1.54120	+	2.30656i	−1.54120	+	2.30656i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.765367			0.765367
$819$		0			0
$820$		0			0
$821$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$821$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$822$		0			0
$823$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$823$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$827$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$828$	−2.84776			−2.84776
$829$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$829$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$830$		0			0
$831$	0.765367			0.765367
$832$	0.382683	+	1.92388i	0.382683	+	1.92388i
$833$		0			0
$834$	3.01367	−	1.24830i	3.01367	−	1.24830i
$835$		0			0
$836$		0			0
$837$	2.30656	−	1.54120i	2.30656	−	1.54120i
$838$		0			0
$839$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$839$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$840$		0			0
$841$	0.216773	−	0.0897902i	0.216773	−	0.0897902i
$842$		0			0
$843$		0			0
$844$	0.382683	+	1.92388i	0.382683	+	1.92388i
$845$		0			0
$846$	4.02734			4.02734
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−2.01367	−	3.01367i	−2.01367	−	3.01367i
$853$	0.324423	−	1.63099i	0.324423	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$853$	0.324423	−	1.63099i	0.324423	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$857$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.707107	−	0.707107i	$-0.750000\pi$
$859$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$863$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$864$	−3.01367	−	2.01367i	−3.01367	−	2.01367i
$865$		0			0
$866$		0			0
$867$	1.92388	−	0.382683i	1.92388	−	0.382683i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	2.72078	+	0.541196i	2.72078	+	0.541196i
$877$	−0.216773	+	0.324423i	−0.216773	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$877$	−0.216773	+	0.324423i	−0.216773	+	0.324423i	0.707107	+	0.707107i	$0.250000\pi$
$878$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$879$		0			0
$880$		0			0
$881$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$881$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$882$		−	2.84776i		−	2.84776i
$883$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−0.382683	−	0.923880i	$-0.625000\pi$
$883$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$887$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$887$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	3.84776			3.84776
$898$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$899$	0.165911	+	0.834089i	0.165911	+	0.834089i
$900$	−2.01367	−	2.01367i	−2.01367	−	2.01367i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$907$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$908$		0			0
$909$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$919$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$920$		0			0
$921$	0.834089	−	2.01367i	0.834089	−	2.01367i
$922$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$923$	2.01367	+	3.01367i	2.01367	+	3.01367i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	0.923880	−	0.617317i	0.923880	−	0.617317i
$929$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$929$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$930$		0			0
$931$		0			0
$932$	0.707107	+	0.292893i	0.707107	+	0.292893i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	4.64466	+	3.10346i	4.64466	+	3.10346i
$937$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$937$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$941$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$942$		0			0
$943$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$944$	0.216773	+	0.324423i	0.216773	+	0.324423i
$945$		0			0
$946$		0			0
$947$	0.923880	+	0.617317i	0.923880	+	0.617317i	0.923880	−	0.382683i	$-0.125000\pi$
$947$	0.923880	+	0.617317i	0.923880	+	0.617317i			1.00000i	$0.5\pi$
$948$		0			0
$949$	−2.72078	−	0.541196i	−2.72078	−	0.541196i
$950$		0			0
$951$	−2.01367	+	0.834089i	−2.01367	+	0.834089i
$952$		0			0
$953$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$953$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$954$		0			0
$955$		0			0
$956$			1.84776i			1.84776i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.414214			0.414214
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.84776	+	0.765367i	1.84776	+	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$967$	1.84776	+	0.765367i	1.84776	+	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$968$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$971$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$972$	−4.64466	+	0.923880i	−4.64466	+	0.923880i
$973$		0			0
$974$	1.84776			1.84776
$975$	2.72078	+	2.72078i	2.72078	+	2.72078i
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$983$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$983$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$984$	−3.84776	+	0.765367i	−3.84776	+	0.765367i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$991$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$992$	0.292893	+	0.707107i	0.292893	+	0.707107i
$993$			3.84776i			3.84776i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.617317	+	0.923880i	0.617317	+	0.923880i	1.00000			$0$
$997$	0.617317	+	0.923880i	0.617317	+	0.923880i	−0.382683	+	0.923880i	$0.625000\pi$
$998$	0.324423	−	1.63099i	0.324423	−	1.63099i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.1.s.a.597.1	✓	8
23.22	odd	2	CM	1472.1.s.a.597.1	✓	8
64.61	even	16	inner	1472.1.s.a.1149.1	yes	8
1472.1149	odd	16	inner	1472.1.s.a.1149.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1472.1.s.a.597.1	✓	8	1.1	even	1	trivial
1472.1.s.a.597.1	✓	8	23.22	odd	2	CM
1472.1.s.a.1149.1	yes	8	64.61	even	16	inner
1472.1.s.a.1149.1	yes	8	1472.1149	odd	16	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1472.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.382683 + 0.923880i) q^{2} +(0.382683 + 1.92388i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(-1.63099 + 1.08979i) q^{6} +(-0.923880 - 0.382683i) q^{8} +(-2.63099 + 1.08979i) q^{9} +(-1.63099 - 1.08979i) q^{12} +(1.63099 + 1.08979i) q^{13} -1.00000i q^{16} +(-2.01367 - 2.01367i) q^{18} +(-0.382683 - 0.923880i) q^{23} +(0.382683 - 1.92388i) q^{24} +(0.382683 - 0.923880i) q^{25} +(-0.382683 + 1.92388i) q^{26} +(-2.01367 - 3.01367i) q^{27} +(1.08979 - 0.216773i) q^{29} +0.765367i q^{31} +(0.923880 - 0.382683i) q^{32} +(1.08979 - 2.63099i) q^{36} +(-1.47247 + 3.55487i) q^{39} +(-0.765367 - 1.84776i) q^{41} +(0.707107 - 0.707107i) q^{46} +(-1.00000 + 1.00000i) q^{47} +(1.92388 - 0.382683i) q^{48} +(0.707107 + 0.707107i) q^{49} +1.00000 q^{50} +(-1.92388 + 0.382683i) q^{52} +(2.01367 - 3.01367i) q^{54} +(0.617317 + 0.923880i) q^{58} +(-0.324423 + 0.216773i) q^{59} +(-0.707107 + 0.292893i) q^{62} +(0.707107 + 0.707107i) q^{64} +(1.63099 - 1.08979i) q^{69} +(1.70711 + 0.707107i) q^{71} +2.84776 q^{72} +(-1.30656 + 0.541196i) q^{73} +(1.92388 + 0.382683i) q^{75} -3.84776 q^{78} +(3.01367 - 3.01367i) q^{81} +(1.41421 - 1.41421i) q^{82} +(0.834089 + 2.01367i) q^{87} +(0.923880 + 0.382683i) q^{92} +(-1.47247 + 0.292893i) q^{93} +(-1.30656 - 0.541196i) q^{94} +(1.08979 + 1.63099i) q^{96} +(-0.382683 + 0.923880i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 8 q^{47} + 8 q^{48} + 8 q^{50} - 8 q^{52} + 8 q^{58} + 8 q^{71} + 8 q^{72} + 8 q^{75} - 16 q^{78} + 8 q^{81} + 8 q^{87}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^47 + 8 * q^48 + 8 * q^50 - 8 * q^52 + 8 * q^58 + 8 * q^71 + 8 * q^72 + 8 * q^75 - 16 * q^78 + 8 * q^81 + 8 * q^87

Introduction

L-functions

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