LMFDB - Embedded newform 624.1.bi.a.77.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,1,Mod(77,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))

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

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

chi := DirichletCharacter("624.77");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	624.bi (of order $4$, degree $2$, 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:	$0.311416567883$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
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:	$D_{8}$
Projective field:	Galois closure of 8.2.3234424946688.6

Embedding invariants

Embedding label			77.2
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	624.77
Dual form			624.1.bi.a.389.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.382683 - 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(-0.541196 + 0.541196i) q^{5} +(-0.382683 + 0.923880i) q^{6} +(0.923880 + 0.382683i) q^{8} +1.00000i q^{9} +O(q^{10})$	\(q+(-0.382683 - 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(-0.541196 + 0.541196i) q^{5} +(-0.382683 + 0.923880i) q^{6} +(0.923880 + 0.382683i) q^{8} +1.00000i q^{9} +(0.707107 + 0.292893i) q^{10} +(-1.30656 + 1.30656i) q^{11} +1.00000 q^{12} +(0.707107 + 0.707107i) q^{13} +0.765367 q^{15} -1.00000i q^{16} +(0.923880 - 0.382683i) q^{18} -0.765367i q^{20} +(1.70711 + 0.707107i) q^{22} +(-0.382683 - 0.923880i) q^{24} +0.414214i q^{25} +(0.382683 - 0.923880i) q^{26} +(0.707107 - 0.707107i) q^{27} +(-0.292893 - 0.707107i) q^{30} +(-0.923880 + 0.382683i) q^{32} +1.84776 q^{33} +(-0.707107 - 0.707107i) q^{36} -1.00000i q^{39} +(-0.707107 + 0.292893i) q^{40} +0.765367i q^{41} +(-1.00000 + 1.00000i) q^{43} -1.84776i q^{44} +(-0.541196 - 0.541196i) q^{45} +1.84776 q^{47} +(-0.707107 + 0.707107i) q^{48} -1.00000 q^{49} +(0.382683 - 0.158513i) q^{50} -1.00000 q^{52} +(-0.923880 - 0.382683i) q^{54} -1.41421i q^{55} +(-0.541196 + 0.541196i) q^{59} +(-0.541196 + 0.541196i) q^{60} +(0.707107 + 0.707107i) q^{64} -0.765367 q^{65} +(-0.707107 - 1.70711i) q^{66} -1.84776i q^{71} +(-0.382683 + 0.923880i) q^{72} +(0.292893 - 0.292893i) q^{75} +(-0.923880 + 0.382683i) q^{78} -1.41421 q^{79} +(0.541196 + 0.541196i) q^{80} -1.00000 q^{81} +(0.707107 - 0.292893i) q^{82} +(0.541196 + 0.541196i) q^{83} +(1.30656 + 0.541196i) q^{86} +(-1.70711 + 0.707107i) q^{88} +1.84776i q^{89} +(-0.292893 + 0.707107i) q^{90} +(-0.707107 - 1.70711i) q^{94} +(0.923880 + 0.382683i) q^{96} +(0.382683 + 0.923880i) q^{98} +(-1.30656 - 1.30656i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{12} + 8 q^{22} - 8 q^{30} - 8 q^{43} - 8 q^{49} - 8 q^{52} + 8 q^{75} - 8 q^{81} - 8 q^{88} - 8 q^{90}+O(q^{100}) $ 8 * q + 8 * q^12 + 8 * q^22 - 8 * q^30 - 8 * q^43 - 8 * q^49 - 8 * q^52 + 8 * q^75 - 8 * q^81 - 8 * q^88 - 8 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$2$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$4$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$5$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$5$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$6$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	0.923880	+	0.382683i	0.923880	+	0.382683i
$9$			1.00000i			1.00000i
$10$	0.707107	+	0.292893i	0.707107	+	0.292893i
$11$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$11$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$12$	1.00000			1.00000
$13$	0.707107	+	0.707107i	0.707107	+	0.707107i
$13$	0.707107	+	0.707107i	0.707107	+	0.707107i
$14$		0			0
$15$	0.765367			0.765367
$16$		−	1.00000i		−	1.00000i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.923880	−	0.382683i	0.923880	−	0.382683i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		−	0.765367i		−	0.765367i
$21$		0			0
$22$	1.70711	+	0.707107i	1.70711	+	0.707107i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$25$			0.414214i			0.414214i
$26$	0.382683	−	0.923880i	0.382683	−	0.923880i
$27$	0.707107	−	0.707107i	0.707107	−	0.707107i
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$33$	1.84776			1.84776
$34$		0			0
$35$		0			0
$36$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		−	1.00000i		−	1.00000i
$40$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$41$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$41$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$42$		0			0
$43$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$43$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$44$		−	1.84776i		−	1.84776i
$45$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$46$		0			0
$47$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$47$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$48$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$49$	−1.00000			−1.00000
$50$	0.382683	−	0.158513i	0.382683	−	0.158513i
$51$		0			0
$52$	−1.00000			−1.00000
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$55$		−	1.41421i		−	1.41421i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$59$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$60$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$		0			0
$64$	0.707107	+	0.707107i	0.707107	+	0.707107i
$65$	−0.765367			−0.765367
$66$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$71$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$72$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	0.292893	−	0.292893i	0.292893	−	0.292893i
$76$		0			0
$77$		0			0
$78$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$79$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$79$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$80$	0.541196	+	0.541196i	0.541196	+	0.541196i
$81$	−1.00000			−1.00000
$82$	0.707107	−	0.292893i	0.707107	−	0.292893i
$83$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$83$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$84$		0			0
$85$		0			0
$86$	1.30656	+	0.541196i	1.30656	+	0.541196i
$87$		0			0
$88$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$89$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$89$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$90$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$95$		0			0
$96$	0.923880	+	0.382683i	0.923880	+	0.382683i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.382683	+	0.923880i	0.382683	+	0.923880i
$99$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$100$	−0.292893	−	0.292893i	−0.292893	−	0.292893i
$101$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$101$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$103$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$104$	0.382683	+	0.923880i	0.382683	+	0.923880i
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$			1.00000i			1.00000i
$109$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$109$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$110$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$118$	0.707107	+	0.292893i	0.707107	+	0.292893i
$119$		0			0
$120$	0.707107	+	0.292893i	0.707107	+	0.292893i
$121$		−	2.41421i		−	2.41421i
$122$		0			0
$123$	0.541196	−	0.541196i	0.541196	−	0.541196i
$124$		0			0
$125$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$126$		0			0
$127$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$127$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$128$	0.382683	−	0.923880i	0.382683	−	0.923880i
$129$	1.41421			1.41421
$130$	0.292893	+	0.707107i	0.292893	+	0.707107i
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$133$		0			0
$134$		0			0
$135$			0.765367i			0.765367i
$136$		0			0
$137$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$137$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$138$		0			0
$139$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$139$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$140$		0			0
$141$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$142$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$143$	−1.84776			−1.84776
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$	0.707107	+	0.707107i	0.707107	+	0.707107i
$148$		0			0
$149$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$149$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$150$	−0.382683	−	0.158513i	−0.382683	−	0.158513i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.707107	+	0.707107i	0.707107	+	0.707107i
$157$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$157$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$158$	0.541196	+	1.30656i	0.541196	+	1.30656i
$159$		0			0
$160$	0.292893	−	0.707107i	0.292893	−	0.707107i
$161$		0			0
$162$	0.382683	+	0.923880i	0.382683	+	0.923880i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$165$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$166$	0.292893	−	0.707107i	0.292893	−	0.707107i
$167$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$167$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$168$		0			0
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$		−	1.41421i		−	1.41421i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$	1.30656	+	1.30656i	1.30656	+	1.30656i
$177$	0.765367			0.765367
$178$	1.70711	−	0.707107i	1.70711	−	0.707107i
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$	0.765367			0.765367
$181$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$181$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	1.00000i		−	1.00000i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	0.541196	+	0.541196i	0.541196	+	0.541196i
$196$	0.707107	−	0.707107i	0.707107	−	0.707107i
$197$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$197$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$198$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$199$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$199$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$200$	−0.158513	+	0.382683i	−0.158513	+	0.382683i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$206$	1.84776	−	0.765367i	1.84776	−	0.765367i
$207$		0			0
$208$	0.707107	−	0.707107i	0.707107	−	0.707107i
$209$		0			0
$210$		0			0
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$		0			0
$213$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$214$		0			0
$215$		−	1.08239i		−	1.08239i
$216$	0.923880	−	0.382683i	0.923880	−	0.382683i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.00000	+	1.00000i	1.00000	+	1.00000i
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−0.414214			−0.414214
$226$		0			0
$227$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$227$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	0.923880	+	0.382683i	0.923880	+	0.382683i
$235$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$236$		−	0.765367i		−	0.765367i
$237$	1.00000	+	1.00000i	1.00000	+	1.00000i
$238$		0			0
$239$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$239$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$240$		−	0.765367i		−	0.765367i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−2.23044	+	0.923880i	−2.23044	+	0.923880i
$243$	0.707107	+	0.707107i	0.707107	+	0.707107i
$244$		0			0
$245$	0.541196	−	0.541196i	0.541196	−	0.541196i
$246$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$247$		0			0
$248$		0			0
$249$		−	0.765367i		−	0.765367i
$250$	−0.414214	+	1.00000i	−0.414214	+	1.00000i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$255$		0			0
$256$	−1.00000			−1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$259$		0			0
$260$	0.541196	−	0.541196i	0.541196	−	0.541196i
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	1.70711	+	0.707107i	1.70711	+	0.707107i
$265$		0			0
$266$		0			0
$267$	1.30656	−	1.30656i	1.30656	−	1.30656i
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$	0.707107	−	0.292893i	0.707107	−	0.292893i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$275$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$276$		0			0
$277$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$277$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$278$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$279$		0			0
$280$		0			0
$281$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$281$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$282$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$	1.30656	+	1.30656i	1.30656	+	1.30656i
$285$		0			0
$286$	0.707107	+	1.70711i	0.707107	+	1.70711i
$287$		0			0
$288$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$289$	1.00000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$293$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$294$	0.382683	−	0.923880i	0.382683	−	0.923880i
$295$		−	0.585786i		−	0.585786i
$296$		0			0
$297$			1.84776i			1.84776i
$298$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$299$		0			0
$300$			0.414214i			0.414214i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$	1.41421	−	1.41421i	1.41421	−	1.41421i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	0.382683	−	0.923880i	0.382683	−	0.923880i
$313$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$313$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$314$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$315$		0			0
$316$	1.00000	−	1.00000i	1.00000	−	1.00000i
$317$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$317$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$318$		0			0
$319$		0			0
$320$	−0.765367			−0.765367
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.707107	−	0.707107i	0.707107	−	0.707107i
$325$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$326$		0			0
$327$		0			0
$328$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$329$		0			0
$330$	1.30656	+	0.541196i	1.30656	+	0.541196i
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$	−0.765367			−0.765367
$333$		0			0
$334$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$335$		0			0
$336$		0			0
$337$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$337$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$338$	0.923880	−	0.382683i	0.923880	−	0.382683i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$	1.00000			1.00000
$352$	0.707107	−	1.70711i	0.707107	−	1.70711i
$353$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$353$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$354$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$355$	1.00000	+	1.00000i	1.00000	+	1.00000i
$356$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$357$		0			0
$358$		0			0
$359$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$359$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$360$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$361$			1.00000i			1.00000i
$362$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$363$	−1.70711	+	1.70711i	−1.70711	+	1.70711i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−0.765367			−0.765367
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$			1.08239i			1.08239i
$376$	1.70711	+	0.707107i	1.70711	+	0.707107i
$377$		0			0
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$382$		0			0
$383$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$383$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$384$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$385$		0			0
$386$		0			0
$387$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$388$		0			0
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$	0.292893	−	0.707107i	0.292893	−	0.707107i
$391$		0			0
$392$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$393$		0			0
$394$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$395$	0.765367	−	0.765367i	0.765367	−	0.765367i
$396$	1.84776			1.84776
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	1.30656	−	0.541196i	1.30656	−	0.541196i
$399$		0			0
$400$	0.414214			0.414214
$401$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$401$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.541196	−	0.541196i	0.541196	−	0.541196i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	−0.224171	+	0.541196i	−0.224171	+	0.541196i
$411$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$412$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$413$		0			0
$414$		0			0
$415$	−0.585786			−0.585786
$416$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$417$	−1.41421			−1.41421
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$			1.84776i			1.84776i
$424$		0			0
$425$		0			0
$426$	1.70711	+	0.707107i	1.70711	+	0.707107i
$427$		0			0
$428$		0			0
$429$	1.30656	+	1.30656i	1.30656	+	1.30656i
$430$	−1.00000	+	0.414214i	−1.00000	+	0.414214i
$431$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$431$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$432$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$433$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$433$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$	0.541196	−	1.30656i	0.541196	−	1.30656i
$441$		−	1.00000i		−	1.00000i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$446$		0			0
$447$	−1.84776			−1.84776
$448$		0			0
$449$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$449$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$450$	0.158513	+	0.382683i	0.158513	+	0.382683i
$451$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$452$		0			0
$453$		0			0
$454$	0.707107	−	1.70711i	0.707107	−	1.70711i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$461$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$462$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		−	1.00000i		−	1.00000i
$469$		0			0
$470$	1.30656	+	0.541196i	1.30656	+	0.541196i
$471$			1.41421i			1.41421i
$472$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$473$		−	2.61313i		−	2.61313i
$474$	0.541196	−	1.30656i	0.541196	−	1.30656i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.292893	+	0.707107i	0.292893	+	0.707107i
$479$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$479$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$480$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.70711	+	1.70711i	1.70711	+	1.70711i
$485$		0			0
$486$	0.382683	−	0.923880i	0.382683	−	0.923880i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$			0.765367i			0.765367i
$493$		0			0
$494$		0			0
$495$	1.41421			1.41421
$496$		0			0
$497$		0			0
$498$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	1.08239			1.08239
$501$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.707107	−	0.707107i	0.707107	−	0.707107i
$508$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$509$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$509$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−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			0
$515$	−1.08239	−	1.08239i	−1.08239	−	1.08239i
$516$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$517$	−2.41421	+	2.41421i	−2.41421	+	2.41421i
$518$		0			0
$519$		0			0
$520$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$523$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		−	1.84776i		−	1.84776i
$529$	−1.00000			−1.00000
$530$		0			0
$531$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$532$		0			0
$533$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$534$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.30656	−	1.30656i	1.30656	−	1.30656i
$540$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$	−1.41421			−1.41421
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$547$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$548$	1.30656	+	1.30656i	1.30656	+	1.30656i
$549$		0			0
$550$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$555$		0			0
$556$			1.41421i			1.41421i
$557$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$557$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$558$		0			0
$559$	−1.41421			−1.41421
$560$		0			0
$561$		0			0
$562$	0.707107	−	0.292893i	0.707107	−	0.292893i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$	1.84776			1.84776
$565$		0			0
$566$		0			0
$567$		0			0
$568$	0.707107	−	1.70711i	0.707107	−	1.70711i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$	1.30656	−	1.30656i	1.30656	−	1.30656i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		−	0.765367i		−	0.765367i
$586$	1.70711	+	0.707107i	1.70711	+	0.707107i
$587$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$587$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$588$	−1.00000			−1.00000
$589$		0			0
$590$	−0.541196	+	0.224171i	−0.541196	+	0.224171i
$591$	−1.84776			−1.84776
$592$		0			0
$593$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$593$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$594$	1.70711	−	0.707107i	1.70711	−	0.707107i
$595$		0			0
$596$			1.84776i			1.84776i
$597$	1.00000	−	1.00000i	1.00000	−	1.00000i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	0.382683	−	0.158513i	0.382683	−	0.158513i
$601$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$601$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.30656	+	1.30656i	1.30656	+	1.30656i
$606$		0			0
$607$	2.00000			2.00000			1.00000			$0$
$607$	2.00000			2.00000			1.00000			$0$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.30656	+	1.30656i	1.30656	+	1.30656i
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$			0.585786i			0.585786i
$616$		0			0
$617$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$617$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$618$	−1.84776	−	0.765367i	−1.84776	−	0.765367i
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.00000			−1.00000
$625$	0.414214			0.414214
$626$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$627$		0			0
$628$	1.41421			1.41421
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$633$		0			0
$634$	0.707107	−	1.70711i	0.707107	−	1.70711i
$635$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$636$		0			0
$637$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$638$		0			0
$639$	1.84776			1.84776
$640$	0.292893	+	0.707107i	0.292893	+	0.707107i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$649$		−	1.41421i		−	1.41421i
$650$	0.382683	+	0.158513i	0.382683	+	0.158513i
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$	0.765367			0.765367
$657$		0			0
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		−	1.41421i		−	1.41421i
$661$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$661$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$662$		0			0
$663$		0			0
$664$	0.292893	+	0.707107i	0.292893	+	0.707107i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.541196	+	0.541196i	0.541196	+	0.541196i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$	0.541196	+	1.30656i	0.541196	+	1.30656i
$675$	0.292893	+	0.292893i	0.292893	+	0.292893i
$676$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	1.84776i		−	1.84776i
$682$		0			0
$683$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$683$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$684$		0			0
$685$	1.00000	+	1.00000i	1.00000	+	1.00000i
$686$		0			0
$687$		0			0
$688$	1.00000	+	1.00000i	1.00000	+	1.00000i
$689$		0			0
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			1.08239i			1.08239i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$703$		0			0
$704$	−1.84776			−1.84776
$705$	1.41421			1.41421
$706$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$707$		0			0
$708$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$709$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$709$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$710$	0.541196	−	1.30656i	0.541196	−	1.30656i
$711$		−	1.41421i		−	1.41421i
$712$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$713$		0			0
$714$		0			0
$715$	1.00000	−	1.00000i	1.00000	−	1.00000i
$716$		0			0
$717$	0.541196	+	0.541196i	0.541196	+	0.541196i
$718$	1.70711	−	0.707107i	1.70711	−	0.707107i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$721$		0			0
$722$	0.923880	−	0.382683i	0.923880	−	0.382683i
$723$		0			0
$724$			1.41421i			1.41421i
$725$		0			0
$726$	2.23044	+	0.923880i	2.23044	+	0.923880i
$727$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$727$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$728$		0			0
$729$		−	1.00000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$	−0.765367			−0.765367
$736$		0			0
$737$		0			0
$738$	0.292893	+	0.707107i	0.292893	+	0.707107i
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$743$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$744$		0			0
$745$			1.41421i			1.41421i
$746$		0			0
$747$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$748$		0			0
$749$		0			0
$750$	1.00000	−	0.414214i	1.00000	−	0.414214i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		−	1.84776i		−	1.84776i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$761$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$762$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$767$	−0.765367			−0.765367
$768$	0.707107	+	0.707107i	0.707107	+	0.707107i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$773$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$774$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−0.765367			−0.765367
$781$	2.41421	+	2.41421i	2.41421	+	2.41421i
$782$		0			0
$783$		0			0
$784$			1.00000i			1.00000i
$785$	1.08239			1.08239
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$			1.84776i			1.84776i
$789$		0			0
$790$	−1.00000	−	0.414214i	−1.00000	−	0.414214i
$791$		0			0
$792$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$801$	−1.84776			−1.84776
$802$	0.707107	+	1.70711i	0.707107	+	1.70711i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	0.585786			0.585786
$821$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$821$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$822$	1.70711	+	0.707107i	1.70711	+	0.707107i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$	−0.765367	+	1.84776i	−0.765367	+	1.84776i
$825$			0.765367i			0.765367i
$826$		0			0
$827$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$827$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$828$		0			0
$829$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$829$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$830$	0.224171	+	0.541196i	0.224171	+	0.541196i
$831$	−1.41421			−1.41421
$832$			1.00000i			1.00000i
$833$		0			0
$834$	0.541196	+	1.30656i	0.541196	+	1.30656i
$835$	0.414214	+	0.414214i	0.414214	+	0.414214i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$839$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$		0			0
$843$	0.541196	−	0.541196i	0.541196	−	0.541196i
$844$		0			0
$845$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$846$	1.70711	−	0.707107i	1.70711	−	0.707107i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		−	1.84776i		−	1.84776i
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$	0.707107	−	1.70711i	0.707107	−	1.70711i
$859$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$859$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$860$	0.765367	+	0.765367i	0.765367	+	0.765367i
$861$		0			0
$862$	0.292893	+	0.707107i	0.292893	+	0.707107i
$863$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$863$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$864$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$865$		0			0
$866$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$867$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$868$		0			0
$869$	1.84776	−	1.84776i	1.84776	−	1.84776i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$	1.84776			1.84776
$880$	−1.41421			−1.41421
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$891$	1.30656	−	1.30656i	1.30656	−	1.30656i
$892$		0			0
$893$		0			0
$894$	0.707107	+	1.70711i	0.707107	+	1.70711i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.707107	+	1.70711i	0.707107	+	1.70711i
$899$		0			0
$900$	0.292893	−	0.292893i	0.292893	−	0.292893i
$901$		0			0
$902$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$903$		0			0
$904$		0			0
$905$			1.08239i			1.08239i
$906$		0			0
$907$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$907$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$908$	−1.84776			−1.84776
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−1.41421			−1.41421
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$919$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$920$		0			0
$921$		0			0
$922$	0.707107	−	1.70711i	0.707107	−	1.70711i
$923$	1.30656	−	1.30656i	1.30656	−	1.30656i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−2.00000			−2.00000
$928$		0			0
$929$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$929$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$937$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$937$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$940$		−	1.41421i		−	1.41421i
$941$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$941$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$942$	1.30656	−	0.541196i	1.30656	−	0.541196i
$943$		0			0
$944$	0.541196	+	0.541196i	0.541196	+	0.541196i
$945$		0			0
$946$	−2.41421	+	1.00000i	−2.41421	+	1.00000i
$947$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$947$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$948$	−1.41421			−1.41421
$949$		0			0
$950$		0			0
$951$		−	1.84776i		−	1.84776i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	0.541196	−	0.541196i	0.541196	−	0.541196i
$957$		0			0
$958$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$959$		0			0
$960$	0.541196	+	0.541196i	0.541196	+	0.541196i
$961$	1.00000			1.00000
$962$		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	$0.5\pi$
$968$	0.923880	−	2.23044i	0.923880	−	2.23044i
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$	−1.00000			−1.00000
$973$		0			0
$974$		0			0
$975$	0.414214			0.414214
$976$		0			0
$977$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$977$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$978$		0			0
$979$	−2.41421	−	2.41421i	−2.41421	−	2.41421i
$980$			0.765367i			0.765367i
$981$		0			0
$982$		0			0
$983$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$983$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$984$	0.707107	−	0.292893i	0.707107	−	0.292893i
$985$			1.41421i			1.41421i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$996$	0.541196	+	0.541196i	0.541196	+	0.541196i
$997$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$997$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.1.bi.a.77.2	✓	8
3.2	odd	2	inner	624.1.bi.a.77.3	yes	8
4.3	odd	2		2496.1.bi.a.1169.3		8
12.11	even	2		2496.1.bi.a.1169.4		8
13.12	even	2	inner	624.1.bi.a.77.3	yes	8
16.5	even	4	inner	624.1.bi.a.389.2	yes	8
16.11	odd	4		2496.1.bi.a.2417.3		8
39.38	odd	2	CM	624.1.bi.a.77.2	✓	8
48.5	odd	4	inner	624.1.bi.a.389.3	yes	8
48.11	even	4		2496.1.bi.a.2417.4		8
52.51	odd	2		2496.1.bi.a.1169.4		8
156.155	even	2		2496.1.bi.a.1169.3		8
208.155	odd	4		2496.1.bi.a.2417.4		8
208.181	even	4	inner	624.1.bi.a.389.3	yes	8
624.155	even	4		2496.1.bi.a.2417.3		8
624.389	odd	4	inner	624.1.bi.a.389.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.1.bi.a.77.2	✓	8	1.1	even	1	trivial
624.1.bi.a.77.2	✓	8	39.38	odd	2	CM
624.1.bi.a.77.3	yes	8	3.2	odd	2	inner
624.1.bi.a.77.3	yes	8	13.12	even	2	inner
624.1.bi.a.389.2	yes	8	16.5	even	4	inner
624.1.bi.a.389.2	yes	8	624.389	odd	4	inner
624.1.bi.a.389.3	yes	8	48.5	odd	4	inner
624.1.bi.a.389.3	yes	8	208.181	even	4	inner
2496.1.bi.a.1169.3		8	4.3	odd	2
2496.1.bi.a.1169.3		8	156.155	even	2
2496.1.bi.a.1169.4		8	12.11	even	2
2496.1.bi.a.1169.4		8	52.51	odd	2
2496.1.bi.a.2417.3		8	16.11	odd	4
2496.1.bi.a.2417.3		8	624.155	even	4
2496.1.bi.a.2417.4		8	48.11	even	4
2496.1.bi.a.2417.4		8	208.155	odd	4

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	624.bi (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.382683 - 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(-0.541196 + 0.541196i) q^{5} +(-0.382683 + 0.923880i) q^{6} +(0.923880 + 0.382683i) q^{8} +1.00000i q^{9} +O(q^{10})\)	\(q+(-0.382683 - 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(-0.707107 + 0.707107i) q^{4} +(-0.541196 + 0.541196i) q^{5} +(-0.382683 + 0.923880i) q^{6} +(0.923880 + 0.382683i) q^{8} +1.00000i q^{9} +(0.707107 + 0.292893i) q^{10} +(-1.30656 + 1.30656i) q^{11} +1.00000 q^{12} +(0.707107 + 0.707107i) q^{13} +0.765367 q^{15} -1.00000i q^{16} +(0.923880 - 0.382683i) q^{18} -0.765367i q^{20} +(1.70711 + 0.707107i) q^{22} +(-0.382683 - 0.923880i) q^{24} +0.414214i q^{25} +(0.382683 - 0.923880i) q^{26} +(0.707107 - 0.707107i) q^{27} +(-0.292893 - 0.707107i) q^{30} +(-0.923880 + 0.382683i) q^{32} +1.84776 q^{33} +(-0.707107 - 0.707107i) q^{36} -1.00000i q^{39} +(-0.707107 + 0.292893i) q^{40} +0.765367i q^{41} +(-1.00000 + 1.00000i) q^{43} -1.84776i q^{44} +(-0.541196 - 0.541196i) q^{45} +1.84776 q^{47} +(-0.707107 + 0.707107i) q^{48} -1.00000 q^{49} +(0.382683 - 0.158513i) q^{50} -1.00000 q^{52} +(-0.923880 - 0.382683i) q^{54} -1.41421i q^{55} +(-0.541196 + 0.541196i) q^{59} +(-0.541196 + 0.541196i) q^{60} +(0.707107 + 0.707107i) q^{64} -0.765367 q^{65} +(-0.707107 - 1.70711i) q^{66} -1.84776i q^{71} +(-0.382683 + 0.923880i) q^{72} +(0.292893 - 0.292893i) q^{75} +(-0.923880 + 0.382683i) q^{78} -1.41421 q^{79} +(0.541196 + 0.541196i) q^{80} -1.00000 q^{81} +(0.707107 - 0.292893i) q^{82} +(0.541196 + 0.541196i) q^{83} +(1.30656 + 0.541196i) q^{86} +(-1.70711 + 0.707107i) q^{88} +1.84776i q^{89} +(-0.292893 + 0.707107i) q^{90} +(-0.707107 - 1.70711i) q^{94} +(0.923880 + 0.382683i) q^{96} +(0.382683 + 0.923880i) q^{98} +(-1.30656 - 1.30656i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{12} + 8 q^{22} - 8 q^{30} - 8 q^{43} - 8 q^{49} - 8 q^{52} + 8 q^{75} - 8 q^{81} - 8 q^{88} - 8 q^{90}+O(q^{100}) \) 8 * q + 8 * q^12 + 8 * q^22 - 8 * q^30 - 8 * q^43 - 8 * q^49 - 8 * q^52 + 8 * q^75 - 8 * q^81 - 8 * q^88 - 8 * q^90

Introduction

L-functions

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