LMFDB - Embedded newform 760.1.b.a.189.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,1,Mod(189,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

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

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

chi := DirichletCharacter("760.189");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	760.b (of order $2$, degree $1$, 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:	$0.379289409601$
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:	$D_{8}$
Projective field:	Galois closure of 8.0.66724352000.2

Embedding invariants

Embedding label			189.4
Root			$0.382683 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	760.189
Dual form			760.1.b.a.189.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.382683 + 0.923880i) q^{2} +0.765367i q^{3} +(-0.707107 - 0.707107i) q^{4} -1.00000i q^{5} +(-0.707107 - 0.292893i) q^{6} +(0.923880 - 0.382683i) q^{8} +0.414214 q^{9} +O(q^{10})$	\(q+(-0.382683 + 0.923880i) q^{2} +0.765367i q^{3} +(-0.707107 - 0.707107i) q^{4} -1.00000i q^{5} +(-0.707107 - 0.292893i) q^{6} +(0.923880 - 0.382683i) q^{8} +0.414214 q^{9} +(0.923880 + 0.382683i) q^{10} -1.41421i q^{11} +(0.541196 - 0.541196i) q^{12} -1.84776i q^{13} +0.765367 q^{15} +1.00000i q^{16} +(-0.158513 + 0.382683i) q^{18} +1.00000i q^{19} +(-0.707107 + 0.707107i) q^{20} +(1.30656 + 0.541196i) q^{22} +(0.292893 + 0.707107i) q^{24} -1.00000 q^{25} +(1.70711 + 0.707107i) q^{26} +1.08239i q^{27} +(-0.292893 + 0.707107i) q^{30} +(-0.923880 - 0.382683i) q^{32} +1.08239 q^{33} +(-0.292893 - 0.292893i) q^{36} +0.765367i q^{37} +(-0.923880 - 0.382683i) q^{38} +1.41421 q^{39} +(-0.382683 - 0.923880i) q^{40} +(-1.00000 + 1.00000i) q^{44} -0.414214i q^{45} -0.765367 q^{48} +1.00000 q^{49} +(0.382683 - 0.923880i) q^{50} +(-1.30656 + 1.30656i) q^{52} -0.765367i q^{53} +(-1.00000 - 0.414214i) q^{54} -1.41421 q^{55} -0.765367 q^{57} +(-0.541196 - 0.541196i) q^{60} +1.41421i q^{61} +(0.707107 - 0.707107i) q^{64} -1.84776 q^{65} +(-0.414214 + 1.00000i) q^{66} +1.84776i q^{67} +(0.382683 - 0.158513i) q^{72} +(-0.707107 - 0.292893i) q^{74} -0.765367i q^{75} +(0.707107 - 0.707107i) q^{76} +(-0.541196 + 1.30656i) q^{78} +1.00000 q^{80} -0.414214 q^{81} +(-0.541196 - 1.30656i) q^{88} +(0.382683 + 0.158513i) q^{90} +1.00000 q^{95} +(0.292893 - 0.707107i) q^{96} -1.84776 q^{97} +(-0.382683 + 0.923880i) q^{98} -0.585786i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 8 q^{24} - 8 q^{25} + 8 q^{26} - 8 q^{30} - 8 q^{36} - 8 q^{44} + 8 q^{49} - 8 q^{54} + 8 q^{66} + 8 q^{80} + 8 q^{81} + 8 q^{95} + 8 q^{96}+O(q^{100}) $ 8 * q - 8 * q^9 + 8 * q^24 - 8 * q^25 + 8 * q^26 - 8 * q^30 - 8 * q^36 - 8 * q^44 + 8 * q^49 - 8 * q^54 + 8 * q^66 + 8 * q^80 + 8 * q^81 + 8 * q^95 + 8 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$3$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$4$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$5$		−	1.00000i		−	1.00000i
$5$		−	1.00000i		−	1.00000i
$6$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	0.923880	−	0.382683i	0.923880	−	0.382683i
$9$	0.414214			0.414214
$10$	0.923880	+	0.382683i	0.923880	+	0.382683i
$11$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$11$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$12$	0.541196	−	0.541196i	0.541196	−	0.541196i
$13$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$13$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$14$		0			0
$15$	0.765367			0.765367
$16$			1.00000i			1.00000i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.158513	+	0.382683i	−0.158513	+	0.382683i
$19$			1.00000i			1.00000i
$19$			1.00000i			1.00000i
$20$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$21$		0			0
$22$	1.30656	+	0.541196i	1.30656	+	0.541196i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	0.292893	+	0.707107i	0.292893	+	0.707107i
$25$	−1.00000			−1.00000
$26$	1.70711	+	0.707107i	1.70711	+	0.707107i
$27$			1.08239i			1.08239i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−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
$33$	1.08239			1.08239
$34$		0			0
$35$		0			0
$36$	−0.292893	−	0.292893i	−0.292893	−	0.292893i
$37$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$37$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$38$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$39$	1.41421			1.41421
$40$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$45$		−	0.414214i		−	0.414214i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−0.765367			−0.765367
$49$	1.00000			1.00000
$50$	0.382683	−	0.923880i	0.382683	−	0.923880i
$51$		0			0
$52$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$53$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$53$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$54$	−1.00000	−	0.414214i	−1.00000	−	0.414214i
$55$	−1.41421			−1.41421
$56$		0			0
$57$	−0.765367			−0.765367
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$61$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$61$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$		0			0
$64$	0.707107	−	0.707107i	0.707107	−	0.707107i
$65$	−1.84776			−1.84776
$66$	−0.414214	+	1.00000i	−0.414214	+	1.00000i
$67$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$67$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.382683	−	0.158513i	0.382683	−	0.158513i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$75$		−	0.765367i		−	0.765367i
$76$	0.707107	−	0.707107i	0.707107	−	0.707107i
$77$		0			0
$78$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	1.00000			1.00000
$81$	−0.414214			−0.414214
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.382683	+	0.158513i	0.382683	+	0.158513i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.00000			1.00000
$96$	0.292893	−	0.707107i	0.292893	−	0.707107i
$97$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$97$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$98$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$99$		−	0.585786i		−	0.585786i
$100$	0.707107	+	0.707107i	0.707107	+	0.707107i
$101$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$101$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$102$		0			0
$103$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$103$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$104$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$105$		0			0
$106$	0.707107	+	0.292893i	0.707107	+	0.292893i
$107$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$107$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$108$	0.765367	−	0.765367i	0.765367	−	0.765367i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	0.541196	−	1.30656i	0.541196	−	1.30656i
$111$	−0.585786			−0.585786
$112$		0			0
$113$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$113$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$114$	0.292893	−	0.707107i	0.292893	−	0.707107i
$115$		0			0
$116$		0			0
$117$		−	0.765367i		−	0.765367i
$118$		0			0
$119$		0			0
$120$	0.707107	−	0.292893i	0.707107	−	0.292893i
$121$	−1.00000			−1.00000
$122$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$123$		0			0
$124$		0			0
$125$			1.00000i			1.00000i
$126$		0			0
$127$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$127$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$128$	0.382683	+	0.923880i	0.382683	+	0.923880i
$129$		0			0
$130$	0.707107	−	1.70711i	0.707107	−	1.70711i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$133$		0			0
$134$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$135$	1.08239			1.08239
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$139$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.61313			−2.61313
$144$			0.414214i			0.414214i
$145$		0			0
$146$		0			0
$147$			0.765367i			0.765367i
$148$	0.541196	−	0.541196i	0.541196	−	0.541196i
$149$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$149$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$150$	0.707107	+	0.292893i	0.707107	+	0.292893i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	0.382683	+	0.923880i	0.382683	+	0.923880i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$	0.585786			0.585786
$160$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$161$		0			0
$162$	0.158513	−	0.382683i	0.158513	−	0.382683i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		−	1.08239i		−	1.08239i
$166$		0			0
$167$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$167$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$168$		0			0
$169$	−2.41421			−2.41421
$170$		0			0
$171$			0.414214i			0.414214i
$172$		0			0
$173$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$173$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$174$		0			0
$175$		0			0
$176$	1.41421			1.41421
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$	−1.08239			−1.08239
$184$		0			0
$185$	0.765367			0.765367
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	0.541196	+	0.541196i	0.541196	+	0.541196i
$193$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$193$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$194$	0.707107	−	1.70711i	0.707107	−	1.70711i
$195$		−	1.41421i		−	1.41421i
$196$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	0.541196	+	0.224171i	0.541196	+	0.224171i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$201$	−1.41421			−1.41421
$202$	1.30656	+	0.541196i	1.30656	+	0.541196i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$207$		0			0
$208$	1.84776			1.84776
$209$	1.41421			1.41421
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$213$		0			0
$214$	1.70711	+	0.707107i	1.70711	+	0.707107i
$215$		0			0
$216$	0.414214	+	1.00000i	0.414214	+	1.00000i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.00000	+	1.00000i	1.00000	+	1.00000i
$221$		0			0
$222$	0.224171	−	0.541196i	0.224171	−	0.541196i
$223$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$223$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$224$		0			0
$225$	−0.414214			−0.414214
$226$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$227$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$227$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$228$	0.541196	+	0.541196i	0.541196	+	0.541196i
$229$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$229$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	0.707107	+	0.292893i	0.707107	+	0.292893i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.00000			−2.00000			−1.00000			$\pi$
$239$	−2.00000			−2.00000			−1.00000			$\pi$
$240$			0.765367i			0.765367i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.382683	−	0.923880i	0.382683	−	0.923880i
$243$			0.765367i			0.765367i
$244$	1.00000	−	1.00000i	1.00000	−	1.00000i
$245$		−	1.00000i		−	1.00000i
$246$		0			0
$247$	1.84776			1.84776
$248$		0			0
$249$		0			0
$250$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$251$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$251$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$255$		0			0
$256$	−1.00000			−1.00000
$257$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$257$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$258$		0			0
$259$		0			0
$260$	1.30656	+	1.30656i	1.30656	+	1.30656i
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	1.00000	−	0.414214i	1.00000	−	0.414214i
$265$	−0.765367			−0.765367
$266$		0			0
$267$		0			0
$268$	1.30656	−	1.30656i	1.30656	−	1.30656i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−0.414214	+	1.00000i	−0.414214	+	1.00000i
$271$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$271$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.41421i			1.41421i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$			0.765367i			0.765367i
$286$	1.00000	−	2.41421i	1.00000	−	2.41421i
$287$		0			0
$288$	−0.382683	−	0.158513i	−0.382683	−	0.158513i
$289$	1.00000			1.00000
$290$		0			0
$291$		−	1.41421i		−	1.41421i
$292$		0			0
$293$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$293$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$294$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$295$		0			0
$296$	0.292893	+	0.707107i	0.292893	+	0.707107i
$297$	1.53073			1.53073
$298$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$299$		0			0
$300$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$301$		0			0
$302$		0			0
$303$	1.08239			1.08239
$304$	−1.00000			−1.00000
$305$	1.41421			1.41421
$306$		0			0
$307$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$307$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$308$		0			0
$309$			0.585786i			0.585786i
$310$		0			0
$311$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$311$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$312$	1.30656	−	0.541196i	1.30656	−	0.541196i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$317$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$318$	−0.224171	+	0.541196i	−0.224171	+	0.541196i
$319$		0			0
$320$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$321$	1.41421			1.41421
$322$		0			0
$323$		0			0
$324$	0.292893	+	0.292893i	0.292893	+	0.292893i
$325$			1.84776i			1.84776i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	1.00000	+	0.414214i	1.00000	+	0.414214i
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$			0.317025i			0.317025i
$334$	0.707107	−	1.70711i	0.707107	−	1.70711i
$335$	1.84776			1.84776
$336$		0			0
$337$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$337$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$338$	0.923880	−	2.23044i	0.923880	−	2.23044i
$339$			0.585786i			0.585786i
$340$		0			0
$341$		0			0
$342$	−0.382683	−	0.158513i	−0.382683	−	0.158513i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$349$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$350$		0			0
$351$	2.00000			2.00000
$352$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$359$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$360$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$361$	−1.00000			−1.00000
$362$		0			0
$363$		−	0.765367i		−	0.765367i
$364$		0			0
$365$		0			0
$366$	0.414214	−	1.00000i	0.414214	−	1.00000i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$371$		0			0
$372$		0			0
$373$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$373$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$374$		0			0
$375$	−0.765367			−0.765367
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$381$			1.41421i			1.41421i
$382$		0			0
$383$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$383$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$384$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$385$		0			0
$386$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$387$		0			0
$388$	1.30656	+	1.30656i	1.30656	+	1.30656i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	1.30656	+	0.541196i	1.30656	+	0.541196i
$391$		0			0
$392$	0.923880	−	0.382683i	0.923880	−	0.382683i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		−	1.00000i		−	1.00000i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	0.541196	−	1.30656i	0.541196	−	1.30656i
$403$		0			0
$404$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$405$			0.414214i			0.414214i
$406$		0			0
$407$	1.08239			1.08239
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$417$	−1.08239			−1.08239
$418$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$429$		−	2.00000i		−	2.00000i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−1.08239			−1.08239
$433$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$433$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$434$		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			$\pi$
$440$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$441$	0.414214			0.414214
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	0.414214	+	0.414214i	0.414214	+	0.414214i
$445$		0			0
$446$	0.707107	−	1.70711i	0.707107	−	1.70711i
$447$	−1.08239			−1.08239
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.158513	−	0.382683i	0.158513	−	0.382683i
$451$		0			0
$452$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$453$		0			0
$454$	1.70711	+	0.707107i	1.70711	+	0.707107i
$455$		0			0
$456$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		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			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	1.00000i		−	1.00000i
$476$		0			0
$477$		−	0.317025i		−	0.317025i
$478$	0.765367	−	1.84776i	0.765367	−	1.84776i
$479$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$479$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$480$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$481$	1.41421			1.41421
$482$		0			0
$483$		0			0
$484$	0.707107	+	0.707107i	0.707107	+	0.707107i
$485$			1.84776i			1.84776i
$486$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$487$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$487$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$488$	0.541196	+	1.30656i	0.541196	+	1.30656i
$489$		0			0
$490$	0.923880	+	0.382683i	0.923880	+	0.382683i
$491$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$491$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$492$		0			0
$493$		0			0
$494$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$495$	−0.585786			−0.585786
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$499$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$500$	0.707107	−	0.707107i	0.707107	−	0.707107i
$501$		−	1.41421i		−	1.41421i
$502$	−1.84776	−	0.765367i	−1.84776	−	0.765367i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−1.41421			−1.41421
$506$		0			0
$507$		−	1.84776i		−	1.84776i
$508$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	0.382683	−	0.923880i	0.382683	−	0.923880i
$513$	−1.08239			−1.08239
$514$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$515$		−	0.765367i		−	0.765367i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.41421			−1.41421
$520$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$523$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$			1.08239i			1.08239i
$529$	1.00000			1.00000
$530$	0.292893	−	0.707107i	0.292893	−	0.707107i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.84776			−1.84776
$536$	0.707107	+	1.70711i	0.707107	+	1.70711i
$537$		0			0
$538$		0			0
$539$		−	1.41421i		−	1.41421i
$540$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$541$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$541$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$542$	0.541196	−	1.30656i	0.541196	−	1.30656i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$547$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$548$		0			0
$549$			0.585786i			0.585786i
$550$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$			0.585786i			0.585786i
$556$	1.00000	−	1.00000i	1.00000	−	1.00000i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$563$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$564$		0			0
$565$		−	0.765367i		−	0.765367i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$571$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$571$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$572$	1.84776	+	1.84776i	1.84776	+	1.84776i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.292893	−	0.292893i	0.292893	−	0.292893i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$579$			1.41421i			1.41421i
$580$		0			0
$581$		0			0
$582$	1.30656	+	0.541196i	1.30656	+	0.541196i
$583$	−1.08239			−1.08239
$584$		0			0
$585$	−0.765367			−0.765367
$586$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	0.541196	−	0.541196i	0.541196	−	0.541196i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.765367			−0.765367
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.585786	+	1.41421i	−0.585786	+	1.41421i
$595$		0			0
$596$	1.00000	−	1.00000i	1.00000	−	1.00000i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			0.765367i			0.765367i
$604$		0			0
$605$			1.00000i			1.00000i
$606$	−0.414214	+	1.00000i	−0.414214	+	1.00000i
$607$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$607$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$608$	0.382683	−	0.923880i	0.382683	−	0.923880i
$609$		0			0
$610$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	1.70711	+	0.707107i	1.70711	+	0.707107i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−0.541196	−	0.224171i	−0.541196	−	0.224171i
$619$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$619$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$620$		0			0
$621$		0			0
$622$	0.541196	−	1.30656i	0.541196	−	1.30656i
$623$		0			0
$624$			1.41421i			1.41421i
$625$	1.00000			1.00000
$626$		0			0
$627$			1.08239i			1.08239i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$631$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$632$		0			0
$633$		0			0
$634$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$635$		−	1.84776i		−	1.84776i
$636$	−0.414214	−	0.414214i	−0.414214	−	0.414214i
$637$		−	1.84776i		−	1.84776i
$638$		0			0
$639$		0			0
$640$	0.923880	−	0.382683i	0.923880	−	0.382683i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.382683	+	0.158513i	−0.382683	+	0.158513i
$649$		0			0
$650$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.292893	−	0.121320i	−0.292893	−	0.121320i
$667$		0			0
$668$	1.30656	+	1.30656i	1.30656	+	1.30656i
$669$		−	1.41421i		−	1.41421i
$670$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$671$	2.00000			2.00000
$672$		0			0
$673$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$673$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$674$	0.292893	−	0.707107i	0.292893	−	0.707107i
$675$		−	1.08239i		−	1.08239i
$676$	1.70711	+	1.70711i	1.70711	+	1.70711i
$677$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$677$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$678$	−0.541196	−	0.224171i	−0.541196	−	0.224171i
$679$		0			0
$680$		0			0
$681$	1.41421			1.41421
$682$		0			0
$683$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$683$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$684$	0.292893	−	0.292893i	0.292893	−	0.292893i
$685$		0			0
$686$		0			0
$687$	−1.08239			−1.08239
$688$		0			0
$689$	−1.41421			−1.41421
$690$		0			0
$691$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$691$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$692$	1.30656	−	1.30656i	1.30656	−	1.30656i
$693$		0			0
$694$		0			0
$695$	1.41421			1.41421
$696$		0			0
$697$		0			0
$698$	1.84776	+	0.765367i	1.84776	+	0.765367i
$699$		0			0
$700$		0			0
$701$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$701$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$702$	−0.765367	+	1.84776i	−0.765367	+	1.84776i
$703$	−0.765367			−0.765367
$704$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$			2.61313i			2.61313i
$716$		0			0
$717$		−	1.53073i		−	1.53073i
$718$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$719$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$719$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$720$	0.414214			0.414214
$721$		0			0
$722$	0.382683	−	0.923880i	0.382683	−	0.923880i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	0.707107	+	0.292893i	0.707107	+	0.292893i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$	0.765367	+	0.765367i	0.765367	+	0.765367i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	0.765367			0.765367
$736$		0			0
$737$	2.61313			2.61313
$738$		0			0
$739$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$739$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$740$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$741$			1.41421i			1.41421i
$742$		0			0
$743$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$743$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$744$		0			0
$745$	1.41421			1.41421
$746$	0.707107	+	0.292893i	0.707107	+	0.292893i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	0.292893	−	0.707107i	0.292893	−	0.707107i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−1.53073			−1.53073
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$	0.923880	−	0.382683i	0.923880	−	0.382683i
$761$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$761$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$762$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.292893	−	0.707107i	0.292893	−	0.707107i
$767$		0			0
$768$		−	0.765367i		−	0.765367i
$769$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$769$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$			1.41421i			1.41421i
$772$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$773$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$773$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$774$		0			0
$775$		0			0
$776$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$781$		0			0
$782$		0			0
$783$		0			0
$784$			1.00000i			1.00000i
$785$		0			0
$786$		0			0
$787$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$787$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.224171	−	0.541196i	−0.224171	−	0.541196i
$793$	2.61313			2.61313
$794$		0			0
$795$		−	0.585786i		−	0.585786i
$796$		0			0
$797$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$

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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	760.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.382683 + 0.923880i) q^{2} +0.765367i q^{3} +(-0.707107 - 0.707107i) q^{4} -1.00000i q^{5} +(-0.707107 - 0.292893i) q^{6} +(0.923880 - 0.382683i) q^{8} +0.414214 q^{9} +O(q^{10})\)	\(q+(-0.382683 + 0.923880i) q^{2} +0.765367i q^{3} +(-0.707107 - 0.707107i) q^{4} -1.00000i q^{5} +(-0.707107 - 0.292893i) q^{6} +(0.923880 - 0.382683i) q^{8} +0.414214 q^{9} +(0.923880 + 0.382683i) q^{10} -1.41421i q^{11} +(0.541196 - 0.541196i) q^{12} -1.84776i q^{13} +0.765367 q^{15} +1.00000i q^{16} +(-0.158513 + 0.382683i) q^{18} +1.00000i q^{19} +(-0.707107 + 0.707107i) q^{20} +(1.30656 + 0.541196i) q^{22} +(0.292893 + 0.707107i) q^{24} -1.00000 q^{25} +(1.70711 + 0.707107i) q^{26} +1.08239i q^{27} +(-0.292893 + 0.707107i) q^{30} +(-0.923880 - 0.382683i) q^{32} +1.08239 q^{33} +(-0.292893 - 0.292893i) q^{36} +0.765367i q^{37} +(-0.923880 - 0.382683i) q^{38} +1.41421 q^{39} +(-0.382683 - 0.923880i) q^{40} +(-1.00000 + 1.00000i) q^{44} -0.414214i q^{45} -0.765367 q^{48} +1.00000 q^{49} +(0.382683 - 0.923880i) q^{50} +(-1.30656 + 1.30656i) q^{52} -0.765367i q^{53} +(-1.00000 - 0.414214i) q^{54} -1.41421 q^{55} -0.765367 q^{57} +(-0.541196 - 0.541196i) q^{60} +1.41421i q^{61} +(0.707107 - 0.707107i) q^{64} -1.84776 q^{65} +(-0.414214 + 1.00000i) q^{66} +1.84776i q^{67} +(0.382683 - 0.158513i) q^{72} +(-0.707107 - 0.292893i) q^{74} -0.765367i q^{75} +(0.707107 - 0.707107i) q^{76} +(-0.541196 + 1.30656i) q^{78} +1.00000 q^{80} -0.414214 q^{81} +(-0.541196 - 1.30656i) q^{88} +(0.382683 + 0.158513i) q^{90} +1.00000 q^{95} +(0.292893 - 0.707107i) q^{96} -1.84776 q^{97} +(-0.382683 + 0.923880i) q^{98} -0.585786i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 8 q^{24} - 8 q^{25} + 8 q^{26} - 8 q^{30} - 8 q^{36} - 8 q^{44} + 8 q^{49} - 8 q^{54} + 8 q^{66} + 8 q^{80} + 8 q^{81} + 8 q^{95} + 8 q^{96}+O(q^{100}) \) 8 * q - 8 * q^9 + 8 * q^24 - 8 * q^25 + 8 * q^26 - 8 * q^30 - 8 * q^36 - 8 * q^44 + 8 * q^49 - 8 * q^54 + 8 * q^66 + 8 * q^80 + 8 * q^81 + 8 * q^95 + 8 * q^96

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