LMFDB - Embedded newform 1305.1.b.b.1304.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,1,Mod(1304,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1305.1304");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1305.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:	$0.651279841486$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.19575.1

Embedding invariants

Embedding label			1304.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1305.1304
Dual form			1305.1.b.b.1304.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{5} +1.00000i q^{7} -1.00000i q^{8} +(0.707107 + 0.707107i) q^{10} +1.00000 q^{11} +1.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} +1.41421i q^{19} -1.00000i q^{22} -1.00000i q^{25} +1.00000 q^{26} -1.00000 q^{29} -1.41421i q^{31} +1.00000 q^{34} +(-0.707107 - 0.707107i) q^{35} +1.41421 q^{37} +1.41421 q^{38} +(0.707107 + 0.707107i) q^{40} -1.00000i q^{47} -1.00000 q^{50} +1.41421 q^{53} +(-0.707107 + 0.707107i) q^{55} +1.00000 q^{56} +1.00000i q^{58} +1.41421i q^{59} -1.41421i q^{61} -1.41421 q^{62} -1.00000 q^{64} +(-0.707107 - 0.707107i) q^{65} -1.00000i q^{67} +(-0.707107 + 0.707107i) q^{70} -1.41421i q^{74} +1.00000i q^{77} +(0.707107 - 0.707107i) q^{80} +(-0.707107 - 0.707107i) q^{85} -1.00000i q^{88} -1.00000 q^{89} -1.00000 q^{91} -1.00000 q^{94} +(-1.00000 - 1.00000i) q^{95} -1.41421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{11} + 4 q^{14} - 4 q^{16} + 4 q^{26} - 4 q^{29} + 4 q^{34} - 4 q^{50} + 4 q^{56} - 4 q^{64} - 4 q^{89} - 4 q^{91} - 4 q^{94} - 4 q^{95}+O(q^{100}) $ 4 * q + 4 * q^11 + 4 * q^14 - 4 * q^16 + 4 * q^26 - 4 * q^29 + 4 * q^34 - 4 * q^50 + 4 * q^56 - 4 * q^64 - 4 * q^89 - 4 * q^91 - 4 * q^94 - 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$5$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$6$		0			0
$7$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$		−	1.00000i		−	1.00000i
$9$		0			0
$10$	0.707107	+	0.707107i	0.707107	+	0.707107i
$11$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$11$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$13$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$14$	1.00000			1.00000
$15$		0			0
$16$	−1.00000			−1.00000
$17$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$17$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$19$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		0			0
$22$		−	1.00000i		−	1.00000i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$		−	1.00000i		−	1.00000i
$26$	1.00000			1.00000
$27$		0			0
$28$		0			0
$29$	−1.00000			−1.00000
$29$	−1.00000			−1.00000
$30$		0			0
$31$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$31$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$32$		0			0
$33$		0			0
$34$	1.00000			1.00000
$35$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$36$		0			0
$37$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$37$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$38$	1.41421			1.41421
$39$		0			0
$40$	0.707107	+	0.707107i	0.707107	+	0.707107i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$47$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$48$		0			0
$49$		0			0
$50$	−1.00000			−1.00000
$51$		0			0
$52$		0			0
$53$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$53$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$56$	1.00000			1.00000
$57$		0			0
$58$			1.00000i			1.00000i
$59$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$59$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$61$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$62$	−1.41421			−1.41421
$63$		0			0
$64$	−1.00000			−1.00000
$65$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$66$		0			0
$67$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$67$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$68$		0			0
$69$		0			0
$70$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		−	1.41421i		−	1.41421i
$75$		0			0
$76$		0			0
$77$			1.00000i			1.00000i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	0.707107	−	0.707107i	0.707107	−	0.707107i
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$86$		0			0
$87$		0			0
$88$		−	1.00000i		−	1.00000i
$89$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−1.00000			−1.00000
$92$		0			0
$93$		0			0
$94$	−1.00000			−1.00000
$95$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$96$		0			0
$97$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$97$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	1.00000			1.00000
$105$		0			0
$106$		−	1.41421i		−	1.41421i
$107$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$107$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$110$	0.707107	+	0.707107i	0.707107	+	0.707107i
$111$		0			0
$112$		−	1.00000i		−	1.00000i
$113$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$113$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.41421			1.41421
$119$	−1.00000			−1.00000
$120$		0			0
$121$		0			0
$122$	−1.41421			−1.41421
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.707107	+	0.707107i
$126$		0			0
$127$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$127$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$128$			1.00000i			1.00000i
$129$		0			0
$130$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$131$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	−1.41421			−1.41421
$134$	−1.00000			−1.00000
$135$		0			0
$136$	1.00000			1.00000
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.00000i			1.00000i
$144$		0			0
$145$	0.707107	−	0.707107i	0.707107	−	0.707107i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$149$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	1.41421			1.41421
$153$		0			0
$154$	1.00000			1.00000
$155$	1.00000	+	1.00000i	1.00000	+	1.00000i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$163$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$167$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		0			0
$170$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$171$		0			0
$172$		0			0
$173$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$173$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$	1.00000			1.00000
$176$	−1.00000			−1.00000
$177$		0			0
$178$			1.00000i			1.00000i
$179$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$179$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$180$		0			0
$181$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$182$			1.00000i			1.00000i
$183$		0			0
$184$		0			0
$185$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$186$		0			0
$187$			1.00000i			1.00000i
$188$		0			0
$189$		0			0
$190$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$193$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$194$			1.41421i			1.41421i
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$200$	−1.00000			−1.00000
$201$		0			0
$202$			1.00000i			1.00000i
$203$		−	1.00000i		−	1.00000i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	1.00000i		−	1.00000i
$209$			1.41421i			1.41421i
$210$		0			0
$211$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$211$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$212$		0			0
$213$		0			0
$214$		−	1.41421i		−	1.41421i
$215$		0			0
$216$		0			0
$217$	1.41421			1.41421
$218$			1.00000i			1.00000i
$219$		0			0
$220$		0			0
$221$	−1.00000			−1.00000
$222$		0			0
$223$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$223$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$224$		0			0
$225$		0			0
$226$	1.00000			1.00000
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$			1.00000i			1.00000i
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	0.707107	+	0.707107i	0.707107	+	0.707107i
$236$		0			0
$237$		0			0
$238$			1.00000i			1.00000i
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$241$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.41421			−1.41421
$248$	−1.41421			−1.41421
$249$		0			0
$250$	0.707107	−	0.707107i	0.707107	−	0.707107i
$251$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$251$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$			1.41421i			1.41421i
$255$		0			0
$256$		0			0
$257$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$257$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$			1.41421i			1.41421i
$260$		0			0
$261$		0			0
$262$		−	1.00000i		−	1.00000i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$266$			1.41421i			1.41421i
$267$		0			0
$268$		0			0
$269$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		−	1.00000i		−	1.00000i
$273$		0			0
$274$		0			0
$275$		−	1.00000i		−	1.00000i
$276$		0			0
$277$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$277$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$278$		−	1.00000i		−	1.00000i
$279$		0			0
$280$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$	1.00000			1.00000
$287$		0			0
$288$		0			0
$289$		0			0
$290$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$291$		0			0
$292$		0			0
$293$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$293$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$294$		0			0
$295$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$296$		−	1.41421i		−	1.41421i
$297$		0			0
$298$	−1.41421			−1.41421
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	1.41421i		−	1.41421i
$305$	1.00000	+	1.00000i	1.00000	+	1.00000i
$306$		0			0
$307$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$307$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$	1.00000	−	1.00000i	1.00000	−	1.00000i
$311$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$311$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$313$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$317$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$	−1.00000			−1.00000
$320$	0.707107	−	0.707107i	0.707107	−	0.707107i
$321$		0			0
$322$		0			0
$323$	−1.41421			−1.41421
$324$		0			0
$325$	1.00000			1.00000
$326$		−	1.41421i		−	1.41421i
$327$		0			0
$328$		0			0
$329$	1.00000			1.00000
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$			1.41421i			1.41421i
$335$	0.707107	+	0.707107i	0.707107	+	0.707107i
$336$		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			0
$339$		0			0
$340$		0			0
$341$		−	1.41421i		−	1.41421i
$342$		0			0
$343$			1.00000i			1.00000i
$344$		0			0
$345$		0			0
$346$			1.41421i			1.41421i
$347$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$347$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		−	1.00000i		−	1.00000i
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.41421			−1.41421
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$			1.00000i			1.00000i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$367$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$	1.00000	+	1.00000i	1.00000	+	1.00000i
$371$			1.41421i			1.41421i
$372$		0			0
$373$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$373$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$374$	1.00000			1.00000
$375$		0			0
$376$	−1.00000			−1.00000
$377$		−	1.00000i		−	1.00000i
$378$		0			0
$379$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$379$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$386$			1.41421i			1.41421i
$387$		0			0
$388$		0			0
$389$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		−	1.00000i		−	1.00000i
$399$		0			0
$400$			1.00000i			1.00000i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	1.41421			1.41421
$404$		0			0
$405$		0			0
$406$	−1.00000			−1.00000
$407$	1.41421			1.41421
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.41421			−1.41421
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	1.41421			1.41421
$419$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$419$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$420$		0			0
$421$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$421$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$422$	−1.41421			−1.41421
$423$		0			0
$424$		−	1.41421i		−	1.41421i
$425$	1.00000			1.00000
$426$		0			0
$427$	1.41421			1.41421
$428$		0			0
$429$		0			0
$430$		0			0
$431$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$431$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		−	1.41421i		−	1.41421i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$440$	0.707107	+	0.707107i	0.707107	+	0.707107i
$441$		0			0
$442$			1.00000i			1.00000i
$443$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$443$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$444$		0			0
$445$	0.707107	−	0.707107i	0.707107	−	0.707107i
$446$	−1.00000			−1.00000
$447$		0			0
$448$		−	1.00000i		−	1.00000i
$449$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.707107	−	0.707107i	0.707107	−	0.707107i
$456$		0			0
$457$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$457$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$463$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$464$	1.00000			1.00000
$465$		0			0
$466$		0			0
$467$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$467$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$468$		0			0
$469$	1.00000			1.00000
$470$	0.707107	−	0.707107i	0.707107	−	0.707107i
$471$		0			0
$472$	1.41421			1.41421
$473$		0			0
$474$		0			0
$475$	1.41421			1.41421
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$			1.41421i			1.41421i
$482$		−	1.00000i		−	1.00000i
$483$		0			0
$484$		0			0
$485$	1.00000	−	1.00000i	1.00000	−	1.00000i
$486$		0			0
$487$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$487$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$488$	−1.41421			−1.41421
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		−	1.00000i		−	1.00000i
$494$			1.41421i			1.41421i
$495$		0			0
$496$			1.41421i			1.41421i
$497$		0			0
$498$		0			0
$499$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$		−	1.00000i		−	1.00000i
$503$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$503$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$	0.707107	−	0.707107i	0.707107	−	0.707107i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$509$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$		0			0
$514$		−	1.41421i		−	1.41421i
$515$		0			0
$516$		0			0
$517$		−	1.00000i		−	1.00000i
$518$	1.41421			1.41421
$519$		0			0
$520$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$523$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.41421			1.41421
$528$		0			0
$529$	−1.00000			−1.00000
$530$	1.00000	+	1.00000i	1.00000	+	1.00000i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$536$	−1.00000			−1.00000
$537$		0			0
$538$			1.00000i			1.00000i
$539$		0			0
$540$		0			0
$541$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$541$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.707107	−	0.707107i	0.707107	−	0.707107i
$546$		0			0
$547$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$547$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$548$		0			0
$549$		0			0
$550$	−1.00000			−1.00000
$551$		−	1.41421i		−	1.41421i
$552$		0			0
$553$		0			0
$554$	1.00000			1.00000
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	0.707107	+	0.707107i	0.707107	+	0.707107i
$561$		0			0
$562$		0			0
$563$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$563$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$564$		0			0
$565$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$569$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$577$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.41421			1.41421
$584$		0			0
$585$		0			0
$586$	−1.00000			−1.00000
$587$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$587$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$	2.00000			2.00000
$590$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$591$		0			0
$592$	−1.41421			−1.41421
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$	0.707107	−	0.707107i	0.707107	−	0.707107i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$	1.00000	−	1.00000i	1.00000	−	1.00000i
$611$	1.00000			1.00000
$612$		0			0
$613$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$613$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$614$		−	1.41421i		−	1.41421i
$615$		0			0
$616$	1.00000			1.00000
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$619$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		−	1.00000i		−	1.00000i
$623$		−	1.00000i		−	1.00000i
$624$		0			0
$625$	−1.00000			−1.00000
$626$	−1.00000			−1.00000
$627$		0			0
$628$		0			0
$629$			1.41421i			1.41421i
$630$		0			0
$631$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$		0			0
$634$	1.00000			1.00000
$635$	1.00000	−	1.00000i	1.00000	−	1.00000i
$636$		0			0
$637$		0			0
$638$			1.00000i			1.00000i
$639$		0			0
$640$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$641$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$643$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$644$		0			0
$645$		0			0
$646$			1.41421i			1.41421i
$647$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$647$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$			1.41421i			1.41421i
$650$		−	1.00000i		−	1.00000i
$651$		0			0
$652$		0			0
$653$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$653$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$656$		0			0
$657$		0			0
$658$		−	1.00000i		−	1.00000i
$659$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$659$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.00000	−	1.00000i	1.00000	−	1.00000i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	0.707107	−	0.707107i	0.707107	−	0.707107i
$671$		−	1.41421i		−	1.41421i
$672$		0			0
$673$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$673$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$674$			1.41421i			1.41421i
$675$		0			0
$676$		0			0
$677$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$677$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$678$		0			0
$679$		−	1.41421i		−	1.41421i
$680$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$681$		0			0
$682$	−1.41421			−1.41421
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	1.00000			1.00000
$687$		0			0
$688$		0			0
$689$			1.41421i			1.41421i
$690$		0			0
$691$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$691$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$		0			0
$694$		−	1.41421i		−	1.41421i
$695$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$			2.00000i			2.00000i
$704$	−1.00000			−1.00000
$705$		0			0
$706$		0			0
$707$		−	1.00000i		−	1.00000i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$			1.00000i			1.00000i
$713$		0			0
$714$		0			0
$715$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$			1.00000i			1.00000i
$723$		0			0
$724$		0			0
$725$			1.00000i			1.00000i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$			1.00000i			1.00000i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		−	1.41421i		−	1.41421i
$735$		0			0
$736$		0			0
$737$		−	1.00000i		−	1.00000i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$	1.41421			1.41421
$743$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$743$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$	1.00000	+	1.00000i	1.00000	+	1.00000i
$746$	−2.00000			−2.00000
$747$		0			0
$748$		0			0
$749$			1.41421i			1.41421i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			1.00000i			1.00000i
$753$		0			0
$754$	−1.00000			−1.00000
$755$		0			0
$756$		0			0
$757$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$757$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$758$	1.41421			1.41421
$759$		0			0
$760$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		−	1.00000i		−	1.00000i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.41421			−1.41421
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−1.41421			−1.41421
$776$			1.41421i			1.41421i
$777$		0			0
$778$		−	1.00000i		−	1.00000i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$787$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.00000			−1.00000
$792$		0			0
$793$	1.41421			1.41421
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	1.00000			1.00000
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		−	1.41421i		−	1.41421i
$807$		0			0
$808$			1.00000i			1.00000i
$809$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$809$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$811$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$812$		0			0
$813$		0			0
$814$		−	1.41421i		−	1.41421i
$815$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$821$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$			1.41421i			1.41421i
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$829$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$830$		0			0
$831$		0			0
$832$		−	1.00000i		−	1.00000i
$833$		0			0
$834$		0			0
$835$	1.00000	−	1.00000i	1.00000	−	1.00000i
$836$		0			0
$837$		0			0
$838$	−1.41421			−1.41421
$839$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$839$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	1.41421			1.41421
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−1.41421			−1.41421
$849$		0			0
$850$		−	1.00000i		−	1.00000i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		−	1.41421i		−	1.41421i
$855$		0			0
$856$		−	1.41421i		−	1.41421i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$859$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$	1.41421			1.41421
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	1.00000	−	1.00000i	1.00000	−	1.00000i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1.00000			1.00000
$872$			1.00000i			1.00000i
$873$		0			0
$874$		0			0
$875$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$			1.00000i			1.00000i
$879$		0			0
$880$	0.707107	−	0.707107i	0.707107	−	0.707107i
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1.00000			−1.00000
$887$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$887$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		−	1.41421i		−	1.41421i
$890$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$891$		0			0
$892$		0			0
$893$	1.41421			1.41421
$894$		0			0
$895$	1.00000	+	1.00000i	1.00000	+	1.00000i
$896$	−1.00000			−1.00000
$897$		0			0
$898$			1.00000i			1.00000i
$899$			1.41421i			1.41421i
$900$		0			0
$901$			1.41421i			1.41421i
$902$		0			0
$903$		0			0
$904$	1.00000			1.00000
$905$	0.707107	−	0.707107i	0.707107	−	0.707107i
$906$		0			0
$907$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$907$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		0			0
$910$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$911$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$911$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$	1.00000			1.00000
$915$		0			0
$916$		0			0
$917$			1.00000i			1.00000i
$918$		0			0
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	1.41421i		−	1.41421i
$926$	−1.00000			−1.00000
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−2.00000			−2.00000
$935$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$936$		0			0
$937$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$937$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$938$		−	1.00000i		−	1.00000i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		−	1.41421i		−	1.41421i
$945$		0			0
$946$		0			0
$947$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$947$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$		0			0
$950$		−	1.41421i		−	1.41421i
$951$		0			0
$952$			1.00000i			1.00000i
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.00000			−1.00000
$962$	1.41421			1.41421
$963$		0			0
$964$		0			0
$965$	1.00000	−	1.00000i	1.00000	−	1.00000i
$966$		0			0
$967$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$967$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$971$	2.00000			2.00000			1.00000			$0$
$971$	2.00000			2.00000			1.00000			$0$
$972$		0			0
$973$			1.00000i			1.00000i
$974$	−2.00000			−2.00000
$975$		0			0
$976$			1.41421i			1.41421i
$977$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$977$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$	−1.00000			−1.00000
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$	−1.00000			−1.00000
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$			1.00000i			1.00000i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.1.b.b.1304.1	yes	4
3.2	odd	2		1305.1.b.a.1304.4	yes	4
5.4	even	2	inner	1305.1.b.b.1304.4	yes	4
15.14	odd	2		1305.1.b.a.1304.1	✓	4
29.28	even	2		1305.1.b.a.1304.3	yes	4
87.86	odd	2	inner	1305.1.b.b.1304.2	yes	4
145.144	even	2		1305.1.b.a.1304.2	yes	4
435.434	odd	2	inner	1305.1.b.b.1304.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.1.b.a.1304.1	✓	4	15.14	odd	2
1305.1.b.a.1304.2	yes	4	145.144	even	2
1305.1.b.a.1304.3	yes	4	29.28	even	2
1305.1.b.a.1304.4	yes	4	3.2	odd	2
1305.1.b.b.1304.1	yes	4	1.1	even	1	trivial
1305.1.b.b.1304.2	yes	4	87.86	odd	2	inner
1305.1.b.b.1304.3	yes	4	435.434	odd	2	inner
1305.1.b.b.1304.4	yes	4	5.4	even	2	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1305.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{5} +1.00000i q^{7} -1.00000i q^{8} +(0.707107 + 0.707107i) q^{10} +1.00000 q^{11} +1.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} +1.41421i q^{19} -1.00000i q^{22} -1.00000i q^{25} +1.00000 q^{26} -1.00000 q^{29} -1.41421i q^{31} +1.00000 q^{34} +(-0.707107 - 0.707107i) q^{35} +1.41421 q^{37} +1.41421 q^{38} +(0.707107 + 0.707107i) q^{40} -1.00000i q^{47} -1.00000 q^{50} +1.41421 q^{53} +(-0.707107 + 0.707107i) q^{55} +1.00000 q^{56} +1.00000i q^{58} +1.41421i q^{59} -1.41421i q^{61} -1.41421 q^{62} -1.00000 q^{64} +(-0.707107 - 0.707107i) q^{65} -1.00000i q^{67} +(-0.707107 + 0.707107i) q^{70} -1.41421i q^{74} +1.00000i q^{77} +(0.707107 - 0.707107i) q^{80} +(-0.707107 - 0.707107i) q^{85} -1.00000i q^{88} -1.00000 q^{89} -1.00000 q^{91} -1.00000 q^{94} +(-1.00000 - 1.00000i) q^{95} -1.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{11} + 4 q^{14} - 4 q^{16} + 4 q^{26} - 4 q^{29} + 4 q^{34} - 4 q^{50} + 4 q^{56} - 4 q^{64} - 4 q^{89} - 4 q^{91} - 4 q^{94} - 4 q^{95}+O(q^{100}) \) 4 * q + 4 * q^11 + 4 * q^14 - 4 * q^16 + 4 * q^26 - 4 * q^29 + 4 * q^34 - 4 * q^50 + 4 * q^56 - 4 * q^64 - 4 * q^89 - 4 * q^91 - 4 * q^94 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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