LMFDB - Embedded newform 3136.1.r.b.2431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,1,Mod(2431,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3136.2431");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3136.r (of order $6$, degree $2$, not 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:	$1.56506787962$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.3136.1

Embedding invariants

Embedding label			2431.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3136.2431
Dual form			3136.1.r.b.3007.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{5} +O(q^{10})$	$q+(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(0.866025 - 0.500000i) q^{11} +1.00000i q^{15} +(0.500000 + 0.866025i) q^{17} +(-0.866025 - 0.500000i) q^{19} +(0.866025 + 0.500000i) q^{23} +1.00000i q^{27} +(0.866025 - 0.500000i) q^{31} +(0.500000 - 0.866025i) q^{33} +(-0.500000 + 0.866025i) q^{37} +(0.866025 + 0.500000i) q^{47} +(0.866025 + 0.500000i) q^{51} +(-0.500000 - 0.866025i) q^{53} +1.00000i q^{55} -1.00000 q^{57} +(0.866025 - 0.500000i) q^{59} +(0.500000 - 0.866025i) q^{61} +(-0.866025 + 0.500000i) q^{67} +1.00000 q^{69} +(-0.500000 - 0.866025i) q^{73} +(0.866025 + 0.500000i) q^{79} +(0.500000 + 0.866025i) q^{81} -1.00000 q^{85} +(-0.500000 + 0.866025i) q^{89} +(0.500000 - 0.866025i) q^{93} +(0.866025 - 0.500000i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5}+O(q^{10}) $ 4 * q - 2 * q^5	$ 4 q - 2 q^{5} + 2 q^{17} + 2 q^{33} - 2 q^{37} - 2 q^{53} - 4 q^{57} + 2 q^{61} + 4 q^{69} - 2 q^{73} + 2 q^{81} - 4 q^{85} - 2 q^{89} + 2 q^{93}+O(q^{100}) $ 4 * q - 2 * q^5 + 2 * q^17 + 2 * q^33 - 2 * q^37 - 2 * q^53 - 4 * q^57 + 2 * q^61 + 4 * q^69 - 2 * q^73 + 2 * q^81 - 4 * q^85 - 2 * q^89 + 2 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

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			0
$2$		0			0
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$5$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$11$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$			1.00000i			1.00000i
$16$		0			0
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$19$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$23$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			1.00000i			1.00000i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$31$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$	0.500000	−	0.866025i	0.500000	−	0.866025i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$47$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	0.866025	+	0.500000i	0.866025	+	0.500000i
$52$		0			0
$53$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$53$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$54$		0			0
$55$			1.00000i			1.00000i
$56$		0			0
$57$	−1.00000			−1.00000
$58$		0			0
$59$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$59$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$67$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$68$		0			0
$69$	1.00000			1.00000
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$73$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$79$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$80$		0			0
$81$	0.500000	+	0.866025i	0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−1.00000			−1.00000
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$89$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.500000	−	0.866025i	0.500000	−	0.866025i
$94$		0			0
$95$	0.866025	−	0.500000i	0.866025	−	0.500000i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$101$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$102$		0			0
$103$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$103$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$107$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$108$		0			0
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$			1.00000i			1.00000i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$131$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$136$		0			0
$137$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	1.00000			1.00000
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$150$		0			0
$151$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$151$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1.00000i			1.00000i
$156$		0			0
$157$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$157$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$163$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$	0.500000	+	0.866025i	0.500000	+	0.866025i
$166$		0			0
$167$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$167$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.500000	−	0.866025i	0.500000	−	0.866025i
$178$		0			0
$179$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$179$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−2.00000			−2.00000			−1.00000			$\pi$
$181$	−2.00000			−2.00000			−1.00000			$\pi$
$182$		0			0
$183$		−	1.00000i		−	1.00000i
$184$		0			0
$185$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$186$		0			0
$187$	0.866025	+	0.500000i	0.866025	+	0.500000i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$191$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$193$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$199$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.00000			−1.00000
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$227$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$228$		0			0
$229$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$229$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$233$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$234$		0			0
$235$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$236$		0			0
$237$	1.00000			1.00000
$238$		0			0
$239$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$239$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$251$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$252$		0			0
$253$	1.00000			1.00000
$254$		0			0
$255$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$256$		0			0
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$263$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	1.00000			1.00000
$266$		0			0
$267$			1.00000i			1.00000i
$268$		0			0
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$271$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$277$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$284$		0			0
$285$	0.500000	−	0.866025i	0.500000	−	0.866025i
$286$		0			0
$287$		0			0
$288$		0			0
$289$		0			0
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.00000			2.00000			1.00000			$0$
$293$	2.00000			2.00000			1.00000			$0$
$294$		0			0
$295$			1.00000i			1.00000i
$296$		0			0
$297$	0.500000	+	0.866025i	0.500000	+	0.866025i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$304$		0			0
$305$	0.500000	+	0.866025i	0.500000	+	0.866025i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−1.00000			−1.00000
$310$		0			0
$311$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$311$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$312$		0			0
$313$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$313$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$317$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	1.00000			1.00000
$322$		0			0
$323$		−	1.00000i		−	1.00000i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.866025	+	0.500000i	0.866025	+	0.500000i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$331$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1.00000i		−	1.00000i
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.500000	−	0.866025i	0.500000	−	0.866025i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$346$		0			0
$347$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$347$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$359$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.00000			1.00000
$366$		0			0
$367$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$367$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$373$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$374$		0			0
$375$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$379$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$383$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$389$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$			1.00000i			1.00000i
$392$		0			0
$393$	1.00000			1.00000
$394$		0			0
$395$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$396$		0			0
$397$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$397$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$401$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.00000			−1.00000
$406$		0			0
$407$			1.00000i			1.00000i
$408$		0			0
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		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			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$431$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$438$		0			0
$439$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$439$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$443$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$444$		0			0
$445$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$446$		0			0
$447$		−	1.00000i		−	1.00000i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$457$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$458$		0			0
$459$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	0.500000	+	0.866025i	0.500000	+	0.866025i
$466$		0			0
$467$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$467$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.866025	+	0.500000i	0.866025	+	0.500000i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$479$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$487$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$	−1.00000			−1.00000
$490$		0			0
$491$		−	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			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$500$		0			0
$501$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	1.00000			1.00000
$506$		0			0
$507$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$508$		0			0
$509$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$509$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	0.500000	−	0.866025i	0.500000	−	0.866025i
$514$		0			0
$515$	0.866025	−	0.500000i	0.866025	−	0.500000i
$516$		0			0
$517$	1.00000			1.00000
$518$		0			0
$519$		−	1.00000i		−	1.00000i
$520$		0			0
$521$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$521$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$523$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.866025	+	0.500000i	0.866025	+	0.500000i
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$536$		0			0
$537$	0.500000	−	0.866025i	0.500000	−	0.866025i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$542$		0			0
$543$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$544$		0			0
$545$	−1.00000			−1.00000
$546$		0			0
$547$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$547$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$556$		0			0
$557$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$557$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.00000			1.00000
$562$		0			0
$563$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$563$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$570$		0			0
$571$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$571$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$572$		0			0
$573$	−1.00000			−1.00000
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$577$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$578$		0			0
$579$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−1.00000			−1.00000
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$593$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.500000	−	0.866025i	0.500000	−	0.866025i
$598$		0			0
$599$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$599$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$607$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$613$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$619$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	−	0.866025i	0.500000	−	0.866025i
$626$		0			0
$627$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$628$		0			0
$629$	−1.00000			−1.00000
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$647$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$648$		0			0
$649$	0.500000	−	0.866025i	0.500000	−	0.866025i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$653$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$654$		0			0
$655$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.00000i		−	1.00000i
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$677$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$682$		0			0
$683$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$683$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$	1.00000			1.00000
$686$		0			0
$687$			1.00000i			1.00000i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$691$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$			1.00000i			1.00000i
$700$		0			0
$701$	−2.00000			−2.00000			−1.00000			$\pi$
$701$	−2.00000			−2.00000			−1.00000			$\pi$
$702$		0			0
$703$	0.866025	−	0.500000i	0.866025	−	0.500000i
$704$		0			0
$705$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$709$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.00000			1.00000
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$718$		0			0
$719$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$719$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		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			0
$733$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$733$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$738$		0			0
$739$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$739$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	0.500000	+	0.866025i	0.500000	+	0.866025i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$751$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$754$		0			0
$755$		−	1.00000i		−	1.00000i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$	0.866025	−	0.500000i	0.866025	−	0.500000i
$760$		0			0
$761$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		−	1.00000i		−	1.00000i
$772$		0			0
$773$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$773$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.00000			−1.00000
$786$		0			0
$787$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$787$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$	0.500000	−	0.866025i	0.500000	−	0.866025i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.866025	−	0.500000i	0.866025	−	0.500000i
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$			1.00000i			1.00000i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.866025	+	0.500000i	0.866025	+	0.500000i
$808$		0			0
$809$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$809$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−1.00000			−1.00000
$814$		0			0
$815$	0.866025	−	0.500000i	0.866025	−	0.500000i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$822$		0			0
$823$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$823$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$829$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$830$		0			0
$831$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	1.73205	+	1.00000i	1.73205	+	1.00000i
$836$		0			0
$837$	0.500000	+	0.866025i	0.500000	+	0.866025i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.500000	−	0.866025i	0.500000	−	0.866025i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$850$		0			0
$851$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$857$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$858$		0			0
$859$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$859$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$863$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$864$		0			0
$865$	0.500000	+	0.866025i	0.500000	+	0.866025i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.00000			1.00000
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$878$		0			0
$879$	1.73205	−	1.00000i	1.73205	−	1.00000i
$880$		0			0
$881$	2.00000			2.00000			1.00000			$0$
$881$	2.00000			2.00000			1.00000			$0$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	0.500000	+	0.866025i	0.500000	+	0.866025i
$886$		0			0
$887$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$887$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.866025	+	0.500000i	0.866025	+	0.500000i
$892$		0			0
$893$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$894$		0			0
$895$			1.00000i			1.00000i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	0.500000	−	0.866025i	0.500000	−	0.866025i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.00000	−	1.73205i	1.00000	−	1.73205i
$906$		0			0
$907$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$907$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	0.866025	+	0.500000i	0.866025	+	0.500000i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$919$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$929$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$934$		0			0
$935$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$936$		0			0
$937$	−2.00000			−2.00000			−1.00000			$\pi$
$937$	−2.00000			−2.00000			−1.00000			$\pi$
$938$		0			0
$939$			1.00000i			1.00000i
$940$		0			0
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$947$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$			1.00000i			1.00000i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	0.866025	−	0.500000i	0.866025	−	0.500000i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.00000			1.00000
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$970$		0			0
$971$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$971$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$978$		0			0
$979$			1.00000i			1.00000i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$983$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$991$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$992$		0			0
$993$	−1.00000			−1.00000
$994$		0			0
$995$			1.00000i			1.00000i
$996$		0			0
$997$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$997$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$	−0.866025	−	0.500000i	−0.866025	−	0.500000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.1.r.b.2431.2		4
4.3	odd	2	inner	3136.1.r.b.2431.1		4
7.2	even	3		3136.1.d.d.1471.2		2
7.3	odd	6		448.1.r.a.319.2		4
7.4	even	3	inner	3136.1.r.b.3007.1		4
7.5	odd	6		3136.1.d.b.1471.1		2
7.6	odd	2		448.1.r.a.191.1		4
8.3	odd	2		1568.1.r.a.863.2		4
8.5	even	2		1568.1.r.a.863.1		4
28.3	even	6		448.1.r.a.319.1		4
28.11	odd	6	inner	3136.1.r.b.3007.2		4
28.19	even	6		3136.1.d.b.1471.2		2
28.23	odd	6		3136.1.d.d.1471.1		2
28.27	even	2		448.1.r.a.191.2		4
56.3	even	6		224.1.r.a.95.2	yes	4
56.5	odd	6		1568.1.d.b.1471.2		2
56.11	odd	6		1568.1.r.a.1439.1		4
56.13	odd	2		224.1.r.a.191.2	yes	4
56.19	even	6		1568.1.d.b.1471.1		2
56.27	even	2		224.1.r.a.191.1	yes	4
56.37	even	6		1568.1.d.a.1471.1		2
56.45	odd	6		224.1.r.a.95.1	✓	4
56.51	odd	6		1568.1.d.a.1471.2		2
56.53	even	6		1568.1.r.a.1439.2		4
112.3	even	12		1792.1.o.b.1663.2		4
112.13	odd	4		1792.1.o.a.639.1		4
112.27	even	4		1792.1.o.a.639.2		4
112.45	odd	12		1792.1.o.a.1663.2		4
112.59	even	12		1792.1.o.a.1663.1		4
112.69	odd	4		1792.1.o.b.639.2		4
112.83	even	4		1792.1.o.b.639.1		4
112.101	odd	12		1792.1.o.b.1663.1		4
168.59	odd	6		2016.1.cd.a.991.2		4
168.83	odd	2		2016.1.cd.a.415.2		4
168.101	even	6		2016.1.cd.a.991.1		4
168.125	even	2		2016.1.cd.a.415.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.1.r.a.95.1	✓	4	56.45	odd	6
224.1.r.a.95.2	yes	4	56.3	even	6
224.1.r.a.191.1	yes	4	56.27	even	2
224.1.r.a.191.2	yes	4	56.13	odd	2
448.1.r.a.191.1		4	7.6	odd	2
448.1.r.a.191.2		4	28.27	even	2
448.1.r.a.319.1		4	28.3	even	6
448.1.r.a.319.2		4	7.3	odd	6
1568.1.d.a.1471.1		2	56.37	even	6
1568.1.d.a.1471.2		2	56.51	odd	6
1568.1.d.b.1471.1		2	56.19	even	6
1568.1.d.b.1471.2		2	56.5	odd	6
1568.1.r.a.863.1		4	8.5	even	2
1568.1.r.a.863.2		4	8.3	odd	2
1568.1.r.a.1439.1		4	56.11	odd	6
1568.1.r.a.1439.2		4	56.53	even	6
1792.1.o.a.639.1		4	112.13	odd	4
1792.1.o.a.639.2		4	112.27	even	4
1792.1.o.a.1663.1		4	112.59	even	12
1792.1.o.a.1663.2		4	112.45	odd	12
1792.1.o.b.639.1		4	112.83	even	4
1792.1.o.b.639.2		4	112.69	odd	4
1792.1.o.b.1663.1		4	112.101	odd	12
1792.1.o.b.1663.2		4	112.3	even	12
2016.1.cd.a.415.1		4	168.125	even	2
2016.1.cd.a.415.2		4	168.83	odd	2
2016.1.cd.a.991.1		4	168.101	even	6
2016.1.cd.a.991.2		4	168.59	odd	6
3136.1.d.b.1471.1		2	7.5	odd	6
3136.1.d.b.1471.2		2	28.19	even	6
3136.1.d.d.1471.1		2	28.23	odd	6
3136.1.d.d.1471.2		2	7.2	even	3
3136.1.r.b.2431.1		4	4.3	odd	2	inner
3136.1.r.b.2431.2		4	1.1	even	1	trivial
3136.1.r.b.3007.1		4	7.4	even	3	inner
3136.1.r.b.3007.2		4	28.11	odd	6	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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3136.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{5} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(0.866025 - 0.500000i) q^{11} +1.00000i q^{15} +(0.500000 + 0.866025i) q^{17} +(-0.866025 - 0.500000i) q^{19} +(0.866025 + 0.500000i) q^{23} +1.00000i q^{27} +(0.866025 - 0.500000i) q^{31} +(0.500000 - 0.866025i) q^{33} +(-0.500000 + 0.866025i) q^{37} +(0.866025 + 0.500000i) q^{47} +(0.866025 + 0.500000i) q^{51} +(-0.500000 - 0.866025i) q^{53} +1.00000i q^{55} -1.00000 q^{57} +(0.866025 - 0.500000i) q^{59} +(0.500000 - 0.866025i) q^{61} +(-0.866025 + 0.500000i) q^{67} +1.00000 q^{69} +(-0.500000 - 0.866025i) q^{73} +(0.866025 + 0.500000i) q^{79} +(0.500000 + 0.866025i) q^{81} -1.00000 q^{85} +(-0.500000 + 0.866025i) q^{89} +(0.500000 - 0.866025i) q^{93} +(0.866025 - 0.500000i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5}+O(q^{10}) \) 4 * q - 2 * q^5	\( 4 q - 2 q^{5} + 2 q^{17} + 2 q^{33} - 2 q^{37} - 2 q^{53} - 4 q^{57} + 2 q^{61} + 4 q^{69} - 2 q^{73} + 2 q^{81} - 4 q^{85} - 2 q^{89} + 2 q^{93}+O(q^{100}) \) 4 * q - 2 * q^5 + 2 * q^17 + 2 * q^33 - 2 * q^37 - 2 * q^53 - 4 * q^57 + 2 * q^61 + 4 * q^69 - 2 * q^73 + 2 * q^81 - 4 * q^85 - 2 * q^89 + 2 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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