LMFDB - Embedded newform 2475.1.t.a.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,1,Mod(274,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2475.274");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2475.t (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:	$1.23518590627$
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 99)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.891.1
Artin image:	$S_3\times C_{12}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} + \cdots)$

Embedding invariants

Embedding label			274.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2475.274
Dual form			2475.1.t.a.1924.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{11} +(0.866025 - 0.500000i) q^{12} +(-0.500000 - 0.866025i) q^{16} +(1.73205 + 1.00000i) q^{23} +1.00000i q^{27} +(0.500000 - 0.866025i) q^{31} -1.00000i q^{33} +1.00000 q^{36} -1.00000i q^{37} -1.00000 q^{44} +(-0.866025 + 0.500000i) q^{47} -1.00000i q^{48} +(0.500000 - 0.866025i) q^{49} +1.00000i q^{53} +(-0.500000 + 0.866025i) q^{59} -1.00000 q^{64} +(0.866025 + 0.500000i) q^{67} +(1.00000 + 1.73205i) q^{69} -1.00000 q^{71} +(-0.500000 + 0.866025i) q^{81} -2.00000 q^{89} +(1.73205 - 1.00000i) q^{92} +(0.866025 - 0.500000i) q^{93} +(-0.866025 + 0.500000i) q^{97} +(0.500000 - 0.866025i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 + 2 * q^9	$ 4 q + 2 q^{4} + 2 q^{9} - 2 q^{11} - 2 q^{16} + 2 q^{31} + 4 q^{36} - 4 q^{44} + 2 q^{49} - 2 q^{59} - 4 q^{64} + 4 q^{69} - 4 q^{71} - 2 q^{81} - 8 q^{89} + 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 + 2 * q^9 - 2 * q^11 - 2 * q^16 + 2 * q^31 + 4 * q^36 - 4 * q^44 + 2 * q^49 - 2 * q^59 - 4 * q^64 + 4 * q^69 - 4 * q^71 - 2 * q^81 - 8 * q^89 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$4$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$	0.500000	+	0.866025i	0.500000	+	0.866025i
$10$		0			0
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$12$	0.866025	−	0.500000i	0.866025	−	0.500000i
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$23$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			1.00000i			1.00000i
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$31$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$32$		0			0
$33$		−	1.00000i		−	1.00000i
$34$		0			0
$35$		0			0
$36$	1.00000			1.00000
$37$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$37$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$	−1.00000			−1.00000
$45$		0			0
$46$		0			0
$47$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$47$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$48$		−	1.00000i		−	1.00000i
$49$	0.500000	−	0.866025i	0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$53$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$59$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$67$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$68$		0			0
$69$	1.00000	+	1.73205i	1.00000	+	1.73205i
$70$		0			0
$71$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000			−2.00000			−1.00000			$\pi$
$89$	−2.00000			−2.00000			−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	1.73205	−	1.00000i	1.73205	−	1.00000i
$93$	0.866025	−	0.500000i	0.866025	−	0.500000i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$97$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$98$		0			0
$99$	0.500000	−	0.866025i	0.500000	−	0.866025i
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		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			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	0.866025	+	0.500000i	0.866025	+	0.500000i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	0.500000	−	0.866025i	0.500000	−	0.866025i
$112$		0			0
$113$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$113$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$137$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$	−1.00000			−1.00000
$142$		0			0
$143$		0			0
$144$	0.500000	−	0.866025i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	0.866025	−	0.500000i	0.866025	−	0.500000i
$148$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$157$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$158$		0			0
$159$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$163$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	0.500000	+	0.866025i	0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$177$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$178$		0			0
$179$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$179$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$			1.00000i			1.00000i
$189$		0			0
$190$		0			0
$191$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$191$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$192$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$193$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$193$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$		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	$0.333333\pi$
$200$		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
$207$			2.00000i			2.00000i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	0.866025	+	0.500000i	0.866025	+	0.500000i
$213$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$223$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$229$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$	0.500000	+	0.866025i	0.500000	+	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.00000			2.00000			1.00000			$0$
$251$	2.00000			2.00000			1.00000			$0$
$252$		0			0
$253$		−	2.00000i		−	2.00000i
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$257$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$268$	0.866025	−	0.500000i	0.866025	−	0.500000i
$269$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$269$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	2.00000			2.00000
$277$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$277$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$278$		0			0
$279$	1.00000			1.00000
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−1.00000
$290$		0			0
$291$	−1.00000			−1.00000
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	0.866025	−	0.500000i	0.866025	−	0.500000i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$310$		0			0
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$312$		0			0
$313$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$313$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$317$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.500000	+	0.866025i	0.500000	+	0.866025i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	0.866025	−	0.500000i	0.866025	−	0.500000i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$337$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$338$		0			0
$339$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$340$		0			0
$341$	−1.00000			−1.00000
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$353$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		0			0
$363$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$364$		0			0
$365$		0			0
$366$		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	$0.5\pi$
$368$		−	2.00000i		−	2.00000i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		−	1.00000i		−	1.00000i
$373$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$373$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.00000			−2.00000			−1.00000			$\pi$
$379$	−2.00000			−2.00000			−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$383$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$			1.00000i			1.00000i
$389$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$389$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$397$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$397$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	−1.00000			−1.00000
$412$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$419$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$420$		0			0
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	0.866025	−	0.500000i	0.866025	−	0.500000i
$433$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$433$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	1.00000			1.00000
$442$		0			0
$443$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$443$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$444$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$457$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$463$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$467$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.500000	+	0.866025i	0.500000	+	0.866025i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.500000	+	0.866025i	0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$487$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$488$		0			0
$489$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000			−1.00000
$497$		0			0
$498$		0			0
$499$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$499$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1.00000i			1.00000i
$508$		0			0
$509$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$509$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	0.866025	+	0.500000i	0.866025	+	0.500000i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$529$	1.50000	+	2.59808i	1.50000	+	2.59808i
$530$		0			0
$531$	−1.00000			−1.00000
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.866025	+	0.500000i	0.866025	+	0.500000i
$538$		0			0
$539$	−1.00000			−1.00000
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$547$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$548$			1.00000i			1.00000i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		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			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$			1.00000i			1.00000i
$574$		0			0
$575$		0			0
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$577$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$578$		0			0
$579$		0			0
$580$		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
$587$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$587$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$588$		−	1.00000i		−	1.00000i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.866025	+	0.500000i	0.866025	+	0.500000i
$598$		0			0
$599$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$			1.00000i			1.00000i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$617$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$618$		0			0
$619$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$619$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$620$		0			0
$621$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	0.866025	−	0.500000i	0.866025	−	0.500000i
$629$		0			0
$630$		0			0
$631$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.500000	+	0.866025i	0.500000	+	0.866025i
$637$		0			0
$638$		0			0
$639$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$640$		0			0
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$642$		0			0
$643$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$643$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$647$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$648$		0			0
$649$	1.00000			1.00000
$650$		0			0
$651$		0			0
$652$	0.866025	+	0.500000i	0.866025	+	0.500000i
$653$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$653$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.00000			−2.00000
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$673$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$674$		0			0
$675$		0			0
$676$	1.00000			1.00000
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$683$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.73205	−	1.00000i	1.73205	−	1.00000i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$691$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		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
$703$		0			0
$704$	0.500000	+	0.866025i	0.500000	+	0.866025i
$705$		0			0
$706$		0			0
$707$		0			0
$708$			1.00000i			1.00000i
$709$	−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
$713$	1.73205	−	1.00000i	1.73205	−	1.00000i
$714$		0			0
$715$		0			0
$716$	0.500000	−	0.866025i	0.500000	−	0.866025i
$717$		0			0
$718$		0			0
$719$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$719$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$725$		0			0
$726$		0			0
$727$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$727$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		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
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$752$	0.866025	+	0.500000i	0.866025	+	0.500000i
$753$	1.73205	+	1.00000i	1.73205	+	1.00000i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$757$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$758$		0			0
$759$	1.00000	−	1.73205i	1.00000	−	1.73205i
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$	1.00000			1.00000
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$772$		0			0
$773$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$773$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.500000	+	0.866025i	0.500000	+	0.866025i
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.500000	−	0.866025i	0.500000	−	0.866025i
$797$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$797$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$802$		0			0
$803$		0			0
$804$	1.00000			1.00000
$805$		0			0
$806$		0			0
$807$	0.866025	+	0.500000i	0.866025	+	0.500000i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$823$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	1.73205	+	1.00000i	1.73205	+	1.00000i
$829$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$829$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.866025	+	0.500000i	0.866025	+	0.500000i
$838$		0			0
$839$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$839$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	0.866025	−	0.500000i	0.866025	−	0.500000i
$849$		0			0
$850$		0			0
$851$	1.00000	−	1.73205i	1.00000	−	1.73205i
$852$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$853$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$853$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$859$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−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
$883$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$883$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.00000			1.00000
$892$			2.00000i			2.00000i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$907$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$911$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	1.00000i		−	1.00000i
$928$		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
$933$	0.866025	−	0.500000i	0.866025	−	0.500000i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$	−2.00000			−2.00000
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	1.00000			1.00000
$945$		0			0
$946$		0			0
$947$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$947$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	2.00000			2.00000
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			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
$965$		0			0
$966$		0			0
$967$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$967$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.00000			2.00000			1.00000			$0$
$971$	2.00000			2.00000			1.00000			$0$
$972$			1.00000i			1.00000i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$977$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$978$		0			0
$979$	1.00000	+	1.73205i	1.00000	+	1.73205i
$980$		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
$991$	2.00000			2.00000			1.00000			$0$
$991$	2.00000			2.00000			1.00000			$0$
$992$		0			0
$993$			1.00000i			1.00000i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$	1.00000			1.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.1.t.a.274.2		4
5.2	odd	4		2475.1.y.a.76.1		2
5.3	odd	4		99.1.h.a.76.1	yes	2
5.4	even	2	inner	2475.1.t.a.274.1		4
9.7	even	3	inner	2475.1.t.a.1924.1		4
11.10	odd	2	CM	2475.1.t.a.274.2		4
15.8	even	4		297.1.h.a.10.1		2
20.3	even	4		1584.1.bf.b.769.1		2
45.7	odd	12		2475.1.y.a.1726.1		2
45.13	odd	12		891.1.c.a.406.1		1
45.23	even	12		891.1.c.b.406.1		1
45.34	even	6	inner	2475.1.t.a.1924.2		4
45.38	even	12		297.1.h.a.208.1		2
45.43	odd	12		99.1.h.a.43.1	✓	2
55.3	odd	20		1089.1.s.a.112.1		8
55.8	even	20		1089.1.s.a.112.1		8
55.13	even	20		1089.1.s.a.40.1		8
55.18	even	20		1089.1.s.a.94.1		8
55.28	even	20		1089.1.s.a.481.1		8
55.32	even	4		2475.1.y.a.76.1		2
55.38	odd	20		1089.1.s.a.481.1		8
55.43	even	4		99.1.h.a.76.1	yes	2
55.48	odd	20		1089.1.s.a.94.1		8
55.53	odd	20		1089.1.s.a.40.1		8
55.54	odd	2	inner	2475.1.t.a.274.1		4
99.43	odd	6	inner	2475.1.t.a.1924.1		4
165.8	odd	20		3267.1.w.a.1927.1		8
165.38	even	20		3267.1.w.a.118.1		8
165.53	even	20		3267.1.w.a.766.1		8
165.68	odd	20		3267.1.w.a.766.1		8
165.83	odd	20		3267.1.w.a.118.1		8
165.98	odd	4		297.1.h.a.10.1		2
165.113	even	20		3267.1.w.a.1927.1		8
165.128	odd	20		3267.1.w.a.820.1		8
165.158	even	20		3267.1.w.a.820.1		8
180.43	even	12		1584.1.bf.b.241.1		2
220.43	odd	4		1584.1.bf.b.769.1		2
495.38	even	60		3267.1.w.a.1207.1		8
495.43	even	12		99.1.h.a.43.1	✓	2
495.83	odd	60		3267.1.w.a.1207.1		8
495.128	odd	60		3267.1.w.a.1909.1		8
495.142	even	12		2475.1.y.a.1726.1		2
495.173	odd	60		3267.1.w.a.3016.1		8
495.178	even	60		1089.1.s.a.403.1		8
495.218	even	60		3267.1.w.a.1855.1		8
495.223	odd	60		1089.1.s.a.475.1		8
495.263	odd	12		297.1.h.a.208.1		2
495.268	odd	60		1089.1.s.a.457.1		8
495.313	odd	60		1089.1.s.a.844.1		8
495.358	even	60		1089.1.s.a.844.1		8
495.373	even	12		891.1.c.a.406.1		1
495.398	odd	60		3267.1.w.a.1855.1		8
495.403	even	60		1089.1.s.a.457.1		8
495.428	odd	12		891.1.c.b.406.1		1
495.439	odd	6	inner	2475.1.t.a.1924.2		4
495.443	even	60		3267.1.w.a.3016.1		8
495.448	even	60		1089.1.s.a.475.1		8
495.488	even	60		3267.1.w.a.1909.1		8
495.493	odd	60		1089.1.s.a.403.1		8
1980.43	odd	12		1584.1.bf.b.241.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.1.h.a.43.1	✓	2	45.43	odd	12
99.1.h.a.43.1	✓	2	495.43	even	12
99.1.h.a.76.1	yes	2	5.3	odd	4
99.1.h.a.76.1	yes	2	55.43	even	4
297.1.h.a.10.1		2	15.8	even	4
297.1.h.a.10.1		2	165.98	odd	4
297.1.h.a.208.1		2	45.38	even	12
297.1.h.a.208.1		2	495.263	odd	12
891.1.c.a.406.1		1	45.13	odd	12
891.1.c.a.406.1		1	495.373	even	12
891.1.c.b.406.1		1	45.23	even	12
891.1.c.b.406.1		1	495.428	odd	12
1089.1.s.a.40.1		8	55.13	even	20
1089.1.s.a.40.1		8	55.53	odd	20
1089.1.s.a.94.1		8	55.18	even	20
1089.1.s.a.94.1		8	55.48	odd	20
1089.1.s.a.112.1		8	55.3	odd	20
1089.1.s.a.112.1		8	55.8	even	20
1089.1.s.a.403.1		8	495.178	even	60
1089.1.s.a.403.1		8	495.493	odd	60
1089.1.s.a.457.1		8	495.268	odd	60
1089.1.s.a.457.1		8	495.403	even	60
1089.1.s.a.475.1		8	495.223	odd	60
1089.1.s.a.475.1		8	495.448	even	60
1089.1.s.a.481.1		8	55.28	even	20
1089.1.s.a.481.1		8	55.38	odd	20
1089.1.s.a.844.1		8	495.313	odd	60
1089.1.s.a.844.1		8	495.358	even	60
1584.1.bf.b.241.1		2	180.43	even	12
1584.1.bf.b.241.1		2	1980.43	odd	12
1584.1.bf.b.769.1		2	20.3	even	4
1584.1.bf.b.769.1		2	220.43	odd	4
2475.1.t.a.274.1		4	5.4	even	2	inner
2475.1.t.a.274.1		4	55.54	odd	2	inner
2475.1.t.a.274.2		4	1.1	even	1	trivial
2475.1.t.a.274.2		4	11.10	odd	2	CM
2475.1.t.a.1924.1		4	9.7	even	3	inner
2475.1.t.a.1924.1		4	99.43	odd	6	inner
2475.1.t.a.1924.2		4	45.34	even	6	inner
2475.1.t.a.1924.2		4	495.439	odd	6	inner
2475.1.y.a.76.1		2	5.2	odd	4
2475.1.y.a.76.1		2	55.32	even	4
2475.1.y.a.1726.1		2	45.7	odd	12
2475.1.y.a.1726.1		2	495.142	even	12
3267.1.w.a.118.1		8	165.38	even	20
3267.1.w.a.118.1		8	165.83	odd	20
3267.1.w.a.766.1		8	165.53	even	20
3267.1.w.a.766.1		8	165.68	odd	20
3267.1.w.a.820.1		8	165.128	odd	20
3267.1.w.a.820.1		8	165.158	even	20
3267.1.w.a.1207.1		8	495.38	even	60
3267.1.w.a.1207.1		8	495.83	odd	60
3267.1.w.a.1855.1		8	495.218	even	60
3267.1.w.a.1855.1		8	495.398	odd	60
3267.1.w.a.1909.1		8	495.128	odd	60
3267.1.w.a.1909.1		8	495.488	even	60
3267.1.w.a.1927.1		8	165.8	odd	20
3267.1.w.a.1927.1		8	165.113	even	20
3267.1.w.a.3016.1		8	495.173	odd	60
3267.1.w.a.3016.1		8	495.443	even	60

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2475.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{11} +(0.866025 - 0.500000i) q^{12} +(-0.500000 - 0.866025i) q^{16} +(1.73205 + 1.00000i) q^{23} +1.00000i q^{27} +(0.500000 - 0.866025i) q^{31} -1.00000i q^{33} +1.00000 q^{36} -1.00000i q^{37} -1.00000 q^{44} +(-0.866025 + 0.500000i) q^{47} -1.00000i q^{48} +(0.500000 - 0.866025i) q^{49} +1.00000i q^{53} +(-0.500000 + 0.866025i) q^{59} -1.00000 q^{64} +(0.866025 + 0.500000i) q^{67} +(1.00000 + 1.73205i) q^{69} -1.00000 q^{71} +(-0.500000 + 0.866025i) q^{81} -2.00000 q^{89} +(1.73205 - 1.00000i) q^{92} +(0.866025 - 0.500000i) q^{93} +(-0.866025 + 0.500000i) q^{97} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 + 2 * q^9	\( 4 q + 2 q^{4} + 2 q^{9} - 2 q^{11} - 2 q^{16} + 2 q^{31} + 4 q^{36} - 4 q^{44} + 2 q^{49} - 2 q^{59} - 4 q^{64} + 4 q^{69} - 4 q^{71} - 2 q^{81} - 8 q^{89} + 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 + 2 * q^9 - 2 * q^11 - 2 * q^16 + 2 * q^31 + 4 * q^36 - 4 * q^44 + 2 * q^49 - 2 * q^59 - 4 * q^64 + 4 * q^69 - 4 * q^71 - 2 * q^81 - 8 * q^89 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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