LMFDB - Embedded newform 900.1.bd.a.707.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,1,Mod(407,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 2, 3]))

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

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

chi := DirichletCharacter("900.407");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	900.bd (of order $12$, degree $4$, 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:	$0.449158511370$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.39366000.2

Embedding invariants

Embedding label			707.2
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	900.707
Dual form			900.1.bd.a.443.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258819 - 0.965926i) q^{2} +(-0.258819 - 0.965926i) q^{3} +(-0.866025 - 0.500000i) q^{4} -1.00000 q^{6} +(-1.67303 - 0.448288i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})$	\(q+(0.258819 - 0.965926i) q^{2} +(-0.258819 - 0.965926i) q^{3} +(-0.866025 - 0.500000i) q^{4} -1.00000 q^{6} +(-1.67303 - 0.448288i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{9} +(-0.258819 + 0.965926i) q^{12} +(-0.866025 + 1.50000i) q^{14} +(0.500000 + 0.866025i) q^{16} +(0.258819 + 0.965926i) q^{18} +1.73205i q^{21} +(-0.258819 - 0.965926i) q^{23} +(0.866025 + 0.500000i) q^{24} +(0.707107 + 0.707107i) q^{27} +(1.22474 + 1.22474i) q^{28} +(-0.866025 - 1.50000i) q^{29} +(0.965926 - 0.258819i) q^{32} +1.00000 q^{36} +(-1.50000 - 0.866025i) q^{41} +(1.67303 + 0.448288i) q^{42} -1.00000 q^{46} +(0.258819 - 0.965926i) q^{47} +(0.707107 - 0.707107i) q^{48} +(1.73205 + 1.00000i) q^{49} +(0.866025 - 0.500000i) q^{54} +(1.50000 - 0.866025i) q^{56} +(-1.67303 + 0.448288i) q^{58} +(-0.500000 - 0.866025i) q^{61} +(1.67303 - 0.448288i) q^{63} -1.00000i q^{64} +(0.448288 + 1.67303i) q^{67} +(-0.866025 + 0.500000i) q^{69} +(0.258819 - 0.965926i) q^{72} +(0.500000 - 0.866025i) q^{81} +(-1.22474 + 1.22474i) q^{82} +(-0.965926 - 0.258819i) q^{83} +(0.866025 - 1.50000i) q^{84} +(-1.22474 + 1.22474i) q^{87} +1.73205 q^{89} +(-0.258819 + 0.965926i) q^{92} +(-0.866025 - 0.500000i) q^{94} +(-0.500000 - 0.866025i) q^{96} +(1.41421 - 1.41421i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{6}+O(q^{10}) $ 8 * q - 8 * q^6	$ 8 q - 8 q^{6} + 4 q^{16} + 8 q^{36} - 12 q^{41} - 8 q^{46} + 12 q^{56} - 4 q^{61} + 4 q^{81} - 4 q^{96}+O(q^{100}) $ 8 * q - 8 * q^6 + 4 * q^16 + 8 * q^36 - 12 * q^41 - 8 * q^46 + 12 * q^56 - 4 * q^61 + 4 * q^81 - 4 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$	0.258819	−	0.965926i	0.258819	−	0.965926i
$2$	0.258819	−	0.965926i	0.258819	−	0.965926i
$3$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$3$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$4$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$5$		0			0
$5$		0			0
$6$	−1.00000			−1.00000
$7$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$7$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$8$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$9$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$13$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$13$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$14$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$15$		0			0
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	0.258819	+	0.965926i	0.258819	+	0.965926i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$			1.73205i			1.73205i
$22$		0			0
$23$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$23$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$24$	0.866025	+	0.500000i	0.866025	+	0.500000i
$25$		0			0
$26$		0			0
$27$	0.707107	+	0.707107i	0.707107	+	0.707107i
$28$	1.22474	+	1.22474i	1.22474	+	1.22474i
$29$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$29$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	0.965926	−	0.258819i	0.965926	−	0.258819i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.00000			1.00000
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$42$	1.67303	+	0.448288i	1.67303	+	0.448288i
$43$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$43$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$44$		0			0
$45$		0			0
$46$	−1.00000			−1.00000
$47$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$47$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$48$	0.707107	−	0.707107i	0.707107	−	0.707107i
$49$	1.73205	+	1.00000i	1.73205	+	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	0.866025	−	0.500000i	0.866025	−	0.500000i
$55$		0			0
$56$	1.50000	−	0.866025i	1.50000	−	0.866025i
$57$		0			0
$58$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$61$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$62$		0			0
$63$	1.67303	−	0.448288i	1.67303	−	0.448288i
$64$		−	1.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	0.448288	+	1.67303i	0.448288	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$67$	0.448288	+	1.67303i	0.448288	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$68$		0			0
$69$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	0.258819	−	0.965926i	0.258819	−	0.965926i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	0.500000	−	0.866025i	0.500000	−	0.866025i
$82$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$83$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$83$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$84$	0.866025	−	1.50000i	0.866025	−	1.50000i
$85$		0			0
$86$		0			0
$87$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$88$		0			0
$89$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$89$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$93$		0			0
$94$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$95$		0			0
$96$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$97$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$97$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$98$	1.41421	−	1.41421i	1.41421	−	1.41421i
$99$		0			0
$100$		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			0		−0.258819	−	0.965926i	$-0.583333\pi$
$103$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$107$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$108$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$109$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$109$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$113$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$113$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$114$		0			0
$115$		0			0
$116$			1.73205i			1.73205i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	−	0.866025i	0.500000	−	0.866025i
$122$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$123$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$124$		0			0
$125$		0			0
$126$		−	1.73205i		−	1.73205i
$127$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$127$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$128$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$	1.73205			1.73205
$135$		0			0
$136$		0			0
$137$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$137$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$138$	0.258819	+	0.965926i	0.258819	+	0.965926i
$139$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$139$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$140$		0			0
$141$	−1.00000			−1.00000
$142$		0			0
$143$		0			0
$144$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$145$		0			0
$146$		0			0
$147$	0.517638	−	1.93185i	0.517638	−	1.93185i
$148$		0			0
$149$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$149$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$150$		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			0		−0.258819	−	0.965926i	$-0.583333\pi$
$157$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.73205i			1.73205i
$162$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	0.866025	+	1.50000i	0.866025	+	1.50000i
$165$		0			0
$166$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$167$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$167$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$168$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$169$	0.866025	−	0.500000i	0.866025	−	0.500000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$173$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$174$	0.866025	+	1.50000i	0.866025	+	1.50000i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.448288	−	1.67303i	0.448288	−	1.67303i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		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
$183$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$184$	0.866025	+	0.500000i	0.866025	+	0.500000i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$189$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$193$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$193$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$194$		0			0
$195$		0			0
$196$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	1.50000	−	0.866025i	1.50000	−	0.866025i
$202$		0			0
$203$	0.776457	+	2.89778i	0.776457	+	2.89778i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.707107	+	0.707107i	0.707107	+	0.707107i
$208$		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			0
$213$		0			0
$214$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$215$		0			0
$216$	−1.00000			−1.00000
$217$		0			0
$218$	0.965926	+	0.258819i	0.965926	+	0.258819i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$223$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$224$	−1.73205			−1.73205
$225$		0			0
$226$		0			0
$227$	−0.517638	+	1.93185i	−0.517638	+	1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$227$	−0.517638	+	1.93185i	−0.517638	+	1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$229$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$	1.67303	+	0.448288i	1.67303	+	0.448288i
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		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
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$242$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$243$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$244$			1.00000i			1.00000i
$245$		0			0
$246$	1.50000	+	0.866025i	1.50000	+	0.866025i
$247$		0			0
$248$		0			0
$249$			1.00000i			1.00000i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$253$		0			0
$254$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$257$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.50000	+	0.866025i	1.50000	+	0.866025i
$262$		0			0
$263$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$263$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$268$	0.448288	−	1.67303i	0.448288	−	1.67303i
$269$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$269$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$270$		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
$276$	1.00000			1.00000
$277$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$277$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$281$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$282$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$283$	1.67303	−	0.448288i	1.67303	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$283$	1.67303	−	0.448288i	1.67303	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.12132	+	2.12132i	2.12132	+	2.12132i
$288$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$289$			1.00000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$293$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$294$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$307$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$313$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$317$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$322$	1.67303	+	0.448288i	1.67303	+	0.448288i
$323$		0			0
$324$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$325$		0			0
$326$		0			0
$327$	0.965926	−	0.258819i	0.965926	−	0.258819i
$328$	1.67303	−	0.448288i	1.67303	−	0.448288i
$329$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	0.707107	+	0.707107i	0.707107	+	0.707107i
$333$		0			0
$334$		−	1.00000i		−	1.00000i
$335$		0			0
$336$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$337$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$337$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$338$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$347$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$348$	1.67303	−	0.448288i	1.67303	−	0.448288i
$349$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$349$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$353$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$354$		0			0
$355$		0			0
$356$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$357$		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
$362$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$363$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$364$		0			0
$365$		0			0
$366$	0.500000	+	0.866025i	0.500000	+	0.866025i
$367$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$367$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$368$	0.707107	−	0.707107i	0.707107	−	0.707107i
$369$	1.73205			1.73205
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$373$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$374$		0			0
$375$		0			0
$376$	0.500000	+	0.866025i	0.500000	+	0.866025i
$377$		0			0
$378$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$382$		0			0
$383$	−0.517638	−	1.93185i	−0.517638	−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$383$	−0.517638	−	1.93185i	−0.517638	−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$384$			1.00000i			1.00000i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$389$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−1.93185	+	0.517638i	−1.93185	+	0.517638i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$403$		0			0
$404$		0			0
$405$		0			0
$406$	3.00000			3.00000
$407$		0			0
$408$		0			0
$409$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$409$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.866025	−	0.500000i	0.866025	−	0.500000i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$421$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$	0.258819	+	0.965926i	0.258819	+	0.965926i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.448288	+	1.67303i	0.448288	+	1.67303i
$428$	0.258819	+	0.965926i	0.258819	+	0.965926i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$	0.500000	−	0.866025i	0.500000	−	0.866025i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	−2.00000			−2.00000
$442$		0			0
$443$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$443$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$444$		0			0
$445$		0			0
$446$	1.50000	+	0.866025i	1.50000	+	0.866025i
$447$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$448$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$449$		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			0
$454$	1.73205	+	1.00000i	1.73205	+	1.00000i
$455$		0			0
$456$		0			0
$457$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$457$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$458$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$459$		0			0
$460$		0			0
$461$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$461$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$462$		0			0
$463$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$463$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$464$	0.866025	−	1.50000i	0.866025	−	1.50000i
$465$		0			0
$466$		0			0
$467$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$467$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$468$		0			0
$469$		−	3.00000i		−	3.00000i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		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
$482$	0.965926	−	0.258819i	0.965926	−	0.258819i
$483$	1.67303	−	0.448288i	1.67303	−	0.448288i
$484$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$485$		0			0
$486$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	0.965926	+	0.258819i	0.965926	+	0.258819i
$489$		0			0
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$	1.22474	−	1.22474i	1.22474	−	1.22474i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.965926	+	0.258819i	0.965926	+	0.258819i
$499$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$499$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$500$		0			0
$501$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$502$		0			0
$503$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$503$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$504$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$505$		0			0
$506$		0			0
$507$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$508$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$509$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$509$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$	0.707107	+	0.707107i	0.707107	+	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$521$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$522$	1.22474	−	1.22474i	1.22474	−	1.22474i
$523$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$523$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$524$		0			0
$525$		0			0
$526$	1.00000	−	1.73205i	1.00000	−	1.73205i
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	−1.73205			−1.73205
$535$		0			0
$536$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$537$		0			0
$538$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$539$		0			0
$540$		0			0
$541$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$541$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$	0.258819	+	0.965926i	0.258819	+	0.965926i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$547$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$	0.866025	+	0.500000i	0.866025	+	0.500000i
$550$		0			0
$551$		0			0
$552$	0.258819	−	0.965926i	0.258819	−	0.965926i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$563$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$563$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$564$	0.866025	+	0.500000i	0.866025	+	0.500000i
$565$		0			0
$566$		−	1.73205i		−	1.73205i
$567$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		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
$573$		0			0
$574$	2.59808	−	1.50000i	2.59808	−	1.50000i
$575$		0			0
$576$	0.500000	+	0.866025i	0.500000	+	0.866025i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	0.965926	+	0.258819i	0.965926	+	0.258819i
$579$		0			0
$580$		0			0
$581$	1.50000	+	0.866025i	1.50000	+	0.866025i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$587$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$588$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$602$		0			0
$603$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$607$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$608$		0			0
$609$	2.59808	−	1.50000i	2.59808	−	1.50000i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$615$		0			0
$616$		0			0
$617$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$617$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$	0.500000	−	0.866025i	0.500000	−	0.866025i
$622$		0			0
$623$	−2.89778	−	0.776457i	−2.89778	−	0.776457i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		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
$641$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$641$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$642$	0.707107	+	0.707107i	0.707107	+	0.707107i
$643$	1.67303	−	0.448288i	1.67303	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$643$	1.67303	−	0.448288i	1.67303	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$644$	0.866025	−	1.50000i	0.866025	−	1.50000i
$645$		0			0
$646$		0			0
$647$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$647$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$648$	0.258819	+	0.965926i	0.258819	+	0.965926i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$653$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$654$		−	1.00000i		−	1.00000i
$655$		0			0
$656$		−	1.73205i		−	1.73205i
$657$		0			0
$658$	1.22474	+	1.22474i	1.22474	+	1.22474i
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$	0.866025	−	0.500000i	0.866025	−	0.500000i
$665$		0			0
$666$		0			0
$667$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$668$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$669$	1.73205			1.73205
$670$		0			0
$671$		0			0
$672$	0.448288	+	1.67303i	0.448288	+	1.67303i
$673$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$673$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$674$		0			0
$675$		0			0
$676$	−1.00000			−1.00000
$677$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$677$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	2.00000			2.00000
$682$		0			0
$683$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$683$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$684$		0			0
$685$		0			0
$686$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$687$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$		0			0
$694$			2.00000i			2.00000i
$695$		0			0
$696$		−	1.73205i		−	1.73205i
$697$		0			0
$698$	0.258819	+	0.965926i	0.258819	+	0.965926i
$699$		0			0
$700$		0			0
$701$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$701$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$709$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$710$		0			0
$711$		0			0
$712$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		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
$722$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$723$	0.707107	−	0.707107i	0.707107	−	0.707107i
$724$	0.866025	+	0.500000i	0.866025	+	0.500000i
$725$		0			0
$726$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$727$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$727$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$	0.965926	−	0.258819i	0.965926	−	0.258819i
$733$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$733$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$734$		0			0
$735$		0			0
$736$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$737$		0			0
$738$	0.448288	−	1.67303i	0.448288	−	1.67303i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$743$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.965926	−	0.258819i	0.965926	−	0.258819i
$748$		0			0
$749$	0.866025	+	1.50000i	0.866025	+	1.50000i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.965926	−	0.258819i	0.965926	−	0.258819i
$753$		0			0
$754$		0			0
$755$		0			0
$756$			1.73205i			1.73205i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$761$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$762$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$763$	0.448288	−	1.67303i	0.448288	−	1.67303i
$764$		0			0
$765$		0			0
$766$	−2.00000			−2.00000
$767$		0			0
$768$	0.965926	+	0.258819i	0.965926	+	0.258819i
$769$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$769$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.67303	−	0.448288i	1.67303	−	0.448288i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.448288	−	1.67303i	0.448288	−	1.67303i
$784$			2.00000i			2.00000i
$785$		0			0
$786$		0			0
$787$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$787$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$788$		0			0
$789$		−	2.00000i		−	2.00000i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$797$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$802$		0			0
$803$		0			0
$804$	−1.73205			−1.73205
$805$		0			0
$806$		0			0
$807$	0.448288	+	1.67303i	0.448288	+	1.67303i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	0.776457	−	2.89778i	0.776457	−	2.89778i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$819$		0			0
$820$		0			0
$821$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$821$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$822$		0			0
$823$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$823$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$827$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$828$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$829$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$829$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$842$	1.93185	−	0.517638i	1.93185	−	0.517638i
$843$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$844$		0			0
$845$		0			0
$846$	1.00000			1.00000
$847$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$848$		0			0
$849$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$853$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$854$	1.73205			1.73205
$855$		0			0
$856$	1.00000			1.00000
$857$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$857$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$	1.50000	−	2.59808i	1.50000	−	2.59808i
$862$		0			0
$863$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$863$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$864$	0.866025	+	0.500000i	0.866025	+	0.500000i
$865$		0			0
$866$		0			0
$867$	0.965926	−	0.258819i	0.965926	−	0.258819i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$877$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$881$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$882$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$883$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$883$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$884$		0			0
$885$		0			0
$886$	0.500000	−	0.866025i	0.500000	−	0.866025i
$887$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$887$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$888$		0			0
$889$	−2.59808	+	1.50000i	−2.59808	+	1.50000i
$890$		0			0
$891$		0			0
$892$	1.22474	−	1.22474i	1.22474	−	1.22474i
$893$		0			0
$894$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$895$		0			0
$896$	1.50000	+	0.866025i	1.50000	+	0.866025i
$897$		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
$907$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$907$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$908$	1.41421	−	1.41421i	1.41421	−	1.41421i
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.500000	+	0.866025i	0.500000	+	0.866025i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$922$	0.448288	+	1.67303i	0.448288	+	1.67303i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$	−2.89778	−	0.776457i	−2.89778	−	0.776457i
$939$		0			0
$940$		0			0
$941$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$941$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$947$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$	0.965926	+	0.258819i	0.965926	+	0.258819i
$964$		−	1.00000i		−	1.00000i
$965$		0			0
$966$		−	1.73205i		−	1.73205i
$967$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$967$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$968$	0.258819	+	0.965926i	0.258819	+	0.965926i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	0.707107	+	0.707107i	0.707107	+	0.707107i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.500000	−	0.866025i	0.500000	−	0.866025i
$977$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$977$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$982$		0			0
$983$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$983$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$984$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$985$		0			0
$986$		0			0
$987$	1.67303	+	0.448288i	1.67303	+	0.448288i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	0.500000	−	0.866025i	0.500000	−	0.866025i
$997$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$997$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.1.bd.a.707.2	yes	8
3.2	odd	2		2700.1.be.a.1907.1		8
4.3	odd	2	inner	900.1.bd.a.707.1	yes	8
5.2	odd	4	inner	900.1.bd.a.743.2	yes	8
5.3	odd	4	inner	900.1.bd.a.743.1	yes	8
5.4	even	2	inner	900.1.bd.a.707.1	yes	8
9.2	odd	6	inner	900.1.bd.a.407.2	yes	8
9.7	even	3		2700.1.be.a.1007.1		8
12.11	even	2		2700.1.be.a.1907.2		8
15.2	even	4		2700.1.be.a.1043.1		8
15.8	even	4		2700.1.be.a.1043.2		8
15.14	odd	2		2700.1.be.a.1907.2		8
20.3	even	4	inner	900.1.bd.a.743.2	yes	8
20.7	even	4	inner	900.1.bd.a.743.1	yes	8
20.19	odd	2	CM	900.1.bd.a.707.2	yes	8
36.7	odd	6		2700.1.be.a.1007.2		8
36.11	even	6	inner	900.1.bd.a.407.1	✓	8
45.2	even	12	inner	900.1.bd.a.443.2	yes	8
45.7	odd	12		2700.1.be.a.143.1		8
45.29	odd	6	inner	900.1.bd.a.407.1	✓	8
45.34	even	6		2700.1.be.a.1007.2		8
45.38	even	12	inner	900.1.bd.a.443.1	yes	8
45.43	odd	12		2700.1.be.a.143.2		8
60.23	odd	4		2700.1.be.a.1043.1		8
60.47	odd	4		2700.1.be.a.1043.2		8
60.59	even	2		2700.1.be.a.1907.1		8
180.7	even	12		2700.1.be.a.143.2		8
180.43	even	12		2700.1.be.a.143.1		8
180.47	odd	12	inner	900.1.bd.a.443.1	yes	8
180.79	odd	6		2700.1.be.a.1007.1		8
180.83	odd	12	inner	900.1.bd.a.443.2	yes	8
180.119	even	6	inner	900.1.bd.a.407.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.1.bd.a.407.1	✓	8	36.11	even	6	inner
900.1.bd.a.407.1	✓	8	45.29	odd	6	inner
900.1.bd.a.407.2	yes	8	9.2	odd	6	inner
900.1.bd.a.407.2	yes	8	180.119	even	6	inner
900.1.bd.a.443.1	yes	8	45.38	even	12	inner
900.1.bd.a.443.1	yes	8	180.47	odd	12	inner
900.1.bd.a.443.2	yes	8	45.2	even	12	inner
900.1.bd.a.443.2	yes	8	180.83	odd	12	inner
900.1.bd.a.707.1	yes	8	4.3	odd	2	inner
900.1.bd.a.707.1	yes	8	5.4	even	2	inner
900.1.bd.a.707.2	yes	8	1.1	even	1	trivial
900.1.bd.a.707.2	yes	8	20.19	odd	2	CM
900.1.bd.a.743.1	yes	8	5.3	odd	4	inner
900.1.bd.a.743.1	yes	8	20.7	even	4	inner
900.1.bd.a.743.2	yes	8	5.2	odd	4	inner
900.1.bd.a.743.2	yes	8	20.3	even	4	inner
2700.1.be.a.143.1		8	45.7	odd	12
2700.1.be.a.143.1		8	180.43	even	12
2700.1.be.a.143.2		8	45.43	odd	12
2700.1.be.a.143.2		8	180.7	even	12
2700.1.be.a.1007.1		8	9.7	even	3
2700.1.be.a.1007.1		8	180.79	odd	6
2700.1.be.a.1007.2		8	36.7	odd	6
2700.1.be.a.1007.2		8	45.34	even	6
2700.1.be.a.1043.1		8	15.2	even	4
2700.1.be.a.1043.1		8	60.23	odd	4
2700.1.be.a.1043.2		8	15.8	even	4
2700.1.be.a.1043.2		8	60.47	odd	4
2700.1.be.a.1907.1		8	3.2	odd	2
2700.1.be.a.1907.1		8	60.59	even	2
2700.1.be.a.1907.2		8	12.11	even	2
2700.1.be.a.1907.2		8	15.14	odd	2

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	900.bd (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.258819 - 0.965926i) q^{2} +(-0.258819 - 0.965926i) q^{3} +(-0.866025 - 0.500000i) q^{4} -1.00000 q^{6} +(-1.67303 - 0.448288i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\)	\(q+(0.258819 - 0.965926i) q^{2} +(-0.258819 - 0.965926i) q^{3} +(-0.866025 - 0.500000i) q^{4} -1.00000 q^{6} +(-1.67303 - 0.448288i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{9} +(-0.258819 + 0.965926i) q^{12} +(-0.866025 + 1.50000i) q^{14} +(0.500000 + 0.866025i) q^{16} +(0.258819 + 0.965926i) q^{18} +1.73205i q^{21} +(-0.258819 - 0.965926i) q^{23} +(0.866025 + 0.500000i) q^{24} +(0.707107 + 0.707107i) q^{27} +(1.22474 + 1.22474i) q^{28} +(-0.866025 - 1.50000i) q^{29} +(0.965926 - 0.258819i) q^{32} +1.00000 q^{36} +(-1.50000 - 0.866025i) q^{41} +(1.67303 + 0.448288i) q^{42} -1.00000 q^{46} +(0.258819 - 0.965926i) q^{47} +(0.707107 - 0.707107i) q^{48} +(1.73205 + 1.00000i) q^{49} +(0.866025 - 0.500000i) q^{54} +(1.50000 - 0.866025i) q^{56} +(-1.67303 + 0.448288i) q^{58} +(-0.500000 - 0.866025i) q^{61} +(1.67303 - 0.448288i) q^{63} -1.00000i q^{64} +(0.448288 + 1.67303i) q^{67} +(-0.866025 + 0.500000i) q^{69} +(0.258819 - 0.965926i) q^{72} +(0.500000 - 0.866025i) q^{81} +(-1.22474 + 1.22474i) q^{82} +(-0.965926 - 0.258819i) q^{83} +(0.866025 - 1.50000i) q^{84} +(-1.22474 + 1.22474i) q^{87} +1.73205 q^{89} +(-0.258819 + 0.965926i) q^{92} +(-0.866025 - 0.500000i) q^{94} +(-0.500000 - 0.866025i) q^{96} +(1.41421 - 1.41421i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{6}+O(q^{10}) \) 8 * q - 8 * q^6	\( 8 q - 8 q^{6} + 4 q^{16} + 8 q^{36} - 12 q^{41} - 8 q^{46} + 12 q^{56} - 4 q^{61} + 4 q^{81} - 4 q^{96}+O(q^{100}) \) 8 * q - 8 * q^6 + 4 * q^16 + 8 * q^36 - 12 * q^41 - 8 * q^46 + 12 * q^56 - 4 * q^61 + 4 * q^81 - 4 * q^96

Introduction

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