LMFDB - Embedded newform 735.1.o.d.704.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,1,Mod(569,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("735.569");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	735.o (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:	$0.366812784285$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.15435.1
Artin image:	$C_3\times D_8$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

Embedding invariants

Embedding label			704.2
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	735.704
Dual form			735.1.o.d.569.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 1.22474i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +1.41421 q^{6} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.707107 + 1.22474i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +1.41421 q^{6} +(-0.500000 - 0.866025i) q^{9} +(0.707107 - 1.22474i) q^{10} +(0.500000 + 0.866025i) q^{12} -1.00000 q^{15} +(0.500000 + 0.866025i) q^{16} +(0.707107 - 1.22474i) q^{18} +(0.707107 + 1.22474i) q^{19} +1.00000 q^{20} +(-0.707107 - 1.22474i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(-0.707107 - 1.22474i) q^{30} +(-0.707107 + 1.22474i) q^{31} +(-0.707107 + 1.22474i) q^{32} +1.00000 q^{36} +(-1.00000 + 1.73205i) q^{38} +(-0.500000 + 0.866025i) q^{45} +(1.00000 - 1.73205i) q^{46} +1.00000 q^{48} -1.41421 q^{50} +(-0.707107 + 1.22474i) q^{53} +(-0.707107 - 1.22474i) q^{54} +1.41421 q^{57} +(0.500000 - 0.866025i) q^{60} +(-0.707107 - 1.22474i) q^{61} -2.00000 q^{62} -1.00000 q^{64} -1.41421 q^{69} +(0.500000 + 0.866025i) q^{75} -1.41421 q^{76} +(0.500000 - 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} -1.41421 q^{90} +1.41421 q^{92} +(0.707107 + 1.22474i) q^{93} +(0.707107 - 1.22474i) q^{95} +(0.707107 + 1.22474i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 - 2 * q^9	$ 4 q + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{9} + 2 q^{12} - 4 q^{15} + 2 q^{16} + 4 q^{20} - 2 q^{25} - 4 q^{27} + 4 q^{36} - 4 q^{38} - 2 q^{45} + 4 q^{46} + 4 q^{48} + 2 q^{60} - 8 q^{62} - 4 q^{64} + 2 q^{75} + 2 q^{80} - 2 q^{81}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 - 2 * q^9 + 2 * q^12 - 4 * q^15 + 2 * q^16 + 4 * q^20 - 2 * q^25 - 4 * q^27 + 4 * q^36 - 4 * q^38 - 2 * q^45 + 4 * q^46 + 4 * q^48 + 2 * q^60 - 8 * q^62 - 4 * q^64 + 2 * q^75 + 2 * q^80 - 2 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$.

$n$	$346$	$442$	$491$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$

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 $

$2$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$2$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$6$	1.41421			1.41421
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	0.707107	−	1.22474i	0.707107	−	1.22474i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	0.500000	+	0.866025i	0.500000	+	0.866025i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−1.00000			−1.00000
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	0.707107	−	1.22474i	0.707107	−	1.22474i
$19$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$19$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$20$	1.00000			1.00000
$21$		0			0
$22$		0			0
$23$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$23$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−1.00000			−1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$31$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$31$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$32$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.00000			1.00000
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$46$	1.00000	−	1.73205i	1.00000	−	1.73205i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	1.00000			1.00000
$49$		0			0
$50$	−1.41421			−1.41421
$51$		0			0
$52$		0			0
$53$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$53$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$54$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$55$		0			0
$56$		0			0
$57$	1.41421			1.41421
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	0.500000	−	0.866025i	0.500000	−	0.866025i
$61$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$61$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$62$	−2.00000			−2.00000
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$	−1.41421			−1.41421
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.500000	+	0.866025i
$76$	−1.41421			−1.41421
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$	0.500000	−	0.866025i	0.500000	−	0.866025i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$	−1.41421			−1.41421
$91$		0			0
$92$	1.41421			1.41421
$93$	0.707107	+	1.22474i	0.707107	+	1.22474i
$94$		0			0
$95$	0.707107	−	1.22474i	0.707107	−	1.22474i
$96$	0.707107	+	1.22474i	0.707107	+	1.22474i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	−2.00000			−2.00000
$107$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$107$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$108$	0.500000	−	0.866025i	0.500000	−	0.866025i
$109$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$109$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$113$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$114$	1.00000	+	1.73205i	1.00000	+	1.73205i
$115$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$	1.00000	−	1.73205i	1.00000	−	1.73205i
$123$		0			0
$124$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$125$	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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.500000	+	0.866025i	0.500000	+	0.866025i
$136$		0			0
$137$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$137$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$138$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$139$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$139$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.500000	−	0.866025i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$151$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$151$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.41421			1.41421
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$	0.707107	+	1.22474i	0.707107	+	1.22474i
$160$	1.41421			1.41421
$161$		0			0
$162$	−1.41421			−1.41421
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.00000			2.00000			1.00000			$0$
$167$	2.00000			2.00000			1.00000			$0$
$168$		0			0
$169$	1.00000			1.00000
$170$		0			0
$171$	0.707107	−	1.22474i	0.707107	−	1.22474i
$172$		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			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$181$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$181$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$	−1.41421			−1.41421
$184$		0			0
$185$		0			0
$186$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$187$		0			0
$188$		0			0
$189$		0			0
$190$	2.00000			2.00000
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$197$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$199$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−2.00000			−2.00000			−1.00000			$\pi$
$211$	−2.00000			−2.00000			−1.00000			$\pi$
$212$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$213$		0			0
$214$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	2.82843			2.82843
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	1.00000			1.00000
$226$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$227$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$227$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$228$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$229$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$229$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$230$	−2.00000			−2.00000
$231$		0			0
$232$		0			0
$233$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$233$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$241$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$241$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$242$	0.707107	−	1.22474i	0.707107	−	1.22474i
$243$	0.500000	+	0.866025i	0.500000	+	0.866025i
$244$	1.41421			1.41421
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.707107	+	1.22474i	0.707107	+	1.22474i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$257$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$263$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$264$		0			0
$265$	1.41421			1.41421
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$271$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$271$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$272$		0			0
$273$		0			0
$274$	2.00000			2.00000
$275$		0			0
$276$	0.707107	−	1.22474i	0.707107	−	1.22474i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	1.00000	+	1.73205i	1.00000	+	1.73205i
$279$	1.41421			1.41421
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$286$		0			0
$287$		0			0
$288$	1.41421			1.41421
$289$	0.500000	+	0.866025i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.00000			−2.00000			−1.00000			$\pi$
$293$	−2.00000			−2.00000			−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1.00000			−1.00000
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$305$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	1.00000	+	1.73205i	1.00000	+	1.73205i
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$317$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$318$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$319$		0			0
$320$	0.500000	+	0.866025i	0.500000	+	0.866025i
$321$	1.41421			1.41421
$322$		0			0
$323$		0			0
$324$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$325$		0			0
$326$		0			0
$327$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$331$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	1.41421	+	2.44949i	1.41421	+	2.44949i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.707107	+	1.22474i	0.707107	+	1.22474i
$339$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$340$		0			0
$341$		0			0
$342$	2.00000			2.00000
$343$		0			0
$344$		0			0
$345$	0.707107	+	1.22474i	0.707107	+	1.22474i
$346$		0			0
$347$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$347$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$348$		0			0
$349$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$349$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$353$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$363$	−1.00000			−1.00000
$364$		0			0
$365$		0			0
$366$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	0.707107	−	1.22474i	0.707107	−	1.22474i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.41421			−1.41421
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	0.500000	−	0.866025i	0.500000	−	0.866025i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	0.707107	+	1.22474i	0.707107	+	1.22474i
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$395$		0			0
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	2.00000			2.00000
$399$		0			0
$400$	−1.00000			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.00000			1.00000
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$409$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$410$		0			0
$411$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$412$		0			0
$413$		0			0
$414$	−2.00000			−2.00000
$415$		0			0
$416$		0			0
$417$	0.707107	−	1.22474i	0.707107	−	1.22474i
$418$		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	$0.5\pi$
$422$	−1.41421	−	2.44949i	−1.41421	−	2.44949i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.41421			−1.41421
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	1.00000	+	1.73205i	1.00000	+	1.73205i
$437$	1.00000	−	1.73205i	1.00000	−	1.73205i
$438$		0			0
$439$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$439$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$443$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.707107	+	1.22474i	0.707107	+	1.22474i
$451$		0			0
$452$	0.707107	−	1.22474i	0.707107	−	1.22474i
$453$		0			0
$454$	−2.82843			−2.82843
$455$		0			0
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	1.00000	−	1.73205i	1.00000	−	1.73205i
$459$		0			0
$460$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	0.707107	−	1.22474i	0.707107	−	1.22474i
$466$	1.00000	−	1.73205i	1.00000	−	1.73205i
$467$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$467$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.41421			−1.41421
$476$		0			0
$477$	1.41421			1.41421
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$	0.707107	−	1.22474i	0.707107	−	1.22474i
$481$		0			0
$482$	2.00000			2.00000
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.41421			−1.41421
$497$		0			0
$498$		0			0
$499$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$499$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$500$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$501$	1.00000	−	1.73205i	1.00000	−	1.73205i
$502$		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
$507$	0.500000	−	0.866025i	0.500000	−	0.866025i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−1.41421			−1.41421
$513$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$514$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$	2.00000			2.00000
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$	1.00000	+	1.73205i	1.00000	+	1.73205i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.707107	−	1.22474i	0.707107	−	1.22474i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−1.00000			−1.00000
$541$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$541$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$542$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$543$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$544$		0			0
$545$	−2.00000			−2.00000
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	0.707107	+	1.22474i	0.707107	+	1.22474i
$549$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$557$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$557$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$558$	1.00000	+	1.73205i	1.00000	+	1.73205i
$559$		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			0
$565$	0.707107	+	1.22474i	0.707107	+	1.22474i
$566$		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	$0.333333\pi$
$570$	1.00000	−	1.73205i	1.00000	−	1.73205i
$571$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.41421			1.41421
$576$	0.500000	+	0.866025i	0.500000	+	0.866025i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−1.41421	−	2.44949i	−1.41421	−	2.44949i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−2.00000			−2.00000
$590$		0			0
$591$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$592$		0			0
$593$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$593$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$598$		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
$601$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$601$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−2.00000			−2.00000
$609$		0			0
$610$	−2.00000			−2.00000
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$617$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$619$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$620$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$621$	0.707107	+	1.22474i	0.707107	+	1.22474i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$634$	1.00000	−	1.73205i	1.00000	−	1.73205i
$635$		0			0
$636$	−1.41421			−1.41421
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	1.00000	+	1.73205i	1.00000	+	1.73205i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$647$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$653$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$654$	1.41421	−	2.44949i	1.41421	−	2.44949i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$661$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$662$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	0.500000	−	0.866025i	0.500000	−	0.866025i
$676$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$677$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$677$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$678$	−2.00000			−2.00000
$679$		0			0
$680$		0			0
$681$	1.00000	+	1.73205i	1.00000	+	1.73205i
$682$		0			0
$683$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$683$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$684$	0.707107	+	1.22474i	0.707107	+	1.22474i
$685$	−1.41421			−1.41421
$686$		0			0
$687$	−1.41421			−1.41421
$688$		0			0
$689$		0			0
$690$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$691$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$691$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$692$		0			0
$693$		0			0
$694$	−2.00000			−2.00000
$695$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$696$		0			0
$697$		0			0
$698$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$699$	−1.41421			−1.41421
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	2.82843			2.82843
$707$		0			0
$708$		0			0
$709$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$709$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.00000			2.00000
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	−1.00000			−1.00000
$721$		0			0
$722$	−1.41421			−1.41421
$723$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$724$	0.707107	−	1.22474i	0.707107	−	1.22474i
$725$		0			0
$726$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$727$		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
$732$	0.707107	−	1.22474i	0.707107	−	1.22474i
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	2.00000			2.00000
$737$		0			0
$738$		0			0
$739$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$739$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$743$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$	1.41421			1.41421
$751$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$751$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.500000	+	0.866025i	0.500000	+	0.866025i
$769$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$769$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$770$		0			0
$771$	2.00000			2.00000
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$776$		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
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	0.707107	−	1.22474i	0.707107	−	1.22474i
$789$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.707107	−	1.22474i	0.707107	−	1.22474i
$796$	0.707107	+	1.22474i	0.707107	+	1.22474i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$	0.707107	+	1.22474i	0.707107	+	1.22474i
$811$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$811$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$	1.41421			1.41421
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	2.00000			2.00000
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	1.00000	−	1.73205i	1.00000	−	1.73205i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$827$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$828$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$829$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$829$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$	2.00000			2.00000
$835$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$836$		0			0
$837$	0.707107	−	1.22474i	0.707107	−	1.22474i
$838$		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
$844$	1.00000	−	1.73205i	1.00000	−	1.73205i
$845$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$846$		0			0
$847$		0			0
$848$	−1.41421			−1.41421
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$	−1.41421			−1.41421
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$859$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$863$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$864$	0.707107	−	1.22474i	0.707107	−	1.22474i
$865$		0			0
$866$		0			0
$867$	1.00000			1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	2.82843			2.82843
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	1.00000	−	1.73205i	1.00000	−	1.73205i
$879$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.707107	+	1.22474i	0.707107	+	1.22474i
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	0.707107	+	1.22474i	0.707107	+	1.22474i
$913$		0			0
$914$		0			0
$915$	0.707107	+	1.22474i	0.707107	+	1.22474i
$916$	1.41421			1.41421
$917$		0			0
$918$		0			0
$919$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$919$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$920$		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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	2.00000			2.00000
$931$		0			0
$932$	1.41421			1.41421
$933$		0			0
$934$	1.41421	−	2.44949i	1.41421	−	2.44949i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		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
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$947$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$948$		0			0
$949$		0			0
$950$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$951$	−1.41421			−1.41421
$952$		0			0
$953$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$953$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$954$	1.00000	+	1.73205i	1.00000	+	1.73205i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.00000			1.00000
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$	0.707107	−	1.22474i	0.707107	−	1.22474i
$964$	0.707107	+	1.22474i	0.707107	+	1.22474i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−1.00000			−1.00000
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.707107	−	1.22474i	0.707107	−	1.22474i
$977$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$977$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−2.00000			−2.00000
$982$		0			0
$983$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$983$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$984$		0			0
$985$	0.707107	+	1.22474i	0.707107	+	1.22474i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$991$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$992$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$993$	2.00000			2.00000
$994$		0			0
$995$	−1.41421			−1.41421
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.1.o.d.704.2		4
3.2	odd	2		735.1.o.c.704.1		4
5.2	odd	4		3675.1.u.f.851.1		8
5.3	odd	4		3675.1.u.f.851.4		8
5.4	even	2		735.1.o.c.704.1		4
7.2	even	3	inner	735.1.o.d.569.2		4
7.3	odd	6		735.1.f.d.344.1	yes	2
7.4	even	3		735.1.f.c.344.1	✓	2
7.5	odd	6		735.1.o.c.569.2		4
7.6	odd	2		735.1.o.c.704.2		4
15.2	even	4		3675.1.u.f.851.4		8
15.8	even	4		3675.1.u.f.851.1		8
15.14	odd	2	CM	735.1.o.d.704.2		4
21.2	odd	6		735.1.o.c.569.1		4
21.5	even	6	inner	735.1.o.d.569.1		4
21.11	odd	6		735.1.f.d.344.2	yes	2
21.17	even	6		735.1.f.c.344.2	yes	2
21.20	even	2	inner	735.1.o.d.704.1		4
35.2	odd	12		3675.1.u.f.1451.4		8
35.3	even	12		3675.1.c.f.1226.4		4
35.4	even	6		735.1.f.d.344.2	yes	2
35.9	even	6		735.1.o.c.569.1		4
35.12	even	12		3675.1.u.f.1451.3		8
35.13	even	4		3675.1.u.f.851.3		8
35.17	even	12		3675.1.c.f.1226.1		4
35.18	odd	12		3675.1.c.f.1226.3		4
35.19	odd	6	inner	735.1.o.d.569.1		4
35.23	odd	12		3675.1.u.f.1451.1		8
35.24	odd	6		735.1.f.c.344.2	yes	2
35.27	even	4		3675.1.u.f.851.2		8
35.32	odd	12		3675.1.c.f.1226.2		4
35.33	even	12		3675.1.u.f.1451.2		8
35.34	odd	2	inner	735.1.o.d.704.1		4
105.2	even	12		3675.1.u.f.1451.1		8
105.17	odd	12		3675.1.c.f.1226.4		4
105.23	even	12		3675.1.u.f.1451.4		8
105.32	even	12		3675.1.c.f.1226.3		4
105.38	odd	12		3675.1.c.f.1226.1		4
105.44	odd	6	inner	735.1.o.d.569.2		4
105.47	odd	12		3675.1.u.f.1451.2		8
105.53	even	12		3675.1.c.f.1226.2		4
105.59	even	6		735.1.f.d.344.1	yes	2
105.62	odd	4		3675.1.u.f.851.3		8
105.68	odd	12		3675.1.u.f.1451.3		8
105.74	odd	6		735.1.f.c.344.1	✓	2
105.83	odd	4		3675.1.u.f.851.2		8
105.89	even	6		735.1.o.c.569.2		4
105.104	even	2		735.1.o.c.704.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.1.f.c.344.1	✓	2	7.4	even	3
735.1.f.c.344.1	✓	2	105.74	odd	6
735.1.f.c.344.2	yes	2	21.17	even	6
735.1.f.c.344.2	yes	2	35.24	odd	6
735.1.f.d.344.1	yes	2	7.3	odd	6
735.1.f.d.344.1	yes	2	105.59	even	6
735.1.f.d.344.2	yes	2	21.11	odd	6
735.1.f.d.344.2	yes	2	35.4	even	6
735.1.o.c.569.1		4	21.2	odd	6
735.1.o.c.569.1		4	35.9	even	6
735.1.o.c.569.2		4	7.5	odd	6
735.1.o.c.569.2		4	105.89	even	6
735.1.o.c.704.1		4	3.2	odd	2
735.1.o.c.704.1		4	5.4	even	2
735.1.o.c.704.2		4	7.6	odd	2
735.1.o.c.704.2		4	105.104	even	2
735.1.o.d.569.1		4	21.5	even	6	inner
735.1.o.d.569.1		4	35.19	odd	6	inner
735.1.o.d.569.2		4	7.2	even	3	inner
735.1.o.d.569.2		4	105.44	odd	6	inner
735.1.o.d.704.1		4	21.20	even	2	inner
735.1.o.d.704.1		4	35.34	odd	2	inner
735.1.o.d.704.2		4	1.1	even	1	trivial
735.1.o.d.704.2		4	15.14	odd	2	CM
3675.1.c.f.1226.1		4	35.17	even	12
3675.1.c.f.1226.1		4	105.38	odd	12
3675.1.c.f.1226.2		4	35.32	odd	12
3675.1.c.f.1226.2		4	105.53	even	12
3675.1.c.f.1226.3		4	35.18	odd	12
3675.1.c.f.1226.3		4	105.32	even	12
3675.1.c.f.1226.4		4	35.3	even	12
3675.1.c.f.1226.4		4	105.17	odd	12
3675.1.u.f.851.1		8	5.2	odd	4
3675.1.u.f.851.1		8	15.8	even	4
3675.1.u.f.851.2		8	35.27	even	4
3675.1.u.f.851.2		8	105.83	odd	4
3675.1.u.f.851.3		8	35.13	even	4
3675.1.u.f.851.3		8	105.62	odd	4
3675.1.u.f.851.4		8	5.3	odd	4
3675.1.u.f.851.4		8	15.2	even	4
3675.1.u.f.1451.1		8	35.23	odd	12
3675.1.u.f.1451.1		8	105.2	even	12
3675.1.u.f.1451.2		8	35.33	even	12
3675.1.u.f.1451.2		8	105.47	odd	12
3675.1.u.f.1451.3		8	35.12	even	12
3675.1.u.f.1451.3		8	105.68	odd	12
3675.1.u.f.1451.4		8	35.2	odd	12
3675.1.u.f.1451.4		8	105.23	even	12

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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	735.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 1.22474i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +1.41421 q^{6} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.707107 + 1.22474i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +1.41421 q^{6} +(-0.500000 - 0.866025i) q^{9} +(0.707107 - 1.22474i) q^{10} +(0.500000 + 0.866025i) q^{12} -1.00000 q^{15} +(0.500000 + 0.866025i) q^{16} +(0.707107 - 1.22474i) q^{18} +(0.707107 + 1.22474i) q^{19} +1.00000 q^{20} +(-0.707107 - 1.22474i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +(-0.707107 - 1.22474i) q^{30} +(-0.707107 + 1.22474i) q^{31} +(-0.707107 + 1.22474i) q^{32} +1.00000 q^{36} +(-1.00000 + 1.73205i) q^{38} +(-0.500000 + 0.866025i) q^{45} +(1.00000 - 1.73205i) q^{46} +1.00000 q^{48} -1.41421 q^{50} +(-0.707107 + 1.22474i) q^{53} +(-0.707107 - 1.22474i) q^{54} +1.41421 q^{57} +(0.500000 - 0.866025i) q^{60} +(-0.707107 - 1.22474i) q^{61} -2.00000 q^{62} -1.00000 q^{64} -1.41421 q^{69} +(0.500000 + 0.866025i) q^{75} -1.41421 q^{76} +(0.500000 - 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} -1.41421 q^{90} +1.41421 q^{92} +(0.707107 + 1.22474i) q^{93} +(0.707107 - 1.22474i) q^{95} +(0.707107 + 1.22474i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 - 2 * q^9	\( 4 q + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{9} + 2 q^{12} - 4 q^{15} + 2 q^{16} + 4 q^{20} - 2 q^{25} - 4 q^{27} + 4 q^{36} - 4 q^{38} - 2 q^{45} + 4 q^{46} + 4 q^{48} + 2 q^{60} - 8 q^{62} - 4 q^{64} + 2 q^{75} + 2 q^{80} - 2 q^{81}+O(q^{100}) \) 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 - 2 * q^9 + 2 * q^12 - 4 * q^15 + 2 * q^16 + 4 * q^20 - 2 * q^25 - 4 * q^27 + 4 * q^36 - 4 * q^38 - 2 * q^45 + 4 * q^46 + 4 * q^48 + 2 * q^60 - 8 * q^62 - 4 * q^64 + 2 * q^75 + 2 * q^80 - 2 * q^81

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