LMFDB - Embedded newform 3675.1.bf.c.668.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3675,1,Mod(68,3675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 10])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3675.68"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3675.bf (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,8,8,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,-16,0,-8,0,0,0,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(61)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.83406392143$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{48})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.13897288125.2

Embedding invariants

Embedding label			668.3
Root			$-0.793353 + 0.608761i$ of defining polynomial
Character	$\chi$	$=$	3675.668
Dual form			3675.1.bf.c.1832.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.198092 - 0.739288i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(0.358719 + 0.207107i) q^{4} +0.765367i q^{6} +(0.765367 - 0.765367i) q^{8} +(0.866025 - 0.500000i) q^{9} +(-0.400100 - 0.107206i) q^{12} +(-0.207107 - 0.358719i) q^{16} +(-0.366025 - 1.36603i) q^{17} +(-0.198092 - 0.739288i) q^{18} +(-0.923880 - 1.60021i) q^{19} +(-1.78480 - 0.478235i) q^{23} +(-0.541196 + 0.937379i) q^{24} +(-0.707107 + 0.707107i) q^{27} +(0.662827 + 0.382683i) q^{31} +(0.739288 - 0.198092i) q^{32} -1.08239 q^{34} +0.414214 q^{36} +(-1.36603 + 0.366025i) q^{38} +(-0.707107 + 1.22474i) q^{46} +(1.36603 + 0.366025i) q^{47} +(0.292893 + 0.292893i) q^{48} +(0.707107 + 1.22474i) q^{51} +(-0.478235 - 1.78480i) q^{53} +(0.382683 + 0.662827i) q^{54} +(1.30656 + 1.30656i) q^{57} +(0.662827 - 0.382683i) q^{61} +(0.414214 - 0.414214i) q^{62} -1.00000i q^{64} +(0.151613 - 0.565826i) q^{68} +1.84776 q^{69} +(0.280144 - 1.04551i) q^{72} -0.765367i q^{76} +(-1.22474 + 0.707107i) q^{79} +(0.500000 - 0.866025i) q^{81} +(1.00000 + 1.00000i) q^{83} +(-0.541196 - 0.541196i) q^{92} +(-0.739288 - 0.198092i) q^{93} +(0.541196 - 0.937379i) q^{94} +(-0.662827 + 0.382683i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{12} + 8 q^{16} + 8 q^{17} - 16 q^{36} - 8 q^{38} + 8 q^{47} + 16 q^{48} - 16 q^{62} + 8 q^{68} + 8 q^{81} + 16 q^{83}+O(q^{100}) $ 16 * q - 8 * q^12 + 8 * q^16 + 8 * q^17 - 16 * q^36 - 8 * q^38 + 8 * q^47 + 16 * q^48 - 16 * q^62 + 8 * q^68 + 8 * q^81 + 16 * q^83

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times$.

$n$	$1177$	$1226$	$2551$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	0.198092	−	0.739288i	0.198092	−	0.739288i	−0.793353	−	0.608761i	$-0.791667\pi$
$2$	0.198092	−	0.739288i	0.198092	−	0.739288i	0.991445	−	0.130526i	$-0.0416667\pi$
$3$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$3$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$4$	0.358719	+	0.207107i	0.358719	+	0.207107i
$5$		0			0
$5$		0			0
$6$			0.765367i			0.765367i
$7$		0			0
$7$		0			0
$8$	0.765367	−	0.765367i	0.765367	−	0.765367i
$9$	0.866025	−	0.500000i	0.866025	−	0.500000i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−0.400100	−	0.107206i	−0.400100	−	0.107206i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	−0.207107	−	0.358719i	−0.207107	−	0.358719i
$17$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$17$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$18$	−0.198092	−	0.739288i	−0.198092	−	0.739288i
$19$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$19$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.793353	−	0.608761i	$-0.791667\pi$
$23$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.991445	+	0.130526i	$0.958333\pi$
$24$	−0.541196	+	0.937379i	−0.541196	+	0.937379i
$25$		0			0
$26$		0			0
$27$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	0.662827	+	0.382683i	0.662827	+	0.382683i	0.793353	−	0.608761i	$-0.208333\pi$
$31$	0.662827	+	0.382683i	0.662827	+	0.382683i	−0.130526	+	0.991445i	$0.541667\pi$
$32$	0.739288	−	0.198092i	0.739288	−	0.198092i
$33$		0			0
$34$	−1.08239			−1.08239
$35$		0			0
$36$	0.414214			0.414214
$37$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$37$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$38$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$47$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$47$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$48$	0.292893	+	0.292893i	0.292893	+	0.292893i
$49$		0			0
$50$		0			0
$51$	0.707107	+	1.22474i	0.707107	+	1.22474i
$52$		0			0
$53$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	−0.608761	−	0.793353i	$-0.708333\pi$
$53$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	0.130526	−	0.991445i	$-0.458333\pi$
$54$	0.382683	+	0.662827i	0.382683	+	0.662827i
$55$		0			0
$56$		0			0
$57$	1.30656	+	1.30656i	1.30656	+	1.30656i
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	0.662827	−	0.382683i	0.662827	−	0.382683i	−0.130526	−	0.991445i	$-0.541667\pi$
$61$	0.662827	−	0.382683i	0.662827	−	0.382683i	0.793353	+	0.608761i	$0.208333\pi$
$62$	0.414214	−	0.414214i	0.414214	−	0.414214i
$63$		0			0
$64$		−	1.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$	0.151613	−	0.565826i	0.151613	−	0.565826i
$69$	1.84776			1.84776
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.280144	−	1.04551i	0.280144	−	1.04551i
$73$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$73$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$74$		0			0
$75$		0			0
$76$		−	0.765367i		−	0.765367i
$77$		0			0
$78$		0			0
$79$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$79$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$80$		0			0
$81$	0.500000	−	0.866025i	0.500000	−	0.866025i
$82$		0			0
$83$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$83$	1.00000	+	1.00000i	1.00000	+	1.00000i			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	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$93$	−0.739288	−	0.198092i	−0.739288	−	0.198092i
$94$	0.541196	−	0.937379i	0.541196	−	0.937379i
$95$		0			0
$96$	−0.662827	+	0.382683i	−0.662827	+	0.382683i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$	1.04551	−	0.280144i	1.04551	−	0.280144i
$103$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$103$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$104$		0			0
$105$		0			0
$106$	−1.41421			−1.41421
$107$	−0.198092	+	0.739288i	−0.198092	+	0.739288i	0.793353	+	0.608761i	$0.208333\pi$
$107$	−0.198092	+	0.739288i	−0.198092	+	0.739288i	−0.991445	+	0.130526i	$0.958333\pi$
$108$	−0.400100	+	0.107206i	−0.400100	+	0.107206i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$113$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$114$	1.22474	−	0.707107i	1.22474	−	0.707107i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$	−0.151613	−	0.565826i	−0.151613	−	0.565826i
$123$		0			0
$124$	0.158513	+	0.274552i	0.158513	+	0.274552i
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.32565	−	0.765367i	−1.32565	−	0.765367i
$137$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.130526	−	0.991445i	$-0.458333\pi$
$137$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.608761	+	0.793353i	$0.291667\pi$
$138$	0.366025	−	1.36603i	0.366025	−	1.36603i
$139$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$139$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$140$		0			0
$141$	−1.41421			−1.41421
$142$		0			0
$143$		0			0
$144$	−0.358719	−	0.207107i	−0.358719	−	0.207107i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$151$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$152$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$153$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$154$		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	$0.416667\pi$
$158$	0.280144	+	1.04551i	0.280144	+	1.04551i
$159$	0.923880	+	1.60021i	0.923880	+	1.60021i
$160$		0			0
$161$		0			0
$162$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$163$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$163$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$164$		0			0
$165$		0			0
$166$	0.937379	−	0.541196i	0.937379	−	0.541196i
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$			1.00000i			1.00000i
$170$		0			0
$171$	−1.60021	−	0.923880i	−1.60021	−	0.923880i
$172$		0			0
$173$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$173$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−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		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$181$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$182$		0			0
$183$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$184$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$185$		0			0
$186$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$187$		0			0
$188$	0.414214	+	0.414214i	0.414214	+	0.414214i
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	0.258819	+	0.965926i	0.258819	+	0.965926i
$193$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$193$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$197$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$198$		0			0
$199$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$199$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$			0.585786i			0.585786i
$205$		0			0
$206$		0			0
$207$	−1.78480	+	0.478235i	−1.78480	+	0.478235i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	0.198092	−	0.739288i	0.198092	−	0.739288i
$213$		0			0
$214$	0.507306	+	0.292893i	0.507306	+	0.292893i
$215$		0			0
$216$			1.08239i			1.08239i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$227$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$227$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$228$	0.198092	+	0.739288i	0.198092	+	0.739288i
$229$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$229$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.78480	+	0.478235i	1.78480	+	0.478235i	0.991445	−	0.130526i	$-0.0416667\pi$
$233$	1.78480	+	0.478235i	1.78480	+	0.478235i	0.793353	+	0.608761i	$0.208333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.00000	−	1.00000i	1.00000	−	1.00000i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.991445	+	0.130526i	$0.0416667\pi$
$241$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.608761	+	0.793353i	$0.291667\pi$
$242$	−0.739288	+	0.198092i	−0.739288	+	0.198092i
$243$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$244$	0.317025			0.317025
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.800199	−	0.214413i	0.800199	−	0.214413i
$249$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.500000	−	0.866025i	0.500000	−	0.866025i
$257$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$257$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.198092	+	0.739288i	0.198092	+	0.739288i	0.991445	+	0.130526i	$0.0416667\pi$
$263$	0.198092	+	0.739288i	0.198092	+	0.739288i	−0.793353	+	0.608761i	$0.791667\pi$
$264$		0			0
$265$		0			0
$266$		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			0
$271$	−1.60021	+	0.923880i	−1.60021	+	0.923880i	−0.608761	+	0.793353i	$0.708333\pi$
$271$	−1.60021	+	0.923880i	−1.60021	+	0.923880i	−0.991445	+	0.130526i	$0.958333\pi$
$272$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$273$		0			0
$274$		−	0.585786i		−	0.585786i
$275$		0			0
$276$	0.662827	+	0.382683i	0.662827	+	0.382683i
$277$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$277$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$278$	−0.151613	+	0.565826i	−0.151613	+	0.565826i
$279$	0.765367			0.765367
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−0.280144	+	1.04551i	−0.280144	+	1.04551i
$283$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$283$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.541196	−	0.541196i	0.541196	−	0.541196i
$289$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$293$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.765367	−	0.765367i	−0.765367	−	0.765367i
$303$		0			0
$304$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$305$		0			0
$306$	−0.937379	+	0.541196i	−0.937379	+	0.541196i
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		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
$316$	−0.585786			−0.585786
$317$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	0.130526	+	0.991445i	$0.458333\pi$
$317$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	−0.608761	+	0.793353i	$0.708333\pi$
$318$	1.36603	−	0.366025i	1.36603	−	0.366025i
$319$		0			0
$320$		0			0
$321$		−	0.765367i		−	0.765367i
$322$		0			0
$323$	−1.84776	+	1.84776i	−1.84776	+	1.84776i
$324$	0.358719	−	0.207107i	0.358719	−	0.207107i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$331$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$332$	0.151613	+	0.565826i	0.151613	+	0.565826i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	0.739288	+	0.198092i	0.739288	+	0.198092i
$339$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$340$		0			0
$341$		0			0
$342$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.937379	+	0.541196i	0.937379	+	0.541196i
$347$	−1.78480	+	0.478235i	−1.78480	+	0.478235i	−0.991445	−	0.130526i	$-0.958333\pi$
$347$	−1.78480	+	0.478235i	−1.78480	+	0.478235i	−0.793353	+	0.608761i	$0.791667\pi$
$348$		0			0
$349$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$349$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$353$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−1.20711	+	2.09077i	−1.20711	+	2.09077i
$362$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$363$	0.707107	+	0.707107i	0.707107	+	0.707107i
$364$		0			0
$365$		0			0
$366$	0.292893	+	0.507306i	0.292893	+	0.507306i
$367$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$367$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$368$	0.198092	+	0.739288i	0.198092	+	0.739288i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.224171	−	0.224171i	−0.224171	−	0.224171i
$373$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$373$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$374$		0			0
$375$		0			0
$376$	1.32565	−	0.765367i	1.32565	−	0.765367i
$377$		0			0
$378$		0			0
$379$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$379$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$			2.61313i			2.61313i
$392$		0			0
$393$		0			0
$394$	0.507306	−	0.292893i	0.507306	−	0.292893i
$395$		0			0
$396$		0			0
$397$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$397$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$398$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$399$		0			0
$400$		0			0
$401$		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
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.47858	+	0.396183i	1.47858	+	0.396183i
$409$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$409$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$410$		0			0
$411$	−0.662827	+	0.382683i	−0.662827	+	0.382683i
$412$		0			0
$413$		0			0
$414$			1.41421i			1.41421i
$415$		0			0
$416$		0			0
$417$	0.739288	−	0.198092i	0.739288	−	0.198092i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$421$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$	1.36603	−	0.366025i	1.36603	−	0.366025i
$424$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.224171	+	0.224171i	−0.224171	+	0.224171i
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	0.400100	+	0.107206i	0.400100	+	0.107206i
$433$		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			0
$437$	0.883663	+	3.29788i	0.883663	+	3.29788i
$438$		0			0
$439$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$439$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.739288	+	0.198092i	0.739288	+	0.198092i	0.608761	−	0.793353i	$-0.291667\pi$
$443$	0.739288	+	0.198092i	0.739288	+	0.198092i	0.130526	+	0.991445i	$0.458333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	0.306223	−	0.0820522i	0.306223	−	0.0820522i
$453$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$454$		0			0
$455$		0			0
$456$	2.00000			2.00000
$457$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$457$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$458$	0.565826	−	0.151613i	0.565826	−	0.151613i
$459$	1.22474	+	0.707107i	1.22474	+	0.707107i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	0.707107	−	1.22474i	0.707107	−	1.22474i
$467$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$467$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$475$		0			0
$476$		0			0
$477$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	1.00000	−	1.00000i	1.00000	−	1.00000i
$483$		0			0
$484$		−	0.414214i		−	0.414214i
$485$		0			0
$486$	0.662827	+	0.382683i	0.662827	+	0.382683i
$487$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$487$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$488$	0.214413	−	0.800199i	0.214413	−	0.800199i
$489$		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
$496$		−	0.317025i		−	0.317025i
$497$		0			0
$498$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$499$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$499$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$503$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$508$		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
$512$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$513$	1.78480	+	0.478235i	1.78480	+	0.478235i
$514$	0.765367	−	1.32565i	0.765367	−	1.32565i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	1.41421i		−	1.41421i
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$523$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$524$		0			0
$525$		0			0
$526$	0.585786			0.585786
$527$	0.280144	−	1.04551i	0.280144	−	1.04551i
$528$		0			0
$529$	2.09077	+	1.20711i	2.09077	+	1.20711i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$541$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$542$	0.366025	+	1.36603i	0.366025	+	1.36603i
$543$	0.478235	+	1.78480i	0.478235	+	1.78480i
$544$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	0.306223	+	0.0820522i	0.306223	+	0.0820522i
$549$	0.382683	−	0.662827i	0.382683	−	0.662827i
$550$		0			0
$551$		0			0
$552$	1.41421	−	1.41421i	1.41421	−	1.41421i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.274552	−	0.158513i	−0.274552	−	0.158513i
$557$	−0.739288	+	0.198092i	−0.739288	+	0.198092i	−0.608761	−	0.793353i	$-0.708333\pi$
$557$	−0.739288	+	0.198092i	−0.739288	+	0.198092i	−0.130526	+	0.991445i	$0.541667\pi$
$558$	0.151613	−	0.565826i	0.151613	−	0.565826i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$564$	−0.507306	−	0.292893i	−0.507306	−	0.292893i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$571$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$577$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$578$	0.198092	+	0.739288i	0.198092	+	0.739288i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−1.32565	+	0.765367i	−1.32565	+	0.765367i
$587$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$587$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$588$		0			0
$589$		−	1.41421i		−	1.41421i
$590$		0			0
$591$	−0.662827	−	0.382683i	−0.662827	−	0.382683i
$592$		0			0
$593$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$593$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.478235	+	1.78480i	−0.478235	+	1.78480i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$601$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$602$		0			0
$603$		0			0
$604$	0.507306	−	0.292893i	0.507306	−	0.292893i
$605$		0			0
$606$		0			0
$607$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$607$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$608$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.151613	−	0.565826i	−0.151613	−	0.565826i
$613$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$613$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$617$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$618$		0			0
$619$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$619$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$620$		0			0
$621$	1.60021	−	0.923880i	1.60021	−	0.923880i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$631$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$632$	−0.396183	+	1.47858i	−0.396183	+	1.47858i
$633$		0			0
$634$	1.22474	+	0.707107i	1.22474	+	0.707107i
$635$		0			0
$636$			0.765367i			0.765367i
$637$		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	$0.666667\pi$
$642$	−0.565826	−	0.151613i	−0.565826	−	0.151613i
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	1.00000	+	1.73205i	1.00000	+	1.73205i
$647$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$647$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$648$	−0.280144	−	1.04551i	−0.280144	−	1.04551i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.739288	−	0.198092i	−0.739288	−	0.198092i	−0.130526	−	0.991445i	$-0.541667\pi$
$653$	−0.739288	−	0.198092i	−0.739288	−	0.198092i	−0.608761	+	0.793353i	$0.708333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−0.662827	−	0.382683i	−0.662827	−	0.382683i	0.130526	−	0.991445i	$-0.458333\pi$
$661$	−0.662827	−	0.382683i	−0.662827	−	0.382683i	−0.793353	+	0.608761i	$0.791667\pi$
$662$		0			0
$663$		0			0
$664$	1.53073			1.53073
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	−0.207107	+	0.358719i	−0.207107	+	0.358719i
$677$	−1.93185	−	0.517638i	−1.93185	−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$677$	−1.93185	−	0.517638i	−1.93185	−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$678$	0.414214	+	0.414214i	0.414214	+	0.414214i
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.478235	+	1.78480i	0.478235	+	1.78480i	0.608761	+	0.793353i	$0.291667\pi$
$683$	0.478235	+	1.78480i	0.478235	+	1.78480i	−0.130526	+	0.991445i	$0.541667\pi$
$684$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$685$		0			0
$686$		0			0
$687$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.60021	−	0.923880i	1.60021	−	0.923880i	0.608761	−	0.793353i	$-0.291667\pi$
$691$	1.60021	−	0.923880i	1.60021	−	0.923880i	0.991445	−	0.130526i	$-0.0416667\pi$
$692$	−0.414214	+	0.414214i	−0.414214	+	0.414214i
$693$		0			0
$694$			1.41421i			1.41421i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$699$	−1.84776			−1.84776
$700$		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
$706$		−	1.53073i		−	1.53073i
$707$		0			0
$708$		0			0
$709$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$709$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$710$		0			0
$711$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$712$		0			0
$713$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$714$		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	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	1.30656	+	1.30656i	1.30656	+	1.30656i
$723$	−1.78480	−	0.478235i	−1.78480	−	0.478235i
$724$	0.382683	−	0.662827i	0.382683	−	0.662827i
$725$		0			0
$726$	0.662827	−	0.382683i	0.662827	−	0.382683i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	1.00000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	−0.306223	+	0.0820522i	−0.306223	+	0.0820522i
$733$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$733$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$734$		0			0
$735$		0			0
$736$	−1.41421			−1.41421
$737$		0			0
$738$		0			0
$739$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$739$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$743$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$744$	−0.717439	+	0.414214i	−0.717439	+	0.414214i
$745$		0			0
$746$		0			0
$747$	1.36603	+	0.366025i	1.36603	+	0.366025i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$751$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$752$	−0.151613	−	0.565826i	−0.151613	−	0.565826i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	1.04551	+	0.280144i	1.04551	+	0.280144i
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.937379	−	0.541196i	−0.937379	−	0.541196i
$767$		0			0
$768$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$769$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$769$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$770$		0			0
$771$	−2.00000			−2.00000
$772$		0			0
$773$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$773$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	1.93185	+	0.517638i	1.93185	+	0.517638i
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$787$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$788$	0.0820522	+	0.306223i	0.0820522	+	0.306223i
$789$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.662827	−	0.382683i	0.662827	−	0.382683i
$797$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$797$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$798$		0			0
$799$		−	2.00000i		−	2.00000i
$800$		0			0
$801$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$811$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$812$		0			0
$813$	1.30656	−	1.30656i	1.30656	−	1.30656i
$814$		0			0
$815$		0			0
$816$	0.292893	−	0.507306i	0.292893	−	0.507306i
$817$		0			0
$818$	1.00000	+	1.00000i	1.00000	+	1.00000i
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	0.151613	+	0.565826i	0.151613	+	0.565826i
$823$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$823$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$827$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$828$	−0.739288	−	0.198092i	−0.739288	−	0.198092i
$829$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$829$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		−	0.585786i		−	0.585786i
$835$		0			0
$836$		0			0
$837$	−0.739288	+	0.198092i	−0.739288	+	0.198092i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$	0.280144	−	1.04551i	0.280144	−	1.04551i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		−	1.08239i		−	1.08239i
$847$		0			0
$848$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	0.414214	+	0.717439i	0.414214	+	0.717439i
$857$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$857$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$858$		0			0
$859$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$859$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.739288	+	0.198092i	0.739288	+	0.198092i	0.608761	−	0.793353i	$-0.291667\pi$
$863$	0.739288	+	0.198092i	0.739288	+	0.198092i	0.130526	+	0.991445i	$0.458333\pi$
$864$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$865$		0			0
$866$		0			0
$867$	0.707107	−	0.707107i	0.707107	−	0.707107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	2.61313			2.61313
$875$		0			0
$876$		0			0
$877$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$877$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$878$	−0.565826	+	0.151613i	−0.565826	+	0.151613i
$879$	1.73205	+	1.00000i	1.73205	+	1.00000i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$	0.292893	−	0.507306i	0.292893	−	0.507306i
$887$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$887$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.676327	−	2.52409i	−0.676327	−	2.52409i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−2.26303	+	1.30656i	−2.26303	+	1.30656i
$902$		0			0
$903$		0			0
$904$		−	0.828427i		−	0.828427i
$905$		0			0
$906$	0.937379	+	0.541196i	0.937379	+	0.541196i
$907$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$907$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	0.198092	−	0.739288i	0.198092	−	0.739288i
$913$		0			0
$914$		0			0
$915$		0			0
$916$			0.317025i			0.317025i
$917$		0			0
$918$	0.765367	−	0.765367i	0.765367	−	0.765367i
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		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	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$	0.541196	+	0.541196i	0.541196	+	0.541196i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.198092	−	0.739288i	0.198092	−	0.739288i	−0.793353	−	0.608761i	$-0.791667\pi$
$947$	0.198092	−	0.739288i	0.198092	−	0.739288i	0.991445	−	0.130526i	$-0.0416667\pi$
$948$	0.565826	−	0.151613i	0.565826	−	0.151613i
$949$		0			0
$950$		0			0
$951$		−	1.84776i		−	1.84776i
$952$		0			0
$953$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$953$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$954$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.207107	−	0.358719i	−0.207107	−	0.358719i
$962$		0			0
$963$	0.198092	+	0.739288i	0.198092	+	0.739288i
$964$	0.382683	+	0.662827i	0.382683	+	0.662827i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−1.04551	−	0.280144i	−1.04551	−	0.280144i
$969$	1.30656	−	2.26303i	1.30656	−	2.26303i
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$	−0.292893	+	0.292893i	−0.292893	+	0.292893i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.274552	−	0.158513i	−0.274552	−	0.158513i
$977$	1.78480	−	0.478235i	1.78480	−	0.478235i	0.793353	−	0.608761i	$-0.208333\pi$
$977$	1.78480	−	0.478235i	1.78480	−	0.478235i	0.991445	+	0.130526i	$0.0416667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$983$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$984$		0			0
$985$		0			0
$986$		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	$0.666667\pi$
$992$	0.565826	+	0.151613i	0.565826	+	0.151613i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$997$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$997$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$998$	−0.280144	−	1.04551i	−0.280144	−	1.04551i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.1.bf.c.668.3		16
3.2	odd	2		3675.1.bf.d.668.2		16
5.2	odd	4	inner	3675.1.bf.c.2432.3		16
5.3	odd	4		3675.1.bf.d.2432.2		16
5.4	even	2		3675.1.bf.d.668.2		16
7.2	even	3	inner	3675.1.bf.c.68.2		16
7.3	odd	6		3675.1.k.a.293.3	yes	8
7.4	even	3		3675.1.k.c.293.3	yes	8
7.5	odd	6		3675.1.bf.d.68.2		16
7.6	odd	2		3675.1.bf.d.668.3		16
15.2	even	4		3675.1.bf.d.2432.2		16
15.8	even	4	inner	3675.1.bf.c.2432.3		16
15.14	odd	2	CM	3675.1.bf.c.668.3		16
21.2	odd	6		3675.1.bf.d.68.3		16
21.5	even	6	inner	3675.1.bf.c.68.3		16
21.11	odd	6		3675.1.k.a.293.2	✓	8
21.17	even	6		3675.1.k.c.293.2	yes	8
21.20	even	2	inner	3675.1.bf.c.668.2		16
35.2	odd	12	inner	3675.1.bf.c.1832.2		16
35.3	even	12		3675.1.k.c.2057.3	yes	8
35.4	even	6		3675.1.k.a.293.2	✓	8
35.9	even	6		3675.1.bf.d.68.3		16
35.12	even	12		3675.1.bf.d.1832.2		16
35.13	even	4	inner	3675.1.bf.c.2432.2		16
35.17	even	12		3675.1.k.a.2057.2	yes	8
35.18	odd	12		3675.1.k.a.2057.3	yes	8
35.19	odd	6	inner	3675.1.bf.c.68.3		16
35.23	odd	12		3675.1.bf.d.1832.3		16
35.24	odd	6		3675.1.k.c.293.2	yes	8
35.27	even	4		3675.1.bf.d.2432.3		16
35.32	odd	12		3675.1.k.c.2057.2	yes	8
35.33	even	12	inner	3675.1.bf.c.1832.3		16
35.34	odd	2	inner	3675.1.bf.c.668.2		16
105.2	even	12		3675.1.bf.d.1832.3		16
105.17	odd	12		3675.1.k.c.2057.3	yes	8
105.23	even	12	inner	3675.1.bf.c.1832.2		16
105.32	even	12		3675.1.k.a.2057.3	yes	8
105.38	odd	12		3675.1.k.a.2057.2	yes	8
105.44	odd	6	inner	3675.1.bf.c.68.2		16
105.47	odd	12	inner	3675.1.bf.c.1832.3		16
105.53	even	12		3675.1.k.c.2057.2	yes	8
105.59	even	6		3675.1.k.a.293.3	yes	8
105.62	odd	4	inner	3675.1.bf.c.2432.2		16
105.68	odd	12		3675.1.bf.d.1832.2		16
105.74	odd	6		3675.1.k.c.293.3	yes	8
105.83	odd	4		3675.1.bf.d.2432.3		16
105.89	even	6		3675.1.bf.d.68.2		16
105.104	even	2		3675.1.bf.d.668.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3675.1.k.a.293.2	✓	8	21.11	odd	6
3675.1.k.a.293.2	✓	8	35.4	even	6
3675.1.k.a.293.3	yes	8	7.3	odd	6
3675.1.k.a.293.3	yes	8	105.59	even	6
3675.1.k.a.2057.2	yes	8	35.17	even	12
3675.1.k.a.2057.2	yes	8	105.38	odd	12
3675.1.k.a.2057.3	yes	8	35.18	odd	12
3675.1.k.a.2057.3	yes	8	105.32	even	12
3675.1.k.c.293.2	yes	8	21.17	even	6
3675.1.k.c.293.2	yes	8	35.24	odd	6
3675.1.k.c.293.3	yes	8	7.4	even	3
3675.1.k.c.293.3	yes	8	105.74	odd	6
3675.1.k.c.2057.2	yes	8	35.32	odd	12
3675.1.k.c.2057.2	yes	8	105.53	even	12
3675.1.k.c.2057.3	yes	8	35.3	even	12
3675.1.k.c.2057.3	yes	8	105.17	odd	12
3675.1.bf.c.68.2		16	7.2	even	3	inner
3675.1.bf.c.68.2		16	105.44	odd	6	inner
3675.1.bf.c.68.3		16	21.5	even	6	inner
3675.1.bf.c.68.3		16	35.19	odd	6	inner
3675.1.bf.c.668.2		16	21.20	even	2	inner
3675.1.bf.c.668.2		16	35.34	odd	2	inner
3675.1.bf.c.668.3		16	1.1	even	1	trivial
3675.1.bf.c.668.3		16	15.14	odd	2	CM
3675.1.bf.c.1832.2		16	35.2	odd	12	inner
3675.1.bf.c.1832.2		16	105.23	even	12	inner
3675.1.bf.c.1832.3		16	35.33	even	12	inner
3675.1.bf.c.1832.3		16	105.47	odd	12	inner
3675.1.bf.c.2432.2		16	35.13	even	4	inner
3675.1.bf.c.2432.2		16	105.62	odd	4	inner
3675.1.bf.c.2432.3		16	5.2	odd	4	inner
3675.1.bf.c.2432.3		16	15.8	even	4	inner
3675.1.bf.d.68.2		16	7.5	odd	6
3675.1.bf.d.68.2		16	105.89	even	6
3675.1.bf.d.68.3		16	21.2	odd	6
3675.1.bf.d.68.3		16	35.9	even	6
3675.1.bf.d.668.2		16	3.2	odd	2
3675.1.bf.d.668.2		16	5.4	even	2
3675.1.bf.d.668.3		16	7.6	odd	2
3675.1.bf.d.668.3		16	105.104	even	2
3675.1.bf.d.1832.2		16	35.12	even	12
3675.1.bf.d.1832.2		16	105.68	odd	12
3675.1.bf.d.1832.3		16	35.23	odd	12
3675.1.bf.d.1832.3		16	105.2	even	12
3675.1.bf.d.2432.2		16	5.3	odd	4
3675.1.bf.d.2432.2		16	15.2	even	4
3675.1.bf.d.2432.3		16	35.27	even	4
3675.1.bf.d.2432.3		16	105.83	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3675.bf (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.198092 - 0.739288i) q^{2} +(-0.965926 + 0.258819i) q^{3} +(0.358719 + 0.207107i) q^{4} +0.765367i q^{6} +(0.765367 - 0.765367i) q^{8} +(0.866025 - 0.500000i) q^{9} +(-0.400100 - 0.107206i) q^{12} +(-0.207107 - 0.358719i) q^{16} +(-0.366025 - 1.36603i) q^{17} +(-0.198092 - 0.739288i) q^{18} +(-0.923880 - 1.60021i) q^{19} +(-1.78480 - 0.478235i) q^{23} +(-0.541196 + 0.937379i) q^{24} +(-0.707107 + 0.707107i) q^{27} +(0.662827 + 0.382683i) q^{31} +(0.739288 - 0.198092i) q^{32} -1.08239 q^{34} +0.414214 q^{36} +(-1.36603 + 0.366025i) q^{38} +(-0.707107 + 1.22474i) q^{46} +(1.36603 + 0.366025i) q^{47} +(0.292893 + 0.292893i) q^{48} +(0.707107 + 1.22474i) q^{51} +(-0.478235 - 1.78480i) q^{53} +(0.382683 + 0.662827i) q^{54} +(1.30656 + 1.30656i) q^{57} +(0.662827 - 0.382683i) q^{61} +(0.414214 - 0.414214i) q^{62} -1.00000i q^{64} +(0.151613 - 0.565826i) q^{68} +1.84776 q^{69} +(0.280144 - 1.04551i) q^{72} -0.765367i q^{76} +(-1.22474 + 0.707107i) q^{79} +(0.500000 - 0.866025i) q^{81} +(1.00000 + 1.00000i) q^{83} +(-0.541196 - 0.541196i) q^{92} +(-0.739288 - 0.198092i) q^{93} +(0.541196 - 0.937379i) q^{94} +(-0.662827 + 0.382683i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{12} + 8 q^{16} + 8 q^{17} - 16 q^{36} - 8 q^{38} + 8 q^{47} + 16 q^{48} - 16 q^{62} + 8 q^{68} + 8 q^{81} + 16 q^{83}+O(q^{100}) \) 16 * q - 8 * q^12 + 8 * q^16 + 8 * q^17 - 16 * q^36 - 8 * q^38 + 8 * q^47 + 16 * q^48 - 16 * q^62 + 8 * q^68 + 8 * q^81 + 16 * q^83

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