LMFDB - Embedded newform 469.1.bl.a.412.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [469,1,Mod(6,469)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(469, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 40]))

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

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

chi := DirichletCharacter("469.6");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 469 = 7 \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	469.bl (of order $66$, degree $20$, 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:	$0.234061490925$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

Embedding invariants

Embedding label			412.1
Root			$-0.786053 + 0.618159i$ of defining polynomial
Character	$\chi$	$=$	469.412
Dual form			469.1.bl.a.181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.601300 + 0.573338i) q^{2} +(-0.0147371 - 0.309371i) q^{4} +(0.235759 - 0.971812i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})$	\(q+(0.601300 + 0.573338i) q^{2} +(-0.0147371 - 0.309371i) q^{4} +(0.235759 - 0.971812i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +(0.839614 + 1.17907i) q^{11} +(0.698939 - 0.449181i) q^{14} +(0.591660 - 0.0564967i) q^{16} +(-0.738471 - 0.380708i) q^{18} +(-0.171148 + 1.19036i) q^{22} +(-1.54370 + 1.21398i) q^{23} +(-0.654861 - 0.755750i) q^{25} +(-0.304124 - 0.0586152i) q^{28} +(-0.235759 + 0.408346i) q^{29} +(-0.467192 - 0.367404i) q^{32} +(0.101300 + 0.292687i) q^{36} +(0.995472 + 1.72421i) q^{37} +(-1.49547 - 0.961081i) q^{43} +(0.352397 - 0.277128i) q^{44} +(-1.62424 - 0.155096i) q^{46} +(-0.888835 - 0.458227i) q^{49} +(0.0395325 - 0.829889i) q^{50} +(1.56199 - 1.00383i) q^{53} +(-0.631192 - 0.886386i) q^{56} +(-0.375883 + 0.110369i) q^{58} +(0.0475819 + 0.998867i) q^{63} +(-0.154861 - 1.07708i) q^{64} +(-0.995472 + 0.0950560i) q^{67} +(0.0552004 + 1.15880i) q^{71} +(-0.452036 + 0.989821i) q^{72} +(-0.389977 + 1.60751i) q^{74} +(1.34378 - 0.537970i) q^{77} +(-0.279486 + 0.0538665i) q^{79} +(0.841254 - 0.540641i) q^{81} +(-0.348202 - 1.43531i) q^{86} +(1.56794 + 0.149720i) q^{88} +(0.398318 + 0.459684i) q^{92} +(-0.271738 - 0.785135i) q^{98} +(-1.13779 - 0.894765i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) $ 20 * q + 2 * q^2 + 3 * q^4 + q^7 - 8 * q^8 - 2 * q^9	$ 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9} - q^{11} - 4 q^{14} + 5 q^{16} + 2 q^{18} - 7 q^{22} - q^{23} - 2 q^{25} - 8 q^{28} - q^{29} + 6 q^{32} - 8 q^{36} - q^{37} - 9 q^{43} - 3 q^{44} - 13 q^{46} + q^{49} + 2 q^{50} + 2 q^{53} - 7 q^{56} + 4 q^{58} + q^{63} + 8 q^{64} + q^{67} - q^{71} + 14 q^{72} - 2 q^{74} - q^{77} + 2 q^{79} - 2 q^{81} - 2 q^{86} - 4 q^{88} - 5 q^{92} + 2 q^{98} - q^{99}+O(q^{100}) $ 20 * q + 2 * q^2 + 3 * q^4 + q^7 - 8 * q^8 - 2 * q^9 - q^11 - 4 * q^14 + 5 * q^16 + 2 * q^18 - 7 * q^22 - q^23 - 2 * q^25 - 8 * q^28 - q^29 + 6 * q^32 - 8 * q^36 - q^37 - 9 * q^43 - 3 * q^44 - 13 * q^46 + q^49 + 2 * q^50 + 2 * q^53 - 7 * q^56 + 4 * q^58 + q^63 + 8 * q^64 + q^67 - q^71 + 14 * q^72 - 2 * q^74 - q^77 + 2 * q^79 - 2 * q^81 - 2 * q^86 - 4 * q^88 - 5 * q^92 + 2 * q^98 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/469\mathbb{Z}\right)^\times$.

$n$	$269$	$337$
$\chi(n)$	$-1$	$e\left(\frac{8}{33}\right)$

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 $

$2$	0.601300	+	0.573338i	0.601300	+	0.573338i	0.928368	−	0.371662i	$-0.121212\pi$
$2$	0.601300	+	0.573338i	0.601300	+	0.573338i	−0.327068	+	0.945001i	$0.606061\pi$
$3$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$3$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$4$	−0.0147371	−	0.309371i	−0.0147371	−	0.309371i
$5$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$5$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$6$		0			0
$7$	0.235759	−	0.971812i	0.235759	−	0.971812i
$7$	0.235759	−	0.971812i	0.235759	−	0.971812i
$8$	0.712591	−	0.822373i	0.712591	−	0.822373i
$9$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$10$		0			0
$11$	0.839614	+	1.17907i	0.839614	+	1.17907i	0.981929	+	0.189251i	$0.0606061\pi$
$11$	0.839614	+	1.17907i	0.839614	+	1.17907i	−0.142315	+	0.989821i	$0.545455\pi$
$12$		0			0
$13$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$13$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$14$	0.698939	−	0.449181i	0.698939	−	0.449181i
$15$		0			0
$16$	0.591660	−	0.0564967i	0.591660	−	0.0564967i
$17$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$17$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$18$	−0.738471	−	0.380708i	−0.738471	−	0.380708i
$19$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$19$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$20$		0			0
$21$		0			0
$22$	−0.171148	+	1.19036i	−0.171148	+	1.19036i
$23$	−1.54370	+	1.21398i	−1.54370	+	1.21398i	−0.654861	+	0.755750i	$0.727273\pi$
$23$	−1.54370	+	1.21398i	−1.54370	+	1.21398i	−0.888835	+	0.458227i	$0.848485\pi$
$24$		0			0
$25$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$26$		0			0
$27$		0			0
$28$	−0.304124	−	0.0586152i	−0.304124	−	0.0586152i
$29$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$29$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$30$		0			0
$31$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$31$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$32$	−0.467192	−	0.367404i	−0.467192	−	0.367404i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.101300	+	0.292687i	0.101300	+	0.292687i
$37$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$37$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$41$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$42$		0			0
$43$	−1.49547	−	0.961081i	−1.49547	−	0.961081i	−0.995472	−	0.0950560i	$-0.969697\pi$
$43$	−1.49547	−	0.961081i	−1.49547	−	0.961081i	−0.500000	−	0.866025i	$-0.666667\pi$
$44$	0.352397	−	0.277128i	0.352397	−	0.277128i
$45$		0			0
$46$	−1.62424	−	0.155096i	−1.62424	−	0.155096i
$47$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$47$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$48$		0			0
$49$	−0.888835	−	0.458227i	−0.888835	−	0.458227i
$50$	0.0395325	−	0.829889i	0.0395325	−	0.829889i
$51$		0			0
$52$		0			0
$53$	1.56199	−	1.00383i	1.56199	−	1.00383i	0.580057	−	0.814576i	$-0.303030\pi$
$53$	1.56199	−	1.00383i	1.56199	−	1.00383i	0.981929	−	0.189251i	$-0.0606061\pi$
$54$		0			0
$55$		0			0
$56$	−0.631192	−	0.886386i	−0.631192	−	0.886386i
$57$		0			0
$58$	−0.375883	+	0.110369i	−0.375883	+	0.110369i
$59$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$59$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$60$		0			0
$61$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$61$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$62$		0			0
$63$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$64$	−0.154861	−	1.07708i	−0.154861	−	1.07708i
$65$		0			0
$66$		0			0
$67$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$67$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.0552004	+	1.15880i	0.0552004	+	1.15880i	0.841254	+	0.540641i	$0.181818\pi$
$71$	0.0552004	+	1.15880i	0.0552004	+	1.15880i	−0.786053	+	0.618159i	$0.787879\pi$
$72$	−0.452036	+	0.989821i	−0.452036	+	0.989821i
$73$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$73$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$74$	−0.389977	+	1.60751i	−0.389977	+	1.60751i
$75$		0			0
$76$		0			0
$77$	1.34378	−	0.537970i	1.34378	−	0.537970i
$78$		0			0
$79$	−0.279486	+	0.0538665i	−0.279486	+	0.0538665i	−0.327068	−	0.945001i	$-0.606061\pi$
$79$	−0.279486	+	0.0538665i	−0.279486	+	0.0538665i	0.0475819	+	0.998867i	$0.484848\pi$
$80$		0			0
$81$	0.841254	−	0.540641i	0.841254	−	0.540641i
$82$		0			0
$83$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$83$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$84$		0			0
$85$		0			0
$86$	−0.348202	−	1.43531i	−0.348202	−	1.43531i
$87$		0			0
$88$	1.56794	+	0.149720i	1.56794	+	0.149720i
$89$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$89$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$90$		0			0
$91$		0			0
$92$	0.398318	+	0.459684i	0.398318	+	0.459684i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	−0.271738	−	0.785135i	−0.271738	−	0.785135i
$99$	−1.13779	−	0.894765i	−1.13779	−	0.894765i
$100$	−0.224156	+	0.213732i	−0.224156	+	0.213732i
$101$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$101$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$102$		0			0
$103$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$103$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$104$		0			0
$105$		0			0
$106$	1.51475	+	0.291945i	1.51475	+	0.291945i
$107$	−0.738471	−	1.61703i	−0.738471	−	1.61703i	−0.786053	−	0.618159i	$-0.787879\pi$
$107$	−0.738471	−	1.61703i	−0.738471	−	1.61703i	0.0475819	−	0.998867i	$-0.484848\pi$
$108$		0			0
$109$	0.428368	+	0.494363i	0.428368	+	0.494363i	0.928368	−	0.371662i	$-0.121212\pi$
$109$	0.428368	+	0.494363i	0.428368	+	0.494363i	−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	0.0845850	−	0.588302i	0.0845850	−	0.588302i
$113$	0.995472	+	0.0950560i	0.995472	+	0.0950560i	0.580057	−	0.814576i	$-0.303030\pi$
$113$	0.995472	+	0.0950560i	0.995472	+	0.0950560i	0.415415	+	0.909632i	$0.363636\pi$
$114$		0			0
$115$		0			0
$116$	0.129805	+	0.0669190i	0.129805	+	0.0669190i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.358193	+	1.03493i	−0.358193	+	1.03493i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.544078	+	0.627899i	−0.544078	+	0.627899i
$127$	0.396666	−	1.63508i	0.396666	−	1.63508i	−0.327068	−	0.945001i	$-0.606061\pi$
$127$	0.396666	−	1.63508i	0.396666	−	1.63508i	0.723734	−	0.690079i	$-0.242424\pi$
$128$	0.179656	−	0.252292i	0.179656	−	0.252292i
$129$		0			0
$130$		0			0
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$		0			0
$134$	−0.653077	−	0.513585i	−0.653077	−	0.513585i
$135$		0			0
$136$		0			0
$137$	0.223734	+	1.55610i	0.223734	+	1.55610i	0.723734	+	0.690079i	$0.242424\pi$
$137$	0.223734	+	1.55610i	0.223734	+	1.55610i	−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$139$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$140$		0			0
$141$		0			0
$142$	−0.631192	+	0.728435i	−0.631192	+	0.728435i
$143$		0			0
$144$	−0.551777	+	0.220898i	−0.551777	+	0.220898i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.518749	−	0.333380i	0.518749	−	0.333380i
$149$	0.959493	+	0.281733i	0.959493	+	0.281733i	0.723734	−	0.690079i	$-0.242424\pi$
$149$	0.959493	+	0.281733i	0.959493	+	0.281733i	0.235759	+	0.971812i	$0.424242\pi$
$150$		0			0
$151$	−0.0913090	+	1.91681i	−0.0913090	+	1.91681i	0.235759	+	0.971812i	$0.424242\pi$
$151$	−0.0913090	+	1.91681i	−0.0913090	+	1.91681i	−0.327068	+	0.945001i	$0.606061\pi$
$152$		0			0
$153$		0			0
$154$	1.11646	+	0.446961i	1.11646	+	0.446961i
$155$		0			0
$156$		0			0
$157$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$157$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$158$	−0.198939	−	0.127850i	−0.198939	−	0.127850i
$159$		0			0
$160$		0			0
$161$	0.815816	+	1.78639i	0.815816	+	1.78639i
$162$	0.815816	+	0.157236i	0.815816	+	0.157236i
$163$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$163$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$167$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$168$		0			0
$169$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$170$		0			0
$171$		0			0
$172$	−0.275291	+	0.476819i	−0.275291	+	0.476819i
$173$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$173$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$174$		0			0
$175$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$176$	0.563380	+	0.650175i	0.563380	+	0.650175i
$177$		0			0
$178$		0			0
$179$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i	−0.995472	−	0.0950560i	$-0.969697\pi$
$179$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i	0.981929	+	0.189251i	$0.0606061\pi$
$180$		0			0
$181$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$181$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$182$		0			0
$183$		0			0
$184$	−0.101682	+	2.13456i	−0.101682	+	2.13456i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.07701	−	0.431171i	1.07701	−	0.431171i	0.235759	−	0.971812i	$-0.424242\pi$
$191$	1.07701	−	0.431171i	1.07701	−	0.431171i	0.841254	+	0.540641i	$0.181818\pi$
$192$		0			0
$193$	1.02951	−	1.18812i	1.02951	−	1.18812i	0.0475819	−	0.998867i	$-0.484848\pi$
$193$	1.02951	−	1.18812i	1.02951	−	1.18812i	0.981929	−	0.189251i	$-0.0606061\pi$
$194$		0			0
$195$		0			0
$196$	−0.128663	+	0.281733i	−0.128663	+	0.281733i
$197$	−0.0947329	−	1.98869i	−0.0947329	−	1.98869i	−0.142315	−	0.989821i	$-0.545455\pi$
$197$	−0.0947329	−	1.98869i	−0.0947329	−	1.98869i	0.0475819	−	0.998867i	$-0.484848\pi$
$198$	−0.171148	−	1.19036i	−0.171148	−	1.19036i
$199$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$199$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$200$	−1.08816			−1.08816
$201$		0			0
$202$		0			0
$203$	0.341254	+	0.325385i	0.341254	+	0.325385i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.13915	−	1.59971i	1.13915	−	1.59971i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.928368	+	0.371662i	−0.928368	+	0.371662i	−0.786053	−	0.618159i	$-0.787879\pi$
$211$	−0.928368	+	0.371662i	−0.928368	+	0.371662i	−0.142315	+	0.989821i	$0.545455\pi$
$212$	−0.333574	−	0.468439i	−0.333574	−	0.468439i
$213$		0			0
$214$	0.483061	−	1.39571i	0.483061	−	1.39571i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.0258596	+	0.542860i	−0.0258596	+	0.542860i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$223$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$224$	−0.467192	+	0.367404i	−0.467192	+	0.367404i
$225$	0.841254	+	0.540641i	0.841254	+	0.540641i
$226$	0.544078	+	0.627899i	0.544078	+	0.627899i
$227$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$227$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$228$		0			0
$229$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$229$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$230$		0			0
$231$		0			0
$232$	0.167814	+	0.484866i	0.167814	+	0.484866i
$233$	−0.370638	−	0.291473i	−0.370638	−	0.291473i	0.415415	−	0.909632i	$-0.363636\pi$
$233$	−0.370638	−	0.291473i	−0.370638	−	0.291473i	−0.786053	+	0.618159i	$0.787879\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$239$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$240$		0			0
$241$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$241$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$242$	−0.808747	+	0.416938i	−0.808747	+	0.416938i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$251$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$252$	0.308319	−	0.0294409i	0.308319	−	0.0294409i
$253$	−2.72747	−	0.800859i	−2.72747	−	0.800859i
$254$	1.17597	−	0.755750i	1.17597	−	0.755750i
$255$		0			0
$256$	−0.815816	+	0.157236i	−0.815816	+	0.157236i
$257$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$257$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$258$		0			0
$259$	1.91030	−	0.560914i	1.91030	−	0.560914i
$260$		0			0
$261$	0.111165	−	0.458227i	0.111165	−	0.458227i
$262$		0			0
$263$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	−0.888835	−	0.458227i	$-0.848485\pi$
$263$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	0.928368	+	0.371662i	$0.121212\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.0440780	+	0.306569i	0.0440780	+	0.306569i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$271$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$272$		0			0
$273$		0			0
$274$	−0.757643	+	1.06396i	−0.757643	+	1.06396i
$275$	0.341254	−	1.40667i	0.341254	−	1.40667i
$276$		0			0
$277$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$277$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.428368	−	1.23769i	0.428368	−	1.23769i	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	0.428368	−	1.23769i	0.428368	−	1.23769i	0.928368	−	0.371662i	$-0.121212\pi$
$282$		0			0
$283$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$283$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$284$	0.357685	−	0.0341548i	0.357685	−	0.0341548i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.551777	+	0.220898i	0.551777	+	0.220898i
$289$	−0.995472	−	0.0950560i	−0.995472	−	0.0950560i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$293$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$294$		0			0
$295$		0			0
$296$	2.12731	+	0.410005i	2.12731	+	0.410005i
$297$		0			0
$298$	0.415415	+	0.719520i	0.415415	+	0.719520i
$299$		0			0
$300$		0			0
$301$	−1.28656	+	1.22673i	−1.28656	+	1.22673i
$302$	−1.15389	+	1.10023i	−1.15389	+	1.10023i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$307$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$308$	−0.186236	−	0.407799i	−0.186236	−	0.407799i
$309$		0			0
$310$		0			0
$311$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$311$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$312$		0			0
$313$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$313$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$314$		0			0
$315$		0			0
$316$	0.0207835	+	0.0856709i	0.0207835	+	0.0856709i
$317$	−1.49547	−	0.770969i	−1.49547	−	0.770969i	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.49547	−	0.770969i	−1.49547	−	0.770969i	−0.995472	+	0.0950560i	$0.969697\pi$
$318$		0			0
$319$	−0.679417	+	0.0648764i	−0.679417	+	0.0648764i
$320$		0			0
$321$		0			0
$322$	−0.533654	+	1.54189i	−0.533654	+	1.54189i
$323$		0			0
$324$	−0.179656	−	0.252292i	−0.179656	−	0.252292i
$325$		0			0
$326$	1.25324	−	0.367986i	1.25324	−	0.367986i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.0748038	−	1.57033i	−0.0748038	−	1.57033i	−0.654861	−	0.755750i	$-0.727273\pi$
$331$	−0.0748038	−	1.57033i	−0.0748038	−	1.57033i	0.580057	−	0.814576i	$-0.303030\pi$
$332$		0			0
$333$	−1.44091	−	1.37391i	−1.44091	−	1.37391i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.0688733	+	0.0656706i	0.0688733	+	0.0656706i	0.723734	−	0.690079i	$-0.242424\pi$
$337$	0.0688733	+	0.0656706i	0.0688733	+	0.0656706i	−0.654861	+	0.755750i	$0.727273\pi$
$338$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$344$	−1.85603	+	0.544979i	−1.85603	+	0.544979i
$345$		0			0
$346$		0			0
$347$	−0.642315	+	0.123796i	−0.642315	+	0.123796i	−0.500000	−	0.866025i	$-0.666667\pi$
$347$	−0.642315	+	0.123796i	−0.642315	+	0.123796i	−0.142315	+	0.989821i	$0.545455\pi$
$348$		0			0
$349$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$349$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$350$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$351$		0			0
$352$	0.0409349	−	0.859330i	0.0409349	−	0.859330i
$353$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$353$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.0621493	+	0.0488747i	−0.0621493	+	0.0488747i
$359$	−0.550294	−	0.353653i	−0.550294	−	0.353653i	0.235759	−	0.971812i	$-0.424242\pi$
$359$	−0.550294	−	0.353653i	−0.550294	−	0.353653i	−0.786053	+	0.618159i	$0.787879\pi$
$360$		0			0
$361$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$367$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$368$	−0.844757	+	0.805475i	−0.844757	+	0.805475i
$369$		0			0
$370$		0			0
$371$	−0.607279	−	1.75462i	−0.607279	−	1.75462i
$372$		0			0
$373$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$373$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.02951	−	0.809616i	1.02951	−	0.809616i	0.0475819	−	0.998867i	$-0.484848\pi$
$379$	1.02951	−	0.809616i	1.02951	−	0.809616i	0.981929	+	0.189251i	$0.0606061\pi$
$380$		0			0
$381$		0			0
$382$	0.894814	+	0.358230i	0.894814	+	0.358230i
$383$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$383$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$384$		0			0
$385$		0			0
$386$	1.30024	−	0.124158i	1.30024	−	0.124158i
$387$	1.70566	+	0.500828i	1.70566	+	0.500828i
$388$		0			0
$389$	−0.607279	+	1.75462i	−0.607279	+	1.75462i	0.0475819	+	0.998867i	$0.484848\pi$
$389$	−0.607279	+	1.75462i	−0.607279	+	1.75462i	−0.654861	+	0.755750i	$0.727273\pi$
$390$		0			0
$391$		0			0
$392$	−1.01021	+	0.404427i	−1.01021	+	0.404427i
$393$		0			0
$394$	1.08323	−	1.25011i	1.08323	−	1.25011i
$395$		0			0
$396$	−0.260046	+	0.365184i	−0.260046	+	0.365184i
$397$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$397$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$398$		0			0
$399$		0			0
$400$	−0.430152	−	0.410149i	−0.430152	−	0.410149i
$401$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$401$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.0186403	+	0.391307i	0.0186403	+	0.391307i
$407$	−1.19715	+	2.62140i	−1.19715	+	2.62140i
$408$		0			0
$409$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$409$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.60215	−	0.308788i	1.60215	−	0.308788i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$419$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$420$		0			0
$421$	−0.154218	−	0.635697i	−0.154218	−	0.635697i	−0.995472	−	0.0950560i	$-0.969697\pi$
$421$	−0.154218	−	0.635697i	−0.154218	−	0.635697i	0.841254	−	0.540641i	$-0.181818\pi$
$422$	−0.771316	−	0.308788i	−0.771316	−	0.308788i
$423$		0			0
$424$	0.287535	−	1.99985i	0.287535	−	1.99985i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.489378	+	0.252292i	−0.489378	+	0.252292i
$429$		0			0
$430$		0			0
$431$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$431$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$432$		0			0
$433$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$433$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$434$		0			0
$435$		0			0
$436$	0.146628	−	0.139810i	0.146628	−	0.139810i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	0.981929	+	0.189251i	0.981929	+	0.189251i
$442$		0			0
$443$	−1.49547	+	0.770969i	−1.49547	+	0.770969i	−0.995472	−	0.0950560i	$-0.969697\pi$
$443$	−1.49547	+	0.770969i	−1.49547	+	0.770969i	−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.08323	−	0.103436i	−1.08323	−	0.103436i
$449$	1.56199	+	0.625325i	1.56199	+	0.625325i	0.981929	−	0.189251i	$-0.0606061\pi$
$449$	1.56199	+	0.625325i	1.56199	+	0.625325i	0.580057	+	0.814576i	$0.303030\pi$
$450$	0.195876	+	0.807410i	0.195876	+	0.807410i
$451$		0			0
$452$	0.0147371	−	0.309371i	0.0147371	−	0.309371i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.13915	−	0.219553i	1.13915	−	0.219553i	0.415415	−	0.909632i	$-0.363636\pi$
$457$	1.13915	−	0.219553i	1.13915	−	0.219553i	0.723734	+	0.690079i	$0.242424\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$461$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$	−0.379436	+	0.532843i	−0.379436	+	0.532843i	−0.959493	−	0.281733i	$-0.909091\pi$
$463$	−0.379436	+	0.532843i	−0.379436	+	0.532843i	0.580057	+	0.814576i	$0.303030\pi$
$464$	−0.116419	+	0.254922i	−0.116419	+	0.254922i
$465$		0			0
$466$	−0.0557520	−	0.387764i	−0.0557520	−	0.387764i
$467$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$467$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$468$		0			0
$469$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.122434	−	2.57021i	−0.122434	−	2.57021i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.21590	+	1.40323i	−1.21590	+	1.40323i
$478$	−1.56554	+	0.459684i	−1.56554	+	0.459684i
$479$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$479$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.325456	+	0.0955625i	0.325456	+	0.0955625i
$485$		0			0
$486$		0			0
$487$	1.76962	+	0.912303i	1.76962	+	0.912303i	0.928368	+	0.371662i	$0.121212\pi$
$487$	1.76962	+	0.912303i	1.76962	+	0.912303i	0.841254	+	0.540641i	$0.181818\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$491$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.13915	+	0.219553i	1.13915	+	0.219553i
$498$		0			0
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$503$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$504$	0.855348	+	0.672653i	0.855348	+	0.672653i
$505$		0			0
$506$	−1.18087	−	2.04532i	−1.18087	−	2.04532i
$507$		0			0
$508$	−0.511691	−	0.0986204i	−0.511691	−	0.0986204i
$509$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$509$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$		0			0
$512$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.47025	+	0.757969i	1.47025	+	0.757969i
$519$		0			0
$520$		0			0
$521$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$521$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$522$	0.329562	−	0.211797i	0.329562	−	0.211797i
$523$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$523$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$524$		0			0
$525$		0			0
$526$	0.0734014	−	0.0293855i	0.0734014	−	0.0293855i
$527$		0			0
$528$		0			0
$529$	0.673501	−	2.77621i	0.673501	−	2.77621i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.631192	+	0.886386i	−0.631192	+	0.886386i
$537$		0			0
$538$		0			0
$539$	−0.205996	−	1.43273i	−0.205996	−	1.43273i
$540$		0			0
$541$	0.195876	−	0.428908i	0.195876	−	0.428908i	−0.786053	−	0.618159i	$-0.787879\pi$
$541$	0.195876	−	0.428908i	0.195876	−	0.428908i	0.981929	+	0.189251i	$0.0606061\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.13915	+	1.59971i	1.13915	+	1.59971i	0.723734	+	0.690079i	$0.242424\pi$
$547$	1.13915	+	1.59971i	1.13915	+	1.59971i	0.415415	+	0.909632i	$0.363636\pi$
$548$	0.478116	−	0.0921493i	0.478116	−	0.0921493i
$549$		0			0
$550$	1.01169	−	0.650175i	1.01169	−	0.650175i
$551$		0			0
$552$		0			0
$553$	−0.0135432	+	0.284307i	−0.0135432	+	0.284307i
$554$	1.41712	+	0.730574i	1.41712	+	0.730574i
$555$		0			0
$556$		0			0
$557$	0.283341	+	0.0270558i	0.283341	+	0.0270558i	0.235759	−	0.971812i	$-0.424242\pi$
$557$	0.283341	+	0.0270558i	0.283341	+	0.0270558i	0.0475819	+	0.998867i	$0.484848\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.967192	−	0.498622i	0.967192	−	0.498622i
$563$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$563$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$568$	0.992301	+	0.780354i	0.992301	+	0.780354i
$569$	0.601300	−	0.573338i	0.601300	−	0.573338i	−0.327068	−	0.945001i	$-0.606061\pi$
$569$	0.601300	−	0.573338i	0.601300	−	0.573338i	0.928368	+	0.371662i	$0.121212\pi$
$570$		0			0
$571$	−1.13779	−	0.894765i	−1.13779	−	0.894765i	−0.142315	−	0.989821i	$-0.545455\pi$
$571$	−1.13779	−	0.894765i	−1.13779	−	0.894765i	−0.995472	+	0.0950560i	$0.969697\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.92837	+	0.371662i	1.92837	+	0.371662i
$576$	0.452036	+	0.989821i	0.452036	+	0.989821i
$577$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$577$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$578$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.49505	+	0.998867i	2.49505	+	0.998867i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$587$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.686393	+	0.963904i	0.686393	+	0.963904i
$593$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$593$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$594$		0			0
$595$		0			0
$596$	0.0730196	−	0.300991i	0.0730196	−	0.300991i
$597$		0			0
$598$		0			0
$599$	0.0883470	+	1.85463i	0.0883470	+	1.85463i	0.415415	+	0.909632i	$0.363636\pi$
$599$	0.0883470	+	1.85463i	0.0883470	+	1.85463i	−0.327068	+	0.945001i	$0.606061\pi$
$600$		0			0
$601$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$601$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$602$	−1.47694			−1.47694
$603$	0.928368	−	0.371662i	0.928368	−	0.371662i
$604$	0.594351			0.594351
$605$		0			0
$606$		0			0
$607$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$607$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.07701	−	0.431171i	1.07701	−	0.431171i	0.235759	−	0.971812i	$-0.424242\pi$
$613$	1.07701	−	0.431171i	1.07701	−	0.431171i	0.841254	+	0.540641i	$0.181818\pi$
$614$		0			0
$615$		0			0
$616$	0.515155	−	1.48844i	0.515155	−	1.48844i
$617$	−1.49547	+	0.961081i	−1.49547	+	0.961081i	−0.500000	+	0.866025i	$0.666667\pi$
$617$	−1.49547	+	0.961081i	−1.49547	+	0.961081i	−0.995472	+	0.0950560i	$0.969697\pi$
$618$		0			0
$619$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$619$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.462997	+	0.0892353i	0.462997	+	0.0892353i	0.415415	−	0.909632i	$-0.363636\pi$
$631$	0.462997	+	0.0892353i	0.462997	+	0.0892353i	0.0475819	+	0.998867i	$0.484848\pi$
$632$	−0.154861	+	0.268227i	−0.154861	+	0.268227i
$633$		0			0
$634$	−0.457201	−	1.32100i	−0.457201	−	1.32100i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.445729	−	0.350525i	−0.445729	−	0.350525i
$639$	−0.379436	−	1.09631i	−0.379436	−	1.09631i
$640$		0			0
$641$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$641$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$642$		0			0
$643$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$643$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$644$	0.540633	−	0.278716i	0.540633	−	0.278716i
$645$		0			0
$646$		0			0
$647$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$647$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$648$	0.154861	−	1.07708i	0.154861	−	1.07708i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.432787	−	0.223117i	−0.432787	−	0.223117i
$653$	−0.0135432	+	0.284307i	−0.0135432	+	0.284307i	0.981929	+	0.189251i	$0.0606061\pi$
$653$	−0.0135432	+	0.284307i	−0.0135432	+	0.284307i	−0.995472	+	0.0950560i	$0.969697\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.379436	−	0.532843i	−0.379436	−	0.532843i	0.580057	−	0.814576i	$-0.303030\pi$
$659$	−0.379436	−	0.532843i	−0.379436	−	0.532843i	−0.959493	+	0.281733i	$0.909091\pi$
$660$		0			0
$661$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$661$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$662$	0.855348	−	0.987125i	0.855348	−	0.987125i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.0787070	−	1.65226i	−0.0787070	−	1.65226i
$667$	−0.131783	−	0.916569i	−0.131783	−	0.916569i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.283341	+	1.97068i	0.283341	+	1.97068i	0.235759	+	0.971812i	$0.424242\pi$
$673$	0.283341	+	1.97068i	0.283341	+	1.97068i	0.0475819	+	0.998867i	$0.484848\pi$
$674$	0.00376206	+	0.0789754i	0.00376206	+	0.0789754i
$675$		0			0
$676$	−0.179656	+	0.252292i	−0.179656	+	0.252292i
$677$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$677$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.379436	+	1.09631i	−0.379436	+	1.09631i	0.580057	+	0.814576i	$0.303030\pi$
$683$	−0.379436	+	1.09631i	−0.379436	+	1.09631i	−0.959493	+	0.281733i	$0.909091\pi$
$684$		0			0
$685$		0			0
$686$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i
$687$		0			0
$688$	−0.939109	−	0.484144i	−0.939109	−	0.484144i
$689$		0			0
$690$		0			0
$691$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$691$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$692$		0			0
$693$	−1.13779	+	0.894765i	−1.13779	+	0.894765i
$694$	−0.457201	−	0.293825i	−0.457201	−	0.293825i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.154861	+	0.268227i	0.154861	+	0.268227i
$701$	0.428368	+	1.23769i	0.428368	+	1.23769i	0.928368	+	0.371662i	$0.121212\pi$
$701$	0.428368	+	1.23769i	0.428368	+	1.23769i	−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$	1.13993	−	1.08692i	1.13993	−	1.08692i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.815816	+	0.157236i	0.815816	+	0.157236i	0.580057	−	0.814576i	$-0.303030\pi$
$709$	0.815816	+	0.157236i	0.815816	+	0.157236i	0.235759	+	0.971812i	$0.424242\pi$
$710$		0			0
$711$	0.252989	−	0.130425i	0.252989	−	0.130425i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.0293408	+	0.00280171i	0.0293408	+	0.00280171i
$717$		0			0
$718$	−0.128129	−	0.528156i	−0.128129	−	0.528156i
$719$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$719$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$720$		0			0
$721$		0			0
$722$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$723$		0			0
$724$		0			0
$725$	0.462997	−	0.0892353i	0.462997	−	0.0892353i
$726$		0			0
$727$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$727$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$728$		0			0
$729$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$733$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$734$		0			0
$735$		0			0
$736$	1.16722			1.16722
$737$	−0.947890	−	1.09392i	−0.947890	−	1.09392i
$738$		0			0
$739$	1.34378	+	1.28129i	1.34378	+	1.28129i	0.928368	+	0.371662i	$0.121212\pi$
$739$	1.34378	+	1.28129i	1.34378	+	1.28129i	0.415415	+	0.909632i	$0.363636\pi$
$740$		0			0
$741$		0			0
$742$	0.640832	−	1.40323i	0.640832	−	1.40323i
$743$	−0.165101	+	0.231852i	−0.165101	+	0.231852i	−0.888835	−	0.458227i	$-0.848485\pi$
$743$	−0.165101	+	0.231852i	−0.165101	+	0.231852i	0.723734	+	0.690079i	$0.242424\pi$
$744$		0			0
$745$		0			0
$746$	−0.0758623	+	0.0222752i	−0.0758623	+	0.0222752i
$747$		0			0
$748$		0			0
$749$	−1.74555	+	0.336426i	−1.74555	+	0.336426i
$750$		0			0
$751$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$751$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.21590	−	0.486774i	−1.21590	−	0.486774i	−0.327068	−	0.945001i	$-0.606061\pi$
$757$	−1.21590	−	0.486774i	−1.21590	−	0.486774i	−0.888835	+	0.458227i	$0.848485\pi$
$758$	1.08323	+	0.103436i	1.08323	+	0.103436i
$759$		0			0
$760$		0			0
$761$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$761$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$	0.581419	−	0.299742i	0.581419	−	0.299742i
$764$	−0.149264	−	0.326842i	−0.149264	−	0.326842i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$769$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$770$		0			0
$771$		0			0
$772$	−0.382741	−	0.300991i	−0.382741	−	0.300991i
$773$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$773$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$774$	0.738471	+	1.27907i	0.738471	+	1.27907i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.37115	+	0.706875i	−1.37115	+	0.706875i
$779$		0			0
$780$		0			0
$781$	−1.31996	+	1.03803i	−1.31996	+	1.03803i
$782$		0			0
$783$		0			0
$784$	−0.551777	−	0.220898i	−0.551777	−	0.220898i
$785$		0			0
$786$		0			0
$787$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$787$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$788$	−0.613846	+	0.0586152i	−0.613846	+	0.0586152i
$789$		0			0
$790$		0			0
$791$	0.327068	−	0.945001i	0.327068	−	0.945001i
$792$	−1.54661	+	0.298084i	−1.54661	+	0.298084i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$797$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$798$		0			0
$799$		0			0
$800$	0.0282804	+	0.593678i	0.0282804	+	0.593678i
$801$		0			0
$802$	−0.171148	−	0.163189i	−0.171148	−	0.163189i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	0.0475819	+	0.998867i	$0.484848\pi$
$809$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	−0.786053	+	0.618159i	$0.787879\pi$
$810$		0			0
$811$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$811$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$812$	0.0956353	−	0.110369i	0.0956353	−	0.110369i
$813$		0			0
$814$	−2.22280	+	0.889875i	−2.22280	+	0.889875i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.0748038	+	1.57033i	−0.0748038	+	1.57033i	0.580057	+	0.814576i	$0.303030\pi$
$821$	−0.0748038	+	1.57033i	−0.0748038	+	1.57033i	−0.654861	+	0.755750i	$0.727273\pi$
$822$		0			0
$823$	−0.235759	−	0.971812i	−0.235759	−	0.971812i	−0.959493	−	0.281733i	$-0.909091\pi$
$823$	−0.235759	−	0.971812i	−0.235759	−	0.971812i	0.723734	−	0.690079i	$-0.242424\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.0748038	+	0.0588264i	−0.0748038	+	0.0588264i	−0.654861	−	0.755750i	$-0.727273\pi$
$827$	−0.0748038	+	0.0588264i	−0.0748038	+	0.0588264i	0.580057	+	0.814576i	$0.303030\pi$
$828$	−0.511691	−	0.328844i	−0.511691	−	0.328844i
$829$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$829$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$839$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$840$		0			0
$841$	0.388835	+	0.673483i	0.388835	+	0.673483i
$842$	0.271738	−	0.470664i	0.271738	−	0.470664i
$843$		0			0
$844$	0.128663	+	0.281733i	0.128663	+	0.281733i
$845$		0			0
$846$		0			0
$847$	0.921310	+	0.592090i	0.921310	+	0.592090i
$848$	0.867451	−	0.682171i	0.867451	−	0.682171i
$849$		0			0
$850$		0			0
$851$	−3.62985	−	1.45317i	−3.62985	−	1.45317i
$852$		0			0
$853$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$853$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$854$		0			0
$855$		0			0
$856$	−1.85603	−	0.544979i	−1.85603	−	0.544979i
$857$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$857$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$858$		0			0
$859$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$859$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$860$		0			0
$861$		0			0
$862$	−1.15389	+	0.338812i	−1.15389	+	0.338812i
$863$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.327068	+	0.945001i	$0.606061\pi$
$863$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.888835	+	0.458227i	$0.848485\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.298173	−	0.284307i	−0.298173	−	0.284307i
$870$		0			0
$871$		0			0
$872$	0.711802			0.711802
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.481929	−	0.676774i	0.481929	−	0.676774i	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	0.481929	−	0.676774i	0.481929	−	0.676774i	0.981929	+	0.189251i	$0.0606061\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$881$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$882$	0.481929	+	0.676774i	0.481929	+	0.676774i
$883$	1.82318	−	0.351390i	1.82318	−	0.351390i	0.841254	−	0.540641i	$-0.181818\pi$
$883$	1.82318	−	0.351390i	1.82318	−	0.351390i	0.981929	+	0.189251i	$0.0606061\pi$
$884$		0			0
$885$		0			0
$886$	−1.34125	−	0.393828i	−1.34125	−	0.393828i
$887$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$887$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$888$		0			0
$889$	−1.49547	−	0.770969i	−1.49547	−	0.770969i
$890$		0			0
$891$	1.34378	+	0.537970i	1.34378	+	0.537970i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.202824	−	0.234072i	−0.202824	−	0.234072i
$897$		0			0
$898$	0.580699	+	1.27155i	0.580699	+	1.27155i
$899$		0			0
$900$	0.154861	−	0.268227i	0.154861	−	0.268227i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.787535	−	0.750914i	0.787535	−	0.750914i
$905$		0			0
$906$		0			0
$907$	−0.473420	−	1.36786i	−0.473420	−	1.36786i	−0.888835	−	0.458227i	$-0.848485\pi$
$907$	−0.473420	−	1.36786i	−0.473420	−	1.36786i	0.415415	−	0.909632i	$-0.363636\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.653077	−	1.43004i	−0.653077	−	1.43004i	−0.888835	−	0.458227i	$-0.848485\pi$
$911$	−0.653077	−	1.43004i	−0.653077	−	1.43004i	0.235759	−	0.971812i	$-0.424242\pi$
$912$		0			0
$913$		0			0
$914$	0.810848	+	0.521101i	0.810848	+	0.521101i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i	0.981929	−	0.189251i	$-0.0606061\pi$
$919$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i	−0.959493	+	0.281733i	$0.909091\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.651174	−	1.88144i	0.651174	−	1.88144i
$926$	−0.533654	+	0.102853i	−0.533654	+	0.102853i
$927$		0			0
$928$	0.260173	−	0.104157i	0.260173	−	0.104157i
$929$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$929$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$930$		0			0
$931$		0			0
$932$	−0.0847111	+	0.118960i	−0.0847111	+	0.118960i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−0.653077	+	0.513585i	−0.653077	+	0.513585i
$939$		0			0
$940$		0			0
$941$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$941$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	1.39998	−	1.61566i	1.39998	−	1.61566i
$947$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$947$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.78153	−	0.523103i	−1.78153	−	0.523103i	−0.786053	−	0.618159i	$-0.787879\pi$
$953$	−1.78153	−	0.523103i	−1.78153	−	0.523103i	−0.995472	+	0.0950560i	$0.969697\pi$
$954$	−1.53565	+	0.146636i	−1.53565	+	0.146636i
$955$		0			0
$956$	0.540633	+	0.278716i	0.540633	+	0.278716i
$957$		0			0
$958$		0			0
$959$	1.56499	+	0.149438i	1.56499	+	0.149438i
$960$		0			0
$961$	−0.786053	+	0.618159i	−0.786053	+	0.618159i
$962$		0			0
$963$	1.16413	+	1.34347i	1.16413	+	1.34347i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$967$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$968$	0.595855	+	1.03205i	0.595855	+	1.03205i
$969$		0			0
$970$		0			0
$971$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$971$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$972$		0			0
$973$		0			0
$974$	0.541015	+	1.56316i	0.541015	+	1.56316i
$975$		0			0
$976$		0			0
$977$	−0.981929	−	0.189251i	−0.981929	−	0.189251i	−0.327068	−	0.945001i	$-0.606061\pi$
$977$	−0.981929	−	0.189251i	−0.981929	−	0.189251i	−0.654861	+	0.755750i	$0.727273\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.550294	−	0.353653i	−0.550294	−	0.353653i
$982$	0.185885	−	0.146182i	0.185885	−	0.146182i
$983$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$983$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.47528	−	0.331849i	3.47528	−	0.331849i
$990$		0			0
$991$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$991$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$992$		0			0
$993$		0			0
$994$	0.559092	+	0.785135i	0.559092	+	0.785135i
$995$		0			0
$996$		0			0
$997$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$997$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$998$	−0.195876	+	0.807410i	−0.195876	+	0.807410i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	469.1.bl.a.412.1	yes	20
7.2	even	3		3283.1.cd.a.2824.1		20
7.3	odd	6		3283.1.bw.a.1685.1		20
7.4	even	3		3283.1.bw.a.1685.1		20
7.5	odd	6		3283.1.cd.a.2824.1		20
7.6	odd	2	CM	469.1.bl.a.412.1	yes	20
67.47	even	33	inner	469.1.bl.a.181.1	✓	20
469.47	odd	66		3283.1.bw.a.717.1		20
469.114	even	33		3283.1.bw.a.717.1		20
469.181	odd	66	inner	469.1.bl.a.181.1	✓	20
469.248	odd	66		3283.1.cd.a.2861.1		20
469.382	even	33		3283.1.cd.a.2861.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
469.1.bl.a.181.1	✓	20	67.47	even	33	inner
469.1.bl.a.181.1	✓	20	469.181	odd	66	inner
469.1.bl.a.412.1	yes	20	1.1	even	1	trivial
469.1.bl.a.412.1	yes	20	7.6	odd	2	CM
3283.1.bw.a.717.1		20	469.47	odd	66
3283.1.bw.a.717.1		20	469.114	even	33
3283.1.bw.a.1685.1		20	7.3	odd	6
3283.1.bw.a.1685.1		20	7.4	even	3
3283.1.cd.a.2824.1		20	7.2	even	3
3283.1.cd.a.2824.1		20	7.5	odd	6
3283.1.cd.a.2861.1		20	469.248	odd	66
3283.1.cd.a.2861.1		20	469.382	even	33

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:	\( N \)	\(=\)	\( 469 = 7 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	469.bl (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.601300 + 0.573338i) q^{2} +(-0.0147371 - 0.309371i) q^{4} +(0.235759 - 0.971812i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})\)	\(q+(0.601300 + 0.573338i) q^{2} +(-0.0147371 - 0.309371i) q^{4} +(0.235759 - 0.971812i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +(0.839614 + 1.17907i) q^{11} +(0.698939 - 0.449181i) q^{14} +(0.591660 - 0.0564967i) q^{16} +(-0.738471 - 0.380708i) q^{18} +(-0.171148 + 1.19036i) q^{22} +(-1.54370 + 1.21398i) q^{23} +(-0.654861 - 0.755750i) q^{25} +(-0.304124 - 0.0586152i) q^{28} +(-0.235759 + 0.408346i) q^{29} +(-0.467192 - 0.367404i) q^{32} +(0.101300 + 0.292687i) q^{36} +(0.995472 + 1.72421i) q^{37} +(-1.49547 - 0.961081i) q^{43} +(0.352397 - 0.277128i) q^{44} +(-1.62424 - 0.155096i) q^{46} +(-0.888835 - 0.458227i) q^{49} +(0.0395325 - 0.829889i) q^{50} +(1.56199 - 1.00383i) q^{53} +(-0.631192 - 0.886386i) q^{56} +(-0.375883 + 0.110369i) q^{58} +(0.0475819 + 0.998867i) q^{63} +(-0.154861 - 1.07708i) q^{64} +(-0.995472 + 0.0950560i) q^{67} +(0.0552004 + 1.15880i) q^{71} +(-0.452036 + 0.989821i) q^{72} +(-0.389977 + 1.60751i) q^{74} +(1.34378 - 0.537970i) q^{77} +(-0.279486 + 0.0538665i) q^{79} +(0.841254 - 0.540641i) q^{81} +(-0.348202 - 1.43531i) q^{86} +(1.56794 + 0.149720i) q^{88} +(0.398318 + 0.459684i) q^{92} +(-0.271738 - 0.785135i) q^{98} +(-1.13779 - 0.894765i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) 20 * q + 2 * q^2 + 3 * q^4 + q^7 - 8 * q^8 - 2 * q^9	\( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9} - q^{11} - 4 q^{14} + 5 q^{16} + 2 q^{18} - 7 q^{22} - q^{23} - 2 q^{25} - 8 q^{28} - q^{29} + 6 q^{32} - 8 q^{36} - q^{37} - 9 q^{43} - 3 q^{44} - 13 q^{46} + q^{49} + 2 q^{50} + 2 q^{53} - 7 q^{56} + 4 q^{58} + q^{63} + 8 q^{64} + q^{67} - q^{71} + 14 q^{72} - 2 q^{74} - q^{77} + 2 q^{79} - 2 q^{81} - 2 q^{86} - 4 q^{88} - 5 q^{92} + 2 q^{98} - q^{99}+O(q^{100}) \) 20 * q + 2 * q^2 + 3 * q^4 + q^7 - 8 * q^8 - 2 * q^9 - q^11 - 4 * q^14 + 5 * q^16 + 2 * q^18 - 7 * q^22 - q^23 - 2 * q^25 - 8 * q^28 - q^29 + 6 * q^32 - 8 * q^36 - q^37 - 9 * q^43 - 3 * q^44 - 13 * q^46 + q^49 + 2 * q^50 + 2 * q^53 - 7 * q^56 + 4 * q^58 + q^63 + 8 * q^64 + q^67 - q^71 + 14 * q^72 - 2 * q^74 - q^77 + 2 * q^79 - 2 * q^81 - 2 * q^86 - 4 * q^88 - 5 * q^92 + 2 * q^98 - q^99

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