LMFDB - Embedded newform 469.1.bl.a.370.1

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			370.1
Root			$0.928368 + 0.371662i$ of defining polynomial
Character	$\chi$	$=$	469.370
Dual form			469.1.bl.a.90.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.195876 - 0.807410i) q^{2} +(0.275291 + 0.141923i) q^{4} +(0.723734 + 0.690079i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})$	\(q+(0.195876 - 0.807410i) q^{2} +(0.275291 + 0.141923i) q^{4} +(0.723734 + 0.690079i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +(-0.469383 + 0.0448206i) q^{11} +(0.698939 - 0.449181i) q^{14} +(-0.344757 - 0.484144i) q^{16} +(0.0395325 + 0.829889i) q^{18} +(-0.0557520 + 0.387764i) q^{22} +(-0.607279 - 0.243118i) q^{23} +(-0.654861 - 0.755750i) q^{25} +(0.101300 + 0.292687i) q^{28} +(-0.723734 - 1.25354i) q^{29} +(0.551777 - 0.220898i) q^{32} +(-0.304124 - 0.0586152i) q^{36} +(-0.580057 + 1.00469i) q^{37} +(0.0800569 + 0.0514495i) q^{43} +(-0.135578 - 0.0542773i) q^{44} +(-0.315247 + 0.442702i) q^{46} +(0.0475819 + 0.998867i) q^{49} +(-0.738471 + 0.380708i) q^{50} +(-1.32254 + 0.849945i) q^{53} +(1.08323 - 0.103436i) q^{56} +(-1.15389 + 0.338812i) q^{58} +(-0.888835 - 0.458227i) q^{63} +(-0.154861 - 1.07708i) q^{64} +(0.580057 + 0.814576i) q^{67} +(1.76962 + 0.912303i) q^{71} +(-0.452036 + 0.989821i) q^{72} +(0.697576 + 0.665138i) q^{74} +(-0.370638 - 0.291473i) q^{77} +(0.0930932 - 0.268975i) q^{79} +(0.841254 - 0.540641i) q^{81} +(0.0572220 - 0.0545611i) q^{86} +(-0.297618 + 0.417947i) q^{88} +(-0.132675 - 0.153115i) q^{92} +(0.815816 + 0.157236i) q^{98} +(0.437742 - 0.175245i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/469\mathbb{Z}\right)^\times$.

$n$	$269$	$337$
$\chi(n)$	$-1$	$e\left(\frac{19}{33}\right)$

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 $

$2$	0.195876	−	0.807410i	0.195876	−	0.807410i	−0.786053	−	0.618159i	$-0.787879\pi$
$2$	0.195876	−	0.807410i	0.195876	−	0.807410i	0.981929	−	0.189251i	$-0.0606061\pi$
$3$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$3$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$4$	0.275291	+	0.141923i	0.275291	+	0.141923i
$5$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$5$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$6$		0			0
$7$	0.723734	+	0.690079i	0.723734	+	0.690079i
$7$	0.723734	+	0.690079i	0.723734	+	0.690079i
$8$	0.712591	−	0.822373i	0.712591	−	0.822373i
$9$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$10$		0			0
$11$	−0.469383	+	0.0448206i	−0.469383	+	0.0448206i	−0.327068	−	0.945001i	$-0.606061\pi$
$11$	−0.469383	+	0.0448206i	−0.469383	+	0.0448206i	−0.142315	+	0.989821i	$0.545455\pi$
$12$		0			0
$13$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$13$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$14$	0.698939	−	0.449181i	0.698939	−	0.449181i
$15$		0			0
$16$	−0.344757	−	0.484144i	−0.344757	−	0.484144i
$17$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$17$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$18$	0.0395325	+	0.829889i	0.0395325	+	0.829889i
$19$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$19$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$20$		0			0
$21$		0			0
$22$	−0.0557520	+	0.387764i	−0.0557520	+	0.387764i
$23$	−0.607279	−	0.243118i	−0.607279	−	0.243118i	0.0475819	−	0.998867i	$-0.484848\pi$
$23$	−0.607279	−	0.243118i	−0.607279	−	0.243118i	−0.654861	+	0.755750i	$0.727273\pi$
$24$		0			0
$25$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$26$		0			0
$27$		0			0
$28$	0.101300	+	0.292687i	0.101300	+	0.292687i
$29$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$29$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$30$		0			0
$31$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$31$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$32$	0.551777	−	0.220898i	0.551777	−	0.220898i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.304124	−	0.0586152i	−0.304124	−	0.0586152i
$37$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$37$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$41$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$42$		0			0
$43$	0.0800569	+	0.0514495i	0.0800569	+	0.0514495i	0.580057	−	0.814576i	$-0.303030\pi$
$43$	0.0800569	+	0.0514495i	0.0800569	+	0.0514495i	−0.500000	+	0.866025i	$0.666667\pi$
$44$	−0.135578	−	0.0542773i	−0.135578	−	0.0542773i
$45$		0			0
$46$	−0.315247	+	0.442702i	−0.315247	+	0.442702i
$47$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$47$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$48$		0			0
$49$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$50$	−0.738471	+	0.380708i	−0.738471	+	0.380708i
$51$		0			0
$52$		0			0
$53$	−1.32254	+	0.849945i	−1.32254	+	0.849945i	−0.995472	−	0.0950560i	$-0.969697\pi$
$53$	−1.32254	+	0.849945i	−1.32254	+	0.849945i	−0.327068	+	0.945001i	$0.606061\pi$
$54$		0			0
$55$		0			0
$56$	1.08323	−	0.103436i	1.08323	−	0.103436i
$57$		0			0
$58$	−1.15389	+	0.338812i	−1.15389	+	0.338812i
$59$		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		−0.995472	−	0.0950560i	$-0.969697\pi$
$61$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$62$		0			0
$63$	−0.888835	−	0.458227i	−0.888835	−	0.458227i
$64$	−0.154861	−	1.07708i	−0.154861	−	1.07708i
$65$		0			0
$66$		0			0
$67$	0.580057	+	0.814576i	0.580057	+	0.814576i
$67$	0.580057	+	0.814576i	0.580057	+	0.814576i
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.76962	+	0.912303i	1.76962	+	0.912303i	0.928368	+	0.371662i	$0.121212\pi$
$71$	1.76962	+	0.912303i	1.76962	+	0.912303i	0.841254	+	0.540641i	$0.181818\pi$
$72$	−0.452036	+	0.989821i	−0.452036	+	0.989821i
$73$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$73$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$74$	0.697576	+	0.665138i	0.697576	+	0.665138i
$75$		0			0
$76$		0			0
$77$	−0.370638	−	0.291473i	−0.370638	−	0.291473i
$78$		0			0
$79$	0.0930932	−	0.268975i	0.0930932	−	0.268975i	−0.888835	−	0.458227i	$-0.848485\pi$
$79$	0.0930932	−	0.268975i	0.0930932	−	0.268975i	0.981929	+	0.189251i	$0.0606061\pi$
$80$		0			0
$81$	0.841254	−	0.540641i	0.841254	−	0.540641i
$82$		0			0
$83$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$83$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$84$		0			0
$85$		0			0
$86$	0.0572220	−	0.0545611i	0.0572220	−	0.0545611i
$87$		0			0
$88$	−0.297618	+	0.417947i	−0.297618	+	0.417947i
$89$		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
$92$	−0.132675	−	0.153115i	−0.132675	−	0.153115i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	0.815816	+	0.157236i	0.815816	+	0.157236i
$99$	0.437742	−	0.175245i	0.437742	−	0.175245i
$100$	−0.0730196	−	0.300991i	−0.0730196	−	0.300991i
$101$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$101$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$102$		0			0
$103$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$103$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$104$		0			0
$105$		0			0
$106$	0.427201	+	1.23432i	0.427201	+	1.23432i
$107$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	0.928368	−	0.371662i	$-0.121212\pi$
$107$	0.0395325	+	0.0865641i	0.0395325	+	0.0865641i	−0.888835	+	0.458227i	$0.848485\pi$
$108$		0			0
$109$	−1.28605	−	1.48418i	−1.28605	−	1.48418i	−0.786053	−	0.618159i	$-0.787879\pi$
$109$	−1.28605	−	1.48418i	−1.28605	−	1.48418i	−0.500000	−	0.866025i	$-0.666667\pi$
$110$		0			0
$111$		0			0
$112$	0.0845850	−	0.588302i	0.0845850	−	0.588302i
$113$	−0.580057	+	0.814576i	−0.580057	+	0.814576i	−0.995472	−	0.0950560i	$-0.969697\pi$
$113$	−0.580057	+	0.814576i	−0.580057	+	0.814576i	0.415415	+	0.909632i	$0.363636\pi$
$114$		0			0
$115$		0			0
$116$	−0.0213315	−	0.447804i	−0.0213315	−	0.447804i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.763617	+	0.147175i	−0.763617	+	0.147175i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.544078	+	0.627899i	−0.544078	+	0.627899i
$127$	1.21769	+	1.16106i	1.21769	+	1.16106i	0.981929	+	0.189251i	$0.0606061\pi$
$127$	1.21769	+	1.16106i	1.21769	+	1.16106i	0.235759	+	0.971812i	$0.424242\pi$
$128$	−0.308319	−	0.0294409i	−0.308319	−	0.0294409i
$129$		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
$134$	0.771316	−	0.308788i	0.771316	−	0.308788i
$135$		0			0
$136$		0			0
$137$	−0.264241	−	1.83784i	−0.264241	−	1.83784i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−0.264241	−	1.83784i	−0.264241	−	1.83784i	0.235759	−	0.971812i	$-0.424242\pi$
$138$		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
$142$	1.08323	−	1.25011i	1.08323	−	1.25011i
$143$		0			0
$144$	0.467192	+	0.367404i	0.467192	+	0.367404i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.302273	+	0.194259i	−0.302273	+	0.194259i
$149$	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
$151$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.723734	−	0.690079i	$-0.242424\pi$
$151$	1.70566	−	0.879330i	1.70566	−	0.879330i	0.981929	−	0.189251i	$-0.0606061\pi$
$152$		0			0
$153$		0			0
$154$	−0.307937	+	0.242165i	−0.307937	+	0.242165i
$155$		0			0
$156$		0			0
$157$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$157$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$158$	−0.198939	−	0.127850i	−0.198939	−	0.127850i
$159$		0			0
$160$		0			0
$161$	−0.271738	−	0.595023i	−0.271738	−	0.595023i
$162$	−0.271738	−	0.785135i	−0.271738	−	0.785135i
$163$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$163$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$167$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$168$		0			0
$169$	0.928368	−	0.371662i	0.928368	−	0.371662i
$170$		0			0
$171$		0			0
$172$	0.0147371	+	0.0255255i	0.0147371	+	0.0255255i
$173$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$173$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$174$		0			0
$175$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$176$	0.183523	+	0.211797i	0.183523	+	0.211797i
$177$		0			0
$178$		0			0
$179$	0.252989	−	1.75958i	0.252989	−	1.75958i	−0.327068	−	0.945001i	$-0.606061\pi$
$179$	0.252989	−	1.75958i	0.252989	−	1.75958i	0.580057	−	0.814576i	$-0.303030\pi$
$180$		0			0
$181$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$181$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$182$		0			0
$183$		0			0
$184$	−0.632675	+	0.326167i	−0.632675	+	0.326167i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.56499	+	1.23072i	1.56499	+	1.23072i	0.841254	+	0.540641i	$0.181818\pi$
$191$	1.56499	+	1.23072i	1.56499	+	1.23072i	0.723734	+	0.690079i	$0.242424\pi$
$192$		0			0
$193$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.327068	+	0.945001i	$0.606061\pi$
$193$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.888835	+	0.458227i	$0.848485\pi$
$194$		0			0
$195$		0			0
$196$	−0.128663	+	0.281733i	−0.128663	+	0.281733i
$197$	−1.03115	−	0.531595i	−1.03115	−	0.531595i	−0.142315	−	0.989821i	$-0.545455\pi$
$197$	−1.03115	−	0.531595i	−1.03115	−	0.531595i	−0.888835	+	0.458227i	$0.848485\pi$
$198$	−0.0557520	−	0.387764i	−0.0557520	−	0.387764i
$199$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$199$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$200$	−1.08816			−1.08816
$201$		0			0
$202$		0			0
$203$	0.341254	−	1.40667i	0.341254	−	1.40667i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.651174	+	0.0621796i	0.651174	+	0.0621796i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.786053	+	0.618159i	0.786053	+	0.618159i	0.928368	−	0.371662i	$-0.121212\pi$
$211$	0.786053	+	0.618159i	0.786053	+	0.618159i	−0.142315	+	0.989821i	$0.545455\pi$
$212$	−0.484710	+	0.0462842i	−0.484710	+	0.0462842i
$213$		0			0
$214$	0.0776362	−	0.0149631i	0.0776362	−	0.0149631i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.45025	+	0.747657i	−1.45025	+	0.747657i
$219$		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	$0.545455\pi$
$224$	0.551777	+	0.220898i	0.551777	+	0.220898i
$225$	0.841254	+	0.540641i	0.841254	+	0.540641i
$226$	0.544078	+	0.627899i	0.544078	+	0.627899i
$227$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$227$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$228$		0			0
$229$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$229$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$230$		0			0
$231$		0			0
$232$	−1.54661	−	0.298084i	−1.54661	−	0.298084i
$233$	1.34378	−	0.537970i	1.34378	−	0.537970i	0.415415	−	0.909632i	$-0.363636\pi$
$233$	1.34378	−	0.537970i	1.34378	−	0.537970i	0.928368	+	0.371662i	$0.121212\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$239$	0.327068	+	0.566498i	0.327068	+	0.566498i	−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	$0.363636\pi$
$242$	−0.0307433	+	0.645381i	−0.0307433	+	0.645381i
$243$		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		0.888835	−	0.458227i	$-0.151515\pi$
$251$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$252$	−0.179656	−	0.252292i	−0.179656	−	0.252292i
$253$	0.295943	+	0.0868967i	0.295943	+	0.0868967i
$254$	1.17597	−	0.755750i	1.17597	−	0.755750i
$255$		0			0
$256$	0.271738	−	0.785135i	0.271738	−	0.785135i
$257$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$257$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$258$		0			0
$259$	−1.11312	+	0.326842i	−1.11312	+	0.326842i
$260$		0			0
$261$	1.04758	+	0.998867i	1.04758	+	0.998867i
$262$		0			0
$263$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	0.0475819	+	0.998867i	$0.484848\pi$
$263$	−0.738471	+	1.61703i	−0.738471	+	1.61703i	−0.786053	+	0.618159i	$0.787879\pi$
$264$		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
$274$	−1.53565	−	0.146636i	−1.53565	−	0.146636i
$275$	0.341254	+	0.325385i	0.341254	+	0.325385i
$276$		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
$281$	−1.28605	+	0.247866i	−1.28605	+	0.247866i	−0.786053	−	0.618159i	$-0.787879\pi$
$281$	−1.28605	+	0.247866i	−1.28605	+	0.247866i	−0.500000	+	0.866025i	$0.666667\pi$
$282$		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	+	0.502299i	0.357685	+	0.502299i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.467192	+	0.367404i	−0.467192	+	0.367404i
$289$	0.580057	−	0.814576i	0.580057	−	0.814576i
$290$		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
$296$	0.412886	+	1.19295i	0.412886	+	1.19295i
$297$		0			0
$298$	0.415415	−	0.719520i	0.415415	−	0.719520i
$299$		0			0
$300$		0			0
$301$	0.0224357	+	0.0924813i	0.0224357	+	0.0924813i
$302$	−0.375883	−	1.54941i	−0.375883	−	1.54941i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$307$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$308$	−0.0606669	−	0.132842i	−0.0606669	−	0.132842i
$309$		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
$316$	0.0638014	−	0.0608346i	0.0638014	−	0.0608346i
$317$	0.0800569	+	1.68060i	0.0800569	+	1.68060i	0.580057	+	0.814576i	$0.303030\pi$
$317$	0.0800569	+	1.68060i	0.0800569	+	1.68060i	−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	0.395893	+	0.555954i	0.395893	+	0.555954i
$320$		0			0
$321$		0			0
$322$	−0.533654	+	0.102853i	−0.533654	+	0.102853i
$323$		0			0
$324$	0.308319	−	0.0294409i	0.308319	−	0.0294409i
$325$		0			0
$326$	−1.48014	+	0.434610i	−1.48014	+	0.434610i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.65033	−	0.850806i	−1.65033	−	0.850806i	−0.995472	−	0.0950560i	$-0.969697\pi$
$331$	−1.65033	−	0.850806i	−1.65033	−	0.850806i	−0.654861	−	0.755750i	$-0.727273\pi$
$332$		0			0
$333$	0.273507	−	1.12741i	0.273507	−	1.12741i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.419102	+	1.72756i	−0.419102	+	1.72756i	0.235759	+	0.971812i	$0.424242\pi$
$337$	−0.419102	+	1.72756i	−0.419102	+	1.72756i	−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
$344$	0.0993585	−	0.0291743i	0.0993585	−	0.0291743i
$345$		0			0
$346$		0			0
$347$	−0.642315	+	1.85585i	−0.642315	+	1.85585i	−0.142315	+	0.989821i	$0.545455\pi$
$347$	−0.642315	+	1.85585i	−0.642315	+	1.85585i	−0.500000	+	0.866025i	$0.666667\pi$
$348$		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
$352$	−0.249094	+	0.128417i	−0.249094	+	0.128417i
$353$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$353$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.37115	−	0.548924i	−1.37115	−	0.548924i
$359$	1.65210	+	1.06174i	1.65210	+	1.06174i	0.928368	+	0.371662i	$0.121212\pi$
$359$	1.65210	+	1.06174i	1.65210	+	1.06174i	0.723734	+	0.690079i	$0.242424\pi$
$360$		0			0
$361$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$367$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$368$	0.0916599	+	0.377827i	0.0916599	+	0.377827i
$369$		0			0
$370$		0			0
$371$	−1.54370	−	0.297523i	−1.54370	−	0.297523i
$372$		0			0
$373$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$373$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.21590	−	0.486774i	−1.21590	−	0.486774i	−0.327068	−	0.945001i	$-0.606061\pi$
$379$	−1.21590	−	0.486774i	−1.21590	−	0.486774i	−0.888835	+	0.458227i	$0.848485\pi$
$380$		0			0
$381$		0			0
$382$	1.30024	−	1.02252i	1.30024	−	1.02252i
$383$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$383$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$384$		0			0
$385$		0			0
$386$	0.894814	+	1.25659i	0.894814	+	1.25659i
$387$	−0.0913090	−	0.0268107i	−0.0913090	−	0.0268107i
$388$		0			0
$389$	−1.54370	+	0.297523i	−1.54370	+	0.297523i	−0.888835	−	0.458227i	$-0.848485\pi$
$389$	−1.54370	+	0.297523i	−1.54370	+	0.297523i	−0.654861	+	0.755750i	$0.727273\pi$
$390$		0			0
$391$		0			0
$392$	0.855348	+	0.672653i	0.855348	+	0.672653i
$393$		0			0
$394$	−0.631192	+	0.728435i	−0.631192	+	0.728435i
$395$		0			0
$396$	0.145378	+	0.0138819i	0.145378	+	0.0138819i
$397$		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
$400$	−0.140124	+	0.577597i	−0.140124	+	0.577597i
$401$	−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
$406$	−1.06891	−	0.551063i	−1.06891	−	0.551063i
$407$	0.227238	−	0.497582i	0.227238	−	0.497582i
$408$		0			0
$409$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$409$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.177754	−	0.513585i	0.177754	−	0.513585i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$419$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$420$		0			0
$421$	1.42131	−	1.35522i	1.42131	−	1.35522i	0.580057	−	0.814576i	$-0.303030\pi$
$421$	1.42131	−	1.35522i	1.42131	−	1.35522i	0.841254	−	0.540641i	$-0.181818\pi$
$422$	0.653077	−	0.513585i	0.653077	−	0.513585i
$423$		0			0
$424$	−0.243458	+	1.69328i	−0.243458	+	1.69328i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.00140244	+	0.0294409i	−0.00140244	+	0.0294409i
$429$		0			0
$430$		0			0
$431$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$431$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$432$		0			0
$433$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$433$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$434$		0			0
$435$		0			0
$436$	−0.143400	−	0.591103i	−0.143400	−	0.591103i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$442$		0			0
$443$	0.0800569	−	1.68060i	0.0800569	−	1.68060i	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	0.0800569	−	1.68060i	0.0800569	−	1.68060i	0.580057	−	0.814576i	$-0.303030\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.631192	−	0.886386i	0.631192	−	0.886386i
$449$	−1.32254	+	1.04006i	−1.32254	+	1.04006i	−0.327068	+	0.945001i	$0.606061\pi$
$449$	−1.32254	+	1.04006i	−1.32254	+	1.04006i	−0.995472	+	0.0950560i	$0.969697\pi$
$450$	0.601300	−	0.573338i	0.601300	−	0.573338i
$451$		0			0
$452$	−0.275291	+	0.141923i	−0.275291	+	0.141923i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.651174	−	1.88144i	0.651174	−	1.88144i	0.235759	−	0.971812i	$-0.424242\pi$
$457$	0.651174	−	1.88144i	0.651174	−	1.88144i	0.415415	−	0.909632i	$-0.363636\pi$
$458$		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
$463$	−1.95496	−	0.186677i	−1.95496	−	0.186677i	−0.959493	−	0.281733i	$-0.909091\pi$
$463$	−1.95496	−	0.186677i	−1.95496	−	0.186677i	−0.995472	+	0.0950560i	$0.969697\pi$
$464$	−0.357383	+	0.782560i	−0.357383	+	0.782560i
$465$		0			0
$466$	−0.171148	−	1.19036i	−0.171148	−	1.19036i
$467$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$467$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$468$		0			0
$469$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.0398833	−	0.0205613i	−0.0398833	−	0.0205613i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.02951	−	1.18812i	1.02951	−	1.18812i
$478$	0.521461	−	0.153115i	0.521461	−	0.153115i
$479$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$479$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.231105	−	0.0678585i	−0.231105	−	0.0678585i
$485$		0			0
$486$		0			0
$487$	0.0552004	+	1.15880i	0.0552004	+	1.15880i	0.841254	+	0.540641i	$0.181818\pi$
$487$	0.0552004	+	1.15880i	0.0552004	+	1.15880i	−0.786053	+	0.618159i	$0.787879\pi$
$488$		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
$497$	0.651174	+	1.88144i	0.651174	+	1.88144i
$498$		0			0
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$503$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$504$	−1.01021	+	0.404427i	−1.01021	+	0.404427i
$505$		0			0
$506$	0.128129	−	0.221926i	0.128129	−	0.221926i
$507$		0			0
$508$	0.170438	+	0.492448i	0.170438	+	0.492448i
$509$		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
$518$	0.0458622	+	0.962766i	0.0458622	+	0.962766i
$519$		0			0
$520$		0			0
$521$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$521$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$522$	1.01169	−	0.650175i	1.01169	−	0.650175i
$523$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$523$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$524$		0			0
$525$		0			0
$526$	1.16096	+	0.912985i	1.16096	+	0.912985i
$527$		0			0
$528$		0			0
$529$	−0.414053	−	0.394798i	−0.414053	−	0.394798i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	1.08323	+	0.103436i	1.08323	+	0.103436i
$537$		0			0
$538$		0			0
$539$	−0.0671040	−	0.466718i	−0.0671040	−	0.466718i
$540$		0			0
$541$	0.601300	−	1.31666i	0.601300	−	1.31666i	−0.327068	−	0.945001i	$-0.606061\pi$
$541$	0.601300	−	1.31666i	0.601300	−	1.31666i	0.928368	−	0.371662i	$-0.121212\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.651174	−	0.0621796i	0.651174	−	0.0621796i	0.235759	−	0.971812i	$-0.424242\pi$
$547$	0.651174	−	0.0621796i	0.651174	−	0.0621796i	0.415415	+	0.909632i	$0.363636\pi$
$548$	0.188087	−	0.543443i	0.188087	−	0.543443i
$549$		0			0
$550$	0.329562	−	0.211797i	0.329562	−	0.211797i
$551$		0			0
$552$		0			0
$553$	0.252989	−	0.130425i	0.252989	−	0.130425i
$554$	−0.0758623	−	1.59255i	−0.0758623	−	1.59255i
$555$		0			0
$556$		0			0
$557$	−0.165101	+	0.231852i	−0.165101	+	0.231852i	−0.888835	−	0.458227i	$-0.848485\pi$
$557$	−0.165101	+	0.231852i	−0.165101	+	0.231852i	0.723734	+	0.690079i	$0.242424\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0517765	+	1.08692i	−0.0517765	+	1.08692i
$563$		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
$567$	0.981929	+	0.189251i	0.981929	+	0.189251i
$568$	2.01127	−	0.805191i	2.01127	−	0.805191i
$569$	0.195876	+	0.807410i	0.195876	+	0.807410i	0.981929	+	0.189251i	$0.0606061\pi$
$569$	0.195876	+	0.807410i	0.195876	+	0.807410i	−0.786053	+	0.618159i	$0.787879\pi$
$570$		0			0
$571$	0.437742	−	0.175245i	0.437742	−	0.175245i	−0.142315	−	0.989821i	$-0.545455\pi$
$571$	0.437742	−	0.175245i	0.437742	−	0.175245i	0.580057	+	0.814576i	$0.303030\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.213947	+	0.618159i	0.213947	+	0.618159i
$576$	0.452036	+	0.989821i	0.452036	+	0.989821i
$577$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$577$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$578$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.582682	−	0.458227i	0.582682	−	0.458227i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$587$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.686393	−	0.0655426i	0.686393	−	0.0655426i
$593$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$593$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$594$		0			0
$595$		0			0
$596$	0.224156	+	0.213732i	0.224156	+	0.213732i
$597$		0			0
$598$		0			0
$599$	1.39734	+	0.720381i	1.39734	+	0.720381i	0.981929	−	0.189251i	$-0.0606061\pi$
$599$	1.39734	+	0.720381i	1.39734	+	0.720381i	0.415415	+	0.909632i	$0.363636\pi$
$600$		0			0
$601$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$601$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$602$	0.0790650			0.0790650
$603$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$604$	0.594351			0.594351
$605$		0			0
$606$		0			0
$607$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$607$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.56499	+	1.23072i	1.56499	+	1.23072i	0.841254	+	0.540641i	$0.181818\pi$
$613$	1.56499	+	1.23072i	1.56499	+	1.23072i	0.723734	+	0.690079i	$0.242424\pi$
$614$		0			0
$615$		0			0
$616$	−0.503813	+	0.0971020i	−0.503813	+	0.0971020i
$617$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	0.580057	+	0.814576i	$0.303030\pi$
$618$		0			0
$619$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$619$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$620$		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
$631$	−0.473420	−	1.36786i	−0.473420	−	1.36786i	−0.888835	−	0.458227i	$-0.848485\pi$
$631$	−0.473420	−	1.36786i	−0.473420	−	1.36786i	0.415415	−	0.909632i	$-0.363636\pi$
$632$	−0.154861	−	0.268227i	−0.154861	−	0.268227i
$633$		0			0
$634$	1.37262	+	0.264550i	1.37262	+	0.264550i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.526429	−	0.210750i	0.526429	−	0.210750i
$639$	−1.95496	−	0.376789i	−1.95496	−	0.376789i
$640$		0			0
$641$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$641$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$642$		0			0
$643$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$643$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$644$	0.00964009	−	0.202370i	0.00964009	−	0.202370i
$645$		0			0
$646$		0			0
$647$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$647$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$648$	0.154861	−	1.07708i	0.154861	−	1.07708i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.0273630	−	0.574420i	−0.0273630	−	0.574420i
$653$	0.252989	−	0.130425i	0.252989	−	0.130425i	−0.327068	−	0.945001i	$-0.606061\pi$
$653$	0.252989	−	0.130425i	0.252989	−	0.130425i	0.580057	+	0.814576i	$0.303030\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.95496	+	0.186677i	−1.95496	+	0.186677i	−0.995472	−	0.0950560i	$-0.969697\pi$
$659$	−1.95496	+	0.186677i	−1.95496	+	0.186677i	−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	$0.909091\pi$
$662$	−1.01021	+	1.16584i	−1.01021	+	1.16584i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.856711	−	0.441665i	−0.856711	−	0.441665i
$667$	0.134750	+	0.937203i	0.134750	+	0.937203i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.165101	−	1.14831i	−0.165101	−	1.14831i	−0.888835	−	0.458227i	$-0.848485\pi$
$673$	−0.165101	−	1.14831i	−0.165101	−	1.14831i	0.723734	−	0.690079i	$-0.242424\pi$
$674$	1.31276	+	0.676774i	1.31276	+	0.676774i
$675$		0			0
$676$	0.308319	+	0.0294409i	0.308319	+	0.0294409i
$677$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$677$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.95496	+	0.376789i	−1.95496	+	0.376789i	−0.959493	+	0.281733i	$0.909091\pi$
$683$	−1.95496	+	0.376789i	−1.95496	+	0.376789i	−0.995472	+	0.0950560i	$0.969697\pi$
$684$		0			0
$685$		0			0
$686$	0.481929	+	0.676774i	0.481929	+	0.676774i
$687$		0			0
$688$	−0.00269127	−	0.0564967i	−0.00269127	−	0.0564967i
$689$		0			0
$690$		0			0
$691$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$691$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$692$		0			0
$693$	0.437742	+	0.175245i	0.437742	+	0.175245i
$694$	1.37262	+	0.882127i	1.37262	+	0.882127i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.154861	−	0.268227i	0.154861	−	0.268227i
$701$	−1.28605	−	0.247866i	−1.28605	−	0.247866i	−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.28605	−	0.247866i	−1.28605	−	0.247866i	−0.786053	+	0.618159i	$0.787879\pi$
$702$		0			0
$703$		0			0
$704$	0.120964	+	0.498622i	0.120964	+	0.498622i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.271738	−	0.785135i	−0.271738	−	0.785135i	−0.995472	−	0.0950560i	$-0.969697\pi$
$709$	−0.271738	−	0.785135i	−0.271738	−	0.785135i	0.723734	−	0.690079i	$-0.242424\pi$
$710$		0			0
$711$	−0.0135432	+	0.284307i	−0.0135432	+	0.284307i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.319369	−	0.448492i	0.319369	−	0.448492i
$717$		0			0
$718$	1.18087	−	1.12595i	1.18087	−	1.12595i
$719$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$719$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$720$		0			0
$721$		0			0
$722$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$723$		0			0
$724$		0			0
$725$	−0.473420	+	1.36786i	−0.473420	+	1.36786i
$726$		0			0
$727$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$727$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$728$		0			0
$729$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$733$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$734$		0			0
$735$		0			0
$736$	−0.388786			−0.388786
$737$	−0.308779	−	0.356349i	−0.308779	−	0.356349i
$738$		0			0
$739$	−0.370638	+	1.52779i	−0.370638	+	1.52779i	0.415415	+	0.909632i	$0.363636\pi$
$739$	−0.370638	+	1.52779i	−0.370638	+	1.52779i	−0.786053	+	0.618159i	$0.787879\pi$
$740$		0			0
$741$		0			0
$742$	−0.542596	+	1.18812i	−0.542596	+	1.18812i
$743$	0.283341	+	0.0270558i	0.283341	+	0.0270558i	0.235759	−	0.971812i	$-0.424242\pi$
$743$	0.283341	+	0.0270558i	0.283341	+	0.0270558i	0.0475819	+	0.998867i	$0.484848\pi$
$744$		0			0
$745$		0			0
$746$	1.41712	−	0.416103i	1.41712	−	0.416103i
$747$		0			0
$748$		0			0
$749$	−0.0311250	+	0.0899299i	−0.0311250	+	0.0899299i
$750$		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
$757$	1.02951	−	0.809616i	1.02951	−	0.809616i	0.0475819	−	0.998867i	$-0.484848\pi$
$757$	1.02951	−	0.809616i	1.02951	−	0.809616i	0.981929	+	0.189251i	$0.0606061\pi$
$758$	−0.631192	+	0.886386i	−0.631192	+	0.886386i
$759$		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
$763$	0.0934441	−	1.96163i	0.0934441	−	1.96163i
$764$	0.256161	+	0.560914i	0.256161	+	0.560914i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$769$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$770$		0			0
$771$		0			0
$772$	−0.533877	+	0.213732i	−0.533877	+	0.213732i
$773$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$773$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$774$	−0.0395325	+	0.0684723i	−0.0395325	+	0.0684723i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.0621493	+	1.30467i	−0.0621493	+	1.30467i
$779$		0			0
$780$		0			0
$781$	−0.871520	−	0.348904i	−0.871520	−	0.348904i
$782$		0			0
$783$		0			0
$784$	0.467192	−	0.367404i	0.467192	−	0.367404i
$785$		0			0
$786$		0			0
$787$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$787$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$788$	−0.208421	−	0.292687i	−0.208421	−	0.292687i
$789$		0			0
$790$		0			0
$791$	−0.981929	+	0.189251i	−0.981929	+	0.189251i
$792$	0.167814	−	0.484866i	0.167814	−	0.484866i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$797$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$798$		0			0
$799$		0			0
$800$	−0.528280	−	0.272347i	−0.528280	−	0.272347i
$801$		0			0
$802$	−0.0557520	+	0.229813i	−0.0557520	+	0.229813i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	−0.888835	−	0.458227i	$-0.848485\pi$
$809$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	0.928368	+	0.371662i	$0.121212\pi$
$810$		0			0
$811$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$811$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$812$	0.293582	−	0.338812i	0.293582	−	0.338812i
$813$		0			0
$814$	−0.357242	−	0.280938i	−0.357242	−	0.280938i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.65033	+	0.850806i	−1.65033	+	0.850806i	−0.654861	+	0.755750i	$0.727273\pi$
$821$	−1.65033	+	0.850806i	−1.65033	+	0.850806i	−0.995472	+	0.0950560i	$0.969697\pi$
$822$		0			0
$823$	−0.723734	+	0.690079i	−0.723734	+	0.690079i	−0.959493	−	0.281733i	$-0.909091\pi$
$823$	−0.723734	+	0.690079i	−0.723734	+	0.690079i	0.235759	+	0.971812i	$0.424242\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.65033	−	0.660694i	−1.65033	−	0.660694i	−0.654861	−	0.755750i	$-0.727273\pi$
$827$	−1.65033	−	0.660694i	−1.65033	−	0.660694i	−0.995472	+	0.0950560i	$0.969697\pi$
$828$	0.170438	+	0.109534i	0.170438	+	0.109534i
$829$		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		0.928368	−	0.371662i	$-0.121212\pi$
$839$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$840$		0			0
$841$	−0.547582	+	0.948440i	−0.547582	+	0.948440i
$842$	−0.815816	−	1.41303i	−0.815816	−	1.41303i
$843$		0			0
$844$	0.128663	+	0.281733i	0.128663	+	0.281733i
$845$		0			0
$846$		0			0
$847$	−0.654218	−	0.420441i	−0.654218	−	0.420441i
$848$	0.867451	+	0.347275i	0.867451	+	0.347275i
$849$		0			0
$850$		0			0
$851$	0.596514	−	0.469104i	0.596514	−	0.469104i
$852$		0			0
$853$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$853$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$854$		0			0
$855$		0			0
$856$	0.0993585	+	0.0291743i	0.0993585	+	0.0291743i
$857$		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		0.327068	−	0.945001i	$-0.393939\pi$
$859$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$860$		0			0
$861$		0			0
$862$	−0.375883	+	0.110369i	−0.375883	+	0.110369i
$863$	1.02951	−	1.18812i	1.02951	−	1.18812i	0.0475819	−	0.998867i	$-0.484848\pi$
$863$	1.02951	−	1.18812i	1.02951	−	1.18812i	0.981929	−	0.189251i	$-0.0606061\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.0316407	+	0.130425i	−0.0316407	+	0.130425i
$870$		0			0
$871$		0			0
$872$	−2.13698			−2.13698
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.827068	−	0.0789754i	−0.827068	−	0.0789754i	−0.327068	−	0.945001i	$-0.606061\pi$
$877$	−0.827068	−	0.0789754i	−0.827068	−	0.0789754i	−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$881$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$882$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i
$883$	0.514186	−	1.48564i	0.514186	−	1.48564i	−0.327068	−	0.945001i	$-0.606061\pi$
$883$	0.514186	−	1.48564i	0.514186	−	1.48564i	0.841254	−	0.540641i	$-0.181818\pi$
$884$		0			0
$885$		0			0
$886$	−1.34125	−	0.393828i	−1.34125	−	0.393828i
$887$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$887$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$888$		0			0
$889$	0.0800569	+	1.68060i	0.0800569	+	1.68060i
$890$		0			0
$891$	−0.370638	+	0.291473i	−0.370638	+	0.291473i
$892$		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
$904$	0.256542	+	1.05748i	0.256542	+	1.05748i
$905$		0			0
$906$		0			0
$907$	0.462997	+	0.0892353i	0.462997	+	0.0892353i	0.415415	−	0.909632i	$-0.363636\pi$
$907$	0.462997	+	0.0892353i	0.462997	+	0.0892353i	0.0475819	+	0.998867i	$0.484848\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.723734	+	0.690079i	$0.242424\pi$
$911$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.0475819	+	0.998867i	$0.484848\pi$
$912$		0			0
$913$		0			0
$914$	−1.39155	−	0.894293i	−1.39155	−	0.894293i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.28656	+	1.22673i	−1.28656	+	1.22673i	−0.327068	+	0.945001i	$0.606061\pi$
$919$	−1.28656	+	1.22673i	−1.28656	+	1.22673i	−0.959493	+	0.281733i	$0.909091\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.13915	−	0.219553i	1.13915	−	0.219553i
$926$	−0.533654	+	1.54189i	−0.533654	+	1.54189i
$927$		0			0
$928$	−0.676245	−	0.531805i	−0.676245	−	0.531805i
$929$		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
$932$	0.446282	+	0.0426148i	0.446282	+	0.0426148i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	0.771316	+	0.308788i	0.771316	+	0.308788i
$939$		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
$946$	−0.0244136	+	0.0281748i	−0.0244136	+	0.0281748i
$947$	−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
$953$	1.50842	+	0.442913i	1.50842	+	0.442913i	0.928368	−	0.371662i	$-0.121212\pi$
$953$	1.50842	+	0.442913i	1.50842	+	0.442913i	0.580057	+	0.814576i	$0.303030\pi$
$954$	−0.757643	−	1.06396i	−0.757643	−	1.06396i
$955$		0			0
$956$	0.00964009	+	0.202370i	0.00964009	+	0.202370i
$957$		0			0
$958$		0			0
$959$	1.07701	−	1.51245i	1.07701	−	1.51245i
$960$		0			0
$961$	0.928368	+	0.371662i	0.928368	+	0.371662i
$962$		0			0
$963$	−0.0623191	−	0.0719200i	−0.0623191	−	0.0719200i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$967$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$968$	−0.423114	+	0.732854i	−0.423114	+	0.732854i
$969$		0			0
$970$		0			0
$971$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$971$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$972$		0			0
$973$		0			0
$974$	0.946439	+	0.182411i	0.946439	+	0.182411i
$975$		0			0
$976$		0			0
$977$	0.327068	+	0.945001i	0.327068	+	0.945001i	0.981929	+	0.189251i	$0.0606061\pi$
$977$	0.327068	+	0.945001i	0.327068	+	0.945001i	−0.654861	+	0.755750i	$0.727273\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.65210	+	1.06174i	1.65210	+	1.06174i
$982$	−0.219539	−	0.0878903i	−0.219539	−	0.0878903i
$983$		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
$989$	−0.0361086	−	0.0507074i	−0.0361086	−	0.0507074i
$990$		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
$994$	1.64665	−	0.157236i	1.64665	−	0.157236i
$995$		0			0
$996$		0			0
$997$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$997$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$998$	−0.601300	−	0.573338i	−0.601300	−	0.573338i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	469.1.bl.a.370.1	yes	20
7.2	even	3		3283.1.bw.a.1844.1		20
7.3	odd	6		3283.1.cd.a.705.1		20
7.4	even	3		3283.1.cd.a.705.1		20
7.5	odd	6		3283.1.bw.a.1844.1		20
7.6	odd	2	CM	469.1.bl.a.370.1	yes	20
67.23	even	33	inner	469.1.bl.a.90.1	✓	20
469.23	even	33		3283.1.cd.a.2971.1		20
469.90	odd	66	inner	469.1.bl.a.90.1	✓	20
469.157	odd	66		3283.1.bw.a.1832.1		20
469.291	even	33		3283.1.bw.a.1832.1		20
469.425	odd	66		3283.1.cd.a.2971.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
469.1.bl.a.90.1	✓	20	67.23	even	33	inner
469.1.bl.a.90.1	✓	20	469.90	odd	66	inner
469.1.bl.a.370.1	yes	20	1.1	even	1	trivial
469.1.bl.a.370.1	yes	20	7.6	odd	2	CM
3283.1.bw.a.1832.1		20	469.157	odd	66
3283.1.bw.a.1832.1		20	469.291	even	33
3283.1.bw.a.1844.1		20	7.2	even	3
3283.1.bw.a.1844.1		20	7.5	odd	6
3283.1.cd.a.705.1		20	7.3	odd	6
3283.1.cd.a.705.1		20	7.4	even	3
3283.1.cd.a.2971.1		20	469.23	even	33
3283.1.cd.a.2971.1		20	469.425	odd	66

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)

\(f(q)\)	\(=\)	\(q+(0.195876 - 0.807410i) q^{2} +(0.275291 + 0.141923i) q^{4} +(0.723734 + 0.690079i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})\)	\(q+(0.195876 - 0.807410i) q^{2} +(0.275291 + 0.141923i) q^{4} +(0.723734 + 0.690079i) q^{7} +(0.712591 - 0.822373i) q^{8} +(-0.959493 + 0.281733i) q^{9} +(-0.469383 + 0.0448206i) q^{11} +(0.698939 - 0.449181i) q^{14} +(-0.344757 - 0.484144i) q^{16} +(0.0395325 + 0.829889i) q^{18} +(-0.0557520 + 0.387764i) q^{22} +(-0.607279 - 0.243118i) q^{23} +(-0.654861 - 0.755750i) q^{25} +(0.101300 + 0.292687i) q^{28} +(-0.723734 - 1.25354i) q^{29} +(0.551777 - 0.220898i) q^{32} +(-0.304124 - 0.0586152i) q^{36} +(-0.580057 + 1.00469i) q^{37} +(0.0800569 + 0.0514495i) q^{43} +(-0.135578 - 0.0542773i) q^{44} +(-0.315247 + 0.442702i) q^{46} +(0.0475819 + 0.998867i) q^{49} +(-0.738471 + 0.380708i) q^{50} +(-1.32254 + 0.849945i) q^{53} +(1.08323 - 0.103436i) q^{56} +(-1.15389 + 0.338812i) q^{58} +(-0.888835 - 0.458227i) q^{63} +(-0.154861 - 1.07708i) q^{64} +(0.580057 + 0.814576i) q^{67} +(1.76962 + 0.912303i) q^{71} +(-0.452036 + 0.989821i) q^{72} +(0.697576 + 0.665138i) q^{74} +(-0.370638 - 0.291473i) q^{77} +(0.0930932 - 0.268975i) q^{79} +(0.841254 - 0.540641i) q^{81} +(0.0572220 - 0.0545611i) q^{86} +(-0.297618 + 0.417947i) q^{88} +(-0.132675 - 0.153115i) q^{92} +(0.815816 + 0.157236i) q^{98} +(0.437742 - 0.175245i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) 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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